  Density functional theory (DFT) is an exact alternative formulation of
quantum mechanics, in which it is possible to calculate the total energy, the
spin and the charge density of many-electron systems in the ground state. In
practice, it is necessary to use uncontrolled approximations that can mainly be
verified against experimental data. Atoms and ions are simple systems, where
the approximations of DFT can be easily tested. We have calculated within DFT
the total energies, spin and higher ionization energies of all the ions of
elements with 1 $\leq$ Z $\leq$ 29. We find the calculations in close agreement
with experiment, with an error of typically less than ca. 1% for 1 $\leq$ Z
$\leq$ 29. Surprisingly, the error depends on the electronic configuration of
the ion in both local spin density approximation (LSDA) and
Perdew-Burke-Ernzerhof general gradient approximation (PBE-GGA) and independent
of both self-interaction correction (SIC) and relativistic corrections. Larger
errors are found for systems in which the spin-spin correlation is significant,
which indicates the possible benefit from an orbital-dependent formulation of
the correlation energy functional.
