Invariant Manifolds and Rate Constants in Driven Chemical Reactions

Feldmaier, Matthias; Schraft, Philippe; Bardakcioglu, Robin; Reiff, Johannes; Lober, Melissa; Tschöpe, Martin; Junginger, Andrej; Main, Jörg; Bartsch, Thomas; Hernandez, Rigoberto

doi:10.1021/acs.jpcb.8b10541

Cited by 32 publications

(48 citation statements)

References 103 publications

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“…For this reason, we start several ensembles at different initial times t and piece together the results of each ensemble to fully resolve k(t) of a specific trajectory on the NHIM with good accuracy everywhere. Whereas in other references, thousands to millions of trajectories had to be propagated to somehow resolve a rate constant, 7,21,30,43 with the methods introduced here we are able to adequately resolve the time-dependent reaction rate k(t) propagating just a few hundreds of trajectories.…”

Section: Instantaneous Ensemble Ratesmentioning

confidence: 90%

Phase-space resolved rates in driven multidimensional chemical reactions

Feldmaier

Bardakcioglu

Reiff

et al. 2019

The Journal of Chemical Physics

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Chemical reactions in multidimensional driven systems are typically described by a time-dependent rank-1 saddle associated with one reaction and several orthogonal coordinates (including the solvent bath). To investigate reactions in such systems, we develop a fast and robust method -viz., local manifold analysis (LMA)-for computing the instantaneous decay rate of reactants. Specifically, it computes the instantaneous decay rates along saddle-bound trajectories near the activated complex by exploiting local properties of the stable and unstable manifold associated with the normally hyperbolic invariant manifold (NHIM). The LMA method offers substantial reduction of numerical effort and increased reliability in comparison to direct ensemble integration. It provides an instantaneous flux that can be assigned to every point on the NHIM and which is associated with a trajectory-regardless of whether it is periodic, quasi-periodic, or chaotic-that is bound on the NHIM. The time average of these fluxes in the driven system corresponds to the average rate through a given local section containing the corresponding point on the NHIM. We find good agreement between the results of the LMA and direct ensemble integration obtained using numerically constructed, recrossing-free dividing surfaces.

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Section: Instantaneous Ensemble Ratesmentioning

confidence: 90%

Phase-space resolved rates in driven multidimensional chemical reactions

Feldmaier

Bardakcioglu

Reiff

et al. 2019

The Journal of Chemical Physics

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“…Without such stabilization, these trajectories would depart exponentially fast from the NHIM to either the reactant or the product side due to the limited precision of numerical calculations, as seen in in Ref. 36. It enables the analysis of the long-time dynamics on the NHIM using a stroboscopic Poincaré surface of section (PSOS).…”

Section: Discussionmentioning

confidence: 99%

“…III, the stabilization restricts the dynamics of the trajectories from the (4 + 1)-dimensional extended phase space (including time) of H(x, v x , y, v y , t), as discussed in Ref. 36, to the (2 + 1)-dimensional subspace of the NHIM, viz.…”

Section: B Stabilization Of Trajectories On the Nhimmentioning

confidence: 99%

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

et al. 2020

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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.

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“…In addition, the LD method based detection of NHIM is simple to implement and quickly provides a lay of the dynamical land which is a preliminirary step in applying phase space transport to problems in physics and chemistry. This method can also be used to set up starting guess for other numerical procedures which rely on good initial guess or can also be used in conjunction with machine learning methods for rendering the smooth pieces of NHIM [48,69]. where V B (x, y) is the potential energy function (1).…”

Section: Discussionmentioning

confidence: 99%

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

Naik

Wiggins

2019

Phys. Rev. E

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We present the capability of Lagrangian descriptors 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 normally hyperbolic invariant manifolds (NHIM) and their stable and unstable manifolds, and act as codimenision-1 barriers to phase space transport. The method is applied to classical two and three degrees-of-freedom Hamiltonian systems which have implications for myriad applications in physics and chemistry.

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Invariant Manifolds and Rate Constants in Driven Chemical Reactions

Cited by 32 publications

References 103 publications

Phase-space resolved rates in driven multidimensional chemical reactions

Phase-space resolved rates in driven multidimensional chemical reactions

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

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

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