Rigorous and unifying physical interpretation of the exchange potential and energy in the local-density approximation.

Abstract

In this paper we explain that the exchange potential and energy in the local-density approximation (LDA) of density-functional theory has a IrigorousP and IunifiedP physical interpretation founded in the work of Harbola and Sahni. Accordingly, the IsourceP charge distribution that gives rise to both the LDA exchange (path-independent) potential IandP energy is the Fermi hole as derived in the gradient-expansion approximation (GEA) to O(\ensuremath{\nabla}). Thus, the LDA exchange potential, or equivalently the functional derivative of the LDA exchange-energy functional of the density, is the work required to bring an electron from infinity to its position at r against the force field of this charge distribution. The LDA exchange energy in turn is the energy of interaction between the electronic density and this charge. However, it is the non-spherically-symmetric component of the source charge that gives rise to the potential but its spherically symmetric component that contributes to the energy. Since the underlying physics of the LDA for exchange lies in its source charge, we next determine the structure of the GEA Fermi hole to O(\ensuremath{\nabla}) for the nonuniform electronic system in atoms and at metallic surfaces. A study of this structure as a function of electron position shows that the errors in the LDA arise because the source charge does not in general reproduce accurately the structure of the exact Fermi hole, that it violates the quantum-mechanical requirement of positivity, and further that it oscillates, albeit with decaying amplitude, far into the classically forbidden region.