\alpha_0}-S_{\alpha \, \alpha_0}^{J\pi} O_{\alpha}(k_\alpha \rho_\alpha)],\end{aligned}$$ where $k_\alpha=\sqrt{2\mu_\alpha |E_\alpha |}/\hbar$, $v_\alpha=\hbar k_\alpha/\mu_\alpha$ and $S_{\alpha \, \alpha_0}^{J\pi}$ is an element of the $S$-matrix (or collision matrix) to be determined. Here $I_{\alpha}(k_\alpha \rho_\alpha)$ and $O_{\alpha}(k_\alpha \rho_\alpha)$ are the incoming and outgoing waves defined by $$\begin{aligned}
I_{\alpha}(k_\alpha \rho_\alpha)=O_{\alpha}(k_\alpha \rho_\alpha)^*=
G_\ell(\eta_{\alpha}, k_\alpha \rho_\alpha)-iF_\ell(\eta_{\alpha}, k_\alpha \rho_\alpha),\end{aligned}$$ with the regular and irregular Coulomb functions $F_\ell$ and $G_\ell$. The Sommerfeld parameter $\eta_{\alpha}$ is $\mu_\alpha Z_a Z_b e^2/\hbar^2
k_\alpha$. For a closed channel $\alpha$ $(E_\alpha < 0)$, the asymptotic form of $g_\alpha$ is given by the Whittaker function $$\begin{aligned}
g_{\alpha}(\rho_\alpha) \propto \rho_\alpha^{-1} 
W_{-\eta_{\alpha},\ell+1/2}(2k_\alpha\rho_\alpha).\end{aligned}$$

The matrix elements $S_{\alpha \alpha_0}^{J\pi}$ are determined by solving a Schrödinger equation with a microscopic Hamiltonian $H$ involving the $A_a + A_b$ nucleons, $$\begin{aligned}
(H+{\cal L} -E)\Psi_{\rm int}^{JM\pi}={\cal L}\Psi_{\rm ext}^{JM\pi},\end{aligned}$$ with the Bloch operator ${\cal L}$ $$\begin{aligned}
{\cal L}=\sum_{\alpha}\frac{\hbar^2}{2\mu_\alpha a}
\vert \phi_{\alpha}^{JM \pi} \rangle \delta(\rho_\alpha-a)
\left(\frac{\partial}{\partial \rho_\alpha}-\frac{b_\alpha}{\rho_\alpha} \right)\rho_\alpha
\langle \phi_{\alpha}^{JM\pi}\vert,
\label{Bloch}\end{aligned}$$ where the channel radius $a$ is chosen to be the same for all channels, and the $b_\alpha$ are arbitrary constants. Here, we choose $b_\alpha=0$ for the open channels and $b_\alpha=2k_\alpha a W'_{-\eta_\alpha,\ell+1/2}(2k_\alpha a)/W_{-\eta_\alpha,\ell+1/2}(2k_\alpha a)$ for the closed channels. The results do not depend on the choices for $b_\alpha$ but these values simplify the calculations. Notice 