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=${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$+${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$=3/4\ensuremath{\pi}${\mathit{r}}_{\mathit{s}}^{3}$=\ensuremath{\pi}${\mathit{k}}_{\mathit{s}}^{6}$/192 and spin polarization \ensuremath{\zeta}=(${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$-${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$)/n, we study the Fourier transform \ensuremath{\rho}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{c}}$(k,${\mathit{r}}_{\mathit{s}}$,\ensuremath{\zeta}) of the correlation hole, as well as the correlation energy ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$,\ensuremath{\zeta})=${\mathcal{F}}_{0}^{\mathrm{\ensuremath{\infty}}}$dk \ensuremath{\rho}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{c}}$/\ensuremath{\pi}. In the high-density (${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}0) limit, we find a simple scaling relation ${\mathit{k}}_{\mathit{s}}$\ensuremath{\rho}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{c}}$/\ensuremath{\pi}${\mathit{g}}^{2}$\ensuremath{\rightarrow}f(z,\ensuremath{\zeta}), where z=k/${\mathit{gk}}_{\mathit{s}}$, g=[(1+\ensuremath{\zeta}${)}^{2/3}$+(1-\ensuremath{\zeta}${)}^{2/3}$]/2, and f(z,1)=f(z,0). The function f(z,\ensuremath{\zeta}) is only weakly \ensuremath{\zeta} dependent, and its small-z expansion -3z/${\mathrm{\ensuremath{\pi}}}^{2}$+4 \ensuremath{\surd}3 ${\mathrm{z}}^{2}$/${\mathrm{\ensuremath{\pi}}}^{2}$+... is also the exact small-wave-vector (k\ensuremath{\rightarrow}0) expansion for any ${\mathit{r}}_{\mathit{s}}$ or \ensuremath{\zeta}. Motivated by these considerations, and by a discussion of the large-wave-vector and low-density limits, we present two Pad\'e representations for \ensuremath{\rho}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{c}}$ at any k, ${\mathit{r}}_{\mathit{s}}$, or \ensuremath{\zeta}, one within and one beyond the random-phase approximation (RPA). We also show that \ensuremath{\rho}\ifmmode\bar\else\textasciimacron\fi{} $_{\mathit{c}}^{\mathrm{RPA}}$ obeys a generalization of Misawa's spin-scaling relation for ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}^{\mathrm{RPA}}$, and that the low-density (${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}\ensuremath{\infty}) limit of ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}^{\mathrm{RPA}}$ is \ensuremath{\sim}${\mathit{r}}_{\mathit{s}}^{\mathrm{\ensuremath{-}}3/4}$.