Analysis of the local-density approximation of density-functional theory

In this paper we perform a configuration-space analysis of the local-density approximation (LDA) for the exchange-correlation energy functional of Kohn-Sham density-functional theory in terms of the corresponding average exchange-correlation charge (hole) and energy densities. According to our analysis, the explanation for the quantitative success of the LDA based on the hole charge-conservation sum rule and the assumed consequent cancellation of errors in the spherical averages of the hole is inadequate. The principal conclusion of our work is that the constraint of charge neutrality is a necessary but not sufficient condition for an approximate energy functional to lead to accurate ground-state energies and ionization potentials. The significant additional requirement for the functional is that it must, at least qualitatively, reproduce correctly the structure of the hole as a function of electron position. We perform our calculations within the exchange-only approximation as applied to atoms and jellium metal surfaces. In atoms the Fermi hole is localized about the nucleus; as a consequence the LDA Fermi hole is accurate only for electron positions close to it.

However, we show that the spherically averaged LDA hole is accurate for electron positions in the shell regions; it is substantially in error in the intershell and classically forbidden regions. The fact that the principal contribution to the exchange energy comes from the inner-shell region of the atom, where the LDA hole is accurate, explains why the errors in the LDA ground-state energies are small. However, the ionization potential, which depends on the structure of the hole in the outer regions of the atom, is substantially in error in the LDA since here the LDA hole differs significantly from the exact one. For metallic surfaces, on the other hand, as an electron is pulled from within the metal to infinity outside, the Fermi hole is delocalized and spread throughout the crystal. As a consequence, the planar-averaged LDA hole, which is accurate for electron positions inside the metal, bears little resemblance to the exact hole for electron positions outside the surface. Thus the planar-averaged LDA energy density in the regions at and outside the surface, from which there is a significant contribution to the energy, is poor, and consequently so is the surface exchange energy. However, the LDA work function is accurate since the surface dipole barrier is fairly insensitive to the choice of the approximate energy functional.

Finally, our calculations exhibit a striking similarity in the structure of the exact spherically averaged Fermi holes for the few-electron-atomic and many-electron-metallic-surface nonuniform systems, thus demonstrating the truly universal nature of the exact energy functional.