representation. We characterize states via electron occupation numbers and determine the value of the effective charge, by requiring the first-order perturbative correction to the energy of the system to vanish. With this we obtain a description of energy spectra, both ground and excited states together with wave functions. We have shown that the analytical description within the effective charge model agrees with the high accuracy with the numerical solution of Dirac-Hartree-Fock (DHF) equations for all ions and atoms of the periodic table.

The article is organized in the following way: in Sec. \[sec:relat-effect-charge\] we introduce the effective charge description and construct a perturbation theory series. In Sec. \[sec:zeroth-order-appr\] we analyze the accuracy of the zeroth- and first-order approximations. Consequently, we start from computing numerical values of effective charges by solving Eq. (\[eq:13\]). After this in Sec. \[sec:ground-state-energ\] we determine the ground state energies for 118 atoms of the periodic table and the electronic densities for some selected atoms. In Sec. \[sec:atom-scatt-fact\] we analyze atomic scattering factors. In Sec. \[sec:highly-charged-ions\] we demonstrate that the leading-order approximation is especially suited for the description of highly charged ions, where our analytical functions are almost coincident with numerical DHF wave functions. In Sec. \[sec:conclusions\] 