Analytic theory of ground-state properties of a three-dimensional electron gas at varying spin polarization

We present an analytic theory of the spin-resolved pair distribution functions $g_{\sigma {\sigma }^{\mathrm{\prime }}}(r)$ and the ground-state energy of an electron gas with an arbitrary degree of spin polarization. We first use the Hohenberg-Kohn variational principle and the von Weizs"{a}cker-Herring ideal kinetic energy functional to derive a zero-energy scattering Schr"{o}dinger equation for $\sqrt{g_{\sigma {\sigma }^{\mathrm{\prime }}}(r)}$. The solution of this equation is implemented within a Fermi-hypernetted-chain approximation which embodies the Hartree-Fock limit and is shown to satisfy an important set of sum rules. We present numerical results for the ground-state energy at selected values of the spin polarization and for $g_{\sigma {\sigma }^{\mathrm{\prime }}}(r)$ in both a paramagnetic and a fully spin-polarized electron gas, in comparison with the available data from Quantum Monte Carlo studies over a wide range of electron density.