Convexity and translational invariance constraint on the exchange-correlation functional

Abstract

Knowledge of the properties of the exchange-correlation functional in the form $\frac 1λv_{xc}([ρ_λ],\frac{\bf r}λ)$, where $ρ_λ({\bf r})=$ $λ^3ρ(λ{\bf r}),$ is important when expressing the exchange-correlation energy as a line integral $% E_{xc}[ρ]=\int_0^1dλ\int d{\bf r}\frac 1λv_{xc}([ρ_λ],\frac{\bf r}λ)\left[ 3ρ({\bf r})+{\bf r.\nabla }ρ({\bf r})\right] $ (van Leeuwen and Baerends, Phys. Rev. A {\bf 51}, 170 (1995)). With this in mind, it is shown that in the low density limit $% \lim_{λ\rightarrow 0}\int ρ({\bf r})\nabla ^2\frac 1λv_{xc}([ρ_λ],\frac{\bf r}λ)\ d^3r\leq 4π\int ρ(% {\bf r})^2d^3r.$ This inequality is violated in the local-density approximation.