Gradient expansion of the exchange-energy density functional: Effect of taking limits in the wrong order

Abstract

We show that if one uses a screened Coulomb potential V(K)=4\ensuremath{\pi}/(${K}^{2}$+${K}_{s}^{2}$) to calculate ${\ensuremath{\gamma}}_{x}$, the exchange-energy density-functional gradient expansion coefficient, one obtains Sham's result if one takes K\ensuremath{\rightarrow}0 before taking ${K}_{s}$\ensuremath{\rightarrow}0. The exchange energy is by its very definition an unscreened quantity so that the correct order of the limits is ${K}_{s}$\ensuremath{\rightarrow}0 before K\ensuremath{\rightarrow}0, in which case ${\ensuremath{\gamma}}_{x}$=1.42${\ensuremath{\gamma}}_{x}^{\mathrm{Sham}}$. .AE