The Redmond Formula with Seniority

As we get to heavier nuclei, we find more states with different seniorities and several states of a given seniority. There is a recursion formula by Redmond that relates an $n \to (n+1)$ coefficient of fractional parentage (cfp) to that of $(n-1) \to n$. However, this involves an {\it overcomplete} set of principal parent (pp) cfp's. For example, for a 3-particle system, we can form basis states $[ [12]^{J_0} 3]^J$, where $J_0$ is the pp; we then antisymmetrize and normalize $\Psi [J_0]=N[J_0] (1-P_{12}-P_{13}) \left[ [12]^{J_0} 3\right]^J$, and form a ppcfp expansion $\Psi[J_0]=\sum_{J_1} [j^2 (J_1) j |\} j^ 3 [J_0] J] \left[ [12]^{J_1} 3\right]^J$. But for, say, $J=j=9/2$, there are five $\Psi[J_0]$'s, but only two independent wave functions, one with seniority 1 and one with seniority 3. We note that $[j^2 (J_0) j |\} j^3 [J_0] J]=1/(3 N[J_0])$. We are able then to obtain the following relation between overcomplete ppcfp's and complete orthonormal cfp's: $A=B=C$, where $$ A=(n+1)[j^n(J_0 v_0) j |\} j^{n+1} [J_0 v_0] J] \; \; [j^n (J_1 v_1) j |\} j^{n+1} [J_0 v_0] J], $$ $$ B=(n+1) \sum_v [j^n (J_0 v_0) j |\} j^{n+1} J v] \; \; [j^n (J_1 v_1) j |\} j^{n+1} J v], $$ $$ C=\delta_{J_0 J_1} \delta_{v_0 v_1} + n (-1)^{J_0+J_1} \sqrt{(2J_0+1)(2J_1+1)} \sum_{v_2 J_2} \begin{Bmatrix} J_2 & j & J_1 \\ J & j & J_0 \end{Bmatrix} \times $$ $$ \times [j^{n-1} (J_2 v_2) j |\} j^n J_0 v_0] \; \; [ j^{n-1} (J_2 v_2) j |\} j^n J_1 v_1]. $$