say, $L_1=L$, $L_2=1$, $L_{12}=L$, $L_3=0$ [@suzuki00; @DGVR]. The simplest choice for the $0^-$ state is to use three global vectors with $L_1=L_2=L_{12}=L_3=1$ [@suzuki00]. In this way, the basis function (\[cgtgv\]) can be versatile enough to describe bound states of not only four- but also more-particle systems with arbitrary $L$ and $\pi$.

To assure the permutation symmetry of the wave function, we have to operate a permutation $P$ on $F$. Since $P$ induces a linear transformation of the coordinate set, a new set of the permuted coordinates, ${{\mbox{\boldmath $x$}}}_P$, is related to the original coordinate set ${{\mbox{\boldmath $x$}}}$ as ${{\mbox{\boldmath $x$}}}_P\!=\!{\cal P}{{\mbox{\boldmath $x$}}}$ with an $(N-1)\times (N-1)$ matrix ${\cal P}$. As before, this permutation does not change the form of the $F$-function: $$\begin{aligned}
& &PF_{L_1 L_2(L_{12}) L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})
\nonumber \\
&&\quad =F_{L_1 L_2 (L_{12})L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}}_P)
\nonumber \\ 
&&\quad =F_{L_1 L_2(L_{12}) L_3 LM}(\widetilde{\cal P}u_1,
\widetilde{\cal P}u_2,\widetilde{\cal P}u_3,\widetilde{\cal P}A{\cal P},
{{\mbox{\boldmath $x$}}}).
\label{trans.cg}\end{aligned}$$ The fact that the functional form of $F$ remains unchanged under the permutation as well as the transformation of coordinates enables one to unify the method of calculating the matrix elements. This unique property is one of the most notable points in the present method.

Calculation of matrix 