Treatments of the exchange energy in density-functional theory

TL;DR

This paper provides a unified derivation of the exchange energy correction in density-functional theory, connecting Kohn-Sham and Hartree-Fock approaches through a common variational framework.

Contribution

It introduces a simple derivation of the density-functional correction to Hartree-Fock equations and explores the formulation of exchange energy as degree-two homogeneous density functionals.

Findings

Unified view of Kohn-Sham, Hartree-Fock-Kohn-Sham, and Schrödinger equations

Expression of exchange energy as a sequence of degree-two homogeneous functionals

Identification of the first element as the classical Coulomb-repulsion energy

Abstract

Following a recent work [Gal, Phys. Rev. A 64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree-Fock equations, the Hartree-Fock-Kohn-Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn-Sham equations, the Hartree-Fock-Kohn-Sham equations and the ground-state Schrodinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn-Sham formulation of the Hartree-Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn-Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N=1,2,...), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.