A practical method of inverting the metric matrices of crystals

Abstract

When orthogonalising the localised basis functions in crystals with Lowdin's symmetrical method (1956), the inverse of the metric matrix plays a crucial role. A practical method of inverting the general metric matrices of crystals is evaluated. The procedure is based on the cluster method, introduced for spherically symmetric states by Lundqvist and Froman (1950), and it exploits completely the rotational symmetry of the crystal. The general equations for solving the elements of the inverse metric matrix are derived, and explicit coefficients are given for s and p states. Calculations have been performed for some alkali halides and magnesium oxide, and numerical results for sodium fluoride are given and considered in some detail. The series expansion approximation for the inverse metric matrix elements is found to be reasonable for alkali halides but not at all suitable for magnesium oxide. However, 'exact' values for inverse metric matrix elements should be used in all accurate theoretical calculations.