Y}_{L_3}(\widetilde{u_3}{{\mbox{\boldmath $x$}}})\right]_{LM},
\label{cgtgv}\end{aligned}$$ where $${\cal Y}_{L_iM_i}(\widetilde{u_i}{{\mbox{\boldmath $x$}}})=
|\widetilde{u_i}{{\mbox{\boldmath $x$}}}|^{L_i}Y_{L_iM_i}(\widehat{\widetilde{u_i}{{\mbox{\boldmath $x$}}}})$$ is a solid spherical harmonics and its argument, $\widetilde{u_i}{{\mbox{\boldmath $x$}}}$, what we call a global vector, is a vector defined through an $(N-1)$-dimensional column vector $u_i$ and ${{\mbox{\boldmath $x$}}}$ as $$\widetilde{u_i}{{\mbox{\boldmath $x$}}}=\sum_{j=1}^{N-1}(u_i)_j{{\mbox{\boldmath $x$}}}_j,
\label{def.ux}$$ where $(u_i)_j$ is the $j$th element of $u_i$. In Eq. (\[cgtgv\]) $A$ is an $(N-1)\times(N-1)$ real and symmetric matrix, and it must be positive-definite for the function $F$ to have a finite norm, but otherwise may be arbitrary. Non-diagonal elements of $A$ can be nonzero.

The matrix $A$ and the vectors $u_1, u_2, u_3$ are parameters to characterize the “shape” of the correlated Gaussian function. The Gaussian function including $A$ describes a spherical motion of the system, while the global vectors are responsible for a rotational motion. The spatial function (\[space.part\]) is found to reduce to the general form (\[cgtgv\]). Suppose that ${{\mbox{\boldmath $x$}}}$ stands for the H-type coordinate. Then a choice of $\widetilde{u_1}$=(1,0,0), $\widetilde{u_2}$=(0,1,0) and $\widetilde{u_3}$=(0,0,1) together with a diagonal matrix $A$ provides us with the basis function (\[space.part\]) employed to represent the configurations of the $2N$+$2N$ channel. On the other hand, the K-type basis function looks like $${\rm exp}\left(-{\frac{1}{2}}{\widetilde{{{\mbox{\boldmath $x$}}}'}} A_K {{\mbox{\boldmath $x$}}}'\right)
\left[[{\cal 