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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 rₒ1 while the low-density calculation of Wigner is valid for rₒ20. For actual metallic densities it is shown that the contribution to the correlation energy from long-wavelength momentum transfers (k<k₀<0. 47{rₒ}^1{2}k₀) 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 () = (-0. 458{^2}{rₒ}+0. 866{^3}{{rₒ}^3{2}}-0. 98{^4}{{rₒ}^2}) (+0. 019{^4}{rₒ}+0. 706{^5}{{rₒ}^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. 47{rₒ}^1{2}k₀k₀) 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 E₂ (-0. 115+0. 031lnrₒ) ry.
Nozières et al. (Tue,) studied this question.