Exchange density-functional gradient expansion

Abstract

The coefficient ${\ensuremath{\gamma}}_{x}$ of the first term in a gradient expansion of the Hartree-Fock (HF) density functional was calculated by Sham to first order in ${e}^{2}$. It is now known that ${\ensuremath{\gamma}}_{x}^{\mathrm{HF}}$ diverges if ${e}^{2}$ is included to all orders. It has recently been claimed that if the exchange energy is defined in terms of density-functional (DF) eigenfunctions, rather than HF eigenfunctions, not only is ${\ensuremath{\gamma}}_{x}^{\mathrm{DF}}$ first order in ${e}^{2}$ but also ${\ensuremath{\gamma}}_{x}^{\mathrm{DF}}={\ensuremath{\gamma}}_{\mathrm{Sham}}$. It is proven in this paper that, in fact, ${\ensuremath{\gamma}}_{x}^{\mathrm{DF}}=\frac{8}{7}{\ensuremath{\gamma}}_{\mathrm{Sham}}$.