Electronic Energy Bands in Sodium Chloride

Abstract

The Wigner and Seitz method of cellular potentials has been applied to the calculation of wave functions in NaCl. A renormalized Hartree field has been used around the Cl and the Prokofjew field around the Na. The relative heights of the potentials are determined by use of Madelung's number. The problem of joining the functions at the cell boundaries has been treated by the Slater method of fitting $\ensuremath{\psi}$ and ${\ensuremath{\psi}}^{\ensuremath{'}}$ at midpoints. For the outer Cl electrons a reasonable approximation is to join at Cl---Cl midpoints only. This gives rise to a face-centered lattice for which solutions of the Slater conditions have been found by Krutter. Several new solutions have been derived which allow fairly accurate energy contours in momentum space to be drawn for the Cl $3p$ band. If the joining is made at Cl---Na midpoints alone, a large number of unsatisfactory zero-width bands arise. When both Cl---Cl and Cl---Na midpoints are used, the boundary conditions can be treated only for special cases. For these they are consistent with the Cl---Cl solutions. Several attempts to calculate the ultraviolet absorption frequency are described and the difficulties involved are discussed.