A phase-based analysis of reaction dynamics

The reaction dynamics of realistic molecular Hamiltonians including both mode coupling and anharmonicity may be profitably explored by classical trajectories. However even if such trajectories begin in phase the energy dependence of individual orbital periods of anharmonic oscillators quickly results in an incoherent ensemble, thereby obscuring organization present in the reaction dynamics of a given Hamiltonian. One solution to these difficulties is to compare trajectories in phase on a cycle by cycle basis (i.e. coherently). In this work a model independent means for phase based, or coherent, comparison was developed utilizing the Hilbert transform. Phase based analysis reveals that in particular unimolecular reactions correlated motion occurs when energy transfer between an orthogonal mode and the reaction coordinate forces synchronization in their motions, resulting in convergence of their phases. Thus a restricted and systematic set of states (i.e. points in phase space) precedes reaction, apparently contradicting the RRKM assumption that all states are equally likely to react. Detailed examination of the fundamental RRKM equation shows that this assumption differs from requiring that all states are equally likely to react per unit of time. States involved in correlated motion react sooner than others, but all states ultimately react, so their reaction probabilities are equal, therefore correlated motion can be consistent with RRKM kinetics. To explore the properties of different distributions of internal energy within a microcanonical ensemble a variant of phase space is proposed wherein every point is indexed by time remaining until reaction. (In the absence of trapping all unimolecular trajectories reside in the bound region of phase space for only a finite duration.) Under this variant points belonging to a particular allocation of internal energy are not scattered randomly across the lifetime distribution mapping (e.g. states in close time proximity to the transition state obviously all share the property of having sufficient energy in the reaction coordinate to clear the barrier). The microcanonical rate constant is an average of those for all component distributions of available internal energy, so the existence of rate constants with mode specificity can also be consistent with RRKM kinetics