Invariant Manifolds and Rate Constants in Driven Chemical Reactions

Chaos in conservative discrete-time systems subjected to parameter drift

Based on the example of a paradigmatic area preserving low-dimensional mapping subjected to different scenarios of parameter drifts, we illustrate that the dynamics can best be understood by following ensembles of initial conditions corresponding to the tori of the initial system. When such ensembles are followed, snapshot tori are obtained, which change their location and shape. Within a time-dependent snapshot chaotic sea, we demonstrate the existence of snapshot stable and unstable foliations. Two easily visualizable conditions for torus breakup are found: one in relation to a discontinuity of the map and the other to a specific snapshot stable manifold, indicating that points of the torus are going to become subjected to strong stretching. In a more general setup, the latter can be formulated in terms of the so-called stable pseudo-foliation, which is shown to be able to extend beyond the instantaneous chaotic sea. The average distance of nearby point pairs initiated on an original torus crosses over into an exponential growth when the snapshot torus breaks up according to the second condition. As a consequence of the strongly non-monotonous change of phase portraits in maps, the exponential regime is found to split up into shorter periods characterized by different finite-time Lyapunov exponents. In scenarios with plateau ending, the divided phase space of the plateau might lead to the Lyapunov exponent averaged over the ensemble of a torus being much smaller than that of the stationary map of the plateau.

Dynamics and decay rates of a time-dependent two-saddle system

The framework of transition state theory (TST) provides a powerful way for analyzing the dynamics of physical and chemical reactions. While TST has already been successfully used to obtain reaction rates for systems with a single time-dependent saddle point, multiple driven saddles have proven challenging because of their fractal-like phase space structure. This paper presents the construction of an approximately recrossing-free dividing surface based on the normally hyperbolic invariant manifold in a time-dependent two-saddle model system. Based on this, multiple methods for obtaining instantaneous (time-resolved) decay rates of the underlying activated complex are presented and their results discussed.

Eyring equation and fluctuation-dissipation far away from equilibrium

Understanding and managing the influence that either external forces or non-equilibrated environments may have on chemical processes is essential for the current and future development of theoretical chemistry. One of the central questions to solve is how to generalize the transition state theory in order to make it applicable in far from equilibrium situations. In this sense, here we propose a way to generalize Eyring's equation based on the definition of an effective thermal energy (temperature) emerging from the coupling of both fast and slow dynamic variables analyzed within the generalized Langevin dynamics scheme. This coupling makes the energy distribution of the fast degrees of freedom not equilibrate because they have been enslaved to the dynamics of the corresponding slow degrees. However, the introduction of the effective thermal energy enables us to restore an effective adiabatic separation of timescales leading to a renormalization of the generalized fluctuation-dissipation theorem. Hence, this procedure opens the possibility to deal with systems far away from equilibrium. A significant consequence of our results is that Eyring's equation is generalized to treat systems under the influence of strong external forces.

Phase space structure and escape time dynamics in a Van der Waals model for exothermic reactions

We study the phase space objects that control the transport in a classical Hamiltonian model for a chemical reaction. This model has been proposed to study the yield of products in an ultracold exothermic reaction. In this model, two features determine the evolution of the system: a Van der Waals force and a short-range force associated with the many-body interactions. In the previous work, small random periodic changes in the direction of the momentum were used to simulate the short-range many-body interactions. In the present work, random Gaussian bumps have been added to the Van der Waals potential energy to simulate the short-range effects between the particles in the system. We compare both variants of the model and explain their differences and similarities from a phase space perspective. To visualize the structures that direct the dynamics in the phase space, we construct a natural Lagrangian descriptor for Hamiltonian systems based on the Maupertuis action S_{0}=∫_{q_{i}}^{q_{f}}p·dq.

Thermal decay rates of an activated complex in a driven model chemical reaction

Recent work has shown that in a nonthermal, multidimensional system, the trajectories in the activated complex possess different instantaneous and time-averaged reactant decay rates. Under dissipative dynamics, it is known that these trajectories, which are bound on the normally hyperbolic invariant manifold (NHIM), converge to a single trajectory over time. By subjecting these dissipative systems to thermal noise, we find fluctuations in the saddle-bound trajectories and their instantaneous decay rates. Averaging over these instantaneous rates results in the decay rate of the activated complex in a thermal system. We find that the temperature dependence of the activated complex decay in a thermal system can be linked to the distribution of the phase space resolved decay rates on the NHIM in the nondissipative case. By adjusting the external driving of the reaction, we show that it is possible to influence how the decay rate of the activated complex changes with rising temperature.

Detecting reactive islands in a system-bath model of isomerization

In this article, we study the conformational isomerization in a solvent using a system-bath model where the phase space structures relevant for the reaction dynamics are revealed. These phase space structures are an integral part of understanding the reaction mechanism, that is the pathways that reactive trajectories undertake, in the presence of a solvent. Our approach involves detecting the analogs of the reactive islands first discussed in the works by Davis, Marston, De Leon, Berne and coauthors in the system-bath model using Lagrangian descriptors. We first present the structure of the reactive islands for the two degrees of freedom system modelling isomerization in the absence of the bath using direct computation of cylindrical (tube) manifolds and verify the Lagrangian descriptor method for detecting the reactive islands. The hierarchy of the reactive islands as indicated in the recent work by Patra and Keshavamurthy is shown to be related to the temporal features in committor probabilities. Next, we investigate the influence of the solvent on the reactive islands that we previously revealed for the two degrees of freedom system and discuss the use of the Lagrangian descriptor in the high-dimensional phase space of the system-bath model.

Identifying reaction pathways in phase space via asymptotic trajectories

In this paper, we revisit the concepts of the reactivity map and the reactivity bands as an alternative to the use of perturbation theory for the determination of the phase space geometry of chemical reactions. We introduce a reformulated metric, called the asymptotic trajectory indicator, and an efficient algorithm to obtain reactivity boundaries. We demonstrate that this method has sufficient accuracy to reproduce phase space structures such as turnstiles for a 1D model of the isomerization of ketene in an external field. The asymptotic trajectory indicator can be applied to higher dimensional systems coupled to Langevin baths as we demonstrate for a 3D model of the isomerization of ketene.

Neural network approach for the dynamics on the normally hyperbolic invariant manifold of periodically driven systems

Chemical reactions in multidimensional systems are often described by a rank-1 saddle, whose stable and unstable manifolds intersect in the normally hyperbolic invariant manifold (NHIM). Trajectories started on the NHIM in principle never leave this manifold when propagated forward or backward in time. However, the numerical investigation of the dynamics on the NHIM is difficult because of the instability of the motion. We apply a neural network to describe time-dependent NHIMs and use this network to stabilize the motion on the NHIM for a periodically driven model system with two degrees of freedom. The method allows us to analyze the dynamics on the NHIM via Poincaré surfaces of section (PSOS) and to determine the transition-state (TS) trajectory as a periodic orbit with the same periodicity as the driving saddle, viz. a fixed point of the PSOS surrounded by near-integrable tori. Based on transition state theory and a Floquet analysis of a periodic TS trajectory we compute the rate constant of the reaction with significantly reduced numerical effort compared to the propagation of a large trajectory ensemble.

Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential

We present a method based on a Lagrangian descriptor for revealing the high-dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include a normally hyperbolic invariant manifold and its stable and unstable manifolds, which act as codimension-1 barriers to phase space transport. In this article, finding the invariant manifolds in high-dimensional phase space will constitute identifying coordinates on these invariant manifolds. The method of Lagrangian descriptor is demonstrated by applying to classical two and three degrees of freedom Hamiltonian systems which have implications for myriad applications in chemistry, engineering, and physics.