Monte Carlo Approach to the Correlation Energy of the Electron Gas

Abstract

The correlation energy of the electron gas is studied for a characteristic range of metallic densities ($2<{\mathcal{r}}_{s}<6$) using a variational wave function of the form $\ensuremath{\Psi}=D\ensuremath{\Pi}{i<j}^{}[f({\mathcal{r}}_{\mathrm{ij}})]$, where $D$ is a determinant of plane waves and spin functions and $f({\mathcal{r}}_{\mathrm{ij}})$ is a correlation function of the form used by Becker et al. The problem is divided into two parts with the help of the method developed by Wu and Feenberg to take into account the antisymmetry of the wave function. First, the ground-state energy of the gas of charged bosons described by states of the form $\ensuremath{\varphi}=\ensuremath{\Pi}{i<j}^{}[f({\mathcal{r}}_{\mathrm{ij}})]$ is calculated by a Monte Carlo method for $0.5<{\mathcal{r}}_{s}<20$. The long-range correlations are taken into account by means of a lattice-summation technique already used by Brush et al. in their treatment of a classical one-component plasma. The resulting ground-state energies lie below the best existing results, and it is conjectured that the present values are an upper bound to the exact ground-state energy of a system of charged bosons. Assuming a zero-spin ground state for the metallic densities, the Wu-Feenberg expansion is then applied up to second order, allowing for two- and three-particle exchanges in the wave function. The resulting correlation energies are situated between the values obtained from the Nozi\`eres-Pines interpolation formula and those given by Singwi et al. in their dielectric approach. The corresponding pair-distribution functions are positive at all separations but do not satisfy the usual normalization condition. Higher-order exchanges in the wave function seem to be important and would tend to lower the correlation energy.