a lot of effort has gone into improving the accuracy of these models, while keeping their complexity low. For example, the inclusion of relativistic corrections [@dolg_relativistic_2012; @nakajima_douglaskrollhess_2012; @marini_relativistic_1981; @waber_relativistic_1975] and inhomogeneity corrections [@chau_systematic_2018] into the TF model was performed. However, solving the TF equation to produce numerically stable results of electronic densities is in general nontrivial problem and is an active area of research [@akgul_constructing_2017; @parand_accurate_2017; @liu_laguerre_2015].

Consequently, finding a set of analytical wave functions, which provide better accuracy than the models described above would be advantageous. As was recently demonstrated for nonrelativistic atoms or ions, this can be achieved within the effective charge model [@skoromnik_analytic_2017], where the hydrogen-like wave functions can be used to analytically describe observable characteristics with high accuracy. In this approach, one specifies a complete and orthonormal hydrogen-like basis set with a single parameter — effective nuclear charge $Z^{*}$ — identical for all single-electron wave functions of a given atom. The basis completeness allows one to perform a transition into the secondary quantized representation, which is especially suited for the description of many-body problems.

Then in order to compute observable characteristics one specifies the set of occupation numbers for the state in question and the charge 