Thermal Properties of the Inhomogeneous Electron Gas

A variational property of the ground-state energy of an electron gas in an external potential $v(\mathrm{r})$, derived by Hohenberg and Kohn, is extended to nonzero temperatures. It is first shown that in the grand canonical ensemble at a given temperature and chemical potential, no two $v(\mathrm{r})$ lead to the same equilibrium density. This fact enables one to define a functional of the density $F[n(\mathrm{r})]$ independent of $v(\mathrm{r})$, such that the quantity $\ensuremath{\Omega}=\ensuremath{\int}v(\mathrm{r})n(\mathrm{r})d\mathrm{r}+F[n(\mathrm{r})]$ is at a minimum and equal to the grand potential when $n(\mathrm{r})$ is the equilibrium density in the grand ensemble in the presence of $v(\mathrm{r})$.