Exchange energy as functional of electronic density from Hartree-Fock theory of inhomogeneous electron gas

Abstract

In a previous paper, the validity of gradient corrections to the Dirac-Slater exchange energy was discussed within the density functional theory, for a local potential. This local potential, by definition, generated the exact charge density in the (infinite) system considered. The present work considers the full Hartree-Fock theory within the same density functional framework. In particular, using a perturbation expansion of the Dirac density matrix, we obtain the exchange energy as an expansion in the displaced charge Δ H . F . ( r ) = ρ H . F . ( r ) − ρ 0 , , p0 being the average electron density and p H.F. (r) the Hartree-Fock spatially dependent density. In contrast to the local potential theory, a basic integral equation must be solved to express the above relation in explicit form. No exact solution of this equation exists to date, but using the work of Kleinman and Langreth on the vertex function we are able to find an explicit approximation for the exchange energy to second-order in Δ H.F. The difficulties encountered in defining gradient series in a local potential theory are not removed in the full Hartree-Fock theory. However, it is possible to sum sub-series of gradient terms to infinite order to give a useful generalization of Dirac-Slater theory. Although it is known in metals that correlation effects have a major influence on the Hartree-Fock results, the results obtained here may well be useful in atomic and molecular problems with a sufficiently large number of electrons, as well as in non-metallic crystals. The influence of electronic screening in the case of metals is briefly referred to.