A variational property of the ground-state energy of an electron gas in an external potential $v\left(\mathrm{r}\right)$, 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 $v\left(\mathrm{r}\right)$ lead to the same equilibrium density. This fact enables one to define a functional of the density $F\left[n\right(\mathrm{r}\left)\right]$ independent of $v\left(\mathrm{r}\right)$, such that the quantity $\Omega =\int v\left(\mathrm{r}\right)n\left(\mathrm{r}\right)d\mathrm{r}+F\left[n\right(\mathrm{r}\left)\right]$ is at a minimum and equal to the grand potential when $n\left(\mathrm{r}\right)$ is the equilibrium density in the grand ensemble in the presence of $v\left(\mathrm{r}\right)$.

• Received 8 October 1964

DOI:https://doi.org/10.1103/PhysRev.137.A1441