**Applications**

**Chapter 2**

**Provisional chapter**

**Modeling with DFT and Chemical Descriptors Approach**

Material properties and process modeling with density functional theory (DFT) is an accurate method to facilitate the study and the design of materials computationally for the development of different electrochemical technologies such as fuel cells, solar cells, and batteries, among others, mainly to achieve alternative ways for energy conversion and storage. Considering the relevance of DFT in the development of these alternative technologies for energy generation and storage, in this chapter, the application of DFT to study catalytic alloys and their reactivity processes to develop polymer membrane fuel cells (PEMFCs) is presented. In this sense, firstly, a brief review of the application of DFT to develop catalysts for PEMFCs and the relation with the concept of chemical descriptors is presented. Secondly, the main chemical descriptors for this task are presented and discussed. Finally, a summary of the main findings of the modeling with DFT and chemical descriptors approach of catalytic alloys for PEMFCs is presented and

The rational design of materials is an important and challenging task, which implies the indepth understanding of determining factors for the desired properties of the materials. These properties emerge from the material chemistry (atoms involved and its electronic structure) and material structure (organization at nano and microscale) [1, 2], for which the understanding at atomistic level is key and could enable the tailoring of materials that improve the efficiency of almost any technology. Particularly, catalyst development is crucial for efficient

**Keywords:** DFT, chemical descriptors, catalytic alloys, PEMFCs

**Modeling with DFT and Chemical Descriptors Approach** 

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

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

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

DOI: 10.5772/intechopen.80922

**for the Development of Catalytic Alloys for PEMFCs**

**for the Development of Catalytic Alloys for PEMFCs**

Alejandro E. Pérez and Rafael Ribadeneira

Alejandro E. Pérez and Rafael Ribadeneira

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.80922

**Abstract**

analyzed.

**1. Introduction**

#### **Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys for PEMFCs Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys for PEMFCs**

DOI: 10.5772/intechopen.80922

Alejandro E. Pérez and Rafael Ribadeneira Alejandro E. Pérez and Rafael Ribadeneira

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.80922

#### **Abstract**

Material properties and process modeling with density functional theory (DFT) is an accurate method to facilitate the study and the design of materials computationally for the development of different electrochemical technologies such as fuel cells, solar cells, and batteries, among others, mainly to achieve alternative ways for energy conversion and storage. Considering the relevance of DFT in the development of these alternative technologies for energy generation and storage, in this chapter, the application of DFT to study catalytic alloys and their reactivity processes to develop polymer membrane fuel cells (PEMFCs) is presented. In this sense, firstly, a brief review of the application of DFT to develop catalysts for PEMFCs and the relation with the concept of chemical descriptors is presented. Secondly, the main chemical descriptors for this task are presented and discussed. Finally, a summary of the main findings of the modeling with DFT and chemical descriptors approach of catalytic alloys for PEMFCs is presented and analyzed.

**Keywords:** DFT, chemical descriptors, catalytic alloys, PEMFCs

#### **1. Introduction**

The rational design of materials is an important and challenging task, which implies the indepth understanding of determining factors for the desired properties of the materials. These properties emerge from the material chemistry (atoms involved and its electronic structure) and material structure (organization at nano and microscale) [1, 2], for which the understanding at atomistic level is key and could enable the tailoring of materials that improve the efficiency of almost any technology. Particularly, catalyst development is crucial for efficient

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

chemical production and clean energy generation and storage technologies such as photoelectrochemical cells, fuel cells, and batteries, which are required for an environmentally responsible economic growth [3].

In the last few decades, density functional theory (DFT) has been successfully used to study heterogeneous catalysis process and the properties of the catalysts, achieving an understanding of reaction mechanisms and the determining factors in the catalytic activity of the materials. For example, it has been found that the electronic structure of the catalytic surface determines its properties [4–6]. Also, some general relations between these factors and the catalytic activity have been developed and used for the search and design of catalytic materials for different applications [7–14].

In the next sections, the application of DFT to study catalytic materials and their reactivity to develop polymer membrane fuel cells (PEMFCs) is presented. The first section is devoted to review briefly and in a general way, the different works, where DFT is applied for studying catalysts for the electrodes of PEMFCs and some determining factors on catalysis found in some of these works are drawn. Secondly, the main factors used for the development of catalysts through the concept of chemical descriptors are presented and discussed. Finally, a summary of the main findings of the modeling with DFT and chemical descriptors approach of catalytic alloys for PEMFCs is presented and analyzed.

#### **2. Using DFT to study heterogeneous catalysis in PEMFCs**

Fuel cells are devices that convert chemical energy of supplied fuel and oxidant into electrical energy. Their operation involves two electrodes, an anode and a cathode, separated by an electrolyte. The anode fuel, such as hydrogen, methanol, and ethanol, which have been proposed for PEMFCs, is oxidized and donates electrons. Electrons travel through the extern circuit, where they perform work and reach the cathode, where oxidant, generally oxygen, is reduced combining with electrons. The electrolyte, which in the case of PEMFCs is a conducting polymer, conducts electricity through the movement of ions, completing the circuit. A schematic representation of a unit cell, illustrating the operation principle explained above is shown in **Figure 1**.

adsorbate-catalyst interactions, which allow determining the potential energy surface and thus the energy barrier of the chemical transformations, also allows calculating the electronic structure of materials, which can help to understand bond-breaking and bond-making process. Also DFT, through thermodynamics approach, facilitates to describe reaction process and energetics of intermediates, reactants, and products under realistic conditions. Additionally, the information from atomistic scale can be used to compute reaction rates. In the next para-

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys…

http://dx.doi.org/10.5772/intechopen.80922

35

Heterogeneous catalytic reactions in PEMFCs involve adsorption of reactants, different bond breaking and/or bond making processes and some electron transfer processes, and desorption of products. The reaction follows the path with minimal energy to go from reactants to product configuration, so this path depends, essentially, on the energetics of the possible elementary steps. So, many studies of catalysts applying DFT is focused in considering each possible reaction step to calculate energetics of reactants and products of each step from which energy barrier for each step can be calculated with different methods such as nudged elastic band [15–17], synchronous transit-guided quasi Newton (STQN) methods [18], among others. With this method, a free-energy diagram for different process can be drawn, an example for dis-

The above described approach was used to study EOR or the ethanol decomposition in other processes on Pt [19, 20], Rh [21], Pd [22], Au [23], and some Pt alloys as PtRh, PtRe, PtRu, PtSn, and PtPd [24]. In some of these works, only some steps or the complete reaction network was considered. For example, it was calculated by Alcalá et al., the transition states for C-C and C-O

graphs, some works about catalysts for PEMFCs are presented.

**Figure 1.** Schematic of a fuel cell.

sociative mechanism of oxygen on Pt is shown in **Figure 2**.

The PEMFCs require the improvement of their performance and the decreasing cost. A principal factor for their performance and its cost are the electrodes, where Pt catalysts are commonly used. So, it required the development of low-cost catalysts, mainly for the oxygen reduction reaction (ORR) at the cathode and hydrogen, methanol or ethanol oxidation reaction (EOR) at anode. The catalysts should reduce the Pt content increasing the rates of the corresponding reactions. Looking for better catalysts is an important challenge, which should be addressed with a comprehensive approach that involves the understanding of the reasons for which a material is an efficient catalyst for a given reaction; in this way, materials can be designed with a "scientific" base.

In this sense, the use of simulations to study catalysis is a great tool, particularly, with the use of atomistic models based on DFT and molecular dynamics. DFT is suited to study Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys… http://dx.doi.org/10.5772/intechopen.80922 35

**Figure 1.** Schematic of a fuel cell.

chemical production and clean energy generation and storage technologies such as photoelectrochemical cells, fuel cells, and batteries, which are required for an environmentally respon-

In the last few decades, density functional theory (DFT) has been successfully used to study heterogeneous catalysis process and the properties of the catalysts, achieving an understanding of reaction mechanisms and the determining factors in the catalytic activity of the materials. For example, it has been found that the electronic structure of the catalytic surface determines its properties [4–6]. Also, some general relations between these factors and the catalytic activity have been developed and used for the search and design of catalytic materi-

In the next sections, the application of DFT to study catalytic materials and their reactivity to develop polymer membrane fuel cells (PEMFCs) is presented. The first section is devoted to review briefly and in a general way, the different works, where DFT is applied for studying catalysts for the electrodes of PEMFCs and some determining factors on catalysis found in some of these works are drawn. Secondly, the main factors used for the development of catalysts through the concept of chemical descriptors are presented and discussed. Finally, a summary of the main findings of the modeling with DFT and chemical descriptors approach

Fuel cells are devices that convert chemical energy of supplied fuel and oxidant into electrical energy. Their operation involves two electrodes, an anode and a cathode, separated by an electrolyte. The anode fuel, such as hydrogen, methanol, and ethanol, which have been proposed for PEMFCs, is oxidized and donates electrons. Electrons travel through the extern circuit, where they perform work and reach the cathode, where oxidant, generally oxygen, is reduced combining with electrons. The electrolyte, which in the case of PEMFCs is a conducting polymer, conducts electricity through the movement of ions, completing the circuit. A schematic representation of a unit cell, illustrating the operation principle explained above is

The PEMFCs require the improvement of their performance and the decreasing cost. A principal factor for their performance and its cost are the electrodes, where Pt catalysts are commonly used. So, it required the development of low-cost catalysts, mainly for the oxygen reduction reaction (ORR) at the cathode and hydrogen, methanol or ethanol oxidation reaction (EOR) at anode. The catalysts should reduce the Pt content increasing the rates of the corresponding reactions. Looking for better catalysts is an important challenge, which should be addressed with a comprehensive approach that involves the understanding of the reasons for which a material is an efficient catalyst for a given reaction; in this way, materials can be

In this sense, the use of simulations to study catalysis is a great tool, particularly, with the use of atomistic models based on DFT and molecular dynamics. DFT is suited to study

sible economic growth [3].

34 Density Functional Theory

als for different applications [7–14].

shown in **Figure 1**.

designed with a "scientific" base.

of catalytic alloys for PEMFCs is presented and analyzed.

**2. Using DFT to study heterogeneous catalysis in PEMFCs**

adsorbate-catalyst interactions, which allow determining the potential energy surface and thus the energy barrier of the chemical transformations, also allows calculating the electronic structure of materials, which can help to understand bond-breaking and bond-making process. Also DFT, through thermodynamics approach, facilitates to describe reaction process and energetics of intermediates, reactants, and products under realistic conditions. Additionally, the information from atomistic scale can be used to compute reaction rates. In the next paragraphs, some works about catalysts for PEMFCs are presented.

Heterogeneous catalytic reactions in PEMFCs involve adsorption of reactants, different bond breaking and/or bond making processes and some electron transfer processes, and desorption of products. The reaction follows the path with minimal energy to go from reactants to product configuration, so this path depends, essentially, on the energetics of the possible elementary steps. So, many studies of catalysts applying DFT is focused in considering each possible reaction step to calculate energetics of reactants and products of each step from which energy barrier for each step can be calculated with different methods such as nudged elastic band [15–17], synchronous transit-guided quasi Newton (STQN) methods [18], among others. With this method, a free-energy diagram for different process can be drawn, an example for dissociative mechanism of oxygen on Pt is shown in **Figure 2**.

The above described approach was used to study EOR or the ethanol decomposition in other processes on Pt [19, 20], Rh [21], Pd [22], Au [23], and some Pt alloys as PtRh, PtRe, PtRu, PtSn, and PtPd [24]. In some of these works, only some steps or the complete reaction network was considered. For example, it was calculated by Alcalá et al., the transition states for C-C and C-O

described in averaged manner as dielectric continuum characterized by a dielectric constant, which is used to estimate free energy of solute-solvent interactions [36–38]; this approach is

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys…

As it is envisioned above, significant progress for the accurate description of electrochemical reactions have been achieved. However, it also can be noted that the complete kinetic description of a given system is quite a demanding task. But, different works show that enough knowledge about catalytic activity could be gained studying some properties of the system at the atomic scale, which can be calculated in a simpler way. These properties are named descriptors, that is, an observable that can be directly linked to a fundamental property [10, 14]; in this case, the catalytic activity. For example, Nørskov et al. show that the oxygen binding energy on metal

of each step is related with the energy change in each process, which in turn is related with stability of adsorbed OOH and OH, respectively. These adsorbed species will be more stable if O binding to the surface is stronger; therefore, the O binding energy can be related with the reaction rates of these two key steps. The reasoning for this relation is as follows: If O binding

to stabilize the OOH to allow the O-O bond to break, so the O binding energy should be moderated. Precisely, in different works, bimetallic Pt and Pd alloys promising as catalysts for ORR looking for materials with moderated oxygen binding energy were identified [39, 40]. The descriptor approach also was used to search materials for hydrogen oxidation reaction on anode of PEMFC. For this reaction, the binding energy of H atom is a key, and according to Sabatier principle, optimal catalytic activity can be achieved on surfaces with moderated binding energies [9, 41]. Sabatier principle states that the bond between the catalyst and the key adsorbate should not be too weak, such that the reactants do not bind, and are not too strong, since this leads to catalyst poisoning [42]. Similarly, in other work, the CO binding energy is considered along the H binding energy to find active catalysts tolerant to CO poisoning [11]. Other property that can be used as a descriptor of catalytic activity is the d-band center of the transition metal surfaces, that is, the average energy of the d states. As is proposed by the d-band model [6, 43], the d-band center is correlated with the binding energy of adsorbates to the surface. The d-band model is fulfilled for bonding of several atoms and molecules, then, as is shown in several works, tuning the electronic structure of a metal by the addition of alloying atom can lead to a better catalyst for ORR [44–46] or catalysts more tolerant to CO [47]. Also, it identified other electronic structure factors, such as transition metal, e.g., band filling and covalency, that can be used as descriptors for ORR on metal

It is worth to mention that the review presented is very brief and there are other extensive reviews that should be consulted [48, 49]. Besides, from the works reviewed above, it should be remarked that it seems that one key property for catalytic activity is the adsorption energy of key intermediate molecules, which can be used to search better catalysts for a determined reaction. Next section reviews the principal concepts in catalysis used for the identification of

to form adsorbed OOH before the

http://dx.doi.org/10.5772/intechopen.80922

37

O. The activation energy

O; if it is weak, it is difficult

electrodes is related to the turnover frequency of the oxygen reduction reaction [30].

ORR involves two key steps, the H transfer to adsorbed O<sup>2</sup>

O-O bond breaking and the H transfer to adsorbed OH to form H<sup>2</sup>

is stronger, it is difficult to destabilize the Pt-OH bond to form H<sup>2</sup>

named implicit solvation method.

oxide catalysts [13].

**Figure 2.** Potential energy surface diagram of dissociative mechanism for oxygen reduction reaction on Pt (1 0 0).

bond cleavage on Pt using constrained optimization, where the length of the bond representing the reaction coordinate is constrained; they identified the steps with lowest activation energy for these bond cleavage reactions. The evaluation of complete reaction network is a formidable task due to the higher quantity of possible intermediate derivatives, but in the ethanol oxidation, reaction network is explored thoroughly identifying key steps for the selectivity on ethanol oxidation and the intermediates, which can lead to C-C bond cleavage in some surfaces.

Also, the mechanistic details of ORR on Pt and Pt-based alloys were investigated using DFT to calculate the intermediate energies and to estimate reaction and barrier energy of each step considered in the reaction pathway [25–33]. Different pathways are considered such as the O2 dissociation mechanism, and the associative mechanism implying the addition of H to the O2 -containing species. Also, it considered two different ways for hydrogen transfer reaction within each of these pathways: the direct addition of H from the electrolyte (Eley-Rideal type) or the H adsorption and subsequent transfer to O<sup>2</sup> -containing species (Langmuir-Hinshelwood type) [27]. For example, in [30], the dissociative mechanism finding that the high stability of oxygen and hydroxyl can imply kinetic limitations in proton/electron transfer to adsorbed oxygen or hydroxyl was studied. In other study, different mechanisms were considered and it is found that under determined conditions, the determining step is the O<sup>2</sup> dissociation, so the associative mechanism is preferred [27].

One of the challenges in the study of electrochemical reactions is considering the effects of the solvation of molecules, the electric field in the interface electrolyte-electrode, and the other adsorbed molecules, nonetheless, some developments have successfully addressed those issues. It is widely accepted that the water layers between the polymer and the catalysts are very important in ORR. It was intended to study the effects of solvation using nine water molecules and one hydronium ion in an ab initio molecular dynamic study [34]. Also, in other studies, adsorption energies of intermediates were calculated adding water adsorbed to the surface or water on top of the adsorbed intermediates [30, 35]. The solvent can be also described in averaged manner as dielectric continuum characterized by a dielectric constant, which is used to estimate free energy of solute-solvent interactions [36–38]; this approach is named implicit solvation method.

As it is envisioned above, significant progress for the accurate description of electrochemical reactions have been achieved. However, it also can be noted that the complete kinetic description of a given system is quite a demanding task. But, different works show that enough knowledge about catalytic activity could be gained studying some properties of the system at the atomic scale, which can be calculated in a simpler way. These properties are named descriptors, that is, an observable that can be directly linked to a fundamental property [10, 14]; in this case, the catalytic activity. For example, Nørskov et al. show that the oxygen binding energy on metal electrodes is related to the turnover frequency of the oxygen reduction reaction [30].

ORR involves two key steps, the H transfer to adsorbed O<sup>2</sup> to form adsorbed OOH before the O-O bond breaking and the H transfer to adsorbed OH to form H<sup>2</sup> O. The activation energy of each step is related with the energy change in each process, which in turn is related with stability of adsorbed OOH and OH, respectively. These adsorbed species will be more stable if O binding to the surface is stronger; therefore, the O binding energy can be related with the reaction rates of these two key steps. The reasoning for this relation is as follows: If O binding is stronger, it is difficult to destabilize the Pt-OH bond to form H<sup>2</sup> O; if it is weak, it is difficult to stabilize the OOH to allow the O-O bond to break, so the O binding energy should be moderated. Precisely, in different works, bimetallic Pt and Pd alloys promising as catalysts for ORR looking for materials with moderated oxygen binding energy were identified [39, 40].

bond cleavage on Pt using constrained optimization, where the length of the bond representing the reaction coordinate is constrained; they identified the steps with lowest activation energy for these bond cleavage reactions. The evaluation of complete reaction network is a formidable task due to the higher quantity of possible intermediate derivatives, but in the ethanol oxidation, reaction network is explored thoroughly identifying key steps for the selectivity on ethanol oxidation and the intermediates, which can lead to C-C bond cleavage in some surfaces.

**Figure 2.** Potential energy surface diagram of dissociative mechanism for oxygen reduction reaction on Pt (1 0 0).

Also, the mechanistic details of ORR on Pt and Pt-based alloys were investigated using DFT to calculate the intermediate energies and to estimate reaction and barrier energy of each step considered in the reaction pathway [25–33]. Different pathways are considered such as

reaction within each of these pathways: the direct addition of H from the electrolyte (Eley-

Hinshelwood type) [27]. For example, in [30], the dissociative mechanism finding that the high stability of oxygen and hydroxyl can imply kinetic limitations in proton/electron transfer to adsorbed oxygen or hydroxyl was studied. In other study, different mechanisms were considered and it is found that under determined conditions, the determining step is the O<sup>2</sup>

One of the challenges in the study of electrochemical reactions is considering the effects of the solvation of molecules, the electric field in the interface electrolyte-electrode, and the other adsorbed molecules, nonetheless, some developments have successfully addressed those issues. It is widely accepted that the water layers between the polymer and the catalysts are very important in ORR. It was intended to study the effects of solvation using nine water molecules and one hydronium ion in an ab initio molecular dynamic study [34]. Also, in other studies, adsorption energies of intermediates were calculated adding water adsorbed to the surface or water on top of the adsorbed intermediates [30, 35]. The solvent can be also

Rideal type) or the H adsorption and subsequent transfer to O<sup>2</sup>

dissociation, so the associative mechanism is preferred [27].

dissociation mechanism, and the associative mechanism implying the addition of H



the O2

to the O2

36 Density Functional Theory

The descriptor approach also was used to search materials for hydrogen oxidation reaction on anode of PEMFC. For this reaction, the binding energy of H atom is a key, and according to Sabatier principle, optimal catalytic activity can be achieved on surfaces with moderated binding energies [9, 41]. Sabatier principle states that the bond between the catalyst and the key adsorbate should not be too weak, such that the reactants do not bind, and are not too strong, since this leads to catalyst poisoning [42]. Similarly, in other work, the CO binding energy is considered along the H binding energy to find active catalysts tolerant to CO poisoning [11].

Other property that can be used as a descriptor of catalytic activity is the d-band center of the transition metal surfaces, that is, the average energy of the d states. As is proposed by the d-band model [6, 43], the d-band center is correlated with the binding energy of adsorbates to the surface. The d-band model is fulfilled for bonding of several atoms and molecules, then, as is shown in several works, tuning the electronic structure of a metal by the addition of alloying atom can lead to a better catalyst for ORR [44–46] or catalysts more tolerant to CO [47]. Also, it identified other electronic structure factors, such as transition metal, e.g., band filling and covalency, that can be used as descriptors for ORR on metal oxide catalysts [13].

It is worth to mention that the review presented is very brief and there are other extensive reviews that should be consulted [48, 49]. Besides, from the works reviewed above, it should be remarked that it seems that one key property for catalytic activity is the adsorption energy of key intermediate molecules, which can be used to search better catalysts for a determined reaction. Next section reviews the principal concepts in catalysis used for the identification of appropriate descriptor, in this sense scaling relations between activation energies and binding energies, Sabatier principle and d-band model are reviewed.

#### **3. Development of catalysts using chemical descriptors**

In the last section, different properties of a material that can be related with its catalytic activity were observed. For instance, binding energies of atoms and molecules can be related with activation energies, so with reaction rates. Sabatier principle shows that a determined value of the adsorption energy of intermediates will lead to higher catalytic activity. Finally, it is mentioned that modification of the electronic structure changes the catalytic activity.

#### **3.1. Correlations between adsorption and activation energies and Sabatier principle**

The central issue in catalysis is the effect of the catalyst on the rate of a chemical reaction, the rate constant of an elementary reaction can be calculated using an Arrhenius expression in terms of a pre-factor (*v*) and an activation energy (*Ea* )as follows:

$$\mathbf{k} = \mathbf{z}\mathbf{e}^{\mathbf{e}\_{f\_{k\mathbf{e}}}} \tag{1}$$

configuration of the products with minimum energy (P) along the reaction path; there is a configuration with the highest energy, which is called transition state (TS). The difference

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys…

and TS in a catalytic process involve adsorbed molecules, then the energy of these configura-

*Ea* = *ETS* − *ER* (2)

Also, it is worth to remark that as transition states have the same basic physics as adsorbed species, then the transition state energy (*ETS*) is correlated with adsorption energies (*Eads*). In fact, linear correlations between transition state and adsorption energies have been found [53–56]. This type of relationship is a well-established approach in the understanding of trends in chemical reactions that dates back to Brønsted in 1928, also Evans and Polanyi in 1938 [57], who stated a linear relationship between activation energies and reaction energies,

The linear relationship between *ETS* of a chemical reaction and a particular *Eads* relevant for that reaction can be understood thinking that TS is an absorbed specie with a configuration relatively close to the adsorption configuration of the molecule X with which is correlated. Then, it is reasonable to express the *ETS* as the sum of two contributions, the first is part of adsorption

the surface and the molecule. The second is the quantity of energy (*β*), which accounts for the reorganization of the molecule X to transform into the transition state molecule. So, the

*ETS* = *αEads* + *β* (3)

It is common to use *Eads* as the energy of products configuration. In this case, parameter *α* is a value between 0 and 1, which indicates the position of the transition state along the reaction coordinate. If alpha is close to 1, the transition state has a configuration similar to the products (late transition state) [56], whereas *β* is a part of the energy involved in the reverse reaction

Using DFT calculations to calculate transition state energies and product energies, it has been possible to establish transition state scaling relations with sufficient statistical significance. This linear relationship provides a straightforward way to study the effect of the surface on the activity for a given reaction, as it enables the estimation of activation energies from adsorption energies, which could be calculated in a simpler way using computational calculations. Transition state scaling relationship implies that the activation energies are lower in surfaces, where the products are more stable (more negative adsorption energies of products) and hence the reaction rate is higher. However, the very strong adsorption of products leads to poison the surface, since the desorption of products is more difficult, hence the rate of desorption is lower. On the other hand, if adsorption of products is very weak, the reaction rate would be lower and the desorption rate higher. Therefore, the adsorption energy of products

(energy to transform products into transition state molecule) [56].

), which accounts, mainly, for the interaction energy between

) [50–52]. The configurations of R, P,

http://dx.doi.org/10.5772/intechopen.80922

39

between the TS and R energies is the activation energy (*Ea*

named Brønsted-Evans-Polanyi (BEP) relationship.

energy of the molecule X (*αEads*−*<sup>X</sup>*

general form of linear relationship is:

tions, and hence activation energy, depends on each surface.

The activation energy is an important quantity, which varies with the change of catalyst, so reviewing the origin of activation energy is useful to understand the effect of catalyst on the reaction rate.

As it is illustrated in **Figure 3**, any chemical reaction can be described as a transition between two local minima on the potential energy surface of the system as a function of the spatial coordinates of all the involved atoms. The reaction path is defined as the minimum energy path, which connects the configuration of the reactants with minimum energy (R) to the

**Figure 3.** Potential energy surface of a chemical reaction process.

configuration of the products with minimum energy (P) along the reaction path; there is a configuration with the highest energy, which is called transition state (TS). The difference between the TS and R energies is the activation energy (*Ea* ) [50–52]. The configurations of R, P, and TS in a catalytic process involve adsorbed molecules, then the energy of these configurations, and hence activation energy, depends on each surface.

$$E\_a = E\_{TS} - E\_{\mathcal{R}} \tag{2}$$

Also, it is worth to remark that as transition states have the same basic physics as adsorbed species, then the transition state energy (*ETS*) is correlated with adsorption energies (*Eads*). In fact, linear correlations between transition state and adsorption energies have been found [53–56]. This type of relationship is a well-established approach in the understanding of trends in chemical reactions that dates back to Brønsted in 1928, also Evans and Polanyi in 1938 [57], who stated a linear relationship between activation energies and reaction energies, named Brønsted-Evans-Polanyi (BEP) relationship.

The linear relationship between *ETS* of a chemical reaction and a particular *Eads* relevant for that reaction can be understood thinking that TS is an absorbed specie with a configuration relatively close to the adsorption configuration of the molecule X with which is correlated. Then, it is reasonable to express the *ETS* as the sum of two contributions, the first is part of adsorption energy of the molecule X (*αEads*−*<sup>X</sup>* ), which accounts, mainly, for the interaction energy between the surface and the molecule. The second is the quantity of energy (*β*), which accounts for the reorganization of the molecule X to transform into the transition state molecule. So, the general form of linear relationship is:

$$E\_{\rm rs} = \alpha E\_{\rm ads} + \beta \tag{3}$$

It is common to use *Eads* as the energy of products configuration. In this case, parameter *α* is a value between 0 and 1, which indicates the position of the transition state along the reaction coordinate. If alpha is close to 1, the transition state has a configuration similar to the products (late transition state) [56], whereas *β* is a part of the energy involved in the reverse reaction (energy to transform products into transition state molecule) [56].

Using DFT calculations to calculate transition state energies and product energies, it has been possible to establish transition state scaling relations with sufficient statistical significance. This linear relationship provides a straightforward way to study the effect of the surface on the activity for a given reaction, as it enables the estimation of activation energies from adsorption energies, which could be calculated in a simpler way using computational calculations.

Transition state scaling relationship implies that the activation energies are lower in surfaces, where the products are more stable (more negative adsorption energies of products) and hence the reaction rate is higher. However, the very strong adsorption of products leads to poison the surface, since the desorption of products is more difficult, hence the rate of desorption is lower. On the other hand, if adsorption of products is very weak, the reaction rate would be lower and the desorption rate higher. Therefore, the adsorption energy of products

**Figure 3.** Potential energy surface of a chemical reaction process.

appropriate descriptor, in this sense scaling relations between activation energies and binding

In the last section, different properties of a material that can be related with its catalytic activity were observed. For instance, binding energies of atoms and molecules can be related with activation energies, so with reaction rates. Sabatier principle shows that a determined value of the adsorption energy of intermediates will lead to higher catalytic activity. Finally, it is

mentioned that modification of the electronic structure changes the catalytic activity.

**3.1. Correlations between adsorption and activation energies and Sabatier principle**

The central issue in catalysis is the effect of the catalyst on the rate of a chemical reaction, the rate constant of an elementary reaction can be calculated using an Arrhenius expression in

*k* = *ve* <sup>−</sup>*Ea* <sup>⁄</sup> *kBT* (1)

The activation energy is an important quantity, which varies with the change of catalyst, so reviewing the origin of activation energy is useful to understand the effect of catalyst on the

As it is illustrated in **Figure 3**, any chemical reaction can be described as a transition between two local minima on the potential energy surface of the system as a function of the spatial coordinates of all the involved atoms. The reaction path is defined as the minimum energy path, which connects the configuration of the reactants with minimum energy (R) to the

)as follows:

energies, Sabatier principle and d-band model are reviewed.

terms of a pre-factor (*v*) and an activation energy (*Ea*

reaction rate.

38 Density Functional Theory

**3. Development of catalysts using chemical descriptors**

and antibonding states; the antibonding states are higher in energy and its filling leads to the

If more of antibonding states are empty, then the surface bonds strongly the molecule. Since electrons fill up all states located below the Fermi energy, in order to estimate the strength of the bond is enough to know how many antibonding states are higher in energy than Fermi

bonding states also would shift up in energy and probably would be emptier. In this sense, the "orange surface" of **Figure 5** should bind the adsorbate stronger than blue surface, since it has d-states with higher energy and so the antibonding states would be above Fermi energy

The average is a representative value for a group the values or a statistical distribution is the average. In this case, energy average of d-states can be used to indicate the energy of all metal d-states, and it is calculated using Eq. (4) [51]. The difference between this value and Fermi energy can be used to compare in an effective way the d-states energy of different metals.

states would be above Fermi energy and the bond would be stronger. This is the called d-band

−∞ <sup>+</sup><sup>∞</sup> *<sup>n</sup>* \_\_\_\_\_\_\_\_\_ *<sup>d</sup>*(*E*)*EdE*

**Figure 5.** Illustration of the d-band model. The interaction of adsorbate states (green) with metal d-states (blue and orange) forms bonding and anti-bonding states. As the d-band center of metal shift down in energy (blue), more

(dotted line) and hence bond is weaker.

∫ −∞ +∞ *nd*

). It has been found that if d-bands of clean surface shifts up in energy, the anti-

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys…

), if it is higher (less negative), then more antibonding

(*E*)*dE* (4)

http://dx.doi.org/10.5772/intechopen.80922

41

weakening of the bond between adsorbate and surface [4, 5, 43, 61].

energy (*EF*

(emptier antibonding states).

antibonding states are below EF

This value is named d-band center (*ε<sup>d</sup>*

model proposed by Hammer and Nørskov [6, 43].

*<sup>ε</sup><sup>d</sup>* <sup>=</sup> <sup>∫</sup>

**Figure 4.** Volcano plot obtained from simple model.

should not be too weak, such that the reaction does not happens, and not too strong since this leads to catalyst poisoning [42]. The above mentioned is precisely the Sabatier principle, a valuable statement for catalyst design.

It allows an explanation of volcano curves, which show the catalytic activity for a given elementary reaction as function of the adsorption energy. As shown in **Figure 4**, at low values of adsorption energy (more positive values), the reaction is slow because the rate of the reaction is rate-limiting, whereas at high values of adsorption energy, desorption becomes the rate-limiting step. Then, the intermediate values of adsorption energies are required in order to obtain the highest activity (top of the volcano) [42, 58–60]. The volcano plot in the figure was obtained from a theoretical calculation, where the rate constants for temperature value of 298 K are obtained from Eq. (1) (*k* = *ve* <sup>−</sup>*Ea* <sup>⁄</sup> *<sup>k</sup> B T* ), the activation energy for the reaction is obtained from Eqs. (2) and (3) (*Ea* <sup>=</sup> *ETS* <sup>−</sup> *ER* , *ETS* <sup>=</sup> *<sup>α</sup>Eads* <sup>+</sup> *<sup>β</sup>*) with *α* and *β* equal to 1, and *ER* equal to −1.2. The activation energy for desorption process is the negative of *Eads*.

As mentioned above, a key issue in catalysis is chemisorption, the bond formed between surface and adsorbate can be described in terms of electronic structure of the surface. For transition metals, it is described in terms of d-bands of metals, as is proposed in the "d-band model," which is presented in the next subsection.

#### **3.2. d-band model**

The electronic structure of transition metals involves a broad s-band and narrow d-band. The interaction of the adsorbate valence electrons with s-band electrons can be assumed to be similar for all the different transition metals; hence, the principal differences in the interaction emerge from d-band electrons [4, 5, 43]. The interaction between the adsorbate and surface d-electrons can be represented with molecular orbitals theory, as illustrated in **Figure 5**, the interaction causes the formation of new shared states, which can be classified into bonding and antibonding states; the antibonding states are higher in energy and its filling leads to the weakening of the bond between adsorbate and surface [4, 5, 43, 61].

If more of antibonding states are empty, then the surface bonds strongly the molecule. Since electrons fill up all states located below the Fermi energy, in order to estimate the strength of the bond is enough to know how many antibonding states are higher in energy than Fermi energy (*EF* ). It has been found that if d-bands of clean surface shifts up in energy, the antibonding states also would shift up in energy and probably would be emptier. In this sense, the "orange surface" of **Figure 5** should bind the adsorbate stronger than blue surface, since it has d-states with higher energy and so the antibonding states would be above Fermi energy (emptier antibonding states).

The average is a representative value for a group the values or a statistical distribution is the average. In this case, energy average of d-states can be used to indicate the energy of all metal d-states, and it is calculated using Eq. (4) [51]. The difference between this value and Fermi energy can be used to compare in an effective way the d-states energy of different metals. This value is named d-band center (*ε<sup>d</sup>* ), if it is higher (less negative), then more antibonding states would be above Fermi energy and the bond would be stronger. This is the called d-band model proposed by Hammer and Nørskov [6, 43].

should not be too weak, such that the reaction does not happens, and not too strong since this leads to catalyst poisoning [42]. The above mentioned is precisely the Sabatier principle, a

It allows an explanation of volcano curves, which show the catalytic activity for a given elementary reaction as function of the adsorption energy. As shown in **Figure 4**, at low values of adsorption energy (more positive values), the reaction is slow because the rate of the reaction is rate-limiting, whereas at high values of adsorption energy, desorption becomes the rate-limiting step. Then, the intermediate values of adsorption energies are required in order to obtain the highest activity (top of the volcano) [42, 58–60]. The volcano plot in the figure was obtained from a theoretical calculation, where the rate constants for temperature value

As mentioned above, a key issue in catalysis is chemisorption, the bond formed between surface and adsorbate can be described in terms of electronic structure of the surface. For transition metals, it is described in terms of d-bands of metals, as is proposed in the "d-band

The electronic structure of transition metals involves a broad s-band and narrow d-band. The interaction of the adsorbate valence electrons with s-band electrons can be assumed to be similar for all the different transition metals; hence, the principal differences in the interaction emerge from d-band electrons [4, 5, 43]. The interaction between the adsorbate and surface d-electrons can be represented with molecular orbitals theory, as illustrated in **Figure 5**, the interaction causes the formation of new shared states, which can be classified into bonding

, *ETS* <sup>=</sup> *<sup>α</sup>Eads* <sup>+</sup> *<sup>β</sup>*) with *α* and *β* equal to 1, and *ER*

), the activation energy for the reaction is obtained

equal to −1.2. The

*B T*

valuable statement for catalyst design.

**Figure 4.** Volcano plot obtained from simple model.

40 Density Functional Theory

of 298 K are obtained from Eq. (1) (*k* = *ve* <sup>−</sup>*Ea* <sup>⁄</sup>*<sup>k</sup>*

activation energy for desorption process is the negative of *Eads*.

model," which is presented in the next subsection.

from Eqs. (2) and (3) (*Ea* <sup>=</sup> *ETS* <sup>−</sup> *ER*

**3.2. d-band model**

$$\varepsilon\_d = \frac{\int\_{\text{--}}^{\text{--}} n\_{dBME}}{\int\_{\text{--}}^{\text{--}} n\_d(E)dE} \tag{4}$$

**Figure 5.** Illustration of the d-band model. The interaction of adsorbate states (green) with metal d-states (blue and orange) forms bonding and anti-bonding states. As the d-band center of metal shift down in energy (blue), more antibonding states are below EF (dotted line) and hence bond is weaker.

The concepts above presented represent the basic theoretical instruments used to investigate catalytic chemical reactions and to provide satisfactory descriptions with a reduced set of properties compared to complete kinetic description.

of the metal with this bond more difficult than with C-H single bond [72]. Probably, breaking of C-H bonds prior to C-C bond cleavage is required, thus the metal could interact with π-orbitals of double C-C bond, which are oriented sideway and have less steric constrains [72]. In fact, it is found that in hydrocarbon reaction on metal surfaces, molecules dehydrogenate

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys…

Experimental works on ethanol oxidation reaction [62, 74–76] shows that ethanol oxidation

more feasible, so it is worth to focus on them. The reaction for C-C bond cleavage in these

*CHx CO*(*ads*) → *CHx*(*ads*) + *CO*(*ads*) (5)

Linear relationship between products energy and transition state energy has been previously developed for this reaction [19, 22, 53, 77]. So, the activation energy would be related with the interaction of the surface with CHX and CO molecules. It is expected that the trends in the reactivity do not changes with the molecule CHXCO chosen, since the adsorption energy of the CHX molecules is expected to be similar [53], that is, if rhodium has higher adsorption

work, to study the C-C bond cleavage in the simplest molecule CHCO, also this step, according to previous works [19, 22], has the lowest activation energy for the C-C bond cleavage on different transition metals. The linear relationship [77], which will be used, is shown in Eq. (6).

*ETS* = *αEP* + *β with α* = 0.88 *β* = 1.07 (6)

the surface far apart to the CH and CO adsorbed on the surface that is expressed in Eq. (7)

*EP* = *Eads*−*CO* + *Eads*−*CH* + *Ediss*−*CHCO* (7)

respect to the energy of reactants (CHCO) in gas phase and the clean slab, that is, the adsorption energy of CHCO. So, the final expression for activation energy, which states the relation-

*Ea* = *α*(*Eads*−*CO* + *Eads*−*CH* + *Ediss*−*CHCO*) + *β* − *Eads*−*CHCO* (8)

oxidation by OH, which is available in alkaline media and in acidic media can be obtained from water activation facilitated by oxyphilic atoms as Sn, Ru, and Mo [20, 75, 78, 79]. So, it seems reasonable to expect that in the appropriated conditions (alkaline media or oxyphilic

metal as co-catalyst), the surfaces with higher CO formation also have higher CO<sup>2</sup>

, which desorbs from surface easily, can be readily obtained from CO

are, respectively, the transition state and product energies respect to the energy of

CO; this molecule could be further

http://dx.doi.org/10.5772/intechopen.80922

43

CO and CHCO. In all the molecules, the C-C bond cleavage is

and CH3

is the energy involved to go from CHCO and

is the energy of *CHCO* adsorbed

formation.

than Ru. So, in this

before the cleaving of C-C bond [73].

CHXCO molecules is written as follows

dehydrogenated forming CH2

*ETS* and *EP*

Considering that CO2

involves dehydrogenation reactions, forming acetyl CH<sup>3</sup>

of CH than Ru, it would have higher adsorption energy of CH<sup>2</sup>

reactants in gas phase and the clean slab. So, *EP*

Activation energy is expressed as in Eq. (2), in this case, *ER*

ship between catalytic activity and adsorption energy, is:

#### **3.3. Summarizing the descriptors approach**

As remarked in the above sections, the descriptor approach is based on the linking of catalytic activity with a limited set of properties, named descriptors. The main step of this approach is to find a suitable reactivity descriptor and the other is to calculate it. As it was remarked, the adsorption energies of key intermediates can be used as descriptors, the identification of particular adsorption energies that are descriptors can be achieved considering the underlying reaction mechanism in order to identify the rate limiting step, also a Sabatier analysis and the known BEP relations can be used to identify the key adsorption energy. Also, an adsorption energy descriptor can be found through an educated guess, resulting in the knowledge on similar reactions [4, 5, 14].

Since adsorbate-surface interaction is a phenomenon originated by electronic interaction, the electronic structure of the systems can be related with adsorption energies, and some characteristics of this electronic structure can be used as descriptor, one of them is the d-band center. But additional research is required to identify more reliable properties, which can be used in other systems different to transition metals. The next section shows the main results of the modeling with chemical descriptors approach of catalytic carbon-carbon bond cleavage in the ethanol oxidation, reaction important for PEMFCs.

## **4. Analysis of C-C bond cleavage using chemical descriptors**

The catalytic cleavage of the C-C bond in ethanol is fundamental for improving the proton exchange membrane fuel cells fueled with ethanol, whose efficiency is still far from obtained with hydrogen fuel cells, which can be partly attributed to the incomplete oxidation of ethanol to *CO* <sup>2</sup> , so energy from ethanol is not totally exploited [62–67]. Catalysts actually used only promotes partial ethanol oxidation to acetic acid, the limiting step for total oxidation to *CO* <sup>2</sup> is, precisely, the C-C bond cleavage. Some works suggest that addition of metals as Ni [68], Rh [69], Ir [70], and Re [71] to Pt helps in C-C breakage at temperatures below 100°C. So, it is of interest to model this reaction step on Pt-based catalysts in order to clarify the effect of different co-catalysts on the C-C bond cleavage and to understand the conditions under which the C-C bond can be broken. It can be achieved using the descriptor approach above presented.

#### **4.1. Reactivity descriptor for C-C bond cleavage**

The C-C cleavage in molecules with more hydrogen atoms attached to the carbon atoms, as ethanol molecule (CH3 CH2 OH) or acetaldehyde (CH3 CHO), has higher activation energies [19, 21, 22]. It is attributed to the high directionality of C-C bond σ-orbital in these molecules, which is constrained along the bond axis, and the several substituents (hydrogen and oxygen atoms) on both carbon atoms constraining sterically this bond, which makes the interaction of the metal with this bond more difficult than with C-H single bond [72]. Probably, breaking of C-H bonds prior to C-C bond cleavage is required, thus the metal could interact with π-orbitals of double C-C bond, which are oriented sideway and have less steric constrains [72]. In fact, it is found that in hydrocarbon reaction on metal surfaces, molecules dehydrogenate before the cleaving of C-C bond [73].

The concepts above presented represent the basic theoretical instruments used to investigate catalytic chemical reactions and to provide satisfactory descriptions with a reduced set of

As remarked in the above sections, the descriptor approach is based on the linking of catalytic activity with a limited set of properties, named descriptors. The main step of this approach is to find a suitable reactivity descriptor and the other is to calculate it. As it was remarked, the adsorption energies of key intermediates can be used as descriptors, the identification of particular adsorption energies that are descriptors can be achieved considering the underlying reaction mechanism in order to identify the rate limiting step, also a Sabatier analysis and the known BEP relations can be used to identify the key adsorption energy. Also, an adsorption energy descriptor can be found through an educated guess, resulting in the knowledge on

Since adsorbate-surface interaction is a phenomenon originated by electronic interaction, the electronic structure of the systems can be related with adsorption energies, and some characteristics of this electronic structure can be used as descriptor, one of them is the d-band center. But additional research is required to identify more reliable properties, which can be used in other systems different to transition metals. The next section shows the main results of the modeling with chemical descriptors approach of catalytic carbon-carbon bond cleavage in the

The catalytic cleavage of the C-C bond in ethanol is fundamental for improving the proton exchange membrane fuel cells fueled with ethanol, whose efficiency is still far from obtained with hydrogen fuel cells, which can be partly attributed to the incomplete oxidation of ethanol

promotes partial ethanol oxidation to acetic acid, the limiting step for total oxidation to *CO* <sup>2</sup>

precisely, the C-C bond cleavage. Some works suggest that addition of metals as Ni [68], Rh [69], Ir [70], and Re [71] to Pt helps in C-C breakage at temperatures below 100°C. So, it is of interest to model this reaction step on Pt-based catalysts in order to clarify the effect of different co-catalysts on the C-C bond cleavage and to understand the conditions under which the C-C bond can be broken. It can be achieved using the descriptor approach above presented.

The C-C cleavage in molecules with more hydrogen atoms attached to the carbon atoms, as

[19, 21, 22]. It is attributed to the high directionality of C-C bond σ-orbital in these molecules, which is constrained along the bond axis, and the several substituents (hydrogen and oxygen atoms) on both carbon atoms constraining sterically this bond, which makes the interaction

OH) or acetaldehyde (CH3

, so energy from ethanol is not totally exploited [62–67]. Catalysts actually used only

is,

CHO), has higher activation energies

properties compared to complete kinetic description.

ethanol oxidation, reaction important for PEMFCs.

**4.1. Reactivity descriptor for C-C bond cleavage**

CH2

ethanol molecule (CH3

**4. Analysis of C-C bond cleavage using chemical descriptors**

**3.3. Summarizing the descriptors approach**

similar reactions [4, 5, 14].

42 Density Functional Theory

to *CO* <sup>2</sup>

Experimental works on ethanol oxidation reaction [62, 74–76] shows that ethanol oxidation involves dehydrogenation reactions, forming acetyl CH<sup>3</sup> CO; this molecule could be further dehydrogenated forming CH2 CO and CHCO. In all the molecules, the C-C bond cleavage is more feasible, so it is worth to focus on them. The reaction for C-C bond cleavage in these CHXCO molecules is written as follows

$$\text{CH}\_{\text{x}}\text{CO}\_{\text{(ads)}} \rightarrow \text{CH}\_{\text{x(ads)}} \star \text{CO}\_{\text{(ads)}}\tag{5}$$

Linear relationship between products energy and transition state energy has been previously developed for this reaction [19, 22, 53, 77]. So, the activation energy would be related with the interaction of the surface with CHX and CO molecules. It is expected that the trends in the reactivity do not changes with the molecule CHXCO chosen, since the adsorption energy of the CHX molecules is expected to be similar [53], that is, if rhodium has higher adsorption of CH than Ru, it would have higher adsorption energy of CH<sup>2</sup> and CH3 than Ru. So, in this work, to study the C-C bond cleavage in the simplest molecule CHCO, also this step, according to previous works [19, 22], has the lowest activation energy for the C-C bond cleavage on different transition metals. The linear relationship [77], which will be used, is shown in Eq. (6).

$$E\_{rs} = \alpha E\_p + \beta \text{ with } \alpha = 0.88 \text{ } \beta = 1.07 \tag{6}$$

*ETS* and *EP* are, respectively, the transition state and product energies respect to the energy of reactants in gas phase and the clean slab. So, *EP* is the energy involved to go from CHCO and the surface far apart to the CH and CO adsorbed on the surface that is expressed in Eq. (7)

$$E\_p = E\_{abs-CO} + E\_{abs-CH} + E\_{diss-CHCO} \tag{7}$$

Activation energy is expressed as in Eq. (2), in this case, *ER* is the energy of *CHCO* adsorbed respect to the energy of reactants (CHCO) in gas phase and the clean slab, that is, the adsorption energy of CHCO. So, the final expression for activation energy, which states the relationship between catalytic activity and adsorption energy, is:

$$E\_a = \alpha (E\_{abs-CO} + E\_{abs-CH} + E\_{diss-CHCO}) + \beta - E\_{abs-CHCO} \tag{8}$$

Considering that CO2 , which desorbs from surface easily, can be readily obtained from CO oxidation by OH, which is available in alkaline media and in acidic media can be obtained from water activation facilitated by oxyphilic atoms as Sn, Ru, and Mo [20, 75, 78, 79]. So, it seems reasonable to expect that in the appropriated conditions (alkaline media or oxyphilic metal as co-catalyst), the surfaces with higher CO formation also have higher CO<sup>2</sup> formation. Therefore, the study of the C-C bond cleavage step can give enough information about which catalysts can helps to obtain higher CO2 . The reactivity of the surfaces for C-C bond cleavage is studied through DFT modeling of adsorption of CH, CO, and CHCO.

The adsorption energies of these molecules were calculated for different Pt<sup>3</sup> M alloys (M = Sn, Re, Ru, Rh, and Ni). Since the alloys with this composition or lower content of M are commonly proposed as a convenient composition to improve the catalytic activity to ethanol oxidation in experimental studies [68, 69, 71, 80–82].

#### **4.2. Calculation of reactivity descriptors**

The adsorption energies of CH, CO, and CHCO are calculated using the slab model to represent the (111) facet of the face-centered cubic metals (Pt3 M) and DFT as is implemented in Quantum Espresso [83]. It used plane waves, exchange correlation energy described by Perdew and Wang (PW91) [84], and ultra-soft pseudopotentials [85]. Convergence was tested with respect to total energy for the Brillouin sampling and for the kinetic energy cut-off in the plane wave expansion, so 3 × 3 × 1 Monkhorst-Pack k-mesh [86] and the kinetic energy cut-off for the plane waves of 544 eV was used.

For slab construction, the lattice parameters obtained from geometric optimization with DFT was used. The effect of the size of the supercell has also been carefully tested for the adsorption energy. Increasing the supercell size from 2 × 2, which we have used, to 4 × 4 changes the adsorption energy by only 0.008 eV, so the description with 2 × 2 is considered good enough considering the computational cost of the simulation of large systems. Five layers of atoms were employed for representing the portion of solid and empty space of 9.00 Å was left above of the atomic surface to avoid self-interaction of the slab. With this empty space, the adsorption energies changed just 0.02 eV compared to the adsorption energies obtained using an empty space of 15.00 Å, indicating that the top of one slab has essentially no effect on the bottom of the next.

The adsorption energy was calculated as total energy of the slab with the i-species adsorbed *Ei*/*s* minus the total energy of clean slab *Es* and the i-species in gas phase *Ei* .

$$E\_{ato-l} = E\_{i|s} - E\_s - E\_i \tag{9}$$

The most stable adsorption configuration for CH adsorption on Pt<sup>3</sup>

found when CHCO is close to hollow-Pt sites, except for Pt<sup>3</sup>

**Figure 6.** CO and CH and CHCO adsorption energies on Pt(111) and Pt3

**Figure 7.** Geometric configurations found for CHCO adsorption on Pt<sup>3</sup>

significantly lower.

hollow sites have similar adsorption energy) with molecular axis of CH normal to the surface and C atom bonded to an atom on the surface in agreement with other theoretical studies [94–96]. It is noted in **Figure 6**, a trend similar to CO adsorption, adsorption on sites with only Pt in the alloy is weaker than pure Pt, whereas the sites with metal atoms

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys…

Pt is shown in **Figure 7**. In this geometry, the CH part is located at bridge position between M and Pt atom (B-M), whereas the CO part is located close to a Pt atom in a Pt-atop-like position (A-Pt). For both groups, the C atom is closer to the surface. This configuration is like other found in theoretical studies for transition metals [77, 97]. Other adsorption configuration is

tion configuration as the above described. In this configuration, the carbon atom linked to the hydrogen atom is bound to the surface in a bridge position between two Pt atoms (B-Pt), whereas the carbon atom linked to oxygen is far from the surface as is shown in **Figure 7**. This

Adsorption energies of former configuration with CH and CO parts bonded to the surface ranges from −3.66 to −3.98 and C-C bond length is stretched by 0.20 ± 0.01 Ǻ, whereas adsorption energies of last configuration ranges from −2.68 to −3.24 eV and C-C bond length is only

M(111) surfaces.

M(111) surfaces.

have higher or similar adsorption energy than pure Pt with the exception of Pt3

The most stable adsorption configuration for CHCO on the alloys (except for Pt<sup>3</sup>

configuration was found in other work for metals considered inactive [97].

M is at hollow site (both

http://dx.doi.org/10.5772/intechopen.80922

Rh, whose Pt sites have an adsorp-

Sn, which is

45

Sn) and pure

Adsorption of *CH*, *CO*, and *CHCO* were tested on several sites, atop sites (A-Pt and A-M), bridge sites (B-Pt and B-M), and hollow sites with and without an atom directly below in the subsequent layer, which are called HCP and FCC hollow sites (HCP-Pt, HCP-M, FCC-Pt, and FCC-M).

The most stable adsorption configuration for CO adsorption on Pt<sup>3</sup> M is at atop site with molecular axis of CO normal to the surface and C atom bonded to the surface in agreement with the experimental findings for Pt-based catalysts [87]. The CO adsorption energies on A-Pt and A-M for the different surfaces are shown in **Figure 6**. It is found that CO adsorption energy on A-Pt sites in the alloys is reduced respect to pure Pt (dotted line) and it is more reduced when the metal atom adsorbs more strongly the CO molecule. Also it is remarkable that adsorption energy on tin atom is only −0.08 eV. It is in agreement with other works, which points out that Pt-CO binding is weakened in the PtM alloy [88–90], whereas the M-CO bond is strengthened with the presence of Pt and CO adsorption on tin is not stable [91–93].

The most stable adsorption configuration for CH adsorption on Pt<sup>3</sup> M is at hollow site (both hollow sites have similar adsorption energy) with molecular axis of CH normal to the surface and C atom bonded to an atom on the surface in agreement with other theoretical studies [94–96]. It is noted in **Figure 6**, a trend similar to CO adsorption, adsorption on sites with only Pt in the alloy is weaker than pure Pt, whereas the sites with metal atoms have higher or similar adsorption energy than pure Pt with the exception of Pt3 Sn, which is significantly lower.

Therefore, the study of the C-C bond cleavage step can give enough information about which

Re, Ru, Rh, and Ni). Since the alloys with this composition or lower content of M are commonly proposed as a convenient composition to improve the catalytic activity to ethanol oxi-

The adsorption energies of CH, CO, and CHCO are calculated using the slab model to rep-

in Quantum Espresso [83]. It used plane waves, exchange correlation energy described by Perdew and Wang (PW91) [84], and ultra-soft pseudopotentials [85]. Convergence was tested with respect to total energy for the Brillouin sampling and for the kinetic energy cut-off in the plane wave expansion, so 3 × 3 × 1 Monkhorst-Pack k-mesh [86] and the kinetic energy cut-off

For slab construction, the lattice parameters obtained from geometric optimization with DFT was used. The effect of the size of the supercell has also been carefully tested for the adsorption energy. Increasing the supercell size from 2 × 2, which we have used, to 4 × 4 changes the adsorption energy by only 0.008 eV, so the description with 2 × 2 is considered good enough considering the computational cost of the simulation of large systems. Five layers of atoms were employed for representing the portion of solid and empty space of 9.00 Å was left above of the atomic surface to avoid self-interaction of the slab. With this empty space, the adsorption energies changed just 0.02 eV compared to the adsorption energies obtained using an empty space of 15.00 Å, indicating that the top of one slab has essentially no effect on the bottom of the next. The adsorption energy was calculated as total energy of the slab with the i-species adsorbed

*Eads*−*<sup>i</sup>* = *Ei*/*<sup>s</sup>* − *Es* − *Ei* (9)

Adsorption of *CH*, *CO*, and *CHCO* were tested on several sites, atop sites (A-Pt and A-M), bridge sites (B-Pt and B-M), and hollow sites with and without an atom directly below in the subsequent layer, which are called HCP and FCC hollow sites (HCP-Pt, HCP-M, FCC-Pt, and FCC-M).

molecular axis of CO normal to the surface and C atom bonded to the surface in agreement with the experimental findings for Pt-based catalysts [87]. The CO adsorption energies on A-Pt and A-M for the different surfaces are shown in **Figure 6**. It is found that CO adsorption energy on A-Pt sites in the alloys is reduced respect to pure Pt (dotted line) and it is more reduced when the metal atom adsorbs more strongly the CO molecule. Also it is remarkable that adsorption energy on tin atom is only −0.08 eV. It is in agreement with other works, which points out that Pt-CO binding is weakened in the PtM alloy [88–90], whereas the M-CO bond is strengthened with the presence of Pt and CO adsorption on tin is not stable [91–93].

The most stable adsorption configuration for CO adsorption on Pt<sup>3</sup>

and the i-species in gas phase *Ei*

is studied through DFT modeling of adsorption of CH, CO, and CHCO.

The adsorption energies of these molecules were calculated for different Pt<sup>3</sup>

. The reactivity of the surfaces for C-C bond cleavage

M alloys (M = Sn,

M) and DFT as is implemented

.

M is at atop site with

catalysts can helps to obtain higher CO2

44 Density Functional Theory

**4.2. Calculation of reactivity descriptors**

for the plane waves of 544 eV was used.

minus the total energy of clean slab *Es*

*Ei*/*s*

dation in experimental studies [68, 69, 71, 80–82].

resent the (111) facet of the face-centered cubic metals (Pt3

The most stable adsorption configuration for CHCO on the alloys (except for Pt<sup>3</sup> Sn) and pure Pt is shown in **Figure 7**. In this geometry, the CH part is located at bridge position between M and Pt atom (B-M), whereas the CO part is located close to a Pt atom in a Pt-atop-like position (A-Pt). For both groups, the C atom is closer to the surface. This configuration is like other found in theoretical studies for transition metals [77, 97]. Other adsorption configuration is found when CHCO is close to hollow-Pt sites, except for Pt<sup>3</sup> Rh, whose Pt sites have an adsorption configuration as the above described. In this configuration, the carbon atom linked to the hydrogen atom is bound to the surface in a bridge position between two Pt atoms (B-Pt), whereas the carbon atom linked to oxygen is far from the surface as is shown in **Figure 7**. This configuration was found in other work for metals considered inactive [97].

Adsorption energies of former configuration with CH and CO parts bonded to the surface ranges from −3.66 to −3.98 and C-C bond length is stretched by 0.20 ± 0.01 Ǻ, whereas adsorption energies of last configuration ranges from −2.68 to −3.24 eV and C-C bond length is only

**Figure 6.** CO and CH and CHCO adsorption energies on Pt(111) and Pt3 M(111) surfaces.

**Figure 7.** Geometric configurations found for CHCO adsorption on Pt<sup>3</sup> M(111) surfaces.

stretched slightly by 0.08 ± 0.01 Ǻ. So, the ensembles with only Pt atoms in Pt<sup>3</sup> M (with exception of Pt3 Rh) have adsorption energies significantly lower than pure Pt and do not stretch significantly the C-C bond, whereas the ensembles with M atom have adsorption energies similar to Pt and stretch significantly the C-C bond. Indicating that the addition of M to Pt decreases the ability of Pt for interacting with CHCO and hence part of the alloy is inactive, except for Pt3 Rh.

Summarizing, the adsorption of CH, CO, and CHCO shows that the addition of M (M = Re, Rh, Ni, Ru, or Sn) to Pt leads to obtain two active sites with different reactivities. The Pt atoms in all these alloys form bonds with the described molecules weaker than Pt atoms in pure Pt, whereas the Re, Rh, and Ru atoms in the alloys have a stronger interaction with CH, CO, and CHCO than pure Pt. This difference explains that adsorption energies of CO on metal atoms in Pt3 M (M = Rh, Re, and Ru) are higher than Pt (111), whereas the adsorption energies of CH on sites with M atoms in Pt3 M (M = Rh, Re, and Ru) are similar to Pt (111), since CH interacts with two Pt and one M atom at hollow site. Then, the stronger interaction of mentioned M atoms should compensate the weaker interaction of Pt in these alloys.

This trend agrees with previous experimental results, where it is reported that Pt-based alloys

**Figure 8.** Final state energy (Ep) relative to reactants (CHCO) in gas phase and clean slab and activation energy on the

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys…

In contrast, addition of Rhenium to Pt improves performance for C-C cleavage, but this improvement is not much higher, it is reported that Re addition to Pt improves the acetaldehyde decomposition, increasing the produced CO (product of C-C bond cleavage) from 10 to 18% [100]. On

But the PtSn alloys are recognized as the better catalyst for EOR, where the enhancement of the activity with this alloy is attributed to the promotion of bifunctional mechanism promoting the

Due to mentioned issues, different authors propose materials with Pt, Sn, and Rh as effective catalysts in splitting the C-C bond and oxidizing intermediates molecules of EOR via bifunctional mechanism. This mix in the appropriate composition will provide high overall

presence of oxidants as OH, which should promote the oxidation of intermediates to acetic acid and decreases the C-C bond cleavage rate [101]. But the unavailability of oxidants limits

considering, in addition of C-C cleavage, the step of oxidation to produce acetic acid and the

Department of Processes and Energy, Facultad de Minas, Universidad Nacional de

higher than Pt and Pt alloys [69, 99].

http://dx.doi.org/10.5772/intechopen.80922

47

than Pt. However, the selectivity can be

remains elusive. It can be attributed to the

, poisoning the surface. Finding a better catalyst for EOR requires

product respect to Pt [70, 98].

with rhodium (PtRh and PtSnRh) show a production of CO2

conversion of ethanol and higher selectivity to CO2

considered low and the total conversion to CO2

Alejandro E. Pérez and Rafael Ribadeneira\*

Colombia-Sede Medellín, Colombia

\*Address all correspondence to: reribade@unal.edu.co

acetic acid production.

different surfaces sites found active.

the conversion of CO to CO2

dehydrogenation step.

**Author details**

the contrary, it is reported that Sn addition to Pt decreases the CO<sup>2</sup>

Besides, Ni and Pt atoms in Pt<sup>3</sup> Ni have slightly weaker interaction than pure Pt (111), so the adsorption energies at different sites in this alloy are slightly lower than pure Pt. On the other hand, Sn and Pt atoms in Pt<sup>3</sup> Sn have a weaker interaction with the molecules than pure Pt (111), suggesting that Pt<sup>3</sup> Sn has low catalytic activity toward C-C bond cleavage, confirming previous experimental and theoretical works that affirms that PtSn alloys have lower catalytic activity for C-C bond cleavage, but they increase the catalytic activity for EOR by promoting bifunctional mechanism and the pathway toward acetic acid [70, 98]. Other remarkable finding is the two configurations for CHCO adsorption, one of them destabilizes the molecule binding it strongly and stretching significantly the C-C bond (around to 0.20 Ǻ) and the other is found in metals inactive, it suggests that part of alloy is inactive, except for Pt<sup>3</sup> Rh.

The adsorption energies obtained can be related with activation energy for C-C bond cleavage through Eq. (8), only the active sites are considered since the linear relationship between ETS and Eads only have sense for them. As is can be expected from Eq. (8), the surfaces with higher adsorption energies of CH and CO have lower activation energy, that is the sites with Re, Rh, and Ru atoms, it can be observed in **Figure 8**, which shows the product energies E<sup>p</sup> (see Eq. (7)) and activation energies for the different active sites.

These results indicate that Pt3 Sn is the catalyst with the lowest activity for C-C bond cleavage, since in this alloy Sn has a weak interaction with all molecules studied and Pt atoms have a weaker interaction than Pt atoms in pure Pt (111). Also, it is expected that Pt<sup>3</sup> Rh alloy has higher reactivity for C-C bond cleavage since all sites, sites with only Pt, and sites with Rh atoms have similar or higher reactivity than pure Pt. Whereas Pt<sup>3</sup> Re and Pt3 Ru have a nonsignificant improvement of the reactivity with respect to pure Pt, since they have one site with higher reactivity (site with M atom) and other with lesser reactivity (site with only Pt atoms); between these ones, Pt<sup>3</sup> Re has the most reactive site. Pt3 Ni has slightly weaker interaction with molecules than Pt. Consequently, the ranking from high to low reactivity for C-C bond cleavage is Pt3 Rh, Pt<sup>3</sup> Re, Pt<sup>3</sup> Ru, Pt, Pt<sup>3</sup> Ni, and Pt<sup>3</sup> Sn.

Modeling with DFT and Chemical Descriptors Approach for the Development of Catalytic Alloys… http://dx.doi.org/10.5772/intechopen.80922 47

**Figure 8.** Final state energy (Ep) relative to reactants (CHCO) in gas phase and clean slab and activation energy on the different surfaces sites found active.

This trend agrees with previous experimental results, where it is reported that Pt-based alloys with rhodium (PtRh and PtSnRh) show a production of CO2 higher than Pt and Pt alloys [69, 99]. In contrast, addition of Rhenium to Pt improves performance for C-C cleavage, but this improvement is not much higher, it is reported that Re addition to Pt improves the acetaldehyde decomposition, increasing the produced CO (product of C-C bond cleavage) from 10 to 18% [100]. On the contrary, it is reported that Sn addition to Pt decreases the CO<sup>2</sup> product respect to Pt [70, 98]. But the PtSn alloys are recognized as the better catalyst for EOR, where the enhancement of the activity with this alloy is attributed to the promotion of bifunctional mechanism promoting the acetic acid production.

Due to mentioned issues, different authors propose materials with Pt, Sn, and Rh as effective catalysts in splitting the C-C bond and oxidizing intermediates molecules of EOR via bifunctional mechanism. This mix in the appropriate composition will provide high overall conversion of ethanol and higher selectivity to CO2 than Pt. However, the selectivity can be considered low and the total conversion to CO2 remains elusive. It can be attributed to the presence of oxidants as OH, which should promote the oxidation of intermediates to acetic acid and decreases the C-C bond cleavage rate [101]. But the unavailability of oxidants limits the conversion of CO to CO2 , poisoning the surface. Finding a better catalyst for EOR requires considering, in addition of C-C cleavage, the step of oxidation to produce acetic acid and the dehydrogenation step.

#### **Author details**

stretched slightly by 0.08 ± 0.01 Ǻ. So, the ensembles with only Pt atoms in Pt<sup>3</sup>

Rh) have adsorption energies significantly lower than pure Pt and do not stretch

significantly the C-C bond, whereas the ensembles with M atom have adsorption energies similar to Pt and stretch significantly the C-C bond. Indicating that the addition of M to Pt decreases the ability of Pt for interacting with CHCO and hence part of the alloy is inactive,

Summarizing, the adsorption of CH, CO, and CHCO shows that the addition of M (M = Re, Rh, Ni, Ru, or Sn) to Pt leads to obtain two active sites with different reactivities. The Pt atoms in all these alloys form bonds with the described molecules weaker than Pt atoms in pure Pt, whereas the Re, Rh, and Ru atoms in the alloys have a stronger interaction with CH, CO, and CHCO than pure Pt. This difference explains that adsorption energies of CO on metal atoms

M (M = Rh, Re, and Ru) are higher than Pt (111), whereas the adsorption energies of CH

with two Pt and one M atom at hollow site. Then, the stronger interaction of mentioned M

adsorption energies at different sites in this alloy are slightly lower than pure Pt. On the other

previous experimental and theoretical works that affirms that PtSn alloys have lower catalytic activity for C-C bond cleavage, but they increase the catalytic activity for EOR by promoting bifunctional mechanism and the pathway toward acetic acid [70, 98]. Other remarkable finding is the two configurations for CHCO adsorption, one of them destabilizes the molecule binding it strongly and stretching significantly the C-C bond (around to 0.20 Ǻ) and the other

The adsorption energies obtained can be related with activation energy for C-C bond cleavage through Eq. (8), only the active sites are considered since the linear relationship between ETS and Eads only have sense for them. As is can be expected from Eq. (8), the surfaces with higher adsorption energies of CH and CO have lower activation energy, that is the sites with Re, Rh, and Ru atoms, it can be observed in **Figure 8**, which shows the product energies E<sup>p</sup>

since in this alloy Sn has a weak interaction with all molecules studied and Pt atoms have

higher reactivity for C-C bond cleavage since all sites, sites with only Pt, and sites with Rh

nificant improvement of the reactivity with respect to pure Pt, since they have one site with higher reactivity (site with M atom) and other with lesser reactivity (site with only Pt atoms);

with molecules than Pt. Consequently, the ranking from high to low reactivity for C-C bond

Sn.

is found in metals inactive, it suggests that part of alloy is inactive, except for Pt<sup>3</sup>

a weaker interaction than Pt atoms in pure Pt (111). Also, it is expected that Pt<sup>3</sup>

Re has the most reactive site. Pt3

Ni, and Pt<sup>3</sup>

(see Eq. (7)) and activation energies for the different active sites.

atoms have similar or higher reactivity than pure Pt. Whereas Pt<sup>3</sup>

Ru, Pt, Pt<sup>3</sup>

atoms should compensate the weaker interaction of Pt in these alloys.

M (M = Rh, Re, and Ru) are similar to Pt (111), since CH interacts

Ni have slightly weaker interaction than pure Pt (111), so the

Sn have a weaker interaction with the molecules than pure Pt

Sn is the catalyst with the lowest activity for C-C bond cleavage,

Re and Pt3

Ni has slightly weaker interaction

Sn has low catalytic activity toward C-C bond cleavage, confirming

tion of Pt3

46 Density Functional Theory

except for Pt3

in Pt3

Rh.

on sites with M atoms in Pt3

Besides, Ni and Pt atoms in Pt<sup>3</sup>

hand, Sn and Pt atoms in Pt<sup>3</sup>

These results indicate that Pt3

between these ones, Pt<sup>3</sup>

Rh, Pt<sup>3</sup>

Re, Pt<sup>3</sup>

cleavage is Pt3

(111), suggesting that Pt<sup>3</sup>

M (with excep-

Rh.

Rh alloy has

Ru have a nonsig-

Alejandro E. Pérez and Rafael Ribadeneira\*

\*Address all correspondence to: reribade@unal.edu.co

Department of Processes and Energy, Facultad de Minas, Universidad Nacional de Colombia-Sede Medellín, Colombia

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[68] Beyhan S, Coutanceau C, Léger JM, Napporn TW, Kadirgan F. Promising anode candidates for direct ethanol fuel cell: Carbon supported PtSn-based trimetallic catalysts prepared by Bönnemann method. International Journal of Hydrogen Energy. 2013;**38**:

[69] Kowal A, Li M, Shao M, Sasaki K, Vukmirovic MB, Zhang J, et al. Ternary Pt/Rh/SnO<sup>2</sup>

[70] Li M, Cullen DA, Sasaki K, Marinkovic NS, More K, Adzic RR. Ternary electrocatalysts for oxidizing ethanol to carbon dioxide: Making ir capable of splitting C-C bond. Journal

[71] Tayal J, Rawat B, Basu S. Effect of addition of rhenium to Pt-based anode catalysts in electro-oxidation of ethanol in direct ethanol PEM fuel cell. International Journal of

[72] Murakami M, Ito Y. Cleavage of carbon—carbon single bonds by transition metals. In: Activation of Unreactive Bonds and Organic Synthesis. Berlin Heidelberg: Springer; 1999

. Nature Materials. 2009;**8**(4):325-330

metal catalysts. Journal of Catalysis. 2001;**197**(2):229-231

electrolytes. Journal of Catalysis. 2011;**278**(2):181-188

cells. International Journal of Hydrogen Energy. 2012;**38**:1-16

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6830-6841


[87] Van Hove MA. The adsorption sites of CO and NO molecules on metal surfaces. Israel Journal of Chemistry. 1998;**38**(4):349-352

**Chapter 3**

**Provisional chapter**

**Density Functional Theory Studies of Catalytic Sites in**

**Density Functional Theory Studies of Catalytic Sites in** 

Theoretical methods have become indispensable tools in many fields of chemistry and materials research. Metal-organic frameworks (MOFs) are porous materials; they have been intensively developed due to their diverse properties suitable for a wide range of applications. Theoretical approaches have thus been frequently employed toward the design and characterization of MOFs. We focus here in particular on theoretical studies of single-site catalytic reactions that occur inside the cavities of MOFs. The density functional method (DFT) has been the main approach used for such studies. We briefly review the uses of DFT to examine the catalytic reactions in MOFs. We note that DFT methods are versatile and can be made to work for different purposes such as, e.g., force-field development for molecular simulations. We shall, however, cover this field only very

**Keywords:** theoretical chemistry, computational chemistry, quantum chemistry, density functional theory (DFT), metal-organic frameworks (MOF), catalysis, single-site

Experiments in chemistry aim primarily at understanding, or insight. These insights lead in turn to "innovation" and new technologies [1]. However, in many instances, such an insight cannot be obtained purely from current experiments. The required fundamental theory has been available for some time. The more recent development of computational methodologies and the dramatic increase in computer power made computational methods an additional

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

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

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

DOI: 10.5772/intechopen.80698

**Metal-Organic Frameworks**

**Metal-Organic Frameworks**

Siwarut Siwaipram, Sarawoot Impeng, Philippe A. Bopp and Sareeya Bureekaew

Siwarut Siwaipram, Sarawoot Impeng, Philippe A. Bopp and Sareeya Bureekaew

http://dx.doi.org/10.5772/intechopen.80698

**Abstract**

catalytic reactions

**1. Introduction**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

succinctly to put it into context with our main topic.


#### **Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks**

DOI: 10.5772/intechopen.80698

Siwarut Siwaipram, Sarawoot Impeng, Philippe A. Bopp and Sareeya Bureekaew Siwarut Siwaipram, Sarawoot Impeng, Philippe A. Bopp and Sareeya Bureekaew

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.80698

#### **Abstract**

[87] Van Hove MA. The adsorption sites of CO and NO molecules on metal surfaces. Israel

[88] Liao MS, Cabrera CR, Ishikawa Y. Theoretical study of CO adsorption on Pt, Ru and

[89] Su HY, Bao XH, Li WX. Modulating the reactivity of Ni-containing Pt(111)-skin catalysts by density functional theory calculations. The Journal of Chemical Physics. 2008;

Pt(111) surface studied by photoemission and thermal desorption. Surface Science.

[91] Shubina TE, Koper MTM. Quantum-chemical calculations of CO and OH interacting

[92] Dupont C, Jugnet Y, Loffreda D. Theoretical evidence of PtSn alloy efficiency for CO oxidation. Journal of the American Chemical Society. 2006;**128**(28):9129-9136

[94] Kua J, Faglioni F, W a G. Thermochemistry for hydrocarbon intermediates chemisorbed

[95] Hahn K, Mavrikakis M. Atomic and molecular adsorption on Re(0001). Topics in

[96] Zhang M, Yang K, Zhang X, Yu Y. Effect of Ni(111) surface alloying by Pt on partial oxidation of methane to syngas: A DFT study. Surface Science. 2014;**630**:236-243 [97] Courtois J, Du W, Wong E, Teng X, Deskins NA. Screening iridium-based bimetallic alloys as catalysts for direct ethanol fuel cells. Applied Catalysis A: General. 2014;**483**:

[98] Alcala R, Shabaker JW, Huber GW, MA S-C, JA D. Experimental and DFT studies of the conversion of ethanol and acetic acid on PtSn-based catalysts. The Journal of Physical

[99] Bach Delpeuch A, Maillard F, Chatenet M, Soudant P, Cremers C. Ethanol oxidation reaction (EOR) investigation on Pt/C, Rh/C, and Pt-based bi- and tri-metallic electrocatalysts: A DEMS and in situ FTIR study. Applied Catalysis B: Environmental. 2016;

[100] Ciftci A, Ligthart DAJM, Sen AO, van Hoof AJF, Friedrich H, Hensen EJM. Pt-Re synergy in aqueous-phase reforming of glycerol and the water-gas shift reaction. Journal

platinum in the direct ethanol fuel cell. Angewandte Chemie, International Edition.

[101] Kavanagh R, Cao XM, Lin WF, Hardacre C, Hu P. Origin of low CO<sup>2</sup>

*<sup>m</sup>* with *n* = 1, 2, 3 on Pt, Ir, Os, Pd, Rh, and Ru. Journal of

with bimetallic surfaces. Electrochimica Acta. 2002;**47**(22-23):3621-3628

[93] Liu P, Logadottir A, Nørskov JK. Modeling the electro-oxidation of CO and H<sup>2</sup>

Pt, Ru, PtRu and Pt3Sn. Electrochimica Acta. 2003;**48**(25-26):3731-3742

adsorption on the Re/

/CO on

selectivity on

Pt-M (M = Ru, Sn, Ge) clusters. Surface Science. 2000;**445**(2-3):267-282

[90] Ramstad A, Strisland F, Raaen S, Borg A, Berg C. CO and O<sup>2</sup>

*m*(CH3 )

the American Chemical Society. 2000;**122**(5):2309-2321

Journal of Chemistry. 1998;**38**(4):349-352

**128**(19):1-7

54 Density Functional Theory

1999;**440**(1-2):290-300

on metal surfaces: CH*n*-

85-96

**181**:672-680

Catalysis. 2014;**57**(1-4):54-68

Chemistry. B. 2005;**109**(6):2074-2085

of Catalysis. 2014;**311**:88-101

2012;**51**(7):1572-1575

Theoretical methods have become indispensable tools in many fields of chemistry and materials research. Metal-organic frameworks (MOFs) are porous materials; they have been intensively developed due to their diverse properties suitable for a wide range of applications. Theoretical approaches have thus been frequently employed toward the design and characterization of MOFs. We focus here in particular on theoretical studies of single-site catalytic reactions that occur inside the cavities of MOFs. The density functional method (DFT) has been the main approach used for such studies. We briefly review the uses of DFT to examine the catalytic reactions in MOFs. We note that DFT methods are versatile and can be made to work for different purposes such as, e.g., force-field development for molecular simulations. We shall, however, cover this field only very succinctly to put it into context with our main topic.

**Keywords:** theoretical chemistry, computational chemistry, quantum chemistry, density functional theory (DFT), metal-organic frameworks (MOF), catalysis, single-site catalytic reactions

#### **1. Introduction**

Experiments in chemistry aim primarily at understanding, or insight. These insights lead in turn to "innovation" and new technologies [1]. However, in many instances, such an insight cannot be obtained purely from current experiments. The required fundamental theory has been available for some time. The more recent development of computational methodologies and the dramatic increase in computer power made computational methods an additional

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

and competitive tool to describe chemical phenomena [2]. This field is broadly known as "modeling." In "molecular modeling," the premise is that the interatomic and intermolecular interactions can be known [3]. DFT has become the dominant tool for the purpose of determining such interactions [4]. In particular, in solid-state chemistry and physics, it has become increasingly popular since appropriate functionals and basis sets [5] were designed to provide a viable balance between the reliability of the numerical results and the computational costs.

quality of the single crystals is not sufficient, the structure can be inferred from powder diffraction patterns [18] which often remains incomplete because of the complicated molecular system, large unit cell, and low symmetry. Molecular modeling methods must thus comple-

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

http://dx.doi.org/10.5772/intechopen.80698

57

The periodic structure of an MOF can be studied by DFT: properties such as cell parameters and elastic properties can be obtained from periodic calculations. However, such calculations consume large amounts of computational resources due to the typically large unit cell of MOFs. Simple functionals such as local-density approximation (LDA) or generalized gradient approximations (GGA) are, however, often sufficient in this case [20]. Studying catalytic effects requires more advanced "hybrid" functionals, e.g., BP86 [21], B3-LYP [22], or M06 series [23]. Molecular mechanics (MM) is a much cheaper computational method to study large systems. The reliability of these calculations depends on the quality of the force fields used to describe the interactions between the atoms in the investigated system. Some high-accuracy force fields for MOFs, such as BTW-FF [24], MOF-FF [25], Quick-FF [26], and others [27, 28]**,** were parametrized from DFT calculations**.** They were successfully used to determine and predict the periodic structures [29, 30], as well as to examine phase transformations [31–33], guest diffusion in the pores and other mechanical properties such as bulk moduli [34], elastic constants [35] and mechanical stability [36]. Note that MM calculations cannot describe electronic structures. Bond breaking and forming cannot be accounted for in such calculations. Thus, it

is not possible to study reaction mechanisms using conventional MM methods [37].

In summary, for structurally well-defined and small enough systems, DFT calculations are a cogent approach especially in the field of localized catalytic reactions. We aim here to outline the suitability of DFT techniques to study catalysis in MOFs materials through examples from

MOFs have well-defined crystal structures with high concentrations of metal centers organized at topical distances, and large pore volumes between these centers. Even though MOFs are less able to withstand high temperatures compared to some other catalytic porous materials, their undeniable qualities such as large internal surface areas and uniform pore and cavity sizes make them attractive for various applications [38]. The pore size/shape in the framework can be tuned for selectivity for a particular reaction. In contrast, the small pore windows of other nanoporous materials, such as zeolites, commonly limit the catalytic transformation of

In principle, various types of active sites can be incorporated into MOFs. Coordinatively unsaturated metal sites (CUSs) and functional groups on the organic linkers (usually acid/ base sites) are the main catalytic sites. In addition, even though the MOF framework itself does not contain active sites, the catalytic reactivity can be enhanced by (a) postsynthetic functionalized modification [40] and (b) encapsulation of catalytic species [41]. The pore volume of MOFs is able to accommodate organic molecules, inorganic nanoparticles, metal complexes,

ment these experiments [19].

the recent literature.

**3. Catalytic MOFs**

large molecules [39].

Metal-organic frameworks (MOFs), also known as porous coordination polymers (PCPs) [6–8], are a class of hybrid organic-inorganic materials with high surface area, a permanent porosity with large internal pore volume and tunable pore sizes. They have a wide range of potential applications such as gas storage [9] and separation [10], catalysis [11], and many others [12–14]. MOFs are constructed by interlinking metal ions or, more generally, metalcontaining units with organic moieties (carboxylates, azolates, imidazolates, pyridyl, etc.) through coordination bonds, thus creating crystalline frameworks. Due to the variety of the structural and chemical elements, some MOFs show unexpected characteristics, which sometimes cannot be fully assessed experimentally. Theoretical approaches have been intensively employed in this particular case to investigate the systems at the atomic level. Beyond the analysis of observed phenomena, the prediction of unknown attributes has been attempted.

In this book-chapter, we review and discuss the current status and challenges of DFT studies of MOFs. We focus in particular on aspects relevant to catalysis at the metal centers found in these structures.

#### **2. DFT calculations on MOFs**

One of the main issues of DFT calculations is that a compromise between the dependability of the results and computational costs must be found, especially for large and complex systems [15]. A primary issue is how to select a functional suitable for the investigated system [16]**.** Different functionals, based on different approximations, may give different (even opposite) results. It is thus important to choose the functionals wisely. No functional devised up to now has proven to be suitable for all systems. Therefore, the validation of any selected functional is crucial**.**

MOFs are crystalline materials with extended frameworks. This needs, in some instances, to be taken into account when, for catalytic applications of MOFs, DFT is used to investigate the reaction mechanism. The reaction enthalpy can be obtained approximately from the energy at 0 K by including a zero-point energy correction calculated from (possibly scaled) vibrational frequencies. To compute the entropy, harmonic or anharmonic low-frequency modes can be used. The temperature-dependent free and activation energies can be calculated by adding thermal enthalpic and entropic effects. The so-obtained free energies of reaction are used to calculate temperaturedependent equilibrium constants. Note that the kinetics of a reaction can also be investigated, e.g., by transition state theory (TST) [17]. However, this is beyond the scope of this chapter.

Since the properties of MOFs depend not only on their chemical composition but also their structures, atomistic structural characterization is essentials. Generally, the structure of a newly synthesized MOF is characterized by single-crystal X-ray diffraction (SC-XRD). If the quality of the single crystals is not sufficient, the structure can be inferred from powder diffraction patterns [18] which often remains incomplete because of the complicated molecular system, large unit cell, and low symmetry. Molecular modeling methods must thus complement these experiments [19].

The periodic structure of an MOF can be studied by DFT: properties such as cell parameters and elastic properties can be obtained from periodic calculations. However, such calculations consume large amounts of computational resources due to the typically large unit cell of MOFs. Simple functionals such as local-density approximation (LDA) or generalized gradient approximations (GGA) are, however, often sufficient in this case [20]. Studying catalytic effects requires more advanced "hybrid" functionals, e.g., BP86 [21], B3-LYP [22], or M06 series [23].

Molecular mechanics (MM) is a much cheaper computational method to study large systems. The reliability of these calculations depends on the quality of the force fields used to describe the interactions between the atoms in the investigated system. Some high-accuracy force fields for MOFs, such as BTW-FF [24], MOF-FF [25], Quick-FF [26], and others [27, 28]**,** were parametrized from DFT calculations**.** They were successfully used to determine and predict the periodic structures [29, 30], as well as to examine phase transformations [31–33], guest diffusion in the pores and other mechanical properties such as bulk moduli [34], elastic constants [35] and mechanical stability [36]. Note that MM calculations cannot describe electronic structures. Bond breaking and forming cannot be accounted for in such calculations. Thus, it is not possible to study reaction mechanisms using conventional MM methods [37].

In summary, for structurally well-defined and small enough systems, DFT calculations are a cogent approach especially in the field of localized catalytic reactions. We aim here to outline the suitability of DFT techniques to study catalysis in MOFs materials through examples from the recent literature.

## **3. Catalytic MOFs**

and competitive tool to describe chemical phenomena [2]. This field is broadly known as "modeling." In "molecular modeling," the premise is that the interatomic and intermolecular interactions can be known [3]. DFT has become the dominant tool for the purpose of determining such interactions [4]. In particular, in solid-state chemistry and physics, it has become increasingly popular since appropriate functionals and basis sets [5] were designed to provide a viable balance between the reliability of the numerical results and the computational costs. Metal-organic frameworks (MOFs), also known as porous coordination polymers (PCPs) [6–8], are a class of hybrid organic-inorganic materials with high surface area, a permanent porosity with large internal pore volume and tunable pore sizes. They have a wide range of potential applications such as gas storage [9] and separation [10], catalysis [11], and many others [12–14]. MOFs are constructed by interlinking metal ions or, more generally, metalcontaining units with organic moieties (carboxylates, azolates, imidazolates, pyridyl, etc.) through coordination bonds, thus creating crystalline frameworks. Due to the variety of the structural and chemical elements, some MOFs show unexpected characteristics, which sometimes cannot be fully assessed experimentally. Theoretical approaches have been intensively employed in this particular case to investigate the systems at the atomic level. Beyond the analysis of observed phenomena, the prediction of unknown attributes has been attempted. In this book-chapter, we review and discuss the current status and challenges of DFT studies of MOFs. We focus in particular on aspects relevant to catalysis at the metal centers found in

One of the main issues of DFT calculations is that a compromise between the dependability of the results and computational costs must be found, especially for large and complex systems [15]. A primary issue is how to select a functional suitable for the investigated system [16]**.** Different functionals, based on different approximations, may give different (even opposite) results. It is thus important to choose the functionals wisely. No functional devised up to now has proven to

MOFs are crystalline materials with extended frameworks. This needs, in some instances, to be taken into account when, for catalytic applications of MOFs, DFT is used to investigate the reaction mechanism. The reaction enthalpy can be obtained approximately from the energy at 0 K by including a zero-point energy correction calculated from (possibly scaled) vibrational frequencies. To compute the entropy, harmonic or anharmonic low-frequency modes can be used. The temperature-dependent free and activation energies can be calculated by adding thermal enthalpic and entropic effects. The so-obtained free energies of reaction are used to calculate temperaturedependent equilibrium constants. Note that the kinetics of a reaction can also be investigated, e.g.,

Since the properties of MOFs depend not only on their chemical composition but also their structures, atomistic structural characterization is essentials. Generally, the structure of a newly synthesized MOF is characterized by single-crystal X-ray diffraction (SC-XRD). If the

be suitable for all systems. Therefore, the validation of any selected functional is crucial**.**

by transition state theory (TST) [17]. However, this is beyond the scope of this chapter.

these structures.

56 Density Functional Theory

**2. DFT calculations on MOFs**

MOFs have well-defined crystal structures with high concentrations of metal centers organized at topical distances, and large pore volumes between these centers. Even though MOFs are less able to withstand high temperatures compared to some other catalytic porous materials, their undeniable qualities such as large internal surface areas and uniform pore and cavity sizes make them attractive for various applications [38]. The pore size/shape in the framework can be tuned for selectivity for a particular reaction. In contrast, the small pore windows of other nanoporous materials, such as zeolites, commonly limit the catalytic transformation of large molecules [39].

In principle, various types of active sites can be incorporated into MOFs. Coordinatively unsaturated metal sites (CUSs) and functional groups on the organic linkers (usually acid/ base sites) are the main catalytic sites. In addition, even though the MOF framework itself does not contain active sites, the catalytic reactivity can be enhanced by (a) postsynthetic functionalized modification [40] and (b) encapsulation of catalytic species [41]. The pore volume of MOFs is able to accommodate organic molecules, inorganic nanoparticles, metal complexes, and enzymes to conduct catalytic reactions. Here, we focus only catalytic reactions taking place on one CUS.

It is because of these functionalities [39] that MOFs were originally proposed for applications in catalysis. From this point of view, one tends to focus on the design and synthesis of MOFs with large pore sizes. The purpose is to allow the reactant to diffuse easily into the framework reaching the catalytic centers. Suitable synthetic techniques make it possible to choose the linkers and the metal-containing nodes to construct MOF suitable for particular reaction. Such a welldesigned MOF catalyst should be highly reactive, selective, and stable. In order to approach this goal, some understanding of the reaction mechanisms is essential. In many cases, the reactants and resulting products tend to be well localized, for a sufficiently long time, in the framework due to host-guest interaction. The catalytic site can thus often be separated from the framework and investigated independently, making it easier to follow the reaction mechanisms [42].

We focus in this brief overview, mostly through a series of examples, on such local analyses of the activity of single-site catalysts. Selected examples of work based on cluster, or truncated, models of reactions catalyzed by metal centers, metal substitutions, and deposited metal complexes are succinctly reviewed.

#### **4. Catalytically reactive metal nodes**

As mentioned above, MOFs are built up from metal ions or metal-containing clusters as the inorganic building node and organic ligands as linkers. In some MOFs, coordinatively unsaturated metal sites (CUSs) have a catalytic effect. The sites are spatially well separated and, ideally, structurally identical. Thus, to investigate the reaction mechanism with DFT methods, truncated structures can often be used to represent the entire MOFs.

larger charge transfer from the CUS to the oxidant O<sup>2</sup>

Adapted from Ref. [44] with permission of Wiley, copyright 2016.

(H<sup>2</sup> BTC)<sup>4</sup>

promote Knoevenagel condensations and Michael additions.

Llabrés i Xamena et al. [50] demonstrated that Cu(2-pymo)<sup>2</sup>

(**Figure 2**) optimized at the PBE-D2 level. The catalytic activities of M<sup>2</sup>

The M-MOF-74 series (M<sup>2</sup>

**Figure 1.** (a) Truncated model of [Fe<sup>2</sup>

propylene oxide is favored over that of carbonylic products (propanal and acetone). Propanal and acetone were formed on the Fe-MOF cluster via the formation of a C▬O bond. Then, the

involved in the formation of ethylene oxide (solid line) and the acetaldehyde (dash line) over the Fe paddle wheel.

Ni, Co, Cu, and Zn) has been proposed as good catalysts for several reactions, owing to their reactive CUSs and large 1D channels, beneficial for the reactants' access to the active site. They are also thermally and chemically very stable [46–48]**.** Valvekens et al. [49] used MOF-74 with various metal ions, i.e., Mg (II), Ni (II), Co (II), Cu (II), and Zn (II) as Lewis acid catalysts to

DFT calculations were performed on truncated models cut from the periodic geometries

respect to the Knoevenagel condensation and Michael additions were examined. The calculations at the B3LYP level with a 6–31 g(d) basis showed that Ni-MOF-74 is the most active catalyst for both reactions. In addition, it was found that the phenolate groups coordinated with the CUSs substantially increase the catalytic performance. The phenolate oxygen proved to be a stronger base than the carboxylate oxygen, resulting in more acidic CUSs, enhancing the catalytic activity.

are 2-hydroxypyrimidinolat and phenylimidazolate, respectively) promote the aerobic oxidation reaction converting tetralin hydrocarbon to ketone and alcohol derivatives. The tetralin was first oxidized to hydroperoxides under oxygen condition and then decomposed to ketone and alcohol under the influence of the Lewis acid sites Cu(II) and Co(II). Ryan et al. [51] studied the mechanism of the hydroperoxide decomposition over three different complexes

(DOBDC), where DOBDC = 2,5-dioxidoterephthalate and M = Mg,

] and (b) energy profile of reactants, intermediates, and transition states

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

and Co(PhIM)<sup>2</sup>

propyleneoxy intermediates and acetone are formed via a 1,2-hydride shift.

. It was found that the production of

http://dx.doi.org/10.5772/intechopen.80698

59

(DOBDC) systems with

(2-pymo and PhIM

#### **4.1. Reactivity of coordinatively unsaturated metal sites**

MOFs with CUSs provide identical active metal sites, which are spatially isolated from each another. The sites are structurally well-characterized, without coupling to their neighbors. In typical dense heterogeneous catalyst, the reaction takes place on the outer surface [43]. One way to enhance the catalytic effect is to increase the accessible surface area. In MOFs, reactions occur not only on the outer surface, easily accessible active sites inside the framework also contribute.

Maihom et al. investigated the epoxidations of ethylene over Fe<sup>3</sup> (BTC)<sup>2</sup> (BTC = 1,3,5-benzenetricarboxylate) using N<sup>2</sup> O as oxidant [44]. **A** truncated model [Fe<sup>2</sup> (H<sup>2</sup> BTC)<sup>4</sup> ] (**Figure 1a**) was used. The reaction mechanism started with the N<sup>2</sup> O decomposition over Fe to generate an active Fe-oxo moiety, requiring an activation energy of 23.7 kcal/mol. An ethylenoxy intermediate, and eventually, ethylene oxide, were obtained as final products rather than acetaldehyde (**Figure 1b**).

An investigation of the epoxidation of propylene using Cu<sup>3</sup> (BTC)<sup>2</sup> and Fe<sup>3</sup> (BTC)<sup>2</sup> was recently carried out by the same group [45]. As mentioned, truncated clusters, [Cu<sup>2</sup> (H<sup>2</sup> BTC)<sup>4</sup> ] and [Fe<sup>2</sup> (H<sup>2</sup> BTC)<sup>4</sup> ], are suitable for these calculations, carried out at the M06-L/6-31G(d,p) level. As expected, the calculations showed that Fe<sup>3</sup> (BTC)<sup>2</sup> is more active than Cu<sup>3</sup> (BTC)<sup>2</sup> due to a

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks http://dx.doi.org/10.5772/intechopen.80698 59

and enzymes to conduct catalytic reactions. Here, we focus only catalytic reactions taking

It is because of these functionalities [39] that MOFs were originally proposed for applications in catalysis. From this point of view, one tends to focus on the design and synthesis of MOFs with large pore sizes. The purpose is to allow the reactant to diffuse easily into the framework reaching the catalytic centers. Suitable synthetic techniques make it possible to choose the linkers and the metal-containing nodes to construct MOF suitable for particular reaction. Such a welldesigned MOF catalyst should be highly reactive, selective, and stable. In order to approach this goal, some understanding of the reaction mechanisms is essential. In many cases, the reactants and resulting products tend to be well localized, for a sufficiently long time, in the framework due to host-guest interaction. The catalytic site can thus often be separated from the framework and investigated independently, making it easier to follow the reaction mechanisms [42].

We focus in this brief overview, mostly through a series of examples, on such local analyses of the activity of single-site catalysts. Selected examples of work based on cluster, or truncated, models of reactions catalyzed by metal centers, metal substitutions, and deposited metal com-

As mentioned above, MOFs are built up from metal ions or metal-containing clusters as the inorganic building node and organic ligands as linkers. In some MOFs, coordinatively unsaturated metal sites (CUSs) have a catalytic effect. The sites are spatially well separated and, ideally, structurally identical. Thus, to investigate the reaction mechanism with DFT methods,

MOFs with CUSs provide identical active metal sites, which are spatially isolated from each another. The sites are structurally well-characterized, without coupling to their neighbors. In typical dense heterogeneous catalyst, the reaction takes place on the outer surface [43]. One way to enhance the catalytic effect is to increase the accessible surface area. In MOFs, reactions occur not only on the outer surface, easily accessible active sites inside the framework also contribute.

O as oxidant [44]. **A** truncated model [Fe<sup>2</sup>

Fe-oxo moiety, requiring an activation energy of 23.7 kcal/mol. An ethylenoxy intermediate, and eventually, ethylene oxide, were obtained as final products rather than acetaldehyde (**Figure 1b**).

], are suitable for these calculations, carried out at the M06-L/6-31G(d,p) level.

(BTC)<sup>2</sup>

carried out by the same group [45]. As mentioned, truncated clusters, [Cu<sup>2</sup>

(BTC)<sup>2</sup>

(H<sup>2</sup> BTC)<sup>4</sup>

(BTC)<sup>2</sup>

is more active than Cu<sup>3</sup>

O decomposition over Fe to generate an active

and Fe<sup>3</sup>

(BTC)<sup>2</sup>

(H<sup>2</sup> BTC)<sup>4</sup>

(BTC)<sup>2</sup>

(BTC = 1,3,5-benzen-

] (**Figure 1a**) was

was recently

] and

due to a

truncated structures can often be used to represent the entire MOFs.

Maihom et al. investigated the epoxidations of ethylene over Fe<sup>3</sup>

An investigation of the epoxidation of propylene using Cu<sup>3</sup>

used. The reaction mechanism started with the N<sup>2</sup>

As expected, the calculations showed that Fe<sup>3</sup>

**4.1. Reactivity of coordinatively unsaturated metal sites**

place on one CUS.

58 Density Functional Theory

plexes are succinctly reviewed.

etricarboxylate) using N<sup>2</sup>

[Fe<sup>2</sup> (H<sup>2</sup> BTC)<sup>4</sup>

**4. Catalytically reactive metal nodes**

**Figure 1.** (a) Truncated model of [Fe<sup>2</sup> (H<sup>2</sup> BTC)<sup>4</sup> ] and (b) energy profile of reactants, intermediates, and transition states involved in the formation of ethylene oxide (solid line) and the acetaldehyde (dash line) over the Fe paddle wheel. Adapted from Ref. [44] with permission of Wiley, copyright 2016.

larger charge transfer from the CUS to the oxidant O<sup>2</sup> . It was found that the production of propylene oxide is favored over that of carbonylic products (propanal and acetone). Propanal and acetone were formed on the Fe-MOF cluster via the formation of a C▬O bond. Then, the propyleneoxy intermediates and acetone are formed via a 1,2-hydride shift.

The M-MOF-74 series (M<sup>2</sup> (DOBDC), where DOBDC = 2,5-dioxidoterephthalate and M = Mg, Ni, Co, Cu, and Zn) has been proposed as good catalysts for several reactions, owing to their reactive CUSs and large 1D channels, beneficial for the reactants' access to the active site. They are also thermally and chemically very stable [46–48]**.** Valvekens et al. [49] used MOF-74 with various metal ions, i.e., Mg (II), Ni (II), Co (II), Cu (II), and Zn (II) as Lewis acid catalysts to promote Knoevenagel condensations and Michael additions.

DFT calculations were performed on truncated models cut from the periodic geometries (**Figure 2**) optimized at the PBE-D2 level. The catalytic activities of M<sup>2</sup> (DOBDC) systems with respect to the Knoevenagel condensation and Michael additions were examined. The calculations at the B3LYP level with a 6–31 g(d) basis showed that Ni-MOF-74 is the most active catalyst for both reactions. In addition, it was found that the phenolate groups coordinated with the CUSs substantially increase the catalytic performance. The phenolate oxygen proved to be a stronger base than the carboxylate oxygen, resulting in more acidic CUSs, enhancing the catalytic activity.

Llabrés i Xamena et al. [50] demonstrated that Cu(2-pymo)<sup>2</sup> and Co(PhIM)<sup>2</sup> (2-pymo and PhIM are 2-hydroxypyrimidinolat and phenylimidazolate, respectively) promote the aerobic oxidation reaction converting tetralin hydrocarbon to ketone and alcohol derivatives. The tetralin was first oxidized to hydroperoxides under oxygen condition and then decomposed to ketone and alcohol under the influence of the Lewis acid sites Cu(II) and Co(II). Ryan et al. [51] studied the mechanism of the hydroperoxide decomposition over three different complexes

**Figure 2.** Periodic structure of M<sup>2</sup> (DOBDC) optimized at the PBE-D2 level of theory (left), and the cluster model M9 (DOBDC)<sup>9</sup> , truncated from the periodic structure (right). Adapted from Ref. [49] with permission of Elsevier, copyright 2014.

possessing copper or cobalt CUSs, i.e., Co(imidazolate)<sup>4</sup> , Cu<sup>2</sup> (2-hydroxypyrimidinolat)<sup>4</sup> , and Cu<sup>2</sup> (acetate)<sup>4</sup> , as illustrated in **Figure 3**.

From DFT calculations (B3LYP/TZVP/6-31G(d,p)), the access of hydroperoxide to the CUSs of Cu<sup>2</sup> (2-hydroxypyrimidinolat)<sup>4</sup> and of Co(imidazolate)<sup>4</sup> was found to be blocked by this bulky ligand. As a consequence, no decomposition of hydroperoxide was observed for the two systems. On the other hand, for Cu<sup>2</sup> (acetate)<sup>4</sup> , i.e., without the bulky ligands, the decomposition should occur. However, the calculations revealed that Cu<sup>2</sup> (acetate)<sup>4</sup> was not active for the decomposition of tetralin. The energy barrier of the O▬O bond cleavage over Cu<sup>2</sup> (acetate)<sup>4</sup> (35.6–36.8 kcal/mol) was almost identical with that of the cleavage in the gas phase without catalyst (36.0–36.7 kcal/mol). The external surface was then considered as the active sites.

by terephthalate linkers to form a 3D framework possessing 1D rhombic channels**.** Each V(IV)-center in this chain coordinates to six oxygen atoms—four from carboxylate and two

with one water molecule (blue) and oxygen▬oxygen bond cleavage without catalyst (green). All energies are in kcal/mol.

structural defects obtained by partially breaking the V▬OCOO bonds and from the hydrolysis

The main products of the cyclohexene oxidation reaction are cyclohexene oxide, cyclohexane-1,2-diol, tert-butyl-2-cyclohexenyl-1-peroxide, and 2-cyclohexen-1-one with a conversion of about 80%. It was found that the catalytic performance of MIL-47(V) was comparable with

when the oxidant was dissolved in the solution. The structures of the MIL-47(V) were maintained intact until the end of reaction. Leus and Vandichel et al. [52] proposed a reaction

The results were in good agreement with EPR and NMR measurement that are interpreted in terms of the existence of (at least) two different catalytic pathways, described as the "direct" and "radical" pathways [52]. The first step of both pathways was the formation of a vanadium hydroperoxide species from TBHP. For the direct pathway, cyclohexene was directly converted to cyclohexene oxide, while the oxidation state of vanadium (V(IV)) did not change during the reaction. In the case of the radical pathway, V(IV) was oxidized to V(V), which further reacted with the oxidant TBHP to give an active species of the cyclohexene epoxidation. Finally, the catalyst was regenerated by interacting with TBHP again. At the B3LYP-D3 and 311 + g(3df,2p) level of theory, the reaction free energy for epoxidation reaction was 37 kJ/mol

Moreover, Matthias et al. [53] studied the cyclohexene oxidation using MIL-47(V)-function-

of experimental and theoretical works. Different catalytic conversions were observed depending on the functionalization of the linkers. DFT calculations on truncated clusters of MIL-47(V) and its derivatives were performed as in the abovementioned example for the radical

pathway from DFT calculations on finite clusters of MIL-47(V) (**Figure 5**).

Gibbs free energy along the reaction coordinate for the proposed reaction cycle over the Cu<sup>2</sup>

alized linkers (with the functional groups ▬OH, ▬F, ▬Cl, ▬Br, ▬CH<sup>3</sup>

O. Typically, MIL-47(V) does not provide CUSs. However, CUSs can be created from

(2-hydroxypyrimidinolat)<sup>3</sup>

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

http://dx.doi.org/10.5772/intechopen.80698

61

**.** Note that no leaching of V(IV) was observed

with one water molecule, (b)

(2-hydroxypyrimidinolat)<sup>3</sup>

, and ▬NH<sup>2</sup>

) by means

from μ<sup>2</sup>

of the V▬OCOO bonds.

that of typical homogeneous catalyst, VO(acac)<sup>2</sup>

**Figure 4.** (a) Model of a cluster representing the exterior of a Cu<sup>2</sup>

Figure adapted from Ref. [51] with permission of Elsevier, copyright 2012.

for the more favorable radical route.

A complex consisting of Cu coordinated by three organic linkers and one water molecule was modeled representing the edge or outer surface of the framework (**Figure 4a**). The energy barrier of the hydroperoxide decomposition on the new model complex decreased to 25.3 kcal/mol, which was considerably lower than the value of the gas-phase reaction without MOFs as shown in **Figure 4b**.

Vanadium-based MIL-47(V), [VO(BDC)] (BDC = benzene-1,4-dicarboxylate), is an active catalyst for the liquid-phase cyclohexene oxidation-reaction using tert-butyl hydroperoxide (TBHP) as an oxidant [52]. MIL-47(V) is built up from linear V-(μ<sup>2</sup> -O)-V chains interconnected

**Figure 3.** The metal cluster models taken from (a) Co(imidazolate)<sup>4</sup> , (b) Cu<sup>2</sup> (2-hydroxypyrimidinolat)<sup>4</sup> , and (c) Cu<sup>2</sup> (acetate)<sup>4</sup> .

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks http://dx.doi.org/10.5772/intechopen.80698 61

**Figure 4.** (a) Model of a cluster representing the exterior of a Cu<sup>2</sup> (2-hydroxypyrimidinolat)<sup>3</sup> with one water molecule, (b) Gibbs free energy along the reaction coordinate for the proposed reaction cycle over the Cu<sup>2</sup> (2-hydroxypyrimidinolat)<sup>3</sup> with one water molecule (blue) and oxygen▬oxygen bond cleavage without catalyst (green). All energies are in kcal/mol. Figure adapted from Ref. [51] with permission of Elsevier, copyright 2012.

possessing copper or cobalt CUSs, i.e., Co(imidazolate)<sup>4</sup>

should occur. However, the calculations revealed that Cu<sup>2</sup>

(TBHP) as an oxidant [52]. MIL-47(V) is built up from linear V-(μ<sup>2</sup>

**Figure 3.** The metal cluster models taken from (a) Co(imidazolate)<sup>4</sup>

, as illustrated in **Figure 3**.

Cu<sup>2</sup>

M9

Cu<sup>2</sup>

Cu<sup>2</sup>

(acetate)<sup>4</sup> .

(acetate)<sup>4</sup>

(DOBDC)<sup>9</sup>

60 Density Functional Theory

copyright 2014.

shown in **Figure 4b**.

(2-hydroxypyrimidinolat)<sup>4</sup>

**Figure 2.** Periodic structure of M<sup>2</sup>

tems. On the other hand, for Cu<sup>2</sup>

, Cu<sup>2</sup>

(DOBDC) optimized at the PBE-D2 level of theory (left), and the cluster model

From DFT calculations (B3LYP/TZVP/6-31G(d,p)), the access of hydroperoxide to the CUSs of

, truncated from the periodic structure (right). Adapted from Ref. [49] with permission of Elsevier,

ligand. As a consequence, no decomposition of hydroperoxide was observed for the two sys-

(35.6–36.8 kcal/mol) was almost identical with that of the cleavage in the gas phase without catalyst (36.0–36.7 kcal/mol). The external surface was then considered as the active sites.

A complex consisting of Cu coordinated by three organic linkers and one water molecule was modeled representing the edge or outer surface of the framework (**Figure 4a**). The energy barrier of the hydroperoxide decomposition on the new model complex decreased to 25.3 kcal/mol, which was considerably lower than the value of the gas-phase reaction without MOFs as

Vanadium-based MIL-47(V), [VO(BDC)] (BDC = benzene-1,4-dicarboxylate), is an active catalyst for the liquid-phase cyclohexene oxidation-reaction using tert-butyl hydroperoxide

and of Co(imidazolate)<sup>4</sup>

decomposition of tetralin. The energy barrier of the O▬O bond cleavage over Cu<sup>2</sup>

(acetate)<sup>4</sup>

(2-hydroxypyrimidinolat)<sup>4</sup>

was found to be blocked by this bulky

was not active for the


(2-hydroxypyrimidinolat)<sup>4</sup>

, and (c)

, i.e., without the bulky ligands, the decomposition

(acetate)<sup>4</sup>

, (b) Cu<sup>2</sup>

, and

(acetate)<sup>4</sup>

by terephthalate linkers to form a 3D framework possessing 1D rhombic channels**.** Each V(IV)-center in this chain coordinates to six oxygen atoms—four from carboxylate and two from μ<sup>2</sup> O. Typically, MIL-47(V) does not provide CUSs. However, CUSs can be created from structural defects obtained by partially breaking the V▬OCOO bonds and from the hydrolysis of the V▬OCOO bonds.

The main products of the cyclohexene oxidation reaction are cyclohexene oxide, cyclohexane-1,2-diol, tert-butyl-2-cyclohexenyl-1-peroxide, and 2-cyclohexen-1-one with a conversion of about 80%. It was found that the catalytic performance of MIL-47(V) was comparable with that of typical homogeneous catalyst, VO(acac)<sup>2</sup> **.** Note that no leaching of V(IV) was observed when the oxidant was dissolved in the solution. The structures of the MIL-47(V) were maintained intact until the end of reaction. Leus and Vandichel et al. [52] proposed a reaction pathway from DFT calculations on finite clusters of MIL-47(V) (**Figure 5**).

The results were in good agreement with EPR and NMR measurement that are interpreted in terms of the existence of (at least) two different catalytic pathways, described as the "direct" and "radical" pathways [52]. The first step of both pathways was the formation of a vanadium hydroperoxide species from TBHP. For the direct pathway, cyclohexene was directly converted to cyclohexene oxide, while the oxidation state of vanadium (V(IV)) did not change during the reaction. In the case of the radical pathway, V(IV) was oxidized to V(V), which further reacted with the oxidant TBHP to give an active species of the cyclohexene epoxidation. Finally, the catalyst was regenerated by interacting with TBHP again. At the B3LYP-D3 and 311 + g(3df,2p) level of theory, the reaction free energy for epoxidation reaction was 37 kJ/mol for the more favorable radical route.

Moreover, Matthias et al. [53] studied the cyclohexene oxidation using MIL-47(V)-functionalized linkers (with the functional groups ▬OH, ▬F, ▬Cl, ▬Br, ▬CH<sup>3</sup> , and ▬NH<sup>2</sup> ) by means of experimental and theoretical works. Different catalytic conversions were observed depending on the functionalization of the linkers. DFT calculations on truncated clusters of MIL-47(V) and its derivatives were performed as in the abovementioned example for the radical

this magnesium-diluted MOFs using N<sup>2</sup>

not change during the reaction.

with different linkers, in particular, ▬CH<sup>3</sup>

consecutive reaction (**Figure 6**): the formation of the Fe(IV)-oxo unit via N<sup>2</sup>

species is indeed an active center for the oxidation of ethane to ethanol.

and the hydroxylation of ethane to ethanol over the Fe(IV)-oxo complex. It is found that the first step is the rate-determining step with an activation energy of 82 kJ/mol. Compared to the uncatalyzed reaction, where nitrous oxide directly oxidizes the ethane to ethanol, it was found that the reaction requires an activation barrier of 280 kJ/mol. This suggests that the Fe(IV)-oxo

In the meantime, Hirrao et al. [57] also carried out combined quantum and molecular mechanics (QM/MM) calculations for the hydroxylation of ethane, in analogy to Mg-MOF-74. This methodology treats the active site by QM and the rest of the system is treated by MM. The accuracy of QM/MM methods certainly depends on the choice of the QM theory level and the set of MM parameters, also on the treatment of electrostatics at the interface between the QM and MM regions. One of three Mg(II) ions was substituted by Fe(II) in the QM region, as in the model suggested by Verma et al. [56]. In the ONIOM [58] scheme, the MM region was treated at the B3LYP/[SDD(Fe),6-31G(d)(others)] level while the UFF interaction model [59] was used for the MM region. The 6–311 + G(df,p) basis set was used for single-point reference calculations. This work also suggested that the spin state (S = 2) of the Fe(IV)-oxo species does

Interestingly, Liao et al. [60] then improved based on Verma et al. model [56] the understanding on the activity of Fe open sites for the oxidation of ethane reaction. They studied MOF-74

**Figure 6.** Enthalpy profile, ΔH298.15 (in kJ/mol), for the intermediates and transition states of the catalytic cycle. The key bond distances are given in Å (color code: orange = Fe, red = O, blue = N, gray = C, and white = H). Reproduced from Ref.

[56] with permission of the American Chemical Society, copyright 2015.

, NH<sup>2</sup>

O as oxidant was proposed to proceed via a two-step

http://dx.doi.org/10.5772/intechopen.80698

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

, ▬COOH, ▬CN, ▬OH, ▬OCH<sup>3</sup>

O decomposition

63

, ▬N(CH<sup>3</sup>

)2 ,

**Figure 5.** Simulated model of V-MIL-47.

pathway. The results showed that, compared to the parent MIL-47(V), the catalytic efficiency significantly increased due to the functional groups. Among these, the ▬OH-functionalized structure showed the largest improvement (lowest activation energy), which was in good agreement with the experimental results. This is because of a strong hydrogen bond between the ▬OH and the alkylperoxo group possibly stabilizing the transition state.

#### **4.2. Reactivity of substituted metal centers as single-site catalysts**

Partial substitution of metal ions while maintaining the structural integrity and the porosity is a promising strategy enabling some otherwise inactive MOFs to show a catalytic activity [54]. The metal substitution usually involves the cleavage and regeneration of coordination bonds between metal ions and organic ligands. The atomic-level understanding of this process is difficult, making room for molecular modeling. Here, we show examples of theoretical studies on the catalytic activity of metal-substituted MOFs.

Xiao et al. [55] reported the oxidation reaction of ethane to ethanol over magnesium-diluted Fe0.1Mg1.9(DOBDC), MOF-74, in which 5% of the redox-inactive Mg(II) were substituted by the active Fe(II). This framework provides wide hexagonal channels beneficial for an easy access of the reactants to the site-isolated open Fe-CUSs. Nitrous oxide (N<sup>2</sup> O) was used as the oxidant, generating highly reactive Fe(IV)-oxo intermediates, which could further activate strong C▬H bonds of alkanes, yielding ethanol and acetaldehyde. Owing to the short lifetime of Fe(IV)-oxo, it could neither be isolated nor characterized by conventional experiments.

Shortly afterward, Verma el al. [56] applied DFT calculations to confirm whether the Fe(IV) oxo species are formed and act as active centers for ethane hydroxylation to ethanol. They modeled the Fe sites in magnesium-diluted Fe<sup>2</sup> (DOBDC) by replacing 1 Mg atom with Fe atom in cluster models comprising 88 atoms of Mg-MOF-74. The oxidation of ethane to ethanol over this magnesium-diluted MOFs using N<sup>2</sup> O as oxidant was proposed to proceed via a two-step consecutive reaction (**Figure 6**): the formation of the Fe(IV)-oxo unit via N<sup>2</sup> O decomposition and the hydroxylation of ethane to ethanol over the Fe(IV)-oxo complex. It is found that the first step is the rate-determining step with an activation energy of 82 kJ/mol. Compared to the uncatalyzed reaction, where nitrous oxide directly oxidizes the ethane to ethanol, it was found that the reaction requires an activation barrier of 280 kJ/mol. This suggests that the Fe(IV)-oxo species is indeed an active center for the oxidation of ethane to ethanol.

In the meantime, Hirrao et al. [57] also carried out combined quantum and molecular mechanics (QM/MM) calculations for the hydroxylation of ethane, in analogy to Mg-MOF-74. This methodology treats the active site by QM and the rest of the system is treated by MM. The accuracy of QM/MM methods certainly depends on the choice of the QM theory level and the set of MM parameters, also on the treatment of electrostatics at the interface between the QM and MM regions. One of three Mg(II) ions was substituted by Fe(II) in the QM region, as in the model suggested by Verma et al. [56]. In the ONIOM [58] scheme, the MM region was treated at the B3LYP/[SDD(Fe),6-31G(d)(others)] level while the UFF interaction model [59] was used for the MM region. The 6–311 + G(df,p) basis set was used for single-point reference calculations. This work also suggested that the spin state (S = 2) of the Fe(IV)-oxo species does not change during the reaction.

Interestingly, Liao et al. [60] then improved based on Verma et al. model [56] the understanding on the activity of Fe open sites for the oxidation of ethane reaction. They studied MOF-74 with different linkers, in particular, ▬CH<sup>3</sup> , NH<sup>2</sup> , ▬COOH, ▬CN, ▬OH, ▬OCH<sup>3</sup> , ▬N(CH<sup>3</sup> )2 ,

pathway. The results showed that, compared to the parent MIL-47(V), the catalytic efficiency significantly increased due to the functional groups. Among these, the ▬OH-functionalized structure showed the largest improvement (lowest activation energy), which was in good agreement with the experimental results. This is because of a strong hydrogen bond between

Partial substitution of metal ions while maintaining the structural integrity and the porosity is a promising strategy enabling some otherwise inactive MOFs to show a catalytic activity [54]. The metal substitution usually involves the cleavage and regeneration of coordination bonds between metal ions and organic ligands. The atomic-level understanding of this process is difficult, making room for molecular modeling. Here, we show examples of theoretical studies

Xiao et al. [55] reported the oxidation reaction of ethane to ethanol over magnesium-diluted Fe0.1Mg1.9(DOBDC), MOF-74, in which 5% of the redox-inactive Mg(II) were substituted by the active Fe(II). This framework provides wide hexagonal channels beneficial for an easy

oxidant, generating highly reactive Fe(IV)-oxo intermediates, which could further activate strong C▬H bonds of alkanes, yielding ethanol and acetaldehyde. Owing to the short lifetime of Fe(IV)-oxo, it could neither be isolated nor characterized by conventional experiments.

Shortly afterward, Verma el al. [56] applied DFT calculations to confirm whether the Fe(IV) oxo species are formed and act as active centers for ethane hydroxylation to ethanol. They

in cluster models comprising 88 atoms of Mg-MOF-74. The oxidation of ethane to ethanol over

O) was used as the

(DOBDC) by replacing 1 Mg atom with Fe atom

the ▬OH and the alkylperoxo group possibly stabilizing the transition state.

access of the reactants to the site-isolated open Fe-CUSs. Nitrous oxide (N<sup>2</sup>

**4.2. Reactivity of substituted metal centers as single-site catalysts**

on the catalytic activity of metal-substituted MOFs.

**Figure 5.** Simulated model of V-MIL-47.

62 Density Functional Theory

modeled the Fe sites in magnesium-diluted Fe<sup>2</sup>

**Figure 6.** Enthalpy profile, ΔH298.15 (in kJ/mol), for the intermediates and transition states of the catalytic cycle. The key bond distances are given in Å (color code: orange = Fe, red = O, blue = N, gray = C, and white = H). Reproduced from Ref. [56] with permission of the American Chemical Society, copyright 2015.

and 4-pyridyl functionalized groups on different linkers such as using methanol, pyridine, formate, benzoate anion, and the imidazole anion. The authors concluded that the population of the d-orbital was significantly influenced by the coordination ligand field. The 3d orbital energy of Fe correlated with the electron-donating strength of the functional groups on the linkers. The results reveal that linkers with ▬NH<sup>2</sup> groups reduce the enthalpic barrier for the most endothermic step in the ethane oxidation pathway. This study illustrates the use of simple models to understand complicated and computationally intensive systems. The activity of CUSs thus might be optimized by selecting suitable ligand environments, which might be useful for upgrading certain hydrocarbon process.

**5. Reactivity of metal nodes deposited as single-site catalysts**

consist of Zr<sup>6</sup>

O)<sup>4</sup> (μ3 -OH)<sup>4</sup> O8

(OH)<sup>4</sup> (H<sup>2</sup> O)<sup>4</sup>

studied intensively [66, 67].

Zr-based MOFs such as MOF-808 [Zr<sup>6</sup>

and UiO-66, known as NU-1000 and [Zr<sup>6</sup>

energy of 28.5 and 26.5 kcal/mol, respectively.

orbital, leading to a lower catalytic activity.

grafted Ni and Co were terminated with active ▬OH and ▬OH<sup>2</sup>

(TBAPy)<sup>2</sup>

In this section, we discuss systems possessing metal-complexes/ions anchored on the inorganic building nodes of the framework, which can be achieved by postsynthetic techniques such as metalation [63], cation exchange [64], or atomic layer deposition [65]. Zr-based MOFs

as a primary node linked with carboxylate linkers. The Zr<sup>6</sup>

nected by 12 linkers result in the UiO-66 series, while the partial connection with 8 and 6 carboxylates produces NU-1000 and MOF-808, respectively. The catalytic reactivity can be enhanced through such modifications. This was often employed for Zr-based MOFs, which are remarkable because of their appealing chemical and thermal stability [40]. Well-known

Ortuño et al. [68] reported on a computational screening of the first row of divalent transition metals (i.e., Fe(II), Co(II), Ni(II), Cu(II), and Zn(II)) supported on NU-1000 (**Figure 8**) for acceptorless alcohol dehydrogenation. The author proposed the reaction taking place via a three-step reaction mechanism, composed of (i) a proton transfer, (ii) β-hydride elimination, and (iii) H▬H bond formation. The Fe(II), Co(II), and Ni(II) complexes, consistent with weakfield oxide ligands, had high-spin ground electronic states as quintet, quartet, and triplet, respectively. The Cu(II) and Zn(II) species were predicted to have doublet and singlet ground states, respectively. It was found that the Co(II) and Ni(II) supported NU-1000 were the two most promising catalysts for the acceptorless alcohol dehydrogenation with an activation free

Later, the same group also studied Ni(II) and Co(II) deposited on Zr-NU-1000 as a catalyst for ethylene dimerization, which converts ethylene to 1-butene and 2-butene [69]. For the structure optimization, the DFT level of theory (M06-L) and def2-TZVPP basis set were employed, whereas complete active space self-consistent field (CASSCF) and second-order perturbation theory (CASPT2) were used for the electronic structure characterization of the reactive species. The NU-1000 models were used as the same as in the previous example (**Figure 8**). The

The Cossee-Arlman mechanism was found as the energetically preferable pathway. Ethylene insertion into the existing metal-ethyl bond was the rate-determining step. Concerning the spin state of the two catalysts, the Co(II) species can have doublet or quartet states and the Ni(II) can have singlet or triplet states. Based on these calculations, Ni(II)-modified NU-1000 (with an activation energy of 15.6 kcal/mol) presents a greater reactivity than the Co(II) modified system (with an activation energy of 24.1 kcal/mol) for the ethylene insertion into the metal-carbon bond. The energetics of ethylene dimerization for the Ni(II) and Co(II) complexes are shown in **Figure 9**. From the most relevant CASSCF molecular orbitals (MOs), the 3*d* orbitals of the low-spin Ni(II) complex hybridized with the *2p* orbitals of the active carbon atom because of the empty fifth orbital. In contrast to the high-spin Co(II) complex, the degree of hybridization in the Ni-case is less able to stabilize the TS structure due to the half-filled 3*d*

(OH)<sup>6</sup> (H<sup>2</sup> O)<sup>6</sup>

(BDC)<sup>6</sup>

] (TBAPy = tetratopic 1,3,6,8-tetrakis(p-benzoate)pyrene)

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

(μ3 -O)<sup>4</sup> (μ3 -OH)<sup>4</sup>

(μ3 -O)<sup>4</sup> (μ3 -OH)<sup>4</sup> O8

http://dx.doi.org/10.5772/intechopen.80698

], NU-1000 [Zr<sup>6</sup>

], known as UiO-66 have been

(BTC)<sup>2</sup>

groups.

nodes con-

(μ3 - 65

The design of Fe(IV)-oxo complexes in MOFs was further reported by Impeng et al. [61]. In this work, we demonstrated the possibility of designing Fe-oxo complexes in MOFs for the activation of alkane C▬H bond by incorporating an Fe ion into a Zn-based cluster derived from Zn<sup>4</sup> O(BDC)<sup>6</sup> , known as MOF-5 [62], and generating the Fe-O unit through N<sup>2</sup> O dissociation on an Fe-substituted Zn-based cluster (Fe-Zn<sup>3</sup> O(pyrazole)<sup>6</sup> ). The calculations with B3LYP-D3 showed that both steps are feasible and that the catalytic activity of Fe-Zn<sup>3</sup> O(pyrazole)<sup>6</sup> for N<sup>2</sup> O decomposition is on a par with the Fe sites in magnesium-diluted Fe<sup>2</sup> (DOBDC). Concerning ethane C▬H bond activation, in addition to σ and π pathways on triplet and quintet surfaces, an alternative unusual pathway, called δ, is also observed on the triplet surface. The σ pathway on the quintet surface (TSσ 5 ) has the lowest activation energy owing to less steric hindrance and favorable d-d interactions on the Fe active site at the transition state, as illustrated in **Figure 7**.

**Figure 7.** Reaction profile for ethane C▬H bond activation over the FeO-Zn<sup>3</sup> O(pyrazole)<sup>6</sup> cluster. The inserted numbers are the relative energies ΔE, together with the enthalpies ΔH298K in parentheses. Energies are given in kcal/mol and distances are expressed in Å. The notations ADS, TS, and INT refer to the adsorption, transition, and intermediate steps, respectively. Superscripts 3 and 5 refer to the triplet and quintet spin states, respectively.

#### **5. Reactivity of metal nodes deposited as single-site catalysts**

and 4-pyridyl functionalized groups on different linkers such as using methanol, pyridine, formate, benzoate anion, and the imidazole anion. The authors concluded that the population of the d-orbital was significantly influenced by the coordination ligand field. The 3d orbital energy of Fe correlated with the electron-donating strength of the functional groups on the

the most endothermic step in the ethane oxidation pathway. This study illustrates the use of simple models to understand complicated and computationally intensive systems. The activity of CUSs thus might be optimized by selecting suitable ligand environments, which might

The design of Fe(IV)-oxo complexes in MOFs was further reported by Impeng et al. [61]. In this work, we demonstrated the possibility of designing Fe-oxo complexes in MOFs for the activation of alkane C▬H bond by incorporating an Fe ion into a Zn-based cluster derived from

, known as MOF-5 [62], and generating the Fe-O unit through N<sup>2</sup>

ethane C▬H bond activation, in addition to σ and π pathways on triplet and quintet surfaces, an alternative unusual pathway, called δ, is also observed on the triplet surface. The σ pathway

favorable d-d interactions on the Fe active site at the transition state, as illustrated in **Figure 7**.

are the relative energies ΔE, together with the enthalpies ΔH298K in parentheses. Energies are given in kcal/mol and distances are expressed in Å. The notations ADS, TS, and INT refer to the adsorption, transition, and intermediate steps,

showed that both steps are feasible and that the catalytic activity of Fe-Zn<sup>3</sup>

decomposition is on a par with the Fe sites in magnesium-diluted Fe<sup>2</sup>

O(pyrazole)<sup>6</sup>

) has the lowest activation energy owing to less steric hindrance and

O(pyrazole)<sup>6</sup>

groups reduce the enthalpic barrier for

). The calculations with B3LYP-D3

O(pyrazole)<sup>6</sup>

(DOBDC). Concerning

cluster. The inserted numbers

O dissociation

 for N<sup>2</sup> O

linkers. The results reveal that linkers with ▬NH<sup>2</sup>

be useful for upgrading certain hydrocarbon process.

on an Fe-substituted Zn-based cluster (Fe-Zn<sup>3</sup>

5

**Figure 7.** Reaction profile for ethane C▬H bond activation over the FeO-Zn<sup>3</sup>

respectively. Superscripts 3 and 5 refer to the triplet and quintet spin states, respectively.

Zn<sup>4</sup>

O(BDC)<sup>6</sup>

64 Density Functional Theory

on the quintet surface (TSσ

In this section, we discuss systems possessing metal-complexes/ions anchored on the inorganic building nodes of the framework, which can be achieved by postsynthetic techniques such as metalation [63], cation exchange [64], or atomic layer deposition [65]. Zr-based MOFs consist of Zr<sup>6</sup> O8 as a primary node linked with carboxylate linkers. The Zr<sup>6</sup> O8 nodes connected by 12 linkers result in the UiO-66 series, while the partial connection with 8 and 6 carboxylates produces NU-1000 and MOF-808, respectively. The catalytic reactivity can be enhanced through such modifications. This was often employed for Zr-based MOFs, which are remarkable because of their appealing chemical and thermal stability [40]. Well-known Zr-based MOFs such as MOF-808 [Zr<sup>6</sup> (μ3 -O)<sup>4</sup> (μ3 -OH)<sup>4</sup> (OH)<sup>6</sup> (H<sup>2</sup> O)<sup>6</sup> (BTC)<sup>2</sup> ], NU-1000 [Zr<sup>6</sup> (μ3 - O)<sup>4</sup> (μ3 -OH)<sup>4</sup> (OH)<sup>4</sup> (H<sup>2</sup> O)<sup>4</sup> (TBAPy)<sup>2</sup> ] (TBAPy = tetratopic 1,3,6,8-tetrakis(p-benzoate)pyrene) and UiO-66, known as NU-1000 and [Zr<sup>6</sup> (μ3 -O)<sup>4</sup> (μ3 -OH)<sup>4</sup> (BDC)<sup>6</sup> ], known as UiO-66 have been studied intensively [66, 67].

Ortuño et al. [68] reported on a computational screening of the first row of divalent transition metals (i.e., Fe(II), Co(II), Ni(II), Cu(II), and Zn(II)) supported on NU-1000 (**Figure 8**) for acceptorless alcohol dehydrogenation. The author proposed the reaction taking place via a three-step reaction mechanism, composed of (i) a proton transfer, (ii) β-hydride elimination, and (iii) H▬H bond formation. The Fe(II), Co(II), and Ni(II) complexes, consistent with weakfield oxide ligands, had high-spin ground electronic states as quintet, quartet, and triplet, respectively. The Cu(II) and Zn(II) species were predicted to have doublet and singlet ground states, respectively. It was found that the Co(II) and Ni(II) supported NU-1000 were the two most promising catalysts for the acceptorless alcohol dehydrogenation with an activation free energy of 28.5 and 26.5 kcal/mol, respectively.

Later, the same group also studied Ni(II) and Co(II) deposited on Zr-NU-1000 as a catalyst for ethylene dimerization, which converts ethylene to 1-butene and 2-butene [69]. For the structure optimization, the DFT level of theory (M06-L) and def2-TZVPP basis set were employed, whereas complete active space self-consistent field (CASSCF) and second-order perturbation theory (CASPT2) were used for the electronic structure characterization of the reactive species. The NU-1000 models were used as the same as in the previous example (**Figure 8**). The grafted Ni and Co were terminated with active ▬OH and ▬OH<sup>2</sup> groups.

The Cossee-Arlman mechanism was found as the energetically preferable pathway. Ethylene insertion into the existing metal-ethyl bond was the rate-determining step. Concerning the spin state of the two catalysts, the Co(II) species can have doublet or quartet states and the Ni(II) can have singlet or triplet states. Based on these calculations, Ni(II)-modified NU-1000 (with an activation energy of 15.6 kcal/mol) presents a greater reactivity than the Co(II) modified system (with an activation energy of 24.1 kcal/mol) for the ethylene insertion into the metal-carbon bond. The energetics of ethylene dimerization for the Ni(II) and Co(II) complexes are shown in **Figure 9**. From the most relevant CASSCF molecular orbitals (MOs), the 3*d* orbitals of the low-spin Ni(II) complex hybridized with the *2p* orbitals of the active carbon atom because of the empty fifth orbital. In contrast to the high-spin Co(II) complex, the degree of hybridization in the Ni-case is less able to stabilize the TS structure due to the half-filled 3*d* orbital, leading to a lower catalytic activity.

**Figure 8.** The "top" and "side" views of a 12-, 8-, and 6-connected Zr<sup>6</sup> oxide node. The arrows indicate the unsaturated 8- and 6-connected nodes, which have potential anchoring points. Reproduced from Ref. [66] with permission of Wiley, copyright 2016.

in line with experimental results. DFT cluster calculations allowed to study the reaction and showed the influence on the reaction of adding Co(II) cations as a secondary metal. The com-

**Figure 10.** Computed enthalpies, ΔH503K (kcal/mol), for propane oxidative dehydrogenation. Reproduced from Ref. [70]

In more detail, the reactive Co(III)▬O˙ moiety was generated by the catalyst regeneration

pane dehydrogenation, and forms isopropyl at the Co(III) active site. The abstraction of the terminal C-H bond by the Co(III)▬O˙ intermediate is the rate-determining step, yielding

**Figure 11.** Calculated enthalpy, ΔH298.15K (kcal/mol), for ethylene hydrogenation to ethane and dimerization to 1-butene

H4 )2

on NU-1000, UiO-67, or DAY zeolites. Reproduced

having a strong interaction with the propane molecules, promotes the pro-

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

http://dx.doi.org/10.5772/intechopen.80698

67

puted enthalpies for the equilibrium structures are shown in **Figure 10**.

with permission of the American Chemical Society, copyright 2017.

catalyzed by supported metal complexes initially present as Rh(C<sup>2</sup>

from Ref. [71] with permission of the American Chemical Society, copyright 2017.

reaction with O<sup>2</sup>

The oxidative dehydrogenation mechanism of propane on an active Co(II)-oxygen cluster anchored on the inorganic node of Zr-NU-1000 [70] was also explored by DFT with the M06-L functional and def2-TZVPP. A highly selective propane dehydrogenation at low temperature (230°C) has been found in experiments. Li et al. [70] found a computed propene formation

**Figure 9.** Reaction coordinate and calculated enthalpy ΔH298.15K for stationary points along the reaction coordinates for ethylene dimerization catalyzed by M(II)-NU-1000 (M = Co and Ni). Reproduced from Ref. [69] with permission of the American Chemical Society, copyright 2016.

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks http://dx.doi.org/10.5772/intechopen.80698 67

**Figure 10.** Computed enthalpies, ΔH503K (kcal/mol), for propane oxidative dehydrogenation. Reproduced from Ref. [70] with permission of the American Chemical Society, copyright 2017.

in line with experimental results. DFT cluster calculations allowed to study the reaction and showed the influence on the reaction of adding Co(II) cations as a secondary metal. The computed enthalpies for the equilibrium structures are shown in **Figure 10**.

In more detail, the reactive Co(III)▬O˙ moiety was generated by the catalyst regeneration reaction with O<sup>2</sup> having a strong interaction with the propane molecules, promotes the propane dehydrogenation, and forms isopropyl at the Co(III) active site. The abstraction of the terminal C-H bond by the Co(III)▬O˙ intermediate is the rate-determining step, yielding

**Figure 11.** Calculated enthalpy, ΔH298.15K (kcal/mol), for ethylene hydrogenation to ethane and dimerization to 1-butene catalyzed by supported metal complexes initially present as Rh(C<sup>2</sup> H4 )2 on NU-1000, UiO-67, or DAY zeolites. Reproduced from Ref. [71] with permission of the American Chemical Society, copyright 2017.

**Figure 9.** Reaction coordinate and calculated enthalpy ΔH298.15K for stationary points along the reaction coordinates for ethylene dimerization catalyzed by M(II)-NU-1000 (M = Co and Ni). Reproduced from Ref. [69] with permission of the

The oxidative dehydrogenation mechanism of propane on an active Co(II)-oxygen cluster anchored on the inorganic node of Zr-NU-1000 [70] was also explored by DFT with the M06-L functional and def2-TZVPP. A highly selective propane dehydrogenation at low temperature (230°C) has been found in experiments. Li et al. [70] found a computed propene formation

8- and 6-connected nodes, which have potential anchoring points. Reproduced from Ref. [66] with permission of Wiley,

oxide node. The arrows indicate the unsaturated

**Figure 8.** The "top" and "side" views of a 12-, 8-, and 6-connected Zr<sup>6</sup>

copyright 2016.

66 Density Functional Theory

American Chemical Society, copyright 2016.

Co(II)▬OH and isopropyl radicals, which are converted into propene. The energy required for the H abstraction producing propene (15.8 kcal/mol) was significantly lower than that for the isopropyl migration (23.1 kcal/mol), the propene formation path is thus preferable.

**Acknowledgements**

**Conflicts of interest**

**Author details**

**References**

science.1230444

**37**(1):191-214. DOI: 10.1039/b618320b

There are no conflicts to declare.

\*Address all correspondence to: sareeya.b@vistec.ac.th

and Technology (VISTEC), Rayong, Thailand

Development Agency, Pathumthani, Thailand

Science and Technology (VISTEC), Rayong, Thailand

The writing and publication of this book chapter was financially supported by Vidyasirimedhi Institute of Science and Technology and the Thailand Research Fund (TRF) (RSA6080068).

Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

http://dx.doi.org/10.5772/intechopen.80698

69

Siwarut Siwaipram¹, Sarawoot Impeng², Philippe A. Bopp¹,³ and Sareeya Bureekaew¹\*

2 National Nanotechnology Center (NANOTEC), National Science and Technology

3 Department of Molecular Science and Engineering (MSE), Vidyasirimedhi Institute of

[1] Anastas PT, Kirchhoff MM. Origins, current status, and future challenges of green chemistry. Accounts of Chemical Research. 2002;**35**(9):686-694. DOI: 10.1021/ar010065m [2] Law AL. Simulation Modeling and Analysis. Vol. 2. New York: McGraw. McGraw-Hill

[3] Whitehead TL. Molecular modeling: Basic principles and applications, 2nd Edition (Hans-Dieter Höltje, Wolfgang Sippl, Didier Rognan, and Gerd Folkers). Journal of Che-

[4] Becke AD. Density-functional thermochemistry. III. The role of exact exchange. The

[5] Pople JA, Gill PMW, Johnson BG. Kohn-Sham density-functional theory within a finite basis set. Chemical Physics Letters. 1992;**199**(6):557-560. DOI: 10.1016/0009-2614(92)85009-y [6] Furukawa H, Cordova KE, O'Keeffe M, Yaghi OM. The chemistry and applications of metal-organic frameworks. Science. 2013;**341**(6149):123044-0-123044-12. DOI: 10.1126/

[7] Férey G. Hybrid porous solids: Past, present, future. Chemical Society Reviews. 2008;

Journal of Chemical Physics. 1993;**98**(7):5648-5652. DOI: 10.1063/1.464913

New York; 2007. Retrieved from https://www.researchgate.net/

mical Education. 2006;**83**(6):851. DOI: 10.1021/ed083p851

1 Department of Energy Science and Engineering (ESE), Vidyasirimedhi Institute of Science

Gates et al. [71] showed that Rh(I) and Ir(I) diethylene complexes can be anchored on the Zr-NU-1000, Hf-NU-1000, and Zr-UiO-67 to catalyze ethylene hydrogenation and dimerization. They also used DAY zeolite as supporting material to compare the catalytic performance. DFT calculations with M06-L and def2-TZVPP were employed. The optimized geometries were in good agreement with the data analyzed from IR spectra and X-ray absorption fine structure (EXAFS) results. The ethylene insertion to the ethyl▬Rh bond (rate-determining step) significantly depended on the nature of the supporting MOF. The catalytic activity was compared between Zr- and Hf-nodes and DAY zeolites. The calculated enthalpies for the ethylene hydrogenation and dimerization were shown in **Figure 11**. The calculations reveal a lower energy barrier of the DAY zeolite as supporting material than Zr-NU-1000, Hf-NU-1000, and Zr-UiO-67 for the hydrogenation and dimerization. This is opposite to the experiments which show a better catalytic activity of the DAY zeolite than the MOFs materials. The authors proposed that this might be because of the spillover effect rather than a direct electronic effect on the Rh(I) complex.

#### **6. Outlook**

We focus in this brief overview on theoretical studies of the catalytic activity of single-site catalytic MOFs. DFT-based approaches can provide a detailed, if only local, description of the chemistry occurring in various environments encountered in such MOFs. This not only helps in the interpretation of experiment data; the abilities of DFT calculations are a crucial characterization method toward a deeper molecular perspective. This technique is now powerful and reliable enough to predict and guide the synthesis of materials.

There are, however, several problems which need to be overcome:


#### **Acknowledgements**

Co(II)▬OH and isopropyl radicals, which are converted into propene. The energy required for the H abstraction producing propene (15.8 kcal/mol) was significantly lower than that for the isopropyl migration (23.1 kcal/mol), the propene formation path is thus preferable.

Gates et al. [71] showed that Rh(I) and Ir(I) diethylene complexes can be anchored on the Zr-NU-1000, Hf-NU-1000, and Zr-UiO-67 to catalyze ethylene hydrogenation and dimerization. They also used DAY zeolite as supporting material to compare the catalytic performance. DFT calculations with M06-L and def2-TZVPP were employed. The optimized geometries were in good agreement with the data analyzed from IR spectra and X-ray absorption fine structure (EXAFS) results. The ethylene insertion to the ethyl▬Rh bond (rate-determining step) significantly depended on the nature of the supporting MOF. The catalytic activity was compared between Zr- and Hf-nodes and DAY zeolites. The calculated enthalpies for the ethylene hydrogenation and dimerization were shown in **Figure 11**. The calculations reveal a lower energy barrier of the DAY zeolite as supporting material than Zr-NU-1000, Hf-NU-1000, and Zr-UiO-67 for the hydrogenation and dimerization. This is opposite to the experiments which show a better catalytic activity of the DAY zeolite than the MOFs materials. The authors proposed that this might be because of the spillover effect rather than a direct

We focus in this brief overview on theoretical studies of the catalytic activity of single-site catalytic MOFs. DFT-based approaches can provide a detailed, if only local, description of the chemistry occurring in various environments encountered in such MOFs. This not only helps in the interpretation of experiment data; the abilities of DFT calculations are a crucial characterization method toward a deeper molecular perspective. This technique is now powerful

**I.** To find a link between a predicted structure and its syntheses still a challenge.

**II.** Sometime nonlocal phenomena such as a stimulus-responsive behavior of the catalytic site itself must be taken into account. Many works[ref] attempt to simulate larger models in order to describe such systems. This seems to require improved methodologies to limit

**III.** The functionals: Since there is no "universal functional" DFT must be always used with a lot of caution. A lot of experience has been accumulated on the suitability of various approximations to investigate MOFs. Yet new, and costly, validations are still required

**IV.** Last but not least, in some cases, a local view as described in this chapter may not be sufficient to catch the main features of a particular process. The overall outcome of the process under study may depend on many factors, in the worst case, none of which dominant. Various "local views" that may have been obtained could be combined as building blocks (e.g., by understanding the molecular motions between "sites"). This

remains largely a task for theoretical improvements in the future.

and reliable enough to predict and guide the synthesis of materials. There are, however, several problems which need to be overcome:

electronic effect on the Rh(I) complex.

the computational requirements.

whenever a "new" system is to be studied.

**6. Outlook**

68 Density Functional Theory

The writing and publication of this book chapter was financially supported by Vidyasirimedhi Institute of Science and Technology and the Thailand Research Fund (TRF) (RSA6080068).

#### **Conflicts of interest**

There are no conflicts to declare.

#### **Author details**

Siwarut Siwaipram¹, Sarawoot Impeng², Philippe A. Bopp¹,³ and Sareeya Bureekaew¹\*

\*Address all correspondence to: sareeya.b@vistec.ac.th

1 Department of Energy Science and Engineering (ESE), Vidyasirimedhi Institute of Science and Technology (VISTEC), Rayong, Thailand

2 National Nanotechnology Center (NANOTEC), National Science and Technology Development Agency, Pathumthani, Thailand

3 Department of Molecular Science and Engineering (MSE), Vidyasirimedhi Institute of Science and Technology (VISTEC), Rayong, Thailand

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[35] Ortiz AU, Boutin A, Fuchs AH, Coudert F-X. Metal-organic frameworks with wine-rack motif: What determines their flexibility and elastic properties? The Journal of Chemical

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[45] Maihom T, Sawangphruk M, Probst M, Limtrakul J. A computational study of the catalytic aerobic epoxidation of propylene over the coordinatively unsaturated metalorganic framework Fe3(btc)2: Formation of propylene oxide and competing reactions. Physical Chemistry Chemical Physics. 2018;**20**(9):6726-6734. DOI: 10.1039/c7cp07550b [46] Xiao DJ, Oktawiec J, Milner PJ, Long JR. Pore Environment Effects on Catalytic Cyclohexane Oxidation in Expanded Fe2 (dobdc) Analogues. Journal of the American Che-

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**Chapter 4**

**Provisional chapter**

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

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

DOI: 10.5772/intechopen.80267

**Electrochemical Activity of Metal Clathrochelates with**

**Electrochemical Activity of Metal Clathrochelates with** 

The energetic needs of a rising human population have led to the search for alternative energy sources. A promising route for the large-scale storage of renewable energy is water electrolysis, which is performed with a proton-conducting polymer electrolyte. However, only platinum group metal electrocatalysts have the adequate properties to minimize the overvoltages associated with either hydrogen or oxygen evolution reactions. Alternative materials based on transition metals are scarce, but molecular electrochemistry offers some alternatives. In particular, transition metal clathrochelates exhibit an interesting activity with regard to the hydrogen evolution reaction (HER). However, such complexes form a vast family, and there is a need to implement screening approaches to identify the most performing ones. Theoretical studies on molecular electrocatalysts are adequate for this purpose, since density functional theory (DFT) has a strong predicting capability to provide clues for the improvement of practical devices. This chapter describes the most recent theoretical methods applied to several members of the clathrochelate family. We describe the computation of their common spectroscopic and electrochemical properties. In addition, DFT analysis is used to decipher the multistep reaction mechanism of a model Co clathrochelate with regard to the hydrogen

**Keywords:** clathrochelate, cage compounds, DFT, electrochemistry, electrocatalysis,

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

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

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

**Regard to the Hydrogen Evolution Reaction in the**

**Regard to the Hydrogen Evolution Reaction in the** 

**Homogeneous Phase**

**Homogeneous Phase**

Manuel Antuch and Pierre Millet

Manuel Antuch and Pierre Millet

http://dx.doi.org/10.5772/intechopen.80267

**Abstract**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

evolution reaction in the homogeneous phase.

hydrogen evolution reaction


#### **The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal Clathrochelates with Regard to the Hydrogen Evolution Reaction in the Homogeneous Phase The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal Clathrochelates with Regard to the Hydrogen Evolution Reaction in the Homogeneous Phase**

DOI: 10.5772/intechopen.80267

Manuel Antuch and Pierre Millet Manuel Antuch and Pierre Millet

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.80267

#### **Abstract**

[60] Liao P, Getman RB, Snurr RQ. Optimizing open iron sites in metal-organic frameworks for ethane oxidation: A first-principles study. ACS Applied Materials & Interfaces.

[61] Impeng S, Siwaipram S, Bureekaew S, Probst M. Ethane C-H bond actIVation on the Fe(IV)-oxo species in a Zn-based cluster of metal-organic frameworks: A density functional theory study. Physical Chemistry Chemical Physics. 2017;**19**(5):3782-3791. DOI:

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[63] Evans JD, Sumby CJ, Doonan CJ. Post-synthetic metalation of metal-organic frameworks. Chemical Society Reviews. 2014;**43**(16):5933-5951. DOI: 10.1039/C4CS00076E [64] Brozek CK, Dincă M. Cation exchange at the secondary building units of metalorganic frameworks. Chemical Society Reviews. 2014;**43**(16):5456-5467. DOI: 10.1039/

[65] Rimoldi M, Bernales V, Borycz J, Vjunov A, Gallington LC, Platero-Prats AE, et al. Atomic layer deposition in a metal-organic framework: Synthesis, characterization, and performance of a solid acid. Chemistry of Materials. 2017;**29**(3):1058-1068. DOI: 10.1021/

[66] Liu T, Vermeulen NA, Howarth AJ, Li P, Sarjeant AA, Hupp JT, et al. Adding to the arsenal of zirconium-based metal-organic frameworks: The topology as a platform for solvent-assisted metal incorporation. European Journal of Inorganic Chemistry.

[67] Bernales V, Ortuño MA, Truhlar DG, Cramer CJ, Gagliardi L. Computational design of functionalized metal–organic framework nodes for catalysis. ACS Central Science.

[68] Ortuño MA, Bernales V, Gagliardi L, Cramer CJ. Computational study of first-row transition metals supported on MOF NU-1000 for catalytic acceptorless alcohol dehydrogenation. Journal of Physical Chemistry C. 2016;**120**(43):24697-24705. DOI: 10.1021/acs.

[69] Bernales V, League AB, Li Z, Schweitzer NM, Peters AW, Carlson RK, et al. Computationally guided discovery of a catalytic cobalt-decorated metal-organic framework for ethylene dimerization. Journal of Physical Chemistry C. 2016;**120**(41):23576-23583. DOI:

[70] Li Z, Peters AW, Bernales V, Ortuño MA, Schweitzer NM, Destefano MR, et al. Metalorganic framework supported cobalt catalysts for the oxidative dehydrogenation of propane at low temperature. ACS Central Science. 2017;**3**(1):31-38. DOI: 10.1021/

[71] Bernales V, Yang D, Yu J, Gümüşlü G, Cramer CJ, Gates BC, et al. Molecular rhodium complexes supported on the metal-oxide-like nodes of metal organic frameworks and on Zeolite HY: Catalysts for ethylene hydrogenation and dimerization. ACS Applied

Materials & Interfaces. 2017;**9**(39):33511-33520. DOI: 10.1021/acsami.7b03858

2017;**9**(39):33484-33492. DOI: 10.1021/acsami.7b02195

2013;**135**(34):12886-12891. DOI: 10.1021/ja4064475

2016;**2016**(27):4349-4352. DOI: 10.1002/ejic.201600627

2017;**4**(1):5-19. DOI: 10.1021/acscentsci.7b00500

10.1039/c6cp07771d

74 Density Functional Theory

C4CS00002A

jpcc.6b06381

10.1021/acs.jpcc.6b07362

acscentsci.6b00290

acs.chemmater.6b03880

The energetic needs of a rising human population have led to the search for alternative energy sources. A promising route for the large-scale storage of renewable energy is water electrolysis, which is performed with a proton-conducting polymer electrolyte. However, only platinum group metal electrocatalysts have the adequate properties to minimize the overvoltages associated with either hydrogen or oxygen evolution reactions. Alternative materials based on transition metals are scarce, but molecular electrochemistry offers some alternatives. In particular, transition metal clathrochelates exhibit an interesting activity with regard to the hydrogen evolution reaction (HER). However, such complexes form a vast family, and there is a need to implement screening approaches to identify the most performing ones. Theoretical studies on molecular electrocatalysts are adequate for this purpose, since density functional theory (DFT) has a strong predicting capability to provide clues for the improvement of practical devices. This chapter describes the most recent theoretical methods applied to several members of the clathrochelate family. We describe the computation of their common spectroscopic and electrochemical properties. In addition, DFT analysis is used to decipher the multistep reaction mechanism of a model Co clathrochelate with regard to the hydrogen evolution reaction in the homogeneous phase.

**Keywords:** clathrochelate, cage compounds, DFT, electrochemistry, electrocatalysis, hydrogen evolution reaction

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

#### **1. Introduction**

Clathrochelate is a term used in coordination chemistry to denote the ligands that can encapsulate metal ions. The word is specifically used here as a generic term to designate a family of macrocyclic organometallic compounds that contain tris-glyoximate ligands, forming a cage that is strongly bound to a central metal atom (typically Fe [1] or Co [2]; (**Figure 1**). Interestingly, such clathrochelates form a wide family of compounds since their apical and ribbed substituents may be varied with ease. Electron donor (such as alkyl) or electron-withdrawing substituents (such as halogens) can be used in the structure to adjust the electronic properties of the metallic center. The XRD structure of the cobalt clathrochelates in their resting state (i.e., neutral molecule with the metal ion in a 2+ oxidation state) shows four N atoms of two glyoximate groups closer to Co, typically at a distance between 1.88 to 1.99 Å, and two other N atoms of the third glyoximate group more elongated at a distance of 2.10 to 2.17 Å [2, 3], reflecting a Jahn-Teller distortion. Another structural parameter employed for their description is the dihedral angle (θ) between the atoms B'N'NB (where the apostrophe distinguishes the two apical extremes of the complex), which expresses the rotation between the two N′··N′··N′ and N··N··N triangles, as illustrated in **Figure 1**. Both Co―N distances and θ depend on the oxidation state of the molecule, reflecting structural variations occurring upon electron gain or loss.

Herein, we illustrate the use of DFT to study the electrochemical properties of these clathrochelates with regard to the HER. First, we provide a concise overview of the theoretical methods currently employed to address these complexes and the typical procedures employed to aid and stabilize self-consistent field (SCF) and geometry optimization convergence. Subsequently, we provide examples of the computation of common spectroscopies, such as infrared (IR) or ultraviolet-visible (UV-Vis). The core of this chapter is then devoted to the computation of electron transfer processes and to unravel the HER electrocatalytic mechanism with a model Co clathrochelate. Finally, we provide an overall conclusion of the state of the art regarding the application of DFT methods to clathrochelates, along with conceivable

The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal…

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77

The choice of the theoretical method is the cornerstone requirement for a successful computation leading to results of predictive value. However, there is no established and unique approach to select the appropriate method for solving a particular problem. Despite this,

For example, DFT studies of organic molecules have overwhelmingly been performed with the B3LYP exchange-correlation functional [6], which is currently a standard procedure joined to Pople's basis sets (the family of 6-31G and so on). At the present time, there are no huge problems with the theoretical treatment of relatively small organic molecules. Conversely, organometallic complexes in general, and clathrochelates in particular, are much more challenging, because of the plethora of functionals and basis sets available. Ultimately, it is the comparison with an experimental quantity that tells whether a theoretical method is satisfactory or not. Typically, the initial manner is to skim the literature to find similar systems which have been treated before and to use such methodology. In the case of Co clathrochelates, the literature is dominated by the use of the B3LYP exchange-correlation functional. We have successfully used the B3LYP functional for geometry optimizations, with excellent agreement to experiment [2]; however, all attempts to use other functionals (M06-L, TPSS, BP86, PBE, or PBE0) yielded unreasonable geometries [7]. Regarding the selection of the basis set, the LANL2DZ with effective core potentials for heavy atoms predicts well the geometry of Co clathrochelates; on the other hand, the use of a more complete and more expensive all-electron basis set cc-pVTZ for Co and cc-pVDZ for all other atoms did not

Regarding geometry optimization convergence, it is reasonable to be generous on the allowed iteration steps; for instance, we have always employed 100 iterations for these kind of problems. Another frequent issue is SCF convergence. We recommend the use of Pulay's direct inversion in the iterative subspace (DIIS) and the damping of the Fock matrix, with the purpose to accelerate and stabilize convergence; alternatively to DIIS, the second-order SCF orbital optimization (SOSCF) method may be used as well. Finally, the computation of SCF may be performed directly or not. This has an impact mostly on the calculation time, and the choice depends on the hardware available for work. In the case of direct SCF, the

**2. Comments on the selection of the calculation methodology**

some general guidelines may be set in order to select the suitable tools.

perspectives for future work on the topic.

improve significantly the results [7].

The relevant property of these clathrochelates that motivates this chapter is their electrocatalytic activity for hydrogen production from protons in the homogeneous phase (i.e., when the compound is dissolved in solution) [2] known as the hydrogen evolution reaction (HER). The versatility of these molecules makes them further suitable for their incorporation in heterogeneous systems for light harvesting [4] or as precursors for the deposition of Co-containing nanoparticles [5], which are also active for the HER.

**Figure 1.** General Lewis structure of a Co clathrochelate. These molecules are versatile since they may bear various apical and ribbed substituents.

Herein, we illustrate the use of DFT to study the electrochemical properties of these clathrochelates with regard to the HER. First, we provide a concise overview of the theoretical methods currently employed to address these complexes and the typical procedures employed to aid and stabilize self-consistent field (SCF) and geometry optimization convergence. Subsequently, we provide examples of the computation of common spectroscopies, such as infrared (IR) or ultraviolet-visible (UV-Vis). The core of this chapter is then devoted to the computation of electron transfer processes and to unravel the HER electrocatalytic mechanism with a model Co clathrochelate. Finally, we provide an overall conclusion of the state of the art regarding the application of DFT methods to clathrochelates, along with conceivable perspectives for future work on the topic.

#### **2. Comments on the selection of the calculation methodology**

**1. Introduction**

76 Density Functional Theory

upon electron gain or loss.

and ribbed substituents.

nanoparticles [5], which are also active for the HER.

Clathrochelate is a term used in coordination chemistry to denote the ligands that can encapsulate metal ions. The word is specifically used here as a generic term to designate a family of macrocyclic organometallic compounds that contain tris-glyoximate ligands, forming a cage that is strongly bound to a central metal atom (typically Fe [1] or Co [2]; (**Figure 1**). Interestingly, such clathrochelates form a wide family of compounds since their apical and ribbed substituents may be varied with ease. Electron donor (such as alkyl) or electron-withdrawing substituents (such as halogens) can be used in the structure to adjust the electronic properties of the metallic center. The XRD structure of the cobalt clathrochelates in their resting state (i.e., neutral molecule with the metal ion in a 2+ oxidation state) shows four N atoms of two glyoximate groups closer to Co, typically at a distance between 1.88 to 1.99 Å, and two other N atoms of the third glyoximate group more elongated at a distance of 2.10 to 2.17 Å [2, 3], reflecting a Jahn-Teller distortion. Another structural parameter employed for their description is the dihedral angle (θ) between the atoms B'N'NB (where the apostrophe distinguishes the two apical extremes of the complex), which expresses the rotation between the two N′··N′··N′ and N··N··N triangles, as illustrated in **Figure 1**. Both Co―N distances and θ depend on the oxidation state of the molecule, reflecting structural variations occurring

The relevant property of these clathrochelates that motivates this chapter is their electrocatalytic activity for hydrogen production from protons in the homogeneous phase (i.e., when the compound is dissolved in solution) [2] known as the hydrogen evolution reaction (HER). The versatility of these molecules makes them further suitable for their incorporation in heterogeneous systems for light harvesting [4] or as precursors for the deposition of Co-containing

**Figure 1.** General Lewis structure of a Co clathrochelate. These molecules are versatile since they may bear various apical

The choice of the theoretical method is the cornerstone requirement for a successful computation leading to results of predictive value. However, there is no established and unique approach to select the appropriate method for solving a particular problem. Despite this, some general guidelines may be set in order to select the suitable tools.

For example, DFT studies of organic molecules have overwhelmingly been performed with the B3LYP exchange-correlation functional [6], which is currently a standard procedure joined to Pople's basis sets (the family of 6-31G and so on). At the present time, there are no huge problems with the theoretical treatment of relatively small organic molecules. Conversely, organometallic complexes in general, and clathrochelates in particular, are much more challenging, because of the plethora of functionals and basis sets available. Ultimately, it is the comparison with an experimental quantity that tells whether a theoretical method is satisfactory or not. Typically, the initial manner is to skim the literature to find similar systems which have been treated before and to use such methodology. In the case of Co clathrochelates, the literature is dominated by the use of the B3LYP exchange-correlation functional. We have successfully used the B3LYP functional for geometry optimizations, with excellent agreement to experiment [2]; however, all attempts to use other functionals (M06-L, TPSS, BP86, PBE, or PBE0) yielded unreasonable geometries [7]. Regarding the selection of the basis set, the LANL2DZ with effective core potentials for heavy atoms predicts well the geometry of Co clathrochelates; on the other hand, the use of a more complete and more expensive all-electron basis set cc-pVTZ for Co and cc-pVDZ for all other atoms did not improve significantly the results [7].

Regarding geometry optimization convergence, it is reasonable to be generous on the allowed iteration steps; for instance, we have always employed 100 iterations for these kind of problems. Another frequent issue is SCF convergence. We recommend the use of Pulay's direct inversion in the iterative subspace (DIIS) and the damping of the Fock matrix, with the purpose to accelerate and stabilize convergence; alternatively to DIIS, the second-order SCF orbital optimization (SOSCF) method may be used as well. Finally, the computation of SCF may be performed directly or not. This has an impact mostly on the calculation time, and the choice depends on the hardware available for work. In the case of direct SCF, the integrals are calculated each time whenever necessary; on the other hand, if direct SCF is not chosen, integrals are calculated only once and then stored in memory. For computers that have mechanical-based hard disks, it is advisable to use direct SCF, because the calculation of integrals by the processor is generally faster than their search in memory. The time spent for the calculation of clathrochelates may be high, of course depending on the functional, the basis set, and the substituents of the molecule. As an illustrative example, by running a parallelized version of GAMESS [8] on 29 processors, it took 1 week for a complete optimization and Hessian computation of complex CoBd<sup>3</sup> (B―nC<sup>4</sup> H9 ) 2 which possesses n-butyl and phenyl substituents in the apical and ribbed positions, respectively, using the B3LYP functional and the LANL2DZ basis set with effective core potentials for Co.

### **3. Computation of typical spectroscopic properties of a model clathrochelate**

A major goal of theoretical chemistry is to be able to predict measurable properties and therefore to provide insights into the behavior of molecular systems. Two of the most ubiquitous techniques available are infrared (IR) and ultraviolet-visible (UV-Vis) spectroscopies, which provide information on the normal modes of vibration (IR) and electronic excitations (UV-Vis). At the end of any geometry optimization, a Hessian calculation should be performed, giving access to the normal modes of the molecule; indeed, all calculated frequencies must be positive in order to verify that the identified stationary point is a true minimum. A typical example of Hessian calculation for the Co(Cl2 Gm)3 (B―PhF<sup>5</sup> )2 molecule (where Gm and PhF<sup>5</sup> stand for the glyoximate and the pentafluorophenyl moieties, respectively) is depicted in **Figure 2a**, along with a comparison to the experimental IR spectrum. The calculation was performed with the B3LYP exchange-correlation functional and using the LANL2DZ basis set for all atoms, including effective core potentials for Co and Cl. Note the excellent agreement between experiment and calculation, showing that most signals observed in the experimental spectrum are not pure normal modes but a complex convolution of multiple vibrations that are close in frequency. In particular, the absorption band highlighted with a (1) is essentially composed by a single normal mode. However, such mode is not located in a small part of the molecule but comprises the whole molecular structure, with coupled vibrations from the ligand cage, and the pentafluorophenyl rings, as depicted on the right side of **Figure 2b**.

polarization function (TZVP) for Co and double-ζ split valence for all other atoms [3]. For our calculations (**Figure 2c** and **d**), we have employed the augmented version of the correlationconsistent polarized valence triple-ζ (aug-cc-pVTZ) basis set for Co and the correlation-consistent polarized valence double-ζ (cc-pVDZ) for all other atoms, and we have computed the first 40 excited states. It should be noted that, for the calculation of UV-Vis spectra of usual small organic molecules, it is often sufficient to calculate only four excited states [6], in open contrast to organometallic compounds, where a large number has to be computed in order to adequately reproduce the spectrum. As for IR spectra, these complex molecules exhibit complex UV-Vis spectra, and the assignment of the different contributions is not straightforward because each transition is composed of several contributions from different orbitals. In particular, two transitions occurring in the visible region are highlighted with numbers (2) and (3) in **Figure 2c**. The main orbitals involved in such transition are depicted on the right side of **Figure 2d**. Certainly, these transitions cannot be classified simply into metal-toligand charge transfer (MLCT) or ligand-to-metal charge transfer (LMCT), because all orbitals involved in the transition have important contributions from the d orbitals of cobalt and from

hessian calculation (red sticks); (b) illustration of the normal mode highlighted with number (1) in item (a), functional B3LYP, and basis set LANL2DZ with ECP for Co and Cl; (c) experimental UV-Vis spectrum superimposed to the theoretical spectrum calculated by TDDFT. Functional B3LYP and aug-cc-pVTZ basis set for Co and cc-pVDZ for all

Gm)3

The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal…

(B―PhF<sup>5</sup> ) 2

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79

molecule and the

**Figure 2.** (a) Comparison of the experimental IR spectrum (blue lines) of the Co(Cl<sup>2</sup>

the glyoxime moieties of the ligand.

other atoms.

Another important property of these molecules that may be described is their optical absorption. In order to calculate the UV-Vis characteristics, time-dependent density functional theory (TDDFT) formalism has to be employed. The use of the B3LYP functional for the calculation of these kind of spectra has been shown to be an adequate choice [3].

However, the basis function should be chosen to be more complete than for geometry optimizations and Hessian calculations. In particular, it is desirable to include diffuse functions in the basis set, because the optical excitations may occur to nonbonding, unfilled orbitals which develop farther away from the molecule. Previous predictions of optical spectroscopy of clathrochelates have employed a mixture of basis sets, such as the triple-ζ with a The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal… http://dx.doi.org/10.5772/intechopen.80267 79

integrals are calculated each time whenever necessary; on the other hand, if direct SCF is not chosen, integrals are calculated only once and then stored in memory. For computers that have mechanical-based hard disks, it is advisable to use direct SCF, because the calculation of integrals by the processor is generally faster than their search in memory. The time spent for the calculation of clathrochelates may be high, of course depending on the functional, the basis set, and the substituents of the molecule. As an illustrative example, by running a parallelized version of GAMESS [8] on 29 processors, it took 1 week for a complete optimization

(B―nC<sup>4</sup>

substituents in the apical and ribbed positions, respectively, using the B3LYP functional and

A major goal of theoretical chemistry is to be able to predict measurable properties and therefore to provide insights into the behavior of molecular systems. Two of the most ubiquitous techniques available are infrared (IR) and ultraviolet-visible (UV-Vis) spectroscopies, which provide information on the normal modes of vibration (IR) and electronic excitations (UV-Vis). At the end of any geometry optimization, a Hessian calculation should be performed, giving access to the normal modes of the molecule; indeed, all calculated frequencies must be positive in order to verify that the identified stationary point is a true minimum.

depicted in **Figure 2a**, along with a comparison to the experimental IR spectrum. The calculation was performed with the B3LYP exchange-correlation functional and using the LANL2DZ basis set for all atoms, including effective core potentials for Co and Cl. Note the excellent agreement between experiment and calculation, showing that most signals observed in the experimental spectrum are not pure normal modes but a complex convolution of multiple vibrations that are close in frequency. In particular, the absorption band highlighted with a (1) is essentially composed by a single normal mode. However, such mode is not located in a small part of the molecule but comprises the whole molecular structure, with coupled vibrations from the ligand cage, and the pentafluorophenyl rings, as depicted on the right

Another important property of these molecules that may be described is their optical absorption. In order to calculate the UV-Vis characteristics, time-dependent density functional theory (TDDFT) formalism has to be employed. The use of the B3LYP functional for the calculation of

However, the basis function should be chosen to be more complete than for geometry optimizations and Hessian calculations. In particular, it is desirable to include diffuse functions in the basis set, because the optical excitations may occur to nonbonding, unfilled orbitals which develop farther away from the molecule. Previous predictions of optical spectroscopy of clathrochelates have employed a mixture of basis sets, such as the triple-ζ with a

**3. Computation of typical spectroscopic properties of a model** 

H9 ) 2

Gm)3

stand for the glyoximate and the pentafluorophenyl moieties, respectively) is

(B―PhF<sup>5</sup>

)2

molecule (where

which possesses n-butyl and phenyl

and Hessian computation of complex CoBd<sup>3</sup>

**clathrochelate**

78 Density Functional Theory

Gm and PhF<sup>5</sup>

side of **Figure 2b**.

the LANL2DZ basis set with effective core potentials for Co.

A typical example of Hessian calculation for the Co(Cl2

these kind of spectra has been shown to be an adequate choice [3].

**Figure 2.** (a) Comparison of the experimental IR spectrum (blue lines) of the Co(Cl<sup>2</sup> Gm)3 (B―PhF<sup>5</sup> ) 2 molecule and the hessian calculation (red sticks); (b) illustration of the normal mode highlighted with number (1) in item (a), functional B3LYP, and basis set LANL2DZ with ECP for Co and Cl; (c) experimental UV-Vis spectrum superimposed to the theoretical spectrum calculated by TDDFT. Functional B3LYP and aug-cc-pVTZ basis set for Co and cc-pVDZ for all other atoms.

polarization function (TZVP) for Co and double-ζ split valence for all other atoms [3]. For our calculations (**Figure 2c** and **d**), we have employed the augmented version of the correlationconsistent polarized valence triple-ζ (aug-cc-pVTZ) basis set for Co and the correlation-consistent polarized valence double-ζ (cc-pVDZ) for all other atoms, and we have computed the first 40 excited states. It should be noted that, for the calculation of UV-Vis spectra of usual small organic molecules, it is often sufficient to calculate only four excited states [6], in open contrast to organometallic compounds, where a large number has to be computed in order to adequately reproduce the spectrum. As for IR spectra, these complex molecules exhibit complex UV-Vis spectra, and the assignment of the different contributions is not straightforward because each transition is composed of several contributions from different orbitals. In particular, two transitions occurring in the visible region are highlighted with numbers (2) and (3) in **Figure 2c**. The main orbitals involved in such transition are depicted on the right side of **Figure 2d**. Certainly, these transitions cannot be classified simply into metal-toligand charge transfer (MLCT) or ligand-to-metal charge transfer (LMCT), because all orbitals involved in the transition have important contributions from the d orbitals of cobalt and from the glyoxime moieties of the ligand.

#### **4. Computation of electrochemical properties**

#### **4.1. Estimation of redox potentials with DFT**

DFT calculations can also be used to analyze the redox properties of molecules. Electron transfer reactions may take place between chemical species dissolved in solution or between a molecule in solution and a solid-state electrode or photoelectrode. In particular, electrochemistry is the science that studies matter transformation upon the passage of electric current. Species that gain electrons decrease their oxidation state (and are said to be reduced), while species providing electrons to others increase their oxidation sate (and are said to be oxidized). These oxidation-reduction reactions, which necessarily involve the exchange of electrons but may involve chemical steps as well, are referred with the shorthand term "redox." Certain molecules may exist in two (or more) oxidation states, and these pairs are known as redox couples, represented as O|R, where O stands for the oxidized form, and R represents the reduced form. The redox reaction O + e− = R occurs at a certain electrode potential (E°), which is mainly observable in electrochemistry. The redox potential is important because the more positive it is, the more oxidizing is O; likewise, the more negative it is, the more reducing is R. Essentially, the redox potential is a thermodynamic quantity that allows to predict the feasibility of redox reactions. Absolute redox potentials for molecules are not measurable, and therefore a proper reference electrode is employed. Arbitrarily, the standard hydrogen electrode (SHE) has been adopted as zero in the potential scale [9]. This relative scale is quantitatively related to the absolute energy scale used in physics, i.e., the electron in a vacuum [10].

DFT may provide an estimation of the absolute redox potential of molecules, following two main methodologies. The first method consists in the calculation of a Born-Haber process, by optimizing all relevant species both in gas phase and in the solvent. This procedure is relatively lengthy and therefore shall not receive further attention in this chapter, but the interested reader is referred to the specialized literature [11–14]. Conversely, the full calculation of the Born-Haber process may be bypassed, and the optimization and Hessian calculation of the relevant redox species, i.e., O and R, in the desired solvent may be undertaken directly [7, 15]. The calculation of the absolute redox potential of the species is found according to Eq. (1):

$$\mathbf{E}^{\prime}\_{\text{ abs}} = -\left(\boldsymbol{\Lambda}\mathbf{G}^{\prime}\_{\text{ abs}}/\mathbf{n}\mathbf{F}\right) \tag{1}$$

assess precision in the calculation, given that all authors provide arguments for their absolute values of the SHE. In order to overcome this drawback, the choice of a reference redox reaction is a safer option. The most commonly used reference reaction is that of the Fc1+|Fc (i.e., ferrocenium|ferrocene) redox couple. When this procedure is used, the optimization and Hessian calculation of both Fc1+ and Fc at exactly the same theoretical level than the O|R couple under study are performed (**Figure 3a**). Then, the absolute potential of Fc1+|Fc is obtained using Eq. (1). Obviously, this is a sort of "redox calibration" of the computational method. The theoretical estimation of the redox potential of the O|R couple vs. the Fc1+|Fc couple is just the

The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal…

http://dx.doi.org/10.5772/intechopen.80267

81

Current DFT methods provide estimates of redox potentials with an error of at most 200 mV [11, 12, 20], which is nowadays tacitly considered as the limit for a reasonable estimation. There are several causes for such discrepancies with experimental observations, such as the fact that (i) the exact functional linking the electron density and the electronic energy is not known, (ii) the use of an incomplete basis set, (iii) the consideration of only one conformation of the molecule for the computation, (iv) the treatment of the solvent as an electrostatic continuum, and (v) the harmonic oscillator approximation to estimate thermodynamic quantities. Despite these sources of errors, DFT is powerful in predicting reactivity trends while

**Figure 3.** (a) Optimized structures for ferrocene redox chemistry, which is the reference reaction allowing the calculation

Gm)3

were optimized with the B3LYP functional and the LANL2DZ basis set, with effective core potentials for Co and Cl in

dimethylformamide (DMF). The number in the upper left represents the spin multiplicity.

(B―PhF<sup>5</sup> ) 2

in Co(II) and Co(I) oxidation states. Geometries

difference between the absolute potentials these two redox pairs.

giving reasonable estimations for observable properties.

of redox potentials; (b) optimized geometries of the Co(Cl2

where F and n represent Faraday's constant and the number of exchanged electrons, respectively, and ΔGabs <sup>0</sup> is the difference in free energy of the reaction O + e− = R. The value of ΔGabs 0 contains (i) the total electronic energy of the molecule (i.e., electron-electron and nucleus-nucleus repulsions plus electron-nucleus attraction), (ii) zero-point energy (ZPE) corrections, (iii) vibrational corrections to the free energy, and (iv) thermal corrections. We note that the electronic energy of the molecule is obtained by a single-point calculation, but all other thermodynamic corrections are only accessible after a Hessian calculation.

This calculated absolute potential may be referred to an estimated value of the absolute potential of the SHE. Unfortunately, the literature is plenty of different values, for example, 4.2 ± 0.4 V [16] to 4.29 ± 0.02 V [17], 4.05 V [18], and 4.44 V [19], which makes it difficult to assess precision in the calculation, given that all authors provide arguments for their absolute values of the SHE. In order to overcome this drawback, the choice of a reference redox reaction is a safer option. The most commonly used reference reaction is that of the Fc1+|Fc (i.e., ferrocenium|ferrocene) redox couple. When this procedure is used, the optimization and Hessian calculation of both Fc1+ and Fc at exactly the same theoretical level than the O|R couple under study are performed (**Figure 3a**). Then, the absolute potential of Fc1+|Fc is obtained using Eq. (1). Obviously, this is a sort of "redox calibration" of the computational method. The theoretical estimation of the redox potential of the O|R couple vs. the Fc1+|Fc couple is just the difference between the absolute potentials these two redox pairs.

**4. Computation of electrochemical properties**

DFT calculations can also be used to analyze the redox properties of molecules. Electron transfer reactions may take place between chemical species dissolved in solution or between a molecule in solution and a solid-state electrode or photoelectrode. In particular, electrochemistry is the science that studies matter transformation upon the passage of electric current. Species that gain electrons decrease their oxidation state (and are said to be reduced), while species providing electrons to others increase their oxidation sate (and are said to be oxidized). These oxidation-reduction reactions, which necessarily involve the exchange of electrons but may involve chemical steps as well, are referred with the shorthand term "redox." Certain molecules may exist in two (or more) oxidation states, and these pairs are known as redox couples, represented as O|R, where O stands for the oxidized form, and R represents the reduced form. The redox reaction O + e− = R occurs at a certain electrode potential (E°), which is mainly observable in electrochemistry. The redox potential is important because the more positive it is, the more oxidizing is O; likewise, the more negative it is, the more reducing is R. Essentially, the redox potential is a thermodynamic quantity that allows to predict the feasibility of redox reactions. Absolute redox potentials for molecules are not measurable, and therefore a proper reference electrode is employed. Arbitrarily, the standard hydrogen electrode (SHE) has been adopted as zero in the potential scale [9]. This relative scale is quantitatively related to the absolute energy scale used in physics, i.e., the electron in

DFT may provide an estimation of the absolute redox potential of molecules, following two main methodologies. The first method consists in the calculation of a Born-Haber process, by optimizing all relevant species both in gas phase and in the solvent. This procedure is relatively lengthy and therefore shall not receive further attention in this chapter, but the interested reader is referred to the specialized literature [11–14]. Conversely, the full calculation of the Born-Haber process may be bypassed, and the optimization and Hessian calculation of the relevant redox species, i.e., O and R, in the desired solvent may be undertaken directly [7, 15]. The calculation of the absolute redox potential of the species is found according to Eq. (1):

abs = −(ΔG°

where F and n represent Faraday's constant and the number of exchanged electrons, respec-

tains (i) the total electronic energy of the molecule (i.e., electron-electron and nucleus-nucleus repulsions plus electron-nucleus attraction), (ii) zero-point energy (ZPE) corrections, (iii) vibrational corrections to the free energy, and (iv) thermal corrections. We note that the electronic energy of the molecule is obtained by a single-point calculation, but all other thermodynamic

This calculated absolute potential may be referred to an estimated value of the absolute potential of the SHE. Unfortunately, the literature is plenty of different values, for example, 4.2 ± 0.4 V [16] to 4.29 ± 0.02 V [17], 4.05 V [18], and 4.44 V [19], which makes it difficult to

<sup>0</sup> is the difference in free energy of the reaction O + e− = R. The value of ΔGabs

abs /nF) (1)

0 con-

**4.1. Estimation of redox potentials with DFT**

a vacuum [10].

80 Density Functional Theory

tively, and ΔGabs

E°

corrections are only accessible after a Hessian calculation.

Current DFT methods provide estimates of redox potentials with an error of at most 200 mV [11, 12, 20], which is nowadays tacitly considered as the limit for a reasonable estimation. There are several causes for such discrepancies with experimental observations, such as the fact that (i) the exact functional linking the electron density and the electronic energy is not known, (ii) the use of an incomplete basis set, (iii) the consideration of only one conformation of the molecule for the computation, (iv) the treatment of the solvent as an electrostatic continuum, and (v) the harmonic oscillator approximation to estimate thermodynamic quantities. Despite these sources of errors, DFT is powerful in predicting reactivity trends while giving reasonable estimations for observable properties.

**Figure 3.** (a) Optimized structures for ferrocene redox chemistry, which is the reference reaction allowing the calculation of redox potentials; (b) optimized geometries of the Co(Cl2 Gm)3 (B―PhF<sup>5</sup> ) 2 in Co(II) and Co(I) oxidation states. Geometries were optimized with the B3LYP functional and the LANL2DZ basis set, with effective core potentials for Co and Cl in dimethylformamide (DMF). The number in the upper left represents the spin multiplicity.

The comparison between theoretical and experimental redox potentials has another subtlety. Indeed, when an aqueous reference electrode, such as the saturated calomel electrode (SCE) or the silver-silver chloride (Ag|AgCl|Cl− ), is used in a nonaqueous environment, there is an inherent liquid junction potential (LJP) [21] which depends on electrolyte composition and biases the measurement. Hence, in order to compare theoretical results with experimental values, the first issue to pay attention to is that an aqueous reference had not been used to make the measurement in a nonaqueous environment. If an aqueous reference has been used in a nonaqueous solvent the Co atom moves to one side, it is still possible to make the comparison with theory, by finding a literature value for the Fc1+|Fc couple and converting the potential scale of the aqueous reference (which was used in a nonaqueous solvent) to the Fc1+|Fc scale. Literature values for Fc1+|Fc have been reported as +0.50 V vs. Ag|AgCl|Cl− [22] or + 0.46 V vs. SCE [2], comprising the LJP.

other N lying farther apart at 2.45 Å; consequently, the Co atom moves to one side of the cage, thus decoordinating from two N atoms and resulting in a square planar geometry for the reduced

The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal…

Ligand non-innocence in the case of clathrochelates may be further investigated because their chemical structure invites to think on multiple sites as putative candidates for protonation.

have extended such approach by further exploring other protonation sites. Indeed, the basic character of N and the wide evidence of metal hydrides as intermediates in the HER have led

Until now, the most complete theoretical study of hydrogen evolution involving clathrochelates is found in Ref. [7]. The work was performed on a clathrochelate having chlorine and methyl moieties as the ribbed and apical substituents, respectively. Henceforward, this epi-

Optimized structures for three protonation possibilities are illustrated in **Figure 4**; the investigated protonation sites are highlighted with a red square (C), blue circle (N), and an orange triangle (Co). The insets on the right side of **Figure 4** show a magnification of the coordination

the concomitant change in the geometry of such C from planar to tetrahedral. Protonation of N has a similar effect on the local geometry of the ligand, which is modified from planar to tetrahedral as well. Direct protonation of Co might seem unlikely at the first sight because the metal appears buried inside the organic cage. However, the movement of Co inside the

**Figure 4.** Magnification of the organic cage surrounding the Co-containing hexachloride clathrochelate Co(Cl<sup>2</sup>

LANL2DZ basis set, with effective core potentials for Co and Cl in dimethylformamide (DMF).

and the different protonation sites available. Geometries were optimized with the B3LYP functional and the

Previous theoretical studies have addressed only protonation on the iminic C(sp<sup>2</sup>

) bonds from 1.30 to 1.28 Å.

http://dx.doi.org/10.5772/intechopen.80267

) brings a hybridization change to (sp3

) [30], but we

83

) with

Gm)3 (B―

complex; furthermore, there is a slight shortening of N―C(sp<sup>2</sup>

us to explore such alternative protonation sites.

graph presents those results in a concise manner.

. Protonation of the iminic C(sp<sup>2</sup>

position of H+

CH3 )2

Finally, it is worth to stress that the redox potential and the energy of molecular orbitals are different things and cannot be equated. Certainly, when representing electron transfer reactions, the educational literature often depicts electrons as being transferred from frontier orbitals [23]. However, such representations might induce to think that the energy of orbitals is somehow linked to redox potentials. As a matter of fact, molecular orbitals are mathematical functions that describe the electron motion in a molecule and are certainly not observable [24], in contrast to the redox potential which is an observable thermodynamic quantity. Despite this, the energetic trend in orbital energies may be commonly paralleled to the oxidizing/reducing power of molecules. For example, with all other things being equal, the more electronegative the substituents in the ligand of a complex, the lower the energy of the molecular orbitals. Hence, the complex will be more likely to gain an electron in the lowest unoccupied molecular orbital (LUMO) than to give it from the highest occupied molecular orbital (HOMO); this is equivalent to say that the complex shall be a better oxidant and its redox potential should be more positive. This kind of trends, linking the energy of molecular orbitals and redox potentials, has been presented several times in the literature [25, 26]. Though we have shown that the CoIII|CoII couple in a series of clathrochelates complies with it, the CoII|CoI redox pair does not follow the orbital trends. Further details may be found in Ref. [2].

#### **4.2. Mechanistic investigation of the electrocatalytic hydrogen evolution reaction and ligand noninnocence**

The origin of the catalytic activity of metal-containing clathrochelates is currently under debate [1]. It is likely that the role of the ligand is essential to electrocatalysis, since it may stabilize low oxidation states in the metal to enable electron transfer or may participate in protonation reactions that may help bring protons together and evolve H2 . That is why the term "ligand non-innocence" has been coined to refer to organometallic complexes in which the organic ligand has a major participation in its redox behavior to account for high reactivity involving small species, such as CO<sup>2</sup> or H+ [27–29].

For example, upon the one-electron reduction process depicted in **Figure 3b**, we have predicted interesting structural modifications within the molecule. The asymmetry inside the Co―N cage is calculated to be reinforced, with four Co―N distances being at around 1.88 Å and the two other N lying farther apart at 2.45 Å; consequently, the Co atom moves to one side of the cage, thus decoordinating from two N atoms and resulting in a square planar geometry for the reduced complex; furthermore, there is a slight shortening of N―C(sp<sup>2</sup> ) bonds from 1.30 to 1.28 Å.

The comparison between theoretical and experimental redox potentials has another subtlety. Indeed, when an aqueous reference electrode, such as the saturated calomel electrode (SCE) or

ent liquid junction potential (LJP) [21] which depends on electrolyte composition and biases the measurement. Hence, in order to compare theoretical results with experimental values, the first issue to pay attention to is that an aqueous reference had not been used to make the measurement in a nonaqueous environment. If an aqueous reference has been used in a nonaqueous solvent the Co atom moves to one side, it is still possible to make the comparison with theory, by finding a literature value for the Fc1+|Fc couple and converting the potential scale of the aqueous reference (which was used in a nonaqueous solvent) to the Fc1+|Fc scale. Literature values for Fc1+|Fc

Finally, it is worth to stress that the redox potential and the energy of molecular orbitals are different things and cannot be equated. Certainly, when representing electron transfer reactions, the educational literature often depicts electrons as being transferred from frontier orbitals [23]. However, such representations might induce to think that the energy of orbitals is somehow linked to redox potentials. As a matter of fact, molecular orbitals are mathematical functions that describe the electron motion in a molecule and are certainly not observable [24], in contrast to the redox potential which is an observable thermodynamic quantity. Despite this, the energetic trend in orbital energies may be commonly paralleled to the oxidizing/reducing power of molecules. For example, with all other things being equal, the more electronegative the substituents in the ligand of a complex, the lower the energy of the molecular orbitals. Hence, the complex will be more likely to gain an electron in the lowest unoccupied molecular orbital (LUMO) than to give it from the highest occupied molecular orbital (HOMO); this is equivalent to say that the complex shall be a better oxidant and its redox potential should be more positive. This kind of trends, linking the energy of molecular orbitals and redox potentials, has been presented several times in the literature [25, 26]. Though we have shown that the CoIII|CoII couple in a series of clathrochelates com-

**4.2. Mechanistic investigation of the electrocatalytic hydrogen evolution reaction** 

protonation reactions that may help bring protons together and evolve H2

or H+

The origin of the catalytic activity of metal-containing clathrochelates is currently under debate [1]. It is likely that the role of the ligand is essential to electrocatalysis, since it may stabilize low oxidation states in the metal to enable electron transfer or may participate in

term "ligand non-innocence" has been coined to refer to organometallic complexes in which the organic ligand has a major participation in its redox behavior to account for high reactivity

 [27–29]. For example, upon the one-electron reduction process depicted in **Figure 3b**, we have predicted interesting structural modifications within the molecule. The asymmetry inside the Co―N cage is calculated to be reinforced, with four Co―N distances being at around 1.88 Å and the two

), is used in a nonaqueous environment, there is an inher-

[22] or + 0.46 V vs. SCE [2], comprising the LJP.

redox pair does not follow the orbital trends. Further details may

. That is why the

the silver-silver chloride (Ag|AgCl|Cl−

82 Density Functional Theory

have been reported as +0.50 V vs. Ag|AgCl|Cl−

plies with it, the CoII|CoI

**and ligand noninnocence**

involving small species, such as CO<sup>2</sup>

be found in Ref. [2].

Ligand non-innocence in the case of clathrochelates may be further investigated because their chemical structure invites to think on multiple sites as putative candidates for protonation. Previous theoretical studies have addressed only protonation on the iminic C(sp<sup>2</sup> ) [30], but we have extended such approach by further exploring other protonation sites. Indeed, the basic character of N and the wide evidence of metal hydrides as intermediates in the HER have led us to explore such alternative protonation sites.

Until now, the most complete theoretical study of hydrogen evolution involving clathrochelates is found in Ref. [7]. The work was performed on a clathrochelate having chlorine and methyl moieties as the ribbed and apical substituents, respectively. Henceforward, this epigraph presents those results in a concise manner.

Optimized structures for three protonation possibilities are illustrated in **Figure 4**; the investigated protonation sites are highlighted with a red square (C), blue circle (N), and an orange triangle (Co). The insets on the right side of **Figure 4** show a magnification of the coordination position of H+ . Protonation of the iminic C(sp<sup>2</sup> ) brings a hybridization change to (sp3 ) with the concomitant change in the geometry of such C from planar to tetrahedral. Protonation of N has a similar effect on the local geometry of the ligand, which is modified from planar to tetrahedral as well. Direct protonation of Co might seem unlikely at the first sight because the metal appears buried inside the organic cage. However, the movement of Co inside the

**Figure 4.** Magnification of the organic cage surrounding the Co-containing hexachloride clathrochelate Co(Cl<sup>2</sup> Gm)3 (B― CH3 )2 and the different protonation sites available. Geometries were optimized with the B3LYP functional and the LANL2DZ basis set, with effective core potentials for Co and Cl in dimethylformamide (DMF).

ligand cage upon the one-electron reductive activation and the associated decoordination from two N makes the metal accessible for protonation as well. The calculated X-H distances (where X stands for Co, N, or Csp2 ) for these three species are 1.09 Å for Csp2 -H, 1.45 Å for Co―H, and 1.03 Å for N―H. Such bond distances show that proton binding to N or C results in the formation of a strong bond, in contrast to the binding to the metal center, where a much more elongated Co(III)―H bond is created, indicating a weaker binding and therefore a more labile H+ .

An energetic analysis of this initial part of the mechanism is illustrated in **Figure 5a**. Indeed, there is an energetic gain upon the reductive activation of the [2 CoL]0 to yield the catalytically active [CoL]1− intermediate. Then, protonation of N or C(sp2 ) is likely, but the most favored protonation site was predicted to be the Co central ion.

We have also addressed the possibility of double protonation in two adjacent N; in a C(sp2 ) and a N; in two adjacent C(sp2 ); in Co and N; and in Co, for both high-spin and low-spin species. However, most optimized structures were largely distorted due to heavy structural changes caused by the sp2 to sp3 hybridization changes that should occur after protonation. As a consequence, the calculated energy of doubly protonated intermediates was prohibitively high, and these species are unlikely to be formed. The most stable doubly protonated species was the one with protons coordinated in two adjacent N atoms. However, it seems unlikely that the mechanism of hydrogen evolution involved initially a mono-protonated species that is a cobalt hydride and that the subsequent species in the mechanism is a double-protonated intermediate with protons in adjacent N atoms.

Considering that the hydrogen production step is of key importance, we have devoted wide efforts to explain how it happens. When the calculations on a mechanism yield high energy intermediates, it is possible that a concerted electron-proton transfer process occurred, in order to avoid energy-demanding paths [15, 20]. Thus, potential energy surface (PES) scans were performed with the Co(II)―(HH) and Co(III)―HH) species at different (fixed) Co―H distances, while the rest of the structure was allowed to optimize freely. These results are summarized in **Figure 6a** and **b**. Interestingly, in the case of Co(II)―(HH), the increase in the Co―H bond distance was linked to a decrease in the H―H distance, approaching the equilibrium

**Figure 6.** Main results from potential energy surface scans showing the relation between the Co―(HH) and the H―H

basis set, with effective core potentials for Co and Cl. The energy was further refined with the more complete all-electron basis set cc-pVTZ for Co and cc-pVDZ for all other atoms in dimethylformamide (DMF). The zero in the energy scale is

Gm)3

(B―CH3 )2

The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal…

decreased to approach the final energy level of the reaction (observe the blue scale on the right

nice picture of two protons that come together in the nearby of Co(II) and bind to one another

Conversely, the same calculations on the Co(III) species gave the same bond-breaking (Co―H) bond-formation (H―H) sequence as just described for Co(II). However, at the same Co―H distance, the H―H separation was consistently higher in the case of Co(III), thus indicating that Co(III) was less effective than Co(II) to promote the reduction and binding of two protons. Moreover, the total energy of the system for the Co(III) ion did not decrease smoothly as in the case of Co(II) (**Figure 6b**). Instead, it was excessively high, and therefore such intermediate was definitively ruled out as a possibility to explain the catalytic activity of this clathrochelate.

The clathrochelate family has received attention due to its multiple applications. In this chapter, we have illustrated the use of density functional theory to describe several properties of

while going away from the metal, with the concomitant decrease in total energy.

, which is 0.74 Å. At the same time, the energy of the system gradually

CoL]0 + H2

. These calculations show a

from the complex, (a) Co(II) and (b)

at the B3LYP level and the LANL2DZ

http://dx.doi.org/10.5772/intechopen.80267

85

bond distance for H2

species.

to form H2

the [2 CoL]0

side of **Figure 6a**), which is depicted in **Figure 5b** as [2

bond distance along with the relative energy involved during the separation of H2

Co(III). These calculations were performed in the complex Co(Cl<sup>2</sup>

**5. Conclusions and perspectives**

The Co(III)―H intermediate is interesting and deserved closer attention. In particular, such species may undergo a further one-electron reduction to yield Co(II)―H, which may also have a role in the mechanism. Certainly, we have predicted that such reduction brings an important energetic gain, but the search for doubly protonated intermediates and the search for a transition state to explain hydrogen evolution were unsuccessful. We were able to locate a transition state of the η<sup>2</sup> ―Co(III)―(HH) type, but again this was far too energetic to be a plausible candidate to explain hydrogen evolution (**Figure 6b**).

**Figure 5.** (a) Energetic diagram of the first electron transfer to produce the catalytically active [CoI ]1− species and its further protonation at different sites of the ligand, as depicted in figure X; (b) representation of the second part of the mechanism, showing the unfeasibility of hydrogen evolution through a transition state of high energy and the alternative possibility of concerted proton-electron transfer.

The Use of Density Functional Theory to Decipher the Electrochemical Activity of Metal… http://dx.doi.org/10.5772/intechopen.80267 85

**Figure 6.** Main results from potential energy surface scans showing the relation between the Co―(HH) and the H―H bond distance along with the relative energy involved during the separation of H2 from the complex, (a) Co(II) and (b) Co(III). These calculations were performed in the complex Co(Cl<sup>2</sup> Gm)3 (B―CH3 )2 at the B3LYP level and the LANL2DZ basis set, with effective core potentials for Co and Cl. The energy was further refined with the more complete all-electron basis set cc-pVTZ for Co and cc-pVDZ for all other atoms in dimethylformamide (DMF). The zero in the energy scale is the [2 CoL]0 species.

Considering that the hydrogen production step is of key importance, we have devoted wide efforts to explain how it happens. When the calculations on a mechanism yield high energy intermediates, it is possible that a concerted electron-proton transfer process occurred, in order to avoid energy-demanding paths [15, 20]. Thus, potential energy surface (PES) scans were performed with the Co(II)―(HH) and Co(III)―HH) species at different (fixed) Co―H distances, while the rest of the structure was allowed to optimize freely. These results are summarized in **Figure 6a** and **b**. Interestingly, in the case of Co(II)―(HH), the increase in the Co―H bond distance was linked to a decrease in the H―H distance, approaching the equilibrium bond distance for H2 , which is 0.74 Å. At the same time, the energy of the system gradually decreased to approach the final energy level of the reaction (observe the blue scale on the right side of **Figure 6a**), which is depicted in **Figure 5b** as [2 CoL]0 + H2 . These calculations show a nice picture of two protons that come together in the nearby of Co(II) and bind to one another to form H2 while going away from the metal, with the concomitant decrease in total energy. Conversely, the same calculations on the Co(III) species gave the same bond-breaking (Co―H) bond-formation (H―H) sequence as just described for Co(II). However, at the same Co―H distance, the H―H separation was consistently higher in the case of Co(III), thus indicating that Co(III) was less effective than Co(II) to promote the reduction and binding of two protons. Moreover, the total energy of the system for the Co(III) ion did not decrease smoothly as in the case of Co(II) (**Figure 6b**). Instead, it was excessively high, and therefore such intermediate was definitively ruled out as a possibility to explain the catalytic activity of this clathrochelate.

#### **5. Conclusions and perspectives**

ligand cage upon the one-electron reductive activation and the associated decoordination from two N makes the metal accessible for protonation as well. The calculated X-H distances

Co―H, and 1.03 Å for N―H. Such bond distances show that proton binding to N or C results in the formation of a strong bond, in contrast to the binding to the metal center, where a much more elongated Co(III)―H bond is created, indicating a weaker binding and therefore a more

An energetic analysis of this initial part of the mechanism is illustrated in **Figure 5a**. Indeed,

We have also addressed the possibility of double protonation in two adjacent N; in a C(sp2

species. However, most optimized structures were largely distorted due to heavy structural

a consequence, the calculated energy of doubly protonated intermediates was prohibitively high, and these species are unlikely to be formed. The most stable doubly protonated species was the one with protons coordinated in two adjacent N atoms. However, it seems unlikely that the mechanism of hydrogen evolution involved initially a mono-protonated species that is a cobalt hydride and that the subsequent species in the mechanism is a double-protonated

The Co(III)―H intermediate is interesting and deserved closer attention. In particular, such species may undergo a further one-electron reduction to yield Co(II)―H, which may also have a role in the mechanism. Certainly, we have predicted that such reduction brings an important energetic gain, but the search for doubly protonated intermediates and the search for a transition state to explain hydrogen evolution were unsuccessful. We were able to locate

there is an energetic gain upon the reductive activation of the [2

active [CoL]1− intermediate. Then, protonation of N or C(sp2

protonation site was predicted to be the Co central ion.

to sp3

plausible candidate to explain hydrogen evolution (**Figure 6b**).

**Figure 5.** (a) Energetic diagram of the first electron transfer to produce the catalytically active [CoI

further protonation at different sites of the ligand, as depicted in figure X; (b) representation of the second part of the mechanism, showing the unfeasibility of hydrogen evolution through a transition state of high energy and the

intermediate with protons in adjacent N atoms.

alternative possibility of concerted proton-electron transfer.

) for these three species are 1.09 Å for Csp2

CoL]0

); in Co and N; and in Co, for both high-spin and low-spin

hybridization changes that should occur after protonation. As

―Co(III)―(HH) type, but again this was far too energetic to be a


]1− species and its

)

to yield the catalytically

) is likely, but the most favored

(where X stands for Co, N, or Csp2

and a N; in two adjacent C(sp2

changes caused by the sp2

a transition state of the η<sup>2</sup>

labile H+ .

84 Density Functional Theory

The clathrochelate family has received attention due to its multiple applications. In this chapter, we have illustrated the use of density functional theory to describe several properties of clathrochelates. We began the presentation by providing tips on the choice of the calculation methodology to address these complex molecules. Then, the spectroscopic properties (IR and UV-Vis) have been reproduced. The possibility to accurately predict spectroscopic properties gives an adequate degree of confidence on the theoretical procedures employed. Subsequently, we have illustrated the procedures to predict the redox potentials of molecules, and we have highlighted some structural changes underwent during the electron transfer process. This chapter has introduced the concept of "ligand non-innocence," which is relevant in complex electrocatalytic reactions involving complexes of transition metals, where the ligand has a major role to play, not only to stabilize unusually low oxidation states but also to act cooperatively to bind small species, such as H+ . Finally, a methodology has been presented to address the complex mechanism of hydrogen evolution. Certainly, the hydrogen production step was particularly intriguing for the Co(Cl2 Gm)3 (B―CH3 ) 2 complex, given that neither doubly protonated intermediates nor plausible transition states could be found. Therefore, a concerted proton-electron transfer step was envisaged, with the Co(II)―(HH) hydride intermediate playing the most relevant role. The calculation of potential energy surface scans fixing the Co(II)―(HH) bond distance showed that the simultaneous elongation of Co―H bond leads to the decrease in H―H distance and the smooth decrease in energy, which nicely explained hydrogen evolution from this complex in the homogeneous phase. Overall, results reported in this chapter could contribute to stimulate theoretical and experimentalist chemists to explore the influence of different substituents in the organic cage on the HER activity, thereby providing new insights to researchers that try to develop and optimize alternative electrocatalysts for the HER. Clearly, the formal theoretical understanding of clathrochelate's electrochemistry is still in its early stages, and the missing piece in this scientific challenge is to apply the theoretical methodology outlined in this chapter to other clathrochelates in order to justify the distinct reactivity observed experimentally among different complexes. Moreover, the approach proposed here should provide guidelines to synthetic chemists in the search for more active compounds. Thus, we foresee that the future theoretical research shall be performed in such direction. This is a stimulating moment to conduct research aiming to find novel and efficient catalysts for hydrogen production, and transition metal clathrochelates are perhaps the family of complexes that has received the least attention from the chemical community.

**Author details**

**References**

of Rh: SrTiO<sup>3</sup>

Manuel Antuch\* and Pierre Millet

Paris-Saclay University, Orsay, France

3436-3439. DOI: 10.1039/c7cc09611a

DOI: 10.1016/j.electacta.2017.10.018

\*Address all correspondence to: manuel.antuch-cubillas@u-psud.fr

Research and Innovation Team in Electrochemistry for Energy (ERIEE), Institute of Molecular and Materials Chemistry (ICMMO), UMR CNRS 8182, Paris-Sud University,

[1] Voloshin YZ, Novikov VV, Nelyubina YV, Belov AS, Roitershtein DM, Savitsky A, Mokhir A, Miehlich ME, Meyer K. Synthesis and characterization of an Fe (I) cage complex with high stability towards strong H-acids. Chemical Communications. 2018;**54**:

[2] Antuch M, Ranjbari A, Grigoriev S, Al-Cheikh J, Villagrá A, Assaud L, Voloshin YA, Millet P. Effect of the ligand framework of cobalt clathrochelates on hydrogen evolution electrocatalysis: Electrochemical, spectroscopic and density functional theory analyses.

[3] Nguyen MTD, Charlot MF, Aukauloo A. Structural, electronic, and theoretical description of a series of cobalt clathrochelate complexes in the Co(III), Co(II) and Co(I) oxidation states. The Journal of Physical Chemistry. A. 2011;**115**:911-922. DOI: 10.1021/jp1103643

[4] Antuch M, Millet P, Iwase A, Kudo A, Grigoriev SA, Voloshin YZ. Characterization

[5] Anxolabéhère-Mallart E, Costentin C, Fournier M, Nowak S, Robert M, Savéant JM. Boron-capped tris(glyoximato) cobalt clathrochelate as a precursor for the electrodeposition of nanoparticles catalyzing H2 evolution in water. Journal of the American

[6] de la Vega-Hernández K, Antuch M, Cuesta-Rubio O, Núñez-Figueredo Y, Pardo-Andreu GL. Discerning the antioxidant mechanism of rapanone : A naturally occurring benzoquinone with iron complexing and radical scavenging activities. Journal of

[7] Antuch M, Millet P. Approach to the mechanism of hydrogen evolution electrocatalyzed by a model Co clathrochelate: A theoretical study by density functional theory.

Inorganic Biochemistry. 2017;**170**:134-147. DOI: 10.1016/j.jinorgbio.2017.02.019

Chemical Society. 2012;**134**:6104-6107. DOI: 10.1021/ja301134e

Chemphyschem. 2018. DOI: 10.1002/cphc.201800383

application to the hydrogen evolution reaction. Electrochimica Acta. 2017;**258**:255-265.

photoelectrodes surface-modified with a cobalt clathrochelate and their

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87

Electrochimica Acta. 2017;**245**:1065-1074. DOI: 10.1016/j.electacta.2017.03.005

#### **Acknowledgements**

Manuel Antuch is thankful to the French ministerial scholarship and the Initiative d'Excellence (IDEX) program for funding.

#### **Conflict of interest**

The authors do not have conflict of interest.

#### **Author details**

clathrochelates. We began the presentation by providing tips on the choice of the calculation methodology to address these complex molecules. Then, the spectroscopic properties (IR and UV-Vis) have been reproduced. The possibility to accurately predict spectroscopic properties gives an adequate degree of confidence on the theoretical procedures employed. Subsequently, we have illustrated the procedures to predict the redox potentials of molecules, and we have highlighted some structural changes underwent during the electron transfer process. This chapter has introduced the concept of "ligand non-innocence," which is relevant in complex electrocatalytic reactions involving complexes of transition metals, where the ligand has a major role to play, not only to stabilize unusually low oxidation states but also to

to address the complex mechanism of hydrogen evolution. Certainly, the hydrogen produc-

doubly protonated intermediates nor plausible transition states could be found. Therefore, a concerted proton-electron transfer step was envisaged, with the Co(II)―(HH) hydride intermediate playing the most relevant role. The calculation of potential energy surface scans fixing the Co(II)―(HH) bond distance showed that the simultaneous elongation of Co―H bond leads to the decrease in H―H distance and the smooth decrease in energy, which nicely explained hydrogen evolution from this complex in the homogeneous phase. Overall, results reported in this chapter could contribute to stimulate theoretical and experimentalist chemists to explore the influence of different substituents in the organic cage on the HER activity, thereby providing new insights to researchers that try to develop and optimize alternative electrocatalysts for the HER. Clearly, the formal theoretical understanding of clathrochelate's electrochemistry is still in its early stages, and the missing piece in this scientific challenge is to apply the theoretical methodology outlined in this chapter to other clathrochelates in order to justify the distinct reactivity observed experimentally among different complexes. Moreover, the approach proposed here should provide guidelines to synthetic chemists in the search for more active compounds. Thus, we foresee that the future theoretical research shall be performed in such direction. This is a stimulating moment to conduct research aiming to find novel and efficient catalysts for hydrogen production, and transition metal clathrochelates are perhaps the family of complexes that has received the least attention from the

Manuel Antuch is thankful to the French ministerial scholarship and the Initiative d'Excellence

Gm)3

(B―CH3 ) 2

. Finally, a methodology has been presented

complex, given that neither

act cooperatively to bind small species, such as H+

tion step was particularly intriguing for the Co(Cl2

chemical community.

86 Density Functional Theory

**Acknowledgements**

(IDEX) program for funding.

The authors do not have conflict of interest.

**Conflict of interest**

Manuel Antuch\* and Pierre Millet

\*Address all correspondence to: manuel.antuch-cubillas@u-psud.fr

Research and Innovation Team in Electrochemistry for Energy (ERIEE), Institute of Molecular and Materials Chemistry (ICMMO), UMR CNRS 8182, Paris-Sud University, Paris-Saclay University, Orsay, France

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[23] Elgrishi N, Rountree KJ, McCarthy BD, Rountree ES, Eisenhart TT, Dempsey JL.A practical Beginner's guide to cyclic voltammetry. Journal of Chemical Education. 2018;**95**:197-

[24] Autschbach J. Orbitals: Some fiction and some facts. Journal of Chemical Education.

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[26] Al-Anber M, Milde B, Alhalasah W, Lang H, Holze R. Electrochemical and DFT-studies

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[28] Lacy DC, Mccrory CCL, Peters JC. Studies of cobalt-mediated electrocatalytic CO<sup>2</sup> reduction using a redox-active ligand. Inorganic Chemistry. 2014;**53**:2-10. DOI: 10.1021/

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**Chapter 5**

Provisional chapter

**The Application of Periodic Density Functional Theory**

DOI: 10.5772/intechopen.79558

With the advent of increased computer capacities, improved computational resources, and easier access to large-scale computer facilities, the use of density functional theory methods has become nowadays a frequently used and highly successful approach for the research of solid-state materials. However, the study of solid materials containing heavy elements as lanthanide and actinide elements is very complex due to the large size of these atoms and the requirement of including relativistic effects. These features impose the availability of large computational resources and the use of high quality relativistic pseudopotentials for the description of the electrons localized in the inner shells of these atoms. The important case of the description of uranyl-containing materials and their properties has been faced recently. The study of these materials is very important in the energetic and environmental disciplines. Uranyl-containing materials are fundamental components of the paragenetic sequence of secondary phases that results from the weathering of uraninite ore deposits and are also prominent phases appearing from the alteration of the spent nuclear fuel. The development of a new norm-conserving relativistic pseudopotential for uranium, the use of energy density functionals specific for solids, and the inclusion of empirical dispersion corrections for describing the long-range interactions present in the structures of these materials have allowed the study of the properties of these materials with an unprecedented accuracy level. This feature is very relevant because these methods provide a safe, accurate, and cheap manner of obtaining these properties for uranium-containing materials which are highly radiotoxic, and their experimental studies demand a careful handling of the samples used. In this work, the results of recent applications of theoretical solid state methods based on density functional theory using plane waves and pseudopotentials to the determination of the thermodynamic properties and stability of uranyl-containing materials are reviewed. The knowledge of these thermodynamic properties is indispensable to model the dynamical behavior of nuclear materials under diverse geochemical conditions. The theoretical methods provide

> © 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.

The Application of Periodic Density Functional Theory

**to the Study of Uranyl-Containing Materials:**

to the Study of Uranyl-Containing Materials:

**Thermodynamic Properties and Stability**

Thermodynamic Properties and Stability

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Francisco Colmenero Ruiz

Francisco Colmenero Ruiz

Abstract

http://dx.doi.org/10.5772/intechopen.79558

#### **The Application of Periodic Density Functional Theory to the Study of Uranyl-Containing Materials: Thermodynamic Properties and Stability** The Application of Periodic Density Functional Theory to the Study of Uranyl-Containing Materials: Thermodynamic Properties and Stability

DOI: 10.5772/intechopen.79558

Francisco Colmenero Ruiz Francisco Colmenero Ruiz

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.79558

#### Abstract

With the advent of increased computer capacities, improved computational resources, and easier access to large-scale computer facilities, the use of density functional theory methods has become nowadays a frequently used and highly successful approach for the research of solid-state materials. However, the study of solid materials containing heavy elements as lanthanide and actinide elements is very complex due to the large size of these atoms and the requirement of including relativistic effects. These features impose the availability of large computational resources and the use of high quality relativistic pseudopotentials for the description of the electrons localized in the inner shells of these atoms. The important case of the description of uranyl-containing materials and their properties has been faced recently. The study of these materials is very important in the energetic and environmental disciplines. Uranyl-containing materials are fundamental components of the paragenetic sequence of secondary phases that results from the weathering of uraninite ore deposits and are also prominent phases appearing from the alteration of the spent nuclear fuel. The development of a new norm-conserving relativistic pseudopotential for uranium, the use of energy density functionals specific for solids, and the inclusion of empirical dispersion corrections for describing the long-range interactions present in the structures of these materials have allowed the study of the properties of these materials with an unprecedented accuracy level. This feature is very relevant because these methods provide a safe, accurate, and cheap manner of obtaining these properties for uranium-containing materials which are highly radiotoxic, and their experimental studies demand a careful handling of the samples used. In this work, the results of recent applications of theoretical solid state methods based on density functional theory using plane waves and pseudopotentials to the determination of the thermodynamic properties and stability of uranyl-containing materials are reviewed. The knowledge of these thermodynamic properties is indispensable to model the dynamical behavior of nuclear materials under diverse geochemical conditions. The theoretical methods provide

© 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.

a profound understanding of the thermodynamic stability of these mineral phases and represent a powerful predictive tool to determine their thermodynamic properties.

for the software packages for the calculation of equilibria in multi-component systems [12, 13, 18, 19, 22, 23, 29–32]. Examples of successful applications of thermodynamic techniques to the computation of equilibrium phase assemblages in geological and planetary systems have been

The Application of Periodic Density Functional Theory to the Study of Uranyl-Containing Materials…

http://dx.doi.org/10.5772/intechopen.79558

93

In the field of nuclear technology, the thermodynamic information is indispensable in order to predict the chemical behavior and dynamics of nuclear materials under diverse environmental conditions. The knowledge of precise thermodynamic data is fundamental for the development of geochemical models for nuclear fuel degradation, to evaluate the origin and evolution of uranium ore bodies, in developing programs for the solution mining of uranium deposits or mine dumps, in the study of spent nuclear fuel (SNF) radioactive waste and in the containment of such waste, and may also be of importance in reactions within breeder reactors [37–41].

The behavior of a deep geologic repository (DGR) of high level radioactive waste (HLWR) will depend mainly on the interaction between the SNF and their surroundings. The hydration and corrosion of the SNF under oxidizing conditions will result in the dissolution of the uranium dioxide composing the SNF matrix and the formation of uranyl secondary mineral phases [42–54]. Therefore, the formation and stability of uranyl minerals will determine the release of U(VI) and other actinide elements from the HLWR container and subsequently from the

The stabilities and dissolution rates of uranyl minerals are functions of the solution composition, temperature, and local conditions (mainly pH and electrochemical potential), and their prediction requires the knowledge of the Gibbs free energy, enthalpy, and entropy thermodynamic functions of formation for each phase of interest and their variation with temperature. The simulation of the release of uranium from DGRs under oxidizing conditions and the mobility of uranium in the environment can be only performed if the thermodynamic properties of the secondary uranyl minerals that may form in the DGR are available. Consequently, the knowledge of the thermodynamic parameters is crucial for predicting DGR performance [71–73]. However, reliable temperature-dependent thermodynamic data are completely lacking, except for the simplest uranyl-containing materials. Therefore, the development of a

The rapid development of the nuclear technology strongly encouraged the research on the field of thermodynamics of nuclear materials and the development of nuclear thermodynamic databases [74, 75]. The great significance of the thermodynamic information of materials containing uranium and related elements in the assessment of the safety of DGRs is reflected by the large number of recent experimental works leading to large reviews and updates of thermodynamic properties of materials [72–77]. Among these studies, we may remark the recent experimental measurements by means of solubility and calorimetry techniques of the thermodynamic properties of uranyl peroxide hydrates [51–54], uranyl carbonate minerals [78], uranyl phosphate and orthophosphate minerals [79], and uranyl silicates [71, 80–84].

Despite of the fast progress in the generation of the nuclear thermodynamic database, there are many uranium-containing materials for which the corresponding data are unreliable due to the large experimental uncertainties [38]. A large amount of effort has been dedicated to the

complete thermodynamic database for these minerals is mandatory.

reported [33–36].

repository to the biosphere [55–70].

Keywords: uranyl-containing minerals, spent nuclear fuel, density functional theory, thermodynamic stability, heat capacities, entropies, enthalpies, Gibbs free energies, thermodynamic properties of formation, Gibbs free energies of reaction

#### 1. Introduction

The basis of thermodynamic theory [1] is known since the end of the nineteenth century, and the fundamental developments carried out in the twentieth century have established this theory as a self-contained system of knowledge. The thermodynamic calculations are mainly used for the description of the changes of state associated with the transfer of matter and energy and are an indispensable part of technical and scientific investigations in various fields such as chemistry, metallurgy, chemical engineering, and the energy and environmental technologies. The reliability of the results of thermochemical calculations depends, in the first instance, on the accuracy of the thermochemical data used and the inclusion of the most important species involved, which are often quite numerous if one desires to obtain a fair description of real systems.

There are different sources of data available for establishing thermodynamic information such as calorimetric and solubility measurements [2, 3], phase equilibrium data [4], experimental data on solid solutions [5], and heat capacities and entropies estimated from lattice vibrational models [6–8]. Reasonably complete sets of basic data needed for the calculation of thermochemical functions are available only for a relatively small number of substances [9], and frequently these sets must be completed with empirical data of lower accuracy obtained by analyzing values from different sources as well as by performing estimations before they can be used for actual calculations.

The data listed in the thermochemical tables generally form databases [10–18, 4]. An internally consistent database is one which permits the computation of phase equilibrium relations as established by experimental studies, and it is at the same time compatible with calorimetric and other measurements of thermochemical properties of the phases involved. The generation of such databases is very complex due to the large uncertainties associated with phase equilibrium studies at high temperature and pressure and because many experiments may be irreversible. The CALPHAD method (CALculation of PHAse Diagrams) [19, 20] may be used in the derivation and assessment of this kind of databases [21].

These databases can be used to carry out thermodynamic multi-component, multi-phase, multi-reaction equilibrium calculations on systems made up of any of the substances present in the database. These calculations are best performed by adopting the method of minimization of the total Gibbs free energy of the closed system [22–28]. A detailed description of the principles and techniques used in the computation of equilibrium assemblages of a closed system can be found in the work of Smith and Missen [24]. Thus, these databases are the basis for the software packages for the calculation of equilibria in multi-component systems [12, 13, 18, 19, 22, 23, 29–32]. Examples of successful applications of thermodynamic techniques to the computation of equilibrium phase assemblages in geological and planetary systems have been reported [33–36].

a profound understanding of the thermodynamic stability of these mineral phases and represent a powerful predictive tool to determine their thermodynamic properties. Keywords: uranyl-containing minerals, spent nuclear fuel, density functional theory, thermodynamic stability, heat capacities, entropies, enthalpies, Gibbs free energies,

The basis of thermodynamic theory [1] is known since the end of the nineteenth century, and the fundamental developments carried out in the twentieth century have established this theory as a self-contained system of knowledge. The thermodynamic calculations are mainly used for the description of the changes of state associated with the transfer of matter and energy and are an indispensable part of technical and scientific investigations in various fields such as chemistry, metallurgy, chemical engineering, and the energy and environmental technologies. The reliability of the results of thermochemical calculations depends, in the first instance, on the accuracy of the thermochemical data used and the inclusion of the most important species involved, which are often quite numerous if one desires to obtain a fair

There are different sources of data available for establishing thermodynamic information such as calorimetric and solubility measurements [2, 3], phase equilibrium data [4], experimental data on solid solutions [5], and heat capacities and entropies estimated from lattice vibrational models [6–8]. Reasonably complete sets of basic data needed for the calculation of thermochemical functions are available only for a relatively small number of substances [9], and frequently these sets must be completed with empirical data of lower accuracy obtained by analyzing values from different sources as well as by performing estimations before they can

The data listed in the thermochemical tables generally form databases [10–18, 4]. An internally consistent database is one which permits the computation of phase equilibrium relations as established by experimental studies, and it is at the same time compatible with calorimetric and other measurements of thermochemical properties of the phases involved. The generation of such databases is very complex due to the large uncertainties associated with phase equilibrium studies at high temperature and pressure and because many experiments may be irreversible. The CALPHAD method (CALculation of PHAse Diagrams) [19, 20] may be used in

These databases can be used to carry out thermodynamic multi-component, multi-phase, multi-reaction equilibrium calculations on systems made up of any of the substances present in the database. These calculations are best performed by adopting the method of minimization of the total Gibbs free energy of the closed system [22–28]. A detailed description of the principles and techniques used in the computation of equilibrium assemblages of a closed system can be found in the work of Smith and Missen [24]. Thus, these databases are the basis

thermodynamic properties of formation, Gibbs free energies of reaction

1. Introduction

92 Density Functional Theory

description of real systems.

be used for actual calculations.

the derivation and assessment of this kind of databases [21].

In the field of nuclear technology, the thermodynamic information is indispensable in order to predict the chemical behavior and dynamics of nuclear materials under diverse environmental conditions. The knowledge of precise thermodynamic data is fundamental for the development of geochemical models for nuclear fuel degradation, to evaluate the origin and evolution of uranium ore bodies, in developing programs for the solution mining of uranium deposits or mine dumps, in the study of spent nuclear fuel (SNF) radioactive waste and in the containment of such waste, and may also be of importance in reactions within breeder reactors [37–41].

The behavior of a deep geologic repository (DGR) of high level radioactive waste (HLWR) will depend mainly on the interaction between the SNF and their surroundings. The hydration and corrosion of the SNF under oxidizing conditions will result in the dissolution of the uranium dioxide composing the SNF matrix and the formation of uranyl secondary mineral phases [42–54]. Therefore, the formation and stability of uranyl minerals will determine the release of U(VI) and other actinide elements from the HLWR container and subsequently from the repository to the biosphere [55–70].

The stabilities and dissolution rates of uranyl minerals are functions of the solution composition, temperature, and local conditions (mainly pH and electrochemical potential), and their prediction requires the knowledge of the Gibbs free energy, enthalpy, and entropy thermodynamic functions of formation for each phase of interest and their variation with temperature. The simulation of the release of uranium from DGRs under oxidizing conditions and the mobility of uranium in the environment can be only performed if the thermodynamic properties of the secondary uranyl minerals that may form in the DGR are available. Consequently, the knowledge of the thermodynamic parameters is crucial for predicting DGR performance [71–73]. However, reliable temperature-dependent thermodynamic data are completely lacking, except for the simplest uranyl-containing materials. Therefore, the development of a complete thermodynamic database for these minerals is mandatory.

The rapid development of the nuclear technology strongly encouraged the research on the field of thermodynamics of nuclear materials and the development of nuclear thermodynamic databases [74, 75]. The great significance of the thermodynamic information of materials containing uranium and related elements in the assessment of the safety of DGRs is reflected by the large number of recent experimental works leading to large reviews and updates of thermodynamic properties of materials [72–77]. Among these studies, we may remark the recent experimental measurements by means of solubility and calorimetry techniques of the thermodynamic properties of uranyl peroxide hydrates [51–54], uranyl carbonate minerals [78], uranyl phosphate and orthophosphate minerals [79], and uranyl silicates [71, 80–84].

Despite of the fast progress in the generation of the nuclear thermodynamic database, there are many uranium-containing materials for which the corresponding data are unreliable due to the large experimental uncertainties [38]. A large amount of effort has been dedicated to the assignment and correction of these uncertainties by means of the implementation of new statistical methods for hypothesis testing and the improvement of the techniques used for measuring the thermodynamic properties of these systems [73]. The need to make available a comprehensive, internationally recognized and quality assured chemical thermodynamic database that meets the modeling requirements for the safety assessment of radioactive waste disposal systems prompted the Radioactive Waste Management Committee (RWMC) of the Organization for Economic Co-operation and Development (OECD) Nuclear Energy Agency (NEA) to launch the Thermochemical Database Project (NEA TDB). The RWMC assigned a high priority to the critical review of relevant chemical thermodynamic data of inorganic species, actinide compounds, and fission products [72, 73]. Besides, the range of conditions of temperature and pressure for which the thermodynamic parameters are available for most nuclear materials is rather limited.

today [44–48, 118, 119]. However, our knowledge of this sequence is only qualitative, and the performance assessment of the DGRs for HLRW and many other applications in nuclear tech-

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The crystal structures of these materials [100, 102, 105, 120–123] were successfully determined by means of density functional theory using plane waves and pseudopotentials [124]. A new norm-conserving relativistic pseudopotential specific for uranium atom was generated from first principles with this purpose [101, 121]. Then, using these optimized structures, the thermodynamic properties of these materials were determined including specific heats, entropies, enthalpies, and Gibbs free energies [99–105]. The computed thermodynamic properties were combined with those of the corresponding elements in order to determine the enthalpy and free energy of formation of these materials and its variation with temperature [102, 104, 105]. The methods used in the computation of these thermodynamic functions are briefly described in Section 2. Additionally, the calculation of the Gibbs free energies of reaction and associated reaction constants is also described in this section. The main results obtained are described in Section 3, including a study of the thermodynamic stability of the secondary phases of SNF

The generalized gradient approximation (GGA) together with PBE functional [125] supplemented with Grimme empirical dispersion correction [126] was used to study the uranyl-containing materials such as studtite, metastudtite, dehydrated schoepite, schoepite, metaschoepite, becquerelite, soddyite, rutherfordine, and gamma uranium trioxide [99–105, 120–123]. The introduction of dispersion corrections improved significantly the computed structural, vibrational, mechanic and thermodynamic properties of studtite, metastudtite, dehydrated schoepite, schoepite, metaschoepite, becquerelite, and soddyite as a consequence of the better description of the hydrogen bonding present in the crystal structures of these materials. However, for the case of rutherfordine and gamma uranium trioxide phases, the specialized version of PBE functional for solid materials, PBEsol [127], provided much better results [99, 100, 121]. The improved description of the structure of properties of anhydrous materials using this functional over the one obtained using PBE was observed in recent calculations [99, 100, 121] and later confirmed by other research groups [128, 129]. This justifies the great amount of effort made in developing density functionals specific for solid materials [130] and emphasizes the need of determining empirical dispersion parameters specific for these functionals. These functionals are implemented in CASTEP program [131], a module of the Materials Studio package [132], which was employed to model the structures of the materials considered. The pseudopotentials used for H, C, O, Si, and Ca atoms in the unit cells of these minerals were standard norm-conserving pseudopotentials [133] given in CASTEP code (00PBE-OP type). The norm-conserving relativistic pseudopotential for U atom was generated from first principles as shown in the previous works [101, 121]. While our uranium atom pseudopotential includes scalar relativistic effects, the corresponding

nology require its quantitative specification.

2. Methods

2.1. Computational methodology

[102, 104, 105]. Finally, the main conclusions are given in Section 4.

While the knowledge of the temperature dependence of these properties for anhydrous uranium oxides is very complete [73–75], the corresponding data for the secondary phases which arise from alteration of SNF under final DGR conditions are surprisingly scarce. For these secondary phases, the thermodynamic parameters are known only for the standard state (298.15 K and 1 bar). The lack of temperature-dependent information for these phases rules out the possibility of performing reliable thermodynamic modeling studies for the performance assessment of DGRs for SNF. Because the corresponding information is available for anhydrous species, thermodynamic computations have been performed for the uraniumoxygen and sodium-uranium-oxygen system [85–95]. A detailed analysis of previous studies [41, 96–105] suggests that first principles methodology is an excellent complement to experimental methodology for determining the thermodynamic functions of these materials.

In this work, the computation of the thermodynamic properties of a large set of uranylcontaining materials is reviewed [99–105]. This set includes the uranyl peroxide studtite [(UO2) O24H2O] and metastudtite [(UO2)O22H2O], the uranyl hydroxide dehydrated schoepite [UO2(OH)2], the uranyl oxyhydroxide schoepite [(UO2)8O2(OH)1212H2O], metaschoepite [(UO2)8O2(OH)1210H2O] and becquerelite [Ca(UO2)6O4(OH)68H2O], the uranyl silicate soddyite [(UO2)2(SiO4)2H2O], the uranyl carbonate rutherfordine [UO2CO3], and gamma uranium trioxide [γ-UO3]. The first eight materials have been identified to be basic components of the paragenetic sequence of secondary phases arising from the alteration of uraninite ore deposits and corrosion of SNF under the final DGR conditions [42–54], and gamma uranium trioxide is the main oxide of hexavalent uranium [100, 106, 107]. Uranyl peroxides appear in the earlier stages of this paragenetic sequence [50–54, 108–111] due to the production of hydrogen peroxide and other oxidants resulting from the radiolysis of water due to the ionizing radiation of the SNF. The uranyl oxyhydroxides also begin to appear soon from the alteration of uranium dioxide [42–48]. Studtite, schoepite, metaschoepite, and becquerelite phases have been observed as alteration products of spent fuel in cooling basins at the Hanford Site (Washington) [112–116] and on Chernobyl "lava" formed during the nuclear accident that occurred in 1986 [117]. The next mineral phases appearing in this sequence are uranyl silicates and, less frequently, uranyl phosphates [42–48]. Uranyl carbonates may precipitate where the evaporation is significant, and the carbon dioxide partial pressure is large [49, 118]. The main ingredients of this paragenetic sequence were inferred by Frondel already in 1956 [42, 43]. The sequence is still widely accepted

today [44–48, 118, 119]. However, our knowledge of this sequence is only qualitative, and the performance assessment of the DGRs for HLRW and many other applications in nuclear technology require its quantitative specification.

The crystal structures of these materials [100, 102, 105, 120–123] were successfully determined by means of density functional theory using plane waves and pseudopotentials [124]. A new norm-conserving relativistic pseudopotential specific for uranium atom was generated from first principles with this purpose [101, 121]. Then, using these optimized structures, the thermodynamic properties of these materials were determined including specific heats, entropies, enthalpies, and Gibbs free energies [99–105]. The computed thermodynamic properties were combined with those of the corresponding elements in order to determine the enthalpy and free energy of formation of these materials and its variation with temperature [102, 104, 105]. The methods used in the computation of these thermodynamic functions are briefly described in Section 2. Additionally, the calculation of the Gibbs free energies of reaction and associated reaction constants is also described in this section. The main results obtained are described in Section 3, including a study of the thermodynamic stability of the secondary phases of SNF [102, 104, 105]. Finally, the main conclusions are given in Section 4.

#### 2. Methods

assignment and correction of these uncertainties by means of the implementation of new statistical methods for hypothesis testing and the improvement of the techniques used for measuring the thermodynamic properties of these systems [73]. The need to make available a comprehensive, internationally recognized and quality assured chemical thermodynamic database that meets the modeling requirements for the safety assessment of radioactive waste disposal systems prompted the Radioactive Waste Management Committee (RWMC) of the Organization for Economic Co-operation and Development (OECD) Nuclear Energy Agency (NEA) to launch the Thermochemical Database Project (NEA TDB). The RWMC assigned a high priority to the critical review of relevant chemical thermodynamic data of inorganic species, actinide compounds, and fission products [72, 73]. Besides, the range of conditions of temperature and pressure for which the thermodynamic parameters are available for most

While the knowledge of the temperature dependence of these properties for anhydrous uranium oxides is very complete [73–75], the corresponding data for the secondary phases which arise from alteration of SNF under final DGR conditions are surprisingly scarce. For these secondary phases, the thermodynamic parameters are known only for the standard state (298.15 K and 1 bar). The lack of temperature-dependent information for these phases rules out the possibility of performing reliable thermodynamic modeling studies for the performance assessment of DGRs for SNF. Because the corresponding information is available for anhydrous species, thermodynamic computations have been performed for the uraniumoxygen and sodium-uranium-oxygen system [85–95]. A detailed analysis of previous studies [41, 96–105] suggests that first principles methodology is an excellent complement to experi-

mental methodology for determining the thermodynamic functions of these materials.

In this work, the computation of the thermodynamic properties of a large set of uranylcontaining materials is reviewed [99–105]. This set includes the uranyl peroxide studtite [(UO2) O24H2O] and metastudtite [(UO2)O22H2O], the uranyl hydroxide dehydrated schoepite [UO2(OH)2], the uranyl oxyhydroxide schoepite [(UO2)8O2(OH)1212H2O], metaschoepite [(UO2)8O2(OH)1210H2O] and becquerelite [Ca(UO2)6O4(OH)68H2O], the uranyl silicate soddyite [(UO2)2(SiO4)2H2O], the uranyl carbonate rutherfordine [UO2CO3], and gamma uranium trioxide [γ-UO3]. The first eight materials have been identified to be basic components of the paragenetic sequence of secondary phases arising from the alteration of uraninite ore deposits and corrosion of SNF under the final DGR conditions [42–54], and gamma uranium trioxide is the main oxide of hexavalent uranium [100, 106, 107]. Uranyl peroxides appear in the earlier stages of this paragenetic sequence [50–54, 108–111] due to the production of hydrogen peroxide and other oxidants resulting from the radiolysis of water due to the ionizing radiation of the SNF. The uranyl oxyhydroxides also begin to appear soon from the alteration of uranium dioxide [42–48]. Studtite, schoepite, metaschoepite, and becquerelite phases have been observed as alteration products of spent fuel in cooling basins at the Hanford Site (Washington) [112–116] and on Chernobyl "lava" formed during the nuclear accident that occurred in 1986 [117]. The next mineral phases appearing in this sequence are uranyl silicates and, less frequently, uranyl phosphates [42–48]. Uranyl carbonates may precipitate where the evaporation is significant, and the carbon dioxide partial pressure is large [49, 118]. The main ingredients of this paragenetic sequence were inferred by Frondel already in 1956 [42, 43]. The sequence is still widely accepted

nuclear materials is rather limited.

94 Density Functional Theory

#### 2.1. Computational methodology

The generalized gradient approximation (GGA) together with PBE functional [125] supplemented with Grimme empirical dispersion correction [126] was used to study the uranyl-containing materials such as studtite, metastudtite, dehydrated schoepite, schoepite, metaschoepite, becquerelite, soddyite, rutherfordine, and gamma uranium trioxide [99–105, 120–123]. The introduction of dispersion corrections improved significantly the computed structural, vibrational, mechanic and thermodynamic properties of studtite, metastudtite, dehydrated schoepite, schoepite, metaschoepite, becquerelite, and soddyite as a consequence of the better description of the hydrogen bonding present in the crystal structures of these materials. However, for the case of rutherfordine and gamma uranium trioxide phases, the specialized version of PBE functional for solid materials, PBEsol [127], provided much better results [99, 100, 121]. The improved description of the structure of properties of anhydrous materials using this functional over the one obtained using PBE was observed in recent calculations [99, 100, 121] and later confirmed by other research groups [128, 129]. This justifies the great amount of effort made in developing density functionals specific for solid materials [130] and emphasizes the need of determining empirical dispersion parameters specific for these functionals. These functionals are implemented in CASTEP program [131], a module of the Materials Studio package [132], which was employed to model the structures of the materials considered. The pseudopotentials used for H, C, O, Si, and Ca atoms in the unit cells of these minerals were standard norm-conserving pseudopotentials [133] given in CASTEP code (00PBE-OP type). The norm-conserving relativistic pseudopotential for U atom was generated from first principles as shown in the previous works [101, 121]. While our uranium atom pseudopotential includes scalar relativistic effects, the corresponding pseudopotentials used for H, C, O, Si, and Ca atoms do not include them. This pseudopotential has been used extensively in the research of uranyl-containing materials [99–105, 120–123].

corresponding X-ray powder pattern is determined [136] and compared with the experimental one. Only if the comparison is satisfactory, the crystal structure is accepted in order to obtain the final vibrational, mechanic, thermodynamic, and optic properties of the material under study. Otherwise, the calculation parameters are made more stringent and the structural optimization starts again. The convergence of this procedure depends on the proximity of the initial input used to the final solution. If it does not converge or converges towards a structure yielding an X-ray powder pattern which does not agree with the experimental one, the procedure should be restarted from a different initial input (atomic positions and cell parameters).

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The methods employed for the calculation of thermodynamic properties of these materials were described in the previous papers [99–105]. The phonon spectrum at the different points of Brillouin zone can be determined by density functional perturbation theory (DFPT) [137–139] as second-order derivatives of the total energy [137]. Phonon dispersion curves and density of states were calculated, and from them, several important thermodynamic quantities in the quasi-harmonic approximation, such as Gibbs free energy, enthalpy, entropy, and specific heat,

The thermodynamic functions of formation at the different temperatures were determined [102] from the calculated enthalpy and entropy functions of the material being considered,

the experimental enthalpy and entropy functions of the elements forming part of the material. The enthalpy and entropy functions for the elements were taken from JANAF tables [11], and the corresponding functions for uranium atom were taken from Barin [12]. The equilibrium constants for the formation reactions were determined in terms of the corresponding calculated Gibbs free energies of formation using the well-known relationship [11], Δ<sup>f</sup> G(T) = R T

The enthalpies and Gibbs free energies of a given reaction at the different temperatures, ΔrH(T) and ΔrG(T), were determined [103] from the Gibbs free energy of formation and entropy

values used of these properties for studtite, metastudtite, dehydrated schoepite, schoepite, metaschoepite, becquerelite, soddyite, rutherfordine, and gamma uranium trioxide were determined in our previous works [102, 104, 105]. The corresponding data for the remaining materials, which do not contain the uranyl ion, SiO2(cr), H2O(l), CO2(g), O2(g), and H2(g), were taken from JANAF tables [11] and the data for H2O2(l) were taken from Barin [12]. The reaction equilibrium constants were determined in terms of the corresponding Gibbs free energies of

calc, the experimental value of its standard enthalpy of formation, ΔfH<sup>0</sup>

(T) and S<sup>i</sup>

(T), i = 1,…,Nmat. The specific

, and

2.3. Enthalpy and Gibbs free energy of formation in terms of the elements

2.2. Thermodynamic properties

were evaluated [140].

calc and ST

reaction, ΔrG(T) = R T Ln Kr.

2.4. Enthalpies and free energies of reaction

functions of the materials entering in the reaction, ΔfG<sup>i</sup>

[HT -H298]

Ln Kf.

The atomic positions and cell parameters were optimized using the Broyden-Fletcher-- Goldfarb-Shanno method [124, 134] with a convergence threshold on atomic forces of 0.01 eV/Å. The kinetic energy cut-off and K-point mesh [135] were chosen to ensure good convergence for computed structures and energies. The structures of the materials considered in this work were optimized in calculations with augmented complexity by increasing these parameters. The precise calculation parameters used to determine the final results may be found in the corresponding articles [99–105, 120–123]. The flow diagram associated to the theoretical treatment used to study a given crystalline material is shown in Figure 1 [101]. The crystal structure is first optimized starting from an initial atomistic model of the corresponding unit cell (lattice parameters and atomic positions) employing trial values of the calculation parameters (kinetic energy cut-off and K-point mesh). The crystal structure is then updated and the initial values of the kinetic cutoff and K-point density (number of k-points divided by the k-point separation) are systematically increased. The geometry optimization is performed again until the variation of the energy is below a given threshold. The variation of the crystal unit cell is then analyzed, and the structure is reoptimized until this variation is small enough. Once the convergence in the computed energy and structure is met, the

Figure 1. Flow diagram associated to the theoretical solid state treatment used to study the uranyl-containing materials considered in this work.

corresponding X-ray powder pattern is determined [136] and compared with the experimental one. Only if the comparison is satisfactory, the crystal structure is accepted in order to obtain the final vibrational, mechanic, thermodynamic, and optic properties of the material under study. Otherwise, the calculation parameters are made more stringent and the structural optimization starts again. The convergence of this procedure depends on the proximity of the initial input used to the final solution. If it does not converge or converges towards a structure yielding an X-ray powder pattern which does not agree with the experimental one, the procedure should be restarted from a different initial input (atomic positions and cell parameters).

#### 2.2. Thermodynamic properties

pseudopotentials used for H, C, O, Si, and Ca atoms do not include them. This pseudopotential has been used extensively in the research of uranyl-containing materials [99–105, 120–123].

The atomic positions and cell parameters were optimized using the Broyden-Fletcher-- Goldfarb-Shanno method [124, 134] with a convergence threshold on atomic forces of 0.01 eV/Å. The kinetic energy cut-off and K-point mesh [135] were chosen to ensure good convergence for computed structures and energies. The structures of the materials considered in this work were optimized in calculations with augmented complexity by increasing these parameters. The precise calculation parameters used to determine the final results may be found in the corresponding articles [99–105, 120–123]. The flow diagram associated to the theoretical treatment used to study a given crystalline material is shown in Figure 1 [101]. The crystal structure is first optimized starting from an initial atomistic model of the corresponding unit cell (lattice parameters and atomic positions) employing trial values of the calculation parameters (kinetic energy cut-off and K-point mesh). The crystal structure is then updated and the initial values of the kinetic cutoff and K-point density (number of k-points divided by the k-point separation) are systematically increased. The geometry optimization is performed again until the variation of the energy is below a given threshold. The variation of the crystal unit cell is then analyzed, and the structure is reoptimized until this variation is small enough. Once the convergence in the computed energy and structure is met, the

Figure 1. Flow diagram associated to the theoretical solid state treatment used to study the uranyl-containing materials

considered in this work.

96 Density Functional Theory

The methods employed for the calculation of thermodynamic properties of these materials were described in the previous papers [99–105]. The phonon spectrum at the different points of Brillouin zone can be determined by density functional perturbation theory (DFPT) [137–139] as second-order derivatives of the total energy [137]. Phonon dispersion curves and density of states were calculated, and from them, several important thermodynamic quantities in the quasi-harmonic approximation, such as Gibbs free energy, enthalpy, entropy, and specific heat, were evaluated [140].

#### 2.3. Enthalpy and Gibbs free energy of formation in terms of the elements

The thermodynamic functions of formation at the different temperatures were determined [102] from the calculated enthalpy and entropy functions of the material being considered, [HT -H298] calc and ST calc, the experimental value of its standard enthalpy of formation, ΔfH<sup>0</sup> , and the experimental enthalpy and entropy functions of the elements forming part of the material. The enthalpy and entropy functions for the elements were taken from JANAF tables [11], and the corresponding functions for uranium atom were taken from Barin [12]. The equilibrium constants for the formation reactions were determined in terms of the corresponding calculated Gibbs free energies of formation using the well-known relationship [11], Δ<sup>f</sup> G(T) = R T Ln Kf.

#### 2.4. Enthalpies and free energies of reaction

The enthalpies and Gibbs free energies of a given reaction at the different temperatures, ΔrH(T) and ΔrG(T), were determined [103] from the Gibbs free energy of formation and entropy functions of the materials entering in the reaction, ΔfG<sup>i</sup> (T) and S<sup>i</sup> (T), i = 1,…,Nmat. The specific values used of these properties for studtite, metastudtite, dehydrated schoepite, schoepite, metaschoepite, becquerelite, soddyite, rutherfordine, and gamma uranium trioxide were determined in our previous works [102, 104, 105]. The corresponding data for the remaining materials, which do not contain the uranyl ion, SiO2(cr), H2O(l), CO2(g), O2(g), and H2(g), were taken from JANAF tables [11] and the data for H2O2(l) were taken from Barin [12]. The reaction equilibrium constants were determined in terms of the corresponding Gibbs free energies of reaction, ΔrG(T) = R T Ln Kr.

#### 3. Results and discussion

#### 3.1. Thermodynamic properties

The computed isobaric heats and entropies at 298.15 K for all the uranyl-containing materials considered in this work [99, 100, 102, 104, 105] are given in Table 1. In this table, the values obtained for rutherfordine, gamma uranium trioxide, and metaschoepite [99, 100, 104] are compared with the corresponding experimental values. For the rest of the materials, there are not experimental values to compare with. As it can be seen, the computed values agree very well with their experimental counterparts. In fact, the differences between the computed and experimental values of these and other thermodynamic properties are frequently smaller than the difference among several different experimental values. From the analysis of the results obtained, the expected accuracy in the computed-specific heats and entropies of studtite, metastudtite, dehydrated schoepite, schoepite, and soddyite is better than 3–5%.

The calculated isobaric specific heat, entropy, and Gibbs free energy functions of rutherfordine, gamma uranium trioxide, and metaschoepite are displayed in Figure 2, where they are compared with the experimental functions of Hemingway [37], Cordfunke and Westrum [143], and Barin [12], respectively. For rutherfordine, the computed thermodynamic functions are compared with those of Hemingway [37] in the temperature range of 298–700 K, and as it can be appreciated, the calculated and experimental curves are nearly parallel. The computed value of Cp at 700 K, near the limit of thermal stability of rutherfordine [37], Cp = 153.3 J K<sup>1</sup> mol<sup>1</sup> , differs from the experimental value at this temperature, 147.6 J <sup>K</sup><sup>1</sup> mol<sup>1</sup> , by only 3.9%.


Similarly, the differences of the computed entropy and Gibbs free energy with respect to Hemingway's experimental values at 700 K are only 2.3 and 1.3%, respectively. Our theoretical calculations allowed to obtain the values of the thermodynamic functions for the low and high temperature ranges 0–300 and 700–1000 K, which were unknown so far and, consequently,

Figure 2. Calculated and experimental isobaric specific heat entropies and Gibbs free energies of rutherfordine, gamma uranium trioxide, and metaschoepite as a function of temperature. The experimental thermodynamic functions of rutherfordine, uranium trioxide, and metaschoepite are from Hemingway [37], Cordfunke and Westrum [143], and Barin

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The computed thermodynamic properties of uranium trioxide are also in excellent agreement with the experimental data of Cordfunke and Westrum [143] in the full range of temperatures considered 0–1000 K. The differences in the specific heat, entropy, and Gibbs free energy functions are 3.9, 1.8, and 0.1% at 100 K and 6.1, 3.6, and 3.5% at 1000 K. The comparison reveals that the low temperature calculated thermodynamic data are also very accurate. It must be emphasized that while the experimental isobaric heat capacity function of gamma uranium trioxide at 1000 K is above the asymptotic Dulong-Petit limit, our computed function

extended the range in which the thermodynamic functions were known to 0–1000 K.

satisfies properly the requirement of being below this limit [100].

[12], respectively.

All the values are given in units of J K<sup>1</sup> mol<sup>1</sup> . The percent difference of the theoretical and experimental results is given in parenthesis for the specific heats and entropies of rutherfordine, gamma uranium trioxide, and metaschoepite.

Table 1. Specific heats and entropies at 298.15 K for the uranyl-containing material studied in this work.

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3. Results and discussion

98 Density Functional Theory

3.1. Thermodynamic properties

The computed isobaric heats and entropies at 298.15 K for all the uranyl-containing materials considered in this work [99, 100, 102, 104, 105] are given in Table 1. In this table, the values obtained for rutherfordine, gamma uranium trioxide, and metaschoepite [99, 100, 104] are compared with the corresponding experimental values. For the rest of the materials, there are not experimental values to compare with. As it can be seen, the computed values agree very well with their experimental counterparts. In fact, the differences between the computed and experimental values of these and other thermodynamic properties are frequently smaller than the difference among several different experimental values. From the analysis of the results obtained, the expected accuracy in the computed-specific heats and entropies of studtite,

The calculated isobaric specific heat, entropy, and Gibbs free energy functions of rutherfordine, gamma uranium trioxide, and metaschoepite are displayed in Figure 2, where they are compared with the experimental functions of Hemingway [37], Cordfunke and Westrum [143], and Barin [12], respectively. For rutherfordine, the computed thermodynamic functions are compared with those of Hemingway [37] in the temperature range of 298–700 K, and as it can be appreciated, the calculated and experimental curves are nearly parallel. The computed value of Cp at 700 K, near the limit of thermal stability of rutherfordine [37], Cp = 153.3 J K<sup>1</sup> mol<sup>1</sup>

Exp. 106.5 [37] (8.0%), 120.1 [141] (5.1%) 142.70 [37] (0.3%), 139 [142] (3.0%)

Exp. 154.40 [12] (8.0%) 167.00 [12] (0.5%)

in parenthesis for the specific heats and entropies of rutherfordine, gamma uranium trioxide, and metaschoepite.

Table 1. Specific heats and entropies at 298.15 K for the uranyl-containing material studied in this work.

Exp. 81.67 [143] (5.3%), 84.72 [37] (8.7%) 96.11 [143] (3.3%), 98.6 [144] (5.7%)

. The percent difference of the theoretical and experimental results is given

,

, by only 3.9%.

metastudtite, dehydrated schoepite, schoepite, and soddyite is better than 3–5%.

differs from the experimental value at this temperature, 147.6 J <sup>K</sup><sup>1</sup> mol<sup>1</sup>

Material Source Cp S Rutherfordine Calc. [99] 115.02 143.11

γ-UO3 Calc. [100] 77.36 92.96

Metaschoepite Calc. [104] 142.01 166.24

Studtite Calc. [102] 219.97, 211.17 [41] 232.12 Metastudtite Calc. [102] 163.14, 155.81 [41] 179.27 Dehydrated schoepite Calc. [102] 103.85 125.18 Schoepite Calc. [104] 150.62 168.75 Soddyite Calc. [102] 275.15 315.95 Becquerelite Calc. [105] 148.40 172.34

All the values are given in units of J K<sup>1</sup> mol<sup>1</sup>

Figure 2. Calculated and experimental isobaric specific heat entropies and Gibbs free energies of rutherfordine, gamma uranium trioxide, and metaschoepite as a function of temperature. The experimental thermodynamic functions of rutherfordine, uranium trioxide, and metaschoepite are from Hemingway [37], Cordfunke and Westrum [143], and Barin [12], respectively.

Similarly, the differences of the computed entropy and Gibbs free energy with respect to Hemingway's experimental values at 700 K are only 2.3 and 1.3%, respectively. Our theoretical calculations allowed to obtain the values of the thermodynamic functions for the low and high temperature ranges 0–300 and 700–1000 K, which were unknown so far and, consequently, extended the range in which the thermodynamic functions were known to 0–1000 K.

The computed thermodynamic properties of uranium trioxide are also in excellent agreement with the experimental data of Cordfunke and Westrum [143] in the full range of temperatures considered 0–1000 K. The differences in the specific heat, entropy, and Gibbs free energy functions are 3.9, 1.8, and 0.1% at 100 K and 6.1, 3.6, and 3.5% at 1000 K. The comparison reveals that the low temperature calculated thermodynamic data are also very accurate. It must be emphasized that while the experimental isobaric heat capacity function of gamma uranium trioxide at 1000 K is above the asymptotic Dulong-Petit limit, our computed function satisfies properly the requirement of being below this limit [100].

Finally, the theoretical results for metaschoepite mineral phase agree very well with the experimental thermodynamic properties reported by Barin [12] even at temperatures of the order of 800 K, the percent differences of the calculated specific heat, entropy, and Gibbs energy with respect to the corresponding experimental values being 5.4%, 3.2%, and 2.0% at 800 K. The present theoretical data have permitted to discriminate between the experimental thermodynamic functions of metaschoepite reported up to date because the experimental functions reported by Tasker et al. [145] deviate from those of Barin [12] and from our theoretical results already at moderate temperatures [104].

The comparisons performed in the previous paragraph have shown that the variation of the computed thermodynamic functions with temperature is excellent. Hence, it may be expected that the theoretical functions obtained for studtite, metastudtite, dehydrated schoepite, soddyite, schoepite, and becquerelite [102, 104, 105] will also be reliable even at low and high temperatures. For rutherfordine and metaschoepite [99, 104], the calculated thermodynamic properties are recommended instead of the experimental functions because they cover the full temperature range going from 0 to 1000 K and they should provide a uniform accuracy at all temperatures. For gamma uranium trioxide, both sets of data are considered to be equally accurate, but the theoretical specific heat function satisfies properly the asymptotic conditions [100].

#### 3.2. Enthalpies and free energies of formation in terms of the elements

The enthalpies and free energies of formation in terms of the elements of the considered mineral phases as a function or temperature were determined [102, 104, 105] from the calculated thermodynamic data, the experimental or estimated [104] standard enthalpy of formation, and the thermodynamic functions of the corresponding elements [11–12]. The calculated Gibbs free energies of formation of rutherfordine, gamma uranium trioxide, and metaschoepite [102, 104] are shown in Figure 3 together with the corresponding experimental data [12, 37, 143].

3.3. Enthalpies and free energies of reaction

tively. For schoepite, there are no experimental data to compare with.

3.3.1. Reactions of formation in terms of oxides Let us first consider the following reactions:

Studtite: UO3ð Þþ cr 4 H2O lðÞþ 1=2O2ð Þ! g ð Þ UO2 O2 � 4H2O cr ð Þ (I)

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Dehydrated schoepite: UO3ð Þþ cr H2O lðÞ! UO2ð Þ OH <sup>2</sup>ð Þ cr (III)

Rutherfordine: UO3ð Þþ cr CO2ð Þ! g UO2CO3ð Þ cr (VII)

Metastudtite: UO3ð Þþ cr 2 H2O lðÞþ 1=2O2ð Þ! g ð Þ UO2 O2 � 2H2O cr ð Þ (II)

Schoepite: UO3ð Þþ cr <sup>9</sup>=4 H2O lðÞ! <sup>1</sup>=8 UO ð Þ<sup>2</sup> <sup>8</sup>O2ð Þ OH <sup>12</sup> � 12H2O cr ð Þ (V)

Metaschoepite: UO3ð Þþ cr 2 H2O lðÞ! <sup>1</sup>=8 UO ð Þ<sup>2</sup> <sup>8</sup>O2ð Þ OH <sup>12</sup> � 10 H2O cr ð Þ (VI)

Soddyite : 2UO3ð Þþ cr SiO2ð Þþ cr 2H2O lðÞ! ð Þ UO2 <sup>2</sup>ð Þ� SiO4 2H2O cr ð Þ (VIII)

Becquerelite: UO3ð Þþ cr 1=6 CaO cr ð Þþ 11=6H2O lðÞ! 1=6 Ca UO ð Þ<sup>2</sup> <sup>6</sup>O4ð Þ OH <sup>6</sup> � 8H2O cr ð Þ (IV)

Figure 3. Calculated Gibbs free energies of formation of rutherfordine, gamma uranium trioxide, and metaschoepite in terms of the elements as a function of temperature. The experimental Gibbs free energies of formation of rutherfordine, uranium trioxide, and metaschoepite are from Hemingway [37], Cordfunke and Westrum [143], and Barin [12], respec-

As it may be observed in Figure 3, and as it occurred with the thermodynamic functions of the pure substances reported in Section 3.1, the calculated thermodynamic properties of formation agree with the experimental functions in an excellent manner. For these three materials, the differences of the calculated and experimental values are lower than 1% at ambient temperature and the differences remain very small at high temperatures. The differences become 1.6%, 1.0%, and 2.0% at 700, 900, and 800 K for rutherfordine, γ-UO3, and metaschoepite, respectively [102, 104]. Since the theoretical solid state treatments used for studtite, metastudtite, dehydrated schoepite, soddyite, schoepite, and becquerelite were essentially the same as those used for these three materials, we expect a similar accuracy level for their calculated thermodynamic parameters of formation. An example of these calculated parameters for a material in which there are no experimental data to compare with is the case of the schoepite mineral phase, and the corresponding results are also shown in Figure 3. The combination of the results for schoepite and metaschoepite allowed to study the thermodynamics of the dehydration transformation of schoepite into metaschoepite [104].

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Figure 3. Calculated Gibbs free energies of formation of rutherfordine, gamma uranium trioxide, and metaschoepite in terms of the elements as a function of temperature. The experimental Gibbs free energies of formation of rutherfordine, uranium trioxide, and metaschoepite are from Hemingway [37], Cordfunke and Westrum [143], and Barin [12], respectively. For schoepite, there are no experimental data to compare with.

#### 3.3. Enthalpies and free energies of reaction

#### 3.3.1. Reactions of formation in terms of oxides

Finally, the theoretical results for metaschoepite mineral phase agree very well with the experimental thermodynamic properties reported by Barin [12] even at temperatures of the order of 800 K, the percent differences of the calculated specific heat, entropy, and Gibbs energy with respect to the corresponding experimental values being 5.4%, 3.2%, and 2.0% at 800 K. The present theoretical data have permitted to discriminate between the experimental thermodynamic functions of metaschoepite reported up to date because the experimental functions reported by Tasker et al. [145] deviate from those of Barin [12]

The comparisons performed in the previous paragraph have shown that the variation of the computed thermodynamic functions with temperature is excellent. Hence, it may be expected that the theoretical functions obtained for studtite, metastudtite, dehydrated schoepite, soddyite, schoepite, and becquerelite [102, 104, 105] will also be reliable even at low and high temperatures. For rutherfordine and metaschoepite [99, 104], the calculated thermodynamic properties are recommended instead of the experimental functions because they cover the full temperature range going from 0 to 1000 K and they should provide a uniform accuracy at all temperatures. For gamma uranium trioxide, both sets of data are considered to be equally accurate, but the theoretical specific heat function satisfies prop-

The enthalpies and free energies of formation in terms of the elements of the considered mineral phases as a function or temperature were determined [102, 104, 105] from the calculated thermodynamic data, the experimental or estimated [104] standard enthalpy of formation, and the thermodynamic functions of the corresponding elements [11–12]. The calculated Gibbs free energies of formation of rutherfordine, gamma uranium trioxide, and metaschoepite [102, 104] are shown in Figure 3 together with the corresponding experimental data

As it may be observed in Figure 3, and as it occurred with the thermodynamic functions of the pure substances reported in Section 3.1, the calculated thermodynamic properties of formation agree with the experimental functions in an excellent manner. For these three materials, the differences of the calculated and experimental values are lower than 1% at ambient temperature and the differences remain very small at high temperatures. The differences become 1.6%, 1.0%, and 2.0% at 700, 900, and 800 K for rutherfordine, γ-UO3, and metaschoepite, respectively [102, 104]. Since the theoretical solid state treatments used for studtite, metastudtite, dehydrated schoepite, soddyite, schoepite, and becquerelite were essentially the same as those used for these three materials, we expect a similar accuracy level for their calculated thermodynamic parameters of formation. An example of these calculated parameters for a material in which there are no experimental data to compare with is the case of the schoepite mineral phase, and the corresponding results are also shown in Figure 3. The combination of the results for schoepite and metaschoepite allowed to study the thermodynamics of the dehydration transformation of schoepite into

and from our theoretical results already at moderate temperatures [104].

3.2. Enthalpies and free energies of formation in terms of the elements

erly the asymptotic conditions [100].

[12, 37, 143].

100 Density Functional Theory

metaschoepite [104].

Let us first consider the following reactions:

$$\text{Studite:}\,\mathrm{UO\_3(cr)} + 4\,\mathrm{H\_2O(l)} + 1/2\mathrm{O\_2(g)} \to (\mathrm{UO\_2})\mathrm{O\_2} \cdot 4\,\mathrm{H\_2O(cr)}\tag{l}$$

$$\text{Metastudtite:}\ \mathrm{UO\_3(cr)} + 2\ \mathrm{H\_2O(l)} + 1/2\mathrm{O\_2(g)} \rightarrow (\mathrm{UO\_2})\mathrm{O\_2} \cdot 2\mathrm{H\_2O(cr)}\tag{\text{II}}$$

$$\text{Dehydration schedule:}\ \text{UO}\_3(\text{cr}) + \text{H}\_2\text{O}(\text{l}) \rightarrow \text{UO}\_2(\text{OH})\_2(\text{cr})\tag{\text{III}}$$

$$\text{Becquerelite:}\,\mathrm{UO\_3(cr)} + 1/6\,\mathrm{CaO(cr)} + 11/6\,\mathrm{H\_2O(l)} \rightarrow 1/6\,\mathrm{Ca(UO\_2)\_6O\_4(OH)\_6}\cdot 8\,\mathrm{H\_2O(cr)}\text{ (IV)}$$

$$\text{Schedepite: } \mathrm{UO\_3(cr)} + 9/4 \,\mathrm{H\_2O(l)} \to 1/8 \,\left[ (\mathrm{UO\_2})\_8 \mathrm{O\_2(OH)\_{12}} \right] \cdot 12 \,\mathrm{H\_2O(cr)}\tag{V}$$

$$\text{Metaschoepite:}\ \text{UO}\_3(\text{cr}) + 2\ \text{H}\_2\text{O}(\text{l}) \rightarrow 1/8\ \left[ (\text{UO}\_2)\_8\text{O}\_2(\text{OH})\_{12} \right] \cdot 10\ \text{H}\_2\text{O}(\text{cr})\tag{\text{VI}}$$

$$\text{Ratherfordine:}\,\text{UO}\_3(\text{cr}) + \text{CO}\_2(\text{g}) \rightarrow \text{UO}\_2\text{CO}\_3(\text{cr})\tag{\text{VII}}$$

$$\text{Soddyite}: 2\,\text{UO}\_3(\text{cr}) + \text{SiO}\_2(\text{cr}) + 2\,\text{H}\_2\text{O}(\text{l}) \rightarrow (\text{UO}\_2)\_2(\text{SiO}\_4) \cdot 2\,\text{H}\_2\text{O}(\text{cr})\tag{\text{VII}}$$

These important reactions represent the formation of the considered uranyl-containing materials in terms of the corresponding oxides. By combining the calculated thermodynamic properties of formation of these materials in terms of the elements [102, 104, 105] with those of the non-uranyl-bearing materials [11] present in reactions (I)–(VIII), we obtained the Gibbs free energies and associated reaction constants displayed in Figure 4 [103–105].

Figure 4 shows that studtite and metastudtite are unstable with respect to the corresponding oxides at the full range of temperatures studied, 250–500 K, since the corresponding Gibbs free energies of reaction are positive everywhere. Therefore, they are metastable phases at normal conditions. The opposite is true for soddyite mineral phase, which is stable at all the temperatures. However, dehydrated schoepite, becquerelite, schoepite, metaschoepite, and rutherfordine mineral phases are stable at ambient temperature and become unstable at the temperatures of 462, 491, 383, 352, and 514 K, respectively, because the Gibbs free energy of reaction becomes positive at these temperatures. The observation of changes of stability for these phases at these relatively low temperatures was unexpected and highlights the great relevance of the availability of accurate temperature-dependent thermodynamic functions [102, 103].

#### 3.3.2. Reactions of transformation of uranyl-containing materials into studtite in the presence of high hydrogen peroxide concentrations

We will now study the thermodynamic properties of the following set of reactions:

$$\rm{UO\_2(OH)\_2(cr) + 2\ H\_2O\_2(l) + H\_2(g) \to (UO\_2)O\_2 \cdot 4H\_2O(cr)}\tag{18}$$

$$2\text{ }1/2\text{ (UO}\_2\text{)}\_2(\text{SiO}\_4)\cdot2\text{H}\_2\text{O}(\text{cr}) + 2\text{ }1\text{H}\_2\text{O}\_2(\text{l}) + \text{H}\_2(\text{g}) \rightarrow (\text{UO}\_2)\text{O}\_2\cdot4\text{H}\_2\text{O}(\text{cr}) + 1/2\text{ SiO}\_2(\text{cr})\text{ (X)}$$

$$\text{UO}\_3(\text{cr}) + 3\,\text{H}\_2\text{O}\_2(\text{l}) + \text{H}\_2(\text{g}) \to (\text{UO}\_2)\text{O}\_2 \cdot 4\text{H}\_2\text{O}(\text{cr}) + 1/2\,\text{O}\_2(\text{g})\tag{\text{XI}}$$

$$\rm{UO\_2CO\_3(cr) + 3\ H\_2O\_2(l) + H\_2(g) \to (UO\_2)O\_2 \cdot 4H\_2O(cr) + CO\_2(g) + 1/2O\_2(g)}\tag{XII}$$

$$\text{(UO}\_2\text{)O}\_2\cdot2\text{H}\_2\text{O(cr)} + \text{H}\_2\text{O}\_2(\text{l}) + \text{H}\_2(\text{g}) \rightarrow \text{(UO}\_2\text{)O}\_2\cdot4\text{H}\_2\text{O(cr)}\tag{XIII}$$

$$\begin{aligned} &1/6\,\mathrm{Ca(UO\_2)}\_{6}\mathrm{O\_4(OH)}\_{6}\cdot8\mathrm{H\_2O(cr)} + 13/6\,\mathrm{H\_2O\_2(l)} \rightarrow \mathrm{(UO\_2)O\_2}\cdot4\mathrm{H\_2O(cr)} + 1/16\,\mathrm{CaO(cr)}\\ &+ 7/12\,\mathrm{O\_2(g)} \end{aligned}$$

$$\text{(xv)}$$

[103–105].

Figure 4. Calculated Gibbs free energies of formation of studtite, metastudtite, dehydrated schoepite, becquerelite, schoepite, metaschoepite, rutherfordine, and soddyite in terms of the corresponding oxides as a function of temperature

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$$\begin{aligned} &1/8 \left[ (\text{UO}\_2)\_8\text{O}\_2(\text{OH})\_{12} \right] \cdot 12\text{H}\_2\text{O}(\text{cr}) + 7/4 \,\text{H}\_2\text{O}\_2(\text{l}) \rightarrow (\text{UO}\_2)\text{O}\_2 \cdot 4\text{H}\_2\text{O}(\text{cr}) + 3/8 \,\text{O}\_2(\text{g}) \quad \text{(XV)}\\ &1/8 \left[ (\text{UO}\_2)\_8\text{O}\_2(\text{OH})\_{12} \right] \cdot 12\text{H}\_2\text{O}(\text{cr}) + 1/8 \,\text{O}\_2(\text{g}) \rightarrow 1/8 \left[ (\text{UO}\_2)\_8\text{O}\_2(\text{OH})\_{12} \right] \cdot 10 \,\text{H}\_2\text{O}(\text{cr})\\ &+ 1/4 \,\text{H}\_2\text{O}\_2(\text{l}) \end{aligned} \tag{XV}$$

Reactions (IX) to (XV) are the reactions of transformation of dehydrated schoepite, soddyite, uranium trioxide, rutherfordine, metastudtite, becquerelite, and schoepite into studtite in the presence of high hydrogen peroxide concentrations (and absence of water) [103–105]. Reaction (XVI) represents the conversion of schoepite into metaschoepite at these conditions. The computed Gibbs free energies of these reactions are shown in Figure 5. As it can be observed, all

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These important reactions represent the formation of the considered uranyl-containing materials in terms of the corresponding oxides. By combining the calculated thermodynamic properties of formation of these materials in terms of the elements [102, 104, 105] with those of the non-uranyl-bearing materials [11] present in reactions (I)–(VIII), we obtained the Gibbs free

Figure 4 shows that studtite and metastudtite are unstable with respect to the corresponding oxides at the full range of temperatures studied, 250–500 K, since the corresponding Gibbs free energies of reaction are positive everywhere. Therefore, they are metastable phases at normal conditions. The opposite is true for soddyite mineral phase, which is stable at all the temperatures. However, dehydrated schoepite, becquerelite, schoepite, metaschoepite, and rutherfordine mineral phases are stable at ambient temperature and become unstable at the temperatures of 462, 491, 383, 352, and 514 K, respectively, because the Gibbs free energy of reaction becomes positive at these temperatures. The observation of changes of stability for these phases at these relatively low temperatures was unexpected and highlights the great relevance of the availability

3.3.2. Reactions of transformation of uranyl-containing materials into studtite in the presence of high

1=2 UO ð Þ<sup>2</sup> <sup>2</sup>ð Þ� SiO4 2H2O cr ð Þþ 2 H2O2ðÞþl H2ð Þ! g ð Þ UO2 O2 � 4H2O cr ð Þþ 1=2 SiO2ð Þ cr (X)

UO2CO3ð Þþ cr 3 H2O2ðÞþl H2ð Þ! g ð Þ UO2 O2 � 4H2O cr ð Þþ CO2ð Þþ g 1=2O2ð Þ g (XII)

� 12H2O cr ð Þþ <sup>7</sup>=4 H2O2ðÞ!<sup>l</sup> ð Þ UO2 O2 � 4H2O cr ð Þþ <sup>3</sup>=8 O2ð Þ <sup>g</sup> (XV)

1=6 Ca UO ð Þ<sup>2</sup> <sup>6</sup>O4ð Þ OH <sup>6</sup> � 8H2O cr ð Þþ 13=6 H2O2ðÞ!l ð Þ UO2 O2 � 4H2O cr ð Þþ 1=16 CaO cr ð Þ

Reactions (IX) to (XV) are the reactions of transformation of dehydrated schoepite, soddyite, uranium trioxide, rutherfordine, metastudtite, becquerelite, and schoepite into studtite in the presence of high hydrogen peroxide concentrations (and absence of water) [103–105]. Reaction (XVI) represents the conversion of schoepite into metaschoepite at these conditions. The computed Gibbs free energies of these reactions are shown in Figure 5. As it can be observed, all

� 12H2O cr ð Þþ <sup>1</sup>=8 O2ð Þ! <sup>g</sup> <sup>1</sup>=8 UO ð Þ<sup>2</sup> <sup>8</sup>O2ð Þ OH <sup>12</sup>

UO2ð Þ OH <sup>2</sup>ð Þþ cr 2 H2O2ðÞþl H2ð Þ! g ð Þ UO2 O2 � 4H2O cr ð Þ (IX)

UO3ð Þþ cr 3 H2O2ðÞþl H2ð Þ! g ð Þ UO2 O2 � 4H2O cr ð Þþ 1=2 O2ð Þ g (XI)

ð Þ UO2 O2 � 2H2O cr ð Þþ H2O2ðÞþl H2ð Þ! g ð Þ UO2 O2 � 4H2O cr ð Þ (XIII)

(XIV)

(XVI)

� 10 H2O cr ð Þ

We will now study the thermodynamic properties of the following set of reactions:

energies and associated reaction constants displayed in Figure 4 [103–105].

of accurate temperature-dependent thermodynamic functions [102, 103].

hydrogen peroxide concentrations

102 Density Functional Theory

þ 7=12 O2ð Þ g

1=8 UO ð Þ<sup>2</sup> <sup>8</sup>O2ð Þ OH <sup>12</sup>

þ 1=4 H2O2ð Þl

1=8 UO ð Þ<sup>2</sup> <sup>8</sup>O2ð Þ OH <sup>12</sup>

Figure 4. Calculated Gibbs free energies of formation of studtite, metastudtite, dehydrated schoepite, becquerelite, schoepite, metaschoepite, rutherfordine, and soddyite in terms of the corresponding oxides as a function of temperature [103–105].

these phases will transform spontaneously into studtite in the presence of high hydrogen peroxide concentrations, since the Gibbs free energy of all these reactions are negative for the full range of temperatures considered, 300–500 K. In the case of metaschoepite, the thermodynamics of reaction (XVI) shows that it will convert into schoepite mineral phase, but according

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Forbes et al. [110] investigated experimentally the transformation of dehydrated schoepite and soddyite into studtite phase in solutions with large concentrations of hydrogen peroxide. They observed that, at ambient temperature, these materials transform into studtite following the reaction stoichiometry. The results of our calculations [103] for the conversion of dehydrated schoepite and soddyite under high hydrogen peroxide concentrations into studtite agree completely with this experimental study. However, our results extend this study because it shows that the same will happen not only at 298.15 K but also at temperature as high as 500 K. The study performed by Kubatko et al. [146] showed that becquerelite mineral phase also transforms into studtite within 8 hours under high hydrogen peroxide concentrations. The thermodynamics of the conversion of becquerelite into studtite under variable concentrations of hydrogen peroxide was studied by our group in a recent paper [105]. The results displayed in Figure 4 [103–105] show that the same will happen for gamma uranium trioxide, rutherfordine, metastudtite, schoepite, and metaschoepite phases. In fact, because the stability of studtite under these conditions is very high, it is likely that the same will happen for most of

Our study of the thermodynamics of these reactions also permits to comprehend why uranyl peroxide hydrates were the unique phases found in a 2-year corrosion experiment of SNF in deionized water [109]. These phases should be the unique phases found not only in deionized water but also in water containing silicate ions, since studtite is much more stable than soddyite and probably more stable than most other uranyl silicate phases under high hydro-

From the thermodynamic data reported in our previous papers [103–105], the order of thermodynamic stability of the uranyl-containing materials considered in this work was evaluated as a function of temperature under three different conditions: (A) under high concentrations of hydrogen peroxide; (B) in the presence of water and hydrogen peroxide; and (C) in the absence of hydrogen peroxide. The stability of these phases at these conditions in the range of temperatures from 300 to 500 K is displayed in Figures 5A and 6B and C. In these three figures, the relative stabilities are given with respect to studtite, metastudtite, and gamma uranium triox-

Figure 6 provides a very clear idea of the temporal evolution of the paragenetic sequence of secondary phases appearing as a result of the corrosion of SNF under final DGR conditions. Uranyl peroxide phase studtite will appear as the prominent phase at the earlier stages of this paragenetic sequence (see Figure 6A) due to the presence of high hydrogen peroxide concentrations caused by the radiolysis of most of the water reaching the surface of SNF. If the hydrogen peroxide concentration diminishes with time, as expected from the decrease of the

to reaction (XV) the last phase also transforms spontaneously into studtite [104].

the other secondary phases of SNF, as it was suggested in 2017 [101].

gen peroxide concentrations.

3.4. Thermodynamic stability

ide, respectively.

Figure 5. Calculated Gibbs free energies of the reactions of transformation of dehydrated schoepite, soddyite, gamma uranium trioxide, rutherfordine, metastudtite, becquerelite, and schoepite into studtite and of metaschoepite into schoepite in the presence of high hydrogen peroxide concentrations as a function of temperature [103–105].

these phases will transform spontaneously into studtite in the presence of high hydrogen peroxide concentrations, since the Gibbs free energy of all these reactions are negative for the full range of temperatures considered, 300–500 K. In the case of metaschoepite, the thermodynamics of reaction (XVI) shows that it will convert into schoepite mineral phase, but according to reaction (XV) the last phase also transforms spontaneously into studtite [104].

Forbes et al. [110] investigated experimentally the transformation of dehydrated schoepite and soddyite into studtite phase in solutions with large concentrations of hydrogen peroxide. They observed that, at ambient temperature, these materials transform into studtite following the reaction stoichiometry. The results of our calculations [103] for the conversion of dehydrated schoepite and soddyite under high hydrogen peroxide concentrations into studtite agree completely with this experimental study. However, our results extend this study because it shows that the same will happen not only at 298.15 K but also at temperature as high as 500 K. The study performed by Kubatko et al. [146] showed that becquerelite mineral phase also transforms into studtite within 8 hours under high hydrogen peroxide concentrations. The thermodynamics of the conversion of becquerelite into studtite under variable concentrations of hydrogen peroxide was studied by our group in a recent paper [105]. The results displayed in Figure 4 [103–105] show that the same will happen for gamma uranium trioxide, rutherfordine, metastudtite, schoepite, and metaschoepite phases. In fact, because the stability of studtite under these conditions is very high, it is likely that the same will happen for most of the other secondary phases of SNF, as it was suggested in 2017 [101].

Our study of the thermodynamics of these reactions also permits to comprehend why uranyl peroxide hydrates were the unique phases found in a 2-year corrosion experiment of SNF in deionized water [109]. These phases should be the unique phases found not only in deionized water but also in water containing silicate ions, since studtite is much more stable than soddyite and probably more stable than most other uranyl silicate phases under high hydrogen peroxide concentrations.

#### 3.4. Thermodynamic stability

Figure 5. Calculated Gibbs free energies of the reactions of transformation of dehydrated schoepite, soddyite, gamma uranium trioxide, rutherfordine, metastudtite, becquerelite, and schoepite into studtite and of metaschoepite into

schoepite in the presence of high hydrogen peroxide concentrations as a function of temperature [103–105].

104 Density Functional Theory

From the thermodynamic data reported in our previous papers [103–105], the order of thermodynamic stability of the uranyl-containing materials considered in this work was evaluated as a function of temperature under three different conditions: (A) under high concentrations of hydrogen peroxide; (B) in the presence of water and hydrogen peroxide; and (C) in the absence of hydrogen peroxide. The stability of these phases at these conditions in the range of temperatures from 300 to 500 K is displayed in Figures 5A and 6B and C. In these three figures, the relative stabilities are given with respect to studtite, metastudtite, and gamma uranium trioxide, respectively.

Figure 6 provides a very clear idea of the temporal evolution of the paragenetic sequence of secondary phases appearing as a result of the corrosion of SNF under final DGR conditions. Uranyl peroxide phase studtite will appear as the prominent phase at the earlier stages of this paragenetic sequence (see Figure 6A) due to the presence of high hydrogen peroxide concentrations caused by the radiolysis of most of the water reaching the surface of SNF. If the hydrogen peroxide concentration diminishes with time, as expected from the decrease of the

Figure 6. Relative thermodynamic stability of a selected set of secondary phases of SNF: (A) under high hydrogen peroxide concentrations; (B) under the presence of water and hydrogen peroxide; and (C) under the absence of hydrogen peroxide.

intensity of radiation fields over time in a DGR [147], the studtite stability will decrease and the formation of other secondary phases will occur. In the presence of water and hydrogen peroxide (see Figure 6B), the uranyl oxyhydroxide phases (schoepite, metaschoepite, and becquerelite) appear to be the most stable ones. Finally, in the absence of hydrogen peroxide, soddyite is the most stable phase and rutherfordine is also more stable than becquerelite for temperatures lower than 492 K (Figure 6C). Thus, at hydrogen peroxide free conditions, uranyl silicates and carbonates must be the most prominent phases of the SNF.

UO2<sup>þ</sup>

<sup>2</sup> aq , CO3

mental values (LogKexp

4. Conclusions

<sup>2</sup>�(aq), Ca2+(aq), and H+

solubility reactions of schoepite, metaschoepite, becquerelite, and rutherfordine.

The calculated solubility products, LogKcalc

The values of <sup>Δ</sup>spG <sup>Δ</sup>rG are in units of kJ�mol�<sup>1</sup>

Gibbs free energies and associated reaction constants of solubility given in Table 2.

.

Material ΔspG (calc.) Log Ksp (calc.) Log Ksp (exp.)

Metaschoepite �34.14 5.98 5.6 � 0.2 [149], 5.52 � 0.04 [150],

The Application of Periodic Density Functional Theory to the Study of Uranyl-Containing Materials…

Becquerelite �287.55 50.38 40.5 � 1.4 [149], 41.2 � 0.52 [156],

Rutherfordine 96.83 �16.96 �14.91 � 0.10 [153], �13.89 � 0.11 [154],

Table 2. Calculated and experimental Gibbs free energies (ΔspG) and associated reaction constants (Log Ksp) of the

Schoepite �26.11 4.57 —

predicted [104]. Schoepite is shown to be more insoluble than metaschoepite.

5.98, 50.38, and �16.96 respectively, are in very good agreement with the most recent experi-

solubility constant of schoepite has not been determined experimentally, its value was

It has been demonstrated [99–105, 120–123] that Periodic Density Functional Theory methods are an extremely powerful tool in the research of uranium-containing compounds. The use of the new relativistic norm-conserving pseudopotential [101, 121] permitted the computation of the structural properties, X-ray powder patterns, vibrational Raman spectra, and mechanical and thermodynamic properties of these materials. These methods are free of the problems involved in the experimental methods associated to the radiotoxicity of these compounds.

The first principles methodology allowed the safe, accurate, and cheap study of secondary phases of SNF definitive geological disposal conditions. The theoretical methods may be used, in conjunction with experimental techniques, as an interpretative tool of the experimental data or as a predictive tool to determine the structural, vibrational, mechanic, and thermodynamic properties of these substances. One of the most successful applications of this methodology has been achieved when studying their fundamental thermodynamic properties [99–105].

The development of empirical dispersion corrections [126] and the development of density functionals specific for solid materials [130] have improved extraordinarily the reliability of the calculated thermodynamic functions and their temperature dependence. The results were

(aq), and liquid water at 298.15 K [148], one obtains the

6.23 � 0.14 [151], 5.9 � 0.1 [152], 5.14 � 0.05 [153], 5.72 � 0.19 [154],

http://dx.doi.org/10.5772/intechopen.79558

107

41.89 � 0.52 [159], 43.70 � 0.47 [159]

5.79 � 0.19 [155]

43.2 [157], 29 � 1 [158],

�13.29 � 0.01 [155]

sp of metaschoepite, becquerelite, and rutherfordine

sp = 5.6 � 0.2 [149], 40.5 � 1.4 [149], �14.91 � 0.10 [153]). Since there

A full evaluation and understanding of the number and relative amount of the secondary phases of spent nuclear fuel present at the conditions of a final geological disposal over time require the realization of complete thermodynamic calculations employing thermochemical data for a significant number of materials, including the most important secondary phases, amorphous phases, and aqueous species, at a wide range of temperature and pressure conditions [103]. The determination of these thermodynamic data, the evaluation of their temperature and pressure dependence, and the realization of the corresponding thermodynamic computations are one of the main objectives of our current research.

#### 3.5. Solubility constants

The important solubility reactions of schoepite, metaschoepite, rutherfordine, and becquerelite may be written, respectively, as follows:

$$1/8\left[\text{(UO}\_2\text{)}\_8\text{O}\_2\text{(OH)}\_{12}\right] \cdot 12\text{H}\_2\text{O(cr)} + 2\text{ H}^+\text{(aq)} \rightarrow \text{UO}\_2^{2+}\text{(aq)} + 13/4\text{H}\_2\text{O(l)}\tag{XVI}$$

$$\text{1/8}\left[\text{(UO}\_2\text{)}\_8\text{O}\_2\text{(OH)}\_{12}\right]\cdot \text{10H}\_2\text{O(cr)} + \text{2H}^+\text{(aq)} \rightarrow \text{UO}\_2^{2+}\text{(aq)} + \text{3H}\_2\text{O(l)}\tag{\text{XVIII}}$$

$$\text{Ca(UO}\_2\text{)}\_6\text{O}\_4\text{(OH)}\_6\cdot 8\text{H}\_2\text{O(cr)} + 14\text{ H} + \text{(aq)} \rightarrow \text{Ca}^{2+}\text{(aq)} + 6\text{ UO}\_2^{2+}\text{(aq)} + 16\text{ H}\_2\text{O(l)}\text{ (XIX)}$$

$$\text{UO}\_2\text{CO}\_3(\text{cr}) \rightarrow \text{UO}\_2^{2+} \text{(aq)} + \text{CO}\_3^{2-} \text{(aq)}\tag{\text{XX}}$$

Using the computed values of the Gibbs free energies of formation of schoepite, metaschoepite, becquerelite, and rutherfordine and the Gibbs free energies of formation of aqueous ions, UO2 2+(aq), The Application of Periodic Density Functional Theory to the Study of Uranyl-Containing Materials… http://dx.doi.org/10.5772/intechopen.79558 107


Table 2. Calculated and experimental Gibbs free energies (ΔspG) and associated reaction constants (Log Ksp) of the solubility reactions of schoepite, metaschoepite, becquerelite, and rutherfordine.

UO2<sup>þ</sup> <sup>2</sup> aq , CO3 <sup>2</sup>�(aq), Ca2+(aq), and H+ (aq), and liquid water at 298.15 K [148], one obtains the Gibbs free energies and associated reaction constants of solubility given in Table 2.

The calculated solubility products, LogKcalc sp of metaschoepite, becquerelite, and rutherfordine 5.98, 50.38, and �16.96 respectively, are in very good agreement with the most recent experimental values (LogKexp sp = 5.6 � 0.2 [149], 40.5 � 1.4 [149], �14.91 � 0.10 [153]). Since there solubility constant of schoepite has not been determined experimentally, its value was predicted [104]. Schoepite is shown to be more insoluble than metaschoepite.

#### 4. Conclusions

intensity of radiation fields over time in a DGR [147], the studtite stability will decrease and the formation of other secondary phases will occur. In the presence of water and hydrogen peroxide (see Figure 6B), the uranyl oxyhydroxide phases (schoepite, metaschoepite, and becquerelite) appear to be the most stable ones. Finally, in the absence of hydrogen peroxide, soddyite is the most stable phase and rutherfordine is also more stable than becquerelite for temperatures lower than 492 K (Figure 6C). Thus, at hydrogen peroxide free conditions,

Figure 6. Relative thermodynamic stability of a selected set of secondary phases of SNF: (A) under high hydrogen peroxide concentrations; (B) under the presence of water and hydrogen peroxide; and (C) under the absence of hydrogen

A full evaluation and understanding of the number and relative amount of the secondary phases of spent nuclear fuel present at the conditions of a final geological disposal over time require the realization of complete thermodynamic calculations employing thermochemical data for a significant number of materials, including the most important secondary phases, amorphous phases, and aqueous species, at a wide range of temperature and pressure conditions [103]. The determination of these thermodynamic data, the evaluation of their temperature and pressure dependence, and the realization of the corresponding thermodynamic

The important solubility reactions of schoepite, metaschoepite, rutherfordine, and becquerelite

! UO2

! Ca2<sup>þ</sup> aq

<sup>2</sup><sup>þ</sup> aq

Using the computed values of the Gibbs free energies of formation of schoepite, metaschoepite, becquerelite, and rutherfordine and the Gibbs free energies of formation of aqueous ions, UO2

! UO2

<sup>þ</sup> CO3

<sup>2</sup><sup>þ</sup> aq

<sup>2</sup>� aq

<sup>þ</sup> 6 UO2

<sup>2</sup><sup>þ</sup> aq

<sup>2</sup><sup>þ</sup> aq

<sup>þ</sup> <sup>13</sup>=4 H2O lð Þ (XVII)

<sup>þ</sup> 3 H2O lð Þ (XVIII)

(XX)

<sup>þ</sup> 16 H2O lð Þ (XIX)

2+(aq),

uranyl silicates and carbonates must be the most prominent phases of the SNF.

computations are one of the main objectives of our current research.

� 12H2O cr ð Þþ 2 H<sup>þ</sup> aq

� 10H2O cr ð Þþ 2 H<sup>þ</sup> aq

UO2CO3ð Þ! cr UO2

3.5. Solubility constants

peroxide.

106 Density Functional Theory

may be written, respectively, as follows:

1=8 UO ð Þ<sup>2</sup> <sup>8</sup>O2ð Þ OH <sup>12</sup>

1=8 UO ð Þ<sup>2</sup> <sup>8</sup>O2ð Þ OH <sup>12</sup>

Ca UO ð Þ<sup>2</sup> <sup>6</sup>O4ð Þ OH <sup>6</sup> � 8H2O cr ð Þþ 14 H þ aq

It has been demonstrated [99–105, 120–123] that Periodic Density Functional Theory methods are an extremely powerful tool in the research of uranium-containing compounds. The use of the new relativistic norm-conserving pseudopotential [101, 121] permitted the computation of the structural properties, X-ray powder patterns, vibrational Raman spectra, and mechanical and thermodynamic properties of these materials. These methods are free of the problems involved in the experimental methods associated to the radiotoxicity of these compounds.

The first principles methodology allowed the safe, accurate, and cheap study of secondary phases of SNF definitive geological disposal conditions. The theoretical methods may be used, in conjunction with experimental techniques, as an interpretative tool of the experimental data or as a predictive tool to determine the structural, vibrational, mechanic, and thermodynamic properties of these substances. One of the most successful applications of this methodology has been achieved when studying their fundamental thermodynamic properties [99–105].

The development of empirical dispersion corrections [126] and the development of density functionals specific for solid materials [130] have improved extraordinarily the reliability of the calculated thermodynamic functions and their temperature dependence. The results were shown to be accurate at very low and high temperatures [99–105]. The description of the temperature dependence of these functions is very difficult from the experimental point of view. The theoretical approach has permitted in some cases, as those of the rutherfordine [99] and metaschoepite [104] mineral phases, to extend the range of temperatures in which the thermodynamic properties were known and to determine the variation with temperature of these properties for a large series of important phases in which it was completely unknown: studtite, metastudtite, dehydrated schoepite, becquerelite, schoepite, and soddyite. Furthermore, the calculated thermodynamic functions satisfy properly the Dulong-Petit asymptotic constraints.

Dedication

Permissions

right (2018).

right (2018).

Author details

Francisco Colmenero Ruiz

CSIC), Madrid, Spain

References

To Beatriz and Ana María, on the occasion of their 12th and 25th birthdays.

from Ref. [103] with permission from American Chemical Society, Copyright (2018).

Rutherfordine. Part of Figure 2 and data contained in Table 1 adapted from Ref. [99] with permission from American Chemical Society, Copyright (2017). Part of Figure 3 adapted from Ref. [102] with permission from American Chemical Society, Copyright (2018). Parts of Figures 4–6 adapted

The Application of Periodic Density Functional Theory to the Study of Uranyl-Containing Materials…

http://dx.doi.org/10.5772/intechopen.79558

109

Gamma uranium trioxide. Part of Figure 2 and data contained in Table 1 adapted from Ref. [100] with permission from American Chemical Society, Copyright (2017). Part of Figure 3 adapted from Ref. [102] with permission from American Chemical Society, Copyright (2018). Parts of Figures 4–6 adapted from Ref. [103] with permission from American Chemical Society, Copy-

Dehydrated schoepite, studtite, metastudtite, and soddyite. Data contained in Table 1 adapted from Ref. [102] with permission from American Chemical Society, Copyright (2018). Parts of Figures 4–6 adapted from Ref. [103] with permission from American Chemical Society, Copy-

Schoepite and metaschoepite. Parts of Figures 2–6 and data contained in Tables 1 and 2 adapted

Becquerelite. Part of Figures 4–6 and data contained in Tables 1 and 2 reproduced from Ref.

Instituto de Estructura de la Materia – Consejo Superior de Investigaciones Científicas (IEM-

[1] Guggenheim EA. Thermodynamics: An Advanced Treatment for Chemists and Physicists.

from Ref. [104] with permission from American Chemical Society.

Address all correspondence to: francisco.colmenero@iem.cfmac.csic.es

Amsterdam: North-Holland; 1967. ISBN: 0444869514

[105] with permission from the Royal Society of Chemistry.

The comparison of the computed heat capacities and entropies with experimental data was very satisfactory in those cases in which there ware experimental data to compare with. The calculated Gibbs free energies of formation of rutherfordine, γ-UO3, and metaschoepite [102, 104] were in good agreement with experiment at ambient temperature, and the differences with the corresponding experimental values were only 1.6%, 1.0%, and 2.0% at 700, 900, and 800 K, respectively. Because the theoretical treatments used for studtite, metastudtite, dehydrated schoepite, soddyite, schoepite, and becquerelite were essentially the same as those used for these three materials, we expect a similar accuracy level for their calculated thermodynamic parameters of formation [102, 104, 105].

As an application of the calculated thermodynamic properties of the considered uranyl materials, the Gibbs free energies and associated reaction constants of a large number of reactions involving these materials were determined. The results provided a deep and clear understanding of the temporal evolution of the paragenetic sequence of secondary phases appearing at the surface of SNF as a result of its corrosion under final DGR conditions [103–105]. Additional work is now in progress to determine the thermodynamic properties of a significant number of additional phases. The use of these thermodynamic parameters in detailed multi-component thermodynamic computations should be pursued in a near future.

#### Acknowledgements

Supercomputer time by the CETA-CIEMAT, CTI-CSIC, and CESGA centers is acknowledged. This work has been carried out in the context of a CSIC–CIEMAT collaboration agreement: "Caracterización experimental y teórica de fases secundarias y óxidos de uranio formados en condiciones de almacenamiento de combustible nuclear." I want to thank Dr. Ana María Fernández, Dr. Vicente Timon, Dr. Laura J. Bonales, Dr. Joaquín Cobos, and Dr. Rafael Escribano for their continuous help and advice during the realization of these studies.

#### Conflict of interest

The author declares that there is no conflict of interest.

### Dedication

shown to be accurate at very low and high temperatures [99–105]. The description of the temperature dependence of these functions is very difficult from the experimental point of view. The theoretical approach has permitted in some cases, as those of the rutherfordine [99] and metaschoepite [104] mineral phases, to extend the range of temperatures in which the thermodynamic properties were known and to determine the variation with temperature of these properties for a large series of important phases in which it was completely unknown: studtite, metastudtite, dehydrated schoepite, becquerelite, schoepite, and soddyite. Furthermore, the calculated thermodynamic functions satisfy properly the Dulong-Petit asymptotic

The comparison of the computed heat capacities and entropies with experimental data was very satisfactory in those cases in which there ware experimental data to compare with. The calculated Gibbs free energies of formation of rutherfordine, γ-UO3, and metaschoepite [102, 104] were in good agreement with experiment at ambient temperature, and the differences with the corresponding experimental values were only 1.6%, 1.0%, and 2.0% at 700, 900, and 800 K, respectively. Because the theoretical treatments used for studtite, metastudtite, dehydrated schoepite, soddyite, schoepite, and becquerelite were essentially the same as those used for these three materials, we expect a similar accuracy level for their calculated thermodynamic

As an application of the calculated thermodynamic properties of the considered uranyl materials, the Gibbs free energies and associated reaction constants of a large number of reactions involving these materials were determined. The results provided a deep and clear understanding of the temporal evolution of the paragenetic sequence of secondary phases appearing at the surface of SNF as a result of its corrosion under final DGR conditions [103–105]. Additional work is now in progress to determine the thermodynamic properties of a significant number of additional phases. The use of these thermodynamic parameters in detailed multi-component thermodynamic computations should be

Supercomputer time by the CETA-CIEMAT, CTI-CSIC, and CESGA centers is acknowledged. This work has been carried out in the context of a CSIC–CIEMAT collaboration agreement: "Caracterización experimental y teórica de fases secundarias y óxidos de uranio formados en condiciones de almacenamiento de combustible nuclear." I want to thank Dr. Ana María Fernández, Dr. Vicente Timon, Dr. Laura J. Bonales, Dr. Joaquín Cobos, and Dr. Rafael Escribano

for their continuous help and advice during the realization of these studies.

The author declares that there is no conflict of interest.

constraints.

108 Density Functional Theory

parameters of formation [102, 104, 105].

pursued in a near future.

Acknowledgements

Conflict of interest

To Beatriz and Ana María, on the occasion of their 12th and 25th birthdays.

#### Permissions

Rutherfordine. Part of Figure 2 and data contained in Table 1 adapted from Ref. [99] with permission from American Chemical Society, Copyright (2017). Part of Figure 3 adapted from Ref. [102] with permission from American Chemical Society, Copyright (2018). Parts of Figures 4–6 adapted from Ref. [103] with permission from American Chemical Society, Copyright (2018).

Gamma uranium trioxide. Part of Figure 2 and data contained in Table 1 adapted from Ref. [100] with permission from American Chemical Society, Copyright (2017). Part of Figure 3 adapted from Ref. [102] with permission from American Chemical Society, Copyright (2018). Parts of Figures 4–6 adapted from Ref. [103] with permission from American Chemical Society, Copyright (2018).

Dehydrated schoepite, studtite, metastudtite, and soddyite. Data contained in Table 1 adapted from Ref. [102] with permission from American Chemical Society, Copyright (2018). Parts of Figures 4–6 adapted from Ref. [103] with permission from American Chemical Society, Copyright (2018).

Schoepite and metaschoepite. Parts of Figures 2–6 and data contained in Tables 1 and 2 adapted from Ref. [104] with permission from American Chemical Society.

Becquerelite. Part of Figures 4–6 and data contained in Tables 1 and 2 reproduced from Ref. [105] with permission from the Royal Society of Chemistry.

### Author details

Francisco Colmenero Ruiz

Address all correspondence to: francisco.colmenero@iem.cfmac.csic.es

Instituto de Estructura de la Materia – Consejo Superior de Investigaciones Científicas (IEM-CSIC), Madrid, Spain

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**Chapter 6**

**Provisional chapter**

**Magnetic Ordering in Ilmenites and Corundum-**

**Magnetic Ordering in Ilmenites and Corundum-**

DOI: 10.5772/intechopen.81772

In the last few years, the multiferroic materials have represented a very important research topic on the design of new technological devices. A better description for this kind of materials involves two or more forms of ferroic orders coupled in a single crystalline structure. The great number of studies in this field is focused on candidates that present the coupling between a magnetic order and ferroelectricity. However, these material classes are a challenging topic on first-principles calculations due to the strong correlation that arose from the unpaired electrons. Furthermore, the partial filling of d or f orbitals reduces a high localization and a strong interaction causing failures on the electronic structure prediction. The investigation of multiferroic materials aims at their application on the development of devices such as actuators, magnetic readers, sensors and data storage. Multiferroic materials are also alternatives to the production of technological applications based on spintronic. Our proposal is to show our experience in DFT simulations for magnetic states applied in oxides of ilmenites and corundum-ordered structures. Theoretical results reported for our group until now showed a good agreement with experimental

results for half-metallicity, reduced band-gap, and/or ferromagnetic ordering.

**Keywords:** magnetism, DFT, co-linear ilmenites, oxides, semiconductor, spin, bulk,

Over the last few decades, the field of magnetism has assumed a remarkable importance in the field of science and technology due to the development of new technologies. Such kind of

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

© 2019 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.

**Ordered Structures**

**Ordered Structures**

Sergio Ricardo De Lazaro,

**Abstract**

**1. Introduction**

Sergio Ricardo De Lazaro,

Renan Augusto Pontes Ribeiro

http://dx.doi.org/10.5772/intechopen.81772

Luis Henrique Da Silveira Lacerda and

Luis Henrique Da Silveira Lacerda and Renan Augusto Pontes Ribeiro

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

clusters, crystalline field theory degenerated


#### **Magnetic Ordering in Ilmenites and Corundum-Ordered Structures Magnetic Ordering in Ilmenites and Corundum-Ordered Structures**

DOI: 10.5772/intechopen.81772

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

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.81772

#### **Abstract**

[152] Bruno J, Sandino A. The solubility of amorphous and crystalline schoepite in neutral to alkaline aqueous solutions. MRS Symposium Proceedings. 1989;127:871-878. DOI:

[153] Meinrath G, Kato Y, Kimura T, Yoshida Z. Solid-aqueous phase equilibria of uranium (VI) under ambient conditions. Radiochimica Acta. 1996;75:159-167. DOI: 10.1524/

[154] Meinrath G, Kimura T. Behaviour of U(V1) solids under conditions of natural aquatic systems. Inorganic Chimica Acta. 1993;204:79-85. DOI: 10.1016/S0020-1693(00)88116-5

[155] Kramer-Schnabel U, Bischoff H, Xi RH, Marx G. Solubility products and complex formation equilibria in the systems uranyl hydroxide and uranyl carbonate at 25C and I = 0.1

[156] Rai BD, Felmy AR, Hess NJ, LeGore VL, McCready DE. Thermodynamics of the U(VI)- Ca2+-Cl–OH–H2O system: Solubility product of becquerelite. Radiochimica Acta. 2002;

[157] Vochten R, Van Haverbeke L. Transformation of schoepite into the uranyl oxide hydrates: Becquerelite, billietite and wölsendorfite. Mineralogy and Petrology. 1990;43:

[158] Casas I, Bruno J, Cera E, Finch RJ, Ewing RC. Characterization and dissolution behavior of a becquerelite from Shinkolobwe. Zaire. Geochimica et Cosmochimica Acta. 1997;61:

[159] Sandino MCA, Grambow B. Solubility Equilibria in the U(VI)-Ca-K-Cl-H2O system: Transformation of Schoepite into Becquerelite and Compreignacite. Radiochimica Acta.

M. Radiochimica Acta. 1992;56:183-188. DOI: 10.1524/ract.1992.56.4.183

90:495-503. DOI: 10.1524/ract.2002.90.9-11\_2002.495

3879-3884. DOI: 10.1016/S0016-7037(97)00195-6

1994;66–67:37-43. DOI: 10.1524/ract.1994.6667.special-issue.37

65-72. DOI: 10.1007/BF01164222

10.1557/PROC-127-871

ract.1996.75.3.159

122 Density Functional Theory

In the last few years, the multiferroic materials have represented a very important research topic on the design of new technological devices. A better description for this kind of materials involves two or more forms of ferroic orders coupled in a single crystalline structure. The great number of studies in this field is focused on candidates that present the coupling between a magnetic order and ferroelectricity. However, these material classes are a challenging topic on first-principles calculations due to the strong correlation that arose from the unpaired electrons. Furthermore, the partial filling of d or f orbitals reduces a high localization and a strong interaction causing failures on the electronic structure prediction. The investigation of multiferroic materials aims at their application on the development of devices such as actuators, magnetic readers, sensors and data storage. Multiferroic materials are also alternatives to the production of technological applications based on spintronic. Our proposal is to show our experience in DFT simulations for magnetic states applied in oxides of ilmenites and corundum-ordered structures. Theoretical results reported for our group until now showed a good agreement with experimental results for half-metallicity, reduced band-gap, and/or ferromagnetic ordering.

**Keywords:** magnetism, DFT, co-linear ilmenites, oxides, semiconductor, spin, bulk, clusters, crystalline field theory degenerated

#### **1. Introduction**

Over the last few decades, the field of magnetism has assumed a remarkable importance in the field of science and technology due to the development of new technologies. Such kind of

© 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 reproduction in any medium, provided the original work is properly cited. © 2019 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.

technologies are based on the capability of control not only the electrical degree of freedom of electrons (charge) but also the magnetic nature associated with the intrinsic angular momentum of a given particle so-called spin [1].

well-defined boundaries. This theory allowed the observation of the Curie temperature (TC) above which the ferromagnetic domains become paramagnetic and the magnetism disap-

Magnetic Ordering in Ilmenites and Corundum-Ordered Structures

http://dx.doi.org/10.5772/intechopen.81772

125

In the Quantum Theory of magnetism, the formalism is based on the quantum angular momentum of electron, so-called spin, associated with open-shell orbitals commonly founded in Transition Metals. In this way, the microscopic origin of the magnetism is the strong electron-electron interaction that arises from the chemical bond inside the crystalline structure. Moreover, the itinerant character associated with the magnetic moments can be localized or delocalized that originates dia- or paramagnetism in solid state materials. However, the collective magnetism in the perspective of ferromagnetism, antiferromagnetism and ferrimagnetism are the fundamental keys to overcome the main questions behind the solid state magnetic ordering [11] . A schematic overview about this kind of magnetic ordering is given

A ferromagnetic (FM—**Figure 1a**) material is characterized by a spontaneous magnetization even in the absence of a magnetic field. At T = 0 K all magnetic moments are parallel ordered due to the stronger exchange interaction between them, differently of paramagnetic species that exhibit non-interacting magnetic moments. The conventional magnetic transition metal elements with ferromagnetic ordering are Mn, Fe, Ni and Co, where Mn and Fe can also show antiferromagnetic order depending upon the crystalline structure. In addition, the thermal energy eventually overcomes the exchange the electronic exchange in ferromagnets, producing a randomizing effect where the saturation magnetization goes to zero due to the disorder

**Figure 1.** Different collective arrangements of magnetic moments for ordered magnetic systems: (a) ferromagnets, (b)

pears [10, 11].

in **Figure 1**.

antiferromagnets, (c) ferrimagnets.

For instance, in recent years, the scientific interest on the development of devices in the field of spintronic has been intensified. The performance of these devices depends on the spin polarization of the current used for information storage; i. e. the development of spintronic devices requires an effective way of control charge-based electronic properties by magnetic field, as well as of controlling the magnetic properties by electric currents [2–4].

In this context, multiferroic materials are essentially the best candidates due to the intrinsic multifunctional features associated with such class of compounds. In a general view, the most complete definition for these materials is based on two or more ferroic ordering (i.e. ferroelectricity, ferroelasticity, ferromagnetism, ferrotoroidicity) coupled in a single crystalline phase. Nowadays, the majority of studies are strictly focused on candidates that combine a magnetic order (ferromagnetism, antiferromagnetism) with ferroelectricity and are known as magnetoelectric multiferroics. The interest on multiferroic materials was rebirthed by theoretical reports about the scarcity of such compounds, which are also responsible for explain the unusual phenomena and successfully predict new candidates. The main advances addressed to such kind of materials are novel devices such as actuators, transducers and storage devices, as well as other potential applications including multiple state memories and novel memory media [5–9].

In order to clarify the unusual phenomena's commonly observed in such complex materials, as well as to design new candidates with feasible properties, the understanding of magnetism in solid state materials plays a fundamental role due to the complexity of the so-called itinerant electrons commonly founded in transition metals.

In this chapter, we propose a theoretical point of view about the magnetism of Ilmenite and Corundum-ordered structures—widespread candidates for multiferroic and spintronic applications—focusing on the relation between crystalline and electronic structure associated with the ground state magnetic ordering attributed to the materials. The following sections are dedicated to explain the (i) magnetism in solid state materials, (ii) the magnetic ordering of Ilmenite and Corundum-ordered materials, and (iii) theoretical approaches to investigate magnetic solid state materials.

#### **1.1. Magnetism in solid state materials: a fundamental tool for new technologies**

Magnetic materials are the ideal candidates for the consolidation of spintronic technology due to the high degree of freedom of the spin in these materials. Magnetism is one of the oldest and most fundamental scientific problems not completely clear until now. Numerous theories are proposed to explain the magnetic behavior of materials, as the well-known classical formalism where all substances have small magnets that align in certain directions depending on the chemical environment. Another theory is domain-based, where a magnetic domain represents a region with uniform magnetization, separated from other domains by well-defined boundaries. This theory allowed the observation of the Curie temperature (TC) above which the ferromagnetic domains become paramagnetic and the magnetism disappears [10, 11].

technologies are based on the capability of control not only the electrical degree of freedom of electrons (charge) but also the magnetic nature associated with the intrinsic angular momen-

For instance, in recent years, the scientific interest on the development of devices in the field of spintronic has been intensified. The performance of these devices depends on the spin polarization of the current used for information storage; i. e. the development of spintronic devices requires an effective way of control charge-based electronic properties by magnetic

In this context, multiferroic materials are essentially the best candidates due to the intrinsic multifunctional features associated with such class of compounds. In a general view, the most complete definition for these materials is based on two or more ferroic ordering (i.e. ferroelectricity, ferroelasticity, ferromagnetism, ferrotoroidicity) coupled in a single crystalline phase. Nowadays, the majority of studies are strictly focused on candidates that combine a magnetic order (ferromagnetism, antiferromagnetism) with ferroelectricity and are known as magnetoelectric multiferroics. The interest on multiferroic materials was rebirthed by theoretical reports about the scarcity of such compounds, which are also responsible for explain the unusual phenomena and successfully predict new candidates. The main advances addressed to such kind of materials are novel devices such as actuators, transducers and storage devices, as well as other potential applications including multiple state memories and novel memory

In order to clarify the unusual phenomena's commonly observed in such complex materials, as well as to design new candidates with feasible properties, the understanding of magnetism in solid state materials plays a fundamental role due to the complexity of the so-called itiner-

In this chapter, we propose a theoretical point of view about the magnetism of Ilmenite and Corundum-ordered structures—widespread candidates for multiferroic and spintronic applications—focusing on the relation between crystalline and electronic structure associated with the ground state magnetic ordering attributed to the materials. The following sections are dedicated to explain the (i) magnetism in solid state materials, (ii) the magnetic ordering of Ilmenite and Corundum-ordered materials, and (iii) theoretical approaches to investigate

Magnetic materials are the ideal candidates for the consolidation of spintronic technology due to the high degree of freedom of the spin in these materials. Magnetism is one of the oldest and most fundamental scientific problems not completely clear until now. Numerous theories are proposed to explain the magnetic behavior of materials, as the well-known classical formalism where all substances have small magnets that align in certain directions depending on the chemical environment. Another theory is domain-based, where a magnetic domain represents a region with uniform magnetization, separated from other domains by

**1.1. Magnetism in solid state materials: a fundamental tool for new technologies**

field, as well as of controlling the magnetic properties by electric currents [2–4].

tum of a given particle so-called spin [1].

ant electrons commonly founded in transition metals.

magnetic solid state materials.

media [5–9].

124 Density Functional Theory

In the Quantum Theory of magnetism, the formalism is based on the quantum angular momentum of electron, so-called spin, associated with open-shell orbitals commonly founded in Transition Metals. In this way, the microscopic origin of the magnetism is the strong electron-electron interaction that arises from the chemical bond inside the crystalline structure. Moreover, the itinerant character associated with the magnetic moments can be localized or delocalized that originates dia- or paramagnetism in solid state materials. However, the collective magnetism in the perspective of ferromagnetism, antiferromagnetism and ferrimagnetism are the fundamental keys to overcome the main questions behind the solid state magnetic ordering [11] . A schematic overview about this kind of magnetic ordering is given in **Figure 1**.

A ferromagnetic (FM—**Figure 1a**) material is characterized by a spontaneous magnetization even in the absence of a magnetic field. At T = 0 K all magnetic moments are parallel ordered due to the stronger exchange interaction between them, differently of paramagnetic species that exhibit non-interacting magnetic moments. The conventional magnetic transition metal elements with ferromagnetic ordering are Mn, Fe, Ni and Co, where Mn and Fe can also show antiferromagnetic order depending upon the crystalline structure. In addition, the thermal energy eventually overcomes the exchange the electronic exchange in ferromagnets, producing a randomizing effect where the saturation magnetization goes to zero due to the disorder

**Figure 1.** Different collective arrangements of magnetic moments for ordered magnetic systems: (a) ferromagnets, (b) antiferromagnets, (c) ferrimagnets.

associated with the magnetic moments. This occurs at a particular temperature called the Curie temperature (TC) [1].

On the other hand, antiferromagnetic materials (AFM—**Figure 1b**) show an antiparallel arrangement between neighboring magnetic moments. In this case, this ordering can be attributed to the existence of negative exchange interaction between the nearest neighbors, as well as for some lattices that can be divided in two ferromagnetic sublattices, which exhibit an antiferromagnetic ordering between them. This class of compounds is the most common magnetic materials in the nature and shows good perspectives regarding the technological applications. In this class, both metals (Mn2 Au, Fex Mny and others) and semiconductors/insulators (NiO, Cr2 O3 , CoO, BiFeO3 , MnF2 and others) are founded [12].

Ferrimagnetism represents an intermediate position between FM and AFM orderings. In this case, the simplest picture related to solid state materials can be understood by the assumption of two magnetic sublattices with antiparallel orientation but with different magnitude of each magnetization, resulting that the total magnetization does not vanish as for the antiferromagnetic case, as represented in **Figure 1c**. Magnetite (Fe3 O4 ) is a well-known ferrimagnetic material, where the spins on the tetrahedral Fe A-sublattice are antiparallel to those on the octahedral Fe B-sublattice, resulting in a net magnetic moment for the B-site [11].

*1.1.2. Indirect coupling*

the magnetic centers.

in **Figure 2b** [1, 11].

*1.1.4. RKKY coupling*

*1.1.3. Double-exchange indirect coupling*

In this case the electrons located in the partially occupied orbitals are separated from the neighboring electrons by an atom or non-magnetic ligand. The coupling force is dependent on the amount of energy transferred between the levels and the Coulomb repulsion between these particles. Another important feature of this model is that the oxidation numbers for the magnetic cations must be equal or have a difference equal to two, so that there is no movement between the levels. This model is present in several materials, such as the MnO shown

**Figure 2.** Representation of magnetic interactions in solid state materials. (a) Bethe-Slater curve for direct coupling. (b) Indirect coupling between Mn3+ mediated by oxygen atoms in MnO. (c) Oxygen-mediated indirect double-exchange coupling for cations with distinct orbital occupations. (d) RKKY coupling and its dependence with the distance between

Magnetic Ordering in Ilmenites and Corundum-Ordered Structures

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127

This coupling is quite similar to that previously mentioned; however the main difference lies in the movement of the electrons between the orbitals. As the oxidation numbers of the metals are different, in this case, the mismatched electrons of the A site can move to the B site, since it

The RKKY coupling was discovered by Ruderman, Kittel, Kasuya and Yosida (RKKY) and characterized as a long-range interaction. This type of coupling is very common in metals where the overlap between orbitals is minimal or zero, causing a polarized spin ion to induce a field on the conducting electrons of the neighboring atoms allowing a magnetic influence on a second polarized neighbor. As well as direct coupling, the orientation of neighbors is dependent on the distance between them as shown in **Figure 2d**. Another important feature

is possible to find unoccupied levels with the same spin orientation (**Figure 2c**) [1, 11].

Let us now briefly introduce the concept of magnetic interactions, which are responsible for the collective ordering presented in **Figure 1**. In magnetic solid state materials different types of magnetic interaction can be observed, being responsible for intriguing properties associated to the fact that magnetic moments interact between them, enabling the long range magnetic ordering. In a general point of view, two main interactions are founded in solid state materials: direct and indirect. In the first case, the electrons of neighboring magnetic atoms directly interact through an overlap between atomic wave-functions called "direct exchange". On the other hand, if the overlap of the involved wave functions is only small the direct exchange does not represent the dominating mechanism for magnetic properties because an additional atom act as a bridge between the magnetic centers, resulting in an indirect exchange interaction is responsible for magnetism. The representation of different kinds of magnetic interactions commonly founded in magnetic solid state materials is depicted in **Figure 2**.

#### *1.1.1. Direct coupling*

This type of coupling occurs when a direct overlap between the orbitals of adjacent magnetic sites is observed. The collective arrangement associated with the signal of direct coupling depends of the distance between species. At short distance the electrons tend to spend most of the time between the atoms giving rise to an antiparallel coupling due to the Pauli Exclusion Principle. However, at long distances the overlap between the orbital is reduced, inducing a minimization of potential energy for this interaction. This variation was named of Bethe-Slater curve, due to the elucidation of the behavior by these researchers. The representation of this curve is shown in **Figure 2a** [1, 11].

**Figure 2.** Representation of magnetic interactions in solid state materials. (a) Bethe-Slater curve for direct coupling. (b) Indirect coupling between Mn3+ mediated by oxygen atoms in MnO. (c) Oxygen-mediated indirect double-exchange coupling for cations with distinct orbital occupations. (d) RKKY coupling and its dependence with the distance between the magnetic centers.

#### *1.1.2. Indirect coupling*

associated with the magnetic moments. This occurs at a particular temperature called the

On the other hand, antiferromagnetic materials (AFM—**Figure 1b**) show an antiparallel arrangement between neighboring magnetic moments. In this case, this ordering can be attributed to the existence of negative exchange interaction between the nearest neighbors, as well as for some lattices that can be divided in two ferromagnetic sublattices, which exhibit an antiferromagnetic ordering between them. This class of compounds is the most common magnetic materials in the nature and shows good perspectives regarding the technological

Au, Fex

Ferrimagnetism represents an intermediate position between FM and AFM orderings. In this case, the simplest picture related to solid state materials can be understood by the assumption of two magnetic sublattices with antiparallel orientation but with different magnitude of each magnetization, resulting that the total magnetization does not van-

well-known ferrimagnetic material, where the spins on the tetrahedral Fe A-sublattice are antiparallel to those on the octahedral Fe B-sublattice, resulting in a net magnetic moment

Let us now briefly introduce the concept of magnetic interactions, which are responsible for the collective ordering presented in **Figure 1**. In magnetic solid state materials different types of magnetic interaction can be observed, being responsible for intriguing properties associated to the fact that magnetic moments interact between them, enabling the long range magnetic ordering. In a general point of view, two main interactions are founded in solid state materials: direct and indirect. In the first case, the electrons of neighboring magnetic atoms directly interact through an overlap between atomic wave-functions called "direct exchange". On the other hand, if the overlap of the involved wave functions is only small the direct exchange does not represent the dominating mechanism for magnetic properties because an additional atom act as a bridge between the magnetic centers, resulting in an indirect exchange interaction is responsible for magnetism. The representation of different kinds of magnetic interactions commonly founded in magnetic solid state materials is

This type of coupling occurs when a direct overlap between the orbitals of adjacent magnetic sites is observed. The collective arrangement associated with the signal of direct coupling depends of the distance between species. At short distance the electrons tend to spend most of the time between the atoms giving rise to an antiparallel coupling due to the Pauli Exclusion Principle. However, at long distances the overlap between the orbital is reduced, inducing a minimization of potential energy for this interaction. This variation was named of Bethe-Slater curve, due to the elucidation of the behavior by these researchers. The representation of

ish as for the antiferromagnetic case, as represented in **Figure 1c**. Magnetite (Fe3

Mny

and others) are founded [12].

and others) and semiconductors/insu-

O4 ) is a

Curie temperature (TC) [1].

lators (NiO, Cr2

126 Density Functional Theory

for the B-site [11].

depicted in **Figure 2**.

*1.1.1. Direct coupling*

this curve is shown in **Figure 2a** [1, 11].

applications. In this class, both metals (Mn2

, CoO, BiFeO3

, MnF2

O3

In this case the electrons located in the partially occupied orbitals are separated from the neighboring electrons by an atom or non-magnetic ligand. The coupling force is dependent on the amount of energy transferred between the levels and the Coulomb repulsion between these particles. Another important feature of this model is that the oxidation numbers for the magnetic cations must be equal or have a difference equal to two, so that there is no movement between the levels. This model is present in several materials, such as the MnO shown in **Figure 2b** [1, 11].

#### *1.1.3. Double-exchange indirect coupling*

This coupling is quite similar to that previously mentioned; however the main difference lies in the movement of the electrons between the orbitals. As the oxidation numbers of the metals are different, in this case, the mismatched electrons of the A site can move to the B site, since it is possible to find unoccupied levels with the same spin orientation (**Figure 2c**) [1, 11].

#### *1.1.4. RKKY coupling*

The RKKY coupling was discovered by Ruderman, Kittel, Kasuya and Yosida (RKKY) and characterized as a long-range interaction. This type of coupling is very common in metals where the overlap between orbitals is minimal or zero, causing a polarized spin ion to induce a field on the conducting electrons of the neighboring atoms allowing a magnetic influence on a second polarized neighbor. As well as direct coupling, the orientation of neighbors is dependent on the distance between them as shown in **Figure 2d**. Another important feature of this class of magnetic couplings is the existence of effects such as Giant Magnetoresistance and Spin Tunneling that can be explained by the use of RKKY Hamiltonian [1, 11, 13–15].

#### *1.1.5. GKA rules*

The Goodenough-Kanamori-Anderson (GKA) rule describe the interatomic spin-spin interactions between two atoms considering the existence of a virtual electron transfers between them (superexchange) and/or between a shared anion (so-called "bridge") and the two atoms. As consequence, in this rule atoms with orthogonal orbitals do not overlap reducing the repulsion between the electrons, following the Hund's rule. Therefore, the exchange interaction between spins is positive and a ferromagnetic ordering arises.

In addition, the GKA rule predicts some interactions that depend on the occupation of interacting orbitals. In the first case, an antiferromagnetic interaction occurs when the virtual electron transfer is between half-filled orbitals. In opposition, they are ferromagnetic once the virtual electron transfer is from a half-filled to an empty/filled orbital.

couplings constants: Intralayer (J<sup>1</sup>

represent A2+, B4+ and O2− ions, respectively.

romagnetic and FeTiO3

(antiferromagnetic) [25, 30].

romagnetism; since J<sup>2</sup>

depends on the distance.

of B4+ site cations.

J 2

of a t2g orbital in one cation with an eg

interactions are mediated by the BO6

, CoTiO3

) and Interlayers (J<sup>2</sup>

parameter depends upon the occupancies of the interacting orbitals making MnTiO<sup>3</sup>

, NiTiO3

In Fe-based ilmenite materials the long-range exchange coupling (J<sup>2</sup>

magnetic exchange that happens between A-O-A atoms and is dominated by the coupling

**Figure 3.** Ilmenite-type conventional unit cell and their exchange coupling constants. Black, orange and red balls

having less contribution of a direct overlap in [001] direction because of the vacancy in cationic sublattice (**Figure 3**) with opposite magnetization directions between adjacent A layers

B-site replacement can control such magnetic ordering. As previously discussed, the ilmenite structure arrangement creates vacancies between adjacent Fe2+ layers that are separated each other by a B-site plane. So, the interlayer magnetic coupling integral can be visualized as a direct coupling between 3d orbitals of adjacent Fe2+ cations. Furthermore, other evidence that the non-magnetic B-site substitution affects the magnetic ordering is: the vacancy formation occurs in this cationic sublattice and the direct exchange coupling

Hf, Si, Ge, Sn) materials by means of Density Functional Theory (DFT) calculations [31, 32]. **Figure 4** shows the relative Energy between AFM and FM structures as function of ionic radii

From these results, it was observed that the FEM state is stabilized for ilmenite materials with large B-site metals (Sn, Hf, Zr); whereas, ilmenite cells contracted (Si, Ge, Ti) exhibit an AFM behavior. This result can be discussed as function of electronic repulsion between Fe2+ atoms

the cationic vacancy induces a coupling in such direction. Therefore, the B-cation volume control the distance among different Fe2+ layers from an angular distortion in O-B-O bonds in axial plane, which causes an increase/decrease in B-O bond distances allowing a higher/

This behavior was confirmed investigating the magnetic ground-state of FeBO<sup>3</sup>

3d orbitals in different layers. In this case, we use the c-axis oriented to 3d<sup>z</sup>

), as shown in **Figure 1**. J<sup>1</sup>

Magnetic Ordering in Ilmenites and Corundum-Ordered Structures

http://dx.doi.org/10.5772/intechopen.81772

orbital in other. Therefore, the signal for exchange

clusters in the intermetallic connection A-O-B-O-A,

ferromagnetic for intralayer coupling (J<sup>1</sup>

depends of intermetallic connection Fe-O-B-O-Fe, the non-magnetic

refers to the

antifer-

129

). In turn,

(B = Ti, Zr,

orbital, once

2

) stabilizes the antifer-

The GKA rules are important due to the introduction of bond character in the evaluation of the magnetic interactions. It is important to point out that the net spin does not change with the covalent component associated with the chemical bonds; however this component can extends the cation wave function out because the interaction between the orbitals is large, resulting in chemical dependent ferromagnetic/antiferromagnetic couplings associated the superexchange electron transfer [16].

#### *1.1.6. Dzyaloshinskii-Moriya rule*

The Dzyaloshinskii-Moriya [17–20] rule states that a low magnetic resultant is observed in antiferromagnetic material due to a long range interaction between magnetic atoms in disordered crystalline structures. This particular behavior is called weak-Ferromagnetism.

#### **1.2. FeBO3 (B = Ti, Zr, Hf, Si, Ge, Sn) materials: insights into interatomic distance and orbital overlap**

Perovskites oxides, which have the general formula ABO<sup>3</sup> , are widely studied by theoretical or experimental efforts because of the large variety of intriguing properties, such as ferroelectricity, piezoelectricity, multiferroism and others [21–23]. The interest in this kind of structure arise from the possibility of to control the existence of different properties from chemical substitution or doping on A- and B-sites [24]. For instance, a very common mineral on earth surface (FeTiO3 ) presents the ilmenite structure can shelter a high compositional diversity of A2+ and B4+ cations that occupy alternate basal-planes along the [001] hexagonal axis of a ordered corundum structure (R3) [25]. The most investigated ilmenite materials are based on Ti atoms in B sites with different atoms in A2+ (A = Mn, Fe, Ni, Co) cations; however, other materials were known in this structure [26–29].

Goodenough and Stickler proved that the magnetic ordering of ilmenite materials, mainly ATiO3 (A = Mn, Fe, Ni, Co), are antiferromagnetic insulators and have two different magnetic

of this class of magnetic couplings is the existence of effects such as Giant Magnetoresistance and Spin Tunneling that can be explained by the use of RKKY Hamiltonian [1, 11, 13–15].

The Goodenough-Kanamori-Anderson (GKA) rule describe the interatomic spin-spin interactions between two atoms considering the existence of a virtual electron transfers between them (superexchange) and/or between a shared anion (so-called "bridge") and the two atoms. As consequence, in this rule atoms with orthogonal orbitals do not overlap reducing the repulsion between the electrons, following the Hund's rule. Therefore, the exchange interac-

In addition, the GKA rule predicts some interactions that depend on the occupation of interacting orbitals. In the first case, an antiferromagnetic interaction occurs when the virtual electron transfer is between half-filled orbitals. In opposition, they are ferromagnetic once the

The GKA rules are important due to the introduction of bond character in the evaluation of the magnetic interactions. It is important to point out that the net spin does not change with the covalent component associated with the chemical bonds; however this component can extends the cation wave function out because the interaction between the orbitals is large, resulting in chemical dependent ferromagnetic/antiferromagnetic couplings associ-

The Dzyaloshinskii-Moriya [17–20] rule states that a low magnetic resultant is observed in antiferromagnetic material due to a long range interaction between magnetic atoms in disor-

 **(B = Ti, Zr, Hf, Si, Ge, Sn) materials: insights into interatomic distance** 

or experimental efforts because of the large variety of intriguing properties, such as ferroelectricity, piezoelectricity, multiferroism and others [21–23]. The interest in this kind of structure arise from the possibility of to control the existence of different properties from chemical substitution or doping on A- and B-sites [24]. For instance, a very common mineral on earth

of A2+ and B4+ cations that occupy alternate basal-planes along the [001] hexagonal axis of a ordered corundum structure (R3) [25]. The most investigated ilmenite materials are based on Ti atoms in B sites with different atoms in A2+ (A = Mn, Fe, Ni, Co) cations; however, other

Goodenough and Stickler proved that the magnetic ordering of ilmenite materials, mainly

(A = Mn, Fe, Ni, Co), are antiferromagnetic insulators and have two different magnetic

) presents the ilmenite structure can shelter a high compositional diversity

, are widely studied by theoretical

dered crystalline structures. This particular behavior is called weak-Ferromagnetism.

tion between spins is positive and a ferromagnetic ordering arises.

virtual electron transfer is from a half-filled to an empty/filled orbital.

ated the superexchange electron transfer [16].

Perovskites oxides, which have the general formula ABO<sup>3</sup>

materials were known in this structure [26–29].

*1.1.6. Dzyaloshinskii-Moriya rule*

**1.2. FeBO3**

**and orbital overlap**

surface (FeTiO3

ATiO3

*1.1.5. GKA rules*

128 Density Functional Theory

**Figure 3.** Ilmenite-type conventional unit cell and their exchange coupling constants. Black, orange and red balls represent A2+, B4+ and O2− ions, respectively.

couplings constants: Intralayer (J<sup>1</sup> ) and Interlayers (J<sup>2</sup> ), as shown in **Figure 1**. J<sup>1</sup> refers to the magnetic exchange that happens between A-O-A atoms and is dominated by the coupling of a t2g orbital in one cation with an eg orbital in other. Therefore, the signal for exchange parameter depends upon the occupancies of the interacting orbitals making MnTiO<sup>3</sup> antiferromagnetic and FeTiO3 , CoTiO3 , NiTiO3 ferromagnetic for intralayer coupling (J<sup>1</sup> ). In turn, J 2 interactions are mediated by the BO6 clusters in the intermetallic connection A-O-B-O-A, having less contribution of a direct overlap in [001] direction because of the vacancy in cationic sublattice (**Figure 3**) with opposite magnetization directions between adjacent A layers (antiferromagnetic) [25, 30].

In Fe-based ilmenite materials the long-range exchange coupling (J<sup>2</sup> ) stabilizes the antiferromagnetism; since J<sup>2</sup> depends of intermetallic connection Fe-O-B-O-Fe, the non-magnetic B-site replacement can control such magnetic ordering. As previously discussed, the ilmenite structure arrangement creates vacancies between adjacent Fe2+ layers that are separated each other by a B-site plane. So, the interlayer magnetic coupling integral can be visualized as a direct coupling between 3d orbitals of adjacent Fe2+ cations. Furthermore, other evidence that the non-magnetic B-site substitution affects the magnetic ordering is: the vacancy formation occurs in this cationic sublattice and the direct exchange coupling depends on the distance.

This behavior was confirmed investigating the magnetic ground-state of FeBO<sup>3</sup> (B = Ti, Zr, Hf, Si, Ge, Sn) materials by means of Density Functional Theory (DFT) calculations [31, 32]. **Figure 4** shows the relative Energy between AFM and FM structures as function of ionic radii of B4+ site cations.

From these results, it was observed that the FEM state is stabilized for ilmenite materials with large B-site metals (Sn, Hf, Zr); whereas, ilmenite cells contracted (Si, Ge, Ti) exhibit an AFM behavior. This result can be discussed as function of electronic repulsion between Fe2+ atoms 3d orbitals in different layers. In this case, we use the c-axis oriented to 3d<sup>z</sup> 2 orbital, once the cationic vacancy induces a coupling in such direction. Therefore, the B-cation volume control the distance among different Fe2+ layers from an angular distortion in O-B-O bonds in axial plane, which causes an increase/decrease in B-O bond distances allowing a higher/

analysis of ionic radius and bond distance for FeBO3

ionic radius and valence orbitals of the B-site non-magnetic metals.

The corundum ordered structure is most commonly called LiNbO3

 **and NiXO3**

group characteristic of ilmenite. In particular, LNO structure has an ABO3

**1.3. Magnetism in corundum-ordered structure (LiNbO3**

**properties in ferroelectric XNiO3**

on LiNbO3

LiNbO3

R3c structures.

, NiTiO3

PbNiO3

The magnetic properties for XNiO3

and NiHfO3

of this type of structure.

increase in ionic radius from Si4+ to Sn4+ results entails on a large distancing among Fe2+ layers in intermetallic connection that drastically reduce the electronic repulsion and stabilize the FM configuration. The information obtained by these theoretical results demonstrates that the control of magnetic ordering in ilmenite materials is based on a complex relation between

material to present such crystalline phase. R3c materials are largely employed in development of memory devices, holographic data storage, electronic, electro-optical and optical devices, photo-induced devices and photocatalytic application due to a unique set of electronic, optical and ferroelectric properties usually observed for these materials. The first work

type; [33] however, in 1952 such information were refuted by Bailey [34, 35] that obtained results proving that, at room conditions, the LNO crystallizes in a R3c group instead of a R3

lattice parameters a = b ≠ c and angles α = β = 90° and γ = 120°; the A and B cations are both surrounded by six O atoms forming two distorted octahedra. Some features of LiNbO<sup>3</sup>

structure are: (i) high distortion degree for octahedra within structure (**Figure 6a**); (ii) alternation between the cations A and B cation and vacancies along the c axis (**Figure 6b**); (iii) highly compacted layer composed by O atoms; (iv) the presence of intrinsic ordering vacancies [36–38]. These features are the responsible for the high ferroelectric properties characteristic

The magnetism in this structure arises from unpaired electron from atoms occupying A or B sites within structure. In particular, the ground state spin ordering for materials in

ture and presents only one magnetic coupling constant (J) that refers to intralayer interaction (J) that occurs between M-O-M in the structure, where M is the magnetic cation in the structure (as observed in **Figure 6c**). At general, the magnetism in Solid State materials can be successfully described by Direct or Indirect interactions, Exchange interactions and Super Exchange interactions or Goodenough-Kanamori-Anderson (GKA) Rule; all of them, considers the unit cell magnetic resultant to determine the magnetism. However, the structural disorder makes such approaches not enough to predict the magnetism in

were evaluated by computational methodologies based on DFT [39, 40]. The obtained results

ferromagnetic; whereas, the other materials are FM. The evaluation of structural properties for such materials evidences a connection between structural regularity and magnetic phase

and NiXO3

for J for such materials were presented in **Figure 7**; as observed, TiNiO3


(LNO) structure were reported in 1949 and discuss this new structure as a ilmenite

(B = Si, Ge, Sn) materials shows that the

http://dx.doi.org/10.5772/intechopen.81772

**-type structure): magnetic** 

(X = Ti, Ge, Zr, Sn, Hf and Pb) in R3c structures

are AFM and, as predicted by DMI interaction, are weak-

, GeNiO<sup>3</sup>

, ZrNiO<sup>3</sup>

,


general formula,


131

 **(X = Ti, Ge, Zr, Sn, Hf and Pb) materials**

Magnetic Ordering in Ilmenites and Corundum-Ordered Structures

**Figure 4.** Energy difference (in meV) between AFM and FM configuration as function of B-site cation ionic radii (in Å).

lower interlayer distance between adjacent Fe2+ layers, affecting the signal of J, as represented in **Figure 5**.

In addition, the interlayer coupling has an additional degree of freedom associated with the Fe-O-B-O-Fe intermetallic connection, which is the responsible to originate a long-range coupling. This behavior can be related to the large overlap between valence orbitals from transition metals and 2p oxygen orbitals, which strengthen the intermetallic connection and, consequently, it induces a large electronic repulsion among unpaired electrons stabilizing the AFM configuration from Pauli Exclusion Principle. In case of the moving from Ti to Hf, it was noted that only FeTiO3 shows a large contribution of 3d overlapped with O 2p orbital suggesting a higher overlap between these states providing strengthens in intermetallic connection that stabilizes AFM ground-state. In contrast, the smaller overlap between Zr(4d)/Hf(5d) and O(2p) orbitals allied to large Zr-O and Hf-O bond distances creates a smaller interlayer electronic repulsion responsible by FEM ordering.

Likewise, for FeBO<sup>3</sup> (B = Si, Ge, Sn) ilmenite materials is expected a FM ordering due to the absence of d valence orbitals. Nevertheless, only Sn-based ilmenite has this configuration indicating a big effect of ionic radius and interlayer distance of the FeO<sup>6</sup> clusters. The

**Figure 5.** Representation of electronic repulsion between adjacent Fe2+ layer in FeBO3 (B = Ti, Zr, Hf, Si, Ge, Sn) as function of ionic radius of B-site cation and its influence on magnetic ordering.

analysis of ionic radius and bond distance for FeBO3 (B = Si, Ge, Sn) materials shows that the increase in ionic radius from Si4+ to Sn4+ results entails on a large distancing among Fe2+ layers in intermetallic connection that drastically reduce the electronic repulsion and stabilize the FM configuration. The information obtained by these theoretical results demonstrates that the control of magnetic ordering in ilmenite materials is based on a complex relation between ionic radius and valence orbitals of the B-site non-magnetic metals.

#### **1.3. Magnetism in corundum-ordered structure (LiNbO3 -type structure): magnetic properties in ferroelectric XNiO3 and NiXO3 (X = Ti, Ge, Zr, Sn, Hf and Pb) materials**

The corundum ordered structure is most commonly called LiNbO3 -type since this is the first material to present such crystalline phase. R3c materials are largely employed in development of memory devices, holographic data storage, electronic, electro-optical and optical devices, photo-induced devices and photocatalytic application due to a unique set of electronic, optical and ferroelectric properties usually observed for these materials. The first work on LiNbO3 (LNO) structure were reported in 1949 and discuss this new structure as a ilmenite type; [33] however, in 1952 such information were refuted by Bailey [34, 35] that obtained results proving that, at room conditions, the LNO crystallizes in a R3c group instead of a R3 group characteristic of ilmenite. In particular, LNO structure has an ABO3 general formula, lattice parameters a = b ≠ c and angles α = β = 90° and γ = 120°; the A and B cations are both surrounded by six O atoms forming two distorted octahedra. Some features of LiNbO<sup>3</sup> -type structure are: (i) high distortion degree for octahedra within structure (**Figure 6a**); (ii) alternation between the cations A and B cation and vacancies along the c axis (**Figure 6b**); (iii) highly compacted layer composed by O atoms; (iv) the presence of intrinsic ordering vacancies [36–38]. These features are the responsible for the high ferroelectric properties characteristic of this type of structure.

lower interlayer distance between adjacent Fe2+ layers, affecting the signal of J, as represented

**Figure 4.** Energy difference (in meV) between AFM and FM configuration as function of B-site cation ionic radii (in Å).

In addition, the interlayer coupling has an additional degree of freedom associated with the Fe-O-B-O-Fe intermetallic connection, which is the responsible to originate a long-range coupling. This behavior can be related to the large overlap between valence orbitals from transition metals and 2p oxygen orbitals, which strengthen the intermetallic connection and, consequently, it induces a large electronic repulsion among unpaired electrons stabilizing the AFM configuration from Pauli Exclusion Principle. In case of the moving from Ti to Hf, it was

ing a higher overlap between these states providing strengthens in intermetallic connection that stabilizes AFM ground-state. In contrast, the smaller overlap between Zr(4d)/Hf(5d) and O(2p) orbitals allied to large Zr-O and Hf-O bond distances creates a smaller interlayer elec-

the absence of d valence orbitals. Nevertheless, only Sn-based ilmenite has this configura-

tion indicating a big effect of ionic radius and interlayer distance of the FeO<sup>6</sup>

**Figure 5.** Representation of electronic repulsion between adjacent Fe2+ layer in FeBO3

function of ionic radius of B-site cation and its influence on magnetic ordering.

shows a large contribution of 3d overlapped with O 2p orbital suggest-

(B = Si, Ge, Sn) ilmenite materials is expected a FM ordering due to

clusters. The

(B = Ti, Zr, Hf, Si, Ge, Sn) as

in **Figure 5**.

130 Density Functional Theory

noted that only FeTiO3

Likewise, for FeBO<sup>3</sup>

tronic repulsion responsible by FEM ordering.

The magnetism in this structure arises from unpaired electron from atoms occupying A or B sites within structure. In particular, the ground state spin ordering for materials in LiNbO3 -type structure is determined by collinear interaction between layers in the structure and presents only one magnetic coupling constant (J) that refers to intralayer interaction (J) that occurs between M-O-M in the structure, where M is the magnetic cation in the structure (as observed in **Figure 6c**). At general, the magnetism in Solid State materials can be successfully described by Direct or Indirect interactions, Exchange interactions and Super Exchange interactions or Goodenough-Kanamori-Anderson (GKA) Rule; all of them, considers the unit cell magnetic resultant to determine the magnetism. However, the structural disorder makes such approaches not enough to predict the magnetism in R3c structures.

The magnetic properties for XNiO3 and NiXO3 (X = Ti, Ge, Zr, Sn, Hf and Pb) in R3c structures were evaluated by computational methodologies based on DFT [39, 40]. The obtained results for J for such materials were presented in **Figure 7**; as observed, TiNiO3 , GeNiO<sup>3</sup> , ZrNiO<sup>3</sup> , PbNiO3 , NiTiO3 and NiHfO3 are AFM and, as predicted by DMI interaction, are weakferromagnetic; whereas, the other materials are FM. The evaluation of structural properties for such materials evidences a connection between structural regularity and magnetic phase

**Figure 6.** Crystalline structures of R3c structure showing the alternation between B, A and vacancies along z axis (a), [AO6 ] and [BO6 ] octahedra within structure (b) and magnetic coupling constant (J) for R3c structures (c).

of the eg

**Figure 7.** Magnetic coupling constant (J) for XNiO<sup>3</sup>

to the AFM materials, while the blue range represents the FM systems.

als the [NiO6

**Figure 8.** DOS projected results for e<sup>g</sup>

represents the broken on degeneration of t2g energy levels.

t

and t2g energy levels of Ni2+. The projected DOS prove that in FM materials were

] clusters have the same electronic configuration, i.e., five α spins and three β

and t2g energy levels localized on Ni2+ in AFM and FM materials. The gray region

and XNiO3

(X = Ti, Ge, Zr, Sn, Hf, Pb) materials. The red region refers

Magnetic Ordering in Ilmenites and Corundum-Ordered Structures

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133

materi-

observed a displacement on degenerated t2g energy levels that creates two new groups of nondegenerated t2g energy levels (**Figure 8**). In particular, the lowest t2g energy levels and highest

spins giving rise to FM state, where all magnetic clusters are clearly at same orientation.

and NiXO3

2g energy levels are occupied by α and β spins, respectively. For all NiXO3

stability, according to the relation between the distortion degree (δ) between [AO6 ] and [BO6 ], as expressed by δ [*AO*6] /δ [*NiO*6] . Thus, for materials where the value of the equation is less than 1.5, the FM phase is more stable, whereas the AFM phase is more stable for materials with disorder ratio higher than 2.

Moreover, the theoretical investigation also provides information regarding to electronic levels of magnetic cations in XNiO3 and NiXO3 . In these materials, FM and AFM ordering are originated from eg and t2g energy levels of Ni2+ cations localized on [NiO6 ] clusters. Furthermore, projected Density of States (DOS) clarifies why magnetic ordering is changed from chemical modifications. In particular, for AFM materials there are [NiO<sup>6</sup> ] clusters magnetically ordered as α and β spins (**Figure 8**); structurally, these clusters are distributed by adjacent layers being each one oriented as one spin channel. Henceforth, α and β spins localized on eg energy levels are responsible to stabilize the AFM state.

However, how to understand the origin of the FM ordering in these materials? The existence of the FM ordering in materials is dependent on the spin orientation being in the same direction on all magnetic clusters. Thus, it is necessary to discuss the electronic configuration

**Figure 7.** Magnetic coupling constant (J) for XNiO<sup>3</sup> and NiXO3 (X = Ti, Ge, Zr, Sn, Hf, Pb) materials. The red region refers to the AFM materials, while the blue range represents the FM systems.

of the eg and t2g energy levels of Ni2+. The projected DOS prove that in FM materials were observed a displacement on degenerated t2g energy levels that creates two new groups of nondegenerated t2g energy levels (**Figure 8**). In particular, the lowest t2g energy levels and highest t 2g energy levels are occupied by α and β spins, respectively. For all NiXO3 and XNiO3 materials the [NiO6 ] clusters have the same electronic configuration, i.e., five α spins and three β spins giving rise to FM state, where all magnetic clusters are clearly at same orientation.

stability, according to the relation between the distortion degree (δ) between [AO6

and t2g energy levels of Ni2+ cations localized on [NiO6

each one oriented as one spin channel. Henceforth, α and β spins localized on eg

and NiXO3

modifications. In particular, for AFM materials there are [NiO<sup>6</sup>

1.5, the FM phase is more stable, whereas the AFM phase is more stable for materials with

**Figure 6.** Crystalline structures of R3c structure showing the alternation between B, A and vacancies along z axis (a),

] octahedra within structure (b) and magnetic coupling constant (J) for R3c structures (c).

Moreover, the theoretical investigation also provides information regarding to electronic levels

projected Density of States (DOS) clarifies why magnetic ordering is changed from chemical

as α and β spins (**Figure 8**); structurally, these clusters are distributed by adjacent layers being

However, how to understand the origin of the FM ordering in these materials? The existence of the FM ordering in materials is dependent on the spin orientation being in the same direction on all magnetic clusters. Thus, it is necessary to discuss the electronic configuration

. Thus, for materials where the value of the equation is less than

. In these materials, FM and AFM ordering are origi-

as expressed by δ

] and [BO6

132 Density Functional Theory

[AO6

nated from eg

disorder ratio higher than 2.

of magnetic cations in XNiO3

[*AO*6] /δ [*NiO*6]

are responsible to stabilize the AFM state.

] and [BO6

energy levels

] clusters. Furthermore,

] clusters magnetically ordered

],

**Figure 8.** DOS projected results for e<sup>g</sup> and t2g energy levels localized on Ni2+ in AFM and FM materials. The gray region represents the broken on degeneration of t2g energy levels.

Moreover, magnetic LNO-type materials have attracted the interest of materials scientist around the world in the last years as promising alternatives as smart and functional materials, mainly as multiferroic (MF) materials. The multiferroism consists on the coupling between magnetic ordering and some ferroic property in the same crystalline phase, but the main form of multiferroism is called of magnetoelectric (ME) coupling and are obtained by coupling between the magnetic ordering and ferroelectric properties. Consequently, electric polarization can be induced by a magnetic field or vice-versa.

results precision and study time [44]. Another factor responsible for wide use of this theory is the high versatility, once presents good results to investigation of magnetic materials, semi-

Magnetic Ordering in Ilmenites and Corundum-Ordered Structures

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135

Several computational approaches can be applied to evaluation of materials properties; for instance: Molecular Dynamics (MD), *ab initio* methods and Semi-Empirical Methods. The MD analyzes the system properties based on the behavior of ball-and-springs models under application of an Force external field to atoms representation, while *ab initio* and Semi-empirical methodologies uses different approaches to solve the Schrödinger Equation and to obtain the system Wave-function (Ψ). In turn, DFT assumes that the system Total Energy is a unique functional of electronic density; this methodology can be simplified in two postulates: [46].

**i.** The ground state properties for a system can be exactly and completely determined by the Density Functional (ρ), which is only dependent of three variables that determines the

**ii.** Any try function for electronic density will have energy greater or equal than the ground

However, the analytical function for electronic density is not known; thus, the electronic density is obtained by Hartree-Fock Equations (HF) for achievement of ρ by a Self Consistent Field (SCF) Method. Although this similarity was observed, the DFT shows a highest precision and computational cost in relation to HF simulations due to smaller number of variable. The HF and Semi-Empirical methods employ a number of variables in the 4n order, where n refers to the number of electrons in the system; whereas, the DFT is dependent on three variables [43, 44]. The DFT formulism was proposed by Kohn and Shan (KS) in 1965 and consists in two equations applied on two different systems; such systems consider that there are no or there are interactions between electrons, respectively. The results for both systems show a significant difference of energy between then, and, aiming to correct this difference, the Exchangecorrelation Term (EXC) was inserted in DFT proposed. The EXC is the sum of kinetic and potential energy difference between both systems. The physical meaning of EXC is the interaction between electrons in the investigated system and refers to 1% of system Total Energy; due to

The description of EXC terms changes according to employed exchange-correlation functional and, hence, the choice of a functional has a giant effect on materials properties evaluation, as offering better results as offering a reduction in computational cost. Among the EXC functional stands out local (LDA and LSDA), non-local (pure GGA, PBE) and hybrid (PBE0, B3PW91, B3LYP, HSE06)

The DFT is widely used on materials modeling since offers an excellent relation time x efficiency, as denoted by results precision on prevision of materials properties in a short time. For instance, a DFT methodology provides thermodynamic properties for bulk or surfaces of

types; the last represents the most powerful alternative to predict materials project.

conductors, proteins, organic compounds and others [45].

**2.1. Density Functional Theory**

position (x, y and z).

state energy for a real system.

this, DFT describes 100% of system Total Energy.

**2.2. Magnetic materials modeling based on DFT**

The ME coupling was first reported in the first years of 1960s decade; thus, the study of this effect keep stable until that, in 2003, a strong coupling between ferroelectric and magnetic properties was observed for TbMnO3 and TbMn2 O5 materials as well as a high ferroelectricity for BiFeO3 (BFO) films. Thenceforth, the study of MF materials increases drastically so that a fast search on *Web of Science* indicates at least 970 manuscripts focused in this class of material only in 2017. The investigation of multiferroic materials aims its application at development of several devices such as actuators, magnetic readers, sensors, tunneling and data storage devices. Moreover, MF materials are also potential alternatives for spintronics. The main representative of this class is the BFO; this material exhibits antiferromagnetic ordering as well as Magnetoelectric coupling at room conditions allied to a unique set of electronic, optical, magnetic and ferroelectric properties. In addition, other multiferroic materials were investigated by means of theoretical and experimental efforts, such as YMnO<sup>3</sup> , BiMnO3 , PbNiO3 (PNO), FeTiO3 ; PbVO<sup>3</sup> , TbMnO3 , TbMn2 O5 , Ca3 CoMnO6 , LuFe2 O4 , BaNiF4 and others [41].

Even that many multiferroic materials are currently known, the development of MF material is this kind of material is delayed by crystalline structure and low Curie temperature (TC); the first due to the fact that only 13 symmetry groups are able to exhibit multiferroic coupling, while the last makes its application impossible in technological purposes [42]. In a very restrict group of crystalline structures and, among then, are observed structures with lower and higher symmetry, such as P2/c, P-1, perovskite, ilmenite (R3) and LiNbO<sup>3</sup> -type structures (R3c). It is important to highlight that the R3c structure is the most common structure observed in known multiferroic materials, such as BFO, PNO, YMnO<sup>3</sup> , BiMnO3 and TbMnO3 .

#### **2. Theoretical approaches to investigate magnetic solid state materials**

Historically, the investigation of new materials aiming technological applications was a hard task that needed a long time of study. In this period, the theoretical-computational method was seen only as tool to study of materials already discovered. Front of the fact that the technological advance is extremely dependent on development of new materials, theoretical methods helped materials chemists and physicist on development materials at higher speed. Ever since, theoretical methods based on quantum mechanical simulations are an important tool to evaluate material properties, mainly at the molecular level [43].

Nowadays, innumerous theoretical methods are available; however, in the last 30 years the Density Functional Theory (DFT) changes the world offering the best relation between the results precision and study time [44]. Another factor responsible for wide use of this theory is the high versatility, once presents good results to investigation of magnetic materials, semiconductors, proteins, organic compounds and others [45].

#### **2.1. Density Functional Theory**

Moreover, magnetic LNO-type materials have attracted the interest of materials scientist around the world in the last years as promising alternatives as smart and functional materials, mainly as multiferroic (MF) materials. The multiferroism consists on the coupling between magnetic ordering and some ferroic property in the same crystalline phase, but the main form of multiferroism is called of magnetoelectric (ME) coupling and are obtained by coupling between the magnetic ordering and ferroelectric properties. Consequently, electric polariza-

The ME coupling was first reported in the first years of 1960s decade; thus, the study of this effect keep stable until that, in 2003, a strong coupling between ferroelectric and magnetic

that a fast search on *Web of Science* indicates at least 970 manuscripts focused in this class of material only in 2017. The investigation of multiferroic materials aims its application at development of several devices such as actuators, magnetic readers, sensors, tunneling and data storage devices. Moreover, MF materials are also potential alternatives for spintronics. The main representative of this class is the BFO; this material exhibits antiferromagnetic ordering as well as Magnetoelectric coupling at room conditions allied to a unique set of electronic, optical, magnetic and ferroelectric properties. In addition, other multiferroic materials were investigated by means of theoretical and experimental efforts,

O5

(BFO) films. Thenceforth, the study of MF materials increases drastically so

; PbVO<sup>3</sup>

, TbMnO3

materials as well as a high ferroelectric-

, TbMn2

, BiMnO3

O5 , Ca3

CoMnO6


and TbMnO3

,

.

and TbMn2

(PNO), FeTiO3

and higher symmetry, such as P2/c, P-1, perovskite, ilmenite (R3) and LiNbO<sup>3</sup>

observed in known multiferroic materials, such as BFO, PNO, YMnO<sup>3</sup>

tool to evaluate material properties, mainly at the molecular level [43].

**2. Theoretical approaches to investigate magnetic solid state** 

Even that many multiferroic materials are currently known, the development of MF material is this kind of material is delayed by crystalline structure and low Curie temperature (TC); the first due to the fact that only 13 symmetry groups are able to exhibit multiferroic coupling, while the last makes its application impossible in technological purposes [42]. In a very restrict group of crystalline structures and, among then, are observed structures with lower

tures (R3c). It is important to highlight that the R3c structure is the most common structure

Historically, the investigation of new materials aiming technological applications was a hard task that needed a long time of study. In this period, the theoretical-computational method was seen only as tool to study of materials already discovered. Front of the fact that the technological advance is extremely dependent on development of new materials, theoretical methods helped materials chemists and physicist on development materials at higher speed. Ever since, theoretical methods based on quantum mechanical simulations are an important

Nowadays, innumerous theoretical methods are available; however, in the last 30 years the Density Functional Theory (DFT) changes the world offering the best relation between the

tion can be induced by a magnetic field or vice-versa.

properties was observed for TbMnO3

, BiMnO3

and others [41].

, PbNiO3

ity for BiFeO3

134 Density Functional Theory

such as YMnO<sup>3</sup>

**materials**

, BaNiF4

LuFe2 O4 Several computational approaches can be applied to evaluation of materials properties; for instance: Molecular Dynamics (MD), *ab initio* methods and Semi-Empirical Methods. The MD analyzes the system properties based on the behavior of ball-and-springs models under application of an Force external field to atoms representation, while *ab initio* and Semi-empirical methodologies uses different approaches to solve the Schrödinger Equation and to obtain the system Wave-function (Ψ). In turn, DFT assumes that the system Total Energy is a unique functional of electronic density; this methodology can be simplified in two postulates: [46].


However, the analytical function for electronic density is not known; thus, the electronic density is obtained by Hartree-Fock Equations (HF) for achievement of ρ by a Self Consistent Field (SCF) Method. Although this similarity was observed, the DFT shows a highest precision and computational cost in relation to HF simulations due to smaller number of variable. The HF and Semi-Empirical methods employ a number of variables in the 4n order, where n refers to the number of electrons in the system; whereas, the DFT is dependent on three variables [43, 44].

The DFT formulism was proposed by Kohn and Shan (KS) in 1965 and consists in two equations applied on two different systems; such systems consider that there are no or there are interactions between electrons, respectively. The results for both systems show a significant difference of energy between then, and, aiming to correct this difference, the Exchangecorrelation Term (EXC) was inserted in DFT proposed. The EXC is the sum of kinetic and potential energy difference between both systems. The physical meaning of EXC is the interaction between electrons in the investigated system and refers to 1% of system Total Energy; due to this, DFT describes 100% of system Total Energy.

The description of EXC terms changes according to employed exchange-correlation functional and, hence, the choice of a functional has a giant effect on materials properties evaluation, as offering better results as offering a reduction in computational cost. Among the EXC functional stands out local (LDA and LSDA), non-local (pure GGA, PBE) and hybrid (PBE0, B3PW91, B3LYP, HSE06) types; the last represents the most powerful alternative to predict materials project.

#### **2.2. Magnetic materials modeling based on DFT**

The DFT is widely used on materials modeling since offers an excellent relation time x efficiency, as denoted by results precision on prevision of materials properties in a short time. For instance, a DFT methodology provides thermodynamic properties for bulk or surfaces of materials directly or using some additional approach as Berry Phase [47, 48] or Bulk Modulus [49, 50]. In turn, in Solid State materials the magnetic interactions are usually described by the Heisenberg Hamiltonian; however, in computer simulations this approach cannot be used because later determinant is not an eigenstate of such Hamiltonian. Thus, it is necessary the application of the Ising model which possibilities the evaluation of J magnetic coupling constant and ground-state magnetic ordering for simple and complex magnetic structure since using the energy difference between AFM and FM magnetic states since the unpaired electrons are well defined in the model Eq. (1). The Ising Model is described by Eq. (2), where N refers to number of magnetic cations in unit cell, Z is the number of magnetic neighbors and Sx represents the spin charge for each magnetic atom in unit cell. Both Eqs. (1) and (2) are combined in Eq. (3). The application of Ising Model was performed after the structural optimization of magnetic materials and requires that the spin orientation of unpaired electrons is well defined in the model. The Ising Model is described by Eqs. (1) and (2), where N refers to number of magnetic cations in unit cell, Z is the number of magnetic neighbors and S<sup>x</sup> represents the spin charge for each magnetic atom in unit cell. The application of Ising Model was performed after the structural optimization of magnetic materials and requires that the spin orientation of unpaired electrons is well defined in the model. Plus, in materials with more than one magnetic and not reproductive site, the Ising model should be applied to each magnetic ordering and thus, applied to ground-state model.

$$
\Delta E\_T = E\_{T\_{\text{sum}}} - E\_{T\_{\text{m}}} \tag{1}
$$

In case of ilmenite ATiO3

distortion in [XO6

materials design.

**Author details**

Paraná, Brazil

**References**

and Renan Augusto Pontes Ribeiro

Materials. 2013;**346**:58-73

Materials. 2008;**9**(1):014101

derivatives, the local structure associated with the magnetic order-

and FeHfO3.

Magnetic Ordering in Ilmenites and Corundum-Ordered Structures

(X = Ti, Ge, Zr, Sn, Hf and Pb) materi-

http://dx.doi.org/10.5772/intechopen.81772

137

not conventional energy

ing of Fe2+ layers was systematic investigated from theoretical calculations, resulting in chemical dependent interactions depending upon the interlayer distance (B-site cation volume) and orbital-resolved electron transfer. The summation between this analysis was helpful to

and NiXO3

] clusters; (ii) arise from Ni t2g and eg

als the theoretical results indicates that the magnetic ordering is: (i) dependent on structural

degeneration and (iii) even antiferromagnetic materials presents a magnetic resultant in consequence of structural distortion characteristic of R3c structure. The employment of theoretical approaches based on DFT possibilities the proposal of new material to technological purpose based in the magnetism as well as provides a complete description of magnetic,

Despite the challenge dealing with magnetic materials, theoretical simulations, mainly DFTbased, can be used to clarify the unusual phenomena's commonly observed in such materials, as well as to predict new candidates with singular properties, improving the perspective of

Chemical Simulation Group, Department of Chemistry State University of Ponta Grossa,

[1] Singh R. Unexpected magnetism in nanomaterials. Journal of Magnetism and Magnetic

[2] Inomata K, Ikeda N, Tezuka N, Goto R, Sugimoto S, Wojcik M, et al. Highly spinpolarized materials and devices for spintronics. Science and Technology of Advanced

[3] Fabian J, Matos-Abiague A, Ertler C, Stano P, Žutić I. Semiconductor spintronics. Acta

[4] Felser C, Fecher Gerhard H, Balke B. Spintronics: A challenge for materials science and solid-state chemistry. Angewandte Chemie International Edition. 2007;**46**(5):668-699

predict new ferromagnetic materials such as FeSnO3, FeZrO<sup>3</sup>

structural and electronic properties in a short time period.

Sergio Ricardo De Lazaro\*, Luis Henrique Da Silveira Lacerda

Physica Slovaca Reviews and Tutorials. 2007;**57**:565

\*Address all correspondence to: srlazaro@uepg.br

Additionally, for corundum ordered XNiO3

] and [NiO6

$$
\hat{\mathbf{H}}\_{\text{today}} = -NZ.\mathbf{S}\_1.\mathbf{S}\_2.f\tag{2}
$$

$$E\_{\mathbf{r}\_{su}} - E\_{\mathbf{r}\_{su}} = -\mathbf{N}Z.\mathbf{S}\_{\mathbf{r}}\mathbf{S}\_{\mathbf{r}}J \tag{3}$$

The Ising Model methodology to prevision of magnetic properties of Solid State materials was employed by Ribeiro and coauthors [51], Chartier and coauthors [52], Feng and Harrison [53] and Lacerda and de Lazaro [39] to investigate materials and different crystalline structures proving the efficiency of this approach. The high efficiency and almost direct measurement offered by Ising Model indicates it as the best theoretical approach to investigation of magnetic solid state materials.

#### **3. Conclusion**

In this chapter, the role of Density Functional Theory calculations for strongly correlated materials was summarized, focusing in the description of the ground-state magnetic ordering for solid state materials, in particular ATiO3 (A = Ti, Zr, Hf, Si, Ge, Sn) ilmenite derivatives and corundum ordered XNiO3 and NiXO3 (X = Ti, Ge, Zr, Sn, Hf and Pb) materials. A review about the theories behind the magnetism in solid state materials was presented in order to clarify the contribution of collective magnetic ordering, local structure and exchange interactions.

In case of ilmenite ATiO3 derivatives, the local structure associated with the magnetic ordering of Fe2+ layers was systematic investigated from theoretical calculations, resulting in chemical dependent interactions depending upon the interlayer distance (B-site cation volume) and orbital-resolved electron transfer. The summation between this analysis was helpful to predict new ferromagnetic materials such as FeSnO3, FeZrO<sup>3</sup> and FeHfO3.

Additionally, for corundum ordered XNiO3 and NiXO3 (X = Ti, Ge, Zr, Sn, Hf and Pb) materials the theoretical results indicates that the magnetic ordering is: (i) dependent on structural distortion in [XO6 ] and [NiO6 ] clusters; (ii) arise from Ni t2g and eg not conventional energy degeneration and (iii) even antiferromagnetic materials presents a magnetic resultant in consequence of structural distortion characteristic of R3c structure. The employment of theoretical approaches based on DFT possibilities the proposal of new material to technological purpose based in the magnetism as well as provides a complete description of magnetic, structural and electronic properties in a short time period.

Despite the challenge dealing with magnetic materials, theoretical simulations, mainly DFTbased, can be used to clarify the unusual phenomena's commonly observed in such materials, as well as to predict new candidates with singular properties, improving the perspective of materials design.

#### **Author details**

(1)

.*J* (2)

.*J* (3)

(A = Ti, Zr, Hf, Si, Ge, Sn) ilmenite derivatives and

(X = Ti, Ge, Zr, Sn, Hf and Pb) materials. A review about

materials directly or using some additional approach as Berry Phase [47, 48] or Bulk Modulus [49, 50]. In turn, in Solid State materials the magnetic interactions are usually described by the Heisenberg Hamiltonian; however, in computer simulations this approach cannot be used because later determinant is not an eigenstate of such Hamiltonian. Thus, it is necessary the application of the Ising model which possibilities the evaluation of J magnetic coupling constant and ground-state magnetic ordering for simple and complex magnetic structure since using the energy difference between AFM and FM magnetic states since the unpaired electrons are well defined in the model Eq. (1). The Ising Model is described by Eq. (2), where N refers to number of magnetic cations in unit cell, Z is the number of magnetic neighbors and Sx represents the spin charge for each magnetic atom in unit cell. Both Eqs. (1) and (2) are combined in Eq. (3). The application of Ising Model was performed after the structural optimization of magnetic materials and requires that the spin orientation of unpaired electrons is well defined in the model. The Ising Model is described by Eqs. (1) and (2), where N refers to number of magnetic cations in unit cell, Z is the number of magnetic neighbors and S<sup>x</sup> represents the spin charge for each magnetic atom in unit cell. The application of Ising Model was performed after the structural optimization of magnetic materials and requires that the spin orientation of unpaired electrons is well defined in the model. Plus, in materials with more than one magnetic and not reproductive site, the Ising model should be applied to each

− *ETFM*

.*S*2.

.*S*2.

= −*N*.*Z*.*S*<sup>1</sup>

The Ising Model methodology to prevision of magnetic properties of Solid State materials was employed by Ribeiro and coauthors [51], Chartier and coauthors [52], Feng and Harrison [53] and Lacerda and de Lazaro [39] to investigate materials and different crystalline structures proving the efficiency of this approach. The high efficiency and almost direct measurement offered by Ising Model indicates it as the best theoretical approach to investigation of mag-

In this chapter, the role of Density Functional Theory calculations for strongly correlated materials was summarized, focusing in the description of the ground-state magnetic ordering

the theories behind the magnetism in solid state materials was presented in order to clarify the contribution of collective magnetic ordering, local structure and exchange interactions.

*Ising* = −*N*.*Z*.*S*<sup>1</sup>

− *ETFM*

magnetic ordering and thus, applied to ground-state model.

*ET* = *ETAFM*

Ĥ

*ETAFM*

for solid state materials, in particular ATiO3

and NiXO3

netic solid state materials.

corundum ordered XNiO3

**3. Conclusion**

136 Density Functional Theory

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

\*Address all correspondence to: srlazaro@uepg.br

Chemical Simulation Group, Department of Chemistry State University of Ponta Grossa, Paraná, Brazil

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**Chapter 7**

**Provisional chapter**

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

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

Molecular computing devices composed of biological substances, such as nucleic acid and ribonucleic acid plays a key role for the logical processing of a variety of inputs and viable outputs in the cellular machinery of all living organisms. These devices are directly dependent on the advancement in DNA and RNA technology. RNA nanoparticles can be engineered into a programmable and logically acting "Ribocomputing Devices"; a breakthrough at the interface of nanotechnology and synthetic biology. It opens a new path to the synthetic biologists to design reliable synthetic biological circuits which can be useful as the electronic circuits. In this emerging field, a number of challenges persist; as how to translate a variety of nucleic acid based logic gates developed by numerous research laboratories into the realm of silicon-based computing. So in this chapter we will discuss the advances in ribonucleic acid (RNA) based computing and it's potential to serve as an alternative to revolutionize silicon-based technology by theoretical means. Also the results of the calculated parameters with computational tools using Density

functional theory and the designed device circuits will be analyzed.

**Keywords:** genetic information, logic gates, toehold switches, energy landscapes,

RNA nanotechnology is the modern and recent field of science which applies the application of top-down or bottom-up approaches to build the artificial architectures of RNA at the nanoscale. The pioneer work on "RNA nanotechnology" was carried out by eminent Scientist Prof. Dr. P. Guo et al. [1–4] RNA nanotechnology involves the characterization of the physical, chemical and biological properties of artificial RNA scaffolds in computing devices.

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

© 2019 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.

DOI: 10.5772/intechopen.80491

**Devices"**

**Devices"**

Ruby Srivastava

Ruby Srivastava

**Abstract**

RNA nanoparticles

**1. Introduction**

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.80491


#### **Role of Density Functional Theory in "Ribocomputing Devices" Role of Density Functional Theory in "Ribocomputing Devices"**

DOI: 10.5772/intechopen.80491

#### Ruby Srivastava Ruby Srivastava

(B =

perovskites (M = Ti,

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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.80491

#### **Abstract**

Molecular computing devices composed of biological substances, such as nucleic acid and ribonucleic acid plays a key role for the logical processing of a variety of inputs and viable outputs in the cellular machinery of all living organisms. These devices are directly dependent on the advancement in DNA and RNA technology. RNA nanoparticles can be engineered into a programmable and logically acting "Ribocomputing Devices"; a breakthrough at the interface of nanotechnology and synthetic biology. It opens a new path to the synthetic biologists to design reliable synthetic biological circuits which can be useful as the electronic circuits. In this emerging field, a number of challenges persist; as how to translate a variety of nucleic acid based logic gates developed by numerous research laboratories into the realm of silicon-based computing. So in this chapter we will discuss the advances in ribonucleic acid (RNA) based computing and it's potential to serve as an alternative to revolutionize silicon-based technology by theoretical means. Also the results of the calculated parameters with computational tools using Density functional theory and the designed device circuits will be analyzed.

**Keywords:** genetic information, logic gates, toehold switches, energy landscapes, RNA nanoparticles

#### **1. Introduction**

RNA nanotechnology is the modern and recent field of science which applies the application of top-down or bottom-up approaches to build the artificial architectures of RNA at the nanoscale. The pioneer work on "RNA nanotechnology" was carried out by eminent Scientist Prof. Dr. P. Guo et al. [1–4] RNA nanotechnology involves the characterization of the physical, chemical and biological properties of artificial RNA scaffolds in computing devices.

© 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 reproduction in any medium, provided the original work is properly cited. © 2019 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.

These molecular computers are natural and/or artificial devices in which macromolecules, including proteins and nucleic acids mediate necessary functions. The three basic operations are similar to the computer operations as: sensing inputs, processing the inputs and generating specific outputs. The schematic representation of RNA nanoparticles with different constructed motifs are given in **Figure 1**.

device. Gate RNA was used to carry out the signal processing. These devices were operated at the post-transcriptional level and used an extended RNA transcript to co-localize all circuit sensing, computation, signal transduction, and output elements in the same self-assembled molecular complex. The advantage of these systems was the reduction in diffusion-mediated signal losses, lowered metabolic cost and improved circuit reliability. These devices utilized programmable RNA molecules, allowed effective in silico designs, composed of precisely designed synthetic RNAs networks, worked at the post-transcriptional level, minimized delays and improve the reliability of signal transduction. Further these circuits co-localized to integrate multiple circuit functions within a single transcript gate RNA [8]. The input RNAs can interact cooperatively with one another to activate the gate RNA for AND logic or they can prevent for NOT logic. Toehold switch designs, which translate an output gene only if a cognate trigger RNA is expressed in the cell, were also optimized to evaluate the AND logic

Role of Density Functional Theory in "Ribocomputing Devices"

http://dx.doi.org/10.5772/intechopen.80491

143

Now the question arises why there is a need to replace the silicon based technology?

The capabilities of digital electronic devices have increased in lock-step with Moore's Law, which states that the number of transistors in a microchip will double approximately every 2 years [10]. This will also end up reaching a limit, as the unending quest to miniaturize transistors, which also come to a halt due to the quantum tunneling effect [11]. When the distance of a gate is scaled down about 10 nm, its electrons will jump spontaneously from source to drain, and the control over the flow of electricity will be lost. To overcome this problem, it is possible to either increase the size of microchips or fabricate them as stacking microchips. Even then it

Further, there are few critical challenges to make a transistor at the atomic level: (i) the accuracy of computing will be affected, because the wires in the circuit will become too close and they will affect each other; (ii) the heat generated in such a small area with too many concentrated transistors will greatly affect the functions of the transistors; (iii) the energy consumption to cool the circuit board would be too high to be a burden. Also the chip temperature will

So the next option is to use the biomolecules, including DNA, RNA and proteins as the major elements in logic gate operations. The enzymatic selectivity (processing of specific chemical function) of biomolecules give them an advantage over silicon-based computing with both specificity and usability in an intracellular environment. As most of the biological reactions controlled by specific enzymes are interconnected with other functional inputs, it is possible to fabricate DNase- and/or RNase-based informational processing units. These units can be scaled-up to fabricate artificial biocomputing networks possessing variety of logic functions. Replacing the silicon computing devices with the molecular computers has the following other advantages: these devices are subjected to the universal physical laws, have a small mass-to-volume ratio and move in viscous media rather than in a vacuum or air. Though they cannot store momentum, kinetic energy or thermal energy for a significant period of time due to their extremely small size, and must operate isothermally. Molecular computing has an enhanced ability to provide parallel computation. These systems can self-assemble and

impact the circuit reliability, energy consumption and system cost.

gate for the ribocomputing devices [9].

will only delay the stalemate.

Nature has given us the best computer in the form of living cell. These cells generate enormous amount of signals (inputs) using a broad range of environmental factors, example: temperature, pH, pressure, nutrients, signaling chemicals, macromolecules, etc. [5–7]. Biological systems have the ability to adapt to new information from an altered environment. They can be self-assemble and self-reproduce, which might provide some economic advantages. Recently Eminent Prof. Alexander A. Green group has designed the ribocomputing devices in which RNA molecules were used as input signals and protein as the output signal. AND, OR, and NOT logic results were obtained from the self-assembly of input and gate RNAs in the

**Figure 1.** Schematic representation of RNA nanoparticles with different constructed motifs. Adapted with permission from Ref. [96].

device. Gate RNA was used to carry out the signal processing. These devices were operated at the post-transcriptional level and used an extended RNA transcript to co-localize all circuit sensing, computation, signal transduction, and output elements in the same self-assembled molecular complex. The advantage of these systems was the reduction in diffusion-mediated signal losses, lowered metabolic cost and improved circuit reliability. These devices utilized programmable RNA molecules, allowed effective in silico designs, composed of precisely designed synthetic RNAs networks, worked at the post-transcriptional level, minimized delays and improve the reliability of signal transduction. Further these circuits co-localized to integrate multiple circuit functions within a single transcript gate RNA [8]. The input RNAs can interact cooperatively with one another to activate the gate RNA for AND logic or they can prevent for NOT logic. Toehold switch designs, which translate an output gene only if a cognate trigger RNA is expressed in the cell, were also optimized to evaluate the AND logic gate for the ribocomputing devices [9].

These molecular computers are natural and/or artificial devices in which macromolecules, including proteins and nucleic acids mediate necessary functions. The three basic operations are similar to the computer operations as: sensing inputs, processing the inputs and generating specific outputs. The schematic representation of RNA nanoparticles with different

Nature has given us the best computer in the form of living cell. These cells generate enormous amount of signals (inputs) using a broad range of environmental factors, example: temperature, pH, pressure, nutrients, signaling chemicals, macromolecules, etc. [5–7]. Biological systems have the ability to adapt to new information from an altered environment. They can be self-assemble and self-reproduce, which might provide some economic advantages. Recently Eminent Prof. Alexander A. Green group has designed the ribocomputing devices in which RNA molecules were used as input signals and protein as the output signal. AND, OR, and NOT logic results were obtained from the self-assembly of input and gate RNAs in the

**Figure 1.** Schematic representation of RNA nanoparticles with different constructed motifs. Adapted with permission

constructed motifs are given in **Figure 1**.

142 Density Functional Theory

from Ref. [96].

Now the question arises why there is a need to replace the silicon based technology?

The capabilities of digital electronic devices have increased in lock-step with Moore's Law, which states that the number of transistors in a microchip will double approximately every 2 years [10]. This will also end up reaching a limit, as the unending quest to miniaturize transistors, which also come to a halt due to the quantum tunneling effect [11]. When the distance of a gate is scaled down about 10 nm, its electrons will jump spontaneously from source to drain, and the control over the flow of electricity will be lost. To overcome this problem, it is possible to either increase the size of microchips or fabricate them as stacking microchips. Even then it will only delay the stalemate.

Further, there are few critical challenges to make a transistor at the atomic level: (i) the accuracy of computing will be affected, because the wires in the circuit will become too close and they will affect each other; (ii) the heat generated in such a small area with too many concentrated transistors will greatly affect the functions of the transistors; (iii) the energy consumption to cool the circuit board would be too high to be a burden. Also the chip temperature will impact the circuit reliability, energy consumption and system cost.

So the next option is to use the biomolecules, including DNA, RNA and proteins as the major elements in logic gate operations. The enzymatic selectivity (processing of specific chemical function) of biomolecules give them an advantage over silicon-based computing with both specificity and usability in an intracellular environment. As most of the biological reactions controlled by specific enzymes are interconnected with other functional inputs, it is possible to fabricate DNase- and/or RNase-based informational processing units. These units can be scaled-up to fabricate artificial biocomputing networks possessing variety of logic functions.

Replacing the silicon computing devices with the molecular computers has the following other advantages: these devices are subjected to the universal physical laws, have a small mass-to-volume ratio and move in viscous media rather than in a vacuum or air. Though they cannot store momentum, kinetic energy or thermal energy for a significant period of time due to their extremely small size, and must operate isothermally. Molecular computing has an enhanced ability to provide parallel computation. These systems can self-assemble and self-reproduce, which might provide some economic advantages. Moreover, cells can be engineered to sense and respond to environmental signals, even under extreme conditions such as: high temperature, high pressure, radioactivity or toxic chemicals, and have the ability to adapt to new information for a changed environment. The ultimate goals of biocomputing are the monitoring and control of biological systems. The diagnosis of diseases, drug screening and to understand the experimental systems can be done by monitoring. It is also necessary to observe the environment to detect multiple disease indicators [11, 12], and cell based biosensors using logic gates to detect arsenic, mercury and copper ion levels [13]. Another utility is to control development, cell differentiation and re-programming, which mainly depends on gene regulatory networks [14, 15], tissue engineering and tissue regeneration [16], and to control the immune system and malign growth. Logic based biological devices are also executed to detect cancer cells (e.g. small-cell lung cancer, prostate cancer, HeLa cells), and to induce selective apoptosis of these cells [11, 17, 18].

The concept of DNA molecular computation was demonstrated by Prof. Leonard Adleman, in which the ability of synthetic DNA oligonucleotides were used to solve a seven-point Hamiltonian path problem by performing a sequence of logical operations [19]. Since then, the possibility of developing a new generation of molecular logic gates and molecular computers based on the advantages of DNA molecules has started [20–29]. The DNA molecules were chosen as they are amenable to well-regulated, programmable folding, and have the unique ability to store genetic information. The massive parallel computation power and colossal memory capacity [30, 31] were also the salient features of DNA technology, though improvement in specificity of interaction and chemical stability is still to be taken into account. DNA selfassembly properties through WC base interactions can also lead to the formation of a variety of structures (tiles) [32–34]. The studies have also been carried out on the nucleobase stacking pairs and coinage metal clusters to observe the Boolean operations [35]. However, the limitations in the application of DNA-based computation is the intolerable level of execution time in performing logic gates operation and in some circuits, it require more time to reach a stable state to achieve the final value (three-layer see-saw circuits). Another challenge is the limitation of functional proteins required for catalyzing the reactions. So the field of RNA technology has been explored. RNA can be served as the polymeric material to build varieties of structures which included nanoparticles, polygons, bundles, membranes, micro sponges and arrays. The different modifications for stable RNA nanoparticles are shown in **Figure 2**. The vital significance of RNA nanotechnology relies on application of RNA biopolymers at the atomic level. In RNA, the base pairing of nucleotides can be canonical as well as non-canonical and the tertiary interactions of RNA mediate multi-way junctions, bulges and internal hairpin loops. Apart from that RNA is thermodynamically more stable due to relatively low in energy. Now RNA is considered as a subdivision of nucleic acid nanotechnology due to its diverse functions.

chemical modification into nucleotides without significant alteration of the RNA property in folding and self-assembly; tuning the immunogenic properties of synthetic RNA constructs for in vivo applications; role of 2D, 3D, 4D structure and intermolecular interaction of RNA molecules; developing methods to control shape, size, and stoichiometry of RNA nanoparticles; regulation and processing functions of RNA in cells; cost in RNA production by biochemical synthesis; and safety of using RNA due to its therapeutic modality for cancer and

**Figure 2.** Different modifications for stable RNA nanoparticles (A) sugar modifications (B) backbone modifications (C) nucleic acid analogs (D) thermodynamic and (E) serum stability. Adapted with permission from Ref. [96].

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RNA molecules have main role of passing information from genome to proteome in all living creatures. The discovery of non-coding RNAs revealed that RNA performs more versatile functions, including gene expression and regulation. The pictorial representation of beacons and resistive biomemory in RNA nanotechnology applications is given in **Figure 3**. The RNA molecules are involved in computational algorithmic processes, including RNA editing (RNA sequence alternation) [36] and RNA-based regulatory networks [35, 36]. In previous studies carried out on RNA computational systems, the inputs were small RNA elements or motifs,

There are various classes of RNA functional molecules, such as ribozymes, RNA aptamers, riboswitches, miRNA and siRNA, orthogonal ribosomes [38, –39] that enable the fabrication of RNA-based nanoparticles to advance the modern nanotechnology [40–42]. The recent advances are endonuclease CRISPR (Clustered regularly interspaced short palindromic repeats) associated Cas9 [43] and the bacterial CRISPR pathway [44], which enables predictable programming of gene expression. Recent efforts in genome mining and programmable RNA based switches have increased the number of parts which further enable predictable design of deeply layered logic circuits. These biological circuits required digital-like charac-

other diseases without affecting the other organs.

and the output was the mRNA [35–37].

teristics as Boolean logic gates with all or nothing responses.

The reason for taking the RNA is twofold: RNA (i) possesses the structural properties of DNA and (ii) mimics the functional properties of proteins [36–40]. Interestingly the device design and the role of DFT used for DNA technology can implement to RNA nanoparticles also due to some similar structural properties. So here we will discuss the role of DFT in RNA technology based on the previous studies being carried out on the DNA technology. RNA nanoparticles also include the following properties based on these factors: introducing

self-reproduce, which might provide some economic advantages. Moreover, cells can be engineered to sense and respond to environmental signals, even under extreme conditions such as: high temperature, high pressure, radioactivity or toxic chemicals, and have the ability to adapt to new information for a changed environment. The ultimate goals of biocomputing are the monitoring and control of biological systems. The diagnosis of diseases, drug screening and to understand the experimental systems can be done by monitoring. It is also necessary to observe the environment to detect multiple disease indicators [11, 12], and cell based biosensors using logic gates to detect arsenic, mercury and copper ion levels [13]. Another utility is to control development, cell differentiation and re-programming, which mainly depends on gene regulatory networks [14, 15], tissue engineering and tissue regeneration [16], and to control the immune system and malign growth. Logic based biological devices are also executed to detect cancer cells (e.g. small-cell lung cancer, prostate cancer, HeLa cells), and to

The concept of DNA molecular computation was demonstrated by Prof. Leonard Adleman, in which the ability of synthetic DNA oligonucleotides were used to solve a seven-point Hamiltonian path problem by performing a sequence of logical operations [19]. Since then, the possibility of developing a new generation of molecular logic gates and molecular computers based on the advantages of DNA molecules has started [20–29]. The DNA molecules were chosen as they are amenable to well-regulated, programmable folding, and have the unique ability to store genetic information. The massive parallel computation power and colossal memory capacity [30, 31] were also the salient features of DNA technology, though improvement in specificity of interaction and chemical stability is still to be taken into account. DNA selfassembly properties through WC base interactions can also lead to the formation of a variety of structures (tiles) [32–34]. The studies have also been carried out on the nucleobase stacking pairs and coinage metal clusters to observe the Boolean operations [35]. However, the limitations in the application of DNA-based computation is the intolerable level of execution time in performing logic gates operation and in some circuits, it require more time to reach a stable state to achieve the final value (three-layer see-saw circuits). Another challenge is the limitation of functional proteins required for catalyzing the reactions. So the field of RNA technology has been explored. RNA can be served as the polymeric material to build varieties of structures which included nanoparticles, polygons, bundles, membranes, micro sponges and arrays. The different modifications for stable RNA nanoparticles are shown in **Figure 2**. The vital significance of RNA nanotechnology relies on application of RNA biopolymers at the atomic level. In RNA, the base pairing of nucleotides can be canonical as well as non-canonical and the tertiary interactions of RNA mediate multi-way junctions, bulges and internal hairpin loops. Apart from that RNA is thermodynamically more stable due to relatively low in energy. Now RNA is considered as a subdivision of nucleic acid nanotechnology due to its diverse functions.

The reason for taking the RNA is twofold: RNA (i) possesses the structural properties of DNA and (ii) mimics the functional properties of proteins [36–40]. Interestingly the device design and the role of DFT used for DNA technology can implement to RNA nanoparticles also due to some similar structural properties. So here we will discuss the role of DFT in RNA technology based on the previous studies being carried out on the DNA technology. RNA nanoparticles also include the following properties based on these factors: introducing

induce selective apoptosis of these cells [11, 17, 18].

144 Density Functional Theory

**Figure 2.** Different modifications for stable RNA nanoparticles (A) sugar modifications (B) backbone modifications (C) nucleic acid analogs (D) thermodynamic and (E) serum stability. Adapted with permission from Ref. [96].

chemical modification into nucleotides without significant alteration of the RNA property in folding and self-assembly; tuning the immunogenic properties of synthetic RNA constructs for in vivo applications; role of 2D, 3D, 4D structure and intermolecular interaction of RNA molecules; developing methods to control shape, size, and stoichiometry of RNA nanoparticles; regulation and processing functions of RNA in cells; cost in RNA production by biochemical synthesis; and safety of using RNA due to its therapeutic modality for cancer and other diseases without affecting the other organs.

RNA molecules have main role of passing information from genome to proteome in all living creatures. The discovery of non-coding RNAs revealed that RNA performs more versatile functions, including gene expression and regulation. The pictorial representation of beacons and resistive biomemory in RNA nanotechnology applications is given in **Figure 3**. The RNA molecules are involved in computational algorithmic processes, including RNA editing (RNA sequence alternation) [36] and RNA-based regulatory networks [35, 36]. In previous studies carried out on RNA computational systems, the inputs were small RNA elements or motifs, and the output was the mRNA [35–37].

There are various classes of RNA functional molecules, such as ribozymes, RNA aptamers, riboswitches, miRNA and siRNA, orthogonal ribosomes [38, –39] that enable the fabrication of RNA-based nanoparticles to advance the modern nanotechnology [40–42]. The recent advances are endonuclease CRISPR (Clustered regularly interspaced short palindromic repeats) associated Cas9 [43] and the bacterial CRISPR pathway [44], which enables predictable programming of gene expression. Recent efforts in genome mining and programmable RNA based switches have increased the number of parts which further enable predictable design of deeply layered logic circuits. These biological circuits required digital-like characteristics as Boolean logic gates with all or nothing responses.

**Figure 3.** RNA nanotechnology applications in beacons (A) and resistive biomemory (B). Adapted with permission from Ref. [96].

#### **2. Application of RNA nanotechnology in computation (in vivo)**

RNA nanotechnology emphasizes its critical role in the process of catalysis in cell signaling and sensing functions. RNAs use the concentrations of specific chemical species as signals to implement the functions of logic to build up the network of multiple layers Boolean networks. Then these molecular computers were able to operate and communicate directly in a biological environment [45, 46] or work inside a living cell [47]. A plasmid with a specifically encoded DNA was inserted and external program was used to control the computation inside the host cell. The mRNA was used to encode a fluorescent protein and the target sequences for small interfering RNA (siRNA) with control level of fluorescence exhibited by the cell. Another in vivo computation programming was observed by a response to both endogenous and exogenous molecular signals. In this work, combination of ribozymes and RNA aptamers were used [48, 49].

method for the QD/STVQD/STV/Bio-3WJ chimera nanoparticle is given in **Figure 4**. RNA bioelectronics can lead to bio computer devices; as information storage devices, logic gates, field effect transistors and computation systems. An autonomous bio molecular system has been suggested to logically regulate the gene system. The system has been used for prostate cancer state detection. The basic diagnostic rule is that if the specific gene related to prostate cancer is over expressed, then ssDNA bind to their mRNA and exhibits the protein synthesis. The logical output is the release of ssDNA with modulation of gene expression, defined by the two states "positive and negative" [12]. An RNAi logical evaluator has also been used to perform Boolean operations for human kidney cells [47] and intracellular device in single

**Figure 4.** Schematic diagram of the experimental design method for the QD/STVQD/STV/Bio-3WJ chimera nanoparticle (a) site-specification, (b) constitution, (c) expected structure, and (d) function. Adapted with permission from Ref. [96].

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The RNA junction database [53], NanoTiler [54], RNA2D3D algorithms [55], RNA dynamics [56, 57] are used to build RNA nanoparticles that incorporate individual RNA motifs to defined user specifications. The energy landscape perspective of bio molecular dynamics provides a quantitative framework to consistently integrate theoretical concepts and experimental observations. Recognizing the potential of this approach, there has been decades of investigation

mammalian cell.

**3. Role of DFT in RNA nanotechnology**

The concept to utilize the cleaving ability of ribozyme to a specific molecule with the computational work was carried out using the concentrations of two proteins as input and output, implemented by studying the mechanism of ribozyme-aptamer molecules using yeast. In another study it was shown that the individual mammalian cells were capable of executing the basic arithmetic functions in a robust manner. A memory module (toggle switch) was designed [50] and a scalable factor (transcriptional/post transcriptional) was used to construct a synthetic regulatory circuit with a HeLa cancer cell 'classifier' to sense the expression levels of a customizable set of endogenous microRNAs to trigger a cellular response that matches the predetermined profile of interest [51]. In gene circuit engineering, RNA mediated regulation employs three major mechanisms, cis-acting RNA structure related modulation of translation, catalytic RNA or ribozyme-mediated cleavage of target transcripts and trans acting antisense small RNAs-mediated regulation of translation [52]. The experimental design

**Figure 4.** Schematic diagram of the experimental design method for the QD/STVQD/STV/Bio-3WJ chimera nanoparticle (a) site-specification, (b) constitution, (c) expected structure, and (d) function. Adapted with permission from Ref. [96].

method for the QD/STVQD/STV/Bio-3WJ chimera nanoparticle is given in **Figure 4**. RNA bioelectronics can lead to bio computer devices; as information storage devices, logic gates, field effect transistors and computation systems. An autonomous bio molecular system has been suggested to logically regulate the gene system. The system has been used for prostate cancer state detection. The basic diagnostic rule is that if the specific gene related to prostate cancer is over expressed, then ssDNA bind to their mRNA and exhibits the protein synthesis. The logical output is the release of ssDNA with modulation of gene expression, defined by the two states "positive and negative" [12]. An RNAi logical evaluator has also been used to perform Boolean operations for human kidney cells [47] and intracellular device in single mammalian cell.

#### **3. Role of DFT in RNA nanotechnology**

**2. Application of RNA nanotechnology in computation (in vivo)**

were used [48, 49].

Ref. [96].

146 Density Functional Theory

RNA nanotechnology emphasizes its critical role in the process of catalysis in cell signaling and sensing functions. RNAs use the concentrations of specific chemical species as signals to implement the functions of logic to build up the network of multiple layers Boolean networks. Then these molecular computers were able to operate and communicate directly in a biological environment [45, 46] or work inside a living cell [47]. A plasmid with a specifically encoded DNA was inserted and external program was used to control the computation inside the host cell. The mRNA was used to encode a fluorescent protein and the target sequences for small interfering RNA (siRNA) with control level of fluorescence exhibited by the cell. Another in vivo computation programming was observed by a response to both endogenous and exogenous molecular signals. In this work, combination of ribozymes and RNA aptamers

**Figure 3.** RNA nanotechnology applications in beacons (A) and resistive biomemory (B). Adapted with permission from

The concept to utilize the cleaving ability of ribozyme to a specific molecule with the computational work was carried out using the concentrations of two proteins as input and output, implemented by studying the mechanism of ribozyme-aptamer molecules using yeast. In another study it was shown that the individual mammalian cells were capable of executing the basic arithmetic functions in a robust manner. A memory module (toggle switch) was designed [50] and a scalable factor (transcriptional/post transcriptional) was used to construct a synthetic regulatory circuit with a HeLa cancer cell 'classifier' to sense the expression levels of a customizable set of endogenous microRNAs to trigger a cellular response that matches the predetermined profile of interest [51]. In gene circuit engineering, RNA mediated regulation employs three major mechanisms, cis-acting RNA structure related modulation of translation, catalytic RNA or ribozyme-mediated cleavage of target transcripts and trans acting antisense small RNAs-mediated regulation of translation [52]. The experimental design

The RNA junction database [53], NanoTiler [54], RNA2D3D algorithms [55], RNA dynamics [56, 57] are used to build RNA nanoparticles that incorporate individual RNA motifs to defined user specifications. The energy landscape perspective of bio molecular dynamics provides a quantitative framework to consistently integrate theoretical concepts and experimental observations. Recognizing the potential of this approach, there has been decades of investigation into the character of protein folding landscapes and now the recent developments occur in the field of bionics. At theoretical level, the objective is to construct appropriate potential energy functions in order to explore the thermodynamic and kinetic behavior of the system. The discrete path sampling (DPS) procedure [58, 59] was employed to explore the energy landscape of the DNA G-quadruplex by HiRE-RNA force field. In order to fold proteins on physiological timescales, the energy landscape associated with folding should possess only small traps, relative to the energetic gap between the folded and unfolded ensembles. So, these landscapes should be minimally frustrated. These software's can demonstrate the biophysical and biochemical activities and also able to answer the complex analytical questions. The designed hybrid systems as (RNA–RNA, RNA–DNA, RNA–PS DNA etc.) are optimized and the reaction pathways, triggering and charge transfer mechanism can be carried out by DFT and TDDFT methods. NUPACK program [60] is used to calculate the thermodynamic analysis of inputs. Generally, three different theories can be used to analyze the results: density functional theory (DFT), and moving on to many-electron GW [61] methods as well as GW-inspired DFT + Ʃ [62] calculations. Further complex band structure (CBS) calculations are used to estimate the tunneling decay constant Ʃ, and Landauer-Buttiker transport calculations are used to compute the transmission spectra directly. Various effects such as device contact geometry, metal/molecule interfaces, molecular anchoring and side groups, inelastic effects, molecular conformation, and stochastic fluctuations are responsible for the results. So the parameters related to these effects are also taken into account. It is anticipated that the lowest conductance histogram peak correspond to single molecular junctions, while higher conductance peaks are attributed to junctions having multiple molecules linking the gold electrodes.

conductance properties [81–83]. The thermopower S, a derivative of the electron transmission

property which measures the charge carrier type and thermoelectric responses. The transport of electron spin in molecular wires, and the effects of solvent properties can also studied using DFT [84–86] and DFT + Σ [87] calculations. Few calculations have also been carried out on the charge transfer in short duplexes DNA/DNA and DNA/RNA with virtually equivalent sequences [84–86]. G09 [88] and Turbomole [89] software's can been used to carried out the calculations using the implicit water solving model (COSMO) (the Conductor like screening model) [90] or PCM (polarizable continuum model) [91]. The DFT method can also be used to study the charge transfer mechanisms, delocalization nature of orbitals, base stacking, electronic coupling and conformational flexibility. Tersoff potential is preferred for carbon

Scientists have demonstrated how living cells can be induced to carry out computations in the form of tiny robots or computers. Synthetic gene networks have been used to construct a wide range of biological devices, including molecular counters, oscillators, toggle switches, logic gates, cell classifiers and analog signal processors [93–95]. On a long-term scale, we believe that merging the research activities with a focus on the RNA as crucial molecular machinery for the cell will provide unprecedented insights into central molecular aspects of the RNA function and dynamics, ultimately enabling us to generate an integrated view of the molecular picture of the processes tuned by the RNA nanoparticles. Many natural limitations to the engineering of biological computers still remain. How do we eliminate or reduce noise in nucleic-acid based computing? An increase in the number of elements in RNA- and DNAbased logic operations is accompanied by a drastic increase in noise [96, 97]. The common approaches for noise reduction may include following: (i) better optimization of individual logic gates, (ii) utilization of network topology [98, 99] and (iii) for even larger networks, the introduction of new network elements to suppress the redundancy of the elements, thereby improving the signal-to-noise ratio. Also, the biochemical reaction time is also a critical aspect in working with a scalable circuit, which needs to be solved. With continued technological advances, it is now time to establish the rigorous physical and chemical understanding of RNA dynamics. Using the theoretical infrastructure developed in the context of protein folding, the field is positioned to extend the application of energy landscape techniques to the study of the large assemblies. Through close integration of computation and experimentation, it will be possible to dissect the complex interplay of structure and energetics during RNA function. We anticipate that such endeavors may ultimately reveal physical principles that govern biological dynamics. Finally we have a long way to go to understand nano-molecular computers in sufficient detail to be able to "reverse engineer" an existing molecular computer

RS acknowledges the financial assistance by the DST WOS-A (SR/WOS-A/CS-69/2018). RS is also thankful to her mentor Dr. Shrish Tiwari, Bioinformatics Department, CSIR-Centre for

, and proportional to the logarithmic F is another important transport

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149

at the Fermi level EF

**4. Conclusion**

or design a relatively new one.

Cellular and Molecular Biology, Hyderabad for the support.

nanotubes-nucleotides-metal cluster interactions [92].

Electron transport mechanisms can be categorized as: coherent tunneling and incoherent hopping [63]. Along with these parameters, thermal transport and thermoelectric properties are also important [64, 65]. The transport properties estimated by GW formalism are calculated in previous studies [66, 67]. The transmission is based on two parameters Seebeck coefficient and thermo-electric figure of merit; and is very sensitive to energy and k-resolution of the calculations. The DFT + Σ method is considered to be more appropriate method to calculate the conductance, but is only applicable to system with small polarizability of molecule and broadened resonance [68, 69]. Another promising method time dependent density functional theory (TDDFT) is used to measure the wavelengths (absorption and emission) for fluorescence, and time dependent current-density functional theory (TDCDFT) for the measurement of the current–voltage characteristics. The conductance of systems is analyzed with the help of dynamical exchange−correlation correction [70–72]. The software tools TranSIESTA [73] and SMEAGOL [74] are used to calculate the electron transmission spectrum using the nonequilibrium Green's function (NEGF) method and SCARLET [75, 76] for the scattering-state method.

The complex band structure calculations are used to investigate tunneling in DNA strands [77]. Further studies were carried out for pyridine-gold junctions, which show less variation in conductance values [78–80]. These studies can provide us useful insights for the RNA-Au junctions also.

Single-molecular junctions are also the potential candidates for thermoelectric applications due to large phonon mode mismatch at the metal molecular interface and tunable electronic conductance properties [81–83]. The thermopower S, a derivative of the electron transmission at the Fermi level EF , and proportional to the logarithmic F is another important transport property which measures the charge carrier type and thermoelectric responses. The transport of electron spin in molecular wires, and the effects of solvent properties can also studied using DFT [84–86] and DFT + Σ [87] calculations. Few calculations have also been carried out on the charge transfer in short duplexes DNA/DNA and DNA/RNA with virtually equivalent sequences [84–86]. G09 [88] and Turbomole [89] software's can been used to carried out the calculations using the implicit water solving model (COSMO) (the Conductor like screening model) [90] or PCM (polarizable continuum model) [91]. The DFT method can also be used to study the charge transfer mechanisms, delocalization nature of orbitals, base stacking, electronic coupling and conformational flexibility. Tersoff potential is preferred for carbon nanotubes-nucleotides-metal cluster interactions [92].

#### **4. Conclusion**

into the character of protein folding landscapes and now the recent developments occur in the field of bionics. At theoretical level, the objective is to construct appropriate potential energy functions in order to explore the thermodynamic and kinetic behavior of the system. The discrete path sampling (DPS) procedure [58, 59] was employed to explore the energy landscape of the DNA G-quadruplex by HiRE-RNA force field. In order to fold proteins on physiological timescales, the energy landscape associated with folding should possess only small traps, relative to the energetic gap between the folded and unfolded ensembles. So, these landscapes should be minimally frustrated. These software's can demonstrate the biophysical and biochemical activities and also able to answer the complex analytical questions. The designed hybrid systems as (RNA–RNA, RNA–DNA, RNA–PS DNA etc.) are optimized and the reaction pathways, triggering and charge transfer mechanism can be carried out by DFT and TDDFT methods. NUPACK program [60] is used to calculate the thermodynamic analysis of inputs. Generally, three different theories can be used to analyze the results: density functional theory (DFT), and moving on to many-electron GW [61] methods as well as GW-inspired DFT + Ʃ [62] calculations. Further complex band structure (CBS) calculations are used to estimate the tunneling decay constant Ʃ, and Landauer-Buttiker transport calculations are used to compute the transmission spectra directly. Various effects such as device contact geometry, metal/molecule interfaces, molecular anchoring and side groups, inelastic effects, molecular conformation, and stochastic fluctuations are responsible for the results. So the parameters related to these effects are also taken into account. It is anticipated that the lowest conductance histogram peak correspond to single molecular junctions, while higher conductance peaks are attributed to junctions having multiple molecules linking the gold electrodes.

Electron transport mechanisms can be categorized as: coherent tunneling and incoherent hopping [63]. Along with these parameters, thermal transport and thermoelectric properties are also important [64, 65]. The transport properties estimated by GW formalism are calculated in previous studies [66, 67]. The transmission is based on two parameters Seebeck coefficient and thermo-electric figure of merit; and is very sensitive to energy and k-resolution of the calculations. The DFT + Σ method is considered to be more appropriate method to calculate the conductance, but is only applicable to system with small polarizability of molecule and broadened resonance [68, 69]. Another promising method time dependent density functional theory (TDDFT) is used to measure the wavelengths (absorption and emission) for fluorescence, and time dependent current-density functional theory (TDCDFT) for the measurement of the current–voltage characteristics. The conductance of systems is analyzed with the help of dynamical exchange−correlation correction [70–72]. The software tools TranSIESTA [73] and SMEAGOL [74] are used to calculate the electron transmission spectrum using the nonequilibrium Green's function (NEGF) method and SCARLET [75, 76] for the scattering-state

The complex band structure calculations are used to investigate tunneling in DNA strands [77]. Further studies were carried out for pyridine-gold junctions, which show less variation in conductance values [78–80]. These studies can provide us useful insights for the RNA-Au

Single-molecular junctions are also the potential candidates for thermoelectric applications due to large phonon mode mismatch at the metal molecular interface and tunable electronic

method.

junctions also.

148 Density Functional Theory

Scientists have demonstrated how living cells can be induced to carry out computations in the form of tiny robots or computers. Synthetic gene networks have been used to construct a wide range of biological devices, including molecular counters, oscillators, toggle switches, logic gates, cell classifiers and analog signal processors [93–95]. On a long-term scale, we believe that merging the research activities with a focus on the RNA as crucial molecular machinery for the cell will provide unprecedented insights into central molecular aspects of the RNA function and dynamics, ultimately enabling us to generate an integrated view of the molecular picture of the processes tuned by the RNA nanoparticles. Many natural limitations to the engineering of biological computers still remain. How do we eliminate or reduce noise in nucleic-acid based computing? An increase in the number of elements in RNA- and DNAbased logic operations is accompanied by a drastic increase in noise [96, 97]. The common approaches for noise reduction may include following: (i) better optimization of individual logic gates, (ii) utilization of network topology [98, 99] and (iii) for even larger networks, the introduction of new network elements to suppress the redundancy of the elements, thereby improving the signal-to-noise ratio. Also, the biochemical reaction time is also a critical aspect in working with a scalable circuit, which needs to be solved. With continued technological advances, it is now time to establish the rigorous physical and chemical understanding of RNA dynamics. Using the theoretical infrastructure developed in the context of protein folding, the field is positioned to extend the application of energy landscape techniques to the study of the large assemblies. Through close integration of computation and experimentation, it will be possible to dissect the complex interplay of structure and energetics during RNA function. We anticipate that such endeavors may ultimately reveal physical principles that govern biological dynamics. Finally we have a long way to go to understand nano-molecular computers in sufficient detail to be able to "reverse engineer" an existing molecular computer or design a relatively new one.

RS acknowledges the financial assistance by the DST WOS-A (SR/WOS-A/CS-69/2018). RS is also thankful to her mentor Dr. Shrish Tiwari, Bioinformatics Department, CSIR-Centre for Cellular and Molecular Biology, Hyderabad for the support.

#### **Author details**

Ruby Srivastava

Address all correspondence to: amitruby1@gmail.com

CSIR-Center for Cellular and Molecular Biology, Hyderabad, Telangana, India

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## *Edited by Daniel Glossman-Mitnik*

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 understanding of the chemical reactivity through descriptors that constitute the basis of the so called Conceptual DFT (sometimes also named as Chemical Reactivity Theory) as well as the application of the theory and its related computational procedures in the determination of the molecular properties of different systems of academic and industrial interest.

Published in London, UK © 2019 IntechOpen © Alexandre Ostyanko / iStock

Density Functional Theory

Density Functional Theory

*Edited by Daniel Glossman-Mitnik*