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Register a new userRoaming dynamics in ion-molecule reactions: phase space reaction pathways and geometrical interpretation
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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.
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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.
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.
Hamiltonian dynamical systems possessing equilibria of ${saddle} times {centre} times...times {centre}$ stability type display emph{reaction-type dynamics} for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow emph{bottlenecks} created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a emph{Normally Hyperbolic Invariant Manifold (NHIM)}, whose emph{stable and unstable manifolds} have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) emph{dividing surface} which locally divides an energy surface into two components (`reactants and `products), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in emph{transition state theory} where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface. We discuss three presentations of the energy surface and the phase space structures contained in it for 2-degree-of-freedom (DoF) systems in the threedimensional space $R^3$, and two schematic models which capture many of the essential features of the dynamics for $n$-DoF systems. In addition, we elucidate the structure of the NHIM.
Recent studies have found an unusual way of dissociation in formaldehyde. It can be characterized by a hydrogen atom that separates from the molecule, but instead of dissociating immediately it roams around the molecule for a considerable amount of time and extracts another hydrogen atom from the molecule prior to dissociation. This phenomenon has been coined roaming and has since been reported in the dissociation of a number of other molecules. In this paper we investigate roaming in Chesnavichs CH$_4^+$ model. During dissociation the free hydrogen must pass through three phase space bottleneck for the classical motion, that can be shown to exist due to unstable periodic orbits. None of these orbits is associated with saddle points of the potential energy surface and hence related to transition states in the usual sense. We explain how the intricate phase space geometry influences the shape and intersections of invariant manifolds that form separatrices, and establish the impact of these phase space structures on residence times and rotation numbers. Ultimately we use this knowledge to attribute the roaming phenomenon to particular heteroclinic intersections.
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