Exchange and correlation potential for a two-dimensional electron gas at finite temperatures

Calculations for an inhomogeneous electron gas using density-functional theory require suitable exchange and correlation potentials. We present here the local exchange and correlation potential ($V_{\mathrm{xc}}$) for the two-dimensional electron gas over a wide range of temperatures and densities based upon summation of ring graphs. A suitable parametric fit for $V_{\mathrm{xc}}$ is presented which is good to within 2% of the exact values. Various limits and graphs of the polarizability are presented.