model of a multi-electron atom'
---

Introduction {#sec:introduction}
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Despite great advances in the accuracy and efficiency of modern numerical methods [@filippin_multiconfiguration_2016; @puchalski_relativistic_2017; @peach_fractional_2015; @jensen2007introduction], simple analytical models for describing multi-electron atoms and ions are still actively developed [@gomez_simple_2019; @tatewaki_relativistic_2018; @karpov_atomic-number_2017; @hey_forms_2017]. The most popular of them include the multi-parametric screening hydrogen orbitals [@hau2012high; @seaton_quantum_1983], Slater orbitals [@slater_atomic_1930] and the semi-classical Thomas-Fermi (TF) model [@thomas_1927; @landau1981quantum]. These models are widely used, whenever certain level of precision has to be sacrificed for improved computation time, e.g. in computational plasma physics [@ciricosta_direct_2012; @starrett_thomas-fermi_2017; @lee_model_1987; @scott_cretinradiative_2001; @chung_flychk:_2005; @dyachkov_region_2016], semiconductors [@schulze-halberg_position-dependent_2013; @smith_electronic_2014], screening effects in the cross sections of bremsstrahlung and pair production [@tsai_pair_1974; @olsen_photon_1959; @davies_theory_1954; @seltzer_bremsstrahlung_1985; @poskus_brems:_2018], X-ray scattering and diffraction [@hau2012high; @feranchuk_new_2002] and crystallography [@toraya_new_2016].

In addition, the wave functions obtained from these simple analytical models are used as the initial approximation for the solution of Hartree-Fock (HF) [@FockA1930naeherungs; @fischer1977hartree; @grant2007relativistic; @jonsson_new_2013] and post HF equations [@bostock_fully_2011], convergence of which is strongly dependent on the choice of the trial functions. Therefore, improving the choice of the initial wave functions can significantly reduce the number of iterations required for obtaining the desired solution or enable the convergence at all.

At the same time 