Correlation energy of 2-D electrons

The correlation energy and the Fermi momentum of an electron gas in 2-D are evaluated explicitly as functions of density. The ring diagram and first- and second-order exchange contributions are treated. In comparison with the 3-D case, the kinetic energy for the same r_s is approximately one-half and the exchange and correlation energies are somewhat larger. The ground state energy plotted against r_s shows a minimum at around r_s = 1.65 with a minimum value of ?0.9858 Ryd. If the third-order ring contribution is added, the curve is shifted upward. The correlation energy is ?0.6258 to order e⁴. The third-order ringw contribution increases this value almost linearly with r_s. The Fermi momentum decreases with r_s due to the contribution. Different from the 3-D case, no ln r_s term appears in the correlation energy within the approximation.