Correlation Energy of a Free Electron Gas

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 rₛ1 while the low-density calculation of Wigner is valid for rₛ20. For actual metallic densities it is shown that the contribution to the correlation energy from long-wavelength momentum transfers (k<βk₀<0.47rₛ1/2k₀) may be accurately calculated in the random phase approximation. This contribution is calculated using the Bohm-Pines extended Hamiltonian, and is shown to be $E({β})=({-}0.458{{{β}}²}{{r}ₛ}+0.866{{{β}}³}{{{r}ₛ}3/2}{-}0.98{{{β}}⁴}{{{r}ₛ}²})(+0.019{{{β}}⁴}{{r}ₛ}+0.706{{{β}}⁵}{{{r}ₛ}5/2}+{⋯})ry.$ 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 (kk₀) 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 (0.47rₛ1/2k₀kk₀) 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 {~}15%. The result for the over-all correlation energy using the interpolation procedure of Pines is Ec(-0.115+0.031lnrₛ)ry.