Energy Bands by the LCAO Cellular Method

Abstract

Conceptually, the most simple method for solving the energy band problem is the cellular method in which one solves the Schrodinger equation within the unit cell and imposes boundary conditions on the solution at the surface of the unit cell thereby introducing the wave vector dependence of the energy. Computationally, the implementation of this method and the subsequent use of the resulting wave functions suffer from complications associated with the shape of the unit cell. In terms of compactness and the functional form of the wave functions, the LCAO method as employed by Lafon and Lin is the most simple. Although the LCAO method is only an approximate method, these authors have shown that the conduction band of lithium can be reproduced using only the Bloch sums of atomic 1s, 2s and 2p functions to an accuracy of about 0.01 rydbergs¹. Furthermore, the atomic functions used in this calculation were each linear combinations of two Slater orbitals with coefficients and orbital exponents chosen so as to approximate the numerical Hartree-Fock orbitals. It is reasonable to expect that considerably greater accuracy could be obtained without increasing the number of Slater orbitals used to represent the Bloch function at a given point in the Brillouin zone by varying the linear coefficients and orbital exponents at each point to minimize the one-electron energy.