Explicit exponential frequency dependence of multiphonon infrared absorption

The nearly exponential frequency dependence of the infrared absorption coefficient $\beta$ observed by Rupprecht, Deutsch, and others has been explained previously by evaluating the individual $n$-phonon contributions to $\beta$, summing the results, and noting that the sum was nearly exponential over a fairly wide range of frequencies. A new derivation of the multiphonon absorption coefficient yields $\beta \sim e^{-\omega \tau }$ directly, rather than as a sum on $n$, and provides a prescription for estimating the range of $\omega$ over which the nearly exponential behavior extends.