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Generalized analytical orthonormal quasi-hydrogenic radial wave functions with free parameters (radius-dependent screened nuclear charge) are introduced. By them the analytical expressions for Slater radial integrals are derived. Using the Racah coefficients of partial parentage, the sequential formulae for branching fractions of equivalent electrons, incorporating LScoupling of angular momenta inside terms of unfilled subshells, are proposed. Racah partial parentage or branching coefficients for the unfilled electron subshells are implicitly generated via the Pauli exclusion principle. The free parameters of the radial Slater integrals in Hamiltonian are proposed to be optimized by the Levenberg–Marquardt best-fit optimization version of the least squares method. To its cost function the virial ratio of kinetic and potential energy is added as a Lagrange constraint term. Thus, the solution of Hartree–Fock eigenvalue equations is proposed to be replaced by a nonlinear optimization method. The integrals in Hamiltonian correspond to the kinetic energy of electrons, their interaction with the atomic nucleus and electrostatic interaction between electrons. This interaction includes the Coulomb and exchange interaction between equivalent and non-equivalent electron pairs, including the multi-configurational interaction contribution. A general compact formula for 6 j-symbols is presented and used to describe the interaction of single excited electrons with electrons in filled or unfilled atomic subshells. It is proposed to realize generalization of the single configuration approach to a multi-configuration mixing in the n-dimensional Euclidean space, where the mixing coefficients are the polar direction cosines.

Analytical formulae for the energy of electron subshells in atoms and their optimization