New Method for Calculating Wave Functions in Crystals and Molecules

TLDR

Crystal wave‑function calculations are simplest in a plane‑wave representation, and the present formalism provides a rigorous formulation of the empirical‑potential approach within the one‑electron framework, comparing favorably with earlier methods. The study seeks rapid convergence by enforcing orthogonality between valence and core electron wave functions. This is achieved by using orthogonalized plane waves as basis functions, exploiting crystal symmetry to construct smooth Bloch‑function components whose governing equation includes an effective repulsive potential, a method that also extends to molecular wave‑function calculations. Analytic estimates of the effective repulsive potential are presented, and examples demonstrate that cancellation of attractive and repulsive terms in the core region yields rapid convergence for s states while explaining the slower convergence of p states, thereby validating the approach against previous empirical potential models.

Abstract

For metals and semiconductors the calculation of crystal wave functions is simplest in a plane wave representation. However, in order to obtain rapid convergence it is necessary that the valence electron wave functions be made orthogonal to the core wave functions. Herring satisfied this requirement by choosing as basis functions "orthogonalized plane waves." It is here shown that advantage can be taken of crystal symmetry to construct wave functions ${\ensuremath{\phi}}_{\ensuremath{\alpha}}$ which are best described as the smooth part of symmetrized Bloch functions. The wave equation satisfied by ${\ensuremath{\phi}}_{\ensuremath{\alpha}}$ contains an additional term of simple character which corresponds to the usual complicated orthogonalization terms and has a simple physical interpretation as an effective repulsive potential. Qualitative estimates of this potential in analytic form are presented. Several examples are worked out which display the cancellation between attractive and repulsive potentials in the core region which is responsible for rapid convergence of orthogonalized plane wave calculations for $s$ states; the slower convergence of $p$ states is also explained. The formalism developed here can also be regarded as a rigorous formulation of the "empirical potential" approach within the one-electron framework; the present results are compared with previous approaches. The method can be applied equally well to the calculation of wave functions in molecules.