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.
Bethe, H. A. and Salpeter, E. E. 1957. Quantum Mechanics of One- and Two-electron Atoms. Springer.
http://dx.doi.org/10.1007/978-3-662-12869-5
Froese Fischer, Ch., Brage, T., and Jönsson, P. 2000. Computational Atomic Structure. Inst. Phys. Publ., Bristol and Phil.
Golovatyj, V. V., Sapar, A. A., Feklistova, T. H., and Kholtygin, A. F. 1991. Atomic Data for Spectroscopy of Rarefied Astrophysical Plasma. Gaseuos Nebulae. Valgus, Tallinn (in Russian).
Golovatyj, V., Sapar, A., Feklistova, T., and Kholtygin, A. 1997. Catalogue of atomic data for low-density astrophysical plasma. Astron. Astrophys. Transactions, 12, 85–241.
http://dx.doi.org/10.1080/10556799708232079
Goudsmit, S. and Bacher, R. F. 1934. Atomic energy relations I. Phys. Rev., 46, 948–969.
http://dx.doi.org/10.1103/PhysRev.46.948
Hartree, D. R. 1957. The Calculation of Atomic Structures. J. Wiley & Sons, New York; Chapman & Hall Press, London.
Johnson, W. R. 2007. Atomic Structure Theory. Springer.
Jucys, A. P. 1952. Fock equation in multiconfigurational approximation. Zh. eksper. teoret. fiz., (JETP) 23, 129–139 (in Russian).
Jucys, A. P. and Bandzaitis, A. A. 1977. Theory of Angular Momentum in Quantum Mechanics. Mokslas Press (in Russian).
Kupliauskis, Z., Matulaityte, A., and Jucys, A. 1971. Application of the non-orthogonal radial orbitals to the ground configuration of the magnesium-type atoms. Lietuvos fizikos rinkinys, 11, 557–563 (in Russian).
Nielson, C. W. and Koster, G. F. 1963. Spectroscopic Coefficients for pn;dn and f n Configurations. MIT Press, Cambridge MA.
Nikitin, A. A., Rudzikas, Z. B., Sapar, A. A., Feklistova, T. H., and Kholtygin, A. F. 1986. Spectra of Planetary Nebulae. Valgus, Tallinn.
Racah, G. 1942a. Theory of complex spectra I. Phys. Rev., 61, 186–197.
http://dx.doi.org/10.1103/PhysRev.61.186
Racah, G. 1942b. Theory of complex spectra II. Phys. Rev., 62, 438–462.
http://dx.doi.org/10.1103/PhysRev.62.438
Racah, G. 1943. Theory of complex spectra III. Phys. Rev., 63, 367–382.
http://dx.doi.org/10.1103/PhysRev.63.367
Racah, G. 1949. Theory of complex spectra IV. Phys. Rev., 76, 1352–1365.
http://dx.doi.org/10.1103/PhysRev.76.1352
Rudzikas, Z. 2000. Self-consistent field method in various approximations: ‘forgotten ideas’. Molecular Phys., 98, 1205–1212.
http://dx.doi.org/10.1080/00268970050080564
Sapar, A. 1973. On methods for solving Hartree–Fock equations. Tartu Astr. Obs. Publ., 40, 119–142.
Sapar, A., Poolamäe, R., and Sapar, L. 2013a. A new high-precision correction method of temperature distribution in model stellar atmospheres. Baltic Astronomy, 22, 145–159.
Sapar, A., Poolamäe, R., and Sapar, L. 2013b. New methods in modelling stellar atmospheres. Baltic Astronomy, 22, 161–179.
Sobelman, I. I. 1972. Introduction to the Theory of Atomic Spectra. Pergamon Press, 1963 (in Russian).
