Relativistic orbit-orbit interaction between core and open-shell electrons

On the effects of the two-body non-fine-structure operators of the Breit - Pauli Hamiltonian

“…The orbit-orbit operator is a different beast and little has been done in the way of a general treatment. References to earlier work and tabulations of matrix elements for configurations with at most one open-shell have been given by Walker (1971), Saxena et al (1972) and Dankwort (1977). Saxena et al (1972) also tabulated the complete orbit-orbit energy for the ground term of all singly-charged ions, neutral atoms and negative ions from He to Kr, as well as for excited terms of the ground configuration for several atoms.…”

Energy contributions of relativistic two-body operators

“…which is valid for all N (not only for closed shells as suggested by Dankwort (1977)); the expression vanishes for s electrons. The exchange, however, involves all four terms; again, use can be made of the properties of the radial integrals to establish some cancellation between them.…”

“…Throughout this letter, we will associate the orbital angular momenta 1 and I' with the radial orbitals a and b, respectively. Radial integrals are defined in accordance with Dankwort (1977); for relations between them, we refer to this paper.…”

“…The orbit-orbit interaction H,, , the most intricate among the Breit-Pauli operators, can be split into four separate terms H,,(l) to H,,(4) (Uylings 1988). Taking the relations between the integrals into account (Dankwort 1977), it can be seen that the terms Hoe( l ) , HO0(2) and HO0(3) cancel for direct matrix elements, leaving the following relatively simple expression valid for both 11' and 1': There is therefore no contribution to the average energy except when a = b ; in the latter case one can find the expression for equivalent electrons by replacing ( -l)L{ f I :…”

“…The exchange, however, involves all four terms; again, use can be made of the properties of the radial integrals to establish some cancellation between them. The final result can be written: I: i } by -2NN'/(41+2)(41'+2), yielding the expression of Dankwort (1977), which is therefore not repeated.…”