zero otherwise. In the second line of Eq. (\[wf.int\]), the relative motion functions of Eq. (\[wf-ch1\]) are expanded in terms of some basis functions as $$\begin{aligned}
\chi_{\alpha m}({{\mbox{\boldmath $\rho$}}}_\alpha) = \sum_n  f_{\alpha n} 
u_{\alpha n}(\rho_\alpha)Y_{\ell m}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha).
\label{chi.expansion}\end{aligned}$$ In what follows we take $$\begin{aligned}
u_{\alpha n}(\rho_\alpha)=\rho_\alpha^{\ell}\exp(-\frac{1}{2}\lambda_n \rho_\alpha^2)
\label{u.functions}\end{aligned}$$ with a suitable set of $\lambda_n$s.

In the external region ($\rho_\alpha\ge a$), the total wave function takes the form $$\begin{aligned}
\Psi^{JM\pi}_{\rm ext}=\sum_{\alpha} g_{\alpha}(\rho_\alpha)
 \phi^{JM\pi}_{\alpha}
\label{wf.ext}.\end{aligned}$$ Note that the antisymmetrization between the clusters is dropped in the external region under the condition that the channel radius $a$ is large enough. The function $g_{\alpha}(\rho_\alpha)$ of Eq. (\[wf.ext\]) is a solution of the equation $$\begin{aligned}
\left[-\frac{\hbar^2}{2\mu_{\alpha}}\left( \frac{d^2}{d\rho_\alpha^2}
+\frac{2}{\rho_\alpha}\,\frac{d}{d\rho_\alpha}-\frac{\ell(\ell+1)}{\rho_\alpha^2} \right)+\frac{Z_aZ_be^2}{\rho_\alpha}\right] g_{\alpha}(\rho_\alpha) =
E_{\alpha} g_{\alpha}(\rho_\alpha),\end{aligned}$$ where $\mu_{\alpha}$ is the reduced mass for the relative motion in channel $\alpha$, $Z_ae$ and $Z_be$ are the charges of $a$ and $b$, and $E_{\alpha}=E-E_a-E_b$ is the energy for the relative motion, where $E$ is the total energy, and $E_a$ and $E_b$ are the internal energies for the clusters $a$ and $b$, respectively. For the scattering initiated through the channel $\alpha_0$, the asymptotic form of $g_{\alpha}$ for the open channel $\alpha$ $(E_{\alpha} \geq 0)$ is $$\begin{aligned}
g_{\alpha}(\rho_\alpha) = v_\alpha^{-1/2} \rho_\alpha^{-1} 
[I_{\alpha}(k_\alpha \rho_\alpha)\delta_{\alpha\,
 