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 $r_s\lesssim 1$ while the low-density calculation of Wigner is valid for $r_s\gtrsim 20$. For actual metallic densities it is shown that the contribution to the correlation energy from long-wavelength momentum transfers ($k<\mathrm{}\beta k_0<0.47\mathrm{}{r_s}^{\frac{1}{2}}{k}_0$) may be accurately calculated in the random phase approximation. This contribution is calculated using the Bohm-Pines extended Hamiltonian, and is shown to be $E\left(\beta \right)=\left(-0.458\mathrm{}\frac{{\beta }^2}{r_s}+0.866\mathrm{}\frac{{\beta }^3}{{r_s}^{\frac{3}{2}}}-0.98\mathrm{}\frac{{\beta }^4}{{r_s}^2}\right)\left(+0.019\mathrm{}\frac{{\beta }^4}{r_s}+0.706\mathrm{}\frac{{\beta }^5}{{r_s}^{\frac{5}{2}}}+\cdots \right)\mathrm{ry}.$ 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 ($k\gtrsim k_0$) 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 ($0.47{r_s}^{\frac{1}{2}}{k}_0\lesssim k\lesssim k_0$) 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 ∼15%. The result for the over-all correlation energy using the interpolation procedure of Pines is $E_c\cong (-0.115+0.031\mathrm{}\ln r_s)\mathrm{ry}.$