We also present the initial expectation value of energy $$\begin{aligned}
\label{AvEDef}
\bar{E}_N = {\left\langle{\Psi_N(\alpha; 0, x_1, x_2) }\right|}\hat{H}_N(\alpha) {\left| \Psi_N(\alpha; 0, x_1, x_2) \right\rangle}, \\
\bar{E}_1 = 0.05911, \quad \bar{E}_2 = -0.3631, \quad \bar{E}_4 = -0.5766 \nonumber\end{aligned}$$ (all values are in atomic units). Rigorously speaking, $\bar{E}_N$ depends on $\alpha$; however, this dependence is very weak because the initial wave function (\[InitialCondition\]), independent of $\alpha$, is mostly localized in the region where the potential barrier, $\alpha V(x_1) + 3V(x_2)$, vanishes.

The probabilities of tunneling, disintegration (see below for the clarification of this term), and reflection are defined as follows: $$\begin{aligned}
P_T^{(N)} (\alpha, \tau) &=& \int_0^{\infty} dx_1 \int_0^{\infty} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2,  \label{Prob_T_deff}\\
P_D^{(N)} (\alpha, \tau) &=& \int_{-\infty}^0 dx_1 \int_0^{\infty} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2 \nonumber\\
    &+& \int_0^{\infty} dx_1 \int_{-\infty}^0 dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2, \label{Prob_D_deff}\\
P_R^{(N)} (\alpha, \tau) &=& \int_{-\infty}^0 dx_1 \int_{-\infty}^0 dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2. \label{Prob_R_deff}\end{aligned}$$

However, since the potential barrier, $\alpha V(x_1) + 3V(x_2)$, has “well” and “hill” regions, we also employ the corresponding “shifted” probabilities to exclude regions were the potential barrier is localized $$\begin{aligned}
p_t^{(N)} (\alpha, \tau) &=& \int_3^{\infty} dx_1 \int_3^{\infty} dx_2 \, 