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 Eₗ₂n, we previously established the coordinate scaling identity Eₗ₂^n=^2Eₗ₂n_ for =^-1, or Eₗ₂^n_=^2Eₗ₂n, where Eₗ₂^n is the exchange-correlation energy functional for electronic charge e and where n_ (x, y, z) =^3n (, , ). This identity is utilized here to derive the low-density limit lim₀ ^-1Eₗ₂n_=inf_ 〈^₄₄〉- (e^2/2) FFn (r₁) n (r₂) /r₁-r₂d^3r₁d^3r₂, which allows us to express the Lieb-Oxford bound in the tighest-possible manner, namely, lim₀ ^-1Eₗ₂n_-Ce^2Fn^4/3 (r) d^3r, where 1. 431. 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 lim₀ ^-1Eₗ₂n_ are found, including the observation that lim₀ ^-1Eₗ₂n_+ (e^2/2) FFn (r₁) n (r₂) /r₁-r₂d^3r₁d^2r₂ 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 (high-density) constraint. Finally, we show that atomic densities are much closer to the high-density than to the low-density limit.