**Meet the editor**

Dr. Daniel Glossman-Mitnik is now a Titular Researcher at the Centro de Investigación en Materiales Avanzados (CIMAV), Chihuahua, Mexico as well as a National Researcher of Level III of the Consejo Nacional de Ciencia y Tecnología, Mexico. His research interest focuses on Computational Chemistry and Molecular Modeling of molecular systems of pharmacological, food, and

alternative energy interests by resorting to DFT and Conceptual DFT. He has authored and coauthored more than 200 peer-reviewed papers and 10 book chapters. He has delivered speeches at many international and domestic conferences. He serves as the referee for more than 60 international journals, books, and research proposals.

Contents

**Preface VII**

**in Conceptual DFT 3**

Balawender and Frank De Proft

**Organic Frameworks 55**

**Homogeneous Phase 75** Manuel Antuch and Pierre Millet

**Properties and Stability 91** Francisco Colmenero Ruiz

Sareeya Bureekaew

Chapter 1 **New Insights and Horizons from the Linear Response Function**

Chapter 2 **Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys for PEMFCs 33**

Chapter 3 **Density Functional Theory Studies of Catalytic Sites in Metal-**

Siwarut Siwaipram, Sarawoot Impeng, Philippe A. Bopp and

**Electrochemical Activity of Metal Clathrochelates with Regard**

Alejandro E. Pérez and Rafael Ribadeneira

Chapter 4 **The Use of Density Functional Theory to Decipher the**

**to the Hydrogen Evolution Reaction in the**

Chapter 5 **The Application of Periodic Density Functional Theory to the Study of Uranyl-Containing Materials: Thermodynamic**

Paul Geerlings, Stijn Fias, Thijs Stuyver, Paul Ayers, Robert

**Section 1 Concepts 1**

**Section 2 Applications 31**

## Contents

**Preface XI**


## Chapter 6 **Magnetic Ordering in Ilmenites and Corundum-Ordered Structures 123**

Sergio Ricardo De Lazaro, Luis Henrique Da Silveira Lacerda and Renan Augusto Pontes Ribeiro

Preface

chemical reactivity of the studied systems.

systems of academic and industrial interest.

cardo De Lázaro and Ruby Srivastava.

cional de Ciencia y Tecnología (CONACYT), Mexico.

of this is presented in the first chapter within the Concepts section.

Density Functional Theory (or DFT for short) is a potent methodology useful for calculating and understanding the molecular and electronic structure of atoms, molecules, clusters, and solids. Its use relies not only in the ability to calculate the molecular properties of the species of interest but also provides interesting concepts that allow a better comprehension of the

This book represents an attempt to present examples on the utility of DFT for the under‐ standing of the chemical reactivity through descriptors that constitute the basis of the so called Conceptual DFT (sometimes also named as Chemical Reactivity Theory). These de‐ scriptors provide a qualitative and quantitative view of the problem and an updated review

The Applications section contains chapters showing the application of the theory and its re‐ lated computational procedures in the determination of the molecular properties of different

I would like to express my sincere gratitude to all authors who contributed to this book: Paul Geerlings, Stijn Fias, Thijs Stuyver, Paul Ayers, Robert Balawender, Frank De Proft, Alejandro E. Pérez, Rafael Ribadeneira, Siwarut Siwaipram, Sarawoot Impeng, Philippe A. Bopp, Sareeya Bureekaew, Manuel Antuch, Pierre Millet, Francisco Colmenero, Sergio Ri‐

Finally, my warmest thanks go to my beloved wife Carmen and to the memories of my late parents, Sofía and Miguel. I am also grateful for the financial support from the Consejo Na‐

> **Dr. Daniel Glossman-Mitnik** Laboratorio Virtual NANOCOSMOS

> > Chihuahua, Mexico

Departamento de Medio Ambiente y Energía Centro de Investigación en Materiales Avanzados

#### Chapter 7 **Role of Density Functional Theory in "Ribocomputing Devices" 141** Ruby Srivastava

## Preface

Chapter 6 **Magnetic Ordering in Ilmenites and Corundum-Ordered**

Chapter 7 **Role of Density Functional Theory in "Ribocomputing**

Sergio Ricardo De Lazaro, Luis Henrique Da Silveira Lacerda and

**Structures 123**

**VI** Contents

**Devices" 141** Ruby Srivastava

Renan Augusto Pontes Ribeiro

Density Functional Theory (or DFT for short) is a potent methodology useful for calculating and understanding the molecular and electronic structure of atoms, molecules, clusters, and solids. Its use relies not only in the ability to calculate the molecular properties of the species of interest but also provides interesting concepts that allow a better comprehension of the chemical reactivity of the studied systems.

This book represents an attempt to present examples on the utility of DFT for the under‐ standing of the chemical reactivity through descriptors that constitute the basis of the so called Conceptual DFT (sometimes also named as Chemical Reactivity Theory). These de‐ scriptors provide a qualitative and quantitative view of the problem and an updated review of this is presented in the first chapter within the Concepts section.

The Applications section contains chapters showing the application of the theory and its re‐ lated computational procedures in the determination of the molecular properties of different systems of academic and industrial interest.

I would like to express my sincere gratitude to all authors who contributed to this book: Paul Geerlings, Stijn Fias, Thijs Stuyver, Paul Ayers, Robert Balawender, Frank De Proft, Alejandro E. Pérez, Rafael Ribadeneira, Siwarut Siwaipram, Sarawoot Impeng, Philippe A. Bopp, Sareeya Bureekaew, Manuel Antuch, Pierre Millet, Francisco Colmenero, Sergio Ri‐ cardo De Lázaro and Ruby Srivastava.

Finally, my warmest thanks go to my beloved wife Carmen and to the memories of my late parents, Sofía and Miguel. I am also grateful for the financial support from the Consejo Na‐ cional de Ciencia y Tecnología (CONACYT), Mexico.

> **Dr. Daniel Glossman-Mitnik** Laboratorio Virtual NANOCOSMOS Departamento de Medio Ambiente y Energía Centro de Investigación en Materiales Avanzados Chihuahua, Mexico

**Section 1**

**Concepts**

**Section 1**

## **Concepts**

**Chapter 1**

Provisional chapter

**New Insights and Horizons from the Linear Response**

DOI: 10.5772/intechopen.80280

An overview is given of our recent work on the linear response function (LRF) χ r;r<sup>0</sup> ð Þ and its congener, the softness kernel s r;r<sup>0</sup> ð Þ, the second functional derivatives of the energy E and the grand potential Ω with respect to the external potential at constant N and μ, respectively. In a first section on new insights into the LRF in the context of conceptual DFT, the mathematical and physical properties of these kernels are scrutinized through the concavity of the <sup>E</sup> <sup>¼</sup> E N½ � ; <sup>v</sup> and <sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup> <sup>μ</sup>; <sup>v</sup> functionals in <sup>v</sup>ð Þ<sup>r</sup> resulting, for example, in the negative semidefiniteness of χ. As an example of the analogy between the CDFT functionals and thermodynamic state functions, the analogy between the stability conditions of the macroscopic Gibbs free energy function and the concavity conditions for Ω is established, yielding a relationship between the global and local softness and the softness kernel. The role of LRF and especially the softness kernel in Kohn's nearsightedness of electronic matter (NEM) principle is highlighted. The first numerical results on the softness kernel for molecules are reported and scrutinized for their nearsightedness, reconciling the physicists' NEM view and the chemists' transferability paradigm. The extension of LRF in the context of spin polarized conceptual DFT is presented. Finally, two sections are devoted to 'new horizons' for the LRF. The role of LRF in (evaluating) alchemical derivatives is stressed, the latter playing a promising role in exploring the chemical compound space. Examples for the transmutation of N2 and the CC ! BN substitution pattern in 2D and 3D carbocyclic systems illustrate the computational efficiency of the use of alchemical derivatives in exploring nearest neighbours in the chemical compound space. As a second perspective, the role of LRF in evaluating and interpreting molecular conductivity is described. Returning to its forerunner, Coulson's atom-atom polarizability, it is shown how in conjugated π systems (and within certain approximations) a remarkable integralintegrand relationship between the atom-atom polarizability and the transmission probability between the atoms/contacts exists, leading to similar trends in both properties. A simple selection rule for transmission probability in alternating hydrocarbons is derived

> © 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.

New Insights and Horizons from the Linear Response

**Function in Conceptual DFT**

Function in Conceptual DFT

Robert Balawender and Frank De Proft

Robert Balawender and Frank De Proft

http://dx.doi.org/10.5772/intechopen.80280

Abstract

Additional information is available at the end of the chapter

based on the sign of the atom-atom polarizability.

Additional information is available at the end of the chapter

Paul Geerlings, Stijn Fias, Thijs Stuyver, Paul Ayers,

Paul Geerlings, Stijn Fias, Thijs Stuyver, Paul Ayers,

#### **New Insights and Horizons from the Linear Response Function in Conceptual DFT** New Insights and Horizons from the Linear Response Function in Conceptual DFT

DOI: 10.5772/intechopen.80280

Paul Geerlings, Stijn Fias, Thijs Stuyver, Paul Ayers, Robert Balawender and Frank De Proft Paul Geerlings, Stijn Fias, Thijs Stuyver, Paul Ayers, Robert Balawender and Frank De Proft

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.80280

#### Abstract

An overview is given of our recent work on the linear response function (LRF) χ r;r<sup>0</sup> ð Þ and its congener, the softness kernel s r;r<sup>0</sup> ð Þ, the second functional derivatives of the energy E and the grand potential Ω with respect to the external potential at constant N and μ, respectively. In a first section on new insights into the LRF in the context of conceptual DFT, the mathematical and physical properties of these kernels are scrutinized through the concavity of the <sup>E</sup> <sup>¼</sup> E N½ � ; <sup>v</sup> and <sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup> <sup>μ</sup>; <sup>v</sup> functionals in <sup>v</sup>ð Þ<sup>r</sup> resulting, for example, in the negative semidefiniteness of χ. As an example of the analogy between the CDFT functionals and thermodynamic state functions, the analogy between the stability conditions of the macroscopic Gibbs free energy function and the concavity conditions for Ω is established, yielding a relationship between the global and local softness and the softness kernel. The role of LRF and especially the softness kernel in Kohn's nearsightedness of electronic matter (NEM) principle is highlighted. The first numerical results on the softness kernel for molecules are reported and scrutinized for their nearsightedness, reconciling the physicists' NEM view and the chemists' transferability paradigm. The extension of LRF in the context of spin polarized conceptual DFT is presented. Finally, two sections are devoted to 'new horizons' for the LRF. The role of LRF in (evaluating) alchemical derivatives is stressed, the latter playing a promising role in exploring the chemical compound space. Examples for the transmutation of N2 and the CC ! BN substitution pattern in 2D and 3D carbocyclic systems illustrate the computational efficiency of the use of alchemical derivatives in exploring nearest neighbours in the chemical compound space. As a second perspective, the role of LRF in evaluating and interpreting molecular conductivity is described. Returning to its forerunner, Coulson's atom-atom polarizability, it is shown how in conjugated π systems (and within certain approximations) a remarkable integralintegrand relationship between the atom-atom polarizability and the transmission probability between the atoms/contacts exists, leading to similar trends in both properties. A simple selection rule for transmission probability in alternating hydrocarbons is derived based on the sign of the atom-atom polarizability.

© 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.

Keywords: conceptual DFT, linear response function, nearsightedness of electronic matter, alchemical derivatives, molecular conductivity

Note that in the context of time-dependent DFT [17–19], the LRF has made its appearance many years ago as it was realized that the poles of its frequency-dependent form are nothing other than the electronic excitation energies. Thanks to Casida's elegant matrix formalism [20], electronic transition frequencies, intensities and assignments are nowadays routinely performed, implemented as they are in standard quantum chemistry packages. However, this evolution was not accompanied by a parallel endeavour on the evaluation, representation and chemical inter-

New Insights and Horizons from the Linear Response Function in Conceptual DFT

http://dx.doi.org/10.5772/intechopen.80280

5

In the past decade, the ALGC group, in collaboration with colleagues from different countries (Canada (Ayers), US (Yang), Spain (Sola), Poland (Balawender), etc.), set out a program aiming at the systematic evaluation, representation and interpretation of the LRF with the following results obtained until 2013, summarized in a review paper in Chemical Society Reviews [16]

1. The LRF can now be routinely calculated at several levels of approximation for which the coupled perturbed Kohn-Sham perturbational approach turns out to be the most attractive approach, also permitting different levels of sophistication depending on the treatment of the exchange correlation potential (vxc) term in the perturbation equations. In its simplest form (neglecting the influence of the external potential variation on the Coulomb and exchange-correlation terms in the perturbational equations), the independent particle

atoms and molecules, or in the case of molecules, after condensation, via a simple atom-

3. An abundance of chemical information was shown to be present in the LRF ranging from the shell structure of atoms, to inductive and mesomeric effects, electron (de)localization

In the present chapter, a synopsis is given of the progress made since then by the ALGC group in collaboration with other groups as witnessed by two of the authors (P.A and R. B.), both on fundamental and applied aspects, that is, on new insights into the properties of the LRF and on new areas where the LRF is at stake. In Section 2 the mathematical/physical properties of the LRF are revised together with those of its congener, the softness kernel s r;r<sup>0</sup> ð Þ, the latter playing a fundamental role in scrutinizing Kohn's nearsightedness of electronic matter (NEM) principle. The extension of LRF in the context of spin polarized DFT is also addressed. In Section 3, we highlight the importance of the LRF in the emerging field of alchemical derivatives when exploring the chemical compound space. We illustrate the potential of alchemical derivatives in exploring the CC ! BN isoelectronic substitution in 2D and 3D unsaturated carbocyclic molecules: benzene and the C60 fullerenes. Finally, in Section 4 we show how the LRF (in fact its forerunner, Coulson's atom-atom polarizability) can be used to predict/interpret the conductivity behaviour of unsaturated hydrocarbons, thus entering the

) as demonstrated for

(no explicit reference to each of the individual constituting studies will be given).

expression, already presented by Ayers and Parr [8, 21], is retrieved.

atom matrix, reminiscent of reporting the results of a population analysis.

2. The representation can be done via contour diagrams (fixing, e.g. r<sup>0</sup>

pretation of the frequency-independent or static LRF.

and (anti)aromaticity in molecules.

vibrant field of molecular electronics.

#### 1. Introduction

A continuous challenge for theoretical and quantum chemists is to see if 'classical' chemical concepts describing bonding, structure and reactivity—the common language of all chemists —can still be retrieved from the nowadays extensive and complex computational results obtained at different levels of complexity with wave function or density functional theory. Conceptual density functional theory (CDFT) [1–6] has played an important role in this endeavour in the past decades. CDFT is a branch of DFT [7, 8] aiming to give precision to often well-known but sometimes vaguely defined chemical concepts (e.g. electronegativity, hardness and softness), affording their numerical evaluation, and to use them either as such or in the context of principles such as Sanderson's electronegativity equalization principle [9] or Pearson's hard and soft acids and bases principle [10]. 'Chemical' DFT or even 'chemical reactivity' DFT would have been a better name for the obvious reason that concepts are essential for all branches of DFT (especially the fundamentals) and that chemical reactivity is one of the main issues addressed in conceptual DFT.

When looking at the basics of CDFT, the energy functional, E ¼ E N½ � ; v , stands out [8]. Why? It is the key ingredient to get (qualitative and quantitative) insight into the eagerness of an atom, or a molecule, to adapt itself to changes in the number of electrons, N, and/or the external potential, vð Þr , that is, the potential felt by the electrons due to the nuclei. These changes are essential in describing (the onset of) a chemical reaction, hence chemical reactivity. The readiness of a system to adapt itself to these new conditions is quantified through response functions, δ<sup>m</sup> ∂<sup>n</sup> <sup>E</sup>=∂N<sup>n</sup> ð Þ=δvð Þ <sup>r</sup><sup>1</sup> …δvð Þ <sup>r</sup><sup>m</sup> , which are the cornerstones of CDFT. Literature on these response functions is abundant, especially on the first-order responses (electronic chemical potential μ [11] ð Þ n ¼ 1; m ¼ 0 and the electron density rð Þr ð Þ n ¼ 0; m ¼ 1 , the cornerstone of DFT itself) and two of the second-order responses (chemical hardness η ð Þ n ¼ 2; m ¼ 0 and its inverse, the chemical softness S [12], and the electronic Fukui function fð Þr ð Þ n ¼ 1; m ¼ 1 [13]). The most prominent of the third-order response functions [14] is the dual descriptor f ð Þ<sup>2</sup> ð Þ<sup>r</sup> [15], the N-derivative of the Fukui function ð Þ n ¼ 1; m ¼ 1 . Remarkably, response functions diagonal in vð Þr ð Þ n ¼ 0; m ¼ 2 � 3 were nearly absent in the CDFT literature until about 10 years ago (see [16] for an overview of this early work). The reasons are obvious; here we concentrate on its simplest member ð Þ n ¼ 0; m ¼ 2 , the linear response function (LRF).

$$\chi(\mathbf{r}, \mathbf{r}') = \left(\delta^2 E / \delta v(\mathbf{r}) \delta v(\mathbf{r}')\right)\_N \tag{1}$$

The calculation of this kernel turns out to be far from trivial, as is the representation of this quantity, a function of six Cartesian coordinates, and by extension its link to 'chemical' concepts. Note that in the context of time-dependent DFT [17–19], the LRF has made its appearance many years ago as it was realized that the poles of its frequency-dependent form are nothing other than the electronic excitation energies. Thanks to Casida's elegant matrix formalism [20], electronic transition frequencies, intensities and assignments are nowadays routinely performed, implemented as they are in standard quantum chemistry packages. However, this evolution was not accompanied by a parallel endeavour on the evaluation, representation and chemical interpretation of the frequency-independent or static LRF.

Keywords: conceptual DFT, linear response function, nearsightedness of electronic

A continuous challenge for theoretical and quantum chemists is to see if 'classical' chemical concepts describing bonding, structure and reactivity—the common language of all chemists —can still be retrieved from the nowadays extensive and complex computational results obtained at different levels of complexity with wave function or density functional theory. Conceptual density functional theory (CDFT) [1–6] has played an important role in this endeavour in the past decades. CDFT is a branch of DFT [7, 8] aiming to give precision to often well-known but sometimes vaguely defined chemical concepts (e.g. electronegativity, hardness and softness), affording their numerical evaluation, and to use them either as such or in the context of principles such as Sanderson's electronegativity equalization principle [9] or Pearson's hard and soft acids and bases principle [10]. 'Chemical' DFT or even 'chemical reactivity' DFT would have been a better name for the obvious reason that concepts are essential for all branches of DFT (especially the fundamentals) and that chemical reactivity is

When looking at the basics of CDFT, the energy functional, E ¼ E N½ � ; v , stands out [8]. Why? It is the key ingredient to get (qualitative and quantitative) insight into the eagerness of an atom, or a molecule, to adapt itself to changes in the number of electrons, N, and/or the external potential, vð Þr , that is, the potential felt by the electrons due to the nuclei. These changes are essential in describing (the onset of) a chemical reaction, hence chemical reactivity. The readiness of a system to adapt itself to these new conditions is quantified through response functions,

<sup>E</sup>=∂N<sup>n</sup> ð Þ=δvð Þ <sup>r</sup><sup>1</sup> …δvð Þ <sup>r</sup><sup>m</sup> , which are the cornerstones of CDFT. Literature on these response functions is abundant, especially on the first-order responses (electronic chemical potential μ [11] ð Þ n ¼ 1; m ¼ 0 and the electron density rð Þr ð Þ n ¼ 0; m ¼ 1 , the cornerstone of DFT itself) and two of the second-order responses (chemical hardness η ð Þ n ¼ 2; m ¼ 0 and its inverse, the chemical softness S [12], and the electronic Fukui function fð Þr ð Þ n ¼ 1; m ¼ 1 [13]). The most

N-derivative of the Fukui function ð Þ n ¼ 1; m ¼ 1 . Remarkably, response functions diagonal in vð Þr ð Þ n ¼ 0; m ¼ 2 � 3 were nearly absent in the CDFT literature until about 10 years ago (see [16] for an overview of this early work). The reasons are obvious; here we concentrate on

The calculation of this kernel turns out to be far from trivial, as is the representation of this quantity, a function of six Cartesian coordinates, and by extension its link to 'chemical' concepts.

E=δvð Þr δv r <sup>0</sup> ð Þ ð Þ<sup>2</sup> ð Þ<sup>r</sup> [15], the

<sup>N</sup> (1)

prominent of the third-order response functions [14] is the dual descriptor f

its simplest member ð Þ n ¼ 0; m ¼ 2 , the linear response function (LRF).

χ r;r <sup>0</sup> ð Þ¼ <sup>δ</sup><sup>2</sup>

matter, alchemical derivatives, molecular conductivity

one of the main issues addressed in conceptual DFT.

1. Introduction

4 Density Functional Theory

δ<sup>m</sup> ∂<sup>n</sup>

In the past decade, the ALGC group, in collaboration with colleagues from different countries (Canada (Ayers), US (Yang), Spain (Sola), Poland (Balawender), etc.), set out a program aiming at the systematic evaluation, representation and interpretation of the LRF with the following results obtained until 2013, summarized in a review paper in Chemical Society Reviews [16] (no explicit reference to each of the individual constituting studies will be given).


In the present chapter, a synopsis is given of the progress made since then by the ALGC group in collaboration with other groups as witnessed by two of the authors (P.A and R. B.), both on fundamental and applied aspects, that is, on new insights into the properties of the LRF and on new areas where the LRF is at stake. In Section 2 the mathematical/physical properties of the LRF are revised together with those of its congener, the softness kernel s r;r<sup>0</sup> ð Þ, the latter playing a fundamental role in scrutinizing Kohn's nearsightedness of electronic matter (NEM) principle. The extension of LRF in the context of spin polarized DFT is also addressed. In Section 3, we highlight the importance of the LRF in the emerging field of alchemical derivatives when exploring the chemical compound space. We illustrate the potential of alchemical derivatives in exploring the CC ! BN isoelectronic substitution in 2D and 3D unsaturated carbocyclic molecules: benzene and the C60 fullerenes. Finally, in Section 4 we show how the LRF (in fact its forerunner, Coulson's atom-atom polarizability) can be used to predict/interpret the conductivity behaviour of unsaturated hydrocarbons, thus entering the vibrant field of molecular electronics.

#### 2. Theoretical developments

#### 2.1. On the negative and positive semidefiniteness of the LRF and the softness kernel and thermodynamic analogies

The properties of the LRF χ r;r<sup>0</sup> ð Þ are intimately related to the concavity/convexity properties of E N½ � ; v that we addressed in recent years [22, 23]. Where E N½ � ; v is convex with respect to (w.r.t.) N [24], it is well established that E N½ � ; v is concave w.r.t. vð Þr following the Jensen's inequality.

$$E[\mathcal{N}, \lambda \boldsymbol{\nu}\_1 + (1 - \lambda)\boldsymbol{\nu}\_2] \ge \lambda E[\mathcal{N}, \boldsymbol{\nu}\_1] + (1 - \lambda)E[\mathcal{N}, \boldsymbol{\nu}\_2] \quad \land \ \lambda \in [0, 1] \tag{2}$$

A direct consequence is that the LRF is negative semidefinite.

<sup>δ</sup>Eð Þ<sup>2</sup> <sup>¼</sup> <sup>1</sup>=<sup>2</sup>

electron depletion at that point, yielding a negative χð Þ r;r .

second-order variation of the energy

linear response function defined on R3 � <sup>R</sup><sup>3</sup>

molecular conductivity. Defined as

derivatives, proving that πAA can be written as

therefore πAA is negative (see also Section 4).

ðð <sup>χ</sup> <sup>r</sup>;<sup>r</sup>

ðð <sup>δ</sup><sup>2</sup>

<sup>0</sup> ð Þθð Þr θ r

where θð Þr is any continuous function [23]. This inequality shows up when considering the

E=δvð Þr δv r <sup>0</sup> ð Þ � �

> <sup>0</sup>χ r <sup>00</sup>;r

showing that the diagonal elements of the linear response function χ r;r<sup>0</sup> ð Þ should be negative or zero. This result links the concavity of the E N½ � ; v functional to the diagonal elements of the

results (non-integrated and condensed) (see for example [29, 30]). Its physical interpretation is straightforward through the definition of χ r;r<sup>0</sup> ð Þ as δrð Þr =δv r<sup>0</sup> ð Þ ð Þ <sup>N</sup>: if the potential at a given point r is increased (made less negative), this electron-unfavourable situation will lead to

In Section 4, we will point out that Coulson's atom-atom polarizability πAB [31] can be considered as a Hückel-theory forerunner of the LRF and exploit its properties in discussing

the analogy emerges as qA is the π-electron charge on atom A, whereas (the change in) the Coulomb integral α<sup>B</sup> is equivalent to a change in the external potential. The case A ¼ B was explored by Coulson using his complex integral formalism for Hückel's π energy and its

ð Þ <sup>Δ</sup>AAð Þ iy <sup>=</sup>Δð Þ iy <sup>2</sup>

þ ð∞

�∞

Here, Δ represents the Hückel secular determinant (or characteristic function) and ΔAA is its counterpart with row A and column A deleted. It turns out that ΔAAð Þ iy =Δð Þ iy is imaginary and

When introducing the E ¼ E N½ � ; v functional, it was stressed that in the LRF the second functional derivative is taken at constant N. In the remaining part of this chapter, it turns out that no less important role in CDFT is played by the softness kernel, which is the second functional derivative with respect to <sup>v</sup>ð Þ<sup>r</sup> of the <sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup> <sup>μ</sup>; <sup>v</sup> � � functional (the grand potential)

<sup>π</sup>AB <sup>¼</sup> <sup>∂</sup>qA=∂α<sup>B</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

πAA ¼ ð Þ 1=π

When adopting the δvð Þ¼ r V0δ r � r<sup>00</sup> ð Þ choice for δvð Þr , where V<sup>0</sup> is a constant one gets

<sup>δ</sup>Eð Þ<sup>2</sup> <sup>¼</sup> <sup>1</sup>=2V<sup>2</sup>

<sup>0</sup> ð Þdrdr

<sup>N</sup>δvð Þr δv r

New Insights and Horizons from the Linear Response Function in Conceptual DFT

<sup>0</sup> ð Þdrdr

<sup>00</sup> ð Þ ≤ 0 (5)

. This negativity was retrieved in all our numerical

E=∂αA∂α<sup>B</sup> ¼ πBA (6)

dy (7)

<sup>0</sup> ≤ 0 (3)

http://dx.doi.org/10.5772/intechopen.80280

7

<sup>0</sup> (4)

(see for example Lieb [25], Eschrig [26], and Helgaker et al. [27]). In Figure 1 (after Helgaker [28]) we illustrate the physical interpretation of this concavity property. For a given potential v1, the associated ground state energy is given by the expectation value ψ1j j H v½ � <sup>1</sup> ψ<sup>1</sup> � �, with ground state wave function ψ1, the red point in the figure. Changing v<sup>1</sup> to v2, with constant ψ, induces changes in energy linear in <sup>δ</sup>vð Þ<sup>r</sup> as a consequence of the term <sup>Ð</sup> rð Þr vð Þr dr in the DFT energy expression. Relaxing the wave function yields an energy lowering (blue arrow) until the energy obtained by applying the variational principle with v ¼ v<sup>2</sup> is reached (with associated wave function ψ2). Consequently, the true energy (black line) will always be found below the tangent line ensuring concavity.

Figure 1. Illustration of the concavity of the E ¼ E v½ � functional (after Helgaker [28]) (Reprinted by permission from Springer Nature, Copyright 2016 [22]).

A direct consequence is that the LRF is negative semidefinite.

2. Theoretical developments

the tangent line ensuring concavity.

Springer Nature, Copyright 2016 [22]).

thermodynamic analogies

6 Density Functional Theory

2.1. On the negative and positive semidefiniteness of the LRF and the softness kernel and

The properties of the LRF χ r;r<sup>0</sup> ð Þ are intimately related to the concavity/convexity properties of E N½ � ; v that we addressed in recent years [22, 23]. Where E N½ � ; v is convex with respect to (w.r.t.) N [24], it is well established that E N½ � ; v is concave w.r.t. vð Þr following the Jensen's inequality.

(see for example Lieb [25], Eschrig [26], and Helgaker et al. [27]). In Figure 1 (after Helgaker [28]) we illustrate the physical interpretation of this concavity property. For a given potential

ground state wave function ψ1, the red point in the figure. Changing v<sup>1</sup> to v2, with constant ψ,

energy expression. Relaxing the wave function yields an energy lowering (blue arrow) until the energy obtained by applying the variational principle with v ¼ v<sup>2</sup> is reached (with associated wave function ψ2). Consequently, the true energy (black line) will always be found below

Figure 1. Illustration of the concavity of the E ¼ E v½ � functional (after Helgaker [28]) (Reprinted by permission from

v1, the associated ground state energy is given by the expectation value ψ1j j H v½ � <sup>1</sup> ψ<sup>1</sup>

induces changes in energy linear in <sup>δ</sup>vð Þ<sup>r</sup> as a consequence of the term <sup>Ð</sup>

E N½ � ; λv<sup>1</sup> þ ð Þ 1 � λ v<sup>2</sup> ≥ λE N½ �þ ; v<sup>1</sup> ð Þ 1 � λ E N½ � ; v<sup>2</sup> ∧ λ ∈½ � 0; 1 (2)

� �, with

rð Þr vð Þr dr in the DFT

$$\iint \chi(\mathbf{r}, \mathbf{r}') \Theta(\mathbf{r}) \Theta(\mathbf{r}') d\mathbf{r} d\mathbf{r}' \le 0 \tag{3}$$

where θð Þr is any continuous function [23]. This inequality shows up when considering the second-order variation of the energy

$$
\delta E^{(2)} = \mathbb{Y}\_2 \iint \left(\delta^2 E / \delta v(\mathbf{r}) \delta v(\mathbf{r'})\right)\_N \delta v(\mathbf{r}) \delta v(\mathbf{r'}) d\mathbf{r} d\mathbf{r'} \tag{4}
$$

When adopting the δvð Þ¼ r V0δ r � r<sup>00</sup> ð Þ choice for δvð Þr , where V<sup>0</sup> is a constant one gets

$$
\delta E^{(2)} = \%{V\_0^2 \chi(\mathbf{r}'', \mathbf{r}'')} \le 0 \tag{5}
$$

showing that the diagonal elements of the linear response function χ r;r<sup>0</sup> ð Þ should be negative or zero. This result links the concavity of the E N½ � ; v functional to the diagonal elements of the linear response function defined on R3 � <sup>R</sup><sup>3</sup> . This negativity was retrieved in all our numerical results (non-integrated and condensed) (see for example [29, 30]). Its physical interpretation is straightforward through the definition of χ r;r<sup>0</sup> ð Þ as δrð Þr =δv r<sup>0</sup> ð Þ ð Þ <sup>N</sup>: if the potential at a given point r is increased (made less negative), this electron-unfavourable situation will lead to electron depletion at that point, yielding a negative χð Þ r;r .

In Section 4, we will point out that Coulson's atom-atom polarizability πAB [31] can be considered as a Hückel-theory forerunner of the LRF and exploit its properties in discussing molecular conductivity. Defined as

$$
\pi\_{AB} = \eth \eta\_A / \eth \alpha\_B = \eth^2 E / \eth \alpha\_A \eth \alpha\_B = \pi\_{BA} \tag{6}
$$

the analogy emerges as qA is the π-electron charge on atom A, whereas (the change in) the Coulomb integral α<sup>B</sup> is equivalent to a change in the external potential. The case A ¼ B was explored by Coulson using his complex integral formalism for Hückel's π energy and its derivatives, proving that πAA can be written as

$$
\pi\_{AA} = (1/\pi) \int\_{-\infty}^{+\infty} \left(\Delta\_{AA}(\dot{\mathbf{u}})/\Delta(\dot{\mathbf{u}})\right)^2 d\mathbf{y} \tag{7}
$$

Here, Δ represents the Hückel secular determinant (or characteristic function) and ΔAA is its counterpart with row A and column A deleted. It turns out that ΔAAð Þ iy =Δð Þ iy is imaginary and therefore πAA is negative (see also Section 4).

When introducing the E ¼ E N½ � ; v functional, it was stressed that in the LRF the second functional derivative is taken at constant N. In the remaining part of this chapter, it turns out that no less important role in CDFT is played by the softness kernel, which is the second functional derivative with respect to <sup>v</sup>ð Þ<sup>r</sup> of the <sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup> <sup>μ</sup>; <sup>v</sup> � � functional (the grand potential) at constant μ. Here, we switch from the canonical ensemble to the grand canonical ensemble [32] connecting two ways of specifying the same physics via functionals of each time two variables differing in one pair that is connected through a Legendre transformation (see for example [33]).

$$\Omega\left[\mu,\upsilon\right] = E\left[\tilde{N}\left[\mu,\upsilon\right],\upsilon\right] - \mu\tilde{N}\left[\mu,\upsilon\right] \tag{8}$$

where GTT, GTP and GPP are the second derivatives of Gibbs free energy, GXY <sup>¼</sup> <sup>∂</sup><sup>2</sup>

be positive, and the condition that

both variables are now extensive [33].

∂2 Ω=∂μ<sup>2</sup> � �

and finally to

GTTGPP � <sup>G</sup><sup>2</sup>

PT � � <sup>¼</sup> <sup>κ</sup>TVCP=<sup>T</sup> � <sup>α</sup><sup>2</sup>

Let us now consider the analogy with Ω μ; v � �. It is well known that

∂2 Ω=∂μ<sup>2</sup> � �

concavity in all directions leads after some algebra (see [23]) to the condition

<sup>0</sup> ð ÞS � sð Þr s r

sð Þ¼ r ∂ð Þ δΩ=δvð Þr <sup>μ</sup>=∂μ � �

<sup>v</sup>Δ<sup>μ</sup> <sup>þ</sup> <sup>Ð</sup> <sup>∂</sup>ð Þ <sup>δ</sup>Ω=δvð Þ<sup>r</sup> <sup>μ</sup>=∂<sup>μ</sup>

ðð <sup>s</sup> <sup>r</sup>;<sup>r</sup>

� ∂ð Þ δΩ=δvð Þr <sup>μ</sup>=∂μ � �

the analogue of (15) where the local softness

δ r<sup>0</sup> � r<sup>00</sup> ð Þ, one obtains the condition

� �<sup>2</sup>

� �

This stability condition implies the negative semidefiniteness of the Hessian matrix. It yields

that is, the isothermal compressibility κ<sup>T</sup> and the heat capacity at constant pressure C<sup>P</sup> should

where α ¼ ð Þ ∂V=∂T <sup>P</sup>=V is the coefficient of thermal expansion. Another classical example of such stability analysis is the entropy written as S U½ � ; V (at constant number of particles) where

where S is the global softness [3]. As the r.h.s. of (16) is negative, concavity for Ω μ; v � � in μ shows up. As discussed above, Ω is also concave in vð Þr , leading to the positive semidefiniteness of s r;r<sup>0</sup> ð Þ (these expressions being the counterparts of (13) and (14)). The condition for

> v Δvð Þr dr

<sup>0</sup> ð Þ ð Þ Δvð Þr drΔv r

has been introduced [37]. Taking again for Δvð Þr and Δv r<sup>0</sup> ð Þ Dirac delta functions δ r � r<sup>00</sup> ð Þ and

This inequality shows that the diagonal elements sð Þ r;r should be positive, as could be inferred from the concavity of Ω μ; v � � but, more importantly, they impose a restriction on the relative

sð Þ r;r S ≥ ð Þ sð Þr

<sup>0</sup> ð Þ ð Þ <sup>μ</sup>=∂μ � �

<sup>v</sup> <sup>∂</sup> <sup>δ</sup>Ω=δ<sup>v</sup> <sup>r</sup>

GPP ¼ ð Þ ∂V=∂P <sup>T</sup> ¼ �κTV ≤ 0 (13)

New Insights and Horizons from the Linear Response Function in Conceptual DFT

GTT ¼ �ð Þ ∂S=∂T <sup>P</sup> ¼ �CP=T ≤ 0 (14)

<sup>v</sup> ¼ �<sup>S</sup> <sup>μ</sup>; <sup>v</sup> � �, (16)

V<sup>2</sup> � � ≥ 0 ⇔ κTCP=T ≥ α<sup>2</sup>

<sup>þ</sup> ÐÐ <sup>∂</sup><sup>2</sup>

v

�

Ω=∂μ<sup>2</sup> � �

ÞΔvð Þr drΔv r

<sup>0</sup> ð Þdr 0

<sup>v</sup> <sup>¼</sup> <sup>∂</sup>rð Þ<sup>r</sup> <sup>=</sup>∂<sup>μ</sup> � �

<sup>v</sup> <sup>δ</sup><sup>2</sup>

<sup>2</sup> ≥ 0: (20)

<sup>0</sup> ð Þdr 0

<sup>Ω</sup>=δvð Þ<sup>r</sup> <sup>δ</sup><sup>v</sup> <sup>r</sup><sup>0</sup> ð Þ � �

≥ 0, (18)

<sup>v</sup> (19)

≥ 0 (17)

μ

G=∂X∂Y � �.

9

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V (15)

where <sup>μ</sup> and <sup>N</sup> are conjugate variables which are related by identities <sup>N</sup><sup>~</sup> <sup>μ</sup>; <sup>v</sup> � � <sup>¼</sup> <sup>N</sup> and μ~½ �¼ N; v μ. Its apparent analogy with classical thermodynamics will be addressed later.

In [23], we pointed out that <sup>Ω</sup> <sup>μ</sup>; <sup>v</sup> � � just as E N½ � ; <sup>v</sup> is concave w.r.t. <sup>v</sup>ð Þ<sup>r</sup> , implying that its second functional derivative at constant μ, the softness kernel (note the negative sign in the definition) [34].

$$s(\mathbf{r}, \mathbf{r}') = -\left(\delta^2 \Omega / \delta v(\mathbf{r}) \delta v(\mathbf{r}')\right)\_{\mu} \tag{9}$$

is positive semidefinite. This property fits the well-known Berkowitz-Parr relationship [33] linking χ r;r<sup>0</sup> ð Þ and s r;r<sup>0</sup> ð Þ (vide infra).

$$s(\mathbf{r}, \mathbf{r}') = -\chi(\mathbf{r}, \mathbf{r}') + f(\mathbf{r})f(\mathbf{r}')/\eta \tag{10}$$

where fð Þr is the Fukui function and η is the hardness. Indeed,

$$\begin{split} \iint \mathbf{s}(\mathbf{r}, \mathbf{r}') \theta(\mathbf{r}) \theta(\mathbf{r}') d\mathbf{r} d\mathbf{r}' &= \iint \left( -\chi(\mathbf{r}, \mathbf{r}') + f(\mathbf{r}) f(\mathbf{r}') / \eta \right) \theta(\mathbf{r}) \theta(\mathbf{r}') d\mathbf{r} d\mathbf{r}' \\ &= -\iint \chi(\mathbf{r}, \mathbf{r}') \theta(\mathbf{r}) \theta(\mathbf{r}') d\mathbf{r} d\mathbf{r}' + 1/\eta \int \left( f(\mathbf{r}) \theta(\mathbf{r}) \right)^2 d\mathbf{r} \ge 0 \end{split} \tag{11}$$

since the hardness is nonnegative and χ r;r<sup>0</sup> ð Þ was shown to be negative semidefinite.

The properties of χ r;r<sup>0</sup> ð Þ and s r;r<sup>0</sup> ð Þ incited us to reconsider the analogy between the DFT functionals E and Ω [32] on the one hand and the macroscopic thermodynamic state functions U ¼ U S½ � ; V , F ¼ F T½ � ; V , H ¼ H S½ � ; P and G ¼ G T½ � ; P on the other hand (internal energy, Helmholtz free energy, enthalpy and Gibbs free energy written as functions of volume (V), temperature (T), pressure (P) and entropy (S)). Parr and Nalewajski extended the notion of intensive and extensive variables ð Þ T; P and ð Þ S; V , respectively, in thermodynamics to the variables in DFT functionals by classifying external variables as properties additive with respect to any partitioning of the electron density rð Þ¼ r rAð Þþ r rBð Þr [32]. In this way, rð Þr and N are clearly extensive, and μ and vð Þr are intensive. The analogy between G T½ � ; P and Ω μ; v � � can now be stressed: both the state function G and the DFT functional Ω contain two intensive variables. This situation leads to a remarkable property when formulating a DFT analogue of the stability analysis in macroscopic thermodynamics [33, 35, 36]. Concavity for G T½ � ; P in all directions then implies that

$$d^2\mathcal{G} = \mathcal{G}\_{TT}(\Delta T)^2 + 2\mathcal{G}\_{TP}\Delta T\Delta P + \mathcal{G}\_{PP}(\Delta P)^2 \le 0,\tag{12}$$

where GTT, GTP and GPP are the second derivatives of Gibbs free energy, GXY <sup>¼</sup> <sup>∂</sup><sup>2</sup> G=∂X∂Y � �. This stability condition implies the negative semidefiniteness of the Hessian matrix. It yields

$$\mathcal{G}\_{PP} = (\partial V / \partial P)\_T = -\kappa\_\Gamma V \le 0 \tag{13}$$

$$G\_{TT} = - (\partial \mathcal{S} / \partial T)\_P = -\mathcal{C}\_\mathbb{P} / T \le 0 \tag{14}$$

that is, the isothermal compressibility κ<sup>T</sup> and the heat capacity at constant pressure C<sup>P</sup> should be positive, and the condition that

$$\left(\left(\mathbf{G}\_{TT}\mathbf{G}\_{\mathrm{PP}} - \mathbf{G}\_{\mathrm{PT}}^2\right) = \left(\kappa\_\mathrm{T}V\mathbf{C}\_\mathrm{P}/T - \alpha^2V^2\right) \geq 0 \Leftrightarrow \kappa\_\mathrm{T}\mathbf{C}\_\mathrm{P}/T \geq \alpha^2V \tag{15}$$

where α ¼ ð Þ ∂V=∂T <sup>P</sup>=V is the coefficient of thermal expansion. Another classical example of such stability analysis is the entropy written as S U½ � ; V (at constant number of particles) where both variables are now extensive [33].

Let us now consider the analogy with Ω μ; v � �. It is well known that

$$\left(\partial^2 \Omega / \partial \mu^2\right)\_v = -\mathcal{S}\left[\mu, v\right]\_\prime \tag{16}$$

where S is the global softness [3]. As the r.h.s. of (16) is negative, concavity for Ω μ; v � � in μ shows up. As discussed above, Ω is also concave in vð Þr , leading to the positive semidefiniteness of s r;r<sup>0</sup> ð Þ (these expressions being the counterparts of (13) and (14)). The condition for concavity in all directions leads after some algebra (see [23]) to the condition

$$
\begin{split} \int \left( \left( \partial^2 \Omega / \partial \mu^2 \right)\_v \Delta \mu + \int \left( \partial (\delta \Omega / \delta v(\mathbf{r}))\_{\mu} / \partial \mu \right)\_v \Delta v(\mathbf{r}) d\mathbf{r} \right)^2 &+ \iint \left( \left( \partial^2 \Omega / \partial \mu^2 \right)\_v \left( \partial^2 \Omega / \delta v(\mathbf{r}) \delta v(\mathbf{r}) \right)\_{\mu} \right. \\ \left. - \left( \partial (\delta \Omega / \delta v(\mathbf{r}))\_{\mu} / \partial \mu \right)\_v \left( \partial (\delta \Omega / \delta v(\mathbf{r}'))\_{\mu} / \partial \mu \right)\_v \right) \Delta v(\mathbf{r}) d\mathbf{r} \Delta v(\mathbf{r}') d\mathbf{r}' \geq 0 \end{split} \tag{17}$$

and finally to

at constant μ. Here, we switch from the canonical ensemble to the grand canonical ensemble [32] connecting two ways of specifying the same physics via functionals of each time two variables differing in one pair that is connected through a Legendre transformation (see for

where <sup>μ</sup> and <sup>N</sup> are conjugate variables which are related by identities <sup>N</sup><sup>~</sup> <sup>μ</sup>; <sup>v</sup> � � <sup>¼</sup> <sup>N</sup> and μ~½ �¼ N; v μ. Its apparent analogy with classical thermodynamics will be addressed later.

In [23], we pointed out that <sup>Ω</sup> <sup>μ</sup>; <sup>v</sup> � � just as E N½ � ; <sup>v</sup> is concave w.r.t. <sup>v</sup>ð Þ<sup>r</sup> , implying that its second functional derivative at constant μ, the softness kernel (note the negative sign in the

is positive semidefinite. This property fits the well-known Berkowitz-Parr relationship [33]

Ω=δvð Þr δv r <sup>0</sup> ð Þ � �

<sup>0</sup> ð Þþ fð Þr f r

<sup>0</sup> ð Þþ fð Þr f r <sup>0</sup> ð Þ ð Þ=η θð Þr θ r

<sup>0</sup> ð Þdrdr

<sup>0</sup> þ 1=η

<sup>G</sup> <sup>¼</sup> GTTð Þ <sup>Δ</sup><sup>T</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>GTPΔTΔ<sup>P</sup> <sup>þ</sup> GPPð Þ <sup>Δ</sup><sup>P</sup> <sup>2</sup> <sup>≤</sup> <sup>0</sup>, (12)

<sup>0</sup> ð Þθð Þr θ r

The properties of χ r;r<sup>0</sup> ð Þ and s r;r<sup>0</sup> ð Þ incited us to reconsider the analogy between the DFT functionals E and Ω [32] on the one hand and the macroscopic thermodynamic state functions U ¼ U S½ � ; V , F ¼ F T½ � ; V , H ¼ H S½ � ; P and G ¼ G T½ � ; P on the other hand (internal energy, Helmholtz free energy, enthalpy and Gibbs free energy written as functions of volume (V), temperature (T), pressure (P) and entropy (S)). Parr and Nalewajski extended the notion of intensive and extensive variables ð Þ T; P and ð Þ S; V , respectively, in thermodynamics to the variables in DFT functionals by classifying external variables as properties additive with respect to any partitioning of the electron density rð Þ¼ r rAð Þþ r rBð Þr [32]. In this way, rð Þr and N are clearly extensive, and μ and vð Þr are intensive. The analogy between G T½ � ; P and Ω μ; v � � can now be stressed: both the state function G and the DFT functional Ω contain two intensive variables. This situation leads to a remarkable property when formulating a DFT analogue of the stability analysis in macroscopic thermodynamics [33, 35, 36]. Concavity for

s r;r

s r;r

where fð Þr is the Fukui function and η is the hardness. Indeed,

<sup>0</sup> ¼ ðð

> ¼ � ðð χ r;r

<sup>0</sup> ð Þdrdr

<sup>0</sup> ð Þ¼� <sup>δ</sup><sup>2</sup>

<sup>0</sup> ð Þ¼�χ r;r

�χ r;r

since the hardness is nonnegative and χ r;r<sup>0</sup> ð Þ was shown to be negative semidefinite.

<sup>Ω</sup> <sup>μ</sup>; <sup>v</sup> � � <sup>¼</sup> <sup>E</sup> <sup>N</sup><sup>~</sup> <sup>μ</sup>; <sup>v</sup> � �; <sup>v</sup>� � <sup>μ</sup>N<sup>~</sup> <sup>μ</sup>; <sup>v</sup> � � � (8)

<sup>μ</sup> (9)

<sup>0</sup> ð Þ=η (10)

<sup>0</sup> ð Þdrdr 0

ð Þ fð Þr θð Þr

2 dr ≥ 0 (11)

ð

example [33]).

8 Density Functional Theory

definition) [34].

ðð s r;r <sup>0</sup> ð Þθð Þr θ r

linking χ r;r<sup>0</sup> ð Þ and s r;r<sup>0</sup> ð Þ (vide infra).

G T½ � ; P in all directions then implies that

d2

$$\iint (s(\mathbf{r}, \mathbf{r}')S - s(\mathbf{r})s(\mathbf{r}')) \Delta v(\mathbf{r}) d\mathbf{r} \Delta v(\mathbf{r}') d\mathbf{r}' \ge 0,\tag{18}$$

the analogue of (15) where the local softness

$$s(\mathbf{r}) = \left(\eth(\delta\Omega/\delta v(\mathbf{r}))\_{\mu}/\partial\mu\right)\_{v} = \left(\eth\rho(\mathbf{r})/\partial\mu\right)\_{v} \tag{19}$$

has been introduced [37]. Taking again for Δvð Þr and Δv r<sup>0</sup> ð Þ Dirac delta functions δ r � r<sup>00</sup> ð Þ and δ r<sup>0</sup> � r<sup>00</sup> ð Þ, one obtains the condition

$$s(\mathbf{r}, \mathbf{r}) \, \mathbf{S} \ge (s(\mathbf{r}))^2 \ge 0. \tag{20}$$

This inequality shows that the diagonal elements sð Þ r;r should be positive, as could be inferred from the concavity of Ω μ; v � � but, more importantly, they impose a restriction on the relative values of the three softness descriptors s r;r<sup>0</sup> ð Þ,S and sð Þr in analogy to the thermodynamic relationship between κT, C<sup>P</sup> and α at given T and P (V ¼ V T½ � ; P ). This result is compatible with the aforementioned Berkowitz-Parr relationship. Indeed, starting from their expression for r<sup>0</sup> ¼ r (see (10), the relations sð Þ¼ r fð Þr =η)

$$s(\mathbf{r}, \mathbf{r}) = -\chi(\mathbf{r}, \mathbf{r}) + s(\mathbf{r})s(\mathbf{r})/S \tag{21}$$

and knowing that χð Þ r;r ≤ 0 (vide supra) one obtains

$$s(\mathbf{r}, \mathbf{r}) - s(\mathbf{r})s(\mathbf{r})/\mathcal{S} \ge 0,\tag{22}$$

δrð Þr =δv r

The Berkowitz-Parr relationship [34] can then be written as

<sup>0</sup> ð Þ ð Þ <sup>N</sup> ¼ δrð Þr =δv r

s r;r

<sup>0</sup> ð Þ¼�χ r;r

δrð Þr =δv r

macroscopic thermodynamics [33]. As <sup>∂</sup>rð Þ<sup>r</sup> <sup>=</sup>∂<sup>μ</sup>

analysed, in order to scrutinize nearsightedness.

from (28)

<sup>0</sup> ð Þ ð Þ <sup>μ</sup> ¼ �s r;r

The intensity of the red and blue regions represents the magnitude of v r<sup>0</sup> ð Þ and rð Þr , respectively.

<sup>0</sup> ð Þ¼ <sup>δ</sup><sup>2</sup>

Figure 2. Pictorial representation of the nearsightedness of electronic matter principle: when δv r<sup>0</sup> ð Þ ≥ δv<sup>2</sup> r<sup>0</sup> ð Þ, with r<sup>0</sup> outside a sphere with radius R around r0, δr0ð Þ r<sup>0</sup> no longer increases—δr0ð Þ¼ r<sup>0</sup> Δrð Þ r0; R no matter how large δv r<sup>0</sup> ð Þ.

<sup>0</sup> ð Þ ð Þ <sup>μ</sup> � <sup>∂</sup>rð Þ<sup>r</sup> <sup>=</sup>∂<sup>μ</sup>

an equation transforming conditions of constant N into constant μ, for taking the functional derivative of rð Þr w.r.t. δvð Þr in analogy with this type of equation for partial derivatives in

is an alternative way, as compared to ð Þ ∂rð Þr =∂N <sup>v</sup> to write the Fukui function f(r), one retrieves

which will be the key equation in this section. As analytical methods are available to evaluate χ r;r<sup>0</sup> ð Þ, fð Þr and η on equal footing [41], s r;r<sup>0</sup> ð Þ can be evaluated, and its difference with χ r;r<sup>0</sup> ð Þ

<sup>0</sup> ð Þþ fð Þr f r

Ω=δvð Þr δv r <sup>0</sup> ð Þ

New Insights and Horizons from the Linear Response Function in Conceptual DFT

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11

<sup>v</sup> δμ=δv r <sup>0</sup> ð Þ

<sup>v</sup> equals the local softness <sup>s</sup>ð Þ<sup>r</sup> and δμ=δ<sup>v</sup> <sup>r</sup><sup>0</sup> ð Þ

<sup>0</sup> ð Þ=η (25)

<sup>μ</sup> (23)

<sup>N</sup> (24)

N

retrieving our conclusions above.

#### 2.2. Kohn's nearsightedness of electronic matter revisited

We now report on our recent explorations [38] on Kohn's NEM principle. Kohn introduced the NEM concept in 1996 [39] and elaborated on it in 2005 with Prodan et al. [40]. In his own words, it can be viewed as 'underlying such important ideas as Pauling's chemical bond, transferability, and Yang's computational principle of divide and conquer' [40]. Certainly in view of the two former issues, this principle, formulated by a physicist, touches the very heart of chemistry and so, in our opinion, it was tempting to look at it with a chemist's eye. Why however is this issue addressed in this chapter; in other words, what is the link between the LRF and nearsightedness?

The quintessence of the NEM principle is as follows (Figure 2): consider a (many) electron system characterized by an electron density function rð Þr with a given electronic chemical potential μ. Now, concentrate on a point r<sup>0</sup> and perturb the system in its external potential vð Þr at point r<sup>0</sup> , outside a sphere with radius R around r<sup>0</sup> at constant electronic potential μ. Then, the NEM principle states that the absolute value of the density change at r0, j j Δrð Þ r<sup>0</sup> , will be lower than a finite maximum value Δr, which depends on r<sup>0</sup> and R, whatever the magnitude of the perturbation. As stated by Kohn, anthropomorphically the particle density rð Þr cannot 'see' any perturbation <sup>v</sup>ð Þ<sup>r</sup> beyond the distance <sup>R</sup> <sup>r</sup>0;Δ<sup>r</sup> within an accuracy <sup>Δ</sup>r, that is, the density shows nearsightedness. Kohn offered evidence that in the case of 1D 'gapped' systems (i.e. with hardness η larger than zero) the decay of Δr as a function of R (i.e. upon increasing j j r � r<sup>0</sup> ) is exponential and that for gapless systems it follows a power law. This suggests that in the molecular world, where η is observed to be always positive, the electron density should only be sensitive to nearby changes in the external potential. In [38], we provided the first numerical confirmation of this nearsightedness principle for real, 3D, molecules.

Again, why address this issue in this LRF chapter? Going back to Kohn's formulation quintessentially a change in density at a given point Δrð Þ r<sup>0</sup> in response to a change in external potential Δv r<sup>0</sup> ð Þ at different points is analysed. These are the typical ingredients of the LRF (change in v produces a change in r), and in this case the process is considered at constant electronic chemical potential μ, in order words δrð Þr =δv r<sup>0</sup> ð Þ ð Þ <sup>μ</sup> is the key quantity. This is nothing else than the softness kernel s r;r<sup>0</sup> ð Þ with a minus sign in front. Indeed,

New Insights and Horizons from the Linear Response Function in Conceptual DFT http://dx.doi.org/10.5772/intechopen.80280 11

Figure 2. Pictorial representation of the nearsightedness of electronic matter principle: when δv r<sup>0</sup> ð Þ ≥ δv<sup>2</sup> r<sup>0</sup> ð Þ, with r<sup>0</sup> outside a sphere with radius R around r0, δr0ð Þ r<sup>0</sup> no longer increases—δr0ð Þ¼ r<sup>0</sup> Δrð Þ r0; R no matter how large δv r<sup>0</sup> ð Þ. The intensity of the red and blue regions represents the magnitude of v r<sup>0</sup> ð Þ and rð Þr , respectively.

$$\left(\delta\rho(\mathbf{r})/\delta v(\mathbf{r}')\right)\_{\mu} = -s(\mathbf{r}, \mathbf{r}') = \left(\delta^2 \Omega/\delta v(\mathbf{r}) \delta v(\mathbf{r}')\right)\_{\mu} \tag{23}$$

The Berkowitz-Parr relationship [34] can then be written as

values of the three softness descriptors s r;r<sup>0</sup> ð Þ,S and sð Þr in analogy to the thermodynamic relationship between κT, C<sup>P</sup> and α at given T and P (V ¼ V T½ � ; P ). This result is compatible with the aforementioned Berkowitz-Parr relationship. Indeed, starting from their expression

We now report on our recent explorations [38] on Kohn's NEM principle. Kohn introduced the NEM concept in 1996 [39] and elaborated on it in 2005 with Prodan et al. [40]. In his own words, it can be viewed as 'underlying such important ideas as Pauling's chemical bond, transferability, and Yang's computational principle of divide and conquer' [40]. Certainly in view of the two former issues, this principle, formulated by a physicist, touches the very heart of chemistry and so, in our opinion, it was tempting to look at it with a chemist's eye. Why however is this issue addressed in this chapter; in other words, what is the link between the

The quintessence of the NEM principle is as follows (Figure 2): consider a (many) electron system characterized by an electron density function rð Þr with a given electronic chemical potential μ. Now, concentrate on a point r<sup>0</sup> and perturb the system in its external potential vð Þr

NEM principle states that the absolute value of the density change at r0, j j Δrð Þ r<sup>0</sup> , will be lower than a finite maximum value Δr, which depends on r<sup>0</sup> and R, whatever the magnitude of the perturbation. As stated by Kohn, anthropomorphically the particle density rð Þr cannot 'see' any perturbation <sup>v</sup>ð Þ<sup>r</sup> beyond the distance <sup>R</sup> <sup>r</sup>0;Δ<sup>r</sup> within an accuracy <sup>Δ</sup>r, that is, the density shows nearsightedness. Kohn offered evidence that in the case of 1D 'gapped' systems (i.e. with hardness η larger than zero) the decay of Δr as a function of R (i.e. upon increasing j j r � r<sup>0</sup> ) is exponential and that for gapless systems it follows a power law. This suggests that in the molecular world, where η is observed to be always positive, the electron density should only be sensitive to nearby changes in the external potential. In [38], we provided the first numerical

Again, why address this issue in this LRF chapter? Going back to Kohn's formulation quintessentially a change in density at a given point Δrð Þ r<sup>0</sup> in response to a change in external potential Δv r<sup>0</sup> ð Þ at different points is analysed. These are the typical ingredients of the LRF (change in v produces a change in r), and in this case the process is considered at constant electronic chemical potential μ, in order words δrð Þr =δv r<sup>0</sup> ð Þ ð Þ <sup>μ</sup> is the key quantity. This is

confirmation of this nearsightedness principle for real, 3D, molecules.

nothing else than the softness kernel s r;r<sup>0</sup> ð Þ with a minus sign in front. Indeed,

, outside a sphere with radius R around r<sup>0</sup> at constant electronic potential μ. Then, the

sð Þ¼� r;r χð Þþ r;r sð Þr sð Þr =S (21)

sð Þ� r;r sð Þr sð Þr =S ≥ 0, (22)

for r<sup>0</sup> ¼ r (see (10), the relations sð Þ¼ r fð Þr =η)

retrieving our conclusions above.

10 Density Functional Theory

LRF and nearsightedness?

at point r<sup>0</sup>

and knowing that χð Þ r;r ≤ 0 (vide supra) one obtains

2.2. Kohn's nearsightedness of electronic matter revisited

$$(\delta\rho(\mathbf{r})/\delta v(\mathbf{r}'))\_{\mathcal{N}} = (\delta\rho(\mathbf{r})/\delta v(\mathbf{r}'))\_{\mu} - \left(\partial\rho(\mathbf{r})/\partial\mu\right)\_v (\delta\mu/\delta v(\mathbf{r}'))\_{\mathcal{N}} \tag{24}$$

an equation transforming conditions of constant N into constant μ, for taking the functional derivative of rð Þr w.r.t. δvð Þr in analogy with this type of equation for partial derivatives in macroscopic thermodynamics [33]. As <sup>∂</sup>rð Þ<sup>r</sup> <sup>=</sup>∂<sup>μ</sup> <sup>v</sup> equals the local softness <sup>s</sup>ð Þ<sup>r</sup> and δμ=δ<sup>v</sup> <sup>r</sup><sup>0</sup> ð Þ N is an alternative way, as compared to ð Þ ∂rð Þr =∂N <sup>v</sup> to write the Fukui function f(r), one retrieves from (28)

$$s(\mathbf{r}, \mathbf{r}') = -\chi(\mathbf{r}, \mathbf{r}') + f(\mathbf{r})f(\mathbf{r}') / \eta \tag{25}$$

which will be the key equation in this section. As analytical methods are available to evaluate χ r;r<sup>0</sup> ð Þ, fð Þr and η on equal footing [41], s r;r<sup>0</sup> ð Þ can be evaluated, and its difference with χ r;r<sup>0</sup> ð Þ analysed, in order to scrutinize nearsightedness.

Figure 3. Atom condensed linear response function and softness kernel of 1,3,5-hexatriene. The curves of the softness kernels using f+ and f� are overlapping [38].

In Figure 3, we depict the atom condensed linear response function and the softness kernel, the matrices χAB and sAB, of 1,3,5-hexatriene. Condensation was performed by integrating the kernels over the domains of atoms A and B, represented by VA and VA, that is,

$$\mathbf{s}\_{AB} = \int\_{V\_A} \int \mathbf{s}(\mathbf{r}, \mathbf{r}') d\mathbf{r} d\mathbf{r}' \tag{26}$$

effect of functionalization dies off much more quickly under constant μ conditions. In fact, the third carbon atom is hardly affected in this—it should be stressed—unsaturated system, offering the possibility for mesomerism. In the LRF, the decay is slow and effects are still relatively important

Figure 4. Alchemical change in density using the linear response (top) and softness kernel (bottom) for the heptatrienyl

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13

Further case studies on '3D' systems (e.g. alchemically changing methylcubane to fluorocubane and on functionalized neopentane) yield similar results. All together, the results on the nearsightedness of the softness kernel found for all systems discussed in [38] are the first and a firm numerical confirmation of Kohn's NEM principle in the molecular world. To put it in chemical terms, these findings provide computational evidence for the transferability of functional groups: molecular systems can be divided into locally interacting subgroups retaining a similar functionality and reactivity that can only be influenced by changes in the direct environment of the functional group. Thus, the physicist's NEM principle and the chemist's transferability principle [45]—at the heart of, for example, the whole of organic chemistry [46]—are reconciled.

As a natural extension of CDFT to the case where spin polarization is included [47–49], spin polarized conceptual DFT was introduced by Galvan et al. [50, 51] (see also Ghanty and Ghosh

five bonds away from the perturbation.

cation to 1,3,5-hexatriene-1-amine [38].

2.3. Extension of χ r;r<sup>0</sup> ð Þ in the context of spin polarized CDFT

From our previous work on polyenes [42], the χAB matrix elements are known to show an alternating behaviour with maxima on mesomeric active atoms (C2,C<sup>4</sup> and C6) and minima for mesomeric passive atoms (C<sup>3</sup> and C5), with the change in vð Þr taking place at C1.The picture illustrates that the softness kernel is more nearsighted than the LRF (all its values are lower) and that s1, <sup>5</sup> and s1, <sup>6</sup> are very close to zero; the effect of the perturbation has died off completely, confirming the nearsightedness of the softness kernel. This effect can be traced back to a cancellation of the LRF, which is non-nearsighted, by the second term in Eq. (25), and accounts for the density changes induced by charge transfer from the electron reservoir to keep the chemical potential constant. Note that this condition, at first sight somewhat strange, is an often more realistic perspective when considering for example the reactivity of molecules in solution where the chemical potential is fixed by the solvent allowing (partial) charge transfer to or from the molecule while keeping its chemical potential constant [43, 44].

As a second example, we show in Figure 4 the change in density of the 1,3,5-heptatrienyl cation when the C atom of one of the terminal CH2 atoms was alchemically replaced by an N atom. The corresponding density difference was evaluated through the alchemical derivatives approach (Section 3) where the carbon atom of the CH<sup>2</sup> group was annihilated and replaced by a nitrogen atom at constant geometry and constant number of electrons. It is clear that the New Insights and Horizons from the Linear Response Function in Conceptual DFT http://dx.doi.org/10.5772/intechopen.80280 13

effect of functionalization dies off much more quickly under constant μ conditions. In fact, the third carbon atom is hardly affected in this—it should be stressed—unsaturated system, offering the possibility for mesomerism. In the LRF, the decay is slow and effects are still relatively important five bonds away from the perturbation.

Further case studies on '3D' systems (e.g. alchemically changing methylcubane to fluorocubane and on functionalized neopentane) yield similar results. All together, the results on the nearsightedness of the softness kernel found for all systems discussed in [38] are the first and a firm numerical confirmation of Kohn's NEM principle in the molecular world. To put it in chemical terms, these findings provide computational evidence for the transferability of functional groups: molecular systems can be divided into locally interacting subgroups retaining a similar functionality and reactivity that can only be influenced by changes in the direct environment of the functional group. Thus, the physicist's NEM principle and the chemist's transferability principle [45]—at the heart of, for example, the whole of organic chemistry [46]—are reconciled.

#### 2.3. Extension of χ r;r<sup>0</sup> ð Þ in the context of spin polarized CDFT

In Figure 3, we depict the atom condensed linear response function and the softness kernel, the matrices χAB and sAB, of 1,3,5-hexatriene. Condensation was performed by integrating the

Figure 3. Atom condensed linear response function and softness kernel of 1,3,5-hexatriene. The curves of the softness

kernels over the domains of atoms A and B, represented by VA and VA, that is,

kernels using f+ and f� are overlapping [38].

12 Density Functional Theory

sAB ¼

molecule while keeping its chemical potential constant [43, 44].

ð

ð

s r;r <sup>0</sup> ð Þdrdr

<sup>0</sup> (26)

VB

From our previous work on polyenes [42], the χAB matrix elements are known to show an alternating behaviour with maxima on mesomeric active atoms (C2,C<sup>4</sup> and C6) and minima for mesomeric passive atoms (C<sup>3</sup> and C5), with the change in vð Þr taking place at C1.The picture illustrates that the softness kernel is more nearsighted than the LRF (all its values are lower) and that s1, <sup>5</sup> and s1, <sup>6</sup> are very close to zero; the effect of the perturbation has died off completely, confirming the nearsightedness of the softness kernel. This effect can be traced back to a cancellation of the LRF, which is non-nearsighted, by the second term in Eq. (25), and accounts for the density changes induced by charge transfer from the electron reservoir to keep the chemical potential constant. Note that this condition, at first sight somewhat strange, is an often more realistic perspective when considering for example the reactivity of molecules in solution where the chemical potential is fixed by the solvent allowing (partial) charge transfer to or from the

As a second example, we show in Figure 4 the change in density of the 1,3,5-heptatrienyl cation when the C atom of one of the terminal CH2 atoms was alchemically replaced by an N atom. The corresponding density difference was evaluated through the alchemical derivatives approach (Section 3) where the carbon atom of the CH<sup>2</sup> group was annihilated and replaced by a nitrogen atom at constant geometry and constant number of electrons. It is clear that the

VA

As a natural extension of CDFT to the case where spin polarization is included [47–49], spin polarized conceptual DFT was introduced by Galvan et al. [50, 51] (see also Ghanty and Ghosh [52] and for a review see [53]). In the so-called Nα; N<sup>β</sup> � � representation, the response of the electronic energy to perturbations in the number of α electrons, Nα, the number of β electrons, Nβ, the external potential acting on the α electrons, vαð Þr , and the external potential acting on the β electrons, vβð Þr , is studied. In this representation, the E ¼ E N½ � ; v functional from Section 2.1 is generalized to the E ¼ E Nα; Nβ; vα; v<sup>β</sup> � � functional. In the second, equivalent, ð Þ <sup>N</sup>; <sup>N</sup><sup>S</sup> representation, in fact the one introduced by Galvan, the E ¼ E N½ � ; v functional is generalized to E ¼ E N½ � ; NS; v; v<sup>S</sup> or E ¼ E N½ � ; NS; v; B where N<sup>S</sup> denotes the electron spin number defined as the difference between the number of α and β electrons, N<sup>S</sup> ¼ N<sup>α</sup> � Nβ, whereas vs is given by <sup>v</sup>Sð Þ¼ <sup>r</sup> <sup>v</sup>αð Þ� <sup>r</sup> <sup>v</sup>βð Þ<sup>r</sup> � �=2.

<sup>v</sup>ð Þ<sup>r</sup> is equal to <sup>v</sup>αð Þþ <sup>r</sup> <sup>v</sup>βð Þ<sup>r</sup> � �=2 and <sup>B</sup> represents an external magnetic field. If the magnetic field is static and uniform along the z axis, one has vαð Þ¼ r vð Þþ r μBBZ and vβð Þ¼ r vð Þ� r μBBZ, vð Þr is the usual external spin-free potential (as used in CDFT without spin polarization), whereas vSð Þr is related to the magnetic field B rð Þ. In this context, and sticking to the ð Þ N; N<sup>S</sup> representation, three linear response functions can now be defined:

$$\chi\_{\rm NN}(\mathbf{r}, \mathbf{r}') = \left(\delta^2 E / \delta v(\mathbf{r}) \delta v(\mathbf{r}')\right)\_{\rm N, N\downarrow, v\rm s} \tag{27}$$

$$
\lambda \chi\_{\rm SS}(\mathbf{r}, \mathbf{r}') = \left(\delta^2 E / \delta v\_{\rm S}(\mathbf{r}) \delta v\_{\rm S}(\mathbf{r}')\right)\_{\rm N, N\_{\rm S}, v} \tag{28}
$$

Figure 5. Contour plots of the radial distribution function of the spin polarized linear response function of Lithium in the [Nα, Nβ] representation. r is represented on the horizontal axis, r<sup>0</sup> on the vertical axis [(a) Lithium χαα, (b) Lithium χαβ, (c) Lithium χαα, (d) Lithium χββ. In the insert, the χββ plot for He (see text) (Reprinted from [48] with the permission of AIP Publishing)].

New Insights and Horizons from the Linear Response Function in Conceptual DFT

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15

$$\begin{split} \chi\_{\rm NS}(\mathbf{r}, \mathbf{r}') &= \left(\delta^2 \mathbf{E}/\delta\mathbf{v}(\mathbf{r}) \delta\mathbf{v}\_{\rm S}(\mathbf{r}')\right)\_{\rm N, N\_{\rm S}} = (\delta\rho\_{\rm N}(\mathbf{r})/\delta\mathbf{v}\_{\rm S}(\mathbf{r}'))\_{\rm N, N\_{\rm S}, v} \\ &= (\delta\rho\_{\rm S}(\mathbf{r}')/\delta\mathbf{v}(\mathbf{r}))\_{\rm N, N\_{\rm S}, v} = \left(\delta^2 \mathbf{E}/\delta\mathbf{v}\mathbf{s}(\mathbf{r}')\delta\mathbf{v}(\mathbf{r})\right)\_{\rm N, N\_{\rm S}} = \chi\_{\rm SN}(\mathbf{r}', \mathbf{r}) \end{split} \tag{29}$$

where rNð Þ¼ r rαð Þþ r rβð Þr and rSð Þ¼ r rαð Þ� r rβð Þr are the total and spin densities, respectively. χNN r;r<sup>0</sup> ð Þ is the analogue of the spin-independent CDFT expression for the LRF χ r;r<sup>0</sup> ð Þ (Eq. (1)).

Space limitations prevent us to go in detail on the results reported in [47, 48]. We only depict in Figure 5 the SPCDFT analogue of the contour plots for χ r;r<sup>0</sup> ð Þ for closed shell atoms as discussed in [47] and the Chem Soc Rev paper [16]. In Figure 5, we plot the LRF, in the Nα; N<sup>β</sup> � � representation, for the ground state of Li ((1s)<sup>2</sup> (2s)<sup>1</sup> ). The structure of χαα r;r<sup>0</sup> ð Þ and χββ r;r<sup>0</sup> ð Þ is similar to the χ r;r<sup>0</sup> ð Þ plots for closed shell atoms: a negative diagonal part (cfr [47] and Section 2.1) and alternating positive and negative parts for r or r<sup>0</sup> = constant in order to obey the trivial equation (cf. [47])

$$\int \chi(\mathbf{r}, \mathbf{r}') d\mathbf{r} = \int (\delta \rho\_{\rm N}(\mathbf{r}) / \delta v(\mathbf{r}'))\_{\rm N} d\mathbf{r} = (\delta \text{N} / \delta v(\mathbf{r}'))\_{\rm N} = \mathbf{0} \tag{30}$$

χαα extends further away from the nucleus than χββ in line with the extra α electron with the higher principal quantum number (n = 2) extending farther away from the nucleus than the n = 1 β electron. χββ looks similar to the χββ plot for He (insert) but contracted more to the origin (due to higher nuclear charge). The χαβ and χαβ plots show some evident symmetry, χαβ r;r<sup>0</sup> ð Þ¼ χβα r<sup>0</sup> ð Þ ;r , and show positive regions along the diagonal. A positive perturbation in New Insights and Horizons from the Linear Response Function in Conceptual DFT http://dx.doi.org/10.5772/intechopen.80280 15

[52] and for a review see [53]). In the so-called Nα; N<sup>β</sup>

representation, three linear response functions can now be defined:

χNN r;r

χSS r;r

E=δvð Þr δvS r <sup>0</sup> ð Þ � �

� � representation, for the ground state of Li ((1s)<sup>2</sup> (2s)<sup>1</sup>

ð

<sup>0</sup> ð Þ¼ <sup>δ</sup><sup>2</sup>

<sup>0</sup> ð Þ¼ <sup>δ</sup><sup>2</sup>

<sup>0</sup> ð Þ ð Þ=δvð Þ<sup>r</sup> N,NS,v <sup>¼</sup> <sup>δ</sup><sup>2</sup>

2.1 is generalized to the E ¼ E Nα; Nβ; vα; v<sup>β</sup>

by <sup>v</sup>Sð Þ¼ <sup>r</sup> <sup>v</sup>αð Þ� <sup>r</sup> <sup>v</sup>βð Þ<sup>r</sup> � �=2.

14 Density Functional Theory

χNS r;r

obey the trivial equation (cf. [47])

ð χ r;r <sup>0</sup> ð Þdr ¼

(Eq. (1)).

Nα; N<sup>β</sup>

<sup>0</sup> ð Þ¼ <sup>δ</sup><sup>2</sup>

¼ δr<sup>S</sup> r

electronic energy to perturbations in the number of α electrons, Nα, the number of β electrons, Nβ, the external potential acting on the α electrons, vαð Þr , and the external potential acting on the β electrons, vβð Þr , is studied. In this representation, the E ¼ E N½ � ; v functional from Section

representation, in fact the one introduced by Galvan, the E ¼ E N½ � ; v functional is generalized to E ¼ E N½ � ; NS; v; v<sup>S</sup> or E ¼ E N½ � ; NS; v; B where N<sup>S</sup> denotes the electron spin number defined as the difference between the number of α and β electrons, N<sup>S</sup> ¼ N<sup>α</sup> � Nβ, whereas vs is given

<sup>v</sup>ð Þ<sup>r</sup> is equal to <sup>v</sup>αð Þþ <sup>r</sup> <sup>v</sup>βð Þ<sup>r</sup> � �=2 and <sup>B</sup> represents an external magnetic field. If the magnetic field is static and uniform along the z axis, one has vαð Þ¼ r vð Þþ r μBBZ and vβð Þ¼ r vð Þ� r μBBZ, vð Þr is the usual external spin-free potential (as used in CDFT without spin polarization), whereas vSð Þr is related to the magnetic field B rð Þ. In this context, and sticking to the ð Þ N; N<sup>S</sup>

> E=δvð Þr δv r <sup>0</sup> ð Þ � �

E=δvSð Þr δvS r <sup>0</sup> ð Þ � �

N,N<sup>S</sup> ¼ δrNð Þr =δvS r

where rNð Þ¼ r rαð Þþ r rβð Þr and rSð Þ¼ r rαð Þ� r rβð Þr are the total and spin densities, respectively. χNN r;r<sup>0</sup> ð Þ is the analogue of the spin-independent CDFT expression for the LRF χ r;r<sup>0</sup> ð Þ

Space limitations prevent us to go in detail on the results reported in [47, 48]. We only depict in Figure 5 the SPCDFT analogue of the contour plots for χ r;r<sup>0</sup> ð Þ for closed shell atoms as discussed in [47] and the Chem Soc Rev paper [16]. In Figure 5, we plot the LRF, in the

χββ r;r<sup>0</sup> ð Þ is similar to the χ r;r<sup>0</sup> ð Þ plots for closed shell atoms: a negative diagonal part (cfr [47] and Section 2.1) and alternating positive and negative parts for r or r<sup>0</sup> = constant in order to

χαα extends further away from the nucleus than χββ in line with the extra α electron with the higher principal quantum number (n = 2) extending farther away from the nucleus than the n = 1 β electron. χββ looks similar to the χββ plot for He (insert) but contracted more to the origin (due to higher nuclear charge). The χαβ and χαβ plots show some evident symmetry, χαβ r;r<sup>0</sup> ð Þ¼ χβα r<sup>0</sup> ð Þ ;r , and show positive regions along the diagonal. A positive perturbation in

<sup>0</sup> ð Þ ð Þ <sup>N</sup>dr ¼ δN=δv r

δrNð Þr =δv r

E=δvS r <sup>0</sup> ð Þδvð Þ<sup>r</sup> � �

<sup>0</sup> ð Þ ð Þ N,NS, <sup>v</sup>

� � representation, the response of the

N,NS, vS (27)

N,NS,v (28)

<sup>0</sup> ð Þ ;r

). The structure of χαα r;r<sup>0</sup> ð Þ and

<sup>0</sup> ð Þ ð Þ <sup>N</sup> ¼ 0 (30)

(29)

N,N<sup>S</sup> ¼ χSN r

� � functional. In the second, equivalent, ð Þ <sup>N</sup>; <sup>N</sup><sup>S</sup>

Figure 5. Contour plots of the radial distribution function of the spin polarized linear response function of Lithium in the [Nα, Nβ] representation. r is represented on the horizontal axis, r<sup>0</sup> on the vertical axis [(a) Lithium χαα, (b) Lithium χαβ, (c) Lithium χαα, (d) Lithium χββ. In the insert, the χββ plot for He (see text) (Reprinted from [48] with the permission of AIP Publishing)].

the α external potential δvαð Þr will cause a depletion of electrons in the vicinity of the perturbation, the β electrons are not affected directly. However, the depletion in α electrons will influence the Coulomb potential and due to the lower electron-electron repulsion, an accumulation in β electrons in the region considered will occur resulting in a positive diagonal χβαð Þ r;r value. The concentration of the χαβ and χαβ isocontours along the r and r<sup>0</sup> axes can be interpreted when referring to the χαα and χββ plots: perturbing vβð Þr at a distance r<sup>0</sup> larger than 3 a.u. clearly has no effect on the β density, the Coulomb potential, the overall density and consequently on the <sup>α</sup> density. This results in <sup>δ</sup>rαð Þ<sup>r</sup> <sup>=</sup>δv<sup>β</sup> <sup>r</sup><sup>0</sup> ð Þ � � <sup>¼</sup> <sup>χ</sup><sup>a</sup><sup>β</sup> <sup>r</sup>;r<sup>0</sup> ð Þ having zero values for r<sup>0</sup> larger than 3 a.u. On the other hand, perturbing vβð Þr close to the nucleus induces a change in the β density, with repercussion on the Coulomb potential and so on the α density farther away from the nucleus (on the r axis) even in regions where the β density is no longer affected. All these features account for the 'partial plane filling' of the χαβ and χαβ plots with a 'demarcation' line at 3 a.u.

To close this section, we mention that once χ r;r<sup>0</sup> ð Þ (or its counterparts in SPCDFT) is known, a local version of the polarizability tensor components, say αxy, namely αxyð Þr can be obtained by straightforward integration:

$$\alpha\_{xy}(\mathbf{r}) = -\int \mathbf{x}(\mathbf{r})\chi(\mathbf{r}, \mathbf{r}')y(\mathbf{r}')d\mathbf{r}' \tag{31}$$

at stake and not the overall polarizability. The parallel between the relationship between local

New Insights and Horizons from the Linear Response Function in Conceptual DFT

http://dx.doi.org/10.5772/intechopen.80280

17

3. The role of the LRF in alchemical derivatives and exploring chemical

The LRF has recently been exploited when investigating Chemical Compound Space [54–56]. Chemists are continuously exploring chemical compound space (CCS) [57, 58], the space populated by all imaginable chemicals with natural nuclear charges and realistic interatomic distances for which chemical interactions exist. Navigating through this space is costly, obviously for synthetic-experimental chemists and also for theoretical and computational chemists who might and should be guides for indicating relevant domains in CCS to their experimental colleagues. Doing even a simple single-point SCF calculation at every imaginable point leads to prohibitively large computing times (not to speak about bookkeeping aspects and manipulation of the computed data). A very promising ansatz was initiated by Von Lilienfeld et al. [59–63] in his alchemical coupling approach where two, isoelectronic molecules in CCS are coupled 'alchemically' through the interpolation of their external potentials (see also the work by Yang and co-workers on designing molecules by optimizing potentials [64]). At the heart of this ansatz are the alchemical derivatives, partial derivatives of the energy w.r.t. one or more nuclear charges at constant number of electrons and geometry. The simplest members of this new family of response functions are:

<sup>A</sup>½ �� <sup>N</sup>;Z; <sup>R</sup> ð Þ <sup>∂</sup>W=∂ZA <sup>N</sup> ,<sup>R</sup> <sup>¼</sup> ð Þ <sup>∂</sup>E=∂ZA <sup>N</sup> ,<sup>R</sup> <sup>þ</sup> ð Þ <sup>∂</sup>Vnn=∂ZA <sup>N</sup> ,<sup>R</sup> <sup>¼</sup> <sup>μ</sup>al, el

<sup>N</sup> ,<sup>R</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

A μal

where W N½ �� ;Z; R E N½ �þ ; v½ � Z; R Vnn½ � Z; R is the total energy of a system,R ¼ ð Þ RA; RB;… denotes constant geometry and Z ¼ ð Þ ZA; ZB;…; denotes the nuclear charges vector. The analogy with the electronic chemical potential and hardness (see Section 1) is striking. Some years ago, one of the present authors (R.B.) presented a strategy to calculate these derivatives for any atom or atom-atom combination analytically and to use them, starting from a single SCF calculation on the parent or reference molecule, to explore the CCS of first neighbours, implying changes of nuclear charges of +1 or �1 [65]. In this way, simple arithmetic can be used,

Vnn=∂ZA∂ZB � �

<sup>A</sup> dZA þ 1=2

<sup>N</sup> ,<sup>R</sup> <sup>¼</sup> <sup>η</sup>al, el

X A

X B ηal

E=∂ZA∂ZB � � <sup>A</sup> <sup>þ</sup> <sup>μ</sup>al,nuc

AB <sup>¼</sup> <sup>η</sup>al

ABdZAdZB (34)

AB <sup>þ</sup> <sup>η</sup>al,nuc

<sup>A</sup> (32)

BA, (33)

and global softness (Section 2.1) is obvious.

compound space

the alchemical potential

and the alchemical hardness

starting from a Taylor expansion

W=∂ZA∂ZB � �

<sup>N</sup> ,<sup>R</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

<sup>Δ</sup>W d½ �¼ <sup>Z</sup> W N<sup>½</sup> ; <sup>Z</sup> <sup>þ</sup> <sup>d</sup>Z; <sup>R</sup>� � W N½ �¼ ;Z; <sup>R</sup> <sup>X</sup>

instead of a new SCF calculation for each transmutant.

μal

AB½ �� <sup>N</sup>;Z; <sup>R</sup> <sup>∂</sup><sup>2</sup>

ηal

3.1. Context

An example is given in Figure 6 [48] where for the atoms Li through Ne the trend of the spherically averaged <sup>α</sup>ð Þ<sup>r</sup> <sup>1</sup> <sup>3</sup> <sup>α</sup>xxð Þþ <sup>r</sup> <sup>α</sup>yyð Þþ <sup>r</sup> <sup>α</sup>zzð Þ<sup>r</sup> � � � � is given. From 2 a.u. on, the trends in αð Þr for Li up to Ne parallel the global polarizability, known to decrease along a period of the periodic table. At lower distances (preceding the valence region), inversions with even negative values occur. The present results are evidently important when, for example, disentangling reaction mechanisms where local polarizabilities, that is, in certain regions of the reagents, are

Figure 6. Plot of the local polarizability α(r) of the atoms Li through Ne via CPKS (see text) (Reprinted by permission of the publisher (Taylor and Francis Ltd.) [48]).

at stake and not the overall polarizability. The parallel between the relationship between local and global softness (Section 2.1) is obvious.

#### 3. The role of the LRF in alchemical derivatives and exploring chemical compound space

#### 3.1. Context

the α external potential δvαð Þr will cause a depletion of electrons in the vicinity of the perturbation, the β electrons are not affected directly. However, the depletion in α electrons will influence the Coulomb potential and due to the lower electron-electron repulsion, an accumulation in β electrons in the region considered will occur resulting in a positive diagonal χβαð Þ r;r value. The concentration of the χαβ and χαβ isocontours along the r and r<sup>0</sup> axes can be interpreted when referring to the χαα and χββ plots: perturbing vβð Þr at a distance r<sup>0</sup> larger than 3 a.u. clearly has no effect on the β density, the Coulomb potential, the overall density and consequently on the <sup>α</sup> density. This results in <sup>δ</sup>rαð Þ<sup>r</sup> <sup>=</sup>δv<sup>β</sup> <sup>r</sup><sup>0</sup> ð Þ � � <sup>¼</sup> <sup>χ</sup><sup>a</sup><sup>β</sup> <sup>r</sup>;r<sup>0</sup> ð Þ having zero values for r<sup>0</sup> larger than 3 a.u. On the other hand, perturbing vβð Þr close to the nucleus induces a change in the β density, with repercussion on the Coulomb potential and so on the α density farther away from the nucleus (on the r axis) even in regions where the β density is no longer affected. All these features account for the 'partial plane filling' of the χαβ and χαβ plots with a

To close this section, we mention that once χ r;r<sup>0</sup> ð Þ (or its counterparts in SPCDFT) is known, a local version of the polarizability tensor components, say αxy, namely αxyð Þr can be obtained by

> xð Þr χ r;r <sup>0</sup> ð Þy r

An example is given in Figure 6 [48] where for the atoms Li through Ne the trend of the

αð Þr for Li up to Ne parallel the global polarizability, known to decrease along a period of the periodic table. At lower distances (preceding the valence region), inversions with even negative values occur. The present results are evidently important when, for example, disentangling reaction mechanisms where local polarizabilities, that is, in certain regions of the reagents, are

Figure 6. Plot of the local polarizability α(r) of the atoms Li through Ne via CPKS (see text) (Reprinted by permission of

<sup>0</sup> ð Þdr

<sup>3</sup> <sup>α</sup>xxð Þþ <sup>r</sup> <sup>α</sup>yyð Þþ <sup>r</sup> <sup>α</sup>zzð Þ<sup>r</sup> � � � � is given. From 2 a.u. on, the trends in

<sup>0</sup> (31)

ð

αxyð Þ¼� r

'demarcation' line at 3 a.u.

16 Density Functional Theory

straightforward integration:

spherically averaged <sup>α</sup>ð Þ<sup>r</sup> <sup>1</sup>

the publisher (Taylor and Francis Ltd.) [48]).

The LRF has recently been exploited when investigating Chemical Compound Space [54–56]. Chemists are continuously exploring chemical compound space (CCS) [57, 58], the space populated by all imaginable chemicals with natural nuclear charges and realistic interatomic distances for which chemical interactions exist. Navigating through this space is costly, obviously for synthetic-experimental chemists and also for theoretical and computational chemists who might and should be guides for indicating relevant domains in CCS to their experimental colleagues. Doing even a simple single-point SCF calculation at every imaginable point leads to prohibitively large computing times (not to speak about bookkeeping aspects and manipulation of the computed data). A very promising ansatz was initiated by Von Lilienfeld et al. [59–63] in his alchemical coupling approach where two, isoelectronic molecules in CCS are coupled 'alchemically' through the interpolation of their external potentials (see also the work by Yang and co-workers on designing molecules by optimizing potentials [64]). At the heart of this ansatz are the alchemical derivatives, partial derivatives of the energy w.r.t. one or more nuclear charges at constant number of electrons and geometry. The simplest members of this new family of response functions are:

the alchemical potential

$$\mu\_A^{\rm al}[\mathbf{N}, \mathbf{Z}, \mathbf{R}] \equiv (\partial \mathbf{W} / \partial \mathbf{Z}\_A)\_{N, \mathbf{R}} = (\partial \mathbf{E} / \partial \mathbf{Z}\_A)\_{N, \mathbf{R}} + (\partial V\_{\mathbf{m} \uparrow} \partial \mathbf{Z}\_A)\_{N, \mathbf{R}} = \mu\_A^{\rm al, el} + \mu\_A^{\rm al, \rm nuc} \tag{32}$$

and the alchemical hardness

$$\boldsymbol{\eta}^{\rm al}\_{\rm AB}[\mathbf{N}, \mathbf{Z}, \mathbf{R}] \equiv \left(\partial^2 \mathbf{W}/\partial \mathbf{Z}\_A \partial \mathbf{Z}\_B\right)\_{\mathcal{N}, \mathbf{R}} = \left(\partial^2 \mathbf{E}/\partial \mathbf{Z}\_A \partial \mathbf{Z}\_B\right)\_{\mathcal{N}, \mathbf{R}} + \left(\partial^2 V\_{\rm nn}/\partial \mathbf{Z}\_A \partial \mathbf{Z}\_B\right)\_{\mathcal{N}, \mathbf{R}} = \eta\_{AB}^{\rm al, \rm el} + \eta\_{AB}^{\rm al, \rm mac} = \eta\_{BA}^{\rm al} \tag{33}$$

where W N½ �� ;Z; R E N½ �þ ; v½ � Z; R Vnn½ � Z; R is the total energy of a system,R ¼ ð Þ RA; RB;… denotes constant geometry and Z ¼ ð Þ ZA; ZB;…; denotes the nuclear charges vector. The analogy with the electronic chemical potential and hardness (see Section 1) is striking. Some years ago, one of the present authors (R.B.) presented a strategy to calculate these derivatives for any atom or atom-atom combination analytically and to use them, starting from a single SCF calculation on the parent or reference molecule, to explore the CCS of first neighbours, implying changes of nuclear charges of +1 or �1 [65]. In this way, simple arithmetic can be used, starting from a Taylor expansion

$$
\Delta W[d\mathbf{Z}] = W[\mathbf{N}, \mathbf{Z} + d\mathbf{Z}, \mathbf{R}] - W[\mathbf{N}, \mathbf{Z}, \mathbf{R}] \\
= \sum\_{A} \mu\_{A}^{\text{al}} dZ\_{A} + 1/2 \sum\_{A} \sum\_{\mathbf{B}} \eta\_{AB}^{\text{al}} dZ\_{A} dZ\_{\mathbf{B}} \tag{34}
$$

instead of a new SCF calculation for each transmutant.

The position of the alchemical derivatives in CDFT was already mentioned: they are response functions, now related to a particular charge in external potential, namely the charge in one or more nuclear charges. As the second derivatives are 'diagonal' in these particular external potential changes, a direct link with the LRF can be expected. Using the chain rule, one easily writes

$$\begin{split} \eta\_{AB}^{\text{al},\text{el}} = \left(\partial^2 E[N, \mathbf{v}[\mathbf{Z}, \mathbf{R}]] / \partial \mathbf{Z}\_A \partial \mathbf{Z}\_B \right)\_{N,\mathbf{R}} = \left\{ \int (\delta\rho(\mathbf{r}) / \delta\mathbf{v}(\mathbf{r}'))\_N (\partial \mathbf{v}(\mathbf{r}) / \partial \mathbf{Z}\_A)\_{N,\mathbf{R}} (\partial \mathbf{v}(\mathbf{r}') / \partial \mathbf{Z}\_B)\_{N,\mathbf{R}} d\mathbf{r} d\mathbf{r}' \right\} \\ = \int \int \chi(\mathbf{r}, \mathbf{r}') (\mathbf{1} / |\mathbf{r} - \mathbf{R}\_A|) (\mathbf{1} / |\mathbf{r}' - \mathbf{R}\_B|) d\mathbf{r} d\mathbf{r}' \end{split} \tag{35}$$

indicating that the alchemical hardness is obtained by integration of the LRF after multiplication by 1=j j r � R<sup>A</sup> and 1= rj j <sup>0</sup> � R<sup>B</sup> . The basic relationship of the alchemical derivatives and the LRF shows how, again, the LRF makes its appearance in sometimes unexpected areas (see also Section 4). In the case of the first derivative, the ð Þ δE=δvð Þr <sup>N</sup> factor in the integrand simplifies to rð Þr (see Section 1), yielding.

$$
\mu\_A^{\text{al,el}} = \int \rho(\mathbf{r}) / |\mathbf{r} - \mathbf{R}\_A| d\mathbf{r} \tag{36}
$$

order to recover the (changes in) core-core and core-valence correlation). The mean absolute error was only 0.034 a.u., the N2 ! COtransmutation being particularly successful with a

New Insights and Horizons from the Linear Response Function in Conceptual DFT

system results in the cancellation of the odd terms in the Taylor expansion. Of course, we do not claim 'chemical accuracy', yet the ordering of the energy of all transmutants came out correctly, indicating that the alchemical procedure (even when stopped at second order) is a

Similar conclusions could be drawn for transmutation of benzene, for example, by the substitution of CC units by their isoelectronic BN units. The replacement of a CC unit in an aromatic molecule by an isoelectronic unit BN has been shown to impart important, interesting electronic, photophysical and chemical properties, often distinct from the parent hydrocarbon [67]. An in-depth study of all azoborines (Figure 8) C6H6 ! C6-2nH6ð Þ BN <sup>n</sup> (n ¼ 1, 2, 3) turned out to reproduce correctly the stability of all possible isomers for a given n value (3,11,3 for n ¼ 1, 2, 3), which is of importance for applications in graphene chemistry where the ð Þ CC <sup>n</sup> ! ð Þ BN <sup>n</sup> substitution is a topic that has received great interest in recent years [68].

As a computational 'tour de force' , and passing from '2D' benzene to '3D' fullerenes, we recently explored the alchemical approach to study the complete CC ð Þ<sup>n</sup> ! ð Þ BN <sup>n</sup> substitution pattern of C60, all the way down to BN ð Þ30. C60 ! C60-2nð Þ BN <sup>n</sup>, n ¼ 1, 2, …, 30, predicting and interpreting via 'alchemical rules' the most stable isomers for each value of n. This study is based on a single SCF calculation on C60 and its alchemical derivatives up to second order, enabling each possible transmutation energy to be evaluated by simple arithmetic (the diago-

> <sup>11</sup> <sup>þ</sup><sup>X</sup> i¼1

where ΔWn is the transmutation energy from C60 to C60-2nð Þ BN <sup>n</sup> and N=B is a vector of the

Figure 8. CC-BN Substitutions in 2D and 3D unsaturated carbocyclic systems (number of isomers for the 2D case in

X j>i

ηal <sup>N</sup>iN<sup>j</sup> <sup>þ</sup> <sup>η</sup>al BiBj � � �<sup>X</sup>

i¼1

X j¼1 ηal NiBj

, (37)

. The transmutation into a neutral

http://dx.doi.org/10.5772/intechopen.80280

19

difference in energy of only 0.004 a.u., that is, 2 kcal mol �<sup>1</sup>

nal elements of the alchemical hardness matrix are equal)

<sup>Δ</sup>Wn½<sup>N</sup> <sup>¼</sup> ð Þ <sup>N</sup>1; …; Nn ; <sup>B</sup> <sup>¼</sup> ð Þ <sup>B</sup>1; …; Bn � ¼ <sup>n</sup>ηal

carbon atoms replaced by the nitrogen/boron atoms.

parentheses).

simple, straightforward road to explore CCS for neighbouring structures.

the electronic potential at the nucleus, well known as the electronic part of the molecular electrostatic potential [66].

#### 3.2. Applications

As a very simple example, we consider the transmutation of the nitrogen molecule. Five chemically relevant mutants can be generated (see Figure 7) as nearest neighbours in CCS (ΔZ = �1) and at constant number of electrons: CO, NO<sup>þ</sup>, O<sup>2</sup><sup>þ</sup> <sup>2</sup> , CN� and C<sup>2</sup>� <sup>2</sup> . The differences between 'vertical' (i.e. exact, via two SCF calculations) and alchemical transmutation energies were evaluated at the B3LYP/cc-pCVTZ level (note the inclusion of additional tight functions 'C in

Figure 7. Transmutation of the nitrogen molecule to its nearest neighbours in chemical compound space (Reprinted with permission from [54]. Copyright (2017) American Chemical Society).

order to recover the (changes in) core-core and core-valence correlation). The mean absolute error was only 0.034 a.u., the N2 ! COtransmutation being particularly successful with a difference in energy of only 0.004 a.u., that is, 2 kcal mol �<sup>1</sup> . The transmutation into a neutral system results in the cancellation of the odd terms in the Taylor expansion. Of course, we do not claim 'chemical accuracy', yet the ordering of the energy of all transmutants came out correctly, indicating that the alchemical procedure (even when stopped at second order) is a simple, straightforward road to explore CCS for neighbouring structures.

The position of the alchemical derivatives in CDFT was already mentioned: they are response functions, now related to a particular charge in external potential, namely the charge in one or more nuclear charges. As the second derivatives are 'diagonal' in these particular external potential changes, a direct link with the LRF can be expected. Using the chain rule, one easily

δrð Þr =δv r

indicating that the alchemical hardness is obtained by integration of the LRF after multiplication by 1=j j r � R<sup>A</sup> and 1= rj j <sup>0</sup> � R<sup>B</sup> . The basic relationship of the alchemical derivatives and the LRF shows how, again, the LRF makes its appearance in sometimes unexpected areas (see also Section 4). In the case of the first derivative, the ð Þ δE=δvð Þr <sup>N</sup> factor in the integrand simplifies to

the electronic potential at the nucleus, well known as the electronic part of the molecular

As a very simple example, we consider the transmutation of the nitrogen molecule. Five chemically relevant mutants can be generated (see Figure 7) as nearest neighbours in CCS (ΔZ = �1)

'vertical' (i.e. exact, via two SCF calculations) and alchemical transmutation energies were evaluated at the B3LYP/cc-pCVTZ level (note the inclusion of additional tight functions 'C in

Figure 7. Transmutation of the nitrogen molecule to its nearest neighbours in chemical compound space (Reprinted with

<sup>0</sup> ð Þ ð Þ <sup>N</sup>ð Þ ∂vð Þr =∂ZA <sup>N</sup> ,<sup>R</sup> ∂v r

<sup>2</sup> , CN� and C<sup>2</sup>�

<sup>0</sup> ð Þð Þ 1=j j r � R<sup>A</sup> 1= r

<sup>0</sup> ð Þ ð Þ=∂ZB <sup>N</sup> ,Rdrdr

0

<sup>2</sup> . The differences between

<sup>0</sup> ð Þ j j � R<sup>B</sup> drdr

rð Þr =j j r � R<sup>A</sup> dr (36)

0

(35)

N,R¼

¼ ð ð χ r;r

μal, el <sup>A</sup> ¼ ð

ð ð

writes

ηal, el AB <sup>¼</sup> <sup>∂</sup><sup>2</sup>

18 Density Functional Theory

E N½ � ; v½ � Z; R =∂ZA∂ZB � �

and at constant number of electrons: CO, NO<sup>þ</sup>, O<sup>2</sup><sup>þ</sup>

permission from [54]. Copyright (2017) American Chemical Society).

rð Þr (see Section 1), yielding.

electrostatic potential [66].

3.2. Applications

Similar conclusions could be drawn for transmutation of benzene, for example, by the substitution of CC units by their isoelectronic BN units. The replacement of a CC unit in an aromatic molecule by an isoelectronic unit BN has been shown to impart important, interesting electronic, photophysical and chemical properties, often distinct from the parent hydrocarbon [67]. An in-depth study of all azoborines (Figure 8) C6H6 ! C6-2nH6ð Þ BN <sup>n</sup> (n ¼ 1, 2, 3) turned out to reproduce correctly the stability of all possible isomers for a given n value (3,11,3 for n ¼ 1, 2, 3), which is of importance for applications in graphene chemistry where the ð Þ CC <sup>n</sup> ! ð Þ BN <sup>n</sup> substitution is a topic that has received great interest in recent years [68].

As a computational 'tour de force' , and passing from '2D' benzene to '3D' fullerenes, we recently explored the alchemical approach to study the complete CC ð Þ<sup>n</sup> ! ð Þ BN <sup>n</sup> substitution pattern of C60, all the way down to BN ð Þ30. C60 ! C60-2nð Þ BN <sup>n</sup>, n ¼ 1, 2, …, 30, predicting and interpreting via 'alchemical rules' the most stable isomers for each value of n. This study is based on a single SCF calculation on C60 and its alchemical derivatives up to second order, enabling each possible transmutation energy to be evaluated by simple arithmetic (the diagonal elements of the alchemical hardness matrix are equal)

$$\Delta\mathcal{W}\_{\text{il}}[\mathbf{N}=(\text{N1},...,\text{Nn}),\mathbf{B}=(\text{B1},...,\text{Bn})] = n\,\eta\_{11}^{\text{al}} + \sum\_{i=1} \sum\_{j>i} \left(\eta\_{\text{NNij}}^{\text{al}} + \eta\_{\text{BBj}}^{\text{al}}\right) - \sum\_{i=1} \sum\_{j>i} \eta\_{\text{NiBj}}^{\text{al}} \tag{37}$$

where ΔWn is the transmutation energy from C60 to C60-2nð Þ BN <sup>n</sup> and N=B is a vector of the carbon atoms replaced by the nitrogen/boron atoms.

Figure 8. CC-BN Substitutions in 2D and 3D unsaturated carbocyclic systems (number of isomers for the 2D case in parentheses).

In (37), it is seen that the linear term drops as μ<sup>i</sup> is unique by symmetry for all atoms in C60 and because Σ ΔZ <sup>i</sup> = 0 for any transmutation. The alchemical hardness matrix η thus completely determines the substitution energy and pattern. To summarize the results (for an in-depth discussion see [55]), the study reveals that the correct sequence of stabilization energies for each n is retrieved by adopting an approach in which for each n value the problem is looked upon without prejudice of the n � 1 result (called the 'simultaneous' approach). The other, simpler method (we called it the 'successive' approach) was shown to fail already after n = 13, be it that at some n values identical values were obtained with both approaches, for example, for the Belt structure (n = 20). Needless to say that even the 'successive' approach might already be prohibitively demanding for standard ab initio or DFT calculations, let it be for the simultaneous method. The sequence of substitutions could be interpreted in terms of a number of 'alchemical' rules of which the two most important are when (referring to earlier work by Kar et al. [69]): (1�) hexagon-hexagon junctions are preferably substituted, in a way that minimizes the homonuclear (BB,NN) bonds and (2�) the higher stability is created by maximizing the number of filled hexagons. For the subsequent, more intricate, rules we refer to [55].

Here, as in Section 2.1, Δ is the Hückel determinant for the isolated molecule, from which in ΔAB its A-th row and B-th column are deleted and β is a measure of the interaction between the contact atom and its molecular neighbour. Again, what is the role of the LRF in this road to

Going back to expression (7) for the diagonal elements of Coulson's atom-atom polarizability,

(the contour integral in the complex plane in Coulson's formalism [31] is hereby reduced to an integral along the imaginary axis). The integrand of πAB thus turns out to be related to the transition probability at the Fermi level Tr TABð Þ0 when the contacts are placed at atoms A and B of the molecule. An intimate relation between πAB and TABð Þ0 thus exists; the precise connection was evaluated by explicit calculation of πAB and TABð Þ0 , some results being visualized in Figure 9. There, we depict both the πAB and TABð Þ0 values for some linear acenes (benzene, naphthalene, anthracene and tetracene), taking always one atom as the reference atom. For the atom-atom polarizability, the reference atom is denoted by a green circle with its area proportional to the self-polarizability of that atom which is as pointed out in Section 2.1 as always negative. Black and red circles on the other atoms denote the atom-atom polarizability values corresponding to a perturbation on the reference atom (varying αA) on the (charge of the) considered atom B(qB). Black circles correspond to πAB > 0, red circles to πAB < 0. For TABð Þ0 , a similar approach is followed. The empty green circle denotes the position of the first contact 'A' (zero 'ipso' transmission [71]), the magnitude of the black circles on the other atoms 'B' denotes the magnitude of the transmission when the second contact is placed at position B; note that no red circles arise as TABð Þ0 , the transmission probability, always lies

Figure 9 shows that the pattern in the two plots is completely analogous and leads to the conjecture that a positive atom-atom polarizability seems to be a necessary condition in these Kekulean benzenoids in order to have transmission for a certain configuration of the contacts on the molecule. If the areas of the circles are considered, no exact proportionality between πAB

This issue was further investigated in the next linear acene and pentacene (Figure 10), by varying the position of the first contact. For a fixed first contact, the highest transmission occurs when the second contact is at the atom with the highest πAB value involving the first atom. From that atom on, a monotonously decreasing probability is noticed in either directions. The sharpest decline in T is in the direction of the neighbour that exhibits the lowest atom-atom polarizability. On the basis of these results, it can indeed be conjectured that a positive atom-atom polarizability is a necessary condition for transmission and that the ten-

ð Þ <sup>Δ</sup>ABð Þ iy <sup>=</sup>Δð Þ iy <sup>2</sup>

New Insights and Horizons from the Linear Response Function in Conceptual DFT

dy (39)

http://dx.doi.org/10.5772/intechopen.80280

21

ðþ<sup>∞</sup> �∞

calculating/understanding molecular conductivity?

and TABð Þ0 can be inferred, but they are correlated.

dencies between the two series of values are similar.

between 0 and 1.

4.2. The atom-atom polarizability as a conductivity indicator

a general element πAB of this forerunner of the LRF can be written as

πAB ¼ ð Þ 1=π

Shortage of space prevents us to comment on our recent results on the evaluation of isolated atom alchemical derivatives up to third order with different techniques, from numerical differentiation (not discussed hitherto in this chapter), via the coupled perturbed Kohn-Sham approach as discussed before, to the March and Parr combined 1=Z and N�1=<sup>3</sup> E N½ � ; Z expansion ansatz [70], permitting a walk through the periodic table on the road to scrutinize periodicity effects [56].

#### 4. The role of the LRF (or its forerunner, the atom-atom polarizability) in evaluating and interpreting molecular electronics

#### 4.1. Context

In this section the role of the LRF in molecular electronics is highlighted [72] has been a vibrant area of research in recent years. An ever-increasing number of papers (both experimental and theoretical) studied the transport properties of typically organic molecules containing π-conjugated systems and considering possible applications for incorporation in molecular electronic devices (MED) [73]. Most of these theoretical studies have been performed at a high level of theory, but this type of calculation does not always lead to simple insights into why some molecules will conduct and which will insulate, and how the positions of the contacts influence this behaviour.

We therefore adopted a simple ansatz based on Ernzerhof's source and sink potential (for details see [74, 75]) in Fowler's tight-binding Hückel approach [76]. One thereby considers only the π electrons of the molecule and cuts the resonance effects between the contact and the molecule after the molecule's nearest neighbour in the contact. In the so-called weak interaction limit (see details in [71]), the transmission probability at the Fermi level Tð Þ0 , which is directly proportional to the conductance, can then be written as

$$T(0) = 4\beta^2 \Delta\_{AB}^2 / \Delta^2 \tag{38}$$

Here, as in Section 2.1, Δ is the Hückel determinant for the isolated molecule, from which in ΔAB its A-th row and B-th column are deleted and β is a measure of the interaction between the contact atom and its molecular neighbour. Again, what is the role of the LRF in this road to calculating/understanding molecular conductivity?

#### 4.2. The atom-atom polarizability as a conductivity indicator

In (37), it is seen that the linear term drops as μ<sup>i</sup> is unique by symmetry for all atoms in C60 and because Σ ΔZ <sup>i</sup> = 0 for any transmutation. The alchemical hardness matrix η thus completely determines the substitution energy and pattern. To summarize the results (for an in-depth discussion see [55]), the study reveals that the correct sequence of stabilization energies for each n is retrieved by adopting an approach in which for each n value the problem is looked upon without prejudice of the n � 1 result (called the 'simultaneous' approach). The other, simpler method (we called it the 'successive' approach) was shown to fail already after n = 13, be it that at some n values identical values were obtained with both approaches, for example, for the Belt structure (n = 20). Needless to say that even the 'successive' approach might already be prohibitively demanding for standard ab initio or DFT calculations, let it be for the simultaneous method. The sequence of substitutions could be interpreted in terms of a number of 'alchemical' rules of which the two most important are when (referring to earlier work by Kar et al. [69]): (1�) hexagon-hexagon junctions are preferably substituted, in a way that minimizes the homonuclear (BB,NN) bonds and (2�) the higher stability is created by maximizing the number of filled hexagons. For the subsequent, more intricate, rules we refer to [55]. Shortage of space prevents us to comment on our recent results on the evaluation of isolated atom alchemical derivatives up to third order with different techniques, from numerical differentiation (not discussed hitherto in this chapter), via the coupled perturbed Kohn-Sham approach as

E N½ � ; Z expansion ansatz [70],

AB=Δ<sup>2</sup> (38)

discussed before, to the March and Parr combined 1=Z and N�1=<sup>3</sup>

evaluating and interpreting molecular electronics

directly proportional to the conductance, can then be written as

<sup>T</sup>ð Þ¼ <sup>0</sup> <sup>4</sup>β<sup>2</sup>

Δ2

4.1. Context

20 Density Functional Theory

permitting a walk through the periodic table on the road to scrutinize periodicity effects [56].

4. The role of the LRF (or its forerunner, the atom-atom polarizability) in

In this section the role of the LRF in molecular electronics is highlighted [72] has been a vibrant area of research in recent years. An ever-increasing number of papers (both experimental and theoretical) studied the transport properties of typically organic molecules containing π-conjugated systems and considering possible applications for incorporation in molecular electronic devices (MED) [73]. Most of these theoretical studies have been performed at a high level of theory, but this type of calculation does not always lead to simple insights into why some molecules will conduct and which will insulate, and how the positions of the contacts influence this behaviour. We therefore adopted a simple ansatz based on Ernzerhof's source and sink potential (for details see [74, 75]) in Fowler's tight-binding Hückel approach [76]. One thereby considers only the π electrons of the molecule and cuts the resonance effects between the contact and the molecule after the molecule's nearest neighbour in the contact. In the so-called weak interaction limit (see details in [71]), the transmission probability at the Fermi level Tð Þ0 , which is Going back to expression (7) for the diagonal elements of Coulson's atom-atom polarizability, a general element πAB of this forerunner of the LRF can be written as

$$
\pi\_{AB} = (1/\pi) \int\_{-\infty}^{+\infty} (\Delta\_{AB}(\text{iy})/\Delta(\text{iy}))^2 dy \tag{39}
$$

(the contour integral in the complex plane in Coulson's formalism [31] is hereby reduced to an integral along the imaginary axis). The integrand of πAB thus turns out to be related to the transition probability at the Fermi level Tr TABð Þ0 when the contacts are placed at atoms A and B of the molecule. An intimate relation between πAB and TABð Þ0 thus exists; the precise connection was evaluated by explicit calculation of πAB and TABð Þ0 , some results being visualized in Figure 9. There, we depict both the πAB and TABð Þ0 values for some linear acenes (benzene, naphthalene, anthracene and tetracene), taking always one atom as the reference atom. For the atom-atom polarizability, the reference atom is denoted by a green circle with its area proportional to the self-polarizability of that atom which is as pointed out in Section 2.1 as always negative. Black and red circles on the other atoms denote the atom-atom polarizability values corresponding to a perturbation on the reference atom (varying αA) on the (charge of the) considered atom B(qB). Black circles correspond to πAB > 0, red circles to πAB < 0. For TABð Þ0 , a similar approach is followed. The empty green circle denotes the position of the first contact 'A' (zero 'ipso' transmission [71]), the magnitude of the black circles on the other atoms 'B' denotes the magnitude of the transmission when the second contact is placed at position B; note that no red circles arise as TABð Þ0 , the transmission probability, always lies between 0 and 1.

Figure 9 shows that the pattern in the two plots is completely analogous and leads to the conjecture that a positive atom-atom polarizability seems to be a necessary condition in these Kekulean benzenoids in order to have transmission for a certain configuration of the contacts on the molecule. If the areas of the circles are considered, no exact proportionality between πAB and TABð Þ0 can be inferred, but they are correlated.

This issue was further investigated in the next linear acene and pentacene (Figure 10), by varying the position of the first contact. For a fixed first contact, the highest transmission occurs when the second contact is at the atom with the highest πAB value involving the first atom. From that atom on, a monotonously decreasing probability is noticed in either directions. The sharpest decline in T is in the direction of the neighbour that exhibits the lowest atom-atom polarizability. On the basis of these results, it can indeed be conjectured that a positive atom-atom polarizability is a necessary condition for transmission and that the tendencies between the two series of values are similar.

a different set this function is even. It is then easily seen that the odd function has no real part along the imaginary axis, leading to a negative ð Þ <sup>Δ</sup>AB=<sup>Δ</sup> <sup>2</sup> value along the y axis and zero at the origin, yielding a negative atom-atom polarizability and zero transmission at the Fermi level (Tð Þ0 ). On the other hand, when A and B are drawn from opposite sets, ΔAB=Δ is an even function yielding real values along the y axis. A positive πAB value results with either insulation or transmission, depending on whether ΔABð Þ0 is zero or not. Note that the analysis for A ¼ B leads to the same result as for A 6¼ B but belonging to the same set yielding a negative πAA value. The following overall conclusion, formulated as a selection rule, can be drawn: negative atom-atom polarizabilities for non-singular alternant hydrocarbons correspond to devices with insulation at the Fermi level. Conduction at the Fermi level requires, but is not

New Insights and Horizons from the Linear Response Function in Conceptual DFT

http://dx.doi.org/10.5772/intechopen.80280

23

The aforementioned properties were used as guiding principle in our later studies towards a chemical interpretation of molecular electronic conductivity [77] leading to a simple, backof-the-envelope determination of quantum interference [78], thus bridging the gap between

The LRF and its congener, the softness kernel, are now in a stage where many of their mathematical and physical properties are well understood. The possibility to evaluate, represent and interpret them puts them on equal footing for their use in conceptual DFT with their already more traditional second-order companions, the chemical hardness and the Fukui function. In view of the 'chemistry' contained in the LRF kernel as shown some years ago, it is not unexpected, but it still remains to be unravelled whether they are major players in very fundamental issues pertaining to the electronic structure of matter as in Kohn's nearsightedness of electronic matter principle, as well as in more applied fields where they are shown to be of great use to explore chemical compound space (through the alchemical derivatives) and to

P.G. and F.D.P. acknowledge the Vrije Universiteit Brussel (VUB) for a Strategic Research Program. F.D.P. also acknowledges the Franqui Foundation for a position as Francqui Research Professor. S.F. acknowledges the Research Foundation Flanders (FWO) and the European Union's Horizon 2020 Marie Sklodowska-Curie grant (N� 706415) for financially supporting his postdoctoral research at the ALGC group. T.S. acknowledges the FWO for a position as research assistant (11ZG615N). P.W.A. thanks the Natural Sciences and Engineering Research Council, the Canada Research Chairs and Compute Canada for financial support. R.B. thanks the Interdisciplinary Centre for Mathematical and Computational Modelling for a computing grant. P.G. thanks Kristina Nikolova for her meticulous help in styling the manu-

guaranteed by, a positive atom-atom polarizability.

chemical reactivity theory and molecular electronics.

predict/interpret molecular conductivity.

script and Tom Bettens for his help in styling Figure 2.

Acknowledgements

5. Conclusions

Figure 9. The atom-atom polarizability (left) and transmission probability at the Fermi level (right) for a single reference atom for benzene, naphthalene, anthracene and tetracene (Reprinted from [71] with permission of AIP Publishing).

Figure 10. The atom-atom polarizability (left) and transmission probability at the Fermi level (right) for pentacene for variable reference atom (Reprinted from [71] with permission of AIP Publishing).

Further analysis of the behaviour of the ð Þ <sup>Δ</sup>AB=<sup>Δ</sup> <sup>2</sup> function in the complex plane was done in the case of alternant non-singular hydrocarbons (for details see [71]). Coulson's and Longuet Higgins' pairing theorem shows that if A and B are drawn from the same partite set (all 'starred' or 'unstarred' atoms, respectively) Δ<sup>2</sup> ABð Þ<sup>ƶ</sup> <sup>=</sup><sup>Δ</sup> ð Þ<sup>ƶ</sup> <sup>2</sup> is odd and if A and B are taken from a different set this function is even. It is then easily seen that the odd function has no real part along the imaginary axis, leading to a negative ð Þ <sup>Δ</sup>AB=<sup>Δ</sup> <sup>2</sup> value along the y axis and zero at the origin, yielding a negative atom-atom polarizability and zero transmission at the Fermi level (Tð Þ0 ). On the other hand, when A and B are drawn from opposite sets, ΔAB=Δ is an even function yielding real values along the y axis. A positive πAB value results with either insulation or transmission, depending on whether ΔABð Þ0 is zero or not. Note that the analysis for A ¼ B leads to the same result as for A 6¼ B but belonging to the same set yielding a negative πAA value. The following overall conclusion, formulated as a selection rule, can be drawn: negative atom-atom polarizabilities for non-singular alternant hydrocarbons correspond to devices with insulation at the Fermi level. Conduction at the Fermi level requires, but is not guaranteed by, a positive atom-atom polarizability.

The aforementioned properties were used as guiding principle in our later studies towards a chemical interpretation of molecular electronic conductivity [77] leading to a simple, backof-the-envelope determination of quantum interference [78], thus bridging the gap between chemical reactivity theory and molecular electronics.

#### 5. Conclusions

The LRF and its congener, the softness kernel, are now in a stage where many of their mathematical and physical properties are well understood. The possibility to evaluate, represent and interpret them puts them on equal footing for their use in conceptual DFT with their already more traditional second-order companions, the chemical hardness and the Fukui function. In view of the 'chemistry' contained in the LRF kernel as shown some years ago, it is not unexpected, but it still remains to be unravelled whether they are major players in very fundamental issues pertaining to the electronic structure of matter as in Kohn's nearsightedness of electronic matter principle, as well as in more applied fields where they are shown to be of great use to explore chemical compound space (through the alchemical derivatives) and to predict/interpret molecular conductivity.

#### Acknowledgements

Further analysis of the behaviour of the ð Þ <sup>Δ</sup>AB=<sup>Δ</sup> <sup>2</sup> function in the complex plane was done in the case of alternant non-singular hydrocarbons (for details see [71]). Coulson's and Longuet Higgins' pairing theorem shows that if A and B are drawn from the same partite set (all

Figure 10. The atom-atom polarizability (left) and transmission probability at the Fermi level (right) for pentacene for

Figure 9. The atom-atom polarizability (left) and transmission probability at the Fermi level (right) for a single reference atom for benzene, naphthalene, anthracene and tetracene (Reprinted from [71] with permission of AIP Publishing).

ABð Þ<sup>ƶ</sup> <sup>=</sup><sup>Δ</sup> ð Þ<sup>ƶ</sup> <sup>2</sup> is odd and if A and B are taken from

'starred' or 'unstarred' atoms, respectively) Δ<sup>2</sup>

22 Density Functional Theory

variable reference atom (Reprinted from [71] with permission of AIP Publishing).

P.G. and F.D.P. acknowledge the Vrije Universiteit Brussel (VUB) for a Strategic Research Program. F.D.P. also acknowledges the Franqui Foundation for a position as Francqui Research Professor. S.F. acknowledges the Research Foundation Flanders (FWO) and the European Union's Horizon 2020 Marie Sklodowska-Curie grant (N� 706415) for financially supporting his postdoctoral research at the ALGC group. T.S. acknowledges the FWO for a position as research assistant (11ZG615N). P.W.A. thanks the Natural Sciences and Engineering Research Council, the Canada Research Chairs and Compute Canada for financial support. R.B. thanks the Interdisciplinary Centre for Mathematical and Computational Modelling for a computing grant. P.G. thanks Kristina Nikolova for her meticulous help in styling the manuscript and Tom Bettens for his help in styling Figure 2.

#### Conflicts of interest

.The authors report no conflicts of interest regarding this publication.

### Author details

Paul Geerlings<sup>1</sup> \*, Stijn Fias1,2, Thijs Stuyver1 , Paul Ayers<sup>2</sup> , Robert Balawender<sup>3</sup> and Frank De Proft1

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\*Address all correspondence to: pgeerlin@vub.be

1 Algemene Chemie, Vrije Universiteit Brussel, Brussels, Belgium

2 Department of Chemistry and Chemical Biology, McMaster University, Hamilton, ON, Canada

3 Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland

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Conflicts of interest

Author details

24 Density Functional Theory

Paul Geerlings<sup>1</sup>

Frank De Proft1

Canada

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03192

28 Density Functional Theory


**Section 2**

**Applications**

**Section 2**
