Pair distribution function of the spin-polarized electron gas: A first-principles analytic model for all uniform densities

We construct analytic formulas that represent the coupling-constant-averaged pair distribution function ${g}_{\mathrm{xc}}{(r}_{s},\ensuremath{\zeta}{,k}_{F}u)$ of a three-dimensional nonrelativistic ground-state electron gas constrained to a uniform density with density parameter ${r}_{s}=(9\ensuremath{\pi}{/4)}^{1/3}{/k}_{F}$ and relative spin polarization $\ensuremath{\zeta}$ over the whole range $0<{r}_{s}<\ensuremath{\infty}$ and $\ensuremath{-}1<\ensuremath{\zeta}<1,$ with energetically unimportant long range $(\stackrel{\ensuremath{\rightarrow}}{u}\ensuremath{\infty})$ oscillations averaged out. The pair distribution function ${g}_{\mathrm{xc}}$ at the physical coupling constant is then given by differentiation with respect to ${r}_{s}.$ Our formulas are constructed using only known theoretical constraints plus the correlation energy ${\ensuremath{\epsilon}}_{c}{(r}_{s},\ensuremath{\zeta}),$ and accurately reproduce the ${g}_{\mathrm{xc}}$ of the quantum Monte Carlo method and of the fluctuation-dissipation theorem with the Richardson-Ashcroft dynamical local-field factor. Our ${g}_{\mathrm{xc}}$ is correct even in the high-density ${(r}_{s}\ensuremath{\rightarrow}0)$ and low-density ${(r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})$ limits. When the spin resolution of ${\ensuremath{\epsilon}}_{c}$ into $\ensuremath{\uparrow}\ensuremath{\uparrow},$ $\ensuremath{\downarrow}\ensuremath{\downarrow},$ and $\ensuremath{\uparrow}\ensuremath{\downarrow}$ contributions is known, as it is in the high- and low-density limits, our formulas also yield the spin resolution of ${g}_{\mathrm{xc}}.$ Because of these features, our formulas may be useful for the construction of density functionals for nonuniform systems. We also analyze the kinetic energy of correlation into contributions from density fluctuations of various wave vectors. The exchange and long-range correlation parts of our ${g}_{\mathrm{xc}}{(r}_{s},\ensuremath{\zeta}{,k}_{F}u)\ensuremath{-}1$ are analytically Fourier transformable, so that the static structure factor ${S}_{\mathrm{xc}}{(r}_{s},\ensuremath{\zeta}{,k/k}_{F})$ is easily evaluated.