Variational calculation for the electron gas at intermediate densities

An approximate variational calculation for the correlation energy of an electron gas is carried out using a Jastrow wave function and a linked-cluster expansion previously derived. The terms included in the variational calculation are the lowest-order direct terms in the potential and kinetic energies, and corresponding ring diagrams that incorporate the long-range screening effect, the lowest-order exchange kinetic energy, the lowest-order exchange potential energy and associated ring diagrams, and second-order ladder diagrams for the potential energy. The second-order energy terms are small compared with the first-order terms, suggesting that the expansion is valid. The pair correlation functions between electrons of parallel and antiparallel spin are also calculated and are non-negative for all $r$. The pair correlation function for electrons of parallel spins satisfies the condition $g\left(0\right)=0\mathrm{}$. The calculation was performed for different correlating factors between electrons of parallel and antiparallel spin, but this had a negligible effect on the correlation energy.