Density-functional theory of the correlation energy in atoms and ions: A simple analytic model and a challenge

A simple and accurate analytic model is derived for the correlation energy of any atom or ion within three densityfunctional approximations based on the uniform electron gas: the local-spin-density approximation (LSD), the selfinteraction corrected (SIC) version of LSD, and the antiparallel-spin LSD of Stoll et al. The last two approximations give good results for the correlation energies of neutral atoms, in contrast to LSD which overestimates these energies by a factor of 2. However, all three approximations show an incorrect $\mathrm{ln}Z$ leading behavior when the nuclear charge $Z$ tends to infinity at fixed electron number $N$. It is hard to see how any a priori electron-gas approximation can reproduce the exact leading behavior, which is constant or linear in $Z$ according to the value of $N$.