Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium

Abstract

The problem of solving the Schr\"odinger equation in a periodic lattice is studied from the point of view of the variation-iteration method. This approach leads to a very compact scheme if the potential $V(r)$ is spherically symmetrical within the inscribed spheres of the atomic polyhedra and constant in the space between them. The band structure of the lattice is then determined by (1) geometrical structure constants, characteristic of the type of lattice and (2) the logarithmic derivatives, at the surface of the inscribed sphere, of the $s, p, d, \dots{}$ functions corresponding to $V(r)$. By far the greater part of the labor is involved in the calculation of (1), which needs to be done only once for each type of lattice; (2) can be obtained by numerical integration or directly from the atomic spectra. Although derived from a different point of view, this scheme turns out to be essentially equivalent to one proposed by Korringa on the basis of the theory of lattice interferences. The present paper also contains an application to the conduction band of metallic lithium.