Electronic Energy Bands in Metals

The method of Wigner and Seitz is extended to the computation of the excited bands of electrons in a metal, with particular applications to sodium. Instead of using simply one $s$ wave function, as Wigner and Seitz do, a combination of eight separate functions is used, one $s$, three $p$, three $d$, and one $f$. Boundary conditions for an arbitrary electron momentum are satisfied at the midpoints of the lines connecting an atom with its eight nearest neighbors. The solution is carried out for an arbitrary direction of propagation in one of the principal planes. Energy levels and wave functions are determined as functions of internuclear distance, leading to the following qualitative results: At the observed distance of separation, energy levels are given with remarkable accuracy by the Fermi-Sommerfeld theory, the gaps fall approximately where they should as computed from de Broglie waves, and the wave functions act accurately like plane waves in the region between atoms, but fluctuate violently, like $s, p, \cdots$ functions, near the nuclei. Gaps in energy are precisely filled up, though in each definite direction of propagation there are gaps. As the internuclear distance increases, gaps in energy appear at definite points, the allowed regions shrinking to zero breadth about the atomic energy levels at infinite separation.