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Register a new userDynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points
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In this paper we analyze a two degree of freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the roaming mechanism whose reaction dynamics are of current interest in the chemistry community.
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A model Hamiltonian for the reaction CH$_4^+ rightarrow$ CH$_3^+$ + H, parametrized to exhibit either early or late inner transition states, is employed to investigate the dynamical characteristics of the roaming mechanism. Tight/loose transition states and conventional/roaming reaction pathways are identified in terms of time-invariant objects in phase space. These are dividing surfaces associated with normally hyperbolic invariant manifolds (NHIMs). For systems with two degrees of freedom NHIMS are unstable periodic orbits which, in conjunction with their stable and unstable manifolds, unambiguously define the (locally) non-recrossing dividing surfaces assumed in statistical theories of reaction rates. By constructing periodic orbit continuation/bifurcation diagrams for two values of the potential function parameter corresponding to late and early transition states, respectively, and using the total energy as another parameter, we dynamically assign different regions of phase space to reactants and products as well as to conventional and roaming reaction pathways. The classical dynamics of the system are investigated by uniformly sampling trajectory initial conditions on the dividing surfaces. Trajectories are classified into four different categories: direct reactive and non reactive trajectories,which lead to the formation of molecular and radical products respectively, and roaming reactive and non reactive orbiting trajectories, which represent alternative pathways to form molecular and radical products. By analysing gap time distributions at several energies we demonstrate that the phase space structure of the roaming region, which is strongly influenced by non-linear resonances between the two degrees of freedom, results in nonexponential (nonstatistical) decay.
A reduced two dimensional model is used to study Ketene isomerization reaction. In light of recent results by Ulusoy textit{et al.} [J. Phys. Chem. A {bf 117}, 7553 (2013)], the present work focuses on the generalization of the roaming mechanism to the Ketene isomerization reaction by applying our phase space approach previously used to elucidate the roaming phenomenon in ion-molecule reactions. Roaming is again found be associated with the trapping of trajectories in a phase space region between two dividing surfaces; trajectories are classified as reactive or nonreactive, and are further naturally classified as direct or non-direct (roaming). The latter long-lived trajectories are trapped in the region of non-linear mechanical resonances, which in turn define alternative reaction pathways in phase space. It is demonstrated that resonances associated with periodic orbits provide a dynamical explanation of the quantum mechanical resonances found in the isomerization rate constant calculations by Gezelter and Miller [J. Chem. Phys. {bf 103}, 7868-7876 (1995)]. Evidence of the trapping of trajectories by `sticky resonant periodic orbits is provided by plotting Poincare surfaces of section, and a gap time analysis is carried out in order to investigate the statistical assumption inherent in transition state theory for Ketene isomerization.
We examine the phase space structures that govern reaction dynamics in the absence of critical points on the potential energy surface. We show that in the vicinity of hyperbolic invariant tori it is possible to define phase space dividing surfaces that are analogous to the dividing surfaces governing transition from reactants to products near a critical point of the potential energy surface. We investigate the problem of capture of an atom by a diatomic molecule and show that a normally hyperbolic invariant manifold exists at large atom-diatom distances, away from any critical points on the potential. This normally hyperbolic invariant manifold is the anchor for the construction of a dividing surface in phase space, which defines the outer or loose transition state governing capture dynamics. We present an algorithm for sampling an approximate capture dividing surface, and apply our methods to the recombination of the ozone molecule. We treat both 2 and 3 degree of freedom models with zero total angular momentum. We have located the normally hyperbolic invariant manifold from which the orbiting (outer) transition state is constructed. This forms the basis for our analysis of trajectories for ozone in general, but with particular emphasis on the roaming trajectories.
We provide a dynamical interpretation of the recently identified `roaming mechanism for molecular dissociation reactions in terms of geometrical structures in phase space. These are NHIMs (Normally Hyperbolic Invariant Manifolds) and their stable/unstable manifolds that define transition states for ion-molecule association or dissociation reactions. The associated dividing surfaces rigorously define a roaming region of phase space, in which both reactive and nonreactive trajectories can be trapped for arbitrarily long times.
The transition states and dividing surfaces used to find rate constants for bimolecular reactions are shown to undergo qualitative changes, known as Morse bifurcations, and to exist for a large range of energies, not just immediately above the critical energy for first connection between reactants and products. Specifically, we consider capture between two molecules and the associated transition states for the case of non-zero angular momentum and general attitudes. The capture between an atom and a diatom, and then a general molecule are presented, providing concrete examples of Morse bifurcations of transition states and dividing surfaces. The reduction of the $n$-body systems representing the reactions is discussed and reviewed with comments on the difficulties associated with choosing appropriate charts and the global geometry of the reduced spaces.
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