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Wigner's Dynamical Transition State Theory in Phase Space

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Wigner's Dynamical Transition State Theory in Phase Space Wigner’s Dynamical Transition State Theory in Phase Space: Classical and Quantum Holger Waalkens1,2, Roman Schubert1, and Stephen Wiggins1 October 15, 2018 1 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK 2 Department of Mathematics, University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands E-mail: [email protected], [email protected], [email protected] Abstract We develop Wigner’s approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our de- velopment is the construction of a normal form for describing the dynamics in the neighborhood of a specific type of saddle point that governs the evolution from re- actants to products in high dimensional systems. In the classical case this is the standard Poincar´e-Birkhoff normal form. In the quantum case we develop a normal form based on the Weyl calculus and an explicit algorithm for computing this quan- tum normal form. The classical normal form allows us to discover and compute the phase space structures that govern classical reaction dynamics. From this knowledge we are able to provide a direct construction of an energy dependent dividing sur- face in phase space having the properties that trajectories do not locally “re-cross” the surface and the directional flux across the surface is minimal. Using this, we are able to give a formula for the directional flux through the dividing surface that goes beyond the harmonic approximation. We relate this construction to the flux- flux autocorrelation function which is a standard ingredient in the expression for the reaction rate in the chemistry community. We also give a classical mechanical interpretation of the activated complex as a normally hyperbolic invariant manifold arXiv:0708.0511v2 [nlin.CD] 12 Dec 2007 (NHIM), and further describe the structure of the NHIM. The quantum normal form provides us with an efficient algorithm to compute quantum reaction rates and we relate this algorithm to the quantum version of the flux-flux autocorrelation function formalism. The significance of the classical phase space structures for the quantum mechanics of reactions is elucidated by studying the phase space distribution of scat- tering states. The quantum normal form also provides an efficient way of computing Gamov-Siegert resonances. We relate these resonances to the lifetimes of the quan- tum activated complex. We consider several one, two, and three degree-of-freedom systems and show explicitly how calculations of the above quantities can be carried out. Our theoretical framework is valid for Hamiltonian systems with an arbitrary number of degrees of freedom and we demonstrate that in several situations it gives rise to algorithms that are computationally more efficient than existing methods. 1 2 Contents 1 Introduction 3 1.1 Transition State Theory: Classical Dynamics . ..... 5 1.2 Transition State Theory: Quantum Dynamics . 11 2 Classical Normal Form Theory 14 2.1 Transformation of Phase Space Functions through Symplectic Coordi- nateTransformations. 15 2.2 Definition and Computation of the Classical Normal Form . ...... 18 2.3 Nature and Computation of the Normal Form in a Neighborhood of an Equilibrium Point of Saddle-Centre- -Centre Stability Type . 25 · · · 2.3.1 Solution of the homological equation . 27 2.3.2 Integrals of the classical motion from the N th order classical normalform............................. 28 3 Quantum Normal Form Theory 30 3.1 The Classical-Quantum Correspondence . 31 3.1.1 Weyl quantisation . 31 3.1.2 TheMoyalbracket .. .. .. .. .. .. .. 33 3.1.3 Localisinginphasespace . 35 3.2 Transformation of Operators through Conjugation with Unitary Op- eratorsUsingtheWeylCalculus . 35 3.3 Definition and Computation of the Quantum Normal Form . 41 3.4 Nature and Computation of the Quantum Normal Form in a Neigh- bourhood of an Equilibrium Point of the Principal Symbol of Saddle- Centre- -CentreType .......................... 45 · · · 3.4.1 Solution of the homological equation . 45 3.4.2 Structure of the Hamilton operator in N th order quantum nor- malform .............................. 47 3.5 Quantum normal form for one-dimensional potential barriers . 49 4 Classical Reaction Dynamics and Reaction Probabilities 54 4.1 Phase Space Structures that Control Classical Reaction Dynamics: An OverviewoftheGeometry. 54 4.2 The Normal Form Coordinates: Phase Space Structures and Trajec- tories of Hamilton’s Equations . 56 4.3 The Normal Form Coordinates: The Foliation of the Reaction Region byLagrangianSubmanifolds. 62 4.4 The Directional Flux Through the Dividing Surface . ..... 65 4.5 The Normal Form Coordinates: Issues Associated with Truncation . 67 4.6 The Global Dynamics Associated with the Manifolds Constructed in theReactionRegion ............................ 68 4.7 The flux-flux autocorrelation function formalism for computing clas- sical reaction probabilities . 72 5 Quantum Reaction Dynamics and Cumulative Reaction Probabili- ties 75 5.1 Quantumnormalform ........................... 75 5.2 Scattering states for one-dimensional systems . ....... 76 5.3 S-matrix and transmission probability for one-dimensional systems . 78 3 5.4 S-matrix and cumulative reaction probability for multi-dimensional systems ................................... 79 5.5 Distribution of the scattering states in phase space . ........ 81 5.6 TheglobalS-matrix ............................ 84 5.7 The flux-flux autocorrelation function formalism to compute quantum reactionprobabilities. 86 5.7.1 Example: 1D parabolic barrier . 87 5.7.2 Example: General barriers in 1D . 89 5.7.3 Example: General barriers in arbitrary dimensions . ..... 90 6 Quantum Resonances 92 6.1 Definition of quantum resonances . 92 6.2 Computation of resonances of the quantum normal form . ..... 94 6.2.1 Resonances of one-dimensional systems . 94 6.2.2 Resonances of multi-dimensional quantum normal form . 95 6.3 Lifetime of the activated complex . 96 6.4 On the relation between the resonances of the Quantum Normal Form andthefullsystem............................. 97 6.5 Distribution of the resonance states in phase space . ....... 98 7 Examples 98 7.1 Examplewith1DoF............................ 99 7.1.1 Computation of the classical and quantum normal forms . 100 7.1.2 Classical reaction dynamics . 101 7.1.3 Quantum reaction dynamics . 101 7.2 Examplewith2DoF............................103 7.2.1 Computation of the classical and quantum normal forms . 104 7.2.2 Classical reaction dynamics . 105 7.2.3 Quantum reaction dynamics . 106 7.3 Examplewith3DoF............................107 7.3.1 Computation of the classical and quantum normal forms . 108 7.3.2 Classical reaction dynamics . 110 7.3.3 Quantum reaction dynamics . 112 8 Conclusions and Outlook 113 A Proof of Lemma 9 118 B Symbols of the quantum normal forms of the systems studied in Section 7 119 C Computation of Quantum Resonances from the Complex Dilation Method 121 1 Introduction The subject of this paper is transition state theory – classical and quantum. Transi- tion state theory (TST) (sometimes also referred to as “activated complex theory” or the “theory of absolute reaction rates”) is widely regarded as the most important theoretical and computational approach to analyzing chemical reactions, both from 4 a qualitative and quantitative point of view. The central ideas of TST are so funda- mental that in recent years TST has been recognized as a very natural and fruitful approach in areas far beyond its origin of conception in chemistry. For example, it has been used in atomic physics [JFU00], studies of the rearrangements of clusters [KB99, KB02], solid state and semi-conductor physics [JTDF84, Eck95], diffusion dynamics in materials [VMG02], cosmology [dOdAST02], and celestial mechanics [JRL+02, WBW05b]. The literature on TST is vast, which befits the importance, utility, breadth, scope, and success of the theory. Searching ISI Web of Knowledge on the phrase “transition state theory” yields more than 17,600 hits. Searching Google with the same phrase gives more than 41,000,000 hits. There have been numerous reviews of TST, and the relatively recent review of [TGK96] is an excellent source for earlier reviews, historical accounts, books, pedagogical articles, and handbook chapters dealing with TST. Moreover, [TGK96] is notable from the point of view of that in little more that 10 years it has attracted more than 458 citations (and it also contains 844 references)! Certainly the existence of this vast literature begs the question “why does there need to be yet another paper on the theoretical foundations of TST, what new could it possibly add?” The one word answer to this questions is, “dynamics”. Ad- vances in experimental techniques over the past twenty years, such as, e.g., femtosec- ond laser spectroscopy, transition state spectroscopy, and single molecule techniques ([Neu92, PZ95, Zew00]) now provide us with ”real time” dynamical information on the progress of a chemical reaction from ”reactants” to ”products”. At the same time, these new experimental techniques, as well as advances in computational capabilities, have resulted in a growing realization among chemists of the ubiquity of non-ergodic behaviour in complex molecular systems, see, e.g. [SY04, BHC05, BHC06, Car05]. All of these results point to a need to develop a framework for studying and under- standing dynamics in high dimensional dynamical systems and recently developed tools in computational and applied dynamical systems
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