  In this article, we consider the extended Kohn-Sham model for atoms subjected
to cylindrically-symmetric external potentials. The variational approximation
of the model and the construction of appropriate discretization spaces are
detailed together with the algorithm to solve the discretized Kohn-Sham
equations used in our code. Using this code, we compute the occupied and
unoccupied energy levels of all the atoms of the first four rows of the
periodic table for the reduced Hartree-Fock (rHF) and the extended Kohn-Sham
X$\alpha$ models. These results allow us to test numerically the assumptions on
the negative spectra of atomic rHF and Kohn-Sham Hamiltonians used in our
previous theoretical works on density functional perturbation theory and
pseudopotentials. Interestingly, we observe accidental degeneracies between s
and d shells or between p and d shells at the Fermi level of some atoms. We
also consider the case of an atom subjected to a uniform electric-field. For
various magnitudes of the electric field, we compute the response of the
density of the carbon atom confined in a large ball with Dirichlet boundary
conditions, and we check that, in the limit of small electric fields, the
results agree with the ones obtained with first-order density functional
perturbation theory.
