A. Burbanks^*, S. Wiggins^*, D. Farrelly⁺ T. Uzer^&, J. Palacian^§, P.Yanguas^§, L. Wiesenfeld^$ and C. Jaffe⁺⁺

^*School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.

⁺Department of Chemistry, Utah State University, Logan, Utah 84322-030, U.S.A.

^& Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, U.S.A.

^§ Departamento de Matematica e Informatica, Universidad Publica de Navarra, 31006-Pamplona,

^$Universite Joseph-Fourier-Grenoble, 38402 Saint-Martin-d'Heres, France

⁺⁺ Department of Chemistry, West Virginia University, Morgantown, WV 26506, U.S.A

The notion of a transition state is one of the grand unifying concepts in chemistry¹. Many theories of chemical reactions explicitly assume that once reactants pass through the transition state then they cannot return². This “no-recrossing rule” serves to define the transition state and is a necessary assumption in transition state theory^{3 ,4}. Despite its ubiquity in chemistry it is only recently, however, that the existence of the transition state in more than two degrees-of-freedom (dof) has been proved⁵. Furthermore no general theory has existed for actually finding the transition state⁶. Here, combining methods of celestial mechanics with recent advances in dynamical systems theory,⁷ we provide a theory which is rigorously valid in an arbitrary number of dof. Equally important, advances in computational power make the method applicable in practice for large systems. Knowledge of the transition state - a phase space object - allows us to differentiate, with exquisite precision, between reactive and nonreactive molecular configurations wherever they lie in phase space.

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