Oscillator phase and the reaction dynamics of HN₃: A model for correlated motion

Abstract

The reaction coordinate for a unimolecular dissociation reaction is intrinsically anharmonic, therefore the period of vibration depends upon the energy in the bond. Consequently, the final few vibrations leading up to and including reaction can occur over very different time scales, depending upon the amount of vibrational energy in the reaction coordinate. In recent studies, we have compared ensembles of reactive trajectories and have observed correlations in molecular motions during reaction leading up to the transition state. If the comparison is made in time, differences in the periods of the reaction coordinate vibration can obscure these correlations. However, if the reactions are compared vibration by vibration (i.e., coherently) clear patterns of vibrational motions emerge in the reaction dynamics. In this paper, we introduce a new application of the Hilbert transform to assign oscillator phases to the motions of anharmonic oscillators, which permits coherent comparison of trajectories. We also demonstrate by several examples that, when compared coherently, reaction dynamics of HN₃ exhibit order which is not evident in time series comparisons. These orderly patterns of motions reflect the intramolecular conditions necessary for reaction to occur. We propose a model to account for the observed correlated motion in terms of the requirements for intramolecular energy transfer. As a consequence of the constraints of energy transfer, the phases of several oscillators have clearly defined relative values. Physically this corresponds to a correlation in the vibrations of several modes such that (for example) two key bonds always extend simultaneously during the course of reaction. Since the motions of atoms preceding reaction are not random, but rather follow a specific pattern, a restricted set of reactant states immediately precede reaction.

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