A Graphical Method for Calculating First Order Transition Matrices for Open-Shell Atoms in the Random Phase Approximation

A graphical procedure is presented for calculating first order transition matrices for a general (open-shell) atom. The first order transition matrix may be used to calculate matrix elements of a general one-body operator of rank λ in orbital space and σ in spin space. In the random phase approximation we obtain a set of N + N' coupled differential equations for N final state radial functions and N' initial state radial functions which completely determine the first order transition matrix for an atomic system having N final state channels. (The relation of N' to N is dependent on the atomic system studied.) These N + N' differential equations reduce to familiar forms in the following cases: (1) When initial state correlations are ignored, we obtain the N coupled differential equations of the Close-Coupling Approximation; (2) When the atom has only closed subshells we obtain N' = N and the 2N coupled differential equations are those obtained in the Chang-Fano version of the Random Phase Approximation.