**Meet the editor**

Dr. Yang is now a full professor of Southwest University (China). He obtained his PhD degree from Chinese Academy of Science in 2006, and after being an associate professor for 5 years in Northeast Forestry University, he moved to Eindhoven University of Technology to continue his studies as a postdoc researcher. His research interests focus on molecular modeling, compu-

tational soil chemistry, and theoretical catalysis. He has coauthored more than 100 peer-reviewed papers and 7 book chapters. He has delivered speeches in many international and domestic conferences. He serves as the editor for *PLOS One*, *Colloid*, and *Surface Science* as well as the referee for more than 40 international journals, books, proposals of European Research Council, etc.

Contents

**Preface VII**

**Section 1 Method Development and Validation 1**

Almossalami and Nageh K. Allam

**Section 2 Spectra and Thermodynamics 77**

Chapter 5 **Spectral Calculations with DFT 101**

**Mechanisms 79** Burkhard Kirste

Zavala

**the Description of Excited States 31**

Chapter 4 **Spectroscopy, Substituent Effects, and Reaction**

Ataf Ali Altaf, Samia Kausar and Amin Badshah

Zuriel Natanael Cisneros-García, David Alejandro Hernández-Velázquez, Francisco J. Tenorio and Jaime Gustavo Rodríguez-

Chapter 6 **DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins 129**

Chapter 1 **The DFT+U: Approaches, Accuracy, and Applications 3**

Sarah A. Tolba, Kareem M. Gameel, Basant A. Ali, Hossam A.

**Functionals for the Prediction of the Chemical Reactivity of the SYBR Green I and Ethidium Bromide Nucleic Acid Stains 63** Norma Flores-Holguín, Juan Frau and Daniel Glossman-Mitnik

Chapter 2 **Constricted Variational Density Functional Theory Approach to**

Florian Senn, Issaka Seidu and Young Choon Park

Chapter 3 **Assessment of the Validity of Some Minnesota Density**

### Contents

#### **Preface XI**


Chapter 6 **DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins 129** Zuriel Natanael Cisneros-García, David Alejandro Hernández-Velázquez, Francisco J. Tenorio and Jaime Gustavo Rodríguez-Zavala

#### **X** Contents

#### **Section 3 Catalysis and Mechanism 151**

Chapter 7 **Catalytic Activation of PVP-Stabilized Gold/Silver Cluster on p-Nitrophenol Reduction: A DFT 153** Madhulata Shukla and Indrajit Sinha

Preface

Density functional theory (DFT) is a method of obtaining approximate solutions to Schrö‐ dinger equation for many-body systems, where electron density rather than wave function is the central variable. Although DFT can be traced back to 1927 when the Thomas-Fermi model was developed, it was first put on a firm theoretical footing in 1964 by Kohn and

DFT owes its popularity in the predictive power for physical and chemical properties and the ability to handle large systems accurately and efficiently. Now, DFT ranks as the most widely used quantum mechanical method, and its success is clearly demonstrated by the overwhelming amount of research articles in the past decades that report results by means of *DFT* methods. A vast number of DFT software packages are available to us, such as Gaussian, VASP, DMOL, ADF, and Wien. Some of them are easy to use, which significantly

There are numerous books and reviews that showcase the contemporary advances of DFT methods, while DFT is always being developed toward higher accuracy and larger systems with less computational costs. In addition, DFT has expanded its business rapidly and plays an increasingly larger role in a number of disciplines such as chemistry, physics, material, biology, and pharmacy. A major goal of this book is to draw together contributors from the various research fields, to spread knowledge of current capabilities and new possibilities, and to stimulate the exchange of information between disparate disciplines. This book is divided into five sections that include original chapters written by experts in their fields. The first section describes the recent developments of DFT methods and the validation of these computational techniques that is equally important. The following sections are the ap‐ plications of DFT methods in the various domains: Section 2 includes spectra and thermody‐ namics that establish direct contacts with experiments such as spectral assignments, Section 3 focuses on catalysis and mechanism that identifies short-time intermediates and presents complicated reaction processes at the atomic level, Section 4 involves material and molecu‐ lar designs that see the power of DFT methods by predicting the properties of existing and new materials with hitherto unprecedented accuracy, and Section 5 (multidisciplinary inte‐ gration) is to broaden the insights with respect to DFT applications. This section contains only one chapter from our lab, showing the tremendous advantages and bright prospects in computational soil science. As a matter of fact, DFT methods have been widely used and bring forth a number of interdisciplines that are probably better known to us, such as com‐

putational biology, molecular pharmacy, and computational nanoscience.

I would like to express my sincere gratitude to all authors who have contributed to this book. They are Ataf Ali Altaf, Amin Badshah, Basant A. Ali, Burkhard Kirste, Chang Zhu, Daniel

Hohenberg in the framework of two Hohenberg-Kohn theorems.

promotes the popularization of DFT methods.

	- **Section 5 Multidisciplinary Integration 243**

### Preface

**Section 3 Catalysis and Mechanism 151**

**VI** Contents

**Thioureas 167**

Jia Fu

**Section 4 Material and Molecuar Design 195**

**Section 5 Multidisciplinary Integration 243**

Chapter 7 **Catalytic Activation of PVP-Stabilized Gold/Silver Cluster on p-**

Chapter 8 **Mechanistic Study on the Formation of Compounds from**

Warjeet S. Laitonjam and Lokendrajit Nahakpam

Chapter 9 **Carbon Nanotubes: Molecular and Electronic Properties of**

Chapter 10 **Elastic Constants and Homogenized Moduli of Monoclinic**

Chapter 11 **Application of Density Functional Theory in Soil Science 245** Jiena Yun, Qian Wang, Chang Zhu and Gang Yang

María Leonor Contreras Fuentes and Roberto Rozas Soto

**Structures Based on Density Functional Theory 219**

**Regular and Defective Structures 197**

**Nitrophenol Reduction: A DFT 153** Madhulata Shukla and Indrajit Sinha

> Density functional theory (DFT) is a method of obtaining approximate solutions to Schrö‐ dinger equation for many-body systems, where electron density rather than wave function is the central variable. Although DFT can be traced back to 1927 when the Thomas-Fermi model was developed, it was first put on a firm theoretical footing in 1964 by Kohn and Hohenberg in the framework of two Hohenberg-Kohn theorems.

> DFT owes its popularity in the predictive power for physical and chemical properties and the ability to handle large systems accurately and efficiently. Now, DFT ranks as the most widely used quantum mechanical method, and its success is clearly demonstrated by the overwhelming amount of research articles in the past decades that report results by means of *DFT* methods. A vast number of DFT software packages are available to us, such as Gaussian, VASP, DMOL, ADF, and Wien. Some of them are easy to use, which significantly promotes the popularization of DFT methods.

> There are numerous books and reviews that showcase the contemporary advances of DFT methods, while DFT is always being developed toward higher accuracy and larger systems with less computational costs. In addition, DFT has expanded its business rapidly and plays an increasingly larger role in a number of disciplines such as chemistry, physics, material, biology, and pharmacy. A major goal of this book is to draw together contributors from the various research fields, to spread knowledge of current capabilities and new possibilities, and to stimulate the exchange of information between disparate disciplines. This book is divided into five sections that include original chapters written by experts in their fields. The first section describes the recent developments of DFT methods and the validation of these computational techniques that is equally important. The following sections are the ap‐ plications of DFT methods in the various domains: Section 2 includes spectra and thermody‐ namics that establish direct contacts with experiments such as spectral assignments, Section 3 focuses on catalysis and mechanism that identifies short-time intermediates and presents complicated reaction processes at the atomic level, Section 4 involves material and molecu‐ lar designs that see the power of DFT methods by predicting the properties of existing and new materials with hitherto unprecedented accuracy, and Section 5 (multidisciplinary inte‐ gration) is to broaden the insights with respect to DFT applications. This section contains only one chapter from our lab, showing the tremendous advantages and bright prospects in computational soil science. As a matter of fact, DFT methods have been widely used and bring forth a number of interdisciplines that are probably better known to us, such as com‐ putational biology, molecular pharmacy, and computational nanoscience.

> I would like to express my sincere gratitude to all authors who have contributed to this book. They are Ataf Ali Altaf, Amin Badshah, Basant A. Ali, Burkhard Kirste, Chang Zhu, Daniel

Glossman-Mitnik, David Alejandro Hernández-Velázquez, Florian Senn, Francisco J. Tenor‐ io, Gang Yang, Hossam A. Almossalami, Issake Seidu, I. Sinha, Jaime Gustavo Rodríguez-Zavala, Jia FU, Jiena Yun, Juan Frau, Kareem M. Gameel, Lokendrajit Nahakpam, Madhulata Shukla, M. Leonor Contreras, Nageh K. Allam, Norma Flores-Holguín, Qian Wang, Roberto Rozas, Samia Kausar, Sarah A. Tolba, Warjeet S. Laitonjam, Young Choon Park, and Zuriel Natanael Cisneros-García. Thank you very much for your excellent contributions!

Finally, my warmest thanks go to my beloved family, especially to my lovely children (Pan‐ Pan and RongRong). Your constant encouragement and support have meant more than any‐ thing. I also thank the financial support from the National Natural Science Foundation of China (21473137).

> **Gang Yang** College of Resources and Environment & Chongqing Key Laboratory of Soil Multi-scale Interfacial Process Southwest University Chongqing 400715, China

**Section 1**

**Method Development and Validation**

**Method Development and Validation**

Glossman-Mitnik, David Alejandro Hernández-Velázquez, Florian Senn, Francisco J. Tenor‐ io, Gang Yang, Hossam A. Almossalami, Issake Seidu, I. Sinha, Jaime Gustavo Rodríguez-Zavala, Jia FU, Jiena Yun, Juan Frau, Kareem M. Gameel, Lokendrajit Nahakpam, Madhulata Shukla, M. Leonor Contreras, Nageh K. Allam, Norma Flores-Holguín, Qian Wang, Roberto Rozas, Samia Kausar, Sarah A. Tolba, Warjeet S. Laitonjam, Young Choon Park, and Zuriel

Finally, my warmest thanks go to my beloved family, especially to my lovely children (Pan‐ Pan and RongRong). Your constant encouragement and support have meant more than any‐ thing. I also thank the financial support from the National Natural Science Foundation of

**Gang Yang**

Southwest University Chongqing 400715, China

College of Resources and Environment &

Chongqing Key Laboratory of Soil Multi-scale Interfacial Process

Natanael Cisneros-García. Thank you very much for your excellent contributions!

China (21473137).

VIII Preface

**Chapter 1**

**Provisional chapter**

**The DFT+U: Approaches, Accuracy, and Applications**

This chapter introduces the Hubbard model and its applicability as a corrective tool for accurate modeling of the electronic properties of various classes of systems. The attainment of a correct description of electronic structure is critical for predicting further electronic-related properties, including intermolecular interactions and formation energies. The chapter begins with an introduction to the formulation of density functional theory (DFT) functionals, while addressing the origin of bandgap problem with correlated materials. Then, the corrective approaches proposed to solve the DFT bandgap problem are reviewed, while comparing them in terms of accuracy and computational cost. The Hubbard model will then offer a simple approach to correctly describe the behavior of highly correlated materials, known as the Mott insulators. Based on Hubbard model, DFT+U scheme is built, which is computationally convenient for accurate calculations of electronic structures. Later in this chapter, the computational and semiempirical methods of optimizing the value of the Coulomb interaction potential (*U*) are discussed, while evaluating the conditions under which it can be most predictive. The chapter focuses on highlighting the use of *U* to correct the description of the physical properties, by reviewing the results of case studies presented in literature for various

**The DFT+U: Approaches, Accuracy, and Applications**

DOI: 10.5772/intechopen.72020

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Density functional theory (DFT) is one of the most convenient computational tools for the prediction of the properties of different classes of materials [1, 2]. Although its accuracy is acceptable as long as structural and cohesive properties are concerned, it dramatically fails in the prediction of electronic and other related properties of semiconductors up to a factor of

**Keywords:** first principles, Hubbard *U* correction, GGA+U, DFT+U, LDA+U, spin

crossover, metal organic framework, solid defects, band structure

Sarah A. Tolba, Kareem M. Gameel, Basant A. Ali,

Sarah A. Tolba, Kareem M. Gameel, Basant A. Ali, Hossam A. Almossalami and Nageh K. Allam

Hossam A. Almossalami and Nageh K. Allam

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72020

**Abstract**

classes of materials.

**1. Introduction**

**Provisional chapter**

### **The DFT+U: Approaches, Accuracy, and Applications**

**The DFT+U: Approaches, Accuracy, and Applications**

DOI: 10.5772/intechopen.72020

Sarah A. Tolba, Kareem M. Gameel, Basant A. Ali, Hossam A. Almossalami and Nageh K. Allam Hossam A. Almossalami and Nageh K. Allam Additional information is available at the end of the chapter

Sarah A. Tolba, Kareem M. Gameel, Basant A. Ali,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72020

#### **Abstract**

This chapter introduces the Hubbard model and its applicability as a corrective tool for accurate modeling of the electronic properties of various classes of systems. The attainment of a correct description of electronic structure is critical for predicting further electronic-related properties, including intermolecular interactions and formation energies. The chapter begins with an introduction to the formulation of density functional theory (DFT) functionals, while addressing the origin of bandgap problem with correlated materials. Then, the corrective approaches proposed to solve the DFT bandgap problem are reviewed, while comparing them in terms of accuracy and computational cost. The Hubbard model will then offer a simple approach to correctly describe the behavior of highly correlated materials, known as the Mott insulators. Based on Hubbard model, DFT+U scheme is built, which is computationally convenient for accurate calculations of electronic structures. Later in this chapter, the computational and semiempirical methods of optimizing the value of the Coulomb interaction potential (*U*) are discussed, while evaluating the conditions under which it can be most predictive. The chapter focuses on highlighting the use of *U* to correct the description of the physical properties, by reviewing the results of case studies presented in literature for various classes of materials.

**Keywords:** first principles, Hubbard *U* correction, GGA+U, DFT+U, LDA+U, spin crossover, metal organic framework, solid defects, band structure

#### **1. Introduction**

Density functional theory (DFT) is one of the most convenient computational tools for the prediction of the properties of different classes of materials [1, 2]. Although its accuracy is acceptable as long as structural and cohesive properties are concerned, it dramatically fails in the prediction of electronic and other related properties of semiconductors up to a factor of

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

two [3]. However, reaching a correct description of electronic structure is critical for predicting further electronic-related properties, including intermolecular interactions and formation energies. In order to solve this problem, computationally heavier jobs must be employed, using either larger basis sets or hybrid functionals, which include the solution of the exact Hartree-Fock (HF) equations, in order to reach relatively higher accuracies [4]. Nevertheless, in some cases, even solving exact HF equations can fail in correctly predicting the bandgap for a certain class of semiconductors that possess strong correlations between electrons, such as Mott insulators [5, 6]. Consistent research efforts have been employed in order to formulate more accurate functionals, by using corrective approaches or alternatives to the density functionals. The applicability of these alternatives and corrections has large dependence on the type of the system studied, its size and complexity, and the computational cost required. One of the corrective approaches employed to relieve the DFT electronic bandgap problem is the DFT+U correction method, which is the focus of this chapter. Compared to the alternative approaches, such as the hybrid functionals and the post-Hartree-Fock methods, DFT+U correction has proved to be as reliable as the other methods, but with a critical advantage of considerably lower computational cost. By successfully correcting the electronic structure of the studied system using the *U* correction, further accurate predictions of intermolecular interactions and formation energies can be reached [6]. In addition, the *U* correction can further enhance the description of physical properties, other than the electronic structure, including magnetic and structural properties of correlated systems, the electron transfer energetics, and chemical reactions. However, one of the drawbacks of the Hubbard method is that it fails in predicting the properties of systems with more delocalized electrons, such as metals. The relative success of the DFT+U method is related to its straightforward approach to account for the underestimated electronic interactions by simply adding a semiempirically tuned numerical parameter "*U*" [5]. This interaction parameter can be easily controlled, making the DFT+U method a tool to give a qualitative assessment of the influence of the electronic correlations on the physical properties of a system.

**2. Theoretical formulation**

**2.2. Mott insulators and the Hubbard model**

Using exact HF or DFT solutions, the aim is always to reach, as close as possible, the exact description of the total energy of the system. Unluckily, reaching this exact energy description is impossible and approximations have to be employed. In DFT, electronic interaction energies are simply described as the sum of classical Columbic repulsion between electronic densities in a mean field kind of way (Hartree term) and an additive term that is supposed to encompass all the correlations and spin interactions [1]. This additive term, namely the exchange and correlation (xc), is founded on approximations that have the responsibility to recover the exact energy description of the system. This approximated xc functional is a function of the electronic charge density of the system, and the accuracy of a DFT calculation is strongly dependent on the descriptive ability of this functional of the energy of the system [2]. It is generally difficult to model the dependence of the xc functional on electronic charge density, and thus, it can inadequately represent the many-body features of the N-electron ground state. For this reason, systems with physical properties that are controlled by many body electronic interactions (correlated systems) are poorly described by DFT calculations. For these systems, incorrect description of the electronic structure induces the so-called "bandgap problem," which in turn, imposes difficulties in utilizing DFT to

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020 5

predict accurate intermolecular interactions, formation energies, and transition states [7].

The problem of DFT to describe correlated systems can be attributed to the tendency of xc functionals to over-delocalize valence electrons and to over-stabilize metallic ground states [5, 6]. That is why DFT fails significantly in predicting the properties of systems whose ground state is characterized by a more pronounced localization of electrons. The reason behind this delocalization is rooted to the inability of the approximated xc to completely cancel out the electronic self-interaction contained in the Hartree term; thus, a remaining "fragment" of the same electron is still there that can induce added self-interaction, consequently inducing an excessive delocalization of the wave functions [5]. For this reason, hybrid functionals were formulated to include a linear combination of a number of xc explicit density and HF exact exchange functionals, that is self-interaction free, by eliminating the extra self-interaction of electrons through the explicit introduction of a Fock exchange term. However, this method is computationally expensive and is not usually practical when larger, more complex systems are studied. Nonetheless, HF method, which describes the electronic structure with variationally optimized single determinant, cannot describe the physics of strongly correlated materials such as the Mott insulators. In order to describe the behavior of these systems, full account of the multideterminant nature of the N-electron wave function and of the many-body terms of the electronic interactions is needed [6]. Therefore, it is predicted that applying DFT calculations using approximate xc functionals, such as LDA or GGA, will poorly describe the physical properties of strongly correlated systems.

According to the conventional band theories, strongly correlated materials are predicted to be conductive, while they show insulating behavior when experimentally measured. This serious

**2.1. Standard DFT problem**

One of the mostly implemented methods in the DFT+U realm is the LDA+U method. It is widely used due to its simple implementation on the existing LDA codes, which makes it only slightly computationally heavier than the standard DFT computations [6]. In this chapter, we discuss the fundamental formulation of the LDA+U method and examine its applicability for practical implementations for different classes of materials, where DFT is usually found to be impractical. Popular cases of DFT shortage are discussed including materials with strong correlations, defective solid-state materials, and organometallics, while reviewing literature case studies that studied these classes of materials with DFT+U calculations. The methodology of optimizing the *U* correction is inspected, where it can be either formulated from first principles or achieved empirically by tuning the *U* value, while seeking an agreement with experimental results of the system's physical properties. In this chapter, we also present a review of the practical implementation of *U*, while assessing its corrective influence on improving the description of a variety of physical properties related to certain classes of materials. In addition, the effect of the calculation parameters on the chosen *U* value is discussed, including the choice of the localized basis set and the type of DFT functional employed.

### **2. Theoretical formulation**

#### **2.1. Standard DFT problem**

two [3]. However, reaching a correct description of electronic structure is critical for predicting further electronic-related properties, including intermolecular interactions and formation energies. In order to solve this problem, computationally heavier jobs must be employed, using either larger basis sets or hybrid functionals, which include the solution of the exact Hartree-Fock (HF) equations, in order to reach relatively higher accuracies [4]. Nevertheless, in some cases, even solving exact HF equations can fail in correctly predicting the bandgap for a certain class of semiconductors that possess strong correlations between electrons, such as Mott insulators [5, 6]. Consistent research efforts have been employed in order to formulate more accurate functionals, by using corrective approaches or alternatives to the density functionals. The applicability of these alternatives and corrections has large dependence on the type of the system studied, its size and complexity, and the computational cost required. One of the corrective approaches employed to relieve the DFT electronic bandgap problem is the DFT+U correction method, which is the focus of this chapter. Compared to the alternative approaches, such as the hybrid functionals and the post-Hartree-Fock methods, DFT+U correction has proved to be as reliable as the other methods, but with a critical advantage of considerably lower computational cost. By successfully correcting the electronic structure of the studied system using the *U* correction, further accurate predictions of intermolecular interactions and formation energies can be reached [6]. In addition, the *U* correction can further enhance the description of physical properties, other than the electronic structure, including magnetic and structural properties of correlated systems, the electron transfer energetics, and chemical reactions. However, one of the drawbacks of the Hubbard method is that it fails in predicting the properties of systems with more delocalized electrons, such as metals. The relative success of the DFT+U method is related to its straightforward approach to account for the underestimated electronic interactions by simply adding a semiempirically tuned numerical parameter "*U*" [5]. This interaction parameter can be easily controlled, making the DFT+U method a tool to give a qualitative assessment of the influence of the electronic correlations on

4 Density Functional Calculations - Recent Progresses of Theory and Application

One of the mostly implemented methods in the DFT+U realm is the LDA+U method. It is widely used due to its simple implementation on the existing LDA codes, which makes it only slightly computationally heavier than the standard DFT computations [6]. In this chapter, we discuss the fundamental formulation of the LDA+U method and examine its applicability for practical implementations for different classes of materials, where DFT is usually found to be impractical. Popular cases of DFT shortage are discussed including materials with strong correlations, defective solid-state materials, and organometallics, while reviewing literature case studies that studied these classes of materials with DFT+U calculations. The methodology of optimizing the *U* correction is inspected, where it can be either formulated from first principles or achieved empirically by tuning the *U* value, while seeking an agreement with experimental results of the system's physical properties. In this chapter, we also present a review of the practical implementation of *U*, while assessing its corrective influence on improving the description of a variety of physical properties related to certain classes of materials. In addition, the effect of the calculation parameters on the chosen *U* value is discussed, including the choice of the localized basis set and the type of

the physical properties of a system.

DFT functional employed.

Using exact HF or DFT solutions, the aim is always to reach, as close as possible, the exact description of the total energy of the system. Unluckily, reaching this exact energy description is impossible and approximations have to be employed. In DFT, electronic interaction energies are simply described as the sum of classical Columbic repulsion between electronic densities in a mean field kind of way (Hartree term) and an additive term that is supposed to encompass all the correlations and spin interactions [1]. This additive term, namely the exchange and correlation (xc), is founded on approximations that have the responsibility to recover the exact energy description of the system. This approximated xc functional is a function of the electronic charge density of the system, and the accuracy of a DFT calculation is strongly dependent on the descriptive ability of this functional of the energy of the system [2]. It is generally difficult to model the dependence of the xc functional on electronic charge density, and thus, it can inadequately represent the many-body features of the N-electron ground state. For this reason, systems with physical properties that are controlled by many body electronic interactions (correlated systems) are poorly described by DFT calculations. For these systems, incorrect description of the electronic structure induces the so-called "bandgap problem," which in turn, imposes difficulties in utilizing DFT to predict accurate intermolecular interactions, formation energies, and transition states [7].

The problem of DFT to describe correlated systems can be attributed to the tendency of xc functionals to over-delocalize valence electrons and to over-stabilize metallic ground states [5, 6]. That is why DFT fails significantly in predicting the properties of systems whose ground state is characterized by a more pronounced localization of electrons. The reason behind this delocalization is rooted to the inability of the approximated xc to completely cancel out the electronic self-interaction contained in the Hartree term; thus, a remaining "fragment" of the same electron is still there that can induce added self-interaction, consequently inducing an excessive delocalization of the wave functions [5]. For this reason, hybrid functionals were formulated to include a linear combination of a number of xc explicit density and HF exact exchange functionals, that is self-interaction free, by eliminating the extra self-interaction of electrons through the explicit introduction of a Fock exchange term. However, this method is computationally expensive and is not usually practical when larger, more complex systems are studied. Nonetheless, HF method, which describes the electronic structure with variationally optimized single determinant, cannot describe the physics of strongly correlated materials such as the Mott insulators. In order to describe the behavior of these systems, full account of the multideterminant nature of the N-electron wave function and of the many-body terms of the electronic interactions is needed [6]. Therefore, it is predicted that applying DFT calculations using approximate xc functionals, such as LDA or GGA, will poorly describe the physical properties of strongly correlated systems.

#### **2.2. Mott insulators and the Hubbard model**

According to the conventional band theories, strongly correlated materials are predicted to be conductive, while they show insulating behavior when experimentally measured. This serious flaw of the band theory was pointed out by Sir Nevil Mott, who emphasized that interelectron forces cannot be neglected, which lead to the existence of the bandgap in these falsely predicted conductors (Mott insulators) [8]. In these "metal-insulators," the bandgap exists between bands of like character i.e., between suborbitals of the same orbitals, such as 3d character, which originates from crystal field splitting or Hund's rule. The insulating character of the ground state stems from the strong Coulomb repulsion between electrons that forces them to localize in atomiclike orbitals (Mott localization). This Coulomb potential, responsible for localization, is described by the term "*U*," and when electrons are strongly localized, they cannot move freely between atoms and rather jump from one atom to another by a "hopping" mechanism between neighbor atoms, with an amplitude *t* that is proportional to the dispersion (the bandwidth) of the valence electronic states. The formation of an energy gap can be settled as the competition between the Coulomb potential *U* between 3d electrons and the transfer integral *t* of the tight-binding approximation of 3d electrons between neighboring atoms. Therefore, the bandgap can be described by the *U*, *t* and an extra z term that denotes the number of nearest neighbor atoms as [6]:

$$\mathbf{E}\_{\rho\rho} = \mathbf{U} - \mathbf{2}zt \tag{1}$$

it is within the realm of DFT, thus does not require significant effort to be implemented in the existing DFT codes and its computational cost is only slightly higher than that of normal DFT computations. This "*U*" correction can be added to the local and semilocal density functionals offering LDA+U and GGA+U computational operations. The basic role of the *U* correction is to treat the strong on-site Coulomb interaction of localized electrons with an additional Hubbard-like term. The Hubbard Hamiltonian describes the strongly correlated electronic states (*d* and *f* orbitals), while treating the rest of the valence electrons by the normal DFT approximations. For practical implementation of DFT+U in computational chemistry, the strength of the on-site interactions is described by a couple of parameters: the on-site Coulomb term *U* and the site exchange term *J*. These parameters "*U* and *J*" can be extracted from *ab initio* calculations, but usually are obtained semiempirically. The implementation of the DFT+U requires a clear understanding of the approximations it is based on and a precise evaluation of the conditions under which it can be expected to provide accurate quantitative

The LDA+U method is widely implemented to correct the approximate DFT xc functional. The LDA+U works in the same way as the standard LDA method to describe the valence electrons, and only for the strongly correlated electronic states (the *d* and *f* orbitals), the Hubbard model is implemented for a more accurate modeling. Therefore, the total energy of the system (ELDA+*U*) is typically the summation of the standard LDA energy functional (EHub) for all the states and the energy of the Hubbard functional that describes the correlated states. Because of the additive Hubbard term, there will be a double counting error for the correlated states; therefore, a "double-counting" term (Edc) must be deducted from the LDA's total energy that

Therefore, it can be understood that the LDA+U is more like a substitution of the mean-field electronic interaction contained in the approximate xc functional. Nonetheless, the Edc term is not uniquely defined for each system and various formulations can be applied to different systems. The most dominant of these formulations is the FLL formulation [10–12]. It is based on the implementation of fully localized limit (FLL) on systems with more localized electrons on atomic orbitals. The reason for this formulation popularity is due to its ability to expand the width of the Kohn Sham (KS) orbitals and to effectively capture Mott localization. Based

> ,*σ*′ *nm <sup>l</sup> nm*′ *l*′ − *Ul* \_\_ <sup>2</sup> *nl*

m are the localized orbitals occupation numbers identified by the atomic site index

I, state index m, and by the spin *σ*. In Eq. (4), the right-hand side second and third terms are the Hubbard and double-counting terms, specified in Eq. (3). The dependency on the occupation number is expected as the Hubbard correction is only applied to the states that are most disturbed by correlation effects. The occupation number is calculated as the projection

(*nl* − 1)

*Ul* \_\_ <sup>2</sup> ∑*<sup>m</sup>*,*σ*≠*m*′ *<sup>I</sup>* }] − Edc[nIσ] (3)

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020 7

] (4)

describes the electronic interactions in a mean field kind of way [5].

ELDA + *U*[ρ(r)] = ELDA[ρ(r)] + EHub[{*nmm*

on this formulation, the LDA+U can be written as:

of occupied KS orbitals on the states of a localized basis set:

*ELDA*+*<sup>U</sup>* [*ρ*(*r*)] + ∑*l*[

where *nm l*

predictions [5, 6].

Since the problem is rooted down to the band model of the systems, alternative models have been formulated to describe the correlated systems. One of the simplest models is the "Hubbard" model [9]. The Hubbard model is able to include the so-called "on-site repulsion," which stems from the Coulomb repulsion between electrons at the same atomic orbitals, and can therefore explain the transition between the conducting and insulating behavior of these systems. Based on this model, new Hamiltonian can be formulated with an additive Hubbard term that explicitly describes electronic interactions. The additive Hubbard Hamiltonian can be written in its simplest form as follows [6]:

$$H\_{\rm halo} = \text{tr}\sum\_{\{\langle\rangle\}\sigma} \left(c\_{\downarrow\sigma}^{\uparrow}c\_{\downarrow\sigma} + \text{h.c.}\right) + \text{LI}\sum\_{i} \mathfrak{n}\_{\downarrow\uparrow}\mathfrak{n}\_{\downarrow\downarrow} \tag{2}$$

As predicted, the Hubbard Hamiltonian should be dependent on the two terms *t* and *U*, with 〈i.j〉 denoting nearest-neighbor atomic sites and ci † , cj , and ni are electronic creation, annihilation, and number operators for electrons of spin on site *i,* respectively. The hopping amplitude *t* is proportional to the bandwidth (dispersion) of the valence electrons, while the on-site Coulomb repulsion term *U* is proportional to the product of the occupation numbers of atomic states on the same site [6]. The system's insulating character develops when electrons do not have sufficient energy to overcome the repulsion potential of other electrons on neighbor sites, i.e., when *t* « *U*. The ability of the DFT scheme to predict electronic properties is fairly accurate when *t* » *U*, while for large *U* values, DFT significantly fails the HF method, which describes the electronic ground state with a variationally optimized single determinant, that cannot capture the physics of Mott insulators.

#### **2.3. DFT+**U

Inspired by the Hubbard model, DFT+U method is formulated to improve the description of the ground state of correlated systems. The main advantage of the DFT+U method is that it is within the realm of DFT, thus does not require significant effort to be implemented in the existing DFT codes and its computational cost is only slightly higher than that of normal DFT computations. This "*U*" correction can be added to the local and semilocal density functionals offering LDA+U and GGA+U computational operations. The basic role of the *U* correction is to treat the strong on-site Coulomb interaction of localized electrons with an additional Hubbard-like term. The Hubbard Hamiltonian describes the strongly correlated electronic states (*d* and *f* orbitals), while treating the rest of the valence electrons by the normal DFT approximations. For practical implementation of DFT+U in computational chemistry, the strength of the on-site interactions is described by a couple of parameters: the on-site Coulomb term *U* and the site exchange term *J*. These parameters "*U* and *J*" can be extracted from *ab initio* calculations, but usually are obtained semiempirically. The implementation of the DFT+U requires a clear understanding of the approximations it is based on and a precise evaluation of the conditions under which it can be expected to provide accurate quantitative predictions [5, 6].

flaw of the band theory was pointed out by Sir Nevil Mott, who emphasized that interelectron forces cannot be neglected, which lead to the existence of the bandgap in these falsely predicted conductors (Mott insulators) [8]. In these "metal-insulators," the bandgap exists between bands of like character i.e., between suborbitals of the same orbitals, such as 3d character, which originates from crystal field splitting or Hund's rule. The insulating character of the ground state stems from the strong Coulomb repulsion between electrons that forces them to localize in atomiclike orbitals (Mott localization). This Coulomb potential, responsible for localization, is described by the term "*U*," and when electrons are strongly localized, they cannot move freely between atoms and rather jump from one atom to another by a "hopping" mechanism between neighbor atoms, with an amplitude *t* that is proportional to the dispersion (the bandwidth) of the valence electronic states. The formation of an energy gap can be settled as the competition between the Coulomb potential *U* between 3d electrons and the transfer integral *t* of the tight-binding approximation of 3d electrons between neighboring atoms. Therefore, the bandgap can be described by

6 Density Functional Calculations - Recent Progresses of Theory and Application

the *U*, *t* and an extra z term that denotes the number of nearest neighbor atoms as [6]:

be written in its simplest form as follows [6]:

*HHub* = *t* ∑⟨*i*,*j*⟩,*<sup>σ</sup>* (*ci*,*<sup>σ</sup>*

with 〈i.j〉 denoting nearest-neighbor atomic sites and ci

nant, that cannot capture the physics of Mott insulators.

**2.3. DFT+**U

E*gap* = *U* − 2*zt* (1)

Since the problem is rooted down to the band model of the systems, alternative models have been formulated to describe the correlated systems. One of the simplest models is the "Hubbard" model [9]. The Hubbard model is able to include the so-called "on-site repulsion," which stems from the Coulomb repulsion between electrons at the same atomic orbitals, and can therefore explain the transition between the conducting and insulating behavior of these systems. Based on this model, new Hamiltonian can be formulated with an additive Hubbard term that explicitly describes electronic interactions. The additive Hubbard Hamiltonian can

As predicted, the Hubbard Hamiltonian should be dependent on the two terms *t* and *U*,

annihilation, and number operators for electrons of spin on site *i,* respectively. The hopping amplitude *t* is proportional to the bandwidth (dispersion) of the valence electrons, while the on-site Coulomb repulsion term *U* is proportional to the product of the occupation numbers of atomic states on the same site [6]. The system's insulating character develops when electrons do not have sufficient energy to overcome the repulsion potential of other electrons on neighbor sites, i.e., when *t* « *U*. The ability of the DFT scheme to predict electronic properties is fairly accurate when *t* » *U*, while for large *U* values, DFT significantly fails the HF method, which describes the electronic ground state with a variationally optimized single determi-

Inspired by the Hubbard model, DFT+U method is formulated to improve the description of the ground state of correlated systems. The main advantage of the DFT+U method is that

† , cj

† *cj*,*<sup>σ</sup>* + *h*.*c*.) + *U* ∑*<sup>i</sup> ni*,<sup>↑</sup> *ni*,<sup>↓</sup> (2)

are electronic creation,

, and ni

The LDA+U method is widely implemented to correct the approximate DFT xc functional. The LDA+U works in the same way as the standard LDA method to describe the valence electrons, and only for the strongly correlated electronic states (the *d* and *f* orbitals), the Hubbard model is implemented for a more accurate modeling. Therefore, the total energy of the system (ELDA+*U*) is typically the summation of the standard LDA energy functional (EHub) for all the states and the energy of the Hubbard functional that describes the correlated states. Because of the additive Hubbard term, there will be a double counting error for the correlated states; therefore, a "double-counting" term (Edc) must be deducted from the LDA's total energy that describes the electronic interactions in a mean field kind of way [5].

$$\mathcal{E}\_{\rm LDA} + \mathcal{L}[\mathfrak{p}(\mathbf{r})] = \mathcal{E}\_{\rm LDA}[\mathfrak{p}(\mathbf{r})] + \mathcal{E}\_{\rm Hub} \left\{ \left\{ \mathfrak{n}\_{nm}^{\rm hs} \right\} \right\} - \mathcal{E}\_{\rm dc}[\mathfrak{n}^{\rm lr}] \tag{3}$$

Therefore, it can be understood that the LDA+U is more like a substitution of the mean-field electronic interaction contained in the approximate xc functional. Nonetheless, the Edc term is not uniquely defined for each system and various formulations can be applied to different systems. The most dominant of these formulations is the FLL formulation [10–12]. It is based on the implementation of fully localized limit (FLL) on systems with more localized electrons on atomic orbitals. The reason for this formulation popularity is due to its ability to expand the width of the Kohn Sham (KS) orbitals and to effectively capture Mott localization. Based on this formulation, the LDA+U can be written as:

$$E\_{\rm LDA+dl} \left[ \rho(r) \right] + \sum\_{l} \left[ \frac{\rm Ll}{2} \sum\_{m,o:u<,o'} n\_{m}^{l\sigma} n\_{m}^{l\sigma'} - \frac{\rm Ll}{2} n'(n^{l} - 1) \right] \tag{4}$$

where *nm l* m are the localized orbitals occupation numbers identified by the atomic site index I, state index m, and by the spin *σ*. In Eq. (4), the right-hand side second and third terms are the Hubbard and double-counting terms, specified in Eq. (3). The dependency on the occupation number is expected as the Hubbard correction is only applied to the states that are most disturbed by correlation effects. The occupation number is calculated as the projection of occupied KS orbitals on the states of a localized basis set:

$$m\_{m,n'}^{l\sigma} = \sum\_{k,\nu} f\_{k\nu}^{\sigma} \left\langle \Psi\_{l\nu}^{\sigma} \mid \varphi\_{n'}^{l} \right\rangle \left\langle \varphi\_{n}^{\nu} \mid \Psi\_{ln}^{\sigma} \right\rangle \tag{5}$$

correction to solve the bandgap problem is necessary for predicting the properties of transition metal oxides. **Figure 1** shows the effect of *U* potential on correcting the failure of DFT to predict correct bandgaps for strongly correlated materials. Note the underestimation of the bandgap in

**Figure 1.** Comparison of theoretical DOS calculated by LDA and LDA+U for (a) MnO and (b) FeO. Adapted with

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020 9

From the case studies and examples presented within this chapter, one can intuitively conclude that corrective functional LDA+U is particularly dependent on the numerical value of the effective potential *U*eff, which is generally referred to in literature as "*U*" for simplicity. However, the *U* value is not known and practically is often tuned semiempirically to make a good agreement with experimental or higher level computational results. However, the semiempirical way of evaluating the *U* parameter fails to capture its dependence on the volume, structure, or the magnetic phase of the crystal, and also does not permit the capturing of changes in the on-site electronic interaction under changing physical conditions, such as chemical reactions. In order to get full advantage of this method, different procedures have been addressed to determine the Hubbard *U* from first principles [13]. In these procedures, the *U* parameter can generally be calculated using a self-consistent and basis set in an independent way. These different *ab initio* approaches for calculating *U* have been applied to different material systems, where the *U* value is calculated for individual atoms. For each atom, the *U* value is found to be dependent on the material specific parameters, including its position in the lattice and the structural and magnetic properties of the crystal, and also dependent on the localized basis set employed to describe the on-site occupation in the Hubbard functional. Therefore, the value of effective interactions should be re-computed for each type of material

case of MnO and the incorrect prediction of the metallic behavior of FeO [15].

**3.1. Optimizing the** *U* **value**

permission from [15].

where the coefficients *f kv σ* represent the occupations of KS states (labeled by k-point, band, and spin indices), determined by the Fermi-Dirac distribution of the corresponding single-particle energy eigen values. According to this formulation, the fractional occupations of localized orbitals is reduced, while assisting the Mott localization of electrons on particular atomic states [5].

Although the above approach described in Eq. (4) is able to capture Mott localization, it is not invariant under rotation of the atomic orbital basis set employed to define the occupation number of *n* in Eq. (5). This variation makes the calculations performed unfavorably dependent on the unitary transformation of the chosen localized basis set. Therefore, "rotationally invariant formulation" is introduced, which is unitary-transformation invariant of LDA+U [12]. In this formulation, the electronic interactions are fully orbital dependent, and thus considered to be the most complete formulation of the LDA+U. However, a simpler formulation that preserves rotational invariance, which is theoretically based on the full rotationally invariant formulation, had proved to work as effectively as the full formulation for most materials [11]. Based on the simplified LDA+U form, it has been customary to utilize, instead of the interaction parameter *U*, an effective *U* parameter: *U*eff = *U*−J, where the "*J*" parameter is known as the exchange interaction term that accounts for Hund's rule coupling. The *U*eff is generally preferred because the *J* parameter is proven to be crucial to describe the electronic structure of certain classes of materials, typically those subject to strong spin-orbit coupling.

#### **3. Practical implementations of the Hubbard correction**

DFT+U is applicable for all open shell orbitals, such as *d* and *f* orbitals for transition metal elements with localized orbitals existing in extended states, as in the case of many strongly correlated materials and perovskites, where localized *3d* or *4f* orbitals are embedded in elongated s-p states. A complicated many-electron problem is made of electrons living in these localized orbitals, where they experience strong correlations among each other and with a subtle coupling with the extended states. Isolating a few degrees of freedom relevant to the correlation is the idea in the Hubbard model, where screened or renormalized Coulomb interaction (*U*) is kept among the localized orbitals' electrons [13]. In other word, the localized orbitals in the bandgap, which are present as localized states (*d*- and *f*-states), are too close to the Fermi energy. From that aspect, the *U* value should be used to push theses states away from the Fermi level, such as that provided by the GGA+U theory, which adds to the Hamiltonian a term that increases the total energy preventing the unwanted delocalization of the *d*- or *f*-electrons, when two *d*- or *f*-electrons are located on the same cation [14]. It is worth mentioning that using too large values of *U* will over-localize the states and lead to an unphysical flattening of the appropriate bands, which unlike fitting to many other properties, will make the fit worse. Also, the increase in the *U* value can cause an overestimation of the lattice constants as well as a wrong estimation of the ground state energy due to the electronic interaction error. Therefore, applying Hubbard

**Figure 1.** Comparison of theoretical DOS calculated by LDA and LDA+U for (a) MnO and (b) FeO. Adapted with permission from [15].

correction to solve the bandgap problem is necessary for predicting the properties of transition metal oxides. **Figure 1** shows the effect of *U* potential on correcting the failure of DFT to predict correct bandgaps for strongly correlated materials. Note the underestimation of the bandgap in case of MnO and the incorrect prediction of the metallic behavior of FeO [15].

#### **3.1. Optimizing the** *U* **value**

*nm*,*m*′

*kv σ*

8 Density Functional Calculations - Recent Progresses of Theory and Application

where the coefficients *f*

strong spin-orbit coupling.

states [5].

*<sup>I</sup>* = ∑*<sup>k</sup>*,*<sup>v</sup> f*

**3. Practical implementations of the Hubbard correction**

*kv σ* ⟨Ψ*kv <sup>σ</sup>* <sup>|</sup>*ϕm*′ *I* ⟩⟨*ϕ<sup>m</sup> <sup>I</sup>* |Ψ*kv σ*

spin indices), determined by the Fermi-Dirac distribution of the corresponding single-particle energy eigen values. According to this formulation, the fractional occupations of localized orbitals is reduced, while assisting the Mott localization of electrons on particular atomic

Although the above approach described in Eq. (4) is able to capture Mott localization, it is not invariant under rotation of the atomic orbital basis set employed to define the occupation number of *n* in Eq. (5). This variation makes the calculations performed unfavorably dependent on the unitary transformation of the chosen localized basis set. Therefore, "rotationally invariant formulation" is introduced, which is unitary-transformation invariant of LDA+U [12]. In this formulation, the electronic interactions are fully orbital dependent, and thus considered to be the most complete formulation of the LDA+U. However, a simpler formulation that preserves rotational invariance, which is theoretically based on the full rotationally invariant formulation, had proved to work as effectively as the full formulation for most materials [11]. Based on the simplified LDA+U form, it has been customary to utilize, instead of the interaction parameter *U*, an effective *U* parameter: *U*eff = *U*−J, where the "*J*" parameter is known as the exchange interaction term that accounts for Hund's rule coupling. The *U*eff is generally preferred because the *J* parameter is proven to be crucial to describe the electronic structure of certain classes of materials, typically those subject to

DFT+U is applicable for all open shell orbitals, such as *d* and *f* orbitals for transition metal elements with localized orbitals existing in extended states, as in the case of many strongly correlated materials and perovskites, where localized *3d* or *4f* orbitals are embedded in elongated s-p states. A complicated many-electron problem is made of electrons living in these localized orbitals, where they experience strong correlations among each other and with a subtle coupling with the extended states. Isolating a few degrees of freedom relevant to the correlation is the idea in the Hubbard model, where screened or renormalized Coulomb interaction (*U*) is kept among the localized orbitals' electrons [13]. In other word, the localized orbitals in the bandgap, which are present as localized states (*d*- and *f*-states), are too close to the Fermi energy. From that aspect, the *U* value should be used to push theses states away from the Fermi level, such as that provided by the GGA+U theory, which adds to the Hamiltonian a term that increases the total energy preventing the unwanted delocalization of the *d*- or *f*-electrons, when two *d*- or *f*-electrons are located on the same cation [14]. It is worth mentioning that using too large values of *U* will over-localize the states and lead to an unphysical flattening of the appropriate bands, which unlike fitting to many other properties, will make the fit worse. Also, the increase in the *U* value can cause an overestimation of the lattice constants as well as a wrong estimation of the ground state energy due to the electronic interaction error. Therefore, applying Hubbard

represent the occupations of KS states (labeled by k-point, band, and

⟩ (5)

From the case studies and examples presented within this chapter, one can intuitively conclude that corrective functional LDA+U is particularly dependent on the numerical value of the effective potential *U*eff, which is generally referred to in literature as "*U*" for simplicity. However, the *U* value is not known and practically is often tuned semiempirically to make a good agreement with experimental or higher level computational results. However, the semiempirical way of evaluating the *U* parameter fails to capture its dependence on the volume, structure, or the magnetic phase of the crystal, and also does not permit the capturing of changes in the on-site electronic interaction under changing physical conditions, such as chemical reactions. In order to get full advantage of this method, different procedures have been addressed to determine the Hubbard *U* from first principles [13]. In these procedures, the *U* parameter can generally be calculated using a self-consistent and basis set in an independent way. These different *ab initio* approaches for calculating *U* have been applied to different material systems, where the *U* value is calculated for individual atoms. For each atom, the *U* value is found to be dependent on the material specific parameters, including its position in the lattice and the structural and magnetic properties of the crystal, and also dependent on the localized basis set employed to describe the on-site occupation in the Hubbard functional. Therefore, the value of effective interactions should be re-computed for each type of material and each type of LDA+U implementation (e.g., based on augmented plane waves, Gaussian functions, etc.). Most programs these days use the method presented by Cococcioni et al. [16], where the values of *U* can be determined through a linear response method [17], in which the response of the occupation of localized states to a small perturbation of the local potential is calculated. The *U* is self-consistently determined, which is fully consistent with the definition of the DFT+U Hamiltonian, making this approach for the potential calculations fully *ab initio*. The value of *U* implemented by Cococcioni et al. is *U*eff = *U*−*J*, where *J* is indirectly assumed to be zero in order to obtain a simplified expression [17]. Nonetheless, *J* can add some additional flexibility to the DFT+U calculations, but it may yield surprising results including reversing the trends previously obtained in the implemented DFT+U calculations [18].

• Within the study of BiMnO<sup>3</sup>

GGA+U method, calculations show that distortions of the MnO6

distances, and thus overly expands the MnO6

number of k-points needs to be increased.

the same system can be realized:

magnetic moment for negative *U*eff [5].

**3.2. Variation of** *U* **with calculation methods and parameters**

sidered the main unit in the crystal structure, are very sensitive to the value of the Coulomb repulsion *U*. The study showed that large *U* value decreases the *3d*-*2p* hybridization, and therefore decreases the bonding effects, which in turn distortion increases the short Mn-O

The parameters assigned for DFT calculations can significantly affect the choice of the optimum *U* value. These parameters include pseudopotentials, basis sets, the cutoff energy, and k-point sampling. As pseudopotentials are used to reduce computational time by replacing the full electron system in the Columbic potential by a system only taking explicitly into account the "valence" electrons [28], the pseudopotential will strongly affect the *U* value. Thus, calculations have to be converged very well with respect to the cutoff energy and k-point sampling, while taking into account that the symmetry used in DFT+U calculations, because adding the *U* parameter often lowers the crystallographic symmetry, thereby the

Not only is the *U* value affected by the parameters applied, but it is also strongly dependent on the DFT method used. In a published review [29], a comparison of different calculated *U* values using different approaches was highlighted for several transition metal oxides. It was reported that with small *U* values, the electrons were still not localized, and that the *U* value depends on the used exchange-correlation functionals (LDA or GGA), the pseudopotential, the fitted experimental properties, and projection operators [30]. In the computational study of strongly correlated systems, it can be usually found in literature that researchers refer to the utilization of (DFT+U) method in their calculations, which may include generalized gradient approximation (GGA+U) [8], local density approximation (LDA+U) [31], or both [32]. To be able to choose the proper method of calculation for a studied system, one should know the limitation of each of the two methods for that specific system and to what extent is each method approved to be closer to the experimental values. Knowing the optimum *U* value can be reached empirically by applying different values of *U* for either GGA or LDA. From the following list of examples from literature, an assessment of the performance of different values of *U* when applied to both GGA and LDA for

• Griffin et al. [33] studied the FeAs crystal comparing the GGA+U and LDA+U levels of accuracy, using *U*eff = −2 to 4 eV. The results showed that for the bond distances and angles in the crystal, the GGA+U gave results close to the experimental values when *U* ≤ 1 eV, whereas using LDA, the structural properties were poorly predicted. It was observed that increasing the value of *U* in the GGA+U increased the stabilization energy for antiferromagnetic ordering. Both GGA+U and LDA+U overestimated the value of magnetic moment. However, only the GGA+U could attain the experimental values of

octahedral [27].

that has strongly distorted perovskite structure with the

octahedral, which is con-

11

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020

Despite the limitations of choosing the *U* value semiempirically for systems, where variations of on-site electronic interactions are present, it is found to be the most common practice used in literature, where the value of *U* is usually compared to the experimental bandgap. This semiempirical trend in practical implementation of *U* is present because of the significant computational cost of *ab initio* calculation of *U*, and in the cases of studying static physical properties, the results of computed *U* are not necessarily found to be better than the empirical ones. Within this practice, however, caution should be taken while pursuing the semiempirical method [19]. If it will be possible to describe all the relevant aspects of a system, except the bandgap, with a reasonable *U*, one might then look into using a scissor operator or rigid shift to the bandgap [20, 21]. However, in particular cases, where calculations aim at understanding catalysis, it is natural to choose *U* to fit the energy of the oxidation-reduction, as catalysis is controlled by energy differences [14]. Conversely, one of the possible solutions is to venture into a negative value for the Hubbard *U* parameter, there is no obvious physical rationale for that yet, but the results may match with experiment for both the magnetic moment and structural properties, as illustrated later in this chapter.

To elaborate the numerical *U* tuning procedure, three quick examples are presented below that can show the correlation between the value of *U* and the predicted physical properties:


• Within the study of BiMnO<sup>3</sup> that has strongly distorted perovskite structure with the GGA+U method, calculations show that distortions of the MnO6 octahedral, which is considered the main unit in the crystal structure, are very sensitive to the value of the Coulomb repulsion *U*. The study showed that large *U* value decreases the *3d*-*2p* hybridization, and therefore decreases the bonding effects, which in turn distortion increases the short Mn-O distances, and thus overly expands the MnO6 octahedral [27].

#### **3.2. Variation of** *U* **with calculation methods and parameters**

and each type of LDA+U implementation (e.g., based on augmented plane waves, Gaussian functions, etc.). Most programs these days use the method presented by Cococcioni et al. [16], where the values of *U* can be determined through a linear response method [17], in which the response of the occupation of localized states to a small perturbation of the local potential is calculated. The *U* is self-consistently determined, which is fully consistent with the definition of the DFT+U Hamiltonian, making this approach for the potential calculations fully *ab initio*. The value of *U* implemented by Cococcioni et al. is *U*eff = *U*−*J*, where *J* is indirectly assumed to be zero in order to obtain a simplified expression [17]. Nonetheless, *J* can add some additional flexibility to the DFT+U calculations, but it may yield surprising results including reversing

Despite the limitations of choosing the *U* value semiempirically for systems, where variations of on-site electronic interactions are present, it is found to be the most common practice used in literature, where the value of *U* is usually compared to the experimental bandgap. This semiempirical trend in practical implementation of *U* is present because of the significant computational cost of *ab initio* calculation of *U*, and in the cases of studying static physical properties, the results of computed *U* are not necessarily found to be better than the empirical ones. Within this practice, however, caution should be taken while pursuing the semiempirical method [19]. If it will be possible to describe all the relevant aspects of a system, except the bandgap, with a reasonable *U*, one might then look into using a scissor operator or rigid shift to the bandgap [20, 21]. However, in particular cases, where calculations aim at understanding catalysis, it is natural to choose *U* to fit the energy of the oxidation-reduction, as catalysis is controlled by energy differences [14]. Conversely, one of the possible solutions is to venture into a negative value for the Hubbard *U* parameter, there is no obvious physical rationale for that yet, but the results may match with experiment for both the magnetic moment and struc-

To elaborate the numerical *U* tuning procedure, three quick examples are presented below that can show the correlation between the value of *U* and the predicted physical properties:

• The compilation of the correlated nature of cobalt *3d* electrons in the theoretical studies of

• From the study made by Lu and Liu [26] on cerium compounds presented some characteristics for *U* values for Ce atoms in different configurations as isolated atoms and ions.

significantly affect the value of *U* and that when ions are isolated, the values are much

Hx O7

clusters, or CeO2

) does not

 gives a good picture of the significant difference in the *U* value with the difference in most of the calculated properties. The variety of *U* values have been used ranging from 2 to 6 for the properties including bandgap [22], oxidation energy [23], and structural parameters [24], which affect the choice of the value of *U* for each of these properties uniquely. The calculated bandgap at the generalized gradient approximation (GGA)+U agrees well with the experimental value of 1.6 eV. On the other hand, the calculated value using the PBE0 hybrid functional (3.42 eV) is highly overestimated, due to neglecting the screening

the trends previously obtained in the implemented DFT+U calculations [18].

10 Density Functional Calculations - Recent Progresses of Theory and Application

tural properties, as illustrated later in this chapter.

problem of the Hartree-Fock approximation [25].

larger (close to 15 for Ce2.5+ and 18 eV for Ce3.5+).

They illustrated that the ion charge (Ce atoms, Ce in Ce3

Co3 O4 The parameters assigned for DFT calculations can significantly affect the choice of the optimum *U* value. These parameters include pseudopotentials, basis sets, the cutoff energy, and k-point sampling. As pseudopotentials are used to reduce computational time by replacing the full electron system in the Columbic potential by a system only taking explicitly into account the "valence" electrons [28], the pseudopotential will strongly affect the *U* value. Thus, calculations have to be converged very well with respect to the cutoff energy and k-point sampling, while taking into account that the symmetry used in DFT+U calculations, because adding the *U* parameter often lowers the crystallographic symmetry, thereby the number of k-points needs to be increased.

Not only is the *U* value affected by the parameters applied, but it is also strongly dependent on the DFT method used. In a published review [29], a comparison of different calculated *U* values using different approaches was highlighted for several transition metal oxides. It was reported that with small *U* values, the electrons were still not localized, and that the *U* value depends on the used exchange-correlation functionals (LDA or GGA), the pseudopotential, the fitted experimental properties, and projection operators [30]. In the computational study of strongly correlated systems, it can be usually found in literature that researchers refer to the utilization of (DFT+U) method in their calculations, which may include generalized gradient approximation (GGA+U) [8], local density approximation (LDA+U) [31], or both [32]. To be able to choose the proper method of calculation for a studied system, one should know the limitation of each of the two methods for that specific system and to what extent is each method approved to be closer to the experimental values. Knowing the optimum *U* value can be reached empirically by applying different values of *U* for either GGA or LDA. From the following list of examples from literature, an assessment of the performance of different values of *U* when applied to both GGA and LDA for the same system can be realized:

• Griffin et al. [33] studied the FeAs crystal comparing the GGA+U and LDA+U levels of accuracy, using *U*eff = −2 to 4 eV. The results showed that for the bond distances and angles in the crystal, the GGA+U gave results close to the experimental values when *U* ≤ 1 eV, whereas using LDA, the structural properties were poorly predicted. It was observed that increasing the value of *U* in the GGA+U increased the stabilization energy for antiferromagnetic ordering. Both GGA+U and LDA+U overestimated the value of magnetic moment. However, only the GGA+U could attain the experimental values of magnetic moment for negative *U*eff [5].

• Cerium oxides (CeO2 and Ce2 O3 ) were tested by Christoph et al. [34] comparing GGA+U and LDA+U level of theory meanwhile studying the effect of the *U*eff value on the calculated properties. It was found that the value of *U*eff is dependent on the property under examination. The sensitivity toward *U*eff values was especially high for properties of Ce2 O3; because it has an electron in the *4f* orbital, which is sensitive to the change in the effective on-site Coulomb repulsion due to the strong localization, in contrary to the CeO2 that has an empty *4f* orbital. The GGA+U showed an acceptable agreement with experiment at lower energies of *U*eff than LDA did, with values of *U*eff 2.5–3.5 eV for LDA+U and 1.5–2.0 eV for GGA+U, which can be due to the more accurate treatment of correlation effects within the GGA potential. On the other hand, the structural properties were better represented by the LDA+U method for CeO2 . Regarding Ce2 O3 electronic structure, both LDA+U and GGA+U results showed a similarly good accuracy, while for the calculated reaction energies, LDA+U results showed better accuracy. [34]

these electrons are observed to be localized in their orbitals due to strong correlations [38], whereas computationally, DFT approximated xc functionals tend to overly delocalize them while over-stabilizing metallic ground states, and thus underestimating the bandgap for semiconductors, and may reach false prediction of metallic behavior for systems like the Mott insulators. *U* can induce electronic localization due to the explicit account for the on-site electronic interactions. Another common problem that DFT calculations can impose is the prediction of the properties of materials with defects, as the underestimation of the bandgap by DFT can cause the conduction band (CB) or the valence band (VB) to kind of mask the true defect states. This is because defects can cause unpaired electrons and holes to form, which are overly delocalized by DFT, as it attempts to reduce the Coulomb repulsion due to self-

that the description of the distribution of electrons in the unit cell, created from oxygen vacancies and hydrogen impurities, is wrongly predicted using GGA-PBE scheme of DFT calculations. In the case of oxygen vacancies, their calculations predicted a 2.6 eV bandgap, which is about 0.6 eV smaller than that reported experimentally. The electrons left in the system upon vacancy formation are completely delocalized over the entire cell. These electrons are incorrectly shared over all the Ti atoms of the cell, and as a result, the atomic displacements around the vacancies are predicted to be symmetric. All these findings indicate the difficulty of DFT methods to describe the properties of defects in wide bandgap metal oxides. Also, the accuracy of the description of the electronic structure of the partially reduced oxide systems

quently studied in literature. Typically, in the anatase and rutile phases, computational studies encountered the problem of considerable underestimation of the bandgap, which presented a barrier in the prediction of further related properties. Titania is widely studied in various photoelectrochemical applications, and accurate theoretical assessment is required to be able to enhance its catalytic properties. In addition, to further improve the properties of TiO2

a photocatalyst, an optimization of the band structure is required, including narrowing the bandgap (Eg) to improve visible light absorption, and proper positioning of the valence band (VB) and the conduction band (CB) [41]. Efforts on narrowing the bandgap of the TiO<sup>2</sup>

been done through doping with metallic and nonmetallic elements that typically replace Ti or O atoms, and thus change the position of the VB and or the CB leading to a change in the bandgap [42]. In the following subsections, titania will be used as an example to assess the effect of *U* correction by presenting results from literature for both pristine and doped cases. We will monitor the behavior of the materials before and after *U* correction, while assessing

 *with U correction*

Regarding the electronic structure of titania, the bandgap was underestimated by the standard DFT, while found to be overestimated when the hybrid functional Heyd-Scuseria-Ernzerhof (HSE06) was applied. However, the bandgap prediction was markedly improved by adding

the significance of the *U* correction for correct prediction of the material's properties.

–both pristine and doped–is one of the examples that is fre-

, where they showed

13

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020

as

have

Ref. [39] discusses an example of this problem studying the anatase TiO2

was reviewed and discussed within the first principle methods [40].

interaction error.

The electronic structure of TiO2

*3.3.1. The Bandgap problem: pristine TiO2*

• Sun et al. [35] studied PuO2 and Pu2 O3 oxides using both GGA+U and LDA+U methods. Although PuO2 is known to be an insulator [36], its ground state was reported experimentally to be an antiferromagnetic phase [37]. For PuO2 , at *U* = 0, the ground state was a ferromagnetic metal, which is different from experimental results. Upon increasing the amplitude of *U* to 1.5 eV, the LDA+*U* and GGA+U calculations correctly predicted the antiferromagnetic insulating ground-state characteristics. For the lattice parameters, it was found that higher values of *U* (*U* = 4 eV) were needed with the LDA+U than for GGA+U. At *U* = 4 eV, it is expected that both LDA+U and the GGA+U would show a satisfactory prediction of the ground-state atomic structure of Pu2 O3 . However, the study showed that above the metallic-insulating transition, the reaction energy decreases with increasing *U* for the LDA and the GGA schemes. Therefore, for both Pu2 O5 and PuO2 , the LDA+U and GGA+U approaches, with *U* as large as 6 eV, failed to describe the electronic structure correctly. When the energy gap increases, the electrons gain more localization that causes a difficulty of making any new reactions, consequently increasing the reaction energy. When *U* exceeds 4 eV, the conduction band electrons approximately considered to be ionized; thus, the atoms (cores or ions) have got a better chance to react with other atoms which, resulting in a reduction in the reaction energy.

As noticed in the previous studies, the *U* value is material dependent, besides being variable among the level of theory used. In general, the more localized the system is, the more sensitive it is to the value of *U*. The estimated value of *U* for a system of material using a specific level of calculation should not be extended to another system; rather, it should be recomputed each time for each material and even upon change of the level of calculation. Researchers will need to perform calculations using different *U* values within different xc functionals to get the best prediction of the calculated properties in comparison with the experimental measurements or with the other computational results as benchmark.

#### **3.3. The effect of** *U* **on pure and defected systems**

The chemical properties of transition-metal systems with localized electrons, mainly within *d* or *f* orbitals, are typically governed by the properties of the valence electrons. Experimentally, these electrons are observed to be localized in their orbitals due to strong correlations [38], whereas computationally, DFT approximated xc functionals tend to overly delocalize them while over-stabilizing metallic ground states, and thus underestimating the bandgap for semiconductors, and may reach false prediction of metallic behavior for systems like the Mott insulators. *U* can induce electronic localization due to the explicit account for the on-site electronic interactions. Another common problem that DFT calculations can impose is the prediction of the properties of materials with defects, as the underestimation of the bandgap by DFT can cause the conduction band (CB) or the valence band (VB) to kind of mask the true defect states. This is because defects can cause unpaired electrons and holes to form, which are overly delocalized by DFT, as it attempts to reduce the Coulomb repulsion due to selfinteraction error.

• Cerium oxides (CeO2

method for CeO2

though PuO2

and Ce2

12 Density Functional Calculations - Recent Progresses of Theory and Application

. Regarding Ce2

to be an antiferromagnetic phase [37]. For PuO2

the ground-state atomic structure of Pu2

reduction in the reaction energy.

and the GGA schemes. Therefore, for both Pu2

with the other computational results as benchmark.

**3.3. The effect of** *U* **on pure and defected systems**

sults showed better accuracy. [34]

• Sun et al. [35] studied PuO2

O3

and LDA+U level of theory meanwhile studying the effect of the *U*eff value on the calculated properties. It was found that the value of *U*eff is dependent on the property under examina-

it has an electron in the *4f* orbital, which is sensitive to the change in the effective on-site

*4f* orbital. The GGA+U showed an acceptable agreement with experiment at lower energies of *U*eff than LDA did, with values of *U*eff 2.5–3.5 eV for LDA+U and 1.5–2.0 eV for GGA+U, which can be due to the more accurate treatment of correlation effects within the GGA potential. On the other hand, the structural properties were better represented by the LDA+U

showed a similarly good accuracy, while for the calculated reaction energies, LDA+U re-

netic metal, which is different from experimental results. Upon increasing the amplitude of *U* to 1.5 eV, the LDA+*U* and GGA+U calculations correctly predicted the antiferromagnetic insulating ground-state characteristics. For the lattice parameters, it was found that higher values of *U* (*U* = 4 eV) were needed with the LDA+U than for GGA+U. At *U* = 4 eV, it is expected that both LDA+U and the GGA+U would show a satisfactory prediction of

O3

metallic-insulating transition, the reaction energy decreases with increasing *U* for the LDA

proaches, with *U* as large as 6 eV, failed to describe the electronic structure correctly. When the energy gap increases, the electrons gain more localization that causes a difficulty of making any new reactions, consequently increasing the reaction energy. When *U* exceeds 4 eV, the conduction band electrons approximately considered to be ionized; thus, the atoms (cores or ions) have got a better chance to react with other atoms which, resulting in a

As noticed in the previous studies, the *U* value is material dependent, besides being variable among the level of theory used. In general, the more localized the system is, the more sensitive it is to the value of *U*. The estimated value of *U* for a system of material using a specific level of calculation should not be extended to another system; rather, it should be recomputed each time for each material and even upon change of the level of calculation. Researchers will need to perform calculations using different *U* values within different xc functionals to get the best prediction of the calculated properties in comparison with the experimental measurements or

The chemical properties of transition-metal systems with localized electrons, mainly within *d* or *f* orbitals, are typically governed by the properties of the valence electrons. Experimentally,

O5

and PuO2

is known to be an insulator [36], its ground state was reported experimentally

tion. The sensitivity toward *U*eff values was especially high for properties of Ce2

Coulomb repulsion due to the strong localization, in contrary to the CeO2

O3

O3

and Pu2

) were tested by Christoph et al. [34] comparing GGA+U

electronic structure, both LDA+U and GGA+U results

oxides using both GGA+U and LDA+U methods. Al-

, at *U* = 0, the ground state was a ferromag-

. However, the study showed that above the

, the LDA+U and GGA+U ap-

O3; because

that has an empty

Ref. [39] discusses an example of this problem studying the anatase TiO2 , where they showed that the description of the distribution of electrons in the unit cell, created from oxygen vacancies and hydrogen impurities, is wrongly predicted using GGA-PBE scheme of DFT calculations. In the case of oxygen vacancies, their calculations predicted a 2.6 eV bandgap, which is about 0.6 eV smaller than that reported experimentally. The electrons left in the system upon vacancy formation are completely delocalized over the entire cell. These electrons are incorrectly shared over all the Ti atoms of the cell, and as a result, the atomic displacements around the vacancies are predicted to be symmetric. All these findings indicate the difficulty of DFT methods to describe the properties of defects in wide bandgap metal oxides. Also, the accuracy of the description of the electronic structure of the partially reduced oxide systems was reviewed and discussed within the first principle methods [40].

The electronic structure of TiO2 –both pristine and doped–is one of the examples that is frequently studied in literature. Typically, in the anatase and rutile phases, computational studies encountered the problem of considerable underestimation of the bandgap, which presented a barrier in the prediction of further related properties. Titania is widely studied in various photoelectrochemical applications, and accurate theoretical assessment is required to be able to enhance its catalytic properties. In addition, to further improve the properties of TiO2 as a photocatalyst, an optimization of the band structure is required, including narrowing the bandgap (Eg) to improve visible light absorption, and proper positioning of the valence band (VB) and the conduction band (CB) [41]. Efforts on narrowing the bandgap of the TiO<sup>2</sup> have been done through doping with metallic and nonmetallic elements that typically replace Ti or O atoms, and thus change the position of the VB and or the CB leading to a change in the bandgap [42]. In the following subsections, titania will be used as an example to assess the effect of *U* correction by presenting results from literature for both pristine and doped cases. We will monitor the behavior of the materials before and after *U* correction, while assessing the significance of the *U* correction for correct prediction of the material's properties.

#### *3.3.1. The Bandgap problem: pristine TiO2 with U correction*

Regarding the electronic structure of titania, the bandgap was underestimated by the standard DFT, while found to be overestimated when the hybrid functional Heyd-Scuseria-Ernzerhof (HSE06) was applied. However, the bandgap prediction was markedly improved by adding the Hubbard *U* correction. The obtained band structures using GGA-PBE showed bandgaps of 2.140 and 1.973 eV for anatase and rutile, respectively. However, upon applying the localization of the excess electronic charge using +U correction, the predicted bandgaps are accurate and in a good agreement with the experimental and the computationally expensive hybrid functional (HSE06) results [43], **Figure 2**. In another study, for rutile TiO2, the prediction of the experimental bandgap is achieved with a *U* value of 10 eV, whereas the crystal and electronic structures were better described with *U* < 5 eV [19].

structural, and optical properties of TiO2

*3.3.2. Doped-TiO2*

The intrinsic defects in TiO2

charge mobility [49].

[48]. The Ti vacancy effect on the bandgap (E<sup>g</sup>

the valence band, and switching the TiO2

 *with U correction*

polymorphs by applying the *U* correction for the

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020 15

oxygen's *2p* orbitals and titanium's *3d* orbitals [46]. In order to correct the bandgap, while avoiding the use of large *U* values and the bonding problem, Ataei et al. [47] reported that with values of 3.5 eV for both O *2p* and Ti *3d*-states, the results for the lattice constants, band-

In a recent study [40], a comparison was performed to elucidate the effect of different *U* values in representing the bandgap states produced by interstitial hydrogen atom and oxygen vacancy within the bulk Ti anatase structure. The dependence on the method used was observed, beside the value of *U* within GGA+U scheme, see **Figure 4**. When the *U* correction was not applied, the bandgap is underestimated, as expected, and the electrons caused by the oxygen vacancy or the hydrogen impurity are fully delocalized and have conduction band character. Upon applying the *U* correction, the states start to localize and are became deeply localized in the gap with increasing the value of *U*. In all these calculations, the Hubbard *U* correction was applied for the Ti *3d* orbitals only; by applying the correction for the *2p* oxygen

**Figure 3.** Calculated c/a ratio vs. U value. Reproduced from [19], with the permission of AIP Publishing.

cheap method to guide researchers in choosing the defect position in the solid crystal. The oxygen vacancy in the rutile crystal was investigated [48] using the DFT+U with *U* value of 4.0 eV, indicating that oxygen vacancy in the rutile crystal introduces four local states with two occupied and two unoccupied states, with no change in the bandgap (2.75 eV)

*U* values of 7.2 eV. It was found that Ti vacancy caused ferromagnetism besides widening

(vacancies) have been computationally studied, providing a fast-

) was also studied [49] using the GGA+U with

from n-type to p-type semiconductor with higher

orbitals with *U* = 3.5, the results were in agreement with previous results [47].

gap, and gap states are in good agreement with the experimental reports.

**Figure 2.** Total DOS of pure TiO<sup>2</sup> (anatase and rutile) [43]. Copyright (2014)—American Chemical Society.

Dompablo et al. [19] compared the effect of the *U* parameter value (0 < *U* < 10 eV) within the LDA+U and the GGA+U on the calculated properties of anatase TiO2 . Both LDA+U and GGA+U required a small value of *U* (3 and 6 eV, respectively) to reproduce the experimental measurements, **Figure 3**. However, using very large *U* values leads to mismatching, where the lattice parameters (a and c) and the volume of unit cell are increasing with increasing *U*, due to the Coulomb repulsion increase. Note that standard DFT and the hybrid functional HSE06 failed to calculate the crystal lattice.

On the other hand, the calculated bandgap within the GGA+U and LDA+U methods was found to be in better agreement with experiments compared to the conventional GGA or LDA, with small difference in the required *U* value. The bandgap was shown to increase by increasing the *U* value till 8.5 eV, which gave a result close to the experimental bandgap and in agreement with those obtained in previous DFT studies [44]. For values of *U* larger than 8.5, the bandgap was overestimated. It is worth to mention that this value (8.5 eV) is considered high when compared to other *U* values for other transition metal oxides [29]. In all these calculations, the Hubbard *U* parameter was used for the *d* or *f* orbitals of transition metals. However, when the Ti-O bonding is considered, while applying the correction only to the *3d*-states, it can be estimated that this correction might have an influence over the Ti−O covalent bonding, where the Ti states are shifted and the *2p* states of oxygen are not changed [45]. In this regard, several first principle calculations were derived to study the electronic, structural, and optical properties of TiO2 polymorphs by applying the *U* correction for the oxygen's *2p* orbitals and titanium's *3d* orbitals [46]. In order to correct the bandgap, while avoiding the use of large *U* values and the bonding problem, Ataei et al. [47] reported that with values of 3.5 eV for both O *2p* and Ti *3d*-states, the results for the lattice constants, bandgap, and gap states are in good agreement with the experimental reports.

**Figure 3.** Calculated c/a ratio vs. U value. Reproduced from [19], with the permission of AIP Publishing.

#### *3.3.2. Doped-TiO2 with U correction*

the Hubbard *U* correction. The obtained band structures using GGA-PBE showed bandgaps of 2.140 and 1.973 eV for anatase and rutile, respectively. However, upon applying the localization of the excess electronic charge using +U correction, the predicted bandgaps are accurate and in a good agreement with the experimental and the computationally expensive hybrid functional (HSE06) results [43], **Figure 2**. In another study, for rutile TiO2, the prediction of the experimental bandgap is achieved with a *U* value of 10 eV, whereas the crystal and electronic

Dompablo et al. [19] compared the effect of the *U* parameter value (0 < *U* < 10 eV) within

(anatase and rutile) [43]. Copyright (2014)—American Chemical Society.

GGA+U required a small value of *U* (3 and 6 eV, respectively) to reproduce the experimental measurements, **Figure 3**. However, using very large *U* values leads to mismatching, where the lattice parameters (a and c) and the volume of unit cell are increasing with increasing *U*, due to the Coulomb repulsion increase. Note that standard DFT and the hybrid functional

On the other hand, the calculated bandgap within the GGA+U and LDA+U methods was found to be in better agreement with experiments compared to the conventional GGA or LDA, with small difference in the required *U* value. The bandgap was shown to increase by increasing the *U* value till 8.5 eV, which gave a result close to the experimental bandgap and in agreement with those obtained in previous DFT studies [44]. For values of *U* larger than 8.5, the bandgap was overestimated. It is worth to mention that this value (8.5 eV) is considered high when compared to other *U* values for other transition metal oxides [29]. In all these calculations, the Hubbard *U* parameter was used for the *d* or *f* orbitals of transition metals. However, when the Ti-O bonding is considered, while applying the correction only to the *3d*-states, it can be estimated that this correction might have an influence over the Ti−O covalent bonding, where the Ti states are shifted and the *2p* states of oxygen are not changed [45]. In this regard, several first principle calculations were derived to study the electronic,

. Both LDA+U and

the LDA+U and the GGA+U on the calculated properties of anatase TiO2

HSE06 failed to calculate the crystal lattice.

**Figure 2.** Total DOS of pure TiO<sup>2</sup>

structures were better described with *U* < 5 eV [19].

14 Density Functional Calculations - Recent Progresses of Theory and Application

In a recent study [40], a comparison was performed to elucidate the effect of different *U* values in representing the bandgap states produced by interstitial hydrogen atom and oxygen vacancy within the bulk Ti anatase structure. The dependence on the method used was observed, beside the value of *U* within GGA+U scheme, see **Figure 4**. When the *U* correction was not applied, the bandgap is underestimated, as expected, and the electrons caused by the oxygen vacancy or the hydrogen impurity are fully delocalized and have conduction band character. Upon applying the *U* correction, the states start to localize and are became deeply localized in the gap with increasing the value of *U*. In all these calculations, the Hubbard *U* correction was applied for the Ti *3d* orbitals only; by applying the correction for the *2p* oxygen orbitals with *U* = 3.5, the results were in agreement with previous results [47].

The intrinsic defects in TiO2 (vacancies) have been computationally studied, providing a fastcheap method to guide researchers in choosing the defect position in the solid crystal. The oxygen vacancy in the rutile crystal was investigated [48] using the DFT+U with *U* value of 4.0 eV, indicating that oxygen vacancy in the rutile crystal introduces four local states with two occupied and two unoccupied states, with no change in the bandgap (2.75 eV) [48]. The Ti vacancy effect on the bandgap (E<sup>g</sup> ) was also studied [49] using the GGA+U with *U* values of 7.2 eV. It was found that Ti vacancy caused ferromagnetism besides widening the valence band, and switching the TiO2 from n-type to p-type semiconductor with higher charge mobility [49].

resulted from the computational calculations is important to predict the small change in elec-

Density functional theory (DFT) has been used to model the MOFs as it allows the "mapping" of a system of N interacting electrons onto a system of N noninteracting electrons having the same ground state charge density in an effective potential. However, DFT fails to describe electrons in open *d*- or *f*-shells [8]. The pure DFT calculations usually wrongly estimate the bandgap and the ferromagnetic (FM) or antiferromagnetic (AFM) coupling for the centered metal in the MOFs. The reason for this wrong estimation is the localized spin and itinerant spin density that are coupled via the Heisenberg exchange interaction [52, 53]. In this interaction, the ferromagnetic sign is assumed if the hybridization of the conduction electrons (dispersive LUMO band) with a doubly occupied or empty *d* orbital of the magnetic center is sufficiently strong. Owing to Hund's rule, in the *d* shell, it is energetically favorable to induce spin polarization parallel to the *d*-shell spin. The itinerant spin density, however, forms at an energy penalty determined by the dispersion of the conduction band; the larger the density of states at the Fermi level, the easier is for the itinerant spin density to form. The addition of an extra interaction term that accounts for the strong on-site coulomb *U* correction has proved to lead to good results [54]. One more advantage of the DFT+U is that it can be used to model systems containing up to few hundred atoms [55]. The *U* parameter affects the predicted electronic structure and magnetic properties; in the following paragraphs, we will discuss some of the MOF applications and how to fit a proper magnitude of *U* in DFT+U calculations:

• The magnetic properties of the MOF of the complex dimethyl ammonium copper format (DMACuF) were predicted correctly [56] using the (GGA+U) with convenient *U* values (*U* = 4–7 eV) for Cu *3d*-states to describe the effect of electron correlation associated with those states. Also, the magnetic properties of MOFs of TCNQ (7,7,8,8-tetracyanoquinodimethane) and two different (Mn and Ni) *3d* transition metal atoms were predicted correctly without synthesizing. But in this case to properly describe the *d* electrons in Ni and Mn metal centers, spin-polarized calculations using the DFT+U with *U* value of (*U* = 4 eV) were performed [53]. It can be claimed that the varying of *U* in the range of 3 to 5 eV does not appreciably change the values of the Ni and Mn magnetic moments, nor the corresponding *3d* level occupations, in particular, that of the Ni (*3dxz*) orbital that crosses the Fermi level [53].

(*U* = 0–6 eV), and it was found that the value of *U* between 2 and 5 eV gives lattice parameters matching with experiment due to the fact that the Co-O bond length decreases with *U*,

• The Cu-BTC [58], a material consisting of copper dimers linked by 1,3,5-benzenetricarbox-

(BTC) units, was studied for its ability to absorb up to 3.5 H<sup>2</sup>

Cu binds to the closest oxygen of the water molecule [59]. The *U* parameter in the meta-GGA+U calculation of the Cu-BTC was adjusted with the experimental crystallographic structure and the bandgap by minimizing the absorption at 2.3 eV. The *U* values gave the best results at 3.08 eV for Cu and 7.05 eV for O because those values reduced the calculated

metal site, increasing the electrostatic contribution to the binding energy [57].

to a Co-MOF-74 was predicted [57] using DFT+U with *U* values

molecule closer to the charged open

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020 17

O per Cu as the

tronic structure upon application of external stimuli [51].

• The binding energy of CO2

ylate C6

O9 H3

since *U* localizes the Co *d*-states, which allows the CO2

**Figure 4.** Total and partial density of states DOS for anatase TiO<sup>2</sup> doped with (oxygen vacancies and interstitial H atom) obtained with GGA-PBE+U. Adapted with permission from [39]; (a) U = 0, (b) U = 2, (c) U = 3, (d) U = 4.

#### **4. Modeling of organometallics using the (+**U**)**

Hubbard correction is a computational tool that can be applied widely not only to crystals but also to the strongly correlated metals attached to other noncorrelated systems such as organic moieties. One of these important systems is metal organic framework (MOF).

#### **4.1. Metal organic frameworks (MOFs)**

MOFs are crystalline nanoporous materials where a centered transition metal is linked to different types of ligands, which provide a very large surface area [50] that can allow their use in supercapacitors and water splitting applications. Most of the MOFs have open metal sites, which are coordinative unsaturated metal sites with no geometric hindrance. While the whole material remains as a solid, the structure allows the complex framework to be used in gas capturing and storage, and the binding energy between the MOFs and the gas or water molecules allows the prediction of the capturing mechanism. The cage shape of the MOFs and the organic moiety allow their use in many applications such as drug delivery and fertilizers, while the magnetic behavior of MOFs allows the researchers to correctly predict how it can be used in applications. Quantum mechanics frame of work is usually used to describe the full interaction between the centered metal ion and the surrounding ligands, due to the fact that the synthesis of these materials is both time and money consuming. The complex geometry resulted from the computational calculations is important to predict the small change in electronic structure upon application of external stimuli [51].

Density functional theory (DFT) has been used to model the MOFs as it allows the "mapping" of a system of N interacting electrons onto a system of N noninteracting electrons having the same ground state charge density in an effective potential. However, DFT fails to describe electrons in open *d*- or *f*-shells [8]. The pure DFT calculations usually wrongly estimate the bandgap and the ferromagnetic (FM) or antiferromagnetic (AFM) coupling for the centered metal in the MOFs. The reason for this wrong estimation is the localized spin and itinerant spin density that are coupled via the Heisenberg exchange interaction [52, 53]. In this interaction, the ferromagnetic sign is assumed if the hybridization of the conduction electrons (dispersive LUMO band) with a doubly occupied or empty *d* orbital of the magnetic center is sufficiently strong. Owing to Hund's rule, in the *d* shell, it is energetically favorable to induce spin polarization parallel to the *d*-shell spin. The itinerant spin density, however, forms at an energy penalty determined by the dispersion of the conduction band; the larger the density of states at the Fermi level, the easier is for the itinerant spin density to form. The addition of an extra interaction term that accounts for the strong on-site coulomb *U* correction has proved to lead to good results [54]. One more advantage of the DFT+U is that it can be used to model systems containing up to few hundred atoms [55]. The *U* parameter affects the predicted electronic structure and magnetic properties; in the following paragraphs, we will discuss some of the MOF applications and how to fit a proper magnitude of *U* in DFT+U calculations:

• The magnetic properties of the MOF of the complex dimethyl ammonium copper format (DMACuF) were predicted correctly [56] using the (GGA+U) with convenient *U* values (*U* = 4–7 eV) for Cu *3d*-states to describe the effect of electron correlation associated with those states. Also, the magnetic properties of MOFs of TCNQ (7,7,8,8-tetracyanoquinodimethane) and two different (Mn and Ni) *3d* transition metal atoms were predicted correctly without synthesizing. But in this case to properly describe the *d* electrons in Ni and Mn metal centers, spin-polarized calculations using the DFT+U with *U* value of (*U* = 4 eV) were performed [53]. It can be claimed that the varying of *U* in the range of 3 to 5 eV does not appreciably change the values of the Ni and Mn magnetic moments, nor the corresponding *3d* level occupations, in particular, that of the Ni (*3dxz*) orbital that crosses the Fermi level [53].

**4. Modeling of organometallics using the (+**U**)**

16 Density Functional Calculations - Recent Progresses of Theory and Application

**Figure 4.** Total and partial density of states DOS for anatase TiO<sup>2</sup>

**4.1. Metal organic frameworks (MOFs)**

Hubbard correction is a computational tool that can be applied widely not only to crystals but also to the strongly correlated metals attached to other noncorrelated systems such as organic

doped with (oxygen vacancies and interstitial H atom)

MOFs are crystalline nanoporous materials where a centered transition metal is linked to different types of ligands, which provide a very large surface area [50] that can allow their use in supercapacitors and water splitting applications. Most of the MOFs have open metal sites, which are coordinative unsaturated metal sites with no geometric hindrance. While the whole material remains as a solid, the structure allows the complex framework to be used in gas capturing and storage, and the binding energy between the MOFs and the gas or water molecules allows the prediction of the capturing mechanism. The cage shape of the MOFs and the organic moiety allow their use in many applications such as drug delivery and fertilizers, while the magnetic behavior of MOFs allows the researchers to correctly predict how it can be used in applications. Quantum mechanics frame of work is usually used to describe the full interaction between the centered metal ion and the surrounding ligands, due to the fact that the synthesis of these materials is both time and money consuming. The complex geometry

moieties. One of these important systems is metal organic framework (MOF).

obtained with GGA-PBE+U. Adapted with permission from [39]; (a) U = 0, (b) U = 2, (c) U = 3, (d) U = 4.


root mean square residual forces on the ions at their experimental fixed positions to its minimum value. The nonzero *U* of oxygen greatly reduces the residual forces, while the value of *U* for Cu ions controls the splitting in the Cu *d* levels, which have a great effect on calculated bandgap [59].

to occur only on the sixfold nitrogen coordinate Fe1

configuration was indicated to be (dXY)

2 (dπ) 3 (dz 2 ) 1

a configuration (dXY)

of the centered metal.

ion, while the Fe2

[74]. Therefore, a computational calculation was important to

using Mossbauer [68–70], magnetic [71] and

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020

four nitrogen and two oxygen from the water molecules. For the dehydrated compounds, the effect of the Au atom caused a difference in the degree of the covalent bonding, which resulted in a distinct behavior of the Au network as compared to the Ag network. The hydrated and dehydrated Ag networks were predicted to exhibit a low spin-high spin transition, whereas the dehydrated Au network was predicted to remain in a high spin state [55].

• Fe-porphyrin molecules were found to have an intermediate spin state. The ground-state

NMR [72, 73] measurements. However, Raman spectroscopy predicted a ground state with

predict the reason for those results. The DFT+U was used to predict the electronic structure and magnetic properties of Fe molecules for a range of Coulomb *U* parameters (*U* = 2–4 eV), which is reasonable for iron [10, 75], and then compared to available data in literature. It was found that GGA+U with *U* value of 4 eV provided an overall better comparison of the structural, electronic, magnetic properties, and energy level diagram of these systems [76, 77].

To summarize, DFT+U are good to predict the correlation in the centered metal in organometallics. The spin change between FM and AFM states or in SCO can all be well predicted by the Hubbard correction, while the pure DFT fails due to the correlation in the *d* or *f* orbitals

Studying surface chemistry is of great significance for enhancing the overall efficiency of many electrochemical applications [78–80]. In catalysis, for example, understanding the adsorption mechanism of species on catalytic surfaces—mainly electrodes—is essential in order to formulate a design principle for the prefect catalyst that can reach the optimum efficiency for a desired electrochemical process [81–83]. Typically, the adsorption of CO on metal surface is widely acknowledged as the prototypical system for studying molecular chemisorption [84–87]. Despite the extensive experimental studies, grasping the complete theoretical description of the "bonding model" has not yet been reached, due to the inability of experimental tools to fully describe the details of molecular orbital interactions and to make a profound population analysis, which is based on studying the electronic structures of the substrate and surface particles [88, 89]. To this end, DFT can be utilized to explicitly describe electronic structures of the system particles in greater details, which can help in extending the conceptual model of CO chemisorption [90–94]. Unfortunately, due to the inherent wrong description of the electronic structure by DFT, wrong predictions of CO preferred adsorption sites are observed that contradict experimental results, especially for the (111) surface facets of transition metals, leading to the so-called "CO adsorption *Puzzle*" [95, 96]. The root of this DFT problem resides on the fact that both local density and generalized gradient approximation functionals underestimate the CO bandgap, predicting wrong positions of the CO frontier orbitals, which results in an overestimated bond

2 (dπ) 2 (dz 2 ) 2

**5. Solving the CO adsorption puzzle with the** *U* **correction**

strength between the substrate and surface molecules [97].

ion coordinate with

19

#### **4.2. Spin-crossover (SCO)**

Spin-crossover (SCO) is a unique feature in which the centered transition metal ion linked to the surrounding ligand has the ability to attain different spin states with different total spin quantum numbers (S), while keeping the same valence state [47]. This property allows MOFs and organometallics generally to reversibly switch between spin states upon application of temperature, pressure, light, or magnetic field, such as changing between low spin and high spin [60, 61]. The SCO can be predicted effectively using the *U* correction as well as the effect of temperature on the SCO. The use of DFT+U to model SCO was first done by Lebègue et al. [62]. SCO is generally appealing for metals that have availability to change between high spin and low spin due to the small difference between the HOMO and LUMO levels. Iron (Fe) is one of the most important examples for this property. Fe is important since it can be found in many ores and can be used in many applications such as solar cells. Besides, Fe can be called the source of life, which is the heme molecule in the blood and which is responsible for the transfer of oxygen and carbon dioxide to and from the cell, respectively, consist of Fe-porphyrin molecule. Modeling of those molecules and their reaction mechanisms provides information about drug reactions inside the blood stream. Unfortunately, the common exchange-correlation functional fails to predict the properties of the deoxygenated active site of hemoglobin and myoglobin and Fe-porphyrin molecules [63, 64]. Some of the examples of SCO in Fe complexes are listed below:


to occur only on the sixfold nitrogen coordinate Fe1 ion, while the Fe2 ion coordinate with four nitrogen and two oxygen from the water molecules. For the dehydrated compounds, the effect of the Au atom caused a difference in the degree of the covalent bonding, which resulted in a distinct behavior of the Au network as compared to the Ag network. The hydrated and dehydrated Ag networks were predicted to exhibit a low spin-high spin transition, whereas the dehydrated Au network was predicted to remain in a high spin state [55].

root mean square residual forces on the ions at their experimental fixed positions to its minimum value. The nonzero *U* of oxygen greatly reduces the residual forces, while the value of *U* for Cu ions controls the splitting in the Cu *d* levels, which have a great effect on

Spin-crossover (SCO) is a unique feature in which the centered transition metal ion linked to the surrounding ligand has the ability to attain different spin states with different total spin quantum numbers (S), while keeping the same valence state [47]. This property allows MOFs and organometallics generally to reversibly switch between spin states upon application of temperature, pressure, light, or magnetic field, such as changing between low spin and high spin [60, 61]. The SCO can be predicted effectively using the *U* correction as well as the effect of temperature on the SCO. The use of DFT+U to model SCO was first done by Lebègue et al. [62]. SCO is generally appealing for metals that have availability to change between high spin and low spin due to the small difference between the HOMO and LUMO levels. Iron (Fe) is one of the most important examples for this property. Fe is important since it can be found in many ores and can be used in many applications such as solar cells. Besides, Fe can be called the source of life, which is the heme molecule in the blood and which is responsible for the transfer of oxygen and carbon dioxide to and from the cell, respectively, consist of Fe-porphyrin molecule. Modeling of those molecules and their reaction mechanisms provides information about drug reactions inside the blood stream. Unfortunately, the common exchange-correlation functional fails to predict the properties of the deoxygenated active site of hemoglobin and myoglobin and Fe-porphyrin molecules [63, 64]. Some of the examples of

calculated bandgap [59].

18 Density Functional Calculations - Recent Progresses of Theory and Application

SCO in Fe complexes are listed below:

]

• Another study on the complex [Fe(pmd)-(H<sup>2</sup>

2−, [CrFe(CN)<sup>6</sup>

the DFT+U with *U*~ 4 eV and J~ 1 eV. The complexes Fe(phen)<sup>2</sup>

through the correlation between the spin state and the structure [62].

]

2−, [MnFe(CN)<sup>6</sup>

been studied [65] using the DFT+U. It was found that high *U* values > 8 eV should be applied to the low spin Fe site, while low *U* value should be applied to the high spin ion. The results showed a great agreement with other DFT calculations. The generally used DFT-GGA failed to predict the high spin of the five coordinate Fe complexes [68], but it could be obtained by

were tested using a *U* value of 2.5 eV [62], with the energy difference between the low spin state and high spin state is in agreement with the experimental values and proved that the *U* coulomb term was needed. The study showed the importance of magneto elastic couplings

M = Ag or Au) showed an interesting SCO behavior according to temperature [66]. This complex forms chain polymers that contain two different Fe(II) ions Fe1 and Fe2. Through hydration/dehydration, temperature changes between 130 and 230 K for the Ag-based coordination polymer changing the SCO reversibly and this change is due to the structure change caused by the water molecules in the network. For the Au-based complexes, only the SCO transition was different in the hydrated framework [66]. Such behavior could be explained using the DFT+*U* calculations [67]. The low spin-high spin transition was found

O)M2

(CN)4

].H2

]

2−, and [CoFe(CN)<sup>6</sup>

]

(NCS)<sup>2</sup>

2− frameworks has

(NCS)<sup>2</sup>

and Fe(btr)2

O (pmd = pyrimidine and

• The SCO of [TiFe(CN)<sup>6</sup>

**4.2. Spin-crossover (SCO)**

• Fe-porphyrin molecules were found to have an intermediate spin state. The ground-state configuration was indicated to be (dXY) 2 (dπ) 2 (dz 2 ) 2 using Mossbauer [68–70], magnetic [71] and NMR [72, 73] measurements. However, Raman spectroscopy predicted a ground state with a configuration (dXY) 2 (dπ) 3 (dz 2 ) 1 [74]. Therefore, a computational calculation was important to predict the reason for those results. The DFT+U was used to predict the electronic structure and magnetic properties of Fe molecules for a range of Coulomb *U* parameters (*U* = 2–4 eV), which is reasonable for iron [10, 75], and then compared to available data in literature. It was found that GGA+U with *U* value of 4 eV provided an overall better comparison of the structural, electronic, magnetic properties, and energy level diagram of these systems [76, 77].

To summarize, DFT+U are good to predict the correlation in the centered metal in organometallics. The spin change between FM and AFM states or in SCO can all be well predicted by the Hubbard correction, while the pure DFT fails due to the correlation in the *d* or *f* orbitals of the centered metal.

#### **5. Solving the CO adsorption puzzle with the** *U* **correction**

Studying surface chemistry is of great significance for enhancing the overall efficiency of many electrochemical applications [78–80]. In catalysis, for example, understanding the adsorption mechanism of species on catalytic surfaces—mainly electrodes—is essential in order to formulate a design principle for the prefect catalyst that can reach the optimum efficiency for a desired electrochemical process [81–83]. Typically, the adsorption of CO on metal surface is widely acknowledged as the prototypical system for studying molecular chemisorption [84–87]. Despite the extensive experimental studies, grasping the complete theoretical description of the "bonding model" has not yet been reached, due to the inability of experimental tools to fully describe the details of molecular orbital interactions and to make a profound population analysis, which is based on studying the electronic structures of the substrate and surface particles [88, 89]. To this end, DFT can be utilized to explicitly describe electronic structures of the system particles in greater details, which can help in extending the conceptual model of CO chemisorption [90–94]. Unfortunately, due to the inherent wrong description of the electronic structure by DFT, wrong predictions of CO preferred adsorption sites are observed that contradict experimental results, especially for the (111) surface facets of transition metals, leading to the so-called "CO adsorption *Puzzle*" [95, 96]. The root of this DFT problem resides on the fact that both local density and generalized gradient approximation functionals underestimate the CO bandgap, predicting wrong positions of the CO frontier orbitals, which results in an overestimated bond strength between the substrate and surface molecules [97].

One of the popular solutions that has been exploited by researchers to resolve the adsorption site prediction puzzle is the DFT+U correction [97, 98]. In this approach, the position of the 2π\* orbital is shifted to higher values, by adding the on-site Coulomb interaction parameter. By doing so, the interaction of CO 2π\* orbital with the metallic *d*-band will no longer be overestimated, bringing the appropriate estimation of the CO adsorption site. Kresse et al. [99] first implemented this method and successfully obtained a site preference in agreement with experiment, emphasizing that the use of such a simple empirical method is able to capture the essential physics of adsorption. DFT calculations utilizing GGA functionals predict adsorption on the threefold hollow site for Cu(111) and in the bridge site on Cu(001), instead of the experimental on-top site preference. Reference [98] implanted Kresse's method to investigate the adsorption of CO on Cu(111) and (001) surfaces with 1/4 monolayer (ML) coverage on different adsorption sites. In that study, the HOMO-LUMO gap of the isolated CO molecule was demonstrated to be increased by increasing the value of *U*. Also, upon changing the *U* value, the corresponding adsorption energies of the CO over the different adsorption sites were calculated.

**6. Summary and outlook**

In this chapter, the corrective capability of the DFT+U is overviewed and evaluated for a number of different classes of materials. Generally, the addition of the on-site Coulomb interaction potential (*U*) to the standard DFT Hamiltonian proved to provide significant changes to the predicted electronic structures, which can solve the inherent DFT bandgap prediction problem. The value of *U* can either be theoretically calculated or semiempirically tuned to match the experimental electronic structure. For the various case studies and applications reviewed, the criticality of correcting the electronic structure predictions was manifested, as it leads to significant improvements for the prediction of further electronic-related properties. Prior to the practical assessment, the theoretical foundation of the DFT+U method is briefly discussed and is verified to be rather simple adding only marginal computational cost to the standard DFT calculations. Compared to other corrective approaches, the DFT+U formulation demonstrated to be simpler in terms of theoretical formulation and practical implementations with considerably lower computational cost, while having nearly the same predictive power; it can even capture properties of certain materials that cannot be captured by other higher level or exact calculations. One of the most popular implementations of the *U* correction is the description of the electronic structure for strongly correlated materials (Mott insulators). The behavior of these types of insulators cannot be captured by applying Hartree-Fock, band theory based, calculations, as the root of this problem resides on the deficiency of the band theory to capture such behavior, as it neglects the interelectron forces. One of the simple models, which explicitly accounts for the on-site repulsion between electrons at the same atomic orbitals, is the Hubbard model. Based on this model, the DFT+U method is formulated to

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020 21

The theoretical and semiempirical techniques of the *U* optimization are discussed. The semiempirical tuning is found to be the most common practice employed by researchers due to the significant computational cost of *ab initio* calculations that *U* can have, and also, the computed *U* is not necessarily being better than the empirical ones. However, the semiempirical evaluation of *U* does not permit the capturing of changes in the on-site electronic interaction under changing physical conditions, such as chemical reactions. The practical implementations of *U* correction are discussed, while assessing the effect of the DFT scheme employed and the calculation parameters assigned on the numerical value of the optimum *U* utilized. The corrective influence of the *U* correction is validated by reviewing different examples and case studies in literature. Starting with the transition metal oxides, the effect of adding the *U* parameter to correctly describe the electronic structure of pure and defected TiO2 is reviewed, showing the different optimum values of *U* utilized for each level of calculation. Then, the implementation of the Hubbard correction to the systems that comprises molecules with solid-state crystals is reviewed, such as organometallics. The addition of *U* to the DFT calculation provides a better understanding of the behavior of the metals inside the organometallic systems. One of the most importantly studied organometallic systems is the metal organic frameworks (MOFs). Different examples in literature are reviewed, showing the effect of the *U* correction and how it can significantly improve the prediction of the magnetic properties of such systems. Also, one of the unique features of organometallics, which can be influenced the *U* correction, is the spin crossover (SCO). This property allows the MOFs and the organometallics generally to reversibly switch between spin states

improve the description of the ground state of correlated systems.

Reviewing the Cu (111) surface results, five different *U* values (0.0, 0.5, 1.0, 1.25, and 1.5 eV) were used in the calculations. It was observed that only 20 meV changes in the adsorption energy (higher coordinated hollow sites) for *U* = 1.25 and 0.03 eV for *U* = 1.5 eV. Nonetheless, the absolute value of adsorption energy decreases linearly with increasing *U*, where the rate of reduction is found to be larger for higher coordinated sites. It was observed that the site preference between top and bridge sites to be reversed around the *U* value of 0.05 eV, while between the top and hollow sites around *U* = 0.45 eV. Concerning the adsorbate (surface) description in the study, the calculated interlayer relaxations were the same as that calculated using the GGA (PW91) functional without the *U* correction. Not only does the *U* correction help in solving the adsorption puzzle dilemma, but it can also enhance the description of other related properties, such as the calculated work function and the vibrational spectra for the CO-metal complexes, which are also demonstrated in Ref. [98] (**Figure 5**).

**Figure 5.** A schematic sketch of the molecular eigenstates of the CO molecule. The DFT+U technique shifts the LUMO orbitals to higher energies, but the energies of the occupied orbitals remain the same.

#### **6. Summary and outlook**

One of the popular solutions that has been exploited by researchers to resolve the adsorption site prediction puzzle is the DFT+U correction [97, 98]. In this approach, the position of the 2π\* orbital is shifted to higher values, by adding the on-site Coulomb interaction parameter. By doing so, the interaction of CO 2π\* orbital with the metallic *d*-band will no longer be overestimated, bringing the appropriate estimation of the CO adsorption site. Kresse et al. [99] first implemented this method and successfully obtained a site preference in agreement with experiment, emphasizing that the use of such a simple empirical method is able to capture the essential physics of adsorption. DFT calculations utilizing GGA functionals predict adsorption on the threefold hollow site for Cu(111) and in the bridge site on Cu(001), instead of the experimental on-top site preference. Reference [98] implanted Kresse's method to investigate the adsorption of CO on Cu(111) and (001) surfaces with 1/4 monolayer (ML) coverage on different adsorption sites. In that study, the HOMO-LUMO gap of the isolated CO molecule was demonstrated to be increased by increasing the value of *U*. Also, upon changing the *U* value, the corresponding adsorption energies of the CO over the different adsorption sites

20 Density Functional Calculations - Recent Progresses of Theory and Application

Reviewing the Cu (111) surface results, five different *U* values (0.0, 0.5, 1.0, 1.25, and 1.5 eV) were used in the calculations. It was observed that only 20 meV changes in the adsorption energy (higher coordinated hollow sites) for *U* = 1.25 and 0.03 eV for *U* = 1.5 eV. Nonetheless, the absolute value of adsorption energy decreases linearly with increasing *U*, where the rate of reduction is found to be larger for higher coordinated sites. It was observed that the site preference between top and bridge sites to be reversed around the *U* value of 0.05 eV, while between the top and hollow sites around *U* = 0.45 eV. Concerning the adsorbate (surface) description in the study, the calculated interlayer relaxations were the same as that calculated using the GGA (PW91) functional without the *U* correction. Not only does the *U* correction help in solving the adsorption puzzle dilemma, but it can also enhance the description of other related properties, such as the calculated work function and the vibrational spectra for

**Figure 5.** A schematic sketch of the molecular eigenstates of the CO molecule. The DFT+U technique shifts the LUMO

orbitals to higher energies, but the energies of the occupied orbitals remain the same.

the CO-metal complexes, which are also demonstrated in Ref. [98] (**Figure 5**).

were calculated.

In this chapter, the corrective capability of the DFT+U is overviewed and evaluated for a number of different classes of materials. Generally, the addition of the on-site Coulomb interaction potential (*U*) to the standard DFT Hamiltonian proved to provide significant changes to the predicted electronic structures, which can solve the inherent DFT bandgap prediction problem. The value of *U* can either be theoretically calculated or semiempirically tuned to match the experimental electronic structure. For the various case studies and applications reviewed, the criticality of correcting the electronic structure predictions was manifested, as it leads to significant improvements for the prediction of further electronic-related properties. Prior to the practical assessment, the theoretical foundation of the DFT+U method is briefly discussed and is verified to be rather simple adding only marginal computational cost to the standard DFT calculations. Compared to other corrective approaches, the DFT+U formulation demonstrated to be simpler in terms of theoretical formulation and practical implementations with considerably lower computational cost, while having nearly the same predictive power; it can even capture properties of certain materials that cannot be captured by other higher level or exact calculations. One of the most popular implementations of the *U* correction is the description of the electronic structure for strongly correlated materials (Mott insulators). The behavior of these types of insulators cannot be captured by applying Hartree-Fock, band theory based, calculations, as the root of this problem resides on the deficiency of the band theory to capture such behavior, as it neglects the interelectron forces. One of the simple models, which explicitly accounts for the on-site repulsion between electrons at the same atomic orbitals, is the Hubbard model. Based on this model, the DFT+U method is formulated to improve the description of the ground state of correlated systems.

The theoretical and semiempirical techniques of the *U* optimization are discussed. The semiempirical tuning is found to be the most common practice employed by researchers due to the significant computational cost of *ab initio* calculations that *U* can have, and also, the computed *U* is not necessarily being better than the empirical ones. However, the semiempirical evaluation of *U* does not permit the capturing of changes in the on-site electronic interaction under changing physical conditions, such as chemical reactions. The practical implementations of *U* correction are discussed, while assessing the effect of the DFT scheme employed and the calculation parameters assigned on the numerical value of the optimum *U* utilized. The corrective influence of the *U* correction is validated by reviewing different examples and case studies in literature. Starting with the transition metal oxides, the effect of adding the *U* parameter to correctly describe the electronic structure of pure and defected TiO2 is reviewed, showing the different optimum values of *U* utilized for each level of calculation. Then, the implementation of the Hubbard correction to the systems that comprises molecules with solid-state crystals is reviewed, such as organometallics. The addition of *U* to the DFT calculation provides a better understanding of the behavior of the metals inside the organometallic systems. One of the most importantly studied organometallic systems is the metal organic frameworks (MOFs). Different examples in literature are reviewed, showing the effect of the *U* correction and how it can significantly improve the prediction of the magnetic properties of such systems. Also, one of the unique features of organometallics, which can be influenced the *U* correction, is the spin crossover (SCO). This property allows the MOFs and the organometallics generally to reversibly switch between spin states upon changing the external parameters. The SCO is proved to be predicted more effectively by applying the *U* correction, as demonstrated in the results presented in literature. Finally, the significance of the DFT+U method is manifested upon describing the adsorption mechanism of CO on transition metal systems. The influence of *U* correction on solving the so-called adsorption *Puzzle* is demonstrated, which leads to the correct prediction of CO adsorption site preference, which was an unresolved problem when DFT calculations are applied alone.

[3] Levy M. Generalized Kohn-Sham schemes and the band-gap problem. Physical Review

The DFT+U: Approaches, Accuracy, and Applications http://dx.doi.org/10.5772/intechopen.72020 23

[4] Sholl D, Steckel J. Density Functional Theory: A Practical Introduction. Density Functional Theory. New York, USA: John Wiley & Sons; 2009. pp. 83-112. DOI: 10.1002/

[5] Himmetoglu B, Floris A, De Gironcoli S, Cococcioni M. Hubbard-corrected DFT energy functionals: The LDA+U description of correlated systems. International Journal of

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[11] Dudarev SL, Botton GA, Savrasov SY, Humphreys CJ, Sutton AP. Electron-energy-loss spectra and the structural stability of nickel oxide. Physical Review B. 1998;**57**(3):1505-

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Forschungszentrum Julich, Germany. 2012;**2**

9780470447710

0305-4470/38/8/B01

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R5467-R5470. DOI: 10.1103/PhysRevB.52.R5467

Upon reviewing the presented applications and different case studies, where the *U* correction significantly improved the estimated results without changing the essential physics of the systems, we can estimate the potential of the Hubbard correction to gain a greater weight in the future of computational chemistry. Despite the convenience of the semiempirical tuning of *U*, the capabilities of the Hubbard correction in this way cannot be fully exploited, as it cannot be used to study systems with variations of on-site electronic interactions. On the other hand, despite the availability of theoretical *U* calculation methods, their computational costs are considerably large, compared to the semiempirical methods. Therefore, further improvements to the *ab initio* calculation of *U* is still required, with lower computational costs, in order to conceive full potential of the *U* correction that is able to capture phase changes and chemical reactions for the studied physical systems.

#### **Acknowledgements**

This work was made possible by NPRP Grant no. NPRP 6-569-1-112 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.

### **Author details**

Sarah A. Tolba† , Kareem M. Gameel† , Basant A. Ali, Hossam A. Almossalami and Nageh K. Allam\*

\*Address all correspondence to: nageh.allam@aucegypt.edu

Energy Materials Laboratory, The American University in Cairo, New Cairo, Egypt

† These authors contributed equally

#### **References**


[3] Levy M. Generalized Kohn-Sham schemes and the band-gap problem. Physical Review B. 1996;**53**(7):3764-3774. DOI: 10.1103/PhysRevB.53.3764

upon changing the external parameters. The SCO is proved to be predicted more effectively by applying the *U* correction, as demonstrated in the results presented in literature. Finally, the significance of the DFT+U method is manifested upon describing the adsorption mechanism of CO on transition metal systems. The influence of *U* correction on solving the so-called adsorption *Puzzle* is demonstrated, which leads to the correct prediction of CO adsorption site preference,

Upon reviewing the presented applications and different case studies, where the *U* correction significantly improved the estimated results without changing the essential physics of the systems, we can estimate the potential of the Hubbard correction to gain a greater weight in the future of computational chemistry. Despite the convenience of the semiempirical tuning of *U*, the capabilities of the Hubbard correction in this way cannot be fully exploited, as it cannot be used to study systems with variations of on-site electronic interactions. On the other hand, despite the availability of theoretical *U* calculation methods, their computational costs are considerably large, compared to the semiempirical methods. Therefore, further improvements to the *ab initio* calculation of *U* is still required, with lower computational costs, in order to conceive full potential of the *U* correction that is able to capture phase changes and chemi-

This work was made possible by NPRP Grant no. NPRP 6-569-1-112 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the

Energy Materials Laboratory, The American University in Cairo, New Cairo, Egypt

[1] Koch W, Holthausen MC. A Chemist's Guide to Density Functional Theory. New York,

[2] Mattsson AE, Schultz PA, Desjarlais MP. Designing meaningful density functional theory calculations in materials science—A primer. Modelling and Simulation in Materials

Science and Engineering. 13 R, 2005;**1**. DOI: 10.1088/0965-0393/13/1/R01

, Basant A. Ali, Hossam A. Almossalami and

which was an unresolved problem when DFT calculations are applied alone.

22 Density Functional Calculations - Recent Progresses of Theory and Application

cal reactions for the studied physical systems.

, Kareem M. Gameel†

\*Address all correspondence to: nageh.allam@aucegypt.edu

USA: John Wiley & Sons; 2001. DOI: 10.1002/3527600043

**Acknowledgements**

responsibility of the authors.

These authors contributed equally

**Author details**

Sarah A. Tolba†

**References**

†

Nageh K. Allam\*


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**Chapter 2**

Provisional chapter

**Constricted Variational Density Functional Theory**

DOI: 10.5772/intechopen.70932

The aim of this chapter is to present constricted variational density functional theory (CV-DFT), a DFT-based method for calculating excited-state energies. This method involves constructing from the ground-state orbitals, a new set of 'occupied' excited-state orbitals. Consequently, a constraint is applied to ensure that exactly one electron is fully transferred from the occupied to the virtual space. This constraint also prevents a collapse to a lower state. With this set of orbitals, one obtains an electron density for the excited-state and therewith the CV-DFT excitation energy. This excitation energy can now be variationally optimized. With our successful applications to systems differing in the type of excitation, namely, charge-transfer, charge-transfer in disguise, and Rydberg excitations, as well as in size, we demonstrate the strengths of the CV-DFT method. Therewith, CV-DFT provides a valid alternative to calculate excited-state properties, especially in cases where TD-DFT has difficulties. Finally, our studies have shown that the difficulties arising in the TD-DFT excited states are not necessarily stemming from the functional used, but from the application of these standard functionals in combination with the linear response theory.

Keywords: CV-DFT, excited state, charge-transfer, Rydberg excitations, ZnBC-BC

The behavior of atoms and polyatomic systems in the excited-state are of immense importance in the studies of several photophysical phenomena. Thus, the search for methods to study systems in their electronically excited state is the subject of ongoing research [1–13]. Resultantly, there are several methods to choose from within certain consideration such as system size, expected level of accuracy and nature of initial and final electronic state of the system under study. Therefore, some background knowledge is necessary for the accurate treatment of excited states with the available methods. These methods fall under different families, and

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Constricted Variational Density Functional Theory

**Approach to the Description of Excited States**

Approach to the Description of Excited States

Florian Senn, Issaka Seidu and Young Choon Park

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70932

Florian Senn, Issaka Seidu and

Young Choon Park

Abstract

1. Introduction


#### **Constricted Variational Density Functional Theory Approach to the Description of Excited States** Constricted Variational Density Functional Theory Approach to the Description of Excited States

DOI: 10.5772/intechopen.70932

Florian Senn, Issaka Seidu and Young Choon Park Florian Senn, Issaka Seidu and

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70932

#### Abstract

Young Choon Park

[94] Andersson MP. CO adsorption energies on metals with correction for high coordination adsorption sites – A density functional study. Surface Science. 2007;**601**:1747-1753.

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1-12. DOI: 10.1103/PhysRevB.76.195440

10.1103/PhysRevB.69.161401

Materials. 2010;**9**(9):741-744. DOI: 10.1038/nmat2806

30 Density Functional Calculations - Recent Progresses of Theory and Application

The aim of this chapter is to present constricted variational density functional theory (CV-DFT), a DFT-based method for calculating excited-state energies. This method involves constructing from the ground-state orbitals, a new set of 'occupied' excited-state orbitals. Consequently, a constraint is applied to ensure that exactly one electron is fully transferred from the occupied to the virtual space. This constraint also prevents a collapse to a lower state. With this set of orbitals, one obtains an electron density for the excited-state and therewith the CV-DFT excitation energy. This excitation energy can now be variationally optimized. With our successful applications to systems differing in the type of excitation, namely, charge-transfer, charge-transfer in disguise, and Rydberg excitations, as well as in size, we demonstrate the strengths of the CV-DFT method. Therewith, CV-DFT provides a valid alternative to calculate excited-state properties, especially in cases where TD-DFT has difficulties. Finally, our studies have shown that the difficulties arising in the TD-DFT excited states are not necessarily stemming from the functional used, but from the application of these standard functionals in combination with the linear response theory.

Keywords: CV-DFT, excited state, charge-transfer, Rydberg excitations, ZnBC-BC

#### 1. Introduction

The behavior of atoms and polyatomic systems in the excited-state are of immense importance in the studies of several photophysical phenomena. Thus, the search for methods to study systems in their electronically excited state is the subject of ongoing research [1–13]. Resultantly, there are several methods to choose from within certain consideration such as system size, expected level of accuracy and nature of initial and final electronic state of the system under study. Therefore, some background knowledge is necessary for the accurate treatment of excited states with the available methods. These methods fall under different families, and

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

the ab initio wave function family of methods includes multi-reference configuration interaction (MRCI) [14], multi-configurational self-consistent field (MCSCF) [15, 16], complete active space self-consistent field (CASSCF) [17], time-dependent Hartree-Fock (TD-HF) [18–21], restricted active space self-consistent field (RASSCF) [22], RASPT2 [23], complete active space second-order perturbation theory (CASPT2) [24], equation-of-motion coupled cluster (EOMCC) [25], n-electron valence state perturbation theory (NEVPT) [26], spectroscopically oriented configuration interaction (SORCI) [27] and coupled cluster (CC) theory [28, 29].

In this chapter, we will explain the idea and theory of CV-DFT, before we have a look at different examples, where CV-DFT has been applied for transitions of Rydberg and charge-

Constricted Variational Density Functional Theory Approach to the Description of Excited States

In this chapter, we review the theoretical framework of CV-DFT in a nutshell. We refer to the

Here, we only consider the excitation from the closed-shell ground state described with single

DFT starts from the ansatz which describes the excitation as an admixture of occupied

matrix, U, only mixes between occupied and virtual orbitals (Uij ¼ Uab ¼ 0) and is skew symmetric (Uai ¼ �Uia). In CV-DFT, we use the exponential expansion of U which leads to

<sup>Y</sup> <sup>¼</sup> expð Þ¼ <sup>U</sup> <sup>X</sup><sup>∞</sup>

ϕ0 occ ϕ0

ϕi 0 <sup>¼</sup> <sup>X</sup><sup>n</sup>occ j

The corresponding excited-state density becomes

vir ! <sup>¼</sup> <sup>X</sup><sup>∞</sup>

Thus, once the transition matrix, U, is determined, a new set of orbitals is obtained over the

k¼0

Due to the properties of the transition matrix, U, the 'occupied' excited-state orbitals can be

Yjiϕ<sup>j</sup> <sup>þ</sup>X<sup>n</sup>vir

a

Uk k! ! <sup>ϕ</sup>occ

<sup>i</sup> is the excited-state orbital and nvir the number of virtual orbitals. The transition

k¼0

Uk k!

ϕvir

�, where <sup>n</sup>occ is the number of occupied orbitals. CV-

Uaiϕ<sup>a</sup> (1)

http://dx.doi.org/10.5772/intechopen.70932

33

: (2)

� �: (3)

Yaiϕa: (4)

…ϕ<sup>n</sup>occ

; <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>n</sup>occ � � and virtual <sup>ϕ</sup>a; <sup>a</sup> <sup>¼</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>n</sup>vir � � ground-state orbitals [105]:

ϕ0 <sup>i</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>vir a

� �

transfer type following the publications [99–103].

original publications [58–62, 104] for a more in-depth description.

2. Theory

ϕi

where ϕ<sup>0</sup>

written as

2.1. The CV-DFT scheme

Slater determinant, <sup>Ψ</sup><sup>0</sup> <sup>¼</sup> <sup>ϕ</sup>1ϕ2…ϕ<sup>i</sup>

the unitary transformation Y:

unitary transformation

� � �

However, the focus of this book is the Kohn-Sham density functional theory (DFT) [30] and methods based on it. In this chapter, our attention is on the calculation of excited states. Excitedstate studies within DFT gained considerable attention owing to the increasing success of DFT in ground-state studies. Significant research effort toward the development of excited-state methods has resulted in a variety of approaches varying in both major and minor details, each method having its own advantages and disadvantages. The result of this endeavor includes self-consistentfield DFT (ΔSCF-DFT) [31–33] with extensions [34–36], time-dependent DFT [40–44], ensemble DFT [37–39, 45–47], constrained orthogonality method (COM) [48–50], restricted open-shell Kohn-Sham (ROKS) [47, 51, 52], constrained DFT (CDFT) [53], 'taking orthogonality constraints into account' (TOCIA) [54, 55], maximum overlap method (MOM) [56, 57], constricted variational density functional theory (CV-DFT) [58] and extensions [59–62], orthogonality constrained DFT (OCDFT) [63] and guided SCF [64] among others. However, the most widely used by both expert and nonexpert is TD-DFT in the form of linear response adiabatic time-dependent DFT [40, 41, 65– 69] (which we will refer to as TD-DFT) due to its successes.

The strengths and weaknesses of TD-DFT are well known and understood through extensive benchmark studies carried out over the years. The strengths explain its wide usage by delivering 'an excellent compromise between computational efficiency and accuracy' [70]. The weaknesses explain the ongoing fundamental studies searching for solutions in the cases where TD-DFT is found lacking. These include its deficiency in describing Rydberg transitions [71–74], chargetransfer (CT) transitions [75–84] and electronic transition with significant double contribution [42, 43, 83–87]. TD-DFT is a formally exact theory; however, its practical application relies on the adiabatic formalism where use is made of the available ground-state exchange-correlation (XC) functionals [71, 82, 88–90]. As a result, one can necessarily trace all the problems encountered in the application of TD-DFT to this approximation. The numerous research attempts to remedy the pitfalls in TD-DFT are classified as follows:


An often-encountered problem with the development of specialized functionals is that it usually performs very well for the purpose for which it was originally developed but unimaginably erratic for any other situation [71–74, 77, 79, 82, 88, 91–98].

Our contribution to this area of research is in the development of the constricted variational DFT (CV-DFT) [58–62], which combines the strengths of ΔSCF-DFT and TD-DFT methods without the need for 'specialized' functionals.

In this chapter, we will explain the idea and theory of CV-DFT, before we have a look at different examples, where CV-DFT has been applied for transitions of Rydberg and chargetransfer type following the publications [99–103].

#### 2. Theory

the ab initio wave function family of methods includes multi-reference configuration interaction (MRCI) [14], multi-configurational self-consistent field (MCSCF) [15, 16], complete active space self-consistent field (CASSCF) [17], time-dependent Hartree-Fock (TD-HF) [18–21], restricted active space self-consistent field (RASSCF) [22], RASPT2 [23], complete active space second-order perturbation theory (CASPT2) [24], equation-of-motion coupled cluster (EOMCC) [25], n-electron valence state perturbation theory (NEVPT) [26], spectroscopically oriented configuration interaction (SORCI) [27] and coupled cluster (CC) theory [28, 29].

However, the focus of this book is the Kohn-Sham density functional theory (DFT) [30] and methods based on it. In this chapter, our attention is on the calculation of excited states. Excitedstate studies within DFT gained considerable attention owing to the increasing success of DFT in ground-state studies. Significant research effort toward the development of excited-state methods has resulted in a variety of approaches varying in both major and minor details, each method having its own advantages and disadvantages. The result of this endeavor includes self-consistentfield DFT (ΔSCF-DFT) [31–33] with extensions [34–36], time-dependent DFT [40–44], ensemble DFT [37–39, 45–47], constrained orthogonality method (COM) [48–50], restricted open-shell Kohn-Sham (ROKS) [47, 51, 52], constrained DFT (CDFT) [53], 'taking orthogonality constraints into account' (TOCIA) [54, 55], maximum overlap method (MOM) [56, 57], constricted variational density functional theory (CV-DFT) [58] and extensions [59–62], orthogonality constrained DFT (OCDFT) [63] and guided SCF [64] among others. However, the most widely used by both expert and nonexpert is TD-DFT in the form of linear response adiabatic time-dependent DFT [40, 41, 65–

The strengths and weaknesses of TD-DFT are well known and understood through extensive benchmark studies carried out over the years. The strengths explain its wide usage by delivering 'an excellent compromise between computational efficiency and accuracy' [70]. The weaknesses explain the ongoing fundamental studies searching for solutions in the cases where TD-DFT is found lacking. These include its deficiency in describing Rydberg transitions [71–74], chargetransfer (CT) transitions [75–84] and electronic transition with significant double contribution [42, 43, 83–87]. TD-DFT is a formally exact theory; however, its practical application relies on the adiabatic formalism where use is made of the available ground-state exchange-correlation (XC) functionals [71, 82, 88–90]. As a result, one can necessarily trace all the problems encountered in the application of TD-DFT to this approximation. The numerous research attempts to remedy the

1. Finding the XC functionals with the correct short- and long-range behavior or going

An often-encountered problem with the development of specialized functionals is that it usually performs very well for the purpose for which it was originally developed but unimag-

Our contribution to this area of research is in the development of the constricted variational DFT (CV-DFT) [58–62], which combines the strengths of ΔSCF-DFT and TD-DFT methods

69] (which we will refer to as TD-DFT) due to its successes.

32 Density Functional Calculations - Recent Progresses of Theory and Application

pitfalls in TD-DFT are classified as follows:

beyond the adiabatic approximation.

without the need for 'specialized' functionals.

2. Developing new DFT-based excited-state methods.

inably erratic for any other situation [71–74, 77, 79, 82, 88, 91–98].

In this chapter, we review the theoretical framework of CV-DFT in a nutshell. We refer to the original publications [58–62, 104] for a more in-depth description.

#### 2.1. The CV-DFT scheme

Here, we only consider the excitation from the closed-shell ground state described with single Slater determinant, <sup>Ψ</sup><sup>0</sup> <sup>¼</sup> <sup>ϕ</sup>1ϕ2…ϕ<sup>i</sup> …ϕ<sup>n</sup>occ � � � � � �, where <sup>n</sup>occ is the number of occupied orbitals. CV-DFT starts from the ansatz which describes the excitation as an admixture of occupied ϕi ; <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>n</sup>occ � � and virtual <sup>ϕ</sup>a; <sup>a</sup> <sup>¼</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>n</sup>vir � � ground-state orbitals [105]:

$$\phi\_i' = \sum\_a^{n\_{\rm vir}} \mathcal{U}\_{ai} \phi\_a \tag{1}$$

where ϕ<sup>0</sup> <sup>i</sup> is the excited-state orbital and nvir the number of virtual orbitals. The transition matrix, U, only mixes between occupied and virtual orbitals (Uij ¼ Uab ¼ 0) and is skew symmetric (Uai ¼ �Uia). In CV-DFT, we use the exponential expansion of U which leads to the unitary transformation Y:

$$\mathbf{Y} = \exp(\mathbf{U}) = \sum\_{k=0}^{\curve{\bullet\,\,\mathrm{w}}} \frac{\mathbf{U}^{k}}{k!}. \tag{2}$$

Thus, once the transition matrix, U, is determined, a new set of orbitals is obtained over the unitary transformation

$$
\begin{pmatrix} \phi\_{\text{occ}}'\\\phi\_{\text{vir}}' \end{pmatrix} = \left(\sum\_{k=0}^{\curve{\circ}} \frac{\mathbf{U}^k}{k!} \right) \begin{pmatrix} \phi\_{\text{occ}}\\\phi\_{\text{vir}} \end{pmatrix}. \tag{3}
$$

Due to the properties of the transition matrix, U, the 'occupied' excited-state orbitals can be written as

$$\left|\phi\_{i}\right\rangle' = \sum\_{j}^{n\_{\rm{ox}}} Y\_{ji}\phi\_{j} + \sum\_{a}^{n\_{\rm{vir}}} Y\_{ai}\phi\_{a}.\tag{4}$$

The corresponding excited-state density becomes

$$\begin{split} \rho'(1,1') &= \sum\_{i}^{n\_{\text{ox}}} \phi'\_i(1)\phi'\_i(1') \\ &= \sum\_{i}^{n\_{\text{ox}}} \phi\_i(1)\phi\_i(1') + \sum\_{a}^{n\_{\text{red}}} \sum\_{i}^{n\_{\text{ox}}} \Delta P\_{ai} [\phi\_i(1)\phi\_a(1') + \phi\_a(1)\phi\_i(1')] \\ &+ \sum\_{i}^{n\_{\text{ox}}} \sum\_{j}^{n\_{\text{ox}}} \Delta P\_{\vec{\eta}} \phi\_i(1)\phi\_j(1') + \sum\_{a}^{n\_{\text{ox}}} \sum\_{b}^{n\_{ab}} \Delta P\_{ab} \phi\_a(1)\phi\_b(1') \end{split} \tag{5}$$

with the change in density matrix (ΔP). Later, one is given by

$$
\Delta P\_{a\dot{j}} = \sum\_{i}^{n\_{\rm occ}} Y\_{ai} Y\_{\dot{j}i} \tag{6}
$$

ϕi 0 <sup>¼</sup> <sup>ϕ</sup><sup>i</sup> <sup>þ</sup>X<sup>n</sup>vir

EKS <sup>r</sup><sup>0</sup> <sup>1</sup>; <sup>1</sup><sup>0</sup> ½ �¼ ð Þ EKS <sup>r</sup><sup>0</sup> � � <sup>þ</sup><sup>X</sup>

þ 1 2 X ai

KC pq,st ¼

KXC KS ð Þ pq,st ¼

KXC HF ð Þ

the different spin states in the excited-state calculation:

leads to the occupied and virtual NTOs as

ð

pq,st ¼ � ð ð <sup>ψ</sup>pð Þ<sup>1</sup> <sup>ψ</sup>qð Þ<sup>1</sup> <sup>1</sup>

to TD-DFT [59, 106] within the Tamm-Dancoff approximation (TDA) [107].

with

and

a

Uaiϕ<sup>a</sup> � <sup>1</sup> 2 Xnocc j

order terms is negligible [60]. The second-order CV-DFT energy expression becomes

ai

Kpq,st <sup>¼</sup> <sup>K</sup><sup>C</sup>

ð ð <sup>ψ</sup>pð Þ<sup>1</sup> <sup>ψ</sup>qð Þ<sup>1</sup> <sup>1</sup>

X bj

Xnvir a

Constricted Variational Density Functional Theory Approach to the Description of Excited States

With these orbitals, some higher order contributions in U can arise in the density and therewith also in the energy, but we only keep up to the second order in U, as the contribution of higher

> UaiU<sup>∗</sup> ai ε<sup>0</sup> <sup>a</sup> � <sup>ε</sup><sup>0</sup> i � � þ<sup>X</sup>

> > 1 2 X ai

pq,st <sup>þ</sup> <sup>K</sup>XC

r12

ψpð Þ r<sup>1</sup> ψqð Þ r<sup>1</sup> f rð Þ<sup>1</sup> ψsð Þ r<sup>1</sup> ψ<sup>t</sup>

r<sup>12</sup>

ψsð Þ2 ψ<sup>t</sup>

<sup>U</sup>σα <sup>¼</sup> <sup>V</sup>σσΣ <sup>W</sup>αα ð Þ<sup>T</sup> (17)

The exchange-correlation integral is further decomposed into the local (KS) and nonlocal (HF):

where f rð Þ<sup>1</sup> represents the regular energy kernel. We have shown that CV 2ð Þ-DFT is equivalent

In the infinite-order theory (CV(∞)-DFT), the new set of excited-state orbitals is obtained taking the sum in Eq. (3) to infinite order. These excited-state orbitals can be written in the convenient form of natural transition orbitals (NTO) [108]. For this, we decompose the transition matrix, U, into its singular values. Here, we also used a spin-adapted form for further description of

where <sup>Σ</sup>ii <sup>¼</sup> <sup>γ</sup><sup>i</sup> and <sup>σ</sup><sup>∈</sup> <sup>α</sup>; <sup>β</sup> � � depend on spin state (mixed and triplet states, respectively). This

ψsð Þ2 ψ<sup>t</sup>

UaiUbjKai,jb þ

where the two-electron integral is composed of a Coulomb and an exchange-correlation part:

UaiUajϕ<sup>j</sup> <sup>þ</sup> O Uð Þ<sup>3</sup> h i : (11)

http://dx.doi.org/10.5772/intechopen.70932

35

ai

X bj U∗ aiU<sup>∗</sup>

UaiU<sup>∗</sup>

bjKai, bj

pq,st (13)

ð Þ2 dν1dν2: (14)

ð Þ r<sup>1</sup> dr<sup>1</sup> (15)

ð Þ2 dν1dν<sup>2</sup> (16)

bjKai,jb <sup>þ</sup> O Uð Þ<sup>3</sup> h i (12)

$$
\Delta P\_{\vec{\mathcal{H}}} = \sum\_{i}^{n\_{\text{occ}}} \left( Y\_{\vec{\mathcal{H}}} Y\_{ki} - \delta\_{\vec{\mathcal{H}}} \right) \tag{7}
$$

$$
\Delta P\_{ab} = \sum\_{i}^{n\_{\rm occ}} Y\_{ai} Y\_{bi}. \tag{8}
$$

In CV-DFT, we apply the important condition that one electron is fully transferred from occupied into virtual spaces. This condition can be written as the following equation:

$$\sum\_{a}^{n\_{\rm vir}} \Delta P\_{\rm at} = 1 \quad \text{and} \quad \sum\_{i}^{n\_{\rm oc}} \Delta P\_{\rm ii} = -1. \tag{9}$$

It should be noted that in CV-DFT we describe the excited state with a single Slater determinant. Thus, we obtain the mixed and triplet states. While this is uncritical for triplet excitations, for the singlet transition energy, we have to account for this by using the relation (which is also referred to as sum rule) [61]

$$
\Delta E\_S = 2\Delta E\_M - \Delta E\_T.\tag{10}
$$

#### 2.2. CV(n)-DFT

The nth-order CV-DFT, CV(n)-DFT, is determined from the maximum order of U in the CV-DFT energy description. To understand how the order of the applied transition matrix, U, affects the excited-state energies, it is beneficial to discuss two extreme cases—second (n ¼ 2) order and infinite (n ¼ ∞)-order CV-DFT.

The second-order CV-DFT (CV 2ð Þ-DFT) limits the U up to the second order in the Kohn-Sham energy description. For simplicity, the occupied excited-state orbitals in Eq. (4) are approximated to the second order in U:

Constricted Variational Density Functional Theory Approach to the Description of Excited States http://dx.doi.org/10.5772/intechopen.70932 35

$$\left[\phi\_i\right]' = \phi\_i + \sum\_{a}^{n\_{\rm vir}} \mathcal{U}\_{di}\phi\_a - \frac{1}{2} \sum\_{j}^{n\_{\rm loc}} \sum\_{a}^{n\_{\rm vir}} \mathcal{U}\_{di} \mathcal{U}\_{aj} \phi\_j + O\left[\mathcal{U}^{(3)}\right] \tag{11}$$

With these orbitals, some higher order contributions in U can arise in the density and therewith also in the energy, but we only keep up to the second order in U, as the contribution of higher order terms is negligible [60]. The second-order CV-DFT energy expression becomes

$$\begin{aligned} \left[E\_{KS}[\rho'(1,1')]\right] &= E\_{KS}[\rho^0] + \sum\_{ai} \mathcal{U}\_{ai} \mathcal{U}\_{ai}^\* (\varepsilon\_a^0 - \varepsilon\_i^0) + \sum\_{ai} \mathcal{U}\_{ai} \mathcal{U}\_{bj}^\* K\_{ai,bj} \\ &+ \frac{1}{2} \sum\_{ai} \sum\_{bj} \mathcal{U}\_{ai} \mathcal{U}\_{bj} K\_{ai,jb} + \frac{1}{2} \sum\_{ai} \sum\_{bj} \mathcal{U}\_{ai}^\* \mathcal{U}\_{bj}^\* K\_{ai,jb} + O\left[\mathcal{U}^{(3)}\right] \end{aligned} \tag{12}$$

where the two-electron integral is composed of a Coulomb and an exchange-correlation part:

$$K\_{pq,st} = K\_{pq,st}^{\mathbb{C}} + K\_{pq,st}^{\text{XC}} \tag{13}$$

with

<sup>r</sup><sup>0</sup> <sup>1</sup>; <sup>1</sup><sup>0</sup> ð Þ¼ <sup>X</sup><sup>n</sup>occ

i ϕ0 i ð Þ1 ϕ<sup>0</sup> <sup>i</sup> 1<sup>0</sup> ð Þ

34 Density Functional Calculations - Recent Progresses of Theory and Application

<sup>¼</sup> <sup>X</sup><sup>n</sup>occ i ϕi

þX<sup>n</sup>occ i

Xnocc j

with the change in density matrix (ΔP). Later, one is given by

Xnvir a

referred to as sum rule) [61]

order and infinite (n ¼ ∞)-order CV-DFT.

mated to the second order in U:

2.2. CV(n)-DFT

ΔPijϕ<sup>i</sup>

ð Þ<sup>1</sup> <sup>ϕ</sup><sup>i</sup> <sup>1</sup><sup>0</sup> ð ÞþX<sup>n</sup>vir

a

ð Þ<sup>1</sup> <sup>ϕ</sup><sup>j</sup> <sup>1</sup><sup>0</sup> ð ÞþX<sup>n</sup>vir

<sup>Δ</sup>Paj <sup>¼</sup> <sup>X</sup><sup>n</sup>occ

i

<sup>Δ</sup>Pab <sup>¼</sup> <sup>X</sup><sup>n</sup>occ

occupied into virtual spaces. This condition can be written as the following equation:

<sup>Δ</sup>Paa <sup>¼</sup> 1 and <sup>X</sup><sup>n</sup>occ

<sup>Δ</sup>Pjk <sup>¼</sup> <sup>X</sup><sup>n</sup>occ

i

i

In CV-DFT, we apply the important condition that one electron is fully transferred from

It should be noted that in CV-DFT we describe the excited state with a single Slater determinant. Thus, we obtain the mixed and triplet states. While this is uncritical for triplet excitations, for the singlet transition energy, we have to account for this by using the relation (which is also

The nth-order CV-DFT, CV(n)-DFT, is determined from the maximum order of U in the CV-DFT energy description. To understand how the order of the applied transition matrix, U, affects the excited-state energies, it is beneficial to discuss two extreme cases—second (n ¼ 2)-

The second-order CV-DFT (CV 2ð Þ-DFT) limits the U up to the second order in the Kohn-Sham energy description. For simplicity, the occupied excited-state orbitals in Eq. (4) are approxi-

YjiYki � δjk

i

Xnocc i

ΔPai ϕ<sup>i</sup>

Xnvir b

a

ð Þ<sup>1</sup> <sup>ϕ</sup><sup>a</sup> <sup>1</sup><sup>0</sup> ð Þþ <sup>ϕ</sup>að Þ<sup>1</sup> <sup>ϕ</sup><sup>i</sup> <sup>1</sup><sup>0</sup> ð Þ � �

YaiYji (6)

� � (7)

YaiYbi: (8)

ΔPii ¼ �1: (9)

ΔES ¼ 2ΔEM � ΔET: (10)

(5)

ΔPabϕað Þ1 ϕ<sup>b</sup> 1<sup>0</sup> ð Þ

$$K\_{pq,st}^{\mathbb{C}} = \int \int \psi\_p(1)\psi\_q(1)\frac{1}{r\_{12}}\psi\_s(2)\psi\_t(2)d\nu\_1 d\nu\_2. \tag{14}$$

The exchange-correlation integral is further decomposed into the local (KS) and nonlocal (HF):

$$K\_{pq,st}^{\rm XC(KS)} = \int \psi\_p(r\_1)\psi\_q(r\_1)f(r\_1)\psi\_s(r\_1)\psi\_t(r\_1)dr\_1\tag{15}$$

and

$$K\_{pq,st}^{X\mathcal{C}(HF)} = -\int \left[\psi\_p(1)\psi\_q(1)\frac{1}{r\_{12}}\psi\_s(2)\psi\_t(2)d\nu\_1 d\nu\_2\right.\tag{16}$$

where f rð Þ<sup>1</sup> represents the regular energy kernel. We have shown that CV 2ð Þ-DFT is equivalent to TD-DFT [59, 106] within the Tamm-Dancoff approximation (TDA) [107].

In the infinite-order theory (CV(∞)-DFT), the new set of excited-state orbitals is obtained taking the sum in Eq. (3) to infinite order. These excited-state orbitals can be written in the convenient form of natural transition orbitals (NTO) [108]. For this, we decompose the transition matrix, U, into its singular values. Here, we also used a spin-adapted form for further description of the different spin states in the excited-state calculation:

$$\mathbf{U}^{\circ \alpha} = \mathbf{V}^{\circ \alpha} \Sigma (\mathbf{W}^{\alpha \alpha})^{\mathrm{T}} \tag{17}$$

where <sup>Σ</sup>ii <sup>¼</sup> <sup>γ</sup><sup>i</sup> and <sup>σ</sup><sup>∈</sup> <sup>α</sup>; <sup>β</sup> � � depend on spin state (mixed and triplet states, respectively). This leads to the occupied and virtual NTOs as

$$\boldsymbol{\phi}\_{i}^{\boldsymbol{\phi}\_{a}} = \sum\_{j}^{n\_{\rm occ}} (\mathbf{W}^{\boldsymbol{\alpha}a})\_{ji} \boldsymbol{\phi}\_{j}^{\boldsymbol{\alpha}} \tag{18}$$

replacement [104], which is used as the ΔSCF-DFT-like scheme within the RSCF-CV(∞)-DFT

Constricted Variational Density Functional Theory Approach to the Description of Excited States

In Eqs. (22, 23), we obtain the excited-state energy of the mixed and triplet state. The transition matrix, U, is the same as one obtains within TD-DFT (and thus the TD-DFT excitation vector is implemented in CV-DFT). In SCF-CV(∞)-DFT, U is optimized with the variational procedure [60]. For this step, we derived the gradient of the mixed and triplet excited state. The detailed procedures can be found in the [59–61, 104]. Further, also the orbitals which do not participate in the excitation can be changed after the excitation. We refer to this change as the relaxation of orbitals. This leads to R-CV(∞)-DFT. To account for this orbital relaxation effect, we introduced R, which is orthogonal to U, and apply it on the orbitals from Eq. (4). Therewith, the 'occupied'

Rciϕcð Þ� <sup>1</sup> <sup>1</sup>

Rakϕkð Þ� <sup>1</sup> <sup>1</sup>

It is possible to combine the approach of the variational optimization of the transition matrix and orbital relaxation, meaning the variational optimization of U and R, resulting in the most general form of CV-DFT (RSCF-CV(∞)-DFT). The excitation energy expression of RSCF-CV(∞)-

� �

� �Δ<sup>r</sup>

� �Δ<sup>r</sup>

Another idea is to restrict the transition matrix, U, in CV-DFT to the case of single NTO excitations, that is, Eq. (17) is approximated to include only one major excitation in the transition matrix. Three different forms of such restrictions on U were shown and discussed in the previous work [104], which referred to as SOR-R-CV(∞)-DFT, COL-RSCF-CV(∞)-DFT

<sup>0</sup> are the ground-state density and Δr<sup>U</sup>,<sup>R</sup> indicates the excited-state density

� �

2 Xnvir c

2 Xnvir c

Xnocc k

Xnocc k

> � <sup>E</sup> <sup>r</sup><sup>α</sup> <sup>0</sup> ; r β 0 h i

> > U,R <sup>M</sup> dν<sup>1</sup>

� <sup>E</sup> <sup>r</sup><sup>α</sup> <sup>0</sup> ; r β 0 h i

> U,R <sup>T</sup> dν<sup>1</sup>

RciRckϕkð Þ1 (24)

http://dx.doi.org/10.5772/intechopen.70932

37

RakRckϕcð Þ1 : (25)

(26)

(27)

formulation. This will be briefly mentioned in the next section.

2.3. SCF-CV(∞)-DFT, R-CV(∞)-DFT and RSCF-CV(∞)-DFT

and 'virtual' orbitals become

ψi

ð Þ¼ 1 ϕ<sup>i</sup>

<sup>ψ</sup>að Þ¼ <sup>1</sup> <sup>ϕ</sup>að Þ� <sup>1</sup> <sup>X</sup><sup>n</sup>occ

DFT can be written for the mixed and triplet state, respectively:

<sup>M</sup> r<sup>α</sup> <sup>0</sup> þ 1 2 Δr U,R <sup>M</sup> ; r β <sup>0</sup> þ 1 2 Δr U,R M

<sup>T</sup> r<sup>α</sup> <sup>0</sup> þ 1 2 Δr U,R <sup>T</sup> ; r β <sup>0</sup> þ 1 2 Δr U,R T

changes including relaxation effect. The FKS is the Kohn-Sham Fock operator.

<sup>Δ</sup>EM <sup>¼</sup> <sup>E</sup><sup>U</sup>,<sup>R</sup>

¼ ð FKS r<sup>α</sup> <sup>0</sup> þ 1 2 Δr U,R <sup>M</sup> ; r β <sup>0</sup> þ 1 2 Δr U,R M

<sup>Δ</sup>ET <sup>¼</sup> <sup>E</sup><sup>U</sup>,<sup>R</sup>

¼ ð FKS r<sup>α</sup> <sup>0</sup> þ 1 2 Δr U,R <sup>T</sup> ; r β <sup>0</sup> þ 1 2 Δr U,R T

where r<sup>α</sup>

<sup>0</sup> and r β ð Þþ <sup>1</sup> <sup>X</sup><sup>n</sup>vir c

k

$$
\phi\_i^{v\_\sigma} = \sum\_a^{n\_{\rm vir}} (\mathbf{V}^{\sigma o})\_{ai} \phi\_a^{o}. \tag{19}
$$

The resulting matrix W rotates ground-state KS orbitals as j runs over the occupied groundstate orbitals to give the corresponding ith 'occupied' NTO orbital (ϕ<sup>o</sup><sup>α</sup> <sup>i</sup> ). For its virtual counterpart (ϕ<sup>v</sup><sup>σ</sup> <sup>i</sup> ), V does the similar role as W with a running over the virtual ground-state orbitals. With these NTOs, we can rewrite Eq. (4) for the 'occupied' excited-state orbitals as

$$\phi\_i^{'} = \cos[\mathcal{V}\_i] \phi\_i^{o\_a} + \sin[\mathcal{V}\_i] \phi\_i^{v\_o}. \tag{20}$$

Also, the condition of exciting exactly one electron (Eq. (9)) is now written as

$$\sum\_{i}^{n\_{\text{occ}}} \sin \left( \eta \boldsymbol{\nu}\_{i} \right)^{2} = 1. \tag{21}$$

With the sum rule in Eq. (10), the excited-state CV(∞)-DFT energy of the mixed state becomes

$$\begin{split} \Delta E\_{M} &= \sum\_{i}^{n\_{\rm{ox}}} \text{sinc}^{2} [\eta \gamma\_{i}] \left( \varepsilon\_{i}^{p\_{a}} - \varepsilon\_{i}^{o} \right) \\ &+ \frac{1}{2} \sum\_{i}^{n\_{\rm{ox}}} \sum\_{j}^{n\_{\rm{ox}}} \text{sinc}^{2} [\eta \gamma\_{i}] \text{sinc}^{2} \left[ \eta \gamma\_{j} \right] \left( K\_{I^{n\_{f}n\_{f}^{no}} f^{n}} K\_{I^{n\_{f}^{ra}f^{n}} f^{n}} - 2 K\_{I^{n\_{f}^{ra}f^{n}} f^{n}} \right) \\ &+ \frac{1}{2} \sum\_{i}^{n\_{\rm{ox}}} \sum\_{j}^{n\_{\rm{ox}}} \sin \left[ \eta \gamma\_{i} \right] \cos \left[ \eta \gamma\_{i} \right] \sin \left[ \eta \gamma\_{j} \right] \left( K\_{I^{n\_{f}n\_{f}^{ra}f^{n}}} + K\_{I^{n\_{f}n\_{f}^{ra}f^{n}} f^{n}} \right) \\ &+ 2 \sum\_{i}^{n\_{\rm{ox}}} \sum\_{j}^{n\_{\rm{ox}}} \sin \left[ \eta \gamma\_{i} \right] \sin \left[ \eta \gamma\_{j} \right] \cos \left[ \eta \gamma\_{j} \right] \left( K\_{I^{n\_{f}n\_{f}^{ra}f^{n}}} - K\_{I^{n\_{f}n\_{f}^{ra}f^{n}}} \right) \end{split} \tag{22}$$

whereas the triplet exited-state energy has a simpler form:

$$\begin{split} \Delta E\_{T} &= \sum\_{i}^{n\_{\text{occ}}} \sin^{2} \left[ \eta \gamma\_{i} \right] \left( \varepsilon\_{i}^{\nu\_{i}} - \varepsilon\_{i}^{a\_{a}} \right) \\ &+ \frac{1}{2} \sum\_{i}^{n\_{\text{occ}}} \sum\_{j}^{n\_{\text{occ}}} \sin^{2} \left[ \eta \gamma\_{i} \right] \sin^{2} \left[ \eta \gamma\_{j} \right] \left( K\_{I^{u\_{I} p\_{u} j n\_{\text{par}}}} K\_{I^{p\_{I}} i^{p\_{j}} j^{\rho\_{j}}} - K\_{I^{u\_{I} p\_{j}} i^{p\_{j}} j^{\rho\_{j}}} \right) \end{split} . \tag{23}$$

The γ values out of Eq. (21) give information about the excitation character [60]. Keeping only the largest γ value in the excitation will give the most general form of single orbital replacement [104], which is used as the ΔSCF-DFT-like scheme within the RSCF-CV(∞)-DFT formulation. This will be briefly mentioned in the next section.

#### 2.3. SCF-CV(∞)-DFT, R-CV(∞)-DFT and RSCF-CV(∞)-DFT

ϕ<sup>o</sup><sup>α</sup> <sup>i</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>occ j

36 Density Functional Calculations - Recent Progresses of Theory and Application

ϕ<sup>v</sup><sup>σ</sup> <sup>i</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>vir a

With these NTOs, we can rewrite Eq. (4) for the 'occupied' excited-state orbitals as

� �ϕ<sup>o</sup><sup>α</sup>

sin ηγ<sup>i</sup>

With the sum rule in Eq. (10), the excited-state CV(∞)-DFT energy of the mixed state becomes

Ki oα i oα j oα j <sup>o</sup>αKi vα i vα j vα j <sup>v</sup><sup>α</sup> � 2Ki oα i oα j vα j vα

> cos ηγ<sup>j</sup> h i

cos ηγ<sup>j</sup> h i

¼ cos γ<sup>i</sup>

Xnocc i

Also, the condition of exciting exactly one electron (Eq. (9)) is now written as

<sup>i</sup> � <sup>ε</sup><sup>o</sup><sup>α</sup> i � �

� �sin<sup>2</sup> ηγ<sup>j</sup>

� �cos ηγ<sup>i</sup>

� �sin ηγ<sup>i</sup>

h i

� �sin ηγ<sup>j</sup> h i

� �sin ηγ<sup>j</sup> h i

state orbitals to give the corresponding ith 'occupied' NTO orbital (ϕ<sup>o</sup><sup>α</sup>

ϕi 0

terpart (ϕ<sup>v</sup><sup>σ</sup>

<sup>Δ</sup>EM <sup>¼</sup> <sup>X</sup><sup>n</sup>occ

þ 1 2 Xnocc i

þ 1 2 Xnocc i

þ 2 Xnocc i

<sup>Δ</sup>ET <sup>¼</sup> <sup>X</sup><sup>n</sup>occ

þ 1 2 Xnocc i

i

i

sin<sup>2</sup> ηγ<sup>i</sup> � � ε<sup>v</sup><sup>α</sup>

> Xnocc j

> Xnocc j

Xnocc j

sin<sup>2</sup> ηγ<sup>i</sup>

sin ηγ<sup>i</sup>

sin ηγ<sup>i</sup>

whereas the triplet exited-state energy has a simpler form:

sin<sup>2</sup> ηγ<sup>i</sup> � � ε vβ <sup>i</sup> � <sup>ε</sup><sup>o</sup><sup>α</sup> i � �

Xnocc j

sin<sup>2</sup> ηγ<sup>i</sup>

� �sin<sup>2</sup> ηγ<sup>j</sup>

h i

Ki oα i oα j oα j <sup>o</sup>αKi vβ i vβ j vβ j <sup>v</sup><sup>β</sup> � 2Ki oα i oα j vβ j vβ

The γ values out of Eq. (21) give information about the excitation character [60]. Keeping only the largest γ value in the excitation will give the most general form of single orbital

<sup>W</sup>αα ð Þjiϕ<sup>α</sup>

<sup>V</sup>σσ ð Þaiϕσ

<sup>i</sup> ), V does the similar role as W with a running over the virtual ground-state orbitals.

<sup>i</sup> þ sin γ<sup>i</sup>

� �ϕ<sup>v</sup><sup>σ</sup>

� �

Ki oα i vα j oα j <sup>v</sup><sup>α</sup> þ Ki oα i vα j vα j oα

Ki vα i vα j oα j <sup>v</sup><sup>α</sup> � Ki oα i oα j oα j vα

The resulting matrix W rotates ground-state KS orbitals as j runs over the occupied ground-

<sup>j</sup> (18)

<sup>a</sup> : (19)

<sup>i</sup> : (20)

� �<sup>2</sup> <sup>¼</sup> <sup>1</sup>: (21)

� �

� �

� � : (23)

<sup>i</sup> ). For its virtual coun-

(22)

In Eqs. (22, 23), we obtain the excited-state energy of the mixed and triplet state. The transition matrix, U, is the same as one obtains within TD-DFT (and thus the TD-DFT excitation vector is implemented in CV-DFT). In SCF-CV(∞)-DFT, U is optimized with the variational procedure [60]. For this step, we derived the gradient of the mixed and triplet excited state. The detailed procedures can be found in the [59–61, 104]. Further, also the orbitals which do not participate in the excitation can be changed after the excitation. We refer to this change as the relaxation of orbitals. This leads to R-CV(∞)-DFT. To account for this orbital relaxation effect, we introduced R, which is orthogonal to U, and apply it on the orbitals from Eq. (4). Therewith, the 'occupied' and 'virtual' orbitals become

$$
\psi\_i(1) = \phi\_i(1) + \sum\_{c}^{n\_{\rm vir}} R\_{ci} \phi\_c(1) - \frac{1}{2} \sum\_{c}^{n\_{\rm vir}} \sum\_{k}^{n\_{\rm sc}} R\_{ci} R\_{ck} \phi\_k(1) \tag{24}
$$

$$
\psi\_a(1) = \phi\_a(1) - \sum\_k^{n\_{\rm osc}} R\_{dk}\phi\_k(1) - \frac{1}{2} \sum\_c^{n\_{\rm vir}} \sum\_k^{n\_{\rm oc}} R\_{dk} R\_{ck} \phi\_c(1). \tag{25}
$$

It is possible to combine the approach of the variational optimization of the transition matrix and orbital relaxation, meaning the variational optimization of U and R, resulting in the most general form of CV-DFT (RSCF-CV(∞)-DFT). The excitation energy expression of RSCF-CV(∞)- DFT can be written for the mixed and triplet state, respectively:

$$\begin{split} \Delta E\_{M} &= E\_{M}^{\mathbf{U},\mathbf{R}} \left[ \rho\_{0}^{a} + \frac{1}{2} \Delta \rho\_{M}^{\mathbf{U},\mathbf{R}}, \rho\_{0}^{\boldsymbol{\delta}} + \frac{1}{2} \Delta \rho\_{M}^{\mathbf{U},\mathbf{R}} \right] - E \left[ \rho\_{0}^{a}, \rho\_{0}^{\boldsymbol{\delta}} \right] \\ &= \int F\_{KS} \left[ \rho\_{0}^{a} + \frac{1}{2} \Delta \rho\_{M}^{\mathbf{U},\mathbf{R}}, \rho\_{0}^{\boldsymbol{\delta}} + \frac{1}{2} \Delta \rho\_{M}^{\mathbf{U},\mathbf{R}} \right] \Delta \rho\_{M}^{\mathbf{U},\mathbf{R}} \, dv\_{1} \end{split} \tag{26}$$
 
$$\Delta E\_{T} = E\_{T}^{\mathbf{U},\mathbf{R}} \left[ \rho\_{0}^{a} + \frac{1}{2} \Delta \rho\_{T}^{\mathbf{U},\mathbf{R}}, \rho\_{0}^{\boldsymbol{\delta}} + \frac{1}{2} \Delta \rho\_{T}^{\mathbf{U},\mathbf{R}} \right] - E \left[ \rho\_{0}^{a}, \rho\_{0}^{\boldsymbol{\delta}} \right] \tag{27}$$
 
$$= \int F\_{KS} \left[ \rho\_{0}^{a} + \frac{1}{2} \Delta \rho\_{T}^{\mathbf{U},\mathbf{R}}, \rho\_{0}^{\boldsymbol{\delta}} + \frac{1}{2} \Delta \rho\_{T}^{\mathbf{U},\mathbf{R}} \right] \Delta \rho\_{T}^{\mathbf{U},\mathbf{R}} \, dv\_{1} \tag{27}$$

where r<sup>α</sup> <sup>0</sup> and r β <sup>0</sup> are the ground-state density and Δr<sup>U</sup>,<sup>R</sup> indicates the excited-state density changes including relaxation effect. The FKS is the Kohn-Sham Fock operator.

Another idea is to restrict the transition matrix, U, in CV-DFT to the case of single NTO excitations, that is, Eq. (17) is approximated to include only one major excitation in the transition matrix. Three different forms of such restrictions on U were shown and discussed in the previous work [104], which referred to as SOR-R-CV(∞)-DFT, COL-RSCF-CV(∞)-DFT


'T' indicates that it is introduced theoretically.

'I' indicates that it is implemented into the code.

Table 1. Variation of CV-DFT applied.

and SVD-RSCF-CV(∞)-DFT. Among the three methods, we have shown that SVD-RSCF-CV (∞)-DFT as rank 1 approximation is the most general form for such a single NTO excitation:

$$\mathbf{U} = \boldsymbol{\upsilon}\_1^{\alpha \boldsymbol{\sigma}} \left(\boldsymbol{\upsilon}\_1^{\alpha \boldsymbol{\alpha}}\right)^{\mathrm{T}} \tag{28}$$

when compared to <sup>V</sup><sup>b</sup> KS

obtained with V~ KS

weakness of V~ KS

V~ KS XC(r XC(r

XC(r

XC(r

similar deviations from <sup>V</sup><sup>b</sup> KS

after ionization: ΔE ϕ<sup>v</sup> ! ϕ<sup>r</sup>

excitation energies [100].

deviation (RMSD).

theory [71–74]. This results in higher occupied orbital (ε

center of mass where it decays exponentially with <sup>r</sup>, while <sup>V</sup><sup>b</sup> KS

XC(r

yield inaccurate results for transitions in which Sia >> 0.

� � = EA(A<sup>+</sup>

!) from an exact functional derived from high-level wave function

XC(r

!). Resultantly, the success of TD-DFT for valence excitations is

, ϕv, ϕr) þ IP(A, ϕv). Thus, errors in the excitation

!) becomes apparent at medium and large separations r from the polyatomic

Constricted Variational Density Functional Theory Approach to the Description of Excited States

!) as opposed to those ((bεi),(bεa)) derived from <sup>V</sup><sup>b</sup> KS

medium to large values of r in the valence region and in the density tail. This is primarily due

noted, the large errors in the individual orbital energies might be canceled after the energy difference is calculated provided that the average potential experienced by ψ<sup>i</sup> and ψ<sup>a</sup> shows

attributable to this phenomenon for transitions ψ<sup>i</sup> ! ψ<sup>a</sup> where the overlap Sia between the two densities r<sup>i</sup> and r<sup>a</sup> is large [92, 114]. However, for cases such as Rydberg transitions [71–74] as well as charge-transfer excitations [77, 79, 82, 90, 96, 97, 115] where Sia is small, the error in

Since Rydberg transitions are characterized by a single orbital replacement ψ<sup>v</sup> ! ψr, RSCF-CV(∞)-DFT will give results very similar to ΔSCF-DFT; this similarity in case of a single NTO transition has been demonstrated in [104]. Although for ΔSCF-DFT, states of the same symmetry as the ground-state almost always decompose to the ground-state, this weakness is absent in RSCF-CV(∞)-DFT. The RSCF-CV(∞)-DFT triplet and singlet transition energies for these single orbital replacement-type excitations are obtained as special case of Eqs. (26) and (27), with the singlet excitation energy given as 2ΔEM � ΔET. In the analysis of Rydberg excitations based on RSCF-CV(∞)-DFT, the excitation energy is considered as a sum of the ionization potential (IP) of a neutral species, A, and the electron affinity (EA) of the resulting cation, Aþ,

energies are due to error in the calculated EAs and IPs. Consequently, a method or 'specialized' XC functional that provide accurate EAs and IPs would in turn afford accurate Rydberg

Shown in Table 2 for comparison with the experimental data are the IPs (N2) and EAs (N<sup>þ</sup>

calculated by RSCF-CV(∞)-DFT (or ΔSCF-DFT). The near-perfect agreement (RMSDs between 0.1 and 0.3 eV) with the experimental data is transferred to the excitation energies afforded by the RSCF-CV(∞)-DFT method. As noted previously by Verma and Bartlett for functionals used within TD-DFT [118–120] and the authors of the work discussed here [100]. A test set including 73 excitations (32 singlet, 41 triplet) from nine different species (N2, 5; CO, 7; CH2O, 8; C2H2, 8; H2O, 10; C2H4, 13; Be, 6; Mg, 6; Zn, 10) has been used. Broken down into the different species, the results are given in Table 3 in terms of mean absolute error (MAE) and root-mean-square

!) gets more pronounced. It is a common practice in the case of small Sia to construct specialized potentials [71–74, 77, 79, 82, 90–98] in which the proper �1=r decay is enforced yielding acceptable results. The disadvantage here is that these parameterized potentials might

Excitation energies in TD-DFT are not necessarily affected by the instability of V~ KS

to the dependence of the excitation energies in TD-DFT to the difference ε

~i) and virtual orbital (ε

XC(r

!) decays as � �1=r.

http://dx.doi.org/10.5772/intechopen.70932

~<sup>a</sup> � ε

~a) energies

39

XC(r !) for

~i. As can be

2 )

!). Additionally, the

where the vσσ <sup>1</sup> and wαα <sup>1</sup> are the vector of the largest singular value Vσσ and Wαα out of Eq. (17). The SVD-RSCF-CV(∞)-DFT was also shown to give the same excitation energies as ΔSCF-DFT within 0.1 eV [104].

As a roundup we list the current different versions of CV-DFT in Table 1.

#### 3. Applications

In this section, we will show examples of excitations where different versions of CV-DFT have been applied successfully. These excitations are of Rydberg type or possess a dominant chargetransfer character; the work has been published in [99–103]. We would like to note that all CV-DFT-calculations presented here were carried out with developers versions of ADF [112, 113] and we refer to the original publications for the technical details.

#### 3.1. Rydberg excitations

It is well understood that the success of TD-DFT directly depends on how well the approximate exchange-correlation density functional used describes the potential (V<sup>b</sup> KS XC(r !)). Further, it is evident that functionals based on the local density approximation (LDA) or the generalized gradient approximation (GGA) result in the potential, V~ KS XC(r !), that is insufficiently stabilizing when compared to <sup>V</sup><sup>b</sup> KS XC(r !) from an exact functional derived from high-level wave function theory [71–74]. This results in higher occupied orbital (ε ~i) and virtual orbital (ε ~a) energies obtained with V~ KS XC(r !) as opposed to those ((bεi),(bεa)) derived from <sup>V</sup><sup>b</sup> KS XC(r !). Additionally, the weakness of V~ KS XC(r !) becomes apparent at medium and large separations r from the polyatomic center of mass where it decays exponentially with <sup>r</sup>, while <sup>V</sup><sup>b</sup> KS XC(r !) decays as � �1=r.

Excitation energies in TD-DFT are not necessarily affected by the instability of V~ KS XC(r !) for medium to large values of r in the valence region and in the density tail. This is primarily due to the dependence of the excitation energies in TD-DFT to the difference ε ~<sup>a</sup> � ε ~i. As can be noted, the large errors in the individual orbital energies might be canceled after the energy difference is calculated provided that the average potential experienced by ψ<sup>i</sup> and ψ<sup>a</sup> shows similar deviations from <sup>V</sup><sup>b</sup> KS XC(r !). Resultantly, the success of TD-DFT for valence excitations is attributable to this phenomenon for transitions ψ<sup>i</sup> ! ψ<sup>a</sup> where the overlap Sia between the two densities r<sup>i</sup> and r<sup>a</sup> is large [92, 114]. However, for cases such as Rydberg transitions [71–74] as well as charge-transfer excitations [77, 79, 82, 90, 96, 97, 115] where Sia is small, the error in V~ KS XC(r !) gets more pronounced. It is a common practice in the case of small Sia to construct specialized potentials [71–74, 77, 79, 82, 90–98] in which the proper �1=r decay is enforced yielding acceptable results. The disadvantage here is that these parameterized potentials might yield inaccurate results for transitions in which Sia >> 0.

and SVD-RSCF-CV(∞)-DFT. Among the three methods, we have shown that SVD-RSCF-CV (∞)-DFT as rank 1 approximation is the most general form for such a single NTO excitation:

Transition U Relaxation R Introduced

CV(2)-DFT Second No No N/A No T [58], I [109] T [60], I [110] CV(4)-DFT Fourth No No N/A No T [93], I [109] T [109], I [109] CV(∞)-DFT ∞ No No N/A No T [59], I [111] T [60], I [60] SCF-CV(∞)-DFT ∞ Yes No N/A No T [59], I [60] T [60], I [104] RSCF-CV(∞)-DFT ∞ Yes No Second Yes T [61], I [61] T [61], I [104] SOR-R-CV(∞)-DFT ∞ No Uai ¼ δabδij Second Yes T [62] T [62], I [62] COL-RSCF-CV(∞)-DFT ∞ Yes Uai ¼ δij Second Yes T [104] T [104], I [104] SVD-RSCF-CV(∞)-DFT ∞ Yes γ<sup>1</sup> ¼ 1 Second Yes T [104] T [104], I [104]

Order n Optimization Restrictions Order Optimization Singlets Triplets

<sup>1</sup> wαα 1

The SVD-RSCF-CV(∞)-DFT was also shown to give the same excitation energies as ΔSCF-DFT

In this section, we will show examples of excitations where different versions of CV-DFT have been applied successfully. These excitations are of Rydberg type or possess a dominant chargetransfer character; the work has been published in [99–103]. We would like to note that all CV-DFT-calculations presented here were carried out with developers versions of ADF [112, 113]

It is well understood that the success of TD-DFT directly depends on how well the approxi-

is evident that functionals based on the local density approximation (LDA) or the generalized

XC(r

mate exchange-correlation density functional used describes the potential (V<sup>b</sup> KS

<sup>1</sup> are the vector of the largest singular value Vσσ and Wαα out of Eq. (17).

� �<sup>T</sup> (28)

XC(r

!), that is insufficiently stabilizing

!)). Further, it

<sup>U</sup> <sup>¼</sup> <sup>v</sup>σσ

As a roundup we list the current different versions of CV-DFT in Table 1.

and we refer to the original publications for the technical details.

gradient approximation (GGA) result in the potential, V~ KS

where the vσσ

within 0.1 eV [104].

3. Applications

3.1. Rydberg excitations

<sup>1</sup> and wαα

'T' indicates that it is introduced theoretically. 'I' indicates that it is implemented into the code.

38 Density Functional Calculations - Recent Progresses of Theory and Application

Table 1. Variation of CV-DFT applied.

Since Rydberg transitions are characterized by a single orbital replacement ψ<sup>v</sup> ! ψr, RSCF-CV(∞)-DFT will give results very similar to ΔSCF-DFT; this similarity in case of a single NTO transition has been demonstrated in [104]. Although for ΔSCF-DFT, states of the same symmetry as the ground-state almost always decompose to the ground-state, this weakness is absent in RSCF-CV(∞)-DFT. The RSCF-CV(∞)-DFT triplet and singlet transition energies for these single orbital replacement-type excitations are obtained as special case of Eqs. (26) and (27), with the singlet excitation energy given as 2ΔEM � ΔET. In the analysis of Rydberg excitations based on RSCF-CV(∞)-DFT, the excitation energy is considered as a sum of the ionization potential (IP) of a neutral species, A, and the electron affinity (EA) of the resulting cation, Aþ, after ionization: ΔE ϕ<sup>v</sup> ! ϕ<sup>r</sup> � � = EA(A<sup>+</sup> , ϕv, ϕr) þ IP(A, ϕv). Thus, errors in the excitation energies are due to error in the calculated EAs and IPs. Consequently, a method or 'specialized' XC functional that provide accurate EAs and IPs would in turn afford accurate Rydberg excitation energies [100].

Shown in Table 2 for comparison with the experimental data are the IPs (N2) and EAs (N<sup>þ</sup> 2 ) calculated by RSCF-CV(∞)-DFT (or ΔSCF-DFT). The near-perfect agreement (RMSDs between 0.1 and 0.3 eV) with the experimental data is transferred to the excitation energies afforded by the RSCF-CV(∞)-DFT method. As noted previously by Verma and Bartlett for functionals used within TD-DFT [118–120] and the authors of the work discussed here [100]. A test set including 73 excitations (32 singlet, 41 triplet) from nine different species (N2, 5; CO, 7; CH2O, 8; C2H2, 8; H2O, 10; C2H4, 13; Be, 6; Mg, 6; Zn, 10) has been used. Broken down into the different species, the results are given in Table 3 in terms of mean absolute error (MAE) and root-mean-square deviation (RMSD).


The results in Table 3 for RSCF-CV(∞)-DFT (or ΔSCF-DFT) are in general better than TD-DFT with the same functionals but at par with TD-DFT results with 'specialized' functionals [71–74]. With this benchmark the suitability of RSCF-CV(∞)-DFT without the need for sophisticated (or 'specialized') functionals for Rydberg excitations has been demonstrated. The origin of this good performance is attributable to the ability of RSCF-CV(∞)-DFT to afford good estimates of IPs and EAs for all functionals [100, 121, 122]. Admittedly, fortuitous error cancelation in IPs and EAs obtained for both RSCF-CV(∞)-DFT and TD-DFT plays a role in the accuracy of the

Constricted Variational Density Functional Theory Approach to the Description of Excited States

http://dx.doi.org/10.5772/intechopen.70932

41

In this subsection we will have a look at excitations with charge-transfer character.

It is well known that TD-DFT applied with standard local exchange and correlation functionals has difficulties for transitions with charge-transfer character between two spatially separated regions [82, 91, 109], a finding nicely explained by Drew, Weisman and Head-Gordon [114]. According to several authors, the reason lies in the exchange and correlation functional [79, 82, 91, 123, 124]. Indeed, a functional like CAM-B3LYP [125] includes a certain Hartree-Fock exchange and results in a clear improvement of TD-DFT excitation energies for transitions involving a charge-transfer character [79, 124, 126]. To further improve the asymptote of the exchange-correlation potential, long-range corrected hybrid scheme like the ones proposed in [76, 95, 98, 127] and asymptotically corrected model potential scheme like in [128, 129] have been designed. Of course modifying the functional is not the only approach, and it is not surprising also that other DFT-based approaches have been suggested, all having their own assets and drawbacks. Several of them have been applied for excitations involving chargetransfer character, for example, constrained orthogonality method (COM) [49, 50], maximum overlap method (MOM) [56], constricted variational density functional theory (CV-DFT) [58] and its extensions [104, 105], constrained density functional theory [130], self-consistent field DFT (ΔSCF-DFT) [131], orthogonality constrained DFT (OCDFT) [63], ensemble DFT [132, 133]

Ziegler et al. showed in [115] how the theoretical framework of CV-DFT is able to cope with excitations including a charge-transfer character and demonstrated this capability with different applications [102, 109, 121]. Here, we will have a look at examples out of three of these

Ethylene tetrafluoroethylene, C2H4xC2F4, is a system well studied in literature [76, 91, 93, 114, 126, 134]. It allows for the study of the dependence of excitation energies on the separation of

For the system C2H4xC2F4, two transitions are of particular interest, the excitations HOMO ! LUMO and HOMO-1 ! LUMO + 1, both resulting in an excited state of b1 symmetry. With these transitions, a charge is transferred between the two molecules C2H4 and C2F4. Although the concrete orbital localization is highly functional dependent, the orbitals HOMO-1, HOMO,

resultant excitation energies.

3.2. Charge-transfer excitations

and subsystem DFT (FDE-ET) [134].

3.2.1. C2H4XC2F4: long-range charge-transfer excitations

the donor and acceptor and test for the expected �1=R behavior.

mentioned types.

a Energies in eV.

b [72].

c Refers to LC functional combined with BP86 and ω = 0.40. d Represents LC functional combined with BP86 and ω = 0.75. e [116]. f [117]. g Evaluated as EA(Aþ, <sup>ϕ</sup>v, <sup>ϕ</sup>r, S) = <sup>Δ</sup>ES(ϕ<sup>v</sup> ! <sup>ϕ</sup>r) � IP(A, <sup>ϕ</sup>v). <sup>h</sup> Evaluated as EA(Aþ, ϕv, ϕr, T) = ΔET(ϕ<sup>v</sup> ! ϕr) � IP(A, ϕv). Data represented in this table was first published in [100].

Table 2. IP<sup>a</sup> of N2 and EA<sup>a</sup> of N<sup>þ</sup> <sup>2</sup> calculated with <sup>Δ</sup>SCF using an extended basis set<sup>b</sup> and five different functionals.


a Energies in eV.

b [72].

c Refers to LC functional combined with BP86 and ω = 0.40.

d Represents LC functional combined with BP86 and ω = 0.75.

e Comprising 12 states.

Data represented in this table was first published in [100].

Table 3. Summary of RMSDs of Rydberg excitation energies<sup>a</sup> calculated with ΔSCF using an extended basis set<sup>b</sup> and five different functionals.

The results in Table 3 for RSCF-CV(∞)-DFT (or ΔSCF-DFT) are in general better than TD-DFT with the same functionals but at par with TD-DFT results with 'specialized' functionals [71–74].

With this benchmark the suitability of RSCF-CV(∞)-DFT without the need for sophisticated (or 'specialized') functionals for Rydberg excitations has been demonstrated. The origin of this good performance is attributable to the ability of RSCF-CV(∞)-DFT to afford good estimates of IPs and EAs for all functionals [100, 121, 122]. Admittedly, fortuitous error cancelation in IPs and EAs obtained for both RSCF-CV(∞)-DFT and TD-DFT plays a role in the accuracy of the resultant excitation energies.

#### 3.2. Charge-transfer excitations

Energy term LDA BP86 B3LYP LCBP86\*<sup>c</sup> LCBP86<sup>d</sup> Expt. IP(N2, σg) 15.63 15.50 15.74 15.96 16.38 15.58e,f IP(N2, πu) 17.46 17.07 16.87 17.22 17.18 17.07<sup>f</sup>

, <sup>σ</sup>g, 3sσg, S) �3.64 �3.61 �3.49 �3.55 �3.44 �3.38<sup>g</sup>

, <sup>σ</sup>g, 3pπu, S) �2.92 �3.01 �2.87 �2.81 �2.77 �2.68<sup>g</sup>

, <sup>σ</sup>g, 3pσu, S) �2.84 �2.95 �2.79 �2.71 �2.66 �2.60<sup>g</sup>

, <sup>π</sup>u, 3sσg, S) �3.77 �3.77 �3.71 �3.73 �3.67 �3.83<sup>g</sup>

, <sup>σ</sup>g, 3sσg, T) �3.82 �3.81 �3.73 �3.79 �3.72 �3.58<sup>h</sup>

Species No. of states LDA BP86 B3LYP LCBP86\*<sup>c</sup> LCBP86<sup>d</sup> N2 5 0.27 0.34 0.05 0.23 0.62 CO 7 0.22 0.43 0.13 0.12 0.37 CH2O 8 0.21 0.28 0.12 0.20 0.34 C2H2 8 0.31 0.50 0.52 0.25 0.24 H2O 10 0.27 0.17 0.14 0.21 0.24 C2H4 13 0.15 0.20 0.28<sup>e</sup> 0.28 0.29 Be 6 0.45 0.60 0.47 0.31 0.23 Mg 6 0.18 0.35 0.19 0.13 0.12 Zn 10 0.18 0.25 0.27 0.34 0.46 RMSD — 0.24 0.32 0.24 0.23 0.32

Table 3. Summary of RMSDs of Rydberg excitation energies<sup>a</sup> calculated with ΔSCF using an extended basis set<sup>b</sup> and five

<sup>2</sup> calculated with <sup>Δ</sup>SCF using an extended basis set<sup>b</sup> and five different functionals.

EA(N2 +

EA(N2 +

EA(N2 +

EA(N2 +

EA(N2 +

Energies in eV.

Refers to LC functional combined with BP86 and ω = 0.40.

Represents LC functional combined with BP86 and ω = 0.75.

40 Density Functional Calculations - Recent Progresses of Theory and Application

Evaluated as EA(Aþ, <sup>ϕ</sup>v, <sup>ϕ</sup>r, S) = <sup>Δ</sup>ES(ϕ<sup>v</sup> ! <sup>ϕ</sup>r) � IP(A, <sup>ϕ</sup>v). <sup>h</sup> Evaluated as EA(Aþ, ϕv, ϕr, T) = ΔET(ϕ<sup>v</sup> ! ϕr) � IP(A, ϕv). Data represented in this table was first published in [100].

Refers to LC functional combined with BP86 and ω = 0.40.

Data represented in this table was first published in [100].

Represents LC functional combined with BP86 and ω = 0.75.

Table 2. IP<sup>a</sup> of N2 and EA<sup>a</sup> of N<sup>þ</sup>

a

b [72]. c

d

e [116]. f [117]. g

a

b [72]. c

d

e

Energies in eV.

Comprising 12 states.

different functionals.

In this subsection we will have a look at excitations with charge-transfer character.

It is well known that TD-DFT applied with standard local exchange and correlation functionals has difficulties for transitions with charge-transfer character between two spatially separated regions [82, 91, 109], a finding nicely explained by Drew, Weisman and Head-Gordon [114]. According to several authors, the reason lies in the exchange and correlation functional [79, 82, 91, 123, 124]. Indeed, a functional like CAM-B3LYP [125] includes a certain Hartree-Fock exchange and results in a clear improvement of TD-DFT excitation energies for transitions involving a charge-transfer character [79, 124, 126]. To further improve the asymptote of the exchange-correlation potential, long-range corrected hybrid scheme like the ones proposed in [76, 95, 98, 127] and asymptotically corrected model potential scheme like in [128, 129] have been designed. Of course modifying the functional is not the only approach, and it is not surprising also that other DFT-based approaches have been suggested, all having their own assets and drawbacks. Several of them have been applied for excitations involving chargetransfer character, for example, constrained orthogonality method (COM) [49, 50], maximum overlap method (MOM) [56], constricted variational density functional theory (CV-DFT) [58] and its extensions [104, 105], constrained density functional theory [130], self-consistent field DFT (ΔSCF-DFT) [131], orthogonality constrained DFT (OCDFT) [63], ensemble DFT [132, 133] and subsystem DFT (FDE-ET) [134].

Ziegler et al. showed in [115] how the theoretical framework of CV-DFT is able to cope with excitations including a charge-transfer character and demonstrated this capability with different applications [102, 109, 121]. Here, we will have a look at examples out of three of these mentioned types.

#### 3.2.1. C2H4XC2F4: long-range charge-transfer excitations

Ethylene tetrafluoroethylene, C2H4xC2F4, is a system well studied in literature [76, 91, 93, 114, 126, 134]. It allows for the study of the dependence of excitation energies on the separation of the donor and acceptor and test for the expected �1=R behavior.

For the system C2H4xC2F4, two transitions are of particular interest, the excitations HOMO ! LUMO and HOMO-1 ! LUMO + 1, both resulting in an excited state of b1 symmetry. With these transitions, a charge is transferred between the two molecules C2H4 and C2F4. Although the concrete orbital localization is highly functional dependent, the orbitals HOMO-1, HOMO,

these excitations, similar energies are reported using the revised Hessian in [93]. In R-CV(∞)- DFT [61], relaxation of orbitals not directly participating is allowed (see Section 2.3), and it is of no surprise that excitation energies decrease. These results still correspond to a �1=R behavior (resulting in fitting coefficients c<sup>1</sup> for the values presented in Figure 2 of 1.1 and 0.9 Eha0). For the HOMO ! LUMO transition, the values agree with those reported in [76] using LC-BLYP (MAD = 0.2 eV, RMSD = 0.2 eV). Thus, the extrapolated infinite separation value, ΔER!<sup>∞</sup> ¼

Constricted Variational Density Functional Theory Approach to the Description of Excited States

http://dx.doi.org/10.5772/intechopen.70932

43

Turning next to the triplet excitations for both CV(∞)-DFT and R-CV(∞)-DFT, similar findings are obtained. At longer distances, no spin interaction is expected; as envisioned the triplet excitation energies match values obtained for the corresponding singlet excitation. Excluding the HOMO ! LUMO triplet excitations with R < 6 Å, a nice �1=R behavior is obtained.

Until now all the applied methods have one thing in common: the transition matrix U has not been optimized. This means the character of the transition itself has not been changed. With CV-DFT being a variational method, the transition matrix U can be optimized with the aim of minimizing the energy (see Section 2.3). In this case the RSCF-CV(∞)-DFT method [59–61] is applied, whose strength and merits have been demonstrated several times [100, 104, 121]. From Figure 2, it can clearly be seen that RSCF-CV(∞)-DFT minimizes the excitation energy at the expense of nearly distance-independent excitation energies and the loss of the �1=R long-range dependence. This energy gain stems from the optimization of the transition matrix U; a thorough explanation is given in [102]. In summary, the charge-transfer transitions, HOMO ! LUMO and HOMO-1 ! LUMO + 1, are dominated by single NTO transitions. Optimizing the transition matrix results in a mix of (mainly) two NTO transitions with (at least one) different participating fragments, meaning that the two charge-transfer excitations, clearly separated before, do mix now. This mixing of the two different excitations leads to a smaller destabilization and a larger stabilization, resulting in a clear reduction of the excitation energy [102]. An additional issue comes now from having a partial charge cA ∈ð Þ 0; 1 located on fragment A and a partial charge 1 � cA on fragment B, even when these two fragments are further apart. Therefore, from a certain distance on this mixing should be suppressed. To block the optimization algorithm from mixing such unwanted excitations in RSCF-CV(∞)-DFT calculations, two different strategies have been proposed in [102]. But while working, they both depend highly on an arbitrarily chosen value for a threshold parameter. It remains to be seen, if a strategy without the need of such a parameter can be

The focus of this subchapter is on polyacenes, a system with an intramolecular charge-transferlike character, also referred to as charge-transfer in disguise [135]. The polyacenes are understood as a number <sup>n</sup><sup>r</sup> of linearly fused benzene rings. Such linear polyacenes possess <sup>π</sup> ! <sup>π</sup><sup>⋆</sup> excitations La (or B2u when the x-axis corresponds to the long molecular axis) and Lb (or B3u) with distinct properties, described, for instance, in [136]. Additionally, these polyacenes have a singlettriplet gap for which a function of n<sup>r</sup> has been proposed. An extrapolation of this function gave rise to a discussion: if polyacenes with a certain size would have a triplet ground state [137–143].

12.7 eV, is close to the ΔER!<sup>∞</sup> ¼ 12.5 eV reported in [76].

found for RSCF-CV(∞)-DFT.

3.2.2. Polyacenes: excitations with hidden charge-transfer character

Figure 1. C2H4xC2F4: Representation of ground-state KS orbitals (LDA) (R ¼ 5.0 Å) (reprinted from [102], with the permission of AIP Publishing).

LUMO and LUMO + 1 are from certain separation distance, dominantly located on one of the fragments, as visible in Figure 1 (see, e.g., [77, 102]). It should be noted that for a classification to one of the aforementioned types, it is sufficient when the mentioned ground-state orbitals contribute the most, not necessarily uniquely.

The results obtained with CV-DFT and selected reference values for comparison are shown in Figure 2.

First, consideration will be given to the singlet and triplet excitation results with different versions of CV-DFT, where the transition matrix, U, is not optimized, before turning to the most general form RSCF-CV(∞)-DFT.

CV(∞)-DFT results in a �1=R-like behavior, or when assuming a ΔE Rð Þ¼�c1=R þ c<sup>0</sup> function, fitting coefficients c<sup>1</sup> for the results presented in Figure 2 of 1.1 and 0.9 Eha0 are obtained. For

Figure 2. C2H4xC2F4 vertical excitation energies for singlets (circles) and triplets (triangles) using CV(∞)-DFT (orange), R-CV(∞)-DFT (red) and RSCF-CV(∞)-DFT (dark red). The values for the revised hessian out of [96] (purple-filled circles), LC-BLYP out of [76] (black-filled circles) and SAC-CI out of [76] (gray-filled circles) are given as reference. The lines serve as a guide for the eyes, and when the excitation is not dominated by one of the charge-transfer excitations, we set its value to zero (and are therewith not visible in the figure). (reprinted from Senn F, Park YC. The Journal of Chemical Physics. 2016;145(24):244108-1 – 10). DOI: 10.1063/1.4972231. with the permission of AIP Publishing. Color specifications refer to the original figure).

these excitations, similar energies are reported using the revised Hessian in [93]. In R-CV(∞)- DFT [61], relaxation of orbitals not directly participating is allowed (see Section 2.3), and it is of no surprise that excitation energies decrease. These results still correspond to a �1=R behavior (resulting in fitting coefficients c<sup>1</sup> for the values presented in Figure 2 of 1.1 and 0.9 Eha0). For the HOMO ! LUMO transition, the values agree with those reported in [76] using LC-BLYP (MAD = 0.2 eV, RMSD = 0.2 eV). Thus, the extrapolated infinite separation value, ΔER!<sup>∞</sup> ¼ 12.7 eV, is close to the ΔER!<sup>∞</sup> ¼ 12.5 eV reported in [76].

Turning next to the triplet excitations for both CV(∞)-DFT and R-CV(∞)-DFT, similar findings are obtained. At longer distances, no spin interaction is expected; as envisioned the triplet excitation energies match values obtained for the corresponding singlet excitation. Excluding the HOMO ! LUMO triplet excitations with R < 6 Å, a nice �1=R behavior is obtained.

Until now all the applied methods have one thing in common: the transition matrix U has not been optimized. This means the character of the transition itself has not been changed. With CV-DFT being a variational method, the transition matrix U can be optimized with the aim of minimizing the energy (see Section 2.3). In this case the RSCF-CV(∞)-DFT method [59–61] is applied, whose strength and merits have been demonstrated several times [100, 104, 121]. From Figure 2, it can clearly be seen that RSCF-CV(∞)-DFT minimizes the excitation energy at the expense of nearly distance-independent excitation energies and the loss of the �1=R long-range dependence. This energy gain stems from the optimization of the transition matrix U; a thorough explanation is given in [102]. In summary, the charge-transfer transitions, HOMO ! LUMO and HOMO-1 ! LUMO + 1, are dominated by single NTO transitions. Optimizing the transition matrix results in a mix of (mainly) two NTO transitions with (at least one) different participating fragments, meaning that the two charge-transfer excitations, clearly separated before, do mix now. This mixing of the two different excitations leads to a smaller destabilization and a larger stabilization, resulting in a clear reduction of the excitation energy [102]. An additional issue comes now from having a partial charge cA ∈ð Þ 0; 1 located on fragment A and a partial charge 1 � cA on fragment B, even when these two fragments are further apart. Therefore, from a certain distance on this mixing should be suppressed. To block the optimization algorithm from mixing such unwanted excitations in RSCF-CV(∞)-DFT calculations, two different strategies have been proposed in [102]. But while working, they both depend highly on an arbitrarily chosen value for a threshold parameter. It remains to be seen, if a strategy without the need of such a parameter can be found for RSCF-CV(∞)-DFT.

#### 3.2.2. Polyacenes: excitations with hidden charge-transfer character

LUMO and LUMO + 1 are from certain separation distance, dominantly located on one of the fragments, as visible in Figure 1 (see, e.g., [77, 102]). It should be noted that for a classification to one of the aforementioned types, it is sufficient when the mentioned ground-state orbitals

Figure 1. C2H4xC2F4: Representation of ground-state KS orbitals (LDA) (R ¼ 5.0 Å) (reprinted from [102], with the

(a) HOMO-1 (b) HOMO (c) LUMO (d) LUMO+1

The results obtained with CV-DFT and selected reference values for comparison are shown in

First, consideration will be given to the singlet and triplet excitation results with different versions of CV-DFT, where the transition matrix, U, is not optimized, before turning to the

CV(∞)-DFT results in a �1=R-like behavior, or when assuming a ΔE Rð Þ¼�c1=R þ c<sup>0</sup> function, fitting coefficients c<sup>1</sup> for the results presented in Figure 2 of 1.1 and 0.9 Eha0 are obtained. For

ΔE /eV

Figure 2. C2H4xC2F4 vertical excitation energies for singlets (circles) and triplets (triangles) using CV(∞)-DFT (orange), R-CV(∞)-DFT (red) and RSCF-CV(∞)-DFT (dark red). The values for the revised hessian out of [96] (purple-filled circles), LC-BLYP out of [76] (black-filled circles) and SAC-CI out of [76] (gray-filled circles) are given as reference. The lines serve as a guide for the eyes, and when the excitation is not dominated by one of the charge-transfer excitations, we set its value to zero (and are therewith not visible in the figure). (reprinted from Senn F, Park YC. The Journal of Chemical Physics. 2016;145(24):244108-1 – 10). DOI: 10.1063/1.4972231. with the permission of AIP Publishing. Color specifications refer to the original figure).

5 6 7 8 9 10

R /Ang (b) HOMO-1 → LUMO+1

contribute the most, not necessarily uniquely.

42 Density Functional Calculations - Recent Progresses of Theory and Application

5 6 7 8 9 10

R /Ang (a) HOMO → LUMO

most general form RSCF-CV(∞)-DFT.

Figure 2.

permission of AIP Publishing).

ΔE /eV

The focus of this subchapter is on polyacenes, a system with an intramolecular charge-transferlike character, also referred to as charge-transfer in disguise [135]. The polyacenes are understood as a number <sup>n</sup><sup>r</sup> of linearly fused benzene rings. Such linear polyacenes possess <sup>π</sup> ! <sup>π</sup><sup>⋆</sup> excitations La (or B2u when the x-axis corresponds to the long molecular axis) and Lb (or B3u) with distinct properties, described, for instance, in [136]. Additionally, these polyacenes have a singlettriplet gap for which a function of n<sup>r</sup> has been proposed. An extrapolation of this function gave rise to a discussion: if polyacenes with a certain size would have a triplet ground state [137–143]. Polyacenes and their derivatives have been used in a plethora of applications; an overview of some of these applications can be found in [144, 145]. Thus, it is not surprising that polyacenes and their excitation energies have been studied extensively. While high-level calculations exist, see, for example, the work presented in [140, 141, 143], considering the size of larger polyacenes TD-DFT calculations is more common. But the latter ones applied with standard functionals have several difficulties. This is why different methods and strategies have been used, each one having its advantages, and we refer to [101] and references therein for more details. Before moving on to the results obtained with CV-DFT, it must be noted that the polyradical character in the ground-state builds up with increasing number of fused acenes, which was deduced by Ibeji et al. [143] and was confirmed by Plasser et al. [146]. This polyradical character gets bigger and for polyacenes larger than hexacene even big enough to lead to a 'breakdown of single reference approximation used to describe the ground-state of polyacenes in conventional DFT' [132]. Within CV-DFT we rely on a DFT ground-state description. The awareness of this limitation is the reason why only polyacenes as large as hexacene have been studied with CV-DFT.

We will now have a look at the singlet excitation energies. As these energies are not directly measurable, we will use the modified experimental values from Grimme and Parac [136] as reference, for simplicity referred to as experimental results.

As visible from Table 4 and Figure 3, CV(∞)-DFT with LDA results in vertical singlet excitation energies in a very good agreement with the experimental values [147], while for R-CV(∞)-DFT [101], the values deviate more from the experimental ones, although still in an acceptable agreement (a discussion of the difference is given in [101]). As can be seen from Table 4 and Figure 3, both versions of CV-DFT obtain a crossover between 11 B2u and 11 B3u for Anthracene onwards, which is in agreement with experimental findings.


Next, take a look at the obtained triplet excitation energies for the studied polyacenes, shown in Table 5. The equivalency of CV 2ð Þ-DFT and TD-DFT with the TDA stated in theory section (Section 2.2) is once again confirmed by the numbers in Table 5. It can also be seen that in the triplet case, the energies obtained with R-CV(∞)-DFT change only slightly in comparison with the values obtained with CV 2ð Þ-DFT, on average by 0.05 eV (for comparison, singlet excitations

DFT (maroon), ΔSCF-DFT (gray), CV-DFT (orange, out of [147]), [143] (dark blue), experimental values (black, out of [136]). The solid lines serve as guides for the eyes. (reprinted with permission from Senn F, Krykunov M. The Journal of Physical Chemistry. A. 2015;119(42):10575-10581. DOI: 10.1021/acs.jpca.5b07075. Copyright 2015 American Chemical Society. Color

2 3 4 5 6

nr acene rings

B2u (circles) and 11

Constricted Variational Density Functional Theory Approach to the Description of Excited States

B2u and 0.13 eV for 11

surprisingly small difference is due to the nature of the excitation, and for a further discussion

As previously pointed out in [104], R-CV(∞)-DFT results in triplet states of excitation energies being lower than the ones obtained by coupled cluster methods. Nevertheless, with a RMSD of 0.31 and 0.29 eV, respectively, when compared to the values given in [140 and 143], the results are in reasonable agreement (we note that coordinates were optimized slightly differently). The nature of the triplet excited states is in agreements with the findings of [148], namely, a

B2u state for the first triplet excitation, T1; for the second triplet excitation, T2; <sup>3</sup>

B1g for Anthracene to Hexacene.

B3u, values out of [101, 147]). This

B3u (crosses) of linear polyacenes: R-CV(∞)-

http://dx.doi.org/10.5772/intechopen.70932

45

B3u for

have a MAD of 0.30 eV for 11

specifications refer to the original figure).

13

0.5

1

1.5

2

2.5

3

E /eV

3.5

4

4.5

5

5.5

Naphthalene; and <sup>3</sup>

of the contributions, we refer to [101].

Figure 3. Vertical singlet excitation energies for the states 11

a Out of [136].

b Out of [147]. c Out of [101].

<sup>d</sup> <sup>Δ</sup><sup>E</sup> <sup>¼</sup> <sup>Δ</sup><sup>E</sup> <sup>1</sup>1B2u � <sup>Δ</sup><sup>E</sup> <sup>1</sup>1B3u .

Table 4. Vertical singlet excitation energies (in eV) for linear polyacenes.

Polyacenes and their derivatives have been used in a plethora of applications; an overview of some of these applications can be found in [144, 145]. Thus, it is not surprising that polyacenes and their excitation energies have been studied extensively. While high-level calculations exist, see, for example, the work presented in [140, 141, 143], considering the size of larger polyacenes TD-DFT calculations is more common. But the latter ones applied with standard functionals have several difficulties. This is why different methods and strategies have been used, each one having its advantages, and we refer to [101] and references therein for more details. Before moving on to the results obtained with CV-DFT, it must be noted that the polyradical character in the ground-state builds up with increasing number of fused acenes, which was deduced by Ibeji et al. [143] and was confirmed by Plasser et al. [146]. This polyradical character gets bigger and for polyacenes larger than hexacene even big enough to lead to a 'breakdown of single reference approximation used to describe the ground-state of polyacenes in conventional DFT' [132]. Within CV-DFT we rely on a DFT ground-state description. The awareness of this limitation is the reason why only polyacenes as large as hexacene

We will now have a look at the singlet excitation energies. As these energies are not directly measurable, we will use the modified experimental values from Grimme and Parac [136] as

As visible from Table 4 and Figure 3, CV(∞)-DFT with LDA results in vertical singlet excitation energies in a very good agreement with the experimental values [147], while for R-CV(∞)-DFT [101], the values deviate more from the experimental ones, although still in an acceptable agreement (a discussion of the difference is given in [101]). As can be seen from Table 4 and

No. acene units Exp.a CV(∞)-DFT<sup>b</sup> R-CV(∞)-DFTc

B3u ΔE<sup>d</sup> 1<sup>1</sup> B2u 1<sup>1</sup>

MAD — —— 0.06 0.11 — 0.18 0.12 RMSD — —— 0.06 0.13 — 0.19 0.15

 4.66 4.13 0.53 4.73 4.39 0.34 4.58 4.42 0.16 3.60 3.64 �0.04 3.68 3.73 �0.05 3.46 3.75 �0.29 2.88 3.39 �0.51 2.91 3.32 �0.41 2.69 3.33 �0.63 2.37 3.12 �0.75 2.35 3.03 �0.68 2.15 3.04 �0.89 2.02 2.87 �0.85 1.93 2.82 �0.89 1.74 2.83 �1.09

B2u and 11

B3u ΔE<sup>d</sup> 1<sup>1</sup>

B3u for Anthracene

B3u ΔE<sup>d</sup>

B2u 11

have been studied with CV-DFT.

reference, for simplicity referred to as experimental results.

44 Density Functional Calculations - Recent Progresses of Theory and Application

Figure 3, both versions of CV-DFT obtain a crossover between 11

onwards, which is in agreement with experimental findings.

11 B2u 1<sup>1</sup>

a Out of [136]. b Out of [147]. c Out of [101]. <sup>d</sup> <sup>Δ</sup><sup>E</sup> <sup>¼</sup> <sup>Δ</sup><sup>E</sup> <sup>1</sup>1B2u

� <sup>Δ</sup><sup>E</sup> <sup>1</sup>1B3u

.

Table 4. Vertical singlet excitation energies (in eV) for linear polyacenes.

Figure 3. Vertical singlet excitation energies for the states 11 B2u (circles) and 11 B3u (crosses) of linear polyacenes: R-CV(∞)- DFT (maroon), ΔSCF-DFT (gray), CV-DFT (orange, out of [147]), [143] (dark blue), experimental values (black, out of [136]). The solid lines serve as guides for the eyes. (reprinted with permission from Senn F, Krykunov M. The Journal of Physical Chemistry. A. 2015;119(42):10575-10581. DOI: 10.1021/acs.jpca.5b07075. Copyright 2015 American Chemical Society. Color specifications refer to the original figure).

Next, take a look at the obtained triplet excitation energies for the studied polyacenes, shown in Table 5. The equivalency of CV 2ð Þ-DFT and TD-DFT with the TDA stated in theory section (Section 2.2) is once again confirmed by the numbers in Table 5. It can also be seen that in the triplet case, the energies obtained with R-CV(∞)-DFT change only slightly in comparison with the values obtained with CV 2ð Þ-DFT, on average by 0.05 eV (for comparison, singlet excitations have a MAD of 0.30 eV for 11 B2u and 0.13 eV for 11 B3u, values out of [101, 147]). This surprisingly small difference is due to the nature of the excitation, and for a further discussion of the contributions, we refer to [101].

As previously pointed out in [104], R-CV(∞)-DFT results in triplet states of excitation energies being lower than the ones obtained by coupled cluster methods. Nevertheless, with a RMSD of 0.31 and 0.29 eV, respectively, when compared to the values given in [140 and 143], the results are in reasonable agreement (we note that coordinates were optimized slightly differently). The nature of the triplet excited states is in agreements with the findings of [148], namely, a 13 B2u state for the first triplet excitation, T1; for the second triplet excitation, T2; <sup>3</sup> B3u for Naphthalene; and <sup>3</sup> B1g for Anthracene to Hexacene.


a Out of [101].

b With LDA as functional.

c Out of [143] and references therein.

d To be understood as the deviation of the values obtained with R-CV(∞)-DFT in comparison to the values of this column as reference values.

Table 5. Vertical and adiabatic triplet excitation energies (in eV) for linear polyacenes.

From Figure 4 one can see the singlet-triplet gap (ST) decreasing, resembling an exponential function. In order to estimate the ST gap for infinitely large polyacenes, giving an indication if there would be a ST crossover, several authors fitted the excitation energies to the function f nð Þ¼ <sup>r</sup> aexpð Þþ �bn<sup>r</sup> c (see [140, 142, 143]). With the results of R-CV(∞)-DFT for the vertical transition, the limes of an infinitely long polyacene EST vertð Þ¼ <sup>n</sup><sup>r</sup> ! <sup>∞</sup> ð Þ <sup>0</sup>:<sup>3</sup> � <sup>4</sup>:<sup>5</sup> kcal mol�<sup>1</sup> have been obtained and for the 'adiabatic' transition EST adð Þ¼ <sup>n</sup><sup>r</sup> ! <sup>∞</sup> (�1.6 � 4.0) kcal mol�<sup>1</sup> [101]. For the 'adiabatic' or well-to-well excitations, results from different methods in literature are controversial about a possible ST gap crossover ([140, 142] versus [143, 149, 150]); for TD-DFT it even depends on the functional used [142]. Therefore, necessarily the findings presented here will agree with some findings, while disagree with others. It should be noted that these energies are very small, actually smaller than the estimated accuracy of the CV-DFT method, and with its error it must be regarded as giving only a tendency for no ST crossover. Two additional points of precaution which puts the value of the extrapolated results into question: (a) it has been shown in [142] how a small change of a single excitation energy can influence the obtained polymeric limit, and (b) one should have in mind the change of the ground-state character with the polyacene length and, thus, the number of fused acenes.

made with available experimental data [152, 153] and high-level ab initio calculations [154–160]. There are several adjustable parameters that can influence the excitation energies. These include the size of the basis set used, functionals used, geometry (optimized structures or experimental geometries), medium (since the complexes are anions), etc. Use was made of experimental structures which lead to higher excitation energies (0.1–0.3 eV) compared to optimized structures. Marginal influence of solvation was found for the three valence excitations; the calculated

R-CVð Þ ∞ -DFT (orange), [143] (dark blue), [140] (light blue), experimental values (black, out of [143] and references therein). The symbols are used to distinguish between vertical transitions (crosses) and adiabatic as well as 'imitated adiabatic' transitions (circles). The lines are the curves fitted to the function f nð Þ¼ <sup>r</sup> a expð Þþ �bn<sup>r</sup> c and serve as guides for the eyes. (reprinted with permission from Senn F, Krykunov M. The Journal of Physical Chemistry. A. 2015;119(42):10575-10581. DOI: 10.1021/acs.jpca.5b07075. Copyright 2015 American Chemical Society. Color specifications refer to the original figure).

2 3 4 5 6

Constricted Variational Density Functional Theory Approach to the Description of Excited States

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47

nr acene rings

B2u states of linear polyacenes: R-CV(∞)-DFT (maroon), ΔSCF-DFT (gray), SVD-

Table 6 displays the RMSD between the first three experimental dipole-allowed transitions

On average, the three functionals B3LYP, PBE0 with an intermediate fraction of HF exchange and LCBP86\* have the lowest RMSD of 0.2 eV, whereas the local functionals (LDA, BP86, BPE) and BHLYP with the highest HF fraction and LCBP86 have a somewhat larger RMSD of 0.3 eV for both 3d and 4d + 5d averages. TD-DFT with the same functionals performs poorly for the 3d complexes but shows good agreement with experiment for the heavier tetraoxo complexes.

COSMO [161, 162] excitation energies lower the energies by 0.01–0.02 eV [163, 164].

and the corresponding values calculated by RSCF-CV(∞)-DFT.

0.5

Figure 4. Triplet excitation energies for the 13

1

1.5

2

2.5

E /eV

3

3.5

4

#### 3.2.3. Charge-transfer excitations in transition metals

The complicated electronic structure of transition metal (TM) complexes [151] makes them ideal for testing the performance of newly developed methods. This section deals with charge-transfer (and hidden charge-transfer) excitations in these complexes, more precisely the 3d complexes MnO4 �, CrO4 <sup>2</sup>� and VO4 <sup>3</sup>�, as well as their 4d congeners RuO4, TcO4 �, MoO4 <sup>2</sup>� and 5d homologues OsO4, ReO4 � and WO4 <sup>2</sup>� [99]. For these systems, the three lowest valence excitations involving transitions from 1t1, 2t<sup>2</sup> to 2e and 3t<sup>2</sup> are considered [99]. The comparison is Constricted Variational Density Functional Theory Approach to the Description of Excited States http://dx.doi.org/10.5772/intechopen.70932 47

Figure 4. Triplet excitation energies for the 13 B2u states of linear polyacenes: R-CV(∞)-DFT (maroon), ΔSCF-DFT (gray), SVD-R-CVð Þ ∞ -DFT (orange), [143] (dark blue), [140] (light blue), experimental values (black, out of [143] and references therein). The symbols are used to distinguish between vertical transitions (crosses) and adiabatic as well as 'imitated adiabatic' transitions (circles). The lines are the curves fitted to the function f nð Þ¼ <sup>r</sup> a expð Þþ �bn<sup>r</sup> c and serve as guides for the eyes. (reprinted with permission from Senn F, Krykunov M. The Journal of Physical Chemistry. A. 2015;119(42):10575-10581. DOI: 10.1021/acs.jpca.5b07075. Copyright 2015 American Chemical Society. Color specifications refer to the original figure).

From Figure 4 one can see the singlet-triplet gap (ST) decreasing, resembling an exponential function. In order to estimate the ST gap for infinitely large polyacenes, giving an indication if there would be a ST crossover, several authors fitted the excitation energies to the function f nð Þ¼ <sup>r</sup> aexpð Þþ �bn<sup>r</sup> c (see [140, 142, 143]). With the results of R-CV(∞)-DFT for the vertical

d To be understood as the deviation of the values obtained with R-CV(∞)-DFT in comparison to the values of this column

R-CV(∞)-DFTa CV 2ð Þ-DFTa TDDFT<sup>b</sup> Ref. [143] Ref. [140] R-CV(∞)-DFTa Exp.<sup>c</sup> Ref. [143]

No. acene units Vertical Adiabatic

46 Density Functional Calculations - Recent Progresses of Theory and Application

 3.16 3.08 3.08 3.34 3.31 2.89 2.64 2.70 2.15 2.09 2.09 2.47 2.47 1.94 1.86 2.06 1.49 1.44 1.44 1.82 1.76 1.31 1.27 1.48 1.02 0.99 0.99 1.37 1.37 0.88 0.86 1.11 0.69 0.66 0.66 1.07 1.00 0.57 0.54 0.83 MAD<sup>d</sup> — 0.05 0.05 0.31 0.28 — 0.09 0.19 RMSD<sup>d</sup> — 0.06 0.06 0.32 0.29 — 0.12 0.20

the 'adiabatic' or well-to-well excitations, results from different methods in literature are controversial about a possible ST gap crossover ([140, 142] versus [143, 149, 150]); for TD-DFT it even depends on the functional used [142]. Therefore, necessarily the findings presented here will agree with some findings, while disagree with others. It should be noted that these energies are very small, actually smaller than the estimated accuracy of the CV-DFT method, and with its error it must be regarded as giving only a tendency for no ST crossover. Two additional points of precaution which puts the value of the extrapolated results into question: (a) it has been shown in [142] how a small change of a single excitation energy can influence the obtained polymeric limit, and (b) one should have in mind the change of the ground-state character with the

The complicated electronic structure of transition metal (TM) complexes [151] makes them ideal for testing the performance of newly developed methods. This section deals with charge-transfer (and hidden charge-transfer) excitations in these complexes, more precisely the 3d complexes

tions involving transitions from 1t1, 2t<sup>2</sup> to 2e and 3t<sup>2</sup> are considered [99]. The comparison is

<sup>3</sup>�, as well as their 4d congeners RuO4, TcO4

vertð Þ¼ <sup>n</sup><sup>r</sup> ! <sup>∞</sup> ð Þ <sup>0</sup>:<sup>3</sup> � <sup>4</sup>:<sup>5</sup> kcal mol�<sup>1</sup> have

�, MoO4

<sup>2</sup>� [99]. For these systems, the three lowest valence excita-

<sup>2</sup>� and 5d

adð Þ¼ <sup>n</sup><sup>r</sup> ! <sup>∞</sup> (�1.6 � 4.0) kcal mol�<sup>1</sup> [101]. For

transition, the limes of an infinitely long polyacene EST

Table 5. Vertical and adiabatic triplet excitation energies (in eV) for linear polyacenes.

polyacene length and, thus, the number of fused acenes.

� and WO4

3.2.3. Charge-transfer excitations in transition metals

<sup>2</sup>� and VO4

MnO4

a Out of [101].

as reference values.

b With LDA as functional. c Out of [143] and references therein.

�, CrO4

homologues OsO4, ReO4

been obtained and for the 'adiabatic' transition EST

made with available experimental data [152, 153] and high-level ab initio calculations [154–160]. There are several adjustable parameters that can influence the excitation energies. These include the size of the basis set used, functionals used, geometry (optimized structures or experimental geometries), medium (since the complexes are anions), etc. Use was made of experimental structures which lead to higher excitation energies (0.1–0.3 eV) compared to optimized structures. Marginal influence of solvation was found for the three valence excitations; the calculated COSMO [161, 162] excitation energies lower the energies by 0.01–0.02 eV [163, 164].

Table 6 displays the RMSD between the first three experimental dipole-allowed transitions and the corresponding values calculated by RSCF-CV(∞)-DFT.

On average, the three functionals B3LYP, PBE0 with an intermediate fraction of HF exchange and LCBP86\* have the lowest RMSD of 0.2 eV, whereas the local functionals (LDA, BP86, BPE) and BHLYP with the highest HF fraction and LCBP86 have a somewhat larger RMSD of 0.3 eV for both 3d and 4d + 5d averages. TD-DFT with the same functionals performs poorly for the 3d complexes but shows good agreement with experiment for the heavier tetraoxo complexes.


STATE RASPT2<sup>b</sup> LDA BP86 PBE B3LYP PBE0 BHLYP LCBP86\*<sup>c</sup> LCBP86<sup>d</sup> Expt.<sup>e</sup>

Constricted Variational Density Functional Theory Approach to the Description of Excited States

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49

<sup>1</sup>T1g(MC) 4.98 5.33 5.14 5.17 4.85 5.25 4.52 4.78 4.97 — <sup>1</sup>T1u(MLCT) 4.50 4.45 4.39 4.40 4.48 4.62 4.68 4.61 4.37 4.44 2<sup>1</sup>T1u(MLCT) 5.42 5.46 5.47 5.47 5.73 5.93 6.20 6.23 5.85 5.48

<sup>1</sup><sup>3</sup>T1g(MC) 4.28<sup>f</sup> 4.72 4.88 4.91 4.54 4.62 4.15 4.18 4.49 — <sup>1</sup><sup>3</sup>T2g(MC) 4.64<sup>f</sup> 4.63 4.59 4.60 4.39 4.45 4.17 4.45 4.59 —

STATE CASPT2<sup>b</sup> LDA BP86 PBE B3LYP PBE0 BHLYP LCBP86\*<sup>c</sup> LCBP86<sup>d</sup> Expt.<sup>e</sup>

T1g(MC) 3.60 4.17 3.72 3.75 3.42 3.35 3.04 3.35 3.64 3.80–3.94 <sup>1</sup>T1u(MLCT) — 5.57 5.57 5.64 6.10 6.44 — — 6.34 5.69–5.89 <sup>1</sup>T1u(MLCT) — 5.83 5.80 5.72 6.25 6.63 — — 6.93 6.20 <sup>1</sup>T2g(MC) 4.33 4.05 3.74 4.12 4.47 4.46 4.47 4.14 4.37 4.43–4.77

1<sup>3</sup>T1g(MC) 2.67 3.60 3.39 3.41 2.98 2.90 2.49 2.56 2.93 2.94

<sup>4</sup> based on the RSCF-CV(∞)-DFT method.

RMSD 0.20<sup>f</sup> 0.42 0.41 0.33 0.25 0.44 0.40<sup>f</sup> 0.29<sup>f</sup> 0.44

RMSD 0.06 0.02 0.04 0.03 0.18 0.34 0.54 0.54 0.27

Table 7. Calculated excitation energies<sup>a</sup> for Cr(CO)6 based on the RSCF-CV(∞)-DFT method.

Represents LC functional combined with BP86 with ω = 0.75.

Refers to LC functional combined with BP86 using ω = 0.4.

Data represented in this table was first published in [103].

Represents LC functional combined with BP86 with ω = 0.75.

Refers to LC functional combined with BP86 using ω = 0.4.

Data represented in this table was first published in [103].

Table 8. Calculated excitation energies<sup>a</sup> for [Fe(CN)6]

Calculated with three excitation energies.

Singlet

Triplet

a

b [154]. c

d

e [152]. f [155].

Singlet

Triplet

a

b [156]. c

d

e [153]. f

Energies in eV.

Energies in eV.

a Root-mean-square deviation.

b The reference is the observed vertical excitation energies for the three first dipole-allowed transitions.

c For MoO4 2\_ and WO4 <sup>2</sup> only, the first two experimental transitions are available.

d Deviations are in eV.

e No TDA was applied.

f Average of the three 3d complexes.

g Average of the six 4d and 5d complexes.

h Average over all complexes.

Data represented in this table was first published in [99].

Table 6. RMSDs for tetraoxo excitation energies based on RSCF-CV(∞)-DFT and a TZ2P basis seta, b, c, d, e.

This shows a clear lack of consistency. However, RSCF-CV(∞)-DFT shows good and consistent performance for all complexes studied here.

Next, the excitation energies of the octahedral TM complexes [103] are presented. The analyses will be primarily focused on Cr(CO)6 and [Fe(CN)6] <sup>4</sup> where experimental excitation energies are available. The first system to be considered is Cr(CO)6; the RSCF-CV(∞)-DFT results are displayed in Table 7. The results afforded by RSCF-CV(∞)-DFT are in good agreement with the experimental data even at the RSCF-CV(∞)-DFT/LDA level of theory. The RSCF-CV(∞)-DFT/ LDA results show performance identical to the TD-DFT/B3LYP [151] and better performance than TD-DFT with LDA and GGAs.

Considered next is the [Fe(CN)6] <sup>4</sup> complex; the results are shown in Table 8. The RMSDs here were calculated with the lower limit of the experimental [153] excitation energies where ranges are applicable. There are some theoretical calculations carried with TD-DFT [151] and other DFT-based approaches [151] as well as some high-level ab initio methods [156]. Again, there are good performances even for the LDA and GGA functionals. The accurate excitation energies afforded by the RSCF-CV(∞)-DFT method when compared to the experimental data are as a result of, to some extent, fortuitous error cancelation.


a Energies in eV.

b [154].

c Represents LC functional combined with BP86 with ω = 0.75.

d Refers to LC functional combined with BP86 using ω = 0.4.

e [152].

f [155].

This shows a clear lack of consistency. However, RSCF-CV(∞)-DFT shows good and consistent

Complex LDA BP86 PBE B3LYP BHLYP PBE0 LCBP86\* LCBP86

48 Density Functional Calculations - Recent Progresses of Theory and Application

<sup>4</sup> 0.41 0.32 0.33 0.15 0.62 0.19 0.24 0.37

<sup>2</sup> 0.40 0.31 0.34 0.09 0.55 0.04 0.22 0.32

3\_ 0.25 0.14 0.16 0.07 0.18 0.14 0.27 0.37 RuO4 0.32 0.28 0.28 0.21 0.44 0.22 0.19 0.31

<sup>4</sup> 0.10 0.13 0.13 0.25 0.13 0.29 0.27 0.17

2\_ 0.14 0.23 0.23 0.06 0.22 0.18 0.13 0.34 OsO4 0.53 0.51 0.50 0.27 0.39 0.31 0.21 0.26

<sup>4</sup> 0.36 0.43 0.43 0.14 0.25 0.16 0.14 0.16

<sup>2</sup> 0.43 0.51 0.51 0.14 0.11 0.07 0.11 0.16

Average 3d<sup>f</sup> 0.35 0.26 0.28 0.10 0.45 0.12 0.24 0.35 Average 4d + 5d<sup>g</sup> 0.31 0.34 0.34 0.19 0.27 0.22 0.19 0.24 Total average 3d<sup>h</sup> 0.33 0.31 0.32 0.16 0.34 0.18 0.21 0.28

The reference is the observed vertical excitation energies for the three first dipole-allowed transitions.

<sup>2</sup> only, the first two experimental transitions are available.

Table 6. RMSDs for tetraoxo excitation energies based on RSCF-CV(∞)-DFT and a TZ2P basis seta, b, c, d, e.

Next, the excitation energies of the octahedral TM complexes [103] are presented. The analyses

are available. The first system to be considered is Cr(CO)6; the RSCF-CV(∞)-DFT results are displayed in Table 7. The results afforded by RSCF-CV(∞)-DFT are in good agreement with the experimental data even at the RSCF-CV(∞)-DFT/LDA level of theory. The RSCF-CV(∞)-DFT/ LDA results show performance identical to the TD-DFT/B3LYP [151] and better performance

were calculated with the lower limit of the experimental [153] excitation energies where ranges are applicable. There are some theoretical calculations carried with TD-DFT [151] and other DFT-based approaches [151] as well as some high-level ab initio methods [156]. Again, there are good performances even for the LDA and GGA functionals. The accurate excitation energies afforded by the RSCF-CV(∞)-DFT method when compared to the experimental data are as a

<sup>4</sup> where experimental excitation energies

<sup>4</sup> complex; the results are shown in Table 8. The RMSDs here

performance for all complexes studied here.

Data represented in this table was first published in [99].

than TD-DFT with LDA and GGAs.

Considered next is the [Fe(CN)6]

MnO\_

CrO4

VO4

TcO\_

MoO4

ReO\_

WO4

a

b

c For MoO4

d

e

f

g

h

Root-mean-square deviation.

Deviations are in eV.

No TDA was applied.

2\_ and WO4

Average of the three 3d complexes.

Average over all complexes.

Average of the six 4d and 5d complexes.

will be primarily focused on Cr(CO)6 and [Fe(CN)6]

result of, to some extent, fortuitous error cancelation.

Data represented in this table was first published in [103].

Table 7. Calculated excitation energies<sup>a</sup> for Cr(CO)6 based on the RSCF-CV(∞)-DFT method.


a Energies in eV.

b [156].

c Represents LC functional combined with BP86 with ω = 0.75.

d Refers to LC functional combined with BP86 using ω = 0.4.

e [153].

f Calculated with three excitation energies.

Data represented in this table was first published in [103].

Table 8. Calculated excitation energies<sup>a</sup> for [Fe(CN)6] <sup>4</sup> based on the RSCF-CV(∞)-DFT method. A look now at the electronic density change that accompanies the electronic excitation. Figure 5 (a and b) shows the plot of the density changes associated with the electronic transitions in Cr (CO)6. The charge redistribution can be seen from the figure, where the density depletion (rocc), the accumulation (rvir) as well as the density change (Δrex ¼ rvir � rocc) occurs, resulting from the total change in density associated with the electronic transition. For the MLCT transition, the rocc (Figure 5a) is situated on the Cr metal center, the area or space spun by the density that is reminiscent of the dyz and the rvir is mostly situated on the equatorial CO ligands. The depletion density is in the yz plane, the accumulation density is situated on the CO ligand, and there is little interaction between them as can be seen from the difference (Figure 5b). The movement of density is from the metal center to the ligands corresponding to an intramolecular charge-transfer transition. It is clear from Figure 5c that this transition has a significant d ! d character. In the density plots that follow, there is a depletion in the density situated on

the metal with some contribution from the CO in the xy-plane and accumulation of density largely on the central Cr metal along the yz-plane with some accumulation on the CO ligands

Constricted Variational Density Functional Theory Approach to the Description of Excited States

seen for Cr(CO)6. The differences in the density plots representing the MC transition; there is more significant accumulation on the CN ligands, and the density accumulation is in the same plane (xy-plane) as the depletion density (dx<sup>2</sup><sup>y</sup><sup>2</sup> –dxy). As for the MLCT, the associated

The benchmark studies on the tetrahedral and octahedral TM complexes probed the ability of RSCF-CV(∞)-DFT to describe CT and hidden CT excitations. Use was made of the tetrahedral d<sup>0</sup> metal oxides as the first benchmark series since the tetra oxides have a long history as a

<sup>4</sup>. Similar features are seen here as

http://dx.doi.org/10.5772/intechopen.70932

51

<sup>6</sup> . (a) the density change associated with the 11T1<sup>g</sup> state. (b) Exemplifies the

density redistribution associated with the 3<sup>1</sup>T1<sup>u</sup> state. (c) Densities accompanying the calculated 31T1<sup>u</sup> state. (a) Is an example of MC-type transition, and (b) and (c) are MLCT-type transitions. Red signifies loss and green shows gain in density. See Seidu I. Excited-State Studies with the Constricted Variational Density Functional Theory (CV-DFT) Method.

Displayed in Figure 6 are the density plots for [Fe(CN)6]

density movement is identical to that of Cr(CO)6 (see Figure 6(b and c)).

in the same plane.

Figure 6. Δr associated with the Fe(CN)<sup>4</sup>

PhD dissertation. University of Calgary; 2016 for coloured pictures.

Figure 5. Δr associated with the CrCO6, red signifies depletion and green shows accumulation of density. (a) The density change associated with the 1<sup>1</sup>T1<sup>u</sup> state. (b) Exemplifies the density redistribution associated with the 2<sup>1</sup>T1<sup>u</sup> state. (c) Densities accompanying the calculated 11T1<sup>g</sup> state. (a) and (b) are MLCT-type transitions, and (c) is an example of MCtype transition. See Seidu I. Excited-State Studies with the Constricted Variational Density Functional Theory (CV-DFT) Method. PhD dissertation. University of Calgary; 2016 for coloured pictures.

the metal with some contribution from the CO in the xy-plane and accumulation of density largely on the central Cr metal along the yz-plane with some accumulation on the CO ligands in the same plane.

A look now at the electronic density change that accompanies the electronic excitation. Figure 5 (a and b) shows the plot of the density changes associated with the electronic transitions in Cr (CO)6. The charge redistribution can be seen from the figure, where the density depletion (rocc), the accumulation (rvir) as well as the density change (Δrex ¼ rvir � rocc) occurs, resulting from the total change in density associated with the electronic transition. For the MLCT transition, the rocc (Figure 5a) is situated on the Cr metal center, the area or space spun by the density that is reminiscent of the dyz and the rvir is mostly situated on the equatorial CO ligands. The depletion density is in the yz plane, the accumulation density is situated on the CO ligand, and there is little interaction between them as can be seen from the difference (Figure 5b). The movement of density is from the metal center to the ligands corresponding to an intramolecular charge-transfer transition. It is clear from Figure 5c that this transition has a significant d ! d character. In the density plots that follow, there is a depletion in the density situated on

50 Density Functional Calculations - Recent Progresses of Theory and Application

Figure 5. Δr associated with the CrCO6, red signifies depletion and green shows accumulation of density. (a) The density change associated with the 1<sup>1</sup>T1<sup>u</sup> state. (b) Exemplifies the density redistribution associated with the 2<sup>1</sup>T1<sup>u</sup> state. (c) Densities accompanying the calculated 11T1<sup>g</sup> state. (a) and (b) are MLCT-type transitions, and (c) is an example of MCtype transition. See Seidu I. Excited-State Studies with the Constricted Variational Density Functional Theory (CV-DFT)

Method. PhD dissertation. University of Calgary; 2016 for coloured pictures.

Displayed in Figure 6 are the density plots for [Fe(CN)6] <sup>4</sup>. Similar features are seen here as seen for Cr(CO)6. The differences in the density plots representing the MC transition; there is more significant accumulation on the CN ligands, and the density accumulation is in the same plane (xy-plane) as the depletion density (dx<sup>2</sup><sup>y</sup><sup>2</sup> –dxy). As for the MLCT, the associated density movement is identical to that of Cr(CO)6 (see Figure 6(b and c)).

The benchmark studies on the tetrahedral and octahedral TM complexes probed the ability of RSCF-CV(∞)-DFT to describe CT and hidden CT excitations. Use was made of the tetrahedral d<sup>0</sup> metal oxides as the first benchmark series since the tetra oxides have a long history as a

Figure 6. Δr associated with the Fe(CN)<sup>4</sup> <sup>6</sup> . (a) the density change associated with the 11T1<sup>g</sup> state. (b) Exemplifies the density redistribution associated with the 3<sup>1</sup>T1<sup>u</sup> state. (c) Densities accompanying the calculated 31T1<sup>u</sup> state. (a) Is an example of MC-type transition, and (b) and (c) are MLCT-type transitions. Red signifies loss and green shows gain in density. See Seidu I. Excited-State Studies with the Constricted Variational Density Functional Theory (CV-DFT) Method. PhD dissertation. University of Calgary; 2016 for coloured pictures.

challenging testing ground for new methods due to their complex electronic structure. In general there is either a comparable performance for RSCF-CV(∞)-DFT and TD-DFT in cases where TD-DFT shows good performances or RSCF-CV(∞)-DFT outperforms TD-DFT.

Acknowledgements

Author details

Florian Senn1

References

37508

3276(08)60541-9

BF00551551

unflinching support until his untimely passing away.

\*, Issaka Seidu<sup>2</sup> and Young Choon Park<sup>3</sup>

1 Department of Chemistry, University of Calgary, Calgary, Alberta, Canada 2 Department of Chemistry, Carleton University, Ottawa, Ontario, Canada

3 Quantum Theory Project, University of Florida, Gainesville, FL, United States

[1] Wolfsberg M, Helmholz L. The Journal of Chemical Physics. 1952;20(5):837-843. DOI:

[2] Hoffmann R. The Journal of Chemical Physics. 1963;39(6):1397-1412. DOI: http://dx.doi.

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[6] Slater JC. In: Löwdin PO, editor. Advances in Quantum Chemistry, Vol. 6 of Advances in Quantum Chemistry: 1-92. Academic Press; 1972. DOI: http://dx.doi.org/10.1016/S0065-

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\*Address all correspondence to: florian.senn@ucalgary.ca

http://dx.doi.org/10.1063/1.1700580

DOI: http://dx.doi.org/10.1063/1.1701475

DOI: http://dx.doi.org/10.1063/1.1680049

320(1541):161-173. DOI: 10.1098/rspa.1970.0203

org/10.1063/1.1734456

The authors would like to express their gratitude to the late Prof. Dr. Tom Ziegler for his

Constricted Variational Density Functional Theory Approach to the Description of Excited States

http://dx.doi.org/10.5772/intechopen.70932

53

Further, the authors are grateful to Dr. Mykhaylo Krykunov for his helpful discussions.

A trend that manifests itself at larger r for the TM complexes is the excitation energies which become more functional dependent and less accurate. Further, the accuracy of RSCF-CV(∞)- DFT for smaller r is not attributable to the ability of our method to afford accurate values of the IP of the complex and the EA of the cation formed alone; some error cancelation occurs when the IPs and EAs are combined to obtain the excitation energy. Finally, it is possible to plot the density change associated with the electronic transitions afforded by our method with regions of density depletion and accumulation supporting a qualitative classification of excitations as MLCT or MC.

#### 4. Conclusion

In this chapter we presented the CV-DFT method and its different variants. While CV 2ð Þ-DFT is consistent with (adiabatic) TD-DFT within the TDA approximation, CV-DFT allows to go to higher order. Indeed, its strength lies in going beyond linear response and therewith obtaining distance-dependent contributions to the excitation energy naturally. Additionally, the theory allows for the calculation of excitation energies for singlet and triplet states on the same footing. Of course as a variational method, CV-DFT relies on an accurate ground-state description. The theoretical framework allows us to apply special restrictions as done in [104] and therewith obtain a numerically stable method being numerically equivalent to ΔSCF-DFT.

How CV-DFT performs has been shown in Section 3 with the aid of selected examples of charge-transfer or Rydberg excitation type. With these examples, we could demonstrate how CV(∞)-DFT is able to reproduce the expected �1=R long-range behavior for charge-transfer excitations. When orbital relaxation is allowed, the excitation energies obtained by R-CV(∞)- DFT with LDA agree nicely with the findings of long-range corrected functionals. For short distance, the optimization of the transition matrix U is clearly beneficial [100, 104, 121]. But for medium- and long-range distances, a notion of care is to be taken as the optimization may lead to an unwanted mixing of transitions as shown in the case of C2H4xC2F4. Also, for excitations with hidden charge-transfer character, meaningful results are obtained with CV-DFT, for example, accurate results for the first singlet excitation energies of polyacenes [101, 147] for polyacenes as large as hexacene. Not only is CV-DFT able to deliver meaningful results, even for the LDA functional, it has an incredible ability to provide a qualitative picture of the nature or type of excitation under consideration. This is seen in the case of the TM complexes, a complicated yet excellent test set for assessing the range of applicability of every newly developed method. In the case of Rydberg excitations, RSCF-CV(∞)-DFT produces meaningful results without the need for sophisticated (or 'specialized') functionals. This good performance is attributable to the ability of our method affords good estimates of IPs and EAs for all functionals [100, 121, 122]. Admittedly, fortuitous error cancelation in IPs and EAs obtained for both RSCF-CV(∞)-DFT and TD-DFT plays a role in the accuracy of the resultant excitation energies.

### Acknowledgements

challenging testing ground for new methods due to their complex electronic structure. In general there is either a comparable performance for RSCF-CV(∞)-DFT and TD-DFT in cases

A trend that manifests itself at larger r for the TM complexes is the excitation energies which become more functional dependent and less accurate. Further, the accuracy of RSCF-CV(∞)- DFT for smaller r is not attributable to the ability of our method to afford accurate values of the IP of the complex and the EA of the cation formed alone; some error cancelation occurs when the IPs and EAs are combined to obtain the excitation energy. Finally, it is possible to plot the density change associated with the electronic transitions afforded by our method with regions of density depletion and accumulation supporting a qualitative classification of excitations as

In this chapter we presented the CV-DFT method and its different variants. While CV 2ð Þ-DFT is consistent with (adiabatic) TD-DFT within the TDA approximation, CV-DFT allows to go to higher order. Indeed, its strength lies in going beyond linear response and therewith obtaining distance-dependent contributions to the excitation energy naturally. Additionally, the theory allows for the calculation of excitation energies for singlet and triplet states on the same footing. Of course as a variational method, CV-DFT relies on an accurate ground-state description. The theoretical framework allows us to apply special restrictions as done in [104] and therewith obtain a numerically stable method being numerically equivalent to ΔSCF-DFT.

How CV-DFT performs has been shown in Section 3 with the aid of selected examples of charge-transfer or Rydberg excitation type. With these examples, we could demonstrate how CV(∞)-DFT is able to reproduce the expected �1=R long-range behavior for charge-transfer excitations. When orbital relaxation is allowed, the excitation energies obtained by R-CV(∞)- DFT with LDA agree nicely with the findings of long-range corrected functionals. For short distance, the optimization of the transition matrix U is clearly beneficial [100, 104, 121]. But for medium- and long-range distances, a notion of care is to be taken as the optimization may lead to an unwanted mixing of transitions as shown in the case of C2H4xC2F4. Also, for excitations with hidden charge-transfer character, meaningful results are obtained with CV-DFT, for example, accurate results for the first singlet excitation energies of polyacenes [101, 147] for polyacenes as large as hexacene. Not only is CV-DFT able to deliver meaningful results, even for the LDA functional, it has an incredible ability to provide a qualitative picture of the nature or type of excitation under consideration. This is seen in the case of the TM complexes, a complicated yet excellent test set for assessing the range of applicability of every newly developed method. In the case of Rydberg excitations, RSCF-CV(∞)-DFT produces meaningful results without the need for sophisticated (or 'specialized') functionals. This good performance is attributable to the ability of our method affords good estimates of IPs and EAs for all functionals [100, 121, 122]. Admittedly, fortuitous error cancelation in IPs and EAs obtained for both RSCF-CV(∞)-DFT and TD-DFT plays a role in the accuracy of the resultant excitation

where TD-DFT shows good performances or RSCF-CV(∞)-DFT outperforms TD-DFT.

52 Density Functional Calculations - Recent Progresses of Theory and Application

MLCT or MC.

4. Conclusion

energies.

The authors would like to express their gratitude to the late Prof. Dr. Tom Ziegler for his unflinching support until his untimely passing away.

Further, the authors are grateful to Dr. Mykhaylo Krykunov for his helpful discussions.

#### Author details

Florian Senn1 \*, Issaka Seidu<sup>2</sup> and Young Choon Park<sup>3</sup>


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**Chapter 3**

Provisional chapter

**Assessment of the Validity of Some Minnesota Density**

DOI: 10.5772/intechopen.70455

**Reactivity of the SYBR Green I and Ethidium Bromide**

This research work has assessed many Minnesota density functionals to find their molecular structure and electronic properties possessed by SYBR green I (SYBRGI) and ethidium bromide (EtBr) nucleic acid stains. In the determination of the global descriptors that come up from conceptual density functional theory (CDFT), the processes include: Self-Consistent Field Energy Differences (ΔSCF) and higher occupied molecular orbital (HOMO) and lower unoccupied molecular orbital (LUMO) frontier orbitals energies. Regarding the deduced outcomes for the conceptual DFT indices, many of the descriptors have been adjusted to achieve the "Koopmans in DFT (KID)" process. It has also been shown that the only density functionals that confirm this approximation are the range-separated hybrids (RSH).

Keywords: computational chemistry, SYBR green I, chemical reactivity theory, molecular

The chemical reactivity theory [also known as the conceptual density functional theory (CDFT)] is a vital technique that is used to predict, evaluate, and interpret the results from

Research done by Parr and his associates [1] reveals that several theories and models have been discovered after the evaluation of the molecular system with the use of DFT. Almost all the discovered theories are helpful in research because they enable scholars to achieve quantitative forecasts of a chemical reactivity system. In addition to this, the theories can further be

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

**Functionals for the Prediction of the Chemical**

Density Functionals for the Prediction of the

Chemical Reactivity of the SYBR Green I and

Ethidium Bromide Nucleic Acid Stains

Assessment of the Validity of Some Minnesota

**Nucleic Acid Stains**

Daniel Glossman-Mitnik

Glossman-Mitnik

Abstract

1. Introduction

chemical processes [1–4].

Norma Flores-Holguín, Juan Frau and

http://dx.doi.org/10.5772/intechopen.70455

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

modeling, conceptual DFT, ethidium bromide

quantified and are generally termed as conceptual DFT descriptors.

Norma Flores-Holguín, Juan Frau and Daniel


**Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical Reactivity of the SYBR Green I and Ethidium Bromide Nucleic Acid Stains** Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical Reactivity of the SYBR Green I and Ethidium Bromide Nucleic Acid Stains

DOI: 10.5772/intechopen.70455

Norma Flores-Holguín, Juan Frau and Daniel Glossman-Mitnik Norma Flores-Holguín, Juan Frau and Daniel

Additional information is available at the end of the chapter Glossman-Mitnik

http://dx.doi.org/10.5772/intechopen.70455 Additional information is available at the end of the chapter

#### Abstract

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10.1021/ic4009625 PMID: 23957772

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10.1002/chem.200903423

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s002140050457

1993;97(47):12220-12228. DOI: 10.1021/j100149a021

and Computation. 2011;7(12):3961-3977. DOI: 10.1021/ct200597h

This research work has assessed many Minnesota density functionals to find their molecular structure and electronic properties possessed by SYBR green I (SYBRGI) and ethidium bromide (EtBr) nucleic acid stains. In the determination of the global descriptors that come up from conceptual density functional theory (CDFT), the processes include: Self-Consistent Field Energy Differences (ΔSCF) and higher occupied molecular orbital (HOMO) and lower unoccupied molecular orbital (LUMO) frontier orbitals energies. Regarding the deduced outcomes for the conceptual DFT indices, many of the descriptors have been adjusted to achieve the "Koopmans in DFT (KID)" process. It has also been shown that the only density functionals that confirm this approximation are the range-separated hybrids (RSH).

Keywords: computational chemistry, SYBR green I, chemical reactivity theory, molecular modeling, conceptual DFT, ethidium bromide

#### 1. Introduction

The chemical reactivity theory [also known as the conceptual density functional theory (CDFT)] is a vital technique that is used to predict, evaluate, and interpret the results from chemical processes [1–4].

Research done by Parr and his associates [1] reveals that several theories and models have been discovered after the evaluation of the molecular system with the use of DFT. Almost all the discovered theories are helpful in research because they enable scholars to achieve quantitative forecasts of a chemical reactivity system. In addition to this, the theories can further be quantified and are generally termed as conceptual DFT descriptors.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

To obtain the quantitative figures of conceptual descriptors, it may be necessary to analyze the Kohn-Sham theory by calculating the energy system, molecular density, and the orbital energies associated with frontier orbitals [i.e., higher occupied molecular orbital (HOMO) and lower unoccupied molecular orbital (LUMO)] [5–10].

[15] that the legitimacy of the Koopmans' theorem may be approximately approved with a

Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical…

http://dx.doi.org/10.5772/intechopen.70455

65

Use of the γ-tuning technique can be useful in upgrading the features projected by the density functionals. This is due to better utilization of the of Koopmans' theorem, leading to a superior understanding of frontier orbitals energies with I and A. An example can be cited from Lima et al. [19], who just demonstrated a better explanation of the optical features of carotenoids

This therefore illustrates that conformity of any given density functional can be determined by assessing how it has adopted the "Koopmans' in DFT" (KID) process, which guides its behavior to be almost equal to the ideal density functional. This is essential for any precise computation of the conceptual DFT descriptors that help in forecasting and analysis of chemical reactivity in molecular systems. Still, the γ-tuning technique for range-separated hybrids density functional is system dependent. This implies that separate density functionals are to be used in the computations of the descriptors for separate molecular features. We are then going to concentrate on part of the density functionals that have displayed the required

The main aim of this study is to do a comparative research relating to the performance of the just identified Minnesota family of density functionals for the account of the chemical reactivity of two nucleic acids intercalating stains, SYBR green I (SYBRGI) [21] and ethidium bromide

well-informed choice of this final parameter.

precision in physics and chemistry [20].

through tuning of some LR density functionals that are linked.

(EtBr) [22]. The molecular structures of the two are shown in Figure 1.

Figure 1. Molecular structures of (a) ethidium bromide (EtBr) and (b) SYBR green I (SYBRGI).

For research on a molecular system, the first activity before proceeding is selection of the model chemistry. Model chemistry is the collection of basic set, density function, together with an implied solvent model that is known to be consistent for the problem under research. Several studies provide insights on the way to choose the model chemistry. A researcher may also decide to preview past studies when choosing the model.

Even though the fundamentals of DFT reveal that universal functional density is present and that computations using this function can be used to obtain all the features of the system, it is always necessary in practical cases that one refers to the estimated density functionals that have been established for the past 3 decades. For the approximate functionals, almost all of them are perfectly fit to be used in estimating some features, while some can be used for estimating other features. In separate scenarios, you can encounter density functionals that are perfectly fit for estimating the features of a given molecular system and a functional group. It is also important to assess separate density functionals for a separate functional group which can be added to the molecular system under research.

When researching on chemical reactivity (which is a process that entails the transfer of electrons), a person performs computations for both ground and open systems, i.e., cation and anion. It is not easy to obtain consistent outcomes using these computations (when diffuse functions should be a part of the basis set) [5–10]. This necessitates adoption of a more consistent technique that provides all the data that a person will require directly from the outcomes of the computations at ground state in the molecular system under research. In addition to this, a person may also want to find the deionization ability together with the electron affinity of any system being researched without having to calculate the radical cation and anion. This can be determined by the Koopmans' theorem [7–10] that relies on Hartree-Fock Theory, which states that the energy of the HOMO (i.e., I = εH) can be used to estimate the ionization potential. Alternatively, the electron affinity can be estimated using the minus the energy of the LUMO (i.e., A = εL).

The legitimacy of the Koopmans' theorem is yet again a contentious issue because of the existing difference between the fundamental band gap and the HOMO and the LUMO gaps. This can be termed as derivative discontinuity. It has again been discovered that an exact physical description may be assigned to Kohn-Sham HOMO using "the Kohn-Sham analogue of Koopmans' theorem in Hartree-Fock theory" (this theory explains that in the exact theory, the KS HOMO is opposite and same as the ionization potential) [11–14]. The effects brought about by the difference between the fundamental band gap and the HOMO and the LUMO gaps have ensured that no Koopmans' theorem creates a direct relationship between the LUMO energy and the electron affinity. To eliminate these effects, a suggestion has been made by scholars to conceive that the ionization potential of the N + 1 electron system (anion) is almost equal to the electron affinity of N electron system [15]. Regarding the range-separated hybrids (RSH) density functionals [16–18], e.g., that the repulsive coulomb potential has to be separated in the long-range (LR) and short-range (SR) terms, e.g., via r<sup>1</sup> = r<sup>1</sup> erf (γr) + r<sup>1</sup> erfc (γr), with γ representing the range-separation parameter, it was highlighted by Kronik et al. [15] that the legitimacy of the Koopmans' theorem may be approximately approved with a well-informed choice of this final parameter.

To obtain the quantitative figures of conceptual descriptors, it may be necessary to analyze the Kohn-Sham theory by calculating the energy system, molecular density, and the orbital energies associated with frontier orbitals [i.e., higher occupied molecular orbital (HOMO) and

For research on a molecular system, the first activity before proceeding is selection of the model chemistry. Model chemistry is the collection of basic set, density function, together with an implied solvent model that is known to be consistent for the problem under research. Several studies provide insights on the way to choose the model chemistry. A researcher may

Even though the fundamentals of DFT reveal that universal functional density is present and that computations using this function can be used to obtain all the features of the system, it is always necessary in practical cases that one refers to the estimated density functionals that have been established for the past 3 decades. For the approximate functionals, almost all of them are perfectly fit to be used in estimating some features, while some can be used for estimating other features. In separate scenarios, you can encounter density functionals that are perfectly fit for estimating the features of a given molecular system and a functional group. It is also important to assess separate density functionals for a separate functional group which

When researching on chemical reactivity (which is a process that entails the transfer of electrons), a person performs computations for both ground and open systems, i.e., cation and anion. It is not easy to obtain consistent outcomes using these computations (when diffuse functions should be a part of the basis set) [5–10]. This necessitates adoption of a more consistent technique that provides all the data that a person will require directly from the outcomes of the computations at ground state in the molecular system under research. In addition to this, a person may also want to find the deionization ability together with the electron affinity of any system being researched without having to calculate the radical cation and anion. This can be determined by the Koopmans' theorem [7–10] that relies on Hartree-Fock Theory, which states that the energy of the HOMO (i.e., I = εH) can be used to estimate the ionization potential. Alternatively, the electron affinity can be estimated using the minus

The legitimacy of the Koopmans' theorem is yet again a contentious issue because of the existing difference between the fundamental band gap and the HOMO and the LUMO gaps. This can be termed as derivative discontinuity. It has again been discovered that an exact physical description may be assigned to Kohn-Sham HOMO using "the Kohn-Sham analogue of Koopmans' theorem in Hartree-Fock theory" (this theory explains that in the exact theory, the KS HOMO is opposite and same as the ionization potential) [11–14]. The effects brought about by the difference between the fundamental band gap and the HOMO and the LUMO gaps have ensured that no Koopmans' theorem creates a direct relationship between the LUMO energy and the electron affinity. To eliminate these effects, a suggestion has been made by scholars to conceive that the ionization potential of the N + 1 electron system (anion) is almost equal to the electron affinity of N electron system [15]. Regarding the range-separated hybrids (RSH) density functionals [16–18], e.g., that the repulsive coulomb potential has to be separated in the long-range (LR) and short-range (SR) terms, e.g., via r<sup>1</sup> = r<sup>1</sup> erf (γr) + r<sup>1</sup> erfc (γr), with γ representing the range-separation parameter, it was highlighted by Kronik et al.

lower unoccupied molecular orbital (LUMO)] [5–10].

64 Density Functional Calculations - Recent Progresses of Theory and Application

can be added to the molecular system under research.

the energy of the LUMO (i.e., A = εL).

also decide to preview past studies when choosing the model.

Use of the γ-tuning technique can be useful in upgrading the features projected by the density functionals. This is due to better utilization of the of Koopmans' theorem, leading to a superior understanding of frontier orbitals energies with I and A. An example can be cited from Lima et al. [19], who just demonstrated a better explanation of the optical features of carotenoids through tuning of some LR density functionals that are linked.

This therefore illustrates that conformity of any given density functional can be determined by assessing how it has adopted the "Koopmans' in DFT" (KID) process, which guides its behavior to be almost equal to the ideal density functional. This is essential for any precise computation of the conceptual DFT descriptors that help in forecasting and analysis of chemical reactivity in molecular systems. Still, the γ-tuning technique for range-separated hybrids density functional is system dependent. This implies that separate density functionals are to be used in the computations of the descriptors for separate molecular features. We are then going to concentrate on part of the density functionals that have displayed the required precision in physics and chemistry [20].

The main aim of this study is to do a comparative research relating to the performance of the just identified Minnesota family of density functionals for the account of the chemical reactivity of two nucleic acids intercalating stains, SYBR green I (SYBRGI) [21] and ethidium bromide (EtBr) [22]. The molecular structures of the two are shown in Figure 1.

Figure 1. Molecular structures of (a) ethidium bromide (EtBr) and (b) SYBR green I (SYBRGI).

#### 2. Theoretical background

Within the context of DFT [2, 23], the chemical potential μ, which estimates the escaping tendency of the electron from the equilibrium point is stated as follows:

$$
\mu = \left(\frac{\partial E}{\partial \mathbf{N}}\right)\_{\mathbf{r}(\mathbf{r})} \tag{1}
$$

3. Settings and computational details

basic set [27, 28].

4. Results and discussion

were used, water being used as a solvent.

In this research project, each of the computations was done using Gaussian 09 programs [26] and the density functional methods as compelled in the computational package. The gradient method was used to obtain the equilibrium geometries of molecules in this research. Additionally, vibration frequencies and the force parameters were estimated through computation of analytical frequencies on still areas after optimization to check whether they were the actual minima. Def2SVP was used in this research project as the basic set for optimization of geometry and frequencies. Computation of the electronic features was achieved using Def2TZVP

Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical…

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67

We chose a group of Minnesota density functionals, which give consistent outcomes when computing the molecular structures and systems during the research activities. The group below were selected: M11, which falls under RSH meta-generalized gradient approximation (GGA) [29]; M11L, which falls under double-range local meta-GGA [30]; MN12L, which falls under nonseparable meta-nonseparable gradient approximation (NGA) [31]; MN12SX, which falls under nonseparable hybrid nonseparable meta-NGA [32]; N12, which falls under nonseparable gradient estimation [33]; N12SX, which falls under RSH nonseparable gradient estimation [32]; SOGGA11, which falls under generalized gradient approximation (GGA) density functional [34]; and SOGGA11X, which falls under generalized gradient approximation density functional [35]. GGA can be explained as a functional whereby the functional relies on the both the up down spin densities and the minimized gradient. Nonseparable gradient approximation (NGA) can be explained as a functional that relies on up down spin densities and minimized gradient and falls under nonseparable. In the current research, each of the computations was done where the solvent used was water and by doing the calculations

in conformity to the Solvation Model based on Density (SMD) solvation standard [36].

Firstly, the molecular structures of SYBRGI and EtBr were first optimized by MOL structures, then by finding the most stable conformers through Avogadro program [37, 38]. This was done through random sampling with molecular mechanic techniques. After the optimization, the resulting conformers were then re-optimized with MN12SX, M11L, M11, N12, SOGGA11X, and SOGGA11 density functionals. In addition, Def2SVP basic set and SMD solvation model

The HOMO and LUMO energies (in eV), the ionization potential I and electron affinity A (in eV), electronegativity χ, chemical hardness η, global electrophilicity ω, electrodonating ω, and electroaccepting ω<sup>+</sup> powers, and the net electrophilicity Δω of the EtBr and SYBRGI molecules calculated with the same density functionals and solvation model are presented in Tables 1 and 2, respectively. The upper part of the tables shows the results derived assuming the validity of the Kid procedure (hence the subscript K), and the lower part of the tables shows the results derived from the calculated vertical I and A through a ΔSCF technique.

where χ represents the electronegativity.

Chemical hardness is represented by η, which is the opposition to charge transfer:

$$\eta = \left(\frac{\partial^2 E}{\partial \mathcal{N}^2}\right)\_{v(r)}\tag{2}$$

Employing a finite difference approximation and the Koopmans's theorem [7–10], we can write the above equation as:

$$
\mu = -\frac{1}{2}(I + A) \approx \frac{1}{2}(\varepsilon \iota + \varepsilon \iota) = -\chi \chi \tag{3}
$$

$$
\eta = (I - A) \approx (\varepsilon\_L - \varepsilon\_H) = \eta\_K \tag{4}
$$

where ε<sup>H</sup> is the HOMO energy and ε<sup>L</sup> is the LUMO energy.

An expression for the electrophilicity index ω is as below:

$$
\omega = \frac{\mu^2}{2\eta} = \frac{\left(I + A\right)^2}{4(I - A)} \approx \frac{\left(\varepsilon\_L + \varepsilon\_H\right)^2}{4\left(\varepsilon\_L - \varepsilon\_H\right)} = \omega\_K \tag{5}
$$

Expressions for electrodonating ω� and electroaccepting ω<sup>+</sup> powers are as below [24]:

$$
\omega^{-} = \frac{\left(3I + A\right)^{2}}{4(I - A)} \approx \frac{\left(3\varepsilon\_{H} + \varepsilon\_{L}\right)^{2}}{16\eta\_{K}} = \omega\_{K}^{-}\tag{6}
$$

and

$$
\omega \omega^{+} = \frac{\left(I + 3A\right)^{2}}{4(I - A)} \approx \frac{\left(\varepsilon\_{H} + 3\varepsilon\_{L}\right)^{2}}{16\eta\_{K}} = \omega\_{K}^{+} \tag{7}
$$

To obtain a comparison for <sup>ω</sup><sup>+</sup> and � <sup>ω</sup>�, the explanation below for net electrophilicity has been suggested [25]:

$$
\Delta \boldsymbol{\omega}^{\pm} = \boldsymbol{\omega}^{+} - (-\boldsymbol{\omega}^{-}) = \boldsymbol{\omega}^{+} + \boldsymbol{\omega}^{-} \approx \boldsymbol{\omega}\_{\boldsymbol{K}}^{+} - \left(-\boldsymbol{\omega}\_{\boldsymbol{K}}^{-}\right) = \boldsymbol{\omega}\_{\boldsymbol{K}}^{+} + \boldsymbol{\omega}\_{\boldsymbol{K}}^{-} = \Delta \boldsymbol{\omega}\_{\boldsymbol{K}}^{\pm} \tag{8}
$$

### 3. Settings and computational details

2. Theoretical background

where χ represents the electronegativity.

write the above equation as:

and

been suggested [25]:

Within the context of DFT [2, 23], the chemical potential μ, which estimates the escaping

v rð Þ

v rð Þ

ð Þ¼� ε<sup>L</sup> þ ε<sup>H</sup> χ<sup>K</sup> (3)

¼ ω<sup>K</sup> (5)

<sup>K</sup> (6)

<sup>K</sup> (7)

<sup>K</sup> (8)

η ¼ ð Þ I � A ≈ ð Þ¼ ε<sup>L</sup> � ε<sup>H</sup> η<sup>K</sup> (4)

2

2

2

¼ ω�

¼ ω<sup>þ</sup>

<sup>K</sup> þ ω�

<sup>K</sup> ¼ Δω�

4ð Þ ε<sup>L</sup> � ε<sup>H</sup>

16η<sup>K</sup>

16η<sup>K</sup>

<sup>K</sup> � �ω� K <sup>¼</sup> <sup>ω</sup><sup>þ</sup> (1)

(2)

<sup>μ</sup> <sup>¼</sup> <sup>∂</sup><sup>E</sup> ∂N 

<sup>η</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>E</sup> ∂N<sup>2</sup> 

ð Þ <sup>I</sup> <sup>þ</sup> <sup>A</sup> <sup>≈</sup> <sup>1</sup>

<sup>2</sup><sup>η</sup> <sup>¼</sup> ð Þ <sup>I</sup> <sup>þ</sup> <sup>A</sup> <sup>2</sup>

<sup>ω</sup>� <sup>¼</sup> ð Þ <sup>3</sup><sup>I</sup> <sup>þ</sup> <sup>A</sup> <sup>2</sup>

<sup>ω</sup><sup>þ</sup> <sup>¼</sup> ð Þ <sup>I</sup> <sup>þ</sup> <sup>3</sup><sup>A</sup> <sup>2</sup>

Δω� ¼ ω<sup>þ</sup> � �ω� ð Þ¼ ω<sup>þ</sup> þ ω� ≈ ω<sup>þ</sup>

Expressions for electrodonating ω� and electroaccepting ω<sup>+</sup> powers are as below [24]:

Employing a finite difference approximation and the Koopmans's theorem [7–10], we can

2

<sup>4</sup>ð Þ <sup>I</sup> � <sup>A</sup> <sup>≈</sup> ð Þ <sup>ε</sup><sup>L</sup> <sup>þ</sup> <sup>ε</sup><sup>H</sup>

<sup>4</sup>ð Þ <sup>I</sup> � <sup>A</sup> <sup>≈</sup> ð Þ <sup>3</sup>ε<sup>H</sup> <sup>þ</sup> <sup>ε</sup><sup>L</sup>

<sup>4</sup>ð Þ <sup>I</sup> � <sup>A</sup> <sup>≈</sup> ð Þ <sup>ε</sup><sup>H</sup> <sup>þ</sup> <sup>3</sup>ε<sup>L</sup>

To obtain a comparison for <sup>ω</sup><sup>+</sup> and � <sup>ω</sup>�, the explanation below for net electrophilicity has

Chemical hardness is represented by η, which is the opposition to charge transfer:

tendency of the electron from the equilibrium point is stated as follows:

66 Density Functional Calculations - Recent Progresses of Theory and Application

<sup>μ</sup> ¼ � <sup>1</sup> 2

where ε<sup>H</sup> is the HOMO energy and ε<sup>L</sup> is the LUMO energy. An expression for the electrophilicity index ω is as below:

<sup>ω</sup> <sup>¼</sup> <sup>μ</sup><sup>2</sup>

In this research project, each of the computations was done using Gaussian 09 programs [26] and the density functional methods as compelled in the computational package. The gradient method was used to obtain the equilibrium geometries of molecules in this research. Additionally, vibration frequencies and the force parameters were estimated through computation of analytical frequencies on still areas after optimization to check whether they were the actual minima. Def2SVP was used in this research project as the basic set for optimization of geometry and frequencies. Computation of the electronic features was achieved using Def2TZVP basic set [27, 28].

We chose a group of Minnesota density functionals, which give consistent outcomes when computing the molecular structures and systems during the research activities. The group below were selected: M11, which falls under RSH meta-generalized gradient approximation (GGA) [29]; M11L, which falls under double-range local meta-GGA [30]; MN12L, which falls under nonseparable meta-nonseparable gradient approximation (NGA) [31]; MN12SX, which falls under nonseparable hybrid nonseparable meta-NGA [32]; N12, which falls under nonseparable gradient estimation [33]; N12SX, which falls under RSH nonseparable gradient estimation [32]; SOGGA11, which falls under generalized gradient approximation (GGA) density functional [34]; and SOGGA11X, which falls under generalized gradient approximation density functional [35]. GGA can be explained as a functional whereby the functional relies on the both the up down spin densities and the minimized gradient. Nonseparable gradient approximation (NGA) can be explained as a functional that relies on up down spin densities and minimized gradient and falls under nonseparable. In the current research, each of the computations was done where the solvent used was water and by doing the calculations in conformity to the Solvation Model based on Density (SMD) solvation standard [36].

#### 4. Results and discussion

Firstly, the molecular structures of SYBRGI and EtBr were first optimized by MOL structures, then by finding the most stable conformers through Avogadro program [37, 38]. This was done through random sampling with molecular mechanic techniques. After the optimization, the resulting conformers were then re-optimized with MN12SX, M11L, M11, N12, SOGGA11X, and SOGGA11 density functionals. In addition, Def2SVP basic set and SMD solvation model were used, water being used as a solvent.

The HOMO and LUMO energies (in eV), the ionization potential I and electron affinity A (in eV), electronegativity χ, chemical hardness η, global electrophilicity ω, electrodonating ω, and electroaccepting ω<sup>+</sup> powers, and the net electrophilicity Δω of the EtBr and SYBRGI molecules calculated with the same density functionals and solvation model are presented in Tables 1 and 2, respectively. The upper part of the tables shows the results derived assuming the validity of the Kid procedure (hence the subscript K), and the lower part of the tables shows the results derived from the calculated vertical I and A through a ΔSCF technique.


Table 1. HOMO and LUMO energies (in eV), the ionization potential I and electron affinity A (in eV), electronegativity χ, chemical hardness η, global electrophilicity ω, electrodonating ω� and electroaccepting ω<sup>+</sup> powers, and the net electrophilicity Δω� of the EtBr molecule.

For examining the outcomes to determine if the KID process is fulfilled, and the drive from past works [15, 19], we have come up with descriptors having the ability to compare the outcomes from HOMO and LUMO computations with those attained using vertical I and A and a ΔSCF technique. It should again be known that we have no plans to form a gap fitting by reducing the descriptor. We plan to determine if the density functionals employed in this research contain the fixed range parameter γ that helps in effective execution of the KID process. It is somehow astonishing that our research at present lacks the parameter γ. We also included a minus of the energy of the LUMO of the neutral system instead of using A as minus of HOMO of the electron system [15, 19].

The initial three descriptors are associated with the basic accomplishment of "Koopmans in DFT" estimation by associating ε<sup>H</sup> with �I, ε<sup>L</sup> with �A, and their responses in explaining the HOMO-LUMO gap:

$$J\_I = \left| \varepsilon\_H + E\_{\mathfrak{g}^\mathfrak{g}}(\mathcal{N} - 1) - E\_{\mathfrak{g}^\mathfrak{g}}(\mathcal{N}) \right| \tag{9}$$

$$J\_A = \left| \varepsilon\_L + E\_{\rm gs}(N1) - E\_{\rm gs}(N+1) \right| \tag{10}$$

$$J\_{\rm HL} = \sqrt{J\_I^2 + J\_A^2} \tag{11}$$

Four separate descriptors will then be used to examine how the density functionals under research will help in forecasting the electronegativity χ, the chemical hardness η, the global electrophilicity ω, and the collection of conceptual DFT descriptors through deliberation of the

Table 2. HOMO and LUMO energies (in eV), the ionization potential I and electron affinity A (in eV), electronegativity χ, chemical hardness η, global electrophilicity ω, electrodonating ω� and electroaccepting ω<sup>+</sup> powers, and the net

Property M11 M11L MN12L MN12SX N12 N12SX SOGGA11 SOGGA11X

Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical…

HOMO �7.593 �5.132 �4.924 �5.325 �4.587 �5.160 �4.779 �6.099 LUMO �0.558 �2.933 �2.569 �2.545 �2.618 �2.460 �2.959 �1.768 χ<sup>K</sup> 4.075 4.033 3.747 3.935 3.603 3.810 3.869 3.933 η<sup>K</sup> 7.034 2.199 2.355 2.780 1.970 2.700 1.820 4.331 ω<sup>K</sup> 1.181 3.698 2.980 2.785 3.295 2.688 4.112 1.786

<sup>K</sup> 4.839 9.550 7.981 7.711 8.514 7.449 10.273 5.809

<sup>K</sup> 0.763 5.517 4.234 3.776 4.911 3.639 6.404 1.876

<sup>K</sup> 5.602 15.067 12.214 11.486 13.425 11.088 16.677 7.685 I 5.407 5.302 5.062 5.263 4.767 5.074 4.980 5.252 A 2.643 2.747 2.400 2.589 2.358 2.521 2.690 2.564 χ 4.025 4.024 3.731 3.926 3.563 3.797 3.835 3.908 η 2.764 2.555 2.661 2.674 2.409 2.553 2.290 2.688 ω 2.931 3.169 2.615 2.882 2.635 2.824 3.210 2.841 ω� 8.048 8.509 7.262 7.894 7.202 7.706 8.481 7.803 ω<sup>+</sup> 4.023 4.485 3.531 3.968 3.639 3.909 4.646 3.895 Δω� 12.071 12.994 10.972 11.862 10.841 11.615 13.128 11.699

> J<sup>η</sup> ¼ η � η<sup>K</sup> � � �

> > J 2 <sup>χ</sup> þ J 2 <sup>η</sup> þ J 2 ω

Finally, we came up with four extra descriptors to determine the success of the density functionals under research in forecasting of electrodonating power ω�, the electroaccepting

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

, the net electrophilicity Δω�, together with the four descriptors combined and

JD<sup>1</sup> ¼

D1 represents the initial collection of conceptual DFT descriptors.

considering the HOMO and LUMO energies or the vertical I and A:

power ω<sup>+</sup>

ω�

ω+

Δω�

J<sup>χ</sup> ¼ j j χ � χ<sup>K</sup> (12)

J<sup>ω</sup> ¼ j j ω � ω<sup>K</sup> (14)

� (13)

http://dx.doi.org/10.5772/intechopen.70455

69

(15)

energies of the HOMO and LUMO or the vertical I and A:

electrophilicity Δω� of the SYBRGI molecule.

Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical… http://dx.doi.org/10.5772/intechopen.70455 69


Table 2. HOMO and LUMO energies (in eV), the ionization potential I and electron affinity A (in eV), electronegativity χ, chemical hardness η, global electrophilicity ω, electrodonating ω� and electroaccepting ω<sup>+</sup> powers, and the net electrophilicity Δω� of the SYBRGI molecule.

Four separate descriptors will then be used to examine how the density functionals under research will help in forecasting the electronegativity χ, the chemical hardness η, the global electrophilicity ω, and the collection of conceptual DFT descriptors through deliberation of the energies of the HOMO and LUMO or the vertical I and A:

$$J\_{\chi} = |\chi - \chi\_{K}|\tag{12}$$

$$J\_{\eta} = |\eta - \eta\_{\mathbb{K}}|\tag{13}$$

$$J\_{\omega} = |\omega - \omega\_{\mathbb{K}}|\tag{14}$$

$$J\_{D1} = \sqrt{\mathbf{j}\_{\chi}^{2} + \mathbf{j}\_{\eta}^{2} + \mathbf{j}\_{\omega}^{2}} \tag{15}$$

D1 represents the initial collection of conceptual DFT descriptors.

For examining the outcomes to determine if the KID process is fulfilled, and the drive from past works [15, 19], we have come up with descriptors having the ability to compare the outcomes from HOMO and LUMO computations with those attained using vertical I and A and a ΔSCF technique. It should again be known that we have no plans to form a gap fitting by reducing the descriptor. We plan to determine if the density functionals employed in this research contain the fixed range parameter γ that helps in effective execution of the KID process. It is somehow astonishing that our research at present lacks the parameter γ. We also included a minus of the energy of the LUMO of the neutral system instead of using A as minus

Table 1. HOMO and LUMO energies (in eV), the ionization potential I and electron affinity A (in eV), electronegativity χ, chemical hardness η, global electrophilicity ω, electrodonating ω� and electroaccepting ω<sup>+</sup> powers, and the net

Property M11 M11L MN12L MN12SX N12 N12SX SOGGA11 SOGGA11X

HOMO �7.535 �4.956 �4.684 �5.187 �4.164 �4.951 �4.151 �6.108 LUMO �0.535 �3.028 �2.613 �2.576 �2.674 �2.511 �3.121 �1.808 χ<sup>K</sup> 4.035 3.992 3.649 3.882 3.419 3.731 3.636 3.958 η<sup>K</sup> 7.000 1.928 2.071 2.611 1.490 2.440 1.030 4.300 ω<sup>K</sup> 1.163 4.133 3.213 2.886 3.924 2.852 6.420 1.822

68 Density Functional Calculations - Recent Progresses of Theory and Application

<sup>K</sup> 4.781 10.383 8.380 7.876 9.651 7.723 14.722 5.891

<sup>K</sup> 0.746 6.391 4.732 3.994 6.232 3.992 11.086 1.933

<sup>K</sup> 5.527 16.774 13.112 11.869 15.883 11.714 25.808 7.824 I 5.585 5.183 4.883 5.216 4.520 4.970 4.819 5.385 A 2.711 2.782 2.408 2.624 2.344 2.579 2.788 2.663 χ 4.148 3.983 3.646 3.920 3.432 3.775 3.804 4.024 η 2.874 2.401 2.475 2.592 2.177 2.392 2.031 2.721 ω 2.994 3.303 2.685 2.965 2.706 2.978 3.562 2.975 ω� 8.241 8.748 7.348 8.051 7.263 7.993 9.152 8.132 ω<sup>+</sup> 4.093 4.786 3.702 4.131 3.831 4.219 5.348 4.108 Δω� 12.335 13.514 11.050 12.182 11.094 12.212 14.500 12.241

The initial three descriptors are associated with the basic accomplishment of "Koopmans in DFT" estimation by associating ε<sup>H</sup> with �I, ε<sup>L</sup> with �A, and their responses in explaining the

> JI <sup>¼</sup> <sup>ε</sup><sup>H</sup> <sup>þ</sup> Egsð Þ� <sup>N</sup> � <sup>1</sup> Egsð Þ <sup>N</sup> � � �

JA <sup>¼</sup> <sup>ε</sup><sup>L</sup> <sup>þ</sup> Egsð Þ� <sup>N</sup><sup>1</sup> Egsð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> � � �

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2 <sup>I</sup> þ J 2 A

JHL ¼

� (9)

� (10)

(11)

of HOMO of the electron system [15, 19].

electrophilicity Δω� of the EtBr molecule.

HOMO-LUMO gap:

ω�

ω+

Δω�

Finally, we came up with four extra descriptors to determine the success of the density functionals under research in forecasting of electrodonating power ω�, the electroaccepting power ω<sup>+</sup> , the net electrophilicity Δω�, together with the four descriptors combined and considering the HOMO and LUMO energies or the vertical I and A:

$$J\_{\omega^{-}} = \left| \omega^{-} - \omega\_{\mathbb{K}}^{-} \right| \tag{16}$$

$$J\_{\omega^{+}} = \left| \omega^{+} - \omega\_{K}^{+} \right| \tag{17}$$

$$J\_{\Delta\omega \pm} = |\Delta\omega \pm - \Delta\omega \pm\_{\mathbb{K}}| \tag{18}$$

$$J\_{\rm D2} = \sqrt{f\_{\omega-}^2 + f\_{\omega+}^2 + f\_{\Delta \omega \pm}^2} \tag{19}$$

It is vital to know that even though the RSH hybrid NGA and RSH meta-NGA density functionalities are necessary when computing the conceptual DFT descriptors, it is a different case for RSH GGA (M11) density functional. According to Tables 1 and 2, this functional doesnot provide enough explanation concerning LUMO energy, and this can be due to an

Table 4. Descriptors JI, JA, JHL, Jχ, Jη, Jω, JD1, J<sup>ω</sup>, J<sup>ω</sup> +, JΔ<sup>ω</sup> and JD2 for the SYBR green I (SYBRGI) molecule calculated from

Descriptor M11 M11L MN12L MN12SX N12 N12SX SOGGA11 SOGGA11X

Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical…

http://dx.doi.org/10.5772/intechopen.70455

71

J1 2.185 0.170 0.137 0.062 0.180 0.086 0.201 0.847 JA 2.085 0.187 0.169 0.044 0.259 0.061 0.269 0.796 JHL 3.020 0.252 0.218 0.076 0.316 0.105 0.336 1.162 J<sup>χ</sup> 0.050 0.009 0.016 0.009 0.040 0.012 0.034 0.025 J<sup>η</sup> 4.270 0.356 0.306 0.106 0.439 0.147 0.470 1.643 J<sup>ω</sup> 1.751 0.529 0.365 0.097 0.660 0.136 0.902 1.055 JD1 4.615 0.638 0.477 0.144 0.794 0.201 1.018 1.952 J<sup>ω</sup> 3.209 1.041 0.719 0.183 1.312 0.257 1.792 1.994 Jω<sup>+</sup> 3.259 1.032 0.703 0.192 1.272 0.270 1.758 2.019 JΔω<sup>+</sup> 6.469 2.073 1.422 0.376 2.584 0.527 3.550 4.013 JD2 7.923 2.539 1.742 0.460 3.165 0.646 4.348 4.915

Weighing on the outcomes from this research work, DFT-based reactivity descriptors like electronegativity, chemical hardness, global electrophilicity, electrodonating, and electroaccepting powers, and net electrophilicity can be used to forecast EtBr's chemical reactivity.

It has also been illustrated that the KID process can effectively be implemented by the RSH meta-NGA (MN12SX) and the RSH NGA (N12SX) density functionalities. They can then be used in place of the tuned density functionals using a gap-fitting process, and we believe that such a trend can be helpful when analyzing the chemical reactivity of bigger molecular

This work has been partially supported by CIMAV, SC, and Consejo Nacional de Ciencia y Tecnología (CONACYT, Mexico) through Grant 219566/2014 for Basic Science Research and Grant 265217/2016 for a Foreign Sabbatical Leave. Daniel Glossman-Mitnik conducted this work while a Sabbatical Fellow at the University of the Balearic Islands from which support is

inaccurate figure of γ in the functional. A fine tuning of γ can handle the issue.

5. Conclusions

the results of Table 2.

systems.

Acknowledgements

D2 represents the second collection of conceptual DFT descriptors.

The results of the calculations of JI, JA, JHL, Jχ, Jη, Jω, JD1, J<sup>ω</sup> �, J<sup>ω</sup> +, JΔω�, and JD2 for the EtBr and SYBRGI are displayed in Tables 3 and 4, respectively.

As shown in Tables 1 and 2, and the outcomes from Tables 3 and 4, the precision provided by the KID process is outstanding for the MN12SX, which falls under RSH meta-NGA, and N12SX, which falls under RSH NGA density functionals. In reality, values for JI, JA and JHL is not zero. However, the values found can satisfactorily be likened to the past studies of Lima et al. [19], whereby the minima were found by selecting a parameter that imposes such a trend.

The outcomes are necessary because they reveal that we should not depend on JI, JA, and JHL alone, i.e., if we depend on outcomes from Jχ, alone, almost all the values will near zero. For the remaining descriptors, only MN12SX and N12SX reveal such trends. This shows that outcomes for J<sup>χ</sup> can be due to elimination of errors.

Authentication of the KID process is not done correctly by the GGA (SOGGA11) and hybrid-GGA (SOGGA11X). Local density functionals like M11L, MN12L, and N12 are also inappropriate.


Table 3. Descriptors JI, JA, JHL, Jχ, Jη, Jω, JD1, J<sup>ω</sup> �, Jω+, JΔω�, and JD2 for the ethidium bromide (EtBr) molecule calculated from the results of Table 1.

Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical… http://dx.doi.org/10.5772/intechopen.70455 71


Table 4. Descriptors JI, JA, JHL, Jχ, Jη, Jω, JD1, J<sup>ω</sup>, J<sup>ω</sup> +, JΔ<sup>ω</sup> and JD2 for the SYBR green I (SYBRGI) molecule calculated from the results of Table 2.

It is vital to know that even though the RSH hybrid NGA and RSH meta-NGA density functionalities are necessary when computing the conceptual DFT descriptors, it is a different case for RSH GGA (M11) density functional. According to Tables 1 and 2, this functional doesnot provide enough explanation concerning LUMO energy, and this can be due to an inaccurate figure of γ in the functional. A fine tuning of γ can handle the issue.

#### 5. Conclusions

J<sup>ω</sup>� ¼ ω� � ω�

J<sup>ω</sup><sup>þ</sup> ¼ ω<sup>þ</sup> � ω<sup>þ</sup>

The results of the calculations of JI, JA, JHL, Jχ, Jη, Jω, JD1, J<sup>ω</sup> �, J<sup>ω</sup> +, JΔω�, and JD2 for the EtBr and

As shown in Tables 1 and 2, and the outcomes from Tables 3 and 4, the precision provided by the KID process is outstanding for the MN12SX, which falls under RSH meta-NGA, and N12SX, which falls under RSH NGA density functionals. In reality, values for JI, JA and JHL is not zero. However, the values found can satisfactorily be likened to the past studies of Lima et al. [19], whereby the minima were found by selecting a parameter that imposes

The outcomes are necessary because they reveal that we should not depend on JI, JA, and JHL alone, i.e., if we depend on outcomes from Jχ, alone, almost all the values will near zero. For the remaining descriptors, only MN12SX and N12SX reveal such trends. This shows that

Authentication of the KID process is not done correctly by the GGA (SOGGA11) and hybrid-GGA (SOGGA11X). Local density functionals like M11L, MN12L, and N12 are also inappro-

Descriptor M11 M11L MN12L MN12SX N12 N12SX SOGGA11 SOGGA11X

Table 3. Descriptors JI, JA, JHL, Jχ, Jη, Jω, JD1, J<sup>ω</sup> �, Jω+, JΔω�, and JD2 for the ethidium bromide (EtBr) molecule calculated

J1 1.950 0.228 0.199 0.028 0.356 0.020 0.668 0.724 JA 2.176 0.246 0.205 0.048 0.331 0.068 0.333 0.855 JHL 2.922 0.335 0.285 0.055 0.486 0.071 0.747 1.120 J<sup>χ</sup> 0.113 0.009 0.003 0.038 0.013 0.044 0.168 0.066 J<sup>η</sup> 4.127 0.473 0.403 0.019 0.687 0.048 1.001 1.579 J<sup>ω</sup> 1.831 0.830 0.528 0.079 1.219 0.126 2.858 1.154 JD1 4.516 0.955 0.665 0.090 1.399 0.142 3.033 1.957 Jω� 3.460 1.635 1.033 0.175 2.388 0.271 5.570 2.242 Jω<sup>+</sup> 3.347 1.626 1.030 0.137 2.401 0.227 5.738 2.176 JΔω<sup>+</sup> 6.808 3.260 2.062 0.313 4.788 0.498 11.307 4.417 JD2 8.338 3.993 2.526 0.384 5.864 0.611 13.849 5.410

� � �

� � �

J 2 <sup>ω</sup>� <sup>þ</sup> <sup>J</sup> 2 <sup>ω</sup><sup>þ</sup> <sup>þ</sup> <sup>J</sup> 2 Δω�

q

JD<sup>2</sup> ¼

D2 represents the second collection of conceptual DFT descriptors.

SYBRGI are displayed in Tables 3 and 4, respectively.

70 Density Functional Calculations - Recent Progresses of Theory and Application

outcomes for J<sup>χ</sup> can be due to elimination of errors.

such a trend.

priate.

from the results of Table 1.

K

K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� (16)

� (17)

(19)

<sup>J</sup><sup>Δ</sup>ω� <sup>¼</sup> j j <sup>Δ</sup><sup>ω</sup> � �Δω�<sup>K</sup> (18)

Weighing on the outcomes from this research work, DFT-based reactivity descriptors like electronegativity, chemical hardness, global electrophilicity, electrodonating, and electroaccepting powers, and net electrophilicity can be used to forecast EtBr's chemical reactivity.

It has also been illustrated that the KID process can effectively be implemented by the RSH meta-NGA (MN12SX) and the RSH NGA (N12SX) density functionalities. They can then be used in place of the tuned density functionals using a gap-fitting process, and we believe that such a trend can be helpful when analyzing the chemical reactivity of bigger molecular systems.

#### Acknowledgements

This work has been partially supported by CIMAV, SC, and Consejo Nacional de Ciencia y Tecnología (CONACYT, Mexico) through Grant 219566/2014 for Basic Science Research and Grant 265217/2016 for a Foreign Sabbatical Leave. Daniel Glossman-Mitnik conducted this work while a Sabbatical Fellow at the University of the Balearic Islands from which support is gratefully acknowledged. This work was also funded by the Ministerio de Economía y Competitividad (MINECO) and the European Fund for Regional Development (FEDER) (CTQ2014-55835-R).

[11] Perdew JP, Parr RG, Levy M, Balduz JLJ. Density-functional theory for fractional particle number: Derivative discontinuities of the energy. Physical Review Letters. 1982;49:

Assessment of the Validity of Some Minnesota Density Functionals for the Prediction of the Chemical…

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73

[12] Almbladh CO, von Barth U. Exact results for the charge and spin densities, exchangecorrelation potentials, and density-functional eigenvalues. Physical Review B. 1985;31:

[13] Perdew JP, Burke K, Ersernhof M. Erratum: Generalized gradient approximation made

[14] Levy M, Perdew JP, Sahni V. Exact differential equation for the density and ionization

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#### Author details

Norma Flores-Holguín<sup>1</sup> , Juan Frau2 and Daniel Glossman-Mitnik<sup>1</sup> \*

\*Address all correspondence to: dglossman@gmail.com

1 Laboratorio Virtual NANOCOSMOS, Departamento de Medio Ambiente y Energía, Centro de Investigación en Materiales Avanzados, Chihuahua, Mexico

2 Departament de Química, Universitat de les Illes Balears, Palma de Mallorca, Spain

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gratefully acknowledged. This work was also funded by the Ministerio de Economía y Competitividad (MINECO) and the European Fund for Regional Development (FEDER)

, Juan Frau2 and Daniel Glossman-Mitnik<sup>1</sup>

2 Departament de Química, Universitat de les Illes Balears, Palma de Mallorca, Spain

1 Laboratorio Virtual NANOCOSMOS, Departamento de Medio Ambiente y Energía, Centro

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Gaussian Basis Sets for Molecular Calculations. Amsterdam: Elsevier; 1984

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**Section 2**

**Spectra and Thermodynamics**

**Spectra and Thermodynamics**

**Chapter 4**

Provisional chapter

**Spectroscopy, Substituent Effects, and Reaction**

DOI: 10.5772/intechopen.70751

Applications of density functional theory (DFT) calculations to organic chemistry are shown, beginning with geometry optimization and the calculation of vibrational frequencies, infrared (IR) intensities, and thermodynamic properties. The isotropic chemical shielding values and anisotropies relevant to nuclear magnetic resonance (NMR) can be calculated using gauge-invariant atomic orbitals (GIAOs); the calculation of spin-spin couplings is possible but time-consuming. For free radicals, hyperfine couplings and g tensors pertaining to EPR can be obtained. Regarding UV/vis spectra, wavelengths and oscillator strengths can be calculated by using a time-dependent Hamiltonian. In addition to gas-phase acidities, approximate pKa values can be obtained, provided that solvation is taken into account. Several sets of substituent parameters have been calculated: Hammett σ and σ<sup>+</sup> parameters and inductive and mesomeric effects. Regarding reaction mechanisms, geometries and energies of intermediates and transition structures have been calculated for pericyclic reactions, nucleophilic aliphatic substitutions, elec-

Keywords: density functional theory, spectroscopy, magnetic resonance, Hammett

Focusing on density functional theory (DFT) calculations with Gaussian 09 [1] and the B3LYP/ 6-311G(d,p) method, several applications to organic chemistry will be shown. After geometry optimization, which yields the total energy, a frequency calculation can be done, yielding the infrared spectrum (wave numbers and intensities) and, if requested, the Raman intensities and

Using a time-dependent Hamiltonian, UV/vis spectra can be calculated (wave lengths and oscillator strengths). Nuclear magnetic resonance (NMR) spectra can be calculated, providing

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Spectroscopy, Substituent Effects, and Reaction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

trophilic aromatic substitutions, additions, and eliminations.

the thermodynamic properties (enthalpy, entropy, and Gibbs free energy).

parameters, reaction mechanisms, pericyclic reactions

http://dx.doi.org/10.5772/intechopen.70751

**Mechanisms**

Mechanisms

Burkhard Kirste

Burkhard Kirste

Abstract

1. Introduction

Provisional chapter

#### **Spectroscopy, Substituent Effects, and Reaction Mechanisms** Spectroscopy, Substituent Effects, and Reaction Mechanisms

DOI: 10.5772/intechopen.70751

#### Burkhard Kirste

Additional information is available at the end of the chapter Burkhard Kirste

http://dx.doi.org/10.5772/intechopen.70751 Additional information is available at the end of the chapter

#### Abstract

Applications of density functional theory (DFT) calculations to organic chemistry are shown, beginning with geometry optimization and the calculation of vibrational frequencies, infrared (IR) intensities, and thermodynamic properties. The isotropic chemical shielding values and anisotropies relevant to nuclear magnetic resonance (NMR) can be calculated using gauge-invariant atomic orbitals (GIAOs); the calculation of spin-spin couplings is possible but time-consuming. For free radicals, hyperfine couplings and g tensors pertaining to EPR can be obtained. Regarding UV/vis spectra, wavelengths and oscillator strengths can be calculated by using a time-dependent Hamiltonian. In addition to gas-phase acidities, approximate pKa values can be obtained, provided that solvation is taken into account. Several sets of substituent parameters have been calculated: Hammett σ and σ<sup>+</sup> parameters and inductive and mesomeric effects. Regarding reaction mechanisms, geometries and energies of intermediates and transition structures have been calculated for pericyclic reactions, nucleophilic aliphatic substitutions, electrophilic aromatic substitutions, additions, and eliminations.

Keywords: density functional theory, spectroscopy, magnetic resonance, Hammett parameters, reaction mechanisms, pericyclic reactions

#### 1. Introduction

Focusing on density functional theory (DFT) calculations with Gaussian 09 [1] and the B3LYP/ 6-311G(d,p) method, several applications to organic chemistry will be shown. After geometry optimization, which yields the total energy, a frequency calculation can be done, yielding the infrared spectrum (wave numbers and intensities) and, if requested, the Raman intensities and the thermodynamic properties (enthalpy, entropy, and Gibbs free energy).

Using a time-dependent Hamiltonian, UV/vis spectra can be calculated (wave lengths and oscillator strengths). Nuclear magnetic resonance (NMR) spectra can be calculated, providing

> © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

isotropic shielding values as well as tensor data (anisotropies) of all magnetic nuclei, using gauge-invariant atomic orbitals (GIAOs). The calculation of spin-spin coupling constants is also possible but requires much more computational time. For free radicals, EPR data can be calculated: isotropic hyperfine coupling constants, hyperfine tensors, and g tensors; in this case, the restricted B3LYP method has to be replaced by the unrestricted UB3LYP method.

For the example molecule (p-methylstyrene), the salient results are as follows:

Thermal correction to energy 0.167687 Thermal correction to enthalpy 0.168631 Thermal correction to Gibbs free energy 0.127033 Sum of electronic and zero-point energies 348.894997 Sum of electronic and thermal energies 348.887196 Sum of electronic and thermal enthalpies 348.886252 Sum of electronic and thermal free energies 348.927850 Total E (thermal) 105.225 kcal/mol Total CV 30.240 cal/mol-K Total S (entropy) 87.551 cal/mol-K

SCF Done: E(RB3LYP) 349.054882687 AU Zero-point correction 0.159885 (Hartree/particle)

Spectroscopy, Substituent Effects, and Reaction Mechanisms

http://dx.doi.org/10.5772/intechopen.70751

81

Most data are given in Hartree (see Appendix A), they refer to the formation from atomic nuclei and electrons. It is fairly easy to calculate the energy of formation from the atoms by subtracting the energies obtained for respective calculations of free atoms. In order to obtain approximate values for standard enthalpies of formation, bond energies and possibly enthalpies of phase changes (to the gas phase) have to be taken into account. It should be mentioned that the accuracy of these data, i.e., the agreement with experimental data, is not very good. It is advisable to restrain to energy (or enthalpy) differences of similar structures. Alternatively, approximate enthalpies of formation can be obtained more easily from semiem-

Energies of some important free atoms (UB3LYP/6-311G(d,p) in Hartree): H, 0.502155930031; C, 37.8559889346; N, 54.5985431427; O, 75.0853856058; F, 99.7538096003; P, 341.280503655;

Hence, the following energy of formation from the atoms is obtained for p-methylstyrene, C9H10, ΔEf atomic = 348.8871969 (37.8559889346)10 (0.502155930031) = 3.161736288290

Enthalpies required to generate free atoms from the elements in the standard state (kJ/mol) [3]:

These values have to be added to the above-given atomic energy of formation (ignoring somewhat the difference between energy and enthalpy), yielding the following energy of

The energy can be converted to the enthalpy by means of Eq. (1), assuming the validity of the

H, 218.00; C, 716.67; N, 472.68; O, 249.17; F, 78.4; P, 314.55; S, 276.98; and Cl, 121.29.

formation for our example: 8301.14 + 9 716.67 + 10 218.00 = 328.89 kJ/mol.

ideal gas law; Δn is the change in the number of moles of gases:

pirical calculations (such as MNDO, AM1, or PM3).

S, 398.132082447; and Cl, 460.166160487.

Hartree = 8301.139 kJ/mol.

Substituent effects such as the σ parameters in the Hammett equation can also be estimated. Although a calculation of changes in the charge distribution might seem to be a promising method for that purpose, it was found that calculated 19F shielding values ("virtual NMR experiments") yielded much more convincing results. The calculation of gas-phase acidities or basicities is straightforward, and the calculated data show a good correlation with experimental data. However, the correlation with pKa values, which refer to aqueous solutions, is very poor. A reasonable correlation was obtained by taking a few water molecules explicitly into account, in addition to the bulk solvent properties of water.

Regarding organic reaction mechanisms, pericyclic reactions are particularly well amenable to DFT calculations. Usually, the transition structure can be obtained which is characterized by a single imaginary frequency, which belongs to the reaction coordinate. For many other reaction types (substitutions, additions, eliminations, and rearrangements), at least an approximation to the transition structure can be calculated. Moreover, starting with such a structure and performing an optimization, the approximate dynamics of the reaction can be followed.

#### 2. Geometries, energies, and thermodynamic data

#### 2.1. Geometry optimization

As a starting point, a reasonable approximation to the geometry of the target molecule is required. Preferably, the coordinate file should be given as Z matrix, and standard bond lengths and angles may be used. A convenient tool for the generation of Z matrices is molden [2]: in the Z-mat editor, start with methane, substitute by phenyl and finally by vinyl, and save as Z matrix (GAMESS). Next, the input file for the quantum-chemical calculation has to be created by supplementing the Z matrix file with the necessary parameters (see Appendix A).

After a successful calculation, the log file contains the energy (in Hartree) and the coordinates of the optimized structure. Again, it is advantageous to use a tool such as molden for analyzing the log file.

#### 2.2. Calculation of thermodynamic properties

For a determination of the thermodynamic properties, it is necessary to calculate the (vibrational) frequencies. In the Gaussian input file, the preliminary coordinates have to be replaced by the optimized ones and the task "Opt" by "Freq". (Actually, the request for "Freq Prop Pop = Full" additionally provides useful information such as charges and dipole moment. By default, the calculation is done for 298 K and 1.000 atm, but a different temperature or pressure may be specified.)


For the example molecule (p-methylstyrene), the salient results are as follows:

isotropic shielding values as well as tensor data (anisotropies) of all magnetic nuclei, using gauge-invariant atomic orbitals (GIAOs). The calculation of spin-spin coupling constants is also possible but requires much more computational time. For free radicals, EPR data can be calculated: isotropic hyperfine coupling constants, hyperfine tensors, and g tensors; in this case, the restricted B3LYP method has to be replaced by the unrestricted UB3LYP method.

Substituent effects such as the σ parameters in the Hammett equation can also be estimated. Although a calculation of changes in the charge distribution might seem to be a promising method for that purpose, it was found that calculated 19F shielding values ("virtual NMR experiments") yielded much more convincing results. The calculation of gas-phase acidities or basicities is straightforward, and the calculated data show a good correlation with experimental data. However, the correlation with pKa values, which refer to aqueous solutions, is very poor. A reasonable correlation was obtained by taking a few water molecules explicitly

Regarding organic reaction mechanisms, pericyclic reactions are particularly well amenable to DFT calculations. Usually, the transition structure can be obtained which is characterized by a single imaginary frequency, which belongs to the reaction coordinate. For many other reaction types (substitutions, additions, eliminations, and rearrangements), at least an approximation to the transition structure can be calculated. Moreover, starting with such a structure and performing an optimization, the approximate dynamics of the reaction can be followed.

As a starting point, a reasonable approximation to the geometry of the target molecule is required. Preferably, the coordinate file should be given as Z matrix, and standard bond lengths and angles may be used. A convenient tool for the generation of Z matrices is molden [2]: in the Z-mat editor, start with methane, substitute by phenyl and finally by vinyl, and save as Z matrix (GAMESS). Next, the input file for the quantum-chemical calculation has to be created by supplementing the Z matrix file with the necessary parameters (see Appendix A). After a successful calculation, the log file contains the energy (in Hartree) and the coordinates of the optimized structure. Again, it is advantageous to use a tool such as molden for analyz-

For a determination of the thermodynamic properties, it is necessary to calculate the (vibrational) frequencies. In the Gaussian input file, the preliminary coordinates have to be replaced by the optimized ones and the task "Opt" by "Freq". (Actually, the request for "Freq Prop Pop = Full" additionally provides useful information such as charges and dipole moment. By default, the calculation is done for 298 K and 1.000 atm, but a different temperature or pressure

into account, in addition to the bulk solvent properties of water.

80 Density Functional Calculations - Recent Progresses of Theory and Application

2. Geometries, energies, and thermodynamic data

2.2. Calculation of thermodynamic properties

2.1. Geometry optimization

ing the log file.

may be specified.)

Most data are given in Hartree (see Appendix A), they refer to the formation from atomic nuclei and electrons. It is fairly easy to calculate the energy of formation from the atoms by subtracting the energies obtained for respective calculations of free atoms. In order to obtain approximate values for standard enthalpies of formation, bond energies and possibly enthalpies of phase changes (to the gas phase) have to be taken into account. It should be mentioned that the accuracy of these data, i.e., the agreement with experimental data, is not very good. It is advisable to restrain to energy (or enthalpy) differences of similar structures. Alternatively, approximate enthalpies of formation can be obtained more easily from semiempirical calculations (such as MNDO, AM1, or PM3).

Energies of some important free atoms (UB3LYP/6-311G(d,p) in Hartree): H, 0.502155930031; C, 37.8559889346; N, 54.5985431427; O, 75.0853856058; F, 99.7538096003; P, 341.280503655; S, 398.132082447; and Cl, 460.166160487.

Hence, the following energy of formation from the atoms is obtained for p-methylstyrene, C9H10, ΔEf atomic = 348.8871969 (37.8559889346)10 (0.502155930031) = 3.161736288290 Hartree = 8301.139 kJ/mol.

Enthalpies required to generate free atoms from the elements in the standard state (kJ/mol) [3]: H, 218.00; C, 716.67; N, 472.68; O, 249.17; F, 78.4; P, 314.55; S, 276.98; and Cl, 121.29.

These values have to be added to the above-given atomic energy of formation (ignoring somewhat the difference between energy and enthalpy), yielding the following energy of formation for our example: 8301.14 + 9 716.67 + 10 218.00 = 328.89 kJ/mol.

The energy can be converted to the enthalpy by means of Eq. (1), assuming the validity of the ideal gas law; Δn is the change in the number of moles of gases:

$$
\Delta H = \Delta E + \Delta \eta RT \tag{1}
$$

3.2. Nuclear magnetic resonance (NMR)

H) = 31.3919 ppm and σ(

For the example molecule p-methylstyrene, the following <sup>1</sup>

Figure 2. 3D model and numbering scheme of 1-methyl-4-vinylbenzene.

to some standard. For <sup>1</sup>

1

obtained by subtraction:

nuclei: σ(

Table 1) [8].

The calculation of isotropic chemical shielding values (σ) or shielding tensors requires some kind of "scaling" of the orbitals, for instance, the use of gauge-invariant atomic orbitals (GIAOs) [6]. In Gaussian 09, the keyword "NMR" automatically invokes the use of GIAOs, and isotropic shielding values, anisotropies, and shielding tensors are calculated. In NMR experiments, however, not the shielding values are measured, but chemical shifts, which refer

and the respective chemical shifts of the three kinds of magnetic nuclei are set to zero (δH-1(TMS) = 0 ppm, δC-13(TMS) = 0 ppm, and δSi-29(TMS) = 0 ppm). Using the hybrid method B3LYP/6-311(d,p), the following average shielding values are obtained for protons and for 13C

Using the abovementioned reference value for protons, the calculated chemical shifts are generally too small by about 0.5 ppm. In a survey of 21 natural products, a better fit for 13C nuclei, on the average, was obtained by using a reference value of 177.0 ppm instead [7].

were calculated (using the above-given calculated shielding values for TMS as reference); experimental 13C chemical shifts were taken from the NMRSHIFTDB database (Figure 2 and

It is also possible to calculate NMR spin-spin coupling constants, i.e., J [Hz] (Gaussian keyword "NMR = SpinSpin"), but at the expense of computational time. The results obtained with

For p-methylstyrene, the following proton-proton spin-spin coupling constants J [Hz] were calculated: J13,14 = 6.86, J15,16 = 7.09 (o), J14,15 = 1.29, J13,16 = 1.26 (m), J13,15 = 0.41, J14,16 = 0.28 (p),

the B3LYP hybrid functional are much better than those of HF ab initio calculations.

H, 13C, and 29Si NMR, tetramethylsilane (TMS) is used as a standard,

13C) = 179.7024 ppm. The chemical shifts are then simply

δ<sup>i</sup> ¼ σref � σ<sup>i</sup> (3)

Spectroscopy, Substituent Effects, and Reaction Mechanisms

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83

H and 13C chemical shifts (δ in ppm)

$$
\Delta H = \Delta E + \Delta n \times 2.479 \frac{\text{kJ}}{\text{mol}} (T = 298.15 \text{ K}) \tag{2}
$$

In the example, 1 mole of product molecules (in the gas phase) is formed from 19 moles of atoms; therefore, Δn is �18 and the correction is �44.62 kJ/mol. Hence, the calculated enthalpy of formation of p-methylstyrene in the gas phase is ΔHf � = 284.27 kJ/mol.

The following experimental value for the enthalpy of formation of liquid p-methylstyrene is given in the literature: ΔHf �(l) = 114.6 kJ/mol [3]; adding the heat of vaporization of 47.6 kJ/mol [4], ΔHf �(g) = 162.2 kJ/mol for the gas phase. Thus, the calculated value deviates by about 120 kJ/mol.

By comparison, a semiempirical AM1 calculation yields an enthalpy of formation of ΔHf � = 140.3 kJ/mol, in better agreement with experiment.

#### 3. Spectroscopy

#### 3.1. Vibrational spectroscopy: infrared and Raman

Vibrational frequencies and hence infrared (IR) and Raman spectra can be calculated (Gaussian keyword "Freq"). In Gaussian 09, the infrared intensities are calculated by default, but the Raman intensities can also be obtained (keyword "Freq = Raman"). The calculated frequencies can be assigned to the respective molecular motions. The visualization of vibrations is easily achieved by tools such as molden.

As an example, p-cyanobenzaldehyde will be considered (Figure 1). The experimental IR data have been taken from the SDBS database [5]. The two most prominent features are the C]O valence vibration at 1788 (exp. 1708) and the CN valence vibration at 2340 (2230) cm�<sup>1</sup> .

Figure 1. Calculated IR spectrum of p-cyanobenzaldehyde.

#### 3.2. Nuclear magnetic resonance (NMR)

ΔH ¼ ΔE þ ΔnRT (1)

� = 284.27 kJ/mol.

�(l) = 114.6 kJ/mol [3]; adding the heat of vaporization of 47.6 kJ/mol [4],

molð Þ <sup>T</sup> <sup>¼</sup> <sup>298</sup>:15 K (2)

� =

.

<sup>Δ</sup><sup>H</sup> <sup>¼</sup> <sup>Δ</sup><sup>E</sup> <sup>þ</sup> <sup>Δ</sup><sup>n</sup> � <sup>2</sup>:<sup>479</sup> kJ

of formation of p-methylstyrene in the gas phase is ΔHf

82 Density Functional Calculations - Recent Progresses of Theory and Application

140.3 kJ/mol, in better agreement with experiment.

3.1. Vibrational spectroscopy: infrared and Raman

in the literature: ΔHf

3. Spectroscopy

achieved by tools such as molden.

Figure 1. Calculated IR spectrum of p-cyanobenzaldehyde.

ΔHf

In the example, 1 mole of product molecules (in the gas phase) is formed from 19 moles of atoms; therefore, Δn is �18 and the correction is �44.62 kJ/mol. Hence, the calculated enthalpy

The following experimental value for the enthalpy of formation of liquid p-methylstyrene is given

By comparison, a semiempirical AM1 calculation yields an enthalpy of formation of ΔHf

Vibrational frequencies and hence infrared (IR) and Raman spectra can be calculated (Gaussian keyword "Freq"). In Gaussian 09, the infrared intensities are calculated by default, but the Raman intensities can also be obtained (keyword "Freq = Raman"). The calculated frequencies can be assigned to the respective molecular motions. The visualization of vibrations is easily

As an example, p-cyanobenzaldehyde will be considered (Figure 1). The experimental IR data have been taken from the SDBS database [5]. The two most prominent features are the C]O valence vibration at 1788 (exp. 1708) and the CN valence vibration at 2340 (2230) cm�<sup>1</sup>

�(g) = 162.2 kJ/mol for the gas phase. Thus, the calculated value deviates by about 120 kJ/mol.

The calculation of isotropic chemical shielding values (σ) or shielding tensors requires some kind of "scaling" of the orbitals, for instance, the use of gauge-invariant atomic orbitals (GIAOs) [6]. In Gaussian 09, the keyword "NMR" automatically invokes the use of GIAOs, and isotropic shielding values, anisotropies, and shielding tensors are calculated. In NMR experiments, however, not the shielding values are measured, but chemical shifts, which refer to some standard. For <sup>1</sup> H, 13C, and 29Si NMR, tetramethylsilane (TMS) is used as a standard, and the respective chemical shifts of the three kinds of magnetic nuclei are set to zero (δH-1(TMS) = 0 ppm, δC-13(TMS) = 0 ppm, and δSi-29(TMS) = 0 ppm). Using the hybrid method B3LYP/6-311(d,p), the following average shielding values are obtained for protons and for 13C nuclei: σ( 1 H) = 31.3919 ppm and σ( 13C) = 179.7024 ppm. The chemical shifts are then simply obtained by subtraction:

$$
\delta\_i = \sigma\_{r\!\!\!\!f} - \sigma\_i \tag{3}
$$

Using the abovementioned reference value for protons, the calculated chemical shifts are generally too small by about 0.5 ppm. In a survey of 21 natural products, a better fit for 13C nuclei, on the average, was obtained by using a reference value of 177.0 ppm instead [7].

For the example molecule p-methylstyrene, the following <sup>1</sup> H and 13C chemical shifts (δ in ppm) were calculated (using the above-given calculated shielding values for TMS as reference); experimental 13C chemical shifts were taken from the NMRSHIFTDB database (Figure 2 and Table 1) [8].

It is also possible to calculate NMR spin-spin coupling constants, i.e., J [Hz] (Gaussian keyword "NMR = SpinSpin"), but at the expense of computational time. The results obtained with the B3LYP hybrid functional are much better than those of HF ab initio calculations.

For p-methylstyrene, the following proton-proton spin-spin coupling constants J [Hz] were calculated: J13,14 = 6.86, J15,16 = 7.09 (o), J14,15 = 1.29, J13,16 = 1.26 (m), J13,15 = 0.41, J14,16 = 0.28 (p),

Figure 2. 3D model and numbering scheme of 1-methyl-4-vinylbenzene.


Table 1. Chemical shifts of p-methylstyrene (δ in ppm) (cf. Figure 2).

J17,19 = 14.72 (trans), J17,18 = 10.11 (cis), and J18,19 = �0.53 (gem). These values are in accordance with those found in similar systems.

It should be mentioned that NMR data provide an excellent and sensitive test for the accuracy of quantum-chemical calculations.

protons are inequivalent. Comparison with experimental data (ethanol, 230 K, in parentheses) [11]: 6.44 (5.84), methyl protons, and 3.56 (3.68) and 2.11 (2.17) MHz, methylene protons. The g tensor has been measured by high-field EPR experiments [12]; again, the experimental values are given in parentheses: gxx = 2.00826 (2.00646), gyy = 2.00601 (2.00542), gzz = 2.00207 (2.00222),

y

Figure 3. Ubisemiquinone-Q1 radical anion. Left: Mulliken spin densities. Right: calculated HFC.

O

Spectroscopy, Substituent Effects, and Reaction Mechanisms

CH3

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H

6.44 MHz

2.11 MHz

85

O

z

O

O

3.56 MHz

x H

In Hückel molecular orbital (HMO) theory, electronic excitation may be viewed as excitation of an electron from an occupied to an unoccupied orbital. The transition with the lowest energy, i.e., the longest wavelength, involves the excitation from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO), although this transition

In DFT, however, the Kohn-Sham orbitals are not suitable for this procedure, and a timedependent Hamiltonian has to be used in the calculation (Gaussian keyword "TD"). The calculation gives the energies and the wavelengths of the excitations, the oscillator strengths f, and reports the orbitals which are involved. The vibrational fine structure and the conse-

In the case of the symmetrical crystal violet cation, the HOMO is represented by two degenerate orbitals, and two excitations have the same wavelength, calculated as 504.7 nm (f = 0.806);

quences of the Franck-Condon principle are not taken into account.

experimental data for the absorption maxima: 591.0 and 540.5 nm.

and giso = 2.00545 (2.00470).

O

0.120

0.118

0.083 0.073

0.254

O

O

0.040 0.050

O

0.242

might be forbidden.

For further examples, see [13].

3.4. Electron spectroscopy (UV/vis)

#### 3.3. Electron paramagnetic resonance (EPR)

In the case of free radicals, unrestricted calculations have to be performed in which different orbitals are assigned to α and β spins. Whereas unrestricted Hartree-Fock (UHF) calculations yield poor results for hyperfine couplings (HFC) because of serious problems due to spin contamination, calculations with the UB3LYP hybrid functional yield fairly acceptable results [9]. The calculations yield Mulliken spin densities (better designated as spin populations), isotropic HFC (Fermi contact coupling constants), and anisotropic hyperfine tensors. g values and g tensors can also be calculated (in the Gaussian system, this requires the "NMR" keyword). The g value is a dimensionless proportionality factor which relates the magnetic moment to the angular momentum; the value for the free electron is ge = 2.00232, and only the electron spin is involved. In molecules, contributions from orbital momentum have to be taken into account, and the phenomenon becomes anisotropic. The calculated HFC and g values may be compared with experimental data from EPR spectroscopy (electron spin resonance, also called electron paramagnetic resonance) [10].

The method will be illustrated using the ubisemiquinone-Q1 radical anion as example, which serves as a model compound for coenzyme Q10.

Figure 3 shows the calculated Mulliken spin densities and the calculated proton HFC of this radical anion. The rotation of the long side chain is hindered; therefore, the two methylene

Figure 3. Ubisemiquinone-Q1 radical anion. Left: Mulliken spin densities. Right: calculated HFC.

protons are inequivalent. Comparison with experimental data (ethanol, 230 K, in parentheses) [11]: 6.44 (5.84), methyl protons, and 3.56 (3.68) and 2.11 (2.17) MHz, methylene protons. The g tensor has been measured by high-field EPR experiments [12]; again, the experimental values are given in parentheses: gxx = 2.00826 (2.00646), gyy = 2.00601 (2.00542), gzz = 2.00207 (2.00222), and giso = 2.00545 (2.00470).

#### 3.4. Electron spectroscopy (UV/vis)

J17,19 = 14.72 (trans), J17,18 = 10.11 (cis), and J18,19 = �0.53 (gem). These values are in accordance

Pos. C calc C exp H calc H pred 129.7 128.6 6.67 7.39 140.2 136.5 – – 129.7 128.6 6.73 7.39 131.0 127.7 6.54 7.59 136.1 136.0 – – 122.8 127.7 7.24 7.59 18.7 20.0 1.71 2.41 139.7 136.3 6.07 6.72 109.8 112.8 4.70 5.25 ' 5.42 5.76

It should be mentioned that NMR data provide an excellent and sensitive test for the accuracy

In the case of free radicals, unrestricted calculations have to be performed in which different orbitals are assigned to α and β spins. Whereas unrestricted Hartree-Fock (UHF) calculations yield poor results for hyperfine couplings (HFC) because of serious problems due to spin contamination, calculations with the UB3LYP hybrid functional yield fairly acceptable results [9]. The calculations yield Mulliken spin densities (better designated as spin populations), isotropic HFC (Fermi contact coupling constants), and anisotropic hyperfine tensors. g values and g tensors can also be calculated (in the Gaussian system, this requires the "NMR" keyword). The g value is a dimensionless proportionality factor which relates the magnetic moment to the angular momentum; the value for the free electron is ge = 2.00232, and only the electron spin is involved. In molecules, contributions from orbital momentum have to be taken into account, and the phenomenon becomes anisotropic. The calculated HFC and g values may be compared with experimental data from EPR spectroscopy (electron spin resonance, also

The method will be illustrated using the ubisemiquinone-Q1 radical anion as example, which

Figure 3 shows the calculated Mulliken spin densities and the calculated proton HFC of this radical anion. The rotation of the long side chain is hindered; therefore, the two methylene

with those found in similar systems.

of quantum-chemical calculations.

3.3. Electron paramagnetic resonance (EPR)

Table 1. Chemical shifts of p-methylstyrene (δ in ppm) (cf. Figure 2).

84 Density Functional Calculations - Recent Progresses of Theory and Application

called electron paramagnetic resonance) [10].

serves as a model compound for coenzyme Q10.

In Hückel molecular orbital (HMO) theory, electronic excitation may be viewed as excitation of an electron from an occupied to an unoccupied orbital. The transition with the lowest energy, i.e., the longest wavelength, involves the excitation from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO), although this transition might be forbidden.

In DFT, however, the Kohn-Sham orbitals are not suitable for this procedure, and a timedependent Hamiltonian has to be used in the calculation (Gaussian keyword "TD"). The calculation gives the energies and the wavelengths of the excitations, the oscillator strengths f, and reports the orbitals which are involved. The vibrational fine structure and the consequences of the Franck-Condon principle are not taken into account.

In the case of the symmetrical crystal violet cation, the HOMO is represented by two degenerate orbitals, and two excitations have the same wavelength, calculated as 504.7 nm (f = 0.806); experimental data for the absorption maxima: 591.0 and 540.5 nm.

For further examples, see [13].

#### 4. Substituent effects

#### 4.1. Hammett σ parameters

The Hammett σ parameters refer to the acidities of substituted benzoic acids which will be considered in Section 4.3. Regarding electrophilic aromatic substitution (see Section 5.3), a modified set has to be used, at least for the para positions (σ<sup>+</sup> parameters). The Hammett equation is

$$\log \frac{k\_i}{k\_0} = \sigma \rho \tag{4}$$

4.2. Estimating inductive and mesomeric effects by virtual 19F NMR

Table 2. Calculated (DFT) and literature data [14] for Hammett σ/σ<sup>+</sup> parameters.

fluoro-1,3-butadienes were chosen, in two conformations (Figure 5).

shielding values at 0 and at 90, σrel(

(e.g., alkyl groups) (see Table 3).

The relative contributions of inductive (I) and mesomeric (M) effects might be inferred from a comparison of the Hammett σ/σ<sup>+</sup> parameters (see Section 4.1) for the para (I + M) and meta (I + M/3) positions. Yet, a different approach is taken here. Using DFT, the obvious target to look for should be the charge density distribution. However, it turned out that the Mulliken charges, at least, did not yield satisfying results. Therefore, calculated isotropic 19F shielding values were used as a probe of local charge density. As a suitable system, 4-substituted (E,E)-1-

Substituent σ meta DFT σ meta lit. σ<sup>+</sup> para DFT σ<sup>+</sup> para lit. H 0.00 0.00 0.00 0.00 Methyl 0.11 0.10 0.37 0.31 t-Butyl 0.23 0.10 0.47 0.31 Phenyl 0.17 0.00 0.67 0.18 Hydroxy 0.09 0.13 0.64 0.92 Methoxy 0.06 0.05 0.87 0.78 Amino 0.29 0.16 1.38 1.30 Dimethylamino 0.47 0.10 1.65 1.70 Fluoro 0.39 0.35 0.00 0.07 Chloro 0.39 0.40 0.06 0.11 Bromo 0.36 0.41 0.03 0.15 Nitro 0.78 0.73 0.88 0.79 Cyano 0.72 0.56 0.62 0.66 Trifluoromethyl 0.57 0.46 0.58 0.53 Acetyl 0.24 0.36 0.33 0.47 Carboxy 0.29 0.32 0.35 0.42 Sulfonyl 0.58 0.64 0.47 0.73

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87

The geometries were optimized for the planar conformation, and the shielding values were calculated for this conformation (0) and for the orthogonal conformation with a dihedral angle of 90 for the central single bond (see Figure 5). The relative shielding values σrel(

90 should be proportional to the inductive (I) effect, and the differences of the relative

mesomeric (M) effect. The reference compound is, of course, the unsubstituted compound (R = H). These data were calibrated against the Hammett σ/σ<sup>+</sup> parameters, yielding a slope of 20.447, i.e., the data have to be divided by this factor. According to the sign convention of the Hammett <sup>σ</sup>/σ<sup>+</sup> parameters, electron-withdrawing groups (EWG, I, <sup>M</sup>) have a positive sign (e.g., nitro and cyano), whereas electron-releasing groups (ERG, +I, +M) have a negative sign

19F)0<sup>σ</sup>rel(

19F) at

19F)90, should be proportional to the

where ρ is the reaction parameter and k<sup>0</sup> and ki are the rate constants for the unsubstituted and substituted compounds, respectively. The σ/σ<sup>+</sup> parameters for electrophilic aromatic substitution have been determined from the relative stabilities of the σ complexes as averages for the following four reactions: protonation, bromination, nitration, and alkylation (by ethyl groups). A linear fit of σ/σ<sup>+</sup> (literature data [14]) versus calculated σ (DFT) was determined for 17 substituents both in meta and in para positions, yielding a squared correlation coefficient of r <sup>2</sup> = 0.932 (see Figure 4 and Table 2).

Figure 4. Plot of literature data for Hammett σ/σ<sup>+</sup> parameters versus calculated values (DFT).



Table 2. Calculated (DFT) and literature data [14] for Hammett σ/σ<sup>+</sup> parameters.

4. Substituent effects

86 Density Functional Calculations - Recent Progresses of Theory and Application

4.1. Hammett σ parameters

<sup>2</sup> = 0.932 (see Figure 4 and Table 2).

r

The Hammett σ parameters refer to the acidities of substituted benzoic acids which will be considered in Section 4.3. Regarding electrophilic aromatic substitution (see Section 5.3), a modified set has to be used, at least for the para positions (σ<sup>+</sup> parameters). The Hammett equation is

where ρ is the reaction parameter and k<sup>0</sup> and ki are the rate constants for the unsubstituted and substituted compounds, respectively. The σ/σ<sup>+</sup> parameters for electrophilic aromatic substitution have been determined from the relative stabilities of the σ complexes as averages for the following four reactions: protonation, bromination, nitration, and alkylation (by ethyl groups). A linear fit of σ/σ<sup>+</sup> (literature data [14]) versus calculated σ (DFT) was determined for 17 substituents both in meta and in para positions, yielding a squared correlation coefficient of

¼ σρ (4)

log ki k0

Figure 4. Plot of literature data for Hammett σ/σ<sup>+</sup> parameters versus calculated values (DFT).

#### 4.2. Estimating inductive and mesomeric effects by virtual 19F NMR

The relative contributions of inductive (I) and mesomeric (M) effects might be inferred from a comparison of the Hammett σ/σ<sup>+</sup> parameters (see Section 4.1) for the para (I + M) and meta (I + M/3) positions. Yet, a different approach is taken here. Using DFT, the obvious target to look for should be the charge density distribution. However, it turned out that the Mulliken charges, at least, did not yield satisfying results. Therefore, calculated isotropic 19F shielding values were used as a probe of local charge density. As a suitable system, 4-substituted (E,E)-1 fluoro-1,3-butadienes were chosen, in two conformations (Figure 5).

The geometries were optimized for the planar conformation, and the shielding values were calculated for this conformation (0) and for the orthogonal conformation with a dihedral angle of 90 for the central single bond (see Figure 5). The relative shielding values σrel( 19F) at 90 should be proportional to the inductive (I) effect, and the differences of the relative shielding values at 0 and at 90, σrel( 19F)0<sup>σ</sup>rel( 19F)90, should be proportional to the mesomeric (M) effect. The reference compound is, of course, the unsubstituted compound (R = H). These data were calibrated against the Hammett σ/σ<sup>+</sup> parameters, yielding a slope of 20.447, i.e., the data have to be divided by this factor. According to the sign convention of the Hammett <sup>σ</sup>/σ<sup>+</sup> parameters, electron-withdrawing groups (EWG, I, <sup>M</sup>) have a positive sign (e.g., nitro and cyano), whereas electron-releasing groups (ERG, +I, +M) have a negative sign (e.g., alkyl groups) (see Table 3).

4.3. Acids and bases: pKa values

Ka was calculated according to

The calculation of gas-phase acidities is straightforward, but they do not correlate well with experimental pKa values [13]. This situation is only partially improved by taking the bulk properties of the solvent (water) into account (Gaussian keyword "SCRF = (Solvent = Water)"). A much better approximation is obtained when additionally a few water molecules are taken into account explicitly, e.g., two water molecules in the case of carboxylic acids (see Figure 6). Thus, Gibbs free energies for benzoic acid and a series of substituted benzoic acids (15 substituents both in meta and in para positions) as well as the respective anions were calculated. ΔG� = G�(anion) � G�(acid) was converted from Hartree to kJ/mol (see Appendix A), and log

The relative pKa values were found to be quite reasonable but have to be scaled. Since the difference log Ka (substituted benzoic acid)-log Ka (benzoic acid) should be equal to the Hammett σ parameter, a linear fit of σ (literature data [14]) versus calculated Δ(log Ka) (DFT)

was determined, yielding a slope of 0.3437 and a squared correlation coefficient of r

¼ �2:3026 RT log Ka (5)

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pKa ¼ � logKa (6)

<sup>2</sup> = 0.967

89

ΔG�

Figure 6. 3D models of dihydrates of benzoic acid and benzoate anion.

Figure 5. 3D models of two conformations of 1-fluoro-5,5-dimethyl-1,3-hexadiene.


Table 3. Calculated inductive (σI) and mesomeric (σM) effects (DFT) (see text).

#### 4.3. Acids and bases: pKa values

Figure 5. 3D models of two conformations of 1-fluoro-5,5-dimethyl-1,3-hexadiene.

88 Density Functional Calculations - Recent Progresses of Theory and Application

Table 3. Calculated inductive (σI) and mesomeric (σM) effects (DFT) (see text).

Substituent σ<sup>I</sup> σ<sup>M</sup> H 0.00 0.00 Nitro 0.47 0.99 Cyano 0.35 0.57 Acetyl 0.16 0.69 Carboxy 0.16 0.75 Methoxycarbonyl 0.12 0.69 Trifluoromethyl 1.02 �0.29 Fluoro 0.39 �0.43 Chloro 0.33 �0.23 Bromo 0.30 �0.21 Methyl �0.04 �0.25 Hydroxy 0.18 �0.74 Methoxy 0.16 �0.74 Amino 0.01 �1.21 Dimethylamino �0.02 �1.17 The calculation of gas-phase acidities is straightforward, but they do not correlate well with experimental pKa values [13]. This situation is only partially improved by taking the bulk properties of the solvent (water) into account (Gaussian keyword "SCRF = (Solvent = Water)"). A much better approximation is obtained when additionally a few water molecules are taken into account explicitly, e.g., two water molecules in the case of carboxylic acids (see Figure 6).

Thus, Gibbs free energies for benzoic acid and a series of substituted benzoic acids (15 substituents both in meta and in para positions) as well as the respective anions were calculated. ΔG� = G�(anion) � G�(acid) was converted from Hartree to kJ/mol (see Appendix A), and log Ka was calculated according to

$$
\Delta G^{\circ} = -2.3026 \, RT \log K\_{\circ} \tag{5}
$$

$$pK\_a = -\log K\_a \tag{6}$$

The relative pKa values were found to be quite reasonable but have to be scaled. Since the difference log Ka (substituted benzoic acid)-log Ka (benzoic acid) should be equal to the Hammett σ parameter, a linear fit of σ (literature data [14]) versus calculated Δ(log Ka) (DFT) was determined, yielding a slope of 0.3437 and a squared correlation coefficient of r <sup>2</sup> = 0.967

Figure 6. 3D models of dihydrates of benzoic acid and benzoate anion.

conservation of orbital symmetry") [15]. Typical examples are sigmatropic reactions such as the Cope rearrangement, cycloadditions such as the Diels-Alder addition, or electrocyclic

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91

Pericyclic reactions are particularly well amenable to DFT calculations; the transition structure can usually be obtained. The transition structure is a saddle point in the energy hyperspace, i.e., the energy has a maximum along the reaction path (the reaction coordinate), but is minimized with respect to all other coordinates. This can be checked by a frequency calculation. Exactly one frequency should be imaginary, namely, the one pertaining to the reaction coordinate. Thus, the

The Cope rearrangement is a [3,3] sigmatropic reaction. As an example, the degenerate Cope reaction of 1,5-hexadiene is shown (see Figures 8 and 9). The calculated activation energy is

The Diels-Alder addition is a [4 + 2] cycloaddition, a diene reacts with a dienophile to form a (substituted) cyclohexene. As an example, the Diels-Alder addition of acrylonitrile to cyclopentadiene leading to the endo-product is shown (see Figures 10 and 11). The calculated

N

N

activation energy is 40 kJ/mol, and the calculated reaction energy is �132 kJ/mol (DFT).

reactions (ring closures or openings).

5.1.1. Cope rearrangement

5.1.2. Diels-Alder addition

129 kJ/mol (DFT).

reaction dynamics can be visualized by looking at that vibration.

Figure 8. Scheme of the degenerate cope reaction of 1,5-hexadiene.

Figure 9. Cope reaction: 3D models of reactant, transition structure, and product.

N

Figure 10. Scheme of the Diels-Alder reaction between cyclopentadiene and acrylonitrile.

+

Figure 7. Plot of Hammett σ parameters versus calculated relative log Ka values (DFT).

(Figure 7). That means, the substituent effect on the calculated pKa value is overestimated by about a factor of 3.

For the calculation of absolute pKa values, the considerable Gibbs free solvation energy of the proton has to be taken into account (ΔG� = �1120.39 kJ/mol). The calculated Gibbs free energy of the dissociation of benzoic acid in the gas phase is ΔG� = 1446.04 kJ/mol. For the hydration (dihydrate model, vide supra), ΔG� = �47.35 kJ/mol and ΔG� = �302.76 kJ/mol are obtained for benzoic acid and benzoate anion, respectively; in total, ΔG� = 1375.80 kJ/mol. Hence, the estimate for the Gibbs free energy of the dissociation of benzoic acid in aqueous solution is ΔG� = 1446.04 � 1375.80 = 70.24 kJ/mol corresponding to a pKa value of 12.31 (exp. 4.19). To put it differently, the model applied here accounts for about 96.7% of the true Gibbs free energy of solvation.

#### 5. Reaction mechanisms

#### 5.1. Pericyclic reactions

In a pericyclic reaction, σ or π bonds change concertedly ("simultaneously") along a perimeter, i.e., a cycle. They have first been studied theoretically by Woodward and Hoffmann ("the conservation of orbital symmetry") [15]. Typical examples are sigmatropic reactions such as the Cope rearrangement, cycloadditions such as the Diels-Alder addition, or electrocyclic reactions (ring closures or openings).

Pericyclic reactions are particularly well amenable to DFT calculations; the transition structure can usually be obtained. The transition structure is a saddle point in the energy hyperspace, i.e., the energy has a maximum along the reaction path (the reaction coordinate), but is minimized with respect to all other coordinates. This can be checked by a frequency calculation. Exactly one frequency should be imaginary, namely, the one pertaining to the reaction coordinate. Thus, the reaction dynamics can be visualized by looking at that vibration.

#### 5.1.1. Cope rearrangement

The Cope rearrangement is a [3,3] sigmatropic reaction. As an example, the degenerate Cope reaction of 1,5-hexadiene is shown (see Figures 8 and 9). The calculated activation energy is 129 kJ/mol (DFT).

Figure 8. Scheme of the degenerate cope reaction of 1,5-hexadiene.

Figure 9. Cope reaction: 3D models of reactant, transition structure, and product.

#### 5.1.2. Diels-Alder addition

(Figure 7). That means, the substituent effect on the calculated pKa value is overestimated by

Figure 7. Plot of Hammett σ parameters versus calculated relative log Ka values (DFT).

90 Density Functional Calculations - Recent Progresses of Theory and Application

For the calculation of absolute pKa values, the considerable Gibbs free solvation energy of the proton has to be taken into account (ΔG� = �1120.39 kJ/mol). The calculated Gibbs free energy of the dissociation of benzoic acid in the gas phase is ΔG� = 1446.04 kJ/mol. For the hydration (dihydrate model, vide supra), ΔG� = �47.35 kJ/mol and ΔG� = �302.76 kJ/mol are obtained for benzoic acid and benzoate anion, respectively; in total, ΔG� = 1375.80 kJ/mol. Hence, the estimate for the Gibbs free energy of the dissociation of benzoic acid in aqueous solution is ΔG� = 1446.04 � 1375.80 = 70.24 kJ/mol corresponding to a pKa value of 12.31 (exp. 4.19). To put it differently, the model applied here accounts for about 96.7% of the true Gibbs free energy of solvation.

In a pericyclic reaction, σ or π bonds change concertedly ("simultaneously") along a perimeter, i.e., a cycle. They have first been studied theoretically by Woodward and Hoffmann ("the

about a factor of 3.

5. Reaction mechanisms

5.1. Pericyclic reactions

The Diels-Alder addition is a [4 + 2] cycloaddition, a diene reacts with a dienophile to form a (substituted) cyclohexene. As an example, the Diels-Alder addition of acrylonitrile to cyclopentadiene leading to the endo-product is shown (see Figures 10 and 11). The calculated activation energy is 40 kJ/mol, and the calculated reaction energy is �132 kJ/mol (DFT).

Figure 10. Scheme of the Diels-Alder reaction between cyclopentadiene and acrylonitrile.

5.2. Nucleophilic aliphatic substitutions

29 kJ/mol.

would be 132 kJ/mol.

H

Figure 14. Scheme of an SN2 reaction.

H

H

Cl

The most important mechanisms for nucleophilic aliphatic substitutions are the single-step SN2 mechanism with backside attack of the nucleophile and a trigonal-bipyramidal transition state (for primary or secondary substrates) and the two-step SN1 mechanism with a carbenium ion intermediate (for secondary or tertiary substrates). Since these are ionic reactions, the progress

Considering first the degenerate SN2 reaction of fluoromethane with fluoride anion in the gas phase, the most stable species is a cluster of these two particles, which is formed in an exothermic reaction and calculated reaction energy �106 kJ/mol. The formation of the symmetric trigonal-bipyramidal transition state from this cluster requires an activation energy of

In the gas-phase reaction of chloromethane with fluoride anion (see Figures 14 and 15), the calculated reaction energy for the formation of fluoromethane and chloride anion at infinite distance is �198 kJ/mol. There is no activation energy for the forward reaction; the energy of the trigonal-bipyramidal transition state is lower by 19 kJ/mol than that of the cluster of chloromethane with fluoride. The most stable species is the cluster of fluoromethane with chloride anion, and the activation energy of the reverse reaction, starting with this cluster,

F Cl

<sup>+</sup> <sup>+</sup>

F Cl F

H

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H H

H H

H

A typical example for an SN1 reaction is the reaction between tert-butanol and hydrogen chloride, yielding tert-butyl chloride and water (or the reverse reaction; see Figures 16 and 17). In the gas phase, this reaction is slightly exothermic with a calculated reaction energy of �8 kJ/mol (DFT). The concurrent elimination reaction (E1), yielding isobutene, water, and hydrogen chloride, is endothermic with a calculated reaction energy of 47 kJ/mol. For the reaction to proceed, it is necessary to form the protonated alcohol. Only the formation of an ion pair of chloride and protonated tert-butanol is conceivable, requiring an estimated activation energy of about 165 kJ/ mol. The formation of a free tert-butyl carbenium ion is not feasible, because the energy required would be about 682 kJ/mol. The reaction should rather proceed by a backside attack similar to

Figure 15. SN2 reaction: 3D models of reactants, transition structure, and products.

in the gas phase may differ considerably from that in a polar solvent.

Figure 11. Diels-Alder reaction: 3D models of reactants, transition structure, and product.

#### 5.1.3. Electrocyclic reactions

In an electrocyclic reaction, an unsaturated cycloalkane is formed from a conjugated polyene, or the reverse reaction occurs. Here, only thermally allowed electrocyclic reactions will be considered. For instance, cyclobutene is opened in a conrotatory manner to form 1,3-butadiene. (The calculated activation energy is 149 kJ/mol, and the calculated reaction energy is �39 kJ/mol, assuming that the most stable conformation of 1,3-butadiene is formed.) The example shown here is the disrotatory ring closure of 1,3,5-hexatriene to form 1,3-cyclohexadiene (see Figures 12 and 13). Starting with the most stable conformer of 1,3,5-hexatriene, the calculated activation energy is 252 kJ/mol, and the calculated reaction energy is �64 kJ/mol (DFT).

Figure 12. Scheme of the electrocyclic ring closure of 1,3,5-hexatriene.

Figure 13. Electrocyclic ring closure of 1,3,5-hexatriene: 3D models of reactant (two conformations), transition structure, and product.

#### 5.2. Nucleophilic aliphatic substitutions

5.1.3. Electrocyclic reactions

(DFT).

and product.

In an electrocyclic reaction, an unsaturated cycloalkane is formed from a conjugated polyene, or the reverse reaction occurs. Here, only thermally allowed electrocyclic reactions will be considered. For instance, cyclobutene is opened in a conrotatory manner to form 1,3-butadiene. (The calculated activation energy is 149 kJ/mol, and the calculated reaction energy is �39 kJ/mol, assuming that the most stable conformation of 1,3-butadiene is formed.) The example shown here is the disrotatory ring closure of 1,3,5-hexatriene to form 1,3-cyclohexadiene (see Figures 12 and 13). Starting with the most stable conformer of 1,3,5-hexatriene, the calculated activation energy is 252 kJ/mol, and the calculated reaction energy is �64 kJ/mol

Figure 13. Electrocyclic ring closure of 1,3,5-hexatriene: 3D models of reactant (two conformations), transition structure,

Figure 11. Diels-Alder reaction: 3D models of reactants, transition structure, and product.

92 Density Functional Calculations - Recent Progresses of Theory and Application

Figure 12. Scheme of the electrocyclic ring closure of 1,3,5-hexatriene.

The most important mechanisms for nucleophilic aliphatic substitutions are the single-step SN2 mechanism with backside attack of the nucleophile and a trigonal-bipyramidal transition state (for primary or secondary substrates) and the two-step SN1 mechanism with a carbenium ion intermediate (for secondary or tertiary substrates). Since these are ionic reactions, the progress in the gas phase may differ considerably from that in a polar solvent.

Considering first the degenerate SN2 reaction of fluoromethane with fluoride anion in the gas phase, the most stable species is a cluster of these two particles, which is formed in an exothermic reaction and calculated reaction energy �106 kJ/mol. The formation of the symmetric trigonal-bipyramidal transition state from this cluster requires an activation energy of 29 kJ/mol.

In the gas-phase reaction of chloromethane with fluoride anion (see Figures 14 and 15), the calculated reaction energy for the formation of fluoromethane and chloride anion at infinite distance is �198 kJ/mol. There is no activation energy for the forward reaction; the energy of the trigonal-bipyramidal transition state is lower by 19 kJ/mol than that of the cluster of chloromethane with fluoride. The most stable species is the cluster of fluoromethane with chloride anion, and the activation energy of the reverse reaction, starting with this cluster, would be 132 kJ/mol.

Figure 15. SN2 reaction: 3D models of reactants, transition structure, and products.

A typical example for an SN1 reaction is the reaction between tert-butanol and hydrogen chloride, yielding tert-butyl chloride and water (or the reverse reaction; see Figures 16 and 17). In the gas phase, this reaction is slightly exothermic with a calculated reaction energy of �8 kJ/mol (DFT). The concurrent elimination reaction (E1), yielding isobutene, water, and hydrogen chloride, is endothermic with a calculated reaction energy of 47 kJ/mol. For the reaction to proceed, it is necessary to form the protonated alcohol. Only the formation of an ion pair of chloride and protonated tert-butanol is conceivable, requiring an estimated activation energy of about 165 kJ/ mol. The formation of a free tert-butyl carbenium ion is not feasible, because the energy required would be about 682 kJ/mol. The reaction should rather proceed by a backside attack similar to

Figure 17. SN1 reaction: 3D models of reactants, intermediate, and products.

the SN2 reaction and stop at a cluster of tert-butyl chloride and water (reaction energy 19 kJ/ mol).

A true SN1 mechanism requires an efficient solvation of the intermediate carbenium ion by polar solvent molecules.

H C l +

Figure 19. Electrophilic chlorination of benzene. Top row: first π complex, approximate first transition structure, and σ

complex. Bottom row: approximate second transition structure and second π complex (of products).

Figure 18. Scheme of the AlCl3-catalyzed electrophilic substitution of benzene by chlorine.

Cl

Cl

Al Cl Cl

Cl H

+

Cl

Al Cl Cl

Cl

Cl

+ Cl Cl + Cl

Al Cl Cl Cl Cl

Spectroscopy, Substituent Effects, and Reaction Mechanisms

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95

Cl

Al Cl Cl Cl <sup>H</sup>

Cl

Al Cl Cl

#### 5.3. Electrophilic aromatic substitutions

In gas-phase reactions of benzene with a reactive cationic electrophile such as H+ , Br<sup>+</sup> , NO2 + , or CH3CH2 <sup>+</sup> (cf. Section 4.1), the reaction proceeds without any energy barrier to the σ complex and stops there. Some kind of π complex is formed on the reaction path, but it is not a true intermediate because it is not characterized by a local energy minimum.

In a more realistic scenario, the electrophile is a less reactive complex, e.g., of a halogen, an alkyl, or an acyl chloride, with a Lewis acid such as aluminum chloride or iron(III) bromide. Now, the reaction will usually stop at the π complex stage. In order to force the reaction to proceed to the σ complex, a strongly activating substituent such as oxido (i.e., phenolate anion) was introduced. After removal or replacement of this substituent, the optimization procedure allowed the study of either the backward reaction or the forward reaction to the products, possibly after a modification of the arrangement of the reaction partners.

As an example, the chlorination of benzene catalyzed by aluminum chloride will be considered in detail (see Figures 18 and 19). The overall reaction in the gas phase, yielding chlorobenzene and hydrogen chloride, is exothermic with a calculated reaction energy of 131 kJ/mol and a Gibbs free reaction energy of ΔrG = 143 kJ/mol (DFT). The reaction involves three intermediates, π complex 1, σ complex, and π complex 2, which were calculated as local minima and two transition structures, which could not yet be identified unambiguously. The crucial energy barrier is most likely the transition state leading to the σ complex. Taking the energies of the separated reactants as reference, the relative energies are as follows: 45 kJ/mol for π complex 1, approximately 208 kJ/mol for transition state 1, 80 kJ/mol for the σ complex, approximately

Spectroscopy, Substituent Effects, and Reaction Mechanisms http://dx.doi.org/10.5772/intechopen.70751 95

Figure 18. Scheme of the AlCl3-catalyzed electrophilic substitution of benzene by chlorine.

the SN2 reaction and stop at a cluster of tert-butyl chloride and water (reaction energy 19 kJ/

CH3

Cl OH2 Cl

H3C CH3

H Cl + <sup>O</sup>

A true SN1 mechanism requires an efficient solvation of the intermediate carbenium ion by

and stops there. Some kind of π complex is formed on the reaction path, but it is not a true

In a more realistic scenario, the electrophile is a less reactive complex, e.g., of a halogen, an alkyl, or an acyl chloride, with a Lewis acid such as aluminum chloride or iron(III) bromide. Now, the reaction will usually stop at the π complex stage. In order to force the reaction to proceed to the σ complex, a strongly activating substituent such as oxido (i.e., phenolate anion) was introduced. After removal or replacement of this substituent, the optimization procedure allowed the study of either the backward reaction or the forward reaction to the products,

As an example, the chlorination of benzene catalyzed by aluminum chloride will be considered in detail (see Figures 18 and 19). The overall reaction in the gas phase, yielding chlorobenzene and hydrogen chloride, is exothermic with a calculated reaction energy of 131 kJ/mol and a Gibbs free reaction energy of ΔrG = 143 kJ/mol (DFT). The reaction involves three intermediates, π complex 1, σ complex, and π complex 2, which were calculated as local minima and two transition structures, which could not yet be identified unambiguously. The crucial energy barrier is most likely the transition state leading to the σ complex. Taking the energies of the separated reactants as reference, the relative energies are as follows: 45 kJ/mol for π complex 1, approximately 208 kJ/mol for transition state 1, 80 kJ/mol for the σ complex, approximately

<sup>+</sup> (cf. Section 4.1), the reaction proceeds without any energy barrier to the σ complex

, Br<sup>+</sup>

CH3

H

H

CH3 CH3 +

> , NO2 + , or

In gas-phase reactions of benzene with a reactive cationic electrophile such as H+

intermediate because it is not characterized by a local energy minimum.

possibly after a modification of the arrangement of the reaction partners.

mol).

CH3CH2

polar solvent molecules.

5.3. Electrophilic aromatic substitutions

OH

94 Density Functional Calculations - Recent Progresses of Theory and Application

Figure 17. SN1 reaction: 3D models of reactants, intermediate, and products.

H3C

Figure 16. Scheme of an SN1 reaction.

H3C

H3C

Figure 19. Electrophilic chlorination of benzene. Top row: first π complex, approximate first transition structure, and σ complex. Bottom row: approximate second transition structure and second π complex (of products).

27 kJ/mol for transition state 2, �169 kJ/mol for π complex 2 (global minimum), and �131 kJ/ mol for the separated products.

#### 5.4. Additions and eliminations

#### 5.4.1. Electrophilic addition

As an example, the addition of bromine to cyclohexene will be considered, yielding trans-1,2 dibromocyclohexane (see Figures 20 and 21). First, a π complex is formed (relative energy �22 kJ/mol with respect to the separated reactants) and, next, a bicyclic bromonium ion, which is more stable than the respective carbenium ion (by roughly 100 kJ/mol). Finally, the diaxial conformer of trans-1,2-dibromocyclohexane is formed, which is actually more stable than the diequatorial conformer by 7 kJ/mol. This finding is somewhat surprising and stands in contrast to previous assumptions. Apparently, electrostatic repulsion favors the diaxial form, whereas in monosubstituted cyclohexanes, the substituent prefers the equatorial position. The reaction energy is �108 kJ/mol, and the Gibbs free reaction energy is �50 kJ/mol; this is due to the unfavorable reaction entropy.

6. Conclusions and outlook

products.

H3C

H3C

H3C

A. Appendix

of orbitals involved, judging from NMR calculations.

OH

Figure 22. Scheme for the E1 elimination of water from tert-butanol.

In the field of molecular chemistry, the use of DFT in combination with efficient software and modern computer equipment allows the development of "virtual chemistry," i.e., the prediction of essentially all molecular properties and of reaction paths. To a certain extent, supramolecular chemistry is also accessible to this method; molecular clusters and microdroplets of solvents can be simulated. It stands to a reason, however, that computational time increases heavily with molecular size (or cluster size). In the case of ab initio calculations, the proportionality is to the fourth power of the size of the basis set; in DFT, the situation might be somewhat better; computational time is proportional to roughly the third power of the number

Figure 23. Elimination of water from tert-butanol: 3D models of reactants, approximate transition state, and π complex of

H3C CH3

H H H

+ H Cl + <sup>O</sup>

Cl

OH2

H

97

H

H3C CH3

Cl H

http://dx.doi.org/10.5772/intechopen.70751

Spectroscopy, Substituent Effects, and Reaction Mechanisms

It should be pointed out that present-day DFT is only an approximate theory. Therefore, it is

The following conversion factors have been used in this study: 1 Hartree (a.u.) = 2625.50 kJ/

Computational details. For all computations in Chapters 2 to 4, Gaussian 09 was used B3LYP/ 6–311(d,p) [1]. For most computations in Chapter 5, deMon2k was used [16]. For the

necessary to check the quality of the computational results against experimental data.

mol, 1 cal = 4.184 J, and pK = ΔG [kJ/mol] /5.708008 (at T = 298.15 K).

Figure 20. Scheme for the electrophilic addition of bromine to cyclohexene.

Figure 21. Electrophilic addition of bromine to cyclohexene: 3D models of reactants, π complex, bromonium ion, and product.

#### 5.4.2. Elimination

In Section 5.2, it was already briefly mentioned that the reaction between tert-butanol and hydrogen chloride might proceed as an elimination instead of a substitution. The mechanism is E1, and isobutene (2-methylpropene) is formed as product (see Figures 22 and 23).

Figure 22. Scheme for the E1 elimination of water from tert-butanol.

Figure 23. Elimination of water from tert-butanol: 3D models of reactants, approximate transition state, and π complex of products.

#### 6. Conclusions and outlook

27 kJ/mol for transition state 2, �169 kJ/mol for π complex 2 (global minimum), and �131 kJ/

As an example, the addition of bromine to cyclohexene will be considered, yielding trans-1,2 dibromocyclohexane (see Figures 20 and 21). First, a π complex is formed (relative energy �22 kJ/mol with respect to the separated reactants) and, next, a bicyclic bromonium ion, which is more stable than the respective carbenium ion (by roughly 100 kJ/mol). Finally, the diaxial conformer of trans-1,2-dibromocyclohexane is formed, which is actually more stable than the diequatorial conformer by 7 kJ/mol. This finding is somewhat surprising and stands in contrast to previous assumptions. Apparently, electrostatic repulsion favors the diaxial form, whereas in monosubstituted cyclohexanes, the substituent prefers the equatorial position. The reaction energy is �108 kJ/mol, and the Gibbs free reaction energy is �50 kJ/mol; this is due to the

In Section 5.2, it was already briefly mentioned that the reaction between tert-butanol and hydrogen chloride might proceed as an elimination instead of a substitution. The mechanism

Figure 21. Electrophilic addition of bromine to cyclohexene: 3D models of reactants, π complex, bromonium ion, and

+

Br

Br

Br Br

is E1, and isobutene (2-methylpropene) is formed as product (see Figures 22 and 23).

Br

Br

Figure 20. Scheme for the electrophilic addition of bromine to cyclohexene.

mol for the separated products.

96 Density Functional Calculations - Recent Progresses of Theory and Application

5.4. Additions and eliminations

5.4.1. Electrophilic addition

unfavorable reaction entropy.

+

Br

Br

5.4.2. Elimination

product.

In the field of molecular chemistry, the use of DFT in combination with efficient software and modern computer equipment allows the development of "virtual chemistry," i.e., the prediction of essentially all molecular properties and of reaction paths. To a certain extent, supramolecular chemistry is also accessible to this method; molecular clusters and microdroplets of solvents can be simulated. It stands to a reason, however, that computational time increases heavily with molecular size (or cluster size). In the case of ab initio calculations, the proportionality is to the fourth power of the size of the basis set; in DFT, the situation might be somewhat better; computational time is proportional to roughly the third power of the number of orbitals involved, judging from NMR calculations.

It should be pointed out that present-day DFT is only an approximate theory. Therefore, it is necessary to check the quality of the computational results against experimental data.

#### A. Appendix

The following conversion factors have been used in this study: 1 Hartree (a.u.) = 2625.50 kJ/ mol, 1 cal = 4.184 J, and pK = ΔG [kJ/mol] /5.708008 (at T = 298.15 K).

Computational details. For all computations in Chapters 2 to 4, Gaussian 09 was used B3LYP/ 6–311(d,p) [1]. For most computations in Chapter 5, deMon2k was used [16]. For the preparation of input files, i.e., the generation of Z matrices, and the visualization of the results, molden [2] was of great help.

[3] Weast RC, editor. Handbook of Chemistry and Physics. 57th ed. Cleveland, Ohio: CRC

Spectroscopy, Substituent Effects, and Reaction Mechanisms

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99

[6] Kaupp M, Bühl M, Malkin VG, editors. Calculation of NMR and EPR Parameters. Weinheim:

[7] Kirste B. 3D Structure Determination of Natural Products by 13C–NMR-Controlled DFT Calculations [Internet]. 2016. Available from: http://kirste.userpage.fu-berlin.de/chemis-

[9] Kirste B. DFT calculations of hyperfine coupling constants of organic π radicals and comparison with empirical equations and experiment. Magnetic Resonance in Chemistry.

[10] Gerson F, Huber W. Electron Spin Resonance Spectroscopy of Organic Radicals. Weinheim:

[11] Kirste B. Untersuchung paramagnetischer organischer Verbindungen in flüssigen Kristallen und in Festkörpern mit der magnetischen Resonanz (EPR/ENDOR) [habilitation thesis]. Berlin; 1985. Available from: http://kirste.userpage.fu-berlin.de/ag/kirste/pdf/

[12] Möbius K, Savitsky A. High-Field EPR Spectroscopy on Proteins and their Model Sys-

[13] Kirste B. Applications of density functional theory to theoretical organic chemistry.

[14] Smith MB, March J. March's Advanced Organic Chemistry. 6th ed. Hoboken, New Jersey:

[15] Woodward RB, Hoffmann R. The conservation of orbital symmetry. Angewandte Chemie

[16] Koster AM, Calaminici P, Casida ME, Dominguez VD, Flores-Moreno, et al. deMon2k, Version 2. Cinvestav, Mexico City: The deMon Developers; 2006 http://www.demon-

Chemical Sciences Journal. 2016;7:127. DOI: 10.4172/2150-3494.1000127

(International Ed. in English). 1969;8:781-853. DOI: 10.1002/anie.196907811

[4] NIST Chemistry WebBook. http://webbook.nist.gov/chemistry/

try/nmr/nmrdft/index.html [Accessed: 2017-07-03]

[8] NMRShiftDB. https://nmrshiftdb.nmr.uni-koeln.de/

2016;54:835-841. DOI: 10.1002/mrc.4467

bkihabil.pdf Accessed: 2017–07-03

tems. Cambridge: RSC Publishing; 2009

[5] Spectral Database for Organic Compounds, SDBS. http://sdbs.db.aist.go.jp/

Press; 1976

Wiley-VCH; 2004

Wiley-VCH; 2003

Wiley; 2007

software.com/

Sample Gaussian input file: methane

%Chk = methane #B3LYP/6–311G(d,p) Opt Methane 0 1 c h 1 hc2 h 1 hc2 2 hch3 h 1 hc2 3 hch3 2 dih4 h 1 hc2 4 hch3 3 dih5 hc2 1.089 hch3 109.47 dih4 �120.0 dih5 120.0

(The first line specifies the checkpoint file; the second the method and the task, in this case the geometry optimization; the fourth the title, the sixth the total charge, here 0; and the multiplicity, usually 1. Then, the Z matrix follows immediately.)

#### Author details

Burkhard Kirste

Address all correspondence to: kirste@chemie.fu-berlin.de

Institute of Chemistry and Biochemistry, Freie Universität Berlin, Germany

#### References


preparation of input files, i.e., the generation of Z matrices, and the visualization of the results,

(The first line specifies the checkpoint file; the second the method and the task, in this case the geometry optimization; the fourth the title, the sixth the total charge, here 0; and the multiplic-

[1] Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, et al. Gaussian 09. Walling-

[2] Schaftenaar G. CMBI. The Netherlands: Molden http://www.cmbi.ru.nl/molden/

ity, usually 1. Then, the Z matrix follows immediately.)

Address all correspondence to: kirste@chemie.fu-berlin.de

Institute of Chemistry and Biochemistry, Freie Universität Berlin, Germany

ford CT: Gaussian Inc.; 2013 http://www.gaussian.com/

Author details

Burkhard Kirste

References

molden [2] was of great help.

%Chk = methane

Methane

c

0 1

h 1 hc2

hc2 1.089 hch3 109.47 dih4 �120.0 dih5 120.0

#B3LYP/6–311G(d,p) Opt

h 1 hc2 2 hch3

h 1 hc2 3 hch3 2 dih4 h 1 hc2 4 hch3 3 dih5

Sample Gaussian input file: methane

98 Density Functional Calculations - Recent Progresses of Theory and Application


**Chapter 5**

Provisional chapter

**Spectral Calculations with DFT**

Spectral Calculations with DFT

Ataf Ali Altaf, Samia Kausar and Amin Badshah

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Spectra calculations are an important branch of theoretical modeling, and due to the significant improvements of high-level computational methods, the calculated spectra can be used directly and sometimes help to correct the errors of experimental observations. On the other hand, theoretical computations assist the experimental assignments. The authors discuss three spectral calculations (UV-Vis, IR and NMR) that are the most widely used. UV-Visible spectrum can be carried out employing time-dependent density functional theory (TDDFT) with B3LYP/631G(d,p) and CAM-B3LYP functional method to illustrate the characteristics of vertical electronic excitations. The vibrational spectra can be generated from a list of frequencies and intensities using a Gaussian broadening function method. NMR chemical shifts can be calculated by density functional theory individual gauge for localized orbitals (DFTIGLO) method and by gauge including

DOI: 10.5772/intechopen.71080

Keywords: spectral calculation, vibrational spectra, DFT NMR calculation, TDDFT for

Early days of quantum chemistry go back to Thomas-Fermi and Thomas-Fermi-Dirac models of the electronic structure of atoms, which gave the concept of articulating some parts or all of the molecular energy as a functional of the electron density, and then comes the traditional Hartree-Fock (HF) theory [1]. HF theory was a simplest ab initio technique and among the first principles of quantum-chemical theories, being attained directly from the Schrodingerwave equation that did not incorporate any pragmatic contemplations. Although such theories and techniques proved beneficial, it was density functional theory (DFT) that laid a demanding theoretical foundation in 1964 by an outstanding result established by Hohenberg

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

http://dx.doi.org/10.5772/intechopen.71080

atomic orbitals (GIAO) approach.

UV-visible spectra

1. Introduction

and Kohn [2].

Ataf Ali Altaf, Samia Kausar and

Amin Badshah

Abstract

**Chapter 5** Provisional chapter

#### **Spectral Calculations with DFT** Spectral Calculations with DFT

Ataf Ali Altaf, Samia Kausar and Amin Badshah Ataf Ali Altaf, Samia Kausar and

Additional information is available at the end of the chapter Amin Badshah

http://dx.doi.org/10.5772/intechopen.71080 Additional information is available at the end of the chapter

#### Abstract

Spectra calculations are an important branch of theoretical modeling, and due to the significant improvements of high-level computational methods, the calculated spectra can be used directly and sometimes help to correct the errors of experimental observations. On the other hand, theoretical computations assist the experimental assignments. The authors discuss three spectral calculations (UV-Vis, IR and NMR) that are the most widely used. UV-Visible spectrum can be carried out employing time-dependent density functional theory (TDDFT) with B3LYP/631G(d,p) and CAM-B3LYP functional method to illustrate the characteristics of vertical electronic excitations. The vibrational spectra can be generated from a list of frequencies and intensities using a Gaussian broadening function method. NMR chemical shifts can be calculated by density functional theory individual gauge for localized orbitals (DFTIGLO) method and by gauge including atomic orbitals (GIAO) approach.

DOI: 10.5772/intechopen.71080

Keywords: spectral calculation, vibrational spectra, DFT NMR calculation, TDDFT for UV-visible spectra

#### 1. Introduction

Early days of quantum chemistry go back to Thomas-Fermi and Thomas-Fermi-Dirac models of the electronic structure of atoms, which gave the concept of articulating some parts or all of the molecular energy as a functional of the electron density, and then comes the traditional Hartree-Fock (HF) theory [1]. HF theory was a simplest ab initio technique and among the first principles of quantum-chemical theories, being attained directly from the Schrodingerwave equation that did not incorporate any pragmatic contemplations. Although such theories and techniques proved beneficial, it was density functional theory (DFT) that laid a demanding theoretical foundation in 1964 by an outstanding result established by Hohenberg and Kohn [2].

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

The Hohenberg-Kohn theorems state that the exact ground state energy of the system is produced by a unique functional of the electron density r. The first theorem demonstrates a many-particle system n(x,y,z) for which one-to-one plotting happens between the ground state electric density and the ground state wave function. The second theorem verifies that total electric energy of the system E[n(x,y,z)] is minimalized by the ground state density [3]. The work of Kohn and Sham instantly surveyed and paved the way for hands-on computational applicability of DFT to physical systems, by linking a reference-state comprising of a set of noninteracting one-particle orbitals with a particular functional. The reference orbitals determined by a set of operative one-particle Schrodinger-wave equations are called Kohn-Sham (KS) equations [4]; one of them is depicted in Eq. (1).

$$\left(-\frac{1}{2}\nabla^2\left(\overrightarrow{r}\right) + V\_{KS}\right)\Psi\_i = \varepsilon\_i \Psi\_i \tag{1}$$

and biochemical systems. It is obvious that local geometrical and electronic structures influence magnetic resonance parameters, i.e., shielding tensors, nuclear spin-spin coupling parameters, hyperfine tensors and g-tensors. For speculations of NMR parameters, a practical approximation of the real system is depicted by a model system in which 50–100 atoms are treated with ab initio techniques. Over the last years, DFT advanced by the capability to include effects of electron correlation in a very effectual means to the forefront of field of

Spectral Calculations with DFT

103

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Competent calculations of excited states properties are emerging field of interest for quantum chemists; hence, they are developing interesting solutions for these properties. Because of advancement of computations based on the time-dependent density functional theory (TD-DFT) [10] in recent years, calculations of electronic structures in the excited states have become a motivation of interest. A commanding method of carrying out exact quantum mechanical calculations of the intervalence absorption spectrum is provided by the time-dependent concept of electronic spectroscopy. A corporeal picture of the effects of the coupling of electronic and nuclear motions became available by employing calculations in the time realm due to reason that the time development of the wave-packet can be tracked and inferred as well [11]. Plentiful successes of this technique have been recently revised as electronic spectra are often calculated with TD-DFT. Vertical excitation energies, absorption wavelength and oscillator

The influence of vibrational spectroscopy as an analytical tool in several fields is obvious. Qualitative association of bands and specific structures or chemical groups is basic interpretation of vibrational spectra. In contrast to nuclear magnetic resonance where nuclear spin is associated with one peak or multiplet is proved advantageous, united motion of all of the nuclei in sample is reason for observed bands in vibrational spectra. For N nuclei, there are at most 3 N-6 experiential fundamental bands and so far the matrix of internuclear force interactions, i.e., the second derivative matrix in harmonic approximation, has (3 N-6)(3 N -5)/2 exclusive terms. The extraction of force constants from vibrational frequencies is yet an undetermined mathematical problem as many bands are not actually observed; e.g., there are overtones, combination bands and nonconformities from the harmonic approximation; hence,

For the ground state's potential energy surface computation, discovery of effectual codes delivers a possible answer to the problem. A technique that will precisely determine the bonding, as well as intermolecular interactions, will be helpful for calculating vibrational spectra. The ground state properties and potential energies are accurately calculated by DFT approaches; therefore, DFT excellently and proficiently calculates the vibrational spectra from first principles [7]. The absolute values of the frequencies are high in contrast to experimental values while using DFT models as they seem to characterize the bonding pretty well and comparison with experimental trends is possible. DFT methods give sufficiently high accuracy of normal mode calculations as the restrictions of the harmonic approximation are often a foremost cause of divergence between theory and experiment.

calculating NMR parameters [8, 9].

strength calculations can be performed by this technique [12].

2.1. Theoretical vibrational spectra analysis

the problem seems obstinate [13].

where VKS is a local one-body potential defined as total density of the noninteracting system and is the same as the density of a real system.

#### 2. Spectral implementations of DFT

The density functional theory (DFT) has become a powerful tool in computational chemistry owing to its usefulness. Spectroscopic analysis of chemical entities by this technique emerged as commanding implementation. Prediction of frequencies and spectral intensities by DFT calculations are indispensable nowadays for interpreting the experimental spectra of complex molecules. Much advancement has been made in the past decade to design new DFT approaches that can be employed into available quantum-chemical computational programs. For calculation of the molecular and electronic structures of ground-state systems and various spectral parameters related to NMR, ESR, UV-Vis and IR, various density functional practices are available now. Functionals available today can strive with best previous ab initio methods [5].

For experimental spectroscopists, theoretical computation of vibrational frequencies has become practically essential these days as specifically in problematic and uncertain cases it assists to assign and interpret experimental infrared/Raman spectra. Previously, the HF method was used in many studies to calculate vibrational frequencies, but it has long been identified that this approach miscalculates these frequencies even occasionally to a disturbing degree. Inadequate handling of electron correlation and anharmonicity of the vibrations are found to be the main contributions to error [6]. DFT largely overcome the errors as theoretical computations helped to interpret experimental conclusions depicting that using DFT these theoretical data can nearly be attained at the harmonic level. Calculation of optimized geometry, IR intensities, vibrational frequencies and Raman scattering activities can be done by employing different density functional approaches [7].

Taking NMR spectroscopy into consideration, DFT-based new methods are developed that are appropriate for the satisfactory hypothetical interpretation of NMR spectra of various chemical and biochemical systems. It is obvious that local geometrical and electronic structures influence magnetic resonance parameters, i.e., shielding tensors, nuclear spin-spin coupling parameters, hyperfine tensors and g-tensors. For speculations of NMR parameters, a practical approximation of the real system is depicted by a model system in which 50–100 atoms are treated with ab initio techniques. Over the last years, DFT advanced by the capability to include effects of electron correlation in a very effectual means to the forefront of field of calculating NMR parameters [8, 9].

Competent calculations of excited states properties are emerging field of interest for quantum chemists; hence, they are developing interesting solutions for these properties. Because of advancement of computations based on the time-dependent density functional theory (TD-DFT) [10] in recent years, calculations of electronic structures in the excited states have become a motivation of interest. A commanding method of carrying out exact quantum mechanical calculations of the intervalence absorption spectrum is provided by the time-dependent concept of electronic spectroscopy. A corporeal picture of the effects of the coupling of electronic and nuclear motions became available by employing calculations in the time realm due to reason that the time development of the wave-packet can be tracked and inferred as well [11]. Plentiful successes of this technique have been recently revised as electronic spectra are often calculated with TD-DFT. Vertical excitation energies, absorption wavelength and oscillator strength calculations can be performed by this technique [12].

#### 2.1. Theoretical vibrational spectra analysis

The Hohenberg-Kohn theorems state that the exact ground state energy of the system is produced by a unique functional of the electron density r. The first theorem demonstrates a many-particle system n(x,y,z) for which one-to-one plotting happens between the ground state electric density and the ground state wave function. The second theorem verifies that total electric energy of the system E[n(x,y,z)] is minimalized by the ground state density [3]. The work of Kohn and Sham instantly surveyed and paved the way for hands-on computational applicability of DFT to physical systems, by linking a reference-state comprising of a set of noninteracting one-particle orbitals with a particular functional. The reference orbitals determined by a set of operative one-particle Schrodinger-wave equations are called Kohn-Sham

þ VKS

where VKS is a local one-body potential defined as total density of the noninteracting system

The density functional theory (DFT) has become a powerful tool in computational chemistry owing to its usefulness. Spectroscopic analysis of chemical entities by this technique emerged as commanding implementation. Prediction of frequencies and spectral intensities by DFT calculations are indispensable nowadays for interpreting the experimental spectra of complex molecules. Much advancement has been made in the past decade to design new DFT approaches that can be employed into available quantum-chemical computational programs. For calculation of the molecular and electronic structures of ground-state systems and various spectral parameters related to NMR, ESR, UV-Vis and IR, various density functional practices are available now. Functionals available today can strive with best previous ab

For experimental spectroscopists, theoretical computation of vibrational frequencies has become practically essential these days as specifically in problematic and uncertain cases it assists to assign and interpret experimental infrared/Raman spectra. Previously, the HF method was used in many studies to calculate vibrational frequencies, but it has long been identified that this approach miscalculates these frequencies even occasionally to a disturbing degree. Inadequate handling of electron correlation and anharmonicity of the vibrations are found to be the main contributions to error [6]. DFT largely overcome the errors as theoretical computations helped to interpret experimental conclusions depicting that using DFT these theoretical data can nearly be attained at the harmonic level. Calculation of optimized geometry, IR intensities, vibrational frequencies and Raman scattering activities can be done by

Taking NMR spectroscopy into consideration, DFT-based new methods are developed that are appropriate for the satisfactory hypothetical interpretation of NMR spectra of various chemical

Ψ<sup>i</sup> ¼ εiΨ<sup>i</sup> (1)

(KS) equations [4]; one of them is depicted in Eq. (1).

102 Density Functional Calculations - Recent Progresses of Theory and Application

and is the same as the density of a real system.

2. Spectral implementations of DFT

employing different density functional approaches [7].

initio methods [5].

� 1 2 ∇<sup>2</sup> r !

> The influence of vibrational spectroscopy as an analytical tool in several fields is obvious. Qualitative association of bands and specific structures or chemical groups is basic interpretation of vibrational spectra. In contrast to nuclear magnetic resonance where nuclear spin is associated with one peak or multiplet is proved advantageous, united motion of all of the nuclei in sample is reason for observed bands in vibrational spectra. For N nuclei, there are at most 3 N-6 experiential fundamental bands and so far the matrix of internuclear force interactions, i.e., the second derivative matrix in harmonic approximation, has (3 N-6)(3 N -5)/2 exclusive terms. The extraction of force constants from vibrational frequencies is yet an undetermined mathematical problem as many bands are not actually observed; e.g., there are overtones, combination bands and nonconformities from the harmonic approximation; hence, the problem seems obstinate [13].

> For the ground state's potential energy surface computation, discovery of effectual codes delivers a possible answer to the problem. A technique that will precisely determine the bonding, as well as intermolecular interactions, will be helpful for calculating vibrational spectra. The ground state properties and potential energies are accurately calculated by DFT approaches; therefore, DFT excellently and proficiently calculates the vibrational spectra from first principles [7]. The absolute values of the frequencies are high in contrast to experimental values while using DFT models as they seem to characterize the bonding pretty well and comparison with experimental trends is possible. DFT methods give sufficiently high accuracy of normal mode calculations as the restrictions of the harmonic approximation are often a foremost cause of divergence between theory and experiment.

It is reasonable to outspread the approach by including molecular interactions after given achievement of the method employed to small molecules. There are two aspects that can be reasons for inconsistency, i.e., anharmonicity and hydrogen bonding, both result in a lower frequency for a specific normal mode than projected by harmonic approximation [14].

#### 2.1.1. Spectral generation via Gaussian broadening function

Computation of vibrational spectra of molecules in their ground and excited states can be done by using the Gaussian program. The program can also designate the dislocations of the molecule as it undergoes normal modes of vibrations along with prediction of spectral frequencies and intensities. The spectra can be produced from a list of frequencies and intensities employing a Gaussian broadening function as depicted in Eq. (2) [13]:

$$I(\nu) = \sum\_{k}^{N} \frac{A\_k}{\sqrt{2\pi\sigma}} \exp\left\{-\frac{(\upsilon - \upsilon\_0)^2}{2\sigma^2}\right\} \tag{2}$$

<sup>H</sup><sup>b</sup> el <sup>þ</sup> VNN � �Ψel <sup>¼</sup> <sup>U</sup>Ψel (4)

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i, j ¼ 1, 2,…::3N (5)

(6)

where Hb el is the one-electron Hamiltonian and VNN is the operator of potential energy between the nuclei of molecular system intended for numerous molecular geometries to discover

The second derivative of energy regarding the position of nuclei is factor on which molecular frequencies hinge on. Hartree-Fock theory, DFT-B3LYP and other approaches make available the analytic second derivative of energy. Input for frequency calculation is given by the optimized energy [16]. These derivatives are estimated at the equilibrium geometry with origin at the middle of mass by manipulating a set of second derivatives of molecular energy U respecting the 3 N nuclear Cartesian coordinates of a coordinate system as in Eq. (5) [15].

equilibrium geometry of the molecule, i.e., geometry optimization.

∂<sup>2</sup>U ∂Xi∂Xj � �

> Fy <sup>¼</sup> <sup>1</sup> mimj � �<sup>1</sup>=<sup>2</sup>

Then, set of 3 N linear equations in 3 N unknowns in Eq. (7) are being solved.

If coefficient of determinant vanishes, this set of consistent equations has a nontrivial explana-

This determinant is of order 3 N and on extension gives a polynomial whose highest power of λ<sup>k</sup>

vk <sup>¼</sup> <sup>λ</sup><sup>1</sup>=<sup>2</sup> k

By solving Eq. (9), six of the λ<sup>k</sup> values originated will be zero giving six frequencies with zero value, in correspondence to the three translational and three rotational degrees of freedom of

<sup>k</sup> , so the determinantal equation will yield 3 N roots (some of which may be the same) for λk.

det Fij � δijλ<sup>k</sup>

Calculation of molecular harmonic vibrational frequencies is now done as follows [15]:

Fij � δijλ<sup>k</sup>

where mi is mass of the nucleus in correspondence to coordinate Xi.

X 3N

j¼1

2.1.2.2. Calculating vibrational frequencies

tion as follows:

is λ<sup>3</sup><sup>N</sup>

Mass-weighted force constant matrix elements give:

eq

∂<sup>2</sup>U ∂Xi∂Xj � �

eq

� �Ijk <sup>¼</sup> <sup>0</sup> i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …::3<sup>N</sup> (7)

� � <sup>¼</sup> <sup>0</sup> (8)

<sup>2</sup><sup>π</sup> (9)

For each of the N vibrational modes calculated, intensity of each band is Ak in km/mol.

#### 2.1.2. Calculating the vibrational frequencies

Study of a molecular system behind calculating the vibrational frequencies [7]. If springs are considered Hookean, for example, equations of motion can readily be solved when force is proportional to the displacement and we can find that vibrational frequencies are associated with force constants and masses of atoms. For instance, in a simple molecule like CO where there is only one spring, the frequency is calculated as in Eq. (3).

$$\nu = \frac{1}{2\pi} \sqrt{k/\mu} \tag{3}$$

where <sup>1</sup> <sup>μ</sup> <sup>¼</sup> <sup>1</sup> mc <sup>þ</sup> <sup>1</sup> ma and k is the spring constant. The value of k can be computed from DFT calculations at equilibrium state, bond length <sup>k</sup> <sup>¼</sup> <sup>∂</sup>2<sup>E</sup> <sup>∂</sup>x<sup>2</sup> . It is essential to perform geometry optimization prior to frequency calculations as these calculations are effective only at fixed points on the potential energy surface.

#### 2.1.2.1. Calculating second derivative of energy

Before calculating the second derivative of energy, the first step in quantum chemical calculations is optimization of molecular geometry. Vibrational frequency calculations are done after optimizing geometry by using the same method and basis set as used for calculating vibrational frequency. Model's development is constructed on spontaneously rational values for interbond angles, bond distances and dihedral angles in the absence of experimental data. It is usually done by solving the following Schrodinger wave equation [15]:

$$\left(\hat{H}\_{el} + V\_{\text{NN}}\right)\Psi\_{el} = \mathcal{U}\Psi\_{el} \tag{4}$$

where Hb el is the one-electron Hamiltonian and VNN is the operator of potential energy between the nuclei of molecular system intended for numerous molecular geometries to discover equilibrium geometry of the molecule, i.e., geometry optimization.

The second derivative of energy regarding the position of nuclei is factor on which molecular frequencies hinge on. Hartree-Fock theory, DFT-B3LYP and other approaches make available the analytic second derivative of energy. Input for frequency calculation is given by the optimized energy [16]. These derivatives are estimated at the equilibrium geometry with origin at the middle of mass by manipulating a set of second derivatives of molecular energy U respecting the 3 N nuclear Cartesian coordinates of a coordinate system as in Eq. (5) [15].

$$\left(\frac{\partial^2 U}{\partial \mathbf{X}\_i \partial \mathbf{X}\_j}\right)\_{eq} \text{ i, j = 1, 2, ..., 3N} \tag{5}$$

Mass-weighted force constant matrix elements give:

It is reasonable to outspread the approach by including molecular interactions after given achievement of the method employed to small molecules. There are two aspects that can be reasons for inconsistency, i.e., anharmonicity and hydrogen bonding, both result in a lower frequency for a specific normal mode than projected by harmonic approxima-

Computation of vibrational spectra of molecules in their ground and excited states can be done by using the Gaussian program. The program can also designate the dislocations of the molecule as it undergoes normal modes of vibrations along with prediction of spectral frequencies and intensities. The spectra can be produced from a list of frequencies and intensities employing a Gaussian broadening function as depicted in

<sup>2</sup>πσ <sup>p</sup> exp � ð Þ <sup>v</sup> � <sup>v</sup><sup>0</sup>

2

(2)

(3)

2σ<sup>2</sup> ( )

2.1.1. Spectral generation via Gaussian broadening function

104 Density Functional Calculations - Recent Progresses of Theory and Application

2.1.2. Calculating the vibrational frequencies

points on the potential energy surface.

2.1.2.1. Calculating second derivative of energy

<sup>I</sup>ð Þ¼ <sup>ν</sup> <sup>X</sup> N

there is only one spring, the frequency is calculated as in Eq. (3).

usually done by solving the following Schrodinger wave equation [15]:

calculations at equilibrium state, bond length <sup>k</sup> <sup>¼</sup> <sup>∂</sup>2<sup>E</sup>

k

Ak ffiffiffiffiffiffiffiffi

For each of the N vibrational modes calculated, intensity of each band is Ak in km/mol.

<sup>ν</sup> <sup>¼</sup> <sup>1</sup> 2π

optimization prior to frequency calculations as these calculations are effective only at fixed

Before calculating the second derivative of energy, the first step in quantum chemical calculations is optimization of molecular geometry. Vibrational frequency calculations are done after optimizing geometry by using the same method and basis set as used for calculating vibrational frequency. Model's development is constructed on spontaneously rational values for interbond angles, bond distances and dihedral angles in the absence of experimental data. It is

Study of a molecular system behind calculating the vibrational frequencies [7]. If springs are considered Hookean, for example, equations of motion can readily be solved when force is proportional to the displacement and we can find that vibrational frequencies are associated with force constants and masses of atoms. For instance, in a simple molecule like CO where

> ffiffiffiffiffiffiffiffi k=μ q

ma and k is the spring constant. The value of k can be computed from DFT

<sup>∂</sup>x<sup>2</sup> . It is essential to perform geometry

tion [14].

Eq. (2) [13]:

where <sup>1</sup>

<sup>μ</sup> <sup>¼</sup> <sup>1</sup> mc <sup>þ</sup> <sup>1</sup>

$$F\_{\mathcal{Y}} = \frac{1}{\left(m\_i m\_j\right)^{1/2}} \left(\frac{\partial^2 U}{\partial X\_i \partial X\_j}\right)\_{eq} \tag{6}$$

where mi is mass of the nucleus in correspondence to coordinate Xi.

Then, set of 3 N linear equations in 3 N unknowns in Eq. (7) are being solved.

$$\sum\_{j=1}^{3N} \left( F\_{i\bar{j}} - \delta\_{i\bar{j}} \lambda\_k \right) I\_{\bar{j}k} = 0 \quad i, j = 1, 2, \dots \\ \text{3N} \tag{7}$$

If coefficient of determinant vanishes, this set of consistent equations has a nontrivial explanation as follows:

$$\det\left(F\_{\vec{\eta}} - \delta\_{\vec{\eta}}\lambda\_k\right) = 0 \tag{8}$$

This determinant is of order 3 N and on extension gives a polynomial whose highest power of λ<sup>k</sup> is λ<sup>3</sup><sup>N</sup> <sup>k</sup> , so the determinantal equation will yield 3 N roots (some of which may be the same) for λk.

#### 2.1.2.2. Calculating vibrational frequencies

Calculation of molecular harmonic vibrational frequencies is now done as follows [15]:

$$
\sigma\_k = \frac{\lambda\_k^{1/2}}{2\pi} \tag{9}
$$

By solving Eq. (9), six of the λ<sup>k</sup> values originated will be zero giving six frequencies with zero value, in correspondence to the three translational and three rotational degrees of freedom of molecule. In practice, one may find six vibrational frequencies with values almost zero because equilibrium geometry never originates with infinite precision. The remaining 36-N vibrational frequencies depict molecular-harmonic vibrational frequencies.

#### 2.1.2.3. Zero-point energy calculations for maximum vibrations

Zero-point energy is defined as the sum over all the vibrational modes for a molecule with maximum number of vibrational modes and calculated as:

$$E\_{\rm ZPE} = \sum\_{i} \frac{1}{2} \hbar v\_i \tag{10}$$

ii. In Method section, select ground state (HF, Restricted or DFT Restricted and B3LYP according to required calculation) and select a basis set, Charge-0 and Spin-Singlet. Insert SCF = Tight in the Additional Keywords section and if your calculation is in vacuo, select

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iii. Open the output file of the optimized structure in Gauss View. Verify that the optimization calculation has converged by checking that the maximum force, RMS force, maxi-

iv. Go to Results and select the 'Vibrations' option. View the IR spectrum by pressing the 'Spectrum' button and check out whether the number of modes are in accordance with theoretical calculations or not. Each vibrational mode can be visualized by highlighting it

NMR is based on the principle that the energy of a system containing nuclear or electron magnetic moments arising from the spin of a particle, in the existence of an external stationary magnetic field, depends upon the direction of the magnetic moment with respect to the external field. One can thus measure the energy difference for different directions of the electronic magnetic moment (electronic Zeeman effect) or of nuclear magnetic moment (nuclear Zeeman effect) by employing a suitable oscillating external magnetic field as a

NMR parameters are considered as quantities which are determined shifts, indirect spin-spin coupling constant and direct dipole-dipole coupling constants. This electronic structure in turn confidentially interrelated to local and global geometry and hence internal flexibility and intramolecular interactions influences NMR parameters. Chemical shifts along with spin-spin coupling constants found an average experimentally in comparison to those values belonging to all geometrical arrangements arising throughout the sequence of NMR experiment. Unfortunately, chemical shifts are generally found to be dependent on the internal dynamics or on the intermolecular interactions in various descriptions. Consequently, for obtaining structural information, most experimental NMR techniques imply coupling constants or the nuclear Overhauser effect instead of chemical shifts. However, ab initio calculations provide the understanding to structure-chemical shifts or spin-spin coupling constant associations, which can make the experimental data interpretation much easier in this sense. That is the reason that highly capable and steadfast computational calculations

As commonly noted that, when results of sufficient quality are attainable by employing various strategies based on wave function properties. These procedures are restricted to small- and medium-sized systems unfortunately. With the advancements of density functional methods, it is possible to acquire pertinent results even for larger molecules, such as fragments of proteins and nucleic acids [22] where the electron correlation effects are

mum displacement and RMS displacement parameters are all converged.

'None' in the Solvation dialog box. Submit the calculation.

in the 'Display Vibrations' table and pressing the 'Start' button.

3. DFT methods for nuclear magnetic resonance (NMR)

probe [20].

are in high demand [21].

#### 2.1.3. Normal coordinate analysis

Normal coordinate analysis gives comprehensive explanation of vibrational modes. It is termed a practice that calculates the vibrational frequencies involving observed frequencies of more preferably infrared and Raman harmonic frequencies to equilibrium geometry, force constants and atomic masses of an oscillating system. In assigning vibrational spectra, normal coordinate analysis proved beneficial, but reliable intramolecular force constants influence its predictive ability [17].

#### 2.1.4. B3LYP density functional method

DFT syndicates accuracy with computational rapidity and user-friendliness for investigating the ground state characteristics in sturdily bound systems. Consistent highly reliable hybrid DFT methods (discussed in Section 3.1) make them more proficient and endearing. B3LYP functional [18] (discussed in hybrid methods of Section 3.1) is the most extensively used among all hybrid density functional methods as it is considered to give most exact vibrational frequencies of compounds only if calculated frequencies are scaled by a uniform scaling factor. Scaling is done to compensate for all probable causes of inaccuracy produced owing to electronic structure method–related inaccuracies, e.g., basis set insufficiencies and estimated handling of electron-correlation and nuclear motion treatment inaccuracies [19].

#### 2.1.5. IR calculations in Gaussian software

Gaussian is a most employed computational chemistry software program, whereas Gauss View is an inexpensive full featured graphical user interface for Gaussian. One can submit inputs to Gaussian and can observe output graphically, which is usually produced by Gaussian software via using Gauss View. For IR spectrum calculations in Gaussian, the following are the steps:

i. After predicting the number of vibrational modes and expected regions for frequencies for molecule by theoretical calculations, build molecule in Gauss View. Go to Calculate from Gauss View toolbar and select Gaussian. In the Job Type dialog box, select Opt + Freq and optimize to a 'Minimum' Calculate Force Constants-'Never,' Compute Raman- 'Default,' deselect any other option.


#### 3. DFT methods for nuclear magnetic resonance (NMR)

molecule. In practice, one may find six vibrational frequencies with values almost zero because equilibrium geometry never originates with infinite precision. The remaining 36-N vibrational

Zero-point energy is defined as the sum over all the vibrational modes for a molecule with

Normal coordinate analysis gives comprehensive explanation of vibrational modes. It is termed a practice that calculates the vibrational frequencies involving observed frequencies of more preferably infrared and Raman harmonic frequencies to equilibrium geometry, force constants and atomic masses of an oscillating system. In assigning vibrational spectra, normal coordinate analysis proved beneficial, but reliable intramolecular force constants influence its

DFT syndicates accuracy with computational rapidity and user-friendliness for investigating the ground state characteristics in sturdily bound systems. Consistent highly reliable hybrid DFT methods (discussed in Section 3.1) make them more proficient and endearing. B3LYP functional [18] (discussed in hybrid methods of Section 3.1) is the most extensively used among all hybrid density functional methods as it is considered to give most exact vibrational frequencies of compounds only if calculated frequencies are scaled by a uniform scaling factor. Scaling is done to compensate for all probable causes of inaccuracy produced owing to electronic structure method–related inaccuracies, e.g., basis set insufficiencies and estimated

Gaussian is a most employed computational chemistry software program, whereas Gauss View is an inexpensive full featured graphical user interface for Gaussian. One can submit inputs to Gaussian and can observe output graphically, which is usually produced by Gaussian software via using Gauss View. For IR spectrum calculations in Gaussian, the following are

i. After predicting the number of vibrational modes and expected regions for frequencies for molecule by theoretical calculations, build molecule in Gauss View. Go to Calculate from Gauss View toolbar and select Gaussian. In the Job Type dialog box, select Opt + Freq and optimize to a 'Minimum' Calculate Force Constants-'Never,' Compute

handling of electron-correlation and nuclear motion treatment inaccuracies [19].

i 1 2

hvi (10)

EZPE <sup>¼</sup> <sup>X</sup>

frequencies depict molecular-harmonic vibrational frequencies.

2.1.2.3. Zero-point energy calculations for maximum vibrations

106 Density Functional Calculations - Recent Progresses of Theory and Application

maximum number of vibrational modes and calculated as:

2.1.3. Normal coordinate analysis

2.1.4. B3LYP density functional method

2.1.5. IR calculations in Gaussian software

Raman- 'Default,' deselect any other option.

the steps:

predictive ability [17].

NMR is based on the principle that the energy of a system containing nuclear or electron magnetic moments arising from the spin of a particle, in the existence of an external stationary magnetic field, depends upon the direction of the magnetic moment with respect to the external field. One can thus measure the energy difference for different directions of the electronic magnetic moment (electronic Zeeman effect) or of nuclear magnetic moment (nuclear Zeeman effect) by employing a suitable oscillating external magnetic field as a probe [20].

NMR parameters are considered as quantities which are determined shifts, indirect spin-spin coupling constant and direct dipole-dipole coupling constants. This electronic structure in turn confidentially interrelated to local and global geometry and hence internal flexibility and intramolecular interactions influences NMR parameters. Chemical shifts along with spin-spin coupling constants found an average experimentally in comparison to those values belonging to all geometrical arrangements arising throughout the sequence of NMR experiment. Unfortunately, chemical shifts are generally found to be dependent on the internal dynamics or on the intermolecular interactions in various descriptions. Consequently, for obtaining structural information, most experimental NMR techniques imply coupling constants or the nuclear Overhauser effect instead of chemical shifts. However, ab initio calculations provide the understanding to structure-chemical shifts or spin-spin coupling constant associations, which can make the experimental data interpretation much easier in this sense. That is the reason that highly capable and steadfast computational calculations are in high demand [21].

As commonly noted that, when results of sufficient quality are attainable by employing various strategies based on wave function properties. These procedures are restricted to small- and medium-sized systems unfortunately. With the advancements of density functional methods, it is possible to acquire pertinent results even for larger molecules, such as fragments of proteins and nucleic acids [22] where the electron correlation effects are indirectly accounted through the exchange-correlation functional. DFT-based NMR calculations have seen a rapid expansion during the last 10 years—that is perhaps best designated with the word explosion. Since the publication of these calculations, methods have already entered the standard repertoire of quantum chemistry in short time length. More comprehensive and technical as well as more general reviews are available as theoretical portrayal of NMR chemical shifts based on the more traditional ab initio procedures has seen a marvelous progress as well [23]. For NMR calculations, density functional theory (DFT) is recently proved to substitute the traditional Hartree-Fock (HF) and post-HF methods [24]. Inclusion of electron correlation effects in a very efficient way to calculate NMR parameters is a rather new field of application in DFT over the last years.

#### 3.1. Calculations of NMR parameters

Total energy E of an n-electron system is expressed exactly on the basis of DFT approach as in Eq. (11) [2].

$$E = \sum\_{i}^{n} \int d\vec{r} \Psi\_{i}^{\*} \left(\frac{p^{2}}{2} + V\_{N}\right) \Psi\_{i} \frac{1}{2} \int d\vec{r}^{\dagger} \vec{r}^{\dagger} \frac{\rho\left(\vec{r}^{\dagger}\right) \rho\left(\vec{r}^{\dagger}\right)}{|\vec{r}^{\dagger} - \vec{r}^{\dagger}|} + E\_{\text{XC}} \tag{11}$$

In Eq. (11), the f g <sup>Ψ</sup><sup>i</sup> is a set of n orthonormal one-electron functions, <sup>r</sup> <sup>=</sup> <sup>P</sup><sup>n</sup> i Ψ<sup>∗</sup> <sup>i</sup> Ψiis electronic

density of the system, VN is the external (nuclear) potential, and p ! is momentum operator. Hence, the kinetic and potential energy of a model system with the same density is signified by the first integral but without incorporating electron-electron communication. The second term represents itself as coulomb interaction of electron density. The XC energy and E proper (EXC) are functionals of the density although exact functional arrangement for EXC is not known (defined through Eq. (11)). Any application of DFT is mainly concerned with the assessment of numerous approximations; hence, Kohn-Sham (KS) equations are typically derived from Eq. (11) [4, 25].

$$
\hbar\_{\rm KS} \Psi\_i = \varepsilon\_i \Psi\_i \tag{12}
$$

where A !

3.1.1. Exchange-correlation functional

approximations used for Exc are as follows:

3.1.1.1. The local density approximation (LDA)

3.1.1.2. The generalized gradient approximations (GGA)

split into its exchange and correlation parts as:

EGGA XC rα; r<sup>β</sup> h i <sup>¼</sup>

Exc in turn written as:

Eq. (15). The ELDA

be generically inscribed as:

is the vector potential of the field. Though this is not the entire division in DFT. Rather,

h i !

!

rð Þr Exc½ � r dr (16)

� �dr (18)

<sup>C</sup> (19)

XC is also typically

<sup>C</sup> (17)

depicted in Eq. (14) [27].

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(15)

109

the XC energy (EXC) becomes a relativistic for current density functional, which interprets in

EXC½ � <sup>p</sup> ����! substitute EXC <sup>p</sup>; <sup>j</sup>

Quality of the description centers exclusively on precision of the approximation to Exc as mentioned earlier that obvious form of the exchange-correlation functional (Exc) is not identified yet. Unfortunately, there is no systematic mode of improving exchange-correlation functionals although the pursuit for improved and better functionals is at the very core of DFT. Prevailing XC functionals can be roughly categorized into three distinct groups [28] as several

The idea of a nonvariating electron gas laid the foundation of this model, which approximates

ð

Exc½ � r is the exchange-correlation energy per particle of a uniform electron gas of density r(r) in

XC ½ � r can be correctly split into exchange and correlation parts as:

The generalized gradient approximations (GGA) for Exc are not lone functions of local density r(r) but also functions of the gradient of charge density ∇r(r) in comparison to LDA. They can

<sup>X</sup> <sup>þ</sup> <sup>E</sup>LDA

f rα; rβ; Δrα; Δr<sup>β</sup>

<sup>X</sup> <sup>þ</sup> <sup>E</sup>GGA

XC ½ �¼ <sup>r</sup> ELDA

ð

XC ½ �¼ <sup>r</sup> <sup>E</sup>GGA

The GGA exchange functional's representatives are functionals by Becke, 1988 (B or B88) [18] and by Perdew and Wang, 1986 (P or PW86) [29] and are the most familiar ones. Perdew and Wang also established the PW91 exchange-correlation functional [30] comprising exchange

ELDA XC ½ �¼ r

ELDA

α and β denote to "up" and "down" spin, respectively, in Eq. (17). Now, EGGA

EGGA

nonrelativistic system to the electron density r and the current density j

$$h\_{KS} = \frac{p^2}{2} + V\_{KS} = \frac{p^2}{2} + V\_N + \int d\vec{r} \frac{\rho\left(\vec{r'}\right)}{|\vec{r'} - \vec{r'}|} + V\_{XC} \tag{13}$$

The XC potential VXC is the functional derivative of the XC energy EXC with respect to the density, r. The inclusion of magnetic fields is another extension, which can be introduced. Obviously, this is essential to all the properties to be introduced. The magnetic field B ! is most accessibly acquaint with supposed minimal coupling as in Eq. (13) [26].

$$
\overrightarrow{p'} = \overrightarrow{p'} + \overrightarrow{A} \,/\circledast \tag{14}
$$

where A ! is the vector potential of the field. Though this is not the entire division in DFT. Rather, the XC energy (EXC) becomes a relativistic for current density functional, which interprets in nonrelativistic system to the electron density r and the current density j ! depicted in Eq. (14) [27].

$$E\_{\rm XC}[p] \xrightarrow{substrate} E\_{\rm XC}\left[p, \overrightarrow{j}^{\cdot}\right] \tag{15}$$

#### 3.1.1. Exchange-correlation functional

indirectly accounted through the exchange-correlation functional. DFT-based NMR calculations have seen a rapid expansion during the last 10 years—that is perhaps best designated with the word explosion. Since the publication of these calculations, methods have already entered the standard repertoire of quantum chemistry in short time length. More comprehensive and technical as well as more general reviews are available as theoretical portrayal of NMR chemical shifts based on the more traditional ab initio procedures has seen a marvelous progress as well [23]. For NMR calculations, density functional theory (DFT) is recently proved to substitute the traditional Hartree-Fock (HF) and post-HF methods [24]. Inclusion of electron correlation effects in a very efficient way to calculate NMR parameters is a rather

Total energy E of an n-electron system is expressed exactly on the basis of DFT approach as in

Ψi 1 2 ð dr<sup>1</sup> ! r 2 ! <sup>r</sup> <sup>r</sup> 1 !� �

Hence, the kinetic and potential energy of a model system with the same density is signified by the first integral but without incorporating electron-electron communication. The second term represents itself as coulomb interaction of electron density. The XC energy and E proper (EXC) are functionals of the density although exact functional arrangement for EXC is not known (defined through Eq. (11)). Any application of DFT is mainly concerned with the assessment of numerous approximations; hence, Kohn-Sham (KS) equations are typically derived from

<sup>2</sup> <sup>þ</sup> VN <sup>þ</sup>

The XC potential VXC is the functional derivative of the XC energy EXC with respect to the density, r. The inclusion of magnetic fields is another extension, which can be introduced.

> ! <sup>þ</sup> <sup>A</sup> !

Obviously, this is essential to all the properties to be introduced. The magnetic field B

p ! <sup>¼</sup> <sup>p</sup> ð d r ! r r 2 !� �

∣r 1 ! � r 2 ! ∣

In Eq. (11), the f g <sup>Ψ</sup><sup>i</sup> is a set of n orthonormal one-electron functions, <sup>r</sup> <sup>=</sup> <sup>P</sup><sup>n</sup>

density of the system, VN is the external (nuclear) potential, and p

<sup>2</sup> <sup>þ</sup> VKS <sup>¼</sup> <sup>p</sup><sup>2</sup>

accessibly acquaint with supposed minimal coupling as in Eq. (13) [26].

r r 2 !� �

hKSΨ<sup>i</sup> ¼ εiΨ<sup>i</sup> (12)

þ EXC (11)

! is momentum operator.

þ VXC (13)

=c (14)

!

is most

<sup>i</sup> Ψiis electronic

i Ψ<sup>∗</sup>

∣r 1 ! � r 2 ! ∣

new field of application in DFT over the last years.

108 Density Functional Calculations - Recent Progresses of Theory and Application

3.1. Calculations of NMR parameters

<sup>E</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup> i

ð d r !Ψ<sup>∗</sup> i p2 <sup>2</sup> <sup>þ</sup> VN � �

hKS <sup>¼</sup> <sup>p</sup><sup>2</sup>

Eq. (11) [2].

Eq. (11) [4, 25].

Quality of the description centers exclusively on precision of the approximation to Exc as mentioned earlier that obvious form of the exchange-correlation functional (Exc) is not identified yet. Unfortunately, there is no systematic mode of improving exchange-correlation functionals although the pursuit for improved and better functionals is at the very core of DFT. Prevailing XC functionals can be roughly categorized into three distinct groups [28] as several approximations used for Exc are as follows:

#### 3.1.1.1. The local density approximation (LDA)

The idea of a nonvariating electron gas laid the foundation of this model, which approximates Exc in turn written as:

$$E\_{\rm XC}^{LDA}[\rho] = \int \rho(r) \epsilon\_{\rm xc}[\rho] dr \tag{16}$$

Exc½ � r is the exchange-correlation energy per particle of a uniform electron gas of density r(r) in Eq. (15). The ELDA XC ½ � r can be correctly split into exchange and correlation parts as:

$$E\_{X\mathbb{C}}^{LDA}[\rho] = E\_X^{LDA} + E\_{\mathbb{C}}^{LDA} \tag{17}$$

#### 3.1.1.2. The generalized gradient approximations (GGA)

The generalized gradient approximations (GGA) for Exc are not lone functions of local density r(r) but also functions of the gradient of charge density ∇r(r) in comparison to LDA. They can be generically inscribed as:

$$E\_{\rm XC}^{\rm GGA}\left[\rho\_a, \rho\_\beta\right] = \int \left(\rho\_a, \rho\_\beta, \Delta\rho\_a, \Delta\rho\_\beta\right) dr\tag{18}$$

α and β denote to "up" and "down" spin, respectively, in Eq. (17). Now, EGGA XC is also typically split into its exchange and correlation parts as:

$$E\_{\rm XC}^{\rm GGA}[\rho] = E\_{\rm X}^{\rm GGA} + E\_{\rm \mathcal{C}}^{\rm GGA} \tag{19}$$

The GGA exchange functional's representatives are functionals by Becke, 1988 (B or B88) [18] and by Perdew and Wang, 1986 (P or PW86) [29] and are the most familiar ones. Perdew and Wang also established the PW91 exchange-correlation functional [30] comprising exchange and correlation contributions; if used separately, both functionals are symbolized by PW91. The exchange fragment of PW91 is analogous to B88 and the correlation part is a modified form of P86. Currently, possibly the most prevalent correlation functional accounts to contributions of Lee et al. (LYP) [31]. Because of presence of some indigenous components, it differs from the other GGA functionals.

#### 3.1.1.3. The hybrid methods

Exchange-correlation functionals in which exchange part is composed of particular Hartree-Fock exchange and pure density functionals for exchange are called hybrid functionals. These functionals consist of a mixture of Hartree-Fock exchange with DFT exchange and correlation in other words as depicted in Eq. (19).

$$E\_{\rm XC}^{hydrid} = \mathcal{c}^{HF} E\_{\rm X}^{HF} + \mathcal{c}^{DFT} E\_{\rm C}^{DFT} \tag{20}$$

for localized orbitals (DFTIGLO) method [9] and gauge including atomic orbitals (GIAO)

One appraises matrix fundamentals of the Hamiltonian in terms of a basis of fielddependent atomic orbitals (AO) in the GIAO method. By insertion of an intricate phase factor denoting the position of basis function, which is usually nucleus, the basic functions are made obviously dependent on the magnetic field. Such orbitals are termed as London atomic orbitals (LAO) or the gauge including atomic orbitals (GIAO). Matrix elements involved in GIAOs only differentiate in the vector potentials is the essence of idea, in that way entirely eradicating the reference to a complete gauge origin. Employment of field-dependent GIAOs as basic functions [33] is possibly the best solution to the gauge

> i 2c B ! xR ! a :<sup>r</sup> ! <sup>χ</sup><sup>a</sup> <sup>r</sup>

function that is adjusted at position. The field-dependent prefactor Eq. (21) guarantees that only differences of position vectors give the impression in expectation values. This eradicates any origin dependence, even for estimated MOs and finite basis sets. For showing the complete basis set limit, correspondence of the GIAO and the simpler common gauge methods to the NMR shielding are depicted. Nucleus-attached basis functions in geometry optimization processes have been equated with specific AOs (Eq. (21)) by assigning a field-dependent phase

> ! !

tion of Eq. (21). The absolute shielding σ is interrelated to the more acquainted chemical shift δ

where σref is the absolute shielding of reference compound (e.g., tetramethylsilane [TMS] for

H, 13C and 29Si NMR). Opposite sign between δ and σ can be noted, whereas GIAO shielding

division is not exceptional as only the total shielding is an apparent quantity, but for the GIAO

has been divided into its paramagnetic and diamagnetic fragments. Generally, this

σ ! ! <sup>¼</sup> <sup>σ</sup><sup>p</sup> ! ! <sup>þ</sup> <sup>σ</sup><sup>p</sup> ! !

! are the conforming GIAO and <sup>χ</sup><sup>a</sup> <sup>r</sup>

! (22)

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111

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! is customary field-free basis

) can be derived from the GIAO formula-

(24)

δ ¼ σref � σ (23)

χ<sup>a</sup> B ! ; r ! <sup>¼</sup> exp

Working equations for the NMR shielding tensor (σ

approach [32].

problem.

where R !

factor to them.

as follows [33]:

tensor is specified in DFT as [34]:

method, it has been defined exclusively.

1

where σ ! !

<sup>a</sup> and <sup>χ</sup><sup>a</sup> <sup>B</sup>

! ; r

3.2.2. The GIAO method

#### 3.1.1.4. Becke's three-parameter hybrid functional (B3LYP) method

An illustrative example of the above-mentioned hybrid representations is Becke's threeparameter hybrid functional (B3), depicted in Eq. (20). [18], and exchange energy is being calculated by using this (B3) functional. If correlation functional LYP or PW91 is assumed for EGGA <sup>C</sup> , B3LYP or B3PW91 will be obtained, respectively. The earlier mentioned is most widely used exchange-correlation functional nowadays, and in many research domains throughout, it was also rigorously active. Since GGAs are better in comparison to LSDA, they could also employ a functional form like this (B3) as follows:

$$E\_{\rm XC}^{\rm B3} = (1 - a)E\_X^{\rm LSDA} + aE\_X^{\rm HF} + b\Delta E\_X^{\rm B88} + (1 - c)E\_\mathcal{C}^{\rm LSDA} + cE\_\mathcal{c}^{\rm GGA} \tag{21}$$

B3LYP uses this formulation, with LYP for ΔEGGA <sup>c</sup> .

#### 3.2. Shielding tensor calculations

The most basic DFT calculations of NMR chemical shifts were done years ago in which very small basis sets were employed and inappropriate approximation to the exchange-correlation (XC) functional was done due to which point of practical applicability was lost. Also in any handling of magnetic fields, another central exertion that surfaces is the so-called gauge problem [28] be it in DFT or otherwise.

#### 3.2.1. Solving gauge problem

The gauge requirement should disappear just as any anticipation value together with NMR possessions that can only hinge on the values of observable quantities. Certainly, this is the case for particular solutions of, for example, the KS equations, Eq. (12a, b) where large (immeasurable) basis sets are employed. A strong dependence on the choice of gauge is of main concern for approximate solutions with smaller (infinite) basis sets. Current applications of DFT for calculating NMR chemical shifts have used density functional theory individual gauge for localized orbitals (DFTIGLO) method [9] and gauge including atomic orbitals (GIAO) approach [32].

#### 3.2.2. The GIAO method

and correlation contributions; if used separately, both functionals are symbolized by PW91. The exchange fragment of PW91 is analogous to B88 and the correlation part is a modified form of P86. Currently, possibly the most prevalent correlation functional accounts to contributions of Lee et al. (LYP) [31]. Because of presence of some indigenous components, it differs

Exchange-correlation functionals in which exchange part is composed of particular Hartree-Fock exchange and pure density functionals for exchange are called hybrid functionals. These functionals consist of a mixture of Hartree-Fock exchange with DFT exchange and correlation

> HFEHF <sup>X</sup> þ c

An illustrative example of the above-mentioned hybrid representations is Becke's threeparameter hybrid functional (B3), depicted in Eq. (20). [18], and exchange energy is being calculated by using this (B3) functional. If correlation functional LYP or PW91 is assumed for

<sup>C</sup> , B3LYP or B3PW91 will be obtained, respectively. The earlier mentioned is most widely used exchange-correlation functional nowadays, and in many research domains throughout, it was also rigorously active. Since GGAs are better in comparison to LSDA, they could also

<sup>X</sup> <sup>þ</sup> <sup>b</sup>ΔEB<sup>88</sup>

<sup>c</sup> .

The most basic DFT calculations of NMR chemical shifts were done years ago in which very small basis sets were employed and inappropriate approximation to the exchange-correlation (XC) functional was done due to which point of practical applicability was lost. Also in any handling of magnetic fields, another central exertion that surfaces is the so-called gauge

The gauge requirement should disappear just as any anticipation value together with NMR possessions that can only hinge on the values of observable quantities. Certainly, this is the case for particular solutions of, for example, the KS equations, Eq. (12a, b) where large (immeasurable) basis sets are employed. A strong dependence on the choice of gauge is of main concern for approximate solutions with smaller (infinite) basis sets. Current applications of DFT for calculating NMR chemical shifts have used density functional theory individual gauge

DFTEDFT

<sup>X</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>E</sup>LSDA

<sup>C</sup> <sup>þ</sup> cEGGA

<sup>c</sup> (21)

<sup>C</sup> (20)

Ehybrid XC ¼ c

<sup>X</sup> <sup>þ</sup> aEHF

3.1.1.4. Becke's three-parameter hybrid functional (B3LYP) method

110 Density Functional Calculations - Recent Progresses of Theory and Application

employ a functional form like this (B3) as follows:

B3LYP uses this formulation, with LYP for ΔEGGA

XC <sup>¼</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>E</sup>LSDA

EB<sup>3</sup>

3.2. Shielding tensor calculations

problem [28] be it in DFT or otherwise.

3.2.1. Solving gauge problem

from the other GGA functionals.

in other words as depicted in Eq. (19).

3.1.1.3. The hybrid methods

EGGA

One appraises matrix fundamentals of the Hamiltonian in terms of a basis of fielddependent atomic orbitals (AO) in the GIAO method. By insertion of an intricate phase factor denoting the position of basis function, which is usually nucleus, the basic functions are made obviously dependent on the magnetic field. Such orbitals are termed as London atomic orbitals (LAO) or the gauge including atomic orbitals (GIAO). Matrix elements involved in GIAOs only differentiate in the vector potentials is the essence of idea, in that way entirely eradicating the reference to a complete gauge origin. Employment of field-dependent GIAOs as basic functions [33] is possibly the best solution to the gauge problem.

$$\chi\_a \left( \overrightarrow{B} \; , \overrightarrow{r} \right) = \exp \left[ \frac{i}{2c} \left( \overrightarrow{B} \mathbf{x} \overrightarrow{R}\_a \right) . \overrightarrow{r} \right] \chi\_a \left( \overrightarrow{r} \right) \tag{22}$$

where R ! <sup>a</sup> and <sup>χ</sup><sup>a</sup> <sup>B</sup> ! ; r ! are the conforming GIAO and <sup>χ</sup><sup>a</sup> <sup>r</sup> ! is customary field-free basis function that is adjusted at position. The field-dependent prefactor Eq. (21) guarantees that only differences of position vectors give the impression in expectation values. This eradicates any origin dependence, even for estimated MOs and finite basis sets. For showing the complete basis set limit, correspondence of the GIAO and the simpler common gauge methods to the NMR shielding are depicted. Nucleus-attached basis functions in geometry optimization processes have been equated with specific AOs (Eq. (21)) by assigning a field-dependent phase factor to them.

Working equations for the NMR shielding tensor (σ ! ! ) can be derived from the GIAO formulation of Eq. (21). The absolute shielding σ is interrelated to the more acquainted chemical shift δ as follows [33]:

$$
\delta = \sigma\_{\text{ref}} - \sigma \tag{23}
$$

where σref is the absolute shielding of reference compound (e.g., tetramethylsilane [TMS] for 1 H, 13C and 29Si NMR). Opposite sign between δ and σ can be noted, whereas GIAO shielding tensor is specified in DFT as [34]:

$$
\stackrel{\rightarrow}{\vec{\sigma}}\_{} = \stackrel{\rightarrow}{\sigma^p}\_{} + \stackrel{\rightarrow}{\sigma^p}\_{} \tag{24}
$$

where σ ! ! has been divided into its paramagnetic and diamagnetic fragments. Generally, this division is not exceptional as only the total shielding is an apparent quantity, but for the GIAO method, it has been defined exclusively.

#### 3.2.3. The IGLO method

The use of disseminated gauge origins was depicted by the individual gauge for localized orbitals method (IGLO) [9], which was the first to empower the organized learning of nuclear shielding in larger systems. Evaluation of shielding tensor in terms of localized molecular orbitals (MO) is the idea of approach. To minimalize the absolute value of paramagnetic involvement, individual gauge origins of shielding tensor are selected. In the IGLO method, local phase factors are involved to molecular orbital in comparison to the GIAO approach where a phase factor is associated to each atomic orbital. In computational analysis point of view, IGLO approach is less challenging in contrast to the GIAO method since the use of distinct approximations of derivative two-electron integrals by this approach is most probable. By using localized orbitals, previously mentioned approximations are empowered; hence, it is important to select an appropriate basis set in an IGLO calculation [35]. According to IGLO approach, it is possible to allot comparable exponential prefactors to other objects, e.g., to localized MOs as a replacement for GIAO method. In IGLO method, specific integrals are easier to estimate analytically. Gauge difficulty has been resolute recently by retaining numerical integration or by techniques that were used in geometry optimization measures.

The portrayal of calculations follows Ref. [37]. By meaning, the nuclear spin-spin coupling tensor JMNuv is the second derivative of the total energy of the system with respect to the spins

IMu¼INv¼0

The orientationally averaged value of the nuclear spin-spin coupling tensor (JMNuv ) of nuclear

Since calculations for TMS are encompassed in many software, DFT-B3LYP-GIAO is the 6-311 + G(2d.p) [38], a suitable basis set, which easily determines the chemical shifts relative to TMS. The use of smaller 6-31 + G(d,p) basis set for calculations is most suitable for students compared to larger basis sets such as the 6-311 + G(2d,p) or even the very large 6-311++G (3df,3pd) basis sets that are employed for research purposes on small molecules. Many of the movements necessitate much less than 3 hours fixed for a standard research laboratory session even while using the larger basic sets. First of all, in this method, the structure is optimized in all the cases, and then to establish that either optimized structure is at least a local minimal or not, vibrational frequencies are determined. After the complete optimization of the structure, shielding constants are calculated using the GIAO method. Calculations for chemical shifts are carried out by subtracting the value of the screening constant (σ), which is calculated from the

value of the screening constant calculated for TMS using the same level of theory [38].

Basic considerations in NMR calculations are as follows: first, how many types of protons are there in molecule and which are they and second, how many NMR signals are expected to see and in which regions. In the case of an NMR spectrum calculation, geometry optimization calculation is needed first of all and then optimized structure is used to perform NMR spec-

i. Follow the calculations in Section 2.1.5 till step i for geometry optimization calculations and then follow steps used to perform an HNMR spectrum calculation of the optimized

ii. Press the Calculate button in the Gauss View toolbar and select Gaussian. In the Job Type dialog box, select 'NMR' and 'GIAO method. Submit the calculation after checking all

parameters. Open the output file and select the NMR option from Results.

u, v ¼ f g x; y; z (25)

http://dx.doi.org/10.5772/intechopen.71080

Spectral Calculations with DFT

113

<sup>3</sup> JMNxx <sup>þ</sup> JMNyy <sup>þ</sup> JMNzz (26)

δ ¼ σTMS\_σCalculated (27)

of the nuclei M and N as depicted in Eq. (24).

JMNuv <sup>¼</sup> <sup>∂</sup><sup>2</sup>E Ið Þ Mu:INv ð Þ <sup>δ</sup>IMu:δINv

spin-spin coupling constant ð Þ JMN is the focus of interest in many cases.

JMN <sup>¼</sup> <sup>1</sup>

3.4. Chemical shift calculation from DFT-B3LYP-GIAO method

3.5. NMR calculation steps using Gaussian software

trum calculation using the same method and basis set.

structure with the same method and basis set.

#### 3.3. Nuclear spin-spin coupling constant calculations

Highly appropriate ab initio calculations of spin-spin coupling constants are still very infrequent and usually deal with the simplest molecules only in comparison to excessive advancement in concept and computer codes for shielding tensor scheming. Shielding tensors are the reasons behind sensitivity of nuclear spin spin coupling constants to the correlation effects and basis set quality [36]. There are four significant contributions which causes other problems to the nuclear spin-spin coupling constants, are enlisted as follows:


HF calculations of the spin-spin coupling constants of large molecules lead to deprived contract with experimental numbers and further empirical scaling has to be done. On the other hand, the use of post–Hartree-Fock methods and prolonged basis sets for demanding handlings of all the four contributions is very inefficient and time-consuming.

In DFT, only a single paper by Fukui [23] published years ago was worthy initially, where the blend of finite perturbation theory (FPT) and the LCAO-Xa method was employed and only calculations of the FC contribution to the spin-spin coupling constant were carried out, which gave somewhat poor results likely as too poor basis set in combination with the Xa potential was used. As in Ref. [37], a new method to calculate the nuclear spin-spin coupling tensor using DFT procedure is developed sideways to the shielding tensor calculations in accordance. The portrayal of calculations follows Ref. [37]. By meaning, the nuclear spin-spin coupling tensor JMNuv is the second derivative of the total energy of the system with respect to the spins of the nuclei M and N as depicted in Eq. (24).

$$J\_{\rm MN\_{uv}} = \left[\frac{\partial^2 E(I\_{\rm Mu}, I\_{\rm Nv})}{(\delta I\_{\rm Mu}, \delta I\_{\rm Nv})}\right]\_{I\_{\rm Mu} = I\_{\rm Nv} = 0} \quad \text{u/v} = \{\mathbf{x}, y, \mathbf{z}\} \tag{25}$$

The orientationally averaged value of the nuclear spin-spin coupling tensor (JMNuv ) of nuclear spin-spin coupling constant ð Þ JMN is the focus of interest in many cases.

$$J\_{\rm MN} = \frac{1}{3} \left( J\_{\rm MN\_{xx}} + J\_{\rm MN\_{yy}} + J\_{\rm MN\_{zz}} \right) \tag{26}$$

#### 3.4. Chemical shift calculation from DFT-B3LYP-GIAO method

3.2.3. The IGLO method

optimization measures.

a. Fermi contact (FC),

c. Spin dipolar (SD)

b. Paramagnetic spin orbit (PSO)

d. Diamagnetic spin orbit (DSO)

3.3. Nuclear spin-spin coupling constant calculations

112 Density Functional Calculations - Recent Progresses of Theory and Application

the nuclear spin-spin coupling constants, are enlisted as follows:

dlings of all the four contributions is very inefficient and time-consuming.

The use of disseminated gauge origins was depicted by the individual gauge for localized orbitals method (IGLO) [9], which was the first to empower the organized learning of nuclear shielding in larger systems. Evaluation of shielding tensor in terms of localized molecular orbitals (MO) is the idea of approach. To minimalize the absolute value of paramagnetic involvement, individual gauge origins of shielding tensor are selected. In the IGLO method, local phase factors are involved to molecular orbital in comparison to the GIAO approach where a phase factor is associated to each atomic orbital. In computational analysis point of view, IGLO approach is less challenging in contrast to the GIAO method since the use of distinct approximations of derivative two-electron integrals by this approach is most probable. By using localized orbitals, previously mentioned approximations are empowered; hence, it is important to select an appropriate basis set in an IGLO calculation [35]. According to IGLO approach, it is possible to allot comparable exponential prefactors to other objects, e.g., to localized MOs as a replacement for GIAO method. In IGLO method, specific integrals are easier to estimate analytically. Gauge difficulty has been resolute recently by retaining numerical integration or by techniques that were used in geometry

Highly appropriate ab initio calculations of spin-spin coupling constants are still very infrequent and usually deal with the simplest molecules only in comparison to excessive advancement in concept and computer codes for shielding tensor scheming. Shielding tensors are the reasons behind sensitivity of nuclear spin spin coupling constants to the correlation effects and basis set quality [36]. There are four significant contributions which causes other problems to

HF calculations of the spin-spin coupling constants of large molecules lead to deprived contract with experimental numbers and further empirical scaling has to be done. On the other hand, the use of post–Hartree-Fock methods and prolonged basis sets for demanding han-

In DFT, only a single paper by Fukui [23] published years ago was worthy initially, where the blend of finite perturbation theory (FPT) and the LCAO-Xa method was employed and only calculations of the FC contribution to the spin-spin coupling constant were carried out, which gave somewhat poor results likely as too poor basis set in combination with the Xa potential was used. As in Ref. [37], a new method to calculate the nuclear spin-spin coupling tensor using DFT procedure is developed sideways to the shielding tensor calculations in accordance. Since calculations for TMS are encompassed in many software, DFT-B3LYP-GIAO is the 6-311 + G(2d.p) [38], a suitable basis set, which easily determines the chemical shifts relative to TMS. The use of smaller 6-31 + G(d,p) basis set for calculations is most suitable for students compared to larger basis sets such as the 6-311 + G(2d,p) or even the very large 6-311++G (3df,3pd) basis sets that are employed for research purposes on small molecules. Many of the movements necessitate much less than 3 hours fixed for a standard research laboratory session even while using the larger basic sets. First of all, in this method, the structure is optimized in all the cases, and then to establish that either optimized structure is at least a local minimal or not, vibrational frequencies are determined. After the complete optimization of the structure, shielding constants are calculated using the GIAO method. Calculations for chemical shifts are carried out by subtracting the value of the screening constant (σ), which is calculated from the value of the screening constant calculated for TMS using the same level of theory [38].

$$
\delta = \sigma\_{\text{TMS\\_off\\_calculated}} \tag{27}
$$

#### 3.5. NMR calculation steps using Gaussian software

Basic considerations in NMR calculations are as follows: first, how many types of protons are there in molecule and which are they and second, how many NMR signals are expected to see and in which regions. In the case of an NMR spectrum calculation, geometry optimization calculation is needed first of all and then optimized structure is used to perform NMR spectrum calculation using the same method and basis set.


iii. In the 'SCF GIAO magnetic shielding dialog box, select 'H' option in Element drop-down list, and in the Reference list, select the calculation method earlier used. The number of peaks of magnetic shielding observed is noted. Infer calculated spectrum and compare it with theoretical calculations. With this, multiplicity of the peaks, temperature at which NMR experiment is simulated and width of the NMR peaks can be discussed.

determination of electronic excitation spectrum in typical dipole approximation [41]. The poles' strengths are provided by oscillator strengths (fi) or by transition dipole moment (μi)

Electronic absorption spectra calculation is foremost applicability of TD-DFT after the geometry got optimized by using linear response theory. First, the Hohenberg-Kohn theorem statement that dynamic linear response of a system of N-electrons is determined by the groundstate charge density was discussed earlier but no applied mean was found to build this response. A productive procedure was provided by TD-DFT from which absorption spectra

Step 1: The calculations comprised of promulgating Kohn-Sham orbital incidence of a trivial

Step 2: Then induced dipole moment, μinduced(t), is calculated by Eq. (28), which depicts the

calculations of dynamic polarizability in Eq. (29) are carried out after applying Fourier trans-

ð

f I ω2

α t � t

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115

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vappliedð Þ¼ r; t εð Þt :r (28)

α ωð Þ¼ μ ωð Þ=Eð Þ ω (30)

rn rð Þ :t rd (29)

<sup>I</sup> � <sup>ω</sup><sup>2</sup> (31)

Imα ωð Þ þ iη (32)

<sup>0</sup> ð Þεð Þt :Final

dynamic perturbation, e.g., considering system being perturbed by Eq. (27).

difference among time-dependent and permanent dipole moments of a system.

Step 3: Dynamic polarizability, <sup>α</sup>, is expressed by linear term, <sup>μ</sup>inducedðÞ¼ <sup>t</sup> <sup>Ð</sup>

Step 4: Sum-over-states (SOS) form of dynamic polarizability has the form:

μinducedðÞ¼� t e

α ωð Þ¼ <sup>X</sup>

Sð Þ¼ ω

I6¼0

where ω<sup>I</sup> and fI are corresponding vertical excitation energies and conforming oscillator strength. Finally, spectrum was obtained employing Eq. (31) also termed as Lorentzian broadened stick

> 2ω π

Such a procedure has been executed in many computer codes, e.g., in Octopus code [43], and has benefited that absorption spectrum of a very huge molecule over a wide range of energies can be calculated even though using only reasonable spectral resolution. This function parallels to optical absorption spectrum, e.g., calculating spectra of benzene [44] by TD-DFT

consistently also termed as intensity of the optical transitions.

4.1. Absorption spectra calculation

can be calculated by following steps [42]:

form convolution theorem.

spectrum:

approach depicted in Figure 1.

iv. Open the log file and search for or scroll to the line "SCF GIAO Magnetic Shielding tensor (ppm)." For simple NMR experiments, the signals are found at the chemical shifts, which appear after the heading "isotropic." All other numbers relate to the directionality of the NMR signals.

### 4. Time-dependent density functional theory (TD-DFT) for UV-vis spectra

As classic DFT is concerned with ground stationary state and applications, i.e., UV-vis spectroscopy, photochemistry, NLO and others, comprise either electronic excited states or timedependent electronic characteristics, which are in agreement with time-dependent density functional theory (TD-DFT). From classic paper of Runge and Gross [39], formal TD-DFT is traced back, which strained to firm up former efforts on the same topic. Therefore, Runge-Gross TD-DFT is two decades younger than Hohenberg Kohn-Sham theory [2], which is about stationary ground state as mentioned earlier. Four theorems proposed in the Runge-Gross paper. First, Runge-Gross theorem states that external potential up to an additive function of time is regulated by time-dependent charge density, r(r,t), in cooperation with preliminary wave function (Ψ0).

Various chemical and physical molecular properties provide basis for electronic spectra. Different chemical and physical effects can easily be computationally investigated by modifying the spectral characteristics of molecules as many stimulating chemical problems are included in both ground and excited states of molecules. As for ground state properties of atoms, molecules and solids, DFT gave an effective explanation; hence, in order to designate photochemical and photophysical procedures, DFT formalism has to be expanded to excited states. So, for excited state calculations, time-dependent DFT (TD-DFT) is established as an operative tool [40] as it provides first principle technique for calculating excitation energies and various response-related characteristics in density functional outline.

There are classically one or more low energy excited states for a molecule that can be designated as valence-MO-valence-MO single electronic excitations and replicated in spectra. Energies of excited determinants are essential for multiplet energy calculation such as employing spin-orbital number (i) to the empty spin-orbital number (j). Accordingly, specific states are denoted as <sup>π</sup> ! <sup>π</sup>\* , n ! <sup>π</sup>\* transitions. Relative easiness of character sometimes smoothes the process of concluding wave functions for these states. The calculation of the dynamic answer of charge density proposes a rigorous direction to time-dependent simplification of DFT formalism. The poles of dynamic polarizability regulate excitation energies permitting the determination of electronic excitation spectrum in typical dipole approximation [41]. The poles' strengths are provided by oscillator strengths (fi) or by transition dipole moment (μi) consistently also termed as intensity of the optical transitions.

#### 4.1. Absorption spectra calculation

iii. In the 'SCF GIAO magnetic shielding dialog box, select 'H' option in Element drop-down list, and in the Reference list, select the calculation method earlier used. The number of peaks of magnetic shielding observed is noted. Infer calculated spectrum and compare it with theoretical calculations. With this, multiplicity of the peaks, temperature at which

NMR experiment is simulated and width of the NMR peaks can be discussed.

4. Time-dependent density functional theory (TD-DFT)

114 Density Functional Calculations - Recent Progresses of Theory and Application

response-related characteristics in density functional outline.

of the NMR signals.

for UV-vis spectra

wave function (Ψ0).

denoted as <sup>π</sup> ! <sup>π</sup>\*

iv. Open the log file and search for or scroll to the line "SCF GIAO Magnetic Shielding tensor (ppm)." For simple NMR experiments, the signals are found at the chemical shifts, which appear after the heading "isotropic." All other numbers relate to the directionality

As classic DFT is concerned with ground stationary state and applications, i.e., UV-vis spectroscopy, photochemistry, NLO and others, comprise either electronic excited states or timedependent electronic characteristics, which are in agreement with time-dependent density functional theory (TD-DFT). From classic paper of Runge and Gross [39], formal TD-DFT is traced back, which strained to firm up former efforts on the same topic. Therefore, Runge-Gross TD-DFT is two decades younger than Hohenberg Kohn-Sham theory [2], which is about stationary ground state as mentioned earlier. Four theorems proposed in the Runge-Gross paper. First, Runge-Gross theorem states that external potential up to an additive function of time is regulated by time-dependent charge density, r(r,t), in cooperation with preliminary

Various chemical and physical molecular properties provide basis for electronic spectra. Different chemical and physical effects can easily be computationally investigated by modifying the spectral characteristics of molecules as many stimulating chemical problems are included in both ground and excited states of molecules. As for ground state properties of atoms, molecules and solids, DFT gave an effective explanation; hence, in order to designate photochemical and photophysical procedures, DFT formalism has to be expanded to excited states. So, for excited state calculations, time-dependent DFT (TD-DFT) is established as an operative tool [40] as it provides first principle technique for calculating excitation energies and various

There are classically one or more low energy excited states for a molecule that can be designated as valence-MO-valence-MO single electronic excitations and replicated in spectra. Energies of excited determinants are essential for multiplet energy calculation such as employing spin-orbital number (i) to the empty spin-orbital number (j). Accordingly, specific states are

process of concluding wave functions for these states. The calculation of the dynamic answer of charge density proposes a rigorous direction to time-dependent simplification of DFT formalism. The poles of dynamic polarizability regulate excitation energies permitting the

, n ! <sup>π</sup>\* transitions. Relative easiness of character sometimes smoothes the

Electronic absorption spectra calculation is foremost applicability of TD-DFT after the geometry got optimized by using linear response theory. First, the Hohenberg-Kohn theorem statement that dynamic linear response of a system of N-electrons is determined by the groundstate charge density was discussed earlier but no applied mean was found to build this response. A productive procedure was provided by TD-DFT from which absorption spectra can be calculated by following steps [42]:

Step 1: The calculations comprised of promulgating Kohn-Sham orbital incidence of a trivial dynamic perturbation, e.g., considering system being perturbed by Eq. (27).

Step 2: Then induced dipole moment, μinduced(t), is calculated by Eq. (28), which depicts the difference among time-dependent and permanent dipole moments of a system.

Step 3: Dynamic polarizability, <sup>α</sup>, is expressed by linear term, <sup>μ</sup>inducedðÞ¼ <sup>t</sup> <sup>Ð</sup> α t � t <sup>0</sup> ð Þεð Þt :Final calculations of dynamic polarizability in Eq. (29) are carried out after applying Fourier transform convolution theorem.

Step 4: Sum-over-states (SOS) form of dynamic polarizability has the form:

$$
\sigma\_{\text{applied}}(r, t) = \varepsilon(t). \mathbf{r} \tag{28}
$$

$$
\mu\_{induced}(t) = -e \int \mathbf{m}(r.t) \mathbf{r} d\mathbf{r} \tag{29}
$$

$$
\alpha(\omega) = \mu(\omega) / \mathcal{E}(\omega) \tag{30}
$$

$$a(\omega) = \sum\_{I \neq 0} \frac{f\_I}{\omega\_I^2 - \omega^2} \tag{31}$$

where ω<sup>I</sup> and fI are corresponding vertical excitation energies and conforming oscillator strength.

Finally, spectrum was obtained employing Eq. (31) also termed as Lorentzian broadened stick spectrum:

$$S(\omega) = \frac{2\omega}{\pi} \text{Im}\alpha(\omega + i\eta) \tag{32}$$

Such a procedure has been executed in many computer codes, e.g., in Octopus code [43], and has benefited that absorption spectrum of a very huge molecule over a wide range of energies can be calculated even though using only reasonable spectral resolution. This function parallels to optical absorption spectrum, e.g., calculating spectra of benzene [44] by TD-DFT approach depicted in Figure 1.

Figure 1. Absorption spectra of benzene.

#### 4.1.1. Calculating the oscillator strength

It is an important point to note that TD-DFT calculations not only give excitation energies (ω) but also provide respective oscillator strengths. Oscillator strengths are actually pure numbers in a complete basis set [42].

$$\,\_{I}f\_{I} = \frac{2m\_{e}}{3h} \omega\_{I} |\langle 0|r|I\rangle|2\tag{33}$$

on oscillator strength carrying subspace so that absorption spectrum is possibly calculated with less excitations in a fixed energy space. A common problem considered is that a large number of excitations have to be estimated with the purpose of covering energy space of concentration, while figure of absorption spectrum is determined by only some of them

Another detailed DFT method is presented, proposed in ref. [46], to calculate UV absorption

where ℏω is the input running radiation energy and ℏωjk is energy difference between ground (j) and excited (k) state. Γjk is termed as line broadening, which is characterized by electron

As high harmonic generation is possible with TD-DFT methods, emission spectra [48] are

ð

Absorption spectrum is the fundamental property of a system. One working on absorption spectra calculations is mainly concerned with the lowest excited states commonly. Though, excitations over a broader energy range may be essential, consequentially requiring very challenging calculations while working for large molecular complexes and high density of states (DOS) materials [49]. Hence, for large systems approximate, precise and efficient computational methods have to be looked for. LR-TD-DFT is extensively employed to compute absorption spectra of larger systems as well [50]. To estimate the absorption spectrum in full LR-TD-DFT context, a two-sided Lanczos process is proposed in Ref. [51]. To acquire an accurate estimation of absorption spectrum, a more standard Lanczos algorithm with an

Density response of a system by applying an external time-dependent perturbation in linear

� � � � dte<sup>i</sup>ω<sup>t</sup> <sup>d</sup><sup>2</sup>

dt<sup>2</sup> d tð Þ

� � � �

2

<sup>μ</sup>jk � �<sup>2</sup> ħωjk � ħω � iΓjk

Ijk ≈ ωjk

vibration broadening and inversely proportional to the period of k state.

σemisionð Þ¼ ω

4.2. Linear response TD-DFT(LR-TD-DFT) and absorption spectrum

response formulation of TD-DFT (LR-TD-DFT) is usually studied as [52]:

appropriately selected inner product was employed.

4.2.1. LR-TD-DFT formulation

! space. Intensity of absorption is in direct

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(35)

(36)

! from ground jth to kth excited state; hence, it can be

because of their greater oscillator strength (fI) [45].

spectra. The integration is completed over real r

4.1.2. Calculating the intensity of absorption

proportion to transition dipole moment μjk

estimated by formula [47]:

4.1.3. Emission spectra calculation

calculated by:

Obviously, ω = EI � E0 is an excitation energy in the case of |I〉〈0| and ω = E0 � EI is a deexcitation in the case of |0〉〈I|. To more simply relate the molar extinction coefficient (ε) in an absorption experimentation, Eq. (32) has been consciously expressed in Gaussian instead of atomic units. In SI units, frequency spectrum is assumed to first approximation [42] by:

$$\epsilon(\upsilon) = \sum\_{l} \frac{N\_{A}e^{2}}{4m\_{c}c \ln\left(10\right)\epsilon\_{0}} S(\upsilon - \upsilon\_{l}) = \left(6.94 \times 10^{+18} \frac{L}{cm} \, s\right) \sum\_{l} S(\upsilon - \upsilon\_{l})\tag{34}$$

where S(ν) is a spectral shape function usually a Gaussian whose full width at half maximal is determined through experiment with area normalized to unity, whereas NA depicts the Avogadro's number.

Absorption spectrum frequently results by excitations having larger oscillator strength because absorption peaks are often scaled with oscillator strength (fI) of the excitation. Excitations with small oscillator strength (fI) are also omitted from final spectrum while fundamentally working on oscillator strength carrying subspace so that absorption spectrum is possibly calculated with less excitations in a fixed energy space. A common problem considered is that a large number of excitations have to be estimated with the purpose of covering energy space of concentration, while figure of absorption spectrum is determined by only some of them because of their greater oscillator strength (fI) [45].

#### 4.1.2. Calculating the intensity of absorption

Another detailed DFT method is presented, proposed in ref. [46], to calculate UV absorption spectra. The integration is completed over real r ! space. Intensity of absorption is in direct proportion to transition dipole moment μjk ! from ground jth to kth excited state; hence, it can be estimated by formula [47]:

$$I\_{jk} \approx \omega\_{jk} \frac{\left(\mu\_{jk}\right)2}{\hbar \omega\_{jk} - \hbar \omega \pm i \Gamma\_{jk}}\tag{35}$$

where ℏω is the input running radiation energy and ℏωjk is energy difference between ground (j) and excited (k) state. Γjk is termed as line broadening, which is characterized by electron vibration broadening and inversely proportional to the period of k state.

#### 4.1.3. Emission spectra calculation

4.1.1. Calculating the oscillator strength

Figure 1. Absorption spectra of benzene.

<sup>E</sup>ð Þ¼ <sup>v</sup> <sup>X</sup> I

Avogadro's number.

NAe<sup>2</sup> 4mec ln 10 ð ÞE<sup>0</sup>

116 Density Functional Calculations - Recent Progresses of Theory and Application

in a complete basis set [42].

It is an important point to note that TD-DFT calculations not only give excitation energies (ω) but also provide respective oscillator strengths. Oscillator strengths are actually pure numbers

Obviously, ω = EI � E0 is an excitation energy in the case of |I〉〈0| and ω = E0 � EI is a deexcitation in the case of |0〉〈I|. To more simply relate the molar extinction coefficient (ε) in an absorption experimentation, Eq. (32) has been consciously expressed in Gaussian instead of atomic units. In SI units, frequency spectrum is assumed to first approximation [42] by:

where S(ν) is a spectral shape function usually a Gaussian whose full width at half maximal is determined through experiment with area normalized to unity, whereas NA depicts the

Absorption spectrum frequently results by excitations having larger oscillator strength because absorption peaks are often scaled with oscillator strength (fI) of the excitation. Excitations with small oscillator strength (fI) are also omitted from final spectrum while fundamentally working

S vð Þ¼ � vI <sup>6</sup>:<sup>94</sup> � <sup>10</sup>þ<sup>18</sup> <sup>L</sup>

<sup>3</sup><sup>h</sup> <sup>ω</sup>I∣h i <sup>0</sup>jrj<sup>I</sup> <sup>∣</sup><sup>2</sup> (33)

cm :<sup>s</sup>

I

S vð Þ � vI (34)

� �X

<sup>f</sup> <sup>I</sup> <sup>¼</sup> <sup>2</sup>me

As high harmonic generation is possible with TD-DFT methods, emission spectra [48] are calculated by:

$$\sigma\_{emision}(\omega) = \left| \int dt e^{i\omega t} \frac{d^2}{dt^2} d(t) \right|^2 \tag{36}$$

#### 4.2. Linear response TD-DFT(LR-TD-DFT) and absorption spectrum

Absorption spectrum is the fundamental property of a system. One working on absorption spectra calculations is mainly concerned with the lowest excited states commonly. Though, excitations over a broader energy range may be essential, consequentially requiring very challenging calculations while working for large molecular complexes and high density of states (DOS) materials [49]. Hence, for large systems approximate, precise and efficient computational methods have to be looked for. LR-TD-DFT is extensively employed to compute absorption spectra of larger systems as well [50]. To estimate the absorption spectrum in full LR-TD-DFT context, a two-sided Lanczos process is proposed in Ref. [51]. To acquire an accurate estimation of absorption spectrum, a more standard Lanczos algorithm with an appropriately selected inner product was employed.

#### 4.2.1. LR-TD-DFT formulation

Density response of a system by applying an external time-dependent perturbation in linear response formulation of TD-DFT (LR-TD-DFT) is usually studied as [52]:

$$
\rho(\mathbf{r},t) = \rho\_0(\mathbf{r}) + \delta(\mathbf{r},t) \tag{37}
$$

excitations. They do so by transforming the ordinary orbital representation into a more com-

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i. First step follows the calculations presented in Section 2.1.5 till step i for geometry optimization calculations and then follow steps used to perform TD-DFT calculation of the optimized structure for UV-Visible spectral calculations. Solvent can be added using

ii. Calculate the absorption spectrum. Compare the calculated absorption spectra (lambda max) with the experiment taking f > 0.0. Basis set/methods can be changed to generate

iii. Run a single point calculation to generate the NTOs for the desired excited state. This

In our work [55], we have synthesized 1,3-diisobutyl thiourea (C9H20N2S) and its vibrational analysis was carried out using DFT methods. Theoretical calculations for vibrational spectra

Figure 3. Comparative vibrational spectra of compound in gaseous state (theoretical, calculated using B3LYP/6–311G

pact particle (occupied) to empty hole (unoccupied) [54].

spectrum in good agreement with theoretical assumption.

ii. Examine the calculation results to determine the excited state of interest.

i. Run an excited state calculation, saving the checkpoint file.

step can be repeated to view focusing excited state.

5. Examples illustrating vibrational calculations

Figure 2. Crystal structure (ORTEP plot) of 1,3-diisobutyl thiourea compound.

method) and in the solid state (experimental).

TD (singlets, nstates).

Generating NTOs:

In chemistry, LR-TD-DFT equations are articulated in matrix arrangement using the Kohn-Sham basic formulation. First-order density response δr(r, t) can be extended on the basis of unperturbed orbitals. Linear response of the electronic density articulated in matrix form is depicted as:

$$\delta\rho\_{\vec{\eta}}(t) = \int\_{t0}^{t} dt' \sum\_{kl} \delta v\_{kl}(t') \chi\_{\vec{\eta}, \cdot} \, \prescript{}{kl}{(\mathbf{r}, t, \mathbf{r}', \mathbf{t}')}\tag{38}$$

In Fourier space, it is represented as:

$$\delta \rho\_{\vec{\eta}}(\omega) = \sum\_{kl} \delta v\_{kl}(\omega) \chi\_{\vec{\eta}\_{\vec{\eta}}} \, \_{kl}(\omega) \tag{39}$$

The connection between Kohn-Sham response function and particular density response function is efficiently stated in terms of the inverse of their conforming Fourier transform time, t2 � t1 as follows:

$$\chi^{-1}(\mathbf{r}, \mathbf{r}', \omega) = \chi\_s^{-1}(\mathbf{r}, \mathbf{r}', \omega) - \frac{1}{|\mathbf{r}\_1 - \mathbf{r}\_2|} - f\_{\text{xc}}(\mathbf{r}\_1, \mathbf{r}\_2 \omega) \tag{40}$$

In summary, locating excitation energies of the interacting system is problematic, and poles of the response function give the way to calculate excitation energies.

$$\{f\_I, \omega\_I\} \Rightarrow poles \ of \ \chi(\omega).$$

Actually, χ(ω) has poles at correct excitation energies ωI. From this point, calculations for absorption spectra follow the route as explained in Section 4.1. Computation of the dynamic dipole polarizability [α(ω)] is a noninteracting response of specific interest in calculating absorption spectra, which is a response function that relays external potential to the change in dipole as depicted in Section 4.1. The Fourier transform of the dynamic dipole polarizability can be written as in Eq. (29). Excitation energies (ωI) are calculated by poles of the dynamic polarizability [53], while the oscillator strengths are determined by residual (fI) [52].

Calculating the absorption spectrum with LR-TD-DFT method also includes resolving a non-Hermitian eigenvalue problem; therefore, the absorption spectrum is attained as [50]:

$$\sigma(\omega) = \frac{1}{3} \sum\_{i} \mathbf{f}\_i^2 |\delta(\omega - \lambda\_i) - \delta(\omega + \lambda\_i)| \tag{41}$$

#### 4.3. Electronic calculations using Gaussian software

TD-DFT method in Gaussian makes it practical to study excited state systems since it produces results that are comparable in accuracy to ground-state DFT calculations. Natural transition orbitals (NTOs) can be a helpful way of obtaining a qualitative description of electronic excitations. They do so by transforming the ordinary orbital representation into a more compact particle (occupied) to empty hole (unoccupied) [54].


Generating NTOs:

rð Þ¼ r; t r0ð Þþ r δð Þ r; t (37)

;t<sup>0</sup> ð Þ (38)

� f xcð Þ r1;r2ω (40)

δvklð Þ ω χij, klð Þ ω (39)

<sup>i</sup> δ ωð Þ� � λ<sup>i</sup> δ ωð Þ þ λ<sup>i</sup> j j (41)

In chemistry, LR-TD-DFT equations are articulated in matrix arrangement using the Kohn-Sham basic formulation. First-order density response δr(r, t) can be extended on the basis of unperturbed orbitals. Linear response of the electronic density articulated in matrix form is

δvkl t

<sup>0</sup> ð Þχij, kl r; t;r<sup>0</sup>

1 j j r<sup>1</sup> � r<sup>2</sup>

δrijðÞ¼ t

118 Density Functional Calculations - Recent Progresses of Theory and Application

<sup>χ</sup>�<sup>1</sup> <sup>r</sup>;r<sup>0</sup> ð Þ¼ ; <sup>ω</sup> <sup>χ</sup>�<sup>1</sup>

the response function give the way to calculate excitation energies.

σ ωð Þ¼ <sup>1</sup> 3 X i f 2

4.3. Electronic calculations using Gaussian software

In Fourier space, it is represented as:

ðt t0 dt<sup>0</sup> X kl

<sup>δ</sup>rijð Þ¼ <sup>ω</sup> <sup>X</sup>

<sup>s</sup> r;r <sup>0</sup> ð Þ� ; ω

f <sup>I</sup>; ω<sup>I</sup>

polarizability [53], while the oscillator strengths are determined by residual (fI) [52].

Hermitian eigenvalue problem; therefore, the absorption spectrum is attained as [50]:

kl

The connection between Kohn-Sham response function and particular density response function is efficiently stated in terms of the inverse of their conforming Fourier transform time,

In summary, locating excitation energies of the interacting system is problematic, and poles of

� � ) poles of χ ωð Þ

Actually, χ(ω) has poles at correct excitation energies ωI. From this point, calculations for absorption spectra follow the route as explained in Section 4.1. Computation of the dynamic dipole polarizability [α(ω)] is a noninteracting response of specific interest in calculating absorption spectra, which is a response function that relays external potential to the change in dipole as depicted in Section 4.1. The Fourier transform of the dynamic dipole polarizability can be written as in Eq. (29). Excitation energies (ωI) are calculated by poles of the dynamic

Calculating the absorption spectrum with LR-TD-DFT method also includes resolving a non-

TD-DFT method in Gaussian makes it practical to study excited state systems since it produces results that are comparable in accuracy to ground-state DFT calculations. Natural transition orbitals (NTOs) can be a helpful way of obtaining a qualitative description of electronic

depicted as:

t2 � t1 as follows:


#### 5. Examples illustrating vibrational calculations

In our work [55], we have synthesized 1,3-diisobutyl thiourea (C9H20N2S) and its vibrational analysis was carried out using DFT methods. Theoretical calculations for vibrational spectra

Figure 2. Crystal structure (ORTEP plot) of 1,3-diisobutyl thiourea compound.

Figure 3. Comparative vibrational spectra of compound in gaseous state (theoretical, calculated using B3LYP/6–311G method) and in the solid state (experimental).

analysis using DFT B3LYP/6–311G method were done in gaseous state, while experimental calculations were done in solid state. Geometry of compound was optimized using Gaussian employing DFT/B3LYP method with the 6–311G basis set. Optimized geometry of compound was compared with crystal structure of compound that depicted the support for the crystal structure (Figure 2). DFT results for vibrational analysis were compared with experimental

Sundaraganesun et al., calculated structure, harmonic frequencies and vibrational mode assignments for 2-chlorobenzoic acid using HF and DFT methods employing the 6-311++G(d,p) basis set. The results of the molecular structure and vibrational frequencies obtained on the basis of calculations in Gaussian are critically equated with experimental IR data recorded in gas

Alver et al. carried out NMR spectroscopic study, and DFT calculations of GIAO NMR

means of B3LYP method, and 6-311++G(d,p) basis set is used. Comparison between the experimental and the theoretical results illustrates that density functional B3LYP method is able to

Nuclei Experimental (ppm) B3LYP (ppm)

deliver suitable results for expecting NMR properties (Table 1, Figures 6 and 7) [57].

C3, C7 28.2 31.1 C4, C6 30.5 34.6 C5 30.6 34.9 C2, C8 33.2 37.9 C1, C9 42.4 48.9 H16, 17, H24, 25 1.3 1.2 H18, 19, H22, 23, H20, 21 1.3 1.3 H30, 31, H32, 33 1.5 1.2 H14, 15, H26, 27 1.5 1.3 H12, 13, H28, 29 2.6 2.5

J(CnHn) Experimental (Hz) B3LYP (Hz)

H NMR chemical shifts (ppm) and <sup>1</sup>

C2H14H15, C8H26H27 122.4 118.2 C3H16H17, C7H24H25 122.9 119.1 C4H18H19, C6H22H23 123.8 119.7 C5H20H21, C1H12H13, C9H28H29 135.2 132.1

J spin-spin coupling constants of 1,9-diaminononane (danon, C9H22N2) were

JCH coupling constants of danon are calculated by

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JCH NMR coupling constants (Hz) of

analysis, which were found in good agreement with each other (Figure 3) [55].

6. Example illustrating calculation of NMR parameters

H, 13C NMR chemical shifts and <sup>1</sup>

phase (Figures 4 and 5) [56].

shieldings and <sup>1</sup>

done. <sup>1</sup>

1

danon.

Table 1. Experimental and calculated 13C, <sup>1</sup>

Figure 4. Numbering system implemented for 2-chlorobenzoic acid.

Figure 5. Graphic correlation between the experimental and calculated wavenumber obtained by the ab initio HF and DFT/B3LYP/6–311++G(d,p) methods for 2-chlorobenzoic acid.

analysis using DFT B3LYP/6–311G method were done in gaseous state, while experimental calculations were done in solid state. Geometry of compound was optimized using Gaussian employing DFT/B3LYP method with the 6–311G basis set. Optimized geometry of compound was compared with crystal structure of compound that depicted the support for the crystal structure (Figure 2). DFT results for vibrational analysis were compared with experimental analysis, which were found in good agreement with each other (Figure 3) [55].

Sundaraganesun et al., calculated structure, harmonic frequencies and vibrational mode assignments for 2-chlorobenzoic acid using HF and DFT methods employing the 6-311++G(d,p) basis set. The results of the molecular structure and vibrational frequencies obtained on the basis of calculations in Gaussian are critically equated with experimental IR data recorded in gas phase (Figures 4 and 5) [56].

#### 6. Example illustrating calculation of NMR parameters

Figure 4. Numbering system implemented for 2-chlorobenzoic acid.

120 Density Functional Calculations - Recent Progresses of Theory and Application

DFT/B3LYP/6–311++G(d,p) methods for 2-chlorobenzoic acid.

Figure 5. Graphic correlation between the experimental and calculated wavenumber obtained by the ab initio HF and

Alver et al. carried out NMR spectroscopic study, and DFT calculations of GIAO NMR shieldings and <sup>1</sup> J spin-spin coupling constants of 1,9-diaminononane (danon, C9H22N2) were done. <sup>1</sup> H, 13C NMR chemical shifts and <sup>1</sup> JCH coupling constants of danon are calculated by means of B3LYP method, and 6-311++G(d,p) basis set is used. Comparison between the experimental and the theoretical results illustrates that density functional B3LYP method is able to deliver suitable results for expecting NMR properties (Table 1, Figures 6 and 7) [57].


Table 1. Experimental and calculated 13C, <sup>1</sup> H NMR chemical shifts (ppm) and <sup>1</sup> JCH NMR coupling constants (Hz) of danon.

7. Conclusion

IR, NMR and UV spectral calculations.

DOI: 10.1103/PhysRev.136.B864

169-219. DOI: 10.1016/S1380-7323(05)80036-6

\*, Samia Kausar<sup>1</sup> and Amin Badshah<sup>2</sup>

1 Department of Chemistry, University of Gujrat, Gujrat, Pakistan

2 Department of Chemistry, Quaid-i-Azam University, Islamabad, Pakistan

Journal of Chemical Physics. 1993;98:1372-1377. DOI: 10.1063/1.464304

Physical Review. 1965;140:A1133. DOI: 10.1103/PhysRev.140.A1133

[1] Becke AD. A new mixing of Hartree-Fock and local density-functional theories. The

[2] Hohenberg P, Kohn W. Inhomogeneous electron gas. Physical Review. 1964;136:B864.

[3] Johnson BG. Development, implementation and applications of efficient methodologies for density functional calculations. Theoretical and Computational Chemistry. 1995;2:

[4] Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects.

[5] Seminario JM. An introduction to density functional theory in chemistry. Theoretical and

Computational Chemistry. 1995;2:1-27. DOI: 10.1016/S1380-7323(05)80031-7

\*Address all correspondence to: atafali.altaf@uog.edu.pk

Author details

Ataf Ali Altaf<sup>1</sup>

References

This chapter focused on the use of DFT-based methods for spectral calculations, i.e., of vibrational, NMR and electronic calculations. Vibrational spectral generation depends upon Gaussian broadening function, while vibrational frequencies are calculated by determining the second derivative of energy after geometry optimization. Vibrational frequencies and zeropoint energy for maximum vibration can be calculated theoretically as well. B3LYP density functional hybrid method is proposed to give more exact values of vibrational frequencies. The calculation of NMR parameters on DFT grounds depends upon exchange correlation functional, i.e., LDA, GGA and mostly employed hybrid functional B3LYP. The GIAO and IGLO methods are used to solving gauge problems in NMR shielding tensor calculations. Nuclear spin-spin coupling constants can be calculated on theoretical grounds, whereas DFT-B3LYP-GIAO method is employed for chemical shift calculations. For electronic spectral calculations, TD-DFT is employed for excited state measurements. Vertical excitations, oscillator strength and intensity of absorption are calculated using TD-DFT theoretical methods. Linear response function incorporation to TD-DFT approach has proved helpful to study electronic density and absorption spectrum calculations of larger systems. Gaussian software is most employed for

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http://dx.doi.org/10.5772/intechopen.71080

Figure 6. Optimized molecular structure of danon.

Figure 7. Plot of the calculated vs. the experimental 13C NMR chemical shifts (ppm) and <sup>1</sup> J coupling constants (Hz) of danon.

#### 6.1. Different software for DFT calculations

Gaussian is most extensively used software as graphical demonstration of variability of results, e.g., simulation of normal modes, molecular orbital, atomic charges, electrostatic potential, NMR shielding density and potential energy surface scans, can be displayed. Besides this, Octopus is a scientific program aimed at the ab initio virtual experimentation. Electrons are described quantum-mechanically within DFT and TDDFT when doing simulations in time. Spartan is a potent software tool that applies the power of molecular mechanics and quantum chemical calculations on chemistry research with sophisticated computational algorithms. Dalton is a powerful molecular electronic structure program, with an extensive functional for the calculation of molecular properties at the HF, DFT, MCSCF and CC levels of theory. DMol3 is a unique, accurate and reliable DFT quantum mechanical code for research in the chemical and pharmaceutical industries. CASTEP is a software package that uses DFT to provide a good atomic level description of all manner of materials and molecules. It can give information about total energies, forces and stresses on an atomic system, as well as calculate optimum geometries, band structures, optical spectra, phonon spectra and much more. Dynamic simulations can be performed also. HyperChem software includes molecular mechanics, molecular dynamics, and semiempirical and ab initio molecular orbital methods. HyperChemData and HyperNMR have been migrated into HyperChem and new features have been added.

#### 7. Conclusion

This chapter focused on the use of DFT-based methods for spectral calculations, i.e., of vibrational, NMR and electronic calculations. Vibrational spectral generation depends upon Gaussian broadening function, while vibrational frequencies are calculated by determining the second derivative of energy after geometry optimization. Vibrational frequencies and zeropoint energy for maximum vibration can be calculated theoretically as well. B3LYP density functional hybrid method is proposed to give more exact values of vibrational frequencies. The calculation of NMR parameters on DFT grounds depends upon exchange correlation functional, i.e., LDA, GGA and mostly employed hybrid functional B3LYP. The GIAO and IGLO methods are used to solving gauge problems in NMR shielding tensor calculations. Nuclear spin-spin coupling constants can be calculated on theoretical grounds, whereas DFT-B3LYP-GIAO method is employed for chemical shift calculations. For electronic spectral calculations, TD-DFT is employed for excited state measurements. Vertical excitations, oscillator strength and intensity of absorption are calculated using TD-DFT theoretical methods. Linear response function incorporation to TD-DFT approach has proved helpful to study electronic density and absorption spectrum calculations of larger systems. Gaussian software is most employed for IR, NMR and UV spectral calculations.

#### Author details

Ataf Ali Altaf<sup>1</sup> \*, Samia Kausar<sup>1</sup> and Amin Badshah<sup>2</sup>

\*Address all correspondence to: atafali.altaf@uog.edu.pk


#### References

6.1. Different software for DFT calculations

danon.

Figure 6. Optimized molecular structure of danon.

122 Density Functional Calculations - Recent Progresses of Theory and Application

Gaussian is most extensively used software as graphical demonstration of variability of results, e.g., simulation of normal modes, molecular orbital, atomic charges, electrostatic potential, NMR shielding density and potential energy surface scans, can be displayed. Besides this, Octopus is a scientific program aimed at the ab initio virtual experimentation. Electrons are described quantum-mechanically within DFT and TDDFT when doing simulations in time. Spartan is a potent software tool that applies the power of molecular mechanics and quantum chemical calculations on chemistry research with sophisticated computational algorithms. Dalton is a powerful molecular electronic structure program, with an extensive functional for the calculation of molecular properties at the HF, DFT, MCSCF and CC levels of theory. DMol3 is a unique, accurate and reliable DFT quantum mechanical code for research in the chemical and pharmaceutical industries. CASTEP is a software package that uses DFT to provide a good atomic level description of all manner of materials and molecules. It can give information about total energies, forces and stresses on an atomic system, as well as calculate optimum geometries, band structures, optical spectra, phonon spectra and much more. Dynamic simulations can be performed also. HyperChem software includes molecular mechanics, molecular dynamics, and semiempirical and ab initio molecular orbital methods. HyperChemData and HyperNMR have

J coupling constants (Hz) of

been migrated into HyperChem and new features have been added.

Figure 7. Plot of the calculated vs. the experimental 13C NMR chemical shifts (ppm) and <sup>1</sup>


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**Chapter 6**

**Provisional chapter**

**DFT Calculations and Statistical Mechanics Applied to**

In the present work, molar fractions were obtained as a function of temperature with different levels of theory for the most representative isomers of three systems belonging to the family of pseudosaccharins. The choice of those three systems was due to the fact that it is known in the scientific literature that these systems present very small differences in their relative energies which make a complicated experimental characterization, in addition these compounds are of interest in the biological area. These systems represent challenges not only from an experimental point of view but also from a theoretical point of view. From the theoretical perspective, this study is also complicated since several possible isomers with very similar energies are presented. The diagrams of species distribution (molar fractions) provide information that cannot be accessed through the electronic structure calculations at T = 0. Here, this tool was useful to identify the most probable isomer from several quasi-degenerate isomers and to discern if thermal effects favor any of them, as well as to find trends despite the different results of each level of theory. Additionally, an analysis was performed on vibrational, rotational and electronic data in order to know the reason of the behavior of molar fractions as function of temperature.

**DFT Calculations and Statistical Mechanics Applied to** 

DOI: 10.5772/intechopen.70933

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Having the same chemical composition, a molecule can present different geometrical arrangements of atoms, leading to distinct chemical and physical properties. Here lies the importance

**Keywords:** molar fractions, pseudosaccharin, isomerism, DFT calculations,

**Isomerization of Pseudosaccharins**

**Isomerization of Pseudosaccharins**

Zuriel Natanael Cisneros-García,

Zuriel Natanael Cisneros-García,

Jaime Gustavo Rodríguez-Zavala

Jaime Gustavo Rodríguez-Zavala

http://dx.doi.org/10.5772/intechopen.70933

Francisco J. Tenorio and

**Abstract**

statistical mechanics

**1. Introduction**

Francisco J. Tenorio and

David Alejandro Hernández-Velázquez,

David Alejandro Hernández-Velázquez,

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Provisional chapter**

#### **DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins Isomerization of Pseudosaccharins**

**DFT Calculations and Statistical Mechanics Applied to** 

DOI: 10.5772/intechopen.70933

Zuriel Natanael Cisneros-García, David Alejandro Hernández-Velázquez, Francisco J. Tenorio and Jaime Gustavo Rodríguez-Zavala David Alejandro Hernández-Velázquez, Francisco J. Tenorio and Jaime Gustavo Rodríguez-Zavala Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70933

Zuriel Natanael Cisneros-García,

#### **Abstract**

In the present work, molar fractions were obtained as a function of temperature with different levels of theory for the most representative isomers of three systems belonging to the family of pseudosaccharins. The choice of those three systems was due to the fact that it is known in the scientific literature that these systems present very small differences in their relative energies which make a complicated experimental characterization, in addition these compounds are of interest in the biological area. These systems represent challenges not only from an experimental point of view but also from a theoretical point of view. From the theoretical perspective, this study is also complicated since several possible isomers with very similar energies are presented. The diagrams of species distribution (molar fractions) provide information that cannot be accessed through the electronic structure calculations at T = 0. Here, this tool was useful to identify the most probable isomer from several quasi-degenerate isomers and to discern if thermal effects favor any of them, as well as to find trends despite the different results of each level of theory. Additionally, an analysis was performed on vibrational, rotational and electronic data in order to know the reason of the behavior of molar fractions as function of temperature.

**Keywords:** molar fractions, pseudosaccharin, isomerism, DFT calculations, statistical mechanics

#### **1. Introduction**

Having the same chemical composition, a molecule can present different geometrical arrangements of atoms, leading to distinct chemical and physical properties. Here lies the importance

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

of knowing the arrangement of the atoms inside molecules. The different atomic arrangements in a molecule are known as isomers and their study gives rise to isomerization or isomerism. Computational chemistry is one of the most used tools in this branch of research. Through the analysis of stabilities by means of the electronic energy, the isomers that could be found in an experiment can be known. However, when the energetic differences between isomers are very small, it is difficult to assure that the lower energy isomer is the only one present in the experiment. Existing models, both in the framework of methods of electronic structure or molecular dynamics, have contributed greatly to the study of the different types of isomerization, and have been improved taking into account, for example, the inclusion of solvents. Despite advances in theory and technology, no current theoretical methodology is 100% accurate. This leads to the fact that it is not sufficient to consider only the results yielded by these methods, mainly in molecules whose isomers have very similar energies, which could be considered as part of the error associated to the level of theory.

why such rearrangement has awakened the interest to elucidate the mechanism of isomerization [3]. Moreover, pseudosaccharins are receptors of a quite potent electronic density

DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins

http://dx.doi.org/10.5772/intechopen.70933

131

Pseudosaccharins are compounds of interest in several fields of research, specifically in biologically active systems [4]. Their usage has been reported in the treatment of cardiac and hepatic diseases [5, 6], and even for the treatment of breast cancer, as antibiotic agents and herbicides [7–9]. Adding to these applications, Hlasta et al. [10] proposed a series of compounds of the pseudosaccharins family as inhibitors of the human leukocyte elastase protein, which causes the destruction of the connective tissue of the lungs in people with pulmonary emphysema. They also found how this series of compounds have therapeutic usage in people with this

Due to the importance of this type of compounds, studies have been carried out to characterize experimentally and theoretically both their structure and properties. A high reactivity can be found mainly in the junction of the heterocyclic ring and **R**, due to an electron-withdrawing effect of the pseudosaccharyl ring system, which is the reason of the Chapman-like rearrangement. The properties of the pseudosaccharins can be modified depending on which structures are combined with saccharin. In the case of tetrazole-saccharin, the properties of tetrazole, which can be very reactive, have been combined with saccharin, resulting in a greater stability of the tetrazole [12]. Furthermore, the combination of an allyl group with saccharin can also

In previous studies [4, 13, 14], the importance of isomerization in these systems has been demonstrated. As the **R** group can be very flexible, it gives rise to several possible isomers. This can lead to challenges in the characterization of these compounds, since there could be a mixture of isomers whose population can become significant during the synthesis. As a result, there are signals in spectrophotometry [4, 12, 13] that cannot be associated with a precise isomer. Theoretical analysis can ease to obtain the different spectra of the possible isomers and the spectra could be weighted by a number or parameter, for example, a calculated molar fraction. However, when there are small energy differences, it is important to consider the methodology used to obtain the most probable isomers to study; that is, having the challenge of solving structures whose differences in energy are small, methodology becomes very important, because a different functional and basis set (BS) could give rise to different relative

Due to all the previously mentioned, in the present chapter a systematic study of the isomerization of three pseudosaccharins, which have very small energy differences between them and that have also been previously studied experimentally and theoretically, is performed. For this study, a variation of functional and BS is carried out, however, considering that a better comparison can be made between experimental and theoretical results if thermal effects are taken into account, molar fractions are obtained for the different isomers of pseudosaccharins. It is important to mention that, unlike previous studies on these compounds, populations are obtained as a function of temperature, taking into account different contributions through the vibrational, rotational and electronic partition functions. From this, it is possible to observe the effects of methodology in the

molar fractions, as well as to find the reason of behavior of molar fractions plots.

that has called attention in research areas of synthesis of organic compounds.

form compounds with the properties mentioned in previous paragraphs.

condition [10, 11].

stabilities between the isomers.

For this reason, given the importance of elucidating the isomerism of a molecular system with very small energy differences, it is important to make use of other tools. In this chapter, we will obtain diagrams of species predominance (molar fractions) as a function of the temperature (assuming that the equilibrium has been reached) for some systems of the pseudosaccharin family that have isomeric structures with small energy differences and are of great importance in various fields of applied sciences.

Pseudosaccharins, also known as benzisothiazoles, are molecular systems with a wide variety of properties. Their structures are derived from saccharin, which is the first synthetic sweetener that is still used with that aim, but also has many industrial applications. Its photosensitive properties have aroused interest in the derivatives of this system for applications of synthesis of coordination compounds and bioorganic synthesis [1]. Its general structure can be seen in **Figure 1**, where **R** can be any other functional group or molecule. Additionally, these structures may present a tautomerism known as Chapman rearrangement [2]; this internal reaction of the structure consists in the movement of the **R** group to the oxygen of the carbonyl group as the temperature increases. Depending on the degrees of freedom that the **R** group possesses, different structures can be formed with different properties, reason

**Figure 1.** General structure for the family of pseudosaccharins.

why such rearrangement has awakened the interest to elucidate the mechanism of isomerization [3]. Moreover, pseudosaccharins are receptors of a quite potent electronic density that has called attention in research areas of synthesis of organic compounds.

of knowing the arrangement of the atoms inside molecules. The different atomic arrangements in a molecule are known as isomers and their study gives rise to isomerization or isomerism. Computational chemistry is one of the most used tools in this branch of research. Through the analysis of stabilities by means of the electronic energy, the isomers that could be found in an experiment can be known. However, when the energetic differences between isomers are very small, it is difficult to assure that the lower energy isomer is the only one present in the experiment. Existing models, both in the framework of methods of electronic structure or molecular dynamics, have contributed greatly to the study of the different types of isomerization, and have been improved taking into account, for example, the inclusion of solvents. Despite advances in theory and technology, no current theoretical methodology is 100% accurate. This leads to the fact that it is not sufficient to consider only the results yielded by these methods, mainly in molecules whose isomers have very similar energies, which could be considered as

For this reason, given the importance of elucidating the isomerism of a molecular system with very small energy differences, it is important to make use of other tools. In this chapter, we will obtain diagrams of species predominance (molar fractions) as a function of the temperature (assuming that the equilibrium has been reached) for some systems of the pseudosaccharin family that have isomeric structures with small energy differences and are of great

Pseudosaccharins, also known as benzisothiazoles, are molecular systems with a wide variety of properties. Their structures are derived from saccharin, which is the first synthetic sweetener that is still used with that aim, but also has many industrial applications. Its photosensitive properties have aroused interest in the derivatives of this system for applications of synthesis of coordination compounds and bioorganic synthesis [1]. Its general structure can be seen in **Figure 1**, where **R** can be any other functional group or molecule. Additionally, these structures may present a tautomerism known as Chapman rearrangement [2]; this internal reaction of the structure consists in the movement of the **R** group to the oxygen of the carbonyl group as the temperature increases. Depending on the degrees of freedom that the **R** group possesses, different structures can be formed with different properties, reason

part of the error associated to the level of theory.

130 Density Functional Calculations - Recent Progresses of Theory and Application

importance in various fields of applied sciences.

**Figure 1.** General structure for the family of pseudosaccharins.

Pseudosaccharins are compounds of interest in several fields of research, specifically in biologically active systems [4]. Their usage has been reported in the treatment of cardiac and hepatic diseases [5, 6], and even for the treatment of breast cancer, as antibiotic agents and herbicides [7–9]. Adding to these applications, Hlasta et al. [10] proposed a series of compounds of the pseudosaccharins family as inhibitors of the human leukocyte elastase protein, which causes the destruction of the connective tissue of the lungs in people with pulmonary emphysema. They also found how this series of compounds have therapeutic usage in people with this condition [10, 11].

Due to the importance of this type of compounds, studies have been carried out to characterize experimentally and theoretically both their structure and properties. A high reactivity can be found mainly in the junction of the heterocyclic ring and **R**, due to an electron-withdrawing effect of the pseudosaccharyl ring system, which is the reason of the Chapman-like rearrangement. The properties of the pseudosaccharins can be modified depending on which structures are combined with saccharin. In the case of tetrazole-saccharin, the properties of tetrazole, which can be very reactive, have been combined with saccharin, resulting in a greater stability of the tetrazole [12]. Furthermore, the combination of an allyl group with saccharin can also form compounds with the properties mentioned in previous paragraphs.

In previous studies [4, 13, 14], the importance of isomerization in these systems has been demonstrated. As the **R** group can be very flexible, it gives rise to several possible isomers. This can lead to challenges in the characterization of these compounds, since there could be a mixture of isomers whose population can become significant during the synthesis. As a result, there are signals in spectrophotometry [4, 12, 13] that cannot be associated with a precise isomer. Theoretical analysis can ease to obtain the different spectra of the possible isomers and the spectra could be weighted by a number or parameter, for example, a calculated molar fraction. However, when there are small energy differences, it is important to consider the methodology used to obtain the most probable isomers to study; that is, having the challenge of solving structures whose differences in energy are small, methodology becomes very important, because a different functional and basis set (BS) could give rise to different relative stabilities between the isomers.

Due to all the previously mentioned, in the present chapter a systematic study of the isomerization of three pseudosaccharins, which have very small energy differences between them and that have also been previously studied experimentally and theoretically, is performed. For this study, a variation of functional and BS is carried out, however, considering that a better comparison can be made between experimental and theoretical results if thermal effects are taken into account, molar fractions are obtained for the different isomers of pseudosaccharins. It is important to mention that, unlike previous studies on these compounds, populations are obtained as a function of temperature, taking into account different contributions through the vibrational, rotational and electronic partition functions. From this, it is possible to observe the effects of methodology in the molar fractions, as well as to find the reason of behavior of molar fractions plots.

#### **2. Theoretical background**

All calculations were performed with Gaussian09 suite of programs [15], while the construction of graphics of molar fractions as a function of temperature was made with a program designed by Professor Slanina et al. [16]. Relaxation of molecular structures, vibrational frequencies, rotational constants and excited states were obtained through B3LYP [17, 18], M06-2X [19] and PBE [20] exchange-correlation (XC) functionals, along with 6-311++G(3df,3pd), 6-31++G(3df,3pd), 6-311+G(d,p) and 6-31+G(d,p) [21, 22] basis. The choice of the methodology was made taking as a starting point the level of theory used in previous studies [4, 13, 14]. Additionally, the other functionals and BSs were used in order to be able to compare the results and analyze the importance of the methodology. It is important to mention that, for selected cases, total electronic energies were refined through the second-order Møller-Plesset perturbation theory, MP2, along with 6-311++G(3df,3pd) and 6-31+G(d,p) level of calculation.

In this work, calculations of molar fractions (*xi* ) including thermal effects for *n* isomers have been obtained through their partition functions *qi* and the enthalpies at the absolute zero temperature or ground-state energies ΔH<sup>0</sup> 0,*i* (i.e. the relative potential energies corrected for the vibrational zero-point energies) by the formula:

\$\text{vibrational zero-point energies})\text{ by the formula:}\\

$$\mathbf{x}\_{i} = \frac{q\_{i}\exp\left(-\Delta H\_{0i}^{0}/RT\right)}{\sum\_{j=1}^{n}q\_{j}\exp\left(-\Delta H\_{0j}^{0}/RT\right)}\tag{1}$$

**3.1. 2-allyl-1,2-benzisothiazol-3(2***H***)-one 1,1-dioxide**

The molecular structure of the ABIOD pseudosaccharin can be considered to be mainly formed by two components; a heterocyclic ring and a flexible allyl substituent. In ABIOD, the molecule can be rotated two dihedral angles and they give rise to different isomers (see **Figure 2(a)**), nevertheless, according to Ref. [4], the dihedral angle formed by atoms 1, 2, 3 and 4 is irrelevant in the formation of isomers because of the sterical hindrance between the allyl group and the rest of the molecule. Consequently, only the dihedral angle formed by atoms 2, 3, 4 and 5 originate different isomers. In the same study, there were found a total of three isomers of ABIOD in a minimum of the potential-energy surface (PES) and they were repre-

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**Figure 2.** Pseudosaccharins studied in the present work. (a) ABIOD, (b) ABID, (c) and (d) tetrazole-saccharyl.

In a first step, molecular optimization was performed through B3LYP functional; after that,

sented using the abbreviations *C* = *cis* ≈ 0°, *Sk* = *skew* ≈ 120°, *Sk*′ = *skew*′ ≈ −120°.

reoptimizations were performed with M06-2X and PBE functionals.

where *R* is the gas constant and *T* the absolute temperature. It is worth to mention that, as it is expected, single point calculations (MP2 results) are not corrected by the vibrational zeropoint energies. Eq. (1) is an exact formula of the isomers, supposing the conditions of the inter-isomeric thermodynamic equilibrium. Rotational-vibrational partition functions were constructed from the calculated structural and vibrational data using the rigid rotator and harmonic oscillator (RRHO) approximation. The electronic partition function was constructed by 10 time-dependent electronic excitation energies. All calculations were performed taking into account a tight convergence in the self-consistent field calculation, and an ultrafine mesh in the integration.

#### **3. Results and discussion**

From previous studies on pseudosaccharins, three isomers were reported for 2-allyl-1,2-benzisothiazol-3(2*H*)-one 1,1-dioxide (ABIOD) [4], five isomers for allyl ether 3-(allyloxy)- 1,2-benzisothiazole1,1-dioxide (ABID) [13] and six isomers for tetrazole-saccharyl [14]. The isomers for each pseudosaccharyl molecule were obtained through the rotation of two dihedral angles as will be explained in the next subsections. It is shown in **Figure 2(a)** the ABIOD pseudosaccharyl, in **Figure 2(b)** the ABID pseudosaccharyl and in **Figure 2(c)** and **(d)** tetrazole saccharine. The numbers are needed to understand the formation of the different isomers through dihedral angles.

DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins http://dx.doi.org/10.5772/intechopen.70933 133

**Figure 2.** Pseudosaccharins studied in the present work. (a) ABIOD, (b) ABID, (c) and (d) tetrazole-saccharyl.

#### **3.1. 2-allyl-1,2-benzisothiazol-3(2***H***)-one 1,1-dioxide**

**2. Theoretical background**

132 Density Functional Calculations - Recent Progresses of Theory and Application

All calculations were performed with Gaussian09 suite of programs [15], while the construction of graphics of molar fractions as a function of temperature was made with a program designed by Professor Slanina et al. [16]. Relaxation of molecular structures, vibrational frequencies, rotational constants and excited states were obtained through B3LYP [17, 18], M06-2X [19] and PBE [20] exchange-correlation (XC) functionals, along with 6-311++G(3df,3pd), 6-31++G(3df,3pd), 6-311+G(d,p) and 6-31+G(d,p) [21, 22] basis. The choice of the methodology was made taking as a starting point the level of theory used in previous studies [4, 13, 14]. Additionally, the other functionals and BSs were used in order to be able to compare the results and analyze the importance of the methodology. It is important to mention that, for selected cases, total electronic energies were refined through the second-order Møller-Plesset perturbation theory,

) including thermal effects for *n* isomers have

(i.e. the relative potential energies corrected for the

and the enthalpies at the absolute zero tem-

<sup>0</sup> /*RT*) (1)

MP2, along with 6-311++G(3df,3pd) and 6-31+G(d,p) level of calculation.

0,*i*

<sup>0</sup> /*RT*) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>∑</sup>*<sup>j</sup>*=1

*<sup>n</sup> qj exp*(−∆*H*0,*<sup>j</sup>*

where *R* is the gas constant and *T* the absolute temperature. It is worth to mention that, as it is expected, single point calculations (MP2 results) are not corrected by the vibrational zeropoint energies. Eq. (1) is an exact formula of the isomers, supposing the conditions of the inter-isomeric thermodynamic equilibrium. Rotational-vibrational partition functions were constructed from the calculated structural and vibrational data using the rigid rotator and harmonic oscillator (RRHO) approximation. The electronic partition function was constructed by 10 time-dependent electronic excitation energies. All calculations were performed taking into account a tight convergence in the self-consistent field calculation, and an ultrafine mesh

From previous studies on pseudosaccharins, three isomers were reported for 2-allyl-1,2-benzisothiazol-3(2*H*)-one 1,1-dioxide (ABIOD) [4], five isomers for allyl ether 3-(allyloxy)- 1,2-benzisothiazole1,1-dioxide (ABID) [13] and six isomers for tetrazole-saccharyl [14]. The isomers for each pseudosaccharyl molecule were obtained through the rotation of two dihedral angles as will be explained in the next subsections. It is shown in **Figure 2(a)** the ABIOD pseudosaccharyl, in **Figure 2(b)** the ABID pseudosaccharyl and in **Figure 2(c)** and **(d)** tetrazole saccharine. The numbers are needed to understand the formation of the different isomers

In this work, calculations of molar fractions (*xi*

perature or ground-state energies ΔH<sup>0</sup>

in the integration.

**3. Results and discussion**

through dihedral angles.

been obtained through their partition functions *qi*

vibrational zero-point energies) by the formula:

*xi* <sup>=</sup> *qi exp*(−∆*H*0,*<sup>i</sup>*

The molecular structure of the ABIOD pseudosaccharin can be considered to be mainly formed by two components; a heterocyclic ring and a flexible allyl substituent. In ABIOD, the molecule can be rotated two dihedral angles and they give rise to different isomers (see **Figure 2(a)**), nevertheless, according to Ref. [4], the dihedral angle formed by atoms 1, 2, 3 and 4 is irrelevant in the formation of isomers because of the sterical hindrance between the allyl group and the rest of the molecule. Consequently, only the dihedral angle formed by atoms 2, 3, 4 and 5 originate different isomers. In the same study, there were found a total of three isomers of ABIOD in a minimum of the potential-energy surface (PES) and they were represented using the abbreviations *C* = *cis* ≈ 0°, *Sk* = *skew* ≈ 120°, *Sk*′ = *skew*′ ≈ −120°.

In a first step, molecular optimization was performed through B3LYP functional; after that, reoptimizations were performed with M06-2X and PBE functionals.

The corrected zero-point relative energies of the three ABIOD isomers are presented in **Table 1**. For B3LYP XC-functional, it can be observed that the most stable isomer is *Sk*, however, it is important to note that energy differences with *Sk*′ and *C* isomers are less than 1 kcal/mol, no matter the BS used. According to M06-2X XC-functional results, the BS that takes into account more polarization and diffuse functions located the *C* isomer as the most stable and the two BSs that do not possess the maximum increases in diffuse functions and polarization found the *Sk* structure as the most stable in a similar way as the B3LYP functional. Again, note that the energies between these isomers are small since they do not reach 1 kcal/mol. Unlike B3LYP and M06-2X, it was noticed that PBE outcame the *Sk*′ isomer as the most stable at T = 0, nonetheless the energetic differences are very small since these do not reach 1 kcal/mol.

As can be observed from DFT calculations at T = 0, the analysis of isomerism is complicated, especially when small energy differences are presented between isomers. Since none of the three functionals (B3LYP, M06-2X and PBE) obtain the same most stable isomer, this quest appears to be quite dependent on the method used. However, it is also worth commenting that the energy difference between the three isomers is very small, regardless of the XC-functional used. Then, it could be considered that the energy differences are within the anticipated error of calculation and therefore, it would be expected that molar fractions as a function of temperature can provide relevant information.

Considering the previous discussion, **Figure 3** shows the molar fractions as a function of temperature for the three isomers of ABIOD pseudosaccharine.


almost at the same value in all the range of temperature studied. The same trends can be seen for the *Sk* isomer since beyond 200 K there is no significant change. The *C* isomer shows the same trend with a small population increase in the four graphs of B3LYP; that is, at the beginning of the studied range, its population is zero but it slowly grows while temperature

**Figure 3.** Molar fractions for ABIOD isomers taking into account three XC-functionals with a combination of four BSs.

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For M06-2X functional, results show that for the first two BSs the graphs are similar and display the same trend in general; however, with 6-31++G(3df,3pd) it is observed that distribution of isomer populations are narrower than in 6-311++G(3df,3pd). Nonetheless, 6-311+G(d,p) and 6-31+G(d,p) present a different trend since the populations change, this behavior could be understood noticing that the energy difference between the three isomers obtained with these methodologies is very close (see **Table 1**), then, the vibrational frequencies, rotational constants and energy levels become important in the construction of the molar fractions. A deeper analysis on vibration frequencies, rotational constants and energy

increases.

levels is presented below.

For B3LYP XC-functional it can be observed that the molar fractions do not vary significantly as the temperature increases, in fact, the line corresponding to the *Sk*′ isomer is maintained

**Table 1.** Corrected zero-point relative energies (kcal/mol) of ABIOD isomers.

DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins http://dx.doi.org/10.5772/intechopen.70933 135

The corrected zero-point relative energies of the three ABIOD isomers are presented in **Table 1**. For B3LYP XC-functional, it can be observed that the most stable isomer is *Sk*, however, it is important to note that energy differences with *Sk*′ and *C* isomers are less than 1 kcal/mol, no matter the BS used. According to M06-2X XC-functional results, the BS that takes into account more polarization and diffuse functions located the *C* isomer as the most stable and the two BSs that do not possess the maximum increases in diffuse functions and polarization found the *Sk* structure as the most stable in a similar way as the B3LYP functional. Again, note that the energies between these isomers are small since they do not reach 1 kcal/mol. Unlike B3LYP and M06-2X, it was noticed that PBE outcame the *Sk*′ isomer as the most stable at T = 0, nonetheless the energetic differences are very small since these do not

As can be observed from DFT calculations at T = 0, the analysis of isomerism is complicated, especially when small energy differences are presented between isomers. Since none of the three functionals (B3LYP, M06-2X and PBE) obtain the same most stable isomer, this quest appears to be quite dependent on the method used. However, it is also worth commenting that the energy difference between the three isomers is very small, regardless of the XC-functional used. Then, it could be considered that the energy differences are within the anticipated error of calculation and therefore, it would be expected that molar fractions as a function of tem-

Considering the previous discussion, **Figure 3** shows the molar fractions as a function of tem-

For B3LYP XC-functional it can be observed that the molar fractions do not vary significantly as the temperature increases, in fact, the line corresponding to the *Sk*′ isomer is maintained

**Isomer 6-311++G(3df,3pd) 6-31++G(3df,3pd) 6-311+G(d,p) 6-31+G(d,p)**

*C* 0.65 0.73 0.71 0.76 *Sk* 0 0 0 0 *Sk′* 0.15 0.16 0.09 0.04

*C* 0 0 0.04 0.03 *Sk* 0.09 0.1 0 0 *Sk′* 0.59 0.58 0.05 0.43

*C* 0.33 0.33 0.5 0.59 *Sk* 0.06 0.06 0.18 0.26 *Sk′* 0 0 0 0

**Table 1.** Corrected zero-point relative energies (kcal/mol) of ABIOD isomers.

reach 1 kcal/mol.

B3LYP

M06-2X

PBE

perature can provide relevant information.

perature for the three isomers of ABIOD pseudosaccharine.

134 Density Functional Calculations - Recent Progresses of Theory and Application

**Figure 3.** Molar fractions for ABIOD isomers taking into account three XC-functionals with a combination of four BSs.

almost at the same value in all the range of temperature studied. The same trends can be seen for the *Sk* isomer since beyond 200 K there is no significant change. The *C* isomer shows the same trend with a small population increase in the four graphs of B3LYP; that is, at the beginning of the studied range, its population is zero but it slowly grows while temperature increases.

For M06-2X functional, results show that for the first two BSs the graphs are similar and display the same trend in general; however, with 6-31++G(3df,3pd) it is observed that distribution of isomer populations are narrower than in 6-311++G(3df,3pd). Nonetheless, 6-311+G(d,p) and 6-31+G(d,p) present a different trend since the populations change, this behavior could be understood noticing that the energy difference between the three isomers obtained with these methodologies is very close (see **Table 1**), then, the vibrational frequencies, rotational constants and energy levels become important in the construction of the molar fractions. A deeper analysis on vibration frequencies, rotational constants and energy levels is presented below.

As mentioned in a previous paragraph, the relative energies obtained with PBE XC-functional place the *Sk*′ isomer as the most stable, therefore, at the beginning of our range of calculation of molar fractions, this was the isomer with the largest population. As observed in (PBE) graphics for the first two BSs in **Figure 3**, the molar fraction plots of each isomer do not cross at any time, which means that the *Sk*′ isomer is the most predominant over all the range of temperature studied here, even though the difference in energy with the second most stable isomer is at most 0.06 kcal/mol. For 6-311+G(d,p) and 6-31+G(d,p) a distinct trend can be observed in comparison with 6-311++G(3df,3pd) and 6-31++G(3df,3pd), since in 6-311 G(d,p) there is an exchange (before 200 K) of the most populated isomer which turns now to be *Sk*. Additionally, it can be noticed that the population of both *Sk* and *Sk′* isomers is around 45%. For 6-31+G(d,p), the intersection in the population of *Sk* and *Sk′* isomer is given approximately at 300 K.

previously relaxed structures with B3LYP, M06-2X and PBE in order to obtain relative stabilities. For B3LYP optimization and single point calculations through MP2, the relative energies were *Sk* = 0, *Sk′* = 0.7 and *C* = 1.2 kcal/mol using 6-311++G(3df,3pd), on the other hand, *Sk* = 0, *Sk′* = 0.68 and *C* = 1.39 kcal/mol using 6-31+G(d,p). For optimization with M06-2X the results were *Sk* = 0, *Sk′* = 0.38 and *C* = 1.21, additionally, *Sk* = 0, *Sk′* = 0.74 and *C* = 1.24 kcal/mol, when 6-311++G(3df,3pd) and 6-31+G(d,p) were, respectively, used. Finally, for PBE optimization, the results were *Sk* = 0, *Sk′* = 0.84 and *C* = 1.31 with 6-311++G(3df,3pd), and *Sk* = 0, *Sk′* = 0.76

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As it can be seen, the *Sk* isomer results as the most stable, as B3LYP functional, however, the energy differences are magnified compared to B3LYP. On the other hand, the narrowest energy difference between the *Sk* and *Sk′* isomers is given by MP2/6-311++G(3df,3pd)// M06-2X/6-311++G(3df,3pd) methodology and the largest energy difference is obtained by MP2/6-31G(d,p)//PBE/6-31G(d,p) methodology. It is worth to mention that, regardless of the functional and BS used to relax the molecular structure, the energy obtained through MP2 resulted in the same order of stability (*Sk* > *Sk′* > *C*). Therefore, it seems to be that MP2 energy treatment could help to homogenize results at T = 0, additionally, these results are in agreement with the previously reported by Gómez-Zavaglia et al. [4], since they predicted the same most stable isomer with B3LYP functional. In consonance with these results, *Sk* can be

Molar fractions as a function of temperature were re-built using the electronic energy data from MP2 calculations on optimized structures with B3LYP, M06-2X and PBE (see **Figure 4**). As expected, the most populated isomer in all the range of temperature was *Sk*, however, M06-2X with 6-311++G(3df,3pd) maintained the *Sk′* isomer as the most populated. From a previous paragraph, it is known that M06-2X got the lowest energy difference between the *Sk* and *Sk′* isomers, and it could be the reason of the exchange in population, however, there could be something else inducing that exchange. In order to find a possible reason of this induced exchange, several things were tested, for example, knowing that the *Sk′* isomer is the one that presents a different behavior in the M06-2X/6-311++G(3df,3pd) methodology (**Figure 4(b)**) in comparison with the other methodologies, rotational constants obtained with M06-2X/6- 311++G(d,p) were inserted instead of those obtained with M06-2X/6-311++G(3df,3pd), however, there was no great modification in the graph (see **Figure 5(b)**). Subsequently, the same exchange was made with the electronic transitions, without finding again great modifications in the graph (see **Figure 5(c)**). But, when the same exchange was made with the vibrational frequencies, it was observed that the graphs of the populations changed(see **Figure 5(a)**), obtaining the same trends as the other methodologies of **Figure 4**, where *Sk* is the most popu-

Having noticed that the vibrational partition function influences in an important way, the vibration frequencies obtained through M06-2X/6-311++G(3df,3pd), were observed for each isomer. Interestingly, it was found that the first frequencies for each isomer are 32, 35 and 14 cm−1, for *C*, *Sk* and *Sk′*, respectively. As it can be seen, *Sk′* has the first frequency of less than half compared to the first frequency of the other two isomers, which could be one reason why the *Sk′* isomer becomes the one with the highest population at very low temperature

and *C* = 1.54 kcal/mol for 6-31+G(d,p).

lated isomer followed by *Sk′* and *C*.

expected to be the isomer with the largest population.

From **Figure 3**, it can be observed that B3LYP and PBE XC-functionals obtain the same trends: the less stable structure (*C* isomer) has the lowest population, while the two most stable isomers (*Sk* and *Sk′*) are maintained with the largest population. In fact, it can be seen how for the first two BSs in B3LYP,that even though the *Sk* isomer is the most stable, *Sk′* is the most populated since an exchange in population appears at very low temperatures, which is in agreement with PBE. Additionally, in the last two BSs of the PBE functional, although *Sk′* is the most stable isomer at T = 0, it is not maintained as the isomer with the largest population when the temperature increases. So far, at least it is stated that the two most stable isomers are the ones with the largest population regardless of the methodology (functional and BS) used. On the other hand, M06-2X/6-311++G(3df,3pd) and M06-2X/6-31++G(3df,3pd) are consistent showing that *Sk′* has the largest population at room temperature, however, the last two BSs are not in agreement with these results since 6-311+G(d,p) presents the *C* isomer as the most populated, while 6-31+G(d,p) presents *Sk* isomers as the most populated. Therefore, a deeper analysis must be performed, that is why the peculiar case of M06-2X/6-311+G(d,p) results will be analyzed below.

Due to the complicated comparison between the different results obtained, mainly with M06-2X, both at T = 0 and varying temperature, it was decided to refine the electronic energy through the MP2 method. As a first step, a comparison of energies of the three isomers MP2 optimized and non-optimized was performed.

Firstly, the molecular structures of the three isomers were optimized with MP2/6- 311++G(3df,3pd) methodology resulting in the following relative energies: *Sk* = 0, *Sk′* = 0.74 and *C* = 1.09 kcal/mol. The isomer with the lowest energy is taken as 0 kcal/mol. After that, a single point calculation with MP2/6-311++G(3df,3pd) methodology on the previously optimized structure through B3LYP/6-311++G(3df,3pd) methodology was performed, getting the following relative energies: *Sk* = 0, *Sk′* = 0.7 and *C* = 1.2 kcal/mol. These outcomes clarify that optimization of the three isomers, not only result in the same order of stability, but also do not represent a great variation in the relative stability. This can give us confidence in performing only single point calculations with MP2 on structures optimized with different methodologies.

In a second step and according to the results discussed in the previous paragraph, single point calculations were performed with MP2 along with 6-311++G(3df,3pd) and 6-31+G(d,p) BSs on previously relaxed structures with B3LYP, M06-2X and PBE in order to obtain relative stabilities. For B3LYP optimization and single point calculations through MP2, the relative energies were *Sk* = 0, *Sk′* = 0.7 and *C* = 1.2 kcal/mol using 6-311++G(3df,3pd), on the other hand, *Sk* = 0, *Sk′* = 0.68 and *C* = 1.39 kcal/mol using 6-31+G(d,p). For optimization with M06-2X the results were *Sk* = 0, *Sk′* = 0.38 and *C* = 1.21, additionally, *Sk* = 0, *Sk′* = 0.74 and *C* = 1.24 kcal/mol, when 6-311++G(3df,3pd) and 6-31+G(d,p) were, respectively, used. Finally, for PBE optimization, the results were *Sk* = 0, *Sk′* = 0.84 and *C* = 1.31 with 6-311++G(3df,3pd), and *Sk* = 0, *Sk′* = 0.76 and *C* = 1.54 kcal/mol for 6-31+G(d,p).

As mentioned in a previous paragraph, the relative energies obtained with PBE XC-functional place the *Sk*′ isomer as the most stable, therefore, at the beginning of our range of calculation of molar fractions, this was the isomer with the largest population. As observed in (PBE) graphics for the first two BSs in **Figure 3**, the molar fraction plots of each isomer do not cross at any time, which means that the *Sk*′ isomer is the most predominant over all the range of temperature studied here, even though the difference in energy with the second most stable isomer is at most 0.06 kcal/mol. For 6-311+G(d,p) and 6-31+G(d,p) a distinct trend can be observed in comparison with 6-311++G(3df,3pd) and 6-31++G(3df,3pd), since in 6-311 G(d,p) there is an exchange (before 200 K) of the most populated isomer which turns now to be *Sk*. Additionally, it can be noticed that the population of both *Sk* and *Sk′* isomers is around 45%. For 6-31+G(d,p), the intersection in the population of *Sk* and *Sk′* isomer is given approxi-

136 Density Functional Calculations - Recent Progresses of Theory and Application

From **Figure 3**, it can be observed that B3LYP and PBE XC-functionals obtain the same trends: the less stable structure (*C* isomer) has the lowest population, while the two most stable isomers (*Sk* and *Sk′*) are maintained with the largest population. In fact, it can be seen how for the first two BSs in B3LYP,that even though the *Sk* isomer is the most stable, *Sk′* is the most populated since an exchange in population appears at very low temperatures, which is in agreement with PBE. Additionally, in the last two BSs of the PBE functional, although *Sk′* is the most stable isomer at T = 0, it is not maintained as the isomer with the largest population when the temperature increases. So far, at least it is stated that the two most stable isomers are the ones with the largest population regardless of the methodology (functional and BS) used. On the other hand, M06-2X/6-311++G(3df,3pd) and M06-2X/6-31++G(3df,3pd) are consistent showing that *Sk′* has the largest population at room temperature, however, the last two BSs are not in agreement with these results since 6-311+G(d,p) presents the *C* isomer as the most populated, while 6-31+G(d,p) presents *Sk* isomers as the most populated. Therefore, a deeper analysis must be performed, that is why the peculiar case of M06-2X/6-311+G(d,p) results will

Due to the complicated comparison between the different results obtained, mainly with M06-2X, both at T = 0 and varying temperature, it was decided to refine the electronic energy through the MP2 method. As a first step, a comparison of energies of the three isomers MP2

Firstly, the molecular structures of the three isomers were optimized with MP2/6- 311++G(3df,3pd) methodology resulting in the following relative energies: *Sk* = 0, *Sk′* = 0.74 and *C* = 1.09 kcal/mol. The isomer with the lowest energy is taken as 0 kcal/mol. After that, a single point calculation with MP2/6-311++G(3df,3pd) methodology on the previously optimized structure through B3LYP/6-311++G(3df,3pd) methodology was performed, getting the following relative energies: *Sk* = 0, *Sk′* = 0.7 and *C* = 1.2 kcal/mol. These outcomes clarify that optimization of the three isomers, not only result in the same order of stability, but also do not represent a great variation in the relative stability. This can give us confidence in performing only single point calculations with MP2 on structures optimized with different methodologies. In a second step and according to the results discussed in the previous paragraph, single point calculations were performed with MP2 along with 6-311++G(3df,3pd) and 6-31+G(d,p) BSs on

mately at 300 K.

be analyzed below.

optimized and non-optimized was performed.

As it can be seen, the *Sk* isomer results as the most stable, as B3LYP functional, however, the energy differences are magnified compared to B3LYP. On the other hand, the narrowest energy difference between the *Sk* and *Sk′* isomers is given by MP2/6-311++G(3df,3pd)// M06-2X/6-311++G(3df,3pd) methodology and the largest energy difference is obtained by MP2/6-31G(d,p)//PBE/6-31G(d,p) methodology. It is worth to mention that, regardless of the functional and BS used to relax the molecular structure, the energy obtained through MP2 resulted in the same order of stability (*Sk* > *Sk′* > *C*). Therefore, it seems to be that MP2 energy treatment could help to homogenize results at T = 0, additionally, these results are in agreement with the previously reported by Gómez-Zavaglia et al. [4], since they predicted the same most stable isomer with B3LYP functional. In consonance with these results, *Sk* can be expected to be the isomer with the largest population.

Molar fractions as a function of temperature were re-built using the electronic energy data from MP2 calculations on optimized structures with B3LYP, M06-2X and PBE (see **Figure 4**). As expected, the most populated isomer in all the range of temperature was *Sk*, however, M06-2X with 6-311++G(3df,3pd) maintained the *Sk′* isomer as the most populated. From a previous paragraph, it is known that M06-2X got the lowest energy difference between the *Sk* and *Sk′* isomers, and it could be the reason of the exchange in population, however, there could be something else inducing that exchange. In order to find a possible reason of this induced exchange, several things were tested, for example, knowing that the *Sk′* isomer is the one that presents a different behavior in the M06-2X/6-311++G(3df,3pd) methodology (**Figure 4(b)**) in comparison with the other methodologies, rotational constants obtained with M06-2X/6- 311++G(d,p) were inserted instead of those obtained with M06-2X/6-311++G(3df,3pd), however, there was no great modification in the graph (see **Figure 5(b)**). Subsequently, the same exchange was made with the electronic transitions, without finding again great modifications in the graph (see **Figure 5(c)**). But, when the same exchange was made with the vibrational frequencies, it was observed that the graphs of the populations changed(see **Figure 5(a)**), obtaining the same trends as the other methodologies of **Figure 4**, where *Sk* is the most populated isomer followed by *Sk′* and *C*.

Having noticed that the vibrational partition function influences in an important way, the vibration frequencies obtained through M06-2X/6-311++G(3df,3pd), were observed for each isomer. Interestingly, it was found that the first frequencies for each isomer are 32, 35 and 14 cm−1, for *C*, *Sk* and *Sk′*, respectively. As it can be seen, *Sk′* has the first frequency of less than half compared to the first frequency of the other two isomers, which could be one reason why the *Sk′* isomer becomes the one with the highest population at very low temperature

value of 34 cm−1, which is a value around the first frequency of the other two isomers. With this modification, it was observed that the molar fractions return to those obtained by the other methodologies; that is, resulting *Sk* as the most populated. According to this analysis, it can be concluded that the vibrational part is very important for the calculation of the molar fractions. In fact, a similar analysis was performed for the molar fractions obtained in **Figure 3** with M06-2X/6-311+G(d,p) results, which showed a different behavior in comparison with the other methodologies. It was observed that the *C* isomer had a frequency of 23 cm−1, whereas the *Sk* and *Sk′* isomers had their first frequencies at 37 and 39 cm−1. A change was then made in the first frequency of the *C* isomer to a value of 38 cm−1, which is close to those of the other two isomers, resulting in a graph of molar fractions where the *Sk* isomer is the most populated for the entire temperature range shown. In summary, when all isomers have very similar first frequency values, enthalpy predicts the isomer with the largest population; however, if one of the isomers presents its first frequency with a very low value in comparison with the correspondent frequency for the other isomers, it is very likely that such isomer presents a larger population when T is increased. From the present analysis, it can be concluded that the choice of a functional that correctly describes the vibration modes of a molecule is of utmost

**Figure 5.** Comparison of molar fractions varying (a) vibrational frequencies, (b) rotational constants and (c) electronic

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transitions.

importance for obtaining the predominant zones through molar fractions.

**3.2. Pseudosaccharyl allyl ether 3-(allyloxy)-1,2-benzisothiazole1,1-dioxide**

The pseudosaccharyl allyl ether 3-(allyloxy)-1,2-benzisothiazole1,1-dioxide (ABID) is a system that has been studied from an experimental and theoretical point of view [13]. In that

**Figure 4.** Molar fractions with MP2 energy.

(**Figure 4(b)**), even though the *Sk* isomer had the lowest energy at T = 0. In order to verify if the low value of the first frequency in the isomers result in a larger population when the temperature increases, several variations were made in the vibration frequencies trying to observe an effect. For example, the 66 frequencies of *Sk′*, (with first frequency at 14 cm−1), were changed by the 66 frequencies of *Sk*, (first frequency at 35 cm−1), obtaining the graph of the molar fractions such as those obtained with the other methodologies; that is, when the *Sk′* isomer has similar frequencies to those of the other isomers, this promotes the *Sk* isomer with the largest population due to its lower energy at T = 0. Given this interesting result, a simple test was then proposed by changing only the first frequency of the *Sk′* isomer (14 cm−1), by a

**Figure 5.** Comparison of molar fractions varying (a) vibrational frequencies, (b) rotational constants and (c) electronic transitions.

value of 34 cm−1, which is a value around the first frequency of the other two isomers. With this modification, it was observed that the molar fractions return to those obtained by the other methodologies; that is, resulting *Sk* as the most populated. According to this analysis, it can be concluded that the vibrational part is very important for the calculation of the molar fractions. In fact, a similar analysis was performed for the molar fractions obtained in **Figure 3** with M06-2X/6-311+G(d,p) results, which showed a different behavior in comparison with the other methodologies. It was observed that the *C* isomer had a frequency of 23 cm−1, whereas the *Sk* and *Sk′* isomers had their first frequencies at 37 and 39 cm−1. A change was then made in the first frequency of the *C* isomer to a value of 38 cm−1, which is close to those of the other two isomers, resulting in a graph of molar fractions where the *Sk* isomer is the most populated for the entire temperature range shown. In summary, when all isomers have very similar first frequency values, enthalpy predicts the isomer with the largest population; however, if one of the isomers presents its first frequency with a very low value in comparison with the correspondent frequency for the other isomers, it is very likely that such isomer presents a larger population when T is increased. From the present analysis, it can be concluded that the choice of a functional that correctly describes the vibration modes of a molecule is of utmost importance for obtaining the predominant zones through molar fractions.

#### **3.2. Pseudosaccharyl allyl ether 3-(allyloxy)-1,2-benzisothiazole1,1-dioxide**

(**Figure 4(b)**), even though the *Sk* isomer had the lowest energy at T = 0. In order to verify if the low value of the first frequency in the isomers result in a larger population when the temperature increases, several variations were made in the vibration frequencies trying to observe an effect. For example, the 66 frequencies of *Sk′*, (with first frequency at 14 cm−1), were changed by the 66 frequencies of *Sk*, (first frequency at 35 cm−1), obtaining the graph of the molar fractions such as those obtained with the other methodologies; that is, when the *Sk′* isomer has similar frequencies to those of the other isomers, this promotes the *Sk* isomer with the largest population due to its lower energy at T = 0. Given this interesting result, a simple test was then proposed by changing only the first frequency of the *Sk′* isomer (14 cm−1), by a

**Figure 4.** Molar fractions with MP2 energy.

138 Density Functional Calculations - Recent Progresses of Theory and Application

The pseudosaccharyl allyl ether 3-(allyloxy)-1,2-benzisothiazole1,1-dioxide (ABID) is a system that has been studied from an experimental and theoretical point of view [13]. In that study, five structures were found in a minimum in the PES and whose energetic differences are small. The differences between the five isomers are particularly due to the values of two dihedral angles in the structure. In **Figure 2(b)** atoms that conform these angles are shown, the first of them is formed by the atoms numbered as 1, 2, 3, 4 and the second by the atoms 2, 3, 4, 5. From Ref. [13] the five isomers were named according to the values of the dihedral angles in question, the value of the angle formed by the atoms 1, 2, 3, 4 was represented using the abbreviations *T = trans* ≈ 180°, *G = gauge* ≈ 60°, *G′ = gauge′* ≈ −60°, for dihedral angle formed by 2, 3, 4, 5 its value was represented by the abbreviations *C = cis* ≈ 0°, *Sk = skew* ≈ 120°, *Sk′ = skew′* ≈ −120°. Four of the five isomers belong to the *C1* point group of symmetry with the exception of the *TC* isomer whose point group is *CS* .

and PBE XC-functional and the proposed BS for the five isomers are shown. It can be observed that B3LYP and PBE got the same order of stability (*TSk* > *GSk* > *TC* > *GSk′* > *GC*) independently of the BS. Although the energy difference between the isomers is not maintained at the

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With M06-2X functional a change in order of stability is shown, in contrast to B3LYP and PBE functional, *TC* isomer is the most stable. In spite of the order of stability changes, it is also possible to observe that the energy differences are narrower than B3LYP and PBE, since with this functional the largest difference of energy between the distinct isomers and taking into account the four BSs is at most of 0.5 kcal/mol. From M06-2X, it is difficult to propose an isomer to work in a theoretical study. Then, considering that energy differences are very small, it would be worthwhile to observe the effects of temperature on populations of the five isomers. The temperature of interest is around 350 K, since these compounds are synthesized at this temperature. Remember that for B3LYP XC-functional and the four BSs, the *TSk* isomer is the most stable and the energy difference with the second and third most stable isomers is small. In **Figure 6**, molar fractions as a function of temperature obtained with B3LYP and the utilized BSs are shown. The results display that all used BSs are consistent, it is observed that the isomer of greatest population for all BSs at the temperature of interest is *TSk*, although it is the predominant isomer, its population at 350 K does not exceed 60%, the second is around 17% and the rest is lower than 12%. As it can be observed in **Figure 6**, populations obtained through PBE functional have the same trends as B3LYP functional, and despite there is only a small difference, it is shown that the second most stable isomer (*GSk*) has a higher population (at around T = 300 K) compared to its popula-

Regarding M06-2X, there is an interesting event with the lowest potential-energy species, in spite of *TC* isomer having actually the lowest energy at absolute zero (see **Table 2**), the most populated is the second lowest potential-energy isomer. Therefore, the molar fraction calculations showed the *TSk* isomer as the most predominant at the temperature of interest and the rest of them with very similar populations no matter the BS used, in fact, from the theoretical point of view, it is worth mentioning that two well-defined horizontal lines begin to form in **Figure 6**, the first one with the population of the most stable isomer and the other one with the populations of the rest of the isomers. The molar fractions determine that the consideration of statistical physics, in order to introduce thermal effects in the isomerization of molecular structures with small energy differences, becomes important, since, as we can see in **Figure 6**, regardless of the greater stability of the *TC* isomer, *TSk* is obtained as the most abundant, even at very low temperatures as the ones used in the present analysis. Based on these results, the importance of performing an analysis of this nature must be

From these results, it can be concluded that, in one hand, both B3LYP and PBE XC-functional remark the same trends and, on the other hand, in spite than M06-2X obtained a more complicated distribution of relative energies at T = 0, when thermal effects are taken into account, the same trends in populations of the five isomers compared to B3LYP and PBE functionals

same rate, the order of stability is not compromised by varying the BS.

tion with B3LYP functional.

considered.

are obtained.

Molar fractions as a function of temperature of the five isomers with the XC-functionals and the four BSs used in previous subsection were obtained. For each level of theory, the necessary frequency vibrations were obtained in order to obtain, on one hand, the molar fractions and, on the other hand, to corroborate that molecular structure is in a minimum of PES; rotational constants for building the rotational partition function; and 10 excited states for obtaining the electronic partition function. In **Table 2**, the relative energies obtained with B3LYP, M06-2X


**Table 2.** Corrected zero-point relative energies (kcal/mol) of ABID isomers.

and PBE XC-functional and the proposed BS for the five isomers are shown. It can be observed that B3LYP and PBE got the same order of stability (*TSk* > *GSk* > *TC* > *GSk′* > *GC*) independently of the BS. Although the energy difference between the isomers is not maintained at the same rate, the order of stability is not compromised by varying the BS.

study, five structures were found in a minimum in the PES and whose energetic differences are small. The differences between the five isomers are particularly due to the values of two dihedral angles in the structure. In **Figure 2(b)** atoms that conform these angles are shown, the first of them is formed by the atoms numbered as 1, 2, 3, 4 and the second by the atoms 2, 3, 4, 5. From Ref. [13] the five isomers were named according to the values of the dihedral angles in question, the value of the angle formed by the atoms 1, 2, 3, 4 was represented using the abbreviations *T = trans* ≈ 180°, *G = gauge* ≈ 60°, *G′ = gauge′* ≈ −60°, for dihedral angle formed by 2, 3, 4, 5 its value was represented by the abbreviations *C = cis* ≈ 0°, *Sk = skew* ≈ 120°,

.

Molar fractions as a function of temperature of the five isomers with the XC-functionals and the four BSs used in previous subsection were obtained. For each level of theory, the necessary frequency vibrations were obtained in order to obtain, on one hand, the molar fractions and, on the other hand, to corroborate that molecular structure is in a minimum of PES; rotational constants for building the rotational partition function; and 10 excited states for obtaining the electronic partition function. In **Table 2**, the relative energies obtained with B3LYP, M06-2X

**Isomer 6-311++G(3df,3pd) 6-31++G(3df,3pd) 6-311+G(d,p) 6-31+G(d,p)**

*GC* 1.46 1.51 1.49 1.67 *GSk* 0.47 0.51 0.38 0.44 *GSk′* 0.91 0.93 0.8 0.89 *TC* 0.5 0.59 0.52 0.67 *TSk* 0 0 0 0

*GC* 0.4 0.32 0.41 0.5 *GSk* 0.18 0.12 0.001 0.004 *GSk′* 0.39 0.34 0.19 0.24 *TC* 0 0 0 0 *TSk* 0.1 0.05 0.05 0.004

*GC* 1.1 1.18 1.18 1.38 *GSk* 0.13 0.33 0.14 0.27 *GSk′* 0.64 0.8 0.64 0.77 *TC* 0.37 0.43 0.36 0.5 *TSk* 0 0 0 0

**Table 2.** Corrected zero-point relative energies (kcal/mol) of ABID isomers.

point group of symmetry with the

*Sk′ = skew′* ≈ −120°. Four of the five isomers belong to the *C1*

140 Density Functional Calculations - Recent Progresses of Theory and Application

exception of the *TC* isomer whose point group is *CS*

B3LYP

M06-2X

PBE

With M06-2X functional a change in order of stability is shown, in contrast to B3LYP and PBE functional, *TC* isomer is the most stable. In spite of the order of stability changes, it is also possible to observe that the energy differences are narrower than B3LYP and PBE, since with this functional the largest difference of energy between the distinct isomers and taking into account the four BSs is at most of 0.5 kcal/mol. From M06-2X, it is difficult to propose an isomer to work in a theoretical study. Then, considering that energy differences are very small, it would be worthwhile to observe the effects of temperature on populations of the five isomers.

The temperature of interest is around 350 K, since these compounds are synthesized at this temperature. Remember that for B3LYP XC-functional and the four BSs, the *TSk* isomer is the most stable and the energy difference with the second and third most stable isomers is small. In **Figure 6**, molar fractions as a function of temperature obtained with B3LYP and the utilized BSs are shown. The results display that all used BSs are consistent, it is observed that the isomer of greatest population for all BSs at the temperature of interest is *TSk*, although it is the predominant isomer, its population at 350 K does not exceed 60%, the second is around 17% and the rest is lower than 12%. As it can be observed in **Figure 6**, populations obtained through PBE functional have the same trends as B3LYP functional, and despite there is only a small difference, it is shown that the second most stable isomer (*GSk*) has a higher population (at around T = 300 K) compared to its population with B3LYP functional.

Regarding M06-2X, there is an interesting event with the lowest potential-energy species, in spite of *TC* isomer having actually the lowest energy at absolute zero (see **Table 2**), the most populated is the second lowest potential-energy isomer. Therefore, the molar fraction calculations showed the *TSk* isomer as the most predominant at the temperature of interest and the rest of them with very similar populations no matter the BS used, in fact, from the theoretical point of view, it is worth mentioning that two well-defined horizontal lines begin to form in **Figure 6**, the first one with the population of the most stable isomer and the other one with the populations of the rest of the isomers. The molar fractions determine that the consideration of statistical physics, in order to introduce thermal effects in the isomerization of molecular structures with small energy differences, becomes important, since, as we can see in **Figure 6**, regardless of the greater stability of the *TC* isomer, *TSk* is obtained as the most abundant, even at very low temperatures as the ones used in the present analysis. Based on these results, the importance of performing an analysis of this nature must be considered.

From these results, it can be concluded that, in one hand, both B3LYP and PBE XC-functional remark the same trends and, on the other hand, in spite than M06-2X obtained a more complicated distribution of relative energies at T = 0, when thermal effects are taken into account, the same trends in populations of the five isomers compared to B3LYP and PBE functionals are obtained.

were named according to the letters *G = gauge* ≈ 60°, *Sk = Skew* ≈ 120°, *G′ = gauge′* ≈ −60°, *Sk′ = Skew′* ≈ −120°, the angle formed by the atoms numbered as 1, 2, 3 and 4 was used to assign the first letter, while the angle formed by the atoms numbered as 2, 3, 4 and 5 were

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Tautomers are called as 1H tautomer and 2H tautomer, the first one is obtained when H is bonded to atom number 5 (**Figure 2(c)**) and the second one is obtained when H is bonded to the nitrogen atom beside to atom number 5 (**Figure 2(d)**) [14]. Relative energies were obtained

In **Table 3**, corrected zero-point relative energies are shown. As it can be seen, B3LYP and PBE are consistent in the sense that they present the *G′Sk′* isomer as the most stable, with the exception of the B3LYP/6-311+G(d,p) methodology, since this gives *GSk′* as the most stable, although it should be noted that, in this case, the energy difference between *G′Sk′* and *GSk′*

**Isomer 6-311++G(3df,3pd) 6-31++G(3df,3pd) 6-311+G(d,p) 6-31+G(d,p)**

*G′Sk′* 0 0 0.06 0 *GSk′* 0.51 0.55 0 0.25 *G′Sk* 1 1.12 0.65 0.9 *GSk* 1.96 2 1.94 1.85 *G′G′* 1.64 1.8 1.37 1.65 *GG′* 1.97 2.17 1.49 1.74

*G′Sk′* 0.48 0.37 1.2 0.78 *GSk′* 0 0 0 0 *G′Sk* 0.47 0.47 0.57 0.46 *GSk* 1.18 1.22 1.93 1.49 *G′G′* 1.75 1.81 1.92 1.73 *GG′* 1.23 1.33 1.09 1.06

*G′Sk′* 0 0 0 0 *GSk′* 0.64 0.67 0.16 0.44 *G′Sk* 1.23 1.34 0.89 1.18 *GSk* 1.75 1.77 1.65 1.62 *G′G′* 1.63 1.78 1.31 1.64 *GG′* 1.71 1.91 1.19 1.48

**Table 3.** Corrected zero-point relative energies (kcal/mol) of the six isomers of tetrazole-saccharyl.

taken as reference for the second letter [14].

B3LYP

M06-2X

PBE

for the same levels of theory as previous subsections.

**Figure 6.** Molar fractions for ABID isomers.

#### **3.3. Tetrazole-saccharyl**

The third system of the pseudosaccharine family studied in this work is tetrazole-saccharyl (see **Figure 2(c) and (d)**). This system, as well as the previous ones, has been studied from an experimental and theoretical approach, for this molecule there are two tautomers and for each one of them several isomers can be generated by the rotation of two dihedral angles. In **Figure 2(c) and (d)** the involved atoms in the dihedral angles are shown, again the structures were named according to the letters *G = gauge* ≈ 60°, *Sk = Skew* ≈ 120°, *G′ = gauge′* ≈ −60°, *Sk′ = Skew′* ≈ −120°, the angle formed by the atoms numbered as 1, 2, 3 and 4 was used to assign the first letter, while the angle formed by the atoms numbered as 2, 3, 4 and 5 were taken as reference for the second letter [14].

Tautomers are called as 1H tautomer and 2H tautomer, the first one is obtained when H is bonded to atom number 5 (**Figure 2(c)**) and the second one is obtained when H is bonded to the nitrogen atom beside to atom number 5 (**Figure 2(d)**) [14]. Relative energies were obtained for the same levels of theory as previous subsections.

In **Table 3**, corrected zero-point relative energies are shown. As it can be seen, B3LYP and PBE are consistent in the sense that they present the *G′Sk′* isomer as the most stable, with the exception of the B3LYP/6-311+G(d,p) methodology, since this gives *GSk′* as the most stable, although it should be noted that, in this case, the energy difference between *G′Sk′* and *GSk′*


**Table 3.** Corrected zero-point relative energies (kcal/mol) of the six isomers of tetrazole-saccharyl.

**3.3. Tetrazole-saccharyl**

**Figure 6.** Molar fractions for ABID isomers.

142 Density Functional Calculations - Recent Progresses of Theory and Application

The third system of the pseudosaccharine family studied in this work is tetrazole-saccharyl (see **Figure 2(c) and (d)**). This system, as well as the previous ones, has been studied from an experimental and theoretical approach, for this molecule there are two tautomers and for each one of them several isomers can be generated by the rotation of two dihedral angles. In **Figure 2(c) and (d)** the involved atoms in the dihedral angles are shown, again the structures is only 0.06 kcal/mol. On the other hand, M06-2X states the *GSk′* isomer as the most stable for all used BSs. As a summary, all methodologies showed the *G′Sk′* and *GSk′* isomers as the most stables at T = 0 K, with the exceptions of M06-2X/6-311+G(d,p) and M06-2X/6-31+G(d,p).

Here it is worth to mention that in ABIOD and ABID molecules, M06-2X reduced the energy differences, while tetrazole-saccharyl M06-2X does not reduce the energy differences in comparison with the results of B3LYP and PBE. The molar fractions as a function of temperature are analyzed below.

Observing **Figure 7**, it can be seen that the trend of B3LYP and PBE with the four BSs is maintained, there are two predominant isomers in all the graphs at room temperature, with the first two BSs the isomer of greatest abundance is the *G′Sk′* and the second is the *GSk′*, with populations of almost 50 and 20%, respectively. For the last two BSs the difference between the isomers of largest population is reduced, even in the case of B3LYP and the third BS, the *GSk′* isomer has the greatest population, although it is important to note that at room temperature the difference is not very significant, in general the same behavior is kept with PBE, however, in the third BS, *G′Sk′* has the largest population but with a small difference in population with *GSk′*, since it is approximately 33 and 30% for *G′Sk′* and *GSk′*, respectively.

Analyzing the molar fractions as a function of temperature for M06-2X functional, it can be observed that the first two BSs found three isomers with significant populations and the rest remained with a very low population, therefore, it is possible to say that the three isomers whose populations are considered important are the same ones that are predicted as the most stable. For the third and fourth BSs, we can observe that the *GSk′* and *G′Sk* isomers maintained a large population; however, the *G′Sk′* isomer, which had the largest population with B3LYP and PBE, disappears from the isomers of greater population. From this, it can be indicated that M06-2X describes in a different way the most populated isomer in comparison to B3LYP and PBE, however, we can also see that for the two BSs with more diffuse BS and greater freedom in the polarization also obtain *G′Sk′* and *GSk′* as the largest population isomers, indicating then, that M06-2X could be used with a large BS.

An important difference between the results obtained by B3LYP and M06-2X is the change of the order in stability of isomers since M06-2X always brought the *GSk′* isomer as the one with the lowest energy and greatest abundance. On the other hand, it is interesting to observe that in the work of Ismael et al. [14], they compared the experimental infra-red spectrum with the theoretical one at B3LYP/6-311++G(3df,3pd) level of theory and although they found *G′Sk′* isomer is the most favored energetically (as this work showed through B3LYP), they took the infra-red spectrum of *GSk′* as the representative one because its theoretical spectrum agreed better with the experimental spectrum, for this reason, it is important to remark that the M06-2X presents the *GSk′* isomer as the most stable at T = 0 K, even for all BSs.

As it can be observed, M06-2X predicts a different isomer as the most populated compared with B3LYP and PBE, however, it seems to be that the *GSk′* isomer is the one present in the experiment according to the explanation in the previous paragraph, if this is the case, then the question arises as to why M06-2X is the one that better describes this pseudosaccharin. In order to answer the question, let us note that in **Figure 2**, it can be seen that tetrazole-saccharyl is bigger (in size) than ABIOD and ABID molecules, in fact, the tetrazole group could interact with the saccharyl group through noncovalent interactions, moreover, it is well-known that the M06-2X functional has shown promising performance for noncovalent interactions [23, 24], thus, it is the reason why M06-2X describes tetrazolesaccharyl molecule in a better way. This once again proves the importance of testing sev-

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145

eral methodologies.

**Figure 7.** Molar fractions of tetrazole-saccharyl.

DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins http://dx.doi.org/10.5772/intechopen.70933 145

**Figure 7.** Molar fractions of tetrazole-saccharyl.

is only 0.06 kcal/mol. On the other hand, M06-2X states the *GSk′* isomer as the most stable for all used BSs. As a summary, all methodologies showed the *G′Sk′* and *GSk′* isomers as the most stables at T = 0 K, with the exceptions of M06-2X/6-311+G(d,p) and M06-2X/6-31+G(d,p). Here it is worth to mention that in ABIOD and ABID molecules, M06-2X reduced the energy differences, while tetrazole-saccharyl M06-2X does not reduce the energy differences in comparison with the results of B3LYP and PBE. The molar fractions as a function of temperature

144 Density Functional Calculations - Recent Progresses of Theory and Application

Observing **Figure 7**, it can be seen that the trend of B3LYP and PBE with the four BSs is maintained, there are two predominant isomers in all the graphs at room temperature, with the first two BSs the isomer of greatest abundance is the *G′Sk′* and the second is the *GSk′*, with populations of almost 50 and 20%, respectively. For the last two BSs the difference between the isomers of largest population is reduced, even in the case of B3LYP and the third BS, the *GSk′* isomer has the greatest population, although it is important to note that at room temperature the difference is not very significant, in general the same behavior is kept with PBE, however, in the third BS, *G′Sk′* has the largest population but with a small difference in population with *GSk′*, since it is approximately 33 and 30% for *G′Sk′* and *GSk′*, respectively. Analyzing the molar fractions as a function of temperature for M06-2X functional, it can be observed that the first two BSs found three isomers with significant populations and the rest remained with a very low population, therefore, it is possible to say that the three isomers whose populations are considered important are the same ones that are predicted as the most stable. For the third and fourth BSs, we can observe that the *GSk′* and *G′Sk* isomers maintained a large population; however, the *G′Sk′* isomer, which had the largest population with B3LYP and PBE, disappears from the isomers of greater population. From this, it can be indicated that M06-2X describes in a different way the most populated isomer in comparison to B3LYP and PBE, however, we can also see that for the two BSs with more diffuse BS and greater freedom in the polarization also obtain *G′Sk′* and *GSk′* as the largest population iso-

An important difference between the results obtained by B3LYP and M06-2X is the change of the order in stability of isomers since M06-2X always brought the *GSk′* isomer as the one with the lowest energy and greatest abundance. On the other hand, it is interesting to observe that in the work of Ismael et al. [14], they compared the experimental infra-red spectrum with the theoretical one at B3LYP/6-311++G(3df,3pd) level of theory and although they found *G′Sk′* isomer is the most favored energetically (as this work showed through B3LYP), they took the infra-red spectrum of *GSk′* as the representative one because its theoretical spectrum agreed better with the experimental spectrum, for this reason, it is important to remark that

As it can be observed, M06-2X predicts a different isomer as the most populated compared with B3LYP and PBE, however, it seems to be that the *GSk′* isomer is the one present in the experiment according to the explanation in the previous paragraph, if this is the case, then the question arises as to why M06-2X is the one that better describes this pseudosaccharin. In order to answer the question, let us note that in **Figure 2**, it can be seen that

the M06-2X presents the *GSk′* isomer as the most stable at T = 0 K, even for all BSs.

mers, indicating then, that M06-2X could be used with a large BS.

are analyzed below.

tetrazole-saccharyl is bigger (in size) than ABIOD and ABID molecules, in fact, the tetrazole group could interact with the saccharyl group through noncovalent interactions, moreover, it is well-known that the M06-2X functional has shown promising performance for noncovalent interactions [23, 24], thus, it is the reason why M06-2X describes tetrazolesaccharyl molecule in a better way. This once again proves the importance of testing several methodologies.

#### **4. Conclusions**

In this work, an analysis of molar fractions as a function of temperature on three pseudosaccharyl systems (ABIOD, ABID and tetrazole-saccharyl) was performed with three XC-functionals and four BSs. In general, for ABIOD and ABID systems, M06-2X functional reduces the energy differences between the isomers, while B3LYP and PBE are in agreement with each other, but different from M06-2X. It is important to observe that in ABID pseudosaccharine, even if the order of stability changes with the M06-2X functional, the molar fractions maintain the same trend, placing the *TSk* isomer as the one with the largest population, in spite of not being the most stable from DFT calculations at T = 0 K. Additionally, considering the results that are shown in **Table 2**, a minimal effect of the BS is evident considering that the same trend is conserved and the results do not vary significantly in ABID. For tetrazole-saccharyl, it seems that M06-2X describes correctly the predominant isomer found experimentally, contrary to B3LYP and PBE results. Although, this could be understood since it is well-known that M06-2X correctly describes systems with noncovalent interactions as can be inferred in the tetrazolesaccharyl system showed in **Figure 2(c)** and **(d)**.

**Author details**

Zuriel Natanael Cisneros-García1

Ciudad de México, México

DOI: 10.1021/ed055p160

10.1016/j.molstruc.2008.05.054

10.1016/j.tet.2012.10.100

**References**

Francisco J. Tenorio1,2 and Jaime Gustavo Rodríguez-Zavala1

\*Address all correspondence to: jgrz@culagos.udg.mx

, David Alejandro Hernández-Velázquez1

1 Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Paseos de la Montaña, Lagos de Moreno, Jalisco, México

[1] Wotiz J. The discovery of sacharin. Journal of Chemical Education. 2015;**55**(3):161-162.

[2] Qiao N, Li M, Schilindwen W, Malek W, Davies A, Trappitt G. Pharmaceutical cocrystals. International Journal of Pharmaceutics. 2011;**419**:1-11. DOI: 10.1016/j.ijpharm.2011.07.037 [3] Kaczor A, Proniewicz L, Almeida R, Gómez-Zavaglia A, Cristiano M, Beja A, Silva M, Fausto R.The chapman-type rearrangement in pseudosaccharin: The case of 3-(methoxy)- 1,2-benzisothiazole 1,1-dioxide. Journal of molecular Structure. 2008;**892**:343-352. DOI:

[4] Gómez-Zavaglia A, Kaczor A, Coelho D, Lourdes S, Cristiano M, Fausto R.Conformational and structural analysis of 2-allyl-1,2-benzisothiazol-3(2H)-one 1,1-dioxide as probed by matrix-insolation spectroscopy and quantum chemical calculations. Journal of Molecular

[5] Cabral L, Maria T, Martelo L, Eusebio M, Cristiano M, Fausto R. The thermal sigmatropic isomerization of pseudosaccharylcrotyl ether. Tetrahedron. 2013;**69**:810-815. DOI:

[6] Lloyd J, Finlay H, Kover A, Johnson J, Pi Z, Jiang J, Neels J, Cavallaro C, Wexler R, Conder M, Shi H, Li D, Sun H, Chimalakonda A, Huang C, Salvati M, Levesque P. Pseudosaccharin amines as potent and selective Kv 1.5 blockers. Bioorganic & Medicinal Chemistry Letters.

[7] Eacho P, Foxworthy-Mason P, Lin H, Lopez J, Mosior M, Richett M. Benzisothiazol-3-one-Carboxylic Acid Amides as Phospholipase Inhibitors. 2009. US Patent. Available

[8] Wang LH, Yang XY, Zhang X, Mihalic K, Fan X, Xiao W, Howard OMZ, Appella E, Maynard AT, Farrar WL. Suppression of breast cancer by chemical modulation of vulnerable zinc fingers in estrogen receptor. Nature Medicine. 2004;**10**:40-47. DOI: 10.1038/nm969

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2 Departamento de Farmacia, Universidad Nacional Autónoma de México,

Structure. 2009;**919**:271-276. DOI: 10.1016/j.molstruc.2008.09.013

2015;**25**:4983-4986. DOI: 10.1016/j.bmcl.2015.02.066

\*

DFT Calculations and Statistical Mechanics Applied to Isomerization of Pseudosaccharins

,

http://dx.doi.org/10.5772/intechopen.70933

147

Even though the molar fractions show a dependence on the enthalpies of the isomers, the most stable structure is not necessarily the one with the greatest population. By taking into account the vibrational, rotational and electronic partition functions in the calculation of the molar fractions, several aspects are involved to obtain the most predominant isomers at T ≠ 0. It was found that vibrational contribution is the most important factor in obtaining the predominant isomer. Therefore, the use of methodologies that correctly describe the vibrational spectra in order to obtain molar fractions is recommended. Additionally, when a theoretical-experimental spectra comparison is carried out, calculated spectra could be weighted by molar fractions at the temperature of interest.

One of the main differences between the results obtained with the different methodologies is that herein calculations can predict a different order of stability and modify the relative energies. However, in order to be able to assume that an isomer will predominate, it is important to consider the contributions of the different partition functions instead of only considering the relative potential energies.

#### **Acknowledgements**

The authors would like to thank PRODEP (formerly PROMEP) for the support provided through the 103.5/13/6900 office. ZNCG would like to acknowledge the financial support from CONACyT (Mexico) (Grant No. 413573). FJTR would like to thank the University of Guadalajara for authorizing sabbatical leave. DAHV would like to thank CONACYT for the support provided by the program "Apoyos para la Incorporación de Investigadores Vinculada a la Consolidación Institucional de Grupos de Investigación y/o Fortalecimiento del Posgrado Nacional". CONACYT through Project 52827 is also acknowledged.

### **Author details**

**4. Conclusions**

saccharyl system showed in **Figure 2(c)** and **(d)**.

146 Density Functional Calculations - Recent Progresses of Theory and Application

molar fractions at the temperature of interest.

the relative potential energies.

**Acknowledgements**

In this work, an analysis of molar fractions as a function of temperature on three pseudosaccharyl systems (ABIOD, ABID and tetrazole-saccharyl) was performed with three XC-functionals and four BSs. In general, for ABIOD and ABID systems, M06-2X functional reduces the energy differences between the isomers, while B3LYP and PBE are in agreement with each other, but different from M06-2X. It is important to observe that in ABID pseudosaccharine, even if the order of stability changes with the M06-2X functional, the molar fractions maintain the same trend, placing the *TSk* isomer as the one with the largest population, in spite of not being the most stable from DFT calculations at T = 0 K. Additionally, considering the results that are shown in **Table 2**, a minimal effect of the BS is evident considering that the same trend is conserved and the results do not vary significantly in ABID. For tetrazole-saccharyl, it seems that M06-2X describes correctly the predominant isomer found experimentally, contrary to B3LYP and PBE results. Although, this could be understood since it is well-known that M06-2X correctly describes systems with noncovalent interactions as can be inferred in the tetrazole-

Even though the molar fractions show a dependence on the enthalpies of the isomers, the most stable structure is not necessarily the one with the greatest population. By taking into account the vibrational, rotational and electronic partition functions in the calculation of the molar fractions, several aspects are involved to obtain the most predominant isomers at T ≠ 0. It was found that vibrational contribution is the most important factor in obtaining the predominant isomer. Therefore, the use of methodologies that correctly describe the vibrational spectra in order to obtain molar fractions is recommended. Additionally, when a theoretical-experimental spectra comparison is carried out, calculated spectra could be weighted by

One of the main differences between the results obtained with the different methodologies is that herein calculations can predict a different order of stability and modify the relative energies. However, in order to be able to assume that an isomer will predominate, it is important to consider the contributions of the different partition functions instead of only considering

The authors would like to thank PRODEP (formerly PROMEP) for the support provided through the 103.5/13/6900 office. ZNCG would like to acknowledge the financial support from CONACyT (Mexico) (Grant No. 413573). FJTR would like to thank the University of Guadalajara for authorizing sabbatical leave. DAHV would like to thank CONACYT for the support provided by the program "Apoyos para la Incorporación de Investigadores Vinculada a la Consolidación Institucional de Grupos de Investigación y/o Fortalecimiento del Posgrado

Nacional". CONACYT through Project 52827 is also acknowledged.

Zuriel Natanael Cisneros-García1 , David Alejandro Hernández-Velázquez1 , Francisco J. Tenorio1,2 and Jaime Gustavo Rodríguez-Zavala1 \*

\*Address all correspondence to: jgrz@culagos.udg.mx

1 Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Paseos de la Montaña, Lagos de Moreno, Jalisco, México

2 Departamento de Farmacia, Universidad Nacional Autónoma de México, Ciudad de México, México

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**Section 3**

**Catalysis and Mechanism**

## **Catalysis and Mechanism**

**Chapter 7**

**Provisional chapter**

**Catalytic Activation of PVP-Stabilized Gold/Silver**

**Catalytic Activation of PVP-Stabilized Gold/Silver** 

DOI: 10.5772/intechopen.72097

Systematic DFT calculations on poly(N-vinyl-2-pyrrolidone) (PVP) stabilization of Ag13 cluster have shown that the former acts not only as a stabilizer but also plays an important role in activating the Ag catalyst by supplying extra electrons to it through its oxygen atoms. Natural Bonding Orbital (NBO) calculations show that weak back donation of electrons from M(dπ) orbital of Ag to antibonding σ\* of one of the N-O bond, facilitates the formation of the nitroso intermediate. Vibrational frequency calculation of PNP association with Ag13-2PVP cluster carried out to understand the extent and the nature of this interaction better. Red shift in the frequencies is result of strong interaction with that

**Keywords:** nanoparticles, p-nitrophenol, DFT calculation, charge distribution, NBO

Recently nanoparticle (NP) research is an area of adoring scientific research due to wide variety of potential application in different fields of physics, chemistry, material science, medicine and biology, as a result of their unique electronic, optical, magnetic, mechanical, physical, chemical and catalytic properties. Nanoparticle is a microscopic particle with at least one dimension less than 100 nm. The intrinsic properties of metal nanoparticles are mainly determined by their size, shape, composition, stability, crystallinity, structure, etc. The properties of many conventional materials change when formed from nanoparticles. This is typically because nanoparticles have a greater surface area per weight than larger particles which causes them to be more reactive to some other molecules. It can be silver, gold, iron, iron oxide, platinum, silica, titanium oxide, etc. Nanoparticles are of great scientific interest as they

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Cluster on p-Nitrophenol Reduction: A DFT**

**Cluster on p-Nitrophenol Reduction: A DFT**

Madhulata Shukla and Indrajit Sinha

Madhulata Shukla and Indrajit Sinha

http://dx.doi.org/10.5772/intechopen.72097

**Abstract**

calculation

**1. Introduction**

Additional information is available at the end of the chapter

of silver cluster present in Ag13-2PVP-PNP model.

Additional information is available at the end of the chapter

**Provisional chapter**

#### **Catalytic Activation of PVP-Stabilized Gold/Silver Cluster on p-Nitrophenol Reduction: A DFT Cluster on p-Nitrophenol Reduction: A DFT**

**Catalytic Activation of PVP-Stabilized Gold/Silver** 

DOI: 10.5772/intechopen.72097

Madhulata Shukla and Indrajit Sinha Madhulata Shukla and Indrajit Sinha Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72097

#### **Abstract**

Systematic DFT calculations on poly(N-vinyl-2-pyrrolidone) (PVP) stabilization of Ag13 cluster have shown that the former acts not only as a stabilizer but also plays an important role in activating the Ag catalyst by supplying extra electrons to it through its oxygen atoms. Natural Bonding Orbital (NBO) calculations show that weak back donation of electrons from M(dπ) orbital of Ag to antibonding σ\* of one of the N-O bond, facilitates the formation of the nitroso intermediate. Vibrational frequency calculation of PNP association with Ag13-2PVP cluster carried out to understand the extent and the nature of this interaction better. Red shift in the frequencies is result of strong interaction with that of silver cluster present in Ag13-2PVP-PNP model.

**Keywords:** nanoparticles, p-nitrophenol, DFT calculation, charge distribution, NBO calculation

#### **1. Introduction**

Recently nanoparticle (NP) research is an area of adoring scientific research due to wide variety of potential application in different fields of physics, chemistry, material science, medicine and biology, as a result of their unique electronic, optical, magnetic, mechanical, physical, chemical and catalytic properties. Nanoparticle is a microscopic particle with at least one dimension less than 100 nm. The intrinsic properties of metal nanoparticles are mainly determined by their size, shape, composition, stability, crystallinity, structure, etc. The properties of many conventional materials change when formed from nanoparticles. This is typically because nanoparticles have a greater surface area per weight than larger particles which causes them to be more reactive to some other molecules. It can be silver, gold, iron, iron oxide, platinum, silica, titanium oxide, etc. Nanoparticles are of great scientific interest as they

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

are effectively a bridge between bulk materials and atomic or molecular structures. Synthesis of M-NPs can proceed by chemical reduction, thermolysis, photochemical decomposition, electro reduction, microwave and sonochemical irradiation. In recent literature there are large number of reports on synthesis, properties and applications of noble metals Au, Ag and Cu NPs [1–3]. This is mainly because all three elements show good localized surface plasmon resonance (LSPR) absorbance in the visible range and also have large number of catalytic applications. However, the high cost of Au and to a lesser extent Ag restrict their applications in many cases. As an alternative of Au and Ag NPs, researchers have investigated Cu nanoparticles. CuNPs are less expensive and exhibit comparably higher electrical conductivity and catalytic activity [4, 5]. The LSPR absorbance of CuNPs in the visible range is comparatively less intense. Another issue is that Cu nanoparticles are easily oxidized [6], yet nanoparticles of copper oxides also have wide applicability as catalysts. Silver NP found to be quite stable and hence chosen for studying the effect of nanostructures on their catalytic activity. In liquid phase nanoparticle synthesis, polymeric molecules are often used for stabilizing nanoparticles against aggregation. The function of the polymers are to avoid the aggregation of the NP in solution and to control the size and shape at the crystallographic level [7–9]. Among various polymeric molecules investigated in literature, polyvinyl pyrrolidone (PVP) has been one of the most frequently used stabilizers since it is non-toxic and soluble in many polar solvents. While such PVP stabilized noble metal nanoparticles have extensively been used as catalysts for various reactions [10, 11], few workers have concentrated in literature on the effect of such stabilizer molecule on the nanoparticle surface on their catalytic properties [12–14]. Nevertheless, only a few research papers have investigated the effect of such stabilization of nanoparticle surfaces on the electronic properties of such nanocomposites. Tsunoyama et al. proposed that electron transfer occurs from the anionic Au cores of Au:PVP into the LUMO (*π*\*) of O<sup>2</sup> which generates superoxo or peroxo like species. Latter plays a key role in the oxidation of alcohol [10]. Similar mechanisms were proposed by some other related experimental studies as well that the adsorption of PVP on to the catalyst surface can also modify the electronic structure of nanoparticles by charge transfer [15, 16]. Yet several aspects remain unclear, such as whether PVP attaches to the metal surface through its O atom or through the N atom [15, 17]. The strength of interaction of PVP with the metal surface is also another aspect that needs to be investigated. Finally, the most important question that in a catalytic reaction how does the interaction between the catalyst surface and the reactant/substrate change in presence of stabilizer. Very few reports are available regarding this topic [18, 19]. The catalytic reduction of nitroarenes to aminoarenes by transfer hydrogenation methodologies is an important class of organic transformations. The reaction does not occur even in the presence of strong reductants like metal hydrides unless catalyzed by a suitable nanocatalyst [12]. Synthesis of silver NP using PVP has been reported in our earlier study [20]. PVP stabilized Ag nanoparticles have often been successfully used as catalysts for such reactions [14, 20]. However, to the best of our knowledge, there is no detailed DFT study on effect of PVP (poly(N-vinyl-2-pyrrolidone)) stabilization of Ag cluster on its catalytic activity with respect to a nitroarene (p-nitrophenol) substrate. In this chapter, we first carry out DFT calculations on Ag/Au cluster along with the monomer of PVP moiety using Gaussian program. Further, the effect of PVP stabilized Ag clusters (catalyst) on p-nitrophenol (PNP) has been investigated in detail. Energy of interaction found by the B3LYP level of calculation elucidates

the stability of the moieties. Mechanism for activation of the nitro group and formation of nitroso intermediate has been proposed from NBO analysis. IR study of the Ag cluster stabi-

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A 13 atom silver and gold cluster is carved out of the FCC Ag/Au lattice constructed using the MAPS software (Scienomics). The Gaussian 03 program [21] package employed for the DFT calculations at the Becke's three parameter functional and Lee–Yang–Parr hybrid functional (B3LYP) level [22, 23] of calculation. LaNL2DZ for Ag/Au and 6-31G++(d,p) basis set for C, H, N and O atoms was used while performing DFT calculation. B3LYP functional calculations give the stable cluster and also reproduce the experimental results [17, 18, 24]. Calculations are carried out for ground state geometry optimization in gaseous phase. The Mulliken charges of each atom are calculated by the Mulliken population analysis. NBO analysis and IR frequency calculation performed to find out the strength of interaction of nitro group of

Au13 and Ag13 cluster (magic number) were optimized using B3LYP method and LaNL2DZ basis set using Gaussian program [25, 26]. Optimized structure of Ag13 and Au13 has been shown in **Figure 1(a)** and **(b)** respectively. Mulliken charge present on each atom is shown clearly. Several literature are available explaining the charge distribution on gold and silver cluster by varying number of atoms attached as well as by varying the shape of the clusters [24, 25]. It has been reported by Chen and Johnston [25] that charges on atoms vary with variation in shape of

**1 (a) 1(b)**

**Figure 1.** Optimized structure of (a) Ag13 cluster and (b) Au13 cluster with their Mulliken charges present on it.

lized by PVP and its effect on PNP has been studied in detail using DFT calculation.

**2. Computational procedure**

PNP with silver cluster.

**3. Results and discussion**

**3.1. Geometry optimization and Mulliken charge distribution**

the stability of the moieties. Mechanism for activation of the nitro group and formation of nitroso intermediate has been proposed from NBO analysis. IR study of the Ag cluster stabilized by PVP and its effect on PNP has been studied in detail using DFT calculation.

#### **2. Computational procedure**

are effectively a bridge between bulk materials and atomic or molecular structures. Synthesis of M-NPs can proceed by chemical reduction, thermolysis, photochemical decomposition, electro reduction, microwave and sonochemical irradiation. In recent literature there are large number of reports on synthesis, properties and applications of noble metals Au, Ag and Cu NPs [1–3]. This is mainly because all three elements show good localized surface plasmon resonance (LSPR) absorbance in the visible range and also have large number of catalytic applications. However, the high cost of Au and to a lesser extent Ag restrict their applications in many cases. As an alternative of Au and Ag NPs, researchers have investigated Cu nanoparticles. CuNPs are less expensive and exhibit comparably higher electrical conductivity and catalytic activity [4, 5]. The LSPR absorbance of CuNPs in the visible range is comparatively less intense. Another issue is that Cu nanoparticles are easily oxidized [6], yet nanoparticles of copper oxides also have wide applicability as catalysts. Silver NP found to be quite stable and hence chosen for studying the effect of nanostructures on their catalytic activity. In liquid phase nanoparticle synthesis, polymeric molecules are often used for stabilizing nanoparticles against aggregation. The function of the polymers are to avoid the aggregation of the NP in solution and to control the size and shape at the crystallographic level [7–9]. Among various polymeric molecules investigated in literature, polyvinyl pyrrolidone (PVP) has been one of the most frequently used stabilizers since it is non-toxic and soluble in many polar solvents. While such PVP stabilized noble metal nanoparticles have extensively been used as catalysts for various reactions [10, 11], few workers have concentrated in literature on the effect of such stabilizer molecule on the nanoparticle surface on their catalytic properties [12–14]. Nevertheless, only a few research papers have investigated the effect of such stabilization of nanoparticle surfaces on the electronic properties of such nanocomposites. Tsunoyama et al. proposed that electron transfer occurs from the anionic Au cores of Au:PVP

154 Density Functional Calculations - Recent Progresses of Theory and Application

which generates superoxo or peroxo like species. Latter plays a

key role in the oxidation of alcohol [10]. Similar mechanisms were proposed by some other related experimental studies as well that the adsorption of PVP on to the catalyst surface can also modify the electronic structure of nanoparticles by charge transfer [15, 16]. Yet several aspects remain unclear, such as whether PVP attaches to the metal surface through its O atom or through the N atom [15, 17]. The strength of interaction of PVP with the metal surface is also another aspect that needs to be investigated. Finally, the most important question that in a catalytic reaction how does the interaction between the catalyst surface and the reactant/substrate change in presence of stabilizer. Very few reports are available regarding this topic [18, 19]. The catalytic reduction of nitroarenes to aminoarenes by transfer hydrogenation methodologies is an important class of organic transformations. The reaction does not occur even in the presence of strong reductants like metal hydrides unless catalyzed by a suitable nanocatalyst [12]. Synthesis of silver NP using PVP has been reported in our earlier study [20]. PVP stabilized Ag nanoparticles have often been successfully used as catalysts for such reactions [14, 20]. However, to the best of our knowledge, there is no detailed DFT study on effect of PVP (poly(N-vinyl-2-pyrrolidone)) stabilization of Ag cluster on its catalytic activity with respect to a nitroarene (p-nitrophenol) substrate. In this chapter, we first carry out DFT calculations on Ag/Au cluster along with the monomer of PVP moiety using Gaussian program. Further, the effect of PVP stabilized Ag clusters (catalyst) on p-nitrophenol (PNP) has been investigated in detail. Energy of interaction found by the B3LYP level of calculation elucidates

into the LUMO (*π*\*) of O<sup>2</sup>

A 13 atom silver and gold cluster is carved out of the FCC Ag/Au lattice constructed using the MAPS software (Scienomics). The Gaussian 03 program [21] package employed for the DFT calculations at the Becke's three parameter functional and Lee–Yang–Parr hybrid functional (B3LYP) level [22, 23] of calculation. LaNL2DZ for Ag/Au and 6-31G++(d,p) basis set for C, H, N and O atoms was used while performing DFT calculation. B3LYP functional calculations give the stable cluster and also reproduce the experimental results [17, 18, 24]. Calculations are carried out for ground state geometry optimization in gaseous phase. The Mulliken charges of each atom are calculated by the Mulliken population analysis. NBO analysis and IR frequency calculation performed to find out the strength of interaction of nitro group of PNP with silver cluster.

#### **3. Results and discussion**

#### **3.1. Geometry optimization and Mulliken charge distribution**

Au13 and Ag13 cluster (magic number) were optimized using B3LYP method and LaNL2DZ basis set using Gaussian program [25, 26]. Optimized structure of Ag13 and Au13 has been shown in **Figure 1(a)** and **(b)** respectively. Mulliken charge present on each atom is shown clearly. Several literature are available explaining the charge distribution on gold and silver cluster by varying number of atoms attached as well as by varying the shape of the clusters [24, 25]. It has been reported by Chen and Johnston [25] that charges on atoms vary with variation in shape of

**Figure 1.** Optimized structure of (a) Ag13 cluster and (b) Au13 cluster with their Mulliken charges present on it.

the cluster. Mulliken charges present in the pure Ag13 Ih (icosahedral) cluster are +0.256 for the central Ag atom and −0.022 for a peripheral Ag atom. The charges in the pure Au13 Oh (cuboctahedral) cluster are +0.379 for the central Au atom and −0.033 for a peripheral Au atom. Hence, the structural order in 13-atom icosahedral (Ih) and cuboctahedral (Oh) clusters also induces charge transfer from the central atom to the peripheral ones. Varying the number of atoms in an alloy changes the property of nanoparticles drastically. It has been shown that with introduction of single Au/Ag atom in Ag13/Au13 cluster, charge present on each atom vary hugely. Stability of nanoalloys comes from a directional charge transfer induced by the structural order, which is added to that induced by the electronegativity difference between unlike atoms. As the Pauling electronegativity of Au (2.4) is greater than that of Ag (1.9), there is a degree of charge transfer from Ag to Au atoms [25]. It is clear from **Figure 1(a)** and **(b)** that charge present on gold is quite higher than that of silver atom. Central atom has huge positive charge as compared to negative charge on the surface. Also one can say that gold is more active catalyst as compared to silver. Okumura et al. [24] had presented the DFT calculation of gold nanoparticle stabilized with PVP at B3LYP level of calculation. Role of PVP on the catalytic activities of gold cluster has been explained very well and in refined way. Presence of PVP not only acts as a stabilizer to prevent aggregation, but also activates the catalyst by supplying charge to it. Calculations have shown that the charge transfer from the adsorbed PVP to Au13 produces negatively charged O2 on Au13-4PVP. Hence one can conclude that the catalytic activities of Au clusters are affected by the adsorbed PVPs. Varying the number of adsorbed PVP, charge present on the catalyst vary drastically. Similar observation was observed for the silver cluster (Ag13) in our calculation. This is the first report explaining the effect of PVP on the silver cluster.

**3.2. Effect of p-nitrophenol on silver cluster surrounded by PVP**

strongly with silver cluster than that of PVP molecule.

**3.3. Electronic effect due to PVP interaction with nanoparticle**

**(a) (b)**

**Figure 3.** Optimized structure of (a) p-nitrophenol and (b) Ag13-2-PVP-PNP moiety.

Among the available nitroarenes, the reduction of p-nitrophenol (PNP) using NaBH4 as the reductant has been studied extensively as a model pollutant and catalytic reduction reaction [27]. Several literatures are available for this catalytic reduction reaction. In recent study, it has been investigated that during catalytic reduction of p-nitrophenol to p-aminophenol (AP) glycerol as the reductant also in a mixture of glycerol and water as the reaction medium [20]. Several experimental literatures are available for this reduction reaction [27] but the actual mechanism for this catalytic reduction of p-nitrophenol to p-aminophenol is still unclear.

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To study the catalytic reduction mechanism, p-nitrophenol was incorporated in the optimized structure of Ag13-2PVP moiety and reoptimized the whole system at same B3LYP level of calculation. Optimized structure of PNP and Ag13-2PVP-PNP has been shown in **Figure 3(a)** and **(b)** respectively. Important geometrical parameters such as bond length, bond angles have been shown in **Table 1**. It has been observed that N-O bond length increases from 1.28 Å (in PNP) to 1.36 Å in Ag13-2-PVP-PNP moiety. N11-C1 bond length decreases by 0.05 Å when PNP interact with Ag13 cluster. <O12-N11-O13 decrease by ~3° when interact with Ag13 cluster. Ag7----C24=O25, Ag4----C41=O42 distance found to be 2.32 and 2.36 Å. This shows strong interaction with silver cluster and PVP molecules. Ag1-O60 and Ag2-O59 bond length calculated to be 2.28 and 2.27 Å, which clearly shows that PNP is much closer and interacting

**Figure 4(a)** and **(b)** shows the Mulliken charge distribution of Ag13 and Ag13-2PVP moiety (shown by different colors). Negative charge on Ag13 is symmetrically distributed among the shell Ag atoms of the cluster. This is balanced by the electropositive central Ag atom (shown by green color). Charges on shell silver atoms of the Ag cluster (before interaction) found to be either −0.09 or −0.15. Charge on central silver atom is detected to be +1.61. This result is similar with that as reported by Li and Chen [29] and Chen and Johnston [25]. To balance the negative

Hence to study the catalytic reduction mechanism DFT proves to be very useful [28].

Optimized structure of Ag13-2PVP with the charge present on each atoms are clearly has been shown in **Figure 2**. Distance between oxygen of PVP and different silver atoms are 2.32 and 2.36 Å. It clearly shows that interaction of PVP with silver cluster is quite strong. Also charge present on bare Ag13 cluster and Ag13 surrounded by 2 PVP moieties are quite different. Higher charge on the surface silver atom of Ag13-2PVP is indicative of the fact that PVP does not just acts as a stabilizer, but activates the catalyst as well, similar as obtained by Okumura et al. for gold cluster [24].

**Figure 2.** Optimized structure of Ag13-2PVP moiety with charge present on each atom.

#### **3.2. Effect of p-nitrophenol on silver cluster surrounded by PVP**

the cluster. Mulliken charges present in the pure Ag13 Ih (icosahedral) cluster are +0.256 for the central Ag atom and −0.022 for a peripheral Ag atom. The charges in the pure Au13 Oh (cuboctahedral) cluster are +0.379 for the central Au atom and −0.033 for a peripheral Au atom. Hence, the structural order in 13-atom icosahedral (Ih) and cuboctahedral (Oh) clusters also induces charge transfer from the central atom to the peripheral ones. Varying the number of atoms in an alloy changes the property of nanoparticles drastically. It has been shown that with introduction of single Au/Ag atom in Ag13/Au13 cluster, charge present on each atom vary hugely. Stability of nanoalloys comes from a directional charge transfer induced by the structural order, which is added to that induced by the electronegativity difference between unlike atoms. As the Pauling electronegativity of Au (2.4) is greater than that of Ag (1.9), there is a degree of charge transfer from Ag to Au atoms [25]. It is clear from **Figure 1(a)** and **(b)** that charge present on gold is quite higher than that of silver atom. Central atom has huge positive charge as compared to negative charge on the surface. Also one can say that gold is more active catalyst as compared to silver. Okumura et al. [24] had presented the DFT calculation of gold nanoparticle stabilized with PVP at B3LYP level of calculation. Role of PVP on the catalytic activities of gold cluster has been explained very well and in refined way. Presence of PVP not only acts as a stabilizer to prevent aggregation, but also activates the catalyst by supplying charge to it. Calculations have shown that the charge transfer from the adsorbed PVP to Au13 produces negatively charged O2

Au13-4PVP. Hence one can conclude that the catalytic activities of Au clusters are affected by the adsorbed PVPs. Varying the number of adsorbed PVP, charge present on the catalyst vary drastically. Similar observation was observed for the silver cluster (Ag13) in our calculation. This

Optimized structure of Ag13-2PVP with the charge present on each atoms are clearly has been shown in **Figure 2**. Distance between oxygen of PVP and different silver atoms are 2.32 and 2.36 Å. It clearly shows that interaction of PVP with silver cluster is quite strong. Also charge present on bare Ag13 cluster and Ag13 surrounded by 2 PVP moieties are quite different. Higher charge on the surface silver atom of Ag13-2PVP is indicative of the fact that PVP does not just acts as a stabilizer, but activates the catalyst as well, similar as obtained by Okumura

is the first report explaining the effect of PVP on the silver cluster.

156 Density Functional Calculations - Recent Progresses of Theory and Application

**Figure 2.** Optimized structure of Ag13-2PVP moiety with charge present on each atom.

et al. for gold cluster [24].

on

Among the available nitroarenes, the reduction of p-nitrophenol (PNP) using NaBH4 as the reductant has been studied extensively as a model pollutant and catalytic reduction reaction [27]. Several literatures are available for this catalytic reduction reaction. In recent study, it has been investigated that during catalytic reduction of p-nitrophenol to p-aminophenol (AP) glycerol as the reductant also in a mixture of glycerol and water as the reaction medium [20]. Several experimental literatures are available for this reduction reaction [27] but the actual mechanism for this catalytic reduction of p-nitrophenol to p-aminophenol is still unclear. Hence to study the catalytic reduction mechanism DFT proves to be very useful [28].

To study the catalytic reduction mechanism, p-nitrophenol was incorporated in the optimized structure of Ag13-2PVP moiety and reoptimized the whole system at same B3LYP level of calculation. Optimized structure of PNP and Ag13-2PVP-PNP has been shown in **Figure 3(a)** and **(b)** respectively. Important geometrical parameters such as bond length, bond angles have been shown in **Table 1**. It has been observed that N-O bond length increases from 1.28 Å (in PNP) to 1.36 Å in Ag13-2-PVP-PNP moiety. N11-C1 bond length decreases by 0.05 Å when PNP interact with Ag13 cluster. <O12-N11-O13 decrease by ~3° when interact with Ag13 cluster. Ag7----C24=O25, Ag4----C41=O42 distance found to be 2.32 and 2.36 Å. This shows strong interaction with silver cluster and PVP molecules. Ag1-O60 and Ag2-O59 bond length calculated to be 2.28 and 2.27 Å, which clearly shows that PNP is much closer and interacting strongly with silver cluster than that of PVP molecule.

#### **3.3. Electronic effect due to PVP interaction with nanoparticle**

**Figure 4(a)** and **(b)** shows the Mulliken charge distribution of Ag13 and Ag13-2PVP moiety (shown by different colors). Negative charge on Ag13 is symmetrically distributed among the shell Ag atoms of the cluster. This is balanced by the electropositive central Ag atom (shown by green color). Charges on shell silver atoms of the Ag cluster (before interaction) found to be either −0.09 or −0.15. Charge on central silver atom is detected to be +1.61. This result is similar with that as reported by Li and Chen [29] and Chen and Johnston [25]. To balance the negative

**Figure 3.** Optimized structure of (a) p-nitrophenol and (b) Ag13-2-PVP-PNP moiety.


(b)

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(d)

(e) (f)

**Figure 4.** Mulliken charge distribution on different atoms of (a) Ag13 cluster and (b) Ag13-2PVP (c) PNP (d) Ag-PNP. Colour representation of Mulliken charge on different atoms of (e) PNP (f) Ag13-PNP. Red colour represents the most

(a)

(c)

electronegative and green colour represents the most electropositive atom.

**Figure 5.** Electrostatic potential charge distribution on Ag13-2PVP-PNP moiety.

**Table 1.** Selected bond length, bond angles of different system obtained from DFT calculation.

charge present on shell atoms, core becomes positively charged to neutralize and stabilized the whole cluster. As PVP interacts with the Ag13 cluster, charge distribution becomes asymmetrical. The charges present on shell Ag atoms interacting with (PVP) O atoms are −0.07 and −0.05. Rest of the shell Ag atoms have charges more or less around −0.30. The central Ag atom is most electron deficient carrying charge of +3.07. Adsorption of the PVP model molecule onto the surface of Ag13 increases the negative charge density on Ag13 cluster, similar as for Au13 cluster explained by Okumura et al. [24]. With adsorption of PVP on the surface, catalyst becomes more active due to increase of negative charge on the Ag13 surface. To reduce the computational cost, PVP has been removed for further interaction study of PNP with Ag13. Optimized structure of PNP and Ag13-PNP has been shown in **Figure 4(c)** and **(d)** along with their respective charges. Variation in charges on different atoms of PNP before and after interaction has been clearly shown by the different colored atoms of **Figure 4(e)** and **(f)** respectively.

These interactions of PNP with that of Ag13 are better understood from electrostatic potential (ESP) charge distribution shown in **Figure 5**. Orange color represents the negative electron density around the electronegative oxygen atoms.

#### **3.4. Natural bonding orbital (NBO) analysis**

Stabilization energy E(2) found to be proportional to the charge transfer energy or charge distribution energy [30]. For each donor NBO(i) and acceptor NBO(j), the stabilization energy E(2) associated with delocalization of electron pair from donor orbital (i) to acceptor orbital (j) and is defined as

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**Figure 4.** Mulliken charge distribution on different atoms of (a) Ag13 cluster and (b) Ag13-2PVP (c) PNP (d) Ag-PNP. Colour representation of Mulliken charge on different atoms of (e) PNP (f) Ag13-PNP. Red colour represents the most electronegative and green colour represents the most electropositive atom.

**Figure 5.** Electrostatic potential charge distribution on Ag13-2PVP-PNP moiety.

charge present on shell atoms, core becomes positively charged to neutralize and stabilized the whole cluster. As PVP interacts with the Ag13 cluster, charge distribution becomes asymmetrical. The charges present on shell Ag atoms interacting with (PVP) O atoms are −0.07 and −0.05. Rest of the shell Ag atoms have charges more or less around −0.30. The central Ag atom is most electron deficient carrying charge of +3.07. Adsorption of the PVP model molecule onto the surface of Ag13 increases the negative charge density on Ag13 cluster, similar as for Au13 cluster explained by Okumura et al. [24]. With adsorption of PVP on the surface, catalyst becomes more active due to increase of negative charge on the Ag13 surface. To reduce the computational cost, PVP has been removed for further interaction study of PNP with Ag13. Optimized structure of PNP and Ag13-PNP has been shown in **Figure 4(c)** and **(d)** along with their respective charges. Variation in charges on different atoms of PNP before and after interaction has been clearly shown by the different colored atoms of **Figure 4(e)** and **(f)** respectively. These interactions of PNP with that of Ag13 are better understood from electrostatic potential (ESP) charge distribution shown in **Figure 5**. Orange color represents the negative electron

**Table 1.** Selected bond length, bond angles of different system obtained from DFT calculation.

**Parameters [bond length (Å)/bond angles (°)] Data obtained from DFT calculation**

N11-O12 1.28 N11-C1 1.46 <O12-N11-O13 123.51 <O12-N11-C1 118.23

158 Density Functional Calculations - Recent Progresses of Theory and Application

N58-O59 1.36 N58-C48 1.41 Ag1-O60 2.28 Ag2-O59 2.27 <O59-N58-O60 120.83 <O59-N58-C48 119.71 Ag7----C24=O25 2.32 Ag4----C41=O42 2.36

Stabilization energy E(2) found to be proportional to the charge transfer energy or charge distribution energy [30]. For each donor NBO(i) and acceptor NBO(j), the stabilization energy E(2) associated with delocalization of electron pair from donor orbital (i) to acceptor orbital

density around the electronegative oxygen atoms.

**3.4. Natural bonding orbital (NBO) analysis**

(j) and is defined as

**p-nitrophenol (PNP)**

**Ag13-2-PVP-PNP**

$$E(\text{2}) = \,\,\,\Delta\,E\_{\psi} = \frac{q\_i F(i,j)^2}{\varepsilon\_i - \varepsilon\_j} \tag{1}$$

IR frequency calculations on the optimized structure of Ag13-2PVP-PNP moiety at same B3LYP level of calculation and same basis set as mention above. To analyze the changes that occurred in the vibrational spectrum due to such association we contrast it with the calculated IR frequencies of the optimized structure of PNP alone. Calculated data has been presented in **Table 3**. For the same vibrational motion, the frequency of the peak present in PNP varies significantly when it is associated with the silver cluster. Scissoring of O-N-O occurs at 621 cm−1 in PNP while it is observed at 588 cm−1 (red shift of 33 cm−1) when there is interaction with Ag cluster. Similarly, a strong red shift of 131 cm−1 is observed when PNP interacts with silver cluster for benzene ring breathing coupled with C-O(H) stretching and O-N-O scissoring motion in PNP. It is due to cou-

with that of the silver cluster. Another strong red shift of 103 cm−1 is observed for

Catalytic Activation of PVP-Stabilized Gold/Silver Cluster on p-Nitrophenol Reduction: A DFT

**Intensity Difference in** 

588 85 621 27 33 O-N-O scissoring in PNP 710 65 841 30 131 Benzene ring breathing

864 58 887 68 23 Out of plane bending of

1113 85 1124 118 11 C-H in plane bending

1167 356 1170 246 3 C-O-H scissoring in PNP

1174 41 1277 341 103 O-N-O symmetric

1192 32 1204 81 12 H-C=C-H scissoring in

1267 53 1287 55 20 C-O(H) stretching in PNP 1322 42 1277 341 45 C-N stretching in PNP 1520 190 1529 32 9 C-H in plane bending in

1645 60 1657 120 12 C=C stretching in PNP 3185 34 3210 10 25 C-H asymmetric

3708 91 3704 81 4 O-H stretching in PNP

**Table 3.** DFT calculated IR frequency of NB and Ag13-2PVP-PNP at B3LYP level.

**(PNP)]**

**wavenumber/(cm−1)/ [Ag13-2PVP-PNP-**

**Assignment of bands**

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coupled with C-O(H) stretching and O-N-O scissoring in PNP

coupled with O-N-O asymmetric stretching

coupled with C-H in plane bending

stretching in PNP

PNP coupled with C-N

stretching in PNP coupled with C-O-H

C-H in PNP

in PNP

PNP

stretching

scissoring

pling of NO2

**Wavenumber/(cm−1)/ Ag13-2PVP-PNP**

**Intensity Wavenumber (cm−1)/PNP**

where *qi* is the donor orbital occupancy, *ε<sup>i</sup>* and *ε<sup>j</sup>* are the diagonal elements (orbital energies). F(i, j) is the interaction element between donor and acceptor orbitals and is known as diagonal NBO Fock matrix element. The delocalization effects can be identified by means of off-diagonal elements of the Fock matrix. The forces of these delocalization interaction, E(2) (kcal/mol), are estimated by second order perturbation theory [31]. E(2) term corresponding to these interactions can also be the total charge transfer energy in the molecule.

To better understand the interaction between Ag13 cluster and PNP molecules, NBO calculation carried out at same B3LYP level of calculation. **Table 2** presents the major interaction present between silver cluster and PNP through both the oxygen atoms of the latter. It has been observed from **Table 2** that in addition to charge transfer from the oxygen (of the nitro group) to Ag atom, there is also a back donation of electron from M(dπ) orbital of Ag2 to antibonding σ\* of N58-O59 [E(2) = 2.06 kcal/mol]. Further, the back donation of electron from M(dπ) orbital of Ag1 to antibonding σ\* of N58-O60 is comparatively much weaker [E(2) = 0.71 kcal/ mol]. This shows that only one of the two N-O bonds is considerably weakened by this back donation of electron from Ag atoms in comparison to the other N-O bond. Hence, from these obtained NBO results, one can predict the subsequent formation of nitroso compound as an intermediate. This mechanism is in agreement with the one proposed by Liu et al. in an experimental study for reduction of nitrobenzene in presence of Ag catalyst [12] and for reduction of PNP in presence of Ag catalyst by Gu et al. [27].

#### **3.5. Calculated vibrational spectra and analysis of calculated IR spectra of PNP and Ag13-2PVP-PNP**


IR frequency calculations prove to be a good tool for predicting the interaction present in a molecule [31]. For better analysis of the interaction of PNP with silver cluster, we perform the

**Table 2.** Significant donor-acceptor NBO interactions in Ag cluster and PNP moiety with calculated second order stabilization energies E(2) (kcal/mol).

IR frequency calculations on the optimized structure of Ag13-2PVP-PNP moiety at same B3LYP level of calculation and same basis set as mention above. To analyze the changes that occurred in the vibrational spectrum due to such association we contrast it with the calculated IR frequencies of the optimized structure of PNP alone. Calculated data has been presented in **Table 3**. For the same vibrational motion, the frequency of the peak present in PNP varies significantly when it is associated with the silver cluster. Scissoring of O-N-O occurs at 621 cm−1 in PNP while it is observed at 588 cm−1 (red shift of 33 cm−1) when there is interaction with Ag cluster. Similarly, a strong red shift of 131 cm−1 is observed when PNP interacts with silver cluster for benzene ring breathing coupled with C-O(H) stretching and O-N-O scissoring motion in PNP. It is due to coupling of NO2 with that of the silver cluster. Another strong red shift of 103 cm−1 is observed for

*<sup>E</sup>*(2) <sup>=</sup> *<sup>Δ</sup> Eij* <sup>=</sup> *qi <sup>F</sup>* (*i*, *<sup>j</sup>*)2

160 Density Functional Calculations - Recent Progresses of Theory and Application

is the donor orbital occupancy, *ε<sup>i</sup>*

of PNP in presence of Ag catalyst by Gu et al. [27].

149. LP\*(6)Ag2 564. BD\*(1)N58-O59 2.06 262. LP(1)O60 142. LP\*(8)Ag1 1.97 264. LP(3)O60 140. LP\*(6)Ag1 5.21 264. LP(3)O60 141. LP\*(7)Ag1 4.68 264. LP(3)O60 142. LP\*(8)Ag1 1.39 261. LP(3)O59 149. LP\*(6)Ag2 6.45 261. LP(3)O59 150. LP\*(7)Ag2 3.17 260. LP(2)O59 150. LP\*(7)Ag2 2.04 259. LP(1)O59 150. LP\*(7)Ag2 4.02 55. BD(1)N58-O60 150. LP\*(7)Ag2 0.71

where *qi*

**Ag13-2PVP-PNP**

stabilization energies E(2) (kcal/mol).

\_\_\_\_\_\_ *ε<sup>i</sup>* − *ε<sup>j</sup>*

are the diagonal elements (orbital ener-

and *ε<sup>j</sup>*

gies). F(i, j) is the interaction element between donor and acceptor orbitals and is known as diagonal NBO Fock matrix element. The delocalization effects can be identified by means of off-diagonal elements of the Fock matrix. The forces of these delocalization interaction, E(2) (kcal/mol), are estimated by second order perturbation theory [31]. E(2) term corresponding

To better understand the interaction between Ag13 cluster and PNP molecules, NBO calculation carried out at same B3LYP level of calculation. **Table 2** presents the major interaction present between silver cluster and PNP through both the oxygen atoms of the latter. It has been observed from **Table 2** that in addition to charge transfer from the oxygen (of the nitro group) to Ag atom, there is also a back donation of electron from M(dπ) orbital of Ag2 to antibonding σ\* of N58-O59 [E(2) = 2.06 kcal/mol]. Further, the back donation of electron from M(dπ) orbital of Ag1 to antibonding σ\* of N58-O60 is comparatively much weaker [E(2) = 0.71 kcal/ mol]. This shows that only one of the two N-O bonds is considerably weakened by this back donation of electron from Ag atoms in comparison to the other N-O bond. Hence, from these obtained NBO results, one can predict the subsequent formation of nitroso compound as an intermediate. This mechanism is in agreement with the one proposed by Liu et al. in an experimental study for reduction of nitrobenzene in presence of Ag catalyst [12] and for reduction

**3.5. Calculated vibrational spectra and analysis of calculated IR spectra of PNP and** 

IR frequency calculations prove to be a good tool for predicting the interaction present in a molecule [31]. For better analysis of the interaction of PNP with silver cluster, we perform the

**Donor orbital (i) Acceptor orbital (j) Second order perturbation stabilization** 

**Table 2.** Significant donor-acceptor NBO interactions in Ag cluster and PNP moiety with calculated second order

**energy E(2)/(kcal/mol)**

to these interactions can also be the total charge transfer energy in the molecule.

(1)


**Table 3.** DFT calculated IR frequency of NB and Ag13-2PVP-PNP at B3LYP level.

O-N-O symmetric stretching in PNP. For other vibrational motions, difference in wavenumbers has been shown in **Table 3**. Red shifts in most of the frequencies are due to strong interaction of PNP with silver cluster present in Ag13-2PVP-PNP model. Hence IR frequency calculation proves to be a good tool in predicting the strength of interaction present in a molecule [32].

[2] Stamplecoskie KG, Scaiano J. Optimal size of silver nanoparticles for surface-enhanced

Catalytic Activation of PVP-Stabilized Gold/Silver Cluster on p-Nitrophenol Reduction: A DFT

http://dx.doi.org/10.5772/intechopen.72097

163

[3] Duan XC, Ma JM, Lian JB, Zheng WJ. The art of using ionic liquids in the synthesis of

[4] Choi CS, Jo YH, Kim MG, Lee HM. Control of chemical kinetics for sub-10 nm Cu nanoparticles to fabricate highly conductive ink below 150 °C. Nanotechnology.

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[6] Muzikansky A, Nanikashvili P, Grinblat J, Zitoun D. Ag dewetting in Cu@Ag monodisperse core–shell nanoparticles. Journal of Physical Chemistry C. 2013;**117**:

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[10] Tsunoyama H, Ichikuni N, Sakurai H, Tsukuda T. Effect of electronic structures of Au clusters stabilized by poly(N-vinyl-2-pyrrolidone) on aerobic oxidation catalysis. Journal

[11] Quintanilla A, Butselaar-Orthlieb VCL, Kwakernaak C, Sloof WG, Kreutzer MT, Kapteijn F. Weakly bound capping agents on gold nanoparticles in catalysis: Surface poison?

[12] Liu X, Cheng H, Cui P. Catalysis by silver nanoparticles/porous silicon for the reduction of nitroaromatics in the presence of sodium borohydride. Applied Surface Science.

[13] Vadakkekara R, Chakraborty M, Parikh PA. Reduction of aromatic nitro compounds on colloidal hollow silver nanospheres. Colloids and Surfaces A: Physicochemical and

[14] Tejamaya M, Romer I, Merrified RC, Lead JR. Stability of citrate, PVP, and PEG coated silver nanoparticles in ecotoxicology media. Environmental Science & Technology. 2012;

[15] Xian J, Jiang QHZ, Ma Y, Huang W. Size-dependent interaction of the poly(N-vinyl-2-pyrrolidone) capping ligand with Pd nanocrystals. Langmuir. 2012;**28**:6736-6741 [16] Aguilar JG, Garcia MN, Murcia AB, Mori K, Kuwahara Y, Yamashita H, Amoros DC. Evolution of the PVP–Pd surface interaction in nanoparticles through the case study of

formic acid decomposition. Langmuir. 2016;**32**:12110-12118

effects on benzene hydrogenation selectivity. Nano Letters. 2007;**7**:3097-3101

aqueous solution. Materials Research Innovations. 2007;**11**:201-206

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**46**:7011-7017

#### **4. Conclusions**

DFT calculations on PVP stabilized Ag13 cluster have shown that the PVP acts not only as a stabilizer but also plays an important role in activating the Ag catalyst by supplying extra charge to it, mainly through oxygen atom. Electrostatic potential (ESP) charge distribution demonstrates that nitro group in p-nitrophenol (PNP) interacts strongly through oxygen end with the Ag cluster. Furthermore, Natural Bonding Orbital (NBO) analysis shows that there is weak back donation of electron from M(dπ) orbital of Ag to antibonding σ\* of one of the N-O bonds while the other N-O bond in PNP is not affected. Therefore, one of the N-O bonds is drastically weakened in comparison to other N-O bond. Hence the formation of the nitroso intermediate will assist its further reduction reaction. Hence stabilization energy calculation can be a good tool in predicting the intermediate in a reaction catalyzed by NPs, which is difficult to predict experimentally. Finally, significant red shifts in calculated IR frequencies are a consequence of strong interaction of PNP with silver cluster present in Ag13-2PVP-PNP model. Hence, IR frequency calculation is a good tool for predicting the strength of interaction present in a molecule.

#### **Acknowledgements**

Computer Centre, BHU and Computer Unit, IIT-BHU is acknowledged for providing the computational facility. SERB (PDF/2017/002589) is acknowledged for providing financial support.

### **Author details**

Madhulata Shukla1,2\* and Indrajit Sinha1

\*Address all correspondence to: madhu1.shukla@gmail.com

1 Department of Chemistry, Indian Institute of Technology (Banaras Hindu University), Varanasi, India

2 G.B. College, Ramgarh, Kaimur, Veer Kunwar Singh University, Kaimur, India

#### **References**

[1] Iravani S, Korbekandi H, Mirmohammadi SV, Zolfaghari B. Synthesis of silver nanoparticles: chemical, physical and biological methods. Research in Pharmaceutical Sciences. 2014;**9**:385-406

[2] Stamplecoskie KG, Scaiano J. Optimal size of silver nanoparticles for surface-enhanced Raman spectroscopy. Journal of Physical Chemistry C. 2011;**115**:1403-1409

O-N-O symmetric stretching in PNP. For other vibrational motions, difference in wavenumbers has been shown in **Table 3**. Red shifts in most of the frequencies are due to strong interaction of PNP with silver cluster present in Ag13-2PVP-PNP model. Hence IR frequency calculation proves to be a good tool in predicting the strength of interaction present in a molecule [32].

162 Density Functional Calculations - Recent Progresses of Theory and Application

DFT calculations on PVP stabilized Ag13 cluster have shown that the PVP acts not only as a stabilizer but also plays an important role in activating the Ag catalyst by supplying extra charge to it, mainly through oxygen atom. Electrostatic potential (ESP) charge distribution demonstrates that nitro group in p-nitrophenol (PNP) interacts strongly through oxygen end with the Ag cluster. Furthermore, Natural Bonding Orbital (NBO) analysis shows that there is weak back donation of electron from M(dπ) orbital of Ag to antibonding σ\* of one of the N-O bonds while the other N-O bond in PNP is not affected. Therefore, one of the N-O bonds is drastically weakened in comparison to other N-O bond. Hence the formation of the nitroso intermediate will assist its further reduction reaction. Hence stabilization energy calculation can be a good tool in predicting the intermediate in a reaction catalyzed by NPs, which is difficult to predict experimentally. Finally, significant red shifts in calculated IR frequencies are a consequence of strong interaction of PNP with silver cluster present in Ag13-2PVP-PNP model. Hence, IR frequency calculation is a good tool for predicting the strength of interaction present in a molecule.

Computer Centre, BHU and Computer Unit, IIT-BHU is acknowledged for providing the computational facility. SERB (PDF/2017/002589) is acknowledged for providing financial support.

1 Department of Chemistry, Indian Institute of Technology (Banaras Hindu University),

[1] Iravani S, Korbekandi H, Mirmohammadi SV, Zolfaghari B. Synthesis of silver nanoparticles: chemical, physical and biological methods. Research in Pharmaceutical Sciences.

2 G.B. College, Ramgarh, Kaimur, Veer Kunwar Singh University, Kaimur, India

**4. Conclusions**

**Acknowledgements**

**Author details**

Varanasi, India

**References**

2014;**9**:385-406

Madhulata Shukla1,2\* and Indrajit Sinha1

\*Address all correspondence to: madhu1.shukla@gmail.com


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**Chapter 8**

Provisional chapter



**Mechanistic Study on the Formation of Compounds**

DOI: 10.5772/intechopen.73664

Mechanistic Study on the Formation of Compounds

Warjeet S. Laitonjam and Lokendrajit Nahakpam

Formation of 2-(N-arylamino)benzothiazole takes place, when N,N<sup>0</sup>

solvent free conditions. However, when N-substituted-N<sup>0</sup>

treated with polymer-supported tribromide or with iodine-alumina as catalyst under

with polymer-supported tribromide or with iodine-alumina as catalyst either under various conditions or under solvent free conditions, decomposition takes place to give the respective benzamides and thiobenzamides. Mechanistic study of the formation of these compounds is studied using DFT calculations. It is found that electron donating group at the para-position of the aryl group of benzoylthiourea favors the formation of benzamide whereas the presence of electron withdrawing group at para-position of the aryl group of benzoylthiourea, formation of thiobenzamide takes place. When the catalyst is changed to diacetoxyiodobenzene (DIB) under similar reaction conditions, benzoxazole amides are formed; expected benzothiazoles or the decomposition products are not obtained. Mechanistic study of the reaction using DFT calculation again shows that the reaction followed through carbodiimide intermediate undergoes the formation of C-O bond in benzoxazole moiety, instead of the expected C-S bond formation of benzothiazole moiety via a sequen-

Keywords: mechanism, benzoxazoles, benzothiazoles, decomposition, thioureas,

Heterocyclic chemistry is the most complex and intriguing branch of organic chemistry, and heterocyclic compounds constitute the largest and most unique family of organic compounds [1–3]. Nitrogen, oxygen, and sulfur are the most common heteroatoms but some other heterocyclic compounds containing selenium, tellurium, phosphorus, arsenic, silicon, boron, etc., are

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Warjeet S. Laitonjam and Lokendrajit Nahakpam

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73664

tial acylation and deacylation process.

DFT calculations

1. Introduction

**from Thioureas**

from Thioureas

Abstract

#### **Mechanistic Study on the Formation of Compounds from Thioureas** Mechanistic Study on the Formation of Compounds from Thioureas

DOI: 10.5772/intechopen.73664

Warjeet S. Laitonjam and Lokendrajit Nahakpam Warjeet S. Laitonjam and Lokendrajit Nahakpam

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73664

#### Abstract

Formation of 2-(N-arylamino)benzothiazole takes place, when N,N<sup>0</sup> -diphenylthioureas are treated with polymer-supported tribromide or with iodine-alumina as catalyst under solvent free conditions. However, when N-substituted-N<sup>0</sup> -benzoylthioureas are treated with polymer-supported tribromide or with iodine-alumina as catalyst either under various conditions or under solvent free conditions, decomposition takes place to give the respective benzamides and thiobenzamides. Mechanistic study of the formation of these compounds is studied using DFT calculations. It is found that electron donating group at the para-position of the aryl group of benzoylthiourea favors the formation of benzamide whereas the presence of electron withdrawing group at para-position of the aryl group of benzoylthiourea, formation of thiobenzamide takes place. When the catalyst is changed to diacetoxyiodobenzene (DIB) under similar reaction conditions, benzoxazole amides are formed; expected benzothiazoles or the decomposition products are not obtained. Mechanistic study of the reaction using DFT calculation again shows that the reaction followed through carbodiimide intermediate undergoes the formation of C-O bond in benzoxazole moiety, instead of the expected C-S bond formation of benzothiazole moiety via a sequential acylation and deacylation process.

Keywords: mechanism, benzoxazoles, benzothiazoles, decomposition, thioureas, DFT calculations

#### 1. Introduction

Heterocyclic chemistry is the most complex and intriguing branch of organic chemistry, and heterocyclic compounds constitute the largest and most unique family of organic compounds [1–3]. Nitrogen, oxygen, and sulfur are the most common heteroatoms but some other heterocyclic compounds containing selenium, tellurium, phosphorus, arsenic, silicon, boron, etc., are

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

also widely known. Heterocyclic compounds are present in many natural and non-naturally occurring compounds. Some examples of such compounds are alkaloids, vitamins (vitamin B series and vitamin C), antibiotics, amino acids, hemoglobin, hormones, pigments, and a large number of synthetic drugs and dyes. Several natural drugs such as morphine, codeine, quinine, penicillin, papaverine, atropine, emetine, reserpine, procaine, theophylline, etc., are examples of heterocyclic compounds. Some of the synthetic drugs have shown several therapeutic uses such as antidiabetic, antitubercular, antidepressant, antitumor, anti-HIV, anthelmintic, antibacterial, antifungal, antiviral, antimalarial, antileishmanial, analgesic, anti-inflammatory, anticonvulsant, anticancer, muscle relaxants, lipid peroxidation inhibitor, herbicidal, trypanocidal, fungicidal, and insecticidal activities. Thus, heterocyclic compounds are receiving more and more significance in recent years, particularly owing to their pharmacological as well as synthetic potential.

2-aminobenzothiazoles are one of the most important structural motifs in pharmaceutically active compounds and natural products [5]. A large number of 2-aminobenzothiazole derivatives are also found to be anticancer active and the 2-aminobenzothiazole moieties act as a privileged pharmacophores as well as valuable reactive intermediates [6–8]. For example, N-aryl substituted 2-aminobenzothiazole (A; R116010) is a potential inhibitor of retinoic acid metabolism for cancer treatment [9]; 6-substituted 2-aminobenzothiazole (B) is found to exhibit antifungal activity [10]. Riluzole (C) is a 2-aminobenzothiazole compound employed in the treatment of amyotrophic lateral sclerosis [11] and N-disubstituted 2-aminobenzothiazole (D, HM13N) is used as anti-HIV agent [12]. 2-(N-acylamino) benzothiazole derivatives, such as trihydroxybenzoyl-2-aminobenzothiazole (E) [12] exhibit significant topoisomerase I inhibitory activity. Moreover, derivatives of 2-aminobenzothiazoles such as benzothiazole-triazole-pyridine conjugated analogs (F) [13] showed better anti-TB activity compared to rifampicin (RIF) (Figure 1).

Mechanistic Study on the Formation of Compounds from Thioureas

http://dx.doi.org/10.5772/intechopen.73664

169

The main objectives of benzothiazoles synthesis are not only for the development of more diverse and complex bioactive compounds for biological activity and structure-activity relationship (SAR) studies but also for other applications, such as preparation of dyes. There are several methods for the synthesis of 2-aminobenzothiazoles. The most versatile and economical method involves the treatment of various substituted arylthioureas (which are synthesized via treatment of an aromatic amine with isothiocyanate) with oxidizing agent or cyclizing

Recently, several methods have been reported which utilize bromine as catalyst. Basically, cyclization with bromine is achieved by oxidation of aniline, substituted aniline, and arylthiourea in acid or chloroform with alkali thiocyanate. Hugerschoff, in early 1900s, synthesized 2-aminobenzothiazole and found that 1, 3-diarylthiourea can be cyclized with liquid

This reaction worked well for symmetrical thioureas giving exclusively one product. But, when the same reaction is performed using unsymmetrical 1,3-diaryl thioureas, there is always uncertainty as to on which aryl ring the intramolecular electrophilic substitution would take place to give aminobenzothiazole. Kamel et al. have reported the synthesis of 6-chloro-4-(trifluoromethyl) 2-aminobenzothiazole by oxidative cyclization of 4-chloro-2-(trifluoromethyl)phenylthiourea with bromine in chloroform to give an intermediate followed by basification with NH3 (Scheme 2) [16]. Jordan et al. have reported the use of benzyltrimethylammoniumtribromide (PhCH2NMe3Br3) which is an electrophilic bromine source for the conversion of substituted arylthiourea to 2 aminobenzothiazoles under mild conditions in different solvents with good yields (Scheme 3)

agent using different reaction conditions to yield 2-aminobenzothiazoles.

bromine in chloroform to form a 2-aminobenzothiazoles (Scheme 1) [14, 15].

Liu et al. have reported the metal-free synthesis of 2-aminobenzothiazoles from N<sup>0</sup>

N-(2-halophenyl) thioureas via a base-promoted cyclization in dioxane (Scheme 4) [18]. However, this reaction requires drastic conditions, like heating the vial which was sealed in an oil

The palladium-catalyzed intramolecular cyclization of 2-bromophenylthioureas to synthesize 2-substituted benzothiazoles was also reported. Castillon et al.reported Pd- catalyzed cyclization of 2-bromophenylthioamides using Pd2(dba)3/(2-biphenyl)P(t-Bu)2 catalytic system (Scheme 5) [19].


[17].

bath.

In recent years, green chemistry has become one of the most important philosophies in chemistry, since it represents a major change in the way we think about practicing chemistry and using chemicals. The emerging area of green chemistry envisages minimum hazard as the performance criteria while designing new chemical processes. The search for new environmentally benign solvents and catalysts that operate efficiently in them and can be easily recycled is of significant academic and industrial interest. There have been several approaches to access to this problem, e.g., the developments of neat reactions that proceed under various conditions such as microwave irradiation, thermal heating, grinding, sonication, etc., or in organic or inorganic solid-media, or in ionic liquid-media under organic solvent-free reactions. Among the proposed solutions, solvent free conditions are becoming more and more popular and it is often claimed that the best solvent from an ecological point of view is, without a doubt, no solvent. The formation of various compounds from thioureas and their derivatives under different catalysts in solvent free condition is highlighted.

The density functional theory (DFT) method has become one of the most prevalent and efficient tools, as compared to the conventional ab initio method (HF), for studying the detailed reaction mechanism in chemical systems during the last two decades. With DFT methods, many catalytic reaction mechanisms have been widely studied in addition to the assignment of experimental spectra. It is to be noted that study of reaction mechanism is important not only for understanding the reaction and its stereochemistry but also for designing new reactions and catalysts. The mechanistic pathways for the decomposition of benzoylthioureas into benzamides and thiobenzamides; and also the conversion of benzoylthioureas into benzoxazoles, instead of forming bezothiazoles, using different catalysts have also been highlighted with the help of DFT calculations.

#### 2. 2-Aminobenzothiazoles from thioureas

Benzothiazoles are an important class of heterocycles that possess a broad range of biological activities [4]. They were studied extensively for their anti-allergic, anti-inflammatory, antitumor, antimicrobial, and analgesic activities. Among those 2-substituted benzothiazole derivatives, the 2-aminobenzothiazoles are one of the most important structural motifs in pharmaceutically active compounds and natural products [5]. A large number of 2-aminobenzothiazole derivatives are also found to be anticancer active and the 2-aminobenzothiazole moieties act as a privileged pharmacophores as well as valuable reactive intermediates [6–8]. For example, N-aryl substituted 2-aminobenzothiazole (A; R116010) is a potential inhibitor of retinoic acid metabolism for cancer treatment [9]; 6-substituted 2-aminobenzothiazole (B) is found to exhibit antifungal activity [10]. Riluzole (C) is a 2-aminobenzothiazole compound employed in the treatment of amyotrophic lateral sclerosis [11] and N-disubstituted 2-aminobenzothiazole (D, HM13N) is used as anti-HIV agent [12]. 2-(N-acylamino) benzothiazole derivatives, such as trihydroxybenzoyl-2-aminobenzothiazole (E) [12] exhibit significant topoisomerase I inhibitory activity. Moreover, derivatives of 2-aminobenzothiazoles such as benzothiazole-triazole-pyridine conjugated analogs (F) [13] showed better anti-TB activity compared to rifampicin (RIF) (Figure 1).

also widely known. Heterocyclic compounds are present in many natural and non-naturally occurring compounds. Some examples of such compounds are alkaloids, vitamins (vitamin B series and vitamin C), antibiotics, amino acids, hemoglobin, hormones, pigments, and a large number of synthetic drugs and dyes. Several natural drugs such as morphine, codeine, quinine, penicillin, papaverine, atropine, emetine, reserpine, procaine, theophylline, etc., are examples of heterocyclic compounds. Some of the synthetic drugs have shown several therapeutic uses such as antidiabetic, antitubercular, antidepressant, antitumor, anti-HIV, anthelmintic, antibacterial, antifungal, antiviral, antimalarial, antileishmanial, analgesic, anti-inflammatory, anticonvulsant, anticancer, muscle relaxants, lipid peroxidation inhibitor, herbicidal, trypanocidal, fungicidal, and insecticidal activities. Thus, heterocyclic compounds are receiving more and more significance in recent years, particularly owing to their

In recent years, green chemistry has become one of the most important philosophies in chemistry, since it represents a major change in the way we think about practicing chemistry and using chemicals. The emerging area of green chemistry envisages minimum hazard as the performance criteria while designing new chemical processes. The search for new environmentally benign solvents and catalysts that operate efficiently in them and can be easily recycled is of significant academic and industrial interest. There have been several approaches to access to this problem, e.g., the developments of neat reactions that proceed under various conditions such as microwave irradiation, thermal heating, grinding, sonication, etc., or in organic or inorganic solid-media, or in ionic liquid-media under organic solvent-free reactions. Among the proposed solutions, solvent free conditions are becoming more and more popular and it is often claimed that the best solvent from an ecological point of view is, without a doubt, no solvent. The formation of various compounds from thioureas and their derivatives

The density functional theory (DFT) method has become one of the most prevalent and efficient tools, as compared to the conventional ab initio method (HF), for studying the detailed reaction mechanism in chemical systems during the last two decades. With DFT methods, many catalytic reaction mechanisms have been widely studied in addition to the assignment of experimental spectra. It is to be noted that study of reaction mechanism is important not only for understanding the reaction and its stereochemistry but also for designing new reactions and catalysts. The mechanistic pathways for the decomposition of benzoylthioureas into benzamides and thiobenzamides; and also the conversion of benzoylthioureas into benzoxazoles, instead of forming bezothiazoles, using different catalysts have also been highlighted with the help of

Benzothiazoles are an important class of heterocycles that possess a broad range of biological activities [4]. They were studied extensively for their anti-allergic, anti-inflammatory, antitumor, antimicrobial, and analgesic activities. Among those 2-substituted benzothiazole derivatives, the

pharmacological as well as synthetic potential.

168 Density Functional Calculations - Recent Progresses of Theory and Application

under different catalysts in solvent free condition is highlighted.

2. 2-Aminobenzothiazoles from thioureas

DFT calculations.

The main objectives of benzothiazoles synthesis are not only for the development of more diverse and complex bioactive compounds for biological activity and structure-activity relationship (SAR) studies but also for other applications, such as preparation of dyes. There are several methods for the synthesis of 2-aminobenzothiazoles. The most versatile and economical method involves the treatment of various substituted arylthioureas (which are synthesized via treatment of an aromatic amine with isothiocyanate) with oxidizing agent or cyclizing agent using different reaction conditions to yield 2-aminobenzothiazoles.

Recently, several methods have been reported which utilize bromine as catalyst. Basically, cyclization with bromine is achieved by oxidation of aniline, substituted aniline, and arylthiourea in acid or chloroform with alkali thiocyanate. Hugerschoff, in early 1900s, synthesized 2-aminobenzothiazole and found that 1, 3-diarylthiourea can be cyclized with liquid bromine in chloroform to form a 2-aminobenzothiazoles (Scheme 1) [14, 15].

This reaction worked well for symmetrical thioureas giving exclusively one product. But, when the same reaction is performed using unsymmetrical 1,3-diaryl thioureas, there is always uncertainty as to on which aryl ring the intramolecular electrophilic substitution would take place to give aminobenzothiazole. Kamel et al. have reported the synthesis of 6-chloro-4-(trifluoromethyl) 2-aminobenzothiazole by oxidative cyclization of 4-chloro-2-(trifluoromethyl)phenylthiourea with bromine in chloroform to give an intermediate followed by basification with NH3 (Scheme 2) [16].

Jordan et al. have reported the use of benzyltrimethylammoniumtribromide (PhCH2NMe3Br3) which is an electrophilic bromine source for the conversion of substituted arylthiourea to 2 aminobenzothiazoles under mild conditions in different solvents with good yields (Scheme 3) [17].

Liu et al. have reported the metal-free synthesis of 2-aminobenzothiazoles from N<sup>0</sup> -substituted N-(2-halophenyl) thioureas via a base-promoted cyclization in dioxane (Scheme 4) [18]. However, this reaction requires drastic conditions, like heating the vial which was sealed in an oil bath.

The palladium-catalyzed intramolecular cyclization of 2-bromophenylthioureas to synthesize 2-substituted benzothiazoles was also reported. Castillon et al.reported Pd- catalyzed cyclization of 2-bromophenylthioamides using Pd2(dba)3/(2-biphenyl)P(t-Bu)2 catalytic system (Scheme 5) [19].

Figure 1. Several N-substituted-2-aminobenzothiazole derivatives reported as biologically active compounds and pharmaceutical products.

However, both a ligand and a base are required to promote the reaction, and the substrates are not readily available. It was reported recently a catalytic synthesis of 2-substituted benzothiazoles from thiobenzanilides in the presence of a palladium catalyst through C-H

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functionalization or C-S bond formation [20].

Scheme 2. Oxidative cyclization of phenylthiourea to give 2-aminobenzothiazole.

Scheme 3. Synthesis of 2-aminobenzothiazoles using benzyltrimethylammoniumtribromide.

Scheme 4. Synthesis of 2-Substituted Benzothiazoles via a Base-Promoted Cyclization.

Scheme 1. Hugerschoff synthesis of 2-aminobenzothiazole from 1, 3-diarylthiourea with liquid bromine and chloroform.

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Scheme 2. Oxidative cyclization of phenylthiourea to give 2-aminobenzothiazole.

Scheme 3. Synthesis of 2-aminobenzothiazoles using benzyltrimethylammoniumtribromide.

Scheme 4. Synthesis of 2-Substituted Benzothiazoles via a Base-Promoted Cyclization.

Scheme 1. Hugerschoff synthesis of 2-aminobenzothiazole from 1, 3-diarylthiourea with liquid bromine and chloroform.

Figure 1. Several N-substituted-2-aminobenzothiazole derivatives reported as biologically active compounds and

170 Density Functional Calculations - Recent Progresses of Theory and Application

pharmaceutical products.

However, both a ligand and a base are required to promote the reaction, and the substrates are not readily available. It was reported recently a catalytic synthesis of 2-substituted benzothiazoles from thiobenzanilides in the presence of a palladium catalyst through C-H functionalization or C-S bond formation [20].

Meanwhile, the ligand-free copper-catalyzed one-pot tandem reactions of 2-halobenzenamines and isothiocyanates were also reported [24, 25]. However, it should be noted that the copper or iron catalyzed one-pot tandem reactions of 2-halobenzenamines with isothiocyanates generally involve organic solvents such as DMSO, DMF, and toluene which are environmentally unfriendly. Moreover, the reactions which are described above might proceed efficiently; they usually suffer from the use of highly toxic and corrosive reagents, high-costing metal catalysts, and specific ligands. There is also possibility to leave toxic traces of metals in the products. More recently, Jiang et al. have reported a metal-free synthesis of 2-aminobenzothiazoles from cyclohexanones and thioureas using catalytic iodine and molecular oxygen as the oxidant

Recently, Patel et al. have reported a one-pot procedure for the preparation of 2-aminoben-

(EDPBT). In this approach, aryl/alkyl isothiocyanate reacts with o-aminothiophenol to form their monothiourea which on desulfurization with EDPBT led to the formation of corresponding 2-

Very recently, polymer-supported tribromide has been used as a new solid phase and recyclable catalyst for the one-pot synthesis of 2-(N-arylamino)benzothiazole under microwave irra-

The probable reaction mechanism for the formation of 2-(N-arylamino) benzothiazoles through polymer-supported tribromide-mediated intramolecular cyclization of thioureas is

Scheme 8. Metal-free synthesis of 2-aminobenzothiazoles via aerobic oxidative cyclization of cyclohexanones and thio-

Scheme 9. One-pot synthesis of 2-aminobenzothiazoles using ditribromide reagent 1, 1<sup>0</sup>


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under mild conditions (Scheme 8) [26].

aminobenzothiazoles (Scheme 9) [27].

diation (Scheme 10) [28].

given in Scheme 11.

ureas.

bistribromide (EDPBT).

zothiazoles using ditribromide reagent 1,1<sup>0</sup>

Scheme 5. Synthesis of 2-substituted benzothiazoles through palladium-catalyzed intramolecular cyclization of 2 bromophenylthioureas.

Scheme 6. Copper-catalyzed tandem reaction of 2-iodoaniline with phenyl isothiocyanate to form 2-aminobenzothiazole.

Recently, the transition-metal (copper or iron)-catalyzed one-pot tandem reactions of 2-halobenzenamines with isothiocyanates for the synthesis of 2-aminobenzothiazoles have received considerable attention because of their efficiency and low costs. For example, Wu et al. described a copper-catalyzed tandem reaction between 2-halobenzenamines and isothiocyanates using the CuI (10 mol%)/1,10-phenanthroline (20 mol%) catalytic system to prepare 2-aminobenzothiazoles (Scheme 6) [21]. Li and Ding's group reported iron-catalyzed tandem reactions of 2 halobenzenamines and isothiocyanates leading to 2-aminobenzothiazoles (Scheme 7) [22, 23].

Scheme 7. FeCl3-catalyzed tandem reaction of 2-iodoaniline with phenyl isothiocyanate in water to give 2-aminobenzothiazole.

Meanwhile, the ligand-free copper-catalyzed one-pot tandem reactions of 2-halobenzenamines and isothiocyanates were also reported [24, 25]. However, it should be noted that the copper or iron catalyzed one-pot tandem reactions of 2-halobenzenamines with isothiocyanates generally involve organic solvents such as DMSO, DMF, and toluene which are environmentally unfriendly. Moreover, the reactions which are described above might proceed efficiently; they usually suffer from the use of highly toxic and corrosive reagents, high-costing metal catalysts, and specific ligands. There is also possibility to leave toxic traces of metals in the products. More recently, Jiang et al. have reported a metal-free synthesis of 2-aminobenzothiazoles from cyclohexanones and thioureas using catalytic iodine and molecular oxygen as the oxidant under mild conditions (Scheme 8) [26].

Recently, Patel et al. have reported a one-pot procedure for the preparation of 2-aminobenzothiazoles using ditribromide reagent 1,1<sup>0</sup> -(ethane-1, 2-diyl)dipyridinium bistribromide (EDPBT). In this approach, aryl/alkyl isothiocyanate reacts with o-aminothiophenol to form their monothiourea which on desulfurization with EDPBT led to the formation of corresponding 2 aminobenzothiazoles (Scheme 9) [27].

Very recently, polymer-supported tribromide has been used as a new solid phase and recyclable catalyst for the one-pot synthesis of 2-(N-arylamino)benzothiazole under microwave irradiation (Scheme 10) [28].

Recently, the transition-metal (copper or iron)-catalyzed one-pot tandem reactions of 2-halobenzenamines with isothiocyanates for the synthesis of 2-aminobenzothiazoles have received considerable attention because of their efficiency and low costs. For example, Wu et al. described a copper-catalyzed tandem reaction between 2-halobenzenamines and isothiocyanates using the CuI (10 mol%)/1,10-phenanthroline (20 mol%) catalytic system to prepare 2-aminobenzothiazoles (Scheme 6) [21]. Li and Ding's group reported iron-catalyzed tandem reactions of 2 halobenzenamines and isothiocyanates leading to 2-aminobenzothiazoles (Scheme 7) [22, 23].

Scheme 7. FeCl3-catalyzed tandem reaction of 2-iodoaniline with phenyl isothiocyanate in water to give 2-aminoben-

Scheme 6. Copper-catalyzed tandem reaction of 2-iodoaniline with phenyl isothiocyanate to form 2-aminobenzothiazole.

Scheme 5. Synthesis of 2-substituted benzothiazoles through palladium-catalyzed intramolecular cyclization of 2-

172 Density Functional Calculations - Recent Progresses of Theory and Application

bromophenylthioureas.

zothiazole.

The probable reaction mechanism for the formation of 2-(N-arylamino) benzothiazoles through polymer-supported tribromide-mediated intramolecular cyclization of thioureas is given in Scheme 11.

$$\underbrace{\begin{array}{c} \text{ $\n$ } \\ \text{ $\n$ } \\ \text{ $\n$ } \\ \text{ $\n$ } \end{array}}\_{\text{ $\text{$ \n $} } \text{$ \n $} \end{array} + \underbrace{\begin{array}{c} \text{$ \n $} \\ \text{$ \n $} \\ \text{$ \n $} \\ \text{$ \n $} \\ \text{$ \n $} \end{array}}\_{\text{$ \n $} \text{$ \n $} \text{$ \n $} \text{$ \n $} \text{$ \n $} \text{$ \n $} \text{$ \n $} \text{$ \n $} \text{($ \n{\n $}$  $)} \\ \text{$ \n $} \text{$ \n $} \text{$ \n $} \text{$ \n $} \text{S} \text{$ \n $} \text{$ \n $} \text{S} \text{$ \n $} \text{$ \n $} \text{S} \\ \text{$ \n $} \text{$ \n $} \text{S} \text{ $ \n $} \text{S} \text{ ($ \n{\n} $} \text{$ \n $} \text{$ \n $} \text{S} \text{)} \text{($ \n{\n} $} \text{)} \text{($ \n{\n} $} \text{)} \text{($ \n{\n} $)} \text{($ \n{\n} $} \text{)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n} $)} \text{($ \n{\n$$

Scheme 8. Metal-free synthesis of 2-aminobenzothiazoles via aerobic oxidative cyclization of cyclohexanones and thioureas.

Scheme 9. One-pot synthesis of 2-aminobenzothiazoles using ditribromide reagent 1, 1<sup>0</sup> -(ethane-1, 2-diyl)dipyridinium bistribromide (EDPBT).

Scheme 10. One-pot synthesis of 2-(N-arylamino)benzothiazoles under microwave irradiation using polymer-supported tribromide.

for the conversion of benzoylthioureas to benzamides and thiobenzamides using iodine-alumina


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Amides and thioamides are an important class of building blocks in modern organic synthesis, with broad applications in advanced materials, pharmaceuticals, agrochemicals, and polymers, etc. They are used for the synthesis of various natural products as well as intermediates of organic compounds. Generally, amides are prepared from their corresponding ketoximes by Beckmann rearrangement, and thioamides are prepared by thionation of the corresponding amide analogues by Lawesson's reagent. Liana Allen et al. have reported the direct coupling of unactivated carboxylic acids with amines in toluene at 110�C in the absence of catalyst. The use of simple zirconium catalysts at 5.0 mol% loading gave amide formation as little as in 4 h (Scheme 14) [30]. Gelens et al. have also reported the microwave assistance in the coupling of carboxylic acids with amines. An array of structurally diverse amides was synthesized efficiently by combining (primary and secondary) amines and carboxylic acids in one-pot under solvent-free microwave

Scheme 13. Decomposition of benzoylthioureas to benzamides and thiobenzamides using iodine-alumina.

Scheme 14. Direct amide formation from unactivated carboxylic acids with amines using zirconium catalysts.

as catalyst without any solvent was described (Scheme 13) [29].

(MW) conditions (Scheme 15) [31].

Scheme 12. Reaction of N,N<sup>0</sup>

vent-free condition

Scheme 11. Plausible reaction mechanism for the formation of 2-(N-arylamino) benzothiazoles through polymer supported tribromide–mediated intramolecular cyclization of thioureas.

#### 3. Decomposition of benzoylthioureas

When the reaction of N,N<sup>0</sup> -diphenylthioureas with iodine-alumina as catalyst was carried out, the expected 2-(N-arylamino)benzothiazoles were obtained. However, when N-substituted-N<sup>0</sup> benzoylthioureas are treated with the above catalyst, the expected benzothiazoles are not obtained (Scheme 12).

Instead, the decomposition of benzoylthioureas to benzamides and thiobenzamides in a single route using iodine-alumina as catalyst under solvent-free condition takes place. When electron donating group, such as methyl or methoxy group, is present at the para-position of the aryl group of benzoylthioureas, benzamides are obtained as major product. When electron withdrawing group, such as chlorine or nitro group, is at para-position of the aryl group of benzoylthioureas, thiobenzamides are the favored product. Thus, a simple and efficient process

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Scheme 12. Reaction of N,N<sup>0</sup> -diphenylthioureas and N-substituted-N<sup>0</sup> -benzoylthioureas with iodine-alumina under solvent-free condition

for the conversion of benzoylthioureas to benzamides and thiobenzamides using iodine-alumina as catalyst without any solvent was described (Scheme 13) [29].

Amides and thioamides are an important class of building blocks in modern organic synthesis, with broad applications in advanced materials, pharmaceuticals, agrochemicals, and polymers, etc. They are used for the synthesis of various natural products as well as intermediates of organic compounds. Generally, amides are prepared from their corresponding ketoximes by Beckmann rearrangement, and thioamides are prepared by thionation of the corresponding amide analogues by Lawesson's reagent. Liana Allen et al. have reported the direct coupling of unactivated carboxylic acids with amines in toluene at 110�C in the absence of catalyst. The use of simple zirconium catalysts at 5.0 mol% loading gave amide formation as little as in 4 h (Scheme 14) [30].

Gelens et al. have also reported the microwave assistance in the coupling of carboxylic acids with amines. An array of structurally diverse amides was synthesized efficiently by combining (primary and secondary) amines and carboxylic acids in one-pot under solvent-free microwave (MW) conditions (Scheme 15) [31].

Scheme 13. Decomposition of benzoylthioureas to benzamides and thiobenzamides using iodine-alumina.

3. Decomposition of benzoylthioureas

supported tribromide–mediated intramolecular cyclization of thioureas.

174 Density Functional Calculations - Recent Progresses of Theory and Application



the expected 2-(N-arylamino)benzothiazoles were obtained. However, when N-substituted-N<sup>0</sup>

Scheme 11. Plausible reaction mechanism for the formation of 2-(N-arylamino) benzothiazoles through polymer

Scheme 10. One-pot synthesis of 2-(N-arylamino)benzothiazoles under microwave irradiation using polymer-supported

benzoylthioureas are treated with the above catalyst, the expected benzothiazoles are not

Instead, the decomposition of benzoylthioureas to benzamides and thiobenzamides in a single route using iodine-alumina as catalyst under solvent-free condition takes place. When electron donating group, such as methyl or methoxy group, is present at the para-position of the aryl group of benzoylthioureas, benzamides are obtained as major product. When electron withdrawing group, such as chlorine or nitro group, is at para-position of the aryl group of benzoylthioureas, thiobenzamides are the favored product. Thus, a simple and efficient process

When the reaction of N,N<sup>0</sup>

tribromide.

obtained (Scheme 12).

Scheme 14. Direct amide formation from unactivated carboxylic acids with amines using zirconium catalysts.

Very recently, Rajeshwer Vanjariet al. have developed a new approach for the synthesis of amides through manganese dioxide-promoted nondirected C-H activation of methylarenes under mild conditions employing N-chloroamines as effective coupling partners (Scheme 16) [32].

Different synthetic methods have been discovered for the synthesis of thioamides. Among these strategies, thionation of amide analogues with Lawesson's reagent is the most common, but this reaction cannot be classified as an atom economical approach because of crucial limitations: only one oxygen atom is replaced by a sulfur atom, and no other new bond was created. Thus, it is worthwhile to provide a practical and environmentally benign method to synthesize thioamides. Recently, some three component reactions have nicely exploited the use of benzylamine [33], aldehydes [34], and alkyne [35], in combination with elemental sulfur and amine for the synthesis of thioamides (Scheme 17).

More recently, Guntreddi et al. reported a new decarboxylative strategy for the synthesis of thioamides via a three-component reaction involving arylacetic or cinnamic acids, amines, and elemental sulfur powder, without the need of a transition metal and an external oxidant (Scheme 18) [36].

Scheme 15. Microwave (MW)-assisted amide formation.

Scheme 16. Synthesis of amides through manganese dioxide promoted nondirected C-H activation of methylarenes.

The Beckmann rearrangement generally requires a strong acid, high reaction temperature, harsh reaction conditions, and production of unwanted by-products. Several methodologies to check the reaction conditions, such as, in liquid phase, in vapor phase, in supercritical water, and

Scheme 18. Synthesis of thioamides via a three-component reaction by decarboxylative method.

Scheme 17. Multicomponent oxidative coupling into thioamides by elemental sulphur under solvent-free condition.

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Very recently, Rajeshwer Vanjariet al. have developed a new approach for the synthesis of amides through manganese dioxide-promoted nondirected C-H activation of methylarenes under mild

Different synthetic methods have been discovered for the synthesis of thioamides. Among these strategies, thionation of amide analogues with Lawesson's reagent is the most common, but this reaction cannot be classified as an atom economical approach because of crucial limitations: only one oxygen atom is replaced by a sulfur atom, and no other new bond was created. Thus, it is worthwhile to provide a practical and environmentally benign method to synthesize thioamides. Recently, some three component reactions have nicely exploited the use of benzylamine [33], aldehydes [34], and alkyne [35], in combination with elemental sulfur and

More recently, Guntreddi et al. reported a new decarboxylative strategy for the synthesis of thioamides via a three-component reaction involving arylacetic or cinnamic acids, amines, and elemental sulfur powder, without the need of a transition metal and an external oxidant

Scheme 16. Synthesis of amides through manganese dioxide promoted nondirected C-H activation of methylarenes.

conditions employing N-chloroamines as effective coupling partners (Scheme 16) [32].

amine for the synthesis of thioamides (Scheme 17).

176 Density Functional Calculations - Recent Progresses of Theory and Application

Scheme 15. Microwave (MW)-assisted amide formation.

(Scheme 18) [36].

Scheme 17. Multicomponent oxidative coupling into thioamides by elemental sulphur under solvent-free condition.

Scheme 18. Synthesis of thioamides via a three-component reaction by decarboxylative method.

The Beckmann rearrangement generally requires a strong acid, high reaction temperature, harsh reaction conditions, and production of unwanted by-products. Several methodologies to check the reaction conditions, such as, in liquid phase, in vapor phase, in supercritical water, and in ionic liquids have been developed. However, the drawbacks in such methods are the use of toxic solvents, expensive reagents, long reaction times, low yields, and the production of considerable amounts of by-products. Literature survey reveals that there were many reports for the synthesis of amides and its sulfur containing analogue, thioamides; however, there is no report for the simultaneous synthesis of benzamides and thiobenzamides from benzoylthiourea [29].

#### 4. Benzoxazole amides from benzoylthioureas

When N-substituted-N<sup>0</sup> -benzoylthioureas are reacted with diacetoxyiodobenzene (DIB) as catalyst, benzoxazole amides are formed; expected benzothiazoles or the decomposition products are not obtained. Thus, when N-benzoylthiourea reacts with hypervalent iodine (III) reagent (DIB), instead of molecular iodine, an unexpected cyclized benzoxazole derivative is formed (Scheme 19) [37]. Unlike molecular iodine, DIB as catalyst renders the formation of C-O bond in benzoxazole moiety of substituted N-benzoxazol-2-yl-amides, instead of the expected C-S bond formation of benzothiazole moiety. Unexpectedly, the reaction follows different pathways leading to C-O bond formation between carbonyl oxygen and orthocarbon of aryl moiety resulting in oxazole ring formation via a sequential acylation and deacylation process.

Benzoxazoles are a class of heterocyclic compounds exhibiting therapeutical activities (Figure 2), such as, antifungal agents [38–40], cytotoxic compounds [41], as anti-inflammatory agents [42], as HIV-1 protease inhibitor [43], as an antibiotic [44], as CpIMPDH inhibitors [45], nonnucleoside HIV-1 reverse transcriptase inhibitors (NNRTI) [46], and antitumour agents [47].

Various methods have been reported in the literature for the synthesis of benzoxazoles starting from 2-aminophenol precursors with carboxylic acid derivatives, such as carboxylic acids, acid chlorides, acid anhydrides, and amides (Scheme 20), or by reacting 2-aminophenols with

Figure 2. Several benzoxazole derivatives reported as biologically active compounds and pharmaceutical products.

In most cases, 2-aminophenols are used as the starting materials for the preparation of 2-arylbenzoxazoles. However, the synthesis of N-benzoxazol-2-yl-amides is very limited [52]


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aldehydes followed by oxidation (Scheme 21) [48–51].

and the synthesis of N-benzoxazol-2-yl-amides starting from N<sup>0</sup>

hypervalent iodine(III) reagents (DIB) is recently reported [37].

Scheme 19. Synthesis of benzoxazole amides by the reaction of N-substituted-N<sup>0</sup> -benzoylthioureas with diacetoxyiodobenzene (DIB) as catalyst.

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in ionic liquids have been developed. However, the drawbacks in such methods are the use of toxic solvents, expensive reagents, long reaction times, low yields, and the production of considerable amounts of by-products. Literature survey reveals that there were many reports for the synthesis of amides and its sulfur containing analogue, thioamides; however, there is no report for the simultaneous synthesis of benzamides and thiobenzamides from benzoylthiourea [29].

catalyst, benzoxazole amides are formed; expected benzothiazoles or the decomposition products are not obtained. Thus, when N-benzoylthiourea reacts with hypervalent iodine (III) reagent (DIB), instead of molecular iodine, an unexpected cyclized benzoxazole derivative is formed (Scheme 19) [37]. Unlike molecular iodine, DIB as catalyst renders the formation of C-O bond in benzoxazole moiety of substituted N-benzoxazol-2-yl-amides, instead of the expected C-S bond formation of benzothiazole moiety. Unexpectedly, the reaction follows different pathways leading to C-O bond formation between carbonyl oxygen and orthocarbon of aryl moiety resulting in oxazole ring formation via a sequential acylation and

Benzoxazoles are a class of heterocyclic compounds exhibiting therapeutical activities (Figure 2), such as, antifungal agents [38–40], cytotoxic compounds [41], as anti-inflammatory agents [42], as HIV-1 protease inhibitor [43], as an antibiotic [44], as CpIMPDH inhibitors [45], nonnucleoside HIV-1 reverse transcriptase inhibitors (NNRTI) [46], and antitumour agents [47].



4. Benzoxazole amides from benzoylthioureas

178 Density Functional Calculations - Recent Progresses of Theory and Application

Scheme 19. Synthesis of benzoxazole amides by the reaction of N-substituted-N<sup>0</sup>

When N-substituted-N<sup>0</sup>

deacylation process.

dobenzene (DIB) as catalyst.

Figure 2. Several benzoxazole derivatives reported as biologically active compounds and pharmaceutical products.

Various methods have been reported in the literature for the synthesis of benzoxazoles starting from 2-aminophenol precursors with carboxylic acid derivatives, such as carboxylic acids, acid chlorides, acid anhydrides, and amides (Scheme 20), or by reacting 2-aminophenols with aldehydes followed by oxidation (Scheme 21) [48–51].

In most cases, 2-aminophenols are used as the starting materials for the preparation of 2-arylbenzoxazoles. However, the synthesis of N-benzoxazol-2-yl-amides is very limited [52] and the synthesis of N-benzoxazol-2-yl-amides starting from N<sup>0</sup> -benzoylthiourea using hypervalent iodine(III) reagents (DIB) is recently reported [37].

5.1. DFT calculations for the formation of benzamides and thiobenzamides

Scheme 22. Plausible mechanism for the formation of benzamides and thiobenzamides.

are formed, except in N-2-pyridinyl-N<sup>0</sup>

benzamide product is formed [29].

The mechanism for the decomposition of benzoylthioureas to benzamides and thiobenzamides in a single route using iodine-alumina as catalyst under solvent-free condition was studied with DFT calculations; all the structures were optimized by hybrid density functional B3LYP [62, 63] using the segmented all-electron relativistically contracted Def2-TZVP(�df) basis set with the help of ORCA [64]. The DFT calculation shows that the formations of both benzamides and thiobenzamides with by-products, viz., isothiocyanate and isocyanate, respectively, are endothermic. The formation of benzamide and isothiocyanate involves lower energy. Thus, it was found that the formation of benzamide product is a thermodynamically favored reaction although it is observed from the experimental results that both the products

The plausible mechanism for the formation of benzamides and thiobenzamides is shown in Scheme 22. To understand the mechanistic pathway, three most probable iodide intermediates A, B, and C formed after reaction with diiodine (I2) molecule were considered (Scheme 22 and Table 1). The I-I bond in diiodine is often known to be perturbed by thiones and form iodides. The formation of iodide intermediate through oxygen atom C is being ruled out because of its


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Scheme 20. Synthesis of benzoxazoles from 2-aminophenol with carboxylic acid derivatives, acid chlorides, and amides.

Scheme 21. Synthesis of benzoxazoles from 2-aminophenol with aldehydes followed by oxidation.

#### 5. DFT calculations

With the help of density functional theory (DFT), the electronic structure of organic compounds could be expressed by electron density functional. DFT calculation is recently applied to the study of various reaction mechanisms, viz. the reaction mechanisms of the Pd(II)-catalyzed oxidative carbocyclization-alkoxycarbonylation of bisallenes to construct seven-membered carbocycles have been theoretically investigated with the aid of density DFT calculations [53]; the coupling reaction mechanisms of the Rh(III)-catalyzed redox-neutral C7-selective aryl C-H functionalization of indolines with alkynes and alkenes have been theoretically investigated [54]; the mechanism of NHC catalyzed annulation reactions involving an α,β-unsaturated acyl azolium and β-naphthol has been studied using DFT methods [55]; DFT calculations have been performed on Rh(III)-catalyzed phosphoryl-directed oxidative C-H activation/cyclization to investigate the detailed mechanism [56]; DFT calculations were also employed to investigate the energetics of several reaction paths for the Fries rearrangement of aryl formates promoted by boron trichloride [57]; the reactions of hypochlorous acid (HOCl) with ammonia, (di)methylamine, and heterocyclic amines have been studied computationally using double-hybrid DFT methods [58]; the mechanisms and chemo- and stereo-selectivities of PBu3-catalyzed intramolecular cyclizations of N-allylic substituted α-amino nitriles leading to functionalized pyrrolidines (5-endo-trig cyclization, Mechanism A) and their competing reaction leading to another kind of pyrrolidine (5-exo-trig cyclization, Mechanism B) have been investigated using DFT [59]; a systematic theoretical study has been carried out to understand the mechanism and stereoselectivity of N-heterocyclic carbene (NHC)-catalyzed intramolecular-crossed benzoin reaction of enolizable ketoaldehyde using DFT calculations [60]. A simple and convenient method for the construction of substituted cycloheptenones from 1-bromoocta-1,7-diene-3-ols has been developed. The reaction involves Pd(0)-catalyzed intramolecular 7-exo-trigcyclization followed by Pd (II)-catalyzed oxidation of cyclic alcohol. The course of the reaction pathway has been evaluated using DFT calculations [61].

#### 5.1. DFT calculations for the formation of benzamides and thiobenzamides

The mechanism for the decomposition of benzoylthioureas to benzamides and thiobenzamides in a single route using iodine-alumina as catalyst under solvent-free condition was studied with DFT calculations; all the structures were optimized by hybrid density functional B3LYP [62, 63] using the segmented all-electron relativistically contracted Def2-TZVP(�df) basis set with the help of ORCA [64]. The DFT calculation shows that the formations of both benzamides and thiobenzamides with by-products, viz., isothiocyanate and isocyanate, respectively, are endothermic. The formation of benzamide and isothiocyanate involves lower energy. Thus, it was found that the formation of benzamide product is a thermodynamically favored reaction although it is observed from the experimental results that both the products are formed, except in N-2-pyridinyl-N<sup>0</sup> -benzoylthiourea where only the energetically favored benzamide product is formed [29].

The plausible mechanism for the formation of benzamides and thiobenzamides is shown in Scheme 22. To understand the mechanistic pathway, three most probable iodide intermediates A, B, and C formed after reaction with diiodine (I2) molecule were considered (Scheme 22 and Table 1). The I-I bond in diiodine is often known to be perturbed by thiones and form iodides. The formation of iodide intermediate through oxygen atom C is being ruled out because of its

5. DFT calculations

using DFT calculations [61].

With the help of density functional theory (DFT), the electronic structure of organic compounds could be expressed by electron density functional. DFT calculation is recently applied to the study of various reaction mechanisms, viz. the reaction mechanisms of the Pd(II)-catalyzed oxidative carbocyclization-alkoxycarbonylation of bisallenes to construct seven-membered carbocycles have been theoretically investigated with the aid of density DFT calculations [53]; the coupling reaction mechanisms of the Rh(III)-catalyzed redox-neutral C7-selective aryl C-H functionalization of indolines with alkynes and alkenes have been theoretically investigated [54]; the mechanism of NHC catalyzed annulation reactions involving an α,β-unsaturated acyl azolium and β-naphthol has been studied using DFT methods [55]; DFT calculations have been performed on Rh(III)-catalyzed phosphoryl-directed oxidative C-H activation/cyclization to investigate the detailed mechanism [56]; DFT calculations were also employed to investigate the energetics of several reaction paths for the Fries rearrangement of aryl formates promoted by boron trichloride [57]; the reactions of hypochlorous acid (HOCl) with ammonia, (di)methylamine, and heterocyclic amines have been studied computationally using double-hybrid DFT methods [58]; the mechanisms and chemo- and stereo-selectivities of PBu3-catalyzed intramolecular cyclizations of N-allylic substituted α-amino nitriles leading to functionalized pyrrolidines (5-endo-trig cyclization, Mechanism A) and their competing reaction leading to another kind of pyrrolidine (5-exo-trig cyclization, Mechanism B) have been investigated using DFT [59]; a systematic theoretical study has been carried out to understand the mechanism and stereoselectivity of N-heterocyclic carbene (NHC)-catalyzed intramolecular-crossed benzoin reaction of enolizable ketoaldehyde using DFT calculations [60]. A simple and convenient method for the construction of substituted cycloheptenones from 1-bromoocta-1,7-diene-3-ols has been developed. The reaction involves Pd(0)-catalyzed intramolecular 7-exo-trigcyclization followed by Pd (II)-catalyzed oxidation of cyclic alcohol. The course of the reaction pathway has been evaluated

Scheme 21. Synthesis of benzoxazoles from 2-aminophenol with aldehydes followed by oxidation.

180 Density Functional Calculations - Recent Progresses of Theory and Application

Scheme 20. Synthesis of benzoxazoles from 2-aminophenol with carboxylic acid derivatives, acid chlorides, and amides.

Scheme 22. Plausible mechanism for the formation of benzamides and thiobenzamides.

relatively high energy compared to those of intermediates A and B (examples of O-I bond formation of diiodine with ketones are not found in the literature).

The results show that the intermediate (A) has the lowest energy which indicates that it is the most probable intermediate and the optimized structure is shown in (Figure 3). The results further show that for all the reactions theoretically considered, the intermediate (A) has the lowest energy, except for p-chlorinated molecule, which indicates that it is the most probable intermediate (Table 1).

> To study the possibility of breaking the molecular backbone, the strength of different bonds were considered based on the Mayer bond order [65], which indicates a number of electron pairs that constitute a bond. When considering the backbone structure, C1-N2 has the least Mayer bond order in intermediate A while C5-C6 has the least bond order in the intermediates

A 0.9092 1.1663 1.5143 1.3996 0.9238 B 1.1027 1.8061 0.9953 1.1750 0.9085 C 1.0134 1.1414 1.0373 1.9322 0.9493

Table 2. Mayer bond order for selected bonds (atom numbering is shown in Figure 3) for parent molecule.

C1-N2 N2-C3 C3-N4 N4-C5 C5-C6

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In case of the p-chlorinated molecule, an electron withdrawing substituent at the para-position of the aryl group, the intermediate B is the energetically most favored intermediate (Table 4). This indicates that the migration of the phenyl group in p-chlorinated molecule to attack the thiocarbonyl carbon is the favored step, which on further rearrangement gives the product thiobenzamide. The proposed steps are supported by the experimental results where the thiobenzamide is the major product. The probable reason for the formation of appreciable amount of the benzamide product, although not the favored step mechanistically, could be that the benzamide product is thermodynamically more stable than the thiobenzamide product. The formation of benzamide occurs through the intermediate A by the migration of the

For other substituted molecules also, C5-C6 has the least bond order in the intermediate B as mentioned earlier. This explains that the formation of the thiobenzamide product is due to the migration of the phenyl group following the similar steps as in p-chlorinated molecule. However, when the electron withdrawing p-chlorinated aryl group is replaced by p-methylated aryl group, the benzamide product is the major one. The reason for the reaction in this case could be that the formation of benzamide product is preferred by breaking the C1-N2 bond in intermediate A than the mechanistically favored step by breaking C5-C6 bond in intermediate B (as in p-chlorinated molecule). This is so because the intermediate A has lower energy than B (Table 3). It is also interesting to note that when an electron donating group methyl is at the ortho-position in the aryl group, the intermediate A which has lowest energy has the least bond order at C5-C6 bond. This makes the breaking of C1-N2 bond in intermediate A less probable, thus rendering the formation of thiobenzamide product as the major product. Similar result is

Thus, the DFT studies showed that the formation of benzamide was due to the migration of the aryl group (in intermediate A) while the formation of thiobenzamide may be due to the migration of the phenyl group (in intermediate B). It was found that the formation of benzamide product is the thermodynamically favored reaction, although it is observed from the experimental results that both the products are formed, except in N-2-pyridinyl-N<sup>0</sup>

benzoylthiourea where only the energetically favored benzamide product is formed.


aryl group as the C1-N2 bond order is the least in intermediate A.

obtained in o-pyridinated molecule.

B and C (Table 2).

Intermediates Mayer bond order

Table 1. Relative energies of different intermediates for parent molecule.

Figure 3. Optimized structure of A.


Table 2. Mayer bond order for selected bonds (atom numbering is shown in Figure 3) for parent molecule.

relatively high energy compared to those of intermediates A and B (examples of O-I bond

The results show that the intermediate (A) has the lowest energy which indicates that it is the most probable intermediate and the optimized structure is shown in (Figure 3). The results further show that for all the reactions theoretically considered, the intermediate (A) has the lowest energy, except for p-chlorinated molecule, which indicates that it is the most probable

Intermediates Structure Relative energy (kcal)

A 0.0

B 0.5

C 35.2

Table 1. Relative energies of different intermediates for parent molecule.

Figure 3. Optimized structure of A.

formation of diiodine with ketones are not found in the literature).

182 Density Functional Calculations - Recent Progresses of Theory and Application

intermediate (Table 1).

To study the possibility of breaking the molecular backbone, the strength of different bonds were considered based on the Mayer bond order [65], which indicates a number of electron pairs that constitute a bond. When considering the backbone structure, C1-N2 has the least Mayer bond order in intermediate A while C5-C6 has the least bond order in the intermediates B and C (Table 2).

In case of the p-chlorinated molecule, an electron withdrawing substituent at the para-position of the aryl group, the intermediate B is the energetically most favored intermediate (Table 4). This indicates that the migration of the phenyl group in p-chlorinated molecule to attack the thiocarbonyl carbon is the favored step, which on further rearrangement gives the product thiobenzamide. The proposed steps are supported by the experimental results where the thiobenzamide is the major product. The probable reason for the formation of appreciable amount of the benzamide product, although not the favored step mechanistically, could be that the benzamide product is thermodynamically more stable than the thiobenzamide product. The formation of benzamide occurs through the intermediate A by the migration of the aryl group as the C1-N2 bond order is the least in intermediate A.

For other substituted molecules also, C5-C6 has the least bond order in the intermediate B as mentioned earlier. This explains that the formation of the thiobenzamide product is due to the migration of the phenyl group following the similar steps as in p-chlorinated molecule. However, when the electron withdrawing p-chlorinated aryl group is replaced by p-methylated aryl group, the benzamide product is the major one. The reason for the reaction in this case could be that the formation of benzamide product is preferred by breaking the C1-N2 bond in intermediate A than the mechanistically favored step by breaking C5-C6 bond in intermediate B (as in p-chlorinated molecule). This is so because the intermediate A has lower energy than B (Table 3). It is also interesting to note that when an electron donating group methyl is at the ortho-position in the aryl group, the intermediate A which has lowest energy has the least bond order at C5-C6 bond. This makes the breaking of C1-N2 bond in intermediate A less probable, thus rendering the formation of thiobenzamide product as the major product. Similar result is obtained in o-pyridinated molecule.

Thus, the DFT studies showed that the formation of benzamide was due to the migration of the aryl group (in intermediate A) while the formation of thiobenzamide may be due to the migration of the phenyl group (in intermediate B). It was found that the formation of benzamide product is the thermodynamically favored reaction, although it is observed from the experimental results that both the products are formed, except in N-2-pyridinyl-N<sup>0</sup> benzoylthiourea where only the energetically favored benzamide product is formed.

5.2. DFT calculations for the formation of benzoxazoles

pathways for the formation of benzoxazole amides (Scheme 23).

by hydrolysis followed by cyclization, to give benzoxazole derivative (D<sup>0</sup>

oxazole ring also is initiated by deprotonation of NH atom as in the first reaction.

from N<sup>0</sup>

formation of carbodiimide from N<sup>0</sup>

Density functional calculations were performed at a B3LYP/Def2-TZVP(�df) level of theory using ORCA to study the reaction mechanism for the formation of benzoxazole amides from benzoylthioureas in presence of DIB as catalyst and the role of substitution with electron withdrawing/donating group at different positions of the phenyl ring on the reaction. The plausible mechanism for the formation of N-benzoxazol-2-yl-amides from benzoylthioureas is shown in Scheme 18. The final product benzoxazole amide is formed after a series of acylation and deacylation occurring in tandem, as illustrated in Scheme 18, with the formation of a number of intermediates. At first, the acylation occurs due to the formation of carbodiimide



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) through oxidation

give 1 mole each of acetic acid, sulfur atom, and phenyl iodide where iodine is reduced from +3 to +1 oxidation state. There are two possible pathways, namely, Path A and Path B, for the

and B, respectively. The attack of sulfur atom of thiocarbonyl on the thiophilic iodine of diacetoxyiodobenzene is initiated by the deprotonation of either of the NH protons lying on either side of the thiocarbonyl group. The deprotonation of N4 atom results in intermediate A, while the deprotonation of N2 atom results in intermediate B. To determine which pathway is the favored one, the relative energies of intermediates A and B were calculated (Table 5). It is observed that the intermediate B is the thermodynamically favored intermediate in most of the reactions except for substitution at ortho-position with chlorine and at meta-position with both electron withdrawing chlorine and donating methyl groups. This shows that the position of substitution plays a bigger role than the type of substituent in determining the pathway to give carbodiimide. It may be mentioned that earlier in our work for iodide intermediate [49], the relative energy of the deprotonated product from N4 atom was slightly less than that obtained from N2 atom. However, the energy difference between the two in the previous work was appreciably small with maximum of 3.0 kcal/mole. In the present case, the difference in relative energy between the two intermediates increases by two fold (Table 5) although the deprotonated product from N2 atom generally becomes more stable as shown in mechanistic

The carbodiimide thus formed is acylated at carbodiimide carbon which after rearrangement gives the acylated intermediate (C). The conversion from the acylated intermediate C to the final product here also can follow either of two possible routes: Route 1, which is deacylation

of deacylated product by another molecule of DIB or Route 2, which undergoes cyclization through oxidation of acylated intermediate C by another molecule of DIB, to give acylated benzoxazole derivative (E) followed by deacylation on hydrolysis. The cyclization to give

To determine which route the reaction follows, one needs theoretical consideration of each step of deacylation then cyclization (Route 1) or cyclization then deacylation (Route 2). One way to determine which route the reaction follows is to compare the energetics of each step undergoing in either route (Table 6). Comparison of energies of reactions for the first step through Route 1 (deacylation) and Route 2 (cyclization by oxidation of DIB) shows that the initial deacylation (Route 1) is exothermic and thermodynamically favored over oxazole cyclization through Route 2

Table 3. Relative energies of different intermediates for p-chlorinated and p-methylated molecules.


Table 4. Mayer bond order for selected bonds for the p-chlorinated, p-methylated o-methylated molecules and the o-pyridinated molecules. (atom numbering is shown in Figure 3).

#### 5.2. DFT calculations for the formation of benzoxazoles

Structure p-Chlorinated molecule p-Methylated molecule

184 Density Functional Calculations - Recent Progresses of Theory and Application

Table 3. Relative energies of different intermediates for p-chlorinated and p-methylated molecules.

A\_p-Cl 0.9138 1.1550 1.5238 1.3918 0.9254 B\_p-Cl 1.1197 1.7945 1.0017 1.1714 0.9103 C\_p-Cl 1.0194 1.1321 1.0436 1.9265 0.9495

A\_p-Me 0.9047 1.1728 1.5131 1.4038 0.9261

A\_o-Me 0.9585 1.1536 1.5098 1.4015 0.9236 B\_o-Me 1.1379 1.7861 0.9979 1.1722 0.9090 C\_o-Me 0.9087 1.1397 1.0361 1.9337 0.9413

A\_o-Py 1.0052 1.1535 1.5311 1.3810 0.9264

Table 4. Mayer bond order for selected bonds for the p-chlorinated, p-methylated o-methylated molecules and the

1.8130 1.1511

1.8032 1.1453

Intermediates Mayer bond order

1.1200 1.0081

1.0588 1.0440

o-pyridinated molecules. (atom numbering is shown in Figure 3).

p-Chlorinated molecule

p-Methylated molecule

o-Methylated molecule

o-Pyridinated molecule

B\_p-Me C\_p-Me

B\_o-Py C\_o-Py Intermediate Relative energy (kcal) Intermediate Relative energy (kcal)

A\_p-Cl 0.2 A\_p-Me 0.0

B\_p-Cl 0.0 B\_p-Me 0.7

C\_p-Cl 35.0 C\_p-Me 35.5

C1-N2 N2-C3 C3-N4 N4-C5 C5-C6

0.9944 1.0359

1.0024 1.0501 1.1719 1.9327

1.1710 1.9346 0.9076 0.9498

0.9111 0.9517 Density functional calculations were performed at a B3LYP/Def2-TZVP(�df) level of theory using ORCA to study the reaction mechanism for the formation of benzoxazole amides from benzoylthioureas in presence of DIB as catalyst and the role of substitution with electron withdrawing/donating group at different positions of the phenyl ring on the reaction. The plausible mechanism for the formation of N-benzoxazol-2-yl-amides from benzoylthioureas is shown in Scheme 18. The final product benzoxazole amide is formed after a series of acylation and deacylation occurring in tandem, as illustrated in Scheme 18, with the formation of a number of intermediates. At first, the acylation occurs due to the formation of carbodiimide from N<sup>0</sup> -benzoylthiourea. During carbodiimide formation, the molecule is oxidized by DIB to give 1 mole each of acetic acid, sulfur atom, and phenyl iodide where iodine is reduced from +3 to +1 oxidation state. There are two possible pathways, namely, Path A and Path B, for the formation of carbodiimide from N<sup>0</sup> -benzoylthiourea through the formation of intermediates A and B, respectively. The attack of sulfur atom of thiocarbonyl on the thiophilic iodine of diacetoxyiodobenzene is initiated by the deprotonation of either of the NH protons lying on either side of the thiocarbonyl group. The deprotonation of N4 atom results in intermediate A, while the deprotonation of N2 atom results in intermediate B. To determine which pathway is the favored one, the relative energies of intermediates A and B were calculated (Table 5). It is observed that the intermediate B is the thermodynamically favored intermediate in most of the reactions except for substitution at ortho-position with chlorine and at meta-position with both electron withdrawing chlorine and donating methyl groups. This shows that the position of substitution plays a bigger role than the type of substituent in determining the pathway to give carbodiimide. It may be mentioned that earlier in our work for iodide intermediate [49], the relative energy of the deprotonated product from N4 atom was slightly less than that obtained from N2 atom. However, the energy difference between the two in the previous work was appreciably small with maximum of 3.0 kcal/mole. In the present case, the difference in relative energy between the two intermediates increases by two fold (Table 5) although the deprotonated product from N2 atom generally becomes more stable as shown in mechanistic pathways for the formation of benzoxazole amides (Scheme 23).

The carbodiimide thus formed is acylated at carbodiimide carbon which after rearrangement gives the acylated intermediate (C). The conversion from the acylated intermediate C to the final product here also can follow either of two possible routes: Route 1, which is deacylation by hydrolysis followed by cyclization, to give benzoxazole derivative (D<sup>0</sup> ) through oxidation of deacylated product by another molecule of DIB or Route 2, which undergoes cyclization through oxidation of acylated intermediate C by another molecule of DIB, to give acylated benzoxazole derivative (E) followed by deacylation on hydrolysis. The cyclization to give oxazole ring also is initiated by deprotonation of NH atom as in the first reaction.

To determine which route the reaction follows, one needs theoretical consideration of each step of deacylation then cyclization (Route 1) or cyclization then deacylation (Route 2). One way to determine which route the reaction follows is to compare the energetics of each step undergoing in either route (Table 6). Comparison of energies of reactions for the first step through Route 1 (deacylation) and Route 2 (cyclization by oxidation of DIB) shows that the initial deacylation (Route 1) is exothermic and thermodynamically favored over oxazole cyclization through Route 2

Scheme 23. Mechanistic pathways for the formation of benzoxazole amides.

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Table 5. Relative energies of intermediates B with respect to A.

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Scheme 23. Mechanistic pathways for the formation of benzoxazole amides.

Entry number Ar Relative energy of B (kcal/mole)

6.230

Density Functional Calculations - Recent Progresses of Theory and Application

7.571

7.236

0.952

7.416

1.695

7.466

0.892

5.000

Table 5. Relative energies of intermediates B with respect to A.


Table 6. Energy of reaction through Route 1 and Route 2.

which is endothermic. The Route 1 is favored more on substitution with both electron withdrawing as well as donating groups at all positions, that is, ortho-, meta- and para-positions. Further, it is observed that the deacylation of chlorinated derivative of C at ortho-position is the most exothermic and the least at meta-position. This is because of higher conjugation at ortho- and parapositions where the electron withdrawing nature of chlorine favors the deacylation more. On the other hand, it is reverse for methylated derivative, that is, the reaction is most exothermic when substituted at meta-position and least at ortho-position. In case of methylated derivative, the electron donating nature of methyl group disfavors the deacylation when substituted at orthoand para-positions where the conjugation is more. On dimethylation, both at ortho- and parapositions of C, the exothermic energy is further reduced. However, the exothermic energy for the dichlorination at ortho- and para-positions of C is less than that for substitution at ortho-position although it is higher than those for substitutions at meta- and para-positions.

energy involved is affected by the substitution with electron withdrawing/donating groups at different positions of the benzene ring. For methyl-substituted derivative, the cyclization energy involved is greatest when substituted at para-position and least at ortho-position; while for chlorinated derivative, the energy involved is highest at meta-position and least at para-position.

N2 N4

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bond breaking between C1 carbon and iodine giving the final product with the release of phenyl iodide and acetic acid. This aromatization to give benzoxazole derivative is exothermic (Table 6). For both electron-withdrawing chlorine and -donating methyl-substituted derivatives, the exo-

There is further interesting result from the reaction when there is substitution at the meta-position

alyst, benzoxazole amides are formed; expected benzothiazoles or the decomposition products are not obtained. Unlike molecular iodine, DIB as catalyst renders the formation of C-O bond in benzoxazole moiety of substituted N-benzoxazol-2-yl-amides, instead of the expected C-S bond formation of benzothiazole moiety. DFT calculations showed the reaction followed through carbodiimide intermediate. The carbodiimide further undergoes a series of acylation,

deacylation, and cyclization in tandem to give the final product, benzoxazole amides.



thermic energy is highest when substituted at ortho-position and least at meta-position.

may be formed for entry 4 (methylated derivative) and 6 (chlorinated derivative).


The aromaticity is regained upon subsequent deprotonation of sp<sup>3</sup>

Table 7. Mulliken atomic charge of the hydrogen atoms attached to N2 and N4 atoms of D.

Entry number Mulliken charge of hydrogen atom bonded to

 0.2982 0.2679 0.3002 0.2657 0.2984 0.2686 0.301 0.2654 0.3011 0.2653 0.3016 0.2675 0.3018 0.2671 0.3061 0.2658 0.3073 0.2671

(30

When N-substituted-N<sup>0</sup>

The deacylation of C gives the deacylated intermediate D that has two acidic hydrogen atoms which are bonded to N2 and N4 nitrogen atoms as before. Considering the acidity of the hydrogen atoms, it is found that the hydrogen atom attached to N2 nitrogen atom is more acidic than that attached to N4 atom as is indicated by the Mulliken atomic charges of the hydrogen atoms (Table 7). This deprotonation of the proton of N2 atom initiates the attack of carbonyl oxygen O7 atom on ortho-carbon of the aryl group (the attack of ortho-carbon atom is discussed in detail later), leading to the formation of cyclized intermediate D<sup>0</sup> containing oxazole ring and the reduction of another molecule of DIB from +3 to +1 oxidation state of iodine by the attack of C1 carbon atom of D on the iodine of DIB. The attack of O7 atom on ortho-carbon and C1 atom on iodine of DIB is such that the two attacks are anti-periplanar. Therefore, the hydrogen attached to the ortho-carbon and iodine bonded to C1 atom lie on the same side of the molecular plane.

Upon cyclization to form oxazole ring of D<sup>0</sup> , the aromaticity of the benzene ring is lost as the hybridizations of the attacked ortho-carbon as well as that of C1 atom change from sp<sup>2</sup> - to sp<sup>3</sup> hybridization. Hence, this cyclization process to give D<sup>0</sup> is endothermic. It is observed that the


Table 7. Mulliken atomic charge of the hydrogen atoms attached to N2 and N4 atoms of D.

which is endothermic. The Route 1 is favored more on substitution with both electron withdrawing as well as donating groups at all positions, that is, ortho-, meta- and para-positions. Further, it is observed that the deacylation of chlorinated derivative of C at ortho-position is the most exothermic and the least at meta-position. This is because of higher conjugation at ortho- and parapositions where the electron withdrawing nature of chlorine favors the deacylation more. On the other hand, it is reverse for methylated derivative, that is, the reaction is most exothermic when substituted at meta-position and least at ortho-position. In case of methylated derivative, the electron donating nature of methyl group disfavors the deacylation when substituted at orthoand para-positions where the conjugation is more. On dimethylation, both at ortho- and parapositions of C, the exothermic energy is further reduced. However, the exothermic energy for the dichlorination at ortho- and para-positions of C is less than that for substitution at ortho-position

Route 1 Route 2

 �22.81 52.02 �79.23 52.23 �84.65 �17.60 �24.66 57.52 �81.42 55.53 �86.40 �17.70 �24.55 53.96 �81.50 52.12 �86.46 �17.75 �24.61 56.30 �79.75 54.34 �84.67 �17.72 �24.57 56.41 �80.15 53.78 �84.54 �17.56 �24.88 57.55 �79.68 55.48 �85.48 �17.02 �24.91 57.46 �80.33 55.33 �85.27 �17.84 �28.23 57.48 �81.85 52.68 �86.76 �18.52 �25.05 57.63 �80.15 55.72 �85.48 �17.81

C to D D to D<sup>0</sup> D<sup>0</sup> to 2 C to E E to E<sup>0</sup> E<sup>0</sup> to 2

The deacylation of C gives the deacylated intermediate D that has two acidic hydrogen atoms which are bonded to N2 and N4 nitrogen atoms as before. Considering the acidity of the hydrogen atoms, it is found that the hydrogen atom attached to N2 nitrogen atom is more acidic than that attached to N4 atom as is indicated by the Mulliken atomic charges of the hydrogen atoms (Table 7). This deprotonation of the proton of N2 atom initiates the attack of carbonyl oxygen O7 atom on ortho-carbon of the aryl group (the attack of ortho-carbon atom is discussed in detail later), leading to the formation of cyclized intermediate D<sup>0</sup> containing oxazole ring and the reduction of another molecule of DIB from +3 to +1 oxidation state of iodine by the attack of C1 carbon atom of D on the iodine of DIB. The attack of O7 atom on ortho-carbon and C1 atom on iodine of DIB is such that the two attacks are anti-periplanar. Therefore, the hydrogen attached to the ortho-carbon and iodine bonded to C1 atom lie on the

hybridizations of the attacked ortho-carbon as well as that of C1 atom change from sp<sup>2</sup>

hybridization. Hence, this cyclization process to give D<sup>0</sup> is endothermic. It is observed that the

, the aromaticity of the benzene ring is lost as the


although it is higher than those for substitutions at meta- and para-positions.

same side of the molecular plane.

Upon cyclization to form oxazole ring of D<sup>0</sup>

Entry number Energy of reaction (kcal/mole)

188 Density Functional Calculations - Recent Progresses of Theory and Application

Table 6. Energy of reaction through Route 1 and Route 2.

energy involved is affected by the substitution with electron withdrawing/donating groups at different positions of the benzene ring. For methyl-substituted derivative, the cyclization energy involved is greatest when substituted at para-position and least at ortho-position; while for chlorinated derivative, the energy involved is highest at meta-position and least at para-position.

The aromaticity is regained upon subsequent deprotonation of sp<sup>3</sup> -hybridized ortho-carbon and bond breaking between C1 carbon and iodine giving the final product with the release of phenyl iodide and acetic acid. This aromatization to give benzoxazole derivative is exothermic (Table 6). For both electron-withdrawing chlorine and -donating methyl-substituted derivatives, the exothermic energy is highest when substituted at ortho-position and least at meta-position.

There is further interesting result from the reaction when there is substitution at the meta-position (30 -position) of the starting material as there are two possible products (Scheme 24) formed due to presence of two ortho-carbon atoms (2<sup>0</sup> and 6<sup>0</sup> carbon atoms) available for attack. The first product (F) is formed due to attack of O7 atom (initiated by deprotonation of N2 proton) on 2<sup>0</sup> ortho-carbon atom while the second product (G) is formed due to attack on 6<sup>0</sup> ortho-carbon. When considering the Mayer bond order between carbon and oxygen, the O7-C2<sup>0</sup> bond of the first product (F) is stronger than the O7-C6<sup>0</sup> bond of the second product (G) for both methylated and chlorinated derivative. However, the first product (F) is energetically slightly favored by 0.85 kcal/mole for the methylated derivative, while the second product (G) is favored by 0.35 kcal/mole for the chlorinated derivative, although in both derivatives the products have almost same energy. As the products have equivalent energies, it is inferred that both products may be formed for entry 4 (methylated derivative) and 6 (chlorinated derivative).

When N-substituted-N<sup>0</sup> -benzoylthioureas are reacted with diacetoxyiodobenzene (DIB) as catalyst, benzoxazole amides are formed; expected benzothiazoles or the decomposition products are not obtained. Unlike molecular iodine, DIB as catalyst renders the formation of C-O bond in benzoxazole moiety of substituted N-benzoxazol-2-yl-amides, instead of the expected C-S bond formation of benzothiazole moiety. DFT calculations showed the reaction followed through carbodiimide intermediate. The carbodiimide further undergoes a series of acylation, deacylation, and cyclization in tandem to give the final product, benzoxazole amides.

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Scheme 24. Two possible products for the meta-substituted derivative.

Thus, the density functional calculations showed the reaction followed through carbodiimide intermediate formed by the oxidation of N<sup>0</sup> -benzoylthiourea by DIB. The carbodiimide intermediate thus formed undergoes a series of acylation and deacylation in tandem, leading to cyclization to form oxazole ring of substituted N-benzoxazol-2-yl-amide, due to C-O bond formation as a result of attack of carbonyl oxygen on ortho-carbon aryl moiety, instead of the expected C-S bond formation to give benzothiazole moiety.

#### Author details

Warjeet S. Laitonjam\* and Lokendrajit Nahakpam

\*Address all correspondence to: warjeet@manipuruniv.ac.in

Department of Chemistry, Manipur University, Imphal, Manipur, India

#### References


Thus, the density functional calculations showed the reaction followed through carbodiimide

mediate thus formed undergoes a series of acylation and deacylation in tandem, leading to cyclization to form oxazole ring of substituted N-benzoxazol-2-yl-amide, due to C-O bond formation as a result of attack of carbonyl oxygen on ortho-carbon aryl moiety, instead of the

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Author details

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expected C-S bond formation to give benzothiazole moiety.

Scheme 24. Two possible products for the meta-substituted derivative.

190 Density Functional Calculations - Recent Progresses of Theory and Application

\*Address all correspondence to: warjeet@manipuruniv.ac.in

Department of Chemistry, Manipur University, Imphal, Manipur, India

Warjeet S. Laitonjam\* and Lokendrajit Nahakpam

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**Section 4**

**Material and Molecuar Design**

**Material and Molecuar Design**

**Chapter 9**

Provisional chapter

**Carbon Nanotubes: Molecular and Electronic Properties**

DOI: 10.5772/intechopen.70934

Carbon Nanotubes: Molecular and Electronic Properties

In this chapter, we describe how structural parameters affect the reactivity of singlewalled carbon nanotubes through global reactivity descriptors obtained by the DFT methods (B3LYP/6-31G(d) with real frequencies in all cases). First, we investigate regular armchair, chiral, and zigzag nanotubes with bumpy defects (five- and sevenmembered rings), finding that regular and defective zigzag nanotubes exhibit the greater conductive ability, reactivity, and capacity of nanotubes to be reduced. The bumpy defects favor those properties with greater intensity in chiral nanotubes. We also investigate how the properties of armchair nanotubes change in the presence of bumpy, haeckelite, Stone-Wales, and zipper defects, and we found that armchair nanotubes with zipper defects show greater reactivity and better conducting abilities enhanced by nitrogen doping and longer nanotubes. In addition, for armchair nanotubes containing bumpy defects, our results reveal, considering B3LYP-D3 correction, that bumpy defects confer a greater ability to physically adsorb hydrogen, with adsorption energies of 0.32 eV/adsorbed H2. That value is considered ideal for the reversible adsorption of hydrogen at room temperature and low pressures and therefore favorable for use as a clean energy source. These results contribute to the future design of novel useful mate-

Keywords: nitrogen-doped carbon nanotubes, conceptual DFT, global reactivity descriptors, hydrogen physisorption energy, saturated nanotubes, nanotube chirality, bumpy defects, zipper defects, haeckelite defects, Stone-Wales defects, conductive

Carbon is a very important chemical element, essential for the existence of multiple types of structures. Organic chemistry, a broad and interesting branch of chemistry, in simple terms, is

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

**of Regular and Defective Structures**

of Regular and Defective Structures

María Leonor Contreras Fuentes and

María Leonor Contreras Fuentes and

http://dx.doi.org/10.5772/intechopen.70934

rials based in carbon nanotubes.

ability, armchair, chiral, zigzag, dispersion forces

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Roberto Rozas Soto

Roberto Rozas Soto

Abstract

1. Introduction

Provisional chapter

#### **Carbon Nanotubes: Molecular and Electronic Properties of Regular and Defective Structures** Carbon Nanotubes: Molecular and Electronic Properties

DOI: 10.5772/intechopen.70934

María Leonor Contreras Fuentes and Roberto Rozas Soto María Leonor Contreras Fuentes and

Additional information is available at the end of the chapter Roberto Rozas Soto

of Regular and Defective Structures

http://dx.doi.org/10.5772/intechopen.70934 Additional information is available at the end of the chapter

#### Abstract

In this chapter, we describe how structural parameters affect the reactivity of singlewalled carbon nanotubes through global reactivity descriptors obtained by the DFT methods (B3LYP/6-31G(d) with real frequencies in all cases). First, we investigate regular armchair, chiral, and zigzag nanotubes with bumpy defects (five- and sevenmembered rings), finding that regular and defective zigzag nanotubes exhibit the greater conductive ability, reactivity, and capacity of nanotubes to be reduced. The bumpy defects favor those properties with greater intensity in chiral nanotubes. We also investigate how the properties of armchair nanotubes change in the presence of bumpy, haeckelite, Stone-Wales, and zipper defects, and we found that armchair nanotubes with zipper defects show greater reactivity and better conducting abilities enhanced by nitrogen doping and longer nanotubes. In addition, for armchair nanotubes containing bumpy defects, our results reveal, considering B3LYP-D3 correction, that bumpy defects confer a greater ability to physically adsorb hydrogen, with adsorption energies of 0.32 eV/adsorbed H2. That value is considered ideal for the reversible adsorption of hydrogen at room temperature and low pressures and therefore favorable for use as a clean energy source. These results contribute to the future design of novel useful materials based in carbon nanotubes.

Keywords: nitrogen-doped carbon nanotubes, conceptual DFT, global reactivity descriptors, hydrogen physisorption energy, saturated nanotubes, nanotube chirality, bumpy defects, zipper defects, haeckelite defects, Stone-Wales defects, conductive ability, armchair, chiral, zigzag, dispersion forces

#### 1. Introduction

Carbon is a very important chemical element, essential for the existence of multiple types of structures. Organic chemistry, a broad and interesting branch of chemistry, in simple terms, is

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

defined as carbon chemistry and is closely related, for example, to biochemistry and biomedicine. The understanding of the structure-property relations is an important pillar on which the development of these branches of knowledge is based. Thus, exchanging the position of two groups on a molecule of a recommended drug may mean that it becomes a poison. An example is the case of thalidomide disaster [1] where many babies were born with phocomelia (malformation of the limbs) when R-thalidomide, a compound suitable to mitigate some troublesome symptoms of pregnancy, was administered to their mother but contaminated with its enantiomer (compound with the same mirror image, but not superimposable); moreover, the enantiomer, S-thalidomide, has teratogenic effects.

Carbon nanotubes (CNTs) are a family of compounds that, due to their size, composition, surface ratio, and molecular structure, present useful properties. Being lighter than steel, CNTs have a higher tensile strength. They may also be better conductors than copper [2, 3]. In addition, CNTs can penetrate cell membranes without breaking them [4]. Basic nanotube units are six-membered rings of sp<sup>2</sup> -hybridized carbon atoms. The diameter of the nanotubes is of great importance in their properties because it modifies the curvature of the surface of the nanotube affecting the distribution of charge density. Also the way these units are arranged, known as chirality of nanotubes, originates conductive or semiconducting nanotubes. It is important to note that here the concept of chirality is not related to the presence of chiral atoms.

Three types of CNTs are known as characterized by the so-called chirality vectors or Hamada indices, as shown in Figure 1. CNTs of type (n,n) or armchair are metallic; those of type (n,m) or chiral and those of type (n,0) or zigzag are semiconductors or metallic; chiral nanotubes are conductors if the difference n-m is a multiple of 3 [5]. In all these cases, the repeating units are hexagons, and here we call regular CNTs (R) to these nanotubes to differentiate them from other types of CNTs in which there are cyclic units that can be of four, five, seven, or eight members. When present in the nanotubes, these units, different from the hexagons, are known as structural defects. There are different types of structural defects that can be produced by the addition of a pair of carbon atoms. Those having an arrangement (7,5,5,7) are known as bumpy (B) if the defect is repeated in a transverse equidistant form from the ends of the nanotube. If the defect is repeated longitudinally, zipper-type nanotubes (Z) are obtained. Figure 2 shows the arrangement of nonhexagonal units in some different types of defects. The arrangement (8,4,8,4) represents one of the defects called haeckelite (HK). In addition, the arrangement (5,7,7,5) of the topological defects known as Stone-Wales (SW) is shown. The absence of some atoms (vacancy) or the substitution of some carbon atoms by other elements (doping) is also considered as defects.

Figure 1. Regular nanotube models of different chirality. Perspective and lateral views for (a) armchair (5,5), (b) (chiral 6,3),

Carbon Nanotubes: Molecular and Electronic Properties of Regular and Defective Structures

http://dx.doi.org/10.5772/intechopen.70934

199

Figure 2. Nanotube defects representation. (a) (7,5,5,7) bumpy and zipper defects; (b) (8,4,8,4) haeckelite defect; and (c)

and (c) zigzag (8,0) nanotube representations.

(5,7,7,5) Stone-Wales defect.

The particular structure of the CNTs (mainly the regular ones) can be of a single-wall, SWCNT, or of multiple concentric walls and every day finds new applications in diverse fields of science and technology, and there are a great number of publications and patents which account for it (e.g., see [6–9]). Nanotubes with defects have shown advantages in some fields. For example, doping with nitrogen decreases the toxicity of nanotubes [10] and gives them excellent catalytic properties in oxygen reduction reactions [11, 12]. The bumpy nanotubes present interesting values of hydrogen physisorption energies (0.26 eV/H2), better than similar regular nanotubes, estimated through theoretical studies based on the density functional theory (DFT) [13]. This physisorption energy value is within the suggested ideal range for using hydrogen as a clean

Carbon Nanotubes: Molecular and Electronic Properties of Regular and Defective Structures http://dx.doi.org/10.5772/intechopen.70934 199

defined as carbon chemistry and is closely related, for example, to biochemistry and biomedicine. The understanding of the structure-property relations is an important pillar on which the development of these branches of knowledge is based. Thus, exchanging the position of two groups on a molecule of a recommended drug may mean that it becomes a poison. An example is the case of thalidomide disaster [1] where many babies were born with phocomelia (malformation of the limbs) when R-thalidomide, a compound suitable to mitigate some troublesome symptoms of pregnancy, was administered to their mother but contaminated with its enantiomer (compound with the same mirror image, but not superimposable); more-

Carbon nanotubes (CNTs) are a family of compounds that, due to their size, composition, surface ratio, and molecular structure, present useful properties. Being lighter than steel, CNTs have a higher tensile strength. They may also be better conductors than copper [2, 3]. In addition, CNTs can penetrate cell membranes without breaking them [4]. Basic nanotube units

great importance in their properties because it modifies the curvature of the surface of the nanotube affecting the distribution of charge density. Also the way these units are arranged, known as chirality of nanotubes, originates conductive or semiconducting nanotubes. It is important to note that here the concept of chirality is not related to the presence of chiral atoms. Three types of CNTs are known as characterized by the so-called chirality vectors or Hamada indices, as shown in Figure 1. CNTs of type (n,n) or armchair are metallic; those of type (n,m) or chiral and those of type (n,0) or zigzag are semiconductors or metallic; chiral nanotubes are conductors if the difference n-m is a multiple of 3 [5]. In all these cases, the repeating units are hexagons, and here we call regular CNTs (R) to these nanotubes to differentiate them from other types of CNTs in which there are cyclic units that can be of four, five, seven, or eight members. When present in the nanotubes, these units, different from the hexagons, are known as structural defects. There are different types of structural defects that can be produced by the addition of a pair of carbon atoms. Those having an arrangement (7,5,5,7) are known as bumpy (B) if the defect is repeated in a transverse equidistant form from the ends of the nanotube. If the defect is repeated longitudinally, zipper-type nanotubes (Z) are obtained. Figure 2 shows the arrangement of nonhexagonal units in some different types of defects. The arrangement (8,4,8,4) represents one of the defects called haeckelite (HK). In addition, the arrangement (5,7,7,5) of the topological defects known as Stone-Wales (SW) is shown. The absence of some atoms (vacancy) or the substitution of some carbon atoms by other elements (doping) is also considered as defects.

The particular structure of the CNTs (mainly the regular ones) can be of a single-wall, SWCNT, or of multiple concentric walls and every day finds new applications in diverse fields of science and technology, and there are a great number of publications and patents which account for it (e.g., see [6–9]). Nanotubes with defects have shown advantages in some fields. For example, doping with nitrogen decreases the toxicity of nanotubes [10] and gives them excellent catalytic properties in oxygen reduction reactions [11, 12]. The bumpy nanotubes present interesting values of hydrogen physisorption energies (0.26 eV/H2), better than similar regular nanotubes, estimated through theoretical studies based on the density functional theory (DFT) [13]. This physisorption energy value is within the suggested ideal range for using hydrogen as a clean


over, the enantiomer, S-thalidomide, has teratogenic effects.

198 Density Functional Calculations - Recent Progresses of Theory and Application

are six-membered rings of sp<sup>2</sup>

Figure 1. Regular nanotube models of different chirality. Perspective and lateral views for (a) armchair (5,5), (b) (chiral 6,3), and (c) zigzag (8,0) nanotube representations.

Figure 2. Nanotube defects representation. (a) (7,5,5,7) bumpy and zipper defects; (b) (8,4,8,4) haeckelite defect; and (c) (5,7,7,5) Stone-Wales defect.

energy media [14]. The specialized literature lacks enough information to have a broader perspective of the utility and possible applications of defective nanotubes, despite some interesting isolated results [15, 16].

EH<sup>2</sup> is the total energy for the hydrogen molecule; n is the number of encapsulated hydrogen molecules; and vdW corresponds to the van der Waals interaction correction term accounting for the dispersion forces calculated by DFT-D3 methods [22, 23] as implemented in Jaguar v9.2. Specialized research studies have found B3LYP-D3 as a very accurate method for treating

Carbon Nanotubes: Molecular and Electronic Properties of Regular and Defective Structures

The electronic chemical potential μ is defined [27] as the energy changes of the system with respect to the number of electrons N at a given external potential ν(r) defined by the nuclei:

After applying the finite difference approximation, the Koopmans theorem [28, 30] and the Kohn-Sham principle [29, 31], global reactivity descriptors are calculated according to Eqs. (4)–(7) for the electronic chemical potential, the chemical hardness η, the global electrophilicity index ω,

<sup>ω</sup> <sup>¼</sup> <sup>μ</sup><sup>2</sup>

where ELUMO and EHOMO are the B3LYP/6-31G(d) energies of the frontier orbitals: the lowest unoccupied molecular orbitals and the highest occupied molecular orbitals, respectively. The bandgap, BG, is calculated as the energy difference between the frontier molecular orbitals

In this section, we analyze the reactivity of armchair, chiral, and zigzag nanotubes, regular and with bumpy defects, using theoretical DFT methods. We study the effects of both nitrogen

The bandgap is the energy difference between the conduction band and the valence band. The smaller is the bandgap, the nanotube tends to have better conductive ability. The regular CNTs (defect free) formed only by six-membered ring units exhibit properties that depend on the ordering of the rings in the nanotube or chirality. Studies with the DFT methods at B3LYP/6- 31G(d) for CNTs of similar diameters (between 6.3 and 6.8 Å) and eight carbon-atom layer (cl) of length revealed that the bandgap value decreases in the order: armchair (5,5) > chiral (6,3) > zigzag (8,0) with values 1.77, 1.35, and 0.42 eV, respectively [25]. Nitrogen-doped CNTs

doping and the total hydrogenation of nanotubes on the electronic properties.

νð Þr

μ ¼ ð Þ ELUMO þ EHOMO =2 (4)

η ¼ ð Þ ELUMO � EHOMO =2 (5)

=2η (6)

http://dx.doi.org/10.5772/intechopen.70934

Ѕ ¼ 1=η (7)

(3)

201

<sup>μ</sup> <sup>¼</sup> <sup>∂</sup><sup>E</sup> ∂N 

and the softness S, based on conceptual DFT, respectively [27–34]:

intermolecular interactions [26].

ELUMO�EHOMO.

3.1. Bandgap

3. Chirality effect

It is therefore interesting and necessary to evaluate how changes in geometric, structural, topological, or doping parameters can affect properties of stability, conductivity, and chemical reactivity of nanotubes and determine if there is any general tendency to predict or modify specifically their behavior in the search for new applications of these advanced materials.

In this chapter, the effects of chirality, chemisorption, number of defects, and the presence of nitrogen on the reactivity of the regular and defective CNTs are analyzed, and results are obtained confirming the good values of hydrogen physisorption energy. This chapter also discusses how the diameter and length of nanotubes, and the presence of nitrogen and the different types of defects (bumpy, haeckelite, Stone-Wales, and zipper) enhance the molecular and electronic properties of CNTs, particularly armchair-type, through molecular global descriptors based on the DFT methods.

#### 2. Methods

Regular and defective nanotubes are built as finite, open, and terminated by hydrogen atoms by using the HyperTube script [17] embedded in Hyperchem [18]. Saturated nanotubes are full exo-hydrogenated. Nitrogen doping is performed by replacing two carbon atoms by nitrogen atoms at an hexagonal ring building a pyrimidine (with the two nitrogen atoms separated by a carbon atom in the way N1-C2-N3) taking care of locating an additional pyrimidine ring opposite to the first one at the same level in the tube. All the structures are fully optimized by the DFT methods at the B3LYP/6-31G(d) level of theory [19, 20] with Jaguar v9.2 [21]. No symmetry restrictions are applied. Energy minima are verified after the harmonic vibrational frequency real values that are obtained by calculations for optimized structures at the same level of theory. Dispersion forces are corrected by means of the DFT-D3 method [22, 23] validated through the DFT nonlocal method [24] as was performed before [25].

The formation energy, EF, is calculated as follows:

$$E\_F = E\_{NT} - \sum\_{i=1}^{n} n\_i E\_i \tag{1}$$

where ENT and Ei are the total energy for the nanotube structure (B3LYP/6-31G(d) full optimized) and for the i elements (for instance, C, H, N), respectively; ni is the number of each element.

The hydrogen physisorption energy, Eph, is calculated according to expression (2):

$$E\_{ph} = E\_{(NT + nH\_2)} - E\_{NT} - nE\_{H\_2} + \sigma dW \tag{2}$$

where E(NT <sup>+</sup> nH2) and ENT are the total energy for the B3LYP/6-31G(d) full optimized nanotube structure containing the encapsulated hydrogen molecules inside and for the nanotube alone; EH<sup>2</sup> is the total energy for the hydrogen molecule; n is the number of encapsulated hydrogen molecules; and vdW corresponds to the van der Waals interaction correction term accounting for the dispersion forces calculated by DFT-D3 methods [22, 23] as implemented in Jaguar v9.2. Specialized research studies have found B3LYP-D3 as a very accurate method for treating intermolecular interactions [26].

The electronic chemical potential μ is defined [27] as the energy changes of the system with respect to the number of electrons N at a given external potential ν(r) defined by the nuclei:

$$
\mu = \left(\frac{\partial E}{\partial \mathbf{N}}\right)\_{\nu(r)}\tag{3}
$$

After applying the finite difference approximation, the Koopmans theorem [28, 30] and the Kohn-Sham principle [29, 31], global reactivity descriptors are calculated according to Eqs. (4)–(7) for the electronic chemical potential, the chemical hardness η, the global electrophilicity index ω, and the softness S, based on conceptual DFT, respectively [27–34]:

$$
\mu = (E\_{\rm LILMO} + E\_{\rm HOMO})/2 \tag{4}
$$

$$
\eta = (E\_{LIMO} - E\_{HOMO})/2 \tag{5}
$$

$$
\omega = \mu^2 / 2\eta \tag{6}
$$

$$S = \mathbf{1}/\eta \tag{7}$$

where ELUMO and EHOMO are the B3LYP/6-31G(d) energies of the frontier orbitals: the lowest unoccupied molecular orbitals and the highest occupied molecular orbitals, respectively. The bandgap, BG, is calculated as the energy difference between the frontier molecular orbitals ELUMO�EHOMO.

#### 3. Chirality effect

In this section, we analyze the reactivity of armchair, chiral, and zigzag nanotubes, regular and with bumpy defects, using theoretical DFT methods. We study the effects of both nitrogen doping and the total hydrogenation of nanotubes on the electronic properties.

#### 3.1. Bandgap

energy media [14]. The specialized literature lacks enough information to have a broader perspective of the utility and possible applications of defective nanotubes, despite some inter-

It is therefore interesting and necessary to evaluate how changes in geometric, structural, topological, or doping parameters can affect properties of stability, conductivity, and chemical reactivity of nanotubes and determine if there is any general tendency to predict or modify specifically their behavior in the search for new applications of these advanced materials.

In this chapter, the effects of chirality, chemisorption, number of defects, and the presence of nitrogen on the reactivity of the regular and defective CNTs are analyzed, and results are obtained confirming the good values of hydrogen physisorption energy. This chapter also discusses how the diameter and length of nanotubes, and the presence of nitrogen and the different types of defects (bumpy, haeckelite, Stone-Wales, and zipper) enhance the molecular and electronic properties of CNTs, particularly armchair-type, through molecular global

Regular and defective nanotubes are built as finite, open, and terminated by hydrogen atoms by using the HyperTube script [17] embedded in Hyperchem [18]. Saturated nanotubes are full exo-hydrogenated. Nitrogen doping is performed by replacing two carbon atoms by nitrogen atoms at an hexagonal ring building a pyrimidine (with the two nitrogen atoms separated by a carbon atom in the way N1-C2-N3) taking care of locating an additional pyrimidine ring opposite to the first one at the same level in the tube. All the structures are fully optimized by the DFT methods at the B3LYP/6-31G(d) level of theory [19, 20] with Jaguar v9.2 [21]. No symmetry restrictions are applied. Energy minima are verified after the harmonic vibrational frequency real values that are obtained by calculations for optimized structures at the same level of theory. Dispersion forces are corrected by means of the DFT-D3 method [22, 23]

validated through the DFT nonlocal method [24] as was performed before [25].

The hydrogen physisorption energy, Eph, is calculated according to expression (2):

EF <sup>¼</sup> ENT �X<sup>n</sup>

where ENT and Ei are the total energy for the nanotube structure (B3LYP/6-31G(d) full optimized) and for the i elements (for instance, C, H, N), respectively; ni is the number of each

where E(NT <sup>+</sup> nH2) and ENT are the total energy for the B3LYP/6-31G(d) full optimized nanotube structure containing the encapsulated hydrogen molecules inside and for the nanotube alone;

i¼1

Eph ¼ Eð Þ NTþnH<sup>2</sup> � ENT � nEH<sup>2</sup> þ vdW (2)

niEi (1)

The formation energy, EF, is calculated as follows:

esting isolated results [15, 16].

200 Density Functional Calculations - Recent Progresses of Theory and Application

descriptors based on the DFT methods.

2. Methods

element.

The bandgap is the energy difference between the conduction band and the valence band. The smaller is the bandgap, the nanotube tends to have better conductive ability. The regular CNTs (defect free) formed only by six-membered ring units exhibit properties that depend on the ordering of the rings in the nanotube or chirality. Studies with the DFT methods at B3LYP/6- 31G(d) for CNTs of similar diameters (between 6.3 and 6.8 Å) and eight carbon-atom layer (cl) of length revealed that the bandgap value decreases in the order: armchair (5,5) > chiral (6,3) > zigzag (8,0) with values 1.77, 1.35, and 0.42 eV, respectively [25]. Nitrogen-doped CNTs (N-CNTs) in the three mentioned cases show a decrease in the bandgap in the order: chiral, armchair, and zigzag with bandgap values of 1.26, 1.16, and 0.0 eV, respectively. Doped and nondoped zigzag nanotubes exhibit the smallest bandgap [25].

CNTs with two bumpy defects and similar dimensions than the mentioned regular CNTs show the following order of bandgap: armchair > zigzag > chiral with values of 1.72, 0.43, and 0.31 eV, respectively. In the case of nitrogen-doped CNTs with bumpy defects, obtained bandgap values increase for zigzag and chiral nanotubes (0.83 and 0.86 eV, respectively), but decreases for armchair nanotubes (1.29 eV) (see 1a in Table 1). These results indicate that the bumpy defect decreases the bandgap in the doped and nondoped chiral nanotubes. In the zigzag and armchair nanotubes, the bumpy effect depends on the presence of nitrogen since in both nondoped cases, the presence of the bumpy defect does not affect too much, whereas in the cases of doped zigzag and armchair nanotubes, the presence of the bumpy defect increases the bandgap. Armchair nanotubes with five bumpy defects in their structure, doped and nondoped, decrease the bandgap to 1.17 eV (see 1e in Table 1).

The hydrogenation of nanotubes with bumpy defects increases the bandgap to levels between 6.12 and 6.57 eV for two bumpy defects and between 6.37 and 6.79 eV for five bumpy defects. This behavior is expected due to the sp3 hybridization that makes the structures of the nanotubes similar to each other with a similar electron density distribution (see 2a and 2e in Table 2). In all cases, it can be observed that nitrogen decreases the bandgap with values between 5.46 and 5.83 eV.

In other words, in terms of conductive ability, our calculations reveal that both regular and bumpy nanotubes show the same trend: zigzag and chiral nanotubes have higher conductive abilities than armchair nanotubes. The incorporation of bumpy defects in nanotubes only increases the conductive ability of chiral nanotubes (doped or nondoped). Nitrogen doping increases the conductive ability of the three types of regular nanotubes and that of the armchair

Table 2. B3LYP/6-31G(d) bandgap (ELUMO�EHOMO) BG, electronic chemical potential μ, chemical hardness η, and global electrophilicity index ω, in eV, for a series of hydrogenated nitrogen-doped and nondoped armchair (5,5), chiral (6,3), and zigzag (8,0) nanotubes, with two and five bumpy defects, all of them having a similar diameter (6.3–6.7 Å) and 8 cl of

Entry Reactivity descriptor Armchair Chiral Zigzag

2a BG 6.29 5.56 6.12 5.62 6.57 5.46 2b μ �2.71 �2.10 �2.71 �2.21 �2.71 �1.82 2c η 3.14 2.78 3.06 2.81 3.28 2.73 2d ω 1.17 0.79 1.20 0.87 1.12 0.61

2e BG 6.53 5.83 6.37 5.79 6.79 5.48 2f μ �2.73 �2.28 �2.75 �2.32 �2.88 �2.04 2g η 3.27 2.92 3.19 2.90 3.39 2.74 2h ω 1.14 0.89 1.18 0.93 1.22 0.76

0N 4N 0N 4N 0N 4N

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The chemical potential μ, is the inverse of electronegativity. It measures the tendency of electrons to escape from an equilibrium system. At higher value of the chemical potential, the molecule behaves as a better electron-donor against an acceptor. In a study of diene and dienophile reactivity in Diels-Alder reactions in the context of conceptual DFT at the B3LYP/ 6-31G(d) level, a compound with μ = �1.85 eV was characterized as electron-donor while

The values of μ for the regular armchair, chiral, and zigzag nanotubes determined at the same level of theory fluctuate between �3.57 and �3.06 eV. The values of μ for the corresponding nitrogen-doped and nondoped nanotubes with two bumpy defects decrease to �3.67 and �3.25 eV, respectively (the only exception is exhibited by nondoped zigzag nanotubes with a μ value of �3.49 eV) and also decrease for armchair nanotubes with five defects (see 1b and 1f in Table 1). Nitrogen doping slightly increases the value of μ. That means, there is not a very marked difference of chemical potential for nanotubes of different chirality, with or without bumpy defects, doped or nondoped defects. The values of μ are intermediates between the electron-donors and the electron acceptors leaving open the potentiality to behave as electron-

another compound with μ = �7.04 eV was considered as electron-acceptor [34].

nanotubes with two bumpy defects.

3.2. Chemical potential

with two bumpy defects

with five bumpy defects

length.

donors or electron acceptors.


Table 1. B3LYP/6-31G(d) bandgap (ELUMO�EHOMO), BG, electronic chemical potential, μ, chemical hardness, η, and global electrophilicity, ω, in eV, for a series of 32 nitrogen-doped and nondoped armchair (5,5), chiral (6,3), and zigzag (8,0) bumpy nanotubes, with two and five bumpy defects, all of them having a similar diameter (6.3–6.7 Å) and 8 cl of length.


Table 2. B3LYP/6-31G(d) bandgap (ELUMO�EHOMO) BG, electronic chemical potential μ, chemical hardness η, and global electrophilicity index ω, in eV, for a series of hydrogenated nitrogen-doped and nondoped armchair (5,5), chiral (6,3), and zigzag (8,0) nanotubes, with two and five bumpy defects, all of them having a similar diameter (6.3–6.7 Å) and 8 cl of length.

abilities than armchair nanotubes. The incorporation of bumpy defects in nanotubes only increases the conductive ability of chiral nanotubes (doped or nondoped). Nitrogen doping increases the conductive ability of the three types of regular nanotubes and that of the armchair nanotubes with two bumpy defects.

#### 3.2. Chemical potential

(N-CNTs) in the three mentioned cases show a decrease in the bandgap in the order: chiral, armchair, and zigzag with bandgap values of 1.26, 1.16, and 0.0 eV, respectively. Doped and

CNTs with two bumpy defects and similar dimensions than the mentioned regular CNTs show the following order of bandgap: armchair > zigzag > chiral with values of 1.72, 0.43, and 0.31 eV, respectively. In the case of nitrogen-doped CNTs with bumpy defects, obtained bandgap values increase for zigzag and chiral nanotubes (0.83 and 0.86 eV, respectively), but decreases for armchair nanotubes (1.29 eV) (see 1a in Table 1). These results indicate that the bumpy defect decreases the bandgap in the doped and nondoped chiral nanotubes. In the zigzag and armchair nanotubes, the bumpy effect depends on the presence of nitrogen since in both nondoped cases, the presence of the bumpy defect does not affect too much, whereas in the cases of doped zigzag and armchair nanotubes, the presence of the bumpy defect increases the bandgap. Armchair nanotubes with five bumpy defects in their structure, doped and

The hydrogenation of nanotubes with bumpy defects increases the bandgap to levels between 6.12 and 6.57 eV for two bumpy defects and between 6.37 and 6.79 eV for five bumpy defects. This behavior is expected due to the sp3 hybridization that makes the structures of the nanotubes similar to each other with a similar electron density distribution (see 2a and 2e in Table 2). In all cases, it can be observed that nitrogen decreases the bandgap with values

In other words, in terms of conductive ability, our calculations reveal that both regular and bumpy nanotubes show the same trend: zigzag and chiral nanotubes have higher conductive

1a BG 1.72 1.29 0.31 0.86 0.43 0.83 1b μ �3.67 �3.25 �3.67 �3.33 �3.49 �3.48 1c η 0.86 0.64 0.16 0.43 0.21 0.41 1d ω 7.82 8.19 43.22 12.83 28.52 14.66

Table 1. B3LYP/6-31G(d) bandgap (ELUMO�EHOMO), BG, electronic chemical potential, μ, chemical hardness, η, and global electrophilicity, ω, in eV, for a series of 32 nitrogen-doped and nondoped armchair (5,5), chiral (6,3), and zigzag (8,0) bumpy nanotubes, with two and five bumpy defects, all of them having a similar diameter (6.3–6.7 Å) and 8 cl of length.

0N 4N 0N 4N 0N 4N

Entry Reactivity descriptor Armchair Chiral Zigzag

nondoped zigzag nanotubes exhibit the smallest bandgap [25].

202 Density Functional Calculations - Recent Progresses of Theory and Application

nondoped, decrease the bandgap to 1.17 eV (see 1e in Table 1).

between 5.46 and 5.83 eV.

with two bumpy defects

with five bumpy defects

1e BG 1.17 1.17 1f μ �3.65 �3.64 1g η 0.59 0.58 1h ω 11.35 11.32 The chemical potential μ, is the inverse of electronegativity. It measures the tendency of electrons to escape from an equilibrium system. At higher value of the chemical potential, the molecule behaves as a better electron-donor against an acceptor. In a study of diene and dienophile reactivity in Diels-Alder reactions in the context of conceptual DFT at the B3LYP/ 6-31G(d) level, a compound with μ = �1.85 eV was characterized as electron-donor while another compound with μ = �7.04 eV was considered as electron-acceptor [34].

The values of μ for the regular armchair, chiral, and zigzag nanotubes determined at the same level of theory fluctuate between �3.57 and �3.06 eV. The values of μ for the corresponding nitrogen-doped and nondoped nanotubes with two bumpy defects decrease to �3.67 and �3.25 eV, respectively (the only exception is exhibited by nondoped zigzag nanotubes with a μ value of �3.49 eV) and also decrease for armchair nanotubes with five defects (see 1b and 1f in Table 1). Nitrogen doping slightly increases the value of μ. That means, there is not a very marked difference of chemical potential for nanotubes of different chirality, with or without bumpy defects, doped or nondoped defects. The values of μ are intermediates between the electron-donors and the electron acceptors leaving open the potentiality to behave as electrondonors or electron acceptors.

In the saturated nondoped nanotubes, μ increases to values of �2.88 to �2.71 eV and in the presence of nitrogen doping, μ increases to �1.82 eV being the zigzag nanotubes the ones with the greater value of μ (see 2b and 2f in Table 2) and likely with a higher ability to be oxidized. Therefore, zigzag saturated nitrogen-doped nanotubes could potentially behave as a catalyst in C-H activation reactions [12].

The values of ω for the regular nanotubes were reported to be dependent on chirality and grow in the order: armchair (5,5) < chiral (6,3) < zigzag (8,0), with values of ω of 6.56, 9.42, and 28.43 eV, respectively. Nitrogen doping slightly decreases the ω value in the armchair (6.42 eV) and chiral (8.46 eV) nanotubes while increases the ω value in zigzag nanotubes (30.07 eV) [25]. The bumpy defects increase the value of ω especially in chiral nanotubes (43.22 eV). The armchair nanotubes with two bumpy defects are the ones with the lowest value of ω (7.82 eV), which rises in nanotubes with five bumpy defects (11.35 eV). Nitrogen-doped nanotubes exhibit lower ω values with the only exception of armchair nanotubes with two bumpy defects

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The saturated nanotubes with two and five bumpy defects, independent of the chirality, decrease the values of ω (1.12–1.22 eV). For the nitrogen-doped nanotubes, the value of ω is even lower (0.61–0.93) with values in the range of the weak electrophiles (see 2d and 2h in

Detailed analysis of the results obtained for this group of regular and bumpy defective nanotubes (with 8 cl of length) reveals that the zigzag nanotubes are those that exhibit greater conductive ability. Also the zigzag nanotubes are the most reactive and exhibit a great capacity to be reduced, to acquire electronic density, or to be strong electrophiles. The armchair nanotubes are at the other end. The presence of nitrogen in armchair regular nanotubes increases their conductive ability. The presence of bumpy defects also enhances their reactivity and their ability to be reduced, especially with five bumpy defects. Bumpy defects, in general, increase the conductivity, the reactivity, and the capacity of nanotubes to be reduced, with

The 8 cl saturated nanotubes, regular and with bumpy defects, exhibit a significant decrease in conductive ability counteracted by the presence of nitrogen. Our calculations reveal that they are less reactive and their ability to acquire electronic density or to behave as electron acceptors decreases when compared to unsaturated nanotubes with a quite good consistence within the

Inspired by the unique properties of nanotubes and driven by the idea of contributing to a clean energy source for vehicular use, several authors have studied the possibility of using CNTs as hydrogen transporters [36–40]. Hydrogen may be chemisorbed (covalently attached to carbon atoms) or physisorbed (adsorbed as molecular hydrogen through noncovalent interactions with the nanotube). The results have generated controversy since they have not been reproducible [41, 42]. A reasonable explanation is probably due to the presence of uncontrolled defects in the nanotube that make vary the curvature of the nanotube, the electronic density distribution, and therefore its properties [43]. For example, (7,5,5,7) SW defects increase the ability to adsorb H2 of the armchair (5,5), whereas (5,7,7,5) SW defects decrease the ability to adsorb H2 of these nanotubes as was established using the DFT methods at the B3LYP/LanL2DZ level [44]. Additionally, the number of defects in the nanotube affects the reactivity [44]. Other

4. Hydrogen adsorption for the carbon nanotubes with defects

(8.19 eV) (see 1d and 1h in Table 1).

greater intensity in the chiral nanotubes.

different reactivity descriptors.

Table 2).

#### 3.3. Hardness

The hardness, η, is a useful reactivity descriptor associated to the resistance of a molecule to exchange electronic density with the environment [27, 34]. It measures the stability of a system. The increase of η is normally associated with a stabilizing process having a negative change of energy.

In the regular nanotubes, η increases in the order: zigzag (8,0) > chiral (6,3) > armchair (5,5), with η values of 0.21, 0.68, and 0.89 eV, respectively, determined at B3LYP/6-31G(d) [25]. When regular zigzag, chiral, and armchair nanotubes are nitrogen-doped, η decreases to values of 0.20, 0.63, and 0.58 eV, respectively [25].

Defective nanotubes exhibit a decrease in η values. The only exception is for doped zigzag nanotubes. Armchair nanotubes with two bumpy defects reveal the highest values of η (0.64 and 0.86 eV for nitrogen-doped and nondoped nanotubes, respectively) compared to chiral and zigzag nanotubes with η values between 0.16 and 0.43 eV including nitrogen-doped and nondoped systems. Nitrogen doping increases η in chiral and zigzag nanotubes, but decreases it in armchair nanotubes with two defects. In the armchair nanotubes with five defects, η decreases also (0.59 eV) unaffected by the presence of nitrogen doping (see 1c and 1g in Table 1). That is, η depends on the chirality. Regular zigzag nanotubes are the most reactive. Bumpy defects increase the reactivity of nanotubes (with the exception of doped zigzag and armchair nanotubes). Nitrogen doping also increases the reactivity of regular nanotubes especially the armchair nanotubes and also increases the reactivity of armchair nanotubes with bumpy defects. The increase in the number of bumpy defects increases the reactivity of the armchair nanotubes.

The saturated nanotubes with two bumpy defects do not show large η differences with chirality (3.06–3.28 eV) nor is it affected too much in nanotubes with five bumpy defects (3.19–3.39 eV). In all cases, nitrogen doping makes η decrease (2.73–2.92 eV) (see 2c and 2 g in Table 2) and consequently, increases nanotube reactivity.

#### 3.4. Electrophilicity index

The electrophilicity index ω, as a descriptor of molecular reactivity, gives a measure of the stabilizing energy of a molecule when it acquires some additional amount of electron density from the environment. The value of ω allows to differentiating the power of the molecules that act as electrophiles. Thus, for organic compounds, strong electrophiles are considered to have ω > 1.5 eV and weak electrophiles have ω < 0.8 eV [35]. According to this classification, the CNTs are strong electrophiles since they exhibit ω values that are much higher than 1.5 eV. Also, a higher value of ω may be associated with a greater trend of the compound to be reduced due to its ability to acquire electronic density.

The values of ω for the regular nanotubes were reported to be dependent on chirality and grow in the order: armchair (5,5) < chiral (6,3) < zigzag (8,0), with values of ω of 6.56, 9.42, and 28.43 eV, respectively. Nitrogen doping slightly decreases the ω value in the armchair (6.42 eV) and chiral (8.46 eV) nanotubes while increases the ω value in zigzag nanotubes (30.07 eV) [25]. The bumpy defects increase the value of ω especially in chiral nanotubes (43.22 eV). The armchair nanotubes with two bumpy defects are the ones with the lowest value of ω (7.82 eV), which rises in nanotubes with five bumpy defects (11.35 eV). Nitrogen-doped nanotubes exhibit lower ω values with the only exception of armchair nanotubes with two bumpy defects (8.19 eV) (see 1d and 1h in Table 1).

In the saturated nondoped nanotubes, μ increases to values of �2.88 to �2.71 eV and in the presence of nitrogen doping, μ increases to �1.82 eV being the zigzag nanotubes the ones with the greater value of μ (see 2b and 2f in Table 2) and likely with a higher ability to be oxidized. Therefore, zigzag saturated nitrogen-doped nanotubes could potentially behave as a catalyst in

The hardness, η, is a useful reactivity descriptor associated to the resistance of a molecule to exchange electronic density with the environment [27, 34]. It measures the stability of a system. The increase of η is normally associated with a stabilizing process having a negative change of

In the regular nanotubes, η increases in the order: zigzag (8,0) > chiral (6,3) > armchair (5,5), with η values of 0.21, 0.68, and 0.89 eV, respectively, determined at B3LYP/6-31G(d) [25]. When regular zigzag, chiral, and armchair nanotubes are nitrogen-doped, η decreases to values of 0.20,

Defective nanotubes exhibit a decrease in η values. The only exception is for doped zigzag nanotubes. Armchair nanotubes with two bumpy defects reveal the highest values of η (0.64 and 0.86 eV for nitrogen-doped and nondoped nanotubes, respectively) compared to chiral and zigzag nanotubes with η values between 0.16 and 0.43 eV including nitrogen-doped and nondoped systems. Nitrogen doping increases η in chiral and zigzag nanotubes, but decreases it in armchair nanotubes with two defects. In the armchair nanotubes with five defects, η decreases also (0.59 eV) unaffected by the presence of nitrogen doping (see 1c and 1g in Table 1). That is, η depends on the chirality. Regular zigzag nanotubes are the most reactive. Bumpy defects increase the reactivity of nanotubes (with the exception of doped zigzag and armchair nanotubes). Nitrogen doping also increases the reactivity of regular nanotubes especially the armchair nanotubes and also increases the reactivity of armchair nanotubes with bumpy defects. The increase in the

The saturated nanotubes with two bumpy defects do not show large η differences with chirality (3.06–3.28 eV) nor is it affected too much in nanotubes with five bumpy defects (3.19–3.39 eV). In all cases, nitrogen doping makes η decrease (2.73–2.92 eV) (see 2c and 2 g in

The electrophilicity index ω, as a descriptor of molecular reactivity, gives a measure of the stabilizing energy of a molecule when it acquires some additional amount of electron density from the environment. The value of ω allows to differentiating the power of the molecules that act as electrophiles. Thus, for organic compounds, strong electrophiles are considered to have ω > 1.5 eV and weak electrophiles have ω < 0.8 eV [35]. According to this classification, the CNTs are strong electrophiles since they exhibit ω values that are much higher than 1.5 eV. Also, a higher value of ω may be associated with a greater trend of the compound to be

number of bumpy defects increases the reactivity of the armchair nanotubes.

Table 2) and consequently, increases nanotube reactivity.

reduced due to its ability to acquire electronic density.

C-H activation reactions [12].

204 Density Functional Calculations - Recent Progresses of Theory and Application

0.63, and 0.58 eV, respectively [25].

3.4. Electrophilicity index

3.3. Hardness

energy.

The saturated nanotubes with two and five bumpy defects, independent of the chirality, decrease the values of ω (1.12–1.22 eV). For the nitrogen-doped nanotubes, the value of ω is even lower (0.61–0.93) with values in the range of the weak electrophiles (see 2d and 2h in Table 2).

Detailed analysis of the results obtained for this group of regular and bumpy defective nanotubes (with 8 cl of length) reveals that the zigzag nanotubes are those that exhibit greater conductive ability. Also the zigzag nanotubes are the most reactive and exhibit a great capacity to be reduced, to acquire electronic density, or to be strong electrophiles. The armchair nanotubes are at the other end. The presence of nitrogen in armchair regular nanotubes increases their conductive ability. The presence of bumpy defects also enhances their reactivity and their ability to be reduced, especially with five bumpy defects. Bumpy defects, in general, increase the conductivity, the reactivity, and the capacity of nanotubes to be reduced, with greater intensity in the chiral nanotubes.

The 8 cl saturated nanotubes, regular and with bumpy defects, exhibit a significant decrease in conductive ability counteracted by the presence of nitrogen. Our calculations reveal that they are less reactive and their ability to acquire electronic density or to behave as electron acceptors decreases when compared to unsaturated nanotubes with a quite good consistence within the different reactivity descriptors.

#### 4. Hydrogen adsorption for the carbon nanotubes with defects

Inspired by the unique properties of nanotubes and driven by the idea of contributing to a clean energy source for vehicular use, several authors have studied the possibility of using CNTs as hydrogen transporters [36–40]. Hydrogen may be chemisorbed (covalently attached to carbon atoms) or physisorbed (adsorbed as molecular hydrogen through noncovalent interactions with the nanotube). The results have generated controversy since they have not been reproducible [41, 42]. A reasonable explanation is probably due to the presence of uncontrolled defects in the nanotube that make vary the curvature of the nanotube, the electronic density distribution, and therefore its properties [43]. For example, (7,5,5,7) SW defects increase the ability to adsorb H2 of the armchair (5,5), whereas (5,7,7,5) SW defects decrease the ability to adsorb H2 of these nanotubes as was established using the DFT methods at the B3LYP/LanL2DZ level [44]. Additionally, the number of defects in the nanotube affects the reactivity [44]. Other DFT studies for armchair (10,10) nanotubes show that SW defects with five- and eight-membered rings have a greater ability to adsorb H2 than SW defects with five- and seven-membered rings [43], which is also valid for nanotubes of different chirality [43, 45].

SW defects have also been studied in relation to metal-decorated nanotubes and their effects on hydrogen adsorption [46, 47] and also in the study of the interactions with metallic particles and ions [48]. The results obtained for armchair (6,6) nanotubes with SW defects at the B3LYP/ 6-31G(d) level focus on the charge transfer to ions and their relation to the electromigration where the SW defects facilitate the migration of positively charged ions (In<sup>+</sup> and In3+) through the nanotube [48].

In relation to hydrogen storage as a potential source of clean energy, one requirement is to achieve that at least 5.5 wt% of hydrogen [49] can be released under ambient pressure and temperature conditions. For achieving that goal and to favor a reversible H2 adsorptiondesorption process, it is required that the interaction energy CNT-H2 be of the order of 0.1– 0.5 eV/H2, according to calculations of first principles and to thermodynamic considerations [14, 39, 50, 51].

Hydrogen chemisorption energy is a chirality-dependent exothermic property. Zigzag nitrogen-doped and nondoped nanotubes reveal as the most exothermic followed by the chiral and armchair nanotubes [25]. On the contrary, hydrogen physisorption energies, Eph, exhibit endothermic values. Eph values for saturated, nitrogen-doped armchair CNTs obtained by the DFT [B3LYP/6-31G(d)] methods are within the ideal mentioned range (0.26 eV/H2 for 12 H2 physisorbed into a (4,4) nanotube) and grow for chiral and zigzag nitrogen-doped nanotubes [25] in excellent agreement with results obtained for armchair, chiral, and zigzag CNTs by means of the DFT methods that use the local density approximation [37]. In addition to the chirality, the diameter and length of the nanotube affect the nanotube-H2 interactions. The diameter changes cause the variation in the nanotube curvature affecting the arrangement of the molecular orbitals [4], and the nanotube length variation affects the diffusivity as was determined for drug adsorption [52]. Eph values decrease for nanotubes with a smaller diameter and a larger length (higher number of carbon-atom layers) but increase with the number of adsorbed H2.

presence of defects that generate suitable spaces, as explained above, can be predicted through DFT calculations. However, the use of conceptual DFT reactivity descriptors (μ, η, ω, and Ѕ) to correlate or predict hydrogen physisorption energies in nanotubes is not viewed as adequate. Their use is related and manifests mainly in molecular interactions in which there is charge transfer wherein nucleophiles, electrophiles, or species that change their oxidation state take part. A correlation of these descriptors with the hydrogen adsorption efficiency in nanotubes, at this time, could only have limited validity and provided it is used in conjunction with other

Figure 3. B3LYP/6-31G(d) fully optimized structure of an hydrogenated armchair (5,5) bumpy, nondoped nanotube, with 16 cl of length having adsorbed 44 hydrogen molecules (stoichiometry C170H278; Eph = 0.32 eV/H2). Front and lateral

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Nitrogen-doped armchair nanotubes are expected to be good catalysts for oxygen reduction reactions compared to their chiral and zigzag nanotubes because of their particular distribution of electric charge [12, 25]. In this section, we analyze the effect of different types of defects in the reactivity of armchair structures by using global descriptors based on the DFT methods. Bumpy, haeckelite, Stone-Wales, and zipper defects are considered (see Figure 4). We also analyze the effects of nitrogen doping, the diameter and length of the nanotubes, the number of defects and the charges on the carbon atom adjacent to the two pyrimidine nitrogen atoms.

descriptors such as diameter, length, number of defects, or others to be set.

5. Armchair nanotubes

views.

Bumpy defects and nitrogen doping favor interaction with nanotube-H2. Saturated armchair (4,4) nitrogen-doped nanotubes with bumpy defects and 20 cl of length can encapsulate 15 H2 molecules (C164N4H210 stoichiometry) with a favorable Eph of 0.11 eV/H2 obtained at the B3LYP/6-31G(d) level [13]. Our calculations, considering dispersion forces correction, reveal that saturated nondoped armchair (5,5) nanotubes with bumpy defects and 16 cl of length can encapsulate 44 H2 molecules (stoichiometry C170H278) with a good Eph value of 0.32 eV/H2. Figure 3 shows the optimized structure of this system having a hydrogen content of 12 wt%.

The actual available experimental and theoretical information related to hydrogen adsorption in nanotubes indicates that the structural effects that allow obtaining hydrogen physisorption energies within the range considered ideal for the reversible storage of hydrogen (0.1–0.4 eV/H2 under environmental conditions) are those that favor the nanotube-H2 electrostatic interactions. Parameters such as the chirality of the nanotube, a suitable diameter and length, and the

Figure 3. B3LYP/6-31G(d) fully optimized structure of an hydrogenated armchair (5,5) bumpy, nondoped nanotube, with 16 cl of length having adsorbed 44 hydrogen molecules (stoichiometry C170H278; Eph = 0.32 eV/H2). Front and lateral views.

presence of defects that generate suitable spaces, as explained above, can be predicted through DFT calculations. However, the use of conceptual DFT reactivity descriptors (μ, η, ω, and Ѕ) to correlate or predict hydrogen physisorption energies in nanotubes is not viewed as adequate. Their use is related and manifests mainly in molecular interactions in which there is charge transfer wherein nucleophiles, electrophiles, or species that change their oxidation state take part. A correlation of these descriptors with the hydrogen adsorption efficiency in nanotubes, at this time, could only have limited validity and provided it is used in conjunction with other descriptors such as diameter, length, number of defects, or others to be set.

#### 5. Armchair nanotubes

DFT studies for armchair (10,10) nanotubes show that SW defects with five- and eight-membered rings have a greater ability to adsorb H2 than SW defects with five- and seven-membered rings

SW defects have also been studied in relation to metal-decorated nanotubes and their effects on hydrogen adsorption [46, 47] and also in the study of the interactions with metallic particles and ions [48]. The results obtained for armchair (6,6) nanotubes with SW defects at the B3LYP/ 6-31G(d) level focus on the charge transfer to ions and their relation to the electromigration where the SW defects facilitate the migration of positively charged ions (In<sup>+</sup> and In3+) through

In relation to hydrogen storage as a potential source of clean energy, one requirement is to achieve that at least 5.5 wt% of hydrogen [49] can be released under ambient pressure and temperature conditions. For achieving that goal and to favor a reversible H2 adsorptiondesorption process, it is required that the interaction energy CNT-H2 be of the order of 0.1– 0.5 eV/H2, according to calculations of first principles and to thermodynamic considerations

Hydrogen chemisorption energy is a chirality-dependent exothermic property. Zigzag nitrogen-doped and nondoped nanotubes reveal as the most exothermic followed by the chiral and armchair nanotubes [25]. On the contrary, hydrogen physisorption energies, Eph, exhibit endothermic values. Eph values for saturated, nitrogen-doped armchair CNTs obtained by the DFT [B3LYP/6-31G(d)] methods are within the ideal mentioned range (0.26 eV/H2 for 12 H2 physisorbed into a (4,4) nanotube) and grow for chiral and zigzag nitrogen-doped nanotubes [25] in excellent agreement with results obtained for armchair, chiral, and zigzag CNTs by means of the DFT methods that use the local density approximation [37]. In addition to the chirality, the diameter and length of the nanotube affect the nanotube-H2 interactions. The diameter changes cause the variation in the nanotube curvature affecting the arrangement of the molecular orbitals [4], and the nanotube length variation affects the diffusivity as was determined for drug adsorption [52]. Eph values decrease for nanotubes with a smaller diameter and a larger length (higher number of carbon-atom layers) but increase with the number of adsorbed H2. Bumpy defects and nitrogen doping favor interaction with nanotube-H2. Saturated armchair (4,4) nitrogen-doped nanotubes with bumpy defects and 20 cl of length can encapsulate 15 H2 molecules (C164N4H210 stoichiometry) with a favorable Eph of 0.11 eV/H2 obtained at the B3LYP/6-31G(d) level [13]. Our calculations, considering dispersion forces correction, reveal that saturated nondoped armchair (5,5) nanotubes with bumpy defects and 16 cl of length can encapsulate 44 H2 molecules (stoichiometry C170H278) with a good Eph value of 0.32 eV/H2. Figure 3 shows the optimized structure of this system having a hydrogen

The actual available experimental and theoretical information related to hydrogen adsorption in nanotubes indicates that the structural effects that allow obtaining hydrogen physisorption energies within the range considered ideal for the reversible storage of hydrogen (0.1–0.4 eV/H2 under environmental conditions) are those that favor the nanotube-H2 electrostatic interactions. Parameters such as the chirality of the nanotube, a suitable diameter and length, and the

[43], which is also valid for nanotubes of different chirality [43, 45].

206 Density Functional Calculations - Recent Progresses of Theory and Application

the nanotube [48].

[14, 39, 50, 51].

content of 12 wt%.

Nitrogen-doped armchair nanotubes are expected to be good catalysts for oxygen reduction reactions compared to their chiral and zigzag nanotubes because of their particular distribution of electric charge [12, 25]. In this section, we analyze the effect of different types of defects in the reactivity of armchair structures by using global descriptors based on the DFT methods. Bumpy, haeckelite, Stone-Wales, and zipper defects are considered (see Figure 4). We also analyze the effects of nitrogen doping, the diameter and length of the nanotubes, the number of defects and the charges on the carbon atom adjacent to the two pyrimidine nitrogen atoms.

Figure 4. B3LYP/6-31G(d) fully optimized armchair (5,5) nanotube structures. Front and lateral views showing (a) regular R; (b) bumpy B, with five defects; (c) haeckelite HK, with one defect doped with eight nitrogen atoms; (d) Stone-Wales SW, with one defect; and (e) zipper Z, with four defects, nanotubes with 16 cl of length each.

#### 5.1. Formation energies

Nitrogen-doped armchair nanotubes in all cases show higher values of EF than nondoped nanotubes, indicating a greater reactivity. The EF values are calculated according to Eq. (1). A lower value of EF indicates greater stability. Bumpy nanotubes are the most reactive. Reactivity also increases as the number of defects in the nanotube increases as shown in Table 3 for 16 cl armchair nanotubes (see 3b and 3c). The corresponding stoichiometry is also shown in Table 3. Regular nanotubes with smaller diameter are more reactive than those with larger diameter and exhibit EF values of �0.72 and �1.26 eV, respectively. Defective nanotubes reveal the same trend. For instance, the armchair (5,5) and (6,6) nanotubes, both with one bumpy defect exhibit EF values of �0.67 and �1.18 eV, respectively. The same trend is shown for the armchair (5,5) and (6,6) nanotubes with one SW defect (see 3d and 3h in Table 3).

Nitrogen doping increases the CNTs conductive ability both in the regular and defective structures, revealing smaller bandgap values for the nitrogen-doped nanotubes than for the nondoped ones. Defects also increase the conductive ability of armchair nanotubes. In the nitrogen-doped (5,5) nanotubes, with 16 cl and 20 cl of length, the zipper nanotubes exhibit the best conductive properties (see 4j in Table 4 and Figure 6) with bandgap values of 0.82 and 0.57 eV, respectively. This behavior of the zipper nanotubes was also observed in doped nanotubes of 8 cl as indicated above with a bandgap value of 0.83 eV (see 1a in Table 1). As can be appreciated from the obtained values, increasing the length of the nanotubes increases

Table 4. B3LYP/6-31G(d) bandgap (ELUMO-EHOMO) BG, electronic chemical potential μ, chemical hardness η, and global electrophilicity index ω, in eV, and softness Ѕ, in 1/eV, for a series of nitrogen-doped and nondoped armchair (5,5) regular,

bumpy, Stone-Wales, and zipper nanotubes, all of them having 16 cl of length.

Entry Type BG μ ηω S 4a R-0N 1.13 �3.57 0.56 11.32 0.89 4b R-4N 0.85 �3.31 0.42 12.92 1.18 4c B-1D-0N 1.15 �3.62 0.57 11.39 0.87 4d B-1D-4N 0.76 �3.28 0.38 14.18 1.32 4e B-5D-0N 0.89 �3.72 0.45 15.52 1.12 4f B-5D-4N 0.67 �3.45 0.34 17.71 1.49 4g SW-1D-0N 0.49 �3.54 0.24 25.78 2.06 4h SW-1D-4N 0.83 �3.20 0.41 12.43 1.21 4i Z-0N 1.05 �3.97 0.53 15.00 0.95 4j Z-4N 0.82 �3.72 0.41 16.86 1.22

Table 3. B3LYP/6-31G(d) formation energy EF, in eV, and stoichiometry, for some nitrogen-doped and nondoped armchair (5,5) and (6,6) regular, bumpy (with one and five defects), Stone-Wales (with one defect), and zipper nanotubes, all of

3a (5,5)-R-16 cl �0.72 �0.51 C160H20 C156N4H20 3b (5,5)-B-1D-16 cl �0.67 �0.45 C162H20 C158N4H20 3c (5,5)-B-5D-16 cl �0.31 �0.13 C170H20 C166N4H20 3d (5,5)-SW-1D-16 cl �0.59 �0.38 C160H20 C156N4H20 3e (5,5)-Z-16 cl �0.57 �0.39 C168H20 C164 N4H20 3f (6,6)-R-16 cl �1.26 �1.03 C192H24 C188N4H24 3g (6,6)-B-1D-16 cl �1.18 �0.95 C194H24 C190 N4H24 3h (6,6)-SW-1D-16 cl �1.14 �0.92 C192H24 C188N4H24

Entry Type EF Stoichiometry

0 N 4 N 0 N 4 N

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the nanotube conductive ability.

them having 16 cl of length.

#### 5.2. Bandgap and chemical potential

Three groups of armchair nanotubes were studied: (1) (5,5) with 16 cl of length (see Table 4); (2) (6,6) with 16 cl of length (see Figure 5); and (3) (5,5) with 20 cl of length (see Figure 6) with bandgap values between 0.54 and 1.27 eV.


Table 3. B3LYP/6-31G(d) formation energy EF, in eV, and stoichiometry, for some nitrogen-doped and nondoped armchair (5,5) and (6,6) regular, bumpy (with one and five defects), Stone-Wales (with one defect), and zipper nanotubes, all of them having 16 cl of length.


5.1. Formation energies

5.2. Bandgap and chemical potential

bandgap values between 0.54 and 1.27 eV.

Nitrogen-doped armchair nanotubes in all cases show higher values of EF than nondoped nanotubes, indicating a greater reactivity. The EF values are calculated according to Eq. (1). A lower value of EF indicates greater stability. Bumpy nanotubes are the most reactive. Reactivity also increases as the number of defects in the nanotube increases as shown in Table 3 for 16 cl armchair nanotubes (see 3b and 3c). The corresponding stoichiometry is also shown in Table 3. Regular nanotubes with smaller diameter are more reactive than those with larger diameter and exhibit EF values of �0.72 and �1.26 eV, respectively. Defective nanotubes reveal the same trend. For instance, the armchair (5,5) and (6,6) nanotubes, both with one bumpy defect exhibit EF values of �0.67 and �1.18 eV, respectively. The same trend is shown for the armchair (5,5)

Figure 4. B3LYP/6-31G(d) fully optimized armchair (5,5) nanotube structures. Front and lateral views showing (a) regular R; (b) bumpy B, with five defects; (c) haeckelite HK, with one defect doped with eight nitrogen atoms; (d) Stone-Wales SW,

Three groups of armchair nanotubes were studied: (1) (5,5) with 16 cl of length (see Table 4); (2) (6,6) with 16 cl of length (see Figure 5); and (3) (5,5) with 20 cl of length (see Figure 6) with

and (6,6) nanotubes with one SW defect (see 3d and 3h in Table 3).

with one defect; and (e) zipper Z, with four defects, nanotubes with 16 cl of length each.

208 Density Functional Calculations - Recent Progresses of Theory and Application

Table 4. B3LYP/6-31G(d) bandgap (ELUMO-EHOMO) BG, electronic chemical potential μ, chemical hardness η, and global electrophilicity index ω, in eV, and softness Ѕ, in 1/eV, for a series of nitrogen-doped and nondoped armchair (5,5) regular, bumpy, Stone-Wales, and zipper nanotubes, all of them having 16 cl of length.

Nitrogen doping increases the CNTs conductive ability both in the regular and defective structures, revealing smaller bandgap values for the nitrogen-doped nanotubes than for the nondoped ones. Defects also increase the conductive ability of armchair nanotubes. In the nitrogen-doped (5,5) nanotubes, with 16 cl and 20 cl of length, the zipper nanotubes exhibit the best conductive properties (see 4j in Table 4 and Figure 6) with bandgap values of 0.82 and 0.57 eV, respectively. This behavior of the zipper nanotubes was also observed in doped nanotubes of 8 cl as indicated above with a bandgap value of 0.83 eV (see 1a in Table 1). As can be appreciated from the obtained values, increasing the length of the nanotubes increases the nanotube conductive ability.

Figure 5. B3LYP/6-31G(d) values of (a) bandgap (ELUMO�EHOMO); (b) electronic chemical potential; (c) chemical hardness; and (d) global electrophilicity index, for nitrogen-doped and nondoped armchair (6,6) regular nanotubes, and armchair (6,6) defective nanotubes containing one bumpy defect and one Stone-Wales defect, all of them having 16 cl of length.

with the exception of (5,5) nanotubes with one SW defect (see 4h vs. 4g in Table 4). For the (5,5) nanotubes of 20 cl, the same previous trends are maintained without exceptions (see Figure 6). Nitrogen-doped zipper-defected (5,5) nanotubes of 20 cl are predicted to be the most reactive with a hardness of 0.28 eV. They are more reactive than nitrogen-doped zipper-defected (5,5) nanotubes of 16 cl with a value of hardness of 0.41 eV (see 4j in Table 4). The effect of nitrogen doping, the defects, and the length of the nanotube on the nanotube hardness are consistent with the effect of these same structural parameters on the bandgap. The armchair nanotubes with zipper defects are perceived as the most reactive nanotubes and the ones having the best

Figure 6. B3LYP/6-31G(d) values of (a) bandgap (ELUMO�EHOMO); (b) electronic chemical potential; (c) chemical hardness; and (d) global electrophilicity index, for some nitrogen-doped and nondoped armchair (5,5) regular, bumpy,

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Softness, Ѕ, is the reciprocal of hardness. A larger value of softness indicates greater ability to hold a charge once acquired by the nanotube. For armchair (5,5) nanotubes of 16 cl, the softness varies between 0.87 and 2.06 eV and grows with the defects and with the number of defects. Nitrogen-doped nanotubes have a value of Ѕ greater than nondoped nanotubes. The only

conductive ability enhanced by nitrogen doping and longer nanotubes.

haeckelite, Stone-Wales, and zipper nanotubes, all of them having 20 cl of length.

exception is for nanotubes with one SW defect (see Table 4).

Chemical potential of the studied nanotubes presents small variations, with values ranging from �3.20 to �3.97 eV. Nitrogen doping, slightly increases the chemical potential of regular and defective armchair nanotubes, for the three groups of armchair nanotubes, thereby increasing the electron-donor ability of the doped nanotubes, which is consistent with the nitrogen atom electronic characteristics.

#### 5.3. Hardness and softness

The presence of defects decreases the hardness of the (5,5) nanotubes making them more reactive. Nitrogen doping also decreases the hardness of the (5,5) and (6,6) nanotubes of 16 cl,

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Figure 6. B3LYP/6-31G(d) values of (a) bandgap (ELUMO�EHOMO); (b) electronic chemical potential; (c) chemical hardness; and (d) global electrophilicity index, for some nitrogen-doped and nondoped armchair (5,5) regular, bumpy, haeckelite, Stone-Wales, and zipper nanotubes, all of them having 20 cl of length.

with the exception of (5,5) nanotubes with one SW defect (see 4h vs. 4g in Table 4). For the (5,5) nanotubes of 20 cl, the same previous trends are maintained without exceptions (see Figure 6). Nitrogen-doped zipper-defected (5,5) nanotubes of 20 cl are predicted to be the most reactive with a hardness of 0.28 eV. They are more reactive than nitrogen-doped zipper-defected (5,5) nanotubes of 16 cl with a value of hardness of 0.41 eV (see 4j in Table 4). The effect of nitrogen doping, the defects, and the length of the nanotube on the nanotube hardness are consistent with the effect of these same structural parameters on the bandgap. The armchair nanotubes with zipper defects are perceived as the most reactive nanotubes and the ones having the best conductive ability enhanced by nitrogen doping and longer nanotubes.

Chemical potential of the studied nanotubes presents small variations, with values ranging from �3.20 to �3.97 eV. Nitrogen doping, slightly increases the chemical potential of regular and defective armchair nanotubes, for the three groups of armchair nanotubes, thereby increasing the electron-donor ability of the doped nanotubes, which is consistent with the nitrogen

Figure 5. B3LYP/6-31G(d) values of (a) bandgap (ELUMO�EHOMO); (b) electronic chemical potential; (c) chemical hardness; and (d) global electrophilicity index, for nitrogen-doped and nondoped armchair (6,6) regular nanotubes, and armchair (6,6)

defective nanotubes containing one bumpy defect and one Stone-Wales defect, all of them having 16 cl of length.

210 Density Functional Calculations - Recent Progresses of Theory and Application

The presence of defects decreases the hardness of the (5,5) nanotubes making them more reactive. Nitrogen doping also decreases the hardness of the (5,5) and (6,6) nanotubes of 16 cl,

atom electronic characteristics.

5.3. Hardness and softness

Softness, Ѕ, is the reciprocal of hardness. A larger value of softness indicates greater ability to hold a charge once acquired by the nanotube. For armchair (5,5) nanotubes of 16 cl, the softness varies between 0.87 and 2.06 eV and grows with the defects and with the number of defects. Nitrogen-doped nanotubes have a value of Ѕ greater than nondoped nanotubes. The only exception is for nanotubes with one SW defect (see Table 4).

5.5. Charges

this charge has a value of 0.14 e.

6. Conclusions

cess at ambient conditions.

to contribute to the future design of novel useful materials.

particular nanotube molecular properties is required.

A possible explanation for the reactivity of armchair nanotubes with zipper defects may be the charge that develops at the C2 carbon atom that is adjacent to both pyrimidine nitrogen atoms. It has been found to be a feature that allows nitrogen-doped CNTs to behave as good catalysts [11, 12]. Our calculations reveal that the C2 carbon atom Mulliken charge on the nitrogendoped armchair nanotubes with zipper defects reaches a high value of 0.26 e, compared with 0.01 e for the corresponding nondoped nanotubes. In armchair nanotubes with bumpy defects,

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Our results, obtained using the DFT methods, contribute with a ranking of molecular structures of carbon nanotubes according to: (1) their conductive properties, (2) their hydrogen adsorption properties, and (3) their reactivity to charge transfer reactions and redox reactions. We believe that these results will help in the selection of different nanotube structures to be used in a variety of applications that consider materials such as: (1) new polymer composites having conductive properties (biodevices technology); (2) electronic devices (an electronic tongue); (3) energy storage devices; (4) suitable surface functionalized nanotubes for environmental remediation; (5) organic semiconductor-carbon nanotubes composites for separation technology (water purification); (6) catalyst for oxygen reduction reactions; (7) drug delivery systems, imaging and photothermal therapy (biomedicine); and others where tailoring of

We have performed a systematic study of the reactivity of carbon nanotubes based on various structural parameters by using molecular descriptors obtained by the DFT methods at the B3LYP/6-31G(d) level. Our results confirm that the reactivity of nanotubes depends on their structure. The chirality, the diameter, and the presence of structural, topological, and doping defects significantly modify the reactivity of the nanotubes. The zigzag nanotubes, both the regular (defect free) and those with bumpy defects, reveal greater reactivity, better conductive ability, and better behavior as good electrophiles when compared with corresponding armchair and chiral nanotubes suggesting a high ability to be reduced. Within the armchair nanotubes, we highlight those with zipper defects over those with bumpy, haeckelite, and Stone-Wales defects. The armchair nanotubes with zipper defects are revealed to be more reactive, with high conductive capacity, good ability to acquire electronic density and to behave as good electrophiles, and potential good catalysts in redox reactions. Nitrogen doping reveals as an important parameter that increases the conductive ability and reactivity of armchair nanotubes of different diameter and length. Saturated nanotubes behave as less reactive than unsaturated nanotubes. Saturated armchair nanotubes with bumpy defects exhibit favorable values of the hydrogen physisorption energy suggesting a reversible hydrogen adsorption-desorption pro-

These results are valuable to increase the understanding about carbon nanotubes activity and

Figure 7. B3LYP/6-31G(d) fully optimized structures of nitrogen-doped armchair nanotubes. Front and lateral views showing (a) regular, R; (b) bumpy B, with five defects; (c) haeckelite HK, with two defects; (d) Stone-Wales SW, with two defects; and (e) zipper Z, with five defects, nanotubes with 20 cl of length each.

#### 5.4. Electrophilicity index

The armchair (5,5) nanotubes of 16 cl exhibit an ω value of 11.32 eV. The presence of defects in these nanotubes increases the value of ω. The highest value of ω corresponds to the nanotubes with a SW defect (25.78 eV). Nitrogen doping also increases the ω value for regular and defective (5,5) and (6,6) nanotubes except for nanotubes with one SW defect (see Table 4 and Figure 5). The (5,5) nanotubes of 20 cl show the same trend regarding the presence of defects and the nitrogen doping, being highlighted the nanotubes with zipper defects that show the highest values of ω of 27.72 and 25.94 eV for the nondoped and doped nanotubes, respectively (see Figures 6 and 7). These high ω values show the great tendency of these nanotubes to acquire electronic density and their high ability to be reduced compared to the other studied nanotubes. This property of armchair nanotubes with zipper defects, coupled with their high reactivity and good conductive ability, opens a range of possibilities as advanced materials in the field of electronics or other fields where good electrophiles or catalysts are required in redox reactions.

#### 5.5. Charges

A possible explanation for the reactivity of armchair nanotubes with zipper defects may be the charge that develops at the C2 carbon atom that is adjacent to both pyrimidine nitrogen atoms. It has been found to be a feature that allows nitrogen-doped CNTs to behave as good catalysts [11, 12]. Our calculations reveal that the C2 carbon atom Mulliken charge on the nitrogendoped armchair nanotubes with zipper defects reaches a high value of 0.26 e, compared with 0.01 e for the corresponding nondoped nanotubes. In armchair nanotubes with bumpy defects, this charge has a value of 0.14 e.

Our results, obtained using the DFT methods, contribute with a ranking of molecular structures of carbon nanotubes according to: (1) their conductive properties, (2) their hydrogen adsorption properties, and (3) their reactivity to charge transfer reactions and redox reactions. We believe that these results will help in the selection of different nanotube structures to be used in a variety of applications that consider materials such as: (1) new polymer composites having conductive properties (biodevices technology); (2) electronic devices (an electronic tongue); (3) energy storage devices; (4) suitable surface functionalized nanotubes for environmental remediation; (5) organic semiconductor-carbon nanotubes composites for separation technology (water purification); (6) catalyst for oxygen reduction reactions; (7) drug delivery systems, imaging and photothermal therapy (biomedicine); and others where tailoring of particular nanotube molecular properties is required.

#### 6. Conclusions

5.4. Electrophilicity index

redox reactions.

The armchair (5,5) nanotubes of 16 cl exhibit an ω value of 11.32 eV. The presence of defects in these nanotubes increases the value of ω. The highest value of ω corresponds to the nanotubes with a SW defect (25.78 eV). Nitrogen doping also increases the ω value for regular and defective (5,5) and (6,6) nanotubes except for nanotubes with one SW defect (see Table 4 and Figure 5). The (5,5) nanotubes of 20 cl show the same trend regarding the presence of defects and the nitrogen doping, being highlighted the nanotubes with zipper defects that show the highest values of ω of 27.72 and 25.94 eV for the nondoped and doped nanotubes, respectively (see Figures 6 and 7). These high ω values show the great tendency of these nanotubes to acquire electronic density and their high ability to be reduced compared to the other studied nanotubes. This property of armchair nanotubes with zipper defects, coupled with their high reactivity and good conductive ability, opens a range of possibilities as advanced materials in the field of electronics or other fields where good electrophiles or catalysts are required in

Figure 7. B3LYP/6-31G(d) fully optimized structures of nitrogen-doped armchair nanotubes. Front and lateral views showing (a) regular, R; (b) bumpy B, with five defects; (c) haeckelite HK, with two defects; (d) Stone-Wales SW, with two

defects; and (e) zipper Z, with five defects, nanotubes with 20 cl of length each.

212 Density Functional Calculations - Recent Progresses of Theory and Application

We have performed a systematic study of the reactivity of carbon nanotubes based on various structural parameters by using molecular descriptors obtained by the DFT methods at the B3LYP/6-31G(d) level. Our results confirm that the reactivity of nanotubes depends on their structure. The chirality, the diameter, and the presence of structural, topological, and doping defects significantly modify the reactivity of the nanotubes. The zigzag nanotubes, both the regular (defect free) and those with bumpy defects, reveal greater reactivity, better conductive ability, and better behavior as good electrophiles when compared with corresponding armchair and chiral nanotubes suggesting a high ability to be reduced. Within the armchair nanotubes, we highlight those with zipper defects over those with bumpy, haeckelite, and Stone-Wales defects. The armchair nanotubes with zipper defects are revealed to be more reactive, with high conductive capacity, good ability to acquire electronic density and to behave as good electrophiles, and potential good catalysts in redox reactions. Nitrogen doping reveals as an important parameter that increases the conductive ability and reactivity of armchair nanotubes of different diameter and length. Saturated nanotubes behave as less reactive than unsaturated nanotubes. Saturated armchair nanotubes with bumpy defects exhibit favorable values of the hydrogen physisorption energy suggesting a reversible hydrogen adsorption-desorption process at ambient conditions.

These results are valuable to increase the understanding about carbon nanotubes activity and to contribute to the future design of novel useful materials.

To expand the range of applications of carbon nanotubes, an interesting future contribution would be to construct a nanotube-scale of reactivity, based on the values of some reactivity descriptors and their relation with experimental values for some known reactions. The aim of predicting the reactivity of different structures of nanotubes or to be able to designing nanostructures that possess a desired property to obtain advanced materials in a rational way is open.

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2012;38:389-395. DOI: 10.1016/j.jmgm.2012.05.001 [18] Hypercube Inc. HyperChem Release 7.5. Gainesville; 2003

10.1103/PhysRevB.37.785

of Chemical Physics. 1993;98:5648-5652. DOI: 10.1063/1.464913

2017;12:1815-1825. DOI: 10.2147/IJN.S127349

images-1/Terrones.pdf [Accessed: 26-07-2017]

9,086,523

#### Acknowledgements

The authors thank the support of both the Scientific and Technological Direction and the Technological Development Society of the University of Santiago de Chile, Usach, Projects 061642CF and CIA 2981, and the computational resources at the central cluster of the Faculty of Chemistry and Biology. They also thank Mr. Ignacio Villarroel for his valuable technical assistance.

#### Author details

María Leonor Contreras Fuentes\* and Roberto Rozas Soto

\*Address all correspondence to: leonor.contreras@usach.cl

Computational Chemistry and Intellectual Property Laboratory, Department of Environmental Sciences, Faculty of Chemistry and Biology, University of Santiago de Chile, USACH, Santiago, Chile

#### References


[6] Tour JM, Dyke CA, Flatt AK. Bulk separation of carbon nanotubes by bandgap. 2011. US Patent Nr 7,939,047

To expand the range of applications of carbon nanotubes, an interesting future contribution would be to construct a nanotube-scale of reactivity, based on the values of some reactivity descriptors and their relation with experimental values for some known reactions. The aim of predicting the reactivity of different structures of nanotubes or to be able to designing nanostructures that

The authors thank the support of both the Scientific and Technological Direction and the Technological Development Society of the University of Santiago de Chile, Usach, Projects 061642CF and CIA 2981, and the computational resources at the central cluster of the Faculty of Chemistry and Biology. They also thank Mr. Ignacio Villarroel for his valuable technical

Computational Chemistry and Intellectual Property Laboratory, Department of Environmental Sciences, Faculty of Chemistry and Biology, University of Santiago de Chile, USACH, Santiago,

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[2] Dekker C. Carbon nanotubes as molecular quantum wires. Physics Today. 1999;52(5):

[3] Pastorin G. Crucial functionalizations of carbon nanotubes for improved drug delivery: A valuable option? Pharmaceutical Research. 2009;26(4):746-769. DOI: 10.1007/s11095-

[4] Lu X, Chen Z. Curved pi-conjugation, aromaticity, and the related chemistry of small fullerenes (C60) and single-walled carbon nanotubes. Chemical Reviews. 2005;105(10):

[5] Das D, Rahaman H. Carbon Nanotube and Graphene Nanoribbon Interconnects. New

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possess a desired property to obtain advanced materials in a rational way is open.

María Leonor Contreras Fuentes\* and Roberto Rozas Soto \*Address all correspondence to: leonor.contreras@usach.cl

214 Density Functional Calculations - Recent Progresses of Theory and Application

thalidomide-defects/ [Accessed: 19-07-2017]

22-28. DOI: 0.1063/1.882658

3643-3696. DOI: 10.1021/cr030093d

Acknowledgements

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**Chapter 10**

Provisional chapter

**Elastic Constants and Homogenized Moduli of**

Elastic Constants and Homogenized Moduli

of Monoclinic Structures Based on Density

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72301

**Theory**

Functional Theory

Abstract

homogenized moduli

1. Introduction

Jia Fu

Jia Fu

**Monoclinic Structures Based on Density Functional**

DOI: 10.5772/intechopen.72301

Elastic constants and homogenized properties of two monoclinic structures (gypsum and tobermorite) were investigated by first-principles method. The gypsum (chemical formula of CaSO4•2H2O) is an evaporite mineral and a kind of hydration product of anhydrite. Besides, the 11 Å tobermorite model (chemical formula: Ca4Si6O14(OH)42H2O) as an initial configuration of C-S-H structure is commonly used. Elastic constants are calculated based on density functional theory (DFT), which can also contribute to provide information for investigating the stability, stiffness, brittleness, ductility, and anisotropy of gypsum and tobermorite polycrystals. In addition, based on elastic constants (13 independent constants) of the monoclinic gypsum crystal, the elastic properties of polycrystals are obtained. The bulk modulus B, shear modulus G, Young's modulus E, and Poisson's ration ν are derived. Therefore, it is fairly meaningful to study the elastic constants to understand the physical, chemical, and mechanical properties of two monoclinic structures. Elastic constants can be used as the measure criterion of the resistance of a crystal to an externally applied stress. The calculated parameters are all in excellent agreement with reference.

Keywords: DFT calculation, single crystal, nano scale, elastic constants,

The density functional theory (DFT) is commonly used to study the crystal structure, lattice energy, the equation of state, the electronic bandgap, and vibration spectra properties [1]. Based on the kinetic energy density functional of Thomas [2] and the exchange-correlation effects of Dirac [3], DFT has been greatly developed by Kohn and Sham (KS) [4], who have established the fundamental approximation theorem on the functional status to describe real

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,


Provisional chapter

#### **Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory** Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory

DOI: 10.5772/intechopen.72301

Jia Fu Jia Fu

[50] Li J, Furuta T, Goto H, Ohashi T, Fujiwara Y, Yip S. Theoretical evaluation of hydrogen storage capacity in pure carbon nanostructures. The Journal of Chemical Physics. 2003;

[51] Lochan RC, Head-Gordon M. Computational studies of molecular hydrogen binding affinities: The role of dispersion forces electrostatics and orbital interactions. Physical

[52] Panczyk T, Wolski P, Lajtar L. Coadsorption of doxorubicin and selected dyes on carbon nanotubes. Theoretical investigation of potential application as a pH-controlled drug delivery system. Langmuir. 2016;32:4719-4728. DOI: 10.1021/acs.langmuir.6b00296

Chemistry Chemical Physics. 2006;8:1357-1370. DOI: 10.1039/B515409J

119:2376. DOI: 10.1063/1.1582831

218 Density Functional Calculations - Recent Progresses of Theory and Application

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72301

#### Abstract

Elastic constants and homogenized properties of two monoclinic structures (gypsum and tobermorite) were investigated by first-principles method. The gypsum (chemical formula of CaSO4•2H2O) is an evaporite mineral and a kind of hydration product of anhydrite. Besides, the 11 Å tobermorite model (chemical formula: Ca4Si6O14(OH)42H2O) as an initial configuration of C-S-H structure is commonly used. Elastic constants are calculated based on density functional theory (DFT), which can also contribute to provide information for investigating the stability, stiffness, brittleness, ductility, and anisotropy of gypsum and tobermorite polycrystals. In addition, based on elastic constants (13 independent constants) of the monoclinic gypsum crystal, the elastic properties of polycrystals are obtained. The bulk modulus B, shear modulus G, Young's modulus E, and Poisson's ration ν are derived. Therefore, it is fairly meaningful to study the elastic constants to understand the physical, chemical, and mechanical properties of two monoclinic structures. Elastic constants can be used as the measure criterion of the resistance of a crystal to an externally applied stress. The calculated parameters are all in excellent agreement with reference.

Keywords: DFT calculation, single crystal, nano scale, elastic constants, homogenized moduli

#### 1. Introduction

The density functional theory (DFT) is commonly used to study the crystal structure, lattice energy, the equation of state, the electronic bandgap, and vibration spectra properties [1]. Based on the kinetic energy density functional of Thomas [2] and the exchange-correlation effects of Dirac [3], DFT has been greatly developed by Kohn and Sham (KS) [4], who have established the fundamental approximation theorem on the functional status to describe real

> © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

systems by electronic structure calculations. The eigenvalues of KS equations have no physical meaning, and the ionization energy is in the opposite state direction [5]. Moreover, one proposed approach is to introduce the eigenstates to calculate multi-body (many-body calculation) on the basis of Monte Carlo calculations [6] and perturbation theory [7]. The calculation of elastic constants is preceded by full geometry optimization and the stress tensor calculation of a number of distorted structures at the atomic scale. Polycrystalline structure constituted by a single crystal structure contains a variety of information (e.g., orientation) and the properties of a single crystal, such as anisotropy. Within the mechanics of typical crystals structures, the transition from the micro- to the meso-scale (homogenization) and vice versa (localization) can be estimated. Homogenization is an idealized description of a statistical distribution inside the actual heterogeneous material. Once the continuity model is admitted, the concept of homogeneity is deduced from it [8]. For quasi brittle materials, Zhu et al. [9] have formulated the anisotropic model in the framework of Eshelby-based homogenization methods. X-rays diffraction measurement is one of the stress assay test methods in physics field, of which the stress is actually determined by the strain [10]. Diffraction-based stress analysis depends critically on the use of the correct diffraction elastic constants [11]. X-ray method to test the material stress and to obtain elastic constants [12] is commonly based on the Reuss model [13]. Elasticity of single crystal and mechanical properties of polycrystalline material have been closely integrated. Various calculations methods are compared to determine homogenized moduli of the polycrystalline material composed of a single crystal, for example, the certain stress of Reuss model [13] and the certain strain of Voigt model [14].

2.1. Equation of the theoretical approximate solution

nuclei, and the various interactions between them.

mechanics, and crystallography.

parameters.

problem using appropriate theorems and approximations.

From a microscopic point of view, Schrödinger equation describing a periodic crystal system composed of atomic nuclei n in mutual interaction and electron spin σ<sup>i</sup> is positioned

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory

Hamiltonian, in simple cases, consists of five terms: the kinetic energy of the electrons and

The possible analytical representation and resolution of such a problem become a difficult task due to the limited memory of the computer tools. However, it is possible to reformulate the

The fundamental principle approaches of mean field theory, in particular the DFT, are that any properties of an interacting particle system can be considered as a functional density in the ground state of the system n0(r). Besides, the scalar function of the position n0(r) essentially determines the wave functions of the system at the ground state and the excited states. Electronic and mechanical properties of a periodic crystal refer to solid state physics, quantum

The crystalline ion movement of the electron is as and assumes that the

According to the Born-Oppenheimer or adiabatic approximation [16], the dynamics of the system (electrons and nuclei) is described. The electrons are assumed to react instantly to ionic motion. In electronic coordinates, the nucleus positions are considered as immobile external

where the last term of the Hamiltonian is constant and has been introduced in order to preserve the neutrality of the system and avoid the divergence of the eigenvalues. Clean the ground state of the system for fixed nuclear positions, total energy is given by the formula:

electron mobility (w) does not depend on the speed nuclei but on their positions.

ð1Þ

221

http://dx.doi.org/10.5772/intechopen.72301

ð2Þ

ð3Þ

ð4Þ

ð5Þ

and respectively.

DFT as a first-principles theory and a solid band theory in quantum mechanics has own a great success in linking physical properties and molecular structure, the calculation with exact accuracy but for low computational efficiency for macromolecular structure, which can be used to calculate elastic constants of anisotropic crystals, the monoclinic gypsum, and tobermorite crystals, for example. The chemical formula of gypsum is CaSO4•2H2O, which is an evaporite mineral and a kind of hydration product of anhydrite (chemical formula: CaSO4). Moreover, the 11 Å tobermorite model (chemical formula: Ca4Si6O14(OH)42H2O) as an initial configuration of C-S-H structure is commonly used. Since Young's modulus parameters of gypsum and C-S-H are important to the multi-scale model [15], elastic constants of the gypsum crystal are investigated. The crystal is monoclinic, with 13 independent constants. For the homogenization of elastic deformation, especially for polycrystalline structures, the traditional Reuss-Voigt-Hill method is used to calculate the elastic moduli of monoclinic structures. Based on the ab initio plane-wave pseudopotential density functional theory method mentioned earlier, we focus on the monoclinic crystals to estimate their homogenized elastic moduli.

#### 2. Theoretical calculation by density functional theory (DFT)

Despite the above advantages of DFT, however, the resolution of a system by Kohn-Sham equations involves difficulties due to an infinite number of electrons. These electrons maybe changed under an effective potential generated by an infinite number of cores or ions.

#### 2.1. Equation of the theoretical approximate solution

systems by electronic structure calculations. The eigenvalues of KS equations have no physical meaning, and the ionization energy is in the opposite state direction [5]. Moreover, one proposed approach is to introduce the eigenstates to calculate multi-body (many-body calculation) on the basis of Monte Carlo calculations [6] and perturbation theory [7]. The calculation of elastic constants is preceded by full geometry optimization and the stress tensor calculation of a number of distorted structures at the atomic scale. Polycrystalline structure constituted by a single crystal structure contains a variety of information (e.g., orientation) and the properties of a single crystal, such as anisotropy. Within the mechanics of typical crystals structures, the transition from the micro- to the meso-scale (homogenization) and vice versa (localization) can be estimated. Homogenization is an idealized description of a statistical distribution inside the actual heterogeneous material. Once the continuity model is admitted, the concept of homogeneity is deduced from it [8]. For quasi brittle materials, Zhu et al. [9] have formulated the anisotropic model in the framework of Eshelby-based homogenization methods. X-rays diffraction measurement is one of the stress assay test methods in physics field, of which the stress is actually determined by the strain [10]. Diffraction-based stress analysis depends critically on the use of the correct diffraction elastic constants [11]. X-ray method to test the material stress and to obtain elastic constants [12] is commonly based on the Reuss model [13]. Elasticity of single crystal and mechanical properties of polycrystalline material have been closely integrated. Various calculations methods are compared to determine homogenized moduli of the polycrystalline material composed of a single crystal, for example, the certain stress of Reuss

DFT as a first-principles theory and a solid band theory in quantum mechanics has own a great success in linking physical properties and molecular structure, the calculation with exact accuracy but for low computational efficiency for macromolecular structure, which can be used to calculate elastic constants of anisotropic crystals, the monoclinic gypsum, and tobermorite crystals, for example. The chemical formula of gypsum is CaSO4•2H2O, which is an evaporite mineral and a kind of hydration product of anhydrite (chemical formula: CaSO4). Moreover, the 11 Å tobermorite model (chemical formula: Ca4Si6O14(OH)42H2O) as an initial configuration of C-S-H structure is commonly used. Since Young's modulus parameters of gypsum and C-S-H are important to the multi-scale model [15], elastic constants of the gypsum crystal are investigated. The crystal is monoclinic, with 13 independent constants. For the homogenization of elastic deformation, especially for polycrystalline structures, the traditional Reuss-Voigt-Hill method is used to calculate the elastic moduli of monoclinic structures. Based on the ab initio plane-wave pseudopotential density functional theory method mentioned earlier, we focus on the monoclinic crystals to estimate their homogenized elastic moduli.

2. Theoretical calculation by density functional theory (DFT)

Despite the above advantages of DFT, however, the resolution of a system by Kohn-Sham equations involves difficulties due to an infinite number of electrons. These electrons maybe

changed under an effective potential generated by an infinite number of cores or ions.

model [13] and the certain strain of Voigt model [14].

220 Density Functional Calculations - Recent Progresses of Theory and Application

From a microscopic point of view, Schrödinger equation describing a periodic crystal system composed of atomic nuclei n in mutual interaction and electron spin σ<sup>i</sup> is positioned and respectively.

$$
\overrightarrow{\phantom{\rule{1em}1em}} \begin{pmatrix} \overrightarrow{\vec{R}}, \overrightarrow{\phantom{\rule{1em}1em}} \end{pmatrix} \quad \overrightarrow{\phantom{\rule{1em}1em}} \begin{pmatrix} \overrightarrow{\phantom{\phantom{\frac{\overrightarrow{\phantom{\frac{\overrightarrow{\phantom{\frac{\overrightarrow{\phantom{\frac{\overrightarrow{\phantom{\overline{\boldsymbol{\alpha}}}}}}}}}}}}} \end{\overrightarrow{\phantom{\cdot}}}} \begin{pmatrix} \overrightarrow{\phantom{\frac{\overrightarrow{\phantom{\cdot}}{\cdot}}} \end{pmatrix}} \end{pmatrix}} \tag{1}
$$

Hamiltonian, in simple cases, consists of five terms: the kinetic energy of the electrons and nuclei, and the various interactions between them.

$$H = T\_{\text{v}} \upharpoonright T\_{\text{v}} \upharpoonright U\_{\text{swv}} \upharpoonright U\_{\text{sw}} \upharpoonright U\_{\text{sw}} \tag{2}$$

The possible analytical representation and resolution of such a problem become a difficult task due to the limited memory of the computer tools. However, it is possible to reformulate the problem using appropriate theorems and approximations.

The fundamental principle approaches of mean field theory, in particular the DFT, are that any properties of an interacting particle system can be considered as a functional density in the ground state of the system n0(r). Besides, the scalar function of the position n0(r) essentially determines the wave functions of the system at the ground state and the excited states. Electronic and mechanical properties of a periodic crystal refer to solid state physics, quantum mechanics, and crystallography.

The crystalline ion movement of the electron is as and assumes that the electron mobility (w) does not depend on the speed nuclei but on their positions.

According to the Born-Oppenheimer or adiabatic approximation [16], the dynamics of the system (electrons and nuclei) is described. The electrons are assumed to react instantly to ionic motion. In electronic coordinates, the nucleus positions are considered as immobile external parameters.

$$
\stackrel{\wedge}{H} = \stackrel{\wedge}{T}\_{\mathcal{C}} + \stackrel{\wedge}{U}\_{\text{new}} + \stackrel{\wedge}{U}\_{\text{new}} + \stackrel{\wedge}{U}\_{\text{new}} \tag{3}
$$

$$
\hat{\boldsymbol{\Pi}}^{\hat{\boldsymbol{\Gamma}}} \boldsymbol{\phi}\_{\boldsymbol{\mathcal{A}}}^{\boldsymbol{\phi}} \left( \overleftarrow{\boldsymbol{r}} \right) - \mathbb{L}^{\boldsymbol{\pi} \circ \boldsymbol{\gamma}} \left( \overleftarrow{\boldsymbol{\mathcal{A}}} \right) \boldsymbol{\phi}\_{\boldsymbol{\mathcal{A}}}^{\boldsymbol{\alpha}} \left( \overleftarrow{\boldsymbol{r}} \right) \left( \overleftarrow{\boldsymbol{r}} \right) \tag{4}
$$

where the last term of the Hamiltonian is constant and has been introduced in order to preserve the neutrality of the system and avoid the divergence of the eigenvalues. Clean the ground state of the system for fixed nuclear positions, total energy is given by the formula:

$$E^{\mathcal{B}O}\left(\overrightarrow{R}\right)\left\langle\boldsymbol{\varphi}\_{\widetilde{\boldsymbol{R}}}^{0}\middle|\hat{\boldsymbol{H}}\middle|\boldsymbol{\varphi}\_{\widetilde{\boldsymbol{R}}}^{0}\right\rangle=\min\left\langle\boldsymbol{\varphi}\_{\widetilde{\boldsymbol{R}}}^{0}\middle|\hat{\boldsymbol{H}}\middle|\boldsymbol{\varphi}\_{\widetilde{\boldsymbol{R}}}^{0}\right\rangle\tag{5}$$

This energy has a surface in the space coordinates that is said to be ionic Born-Oppenheimer surface. The ions move according to the effective potential energy, including Coulomb repulsion and the anchoring effect of the electron, which are as follows:

$$
\stackrel{\circ}{H}^{\mathfrak{so}} = \stackrel{\circ}{T} + \mathbb{E}^{\mathfrak{so}} \left( \stackrel{\circ}{\mathsf{K}} \right) \tag{6}
$$

2.2. The approximation approach of Kohn-Sham

self-defined electron density.

the variational equation:

for Schrödinger equations:

ε<sup>i</sup> represents the eigenvalues, and H

δEKS δw<sup>∗</sup>

With the constraint of orthonormalization w<sup>i</sup> w<sup>j</sup>

<sup>i</sup> ð Þ<sup>r</sup> <sup>¼</sup> <sup>δ</sup>TS δw<sup>∗</sup> <sup>i</sup> ð Þr þ

The approach of Kohn-Sham system substitutes the interacting particles, which obeys the Hamiltonian in Eq. (3), by a less complex system easily solved. This approach assumes that the density in the ground state of the system is equal to that in some systems composed of non-interacting particles. This involves independent particle equations for the non-interacting system, gathering all the terms complicated and difficult to assess, in a functional exchange-correlation EXCð Þ n .

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory

T is the kinetic energy of a system of particles (electrons) independently (non-interacting)

The Hartree energy or energy of interaction is associated with the Coulomb interaction of the

n rð Þ¼ <sup>X</sup> Ne

δEex <sup>δ</sup>n rð Þ <sup>þ</sup>

H ∧ KS � ε<sup>i</sup>

KSð Þ¼� r

VKSð Þ¼ r Vexð Þþ r

∧

H ∧

i¼1

Solving the auxiliary Kohn and Sham system for the ground state can be seen as a minimization problem while respecting the density n(r). Apart from orbital function TS, all other terms depend on the density. Therefore, it is possible to vary the functions of the wave and to derive

> � � � E ¼ δi,j

1 2

D

w<sup>i</sup> j j ð Þr

δEHartree <sup>δ</sup>n rð Þ <sup>þ</sup>

<sup>δ</sup>n rð Þ � � <sup>δ</sup>n rð Þ

KS is the effective Hamiltonian H,

δEHartree <sup>δ</sup>n rð Þ <sup>þ</sup>

δExc

δw<sup>∗</sup>

� �wið Þ¼ <sup>r</sup> <sup>0</sup> (16)

δExc

<sup>∇</sup><sup>2</sup> <sup>þ</sup> VKSð Þ<sup>r</sup> (17)

<sup>δ</sup>n rð Þ (18)

embedded in an effective potential which is no other than the real system,

ð11Þ

223

ð12Þ

ð13Þ

<sup>2</sup> (14)

http://dx.doi.org/10.5772/intechopen.72301

<sup>i</sup> ð Þ<sup>r</sup> <sup>¼</sup> <sup>0</sup> (15)

, this implies the form of Kohn-Sham

$$
\hat{\vec{\mathcal{H}}}^{\text{SO}} \propto \left(\vec{\mathcal{R}}\right) - \mathbb{E} \mathcal{X}\left(\vec{\mathcal{R}}\right) \tag{7}
$$

The dissociation degrees of freedom of electrons from those of nucleons, obtained through the adiabatic approximation, are very important, because if the electrons must be treated by quantum mechanics, degrees of freedom of ions in most cases are processed in a conventional manner.

This theorem/approach of Hohenberg and Kohn tries to make an exact DFT theory for manybody systems. This formulation applies to any system of mutually interacting particles in an external potential , where the Hamiltonian is written as.

$$\hat{H} = -\frac{\hbar}{2m\_e} \sum\_{\star} \nabla\_{\star}^{\star} - \sum\_{\star} \Gamma\_{\star\star}^{\star}(r\_{\star}) - \frac{1}{2} \sum\_{\star \star} \frac{\mathbf{c}^{\uparrow}}{|r\_{\star} - r\_{\star}|} \tag{8}$$

DFT and its founding principle are summarized in two theorems, first introduced by Hohenberg and Kohn [17], which refer to the set of potential and the density minimizing of Eq. (5).

The total energy of the ground state of a system for interacting electrons is functional (unknown) of the single electron density

$$E\_{\rm HK}[n] = T[n] + E\_{\rm int}[n] + \int d^3r V\_{\rm ex}(r) + E\_{\rm nn}\left(\overrightarrow{R}\right) \cdot F\_{\rm HK}[n] + \int d^3r V\_{\rm ex}(r) + E\_{\rm nn}\left(\overrightarrow{R}\right) \tag{9}$$

As a result, the density n0(r) minimizing the energy associated with the Hamiltonian (9) is obtained and used to evaluate the energy of the ground state of the system.

The principle established in the second theorem of Hohenberg and Kohn specifies that the density that minimizes the energy is the energy of the ground state

$$E^{\rm BO} \left[ \overrightarrow{R} \right] = \min E \left( \overrightarrow{R}, n \left( \overrightarrow{r} \right) \right) \tag{10}$$

Because the ground state is concerned, it is possible to replace the wave system function by the electron charge density, which therefore becomes the fundamental quantity of the problem. In principle, the problem boils down to minimize the total energy of the system in accordance with the variations in the density governed by the constraint on the number of particles . In this stage, the DFT can reformulate the problem rather than solve an uncertain functional FHK(n).

#### 2.2. The approximation approach of Kohn-Sham

This energy has a surface in the space coordinates that is said to be ionic Born-Oppenheimer surface. The ions move according to the effective potential energy, including Coulomb repul-

The dissociation degrees of freedom of electrons from those of nucleons, obtained through the adiabatic approximation, are very important, because if the electrons must be treated by quantum mechanics, degrees of freedom of ions in most cases are processed in a conventional manner. This theorem/approach of Hohenberg and Kohn tries to make an exact DFT theory for manybody systems. This formulation applies to any system of mutually interacting particles in an

DFT and its founding principle are summarized in two theorems, first introduced by Hohenberg and Kohn [17], which refer to the set of potential and the density minimizing

The total energy of the ground state of a system for interacting electrons is functional

As a result, the density n0(r) minimizing the energy associated with the Hamiltonian (9) is

The principle established in the second theorem of Hohenberg and Kohn specifies that the

<sup>¼</sup> min E R!

Because the ground state is concerned, it is possible to replace the wave system function by the electron charge density, which therefore becomes the fundamental quantity of the problem. In principle, the problem boils down to minimize the total energy of the system in accordance with the variations in the density governed by the constraint on the number of particles

; n r! � � � �

. In this stage, the DFT can reformulate the problem rather than solve an uncertain

� �!

� FHK½ �þ n

ð d3

rVexð Þþ r Enn R

� �!

rVexð Þþ r Enn R

ð6Þ

ð7Þ

ð8Þ

(9)

(10)

sion and the anchoring effect of the electron, which are as follows:

222 Density Functional Calculations - Recent Progresses of Theory and Application

external potential , where the Hamiltonian is written as.

ð d3

density that minimizes the energy is the energy of the ground state

obtained and used to evaluate the energy of the ground state of the system.

EBO R h i !

(unknown) of the single electron density

EHK½ �¼ n T n½ �þ Eint½ �þ n

of Eq. (5).

functional FHK(n).

The approach of Kohn-Sham system substitutes the interacting particles, which obeys the Hamiltonian in Eq. (3), by a less complex system easily solved. This approach assumes that the density in the ground state of the system is equal to that in some systems composed of non-interacting particles. This involves independent particle equations for the non-interacting system, gathering all the terms complicated and difficult to assess, in a functional exchange-correlation EXCð Þ n .

$$E\_{\rm KS} = F[n] + \int d^3r V\_{\rm ex}\left(r\right) = T\_{\rm S}[n] + E\_H\left[n\right] + E\_{\rm JC}\left[n\right] + \int d^3r V\_{\rm ex}\left(r\right) \tag{11}$$

T is the kinetic energy of a system of particles (electrons) independently (non-interacting) embedded in an effective potential which is no other than the real system,

$$T\_S \{ \nu \} = \left\langle \nu\_M \left| \hat{T}\_e \right| \nu\_N \right\rangle = \sum\_{i=1}^{N\_e} \left\langle \varphi\_i \left| -\frac{1}{2} \nabla^2 \right| \varphi\_i \right\rangle \tag{12}$$

The Hartree energy or energy of interaction is associated with the Coulomb interaction of the self-defined electron density.

$$E\_{Hottree}[n] = \frac{1}{2} \int d^3r d^3r' \frac{n(r)n(r')}{|r-r|} \tag{13}$$

$$m(r) = \sum\_{i=1}^{N\_\varepsilon} |\varphi\_i(r)|^2 \tag{14}$$

Solving the auxiliary Kohn and Sham system for the ground state can be seen as a minimization problem while respecting the density n(r). Apart from orbital function TS, all other terms depend on the density. Therefore, it is possible to vary the functions of the wave and to derive the variational equation:

$$\frac{\delta E\_{\rm KS}}{\delta \varphi\_i^\*(r)} = \frac{\delta T\_{\rm S}}{\delta \varphi\_i^\*(r)} + \left[ \frac{\delta E\_{\rm ex}}{\delta n(r)} + \frac{\delta E\_{\rm Hartree}}{\delta n(r)} + \frac{\delta E\_{\rm xc}}{\delta n(r)} \right] \frac{\delta n(r)}{\delta \varphi\_i^\*(r)} = 0 \tag{15}$$

With the constraint of orthonormalization w<sup>i</sup> w<sup>j</sup> � � � E ¼ δi,j D , this implies the form of Kohn-Sham for Schrödinger equations:

$$\left(\overset{\wedge}{H}\_{\text{KS}} - \varepsilon\_i\right)\varphi\_i(r) = 0 \tag{16}$$

ε<sup>i</sup> represents the eigenvalues, and H ∧ KS is the effective Hamiltonian H,

$$
\stackrel{\wedge}{H}\_{\rm KS}(r) = -\frac{1}{2}\nabla^2 + V\_{\rm KS}(r) \tag{17}
$$

$$V\_{KS}(r) = V\_{ex}(r) + \frac{\delta E\_{Hatter}}{\delta n(r)} + \frac{\delta E\_{xc}}{\delta n(r)}\tag{18}$$

Eqs. (16)–(18) are known equations of Kohn-Sham, the density n(r) and the resulting total energy EKS. These equations are independent of any approximation on the functional EXC(n), resolution provides the exact values of the density and the energy of the ground state of the interacting system, provided that EXC(n) is exactly known. The latter can be described in terms of Hohenberg Kohn function in Eq. (8)

$$E\_{\rm xc}[\mathfrak{n}] = F\_{\rm HK}[\mathfrak{n}] - \left(T\_S[\mathfrak{n}] + E\_{\rm Hartree}[\mathfrak{n}]\right) \tag{19}$$

This notion of GGA is the choice of functions, which allows us a better adaptation to wide variations so as to maintain the desired properties. The energy is written in its general

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory

r ¼ ð

forms of FXC, the most used are those introduced by Becke [22] and Perdew [23, 24].

Different states of the Schrödinger equation for an independent particle in a system. By Kohn

The effective potential has the periodicity of the crystal and may be expressed using Fourier

<sup>∇</sup><sup>2</sup> <sup>þ</sup> Veffð Þ<sup>r</sup> � �ψ<sup>i</sup>

n rð Þεbom

<sup>x</sup> is the exchange energy of an unpolarized density n(r) system. There are many

<sup>x</sup> ð Þ <sup>n</sup> Fxc½ � <sup>n</sup>; j j <sup>∇</sup><sup>n</sup> ;… <sup>d</sup><sup>3</sup>

http://dx.doi.org/10.5772/intechopen.72301

ð Þ¼ r εiψ<sup>i</sup>

Veffð Þ Gm exp ð Þ iGmr (25)

Veffð Þr exp ð Þ �iGmr dr (26)

r (23)

225

ð Þr (24)

(27)

form [21]:

where εbom

series, in a periodic system:

periodic function:

Gm is the reciprocal lattice vector:

where Ωsell is the volume of the original mesh.

EGGA xc ½ �¼ n ð

2.3. Parameters of Bloch theorem and Brillouin zone

H ∧ effð Þr ψ<sup>i</sup>

and Sham equations, responding to eigenvalue equation is as:

where the electrons are immersed in an effective potential .

ð Þ¼ � r

Veffð Þ¼ <sup>r</sup> <sup>X</sup>

Veffð Þ¼ <sup>G</sup> <sup>1</sup>

primitive cells in each direction (Ni ! ∞ in the case of perfect crystal).

m

Ωsell ð

Ωsell

As the translational symmetry, it is that states are orthogonal and conditioned by the limits of the crystal (infinite volume). In this case, the Eigen functions of KS are governed by the Bloch theorem: they have two quantum numbers: the wave vector k in the Brillouin zone (BZ) and the band index i, and this can be expressed by a product of a plane wave exp.(ik, r) and a

> ψi, <sup>k</sup>ð Þ¼ r exp ð Þ ik;r ui, <sup>k</sup>ð Þr ui, <sup>k</sup>ð Þ¼ r þ R ui, <sup>k</sup>ð Þr <sup>R</sup> <sup>¼</sup> <sup>P</sup>niai, ni <sup>¼</sup> <sup>1</sup>…Ni

where R is the vector of direct space defined by ai with i∈ f g 1; 2; 3 and Ni is the number of

ℏ2 2m

n rð Þεxc½ � <sup>n</sup>; j j <sup>∇</sup><sup>n</sup> ;… <sup>d</sup><sup>3</sup>

or more precisely,

$$E\_{\rm xc}[n] = \left\langle \stackrel{\wedge}{T} \right\rangle - T\_S[n] + \left\langle \stackrel{\wedge}{V}\_{\rm int} \right\rangle - E\_{\rm Hartree}[n] \tag{20}$$

This energy is related to potential exchange-correlation Vxc <sup>¼</sup> <sup>∂</sup>Exc <sup>∂</sup>n rð Þ.

For the exchange-correlation functional, the only ambiguity in the approach of Kohn and Sham (KS) is the exchange-correlation term. It is subject to functional approximations of local or near local order of density that said energy EKS can be written as

$$E\_{\rm xc}[n] = \int n(r)\varepsilon\_{\rm xc}([n], r)d^3r\tag{21}$$

where εxc([n], r) is the exchange-correlation energy per electron at point r, it depends on n(r) in the vicinity of r. These approximations have made enormous progress in the field.

1. The approximation of the local density (LDA)

The use of the local density approximation (LDA) in which the exchange-correlation energy ELDA xc ½ � n is another integral over all space, assuming that εxc([n], r) is the exchangecorrelation energy per particle of a homogeneous electron gas of density n

$$E\_{\rm xc}^{LDA}[n] = \int n(r) \varepsilon\_{\rm xc}^{hom}[n(r)] d^3r = \int n(r) \left\{ \varepsilon\_{\rm x}^{hom}[n(r)] + \varepsilon\_{\rm c}^{hom}[n(r)] \right\} d^3r \tag{22}$$

The exchange term Ebom <sup>x</sup> ½ � n rð Þ can be expressed analytically, while the correlation term is computed accurately using the Monte Carlo by Ceperley Alder [18] and then set in different shapes [19].

This approximation has been particularly checked to deal with non-homogeneous systems.

2. The generalized gradient approximation (GGA)

The generalized gradient approximation (GGA) involves the local density approximation providing a substantial improvement and better adaptation to the systems. This approximation is equal to the exchange-correlation term only as a function of the density. A first approach (GEA) was introduced by Kohn and Sham then used by the authors of Herman et al. [20].

This notion of GGA is the choice of functions, which allows us a better adaptation to wide variations so as to maintain the desired properties. The energy is written in its general form [21]:

$$E\_{\rm xc}^{\rm GGA}[n] = \int n(r)\varepsilon\_{\rm xc}[n, |\nabla\_n|, \dots] d^3r = \int n(r)\varepsilon\_{\rm x}^{\rm bym}(n)F\_{\rm xc}[n, |\nabla\_n|, \dots] d^3r \tag{23}$$

where εbom <sup>x</sup> is the exchange energy of an unpolarized density n(r) system. There are many forms of FXC, the most used are those introduced by Becke [22] and Perdew [23, 24].

#### 2.3. Parameters of Bloch theorem and Brillouin zone

Different states of the Schrödinger equation for an independent particle in a system. By Kohn and Sham equations, responding to eigenvalue equation is as:

$$\overset{\triangle}{H}\_{\ell\overline{\mathcal{H}}}(r)\psi\_i(r) = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V\_{\ell\overline{\mathcal{H}}}(r) \right] \psi\_i(r) = \varepsilon\_i \psi\_i(r) \tag{24}$$

where the electrons are immersed in an effective potential .

The effective potential has the periodicity of the crystal and may be expressed using Fourier series, in a periodic system:

$$V\_{\rm eff}(r) = \sum\_{m} V\_{\rm eff}(G\_m) \exp\left(iG\_m r\right) \tag{25}$$

Gm is the reciprocal lattice vector:

Eqs. (16)–(18) are known equations of Kohn-Sham, the density n(r) and the resulting total energy EKS. These equations are independent of any approximation on the functional EXC(n), resolution provides the exact values of the density and the energy of the ground state of the interacting system, provided that EXC(n) is exactly known. The latter can be described in terms

� TS½ �þ n Vint

For the exchange-correlation functional, the only ambiguity in the approach of Kohn and Sham (KS) is the exchange-correlation term. It is subject to functional approximations of local or near

where εxc([n], r) is the exchange-correlation energy per electron at point r, it depends on n(r) in

The use of the local density approximation (LDA) in which the exchange-correlation

computed accurately using the Monte Carlo by Ceperley Alder [18] and then set in

This approximation has been particularly checked to deal with non-homogeneous systems.

The generalized gradient approximation (GGA) involves the local density approximation providing a substantial improvement and better adaptation to the systems. This approximation is equal to the exchange-correlation term only as a function of the density. A first approach (GEA) was introduced by Kohn and Sham then used by the authors of Herman

xc ½ � n is another integral over all space, assuming that εxc([n], r) is the exchange-

n rð Þ <sup>ε</sup>bom

<sup>x</sup> ½ �þ n rð Þ <sup>ε</sup>bom <sup>c</sup> ½ � n rð Þ � �d<sup>3</sup>

<sup>x</sup> ½ � n rð Þ can be expressed analytically, while the correlation term is

n rð Þεxcð Þ ½ � <sup>n</sup> ;<sup>r</sup> <sup>d</sup><sup>3</sup>

<sup>∧</sup> � �

Exc½ �¼ n FHK½ �� n ð Þ TS½ �þ n EHartree½ � n (19)

<sup>∂</sup>n rð Þ.

� EHartree½ � n (20)

r (21)

r (22)

of Hohenberg Kohn function in Eq. (8)

Exc½ �¼ n T

224 Density Functional Calculations - Recent Progresses of Theory and Application

This energy is related to potential exchange-correlation Vxc <sup>¼</sup> <sup>∂</sup>Exc

local order of density that said energy EKS can be written as

1. The approximation of the local density (LDA)

ð

2. The generalized gradient approximation (GGA)

n rð Þεbom

<sup>∧</sup> � �

Exc½ �¼ n

ð

the vicinity of r. These approximations have made enormous progress in the field.

correlation energy per particle of a homogeneous electron gas of density n

r ¼ ð

xc ½ � n rð Þ <sup>d</sup><sup>3</sup>

or more precisely,

energy ELDA

ELDA xc ½ �¼ n

The exchange term Ebom

different shapes [19].

et al. [20].

$$V\_{\rm eff}(G) = \frac{1}{\Omega\_{\rm self}} \int\_{\Omega\_{\rm self}} V\_{\rm eff}(r) \exp\left(-i\mathcal{G}\_{m}r\right) dr\tag{26}$$

where Ωsell is the volume of the original mesh.

As the translational symmetry, it is that states are orthogonal and conditioned by the limits of the crystal (infinite volume). In this case, the Eigen functions of KS are governed by the Bloch theorem: they have two quantum numbers: the wave vector k in the Brillouin zone (BZ) and the band index i, and this can be expressed by a product of a plane wave exp.(ik, r) and a periodic function:

$$\begin{aligned} \psi\_{i,k}(r) &= \exp\left(ik, r\right) u\_{i,k}(r) \\ u\_{i,k}(r+\mathbb{R}) &= u\_{i,k}(r) \\ \mathbb{R} &= \sum n\_{i}a\_{i\prime}n\_{i} = 1...N\_{i} \end{aligned} \tag{27}$$

where R is the vector of direct space defined by ai with i∈ f g 1; 2; 3 and Ni is the number of primitive cells in each direction (Ni ! ∞ in the case of perfect crystal).

Solving Eq. (24) is equivalent to increase the periodic function ui, <sup>k</sup>ð Þr , in a database-dependent functions points k : f<sup>k</sup> <sup>j</sup>ð Þr j g j ¼ 1…Nbasð Þk n :

$$\mu\_{i,k}(r) = \sum\_{j} \mathbb{C}\_{i}^{j} \phi\_{j}^{k}(r) \tag{28}$$

where

dictions.

aP,T <sup>x</sup> aP,T

cP,T <sup>x</sup> cP,T

bP,T <sup>x</sup> <sup>b</sup>P,T

<sup>y</sup> aP,T <sup>z</sup>

<sup>F</sup> <sup>¼</sup> <sup>E</sup> <sup>þ</sup><sup>X</sup>

i Fth

coefficients of the polynomial are the elastic coefficient:

� � <sup>¼</sup> <sup>r</sup>0<sup>F</sup> <sup>η</sup>ij; <sup>T</sup>

� � <sup>þ</sup>

The components of the stress tensor can be extracted by <sup>σ</sup><sup>i</sup> <sup>¼</sup> <sup>P</sup>

the total energy variation of the system can be expressed as

Taylor expansion of Helmholtz free energy with the strain,

where ηij,ηkl, and ηmn are the coefficients of Lagrange deformation tensor, c<sup>T</sup>

<sup>Δ</sup><sup>E</sup> <sup>¼</sup> <sup>V</sup> 2 X 6

c T ijkl ¼ r<sup>0</sup>

i¼1

The second-order elastic coefficients can be obtained by the coefficient of the second-order

∂2 F ηij; T � �

X 6

j¼1

∂ηij∂ηkl

1 2 X ijkl c T

r0F ηij; T

first-order elastic coefficients, and F ηij; T

2 6 4

3.1. Calculation of elastic constants for single crystal structure

α1ð Þ P; T α6ð Þ P; T α5ð Þ P; T α6ð Þ P; T α2ð Þ P; T α4ð Þ P; T α5ð Þ P; T α4ð Þ P; T α3ð Þ P; T

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory

configuration tensor and the deformation tensor at temperature T (K) under the pressure P (GPa).

A multi-particle electronic structure satisfies the Schrödinger equation. As in [29], Kohn-Sham equation as an approximation to simplify Schrödinger equation is described. For crystal com-

> 1 2

Helmholtz free energy can be calculated for all the thermodynamic quantities. DFT-QHA (quasi-harmonic approximation) is a precise calculation method to calculate thermodynamic properties of solid materials elastic constants and Debye temperature with the accurate pre-

According to the theory of elasticity, under the isothermal strain, the elastic modulus of Helmholtz free energy can be described by the form of the Taylor expansion, of which the

ℏwi þ kBT

ijklηijηkl þ … þ

� � is the Helmholtz free energy.

X i

In 1 � e

1 n! X ijkl… c T

6 j¼1 cijε<sup>j</sup> �ℏwi kBT

posed by vibrator with the vibration frequency wi, the total Helmholtz free energy is

i

<sup>i</sup> <sup>¼</sup> <sup>U</sup> <sup>þ</sup><sup>X</sup>

3 7

<sup>5</sup> separately represent the cell

227

http://dx.doi.org/10.5772/intechopen.72301

� � (32)

ijkl…ηijηkl… (33)

, after the applied strain,

cijeiej (34)

ijklis the isothermal

(35)

<sup>y</sup> bP,T z

<sup>y</sup> cP,T <sup>z</sup>

where f<sup>k</sup> <sup>j</sup> is the wave function developed in a space of infinite dimensions; this means that j should be in principle infinite. But, in practice, we work with a limited set of basic functions, which imply that the description of f<sup>k</sup> <sup>j</sup> will approximate. That the selected database simply solves the system:

$$\begin{aligned} \sum\_{m'} H\_{m,m'}(k) \mathbf{C}\_{i,m'}(k) &= \varepsilon\_i(k) \mathbf{C}\_{i,m}(k) \\ H\_{m,m'}(k) &= \left\langle \boldsymbol{\varphi}\_{m,k}^{j} \middle| \hat{H}\_{\text{eff}} \middle| \boldsymbol{\varphi}\_{m,k}^{j} \right\rangle \end{aligned} \tag{29}$$

where each point is a set of k eigenstates, the label having i = 1, 2, … obtained by diagonalization of the Hamiltonian in Eq. (29).

It is necessary to integrate the points k in the Brillouin zone. For a function fi(k) where i defines the band index, the average value is

$$\overline{f\_i} = \frac{1}{N\_k} \sum\_k f\_i(k) \to \frac{\Omega\_{\text{cell}}}{(2\pi)^d} \int\_{\text{BZ}} f\_i(k) dk \tag{30}$$

Ωcell is the cell volume of the original mesh in the real space and 2ð Þ π d =Ωcell of the cell volume of the Brillouin zone are determined using a sampling points k. Several election procedures exist for these points. Particularly those of Baldereschi [25], Chadi and Kohen [26], and Monkhorst and Pack [27] are the most frequently used.

#### 3. Elastic constants and homogenized moduli of monoclinic structure

According to the crystal theory [28], any crystal lattice system contains six independent variables, namely the cell side length a, b, and c; unit cell angle α, β, and γ. Generally, the crystal under a certain deformation, temperature, and pressure can be described by the corresponding six-dimensional deformation tensor. The temperature and pressure will cause cell-deformed configuration tensor as

$$\mathbf{X}^{P,T} = \begin{bmatrix} a\_x^{p,T} & a\_y^{p,T} & a\_z^{p,T} \\ b\_x^{p,T} & b\_y^{p,T} & b\_z^{p,T} \\ c\_x^{p,T} & c\_y^{p,T} & c\_z^{p,T} \end{bmatrix} = \begin{bmatrix} a\_x^{0,0} & a\_y^{0,0} & a\_z^{0,0} \\ b\_x^{0,0} & b\_y^{0,0} & b\_z^{0,0} \\ c\_x^{0,0} & c\_y^{0,0} & c\_z^{0,0} \end{bmatrix} \left( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} a\_1(P,T) & a\_6(P,T) & a\_7(P,T) \\ a\_6(P,T) & a\_2(P,T) & a\_4(P,T) \\ a\_5(P,T) & a\_4(P,T) & a\_3(P,T) \end{bmatrix} \right) \tag{31}$$

$$\text{where}\begin{bmatrix}a\_{\mathbf{x}}^{p,T} & a\_{\mathbf{y}}^{p,T} & a\_{\mathbf{z}}^{p,T} \\ b\_{\mathbf{x}}^{p,T} & b\_{\mathbf{y}}^{p,T} & b\_{\mathbf{z}}^{p,T} \\ c\_{\mathbf{x}}^{p,T} & c\_{\mathbf{y}}^{p,T} & c\_{\mathbf{z}}^{p,T} \end{bmatrix} \text{and}\begin{bmatrix}a\_1(\mathbf{P},T) & a\_6(\mathbf{P},T) & a\_5(\mathbf{P},T) \\ a\_6(\mathbf{P},T) & a\_2(\mathbf{P},T) & a\_4(\mathbf{P},T) \\ a\_5(\mathbf{P},T) & a\_4(\mathbf{P},T) & a\_5(\mathbf{P},T) \end{bmatrix} \text{ separately represent the cell}$$

configuration tensor and the deformation tensor at temperature T (K) under the pressure P (GPa).

#### 3.1. Calculation of elastic constants for single crystal structure

Solving Eq. (24) is equivalent to increase the periodic function ui, <sup>k</sup>ð Þr , in a database-dependent

should be in principle infinite. But, in practice, we work with a limited set of basic functions,

Hm,m0ð Þk Ci,m0ð Þ¼ k εið Þk Ci,mð Þk

� � �wj m, k

m, <sup>k</sup> H ∧ eff � � �

where each point is a set of k eigenstates, the label having i = 1, 2, … obtained by diagonaliza-

It is necessary to integrate the points k in the Brillouin zone. For a function fi(k) where i defines

ð Þ!<sup>k</sup> <sup>Ω</sup>cell ð Þ 2π d ð BZ f i

of the Brillouin zone are determined using a sampling points k. Several election procedures exist for these points. Particularly those of Baldereschi [25], Chadi and Kohen [26], and

According to the crystal theory [28], any crystal lattice system contains six independent variables, namely the cell side length a, b, and c; unit cell angle α, β, and γ. Generally, the crystal under a certain deformation, temperature, and pressure can be described by the corresponding six-dimensional deformation tensor. The temperature and pressure will cause cell-deformed

2 6 4

0 B@

100 010 001

2 6 4

3. Elastic constants and homogenized moduli of monoclinic structure

<sup>y</sup> a<sup>0</sup>, <sup>0</sup> z

<sup>y</sup> <sup>b</sup><sup>0</sup>, <sup>0</sup> z

<sup>y</sup> c<sup>0</sup>,<sup>0</sup> z

j Cj i fk

<sup>j</sup> is the wave function developed in a space of infinite dimensions; this means that j

<sup>j</sup>ð Þr (28)

ð Þk dk (30)

α1ð Þ P; T α6ð Þ P; T α5ð Þ P; T α6ð Þ P; T α2ð Þ P; T α4ð Þ P; T α5ð Þ P; T α4ð Þ P; T α3ð Þ P; T

3 7 5

(31)

1 CA

=Ωcell of the cell volume

d

<sup>j</sup> will approximate. That the selected database simply

� � (29)

:

ui, <sup>k</sup>ð Þ¼ <sup>r</sup> <sup>X</sup>

<sup>j</sup>ð Þr j g j ¼ 1…Nbasð Þk

226 Density Functional Calculations - Recent Progresses of Theory and Application

X m0

<sup>f</sup> <sup>i</sup> <sup>¼</sup> <sup>1</sup> Nk X k f i

Ωcell is the cell volume of the original mesh in the real space and 2ð Þ π

Monkhorst and Pack [27] are the most frequently used.

Hm,m0ð Þ¼ <sup>k</sup> <sup>w</sup><sup>j</sup>

functions points k : f<sup>k</sup>

where f<sup>k</sup>

solves the system:

n

which imply that the description of f<sup>k</sup>

tion of the Hamiltonian in Eq. (29).

the band index, the average value is

configuration tensor as

bP,T <sup>x</sup> <sup>b</sup>P,T

aP,T <sup>x</sup> aP,T <sup>y</sup> aP,T <sup>z</sup>

cP,T <sup>x</sup> cP,T <sup>y</sup> cP,T <sup>z</sup>

<sup>y</sup> <sup>b</sup>P,T z

a<sup>0</sup>, <sup>0</sup> <sup>x</sup> a<sup>0</sup>, <sup>0</sup>

b<sup>0</sup>, <sup>0</sup> <sup>x</sup> <sup>b</sup><sup>0</sup>, <sup>0</sup>

c<sup>0</sup>,<sup>0</sup> <sup>x</sup> c<sup>0</sup>,<sup>0</sup>

<sup>X</sup>P,T <sup>¼</sup>

A multi-particle electronic structure satisfies the Schrödinger equation. As in [29], Kohn-Sham equation as an approximation to simplify Schrödinger equation is described. For crystal composed by vibrator with the vibration frequency wi, the total Helmholtz free energy is

$$F = E + \sum\_{i} F\_i^{th} = \mathcal{U} + \sum\_{i} \frac{1}{2} \hbar w\_i + k\_{\mathbb{B}} T \sum\_{i} \text{In} \left( 1 - e^{\frac{\hbar w\_i}{k\_{\mathbb{B}} T}} \right) \tag{32}$$

Helmholtz free energy can be calculated for all the thermodynamic quantities. DFT-QHA (quasi-harmonic approximation) is a precise calculation method to calculate thermodynamic properties of solid materials elastic constants and Debye temperature with the accurate predictions.

According to the theory of elasticity, under the isothermal strain, the elastic modulus of Helmholtz free energy can be described by the form of the Taylor expansion, of which the coefficients of the polynomial are the elastic coefficient:

$$\rho\_0 \boldsymbol{F}(\boldsymbol{\eta}\_{\boldsymbol{i}\boldsymbol{j}}, \boldsymbol{T}) = \rho\_0 \boldsymbol{F}(\boldsymbol{\eta}\_{\boldsymbol{i}\boldsymbol{j}}, \boldsymbol{T}) + \frac{1}{2} \sum\_{\boldsymbol{i}\boldsymbol{k}\boldsymbol{l}} \boldsymbol{c}\_{\boldsymbol{i}\boldsymbol{k}\boldsymbol{l}}^{\boldsymbol{T}} \boldsymbol{\eta}\_{\boldsymbol{i}\boldsymbol{j}} \boldsymbol{\eta}\_{\boldsymbol{k}\boldsymbol{l}} + \dots + \frac{1}{n!} \sum\_{\boldsymbol{i}\boldsymbol{k}\boldsymbol{l}\ldots\boldsymbol{i}} \boldsymbol{c}\_{\boldsymbol{i}\boldsymbol{k}\boldsymbol{l}\ldots\boldsymbol{i}}^{\boldsymbol{T}} \boldsymbol{\eta}\_{\boldsymbol{i}\boldsymbol{l}} \boldsymbol{\eta}\_{\boldsymbol{i}\boldsymbol{k}} \dots \tag{33}$$

where ηij,ηkl, and ηmn are the coefficients of Lagrange deformation tensor, c<sup>T</sup> ijklis the isothermal first-order elastic coefficients, and F ηij; T � � is the Helmholtz free energy.

The components of the stress tensor can be extracted by <sup>σ</sup><sup>i</sup> <sup>¼</sup> <sup>P</sup> 6 j¼1 cijε<sup>j</sup> , after the applied strain, the total energy variation of the system can be expressed as

$$
\Delta E = \frac{V}{2} \sum\_{i=1}^{6} \sum\_{j=1}^{6} c\_{ij} c\_i c\_j \tag{34}
$$

The second-order elastic coefficients can be obtained by the coefficient of the second-order Taylor expansion of Helmholtz free energy with the strain,

$$c\_{ijkl}^T = \rho\_0 \frac{\partial^2 F\left(\eta\_{ij}, T\right)}{\partial \eta\_{ij} \partial \eta\_{kl}}\tag{35}$$

Here, strain and thermodynamics deformation are symmetric. There is only six independent deformation tensor in the nine-dimensional deformation tensor. LCEC is a second-order linear combination of independent elastic coefficients corresponding to Helmholtz free energy coefficient under some deformation mode [30, 31]. For all directions under monoclinic crystals, if a strain is added, the corresponding simultaneous equations can be solved to determine all elastic coefficients.

3.3. Homogenization of monoclinic polycrystals by RVH estimation

are the normal strain and shear strain in each direction, respectively.

σ<sup>11</sup> σ<sup>22</sup> σ<sup>33</sup> σ<sup>12</sup> σ<sup>13</sup> σ<sup>23</sup>

¼

GV <sup>¼</sup> <sup>1</sup>

BR ¼ Ω c33c<sup>55</sup> � c

2

<sup>15</sup> c22c<sup>33</sup> � c

f ¼ c<sup>11</sup> c22c<sup>55</sup> � c

� c 2 2

2

þ2ð Þ c13c<sup>25</sup> � c15c<sup>23</sup> ð Þþ c<sup>25</sup> � c<sup>15</sup> f �

g ¼ c11c22c<sup>33</sup> � c11c

2

2 <sup>23</sup> � � <sup>þ</sup> <sup>c</sup> 2 <sup>23</sup> � c22c 2 <sup>13</sup> � c33c 2

<sup>25</sup> c11c<sup>33</sup> � c

C66 Cl5, C25, C35, and C46. The criteria for mechanical stability are given by Wu [32]:

c44c<sup>66</sup> � c

Ω ¼ 2½ � c15c25ðc33c<sup>12</sup> � c13c23Þ þ c15c35ðc22c<sup>13</sup> � c12c23Þ þ c25c35ð Þ c11c<sup>23</sup> � c12c<sup>13</sup>

For monoclinic crystal structure, elastic constants include C11, C22, C33, Cl2, C13, C23, C44, C55,

2 <sup>13</sup> � � <sup>þ</sup> <sup>c</sup>

2

GR ¼ 15 4 c33c<sup>55</sup> � c

elastic stiffness parameters [32]:

Stress-strain relation in an orthotropic monoclinic crystal can be defined by the independent

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory

c<sup>11</sup> c<sup>12</sup> c<sup>13</sup> 0 c<sup>15</sup> 0 c<sup>12</sup> c<sup>22</sup> c<sup>23</sup> 0 c<sup>25</sup> 0 c<sup>13</sup> c<sup>23</sup> c<sup>33</sup> 0 c<sup>35</sup> 0 000 c<sup>44</sup> 0 c<sup>46</sup> c<sup>15</sup> c<sup>25</sup> c<sup>35</sup> 0 c<sup>55</sup> 0 000 c<sup>46</sup> 0 c<sup>66</sup>

where σ represents the normal stress and shear stress in each direction (unit: nN/nm<sup>2</sup>

The homogenized elastic properties of polycrystals can be calculated, of which elastic moduli and Poisson's ratio can be obtained by calculating Voigt and Reuss bounds and averaging term as [32]

> <sup>35</sup> � �ðc<sup>11</sup> <sup>þ</sup> <sup>c</sup><sup>22</sup> <sup>þ</sup> <sup>c</sup>12Þ þ ð Þ <sup>c</sup>23c<sup>55</sup> � <sup>c</sup>25c<sup>35</sup> ð Þ <sup>c</sup><sup>11</sup> � <sup>c</sup><sup>12</sup> � <sup>c</sup><sup>23</sup> � � þð Þ c13c<sup>35</sup> � c15c<sup>33</sup> ð Þþ c<sup>15</sup> þ c<sup>25</sup> ðc13c<sup>55</sup> � c15c35Þ � ð Þ c<sup>22</sup> � c<sup>12</sup> � c<sup>23</sup> � c<sup>13</sup> þð Þ c13c<sup>25</sup> � c15c<sup>23</sup> ð Þþ c<sup>15</sup> � c<sup>25</sup> f �=Ωþ3 g=Ω þ ð Þ c<sup>44</sup> þ c<sup>66</sup> = c44c<sup>66</sup> � c

<sup>35</sup> � �ðc<sup>11</sup> <sup>þ</sup> <sup>c</sup><sup>22</sup> � <sup>2</sup>c12Þ þ ð Þ <sup>c</sup>23c<sup>55</sup> � <sup>c</sup>25c<sup>35</sup> ð Þ <sup>2</sup>c<sup>12</sup> � <sup>2</sup>c<sup>11</sup> � <sup>c</sup><sup>23</sup> � þðc13c<sup>35</sup> � c15c33Þ � ð Þþ c<sup>15</sup> � 2c<sup>25</sup> ð Þ c13c<sup>55</sup> � c15c<sup>35</sup> ð Þ 2c<sup>12</sup> þ 2c<sup>23</sup> � c<sup>13</sup> � 2c<sup>22</sup>

<sup>25</sup> � � � <sup>c</sup>12ðc12c<sup>55</sup> � <sup>c</sup>15c25Þ þ <sup>c</sup>15ðc12c<sup>25</sup> � <sup>c</sup>15c22Þ þ <sup>c</sup>25ð Þ <sup>c</sup>23c<sup>35</sup> � <sup>c</sup>25c<sup>33</sup> (41)

2

<sup>12</sup> � � � � <sup>þ</sup> gc<sup>55</sup> (43)

<sup>35</sup> c11c<sup>22</sup> � c

2

cij > 0ð Þ i ¼ 1; 2; 3; 4; 5; 6 (44)

<sup>46</sup> � � <sup>&</sup>gt; <sup>0</sup> (45)

<sup>15</sup> ½ � <sup>c</sup><sup>11</sup> <sup>þ</sup> <sup>c</sup><sup>22</sup> <sup>þ</sup> <sup>c</sup><sup>33</sup> <sup>þ</sup> <sup>3</sup>ðc<sup>44</sup> <sup>þ</sup> <sup>c</sup><sup>55</sup> <sup>þ</sup> <sup>c</sup>66Þ � ð Þ <sup>c</sup><sup>12</sup> <sup>þ</sup> <sup>c</sup><sup>13</sup> <sup>þ</sup> <sup>c</sup><sup>23</sup> (37)

BV ¼ ½ � c<sup>11</sup> þ c<sup>22</sup> þ c<sup>33</sup> þ 2ð Þ c<sup>12</sup> þ c<sup>13</sup> þ c<sup>23</sup> =9 (39)

ε<sup>11</sup> ε<sup>22</sup> ε<sup>33</sup> γ<sup>12</sup> γ<sup>13</sup> γ<sup>23</sup>

http://dx.doi.org/10.5772/intechopen.72301

(36)

229

); ε and γ

2

<sup>46</sup> � � ����<sup>1</sup> (38)

�<sup>1</sup> (40)

<sup>12</sup> þ 2c12c13c<sup>23</sup> (42)

#### 3.2. The energy-volume relationship of the monoclinic crystal

Deformation tensors to calculate independent Cij constants of monoclinic crystal are listed in Table 1.

For monoclinic crystal, elastic constants include C11, C22, C33, Cl2, C13, C23, C44, C55, C66 Cl5, C25, C35, and C46; the strain energy-volume relation and elastic moduli of monoclinic symmetry based on E-V method can be obtained. The calculated E-δ points are fitted to second-order polynomials E(V, δ). For all strains, different strain forms δ are taken to calculate the total energies E for the strained crystal structure. By applying a series of δ strain amplitude, the independent elastic constants of monoclinic crystal by these simultaneous ΔE-δ equations can be obtained.


Table 1. Deformation tensors to calculate independent elastic constants of monoclinic crystal [30, 31].

#### 3.3. Homogenization of monoclinic polycrystals by RVH estimation

Here, strain and thermodynamics deformation are symmetric. There is only six independent deformation tensor in the nine-dimensional deformation tensor. LCEC is a second-order linear combination of independent elastic coefficients corresponding to Helmholtz free energy coefficient under some deformation mode [30, 31]. For all directions under monoclinic crystals, if a strain is added, the corresponding simultaneous equations can be solved to determine all

Deformation tensors to calculate independent Cij constants of monoclinic crystal are listed in

For monoclinic crystal, elastic constants include C11, C22, C33, Cl2, C13, C23, C44, C55, C66 Cl5, C25, C35, and C46; the strain energy-volume relation and elastic moduli of monoclinic symmetry based on E-V method can be obtained. The calculated E-δ points are fitted to second-order polynomials E(V, δ). For all strains, different strain forms δ are taken to calculate the total energies E for the strained crystal structure. By applying a series of δ strain amplitude, the independent elastic constants of monoclinic crystal by these simultaneous ΔE-δ equations can

3.2. The energy-volume relationship of the monoclinic crystal

228 Density Functional Calculations - Recent Progresses of Theory and Application

Deformation tensor ΔE-V relation of LCEC LCEC

<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>12</sup> <sup>þ</sup> <sup>c</sup><sup>22</sup> 2

<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>23</sup> <sup>þ</sup> <sup>c</sup><sup>33</sup> 2

<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>13</sup> <sup>þ</sup> <sup>c</sup><sup>33</sup> 2

<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>45</sup> <sup>þ</sup> <sup>c</sup><sup>55</sup> 2

<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>16</sup> <sup>þ</sup> <sup>c</sup><sup>66</sup> 2

<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>26</sup> <sup>þ</sup> <sup>c</sup><sup>66</sup> 2

<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>36</sup> <sup>þ</sup> <sup>c</sup><sup>66</sup> 2

<sup>2</sup> � <sup>c</sup><sup>12</sup> <sup>þ</sup> <sup>c</sup><sup>22</sup> 2

<sup>2</sup> � <sup>c</sup><sup>13</sup> <sup>þ</sup> <sup>c</sup><sup>33</sup> 2

<sup>2</sup> � <sup>c</sup><sup>23</sup> <sup>þ</sup> <sup>c</sup><sup>33</sup> 2

<sup>2</sup> þ 2c<sup>15</sup> þ 2c<sup>55</sup>

<sup>2</sup> � 2c<sup>15</sup> þ 2c<sup>55</sup>

<sup>2</sup> þ 2c<sup>25</sup> þ 2c<sup>55</sup>

<sup>2</sup> þ 2c<sup>35</sup> þ 2c<sup>55</sup>

Table 1. Deformation tensors to calculate independent elastic constants of monoclinic crystal [30, 31].

δ<sup>2</sup> c11+ c22 + 2c12

δ<sup>2</sup> c22 + c33 + 2c23

δ<sup>2</sup> c11 + c33 + 2c13

δ<sup>2</sup> c44 + c55 + 2c45

δ<sup>2</sup> c11 + c66 + 2c16

δ<sup>2</sup> c22 + c66 + 2c26

δ<sup>2</sup> c33 + c66 + 2c36

<sup>δ</sup><sup>2</sup> c11 + c22 � 2c12

<sup>δ</sup><sup>2</sup> c11 + c33 � 2c13

<sup>δ</sup><sup>2</sup> c22 + c33 � 2c23

δ<sup>2</sup> c11 + 4c55 + 4c15

<sup>δ</sup><sup>2</sup> c11 + 4c55 � 4c15

δ<sup>2</sup> c22 + 4c55 + 4c25

δ<sup>2</sup> c33 + 4c55 + 4c35

δ<sup>2</sup> c11 + c22 + c33 + 2c12 + 2c13 + 2c23

<sup>2</sup> þ c<sup>12</sup> þ c<sup>13</sup> þ c<sup>23</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>c44δ<sup>2</sup> 4c44

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>22</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>44</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>22</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>33</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup> <sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>22</sup> <sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>33</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>22</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>11</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>22</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>33</sup>

elastic coefficients.

Table 1.

be obtained.

<sup>e</sup> <sup>¼</sup> <sup>δ</sup>; �δ; <sup>δ</sup><sup>2</sup>

<sup>e</sup> <sup>¼</sup> <sup>δ</sup>; <sup>δ</sup><sup>2</sup>

<sup>e</sup> <sup>¼</sup> <sup>δ</sup><sup>2</sup>

<sup>e</sup> <sup>¼</sup> <sup>δ</sup><sup>2</sup>

e ¼ ð Þ δ; δ; 0; 0; 0; 0 <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ 0; δ; δ; 0; 0; 0 <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ δ; 0; δ; 0; 0; 0 <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ 0; 0; 0; δ; δ; 0 <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ δ; 0; 0; 0; 0; δ <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ 0; δ; 0; 0; 0; δ <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ 0; 0; δ; 0; 0; δ <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ δ; δ; δ; 0; 0; 0 <sup>Δ</sup><sup>E</sup>

<sup>=</sup> <sup>1</sup> � <sup>δ</sup><sup>2</sup> ; <sup>0</sup>; <sup>0</sup>; <sup>0</sup> <sup>Δ</sup><sup>E</sup>

<sup>=</sup> <sup>1</sup> � <sup>δ</sup><sup>2</sup> ; �δ; <sup>0</sup>; <sup>0</sup>; <sup>0</sup> <sup>Δ</sup><sup>E</sup>

<sup>=</sup> <sup>1</sup> � <sup>δ</sup><sup>2</sup> ; <sup>δ</sup>; �δ; <sup>0</sup>; <sup>0</sup>; <sup>0</sup> <sup>Δ</sup><sup>E</sup>

<sup>=</sup> <sup>1</sup> � <sup>δ</sup><sup>2</sup> ; <sup>0</sup>; <sup>0</sup>; <sup>2</sup>δ; <sup>0</sup>; <sup>0</sup> <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ δ; 0; 0; 0; 2δ; 0 <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ δ; 0; 0; 0; �2δ; 0 <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ 0; δ; 0; 0; 2δ; 0 <sup>Δ</sup><sup>E</sup>

e ¼ ð Þ 0; 0; δ; 0; 2δ; 0 <sup>Δ</sup><sup>E</sup>

Stress-strain relation in an orthotropic monoclinic crystal can be defined by the independent elastic stiffness parameters [32]:

$$
\begin{bmatrix}
\sigma\_{11} \\
\sigma\_{22} \\
\sigma\_{33} \\
\sigma\_{12} \\
\sigma\_{13} \\
\sigma\_{23} \\
\sigma\_{23}
\end{bmatrix} = \begin{bmatrix}
c\_{11} & c\_{12} & c\_{13} & 0 & c\_{15} & 0 \\
c\_{12} & c\_{22} & c\_{23} & 0 & c\_{25} & 0 \\
c\_{13} & c\_{23} & c\_{33} & 0 & c\_{35} & 0 \\
0 & 0 & 0 & c\_{44} & 0 & c\_{46} \\
c\_{15} & c\_{25} & c\_{35} & 0 & c\_{55} & 0 \\
0 & 0 & 0 & c\_{46} & 0 & c\_{66}
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{11} \\
\varepsilon\_{22} \\
\varepsilon\_{33} \\
\gamma\_{12} \\
\gamma\_{13} \\
\gamma\_{13} \\
\gamma\_{23}
\end{bmatrix} \tag{36}
$$

where σ represents the normal stress and shear stress in each direction (unit: nN/nm<sup>2</sup> ); ε and γ are the normal strain and shear strain in each direction, respectively.

The homogenized elastic properties of polycrystals can be calculated, of which elastic moduli and Poisson's ratio can be obtained by calculating Voigt and Reuss bounds and averaging term as [32]

$$\mathbf{G}\mathbf{v} = \frac{1}{15} \left[ \mathbf{c}\_{11} + \mathbf{c}\_{22} + \mathbf{c}\_{33} + \mathbf{3} (\mathbf{c}\_{44} + \mathbf{c}\_{55} + \mathbf{c}\_{66}) - (\mathbf{c}\_{12} + \mathbf{c}\_{13} + \mathbf{c}\_{23}) \right] \tag{37}$$

$$G\_{R} = 15\left\{4\left[ (c\_{33}c\_{55} - c\_{35}^2)(c\_{11} + c\_{22} + c\_{12}) + (c\_{23}c\_{55} - c\_{25}c\_{35})(c\_{11} - c\_{12} - c\_{23}) \right] \right.$$

$$+ (c\_{13}c\_{35} - c\_{15}c\_{33})(c\_{15} + c\_{25}) + (c\_{13}c\_{55} - c\_{15}c\_{35})\cdot(c\_{22} - c\_{12} - c\_{23} - c\_{13})$$

$$+ (c\_{13}c\_{25} - c\_{15}c\_{23})(c\_{15} - c\_{25}) + f\right]/\Omega + \mathfrak{Z}\left[g/\Omega + (c\_{44} + c\_{66})/\left(c\_{44}c\_{66} - c\_{46}^2\right)\right]^{-1} \tag{38}$$

$$B\_V = \left[c\_{11} + c\_{22} + c\_{33} + 2(c\_{12} + c\_{13} + c\_{23})\right] / 9 \tag{39}$$

$$B\_R = \Omega \left[ (c\_{33}c\_{55} - c\_{35}^2)(c\_{11} + c\_{22} - 2c\_{12}) + (c\_{23}c\_{55} - c\_{25}c\_{35})(2c\_{12} - 2c\_{11} - c\_{23}) \right. $$

$$+ (c\_{13}c\_{35} - c\_{15}c\_{33}) \cdot (c\_{15} - 2c\_{25}) + (c\_{13}c\_{55} - c\_{15}c\_{35})(2c\_{12} + 2c\_{23} - c\_{13} - 2c\_{22}) $$

$$+ 2(c\_{13}c\_{25} - c\_{15}c\_{23})(c\_{25} - c\_{15}) + f \right]^{-1} \tag{40}$$

$$f = c\_{11}(c\_{22}c\_{55} - c\_{25}^2) - c\_{12}(c\_{12}c\_{55} - c\_{15}c\_{25}) + c\_{15}(c\_{12}c\_{25} - c\_{15}c\_{22}) + c\_{25}(c\_{23}c\_{35} - c\_{25}c\_{33})\tag{41}$$

$$g = c\_{11}c\_{22}c\_{33} - c\_{11}c\_{23}^2 - c\_{22}c\_{13}^2 - c\_{33}c\_{12}^2 + 2c\_{12}c\_{13}c\_{23} \tag{42}$$

$$\Omega = 2\left[c\_{15}c\_{25}(c\_{33}c\_{12} - c\_{13}c\_{23}) + c\_{15}c\_{35}(c\_{22}c\_{13} - c\_{12}c\_{23}) + c\_{25}c\_{35}(c\_{11}c\_{23} - c\_{12}c\_{13})\right]$$

$$-\left[c\_{15}^2(c\_{22}c\_{33} - c\_{23}^2) + c\_{25}^2(c\_{11}c\_{33} - c\_{13}^2) + c\_{35}^2(c\_{11}c\_{22} - c\_{12}^2)\right] + gc\_{55}\tag{43}$$

For monoclinic crystal structure, elastic constants include C11, C22, C33, Cl2, C13, C23, C44, C55, C66 Cl5, C25, C35, and C46. The criteria for mechanical stability are given by Wu [32]:

$$\mathbf{c}\_{\vec{\eta}} > \mathbf{0} (i = 1, 2, 3, 4, 5, 6) \tag{44}$$

$$\left(c\_{44}c\_{66} - c\_{46}^2\right) > 0\tag{45}$$

$$\left(c\_{33}c\_{55} - c\_{35}^2\right) > 0\tag{46}$$

$$(c\_{22} + c\_{33} - 2c\_{23}) > 0\tag{47}$$

$$[c\_{11} + c\_{22} + c\_{33} + 2(c\_{12} + c\_{13} + c\_{23})] > 0\tag{48}$$

$$\left[c\_{22}(c\_{33}c\_{55} - c\_{35}^2) + 2c\_{23}c\_{25}c\_{35} - c\_{23}^2c\_{55} - c\_{25}^2c\_{33}\right] > 0\tag{49}$$

$$\begin{aligned} \left\{ 2\left[ \mathbf{c}\_{15} \mathbf{c}\_{25} (\mathbf{c}\_{33} \mathbf{c}\_{12} - \mathbf{c}\_{13} \mathbf{c}\_{23}) + \mathbf{c}\_{15} \mathbf{c}\_{35} (\mathbf{c}\_{22} \mathbf{c}\_{13} - \mathbf{c}\_{12} \mathbf{c}\_{23}) + \mathbf{c}\_{25} \mathbf{c}\_{35} (\mathbf{c}\_{11} \mathbf{c}\_{23} - \mathbf{c}\_{12} \mathbf{c}\_{13}) \right] \right. \\ \left. - \left[ \mathbf{c}\_{15}^{2} (\mathbf{c}\_{22} \mathbf{c}\_{33} - \mathbf{c}\_{23}^{2}) + \mathbf{c}\_{25}^{2} \left( \mathbf{c}\_{11} \mathbf{c}\_{33} - \mathbf{c}\_{13}^{2} \right) + \mathbf{c}\_{35}^{2} \left( \mathbf{c}\_{11} \mathbf{c}\_{22} - \mathbf{c}\_{12}^{2} \right) \right] + \mathbf{g} \mathbf{c}\_{55} \right\} > 0 \end{aligned} \tag{50}$$

Young's modulus and Poisson's ratio can be rewritten based on the Voigt-Reuss-Hill approximation [33]. In terms of the Voigt-Reuss-Hill approximations [34], MH = (1/2)(MR + MV), M refers to B or G. Thus, Young's modulus E and Possion's ratio μ are obtained as

$$E = \frac{9BG}{3B + G} = \frac{9(B\_V/2 + B\_R/2)(G\_V/2 + G\_R/2)}{3(B\_V/2 + B\_R/2) + (G\_V/2 + G\_R/2)}\tag{51}$$

$$\mu = \frac{3B - 2G}{2(3B + G)} = \frac{3(B\_V/2 + B\_R/2) - 2(G\_V/2 + G\_R/2)}{6(B\_V/2 + B\_R/2) + 2(G\_V/2 + G\_R/2)}\tag{52}$$

Then, Voigt-Reuss-Hill average [32] will be determined, and Young's modulus can be calculated.

#### 4. Modeling and homogenized elastic moduli of gypsum structure

#### 4.1. Nanoscale modeling of monoclinic crystals

#### 4.1.1. Nanoscale modeling of monoclinic gypsum crystal

The gypsum morphology is monoclinic, and the initial lattice is as a = 5.677Å, b = 15.207Å, c = 6.528Å, α = β = 90�, and γ = 118.49�, its structure is monoclinic with space group I 2/a [35].

In Figure 1, the gypsum crystal can be summarized as follows: (1) the two hydrogen atoms of water molecules formed weak hydrogen bonds with the O atoms of Ca and S polyhedra; (2) a stacking sequence of CaO8 and SO4 chains in the (010) plane alternates with water layers along the b-axis; and (3) in (010) plane, the sulfate tetrahedra and CaO8 polyhedra alternate to form edge-sharing chains along [100] and zigzag chains along [001] direction [36] (Table 2).

#### 4.1.2. Nanoscale modeling of monoclinic 11 Å tobermorite crystal

Hamid model [37] as the 11 Å tobermorite (formula: Ca4Si6O14(OH)4�2H2O) as an initial configuration is commonly used. The morphology is monoclinic, and the initial lattice is [37]: a = 6.69 Å, b = 7.39 Å, c = 22.779 Å, α = β = 90�, and γ = 123.49�, space group P21. Modeling of 11 Å tobermorite is shown in Figure 2.

In Figure 2(a), the 11 Å tobermorite crystal can be summarized as follows: (1) the structure is basically a layered structure. (2) The central part is a Ca-O sheet (with an empirical formula: CaO2, of which the oxygen in CaO2 also includes that of the silicate tetrahedron part). (3) Silicate chains envelope the Ca-O sheet on both sides. (4) Ca2+ and H2O are filled between individual layers to balance the charges. The infinite layers of calcium polyhedra are parallel to

Figure 2. Modeling of 11 Å tobermorite crystal. Silicate chains, calcium octahedral, and oxygen atoms are shown as yellow tetrahedra, green spheres, and red spheres. (a) 11 Å Tob monoclinic crystal; (b) in x-direction; (c) in y-direction; and

Figure 1. Modeling of gypsum crystal. (a) Gypsum structure [36] along [001]; (b) the real cell; (c) in x-direction; (d) in

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Atom x yz Occupancy rate Uiso or Ueq

Ca 0.5000 0.0786 0.2500 1.00 1.00 S 0.0000 0.0787 0.7500 1.00 1.00 O1 0.0384 0.1326 0.5512 1.00 1.00 O2 0.2429 0.0215 0.8347 1.00 1.00 Ow 0.3784 0.1825 0.4554 1.00 1.00 H1 0.2504 0.1615 0.5009 1.00 1.00 H2 0.4022 0.2435 0.4900 1.00 1.00

Table 2. Atomic coordinates and displacement parameters of gypsum [36].

y-direction; and (e) in z-direction.

(d) in z-direction.

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory http://dx.doi.org/10.5772/intechopen.72301 231

Figure 1. Modeling of gypsum crystal. (a) Gypsum structure [36] along [001]; (b) the real cell; (c) in x-direction; (d) in y-direction; and (e) in z-direction.


Table 2. Atomic coordinates and displacement parameters of gypsum [36].

c33c<sup>55</sup> � c

c<sup>22</sup> c33c<sup>55</sup> � c

230 Density Functional Calculations - Recent Progresses of Theory and Application

2 23 <sup>þ</sup> <sup>c</sup>

<sup>E</sup> <sup>¼</sup> <sup>9</sup>BG

<sup>μ</sup> <sup>¼</sup> <sup>3</sup><sup>B</sup> � <sup>2</sup><sup>G</sup>

4.1. Nanoscale modeling of monoclinic crystals

4.1.1. Nanoscale modeling of monoclinic gypsum crystal

4.1.2. Nanoscale modeling of monoclinic 11 Å tobermorite crystal

11 Å tobermorite is shown in Figure 2.

� c 2

<sup>15</sup> c22c<sup>33</sup> � c

2 35 <sup>þ</sup> <sup>2</sup>c23c25c<sup>35</sup> � <sup>c</sup>

2

<sup>25</sup> c11c<sup>33</sup> � c

refers to B or G. Thus, Young's modulus E and Possion's ratio μ are obtained as

2 35

f2½ � c15c25ðc33c<sup>12</sup> � c13c23Þ þ c15c35ðc22c<sup>13</sup> � c12c23Þ þ c25c35ð Þ c11c<sup>23</sup> � c12c<sup>13</sup>

Young's modulus and Poisson's ratio can be rewritten based on the Voigt-Reuss-Hill approximation [33]. In terms of the Voigt-Reuss-Hill approximations [34], MH = (1/2)(MR + MV), M

<sup>3</sup><sup>B</sup> <sup>þ</sup> <sup>G</sup> <sup>¼</sup> <sup>9</sup>ð Þ BV=<sup>2</sup> <sup>þ</sup> BR=<sup>2</sup> ð Þ GV=<sup>2</sup> <sup>þ</sup> GR=<sup>2</sup>

2 3ð Þ <sup>B</sup> <sup>þ</sup> <sup>G</sup> <sup>¼</sup> <sup>3</sup>ðBV=<sup>2</sup> <sup>þ</sup> BR=2Þ � <sup>2</sup>ð Þ GV=<sup>2</sup> <sup>þ</sup> GR=<sup>2</sup>

Then, Voigt-Reuss-Hill average [32] will be determined, and Young's modulus can be calculated.

The gypsum morphology is monoclinic, and the initial lattice is as a = 5.677Å, b = 15.207Å, c = 6.528Å, α = β = 90�, and γ = 118.49�, its structure is monoclinic with space group I 2/a [35]. In Figure 1, the gypsum crystal can be summarized as follows: (1) the two hydrogen atoms of water molecules formed weak hydrogen bonds with the O atoms of Ca and S polyhedra; (2) a stacking sequence of CaO8 and SO4 chains in the (010) plane alternates with water layers along the b-axis; and (3) in (010) plane, the sulfate tetrahedra and CaO8 polyhedra alternate to form

edge-sharing chains along [100] and zigzag chains along [001] direction [36] (Table 2).

Hamid model [37] as the 11 Å tobermorite (formula: Ca4Si6O14(OH)4�2H2O) as an initial configuration is commonly used. The morphology is monoclinic, and the initial lattice is [37]: a = 6.69 Å, b = 7.39 Å, c = 22.779 Å, α = β = 90�, and γ = 123.49�, space group P21. Modeling of

In Figure 2(a), the 11 Å tobermorite crystal can be summarized as follows: (1) the structure is basically a layered structure. (2) The central part is a Ca-O sheet (with an empirical formula:

4. Modeling and homogenized elastic moduli of gypsum structure

2 13 <sup>þ</sup> <sup>c</sup>

> 0 (46)

ð Þ c<sup>22</sup> þ c<sup>33</sup> � 2c<sup>23</sup> > 0 (47)

2 12

<sup>3</sup>ðBV=<sup>2</sup> <sup>þ</sup> BR=2Þ þ ð Þ GV=<sup>2</sup> <sup>þ</sup> GR=<sup>2</sup> (51)

<sup>6</sup>ðBV=<sup>2</sup> <sup>þ</sup> BR=2Þ þ <sup>2</sup>ð Þ GV=<sup>2</sup> <sup>þ</sup> GR=<sup>2</sup> (52)

½ � c<sup>11</sup> þ c<sup>22</sup> þ c<sup>33</sup> þ 2ð Þ c<sup>12</sup> þ c<sup>13</sup> þ c<sup>23</sup> > 0 (48)

2 <sup>23</sup>c<sup>55</sup> � c 2 <sup>25</sup>c<sup>33</sup>

2

<sup>þ</sup> gc55<sup>g</sup> <sup>&</sup>gt; 0 (50)

> 0 (49)

<sup>35</sup> c11c<sup>22</sup> � c

Figure 2. Modeling of 11 Å tobermorite crystal. Silicate chains, calcium octahedral, and oxygen atoms are shown as yellow tetrahedra, green spheres, and red spheres. (a) 11 Å Tob monoclinic crystal; (b) in x-direction; (c) in y-direction; and (d) in z-direction.

CaO2, of which the oxygen in CaO2 also includes that of the silicate tetrahedron part). (3) Silicate chains envelope the Ca-O sheet on both sides. (4) Ca2+ and H2O are filled between individual layers to balance the charges. The infinite layers of calcium polyhedra are parallel to (001), with tetrahedral chains of wollastonite-type along b and the composite layers stacked along c and connected through the formation of double tetrahedral chains [38]. Atomic coordinates and displacement parameters are seen in Table 3.

From Figure 3, elastic constants at 0 GPa are given as c<sup>11</sup> = 82.464 GPa, c<sup>12</sup> = 34.751 GPa, c<sup>13</sup> = 33.643 GPa, c<sup>15</sup> = 1.987 GPa, c<sup>22</sup> = 63.046 GPa, c<sup>23</sup> = 34.920 GPa, c<sup>25</sup> = 8.071 GPa, c<sup>33</sup> = 57.549 GPa, c<sup>35</sup> = 3.054 GPa, c<sup>44</sup> = 20.863 GPa, c<sup>46</sup> = 4.688 GPa, c<sup>55</sup> = 28.062 GPa, and c<sup>66</sup> = 28.556 GPa. It is found that the oxygen atom of the water molecule did not change its position or occupancy under pressure conditions. A simple pressure increase at an ambient temperature cannot induce dehydration because of the unchange of water molecular in the

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Elastic constants of gypsum crystal model based on DFT are calculated, and parameters are

Initial conditions of tobermorite are quite the same with that of gypsum crystal. Elastic constants of 11 Å tobermorite crystal under 0–1.0 GPa are shown in Figure 4. Elastic constants are

A comparisonal results of Shahsavari [39] are provided. Elastic constants at 0 GPa are as follows: c<sup>11</sup> = 106.63 GPa, c<sup>12</sup> = 50.37 GPa, c<sup>13</sup> = 41.09 GPa, c<sup>15</sup> = 3.50 GPa, c<sup>22</sup> = 131.67 GPa, c<sup>23</sup> = 22.78 GPa, c<sup>25</sup> = 0.78 GPa, c<sup>33</sup> = 71.45 GPa, c<sup>35</sup> = 0.83 GPa, c<sup>44</sup> = 26.03 GPa, c<sup>46</sup> = 0.02 GPa, c<sup>55</sup> = 27.61 GPa, and c<sup>66</sup> = 45.26 GPa. Thus, elastic modulus can be homogenized to compare with

Based on elastic constants, the elastic moduli of gypsum at 0 GPa are verified and averaged in

P C11 C12 C13 C15 C22 C23 C25 C33 C35 C44 C46 C55 C66 10–4[36] ———— ——— —— —— —— 0.0 82.46 34.75 33.64 1.99 63.05 34.92 8.07 57.55 3.05 20.86 4.69 28.06 28.56 0.1 79.82 32.64 29.2 1.8 71.04 29.61 7.54 61.88 3.22 20.13 3.06 26.19 27.7 0.2 82.93 37.75 34.59 1.09 63.62 32.42 7.35 50.64 4.37 21.32 1.1 25.8 17.8 0.3 82.82 39.77 32.81 0.17 65.64 29.61 7.23 57.31 4.45 26.43 5.57 23.17 23.39 0.4 84.47 38.6 32.25 1.27 69.03 32.31 8.51 53.41 2.21 20.8 2.03 28.41 22.34 0.5 75.84 43.68 29.39 0.57 68.7 33.18 8.36 56.08 2.52 29.7 3.88 27.35 22.4 0.6 74.22 43.11 28.77 2.22 69.52 28.87 7.81 53.19 2.68 28.97 1.49 23.24 15.53 0.7 88.37 41.74 32.85 2.25 70.09 32.28 9.14 55.48 4.28 24.66 2.76 27.25 22.58 0.8 88.53 39.65 35.29 2.96 73.28 33.84 8.02 62.22 3.73 24.73 3.44 26.37 24.39 0.9 88.7 45.09 37.97 4.54 66.78 36.02 10.4 61.98 1.2 25.15 4.92 28.93 26.82 1.0 90.12 39.79 34.63 2.7 75.99 34.92 8.74 68.31 3.46 26.32 5.83 28.15 30.19

the results of LD C-S-H phase in nano-indentation test by Vandamme and Ulm [40].

4.3. Homogenized elastic moduli of typical monoclinic structures

4.3.1. Elastic modulus of monoclinic gypsum structure

Table 4. Elastic coefficient Cij (GPa) of gypsum by DFT.

gypsum structure within pressure range [36].

4.2.2. Initial conditions and elastic constants of tobermorite

detailed in Table 4.

shown in Table 5.

Figure 5.

#### 4.2. Initial conditions and elastic constants of monoclinic crystals

#### 4.2.1. Initial conditions and elastic constants of gypsum

The initial conditions are as follows: the pressure region of 0–1 GPa is used. Besides, a planewave basis set and ultrasoft pseudopotentials using GGA are used with a plane-wave cutoff energy of 400 eV. Brillouin zone is 6 6 4. Self-consistent convergence of the total energy per atom is chosen as 10<sup>4</sup> eV. Elastic constants of monoclinic gypsum crystal under 0–1.0 GPa are shown in Figure 3.


Table 3. Atomic coordinates and displacement parameters of 11 Å tobermorite [38].

Figure 3. Gypsum monoclinic crystal under pressure 0–1.0 GPa by DFT. (a) Relative change of a, b, c, and Vand (b) elastic constants.

From Figure 3, elastic constants at 0 GPa are given as c<sup>11</sup> = 82.464 GPa, c<sup>12</sup> = 34.751 GPa, c<sup>13</sup> = 33.643 GPa, c<sup>15</sup> = 1.987 GPa, c<sup>22</sup> = 63.046 GPa, c<sup>23</sup> = 34.920 GPa, c<sup>25</sup> = 8.071 GPa, c<sup>33</sup> = 57.549 GPa, c<sup>35</sup> = 3.054 GPa, c<sup>44</sup> = 20.863 GPa, c<sup>46</sup> = 4.688 GPa, c<sup>55</sup> = 28.062 GPa, and c<sup>66</sup> = 28.556 GPa. It is found that the oxygen atom of the water molecule did not change its position or occupancy under pressure conditions. A simple pressure increase at an ambient temperature cannot induce dehydration because of the unchange of water molecular in the gypsum structure within pressure range [36].

Elastic constants of gypsum crystal model based on DFT are calculated, and parameters are detailed in Table 4.

#### 4.2.2. Initial conditions and elastic constants of tobermorite

(001), with tetrahedral chains of wollastonite-type along b and the composite layers stacked along c and connected through the formation of double tetrahedral chains [38]. Atomic coor-

The initial conditions are as follows: the pressure region of 0–1 GPa is used. Besides, a planewave basis set and ultrasoft pseudopotentials using GGA are used with a plane-wave cutoff energy of 400 eV. Brillouin zone is 6 6 4. Self-consistent convergence of the total energy per atom is chosen as 10<sup>4</sup> eV. Elastic constants of monoclinic gypsum crystal under 0–1.0 GPa are

> Atomic species

XYZ Occupancy

rate

Uiso or Ueq

Uiso or Ueq

Si1 0.7710 0.3830 0.1578 1 0.031 O8 0.7690 0.8430 0.0953 1 0.027 Si2 0.9250 0.7500 0.0721 1 0.030 O9 0.5370 0.7980 0.1968 1 0.036 Si3 0.7720 0.9620 0.1596 1 0.015 O10 0.0040 0.0420 0.2008 1 0.034 O1 0.7740 0.4950 0.0932 1 0.039 O11 0.4330 0.2230 0.0250 0.5 0.072 O2 0.7620 0.1690 0.1305 1 0.019 O12 0.9490 0.2560 0.0000 1 0.080 O3 0.0020 0.5270 0.2000 1 0.032 O13 0.4300 0.7700 0.0220 0.5 0.090 O4 0.5360 0.3040 0.1926 1 0.035 Cal 0.2770 0.4257 0.2083 1 0.024 O5 0.9100 0.7470 0.0000 1 0.034 Ca2 0.7630 0.9160 0.2951 1 0.027 O6 0.2020 0.8870 0.0942 1 0.053 Ca3 0.5620 0.0640 0.0450 0.25 0.038 O7 0.2890 0.4360 0.0940 1 0.076 — ——— — —

Figure 3. Gypsum monoclinic crystal under pressure 0–1.0 GPa by DFT. (a) Relative change of a, b, c, and Vand (b) elastic

dinates and displacement parameters are seen in Table 3.

232 Density Functional Calculations - Recent Progresses of Theory and Application

4.2.1. Initial conditions and elastic constants of gypsum

XYZ Occupancy

rate

Table 3. Atomic coordinates and displacement parameters of 11 Å tobermorite [38].

shown in Figure 3.

Atomic species

constants.

4.2. Initial conditions and elastic constants of monoclinic crystals

Initial conditions of tobermorite are quite the same with that of gypsum crystal. Elastic constants of 11 Å tobermorite crystal under 0–1.0 GPa are shown in Figure 4. Elastic constants are shown in Table 5.

A comparisonal results of Shahsavari [39] are provided. Elastic constants at 0 GPa are as follows: c<sup>11</sup> = 106.63 GPa, c<sup>12</sup> = 50.37 GPa, c<sup>13</sup> = 41.09 GPa, c<sup>15</sup> = 3.50 GPa, c<sup>22</sup> = 131.67 GPa, c<sup>23</sup> = 22.78 GPa, c<sup>25</sup> = 0.78 GPa, c<sup>33</sup> = 71.45 GPa, c<sup>35</sup> = 0.83 GPa, c<sup>44</sup> = 26.03 GPa, c<sup>46</sup> = 0.02 GPa, c<sup>55</sup> = 27.61 GPa, and c<sup>66</sup> = 45.26 GPa. Thus, elastic modulus can be homogenized to compare with the results of LD C-S-H phase in nano-indentation test by Vandamme and Ulm [40].

#### 4.3. Homogenized elastic moduli of typical monoclinic structures

#### 4.3.1. Elastic modulus of monoclinic gypsum structure

Based on elastic constants, the elastic moduli of gypsum at 0 GPa are verified and averaged in Figure 5.


Table 4. Elastic coefficient Cij (GPa) of gypsum by DFT.

Figure 4. 11 Å tobermorite monoclinic crystal under pressure 0–1.0 GPa by DFT. (a) Relative change of a, b, c, and V and (b) elastic constants.


Table 5. Elastic coefficient Cij (GPa) of 11 Å tobermorite by DFT.

As gypsum shows anisotropic compressibility along three crystallographic axes with b > c > a below 5 GPa [44], the pressure region of 0–1.0 GPa is used to verify whether the performance of model under low pressure is stable. Mechanical moduli of gypsum polycrystalline are listed in Table 6.

[44] E = 50 GPa, μ = 0.45. By comparison of gypsum crystal and CH crystal, axial moduli of gypsum in x, y, and z directions are 57.75, 37.22, and 34.91 GPa, while axial moduli of Ca(OH)2 in x, y, and z directions are 93.75, 93.75, and 42.39 GPa, showing that gypsum crystal is much less

Pressure (GPa) Gv (GPa) Bv (GPa) Gr (GPa) Br (GPa) B (GPa) G (GPa) E (GPa) μ Reference [43] 26.5333 39.2556 24.8077 39.2381 25.6705 39.2469 63.2265 0.2315 0.0 22.1459 45.5208 19.7054 43.8224 20.9257 44.6716 54.2985 0.2974 0.1 22.8896 43.9624 21.7569 42.9250 22.3233 43.4437 57.1766 0.2806 0.2 19.1472 45.1906 17.4395 41.9064 18.2934 43.5485 48.1394 0.3158 0.3 21.5041 45.5718 19.3501 43.6675 20.4271 44.6197 53.1678 0.3014 0.4 21.2263 45.9146 19.5324 43.2463 20.3794 44.5805 53.0537 0.3017 0.5 22.1809 45.9026 19.4980 43.7566 20.8395 44.8296 54.1306 0.2988 0.6 19.9615 44.2709 17.7554 41.5818 18.8585 42.9264 49.3486 0.3084 0.7 22.0356 47.5189 20.1833 44.0623 21.1095 45.7906 54.8931 0.3002 0.8 22.7817 49.0659 21.3745 47.0681 22.0781 48.0670 57.4399 0.3008 0.9 22.7399 50.6242 19.4094 48.1542 21.0747 49.3892 55.3510 0.3132 1.0 25.2707 50.3430 23.4785 48.9327 24.3746 49.6379 62.8382 0.2890

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Based on elastic constants of 11 Å tobermorite crystal using GGA calculation method by DFT, bulk modulus B and shear modulus G are separately calculated by Eqs. (37)–(50) (Figure 6).

anisotropic than hydrogen-bonded layered Ca(OH)2 structure [42].

Table 6. Mechanical moduli of gypsum polycrystalline by different methods.

4.3.2. Elastic modulus of monoclinic tobermorite structure

Figure 5. Elastic moduli of gypsum crystal under pressure 0–1.0 GPa.

As an acoustic method [41] and mechanical properties [42] have been investigated, according to elastic constants of gypsum crystal [43], elastic moduli by experiment can be calculated, as shown in Table 6. Elastic moduli are as follows: Gv = 22.146 GPa, Gr = 19.705 GPa, Bv = 45.521 GPa, Br = 43.822 GPa, B = 44.672 GPa, G = 20.926 GPa, E = 54.299 GPa, and μ = 0.2974. These results are close to the plane-strain value of Young's modulus by reference

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory http://dx.doi.org/10.5772/intechopen.72301 235

Figure 5. Elastic moduli of gypsum crystal under pressure 0–1.0 GPa.


Table 6. Mechanical moduli of gypsum polycrystalline by different methods.

[44] E = 50 GPa, μ = 0.45. By comparison of gypsum crystal and CH crystal, axial moduli of gypsum in x, y, and z directions are 57.75, 37.22, and 34.91 GPa, while axial moduli of Ca(OH)2 in x, y, and z directions are 93.75, 93.75, and 42.39 GPa, showing that gypsum crystal is much less anisotropic than hydrogen-bonded layered Ca(OH)2 structure [42].

#### 4.3.2. Elastic modulus of monoclinic tobermorite structure

As gypsum shows anisotropic compressibility along three crystallographic axes with b > c > a below 5 GPa [44], the pressure region of 0–1.0 GPa is used to verify whether the performance of model under low pressure is stable. Mechanical moduli of gypsum polycrystalline are listed

Table 5. Elastic coefficient Cij (GPa) of 11 Å tobermorite by DFT.

234 Density Functional Calculations - Recent Progresses of Theory and Application

Figure 4. 11 Å tobermorite monoclinic crystal under pressure 0–1.0 GPa by DFT. (a) Relative change of a, b, c, and V and

P/GPa C11 C12 C13 C15 C22 C23 C25 C33 C35 C44 C46 C55 C66 SHA[39] 102.65 41.68 27.70 1.25 125.05 18.83 4.10 83.80 3.38 22.90 11.93 23.25 50.20 0.0 106.63 50.37 41.09 3.50 131.67 22.78 0.78 71.45 0.83 26.03 0.02 27.61 45.26 0.1 118.37 45.40 35.91 3.52 129.18 17.19 0.11 67.84 0.55 32.51 3.90 32.74 40.07 0.2 109.13 45.84 35.63 3.22 136.79 23.05 0.03 82.75 0.06 28.88 1.21 22.40 45.69 0.3 115.53 46.36 40.17 4.46 142.59 27.65 0.04 95.03 0.02 31.08 0.49 32.38 50.57 0.4 102.65 35.38 38.73 6.32 123.43 18.11 1.92 74.28 0.05 18.14 0.83 22.38 40.44 0.5 100.08 42.58 36.10 4.52 137.56 21.68 0.26 90.87 0.36 29.92 0.46 29.66 51.82 0.6 97.87 44.09 28.76 5.85 162.17 25.77 0.19 93.71 0.14 24.89 1.20 26.63 40.26 0.7 108.73 48.60 34.07 4.55 147.09 26.78 0.14 92.64 0.01 21.56 2.06 44.25 41.23 0.8 122.87 55.30 40.62 4.05 155.75 29.54 0.25 103.3 0.57 24.90 0.72 33.31 42.67 0.9 120.77 44.19 45.41 4.82 139.59 13.68 0.09 88.25 0.19 27.18 0.35 26.85 53.22 1.0 127.01 41.78 45.00 4.47 143.72 23.65 0.02 98.30 0.12 29.98 0.71 32.08 45.68

As an acoustic method [41] and mechanical properties [42] have been investigated, according to elastic constants of gypsum crystal [43], elastic moduli by experiment can be calculated, as shown in Table 6. Elastic moduli are as follows: Gv = 22.146 GPa, Gr = 19.705 GPa, Bv = 45.521 GPa, Br = 43.822 GPa, B = 44.672 GPa, G = 20.926 GPa, E = 54.299 GPa, and μ = 0.2974. These results are close to the plane-strain value of Young's modulus by reference

in Table 6.

(b) elastic constants.

Based on elastic constants of 11 Å tobermorite crystal using GGA calculation method by DFT, bulk modulus B and shear modulus G are separately calculated by Eqs. (37)–(50) (Figure 6).

Elastic moduli at 0 GPa are verified and averaged as Gv = 32.815 GPa, Bv = 59.803 GPa, Gr = 29.908 GPa, Br = 54.276 GPa, E = 79.512 GPa, and μ = 0.268. Young's modulus is about 79.512 GPa by Reuss-Voigt-Hill estimation, which is close to the simulation result of 89 GPa [45] by Pellenq and result of 78.939 GPa [39] by Shahsavari. Mechanical moduli by different methods are listed in Table 7.

5. Conclusions

Results are as follows:

μ = 0.297.

GPa, and μ = 0.268.

Acknowledgements

Author details

Jia Fu1,2\*

c<sup>55</sup> = 28.062 GPa, and c<sup>66</sup> = 28.556 GPa.

c<sup>55</sup> = 27.61 GPa, and c<sup>66</sup> = 45.26 GPa.

Thanks to Qiufeng Wang for her proofreading.

\*Address all correspondence to: fujia@xsyu.edu.cn

1 Xi'an Shiyou University, Xi 'an, China

2 INSA de Rennes, Rennes, France

Elastic constants of gypsum and tobermorite structures under a certain pressure region are calculated by DFT method, which has a certain value for both application and reference.

Elastic Constants and Homogenized Moduli of Monoclinic Structures Based on Density Functional Theory

http://dx.doi.org/10.5772/intechopen.72301

237

1. For monoclinic gypsum and tobermorite crystals, elastic coefficients are obtained in 0–1- GPa pressure range to verify the reliability of the model by comparing other literatures.

2. Elastic constants of gypsum single crystal at 0 GPa are given as follows: c<sup>11</sup> = 82.464 GPa, c<sup>12</sup> = 34.751 GPa, c<sup>13</sup> = 33.643 GPa, c<sup>15</sup> = 1.987 GPa, c<sup>22</sup> = 63.046 GPa, c<sup>23</sup> = 34.920 GPa, c<sup>25</sup> = 8.071 GPa, c<sup>33</sup> = 57.549 GPa, c<sup>35</sup> = 3.054 GPa, c<sup>44</sup> = 20.863 GPa, c<sup>46</sup> = 4.688 GPa,

3. Elastic constants of 11Å tobermorite single crystal at 0 GPa are as follows: c<sup>11</sup> = 106.63 GPa, c<sup>12</sup> = 50.37 GPa, c<sup>13</sup> = 41.09 GPa, c<sup>15</sup> = 3.50 GPa, c<sup>22</sup> = 131.67 GPa, c<sup>23</sup> = 22.78 GPa, c<sup>25</sup> = 0.78 GPa, c<sup>33</sup> = 71.45 GPa, c<sup>35</sup> = 0.83 GPa, c<sup>44</sup> = 26.03 GPa, c<sup>46</sup> = 0.02 GPa,

4. Young's modulus of gypsum is about 54.299 GPa. Elastic moduli at 0 GPa are as follows: Gv = 22.146 GPa, Gr = 19.705 GPa, Bv = 45.521 GPa, Br = 43.822 GPa, E = 54.299 GPa, and

5. Young's modulus of 11Å tobermorite is about 79.512 GPa. Elastic moduli at 0 GPa are as follows: Gv = 32.815 GPa, Bv = 59.803 GPa, Gr = 29.908 GPa, Br = 54.276 GPa, E = 79.512

Structural, elastic properties of monoclinic crystals are investigated, and Cij determination is given by DFT method. Reuss-Voigt-Hill estimation has been used for polycrystal structures

The authors greatly acknowledge the financial support for this work provided by the China Scholarship Council (CSC) and the support of start-up foundation of Xi'an Shiyou University.

and can be seen as an intermediate step in the homogenization of elastic properties.

However, these values considering the ordered Si-chain at a long range are far away from the nanoindentation experiment performed on the C-S-H phase [40, 46]. It confirms the absence of order at a long range in this phase and that the up-scaling to polycrystals cannot be done with the tobermorite model. Modeling of C-S-H structure with disordered Si chain should be fairly considered.

Figure 6. Elastic moduli of 11 Å tobermorite crystal under pressure 0–1.0 GPa.


Table 7. Mechanical moduli of 11Å tobermorite polycrystalline by different methods.

### 5. Conclusions

Elastic moduli at 0 GPa are verified and averaged as Gv = 32.815 GPa, Bv = 59.803 GPa, Gr = 29.908 GPa, Br = 54.276 GPa, E = 79.512 GPa, and μ = 0.268. Young's modulus is about 79.512 GPa by Reuss-Voigt-Hill estimation, which is close to the simulation result of 89 GPa [45] by Pellenq and result of 78.939 GPa [39] by Shahsavari. Mechanical moduli by different

However, these values considering the ordered Si-chain at a long range are far away from the nanoindentation experiment performed on the C-S-H phase [40, 46]. It confirms the absence of order at a long range in this phase and that the up-scaling to polycrystals cannot be done with the tobermorite

model. Modeling of C-S-H structure with disordered Si chain should be fairly considered.

Pressure (GPa) Bv (GPa) Br (GPa) Gv (GPa) Gr (GPa) B (GPa) G (GPa) E (GPa) μ Reference [191] 54.2133 51.6976 34.1560 28.9168 52.9555 31.5364 78.9391 0.2516 0.0 59.8066 54.2778 32.8140 29.9063 57.0399 31.3615 79.5121 0.2677 0.1 56.9306 50.1535 35.5205 33.4468 53.5438 34.4843 85.1689 0.2349 0.2 59.7454 56.4236 34.3364 31.4164 58.0846 32.8763 82.9743 0.2619 0.3 64.6138 62.4671 38.7385 36.7029 63.5388 37.7202 94.4669 0.2522 0.4 53.8665 50.9999 30.0678 26.1841 52.4333 28.1257 71.5786 0.2725 0.5 58.8039 56.9635 37.4898 34.7400 57.8831 36.1145 89.6903 0.2417 0.6 61.2236 57.0694 35.3648 32.2837 59.1442 33.8242 85.2259 0.2598 0.7 63.0391 60.0785 37.3432 34.0050 61.5598 35.6744 89.6966 0.2572 0.8 70.3213 67.4998 37.2739 34.9015 68.9022 36.0849 92.1653 0.2771 0.9 61.6837 57.8325 37.8067 33.5164 59.7599 35.6606 89.2325 0.2511 1.0 65.5442 63.4872 38.6855 36.6712 64.5147 37.7292 94.7226 0.2553

Figure 6. Elastic moduli of 11 Å tobermorite crystal under pressure 0–1.0 GPa.

Table 7. Mechanical moduli of 11Å tobermorite polycrystalline by different methods.

methods are listed in Table 7.

236 Density Functional Calculations - Recent Progresses of Theory and Application

Elastic constants of gypsum and tobermorite structures under a certain pressure region are calculated by DFT method, which has a certain value for both application and reference. Results are as follows:


Structural, elastic properties of monoclinic crystals are investigated, and Cij determination is given by DFT method. Reuss-Voigt-Hill estimation has been used for polycrystal structures and can be seen as an intermediate step in the homogenization of elastic properties.

#### Acknowledgements

The authors greatly acknowledge the financial support for this work provided by the China Scholarship Council (CSC) and the support of start-up foundation of Xi'an Shiyou University. Thanks to Qiufeng Wang for her proofreading.

### Author details

Jia Fu1,2\*

\*Address all correspondence to: fujia@xsyu.edu.cn


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**Section 5**

**Multidisciplinary Integration**

## **Multidisciplinary Integration**

**Chapter 11**

Provisional chapter

**Application of Density Functional Theory in Soil**

Application of Density Functional Theory in Soil Science

DOI: 10.5772/intechopen.74079

Soil is the basis for life and soil science is regarded as the final frontier; however, as compared to chemistry, physics, biology, and other disciplines, soil science undergoes an obviously slower development and remains almost stagnant in the past few decades, mainly due to two reasons: (1) wrong and outdated perceptions for a large portion of soil researchers; (2) complexity of soil systems that are difficult to characterize by current experimental techniques. Computer simulations have unique advantages to handle complex systems while currently, its role during soil researches is far from being recognized. In this chapter, several examples are given with respect to application of density functional theory (DFT) calculations to soil science, focusing on the adsorption of uranyl ion and SO2 onto mineral surfaces and reaction mechanisms to form acid rain. In this way, insightful clues at the atomic level are provided for the adsorption, interaction, and reactions regarding soil systems. We believe that computer simulations including DFT are the right key to unravel the complicated processes occurring in soils. More efforts of computer simulations are anticipated for soil science with aim to decipher the experimental results and probe the uncharted principles that may result

Keywords: soil science, computer simulations, density functional theory, interfacial

According to Natural Resources Conservation Service (NRCS), soil is defined as a natural body comprised of solids (mainly minerals and organic matters), liquids, and gases that occurs at the intermediate surface of the Earth, occupies space and is characterized by one or both of the following properties: horizons and layers, which are distinguishable from the initial materials as a result of addition, loss, transfer, and transformation of energy and matter or the ability to

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Jiena Yun, Qian Wang, Chang Zhu and Gang Yang

Jiena Yun, Qian Wang, Chang Zhu and Gang Yang

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74079

in a revolutionary in the near future.

adsorption, reaction mechanism, complex systems

Abstract

1. Introduction

**Science**

Provisional chapter

#### **Application of Density Functional Theory in Soil Science** Application of Density Functional Theory in Soil Science

DOI: 10.5772/intechopen.74079

Jiena Yun, Qian Wang, Chang Zhu and Gang Yang Jiena Yun, Qian Wang, Chang Zhu and Gang Yang

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74079

#### Abstract

Soil is the basis for life and soil science is regarded as the final frontier; however, as compared to chemistry, physics, biology, and other disciplines, soil science undergoes an obviously slower development and remains almost stagnant in the past few decades, mainly due to two reasons: (1) wrong and outdated perceptions for a large portion of soil researchers; (2) complexity of soil systems that are difficult to characterize by current experimental techniques. Computer simulations have unique advantages to handle complex systems while currently, its role during soil researches is far from being recognized. In this chapter, several examples are given with respect to application of density functional theory (DFT) calculations to soil science, focusing on the adsorption of uranyl ion and SO2 onto mineral surfaces and reaction mechanisms to form acid rain. In this way, insightful clues at the atomic level are provided for the adsorption, interaction, and reactions regarding soil systems. We believe that computer simulations including DFT are the right key to unravel the complicated processes occurring in soils. More efforts of computer simulations are anticipated for soil science with aim to decipher the experimental results and probe the uncharted principles that may result in a revolutionary in the near future.

Keywords: soil science, computer simulations, density functional theory, interfacial adsorption, reaction mechanism, complex systems

#### 1. Introduction

According to Natural Resources Conservation Service (NRCS), soil is defined as a natural body comprised of solids (mainly minerals and organic matters), liquids, and gases that occurs at the intermediate surface of the Earth, occupies space and is characterized by one or both of the following properties: horizons and layers, which are distinguishable from the initial materials as a result of addition, loss, transfer, and transformation of energy and matter or the ability to

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

support rooted plants in natural circumstances. Soil constitutes the basis for life and bridges the biosphere, atmosphere, hydrosphere, and geosphere. Despite these facts, apparently less attention has been given to soil science than to other disciplines such as physics, chemistry, and biology. As said by Gardner (the past president of Soil Science Society of America) [1], "not a few people mistakenly perceive that everything worth knowing about soils has already been understood, and all we need to do is merely to apply that knowledge properly. Even knowledgeable scientists assume that principles and theories developed from other disciplines can be applied to the researches of soil science in a straightforward way, requiring little imagination or creativity. In their opinions, soil science is just one of expressions for the applied physics, chemistry, or biology."

The relationship between system size and computational accuracy for representative theoretical levels is shown in Figure 1. For huge systems (100,000 or even more atoms), classical methods (Monte Carlo and Molecular Dynamics) seems to be the good choice, although the computational accuracy is relatively low; In contrast, ab initio quantum mechanical methods are restricted to relatively small systems (up to several hundred atoms or even fewer atoms) while achieve high accuracy. The computational accuracy of quantum mechanics/molecular mechanics (QM/MM) [9] and semi-empirical methods fall in-between, and QM/MM methods have recently become increasingly popular due to the satisfying computational accuracy (active sites handled by QM methods) and the easy extension to large systems (all atoms except active sites disposed by classical methods). 2013 Noble Prize in Chemistry was awarded to Karplus, Levitt, and

Let us look back to ab initio quantum mechanical methods. Strictly speaking, a large portion of DFT methods fall outside this scope; e.g., B3LYP is probably the most popular DFT method

<sup>x</sup> <sup>þ</sup> bEB<sup>88</sup>

There is a five-rung Jacob's ladder of common density functional approximations, as local spin-density approximation (LSDA), generalized-gradient approximation (GGA), meta-GGA, hyper-GGA, and random phase approximation (PRA-like functionals), see more details in [12]. As far as we know, the most widely density functionals are at GGA (e.g., B3LYP, PBE) and

Computer simulations, as discussed above, are particularly useful to handle the complex soil systems, and two applications were illustrated in this chapter. Section 2 summarized DFT

Figure 1. The theoretical landscape regarding to the balance between system size and computational accuracy.

<sup>x</sup> <sup>þ</sup> cELYP

<sup>c</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>c</sup> ELSDA

c

Application of Density Functional Theory in Soil Science http://dx.doi.org/10.5772/intechopen.74079 247

Warshel for the development of "multiscale methods for complex systems."

<sup>x</sup> <sup>þ</sup> aEHF

and its exchange-correlation functional is written as [10, 11]

xc <sup>¼</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>E</sup>LSDA

EB3LYP

meta-GGA (e.g., M06 L, M06-2X) levels.

where a, b, and c are empirical parameters.

The situation of soil science research is alarming. The core concepts of current soil science textbooks remain almost unchanged as compared to those of half a century ago. Obsolete or even incorrect standpoints are a commonplace [2]. Fortunately, a few researchers have recognized such a crisis. On the other hand, because of the complexity of systems and co-function of multiple factors, it seems challenging for experimental techniques to in-situ characterize "real" soil properties and processes; in addition, the experimental results from one lab may not be reproducible by others, since soil samples of different areas or even different batches may vary significantly. Computer simulations have unique advantages within this context: (1) probing the various influencing factors one by one; e.g., six factors (identity of heteroatoms, crystallographically distinct T sites, structural alterations, quantity of negative charges, distance from charge centers to metal ions and source of negative charges) were identified to affect the adsorption of metal ions at clay surfaces, and their respective contributions were estimated by density functional theory (DFT) calculations. The quantity of negative charges is the foremost factor that controls the adsorption processes, while other factors in certain circumstances can also play a critical role [3]. The adsorption strengths and numbers of all metal ions increase in a direct proportion to the intensities of electric fields [4]; (2) providing useful and detailed information at the femtosecond scale such as how ions from aqueous solutions diffuse toward to clay surfaces [5]; (3) understanding the adsorption, interaction and reaction processes at the atomic level such as how metal ions interact with surface-O atoms and respond to the increase of electric fields. Based on Hirshfeld, Mulliken, and NBO charge analyses, we found that polarization rather than electrostatic interactions are more likely to result in the pronounced cation-specific effects at clay surfaces [6]; (4) unraveling the exact reaction mechanism by comparing the structural and (especially) activation barriers of competing paths. This can be considered as an extension of (3). There are a plethora of competing reactions occurring in soils; e.g., with DFT calculations, it was clarified that Mn4+ rather than Mn3+ sites are more reactive for the oxidation of As3+ and the oxidation processes are significantly blocked by As5+ complexes [7]. A more convictive example is the mechanistic study of Brönsted acid-catalyzed conversion of biomass sugars [8]. More than 120 reaction paths were explored, and the low reactivity and selectivity of glucose conversions were clearly addressed: unlike fructose that prefers to dehydrate at the anomeric O2H group and initiates a sequence of facile reaction steps toward 5-hydroxymethyl-2-furfural (HMF), the less reactive sites in glucose (O2H and O3H) produces levulic acid not involving fructose and HMF intermediates, while the most reactive O1H site leads to humin precursors or reversion products [8].

The relationship between system size and computational accuracy for representative theoretical levels is shown in Figure 1. For huge systems (100,000 or even more atoms), classical methods (Monte Carlo and Molecular Dynamics) seems to be the good choice, although the computational accuracy is relatively low; In contrast, ab initio quantum mechanical methods are restricted to relatively small systems (up to several hundred atoms or even fewer atoms) while achieve high accuracy. The computational accuracy of quantum mechanics/molecular mechanics (QM/MM) [9] and semi-empirical methods fall in-between, and QM/MM methods have recently become increasingly popular due to the satisfying computational accuracy (active sites handled by QM methods) and the easy extension to large systems (all atoms except active sites disposed by classical methods). 2013 Noble Prize in Chemistry was awarded to Karplus, Levitt, and Warshel for the development of "multiscale methods for complex systems."

Let us look back to ab initio quantum mechanical methods. Strictly speaking, a large portion of DFT methods fall outside this scope; e.g., B3LYP is probably the most popular DFT method and its exchange-correlation functional is written as [10, 11]

$$E\_{\rm xc}^{B3LYP} = (1 - a)E\_{\rm x}^{LSDA} + aE\_{\rm x}^{HF} + bE\_{\rm x}^{B88} + cE\_{\rm c}^{LYP} + (1 - c)E\_{\rm c}^{LSDA}$$

where a, b, and c are empirical parameters.

support rooted plants in natural circumstances. Soil constitutes the basis for life and bridges the biosphere, atmosphere, hydrosphere, and geosphere. Despite these facts, apparently less attention has been given to soil science than to other disciplines such as physics, chemistry, and biology. As said by Gardner (the past president of Soil Science Society of America) [1], "not a few people mistakenly perceive that everything worth knowing about soils has already been understood, and all we need to do is merely to apply that knowledge properly. Even knowledgeable scientists assume that principles and theories developed from other disciplines can be applied to the researches of soil science in a straightforward way, requiring little imagination or creativity. In their opinions, soil science is just one of expressions for the applied physics,

246 Density Functional Calculations - Recent Progresses of Theory and Application

The situation of soil science research is alarming. The core concepts of current soil science textbooks remain almost unchanged as compared to those of half a century ago. Obsolete or even incorrect standpoints are a commonplace [2]. Fortunately, a few researchers have recognized such a crisis. On the other hand, because of the complexity of systems and co-function of multiple factors, it seems challenging for experimental techniques to in-situ characterize "real" soil properties and processes; in addition, the experimental results from one lab may not be reproducible by others, since soil samples of different areas or even different batches may vary significantly. Computer simulations have unique advantages within this context: (1) probing the various influencing factors one by one; e.g., six factors (identity of heteroatoms, crystallographically distinct T sites, structural alterations, quantity of negative charges, distance from charge centers to metal ions and source of negative charges) were identified to affect the adsorption of metal ions at clay surfaces, and their respective contributions were estimated by density functional theory (DFT) calculations. The quantity of negative charges is the foremost factor that controls the adsorption processes, while other factors in certain circumstances can also play a critical role [3]. The adsorption strengths and numbers of all metal ions increase in a direct proportion to the intensities of electric fields [4]; (2) providing useful and detailed information at the femtosecond scale such as how ions from aqueous solutions diffuse toward to clay surfaces [5]; (3) understanding the adsorption, interaction and reaction processes at the atomic level such as how metal ions interact with surface-O atoms and respond to the increase of electric fields. Based on Hirshfeld, Mulliken, and NBO charge analyses, we found that polarization rather than electrostatic interactions are more likely to result in the pronounced cation-specific effects at clay surfaces [6]; (4) unraveling the exact reaction mechanism by comparing the structural and (especially) activation barriers of competing paths. This can be considered as an extension of (3). There are a plethora of competing reactions occurring in soils; e.g., with DFT calculations, it was clarified that Mn4+ rather than Mn3+ sites are more reactive for the oxidation of As3+ and the oxidation processes are significantly blocked by As5+ complexes [7]. A more convictive example is the mechanistic study of Brönsted acid-catalyzed conversion of biomass sugars [8]. More than 120 reaction paths were explored, and the low reactivity and selectivity of glucose conversions were clearly addressed: unlike fructose that prefers to dehydrate at the anomeric O2H group and initiates a sequence of facile reaction steps toward 5-hydroxymethyl-2-furfural (HMF), the less reactive sites in glucose (O2H and O3H) produces levulic acid not involving fructose and HMF intermediates, while the most reactive

O1H site leads to humin precursors or reversion products [8].

chemistry, or biology."

There is a five-rung Jacob's ladder of common density functional approximations, as local spin-density approximation (LSDA), generalized-gradient approximation (GGA), meta-GGA, hyper-GGA, and random phase approximation (PRA-like functionals), see more details in [12]. As far as we know, the most widely density functionals are at GGA (e.g., B3LYP, PBE) and meta-GGA (e.g., M06 L, M06-2X) levels.

Computer simulations, as discussed above, are particularly useful to handle the complex soil systems, and two applications were illustrated in this chapter. Section 2 summarized DFT

Figure 1. The theoretical landscape regarding to the balance between system size and computational accuracy.

calculated results of uranyl adsorption onto mineral surfaces and Section 3 elaborately discussed DFT calculated results of SO2 (one of the main precursors for acid rain) adsorption and formation mechanisms of acid rain. In the end, concluding remarks were provided and some suggestions for DFT applications to the future soil researches were presented as well. A more critical role of computer simulations should have played in soil science, and this chapter aims to arouse the attention of general soil researchers regarding to the applications of computer simulations. More efforts in this regard are greatly beneficial to decipher the experimental results and probe the uncharted principles that will result in a revolutionary for soil science in the near future.

(OH and CO3

plex, where UO2

and water molecules are in circles.

2.930 Å for [UO2(OH)3]

molecules and uranyl ion (UO2

<sup>2</sup>) than H2O in the first coordination sphere. The bond lengths of the U6+ center

2+) that strengthen the adsorption processes.

2+ bonds directly to the lying surface-OH group and has an adsorption energy

2+, 2.808 and

249

<sup>2</sup>, suggesting that the adsorption

Application of Density Functional Theory in Soil Science http://dx.doi.org/10.5772/intechopen.74079

2+ forms two direct bonds with

and two Ob atoms are respectively calculated at 2.248 and 2.333 Å for [UO2(H2O)3]

, 2.852 and 2.960 Å for [UO2(CO3)2]

strengths of the uranyl ion with rutile(110) surface are significantly impaired in the latter two cases. Adsorption of these uranyl complexes on partially hydrated surfaces were then studied, which exhibit stronger interactions and are attributed to the formation of H-bonds between H2O

Kaolinite is the major constituent of sedimentary clay rocks, and adsorption of uranyl species onto its (001) surface [20, 21] and (010) edge surface [22, 23] have been theoretically studied. The adsorption of uranyl species on tetrahedral SiO4 surfaces is thermodynamically unfavorable, with the adsorption energies of inner- and outer-sphere complexes being, respectively, 239 and 206 kJ/mol [20]; in contrast, the adsorption at octahedral AlO6 surfaces is preferred due to the presence of upright (perpendicular to the surface) and lying (parallel to the surface) OH groups. The most stable configuration corresponds to an inner-sphere monodentate com-

of 155 kJ/mol. Martorell et al. [21] continued to study the adsorption of uranyl ion on the bare

Figure 3. (a) Three sites for uranyl adsorption on gibbsite(001) surface and (b) top and (c) side views of the optimal adsorption complex. U, Al, O, and H are, respectively, in blue, pink, red, and white, and H-bonds are in black dash lines

and solvated octahedral AlO6 surfaces of kaolinite, where UO2

#### 2. Uranyl adsorption onto mineral surfaces

Release of radionuclides into the environment seriously threatens the ecosystem and human health, and adsorption of radionuclides onto mineral surfaces significantly affects their migration and transport into the environment [13]. Accordingly, knowledge about the interaction between radionuclides and minerals is essential for the long-term risk assessment of radioactive waste repositories. Uranium is usually present in the uranyl (UO2 2+) form [14, 15] and DFT calculations have been conducted to understand the adsorption of UO2 2+ onto mineral surfaces [16–25].

Perron et al. [16] examined UO2 2+ adsorption onto rutile(110) surface, the most stable face of natural rutile reported by Jones and Hockey [26, 27]. UO2 2+ forms a bidentate inner sphere complex with three H2O molecules to fill its first hydration shell, see Figure 2. There exist two types of surface-O atoms, as Ot (terminal-O) and Ob (bridging-O) that are singly and doubly coordinated, respectively. As a result, a total of three adsorption structures are produced: UO2 2+ coordinated to two Ob atoms (bb), two Ot atoms (tt) and both of Ob and Ot atoms (bt). The bb mode is the most preferred and has a lower energy than the bt and tt modes by 5.0 and 13.6 kJ/mol, respectively.

In addition to the [UO2(H2O)n] 2+ adsorption complexes, other uranyl species containing anionic ligands such as OH and CO3 <sup>2</sup> also play a critical role in the environmental circumstances. Pan and co-workers [17] investigated the effects of different ligands (H2O, OH, CO3 <sup>2</sup>) on the adsorption on rutile(110) surface. Note that only the most stable adsorption mode (bb) has been taken into account therein. The uranyl ion (UO2 2+) interacts more strongly with anionic ligands

Figure 2. Adsorption structures for uranyl ions on rutile(110) surface: tt (Left), bb (Middle), and bt (Right). U, Ti, O, and H are respectively in yellow, blue, red, and white, and H atoms to saturate surface-O atoms are not shown for clarity.

(OH and CO3 <sup>2</sup>) than H2O in the first coordination sphere. The bond lengths of the U6+ center and two Ob atoms are respectively calculated at 2.248 and 2.333 Å for [UO2(H2O)3] 2+, 2.808 and 2.930 Å for [UO2(OH)3] , 2.852 and 2.960 Å for [UO2(CO3)2] <sup>2</sup>, suggesting that the adsorption strengths of the uranyl ion with rutile(110) surface are significantly impaired in the latter two cases. Adsorption of these uranyl complexes on partially hydrated surfaces were then studied, which exhibit stronger interactions and are attributed to the formation of H-bonds between H2O molecules and uranyl ion (UO2 2+) that strengthen the adsorption processes.

calculated results of uranyl adsorption onto mineral surfaces and Section 3 elaborately discussed DFT calculated results of SO2 (one of the main precursors for acid rain) adsorption and formation mechanisms of acid rain. In the end, concluding remarks were provided and some suggestions for DFT applications to the future soil researches were presented as well. A more critical role of computer simulations should have played in soil science, and this chapter aims to arouse the attention of general soil researchers regarding to the applications of computer simulations. More efforts in this regard are greatly beneficial to decipher the experimental results and probe the uncharted principles that will result in a revolutionary for soil science

Release of radionuclides into the environment seriously threatens the ecosystem and human health, and adsorption of radionuclides onto mineral surfaces significantly affects their migration and transport into the environment [13]. Accordingly, knowledge about the interaction between radionuclides and minerals is essential for the long-term risk assessment of radioactive waste

complex with three H2O molecules to fill its first hydration shell, see Figure 2. There exist two types of surface-O atoms, as Ot (terminal-O) and Ob (bridging-O) that are singly and doubly coordinated, respectively. As a result, a total of three adsorption structures are produced: UO2

coordinated to two Ob atoms (bb), two Ot atoms (tt) and both of Ob and Ot atoms (bt). The bb mode is the most preferred and has a lower energy than the bt and tt modes by 5.0 and 13.6 kJ/mol,

adsorption on rutile(110) surface. Note that only the most stable adsorption mode (bb) has been

Figure 2. Adsorption structures for uranyl ions on rutile(110) surface: tt (Left), bb (Middle), and bt (Right). U, Ti, O, and H are respectively in yellow, blue, red, and white, and H atoms to saturate surface-O atoms are not shown for clarity.

and co-workers [17] investigated the effects of different ligands (H2O, OH, CO3

2+) form [14, 15] and DFT calculations

2+ forms a bidentate inner sphere

2+

<sup>2</sup>) on the

2+ onto mineral surfaces [16–25].

2+ adsorption onto rutile(110) surface, the most stable face of

2+ adsorption complexes, other uranyl species containing anionic

<sup>2</sup> also play a critical role in the environmental circumstances. Pan

2+) interacts more strongly with anionic ligands

in the near future.

2. Uranyl adsorption onto mineral surfaces

248 Density Functional Calculations - Recent Progresses of Theory and Application

repositories. Uranium is usually present in the uranyl (UO2

have been conducted to understand the adsorption of UO2

natural rutile reported by Jones and Hockey [26, 27]. UO2

Perron et al. [16] examined UO2

In addition to the [UO2(H2O)n]

ligands such as OH and CO3

taken into account therein. The uranyl ion (UO2

respectively.

Kaolinite is the major constituent of sedimentary clay rocks, and adsorption of uranyl species onto its (001) surface [20, 21] and (010) edge surface [22, 23] have been theoretically studied. The adsorption of uranyl species on tetrahedral SiO4 surfaces is thermodynamically unfavorable, with the adsorption energies of inner- and outer-sphere complexes being, respectively, 239 and 206 kJ/mol [20]; in contrast, the adsorption at octahedral AlO6 surfaces is preferred due to the presence of upright (perpendicular to the surface) and lying (parallel to the surface) OH groups. The most stable configuration corresponds to an inner-sphere monodentate complex, where UO2 2+ bonds directly to the lying surface-OH group and has an adsorption energy of 155 kJ/mol. Martorell et al. [21] continued to study the adsorption of uranyl ion on the bare and solvated octahedral AlO6 surfaces of kaolinite, where UO2 2+ forms two direct bonds with

Figure 3. (a) Three sites for uranyl adsorption on gibbsite(001) surface and (b) top and (c) side views of the optimal adsorption complex. U, Al, O, and H are, respectively, in blue, pink, red, and white, and H-bonds are in black dash lines and water molecules are in circles.

the surface-O atoms of deprotonated OH groups. Two different adsorption structures are assigned according to the coordination of surface-O atoms: two surface-O atoms connected to one and two Al atoms are designated to be AlOO (short-bridge site) and AlO-AlO (long-bridge site), respectively. The formation of adsorption complex at the short-bridge site (AlOO) requires an energy of 195 kJ/mol and is obviously less than at the long-bridge site (AlO-AlO, 261 kJ/mol). Thus, the uranyl ion prefers to adsorption at the short-bridge site; furthermore, similar trends for uranyl adsorption remain when including solvent effects by adding a monolayer of water molecules.

proposed. During 1982–1999, SO2 emissions have reduced by approximately 65% in Europe and 40% in the United States, and SO2 emissions in China decline in the late 1990s while again increase after then. DFT calculations provide useful information about the adsorption of SO2 onto mineral surfaces as well as reaction mechanisms that seem difficult to capture by current experimental techniques [34–45], which are, however, critical to understand the formation of acid rain at the molecular level and to remediate the ecosystem. Clay minerals, such as alumina (Al2O3), iron oxides (FexOy), are good candidates for the adsorption of acid components from acid rain and then convert them into less hazardous compounds. Lo et al. [35] studied the adsorption of SO2 on clean (100), dehydrated (110), and hydrated (110) surfaces of γ-Al2O3, finding that significant adsorption differences exist for the various surfaces and the calculated adsorption energies (13 to –85 kcal/mol) are consistent with experimental results. The γ-Al2O3(100) surface is composed by bridging-O and five-coordinated Al atoms, and a total of five stable configurations are produced for SO2 adsorption (Figure 4). The feeble interaction between S and surface-O atoms results in the physisorption configuration (CM3) with the S-O distance of 2.915 Å, and the corresponding binding energy is very small (2.0 kcal/mol). The interaction between O@SO2 and Al atoms leads to a chemisorption state named CM4, and the O-Al bond distance and adsorption energy are 2.123 Å and 23.9 kcal/mol, respectively. The other three configurations are also ascribed to chemisorption. In CM5, one O@SO2 atom is coordinated to two Al atoms in the vicinity of an octahedral vacancy, and

Application of Density Functional Theory in Soil Science http://dx.doi.org/10.5772/intechopen.74079 251

Figure 4. Optimized configurations of SO2 adsorption on (a) dehydrated (100) and (b) hydrated (110) surfaces of γ-Al2O3,

where Al, H, O, S, and O attached to S are presented in green, pink, white, blue, and red balls, respectively.

Gibbsite is a primary mineral form of aluminum hydroxide, and adsorption of the uranyl ion onto its (001) surface [18, 24] and edge surface [25] have been investigated, similar to the situation of kaolinite discussed above. Figure 3a depicts three potential adsorption sites for uranyl adsorption: Sites I and III hold almost the same O-O distances (about 2.7 Å), while site II shows an apparently elongated O-O distance (about 3.4 Å). Site I represents the lowestenergy adsorption structure (Figure 3), and the adsorption structures of sites II and III are less stable with relative energies being 22.2 and 46.3 kJ/mol, respectively. Three H-bonds constructed between surface-OH groups and UO2 2+ center as shown in Figure 3 stabilize the interactions between uranyl ion and gibbsite surfaces.

#### 3. Acid rain

In 1852, Smith demonstrated the relationship between acid rain and atmospheric pollution in Manchester, and after 20 years (i.e., 1872), he coined the term "acid rain." Now acid rain has become a popular term and one of the world's biggest environmental concerns, especially in North America, Europe, and China [28]. Acid rain refers to any form of precipitations with acidic components that fall to the ground from the atmosphere, including rain, snow, flog, hail, or even dust that is acidic. It results mainly from SO2 and NOx emissions to the atmosphere and the further transport by wind and air currents, during when SO2 and NOx react with water, oxygen, and other substances leading to the formation of sulfuric (H2SO4) and nitric (HNO3) acids. The pH of acid rain is approximately 5.6 and since 1940s, researchers began to recognize its strong impacts on the ecosystem and human health so that soils, freshwaters, forests, and buildings will be damaged. With regard to soils, acid rain inhibits the decomposition of organic matter [29], fixation of nitrogen [30], elution of calcium (Ca2+), magnesium (Mg2+), potassium (K+ ), and other nutrients [31]. As a result, soil fertility and microbial activity show an obvious reduction [32]. Geochemical modeling indicated that Ca2+ leaching in marble due to acid rain neutralization approximates 0.158 mmol/L, in contrast to 10.5 mmol/L by dry deposition, and the corresponding Cu2+ losses in bronze are ca. 0.21 and 47.3 mmol/L, respectively [33].

As aforementioned, SO2 emissions are one of the principal causes of acid rain and also represent a primary source of atmospheric aerosols, which can lead to respiratory diseases, premature deaths, and even climate changes by affecting the properties of clouds and the balance of solar radiation. Therefore, it is of great significance to convert SO2 to other less contaminated compounds, and a number of measures to control SO2 emissions have been proposed. During 1982–1999, SO2 emissions have reduced by approximately 65% in Europe and 40% in the United States, and SO2 emissions in China decline in the late 1990s while again increase after then. DFT calculations provide useful information about the adsorption of SO2 onto mineral surfaces as well as reaction mechanisms that seem difficult to capture by current experimental techniques [34–45], which are, however, critical to understand the formation of acid rain at the molecular level and to remediate the ecosystem. Clay minerals, such as alumina (Al2O3), iron oxides (FexOy), are good candidates for the adsorption of acid components from acid rain and then convert them into less hazardous compounds. Lo et al. [35] studied the adsorption of SO2 on clean (100), dehydrated (110), and hydrated (110) surfaces of γ-Al2O3, finding that significant adsorption differences exist for the various surfaces and the calculated adsorption energies (13 to –85 kcal/mol) are consistent with experimental results.

the surface-O atoms of deprotonated OH groups. Two different adsorption structures are assigned according to the coordination of surface-O atoms: two surface-O atoms connected to one and two Al atoms are designated to be AlOO (short-bridge site) and AlO-AlO (long-bridge site), respectively. The formation of adsorption complex at the short-bridge site (AlOO) requires an energy of 195 kJ/mol and is obviously less than at the long-bridge site (AlO-AlO, 261 kJ/mol). Thus, the uranyl ion prefers to adsorption at the short-bridge site; furthermore, similar trends for uranyl adsorption remain when including solvent effects by adding a mono-

Gibbsite is a primary mineral form of aluminum hydroxide, and adsorption of the uranyl ion onto its (001) surface [18, 24] and edge surface [25] have been investigated, similar to the situation of kaolinite discussed above. Figure 3a depicts three potential adsorption sites for uranyl adsorption: Sites I and III hold almost the same O-O distances (about 2.7 Å), while site II shows an apparently elongated O-O distance (about 3.4 Å). Site I represents the lowestenergy adsorption structure (Figure 3), and the adsorption structures of sites II and III are less stable with relative energies being 22.2 and 46.3 kJ/mol, respectively. Three H-bonds

In 1852, Smith demonstrated the relationship between acid rain and atmospheric pollution in Manchester, and after 20 years (i.e., 1872), he coined the term "acid rain." Now acid rain has become a popular term and one of the world's biggest environmental concerns, especially in North America, Europe, and China [28]. Acid rain refers to any form of precipitations with acidic components that fall to the ground from the atmosphere, including rain, snow, flog, hail, or even dust that is acidic. It results mainly from SO2 and NOx emissions to the atmosphere and the further transport by wind and air currents, during when SO2 and NOx react with water, oxygen, and other substances leading to the formation of sulfuric (H2SO4) and nitric (HNO3) acids. The pH of acid rain is approximately 5.6 and since 1940s, researchers began to recognize its strong impacts on the ecosystem and human health so that soils, freshwaters, forests, and buildings will be damaged. With regard to soils, acid rain inhibits the decomposition of organic matter [29],

fixation of nitrogen [30], elution of calcium (Ca2+), magnesium (Mg2+), potassium (K+

ponding Cu2+ losses in bronze are ca. 0.21 and 47.3 mmol/L, respectively [33].

nutrients [31]. As a result, soil fertility and microbial activity show an obvious reduction [32]. Geochemical modeling indicated that Ca2+ leaching in marble due to acid rain neutralization approximates 0.158 mmol/L, in contrast to 10.5 mmol/L by dry deposition, and the corres-

As aforementioned, SO2 emissions are one of the principal causes of acid rain and also represent a primary source of atmospheric aerosols, which can lead to respiratory diseases, premature deaths, and even climate changes by affecting the properties of clouds and the balance of solar radiation. Therefore, it is of great significance to convert SO2 to other less contaminated compounds, and a number of measures to control SO2 emissions have been

2+ center as shown in Figure 3 stabilize the

), and other

layer of water molecules.

3. Acid rain

constructed between surface-OH groups and UO2

interactions between uranyl ion and gibbsite surfaces.

250 Density Functional Calculations - Recent Progresses of Theory and Application

The γ-Al2O3(100) surface is composed by bridging-O and five-coordinated Al atoms, and a total of five stable configurations are produced for SO2 adsorption (Figure 4). The feeble interaction between S and surface-O atoms results in the physisorption configuration (CM3) with the S-O distance of 2.915 Å, and the corresponding binding energy is very small (2.0 kcal/mol). The interaction between O@SO2 and Al atoms leads to a chemisorption state named CM4, and the O-Al bond distance and adsorption energy are 2.123 Å and 23.9 kcal/mol, respectively. The other three configurations are also ascribed to chemisorption. In CM5, one O@SO2 atom is coordinated to two Al atoms in the vicinity of an octahedral vacancy, and

Figure 4. Optimized configurations of SO2 adsorption on (a) dehydrated (100) and (b) hydrated (110) surfaces of γ-Al2O3, where Al, H, O, S, and O attached to S are presented in green, pink, white, blue, and red balls, respectively.

CM7 and CM8 can be considered to generate from CM5 conversion and recombination. As compared to CM5 and CM7, the adsorption configuration CM8 possesses a superior symmetry, and both O@SO2 atoms participate in the formation of direct bonds with the Al atoms. The adsorption energies of SO2 are calculated to be 39.2, 15.1, and 45.2 kcal/mol, respectively, for CM5, CM7, and CM8. In consequence, three types of sulfite (SO3 <sup>2</sup>) are produced during the adsorption of SO2 onto γ-Al2O3(100) surface. For all adsorption configurations including three with positive adsorption energies (CM1: 1.0 kcal/mol, CM2: 2.7 kcal/mol, and CM6: 21.4 kcal/mol), no direct coupling is detected between S@SO2 and Al atoms. When γ-Al2O3(110) surface is hydrated, five stable adsorption configurations arise that are distinct from dehydrated condition: two physisorption modes (HM1 and HM2) and three chemisorption modes (HM3, HM4, and HM5), see Figure 4. HM1 and HM2 are structurally similar in that their S atoms are coordinated to a surface hydroxyl, while the coordination numbers of their Al atoms are different from each other. HM3 is produced by interaction of O@SO2 atom with five-fold Al sites. HM4 and HM5 contain the sulfite species where the S atom is coordinated with surface-O atoms. The adsorption energies are calculated to be 20.4, 25.3, 31.1, 17.5, and 35.0 kcal/mol, respectively, for HM1, HM2, HM3, HM4, and HM5. In consequence, HM5 with formation of the sulfite species represents the lowest-energy adsorption configuration, which is the same as in dehydrated condition (CM8). Two IR peaks at 1214 and 1349 cm<sup>1</sup> are assigned to the sulfate species, which can be been finely interpreted by DFT calculated results.

Goethite (α-FeO(OH)), which can be found in soils and other low-temperature environments, is an iron-bearing hydroxide. Because of the considerable adsorption capacity for organic acids and anions, goethite has also been widely used in environmental remediation and protection [35]. Zubieta et al. [37] investigated the adsorption of SO2 on partially and fully hydrated (110) surfaces of goethite and obtained eight stable products: six sulfite, one bisulfate, and one sulfate (Figure 5). The six adsorption structures containing sulfite species, created only on two types of partially dehydrated goethite surfaces, are further divided into two monodentric mononuclear (MdMn) and four bidentate (Bd) configurations. In the MdMn configurations (I and II), the S-OFe distances are elongated as compared to the other S-O distances, and they display two symmetrical stretching modes (OSO and OSOFe) centered at ca. 1126 and 976 cm<sup>1</sup> . In BdBn configurations (I and II), the two S-OFe distances are approximately 1.62 Å, and although with similar geometries and stretching modes, the vibrational frequencies deviate significantly from those of MdMn configurations and fall at around 672 and 661 cm<sup>1</sup> . In BdPn configurations (I and II), one S-OFe distance is optimized at 1.75 Å and lengthened as compared to those of BdPn configurations, while the other S-OFe distance equals 1.55 Å and is obviously contracted. The Bader analyses indicate that all sulfite species carry approximately 1.4 |e| charges.

surface of goethite. In this way, two O atoms are singly coordinated and one O atom is threefold coordinated with the Fe ions of goethite(110) surface. The sulfate species is corroborated by one H-bond (1.81 Å) and carries 1.6 |e| charges according to the Bader charge analyses. A stable sulfite structure can be produced under identical conditions, whereas its adsorption energy is obviously less, indicating that the formation of the sulfate species is significantly

Figure 5. Optimized configurations for the adsorption of SO2 on partially hydrated goethite(110) surfaces forming the

Application of Density Functional Theory in Soil Science http://dx.doi.org/10.5772/intechopen.74079 253

The adsorption of SO2 onto the Cu(100), MgO and carbon surfaces was discussed as well, which may provide insightful clues for resembling processes onto mineral surfaces. It was proposed that SO2 and H2O are co-adsorbed onto Cu(100) surfaces [30], through the direct coupling of Cu atom with S and O@H2O atoms. The Cu-O distances ascend in the order of coadsorption of SO2 and H2O < adsorption of only SO2 < < adsorption of only H2O. Accordingly, the interaction between SO2 and Cu(100) surface is stronger than that of H2O, and the coadsorption of SO2 conduces to the enhanced interaction of H2O with Cu(100) surface. At the same time, the Cu-S distance of the co-adsorption configuration is optimized at 2.385 Å and is shorter than that with only SO2 adsorption. That is, water exhibits a promoting effect for the adsorption of SO2 on Cu(100) surface, as corroborated by the calculated adsorption energies. Eid and collaborators [41] found that as compared to regular MgO surface, the adsorption capacity of SO2 at MgO(Fs-center) defects is higher, and MgO(Fs-center) corresponds to an enhanced catalytic activity. With regard to pure carbon materials, SO2 is physisorbed and van der Waals (vdW) is the driving force therein [42]. When carbon materials are modified with functional groups such as carboxyl, lactone, or/and phenolic hydroxyl, the adsorption strengths of SO2 are enhanced pronouncedly, especially for the sites at edge surfaces. In addition, these functional groups show little effects on SO2 adsorption, suggesting that the enhanced adsorp-

tion is mainly due to regulation of carbon surface properties.

sulfite, bisulfite and sulfate species (violet = Fe, red = O, white = H, yellow = S).

preferred.

The bisulfate species is formed when the S atom constructs direct bonds with μ1-OH (Figure 5), which is further stabilized by two H-bonds (1.77 and 1.81 Å). A similar bisulfate species was observed by Liu et al. [35] during the adsorption of SO2 on γ-Al2O3 surfaces. The Bader analyses indicate that the HSO3 species (bisulfate) carries 0.8 |e| charges. The computational sulfite and bisulfate species have IR spectra that are consistent with experimental results [37, 38]. The sulfate species emerges only when two water molecules are deprived from the (110)

CM7 and CM8 can be considered to generate from CM5 conversion and recombination. As compared to CM5 and CM7, the adsorption configuration CM8 possesses a superior symmetry, and both O@SO2 atoms participate in the formation of direct bonds with the Al atoms. The adsorption energies of SO2 are calculated to be 39.2, 15.1, and 45.2 kcal/mol, respectively,

the adsorption of SO2 onto γ-Al2O3(100) surface. For all adsorption configurations including three with positive adsorption energies (CM1: 1.0 kcal/mol, CM2: 2.7 kcal/mol, and CM6: 21.4 kcal/mol), no direct coupling is detected between S@SO2 and Al atoms. When γ-Al2O3(110) surface is hydrated, five stable adsorption configurations arise that are distinct from dehydrated condition: two physisorption modes (HM1 and HM2) and three chemisorption modes (HM3, HM4, and HM5), see Figure 4. HM1 and HM2 are structurally similar in that their S atoms are coordinated to a surface hydroxyl, while the coordination numbers of their Al atoms are different from each other. HM3 is produced by interaction of O@SO2 atom with five-fold Al sites. HM4 and HM5 contain the sulfite species where the S atom is coordinated with surface-O atoms. The adsorption energies are calculated to be 20.4, 25.3, 31.1, 17.5, and 35.0 kcal/mol, respectively, for HM1, HM2, HM3, HM4, and HM5. In consequence, HM5 with formation of the sulfite species represents the lowest-energy adsorption configuration, which is the same as in dehydrated condition (CM8). Two IR peaks at 1214 and 1349 cm<sup>1</sup> are assigned to the sulfate species, which can be been finely interpreted by DFT

Goethite (α-FeO(OH)), which can be found in soils and other low-temperature environments, is an iron-bearing hydroxide. Because of the considerable adsorption capacity for organic acids and anions, goethite has also been widely used in environmental remediation and protection [35]. Zubieta et al. [37] investigated the adsorption of SO2 on partially and fully hydrated (110) surfaces of goethite and obtained eight stable products: six sulfite, one bisulfate, and one sulfate (Figure 5). The six adsorption structures containing sulfite species, created only on two types of partially dehydrated goethite surfaces, are further divided into two monodentric mononuclear (MdMn) and four bidentate (Bd) configurations. In the MdMn configurations (I and II), the S-OFe distances are elongated as compared to the other S-O distances, and they display two symmetrical stretching modes (OSO and OSOFe) centered at ca. 1126 and

. In BdBn configurations (I and II), the two S-OFe distances are approximately 1.62 Å,

and although with similar geometries and stretching modes, the vibrational frequencies deviate significantly from those of MdMn configurations and fall at around 672 and 661 cm<sup>1</sup>

BdPn configurations (I and II), one S-OFe distance is optimized at 1.75 Å and lengthened as compared to those of BdPn configurations, while the other S-OFe distance equals 1.55 Å and is obviously contracted. The Bader analyses indicate that all sulfite species carry approximately

The bisulfate species is formed when the S atom constructs direct bonds with μ1-OH (Figure 5), which is further stabilized by two H-bonds (1.77 and 1.81 Å). A similar bisulfate species was observed by Liu et al. [35] during the adsorption of SO2 on γ-Al2O3 surfaces. The Bader analyses indicate that the HSO3 species (bisulfate) carries 0.8 |e| charges. The computational sulfite and bisulfate species have IR spectra that are consistent with experimental results [37, 38]. The sulfate species emerges only when two water molecules are deprived from the (110)

<sup>2</sup>) are produced during

. In

for CM5, CM7, and CM8. In consequence, three types of sulfite (SO3

252 Density Functional Calculations - Recent Progresses of Theory and Application

calculated results.

976 cm<sup>1</sup>

1.4 |e| charges.

Figure 5. Optimized configurations for the adsorption of SO2 on partially hydrated goethite(110) surfaces forming the sulfite, bisulfite and sulfate species (violet = Fe, red = O, white = H, yellow = S).

surface of goethite. In this way, two O atoms are singly coordinated and one O atom is threefold coordinated with the Fe ions of goethite(110) surface. The sulfate species is corroborated by one H-bond (1.81 Å) and carries 1.6 |e| charges according to the Bader charge analyses. A stable sulfite structure can be produced under identical conditions, whereas its adsorption energy is obviously less, indicating that the formation of the sulfate species is significantly preferred.

The adsorption of SO2 onto the Cu(100), MgO and carbon surfaces was discussed as well, which may provide insightful clues for resembling processes onto mineral surfaces. It was proposed that SO2 and H2O are co-adsorbed onto Cu(100) surfaces [30], through the direct coupling of Cu atom with S and O@H2O atoms. The Cu-O distances ascend in the order of coadsorption of SO2 and H2O < adsorption of only SO2 < < adsorption of only H2O. Accordingly, the interaction between SO2 and Cu(100) surface is stronger than that of H2O, and the coadsorption of SO2 conduces to the enhanced interaction of H2O with Cu(100) surface. At the same time, the Cu-S distance of the co-adsorption configuration is optimized at 2.385 Å and is shorter than that with only SO2 adsorption. That is, water exhibits a promoting effect for the adsorption of SO2 on Cu(100) surface, as corroborated by the calculated adsorption energies. Eid and collaborators [41] found that as compared to regular MgO surface, the adsorption capacity of SO2 at MgO(Fs-center) defects is higher, and MgO(Fs-center) corresponds to an enhanced catalytic activity. With regard to pure carbon materials, SO2 is physisorbed and van der Waals (vdW) is the driving force therein [42]. When carbon materials are modified with functional groups such as carboxyl, lactone, or/and phenolic hydroxyl, the adsorption strengths of SO2 are enhanced pronouncedly, especially for the sites at edge surfaces. In addition, these functional groups show little effects on SO2 adsorption, suggesting that the enhanced adsorption is mainly due to regulation of carbon surface properties.

Adsorption of SO2 is the first step for the formation of acid rain. According to our preliminary studies, the reaction mechanisms of gas phase and mineral surfaces resemble each other, and hence the gas-phase results are beneficial to understand acid rain formation at mineral surfaces. The reaction of SO2 and H2O produces two isomers that have close electronic energies [43]; in addition, the two isomers have apparently lower electronic energies than H2SO3. Accordingly, the gaseous SO2 and H2O mixture is likely to exist as the SO2�H2O complex rather than H2SO3. The activation barrier of SO2 reacting with H2O to form H2SO3 is so high (146.7 kJ/mol) that it becomes very difficult to produce the sulfite species (H2SO3) in gas phase. Five years later, Stirling [44] investigated the hydrolysis of SO2 in aqueous solutions, finding that hydrated SO2 forms the bisulfite anion and hydronium ion after overcoming an energy barrier of about 17 kcal/mol, while the one-step formation of H2SO3 has not been detected. The orientation of water molecules in the hydration shell of SO2 implies a more facile formation of the bisulfite anion rather than H2SO3 [45], in line with the results of meta-dynamics calculations [46]. When HO2 participates in the reaction, the S atom constructs a new bond with O@H2O atom (S-O: 1.716 Å). The energy barrier reduces considerably and equals 56.6 kJ/mol. HO2 exists widely in the atmosphere and participates in a variety of chemical reactions [47]. In addition to HO2, a number of computational studies explicitly indicated that acidic substances in the atmosphere play similar catalytic effects and reduce significantly the energy barriers for the hydrolysis of SO2 [48–50]. The effects of H2SO3 and H2O on the hydrolysis of SO2 were studied by Liu et al. [48], showing that the energy barriers ascend in the order of H2SO3 + SO2�H2O (26.4 kJ/mol) < H2O + SO2�H2O (84.0 kJ/mol) < SO2�H2O (154.6 kJ/mol). The catalytic effect of H2SO3 is obviously more pronounced than that of H2O. As reflected by NBO analyses, the reactant complexes in presence of sulfurous acid have the enhanced second-order stabilization energies as compared to those with addition of only water molecules, which provides strong supports for the pronounced catalytic effect of H2SO3. The kinetic calculations further show that the hydrolysis of SO2 is a nearly autocatalytic reaction.

SO2 + O3 ! SO3 + O2 SO2 + H2O2 ! H2SO4 SO2 and O3 interact mainly through the S and O@O3 atoms, and their reaction causes the formation of S-O bond (1.708 Å) and the rupture of O-O bond in O3. This is one-step process and requires to overcome an energy barrier of 46.5 kJ/mol. The reaction of SO2 with H2O2 proceeds via the OH-abstraction mechanism, and two OH radicals generated from the dissociation of H2O2 are appended to SO2 forming H2SO4. However, the energy barrier of this reaction is 299.5 kJ/mol, which is extremely difficult to proceed at normal conditions. Chen et al. [46] reported the reaction mechanism between SO2 and HO2 and showed that there exist two types of SO2�HO2 complexes: one is to combine the terminal O@HO2 and S atoms. In this reaction, the O-S distance decreases from 2.966 Å (the initial complex: SO2�HO2) to 1.598 Å while the O-O bond of HO2 is gradually elongated until broken. The reaction of SO2 and H2O is divided into three stages, and the energy barrier of SO3�OH formation is so small (1.7 kJ/mol) that can be considered negligible. The final product of this reaction is HSO4. The other complex is characterized by two types of intermolecular H-bonds that form between the terminal O@HO2 and S atoms and between the H@HO2 and O@SO2 atoms. It is a one-step process and the product for this reaction is HOSO�O2. When a single water is added, it has a combined effect on the overall process, not only accelerating the reaction of generating HSO4 by reducing the activation barrier of the second step from 48.8 to 44.8 kJ/mol but also inhibiting the reaction between SO2 and HO2 to produce HOSO due to blocking the interaction between the H@HO2

Application of Density Functional Theory in Soil Science http://dx.doi.org/10.5772/intechopen.74079 255

It can be seen from the above discussions that the SO3 and water reaction is an integral section for the formation of acid rain. The direct reaction of SO3 with one water to produce sulfuric acid (H2SO4) requires to overcome a large energy barrier (28.7 kcal/mol) and seems difficult to occur at normal conditions [55–57]. When the second water participates, the energy barrier reduces substantially and the reaction becomes almost barrierless, which is probably due to the formation of a stable six-membered cyclic transition state, see Figure 6 [58]. The two water molecules transfer their protons in a concerted manner, and a new S-O bond is formed between the newly generated OH and SO3 fragments. In consequence, multiple reaction pathways may co-exist as

Figure 6. Optimized transition states for the formation of sulfuric acid from SO3 in presence of one (left) and two (right) water molecules. Arrows indicate the reaction coordinate vectors, and some important distances (Å) and angles (degrees)

and O@SO2 atoms.

illustrated below

are shown.

Alternatively, acid rain can be formed by release of SO2 into the atmosphere and oxidization of SO3, which then reacts with H2O to form sulfuric acid (H2SO4). There are many oxidants that are able to convert SO2 to SO3, and Calvert [51] have provided the formation mechanism of sulfuric acid (H2SO4)

$$\text{SO}\_2 + \text{OH} \rightarrow \text{HSO}\_3$$

$$\text{HSO}\_3 + \text{O}\_3 \rightarrow \text{SO}\_3 + \text{HO}\_2$$

$$\text{SO}\_3 + \text{H}\_2\text{O} \rightarrow \text{H}\_2\text{SO}\_4$$

Interaction between SO2 and the OH radical (OH• ) has been addressed by ab initio electronic structure calculations [52], and the optimized structure of the HOSO2 radical show differences with that predicted using a small basis set (MP4/6-31G\*\*//HF/3-21G\*) [53]: Inclusion of correlation effects results in an increase of S-O distances and OSO angle, whereas the HOS angle shows a substantial decrease. Although geometrically stable at all levels of theory, stability of the HOSO2 radical is estimated to be 104–110 kJ/mol, suggesting that the direct dissociation to SO3 is almost infeasible. O3 and H2O2 are two oxidizing substances in atmosphere chemistry, and their reaction mechanisms with SO2 were studied by Jiang et al. [54].

SO2 + O3 ! SO3 + O2 SO2 + H2O2 ! H2SO4

Adsorption of SO2 is the first step for the formation of acid rain. According to our preliminary studies, the reaction mechanisms of gas phase and mineral surfaces resemble each other, and hence the gas-phase results are beneficial to understand acid rain formation at mineral surfaces. The reaction of SO2 and H2O produces two isomers that have close electronic energies [43]; in addition, the two isomers have apparently lower electronic energies than H2SO3. Accordingly, the gaseous SO2 and H2O mixture is likely to exist as the SO2�H2O complex rather than H2SO3. The activation barrier of SO2 reacting with H2O to form H2SO3 is so high (146.7 kJ/mol) that it becomes very difficult to produce the sulfite species (H2SO3) in gas phase. Five years later, Stirling [44] investigated the hydrolysis of SO2 in aqueous solutions, finding that hydrated SO2 forms the bisulfite anion and hydronium ion after overcoming an energy barrier of about 17 kcal/mol, while the one-step formation of H2SO3 has not been detected. The orientation of water molecules in the hydration shell of SO2 implies a more facile formation of the bisulfite anion rather than H2SO3 [45], in line with the results of meta-dynamics calculations [46]. When HO2 participates in the reaction, the S atom constructs a new bond with O@H2O atom (S-O: 1.716 Å). The energy barrier reduces considerably and equals 56.6 kJ/mol. HO2 exists widely in the atmosphere and participates in a variety of chemical reactions [47]. In addition to HO2, a number of computational studies explicitly indicated that acidic substances in the atmosphere play similar catalytic effects and reduce significantly the energy barriers for the hydrolysis of SO2 [48–50]. The effects of H2SO3 and H2O on the hydrolysis of SO2 were studied by Liu et al. [48], showing that the energy barriers ascend in the order of H2SO3 + SO2�H2O (26.4 kJ/mol) < H2O + SO2�H2O (84.0 kJ/mol) < SO2�H2O (154.6 kJ/mol). The catalytic effect of H2SO3 is obviously more pronounced than that of H2O. As reflected by NBO analyses, the reactant complexes in presence of sulfurous acid have the enhanced second-order stabilization energies as compared to those with addition of only water molecules, which provides strong supports for the pronounced catalytic effect of H2SO3. The kinetic calculations

254 Density Functional Calculations - Recent Progresses of Theory and Application

further show that the hydrolysis of SO2 is a nearly autocatalytic reaction.

and their reaction mechanisms with SO2 were studied by Jiang et al. [54].

Interaction between SO2 and the OH radical (OH•

sulfuric acid (H2SO4)

Alternatively, acid rain can be formed by release of SO2 into the atmosphere and oxidization of SO3, which then reacts with H2O to form sulfuric acid (H2SO4). There are many oxidants that are able to convert SO2 to SO3, and Calvert [51] have provided the formation mechanism of

> SO2 + OH ! HSO3 HSO3 + O3 ! SO3 + HO2 SO3 + H2O ! H2SO4

structure calculations [52], and the optimized structure of the HOSO2 radical show differences with that predicted using a small basis set (MP4/6-31G\*\*//HF/3-21G\*) [53]: Inclusion of correlation effects results in an increase of S-O distances and OSO angle, whereas the HOS angle shows a substantial decrease. Although geometrically stable at all levels of theory, stability of the HOSO2 radical is estimated to be 104–110 kJ/mol, suggesting that the direct dissociation to SO3 is almost infeasible. O3 and H2O2 are two oxidizing substances in atmosphere chemistry,

) has been addressed by ab initio electronic

SO2 and O3 interact mainly through the S and O@O3 atoms, and their reaction causes the formation of S-O bond (1.708 Å) and the rupture of O-O bond in O3. This is one-step process and requires to overcome an energy barrier of 46.5 kJ/mol. The reaction of SO2 with H2O2 proceeds via the OH-abstraction mechanism, and two OH radicals generated from the dissociation of H2O2 are appended to SO2 forming H2SO4. However, the energy barrier of this reaction is 299.5 kJ/mol, which is extremely difficult to proceed at normal conditions. Chen et al. [46] reported the reaction mechanism between SO2 and HO2 and showed that there exist two types of SO2�HO2 complexes: one is to combine the terminal O@HO2 and S atoms. In this reaction, the O-S distance decreases from 2.966 Å (the initial complex: SO2�HO2) to 1.598 Å while the O-O bond of HO2 is gradually elongated until broken. The reaction of SO2 and H2O is divided into three stages, and the energy barrier of SO3�OH formation is so small (1.7 kJ/mol) that can be considered negligible. The final product of this reaction is HSO4. The other complex is characterized by two types of intermolecular H-bonds that form between the terminal O@HO2 and S atoms and between the H@HO2 and O@SO2 atoms. It is a one-step process and the product for this reaction is HOSO�O2. When a single water is added, it has a combined effect on the overall process, not only accelerating the reaction of generating HSO4 by reducing the activation barrier of the second step from 48.8 to 44.8 kJ/mol but also inhibiting the reaction between SO2 and HO2 to produce HOSO due to blocking the interaction between the H@HO2 and O@SO2 atoms.

It can be seen from the above discussions that the SO3 and water reaction is an integral section for the formation of acid rain. The direct reaction of SO3 with one water to produce sulfuric acid (H2SO4) requires to overcome a large energy barrier (28.7 kcal/mol) and seems difficult to occur at normal conditions [55–57]. When the second water participates, the energy barrier reduces substantially and the reaction becomes almost barrierless, which is probably due to the formation of a stable six-membered cyclic transition state, see Figure 6 [58]. The two water molecules transfer their protons in a concerted manner, and a new S-O bond is formed between the newly generated OH and SO3 fragments. In consequence, multiple reaction pathways may co-exist as illustrated below

Figure 6. Optimized transition states for the formation of sulfuric acid from SO3 in presence of one (left) and two (right) water molecules. Arrows indicate the reaction coordinate vectors, and some important distances (Å) and angles (degrees) are shown.

$$\rm SO\_3 + (H\_2O)\_2 \to H\_2SO\_4 + H\_2O$$

$$\rm SO\_3\cdot H\_2O + H\_2O \to H\_2SO\_4 + H\_2O$$

$$\rm SO\_3 + (H\_2O)\_2 \to H\_2SO\_4 + H\_2O$$

accuracy (≤1 kJ/mol). In consequence, computer simulations including DFT are the right key to unravel the complicated phenomena and processes occurring within soils; e.g., with DFT calculations, the aggregation mechanisms of "real" soils and the driving force therein were

Application of Density Functional Theory in Soil Science http://dx.doi.org/10.5772/intechopen.74079 257

In addition to DFT methods, there are a number of other computational methods, such as QM/ MM and Molecular Dynamics (MD). The choice of suitable computational methods is strongly recommended. We are pleased to see the birth of the ClayFF force-field [66] that was developed specially for clay minerals and the capability of the ReaxFF force-field [67] to handle reaction mechanisms. The 2013 Noble Prize in Chemistry was awarded to the development of "multiscale methods for complex systems," and now it is time to apply these methods to tackle

This work was sponsored by the National Natural Science Foundation of China (21473137), the Fourth Excellent Talents Program of Higher Education in Chongqing (2014-03) and the Natural

College of Resources and Environment and Chongqing Key Laboratory of Soil Multi-Scale

[1] Sposito G, Reginato RJ, Luxmoore RJ, editors. Opportunities in Basic Soil Science Research. Madison, Wisconsin, USA: Soil Science Society of America, Inc.; 1992

[2] Li H, Yang G. Rethink the methodologies in basic soil science research: From the perspec-

[3] Wang Q, Zhu C, Yun JN, Yang G. Isomorphic substitutions in clay materials and adsorption of metal ions onto external surfaces: A DFT investigation. Journal of Physical Chem-

[4] Li X, Li H, Yang G. Electric fields within clay materials: How to affect the adsorption of

Science Foundation Project of CQ CSTC, China (cstc2017jcyjAX0145).

Jiena Yun, Qian Wang, Chang Zhu and Gang Yang\*

istry C. 2017;121:26722-26732

\*Address all correspondence to: theobiochem@gmail.com

Interfacial Process, Southwest University, Chongqing, China

tive of soil chemistry. Acta Pedologica Sinica. 2017;54:819-826

metal ions. Journal of Colloid and Interface Science. 2017;501:54-59

unveiled at an atomic level [6].

complex soil systems.

Acknowledgements

Author details

References

The energy barriers for the reaction of SO3 and water clusters (H2O)n to form H2SO4�(H2O)n-1 are calculated to be 28.7, 11.1, and 4.6 kcal/mol for n = 1, 2, 3, respectively [59]. With increase of water numbers, more water molecules are available to solvate and stabilize the charged transition state complexes, which further reduce the energy barriers. That is, it is favorable to hydrolyze SO3 to form sulfuric acid in presence of sufficient water vapor. The second molecule acts as a good catalyst that promotes proton transfer from water to SO3, and according to both experimental and theoretical reports, the rate constant of SO3 + 2H2O reaction is approximately 10�<sup>15</sup> cm3 �molecule�<sup>1</sup> �s �<sup>1</sup> [60–62]. HO2 [63], HCOOH [62, 64], H2SO4 [49], HNO3 [65] can replace the role of the second water molecule and exhibit a similar catalytic effect on the formation of sulfuric acid (H2SO4).

#### 4. Concluding remarks

In contrast to the rapid development of chemistry, physics, and biology and other disciplines, soil science remains almost stagnant in the past few decades, and to best of our knowledge, no breakthroughs have been reported for rather a long time. Despite that, no one can deny the vital significance of soil to our life, and soil science has been widely acknowledged as the final frontier.

The slow progresses for soil science, we think, should be attributed to two reasons: (1) wrong and outdated perceptions. A majority of soil researchers mistakenly believe that all knowledge worth knowing about soils has already been understood and no revolutionary progresses would take place; (2) complex systems. Soils are very structurally complicated and there are multiple factors to co-function, which makes it very difficult to characterize by experimental techniques. Computer simulations have unique advantages to handle complex systems, while currently its role in soil science is far from being recognized. In this chapter, two examples are elaborately discussed with regard to application of DFT calculations in soil science: one focuses on the adsorption of uranyl onto mineral surfaces, and the other involves the adsorption of SO2 onto mineral surfaces and reaction mechanisms to form acid rain. It can be seen from these discussions that DFT calculations are able to provide useful and detailed information about the adsorption, interaction and reactions at the atomic level that greatly promote our understanding about soil science.

With advent of high-performance computing platforms, the same DFT calculation tasks of 10 years ago can now finish within a remarkably shorter time, even if you increase model size (periodic model is also an option), consider solvent effects by adding explicit solvent molecules or/and choose more accurate methods. The methodological developments regarding to DFT calculations have also made remarkable progresses over the recent three decades, and as a result, thermodynamics and reaction barriers can now be predicted with nearly chemical accuracy (≤1 kJ/mol). In consequence, computer simulations including DFT are the right key to unravel the complicated phenomena and processes occurring within soils; e.g., with DFT calculations, the aggregation mechanisms of "real" soils and the driving force therein were unveiled at an atomic level [6].

In addition to DFT methods, there are a number of other computational methods, such as QM/ MM and Molecular Dynamics (MD). The choice of suitable computational methods is strongly recommended. We are pleased to see the birth of the ClayFF force-field [66] that was developed specially for clay minerals and the capability of the ReaxFF force-field [67] to handle reaction mechanisms. The 2013 Noble Prize in Chemistry was awarded to the development of "multiscale methods for complex systems," and now it is time to apply these methods to tackle complex soil systems.

#### Acknowledgements

SO3 + (H2O)2 ! H2SO4 + H2O SO3�H2O+H2O ! H2SO4 + H2O SO3 + (H2O)2 ! H2SO4 + H2O The energy barriers for the reaction of SO3 and water clusters (H2O)n to form H2SO4�(H2O)n-1 are calculated to be 28.7, 11.1, and 4.6 kcal/mol for n = 1, 2, 3, respectively [59]. With increase of water numbers, more water molecules are available to solvate and stabilize the charged transition state complexes, which further reduce the energy barriers. That is, it is favorable to hydrolyze SO3 to form sulfuric acid in presence of sufficient water vapor. The second molecule acts as a good catalyst that promotes proton transfer from water to SO3, and according to both experimental and theoretical reports, the rate constant of SO3 + 2H2O reaction is approximately

replace the role of the second water molecule and exhibit a similar catalytic effect on the

In contrast to the rapid development of chemistry, physics, and biology and other disciplines, soil science remains almost stagnant in the past few decades, and to best of our knowledge, no breakthroughs have been reported for rather a long time. Despite that, no one can deny the vital significance of soil to our life, and soil science has been widely acknowledged as the final

The slow progresses for soil science, we think, should be attributed to two reasons: (1) wrong and outdated perceptions. A majority of soil researchers mistakenly believe that all knowledge worth knowing about soils has already been understood and no revolutionary progresses would take place; (2) complex systems. Soils are very structurally complicated and there are multiple factors to co-function, which makes it very difficult to characterize by experimental techniques. Computer simulations have unique advantages to handle complex systems, while currently its role in soil science is far from being recognized. In this chapter, two examples are elaborately discussed with regard to application of DFT calculations in soil science: one focuses on the adsorption of uranyl onto mineral surfaces, and the other involves the adsorption of SO2 onto mineral surfaces and reaction mechanisms to form acid rain. It can be seen from these discussions that DFT calculations are able to provide useful and detailed information about the adsorption, interaction and reactions at the atomic level that greatly promote our understand-

With advent of high-performance computing platforms, the same DFT calculation tasks of 10 years ago can now finish within a remarkably shorter time, even if you increase model size (periodic model is also an option), consider solvent effects by adding explicit solvent molecules or/and choose more accurate methods. The methodological developments regarding to DFT calculations have also made remarkable progresses over the recent three decades, and as a result, thermodynamics and reaction barriers can now be predicted with nearly chemical

�<sup>1</sup> [60–62]. HO2 [63], HCOOH [62, 64], H2SO4 [49], HNO3 [65] can

10�<sup>15</sup> cm3

frontier.

ing about soil science.

�molecule�<sup>1</sup>

formation of sulfuric acid (H2SO4).

4. Concluding remarks

�s

256 Density Functional Calculations - Recent Progresses of Theory and Application

This work was sponsored by the National Natural Science Foundation of China (21473137), the Fourth Excellent Talents Program of Higher Education in Chongqing (2014-03) and the Natural Science Foundation Project of CQ CSTC, China (cstc2017jcyjAX0145).

#### Author details

Jiena Yun, Qian Wang, Chang Zhu and Gang Yang\*

\*Address all correspondence to: theobiochem@gmail.com

College of Resources and Environment and Chongqing Key Laboratory of Soil Multi-Scale Interfacial Process, Southwest University, Chongqing, China

#### References


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[19] Sebbari K, Roques J, Simoni E, Domain C, Perron H, Catalette H. First-principles molecular dynamics simulations of uranyl ion interaction at the water/rutile TiO2(110) interface.

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[22] Kremleva A, Krüger S, Rösch N. Quantum chemical modeling of uranyl adsorption on

[23] Kremleva A, Krüger S, Rösch N. Uranyl adsorption at (010) edge surfaces of kaolinite: A density functional study. Geochimica et Cosmochimica Acta. 2011;75:706-718

[24] Veilly E, Roques J, Jodin-Caumon M-C, Humbert B, Drot R, Simoni E. Uranyl interaction with the hydrated (001) basal face of gibbsite: A combined theoretical and spectroscopic

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### *Edited by Gang Yang*

Density functional theory (DFT) ranks as the most widely used quantum mechanical method and plays an increasingly larger role in a number of disciplines such as chemistry, physics, material, biology, and pharmacy. DFT has long been used to complement experimental investigations, while now it is also regarded as an indispensable and powerful tool for researchers of different fields. This book is divided into five sections that include original chapters written by experts in their fields: "Method Development and Validation," "Spectra and Thermodynamics," "Catalysis and Mechanism," "Material and Molecular Design," and "Multidisciplinary Integration." I would like to express my sincere gratitude to all contributors and recommend this book to both beginners and experienced researchers.

Published in London, UK © 2018 IntechOpen © Nathan Anderson / unsplash

Density Functional Calculations - Recent Progresses of Theory and Application

Density Functional

Calculations

Recent Progresses of Theory and Application

*Edited by Gang Yang*