1
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Hillebrand M, Zimper S, Ngapasare A, Katsanikas M, Wiggins S, Skokos C. Quantifying chaos using Lagrangian descriptors. CHAOS (WOODBURY, N.Y.) 2022; 32:123122. [PMID: 36587363 DOI: 10.1063/5.0120889] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2022] [Accepted: 11/03/2022] [Indexed: 06/17/2023]
Abstract
We present and validate simple and efficient methods to estimate the chaoticity of orbits in low-dimensional conservative dynamical systems, namely, autonomous Hamiltonian systems and area-preserving symplectic maps, from computations of Lagrangian descriptors (LDs) on short time scales. Two quantities are proposed for determining the chaotic or regular nature of orbits in a system's phase space, which are based on the values of the LDs of these orbits and of nearby ones: The difference and ratio of neighboring orbits' LDs. Using as generic test models the prototypical two degree of freedom Hénon-Heiles system and the two-dimensional standard map, we find that these indicators are able to correctly characterize the chaotic or regular nature of orbits to better than 90% agreement with results obtained by implementing the Smaller Alignment Index (SALI) method, which is a well-established chaos detection technique. Further investigating the performance of the two introduced quantities, we discuss the effects of the total integration time and of the spacing between the used neighboring orbits on the accuracy of the methods, finding that even typical short time, coarse-grid LD computations are sufficient to provide reliable quantification of the systems' chaotic component, using less CPU time than the SALI. In addition to quantifying chaos, the introduced indicators have the ability to reveal details about the systems' local and global chaotic phase space structure. Our findings clearly suggest that LDs can also be used to quantify and investigate chaos in continuous and discrete low-dimensional conservative dynamical systems.
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Affiliation(s)
- M Hillebrand
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - S Zimper
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - A Ngapasare
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - M Katsanikas
- Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4, Athens GR-11527, Greece
| | - S Wiggins
- School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom
| | - Ch Skokos
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
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2
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Lei YK, Zhang Z, Han X, Yang YI, Zhang J, Gao YQ. Locating Transition Zone in Phase Space. J Chem Theory Comput 2022; 18:6124-6133. [PMID: 36135927 DOI: 10.1021/acs.jctc.2c00385] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Understanding the reaction mechanism is required for better control of chemical reactions and is usually achieved by locating transition states (TSs) along a proper one-dimensional coordinate called reaction coordinate (RC). The identification of RC can be very difficult for high-dimensional realistic systems. A number of methods have been proposed to tackle this problem. A machine learning method is developed here to incorporate the influence of velocity on the reaction process. The method is also free of the unbalanced label problem resulting from the rather low fraction of configurations near the TS and can be easily extended to large systems. It locates the transition zone in the phase space and defines the dividing surface with a high transmission coefficient. Moreover, considering that the reaction environment can not only change the reaction path but also activate the reactive mode through energy transfer, we devise two measures to quantify the influence of these two factors on the reaction process and find that solvents can assist the reaction by directly doing work along the reactive mode. Not surprisingly, there is a positive correlation between the efficiency of energy transfer into the reactive mode and the reaction rate.
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Affiliation(s)
- Yao-Kun Lei
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China.,Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China
| | - Zhen Zhang
- School of Physics and Technology, Tangshan Normal University, 063000 Tangshan, China
| | - Xu Han
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China.,Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China
| | - Yi Isaac Yang
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China
| | - Jun Zhang
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China
| | - Yi Qin Gao
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China.,Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China.,Biomedical Pioneering Innovation Center, Peking University, 100871 Beijing, China
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3
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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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4
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Reiff J, Feldmaier M, Main J, Hernandez R. Dynamics and decay rates of a time-dependent two-saddle system. Phys Rev E 2021; 103:022121. [PMID: 33736042 DOI: 10.1103/physreve.103.022121] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2020] [Accepted: 01/20/2021] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- Johannes Reiff
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Matthias Feldmaier
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Jörg Main
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA.,Departments of Chemical & Biomolecular Engineering, and Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
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5
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Naik S, Wiggins S. Detecting reactive islands in a system-bath model of isomerization. Phys Chem Chem Phys 2020; 22:17890-17912. [PMID: 32478352 DOI: 10.1039/d0cp01362e] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
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.
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Affiliation(s)
- Shibabrat Naik
- School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK.
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6
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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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7
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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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8
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Naik S, Wiggins S. Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential. Phys Rev E 2019; 100:022204. [PMID: 31574621 DOI: 10.1103/physreve.100.022204] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2019] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- Shibabrat Naik
- School of Mathematics, University of Bristol, University Walk, Clifton BS8 1TW, Bristol, United Kingdom
| | - Stephen Wiggins
- School of Mathematics, University of Bristol, University Walk, Clifton BS8 1TW, Bristol, United Kingdom
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9
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Revuelta F, Benito RM, Borondo F. Unveiling the chaotic structure in phase space of molecular systems using Lagrangian descriptors. Phys Rev E 2019; 99:032221. [PMID: 30999489 DOI: 10.1103/physreve.99.032221] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2018] [Indexed: 06/09/2023]
Abstract
We explore here the feasibility of using the recently introduced Lagrangian descriptors [A. M. Mancho et al., Commun. Nonlinear Sci. Numer. Simul. 18, 3530 (2013)1007-570410.1016/j.cnsns.2013.05.002] to unveil the usually rich dynamics taking place in the vibrations of molecular systems, especially if they are floppy. The principal novelty of our work is the inclusion of p norms in the definition of the descriptors in this kind of system, which greatly enhances their power to discern among the different structures existing in the phase space. As an illustration we use the LiCN molecule described by realistic potentials in two and three dimensions, which exhibits chaotic motion within a mixed phase space in the isomerization between the two wells corresponding to the linear isomer stable configurations, LiNC and LiCN. In particular, we pay special attention to the manifolds emerging from the unstable fixed point between the corresponding isomer wells, and also to the marginally stable structures around a parabolic point existing near the LiNC well.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avda. Puerta de Hierro 2-4, 28040 Madrid, Spain
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
| | - R M Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avda. Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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10
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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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11
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Garcia-Meseguer R, Carpenter BK. Re-Evaluating the Transition State for Reactions in Solution. European J Org Chem 2018. [DOI: 10.1002/ejoc.201800841] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Affiliation(s)
| | - Barry K. Carpenter
- School of Chemistry; Cardiff University; CF10 3AT Cardiff United Kingdom
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12
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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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13
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Revuelta F, Craven GT, Bartsch T, Borondo F, Benito RM, Hernandez R. Transition state theory for activated systems with driven anharmonic barriers. J Chem Phys 2017; 147:074104. [DOI: 10.1063/1.4997571] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- F. Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
| | - Galen T. Craven
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104,
USA
| | - Thomas Bartsch
- Centre for Nonlinear Mathematics and Applications, Department of Mathematical Sciences, Loughborough University, Loughborough
LE11 3TU, United Kingdom
| | - F. Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
| | - R. M. Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
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14
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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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15
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Craven GT, Junginger A, Hernandez R. Lagrangian descriptors of driven chemical reaction manifolds. Phys Rev E 2017; 96:022222. [PMID: 28950601 DOI: 10.1103/physreve.96.022222] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2017] [Indexed: 06/07/2023]
Abstract
The persistence of a transition state structure in systems driven by time-dependent environments allows the application of modern reaction rate theories to solution-phase and nonequilibrium chemical reactions. However, identifying this structure is problematic in driven systems and has been limited by theories built on series expansion about a saddle point. Recently, it has been shown that to obtain formally exact rates for reactions in thermal environments, a transition state trajectory must be constructed. Here, using optimized Lagrangian descriptors [G. T. Craven and R. Hernandez, Phys. Rev. Lett. 115, 148301 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.148301], we obtain this so-called distinguished trajectory and the associated moving reaction manifolds on model energy surfaces subject to various driving and dissipative conditions. In particular, we demonstrate that this is exact for harmonic barriers in one dimension and this verification gives impetus to the application of Lagrangian descriptor-based methods in diverse classes of chemical reactions. The development of these objects is paramount in the theory of reaction dynamics as the transition state structure and its underlying network of manifolds directly dictate reactivity and selectivity.
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Affiliation(s)
- Galen T Craven
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Andrej Junginger
- 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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16
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Mauguière FA, Collins P, Kramer ZC, Carpenter BK, Ezra GS, Farantos SC, Wiggins S. Roaming: A Phase Space Perspective. Annu Rev Phys Chem 2017; 68:499-524. [DOI: 10.1146/annurev-physchem-052516-050613] [Citation(s) in RCA: 46] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
| | - Peter Collins
- School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom;, ,
| | - Zeb C. Kramer
- Department of Chemistry and Biochemistry, La Salle University, Philadelphia, Pennsylvania 19141
| | - Barry K. Carpenter
- School of Chemistry, Cardiff University, Cardiff CF10 3AT, United Kingdom
| | - Gregory S. Ezra
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853
| | - Stavros C. Farantos
- Department of Chemistry, University of Crete, Heraklion 700 13, Greece
- Institute of Electronic Structure and Laser, Foundation for Research and Technology—Hellas, Heraklion 711 10, Greece
| | - Stephen Wiggins
- School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom;, ,
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17
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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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18
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Ruiz-Herrera A. Performance of Lagrangian descriptors and their variants in incompressible flows. CHAOS (WOODBURY, N.Y.) 2016; 26:103116. [PMID: 27802675 DOI: 10.1063/1.4966176] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
The method of Lagrangian Descriptors has been applied in many different contexts, specially in geophysical flows. In this paper, we analyze their performance in incompressible flows. We construct broad families of systems where this diagnostic fails in the detection of barriers to transport. Another aim of this manuscript is to illustrate the same deficiencies in the recent diagnostic proposed by Craven and Hernández.
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Junginger A, Craven GT, Bartsch T, Revuelta F, Borondo F, Benito RM, Hernandez R. Transition state geometry of driven chemical reactions on time-dependent double-well potentials. Phys Chem Chem Phys 2016; 18:30270-30281. [DOI: 10.1039/c6cp02519f] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The minimum contour in the forward Lagrangian descriptor overlaps the invariant manifold (in green) dividing reactant and product regions.
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Affiliation(s)
- Andrej Junginger
- Center for Computational Molecular Science and Technology
- School of Chemistry and Biochemistry
- Georgia Institute of Technology
- Atlanta
- USA
| | - Galen T. Craven
- Center for Computational Molecular Science and Technology
- School of Chemistry and Biochemistry
- Georgia Institute of Technology
- Atlanta
- USA
| | - Thomas Bartsch
- Department of Mathematical Sciences
- Loughborough University
- Loughborough LE11 3TU
- UK
| | - F. Revuelta
- Grupo de Sistemas Complejos
- Escuela Técnica Superior de Ingeniería Agronómica
- Alimentaria y de Biosistemas
- Universidad Politécnica de Madrid
- Madrid
| | - F. Borondo
- Instituto de Ciencias Matemáticas (ICMAT)
- Cantoblanco
- Spain
- Departamento de Química
- Universidad Autónoma de Madrid
| | - R. M. Benito
- Grupo de Sistemas Complejos
- Escuela Técnica Superior de Ingeniería Agronómica
- Alimentaria y de Biosistemas
- Universidad Politécnica de Madrid
- Madrid
| | - Rigoberto Hernandez
- Center for Computational Molecular Science and Technology
- School of Chemistry and Biochemistry
- Georgia Institute of Technology
- Atlanta
- USA
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Nagahata Y, Maeda S, Teramoto H, Horiyama T, Taketsugu T, Komatsuzaki T. Deciphering Time Scale Hierarchy in Reaction Networks. J Phys Chem B 2015; 120:1961-71. [DOI: 10.1021/acs.jpcb.5b09941] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Yutaka Nagahata
- Graduate
School of Life Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0812, Japan
| | - Satoshi Maeda
- Department
of Chemistry, Faculty of Science, Hokkaido University, Kita 10,
Nishi 8, Kita-ku, Sapporo 060-0810, Japan
| | - Hiroshi Teramoto
- Graduate
School of Life Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0812, Japan
- Molecule
and Life Nonlinear Sciences Laboratory, Research Institute for Electronic
Science, Hokkaido University, Kita 20, Nishi 10, Kita-ku, Sapporo 001-0020, Japan
| | - Takashi Horiyama
- Graduate
School of Science and Engineering, Saitama University, Shimo-Ookubo
255, Sakura-ku, Saitama 338-8570, Japan
| | - Tetsuya Taketsugu
- Department
of Chemistry, Faculty of Science, Hokkaido University, Kita 10,
Nishi 8, Kita-ku, Sapporo 060-0810, Japan
| | - Tamiki Komatsuzaki
- Graduate
School of Life Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0812, Japan
- Molecule
and 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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