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 Exc[n], we previously established the coordinate scaling identity Exc^α[n]=α²{E}xc$[${n}_{{λ}}] for α={{λ}}^{{{-}}1}$, or ${E}xc^{{λ}}$[${n}_{{λ}}$]=${{λ}}²Exc[n], where Exc^α[n] is the exchange-correlation energy functional for electronic charge {}{α} e and where n_λ(x,y,z)=λ³n({λ}x,{λ}y,{λ}z). This identity is utilized here to derive the low-density limit lim_λ→0 λ^-1{E}xc$[${n}_{{λ}}$]=${inf}_{{{Ψ}}{→}n}〈ΨV{^}ₑₑΨ〉-({e}²$/2)FF[n(${r}₁$)n(${r}₂) /{r}₁$-${r}₂]{d}³r₁{d}³r₂, which allows us to express the Lieb-Oxford bound in the tighest-possible manner, namely, lim_λ→0 λ^-1{E}xc$[${n}_{{λ}}]≥-{Ce}²F{n}4/3$(r)${d}³$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_λ→0 λ^-1{E}xc$[${n}_{{λ}}$] are found, including the observation that ${lim}_{{λ}{→}0}$ ${{λ}}^{{{-}}1}Exc[n_λ]+(e²/2)FF[n(r₁)n(r₂) /{}r₁-r₂{}]d³{r}₁d²{r}₂$ 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.