Electron correlations in a two-dimensional electron gas

The pair-correlation function $g\left(r\right)\mathrm{}$ and the correlation energy of a two-dimensional electron gas have been investigated in the Hartree-Fock and the Lundqvist-Overhauser theory of the random-phase approximation. At carrier densities lower than 4 × ${10}^{12}$ ${\mathrm{cm}}^{-2\mathrm{}}$ in the inversion layer of Si, $g\left(r\right)\mathrm{}$ is found to be negative for small separation between particles. For higher densities, $g\left(r\right)\mathrm{}$ remains positive for all values of $r$ although it tends to become unreasonably large for small particle separations at very high carrier densities. The consequences of these unrealistic values of $g\left(r\right)\mathrm{}$ on the various other properties calculated within these approximations are discussed.