state – odd, the second excited state – even, etc.; therefore, according to Eq. (\[W\_equivalent\_particles\]), if $$\begin{aligned}
\label{Condition_not_forcing_transitions}
V_1(q) = V_2(q),\end{aligned}$$ then the transition from the ground state to the first excited state is forbidden, but if $$\begin{aligned}
\label{Condition_forcing_transitions}
V_1(q) = -V_2(q),\end{aligned}$$ then the transition is allowed. Paraphrasing, we note that if condition (\[Condition\_not\_forcing\_transitions\]) takes place then the interparticle degree of freedom may stay the same while the center of mass traverses the barriers, but if condition (\[Condition\_forcing\_transitions\]) holds then the state of the interparticle degree of freedom can change. In essence, this is the core of the observed phenomenon in Sec. \[Sec2\].

Indeed, condition (\[Condition\_forcing\_transitions\]) is satisfied for Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](f). Hence, the tunneling probability is less in these cases than in cases Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](e) for which equality (\[Condition\_not\_forcing\_transitions\]) takes place. On the whole, the same conclusion is valid as long as the potential $U_N(\rho)$ can have at lest two bound states. In Figs. \[Fig\_potentials\](a) and \[Fig\_potentials\](b), when there is only a single bound state supported by the intraparticle interaction, the simple intuitive picture holds. Why is this the case? After all, there are also continuum states of the intraparticle motion available for the excitation.

To 