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

Abstract

For a uniform electron gas of density n=${\mathit{n}}_{\mathrm{\uparrow }}$+${\mathit{n}}_{\mathrm{\downarrow }}$=3/4π${\mathit{r}}_{\mathit{s}}^3$=π${\mathit{k}}_{\mathit{s}}^6$/192 and spin polarization ζ=(${\mathit{n}}_{\mathrm{\uparrow }}$-${\mathit{n}}_{\mathrm{\downarrow }}$)/n, we study the Fourier transform ρ${\mathrm{¯}}_{\mathit{c}}$(k,${\mathit{r}}_{\mathit{s}}$,ζ) of the correlation hole, as well as the correlation energy ${\mathrm{\varepsilon }}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$,ζ)=${\mathcal{F}}_0^{\mathrm{\infty }}$dk ρ${\mathrm{¯}}_{\mathit{c}}$/π. In the high-density (${\mathit{r}}_{\mathit{s}}$→0) limit, we find a simple scaling relation ${\mathit{k}}_{\mathit{s}}$ρ${\mathrm{¯}}_{\mathit{c}}$/π${\mathit{g}}^2$→f(z,ζ), where z=k/${\mathit{gk}}_{\mathit{s}}$, 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/${\mathrm{\pi }}^2$+4 √3 ${\mathrm{z}}^2$/${\mathrm{\pi }}^2$+... is also the exact small-wave-vector (k→0) expansion for any ${\mathit{r}}_{\mathit{s}}$ 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${\mathrm{¯}}_{\mathit{c}}$ at any k, ${\mathit{r}}_{\mathit{s}}$, or ζ, one within and one beyond the random-phase approximation (RPA). We also show that ρ¯ ${}_{\mathit{c}}{}^{\mathrm{RPA}}{}_{}{}^{}$ obeys a generalization of Misawa’s spin-scaling relation for ${\mathrm{\varepsilon }}_{\mathit{c}}^{\mathrm{RPA}}$, and that the low-density (${\mathit{r}}_{\mathit{s}}$→∞) limit of ${\mathrm{\varepsilon }}_{\mathit{c}}^{\mathrm{RPA}}$ is ∼${\mathit{r}}_{\mathit{s}}^{\mathrm{-}3/4}$.