angle coordinates. In the second stage the remaining terms should contain ${{\mbox{\boldmath $e$}}}_i$s of degree $L_i$ as well. Hence any term with $\lambda_i^2 {{\mbox{\boldmath $e$}}}_i\!\cdot\!{{\mbox{\boldmath $e$}}}_i\!
=\!\lambda_i^2$ etc. can be omitted because the degree of ${{\mbox{\boldmath $e$}}}_i$ becomes smaller than that of $\lambda_i$.

We will explain the above procedures for the case of an overlap matrix element. The matrix element between the generating functions is $$\begin{aligned}
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert 
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right> 
=\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\exp\left(\frac{1}{2}\widetilde{{{\mbox{\boldmath $z$}}}}B^{-1}{{\mbox{\boldmath $z$}}}\right)\end{aligned}$$ with $$\begin{aligned}
B=A'+A,\qquad {{\mbox{\boldmath $z$}}}={{\mbox{\boldmath $s$}}}+{{\mbox{\boldmath $s$}}}'=\sum_{i=1}^6\lambda_i{{\mbox{\boldmath $e$}}}_iu_i.\end{aligned}$$ To perform the operation in the second stage we note that $$\frac{1}{2}\widetilde{{{\mbox{\boldmath $z$}}}}B^{-1}{{\mbox{\boldmath $z$}}}=\frac{1}{2}\sum_{i,j=1}^6
\rho_{ij}\lambda_i\lambda_j {{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j$$ with $$\rho_{ij}=\widetilde{u_i}B^{-1}u_j.
\label{def.rho}$$ As mentioned above, here we can drop the diagonal terms, $\lambda_i^2
{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_i$, and we get $$\begin{aligned}
&&\left(\prod_{i=1}^6\frac{\partial^{L_i}}{\partial \lambda_i^{L_i}}\right)
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert 
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>\Big\vert_{\lambda_i=0} \nonumber\\
&&\quad =\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}\prod_{i=1}^6L_i!
\prod_{i<j}^6\frac{\left(\rho_{ij}{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j\right)^{n_{ij}}}{n_{ij}!}.\end{aligned}$$ Here the non-negative integers $n_{ij}$ must satisfy the following equations in order to assure the degree $L_i$ for ${{\mbox{\boldmath $e$}}}_i$ in the different terms, $$\begin{aligned}
&&n_{12}+n_{13}+n_{14}+n_{15}+n_{16}=L_1, \nonumber\\
&&n_{12}+n_{23}+n_{24}+n_{25}+n_{26}=L_2, \nonumber\\
&&n_{13}+n_{23}+n_{34}+n_{35}+n_{36}=L_3, \nonumber\\
&&n_{14}+n_{24}+n_{34}+n_{45}+n_{46}=L_4, \nonumber\\
&&n_{15}+n_{25}+n_{35}+n_{45}+n_{56}=L_5, \nonumber\\
&&n_{16}+n_{26}+n_{36}+n_{46}+n_{56}=L_6. 
\label{nij}\end{aligned}$$

The last step is to recouple the angular momenta arising from the various terms. Since we have to couple ${{\mbox{\boldmath $e$}}}_i$s to the angular 