Variational Calculation of the Electron-Gas Correlation Energy

Abstract

The ground-state correlation energy of the electron gas is calculated in the region of intermediate densities using the variational method of Becker, Broyles, and Dunn for two trial wave functions. Each trial function is taken to be a product of two factors, one factor being the ground-state wave function for the ideal gas of spin-1/2 particles and the other being a product of pair functions in the relative coordinates of the electrons. In one trial function a single pair function is used; in the other, the pair functions between parallel and antiparallel spins are allowed to differ. The pair functions are parametrized and approximations to the energy minimized. The three-particle correlation functions appearing in the kinetic energy are replaced by either the Kirkwood superposition approximation (KSA) or the convolution approximation (CA) to give two approximate energy functionals for each wave function. The ideal-gas $N$-particle probability density is approximated by a Boltzmann factor with an effective pair potential. This effective potential is obtained by inverting the hypernetted chain equation for the known pair-correlation function of the ideal Fermi gas. The pair-correlation functions for the interacting system are then calculated by means of the hypernetted-chain equation. The CA correlation energies join smoothly with both the high- and low-density expansions. The CA and KSA correlation energies differ by less than 4% everywhere in the intermediate-density region. The pair-correlation functions exhibit generally reasonable physical behavior.