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Register a new userGeometrical Models of the Phase Space Structures Governing Reaction Dynamics
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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.
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Recently the phase space structures governing reaction dynamics in Hamiltonian systems have been identified and algorithms for their explicit construction have been developed. These phase space structures are induced by saddle type equilibrium points which are characteristic for reaction type dynamics. Their construction is based on a Poincar{e}-Birkhoff normal form. Using tools from the geometric theory of Hamiltonian systems and their reduction we show in this paper how the construction of these phase space structures can be generalized to the case of the relative equilibria of a rotational symmetry reduced $N$-body system. As rotations almost always play an important role in the reaction dynamics of molecules the approach presented in this paper is of great relevance for applications.
The collinear hydrogen exchange reaction is a paradigm system for understanding chemical reactions. It is the simplest imaginable atomic system with $2$ degrees of freedom modeling a chemical reaction, yet it exhibits behaviour that is still not well understood - the reaction rate decreases as a function of energy beyond a critical value. Using lobe dynamics we show how invariant manifolds of unstable periodic orbits guide trajectories in phase space. From the structure of the invariant manifolds we deduce that insufficient transfer of energy between the degrees of freedom causes a reaction rate decrease. In physical terms this corresponds to the free hydrogen atom repelling the whole molecule instead of only one atom from the molecule. We further derive upper and lower bounds of the reaction rate, which are desirable for practical reasons.
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.
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.
Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.
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