Localized One-Electron States in Perfect Crystals as a Consequence of the Thermal Single-Determinant Approximation

Abstract

A new extension of Hartree-Fock (HF) theory to nonzero temperature $T$, namely, the thermal single-determinant approximation (TSDA)—based on the variational principle of statistical mechanics—is applied to a model of a crystal of widely separated atoms. It is shown that, in this TSDA, one type of solution to the equations of stationarity of the free energy (TSD equations) consists of one-electron functions that are extended throughout the crystal (like Bloch functions), and another type of solution consists of localized one-electron states (particular Wannier functions), whereas it appears that in the usual, or standard, thermal HF approximation (THFA), only extended solutions are possible at finite atomic separation. (A previous argument that led to results contradictory to the latter statement is shown to be invalid.) Further, in the TSDA at $T>\mathrm{}0\mathrm{}$, the localized solutions give a lower free energy than that corresponding to the extended solutions, as well as a lower free energy than that obtained in the THFA. As far as we know, this is the first calculation in which a strictly variational requirement has rejected this class of spatially extended one-electron functions in favor of localized functions in a perfect crystal (i.e., in a system with translational symmetry).