  A full-dimensional \emph{ab initio} potential energy surface of spectroscopic
quality is developed for the van-der-Waals complex of a methane molecule and an
argon atom. Variational vibrational states are computed on this surface
including all twelve (12) vibrational degrees of freedom of the methane-argon
complex using the GENIUSH computer program and the Smolyak sparse grid method.
The full-dimensional computations make it possible to study fine details of the
interaction and distortion effects and to make a direct assessment of the
reduced-dimensionality models often used in the quantum dynamics study of
weakly-bound complexes. A 12-dimensional (12D) vibrational computation
including only a single harmonic oscillator basis function (9D) to describe the
methane fragment (for which we use the ground-state effective structure as the
reference structure) has a 0.40 cm$^{-1}$ root-mean-square error (rms) with
respect to the converged 12D bound-state excitation energies, which is less
than half of the rms of the 3D model set up with the $\langle r \rangle_0$
methane structure. Allowing 10 basis functions for the methane fragment, the
rms of the bound state vibrational energies is reduced to 0.07 cm$^{-1}$, which
is much better than the 3D models. The full-dimensional potential energy
surface correctly describes the dissociation of the system, which together with
further development of the variational (ro)vibrational methodology opens the
route for the study of the role of dispersion forces on the excited methane
vibrations and the energy transfer from the intra- to the intermolecular
vibrational modes.
