Interaction potential and IR absorption of endohedral H$_2$ in C$_{60}$

We measured the IR spectra of a H$_2$ molecule trapped inside a C$_{60}$ cage at temperatures from 6 to 300\,K and analyzed the spectra by using a model of a vibrating rotor in a spherical potential. The electric dipole moment of IR transitions is induced by the translational motion of H$_2$. The rotation of H$_2$ is unhindered but coupled to the translational motion. The isotropic and translation-rotation coupling part of the potential are anharmonic and different in the ground and excited vibrational states of H$_2$. The vibrational frequency and the rotational constant of endohedral H$_2$ are smaller than in the gas phase. The assignment of IR lines to ortho- and para-H$_2$ is confirmed by measuring spectra of a para enriched H$_2$@C$_{60}$ and is consistent with the earlier interpretation of the low temperature IR spectra [ S. Mamone \textit{et al.}, J. Chem. Phys.\textbf{ 130}, 081103 (2009) ].