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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., Comment: 35 pages, 3 figures

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 Poincar\'e 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., Comment: 22 pages, 9 figures

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., Comment: 20 pages, 6 figures