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 $-0.73\mathrm{}{r_s}^{-2\mathrm{}}$ rydbergs, with $r_s$ 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 $E=E_{\exp }-1.792\mathrm{}{r_s}^{-1\mathrm{}}+2.65\mathrm{}{r_s}^{-\frac{3}{2}}-0.73\mathrm{}{r_s}^{-2\mathrm{}}+O\left({r_s}^{-\frac{5}{2}}\right)$, where $E_{\exp }$ comes from the overlapping of electronic wave functions and falls off exponentially with ${r_s}^{\frac{1}{2}}$; while the ${r_s}^{-1\mathrm{}}$ and ${r_s}^{-\frac{3}{2}}$ terms are, respectively, the Coulomb energy of a bcc lattice and the zero-point energy of the electrons.