Inhomogeneous Electron Gas

This paper deals with the ground state of an interacting electron gas in an external potential $v\left(\mathrm{r}\right)$. It is proved that there exists a universal functional of the density, $F\left[n\right(\mathrm{r}\left)\right]$, independent of $v\left(\mathrm{r}\right)$, such that the expression $E\equiv \int v\left(\mathrm{r}\right)n\left(\mathrm{r}\right)d\mathrm{r}+F\left[n\right(\mathrm{r}\left)\right]$ has as its minimum value the correct ground-state energy associated with $v\left(\mathrm{r}\right)$. The functional $F\left[n\right(\mathrm{r}\left)\right]$ is then discussed for two situations: (1) $n\left(\mathrm{r}\right)=n_0+\tilde{n}\left(\mathrm{r}\right)$, $\frac{\tilde{n}}{n_0}\ll 1$, and (2) $n\left(\mathrm{r}\right)=\varphi \left(\frac{\mathrm{r}}{r_0}\right)$ with $\varphi$ arbitrary and $r_0\to \infty$. In both cases $F$ can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.