Theories of intramolecular vibrational energy transfer

Abstract

T. Uzer, Theories of intramolecular vibrational energy transfer 75 PK(t) =~'~~' U~1~b1(0)j+~UkJUlbJ(0)b~(0).(2.8)The second sum vanishes if the terms in it have uncorrelated phases, leaving the coarse-grained level population asIf only the coarse-grained level populations P~, =~P1 are measurable, the same "random phase" argument as before allowswhere N~is the number of states in one level. This matrix Y, under fairly general conditions, can be expressed [Quack 1979a] as an exponential involving a time-independent matrix K [Quack 1978], Y(t) = exp[K(tta)], (2.11) 80 T. Uzer. Theories of intramolecular vibrational energy transfer and p(t) = Y(t)p(0). This can be reduced [Quack 1978, 1981a] to the differential form of the Pauhi master equation [Pauli 1928], Pz~Kp. (2.12)In contrast to the oscillatory solutions of eqs. (2.1)-(2.4), these equations, (2.11) and (2.12), have decaying solutions, denoting relaxation. For sufficiently short times the exponential function can be expanded and one obtains [Quack 1978]where WKJI2 stands for the average square coupling element between the states in levels K and J, and K is the average frequency separation of states in level K (1 1~K needs to be replaced by the density Pk in the case of the continuum). The preceding development makes clear the factors that need to be considered carefully when studying intramolecular relaxation: the definition of what is being observed (e.g., what is relaxing into what), the time scale over which the observation is taking place, which Hamiltonian and which basis set is being used for the description. Moreover, the preceding development is valuable in connecting Golden-Rule type of first-order perturbation theory prescriptions, which are widely used in IVR, to a fundamental view of the process [Voth 1987].Of course, intramolecular relaxation is a more general phenomenon than the somewhat restricted Pauhi master equation would suggest, i.e., there are selections of course-graining which make oscillatory behavior possible [Quack 1981a], which are, however, the exception rather than the rule. We will see examples of such exponential behavior as well as coherences in the following sections. The Pauhi master equation is merely one (and the earliest) attempt to describe relaxation phenomena in a quantummechanical setting [Van Hove 1962]; it proceeds by reducing the full density-matrix equation to a system of coupled equations for the diagonal density-matrix elements alone. The derivation, as well as the set of companion equations for the off-diagonal matrix elements, the "coherences", have been critically studied by Fox [1978,1989]. The effect of these coherences proves valuable, e.g., in the theory of spectral lineshapes [Faid and Fox 1988]. General molecular model for intramolecular vibrational energy redistribution ([VR)

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