Correlation Energy of a Free Electron Gas

Abstract

The limits of validity of the correlation-energy calculations in the regions of high density, low density, and actual metallic electron densities are discussed. Simple physical arguments are given which show that the high-density calculation of Gell-Mann and Brueckner is valid for ${r}_{s}\ensuremath{\lesssim}1$ while the low-density calculation of Wigner is valid for ${r}_{s}\ensuremath{\gtrsim}20$. For actual metallic densities it is shown that the contribution to the correlation energy from long-wavelength momentum transfers ($k<\ensuremath{\beta}{k}_{0}<0.47{{r}_{s}}^{\frac{1}{2}}{k}_{0}$) may be accurately calculated in the random phase approximation. This contribution is calculated using the Bohm-Pines extended Hamiltonian, and is shown to be $E(\ensuremath{\beta})=\left(\ensuremath{-}0.458\frac{{\ensuremath{\beta}}^{2}}{{r}_{s}}+0.866\frac{{\ensuremath{\beta}}^{3}}{{{r}_{s}}^{\frac{3}{2}}}\ensuremath{-}0.98\frac{{\ensuremath{\beta}}^{4}}{{{r}_{s}}^{2}}\right)\left(+0.019\frac{{\ensuremath{\beta}}^{4}}{{r}_{s}}+0.706\frac{{\ensuremath{\beta}}^{5}}{{{r}_{s}}^{\frac{5}{2}}}+\ensuremath{\cdots}\right)\mathrm{ry}.$ An identical result is obtained by a suitable expansion of the result of Gell-Mann and Brueckner; the validity of the Bohm-Pines neglect of subsidiary conditions in the calculation of the ground-state energy is thereby explicitly established. The contribution to the correlation energy from sufficiently high momentum transfers ($k\ensuremath{\gtrsim}{k}_{0}$) will arise only from the interaction between electrons of antiparallel spin, and may be estimated using second-order perturbation theory. The contribution arising from intermediate momentum transfers ($0.47{{r}_{s}}^{\frac{1}{2}}{k}_{0}\ensuremath{\lesssim}k\ensuremath{\lesssim}{k}_{0}$) cannot be calculated analytically; the interpolation procedures for this domain proposed by Pines and Hubbard are shown to be nearly identical, and their accuracy is estimated as \ensuremath{\sim}15%. The result for the over-all correlation energy using the interpolation procedure of Pines is ${E}_{c}\ensuremath{\cong}(\ensuremath{-}0.115+0.031\mathrm{ln}{r}_{s})\mathrm{ry}.$