For the purpose of approximating the exact density-functional exchange-correlation energy ${\mathit{E}}_{\mathrm{xc}}$[n], we previously established the coordinate scaling identity ${\mathit{E}}_{\mathrm{xc}}^{\mathrm{\alpha }}$[n]=${\mathrm{\alpha }}^2$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$] for α=${\lambda }^{\mathrm{-}1}$, or ${\mathit{E}}_{\mathrm{xc}}^{\lambda }$[${\mathit{n}}_{\lambda }$]=${\lambda }^2$${\mathit{E}}_{\mathrm{xc}}$[n], where ${\mathit{E}}_{\mathrm{xc}}^{\mathrm{\alpha }}$[n] is the exchange-correlation energy functional for electronic charge √α e and where ${\mathit{n}}_{\lambda }$(x,y,z)=${\lambda }^3$n(λx,λy,λz). This identity is utilized here to derive the low-density limit ${\lim }_{\lambda \to 0}$ ${\lambda }^{\mathrm{-}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$]=${\inf }_{\mathrm{\Psi }\to \mathit{n}}$ 〈Ψ‖V${\mathrm{^}}_{\mathit{e}\mathit{e}}$‖Ψ〉-(${\mathit{e}}^2$/2)FF[n(${\mathbf{r}}_1$)n(${\mathbf{r}}_2$) /‖${\mathbf{r}}_1$-${\mathbf{r}}_2$‖]${\mathit{d}}^3$${\mathit{r}}_1$${\mathit{d}}^3$${\mathit{r}}_2$, which allows us to express the Lieb-Oxford bound in the tighest-possible manner, namely, ${\lim }_{\lambda \to 0}$ ${\lambda }^{\mathrm{-}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$]≥-${\mathit{Ce}}^2$F${\mathit{n}}^{4/3}$(r)${\mathit{d}}^3$r, where 1.43≤C≤1.68. Meaningful adherences to and violations of the bound are presented to demonstrate that it is surprisingly tight and thus quite useful. Other key properties of ${\lim }_{\lambda \to 0}$ ${\lambda }^{\mathrm{-}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$] are found, including the observation that ${\lim }_{\lambda \to 0}$ ${\lambda }^{\mathrm{-}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$]+(${\mathit{e}}^2$/2)FF[n(${\mathbf{r}}_1$)n(${\mathbf{r}}_2$) /‖${\mathbf{r}}_1$-${\mathbf{r}}_2$‖]${\mathit{d}}^3$${\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 λ→∞ (high-density) constraint. Finally, we show that atomic densities are much closer to the high-density than to the low-density limit.

• Received 11 June 1993

DOI:https://doi.org/10.1103/PhysRevB.48.11638