Multiplet Auger transition rate for partially filled shells

A previous derivation of the generalized Auger transition for inner-shell multiplets is shown to be incorrect. The previous derivation was based on deducing a coupling scheme from $6-j$ symbols in the transition-rate expression, then transforming the coupling coefficients. It was expected that the transformed transition rate would be correct up to a phase factor. It is shown here that this expectation is incorrect. The transformed transition-rate expression is structurally incorrect (i.e., incorrect irrespective of phase factor). Correct expressions are obtained by direct manipulation of $6-j$ and $9-j$ symbols. It is shown that the correct expression can be found using the coupling transformation provided a modified matrix element expression is used.