$Z^{-1}$ expansion gives reasonable results for highly charged ions. However, it is less accurate than the effective charge model by at least $50\%$.

Conclusions and Outlook {#sec:conclusions}
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We have demonstrated that the relativistic effective charge model describes multi-electron atoms and ions and provides analytic expressions for energies, electronic densities and scattering factors with accuracy comparable to the numerical solutions of Dirac-Hartree-Fock equations. We have also shown that our approach is suitable for the description of arbitrary excited states and provides accuracy independent of the number of electrons already in the zeroth-order approximation. Furthermore, the zeroth-order approximation can be easily modified to include interactions with external fields.

[|c|c|c|c|c|c|c|]{} Configuration, $J^{\pm}$ & $Z^{*}$ & $E^{(0)}$ & $E_{\mathrm{DHF}}$ & $\Delta E/E_{\mathrm{DHF}} \cdot 100\%$ & CI\
$1s^2,J=0^+$ & 91.7130 & -9651.35 & -9651.39 & 0.0004$\%$&-9651.45\
$1s^1 2s^1,J=1^+$ & 91.8623 &-6096.96 & -6097.01 & 0.0008$\%$&-6097.01\
$1s^1 2s^1,J=0^+$ & 91.8404 &-6093.51 & -6093.41 & -0.0016$\%$&-6097.01\
$1s^1 2p^1_{1/2},J=0^-$ & 91.8148 & -6089.50 & -6090.16 &0.0108$\%$ & -6090.17\
$1s^1 2p^1_{1/2},J=1^-$ & 91.8113 & -6088.95 & -6089.10 & 0.0025$\%$ & -6089.11\
$1s^1 2p^1_{3/2},J=2^-$ & 91.8487 & -5928.36 & -5928.44 & 0.0013$\%$ & -5928.44\
$1s^1 2p^1_{3/2},J=1^-$ & 91.8395 & -5927.00 & -5926.68 &-0.0054$\%$ & -5928.44\
$1s^1 3s^1,J=1^+$ & 91.9086 