Electronic Energy Bands in Crystals

Abstract

A study is made of the feasibility of calculating valence and excited electronic energy bands in crystals by making use of one-electron Bloch wave functions. The elements of the secular determinant for this method consist of Bloch sums of overlap and energy integrals. Although often used in evaluating these sums, the approximation of tight binding, which consists of neglecting integrals between non-neighboring atoms of the crystal, is very poor for metals, semiconductors, and valence crystals. By partially expanding each Bloch wave function in a three-dimensional Fourier series, these slowly convergent sums over ordinary space can be transformed into extremely rapidly convergent sums over momentum space. It can then be shown that, to an excellent approximation, the secular determinant vanishes identically. This peculiar behavior results from the poorness of the atomic correspondence for valence electrons. By a suitable transformation, a new secular determinant can be formed which does not vanish identically and which is suitable for numerical calculations. It is found that this secular determinant is identical with that obtained in the method of orthogonalized plane waves (plane waves made orthogonal to the inner-core Bloch wave functions).