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 $\gxcav(r_s,ζ, k_Fu)$ of a uniform electron gas with density parameter $r_s =(9π/4)^{1/3}/k_F$ and relative spin polarization $ζ$ over the whole range $0<r_s<\infty$ and $-1<ζ<1$, with energetically-unimportant long range ($u\to \infty$) oscillations averaged out. The pair distribution function $g_{xc}$ at the physical coupling constant is then given by differentiation with respect to $r_s$. Our formulas are constructed using {\em only} known theoretical constraints plus the correlation energy $\ec(r_s,ζ)$, and accurately reproduce the $g_{xc}$ of the Quantum Monte Carlo method and of the fluctuation-dissipation theorem with the Richardson-Ashcroft dynamical local-field factor. Our $g_{xc}$ seems to be correct even in the high-density ($r_s\to 0$) and low-density ($r_s \to \infty$) limits. When the spin resolution of $\ec$ into $\uu$, $\dd$, and $\ud$ contributions is known, as it is in the high- and low-density limits, our formulas also yield the spin resolution of $g_{xc}$. We also analyze the kinetic energy of correlation into contributions from density fluctuations of various wavevectors.