  Without invoking the Markov approximation, we derive formulas for vibrational
energy relaxation (VER) and dephasing for an anharmonic system oscillator using
a time-dependent perturbation theory. The system-bath Hamiltonian contains more
than the third order coupling terms since we take a normal mode picture as a
zeroth order approximation. When we invoke the Markov approximation, our theory
reduces to the Maradudin-Fein formula which is used to describe VER properties
of glass and proteins. When the system anharmonicity and the renormalization
effect due to the environment vanishes, our formulas reduce to those derived by
Mikami and Okazaki invoking the path-integral influence functional method [J.
Chem. Phys. 121 (2004) 10052]. We apply our formulas to VER of the amide I mode
of a small amino-acide like molecule, N-methylacetamide, in heavy water.
