supported by the Hamiltonian $-1/(2\mu) \partial^2/\partial \rho^2 + U_N(\rho)$. The energies of these states are $-0.955$ for $U_1$; $-1.377$ and $-0.372$ for $U_2$; and, finally, $-1.590$, $-0.856$, $-0.308$, and $-0.012$ for $U_4$.

Following Ref. [@Volkova2006], we study the tunneling within the time-dependent approach solving the time-dependent Schrödinger equation, $$\begin{aligned}
\label{TimDepSchEq}
\left[ i\partial/\partial t - \hat{H}_N(\alpha)\right] \Psi_N(\alpha; t, x_1, x_2) = 0,\end{aligned}$$ with the initial condition at $t=0$ that reads in the coordinates (\[SimpleCMandRelativeCoord\]) as $$\begin{aligned}
\label{InitialCondition}
\Psi_N(\alpha; 0, R, \rho) = C \phi_g(\rho) e^{ -(R-\bar{R})^2/\left(2\sigma_R^2\right) + i\sqrt{2M E_{cm}}R  }.\end{aligned}$$ Here $C$ is a normalization constant and $\phi_g(\rho)$ is the ground state of the interparticle Hamiltonian, $-1/(2\mu) \partial^2/\partial \rho^2 + U_N(\rho)$. In all our studies, we set $m=1$, $\bar{R} = -55$, $\sigma_R = 3$, and $E_{cm} = 1$ (all values are in atomic units).

The initial wave function (\[InitialCondition\]) is localized in the third quadrant, and we calculate the probability of finding the particle in the first quadrant, i.e., the probability of tunneling at later time $\tau$. The reason for using $\phi_g(\rho)$ as the relative coordinate part of the initial wave function is that we wanted to avoid spreading of the wave packet along $\rho$ before it reached the potential barrier.

