Exchange and correlation potentials for electron-ion systems at finite temperatures

We have calculated the exchange and correlation corrections to the free energy ($F_{\mathrm{xc}}$) and the chemical potential (${\mu }_{\mathrm{xc}}$) of an electron system at arbitrary degeneracies and temperatures. These are needed for density-functional, average-atom-type calculations of properties of plasmas and liquid metals. The problem of inverting the parametric relation between ${\mu }_{\mathrm{xc}}$ and the density correction $n_{\mathrm{xc}}$ is considered. We present easy-to-use analytic fits to $F_{\mathrm{xc}}$ and ${\mu }_{\mathrm{xc}}$ as functions of the temperature and density. We also consider the effect of a lineraly responding ion background on the electron chemical potential as well as the ion chemical potential, as these are needed in calculations of ionelectron systems at arbitrary degeneracies and densities.