VOLUME 86, NUMBER 24 PHYSICAL REVIEW LETTERS 11 JUNE 2001 Impenetrable Barriers in Phase-Space S. Wiggins, 1 L. Wiesenfeld, 2 C. Jaffé, 3 and T. Uzer 4 1 School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom 2 Laboratoire de Spectrométrie Physique, Université Joseph-Fourier-Grenoble, 38402 Saint-Martin-d’Hères, France 3 Department of Chemistry, West Virginia University, Morgantown, West Virginia 26506-6049 4 Center for Nonlinear Sciences, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 (Received 29 December 2000) Dynamical systems theory is used to construct a general phase-space version of transition state theory. Special multidimensional separatrices are found which act as impenetrable barriers in phase-space be- tween reacting and nonreacting trajectories. The elusive momentum-dependent transition state between reactants and products is thereby characterized. A practical algorithm is presented and applied to a strongly coupled Hamiltonian. DOI: 10.1103/PhysRevLett.86.5478 PACS numbers: 05.45.– a, 34.10.+x, 45.20.Jj, 82.20.Db Introduction. —Transition state theory (TST) was developed in the 1930s [1] as a way to determine absolute chemical reaction rates. An essentially thermodynamic picture emerged from the original research of Eyring [2] and this continued to be the dominant formulation of the theory for many decades. In parallel with Eyring’s work, a dynamical picture was also being developed by Wigner [3] and this turns out to have considerable advantages. Not the least of these is that Wigner’s formulation quickly leads to the recognition that the transition state (TS) is actually a general property of all dynamical systems, provided that they evolve from “reactants” to “products.” The TS, there- fore, is not confined to chemical reaction dynamics [4], but it also controls rates in a multitude of interesting systems, including, e.g., the rearrangements of clusters [5], the ionization of atoms [6], conductance due to ballistic electron transport through microjunctions [7], and diffu- sion jumps in solids [8]. Since transition state theory is fundamental for transformations in n degree-of-freedom systems, the work summarized here represents a general formulation of the nonlinear dynamics and geometry of classical reaction dynamics. It hinges on finding, for the first time, the dynamically exact higher-dimensional structures (separatrices, dividing surfaces) which regulate transport between qualitatively different states (reagents and products) in three or more degrees of freedom. Stated succinctly, TST postulates the existence of a minimal set of states that all reactive trajectories must cross and which are never encountered by nonreactive trajectories. While the original idea of a TS was expressed as a dividing surface in coordinate space, Wigner rec- ognized [9] that a rigorous treatment must seek dividing surfaces in phase space which separate reactants from products and which no trajectory passes through more than once. Enforcing this “no recrossing” requirement has been a major obstacle to applying TST in strongly coupled, multidimensional systems. Consequently, TST has remained a configuration-space theory that has further been confined to low dimensions for which the dividing surfaces can be found in practice. The problem solved in this Letter is the construction of hypersurfaces of no return in the phase-space of strongly coupled, multidimensional systems. Our solution leads naturally to the multidimensional generalization of a saddle “point” and its associated separatrices. Until very recently, neither theoretical understanding nor computing power was adequate to explore phase-space transport beyond the well-known two degrees-of-freedom (“2dof”) limit [10]. However, with recent advances in dy- namical systems theory [11], especially concerning nor- mally hyperbolic invariant manifolds (NHIM) [12], the classical theory of chemical reactions can now be an- chored rigorously in nonlinear dynamics. Indeed, this Letter makes explicit the long-sought classical structures that act as transition states in phase-space beyond 2dof. As we will show, the rigorous way to describe the notion of a “barrier” in phase-space is through invariant mani- folds. “Invariance” signifies that trajectories starting on the manifold must remain on the manifold for the future and throughout the past. Hence, no trajectory can cross an invariant manifold. These manifolds, as indeed all other multidimensional structures in this Letter, become famil- iar objects when viewed in 2dof: For example, NHIM’s reduce to none other than periodic orbits. Our treatment is general, and because it relies on recent advances in dy- namical systems theory, we provide a “user’s guide” which we then apply to a realistic system. The following review of 2dof systems will set the terms for extension to more degrees of freedom. Lower-dimensional theory. — Current understanding of the transition state as a geometrical structure in 2dof sys- tems has been greatly aided by the discovery [13] that projection of an unstable periodic orbit into configura- tion space defines a surface separating reactants and prod- ucts and is therefore a “periodic orbit dividing surface” or PODS. In phase-space, the stable and unstable manifolds of this orbit partition the energy shell. This partitioning is invariant and separates the reactive and nonreactive dy- namics. It is not generally recognized that the defining periodic orbit also bounds a surface in the energy shell. 5478 0031-9007 01  86(24)  5478(4)$15.00 © 2001 The American Physical Society