New identities for matrix elements of spin-dependent interactions in lNl' configurations.

Abstract

Formulas for the matrix elements of the spin-dependent interactions, such as spin-spin, spin-other-orbit, and effective electrostatic spin-orbit, were constructed for ${l}^{N}$l' configurations, including the two special cases N=1 and 4l+1. Since these interactions are described by two-electron operators, they comprise, in analogy with the electrostatic interaction, both direct and exchange parts. For each of these interactions, 12 angular momenta participate in the orbital part, as well as in the spin part, of the corresponding matrix element. In the formula representing the direct part, both orbital and spin angular momenta are connected according to the identity of Arima, Horie, and Tanabe; in this identity, a double sum of three 6j symbols and one 9j symbol is expressed as a simple product of one 6j symbol and one 9j symbol. In the exchange part, all orbital (spin) angular momenta are grouped in one 12j symbol of the first kind, according to a newly discovered identity; in this identity, the 12j symbol is expressed as a double sum of three 6j symbols and one 9j symbol. All the above-mentioned results were also reproduced by using graphical methods. It can thus be concluded that the identity of Arima, Horie, and Tanabe and the new identity, respectively, represent the symmetry properties of the direct and exchange parts of the two-electron spin-dependent interactions. For the purpose of obtaining simple and closed formulas in the special case N=4l+1, namely, for configurations comprising a hole and an electron, additional new identities were constructed.