A Method for Obtaining Electronic Eigenfunctions and Eigenvalues in Solids with An Application to Sodium

Abstract

As in the Wigner-Seitz and Slater cellular approximation it is assumed that the valence electron potential is spherically symmetrical within a lattice cell. Then the Schr\"odinger equation becomes separable in spherical coordinates so that the eigenfunction can be expanded in terms which are products of spherical harmonics and solutions to the radial equation. The method differs from the previous ones in that: (1) eigenfunctions belonging to special wave vectors are first constructed by suitable linear combinations of spherical harmonics so as to satisfy symmetry requirements and thus, essentially, including more terms in the expansion without increase in labor and (2) surface boundary conditions are satisfied exactly, in effect, at points which are more representative of the cell surface. The method was tested by applying it to the Shockley empty lattice (body centered cubic) for which the eigenvalues are known exactly. The lowest four eigenvalues belonging to the reduced wave vector (0,0,0) and the lowest three belonging to ($0,0,\frac{\ensuremath{\pi}}{a}$) showed errors of one percent or less in energy using from two to four terms in the eigenfunction expansion. When applied to sodium results showed that electrons in the first few Brillouin zones are indeed very nearly free, having free electron energies within a few percent even at points near the center and corners of reduced wave vector space. Furthermore on the boundary of reduced wave vector space in the (0,0,1) direction there is no energy gap between the first and second Brillouin zone eigenvalues since they "stick together" at that and equivalent points.