$a$ and $b$, and ${\cal A}$ is an operator that antisymmetrizes between the clusters. The square bracket $[{I_a} \ {I_b}]_I$ denotes the angular momentum coupling. The coordinate ${{\mbox{\boldmath $\rho$}}}_\alpha$ in the relative motion function $\chi^J_{\alpha}$ is the relative distance vector of the clusters. The channel spin $I$ and the relative angular momentum $\ell$ in $\alpha$ are coupled to give the total angular momentum $J$. The relative-motion functions $\chi_{\alpha}$ also depend on $J$ and $\pi$. For simplicity, this dependence is not displayed explicitly in the notation for $\chi_{\alpha}$ as well as for some other quantities below.

The configuration space is divided into two regions, internal and external, by the channel radius $a$. In the internal region ($\rho_\alpha\le a$), the total wave function may be expressed in terms of a combination of various $\Psi_{\alpha}^{JM\pi}$s $$\begin{aligned}
\Psi^{JM\pi}_{\rm int}&=&\sum_{\alpha} \Psi^{JM\pi}_{\alpha}\nonumber \\
&=&\sum_{\alpha}\sum_{n}f_{\alpha n} {\cal A} 
u_{\alpha n}(\rho_\alpha) \phi^{JM\pi}_{\alpha},
\label{wf.int}\end{aligned}$$ with $$\begin{aligned}
\phi^{JM\pi}_{\alpha} = \frac{1}{\sqrt{(1+\delta_{I_aI_b}\delta_{ab})(1+\delta_{ab})}} \left\{
\left[\left[\Phi^{a}_{I_a}\Phi^{b}_{I_b}\right]_I Y_{\ell}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM} 
\right. \nonumber \\ \left.
+(-1)^{A_a+I_a+I_b-I+\ell}
\left[\left[\Phi^{b}_{I_b}\Phi^{a}_{I_a}\right]_I Y_{\ell}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM} \delta_{ab}
\right\},
\label{wf.channel}\end{aligned}$$ where $A_a$ is the number of nucleons in cluster $a$, $\delta_{ab}$ is unity if $a$ and $b$ are identical clusters and zero otherwise, and $\delta_{I_aI_b}$ is unity if the clusters are in identical states and 