Some questions on the exchange contribution to the effective potential of the Kohn–Sham theory

Abstract

The current success of Density Functional Theory applications hinges upon the availability of explicitly density-dependent functionals to self-consistently solve a set of one-electron equations, the Kohn–Sham (KS) equations, which determine the occupied orbitals and its associated electronic density. In KS theory, a local exchange potential is proposed as part of an effective potential. This potential is compared to the exchange operator of the Hartree–Fock theory, which is of a non-local nature. The present paper discusses the variational framework of the KS equations, and the equivalence between both exchange potentials within a correlation-free theory. The common difficulties of current local exchange functionals to correctly simulate the non-locality of the exchange energy density in chemical systems are also analyzed and explained through an exactly solvable model. We give then numerical arguments and conclude by analyzing the performance of various commonly used approximations to exchange functionals.