The problem of the damping of molecular exciton states in a crystal is considered with respect to the trapping of the exciton at a particular site, and the subsequent radiationless transition within a molecule at the site. Damping considered in this manner has a close connection with the proposals concerning radiationless transitions put forward by Robinson and Frosch. In this paper we make use of the Robinson and Frosch proposals together with the harmonic oscillator representation of the exciton problem to show that damping at a particular site leads to damping of the exciton.