Unimolecular reactions and phase space bottlenecks

Toward an Accurate Black-Box Tool for the Kinetics of Gas-Phase Reactions Involving Barrier-less Elementary Steps

An enhanced computational protocol has been devised for the accurate characterization of gas-phase barrier-less reactions in the framework of the reaction-path (RP) and variable reaction coordinate variational transition-state theory. In particular, the synergistic combination of density functional theory and Monte Carlo sampling to optimize reactive fluxes led to a reliable yet effective computational workflow. A black-box strategy has been developed for selecting the most suited density functional with reference to a high-level one-dimensional reference potential. At the same time, different descriptions of hindered rotations are automatically selected, depending on the corresponding harmonic frequencies along the RP. The performance of the new tool is investigated by means of two prototypical reactions involving different degrees of static and dynamic correlation, namely, H2S + Cl and CH3 + CH3. The remarkable agreement of the computed kinetic parameters with the available experimental data confirms the accuracy and robustness of the proposed approach. Together with their intrinsic interest, these results also pave the way toward systematic investigations of gas-phase reactions involving barrier-less elementary steps by a reliable, user-friendly tool, which can be confidently used also by nonspecialists.

Reactive Vortexes in a Naturally Activated Process: Non-Diffusive Rotational Fluxes at Transition State Uncovered by Persistent Homology

The dynamics of reaction coordinates during barrier-crossing are key to understanding activated processes in complex systems such as proteins. The default assumption from Kramers' physical intuition is that of a diffusion process. However, the dynamics of barrier-crossing in natural complex molecules are largely unexplored. Here we investigate the transition dynamics of alanine dipeptide isomerization, the simplest complex system with a large number of non-reaction coordinates that can serve as an adequate thermal bath feeding energy into the reaction coordinates. We separate conformations along the time axis and construct the dynamic probability surface of reaction. We quantify its topological structure and rotational flux using persistent homology and differential form. Our results uncovered a region with a strong reactive vortex in the configuration-time space, where the highest probability peak and the transition state ensemble are located. This reactive region contains strong rotational fluxes: Most reactive trajectories swirl multiple times around this region in the subspace of the two most important reaction coordinates. Furthermore, the rotational fluxes result from cooperative movement along the isocommitter surfaces and orthogonal barrier-crossing. Overall, our findings offer a first glimpse into the reactive vortex regions that characterize the non-diffusive dynamics of barrier-crossing of a naturally occurring activation process.

Phase space geometry of isolated to condensed chemical reactions

The complexity of gas and condensed phase chemical reactions has generally been uncovered either approximately through transition state theories or exactly through (analytic or computational) integration of trajectories. These approaches can be improved by recognizing that the dynamics and associated geometric structures exist in phase space, ensuring that the propagator is symplectic as in velocity-Verlet integrators and by extending the space of dividing surfaces to optimize the rate variationally, respectively. The dividing surface can be analytically or variationally optimized in phase space, not just over configuration space, to obtain more accurate rates. Thus, a phase space perspective is of primary importance in creating a deeper understanding of the geometric structure of chemical reactions. A key contribution from dynamical systems theory is the generalization of the transition state (TS) in terms of the normally hyperbolic invariant manifold (NHIM) whose geometric phase-space structure persists under perturbation. The NHIM can be regarded as an anchor of a dividing surface in phase space and it gives rise to an exact non-recrossing TS theory rate in reactions that are dominated by a single bottleneck. Here, we review recent advances of phase space geometrical structures of particular relevance to chemical reactions in the condensed phase. We also provide conjectures on the promise of these techniques toward the design and control of chemical reactions.

Intramolecular vibrational energy redistribution and the quantum ergodicity transition: a phase space perspective

The onset of facile intramolecular vibrational energy flow can be related to features in the connected network of anharmonic resonances in the classical phase space.

Open quantum maps from complex scaling of kicked scattering systems

We derive open quantum maps from periodically kicked scattering systems and discuss the computation of their resonance spectra in terms of theoretically grounded methods, such as complex scaling and sufficiently weak absorbing potentials. In contrast, we also show that current implementations of open quantum maps, based on strong absorptive or even projective openings, fail to produce the resonance spectra of kicked scattering systems. This comparison pinpoints flaws in current implementations of open quantum maps, namely, the inability to separate resonance eigenvalues from the continuum as well as the presence of diffraction effects due to strong absorption. The reported deviations from the true resonance spectra appear, even if the openings do not affect the classical trapped set, and become appreciable for shorter-lived resonances, e.g., those associated with chaotic orbits. This makes the open quantum maps, which we derive in this paper, a valuable alternative for future explorations of quantum-chaotic scattering systems, for example, in the context of the fractal Weyl law. The results are illustrated for a quantum map model whose classical dynamics exhibits key features of ionization and a trapped set which is organized by a topological horseshoe.

Using the small alignment index chaos indicator to characterize the vibrational dynamics of a molecular system: LiNC-LiCN

A study of the dynamical characteristics of the phase space corresponding to the vibrations of the LiNC-LiCN molecule using an analysis based on the small alignment index (SALI) is presented. SALI is a good indicator of chaos that can easily determine whether a given trajectory is regular or chaotic regardless of the dimensionality of the system, and can also provide a wealth of dynamical information when conveniently implemented. In two-dimensional (2D) systems SALI maps are computed as 2D phase space representations, where the SALI asymptotic values are represented in color scale. We show here how these maps provide full information on the dynamical phase space structure of the LiNC-LiCN system, even quantifying numerically the volume of the different zones of chaos and regularity as a function of the molecule excitation energy.

Crossing the dividing surface of transition state theory. III. Once and only once. Selecting reactive trajectories

The purpose of the present work is to determine initial conditions that generate reacting, recrossing-free trajectories that cross the conventional dividing surface of transition state theory (i.e., the plane in configuration space passing through a saddle point of the potential energy surface and perpendicular to the reaction coordinate) without ever returning to it. Local analytical equations of motion valid in the neighborhood of this planar surface have been derived as an expansion in Poisson brackets. We show that the mere presence of a saddle point implies that reactivity criteria can be quite simply formulated in terms of elements of this series, irrespective of the shape of the potential energy function. Some of these elements are demonstrated to be equal to a sum of squares and thus to be necessarily positive, which has a profound impact on the dynamics. The method is then applied to a three-dimensional model describing an atom-diatom interaction. A particular relation between initial conditions is shown to generate a bundle of reactive trajectories that form reactive cylinders (or conduits) in phase space. This relation considerably reduces the phase space volume of initial conditions that generate recrossing-free trajectories. Loci in phase space of reactive initial conditions are presented. Reactivity is influenced by symmetry, as shown by a comparative study of collinear and bent transition states. Finally, it is argued that the rules that have been derived to generate reactive trajectories in classical mechanics are also useful to build up a reactive wave packet.

Roles of dynamical symmetry breaking in driving oblate-prolate transitions of atomic clusters

This paper explores the driving mechanisms for structural transitions of atomic clusters between oblate and prolate isomers. We employ the hyperspherical coordinates to investigate structural dynamics of a seven-atom cluster at a coarse-grained level in terms of the dynamics of three gyration radii and three principal axes, which characterize overall mass distributions of the cluster. Dynamics of gyration radii is governed by two kinds of forces. One is the potential force originating from the interactions between atoms. The other is the dynamical forces called the internal centrifugal forces, which originate from twisting and shearing motions of the system. The internal centrifugal force arising from twisting motions has an effect of breaking the symmetry between two gyration radii. As a result, in an oblate isomer, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two largest gyration radii is crucial in triggering structural transitions into prolate isomers. In a prolate isomer, on the other hand, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two smallest gyration radii is crucial in triggering structural transitions into oblate isomers. Activation of a twisting motion that switches the movement patterns of three principal axes is also important for the onset of structural transitions between oblate and prolate isomers. Based on these trigger mechanisms, we finally show that selective activations of specific gyration radii and twisting motions, depending on the isomer of the cluster, can effectively induce structural transitions of the cluster. The results presented here could provide further insights into the control of molecular reactions.

Crossing the dividing surface of transition state theory. II. Recrossing times for the atom-diatom interaction

We consider a triatomic system with zero total angular momentum and demonstrate that, no matter how complicated the anharmonic part of the potential energy function, classical dynamics in the vicinity of a saddle point is constrained by symmetry properties. At short times and at not too high energies, recrossing dynamics is largely determined by elementary local structural parameters and thus can be described in configuration space only. Conditions for recrossing are given in the form of inequalities involving structural parameters only. Explicit expressions for recrossing times, valid for microcanonical ensembles, are shown to obey interesting regularities. In a forward reaction, when the transition state is nonlinear and tight enough, one-fourth of the trajectories are expected to recross the plane R = R* (where R* denotes the position of the saddle point) within a short time. Another fourth of them are expected to have previously recrossed at a short negative time, i.e., close to the saddle point. These trajectories do not contribute to the reaction rate. The reactive trajectories that obey the transition state model are to be found in the remaining half. However, no conclusion can be derived for them, except that if recrossings occur, then they must either take place in the distant future or already have taken place in the remote past, i.e., far away from the saddle point. Trajectories that all cross the plane R = R* at time t = 0, with the same positive translational momentum P(R*) can be partitioned into two sets, distinguished by the parity of their initial conditions; both sets have the same average equation of motion up to and including terms cubic in time. Coordination is excellent in the vicinity of the saddle point but fades out at long (positive or negative) times, i.e., far away from the transition state.

Topological analysis of chaotic transport through a ballistic atom pump

We examine a system consisting of two reservoirs of particles connected by a channel. In the channel are two oscillating repulsive potential-energy barriers. It is known that such a system can transport particles from one reservoir to the other, even when the chemical potentials in the reservoirs are equal. We use computations and the theory of chaotic transport to study this system. Chaotic transport is described by passage around or through a heteroclinic tangle. Topological properties of the tangle are described using a generalization of homotopic lobe dynamics, which is a theory that gives some properties of intermediate-time behavior from properties of short-time behavior. We compare these predicted properties with direct computation of trajectories.

Reactivity boundaries for chemical reactions associated with higher-index and multiple saddles

Reactivity boundaries that divide the origin and destination of trajectories are of crucial importance to reveal the mechanism of reactions, which was recently found to exist robustly even at high energies for index 1 saddles [Phys. Rev. Lett. 105, 048304 (2010)]. Here we revisit the concept of the reactivity boundary and propose a more general definition that can involve a single reaction associated with a bottleneck composed of higher-index saddles and/or several saddle points with different indices, where the normal form theory, based on expansion around a single stationary point, does not work. We numerically demonstrate the reactivity boundary by using a reduced model system of the H(5)(+) cation where the proton exchange reaction takes place through a bottleneck composed of two index 2 saddle points and two index 1 saddle points. The cross section of the reactivity boundary in the reactant region of the phase space reveals which initial conditions are effective in making the reaction happen and thus sheds light on the reaction mechanism.

Isomerization dynamics of a buckled nanobeam

We analyze the dynamics of a model of a nanobeam under compression. The model is a two-mode truncation of the Euler-Bernoulli beam equation subject to compressive stress applied at both ends. We consider parameter regimes where the first mode is unstable and the second mode can be either stable or unstable, and the remaining modes (neglected) are always stable. Material parameters used correspond to a silicon nanobeam. The two-mode model Hamiltonian is the sum of a (diagonal) kinetic energy term and a potential energy term. The form of the potential energy function suggests an analogy with isomerization reactions in chemistry, where "isomerization" here corresponds to a transition between two stable beam configurations. We therefore study the dynamics of the buckled beam using the conceptual framework established for the theory of isomerization reactions. When the second mode is stable the potential energy surface has an index one saddle, and when the second mode is unstable the potential energy surface has an index two saddle and two index one saddles. Symmetry of the system allows us to readily construct a phase space dividing surface between the two "isomers" (buckled states); we rigorously prove that, in a specific energy range, it is a normally hyperbolic invariant manifold. The energy range is sufficiently wide that we can treat the effects of the index one and index two saddles on the isomerization dynamics in a unified fashion. We have computed reactive fluxes, mean gap times, and reactant phase space volumes for three stress values at several different energies. In all cases the phase space volume swept out by isomerizing trajectories is considerably less than the reactant density of states, proving that the dynamics is highly nonergodic. The associated gap time distributions consist of one or more "pulses" of trajectories. Computation of the reactive flux correlation function shows no sign of a plateau region; rather, the flux exhibits oscillatory decay, indicating that, for the two-mode model in the physical regime considered, a rate constant for isomerization does not exist.

Bottlenecks to vibrational energy flow in carbonyl sulfide: structures and mechanisms

Finding the causes for the nonstatistical vibrational energy relaxation in the planar carbonyl sulfide (OCS) molecule is a longstanding problem in chemical physics: Not only is the relaxation incomplete long past the predicted statistical relaxation time but it also consists of a sequence of abrupt transitions between long-lived regions of localized energy modes. We report on the phase space bottlenecks responsible for this slow and uneven vibrational energy flow in this Hamiltonian system with three degrees of freedom. They belong to a particular class of two-dimensional invariant tori which are organized around elliptic periodic orbits. We relate the trapping and transition mechanisms with the linear stability of these structures.

Conformational isomerization kinetics of pent-1-en-4-yne with 3,330 cm-1 of internal energy measured by dynamic rotational spectroscopy

We demonstrate the application of molecular rotational spectroscopy to measure the conformation isomerization rate of vibrationally excited pent-1-en-4-yne (pentenyne). The rotational spectra of single quantum states of pentenyne are acquired by using a combination of IR-Fourier transform microwave double-resonance spectroscopy and high-resolution, single-photon IR spectroscopy. The quantum states probed in these experiments have energy eigenvalues of approximately 3,330 cm(-1) and lie above the barrier to conformational isomerization. At this energy, the presence of intramolecular vibrational energy redistribution (IVR) is indicated through the extensive local perturbations found in the high-resolution rotation-vibration spectrum of the acetylenic C-H stretch normal-mode fundamental. The fact that the IVR process produces isomerization is deduced through a qualitatively different appearance of the excited-state rotational spectra compared with the pure rotational spectra of pentenyne. The rotational spectra of the vibrationally excited molecular eigenstates display coalescence between the characteristic rotational frequencies of the stable cis and skew conformations of the molecule. This coalescence is observed for quantum states prepared from laser excitation originating in the ground vibrational state of either of the two stable conformers. Experimental isomerization rates are extracted by using a three-state Bloch model of the dynamic rotational spectra that includes the effects of chemical exchange between the stable conformations. The time scale for the conformational isomerization rate of pentenyne at total energy of 3,330 cm(-1) is approximately 25 ps and is 50 times slower than the microcanonical isomerization rate predicted by the statistical Rice-Ramsperger-Kassel-Marcus theory.

Measuring picosecond isomerization kinetics via broadband microwave spectroscopy

The rotational spectrum of a highly excited molecule is qualitatively different from its pure rotational spectrum and contains information about the intramolecular dynamics. We have developed a broadband Fourier transform microwave spectrometer that uses chirped-pulse excitation to measure a rotational spectrum in the 7.5- to 18.5-gigahertz range in a single shot and thereby reduces acquisition time sufficiently to couple molecular rotational spectroscopy with tunable laser excitation. After vibrationally exciting a single molecular conformation of cyclopropane carboxaldehyde above the barrier to C-C single-bond isomerization, we applied line-shape analysis of the dynamic rotational spectrum to reveal a product yield and picosecond reaction rate that were significantly different from statistical predictions. The technique should be widely applicable to dynamical studies of radical intermediates, molecular complexes, and conformationally flexible molecules with biological interest.

Escape time statistics for mushroom billiards

Chaotic orbits of the mushroom billiards display intermittent behaviors. We investigate statistical properties of this system by constructing an infinite partition on the chaotic part of a Poincaré surface, which illustrates details of chaotic dynamics. Each piece of the infinite partition has a unique escape time from the half disk region, and from this result it is shown that, for fixed values of the system parameters, the escape time distribution obeys a power law 1/t(esc)(3).

Nonstatistical effects in the dissociation of ethyl radical: Finding order in chaos

How does one identify order in complex dynamical systems? A Born-Oppenheimer molecular dynamics simulation of the dissociation of ethyl radical, C(2)H(5), produces an ensemble of classical trajectories which are decomposed in the time-frequency domain using wavelets. A time-dependent scalar metric, the normalized instantaneous orbital complexity, is constructed and shown to correlate not only to the more conventional Lyapunov exponents but also to the dissociation time for an individual trajectory. The analysis of the ensemble of trajectories confirms that the long-lived trajectories are associated with a low degree of ergodicity. While the analysis of molecular dissociation dynamics is the narrow focus of the present work, the method is more general for discovery and identification of ordered regimes within large sets of chaotic data.

Escape of trajectories from a vase-shaped cavity

We consider the escape of ballistic trajectories from an open, vase-shaped cavity. Such a system serves as a model for microwaves escaping from a cavity or electrons escaping from a microjunction. Fixing the initial position of a particle and recording its escape time as a function of the initial launch direction, the resulting escape-time plot shows "epistrophic fractal" structure--repeated structure within structure at all levels of resolution with new features appearing in the fractal at longer time scales. By launching trajectories simultaneously in all directions (modeling an outgoing wave), a detector placed outside the cavity would measure a train of escaping pulses. We connect the structure of this chaotic pulse train with the fractal structure of the escape-time plot.

Phase-space reaction network on a multisaddle energy landscape: HCN isomerization

By using the HCN/CNH isomerization reaction as an illustrative vehicle of chemical reactions on multisaddle energy landscapes, we give explicit visualizations of molecular motions associated with a straight-through reaction tube in the phase space inside which all reactive trajectories pass from one basin to another, with eliminating recrossing trajectories in the configuration space. This visualization provides us with a chemical intuition of how chemical species "walk along" the reaction-rate slope in the multidimensional phase space compared with the intrinsic reaction path in the configuration space. The distinct nonergodic features in the two different HCN and CNH wells can be easily demonstrated by a section of Poincare surface of section in those potential minima, which predicts in a priori the pattern of trajectories residing in the potential well. We elucidate the global phase-space structure which gives rise to the non-Markovian dynamics or the dynamical correlation of sequential multisaddle chemical reactions. The phase-space structure relevant to the controllability of the product state in chemical reactions is also discussed.

A Quasiclassical Trajectory Study of the Reaction H + O2 ⇔ OH + O with the O2 Reagent Vibrationally Excited

Quasiclassical trajectories have been computed on the Melius-Blint (MB) Potential Energy Surface (PES) and on the Double Many-Body Expansion (DMBE) IV PES of Pastrana et al. describing the H + O(2) <==> OH + O reaction with the nonrotating (J = 0) O(2) reagent vibrationally excited to levels v = 6, 7, 8, 9, and 10 at four temperatures: 1000, 2000, 3000, and 4000 K. The vibrational energy levels were selected by using a semiclassical Einstein-Brillouin-Keller (EBK) quantization procedure while the relative translational energy was sampled from a Boltzmann weighted distribution. The rate coefficient for the formation of the OH + O products is seen to increase monotonically with quantum number and nearly monotonically with temperature. On the MB PES, at T = 1000 K, the total rate coefficient increases by a factor of 5.2 as the initial vibrational quantum number of the O(2) diatom increases from v = 6 to v = 10. For T = 2000 K, this factor drops to 3.3, to 2.9 for T = 3000 K, and to 2.5 for T = 4000 K. On the DMBE IV PES, at T = 1000 K the total rate coefficient increases by a factor of 4.1 as the initial vibrational quantum number of the O(2) diatom increases from v = 6 to v = 10. For T = 2000 K, this factor drops to 3.5, to 2.1 for T = 3000 K, and to 2.0 for T = 4000 K. The less-direct group (defined below) of trajectories is sensitive to the initial O(2) vibrational excitation in several different temperature ranges, apparently retaining the effect of reagent vibrational excitation. The more-direct group (defined below) of trajectories does not exhibit this behavior. Reagent vibrational excitation does not increase the total rate coefficients for the title reaction more than the increase due to a simple temperature increase. The less-direct and more-direct groups of trajectories differ in their contribution to the rate coefficient for the title reaction. In particular, at T = 4000 K, the two PESs used in this work differ dramatically in the roles of the less-direct and more-direct trajectories. The behavior of the more-direct and less-direct groups of trajectories can be understood in terms of the efficiency of intramolecular vibrational energy transfer. This work utilizes the recently introduced PES Library, POTLIB 2001, which made the comparisons between the two PESs discussed in this work possible in a very straightforward way.

NONSTATISTICAL DYNAMICS IN THERMAL REACTIONS OF POLYATOMIC MOLECULES

A brief review is presented of post-RRKM models for unimolecular reaction kinetics. The dynamics of the gas-phase S(N)2 reaction are discussed, and the important role of phase-space bottlenecks is highlighted. The remainder of the review is devoted to experimental and trajectory simulation results on thermal reactions of organic molecules that exhibit nonstatistical dynamics quite unlike that seen in the S(N)2 reaction. Specifically, the intermediates generated in these reactions decay much faster than RRKM theory would predict, and often with bimodal or multimodal lifetime distributions. A qualitative model for this behavior based on overlaps of transitional regions in the molecular phase space is discussed.

Chaos-induced pulse trains in the ionization of hydrogen

We predict that a hydrogen atom in parallel electric and magnetic fields, excited by a short laser pulse to an energy above the classical saddle, ionizes via a train of electron pulses. These pulses are a consequence of classical chaos induced by the magnetic field. We connect the structure of this pulse train (e.g., pulse size and spacing) to fractal structure in the classical dynamics. This structure displays a weak self-similarity, which we call "epistrophic self-similarity." We demonstrate how this self-similarity is reflected in the pulse train.

Geometry and topology of escape. I. Epistrophes

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an "escape-time plot." For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set. In this article we prove the existence of regular repeated sequences, called "epistrophes," which occur at all levels of resolution within the escape-time plot. (The word "epistrophe" comes from rhetoric and means "a repeated ending following a variable beginning.") The epistrophes give the escape-time plot a certain self-similarity, called "epistrophic" self-similarity, which need not imply either strict or asymptotic self-similarity.