Energy, Specific Heat, and Magnetic Properties of the Low-Density Electron Gas

A perturbation expansion in powers of ${r_s}^{-\frac{1}{2}}$ has been used to investigate the ground-state energy of a dilute electron gas, the result being, in rydberg units per particle, $E=-\frac{1.792}{r_s}+\frac{2.66}{{r_s}^{\frac{3}{2}}}+\frac{b}{{r_s}^2}+O\left(\frac{1}{{r_s}^{\frac{5}{2}}}\right)+$terms falling off exponentially with ${r_s}^{\frac{1}{2}}$. The dimensionless parameter $r_s$ is the radius of the unit sphere in Bohr radii. The term in ${r_s}^{-1\mathrm{}}$ is the energy of a body-centered cubic lattice of electrons as calculated by Fuchs; the ${r_s}^{-\frac{3}{2}}$ term is the zero-point vibrational energy of the lattice, as obtained from a calculation of the normal modes, the result differing only by a small amount from the values estimated by Wigner; and $b{r_s}^{-2\mathrm{}}$ is the first-order effect of anharmonicities in the vibration. The constant $b$ has been estimated, its magnitude being smaller than unity.