Short-range correlation in the uniform electron gas: Extended Overhauser model

We use the two-electron wave functions (geminals) and the simple screened Coulomb potential proposed by Overhauser [Can. J. Phys. 73, 683 (1995)] to compute the pair-distribution function $g\left(r\right)\mathrm{}$ for a uniform electron gas, finding the exact $g\left(0\right)\mathrm{}$ for this model and extending the results from $g\left(0\right)\mathrm{}$ to $g\left(r\right).$ We find that the short-range $(r<\mathrm{}{r}_s)$ part of this $g\left(r\right)\mathrm{}$ is in excellent agreement with quantum Monte Carlo simulations for a wide range of electron densities. We are thus able to estimate the value of the second-order ${(r}^2)$ coefficient of the small interelectronic-distance expansion of the pair-distribution function. The coefficients of the small-$r$ expansion of the spin-resolved $g_{\sigma {\sigma }^{\prime }}\mathrm{}\left(r\right)\mathrm{}$ have density or $r_s$ dependencies which we parametrize in a way that makes it easy to find their coupling-constant averages. Their spin-polarization or $\zeta$ dependencies are estimated from a proposed spin-scaling relation.