Knowledge of the properties of the exchange-correlation functional in the form $\frac 1\lambda v_{ xc}([\rho _\lambda ],\frac{ { \bf r}}\lambda )$, where $\rho _\lambda ({ \bf r})=$ $\lambda ^3\rho (\lambda { \bf r}),$ is important when expressing the exchange-correlation energy as a line integral $% E_{ xc}[\rho ]=\int_0^1d\lambda \int d{ \bf r}\frac 1\lambda v_{ xc}([\rho _\lambda ],\frac{ { \bf r}}\lambda )\left[ 3\rho ({ \bf r})+{ \bf r.\nabla }\rho ({ \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 $% \lim_{ \lambda \rightarrow 0}\int \rho ({ \bf r})\nabla ^2\frac 1\lambda v_{ xc}([\rho _\lambda ],\frac{ { \bf r}}\lambda )\ d^3r\leq 4\pi \int \rho (% { \bf r})^2d^3r.$ This inequality is violated in the local-density approximation.
Categories: cond-mat