  The complex of a methane molecule and a fluoride anion represents a
12-dimensional (12D), four-well vibrational problem with multiple
large-amplitude motions, which has challenged the quantum dynamics community
for years. The present work reports vibrational band origins and tunneling
splittings obtained in a full-dimensional variational vibrational computation
using the GENIUSH program and the Smolyak quadrature scheme. The converged 12D
vibrational band origins and tunneling splittings confirm complementary aspects
of the earlier full- and reduced-dimensionality studies: (1) the tunneling
splittings are smaller than 0.02 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 thus, (3) a full-dimensional treatment
appears to be necessary. The present computations extend to a higher energy
range than earlier work, show that the tunneling splittings increase upon
vibrational excitation of the complex, and indicate non-negligible `heavy-atom'
tunneling.
