  We first review how to determine the rate of vibrational energy relaxation
(VER) using perturbation theory. We then apply those theoretical results to the
problem of VER of a CD stretching mode in the protein cytochrome c. We model
cytochrome c in vacuum as a normal mode system with the lowest-order anharmonic
coupling elements. We find that, for the ``lifetime'' width parameter $\gamma=3
\sim 30$ cm$^{-1}$, the VER time is $0.2 \sim 0.3$ ps, which agrees rather well
with the previous classical calculation using the quantum correction factor
method, and is consistent with spectroscopic experiments by Romesberg's group.
We decompose the VER rate into separate contributions from two modes, and find
that the most significant contribution, which depends on the ``lifetime'' width
parameter, comes from those modes most resonant with the CD vibrational mode.
