We develop a theory, based on Fermi’s golden rule, for state-to-state vibrational multiphonon relaxation rates for a diatomic molecule in a rare gas crystal. The interaction between the diatomic and each of the host atoms is modeled with a Lennard-Jones potential. As such, the coupling between the anharmonic vibration of the diatomic and the acoustic phonons of the crystal is nonlinear (and nonexponential) in the atomic displacements from equilibrium. We compare our theory for the relaxation rates with experimental results from a sequence of vibrational transitions of O-2 in Ar crystal.