Exchange density-functional gradient expansion

Abstract

The coefficient ${\gamma }_x$ 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 ${\gamma }_x^{\mathrm{HF}}$ 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 ${\gamma }_x^{\mathrm{DF}}$ first order in $e^2$ but also ${\gamma }_x^{\mathrm{DF}}={\gamma }_{\mathrm{Sham}}$. It is proven in this paper that, in fact, ${\gamma }_x^{\mathrm{DF}}=\frac{8}{7}{\gamma }_{\mathrm{Sham}}$.