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Yamashita T, Miyamura N, Kawai S. Classification of the HCN isomerization reaction dynamics in Ar buffer gas via machine learning. J Chem Phys 2023; 159:124116. [PMID: 38127399 DOI: 10.1063/5.0156313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2023] [Accepted: 09/07/2023] [Indexed: 12/23/2023] Open
Abstract
The effect of the presence of Ar on the isomerization reaction HCN ⇄ CNH is investigated via machine learning. After the potential energy surface function is developed based on the CCSD(T)/aug-cc-pVQZ level ab initio calculations, classical trajectory simulations are performed. Subsequently, with the aim of extracting insights into the reaction dynamics, the obtained reactivity, that is, whether the reaction occurs or not under a given initial condition, is learned as a function of the initial positions and momenta of all the atoms in the system. The prediction accuracy of the trained model is greater than 95%, indicating that machine learning captures the features of the phase space that affect reactivity. Machine learning models are shown to successfully reproduce reactivity boundaries without any prior knowledge of classical reaction dynamics theory. Subsequent analyses reveal that the Ar atom affects the reaction by displacing the effective saddle point. When the Ar atom is positioned close to the N atom (resp. the C atom), the saddle point shifts to the CNH (HCN) region, which disfavors the forward (backward) reaction. The results imply that analyses aided by machine learning are promising tools for enhancing the understanding of reaction dynamics.
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Affiliation(s)
- Takefumi Yamashita
- Laboratory for Systems Biology and Medicine, Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
- Department of Physical Chemistry, School of Pharmacy and Pharmaceutical Sciences, Hoshi University, 2-4-41 Ebara, Shinagawa-ku, Tokyo 142-8501, Japan
| | - Naoaki Miyamura
- Laboratory for Systems Biology and Medicine, Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
| | - Shinnosuke Kawai
- Department of Chemistry, Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan
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2
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Nagahata Y, Hernandez R, Komatsuzaki T. Phase space geometry of isolated to condensed chemical reactions. J Chem Phys 2021; 155:210901. [PMID: 34879678 DOI: 10.1063/5.0059618] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
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.
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Affiliation(s)
- Yutaka Nagahata
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Tamiki Komatsuzaki
- Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0 020, Japan
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Nagahata Y, Borondo F, Benito RM, Hernandez R. Identifying reaction pathways in phase space via asymptotic trajectories. Phys Chem Chem Phys 2020; 22:10087-10105. [PMID: 32342955 DOI: 10.1039/c9cp06610a] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
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.
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Affiliation(s)
- Yutaka Nagahata
- Department of Chemistry, Johns Hopkins University, Baltimore, MD 21218, USA.
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Tschöpe M, Feldmaier M, Main J, Hernandez R. Neural network approach for the dynamics on the normally hyperbolic invariant manifold of periodically driven systems. Phys Rev E 2020; 101:022219. [PMID: 32168686 DOI: 10.1103/physreve.101.022219] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2019] [Accepted: 01/29/2020] [Indexed: 05/21/2023]
Abstract
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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Affiliation(s)
- Martin Tschöpe
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Matthias Feldmaier
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Jörg Main
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
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Feldmaier M, Schraft P, Bardakcioglu R, Reiff J, Lober M, Tschöpe M, Junginger A, Main J, Bartsch T, Hernandez R. Invariant Manifolds and Rate Constants in Driven Chemical Reactions. J Phys Chem B 2019; 123:2070-2086. [DOI: 10.1021/acs.jpcb.8b10541] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Matthias Feldmaier
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Philippe Schraft
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Robin Bardakcioglu
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Johannes Reiff
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Melissa Lober
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Martin Tschöpe
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Andrej Junginger
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Jörg Main
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Thomas Bartsch
- Centre for Nonlinear Mathematics and Applications, Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, United States
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Schraft P, Junginger A, Feldmaier M, Bardakcioglu R, Main J, Wunner G, Hernandez R. Neural network approach to time-dependent dividing surfaces in classical reaction dynamics. Phys Rev E 2018; 97:042309. [PMID: 29758767 DOI: 10.1103/physreve.97.042309] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Indexed: 05/21/2023]
Abstract
In a dynamical system, the transition between reactants and products is typically mediated by an energy barrier whose properties determine the corresponding pathways and rates. The latter is the flux through a dividing surface (DS) between the two corresponding regions, and it is exact only if it is free of recrossings. For time-independent barriers, the DS can be attached to the top of the corresponding saddle point of the potential energy surface, and in time-dependent systems, the DS is a moving object. The precise determination of these direct reaction rates, e.g., using transition state theory, requires the actual construction of a DS for a given saddle geometry, which is in general a demanding methodical and computational task, especially in high-dimensional systems. In this paper, we demonstrate how such time-dependent, global, and recrossing-free DSs can be constructed using neural networks. In our approach, the neural network uses the bath coordinates and time as input, and it is trained in a way that its output provides the position of the DS along the reaction coordinate. An advantage of this procedure is that, once the neural network is trained, the complete information about the dynamical phase space separation is stored in the network's parameters, and a precise distinction between reactants and products can be made for all possible system configurations, all times, and with little computational effort. We demonstrate this general method for two- and three-dimensional systems and explain its straightforward extension to even more degrees of freedom.
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Affiliation(s)
- Philippe Schraft
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Andrej Junginger
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Matthias Feldmaier
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Robin Bardakcioglu
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Jörg Main
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Günter Wunner
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
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7
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Patra S, Keshavamurthy S. Detecting reactive islands using Lagrangian descriptors and the relevance to transition path sampling. Phys Chem Chem Phys 2018; 20:4970-4981. [PMID: 29387842 DOI: 10.1039/c7cp05912d] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
It has been known for sometime now that isomerization reactions, classically, are mediated by phase space structures called reactive islands (RI). RIs provide one possible route to correct for the nonstatistical effects in the reaction dynamics. In this work, we map out the reactive islands for the two dimensional Müller-Brown model potential and show that the reactive islands are intimately linked to the issue of rare event sampling. In particular, we establish the sensitivity of the so called committor probabilities, useful quantities in the transition path sampling technique, to the hierarchical RI structures. Mapping out the RI structure for high dimensional systems, however, is a challenging task. Here, we show that the technique of Lagrangian descriptors is able to effectively identify the RI hierarchy in the model system. Based on our results, we suggest that the Lagrangian descriptors can be useful for detecting RIs in high dimensional systems.
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Affiliation(s)
- Sarbani Patra
- Department of Chemistry, Indian Institute of Technology, Kanpur, U.P. 208 016, India.
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8
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Feldmaier M, Junginger A, Main J, Wunner G, Hernandez R. Obtaining time-dependent multi-dimensional dividing surfaces using Lagrangian descriptors. Chem Phys Lett 2017. [DOI: 10.1016/j.cplett.2017.09.008] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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9
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Junginger A, Duvenbeck L, Feldmaier M, Main J, Wunner G, Hernandez R. Chemical dynamics between wells across a time-dependent barrier: Self-similarity in the Lagrangian descriptor and reactive basins. J Chem Phys 2017; 147:064101. [DOI: 10.1063/1.4997379] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Andrej Junginger
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Lennart Duvenbeck
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Matthias Feldmaier
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Jörg Main
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Günter Wunner
- Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
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10
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Junginger A, Hernandez R. Lagrangian descriptors in dissipative systems. Phys Chem Chem Phys 2016; 18:30282-30287. [PMID: 27327139 DOI: 10.1039/c6cp02532c] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The reaction dynamics of time-dependent systems can be resolved through a recrossing-free dividing surface associated with the transition state trajectory-that is, the unique trajectory which is bound to the barrier region for all time in response to a given time-dependent potential. A general procedure based on the minimization of Lagrangian descriptors has recently been developed by Craven and Hernandez [Phys. Rev. Lett., 2015, 115, 148301] to construct this particular trajectory without requiring perturbative expansions relative to the naive transition state point at the top of the barrier. The extension of the method to account for dissipation in the equations of motion requires additional considerations established in this paper because the calculation of the Lagrangian descriptor involves the integration of trajectories in forward and backward time. The two contributions are in general very different because the friction term can act as a source (in backward time) or sink (in forward time) of energy, leading to the possibility that information about the phase space structure may be lost due to the dominance of only one of the terms. To compensate for this effect, we introduce a weighting scheme within the Lagrangian descriptor and demonstrate that for thermal Langevin dynamics it preserves the essential phase space structures, while they are lost in the nonweighted case.
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Affiliation(s)
- Andrej Junginger
- Center for Computational and Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332-0400, Georgia.
| | - Rigoberto Hernandez
- Center for Computational and Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332-0400, Georgia.
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11
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Junginger A, Hernandez R. Uncovering the Geometry of Barrierless Reactions Using Lagrangian Descriptors. J Phys Chem B 2015; 120:1720-5. [DOI: 10.1021/acs.jpcb.5b09003] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Andrej Junginger
- Center for Computational
and Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332, United States
| | - Rigoberto Hernandez
- Center for Computational
and Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332, United States
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12
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Kawai S. On the environmental modes for the generalized Langevin equation. J Chem Phys 2015; 143:094101. [DOI: 10.1063/1.4929710] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Shinnosuke Kawai
- Department of Chemistry, Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan
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13
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Classical-quantum correspondence in a model for conformational dynamics: Connecting phase space reactive islands with rare events sampling. Chem Phys Lett 2015. [DOI: 10.1016/j.cplett.2015.05.011] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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14
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Teramoto H, Toda M, Komatsuzaki T. A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region. Theor Chem Acc 2014. [DOI: 10.1007/s00214-014-1571-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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15
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Popov AV, Hernandez R. Temperature-driven irreversible generalized Langevin equation can capture the nonequilibrium dynamics of two dissipated coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032145. [PMID: 24125251 DOI: 10.1103/physreve.88.032145] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2013] [Indexed: 06/02/2023]
Abstract
Kawai and Komatsuzaki [J. Chem. Phys. 134, 114523 (2011)] recently derived the nonequilibrium generalized Langevin equation (GLE) for a nonstationary system using the projection operator technique. In the limit when the environment is slowly changing (that is, a quasi-equilibrium bath), it should reduce to the irreversible GLE approach (iGLE) [J. Chem. Phys. 111, 7701 (1999)]. Kawai and Komatsuzaki, however, found that the driven harmonic oscillator, an example of a nonequilibrium system does not obey the iGLE presumably because it did not quite satisfy the limiting conditions of the latter. Notwithstanding the lack of a massive quasi-equilibrium bath (one of the conditions under which the iGLE had been derived earlier), we found that the temperature-driven iGLE (T-iGLE) [J. Chem. Phys. 126, 244506 (2007)] can reproduce the nonequilibrium dynamics of a driven dissipated pair of harmonic oscillators. It requires a choice of the function representing the coupling between the oscillator coordinate and the bath and shows that the T-iGLE representation is consistent with the projection operator formalism if only dominant bath modes are taken into account. Moreover, we also show that the more readily applicable phenomenological iGLE model is recoverable from the Kawai and Komatsuzaki model beyond the adiabatic limit used in the original T-iGLE theory.
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Affiliation(s)
- Alexander V Popov
- Center for Computational and Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, USA
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Makarov DE. Interplay of non-Markov and internal friction effects in the barrier crossing kinetics of biopolymers: Insights from an analytically solvable model. J Chem Phys 2013; 138:014102. [DOI: 10.1063/1.4773283] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
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17
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Kawai S, Komatsuzaki T. Laser Control of Chemical Reactions by Phase Space Structures. BULLETIN OF THE CHEMICAL SOCIETY OF JAPAN 2012. [DOI: 10.1246/bcsj.20120085] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/30/2023]
Affiliation(s)
- Shinnosuke Kawai
- Molecule and Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University
| | - Tamiki Komatsuzaki
- Molecule and Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University
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18
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Bartsch T, Revuelta F, Benito RM, Borondo F. Reaction rate calculation with time-dependent invariant manifolds. J Chem Phys 2012; 136:224510. [DOI: 10.1063/1.4726125] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Kawai S, Komatsuzaki T. Derivation of the generalized Langevin equation in nonstationary environments. J Chem Phys 2011; 134:114523. [DOI: 10.1063/1.3561065] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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20
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Kawai S, Komatsuzaki T. Phase space geometry of dynamics passing through saddle coupled with spatial rotation. J Chem Phys 2011; 134:084304. [DOI: 10.1063/1.3554906] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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21
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Kawai S, Komatsuzaki T. Quantum reaction boundary to mediate reactions in laser fields. J Chem Phys 2011; 134:024317. [DOI: 10.1063/1.3528937] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Kawai S, Komatsuzaki T. Why and how do systems react in thermally fluctuating environments? Phys Chem Chem Phys 2011; 13:21217-29. [DOI: 10.1039/c1cp22504a] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Kawai S, Komatsuzaki T. Nonlinear dynamical effects on reaction rates in thermally fluctuating environments. Phys Chem Chem Phys 2010; 12:7636-47. [DOI: 10.1039/b922596j] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022]
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Kawai S, Komatsuzaki T. Hierarchy of reaction dynamics in a thermally fluctuating environment. Phys Chem Chem Phys 2010; 12:7626-35. [DOI: 10.1039/b922080a] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Affiliation(s)
- Shinnosuke Kawai
- Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan.
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Kawai S, Komatsuzaki T. Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations. Phys Chem Chem Phys 2010; 12:15382-91. [DOI: 10.1039/c0cp00543f] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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26
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Kawai S, Komatsuzaki T. Dynamic pathways to mediate reactions buried in thermal fluctuations. II. Numerical illustrations using a model system. J Chem Phys 2009; 131:224506. [DOI: 10.1063/1.3268622] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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