Coupling coefficients in non-symmetry adapted graphs

We partition the shape formed by two different paths into three parts. A head in which both paths overlap, a loop where the paths separate, run parallel and meet again and a tail in which the paths overlap again. The value of the coupling coefficients is solely determined (Slater s rule) by the number of spinors niooj that are occupied in the loop part of the diagram 

The wavy lines represent all possible paths that connect the begin and end vertex of the line. Indices and v are given by 

To illustrate the procedure we take the calculation of the non-zero coupling coefficients with P=2 and Q=5 for the diagram given in Fig. 4. We see that vertex combinations (l,0)-(5,2), (l,0)-(5,3), (l.l)-(5,2) and (l,l)-(5,3) give rise to loops. The corresponding coupling coefficients are given in Table 1. 

We have 6 non-zero coupling coefficients, that are represented by 24 numbers (signs and the partial T-values). Representation by the summed numbers as is done in the last three lines (16 numbers) would be more advantageous in this case because it requires only 16 numbers, but one can easily imagine that the partitioned representation is favorable in larger spaces. 

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