Ground-State Energy of a High-Density Electron Gas

Abstract

The terms of $O\left(r_s\ln r_s\right)$ and $O\left(r_s\right)$ 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 $r_s\ln r_s$ 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 $E=2.21\mathrm{}{r_s}^{-2\mathrm{}}-0.916\mathrm{}{r_s}^{-1\mathrm{}}+0.0622\mathrm{}\ln r_s-0.096+0.018\mathrm{}{r}_s\ln r_s+({E_3}^{\prime }-0.036)r_s+O\left({r_s}^2\ln r_s\right),$ where ${E_3}^{\prime }$ is a sum of twelve dimensional integrals. Although ${E_3}^{\prime }$ has not been evaluated it is shown with the aid of the virial theorem that no reasonable value of ${E_3}^{\prime }$ can make the series expansion rapidly convergent beyond $r_s\sim 1$. Under the rather arbitrary assumption that ${E_3}^{\prime }{r}_s$ as well as higher order terms can be neglected below $r_s=1\mathrm{}$, 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.