Ground-State Energy of a High-Density Electron Gas

The terms of $O({r}_{s}\mathrm{ln}{r}_{s})$ and $O({r}_{s})$ in the expansion of the ground-state energy of the high-density electron gas are studied in this paper. The value of the coefficient of ${r}_{s}\mathrm{ln}{r}_{s}$ is evaluated, and it is found to differ from the value obtained by DuBois. The result of the present calculation for the energy per electron is $E=2.21{{r}_{s}}^{\ensuremath{-}2}\ensuremath{-}0.916{{r}_{s}}^{\ensuremath{-}1}+0.0622\mathrm{ln}{r}_{s}\ensuremath{-}0.096+0.018{r}_{s}\mathrm{ln}{r}_{s}+({{E}_{3}}^{\ensuremath{'}}\ensuremath{-}0.036){r}_{s}+O({{r}_{s}}^{2}\mathrm{ln}{r}_{s}),$ where ${{E}_{3}}^{\ensuremath{'}}$ is a sum of twelve dimensional integrals. Although ${{E}_{3}}^{\ensuremath{'}}$ has not been evaluated it is shown with the aid of the virial theorem that no reasonable value of ${{E}_{3}}^{\ensuremath{'}}$ can make the series expansion rapidly convergent beyond ${r}_{s}\ensuremath{\sim}1$. Under the rather arbitrary assumption that ${{E}_{3}}^{\ensuremath{'}}{r}_{s}$ as well as higher order terms can be neglected below ${r}_{s}=1$, an interpolation between the present result and the low-density expansion is carried out, and values of the correlation energy in the region of metallic densities are estimated.