of the nucleus. This allows one to determine the effective charge from the vanishing first-order correction to the energy of the system analytically. Then observable characteristics are expressed through the expectation values, computed with the hydrogen-like wave functions of this charge.

We point out here that perturbation theory in $Z^{-1}$ was developed in many works (see [@vainshtein_energy_1985; @fischer1977hartree] and citations therein), however, exactly the introduction of the effective charge $Z^{*}$, instead of the usage of the nuclear charge $Z$, significantly increases the accuracy of the leading zeroth-order approximation, while keeping the complexity of all calculations low as it contains the first-order correction implicitly.

As a result, it was demonstrated that the analytical leading-order approximation describes the whole spectrum of a nonrelativistic multi-electron atom or ion and the associated perturbation theory series was constructed, which converges fast with the rate of $\sim 1/10$. The accuracy of these results does not depend on the number of electrons in an atom, i.e., the effective charge model is uniformly available for all states, both ground and excited ones, of all atoms or ions. Moreover, the results via second-order perturbation theory are comparable with those via multi-configuration HF (MCHF) [@fischer1977hartree].

We emphasize that the effective 