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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.