  Hydrogen is one of the few molecules which has been incarcerated in the
molecular cage of C$_{60}$ and forms endohedral supramolecular complex
H$_2$@C$_{60}$. In this confinement hydrogen acquires new properties. Its
translational motion becomes quantized and is correlated with its rotations. We
applied infrared spectroscopy to study the dynamics of hydrogen isotopologs
H$_2$, D$_2$ and HD incarcerated in C$_{60}$. The translational and rotational
modes appear as side bands to the hydrogen vibrational mode in the mid infrared
part of the absorption spectrum. Because of the large mass difference of
hydrogen and C$_{60}$ and the high symmetry of C$_{60}$ the problem is
identical to a problem of a vibrating rotor moving in a three-dimensional
spherical potential. The translational motion within the C$_{60}$ cavity breaks
the inversion symmetry and induces optical activity of H$_2$. We derive
potential, rotational, vibrational and dipole moment parameters from the
analysis of the infrared absorption spectra. Our results were used to derive
the parameters of a pairwise additive five-dimensional potential energy surface
for H$_2$@C$_{60}$. The same parameters were used to predict H$_2$ energies
inside C$_{70}$[Xu et al., J. Chem. Phys., {\bf 130}, 224306 (2009)]. We
compare the predicted energies and the low temperature infrared absorption
spectra of H$_2$@C$_{70}$.
