Orthogonal scalar operators for p^Nd and pd^N

The concept of operator orthogonality is examined and discussed in the context of mixed configurations of p and d electrons. Operators are constructed in the scalar form P^{( kappa k)}.D^{( kappa k)}, where P^{( kappa k)} denotes a collection of annihilation and creation operators for the p electrons whose resultant spin and orbital ranks are kappa and k, and where D^{( kappa k)} is similarly defined for d electrons. In addition to kappa and k, the irreducible representations of the symplectic groups Sp(6) and Sp(10), as well as those of the special orthogonal group SO(5), are introduced to distinguish the various three-electron operators t_i that can arise when configuration interaction is taken to second order in perturbation theory. Two methods are described for evaluating the matrix elements of the t_i. Tables of numerical results are given for pd, p²d, pd², and p³d. Quasispin ranks K_p and K_d are assigned to the operators so that the conjugate configurations (such as p⁴d, p²d⁹, and p⁴d⁹, the conjugates of p²d) can be treated.