Correlation hole of the spin-polarized electron gas, with exact small-wave-vector and high-density scaling

For a uniform electron gas of density n=n_↑+n_↓=3/4πr_s³=πk_s⁶/192 and spin polarization ζ=(n_↑-n_↓)/n, we study the Fourier transform ρ¯_c(k,r_s,ζ) of the correlation hole, as well as the correlation energy ɛ_c(r_s,ζ)=F₀^∞dk ρ¯_c/π. In the high-density (r_s→0) limit, we find a simple scaling relation k_sρ¯_c/πg²→f(z,ζ), where z=k/gk_s, g=[(1+ζ)^2/3+(1-ζ)^2/3]/2, and f(z,1)=f(z,0). The function f(z,ζ) is only weakly ζ dependent, and its small-z expansion -3z/π²+4 √3 z²/π²+... is also the exact small-wave-vector (k→0) expansion for any r_s or ζ. Motivated by these considerations, and by a discussion of the large-wave-vector and low-density limits, we present two Padé representations for ρ¯_c at any k, r_s, or ζ, one within and one beyond the random-phase approximation (RPA). We also show that ρ¯ _c^RPA obeys a generalization of Misawa’s spin-scaling relation for ɛ_c^RPA, and that the low-density (r_s→∞) limit of ɛ_c^RPA is ∼r_s^-3/4.