Coulomb and spin-orbit interaction matrix elements in d^2d' configuration

Abstract

The $d^2d'$ configuration is analysed in group-theoretical terms. Starting from the table given by Condon and Odabasi (1980) for the configuration $d^2d'$, we determine a set of convenient group-theoretical basis states, and rewrite the Coulomb matrix elements in terms of this new basis. Linear combinations from the different parts of the Coulomb operators are formed such that they have simple group transformation properties in our scheme. The sequence of groups that we use is $U(20) \supset SO_T(3) \times U(10) \supset SO_T(3) \times SO_S(3) \times U(5) \supset SO_T(3) \times SO_S(3) \times SO(5) \supset SO_T(3) \times SO_S(3) \times SO_L(3)$, where $T$ denotes the {\em isospin} of \v Simonis et al (1984), in which electrons with the same angular momentum $l$ but different principle quantum numbers $n$ are accommodated by introducing the eigenvalue $M_T$ of $T_0$. Using the Wigner-Eckart theorem and selection rules on the higher symmetry groups, the tables of the Coulomb and spin-orbit matrix elements for the reconstituted operators (with simple group transformation properties) are much simplified in terms of these basis states.