Empirical Classification of Trajectory Data: An Opportunity for the Use of Machine Learning in Molecular Dynamics.

Classical Hamiltonian trajectories are initiated at random points in phase space on a fixed energy shell of a model two degrees of freedom potential, consisting of two interacting minima in an otherwise flat energy plane of infinite extent. Below the energy of the plane, the dynamics are demonstrably chaotic. However, most of the work in this paper involves trajectories at a fixed energy that is 1% above that of the plane, in which regime the dynamics exhibit behavior characteristic of chaotic scattering. The trajectories are analyzed without reference to the potential, as if they had been generated in a typical direct molecular dynamics simulation. The questions addressed are whether one can recover useful information about the structures controlling the dynamics in phase space from the trajectory data alone, and whether, despite the at least partially chaotic nature of the dynamics, one can make statistically meaningful predictions of trajectory outcomes from initial conditions. It is found that key unstable periodic orbits, which can be identified on the analytical potential, appear by simple classification of the trajectories, and that the specific roles of these periodic orbits in controlling the dynamics are also readily discerned from the trajectory data alone. Two different approaches to predicting trajectory outcomes from initial conditions are evaluated, and it is shown that the more successful of them has ∼90% success. The results are compared with those from a simple neural network, which has higher predictive success (97%) but requires the information obtained from the "by-hand" analysis to achieve that level. Finally, the dynamics, which occur partly on the very flat region of the potential, show characteristics of the much-studied phenomenon called "roaming." On this potential, it is found that roaming trajectories are effectively "failed" periodic orbits and that angular momentum can be identified as a key controlling factor, despite the fact that it is not a strictly conserved quantity. It is also noteworthy that roaming on this potential occurs in the absence of a "roaming saddle," which has previously been hypothesized to be a necessary feature for roaming to occur.

Roaming: A Phase Space Perspective.

In this review we discuss the recently described roaming mechanism for chemical reactions from the point of view of nonlinear dynamical systems in phase space. The recognition of the roaming phenomenon shows the need for further developments in our fundamental understanding of basic reaction dynamics, as is made clear by considering some questions that cut across most studies of roaming: Is the dynamics statistical? Can transition state theory be applied to estimate roaming reaction rates? What role do saddle points on the potential energy surface play in explaining the behavior of roaming trajectories? How do we construct a dividing surface that is appropriate for describing the transformation from reactants to products for roaming trajectories? How should we define the roaming region? We show that the phase space perspective on reaction dynamics provides the setting in which these questions can be properly framed and answered. We illustrate these ideas by considering photodissociation of formaldehyde. The phase-space formulation allows an unambiguous description of all possible reactive events, which also allows us to uncover the phase space mechanism that explains which trajectories roam, as opposed to evolving toward a different reactive event.

Toward Understanding the Roaming Mechanism in H + MgH → Mg + HH Reaction.

The roaming mechanism in the reaction H + MgH →Mg + HH is investigated by classical and quantum dynamics employing an accurate ab initio three-dimensional ground electronic state potential energy surface. The reaction dynamics are explored by running trajectories initialized on a four-dimensional dividing surface anchored on three-dimensional normally hyperbolic invariant manifold associated with a family of unstable orbiting periodic orbits in the entrance channel of the reaction (H + MgH). By locating periodic orbits localized in the HMgH well or involving H orbiting around the MgH diatom, and following their continuation with the total energy, regions in phase space where reactive or nonreactive trajectories may be trapped are found. In this way roaming reaction pathways are deduced in phase space. Patterns similar to periodic orbits projected into configuration space are found for the quantum bound and resonance eigenstates. Roaming is attributed to the capture of the trajectories in the neighborhood of certain periodic orbits. The complex forming trajectories in the HMgH well can either return to the radical channel or "roam" to the MgHH minimum from where the molecule may react.

Phase space barriers and dividing surfaces in the absence of critical points of the potential energy: Application to roaming in ozone.

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 degrees 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.

Phase Space Structures Explain Hydrogen Atom Roaming in Formaldehyde Decomposition.

We re-examine the prototypical roaming reaction--hydrogen atom roaming in formaldehyde decomposition--from a phase space perspective. Specifically, we address the question "why do trajectories roam, rather than dissociate through the radical channel?" We describe and compute the phase space structures that define and control all possible reactive events for this reaction, as well as provide a dynamically exact description of the roaming region in phase space. Using these phase space constructs, we show that in the roaming region, there is an unstable periodic orbit whose stable and unstable manifolds define a conduit that both encompasses all roaming trajectories exiting the formaldehyde well and shepherds them toward the H2···CO well.

Reaction Path Bifurcation in an Electrocyclic Reaction: Ring-Opening of the Cyclopropyl Radical.

Following previous work [J. Chem. Phys. 2013, 139, 154108] on a simple model of a reaction with a post-transition state valley ridge inflection point, we study the chemically important example of the electrocyclic cyclopropyl radical ring-opening reaction using direct dynamics and a reduced dimensional potential energy surface. The overall reaction requires con- or disrotation of the methylenes, but the initial stage of the ring-opening involves substantial internal rotation of only one methylene. The reaction path bifurcation is then associated with the relative sense of rotation of the second methylene. Clear deviations of reactive trajectories from the disrotatory intrinsic reaction coordinate (IRC) for the ring-opening are observed and the dynamical mechanism is discussed. Several features observed in the model system are found to be preserved in the more complex and higher dimensional ring-opening reaction. Most notable is the sensitivity of the reaction mechanism to the shape of the potential manifested as a Newtonian kinetic isotope effect upon deuterium substitution of one of the methylene hydrogens. Dependence of the product yield on frictional dissipation representing external environmental effects is also presented. The dynamics of the post-transition state cyclopropyl radical ring-opening are discussed in detail, and the use of low dimensional models as tools to analyze complicated organic reaction mechanisms is assessed in the context of this reaction.

Nonstatistical dynamics on the caldera.

We explore both classical and quantum dynamics of a model potential exhibiting a caldera: that is, a shallow potential well with two pairs of symmetry related index one saddles associated with entrance/exit channels. Classical trajectory simulations at several different energies confirm the existence of the "dynamical matching" phenomenon originally proposed by Carpenter, where the momentum direction associated with an incoming trajectory initiated at a high energy saddle point determines to a considerable extent the outcome of the reaction (passage through the diametrically opposing exit channel). By studying a "stretched" version of the caldera model, we have uncovered a generalized dynamical matching: bundles of trajectories can reflect off a hard potential wall so as to end up exiting predominantly through the transition state opposite the reflection point. We also investigate the effects of dissipation on the classical dynamics. In addition to classical trajectory studies, we examine the dynamics of quantum wave packets on the caldera potential (stretched and unstretched). These computations reveal a quantum mechanical analogue of the "dynamical matching" phenomenon, where the initial expectation value of the momentum direction for the wave packet determines the exit channel through which most of the probability density passes to product.

Will water act as a photocatalyst for cluster phase chemical reactions? Vibrational overtone-induced dehydration reaction of methanediol.

The possibility of water catalysis in the vibrational overtone-induced dehydration reaction of methanediol is investigated using ab initio dynamical simulations of small methanediol-water clusters. Quantum chemistry calculations employing clusters with one or two water molecules reveal that the barrier to dehydration is lowered by over 20 kcal/mol because of hydrogen-bonding at the transition state. Nevertheless, the simulations of the reaction dynamics following OH-stretch excitation show little catalytic effect of water and, in some cases, even show an anticatalytic effect. The quantum yield for the dehydration reaction exhibits a delayed threshold effect where reaction does not occur until the photon energy is far above the barrier energy. Unlike thermally induced reactions, it is argued that competition between reaction and the irreversible dissipation of photon energy may be expected to raise the dynamical threshold for the reaction above the transition state energy. It is concluded that quantum chemistry calculations showing barrier lowering are not sufficient to infer water catalysis in photochemical reactions, which instead require dynamical modeling.

Water catalysis and anticatalysis in photochemical reactions: observation of a delayed threshold effect in the reaction quantum yield.

The possible catalysis of photochemical reactions by water molecules is considered. Using theoretical simulations, we investigate the HF-elimination reaction of fluoromethanol in small water clusters initiated by the overtone excitation of the hydroxyl group. The reaction occurs in competition with the process of water evaporation that dissipates the excitation and quenches the reaction. Although the transition state barrier is stabilized by over 20 kcal/mol through hydrogen bonding with water, the quantum yield versus energy shows a pronounced delayed threshold that effectively eliminates the catalytic effect. It is concluded that the quantum chemistry calculations of barrier lowering are not sufficient to infer water catalysis in some photochemical reactions, which instead require dynamical modeling.

Vibrational overtone induced elimination reactions within hydrogen-bonded molecular clusters: the dynamics of water catalyzed reactions in CH2FOH.H2On.

The dynamics of the light initiated OH-overtone induced elimination reactions CH(2)FOH.(H(2)O)(n) + hnu--> HF + CH(2)O + n(H(2)O), n = 1-3, are studied using classical trajectory simulations where the ab initio potential energy surface is computed "on-the-fly". Hydrogen bonding to the water is found to lower the barrier to reaction by over 20 kcal mol(-1) and modifies the mechanism to a concerted multiple H-atom transfer process. The reaction process is found to occur on a rapid timescale, <100 fs, and involves the hydronium ion as an intermediate. An essential aspect of dynamics is the successful competition of reaction with energy dissipation through water evaporation from the cluster.