Electronic Properties of Solids

Abstract

In this chapter we present some methods that are employed in performing electronic structure calculations. We start by presenting a general quantum mechanical framework to describe a molecule or a solid. We then introduce the Born–Oppenheimer approximation (also called the adiabatic approximation), which allows us to reduce the problem to its corresponding electronic part.

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1 Introduction

In this chapter we present some methods that are employed in performing electronic structure calculations. We start by presenting a general quantum mechanical framework to describe a molecule or a solid. We then introduce the Born–Oppenheimer approximation (also called the adiabatic approximation), which allows us to reduce the problem to its corresponding electronic part. In the following, we introduce the independent-electron approximation and the main methods commonly used to solve the electronic problem.

2 Hamiltonian of the System

The system Hamiltonian is the starting point for calculating the electronic structure of molecules and solids. The Hamiltonian can be generally written as: Martin 2004

$$\begin{aligned} \hat{H}=-\frac{\hbar ^2}{2m_e}\sum _{i}\nabla ^2_i-\frac{1}{4\pi \varepsilon _0}\sum _{i}\sum _{I}\frac{Z_Ie^2}{|\mathbf {r}_i-\mathbf {R}_I|}+\frac{1}{8\pi \varepsilon _0}\sum _{i}\sum _{j\ne i}\frac{e^2}{|\mathbf {r}_i-\mathbf {r}_j|}\nonumber \\ -\sum _{I}\frac{\hbar ^2}{2M_I}\nabla ^2_I+\frac{1}{8\pi \varepsilon _0}\sum _{I}\sum _{J\ne I}\frac{Z_IZ_Je^2}{|\mathbf {R}_I-\mathbf {R}_J|} \end{aligned}$$

(2.1)

where i, j label the electrons and I, J label the related atomic nuclei . The quantities $M_I$, $Z_I$ and $\mathbf {R}_I$ are, respectively, the mass, the atomic number, and the position of nucleus I. The electron charge and mass are, respectively, written as $-e$ and $m_e$, and $\mathbf {r}_i$ represents the position of the electron i. The first three terms in (2.1) account, respectively, for the kinetic energy of the electrons , the electron-nucleon Coulomb interaction , and the electron-electron Coulomb interaction . The fourth term describes the kinetic energy of the nuclei and the last term is the nucleon-nucleon Coulomb interaction.

An important aspect of this problem is the fact that the nuclei are much more massive than the electrons. This makes the kinetic energy of the nuclei small compared to the other contributions to the Hamiltonian in (2.1). This allows us to decouple the nuclear and electronic contributions to (2.1) and to work on them separately. This constitutes a practical and useful simplification which makes both the structural optimization and electronic structure calculations easier to perform, but still provides sufficient accuracy for many physical problems. This is the so-called Born–Oppenheimer (BO) Martin 2004 or adiabatic approximation , which is a useful approximation for many purposes, such as for the calculation of the vibrational modes in solids. We should point out that the limitation of the BO approximation has been recently discussed for graphene and carbon nanotubes . A widely used approach is to consider the energy from the remaining electronic problem as an extra term added to the ion-ion interaction and to perform a subsequent geometrical optimization based on this effective interatomic potential. By considering the BO approximation, the problem is reduced to the electronic Hamiltonian $\hat{H}_e$ given by:

$$\begin{aligned} \hat{H}_e=-\frac{\hbar ^2}{2m_e}\sum _{i}\nabla ^2_i-\frac{1}{4\pi \varepsilon _0}\sum _{i}\sum _{I}\frac{Z_Ie^2}{|\mathbf {r}_i-\mathbf {R}_I|}+\frac{1}{8\pi \varepsilon _0}\sum _{i}\sum _{j\ne i}\frac{e^2}{|\mathbf {r}_i-\mathbf {r}_j|}\qquad \end{aligned}$$

(2.2)

where the atomic positions entered as parameters.

3 The Electronic Problem

By considering the BO approximation, the electronic problem is simplified compared to the initial problem. However, it is still not easy to solve this problem within the BO approximation. A set of widely used strategies employs the independent-electron approximation. This mean-field approximation consists of defining one-electron wavefunctions that can be obtained from a one-electron Schr$\ddot{\mathrm{o}}$dinger equation. This is a significant simplification but gives very satisfactory results for many interesting physical systems and is used in most theoretical calculations of the electronic structure of molecules and solid state materials.

3.1 The Hartree Method

Hartree was a pioneer in developing the first quantitative electronic calculations for multi-electron systems (Hartree 1928). The Hartree method starts with the one electron equation which is written as:

$$\begin{aligned} \hat{H}_{HT}\psi ^\sigma _i(\mathbf {r})=-\frac{\hbar ^2}{2m_e}\nabla ^2\psi ^\sigma _i(\mathbf {r})+V_{HT}\psi ^\sigma _i(\mathbf {r})=\varepsilon ^\sigma _i\psi ^\sigma _i(\mathbf {r}) \end{aligned}$$

(2.3)

where $\sigma$ represents the spin and $V_{HT}$ is an effective Hartree potential for each electron in the presence of the others. A different potential is defined for each electron in order to avoid self-interaction of the electron with itself. In order to obtain the ground state of the system, one fills the electronic states starting from the lowest energy levels, though always obeying the Pauli exclusion principle . (see Slater 1930)

3.2 Hartree–Fock (HF) Method

In 1930, Fock expanded Hartree’s method by using an anti-symmetric wavefunction in terms of a Slater determinant written using one-electron Schrödinger wavefunctions. (see Fock 1930) The one-electron equations are then obtained by finding the corresponding wavefunctions that minimize the total energy obtained as the expectation energy $\varepsilon ^\sigma _i$ for the full Hamiltonian. This process yields the following equations:

$$\begin{aligned} \hat{H}_{HF}\psi ^\sigma _i(\mathbf {r})= & {} \Big [-\frac{\hbar ^2}{2m_e}\nabla ^2+V_{ext}(\mathbf {r})+V_{HT}(\mathbf {r})+V_{xc}(\mathbf {r})\Big ]\psi ^\sigma _i(\mathbf {r})\nonumber \\= & {} \varepsilon ^\sigma _i\psi ^\sigma _i(\mathbf {r}) \end{aligned}$$

(2.4)

in which $V_{ext}$ is the external potential, $V_{HT}$ is the Hartree potential and $V_{xc}$ is the exchange potential. $V_{HT}$ and $V_{xc}$ are written as:

$$\begin{aligned} V_{HT}(\mathbf {r})= & {} \frac{e^2}{4\pi \varepsilon _0}\sum _{j,\sigma _j}\int d\mathbf {r}'\frac{\psi ^{\sigma _j*}_j(\mathbf {r}')\psi ^{\sigma _j}_j(\mathbf {r}')}{|\mathbf {r}-\mathbf {r}'|}\end{aligned}$$

(2.5)

$$\begin{aligned} V_{xc}(\mathbf {r})= & {} -\frac{e^2}{4\pi \varepsilon _0}\sum _{j}\int d\mathbf {r}'\frac{\psi ^{\sigma *}_j(\mathbf {r}')\psi ^{\sigma }_i(\mathbf {r}')}{|\mathbf {r}-\mathbf {r}'|}\frac{\psi ^\sigma _j(\mathbf {r})}{\psi ^\sigma _i(\mathbf {r})}. \end{aligned}$$

(2.6)

Note that, unlike the original Hartree approach, the mean Coulomb interaction in the Hartree–Fock approach $V_{HT}$ includes a self-interaction contribution. The additional exchange term $V_{xc}$, which does not have a classical analogue, also contains such a self-interaction energy, but with an opposite sign so that the final result does not depend on self-interactions. The presence of the exchange potential $V_{xc}$ is the main difference between the HF and the Hartree approaches. (see Fock 1930)

The meaning of the exchange term $V_{xc}(\mathbf {r})$ is not easy to understand at first because there is no classical analog. The physics behind it relies on the foundations of the independent electron approximation. Without considering any approximation, the solution for the electronic problem consists of a multi-electron wavefunction $\varPsi$, which is a function of the coordinates for all the electrons $\mathbf {r}_i$, $i=1,2,3,\ldots ,N$. Since the electrons are Fermions, they should obey Pauli’s exclusion principle which is reflected by the fact that the electronic wavefunction has to be anti-symmetric with respect to any permutation involving the positions of any two electrons i and j, such that:

$$\begin{aligned} \varPsi (\mathbf {r}_1,\mathbf {r}_2,\ldots ,\mathbf {r}_i,\ldots ,\mathbf {r}_j,\ldots ,\mathbf {r}_{N-1},\mathbf {r}_N)=-\varPsi (\mathbf {r}_1,\mathbf {r}_2,\ldots ,\mathbf {r}_j,\ldots ,\mathbf {r}_i,\ldots ,\mathbf {r}_{N-1},\mathbf {r}_N).\qquad \end{aligned}$$

(2.7)

In addition, we expect the wavefunction $\varPsi$ dependence on the different $\mathbf {r}_i$ coordinates to be correlated in a more general way, so that the behavior of $\varPsi$ relative to a given $\mathbf {r}_i$ depends on the values of the $\mathbf {r}_{j}$ with $j\ne i$. However, when we use the independent electron approximation, we are restricting $\varPsi$ to have the form given by a Slater determinant . (see Slater 1929) By doing that, we are intrinsically losing information which can be directly associated with the electronic correlation. While the Hartree potential, in both the Hartree and HF approaches, represents the interaction of any electron with the system’s electronic cloud, the correlation between electrons is related to the specific interaction of a given electron with any single electron in the system. This is not a simple problem to solve and accounting for this correlation is a central problem in the electronic structure research field. The HF method, however, is a first step in this direction because the exchange potential $V_{xc}$ in (2.6) represents two aspects of such a correlation:

 

• 1. Self-interaction contributions are removed;

• 2. Short range interactions related to the Pauli’s exclusion principle are accounted for.

As can be seen from (2.6), $V_{xc}$ lowers the energy and $V_{xc}$ can be interpreted as the interaction of the electron with an agent usually referred to as an “exchange hole”. According to the expression for the exchange potential, this positive “exchange charge density” is determined by the electronic density (which is a sum over the j states) surrounding the electron i, and $V_{ext}$ favors a ferromagnetic ordering of the electronic spins, since this interaction involves only electrons with the same spin . This is a consequence of Hund’s rule which states that as the number of electrons start to fill a set of degenerate atomic states, the electrons will evenly fill the available states so as to maximize the total spin as much as possible, only starting to occupy orbitals with opposite spin when there are no available empty states in the first spin state . Note also that there is no energy lowering associated with two electrons with the same spin occupying the same electron orbital, since the $j=i$ contribution in the sum in (2.6) cancels with the corresponding self-interacting term in the sum from (2.5), constituting a clear manifestation of Pauli’s exclusion principle. (see Pauli 1925)

3.3 Density Functional Theory

Not all correlation effects are accounted for by the exchange energy term $V_{xc}$ which is added in the HF approach discussed in Sect. 2.3.2. In this regard, the introduction of Density Functional Theory (DFT) represents an important advancement in the field of electronic band structure calculations. (see Hohenberg and Kohn 1964) DFT has, in fact, become a standard tool for condensed matter physics and is widely accessible for many applications.

As we will see, DFT allows us to recast the many-electron problem into a set of one-electron Schr$\ddot{\mathrm{o}}$dinger-like equations. However, in contrast to the Hartree and HF approaches, the DFT approach is a more complete theory but its practical implementation demands other approximations to be made. The electronic density $n(\mathbf {r})$ is the main parameter in DFT and its key role can be understood in terms of the two Hohenberg–Kohn theorems given below, which constitute the basis of DFT: (see Hohenberg and Kohn 1964)

$1^{st}$ Theorem: If a system of interacting electrons is immersed in an external potential $V_{ext}$, this potential is uniquely determined (except by a constant) by the electronic density $n_0$ of the ground state (GS).

$2^{nd}$ Theorem: Let E[n] be the functional for the energy relative to the electronic density $n(\mathbf r)$ for a given $V_{ext}(\mathbf r)$. Then this functional has its global minimum (GS energy) for the exact electronic density $n_0(\mathbf r)$ corresponding to the ground state.

The first theorem states that all the system properties are determined by the electronic density for the ground state since $n_0$ determines $V_{ext}$, which determines the Hamiltonian, which in turn defines the ground state and all the excited states. Also, we can use the energy functional E[n] (see (2.8) where this functional is written as $E_{el}[n]$) to determine the exact ground state energy and density. It is important to note, however, that DFT is not only a ground state theory, but instead gives the system’s Hamiltonian which is, in principle, all we need to obtain the ground state and all the excited states. However, the ground state can be obtained in a systematic way within DFT. The energy functional $E_{HK}[n]$ in the Hohenberg–Kohn approach is written as:

$$\begin{aligned} E_{HK}[n]=T[n]+E_{el}[n]+\int d\mathbf {r}V_{ext}(\mathbf {r})n(\mathbf {r}) \end{aligned}$$

(2.8)

where T[n] is the kinetic energy functional, $V_{ext}$ is the external potential felt by the electrons (including the contribution from the nuclei) and the electronic energy  $E_{el}[n]$ accounts for all the  electron-electron interactions.

Despite having the correct tool to obtain the electronic ground state (i.e., the minimization of E[n] relative to n), it is still not clear how to proceed in using this tool. The necessary recipe is given by the Kohn–Sham ansatz. (see Kohn and Sham 1965) According to this ansatz, the ground state electronic density of our system can be written as the ground state of an auxiliary system of non-interacting electrons. The one-electron wavefunctions  $\psi ^{\sigma }_i$ for this auxiliary system are determined by Schr$\ddot{\mathrm{o}}$dinger-like equations of the form:

$$\begin{aligned} \hat{H}^\sigma _{aux}\psi ^\sigma _i(\mathbf {r})=-\frac{\hbar ^2}{2m_e}\nabla ^2\psi ^\sigma _i(\mathbf {r})+V^\sigma \psi ^\sigma _i(\mathbf {r})=\varepsilon ^\sigma _i\psi ^\sigma _i(\mathbf {r}) \end{aligned}$$

(2.9)

where $\sigma$ labels the electron spin. The electronic density $n(\mathbf r)$ is written as:

$$\begin{aligned} n(\mathbf {r})=\sum _{\sigma }\sum ^{N^{\sigma }}_{i=1}|\psi ^\sigma _i(\mathbf {r})|^2 \end{aligned}$$

(2.10)

where the $N^{\sigma }$ is the total number of electrons in each spin state $\sigma$. The corresponding auxiliary kinetic energy is:

$$\begin{aligned} T_{aux}=-\frac{\hbar ^2}{2m_e}\sum _\sigma \sum ^{N^\sigma }_{i=1}\langle \psi ^\sigma _i|\nabla ^2|\psi ^\sigma _i\rangle =\frac{\hbar ^2}{2m_e}\sum _\sigma \sum ^{N^\sigma }_{i=1}\int d\mathbf {r}|\nabla \psi ^\sigma _i(\mathbf {r})|^2. \end{aligned}$$

(2.11)

The classical Coulomb interaction for the electron-electron repulsion $E_{CI}$ is given by

$$\begin{aligned} E_{CI}[n]=\frac{1}{8\pi \varepsilon _0}\int \int d\mathbf {r}d\mathbf {r}'\frac{n(\mathbf {r})n(\mathbf {r}')}{|\mathbf {r}-\mathbf {r}'|}, \end{aligned}$$

(2.12)

where the $\varepsilon _0$ is the vacuum dielectric permittivity. By summing up these terms, the expression for the Kohn–Sham energy functional reads:

$$\begin{aligned} E_{KS}[n]=T_{aux}[n]+\int d\mathbf {r}V_{ext}(\mathbf {r})n(\mathbf {r})+E_{CI}[n]+E_{xc}[n] \end{aligned}$$

(2.13)

where $V_{ext}$ is the external potential (including the contribution from the nuclei) and $E_{xc}$ is the energy functional which accounts for the exchange and all the correlation effects. If we consider the Hohenberg–Kohn energy $E_{HK}=E_{KS}$, we have:

$$\begin{aligned} E_{xc}[n]=T[n]-T_{aux}[n]+E_{el}[n]-E_{CI}[n] \end{aligned}$$

(2.14)

which indicates that $E_{xc}$ contains the exchange contribution and all the other correlation effects related to both the kinetic energy and the electron-electron interactions. Here lies the main problem of DFT: we do not know the exact form of $E_{xc}$. Even though DFT yields the exact solution for the electronic problem, its practical implementation requires an approximation regarding the form of the exchange and correlation energy terms. The usual approach is to write this energy as:

$$\begin{aligned} E_{xc}[n]=\int d\mathbf {r} n(\mathbf {r})\varepsilon _{xc}([n],\mathbf {r}) \end{aligned}$$

(2.15)

where $\varepsilon _{xc}([n],\mathbf {r})$ is the exchange-correlation energy per electron at the position $\mathbf {r}$ for a given density $n(\mathbf {r})$. The minimization of the energy functional is obtained by varying the one-electron wavefunctions $\psi ^\sigma _i$ and using the Lagrange multipliers $\varepsilon ^\sigma _i$ corresponding to the normalization constraint $\langle \psi ^\sigma _i|\psi ^\sigma _i\rangle =1$:

$$\begin{aligned}&\frac{\delta }{\delta \psi ^{\sigma *}_i}\Big (E_{KS}[n]-\sum _\sigma \sum ^{N_\sigma }_{j=1}\varepsilon ^\sigma _j(\int d\mathbf {r} \psi ^{\sigma *}_j\psi ^{\sigma }_j-1)\Big )=\nonumber \\ {}= & {} \frac{\delta T_{aux}[n]}{\delta \psi ^{\sigma *}_i}+\frac{\delta E_{ext}[n]}{\delta \psi ^{\sigma *}_i}+\frac{\delta E_{HT}[n]}{\delta \psi ^{\sigma *}_i}+\frac{\delta E_{xc}[n]}{\delta \psi ^{\sigma *}_i}-\varepsilon ^\sigma _i\psi ^\sigma _i\nonumber \\= & {} -\frac{\hbar ^2}{2m_e}\nabla ^2\psi ^\sigma _i(\mathbf {r})+\Bigg (\frac{\delta E_{ext}[n]}{\delta n(\mathbf {r})}+\frac{\delta E_{HT}[n]}{\delta n(\mathbf {r})}+\frac{\delta E_{xc}[n]}{\delta n(\mathbf {r})}\Bigg )\frac{\delta n(\mathbf {r})}{\delta \psi ^{\sigma *}_i}-\varepsilon ^\sigma _i\psi ^\sigma _i=0 \end{aligned}$$

(2.16)

This minimization of the energy functional is carried out so that the electrons obey:

$$\begin{aligned} -\frac{\hbar ^2}{2m_e}\nabla ^2\psi ^\sigma _i(\mathbf {r})+\Bigg (V_{ext}(\mathbf {r})+V_{HT}(\mathbf {r})+\varepsilon _{xc}(\mathbf {r})\Bigg )\psi ^\sigma _i=\varepsilon ^\sigma _i\psi ^\sigma _i. \end{aligned}$$

(2.17)

Here the Hartree potential $V_{HT}({\mathbf r})$ which represents the interaction of any electron with its surrounding electronic cloud, is given by:

$$\begin{aligned} V_{HT}[n]=\frac{1}{8\pi \varepsilon _0}\int d\mathbf {r}'\frac{n(\mathbf {r})}{|\mathbf {r}-\mathbf {r}'|} \end{aligned}$$

(2.18)

in which $V_{ext}$ and $\varepsilon _{ext}$ are the external potential and exchange-correlation energy per electron , respectively. Equation (2.17) is the well-known Kohn–Sham equation for the auxiliary problem. This is the basis for many theoretical calculations of the electronic structure that have been performed on molecules and solids.

4 Plane Wave and Localized Basis Sets

In order to solve Schr$\ddot{\mathrm{o}}$dinger’s equation for a molecule or solid, one first has to choose a basis-set to use for the electronic wavefunctions. In order to obtain precise results, the first property we expect from a basis set is completeness:

$$\begin{aligned} \sum _i|\phi _i\rangle \langle \phi _i|=1. \end{aligned}$$

(2.19)

It turns out that, in practice, it is never possible to use such a complete set. Plane waves, for instance, constitute a basis set which is naturally complete, but only as long as an infinite number of plane waves are explicitly included. However, in a numerical implementation, one always has to use a finite discretized set of states which is a subset of the total plane wave set. In this case, the systematic way to improve the accuracy of the calculation is to increase the number of functions in the basis set. Such an improvement is not boundless, since computational resources have a finite processing capability.

One alternative to plane waves is to use the Linear Combination of Atomic Orbitals (LCAO) method . Here, the basis consists of functions corresponding to the electronic states from the isolated atoms. Despite its simplicity, this method yields quite accurate results, and it constitutes the foundation of several computational packages and studies in the literature. The main advantages of this method are the reduced computational cost and the easy association of the molecular levels with the atomic orbitals. One major drawback of this approach is the difficulty in assessing its validity, given the impossibility to systematically improve the basis set, and to calculate the remaining error.

Let us examine the use of the LCAO by expanding the electronic wavefunctions from a crystal in terms of a local orbital basis. In such a basis, each orbital basis function is associated with an atom in the structure. One appropriate choice for these orbitals are functions centered on the atomic sites. These functions can be written as:

$$\begin{aligned} \phi _j(\mathbf {r}-\mathbf {R})=\phi _\alpha (\mathbf {r}-\mathbf {r}_P-\mathbf {R})=\phi ^{\mathbf {R}}_{n_jl_jm_j}(\mathbf {\rho })=\phi ^{\mathbf {R}}_{n_jl_j}(\rho )Y_{l_jm_j}(\hat{\rho }) \end{aligned}$$

(2.20)

where the coordinate $\rho$ is

$$\begin{aligned} \mathbf {\rho }=\mathbf {r}-\mathbf {r}_P-\mathbf {R} \end{aligned}$$

(2.21)

and $\mathbf {r}_P$ is the position of the $P^{th}$ atom in the crystal unit cell (relative to the origin of the unit cell), $\alpha$ enumerates the atomic orbitals centered at P, and $\mathbf {R}$ is a lattice vector from the Bravais lattice (which localizes the origin of its corresponding cell). In this terminology, we define j to represent the $(P,\alpha )$ pair. The $l_j$, $m_j$ and $n_j$ in (2.20) represent the angular momentum, its projection on a given axis, and the number of different functions with the same angular momentum, respectively. Also, the $Y_{l_jm_j}(\hat{\rho })$ functions denote the spherical harmonics , which provide basis functions for the ($l_j,m_j$) states. We list the spherical harmonics for $l_j=0,1,2$ below:

$$\begin{aligned} Y_{0,0}(\theta ,\phi )= & {} \frac{1}{2}\sqrt{\frac{1}{\pi }}\end{aligned}$$

(2.22)

$$\begin{aligned} Y_{1,0}(\theta ,\phi )= & {} \frac{1}{2}\sqrt{\frac{3}{2\pi }}\cos \theta \end{aligned}$$

(2.23)

$$\begin{aligned} Y_{1,\pm 1}(\theta ,\phi )= & {} \mp \frac{1}{2}\sqrt{\frac{3}{2\pi }}e^{\pm i\phi }\sin \theta \end{aligned}$$

(2.24)

$$\begin{aligned} Y_{2,0}(\theta ,\phi )= & {} \frac{1}{4}\sqrt{\frac{5}{\pi }}(3\cos ^2\theta -1)\end{aligned}$$

(2.25)

$$\begin{aligned} Y_{2,\pm 1}(\theta ,\phi )= & {} \mp \frac{1}{2}\sqrt{\frac{15}{2\pi }}e^{\pm i\phi }\sin \theta \cos \theta \end{aligned}$$

(2.26)

$$\begin{aligned} Y_{2,\pm 2}(\theta ,\phi )= & {} \frac{1}{4}\sqrt{\frac{15}{2\pi }}e^{\pm 2i\phi }\sin ^2\theta . \end{aligned}$$

(2.27)

Orbitals with $m=0$ are real, while real orbitals for the $m\ne 0$ cases can be obtained by the following transformation:

$$\begin{aligned} Y^\pm _{l,|m|}=\frac{1}{2}(Y_{l,m}\pm Y_{l,-m}). \end{aligned}$$

(2.28)

Plots for the individual $Y^{\pm }_{l,|m|}$ are shown in Fig. 2.1.

Fig. 2.1

Spherical harmonics in the $Y^{\pm }_{l,|m|}$ form. These harmonics are used to describe s- (l=0,m=0); p- (l=1, m=-1,0,+1) and d-states (l=2, m=-2,-1,0,+1,+2)

It is often more convenient to use the complex $e^{\pm i m\phi }$ expressions in actual calculations since this functional form allows us to simplify the calculation of the two-centered integrals contributing to the Hamiltonian matrix elements as discussed in the next section.

5 Hamiltonian Matrix Elements

The Hamiltonian matrix elements are written in terms of the localized basis functions used in the form of angular momentum spherical harmonics in Sect. 2.4:

$$\begin{aligned} H_{j,l}(\mathbf {R}',\mathbf {R}'')=\int {dr^3}\phi ^*_j(\mathbf {r}-\mathbf {R}')\hat{H}\phi _l(\mathbf {r}-\mathbf {R}''). \end{aligned}$$

(2.29)

In addition, the translational crystal symmetry allows us to write:

$$\begin{aligned} H_{j,l}(\mathbf {R}'-\mathbf {R}''',\mathbf {R}''-\mathbf {R}''')=H_{j,l}(\mathbf {R}',\mathbf {R}'') \end{aligned}$$

(2.30)

so that the $\mathbf {R}'$ and $\mathbf {R}''$ dependence of $H_{j,l}(\mathbf {R}',\mathbf {R}'')$ is determined exclusively by the difference between lattice vectors $\mathbf {R}=\mathbf {R}''-\mathbf {R}'$. Using this fact, we can refer all the lattice vectors to a common origin by writing

$$\begin{aligned} H_{j,l}(\mathbf {R}',\mathbf {R}'')=H_{j,l}(\mathbf {0},\mathbf {R} ''-\mathbf {R}')=H_{j,l}(\mathbf {R})=\int {dr^3}\phi ^*_j(\mathbf {r})\hat{H}\phi _l(\mathbf {r}-\mathbf {R}), \end{aligned}$$

(2.31)

where the matrix elements only involve the lattice vector to the common origin. Similarly, the wavefunction overlap terms are given by:

$$\begin{aligned} S_{j,l}(\mathbf {R})=\int {dr^3}\phi ^*_j(\mathbf {r})\phi _l(\mathbf {r}-\mathbf {R}). \end{aligned}$$

(2.32)

The one electron Hamiltonian operator then has the form:

$$\begin{aligned} \hat{H}=\hat{T}+\sum _{p,\mathbf {R}}V(|\mathbf {r}-\mathbf {r}_p-\mathbf {R}|) \end{aligned}$$

(2.33)

where $\hat{T}$ is the one-electron kinetic energy operator and $V(|\mathbf {r}-\mathbf {r}_p-\mathbf {R}|)$ is the potential energy decomposed into a sum of spherically symmetric terms centered at the atoms located at positions $\mathbf {r}_p$ relative to the unit cell located at $\mathbf {R}$. The kinetic energy contribution to the Hamiltonian matrix elements can be composed of one- or two-centered integrals depending on whether or not the orbitals i and j are centered at the same atom. Since the potential can be viewed as a sum of spherically symmetric terms, the contributions of the potential to the Hamiltonian matrix element can also have three-center integrals as well as one- and two-center integrals. We can readily notice four different types of potential energy contributions:

• One-center: when both orbitals and the potential are centered on the same atom;

• Two-center 1: when the orbitals are centered on different atoms and the potential is on one of these atoms;

• Two-center 2: when both orbitals are centered on the same atom and the potential is on another atom;

• Three-center: when both orbitals and the potential are all centered on different atoms.

The overlap terms are always composed of one- or two-center integrals. The important aspects of the integration can be easily addressed for the two-center integrals. Let $M_{lm,l'm'}$ be a two-center integral, between two orbitals from different atoms, corresponding to the kinetic or potential energy contributions to a Hamiltonian matrix element or to an overlap matrix element. For simplicity, let us suppose that the line joining the two centers corresponds to the z-axis. We can then write:

$$\begin{aligned} M_{lm,l'm'}=\int f_1(\rho _1)f_2(\rho _2)Y^*_{l,m}(\hat{\rho }_1)Y_{l',m'}(\hat{\rho }_2)d^3\mathbf {r} \end{aligned}$$

(2.34)

with

$$\begin{aligned} \hat{\rho }_i=\frac{\mathbf {r}-\mathbf {r}_i}{|\mathbf {r}-\mathbf {r}_i|}\quad i=1,2. \end{aligned}$$

(2.35)

The $\phi$ dependence from this integral can be isolated so that:

$$\begin{aligned} M_{lm,l'm'}=\frac{M_{ll'm}}{2\pi }\int ^{2\pi }_{0}e^{-im\phi }e^{im'\phi }d\phi =M_{ll'm}\delta _{mm'}. \end{aligned}$$

(2.36)

Equation 2.36 represents a significant simplification for the calculation of these angular momentum terms. The usual nomenclature for such quantities is to denote orbital quantum number $l=0,1,2,3,\ldots$ by $s,p,d,f,\ldots$ and $m=0,\pm 1,\pm 2,\ldots$ by $\sigma ,\pi ,\delta ,\ldots$. In Fig. 2.2 we illustrate the different integral schemes for $l=0,1,2$.

Fig. 2.2

Different two-centered integral schemes for the Hamiltonian matrix elements using a localized basis, and using the following notation: $l=0,1,2,3,\ldots$ denotes respectively $s,p,d,f,\ldots$ and $m=0,\pm 1,\pm 2,\ldots$ denotes respectively $\sigma , \pi , \delta$ angular momentum states

6 Bloch Functions

Even with the simplifications introduced in the last section, it is impractical to work with the Hamiltonian in the simple atomic orbitals representation for a periodic solid. Instead we take the advantage of the periodicity V(r) of a crystalline lattice. Bloch’s theorem (see Bloch 1928) indicates that we can write the eigenfunctions $\phi _{j\mathbf {k}}(\mathbf {r})$ for electrons in a periodic potential and in a single unit cell as:

$$\begin{aligned} \phi _{j\mathbf {k}}(\mathbf {r})=N_{j\mathbf {k}}\sum _{\mathbf {R}}e^{i\mathbf {k}\cdot \mathbf {R}}\phi _{j}(\mathbf {r}-\mathbf {R}) \end{aligned}$$

(2.37)

where $N_{j\mathbf {k}}$ is a normalization constant. In the case of an infinite crystal, $\mathbf {k}$ is a vector which can assume any value within the first Brillouin Zone (BZ) . Within this approach we redirect our attention from a set of vectors $\mathbf {R}$ which extend along the infinite real space to a set of lattice vectors $\mathbf {k}$ in reciprocal space which are contained within a finite portion (determined by the BZ) of reciprocal space. By utilizing the periodicity of the crystal in Bloch’s theorem (2.37), it is straightforward to show that the Hamiltonian elements $H_{j,l}(\mathbf {k})$ for an electron in a periodic potential can be written as

$$\begin{aligned} H_{j,l}(\mathbf {k})=\sum _{\mathbf {R}}e^{i\mathbf {k}\cdot \mathbf {R}}H_{j,l}(\mathbf {R}). \end{aligned}$$

(2.38)

Analogously, for the overlap matrix elements $S_{i,j}$, we can also utilize Bloch’s theorem to write

$$\begin{aligned} S_{j,l}(\mathbf {k})=\sum _{\mathbf {R}}e^{i\mathbf {k}\cdot \mathbf {R}}S_{j,l}(\mathbf {R}). \end{aligned}$$

(2.39)

Now, we can then expand the electronic eigenfunctions $\psi$ in terms of the eigenfunctions $\phi _{j\mathbf {k}}(\mathbf {r})$ as:

$$\begin{aligned} \varPsi _\alpha (\mathbf {r})=\sum _jc_{j\alpha }\phi _{j\mathbf {k}}(\mathbf {r}). \end{aligned}$$

(2.40)

Hence, the Schr$\ddot{\mathrm{o}}$dinger equation now reads:

$$\begin{aligned} \hat{H}\varPsi _\alpha (\mathbf {r})=E_\alpha \varPsi _\alpha (\mathbf {r}) \end{aligned}$$

(2.41)

or alternatively, when using the LCAO basis we can write:

$$\begin{aligned} \sum _jc_{j\alpha }\hat{H}\phi _{j\mathbf {k}}(\mathbf {r})=E_\alpha \sum _jc_{j\alpha }\phi _{j\mathbf {k}}(\mathbf {r}). \end{aligned}$$

(2.42)

If we multiply on the left by $\phi ^*_{l\mathbf {k}}(\mathbf {r})$ and integrate over space, we end up with:

$$\begin{aligned} \sum _jc_{j\alpha }\int \phi ^*_{l\mathbf {k}}(\mathbf {r})\hat{H}\phi _{j\mathbf {k}}(\mathbf {r})d^3r=E_\alpha \sum _jc_{j\alpha }\int \phi ^*_{l\mathbf {k}}(\mathbf {r})\phi _{j\mathbf {k}}(\mathbf {r})d^3r \end{aligned}$$

(2.43)

$$\begin{aligned} \sum _jc_{j\alpha }H_{l,j}(\mathbf {k})=\sum _jc_{j\alpha }E_\alpha S_{l,j}(\mathbf {k}) \end{aligned}$$

(2.44)

$$\begin{aligned} \mathbf {H}\mathbf {c}_\alpha =E_\alpha \mathbf {S}\mathbf {c}_\alpha , \end{aligned}$$

(2.45)

where $\mathbf {H}$ and $\mathbf {S}$ are square matrices with elements $H_{l,j}(\mathbf {k})$ and $S_{l,j}(\mathbf {k})$, respectively, and $\mathbf {c}_\alpha$ is a column vector with matrix elements $c_{j\alpha }$. The energy eigenvalues are then obtained algebraically by the secular equation:

$$\begin{aligned} |\mathbf {H}-E_\alpha \mathbf {S}|=0 \end{aligned}$$

(2.46)

where $|\quad |$ denotes the determinant commonly used to solve the eigenvalue problem explicitly.

A computational procedure is then used to calculate the electronic structure $E(\mathbf {k})$, where we use a discrete set of vectors $\mathbf {k}_i$, $i=1,2,3,\ldots ,N$. These $\mathbf {k}$ vectors represent how the electronic states behave as a function of $\mathbf {k}$. Since the Hamiltonian matrix elements coupling different $\mathbf {k}$ vectors are zero, we can write the secular (2.46) in block diagonal form in terms of the energy levels $E_i$ for the various i eigenvalues each of which can have degenerate energy levels depending on the symmetry of the crystal structures. More explicitly

$$\begin{aligned} \left| \begin{array}{ccccc} \mathbf {H}(\mathbf {k}_1)-E_\alpha \mathbf {S}(\mathbf {k}_1) &{} 0 &{} 0 &{} \ldots &{} 0\\ 0 &{} \mathbf {H}(\mathbf {k}_2)-E_\alpha \mathbf {S}(\mathbf {k}_2) &{} 0 &{} \ldots &{} 0\\ 0 &{} 0 &{} \mathbf {H}(\mathbf {k}_3)-E_\alpha \mathbf {S}(\mathbf {k}_3) &{} \ldots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \ldots &{} \mathbf {H}(\mathbf {k}_N)-E_\alpha \mathbf {S}(\mathbf {k}_N) \\ \end{array}\right| =0\nonumber \\ \end{aligned}$$

(2.47)

and such a matrix diagonalization can be broken into smaller sub-blocks:

$$\begin{aligned} |\mathbf {H}(\mathbf {k}_i)-E_\alpha \mathbf {S}(\mathbf {k}_i)|=0,\qquad i=1,2,3,\ldots ,N \end{aligned}$$

(2.48)

for each $\mathbf {k}$-point $\mathbf {k}_i$, and at high symmetry points the various blocks will show the appropriate degeneracies satisfying the symmetry requirements of the potential. For these k-points where degeneracies in energy occur, we select appropriate linear combinations of the wave functions which are each orthogonal to one another.

7 The Slater–Koster Approach

Felix Bloch introduced the concept of an electronic energy band structure $E(\mathbf {k})$ and his famous “Bloch’s Theorem” to handle the symmetry of a periodic lattice. (see Bloch 1928) Later, Jones and co-workers were the first to expand the original s-symmetry-only approach to take into account a basis of different orbitals. (see Jones et al. 1934) However, the Tight-Binding (TB) model in the form it is widely used today was presented by Slater and Koster. (see Slater and Koster 1954) This is the simplest model to solve the electronic problem of periodic systems and, despite its simplicity, it gives excellent results and deep insight into the solid state lattice periodicity and surface phenomena. In this TB approach, one uses a basis of highly localized atomic orbitals and considers the Hamiltonian matrix elements of the system using empirical parameters that work well for rapid calculations of real materials. (see M. Martin 1970)

The TB parameters are further simplified by discarding three-center-integral contributions to the Hamiltonian martix elements. (see Slater and Koster 1954) We are then restricted to considering only the one-center and two-center contributions. The two-center integrals are then simplified using (2.36). However when applying (2.36), one can argue that the choice for the axis will in general not coincide with the line joining the atoms. However it is always possible to write the spherical harmonics relative to the bond line as a linear combinations of the spherical harmonics relative to the z-axis. Using these transformations we can write the two-center-integral contributions to the Hamiltonian matrix elements as a linear combination of the $M_{ll'm}$ terms using (2.36). Slater and Koster came up with expressions for the elements involving the s, p and d orbitals. (see Slater and Koster 1954) Below we reproduce these relations for the case of s and p orbitals, using the same notation used in (2.36) and are described in the caption to Fig. 2.2.

$$\begin{aligned} M_{s,s}= & {} M_{s,s,\sigma }\end{aligned}$$

(2.49)

$$\begin{aligned} M_{s,p_z}= & {} z^2M_{s,p,\sigma }\end{aligned}$$

(2.50)

$$\begin{aligned} M_{p_x,p_x}= & {} x^2M_{p,p,\sigma }+(1-x^2)M_{p,p,\pi }\end{aligned}$$

(2.51)

$$\begin{aligned} M_{p_x,p_y}= & {} xy(M_{p,p,\sigma }-M_{p,p,\pi }). \end{aligned}$$

(2.52)

The TB parameters are fitted to reproduce the crystal properties (such as electronic energy bands or lattice parameters) of a given model system. In addition, one also has to define a cutoff radius for the distance between the atoms so that the Hamiltonian matrix elements for the atomic orbitals are zero when the atoms are separated by a distance larger than the specified cutoff.