Memory functions and recurrences in intramolecular processes

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Abstract
The exact amplitudes of the initial state in some models used in the theory of intramolecular processes (both sequential and nonsequential but all involving a set of equidistant discrete states and one or more continua) are determined. The method is based on the derivation of a memory function for the initial state amplitude. An important feature of the formulas is that although they involve an infinite sum, for any finite time only a finite number of terms is effective, owing to the presence of step functions depending on times that are retarded by a integer number of recurrence periods h/ε, where ε is the level separation in the discrete set. The conditions for the validity of the exponential law can be given more precisely than in the previous discussions of this problem: (a) The decay is rigorously exponential for an observation time t smaller than the recurrence time, whatever values are given to the six parameters defining the models; and (b) if the damping constant of the discrete levels is much larger than the level separation, the law is exponential for an arbitrary t, in contrast with previous assertions where t was limited to t ≪τmax, where τmax is the lifetime of one of the discrete levels. The role of the first recurrence is examined. The term contributed to the amplitude interferes with the initial exponential either in a destructive or in a constructive way. Under some conditions it is possible for the recurrence effect to shorten instead of lengthen the decay of the initial state.