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Solid State

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Topological Approaches to the Chemical Bond

Part of the book series: Theoretical Chemistry and Computational Modelling ((TCCM))

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Abstract

We will examine in this section some of the special features of the application of the topology to condensed phase systems, in particular to periodic crystals. We have already presented what type of information can be extracted from the topological analysis in isolated molecules. The conclusions obtained thereby are basically transferable to the study of molecular crystals, as long as we restrict ourselves to the molecules that constitute the condensed phase system. Much more interesting is the study of the intermolecular behavior in molecular crystals, or the interatomic one in non-molecular systems. As we will see, the application of the theory to the latter regenerates the geometrization of the chemical physics of solids so common in the the first half of the last century. The study of intermolecular interactions using topology is relatively novel, partly due to the difficulty with which such systems are handled by the available ab initio methods, and partly because of the lack of existing chemical intuition as far as their characterization and bond properties are concerned. We will start by examining some essential characteristics of the topological method in periodic domains. Following the same approach as in the previous chapter, we will then analyze structure and reactivity in crystals, characterizing prototype examples and then bonding and coordination changes during phase transitions.

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Notes

  1. 1.

    In geometry, the dual polyhedron is a figure where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Hence, the dual of a tetrahedron is another tetrahedron (inverted with respect to the first one); the dual polyhedron of a cube is an octahedron (and vice versa), etc.

  2. 2.

    Note that since the position of the bond charge will vary, this constant will not be the same for all zinc-blende solids in this BCM approach.

  3. 3.

    Recall that Pauli kinetic energy density is given by

    $$\begin{aligned} t_P(\textbf{r})=t(\textbf{r})-\frac{1}{8}\frac{|\nabla \rho (\boldsymbol{r})|^2}{\rho (\boldsymbol{r})}. \end{aligned}$$

    Using Eq. 7.21,

    $$\begin{aligned} t_P(\boldsymbol{r}) \simeq \frac{3}{10}(3\pi ^2)^{2/3}[\rho (\vec {r})]^{5/3}+\frac{1}{72}\frac{|\nabla \rho (\vec {r})|^2}{\rho (\vec {r})}+\frac{1}{6}\nabla ^2\rho (\vec {r})-\frac{1}{8}\frac{|\nabla \rho (\boldsymbol{r})|^2}{\rho (\boldsymbol{r})} \end{aligned}$$
    $$\begin{aligned} =\frac{3}{10}(3\pi ^2)^{2/3}[\rho (\boldsymbol{r})]^{5/3}-\frac{1}{9}\frac{|\nabla \rho (\boldsymbol{r})|^2}{\rho (\boldsymbol{r})}+\frac{1}{6}\nabla ^2\rho (\boldsymbol{r}) \end{aligned}$$

    .

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Authors and Affiliations

  1. Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain

    Ángel Martín Pendás

  2. Laboratoire de Chimie Théorique, Sorbonne Université, Paris, France

    Julia Contreras-García

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  1. Ángel Martín Pendás
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  2. Julia Contreras-García
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Correspondence to Ángel Martín Pendás .

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Martín Pendás, Á., Contreras-García, J. (2023). Solid State. In: Topological Approaches to the Chemical Bond. Theoretical Chemistry and Computational Modelling. Springer, Cham. https://doi.org/10.1007/978-3-031-13666-5_7

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