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

The pair-distribution function g describes physical correlations between electrons, while its average g¯ over coupling constant generates the exchange-correlation energy. The former is found from the latter by g=(1-${\mathit{a}}_0$∂/∂${\mathit{a}}_0$)g¯, where ${\mathit{a}}_0$ is the Bohr radius. We present an analytic representation of g¯ (and hence g) in real space for a uniform electron gas with density parameter ${\mathit{r}}_{\mathit{s}}$ and spin polarization ζ. 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 (${\mathit{r}}_{\mathit{s}}$→0) and nonoscillatory long-range (R→∞) limits. (3) The value and cusp are properly described in the short-range (R→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.