5.3: Bloch Functions for L.C.A.O. Methods

If the crystalline potential is not weak throughout the crystal, then plane waves may not be the best starting point for eigenfunctions of the Schrödinger equation. This is especially true for electronic states in insulating, semiconducting, and transition metal compounds. For these crystalline solids, solutions to the Schrödinger equation use the Tight-Binding or Linear Combination of Atomic Orbitals (L.C.A.O.) approximations. Therefore, it is important to construct Bloch functions using atomic orbitals. We demonstrate how this is accomplished by using a 1-d linear chain of atoms, each separated by a distance a.

The atoms sit at lattice points, and the group of 1-d lattice translation operations \(\left\{ \boldsymbol{T}_{m} \equiv \left( 1 \middle| ma \right);m = 0,\ldots,N - 1 \right\}\) is the group of the Hamiltonian. The periodic boundary condition is applied to \(N\) atoms (lattice points).

Figure 5.23

We start by selecting valence atomic orbital (AO) as an atomic basis function \(\varphi_{n}(x)\). Then, the symmetry-adapted linear combination (SALC) \(\psi_{nk}(x)\) of this AO for the wavevector \(k\) in the first Brillouin zone can be obtained by using the projection operator \(P^{(k)}\) for the IR \(\Gamma^{(k)}\) of the translation group:

\[P^{(k)} = \frac{1}{\sqrt{N}}\sum_{m}^{}{\left\lbrack \Gamma^{(k)}\left( 1 \middle| ma \right) \right\rbrack^{*}\left( 1 \middle| ma \right)} = \frac{1}{\sqrt{N}}\sum_{m}^{}{\left\lbrack e^{- ikma} \right\rbrack^{*}\left( 1 \middle| ma \right)} = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{ikma}\ \left( 1 \middle| ma \right)} \nonumber \]

This operator is a summation over all \(N\) translation operators of the lattice group. Thus, \(\frac{1}{\sqrt{N}}\) is a normalization factor. Applying \(P^{(k)}\) to the AO \(\varphi_{n}(x)\) forms the Bloch function:

\[\psi_{nk}(x) = P^{(k)}\ \varphi_{n}(x) = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{ikma}\ \left( 1 \middle| ma \right)\varphi_{n}(x)} = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{ikma}\ \varphi_{n}(x - ma)}. \nonumber \]

According to this expression, the Bloch function \(\psi_{nk}(x)\) is delocalized throughout the entire crystal and, for each lattice step, the phase of the atomic basis function changes by \(e^{ika}\). The Bloch functions for every allowed wavevector \(k\) in the first Brillouin zone are mutually orthogonal because they are basis functions for distinct IRs of the translation group. As a result, they are also eigenfunctions for the Hamiltonian operator of this 1-d problem and called crystal orbitals.

Now, the expectation value of the Hamiltonian for the Bloch function \(\psi_{nk}(x)\) gives the wavevector-dependent energy (eigenvalue) \(E_{n}(k)\):

\[E_{n}(k) = \left\langle \psi_{nk} \middle| H \middle| \psi_{nk} \right\rangle = \frac{1}{N}\sum_{m'}^{}{\sum_{m}^{}{e^{ik\left( m - m' \right)a}\left\langle \varphi_{n}(x - m'a) \middle| H \middle| \varphi_{n}(x - ma) \right\rangle}}, \nonumber \]

which contains Hamiltonian integrals between atomic basis functions at different sites in the 1-d crystal. These integrals can be either calculated via first principles methods or assigned numerical values using some type of (semi)empirical approach. A relatively simple method for calculating electronic structures is Hückel theory, which considers just nearest neighbor orbital interactions. In this approximation, two types of integrals are assigned nonzero values:

1. Coulomb Integrals (\(m'= m\)): \(\alpha \equiv \left\langle \varphi_{n}(x - ma) \middle| H \middle| \varphi_{n}(x - ma) \right\rangle\);
2. Resonance Integrals (\(m'= m \pm 1\)): \(\beta \equiv \left\langle \varphi_{n}(x - (m \pm 1)a) \middle| H \middle| \varphi_{n}(x - ma) \right\rangle\).

Coulomb integrals \(\alpha\) are on-site integrals corresponding to the AO energy. They are negative values (\(\alpha < 0\)) because valence electrons occupying these valence AOs are bound in the free atoms. Resonance integrals (also called hopping or transfer integrals) represent interaction energies between AOs on adjacent sites in the structure. These values can be positive (repulsive) or negative (attractive) depending on the through-space orbital overlap. Another feature of Hückel theory is that the atomic basis functions are set to be spatially orthonormal:

\[S_{m'm} \equiv \left\langle \varphi_{n}(x - m'a) \middle| \varphi_{n}(x - ma) \right\rangle = \delta_{m'm} \tag{overlap integral} \]

Using these assignments, the wavevector-dependent energy \(E_{n}(k)\) for the 1-d chain of atoms is

\[E_{n}(k) = \frac{1}{N}\left\lbrack N\alpha + Ne^{ika}\beta + Ne^{- ika}\beta \right\rbrack = \alpha + 2\beta\ \cos{ka}. \nonumber \]

Since the allowed wavevectors form a quasi-continuous set and \(E_{n}( - k) = E_{n}(k)\), \(E_{n}(k)\) is a quasi-continuous energy band that is plotted for wavevectors \(0 \leq k \leq \frac{\pi}{a}\). In the following discussion, we apply these concepts for valence s and p AOs at the atoms. To simplify various mathematical expressions, \(\varphi_{nk}(ma)\) is substituted for \(\varphi_{nk}(x - ma)\).