example we evaluated energies of ground and excited states of He-, Li- and B-like uranium ($Z = 92$) ions.

We want to stress here again, that the effective charge $Z^{*}$ is identical for all single particle states from a set that specifies the state. However, two different states will have different charges. For example, the He-like uranium ground state $1s_{\uparrow}1s_{\downarrow}$ and the He-like uranium excited state $1s_{\uparrow}2s_{\downarrow}$ possess unequal effective charges.

The results of the calculation of electronic densities are presented in Fig. \[fig:HCI\]. The electronic densities, despite being analytical expressions in the zeroth-order approximation, coincide remarkably well with the ones obtained from numerical solutions of the DHF equations.

Furthermore, the energy eigenvalues are presented in Tab. \[tab:Ions\]. It is clear that the accuracy is significantly better (below $0.02\%$) than for neutral atoms and in fact sufficient to obtain correct ordering, even for very closely spaced excited states. It is worth pointing out that the accuracy can be further improved by forming linear combinations of all sub-configurations arising from the single $JLS$ configuration and subsequently diagonalizing the Hamiltonian exactly in such finite basis — a technique that has been successfully employed in the nonrelativistic case [@skoromnik_analytic_2017].

Finally, we note that 