to the nonrelativistic case, the optimal choice of the occupation numbers is given according to the “Aufbau” or Madelung–Janet–Klechkovskii rule [@Madelung1936; @KlechkovskiiA1962justification; @doi:10.1021/ed056p714], as it provides a simple and consistent choice of occupation numbers for any number of electrons, while resulting in the lowest zeroth-order ground state energies in almost all cases. For example, from the three sets of occupation numbers $[\mathrm{Xe}]6\mathrm{s}^{1}$, $[\mathrm{Xe}]4\mathrm{f}^{1}$ and $[\mathrm{Xe}]5\mathrm{d}^{1}$ for cesium ($Z=55$), the first state possesses the lowest energy, while the remaining two are excited states. Compare the values for the energy $E_{[\mathrm{Xe}]6\mathrm{s}^{1}} = -7361.18$ a.u versus $E_{[\mathrm{Xe}]4\mathrm{f}^{1}} = -7346.19$ a.u and $E_{[\mathrm{Xe}]5\mathrm{d}^{1}} = -7354.40$ a.u, which corresponds to the “Aufbau” rule.

The accuracy of the zeroth-order calculation is presented in Fig. \[accuracy\] and numerical values of effective charges and energies are given in Appendix \[sec:values-ground-state\] in Tab. \[tab:ground\_state\_energies\]. We compare our results with solutions of DHF equations from Ref. [@DESCLAUX1973311]. As can be concluded from Fig. \[accuracy\] the effective charge description leads to a uniform approximation, i.e., the relative accuracy is independent of the number of electrons of an atom and $\sim 6.5\%$ with respect to DHF for all elements of the periodic table. This is considerably better than the TF approximation [@landau1981quantum].

