Pair-distribution function and its coupling-constant average for the spin-polarized electron gas

Abstract

The pair-distribution function g describes physical correlations between electrons, while its average g\ifmmode\bar\else\textasciimacron\fi{} over coupling constant generates the exchange-correlation energy. The former is found from the latter by g=(1-${\mathit{a}}_{0}$\ensuremath{\partial}/\ensuremath{\partial}${\mathit{a}}_{0}$)g\ifmmode\bar\else\textasciimacron\fi{}, where ${\mathit{a}}_{0}$ is the Bohr radius. We present an analytic representation of g\ifmmode\bar\else\textasciimacron\fi{} (and hence g) in real space for a uniform electron gas with density parameter ${\mathit{r}}_{\mathit{s}}$ and spin polarization \ensuremath{\zeta}. This expression has the following attractive features: (1) The exchange-only contribution is treated exactly, apart from oscillations we prefer to ignore. (2) The correlation contribution is correct in the high-density (${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}0) and nonoscillatory long-range (R\ensuremath{\rightarrow}\ensuremath{\infty}) limits. (3) The value and cusp are properly described in the short-range (R\ensuremath{\rightarrow}0) limit. (4) The normalization and energy integrals are respected. The result is found to agree with the pair-distribution function g from Ceperley's quantum Monte Carlo calculation. Estimates are also given for the separate contributions from parallel and antiparallel spin correlations, and for the long-range oscillations at a high finite density.