Transition-State Theory Rate Calculations with a Recrossing-Free Moving Dividing Surface ² Thomas Bartsch* Department of Mathematical Sciences, Loughborough UniVersity, Loughborough LE11 3TU, UK T. Uzer Center for Nonlinear Science, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 Jeremy M. Moix Chemical Physics Department, The Weizmann Institute of Science, 76100 RehoVot, Israel Rigoberto Hernandez Center for Computational Molecular Sciences & Technology, School of Chemistry & Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400 ReceiVed: July 16, 2007; In Final Form: August 7, 2007 Two different methods for transition-state theory (TST) rate calculations are presented that use the recently developed notions of the moving dividing surface and the associated moving separatrices: one is based on the flux-over-population approach and the other on the calculation of the reactive flux. The flux-over-population rate can be calculated in two ways by averaging the flux first over the noise and then over the initial conditions or vice versa. The former entails the calculation of reaction probabilities and is closely related to previous TST rate derivations. The latter results in an expression for the transmission factor as the noise average of a stochastic variable that is given explicitly as a function of the moving separatrices. Both the reactive-flux and flux-over-population methods suggest possible new ways of calculating approximate rates in anharmonic systems. In particular, numerical simulations of harmonic and anharmonic systems have been used to calculate reaction rates based on the reactive flux calculation using the fixed and moving dividing surfaces so as to illustrate the computational advantages of the latter. I. Introduction Transition-state theory (TST) has proven to be a powerful method for predicting the rate of activated processes. While originally formulated as a description of the chemical reaction rates for small molecules, 1-3 it has been shown that any system that evolves from well-defined “reactant” to “product” states can be treated within this framework. 4-10 TST is based on the observation that large barriers hindering reactive events lead to bottlenecks in phase space. It is then possible to define a surface that separates reactants from products. The (classical) reaction rate is given simply by the steady-state flux through that surface. This approach yields accurate approximations for isolated systems if the dividing surface is chosen well. However, coupling with the environment, as occurs in almost all chemical reactions in solution and many other systems of practical interest, leads to recrossings of the dividing surface. These deviations result in an overestimate of the true rate by traditional TST calculations. The generalized Langevin equation has been used extensively as a prototypical model for simple chemical and physical reaction processes that are coupled to the surrounding thermal environment. 11-13 As mentioned above, the corresponding rate for such systems must necessarily be modified from that of the bare gas-phase result predicted by traditional TST because of interactions with the external bath. 14,15 An early attempt to formulate a rate theory in the Langevin setting was provided by Kramers’ seminal work. 16 He was able to derive explicit solutions for the rate over a parabolic barrier separately in the limits of weak and strong damping based on a flux-over- population calculation. However, a general solution valid for the entire friction regime (the famous “Kramers’ Turnover” problem) was not found until much later 17,18 by taking advantage of the equivalence 19 of the generalized Langevin equation to a Hamiltonian model of a subsystem bilinearly coupled to a bath of harmonic oscillators. Here, results are presented for the strong coupling limit that lead to the well-known Kramers-Grote- Hynes correction to the TST rate constant. 20-22 However, the present formalism remains within the Langevin representation in the spirit of Kramers’ original work without resorting to the Hamiltonian model. There are many advantages associated with this methodology. It offers an alternative geometrical interpreta- tion of the reaction process, which is convenient and intuitive from both a theoretical and computational perspective, along with the ability to observe the contributing factors to a particular rate process at a trajectory level. In a recent series of papers, a new framework for rate theory calculations has been proposed. 23-25 The Langevin equation ² Part of the James T. (Casey) Hynes Festschrift. * Corresponding author. 206 J. Phys. Chem. B 2008, 112, 206-212 10.1021/jp0755600 CCC: $40.75 © 2008 American Chemical Society Published on Web 10/13/2007