Fourth-order gradient corrections to the exchange-only energy functional: Importance of

Abstract

The pertinence of four-order gradient corrections to the exchange-only energy functional ${\mathit{E}}_{\mathit{x}}$[n] for producing accurate exchange potentials ${\mathit{v}}_{\mathit{x}}$(r) is examined utilizing the exact ${\mathit{v}}_{\mathit{x}}$(r)'s obtained from the optimized-potential model (OPM) for spherical atoms and jellium spheres (whose inner regions have slowly varying densities for which the gradient expansion should be valid). It is found that the fourth-order contributions containing ${\mathrm{\ensuremath{\nabla}}}^{2}$n represent important ingredients of ${\mathit{E}}_{\mathit{x}}$[n] which should be included in gradient-based nonlocal extensions of the local-density approximation. In accordance with this observation the ${\mathit{v}}_{\mathit{x}}$(r)'s resulting from the fourth-order gradient expansion are closer to the exact ${\mathit{v}}_{\mathit{x}}^{\mathrm{OPM}}$(r)'s than those from generalized gradient approximations which do not contain ${\mathrm{\ensuremath{\nabla}}}^{2}$n contributions.