Full-dimensional (12D) variational vibrational states of CH$_4\cdot$F$^-$: interplay of anharmonicity and tunneling

The complex of a methane molecule and a fluoride anion represents a 12-dimensional (12D) vibrational problem with multiple large-amplitude motions, which has challenged the quantum dynamics community for years. The present work reports the near 70 lowest-energy vibrational states, up to 730 cm$^{-1}$ above the zero-point vibrational energy, obtained in a full-dimensional variational vibrational computation using the GENIUSH program and the Smolyak quadrature scheme. The vibrational energies and tunneling splittings are estimated to be converged better than 1 cm$^{-1}$ and 0.05 cm$^{-1}$, respectively. The vibrational level structure confirms complementary aspects of the earlier full- and reduced-dimensionality computations of this six-atomic, four-well system: (1) the tunneling splittings, in the computed range, are smaller than 0.05 cm$^{-1}$; (2) a single-well treatment is not sufficient (except perhaps the zero-point vibration) due to a significant anharmonicity over the wells; and as a result (3) a full-dimensional treatment appears to be necessary. With further development of the quantum dynamics methodology and the potential energy surface, it will become possible to study highly-excited tunneling manifolds, with perhaps larger splittings indicated by reduced-dimensionality results, as well as predissociation phenomena.