We analytically evaluate the generating integralfor all values of the parameters q and q ′ , when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions.

A relativistic version of the effective charge model (ECM) for computation of observable characteristics of multi-electron atoms and ions is developed. A complete and orthogonal Dirac hydrogen basis set, depending on one parameter-effective nuclear charge Z * -identical for all single-electron wave functions of a given atom or ion, is employed for the construction of the secondary-quantized representation. The effective charge is uniquely determined by the charge of the nucleus and a set of occupied single-electron orbitals for a given state. We thoroughly study the accuracy of the leading-order approximation for the total binding energy and demonstrate that it is independent of the number of electrons of a multi-electron atom. In addition, it is shown that the fully analytical leading-order approximation is especially suited for the description of highly charged ions since our wave functions are almost coincident with the Dirac-Hartree-Fock ones for the complete spectrum. Finally, we evaluate various atomic characteristics, such as scattering factors and photoionization cross-sections, and thus envisage that the ECM can replace other models of comparable complexity, such as the Thomas-Fermi-Dirac model for all applications where it is still utilized.

A relativistic effective charge model has been developed for computation of observable characteristics of multi-electron atoms and ions. A complete and orthogonal Dirac hydrogen basis set, depending on one parameter -effective nuclear charge Z * -identical for all single-electron wave functions of a given atom or ion, is employed for the construction of the secondary-quantized representation. The effective charge is uniquely determined by the charge of the nucleus and a set of electron occupation numbers for a given state. We thoroughly study the accuracy of the leadingorder approximation for the total binding energy and demonstrate that it is independent of the number of electrons of a multi-electron atom. In addition, it is shown that the fully analytical leading-order approximation is especially suited for the description of highly charged ions since our wave functions are almost coincident with the Dirac-Hartree-Fock ones for the complete spectrum. Finally, we envision that our effective charge model is more accurate and thus can replace the Thomas-Fermi-Dirac model for all applications where it is still utilized.