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University of Groningen Phase Space Conduits for Reaction in Multidimensional Systems Waalkens, Holger; Burbanks, Andrew; Wiggin

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University of Groningen Phase Space Conduits for Reaction in Multidimensional Systems Waalkens, Holger; Burbanks, Andrew; Wiggin University of Groningen Phase space conduits for reaction in multidimensional systems Waalkens, Holger; Burbanks, Andrew; Wiggins, Stephen Published in: The Journal of Chemical Physics DOI: 10.1063/1.1789891 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2004 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Waalkens, H., Burbanks, A., & Wiggins, S. (2004). Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions. The Journal of Chemical Physics, 121(13), 6207-6225. https://doi.org/10.1063/1.1789891 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 04-10-2021 JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 13 1 OCTOBER 2004 Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions Holger Waalkens, Andrew Burbanks, and Stephen Wiggins School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom ͑Received 1 July 2004; accepted 16 July 2004͒ The three-dimensional hydrogen cyanide/isocyanide isomerization problem is taken as an example to present a general theory for computing the phase space structures which govern classical reaction dynamics in systems with an arbitrary ͑finite͒ number of degrees of freedom. The theory, which is algorithmic in nature, comprises the construction of a dividing surface of minimal flux which is locally a ‘‘surface of no return.’’ The theory also allows for the computation of the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold ͑NHIM͒.A detailed description of the geometrical structures and the resulting constraints on reaction dynamics is given, with particular emphasis on the three degrees of freedom case. A procedure is given which uses these structures to compute orbits homoclinic to, and heteroclinic between, NHIMs. The role of homoclinic and heteroclinic orbits in global recrossings of dividing surfaces and transport in complex systems is explained. The complete description provided here is inherently one within phase space; it cannot be inferred from a configuration space picture. A complexification of the classical phase space structures to incorporate quantum effects is also discussed. The results presented here call into question certain assumptions routinely made on the global dynamics; this paper provides methods that enable one to understand and quantify the phase space dynamics of reactions without making such assumptions. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1789891͔ I. INTRODUCTION the ‘‘the full family of configurations through which the re- acting particles evolve en route from reagents to products.’’ Transition state theory has a long and illustrious place in In this paper, we present a theory of transition structures the field of chemistry, and in recent years its utility and ap- that is firmly rooted in the dynamical arena of phase space. plications have gone far beyond its origins. Following The geometrical picture we describe below has been formu- Truhlar,1 ‘‘transition state theory is the general name for any lated over the years in several works;4–7 and we apply it to a theory based in whole or in part on the fundamental assump- well-known problem, HCN isomerization. For Hamiltonian tion of transition state theory, or some quantum mechanical systems with an arbitrary ͑but finite͒ number of degrees of generalization of this assumption’’; the ‘‘fundamental as- freedom, the theory provides an algorithm for the direct con- sumption’’ of the previous sentence is the existence of a hy- struction of the transition structures which embody the dy- persurface that ͑locally͒ divides phase space into two re- namical features of the definition proposed by Polanyi and gions, corresponding to reactants and products. In what Zewail.3 This includes the construction of a dividing surface follows, we will refer to this hypersurface as the dividing which ͑locally͒ separates the energy surface into disjoint surface. In order to be useful, the dividing surface should components, corresponding to reactants and products. have certain properties that can be difficult to realize, in The dividing surface that we present is locally a ‘‘surface practice, and this has led to numerous versions of transition of no return,’’ i.e., it is free of locally recrossing trajectories state theory. Inevitably, this leads to occasional confusion whose presence would lead to an overestimation of reaction and ambiguity, with the situation being exacerbated by mod- rates. In particular, this dividing surface can be shown to ern experimental techniques which reveal detailed real-time minimize the flux;8 thus, it is the optimal dividing surface dynamical information about molecules as they undergo sought for in variational transition state theory, first proposed transitions from reactants to products. This is described viv- by Wigner,9 with recent developments discussed in Refs. 10 idly by Marcus2—‘‘for a brief moment, the participants in and 11. For the computation of the flux, we provide an ex- the reaction may look like one large molecule ready to fall plicit formula that takes into account the full effect of non- apart.’’ The dividing surface, in addition to being a surface linearity and dynamics. through which a trajectory passes during the reaction, repre- The construction of the dividing surface makes use of sents a collection of possible dynamical states in its own the existence of a so-called normally hyperbolic invariant right. This observation is amply born out in the rapidly de- manifold ͑NHIM͒͑Ref. 12͒, which can be shown to exist veloping experimental field known as transition state spec- near equilibrium points of saddle-center-...-center stability troscopy ͓see Ref. 3͔ which enables a direct observation of type. Such equilibria are characteristic of most kinds of ‘‘re- 0021-9606/2004/121(13)/6207/19/$22.006207 © 2004 American Institute of Physics Downloaded 11 Apr 2008 to 129.125.25.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 6208 J. Chem. Phys., Vol. 121, No. 13, 1 October 2004 Waalkens, Burbanks, and Wiggins action type’’ dynamics in a large and diverse number of ap- It is worth mentioning that the theory that we present is plications in chemistry and physics, as we will explain below not restricted to Hamiltonians of the type ‘‘kinetic plus po- in more detail. Normal hyperbolicity means that the rates of tential energy,’’ i.e., it also applies to Hamiltonians which contraction and expansion for the invariant motion on the contain magnetic or Coriolis terms due to rotations. More- NHIM are dominated by those in the directions transverse to over, it is important to emphasize that our transition struc- the NHIM. It is this property of the NHIM which largely tures, which contain the complete information about the tran- governs the dynamics nearby. The NHIM can be identified sition geometry and dynamics, are inherently phase space with the energy surface of the so-called ‘‘activated complex’’ objects. The complete information which these structures ͑see the orginal work by Eyring13 and, e.g., Miller14 for a embody cannot be obtained from theories which are based on more recent perspective͒ which, as an unstable invariant a configuration space picture. ‘‘supermolecule,’’15 is located between reactants and prod- In the spirit of classical transition state theory we set out ucts. our theory in the framework of classical mechanics. How- It is useful to point out that there is some confusion in ever, the geometry of the transition structures in classical terminology in the literature concerning the notion of transi- phase space, and their complexification, provides a natural tion state: this term is sometimes used to refer to the dividing setting for the study of quantum effects, as we will describe. surface through which reacting trajectories pass, and some- This paper consists of two main parts: Sec. II which times to the activated complex, which is mathematically re- contains a detailed description of the general theory and Sec. alized by the NHIM, and through which trajectories never III in which the theory is applied to the HCN/CNH isomer- pass. In the latter case, where the phrase ‘‘crosses the transi- ization problem. These main parts are organized as follows. tion state’’ appears in the literature, it actually refers to tra- In Sec. II A we stress the importance of saddle-center-...- jectories crossing the projection of the NHIM into configu- center equilibrium points, or ‘‘saddles’’ for short, in diverse ration space; trajectories cannot cross the NHIM ͑in phase applications. Section II B introduces the phase space struc- space͒ because it is an invariant manifold. For clarity, in tures which can be shown to exist near saddles and which what follows, we will refer explicitly to the dividing surface control the transport ‘‘across’’ saddles.
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