Is there a density-gradient expansion for the electronic exchange energy?
An old but recurring question in ground-state density functional theory, relevant to the construction of density functional approximations, is the existence of a second-order gradient expansion for the exchange energy that is exact for densities that vary slowly over space. Here we will argue that there is such an expansion, with a gradient coefficient 10/7 of Sham's, as suggested by theoretical/mathematical derivations and by numerical calculations on model slowly-varying metallic densities, although some standard derivations do not quite capture the correct gradient coefficient. We also discuss long-range anomalies in the gradient expansions for the exchange and correlation holes around an electron. It is plausible that, as suggested earlier and recently, standard derivations are still correct for the coefficient of the second-order gradient expansion for exchange and correlation together, in which case the correct gradient coefficient for correlation is $C_{\mathrm{c}}$ = $C_{\mathrm{xc}}$ - $C_{\mathrm{x}}$. We propose possible numerical evaluations of $C_{\mathrm{xc}}$ and $C_{\mathrm{x}}$ that are free from long-range anomalies, and discuss implications for density functional approximations. Axel Becke's 1988 generalized gradient approximation (the B88 GGA) for the exchange energy brought density functional theory into chemistry, with a gradient coefficient for exchange that is optimal at the GGA level for atoms and molecules but too large for solids. Meta-GGAs seem to work as well for molecules as for solids, but may not be perfected yet. The article concludes with a few memories that the first author has of discussions with Axel Becke.