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 E_xc[n], we previously established the coordinate scaling identity E_xc^α[n]=α²E_xc[n_λ] for α=λ^-1, or E_xc^λ[n_λ]=λ²E_xc[n], where E_xc^α[n] is the exchange-correlation energy functional for electronic charge √α e and where n_λ(x,y,z)=λ³n(λx,λy,λz). This identity is utilized here to derive the low-density limit lim_λ→0 λ^-1E_xc[n_λ]=inf_Ψ→n 〈Ψ‖V^_ee‖Ψ〉-(e²/2)FF[n(r₁)n(r₂) /‖r₁-r₂‖]d³r₁d³r₂, which allows us to express the Lieb-Oxford bound in the tighest-possible manner, namely, lim_λ→0 λ^-1E_xc[n_λ]≥-Ce²Fn^4/3(r)d³r, 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^-1E_xc[n_λ] are found, including the observation that lim_λ→0 λ^-1E_xc[n_λ]+(e²/2)FF[n(r₁)n(r₂) /‖r₁-r₂‖]d³r₁d²r₂ 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.