uniquely coupled to the maximum possible angular momentum. The same rule applies to $\lambda_2$, ${{\mbox{\boldmath $e$}}}_2$ and $\lambda_3$, ${{\mbox{\boldmath $e$}}}_3$ as well.

We outline a method of calculating the matrix element for an operator ${\cal O}$ $$\langle F_{L_4 L_5(L_{45}) L_6 L'M'}(u_4, u_5, u_6, A',{{\mbox{\boldmath $x$}}})\vert
 {\cal O}\vert F_{L_1 L_2(L_{12}) L_3 LM}(u_1, u_2, u_3, A,{{\mbox{\boldmath $x$}}})\rangle. 
\label{meofop}$$ In what follows this matrix element is abbreviated as $\langle F'\vert {\cal O}\vert F\rangle $. Using Eq. (\[gfn\]) in Eq. (\[meofop\]) enables one to relate the matrix element to that between the generating functions: $$\begin{aligned}
\left<F'\vert {\cal O}  \vert F \right> 
&\!=\!&\left(\prod_{i=1}^6 \frac{B_{L_i}}{L_i !}\int d{{\mbox{\boldmath $e$}}}_i\right)
\left[\left[Y_{L_4}({{\mbox{\boldmath $e$}}}_4)Y_{L_5}({{\mbox{\boldmath $e$}}}_5)\right]_{L_{45}}
Y_{L_6}({{\mbox{\boldmath $e$}}}_6)\right]_{L'M'}^* \nonumber\\
&\!\times\!&\left[\left[Y_{L_1}({{\mbox{\boldmath $e$}}}_1)Y_{L_2}({{\mbox{\boldmath $e$}}}_2)\right]_{L_{12}}
Y_{L_3}({{\mbox{\boldmath $e$}}}_3)\right]_{LM}
\nonumber\\
&\!\times\!&\left(\prod_{i=1}^6
\frac{\partial^{L_i}}{\partial \lambda_i^{L_i}}\right)
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert {\cal O}  \vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>\Big\vert_{\lambda_i=0},
\label{grandformula}\end{aligned}$$ with $${{\mbox{\boldmath $s$}}}={\lambda}_1u_1{{\mbox{\boldmath $e$}}}_1 \!+\!{\lambda}_2u_2{{\mbox{\boldmath $e$}}}_2
\!+\!{\lambda}_3u_3{{\mbox{\boldmath $e$}}}_3,\hspace*{1cm}
{{\mbox{\boldmath $s$}}}'={\lambda}_4u_4{{\mbox{\boldmath $e$}}}_4 \!+\!{\lambda}_5u_5{{\mbox{\boldmath $e$}}}_5
\!+\!{\lambda}_6u_6{{\mbox{\boldmath $e$}}}_6.$$ The calculation of the matrix element consists of three stages: (1) Evaluate the matrix element between the generating functions, $\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert {\cal O}  \vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>$. (2) Expand that matrix element in powers of $\lambda_i$ and keep only those terms of degree $L_i$ for each $i$. (3) Recouple the vectors ${{\mbox{\boldmath $e$}}}_i$ and integrate over the 