elements {#sect.4}
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Calculations of matrix elements with the correlated Gaussian $F$ are greatly facilitated with the aid of the generating function $g$ [@vs95; @book] $$\begin{aligned}
g({{\mbox{\boldmath $s$}}}; A, {{\mbox{\boldmath $x$}}})=
\exp\Big(-{\frac{1}{2}}{\widetilde{{\mbox{\boldmath $x$}}}}A{{\mbox{\boldmath $x$}}}+
\widetilde{{\mbox{\boldmath $s$}}}{{\mbox{\boldmath $x$}}}\Big),\end{aligned}$$ with $\widetilde{{\mbox{\boldmath $s$}}}=({{\mbox{\boldmath $s$}}}_1,{{\mbox{\boldmath $s$}}}_2,\ldots,{{\mbox{\boldmath $s$}}}_{N-1})$, where ${{\mbox{\boldmath $s$}}}_i=\sum_{j=1}^3\lambda_j(u_j)_i{{\mbox{\boldmath $e$}}}_j$, ${{\mbox{\boldmath $e$}}}_j$ is a 3-dimensional unit vector (${{\mbox{\boldmath $e$}}}_j\cdot{{\mbox{\boldmath $e$}}}_j=1$), and $\lambda_j$ is a scalar parameter. More explicitly $$\widetilde{{\mbox{\boldmath $s$}}}{{\mbox{\boldmath $x$}}}=\sum_{i=1}^{N-1}{{\mbox{\boldmath $s$}}}_i\cdot{{\mbox{\boldmath $x$}}}_i=
\sum_{i=1}^{N-1}\sum_{j=1}^3\lambda_j
(u_{j})_i {{\mbox{\boldmath $e$}}}_j \cdot{{\mbox{\boldmath $x$}}}_i
=\sum_{j=1}^3 \lambda_j{{\mbox{\boldmath $e$}}}_j\cdot (\widetilde{u_j}{{\mbox{\boldmath $x$}}}).$$ The correlated Gaussian $F$ is generated as follows: $$\begin{aligned}
&&F_{L_1 L_2(L_{12}) L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\nonumber\\
&&\quad =\left(\prod_{i=1}^3 \frac{B_{L_i}}{L_i !}\int d{{\mbox{\boldmath $e$}}}_i\right)
\left[\left[Y_{L_1}({{\mbox{\boldmath $e$}}}_1)Y_{L_2}({{\mbox{\boldmath $e$}}}_2)\right]_{L_{12}}
Y_{L_3}({{\mbox{\boldmath $e$}}}_3)\right]_{LM}\nonumber\\
&&\quad \times \left(\frac{\partial^{L_1+L_2+L_3}}
{\partial \lambda_1^{L_1} \partial \lambda_2^{L_2} \partial \lambda_3^{L_3} }
\,g({{\mbox{\boldmath $s$}}};A,{{\mbox{\boldmath $x$}}})\right)\Bigg\vert_{\lambda_1=\lambda_2=\lambda_3=0},\label{gfn}\end{aligned}$$ where $$\begin{aligned}
\hspace*{-2cm} B_L&=&\frac{(2L+1)!!}{4\pi}.\end{aligned}$$ When $g({{\mbox{\boldmath $s$}}}; A,{{\mbox{\boldmath $x$}}})$ is expanded in powers of $\lambda_1$, only the term of degree $\lambda_{1}^{L_1}$ contributes in Eq. (\[gfn\]), and this term contains the $L_1$th degree ${{\mbox{\boldmath $e$}}}_1$ because $\lambda_1$ and ${{\mbox{\boldmath $e$}}}_1$ always appear simultaneously. In order for the term to contribute to the integration over ${{\mbox{\boldmath $e$}}}_1$, these $L_1$ vectors ${{\mbox{\boldmath $e$}}}_1$ must couple to the angular momentum $L_1$ because of the orthonormality of the spherical harmonics $Y_{L_1M_1}({{\mbox{\boldmath $e$}}}_1)$, that is, they are 