Ground-State Energy of a High-Density Electron Gas

The terms of O(rₛlnrₛ) and O(rₛ) 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 rₛlnrₛ 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 $E=2.21{{r}ₛ}^{{-}2}{-}0.916{{r}ₛ}^{{-}1}+0.0622ln{r}ₛ{-}0.096+0.018{r}ₛln{r}ₛ+({{E}₃}^{{'}}{-}0.036){r}ₛ+O({{r}ₛ}²ln{r}ₛ),$ where ${{E}₃}^{{'}}$ is a sum of twelve dimensional integrals. Although ${{E}₃}^{{'}}$ has not been evaluated it is shown with the aid of the virial theorem that no reasonable value of ${{E}₃}^{{'}}$ can make the series expansion rapidly convergent beyond ${r}ₛ{~}1$. Under the rather arbitrary assumption that ${{E}₃}^{{'}}{r}ₛ$ 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.