Anharmonic Contribution to the Energy of a Dilute Electron Gas—Interpolation for the Correlation Energy

The first anharmonic contribution to the ground-state energy of a body-centered cubic lattice of electrons, oscillating in a uniform background positive charge, has been calculated. The result is -0.73rₛ^-2 rydbergs, with rₛ the radius, in Bohr units, of the sphere equivalent in volume to that occupied per electron. Combining this term with previous results gives for the ground-state energy of a dilute electron gas the expression E=Eₑₓₚ-1.792rₛ^-1+2.65rₛ^-3/2-0.73rₛ^-2+O(rₛ^-5/2), where Eₑₓₚ comes from the overlapping of electronic wave functions and falls off exponentially with rₛ1/2; while the rₛ^-1 and rₛ^-3/2 terms are, respectively, the Coulomb energy of a bcc lattice and the zero-point energy of the electrons.The "correlation" energy corresponding to the above expression, as well as the kinetic and potential parts, has been plotted and an interpolation has been made between the low-density curve and the high-density expression of Gell-Mann and Brueckner. The interpolated curves give strong evidence that the next term in the above low-density expansion for E is approximately -0.8rₛ^-5/2. If the high-density expression is rapidly converging near rₛ=1, it also is predicted that the rₛ term in the high-density expansion will be approximately -0.02rₛ.