Tight bound and convexity constraint on the exchange-correlation-energy functional in the low-density limit, and other formal tests of generalized-gradient approximations

For the purpose of approximating the exact density-functional exchange-correlation energy ${\mathit{E}}_{\mathrm{xc}}$[n], we previously established the coordinate scaling identity ${\mathit{E}}_{\mathrm{xc}}^{\mathrm{\ensuremath{\alpha}}}$[n]=${\mathrm{\ensuremath{\alpha}}}^{2}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}$] for \ensuremath{\alpha}=${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$, or ${\mathit{E}}_{\mathrm{xc}}^{\ensuremath{\lambda}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]=${\ensuremath{\lambda}}^{2}$${\mathit{E}}_{\mathrm{xc}}$[n], where ${\mathit{E}}_{\mathrm{xc}}^{\mathrm{\ensuremath{\alpha}}}$[n] is the exchange-correlation energy functional for electronic charge \ensuremath{\surd}\ensuremath{\alpha} e and where ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,z)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z). This identity is utilized here to derive the low-density limit ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}0}$ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]=${\mathrm{inf}}_{\mathrm{\ensuremath{\Psi}}\ensuremath{\rightarrow}\mathit{n}}$ 〈\ensuremath{\Psi}\ensuremath{\Vert}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$\ensuremath{\Vert}\ensuremath{\Psi}〉-(${\mathit{e}}^{2}$/2)FF[n(${\mathbf{r}}_{1}$)n(${\mathbf{r}}_{2}$) /\ensuremath{\Vert}${\mathbf{r}}_{1}$-${\mathbf{r}}_{2}$\ensuremath{\Vert}]${\mathit{d}}^{3}$${\mathit{r}}_{1}$${\mathit{d}}^{3}$${\mathit{r}}_{2}$, which allows us to express the Lieb-Oxford bound in the tighest-possible manner, namely, ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}0}$ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]\ensuremath{\ge}-${\mathit{Ce}}^{2}$F${\mathit{n}}^{4/3}$(r)${\mathit{d}}^{3}$r, where 1.43\ensuremath{\le}C\ensuremath{\le}1.68. Meaningful adherences to and violations of the bound are presented to demonstrate that it is surprisingly tight and thus quite useful. Other key properties of ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}0}$ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}$] are found, including the observation that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}0}$ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]+(${\mathit{e}}^{2}$/2)FF[n(${\mathbf{r}}_{1}$)n(${\mathbf{r}}_{2}$) /\ensuremath{\Vert}${\mathbf{r}}_{1}$-${\mathbf{r}}_{2}$\ensuremath{\Vert}]${\mathit{d}}^{3}$${\mathit{r}}_{1}$${\mathit{d}}^{2}$${\mathit{r}}_{2}$ is convex, which is a severe constraint. We discuss these and other exact relationships as formal tests of generalized-gradient approximations (GGA's) for exchange and correlation.We find that the Perdew-Wang 1991 (PW91) GGA respects many of the known exact relationships, including those respected by the local-density approximation plus many others that are violated by the local-density approximation. We present a minor variant of the PW91 correlation-energy functional which additionally satisfies a strong \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} (high-density) constraint. Finally, we show that atomic densities are much closer to the high-density than to the low-density limit.