Density-gradient analysis for density functional theory: Application to atoms

We present an analysis of local or semilocal density functionals for the exchange-correlation energy by decomposing them into their gradients rs (local Seitz radius), ζ (relative spin polarization), and s (reduced density gradient). We explain the numerical method pertaining to this kind of analysis and present results for a few atoms and ions. The atomic shell structure is prominent, and only the ranges 0 < rs < 10 and 0 < s < 3 are important. The low-density and large-gradient domains, where the approximations for the exchange-correlation energy are least trustworthy, have very little weight.