Accurate and simple analytic representation of the electron-gas correlation energy

We propose a simple analytic representation of the correlation energy ɛ_c for a uniform electron gas, as a function of density parameter r_s and relative spin polarization ζ. Within the random-phase approximation (RPA), this representation allows for the r_s^-3/4 behavior as r_s→∞. Close agreement with numerical RPA values for ɛ_c(r_s,0), ɛ_c(r_s,1), and the spin stiffness α_c(r_s)=∂²ɛ_c(r_s, ζ=0)/δζ², and recovery of the correct r_slnr_s term for r_s→0, indicate the appropriateness of the chosen analytic form. Beyond RPA, different parameters for the same analytic form are found by fitting to the Green’s-function Monte Carlo data of Ceperley and Alder [Phys. Rev. Lett. 45, 566 (1980)], taking into account data uncertainties that have been ignored in earlier fits by Vosko, Wilk, and Nusair (VWN) [Can. J. Phys. 58, 1200 (1980)] or by Perdew and Zunger (PZ) [Phys. Rev. B 23, 5048 (1981)]. While we confirm the practical accuracy of the VWN and PZ representations, we eliminate some minor problems with these forms. We study the ζ-dependent coefficients in the high- and low-density expansions, and the r_s-dependent spin susceptibility. We also present a conjecture for the exact low-density limit. The correlation potential μ_c^σ(r_s,ζ) is evaluated for use in self-consistent density-functional calculations.