$H^{+}_{2}$ in a lattice of cavities: ammonia-like splitting of the lowest level

It is shown that reconstruction of the lowlying energy levels of the $H^{+}_{2}$ ion under conditions of confinement in a given spatial volume is substantially more significant as compared to the confinement by simple potential barrier. Depending on the cavity parameters, the ground-state binding energy of $H^{+}_{2}$ could be significantly greater than that of the free ion, while the behavior of the lowest molecular term $^{2}\Sigma^{+}_{g}$ of the ion turns out to be quite different when considered as a function of the internuclear distance. In particular, two minima might occur in $^{2}\Sigma^{+}_{g}$, while the relationship between them could be sufficiently different depend ing on the cavity parameters, which is shown on the phase diagram that was obtained in the work. We studied the case of a <<mexican hat>> structure of the effective ion potential in detail. As a result, the lowest electronic level of ion splits into the ground state level and the first excited one, with difference between them being as small as ~10$^{–4}$ eV. As in the NH$_3$ molecule, in the last case the lowest level gives rise to an effective two-level system that is separated from the vibrational and rotational modes by a wide energy gap. More concretely, calculation using the Neumann conditions that actually reproduce the confinement on a lattice of similar cavities shows that the splitting of ~10$^{–4}$ eV occurs for the linear sizes of the confinement area of the order of some $a_{B}$ and a shell potential magnitude of ~10 eV.