Spin scaling of the electron-gas correlation energy in the high-density limit

Abstract

The ground-state correlation energy per particle in a uniform electron gas with spin densities ${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$ and ${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$ may be expressed as ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(\ensuremath{\zeta},${\mathit{r}}_{\mathit{s}}$)=I(\ensuremath{\zeta},${\mathit{r}}_{\mathit{s}}$)${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(0,${\mathit{r}}_{\mathit{s}}$), where ${\mathit{r}}_{\mathit{s}}$=[3/4\ensuremath{\pi}(${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$+${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$)${]}^{1/3}$ is the density parameter and \ensuremath{\zeta}=(${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$-${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$)/(${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$+${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$) is the relative spin polarization. We find an analytic expression for the spin-scaling factor (SSF) I(\ensuremath{\zeta},${\mathit{r}}_{\mathit{s}}$) in the high-density limit ${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}0. It decreases from the value 1 at \ensuremath{\zeta}=0, approaching the value 1/2 with slope -\ensuremath{\infty} as \ensuremath{\zeta} approaches 1. A simple approximation to this SSF which displays the correct qualitative behavior is ${\mathit{g}}^{3}$(\ensuremath{\zeta}), where g(\ensuremath{\zeta})=[(1+\ensuremath{\zeta}${)}^{2/3}$+(1-\ensuremath{\zeta}${)}^{2/3}$]/2. We find that g(\ensuremath{\zeta}) is the SSF for the coefficient of the \ensuremath{\Vert}\ensuremath{\nabla}n${\mathrm{\ensuremath{\Vert}}}^{2}$/${\mathit{n}}^{4/3}$ term of the spin-density gradient expansion of the exchange energy, and a good approximation to the SSF for that of correlation: ${\mathit{scrC}}_{\mathit{x}}$(\ensuremath{\zeta})/${\mathit{scrC}}_{\mathit{x}}$(0)=g(\ensuremath{\zeta}) and ${\mathit{scrC}}_{\mathit{c}}$(\ensuremath{\zeta},${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}0)/${\mathit{scrC}}_{\mathit{c}}$(0, ${\mathrm{r}}_{\mathrm{s}}$\ensuremath{\rightarrow}0)\ensuremath{\approxeq}g(\ensuremath{\zeta}). We also find that the \ensuremath{\Vert}\ensuremath{\nabla}\ensuremath{\zeta}${\mathrm{\ensuremath{\Vert}}}^{2}$ contribution to the correlation energy is always negligible.