Electronic Energy Bands in Crystals

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). Calculations are made on the lithium crystal in order to test how rapidly the energy converges to its limiting value as the order of the secular determinant is increased. For the valence band, this convergence is rapid. The effective mass of the electron at the bottom of the valence band is found to be closer to that of the free electron than are those of previous calculations on lithium. This is probably because of the use of a crystal potential here rather than an atomic potential. The former varies less rapidly than the latter over most of the unit cell of the crystal, and thus should result in a value of effective mass more nearly free-electron-like. Unlike previous calculations on lithium, the computed value of the width of the filled portion of the valence band agrees excellently with experiment. By making use of calculated transition probabilities between the valence band and the 1s level, a theoretical curve is drawn of the shape of the soft x-ray K emission band of lithium. The comparison with the shape of the experimental curve is only fair.