We have compared, for atomic systems, the spherically averaged Fermi-hole correlation function [Formula: see text] in the Hartree–Fock theory with the corresponding function [Formula: see text] employed in local density functional theory. It is shown that, in contrast to [Formula: see text], the function [Formula: see text] behaves qualitatively incorrectly at positions r1 of the reference electron far from the nucleus. Furthermore, we have shown that the qualitatively incorrect behaviour of [Formula: see text] can be remedied by an approximate expansion of [Formula: see text] in powers of s, where s is the inter-electronic distance. However, such an expansion must be conducted in two regions due to the discontinuity of [Formula: see text] as a function of s at the atomic nucleus. Based on the two-region expansion of [Formula: see text] we have developed an alternative approximate density functional expansion [Formula: see text] for the spherically averaged Fermi-hole correlation function. The corresponding exchange energy density functional yields values for the exchange energies of atoms in good agreement with Hartree–Fock results. Keywords: atomic exchange energy, density functional theory, Fermi hole.
Exact-exchange energy density in the gauge of a semilocal density-functional approximation
