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Abstract

The recently proposed effective-charge model (ECM) is a novel approach to producing fully
analytical approximations to the observable characteristics of multi-electron atoms and ions.
It employs a hydrogen-like basis set with a single parameter, called effective charge, for
the construction of perturbation theory. The associated perturbation series converges fast,
includes correlation effects in a natural way and enables an efficient calculation of all subsequent
corrections. This work compares the accuracy of the analytical approximations
produced by the ECM to results of other commonly used methods, such as the Hartree-
Fock method and the Thomas-Fermi model, within both relativistic and non-relativistic
quantum mechanics. For this purpose, ground state, excited state and ionization energies
are evaluated, as well as a wide range of other atomic characteristics, such as electronic
densities, scattering factors, photoionization cross-sections and transition probabilities, for
a wide range of systems, from neutral atoms to highly charged ions. It is also shown how
the Green’s function of the hydrogen atom can be integrated analytically, allowing for an
efficient calculation of the second-order ECM corrections. Finally, various additional effects
that correct the accuracy of the ECM approximations are investigated, in particular those
originating from the Breit interaction, finite-nuclear-size effects and vacuum polarization.
Given that the accuracy of the second-order ECM approximations is already comparable
with results obtained using a multi-configuration Hartree-Fock approach, we envisage that
the ECM can replace the Thomas-Fermi model for all applications where it is still utilized.