Path integrals, reaction dynamics

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

The Melnikov integral also offers a method to estimate the reaction rates for systems with two degrees of freedom. This idea comes from the work of Davis and Gray [9]. However, their idea breaks down for systems with more than two degrees of freedom because of tangency [11,14]. This breakdown requires a new conceptual structure to describe the reaction dynamics from the viewpoint of multiple-dimensional chaos. What we propose for this new concept is the skeleton of reaction paths, where the connections among NHIMs are the focus of our study. 

In the Fourier method each path contributing to Eq. (4.13) is expanded in a Fourier series and the sum over all contributing paths is replaced by an equivalent Riemann integration over all Fourier coefficients. This method was first introduced by Feynman and Hibbs to determine analytic expressions for the harmonic oscillator propagator and has been used by Miller in the context of chemical reaction dynamics. We have further developed the approach for use in finite-temjjerature Monte Carlo studies of quantum sys-tems, and we have found the method to be very useful in the cluster studies discussed in this chapter. 

Hwang et al.131 were the first to calculate the contribution of tunneling and other nuclear quantum effects to enzyme catalysis. Since then, and in particular in the past few years, there has been a significant increase in simulations of QM-nuclear effects in enzyme reactions. The approaches used range from the quantized classical path (QCP) (e.g., Refs. 4,57,136), the centroid path integral approach,137,138 and vibrational TS theory,139 to the molecular dynamics with quantum transition (MDQT) surface hopping method.140 Most studies did not yet examine the reference water reaction, and thus could only evaluate the QM contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g., Refs. 4,57,136) concluded that the QM contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

Mathematical and Physical Sciences, Vol. 170, D. C. Clary, Ed., D. Reidel, Boston, 1986, pp. 27—45. Reaction Path Models for Polyatomic Reaction Dynamics—From Transition State Theory to Path Integrals. 

REACTION PATH MODELS FOR POLYATOMIC REACTION DYNAMICS— FROM TRANSITION STATE THEORY TO PATH INTEGRALS... 

ABSTRACT. The reaction path Hamiltonian model for the dynamics of general polyatomic systems is reviewed. Various dynamical treatments based on it are discussed, from the simplest statistical approximations (e.g., transition state theory, RRKM, etc.) to rigorous path integral computational approaches that can be applied to chemical reactions in polyatomic systems. Examples are presented which illustrate this menu of dynamical possibilities. 

Once a Hamiltonian is constructed in terms of these coordinates and their conjugate momenta—the reaction path Hamiltonian —one needs dynamical theories to describe the reaction dynamics. Section II first discusses the form of the reaction path Hamiltonian, and then Section III describes the variety of dynamical models that have been based on it. These range from the simplest, statistical models (i.e., transition state theory) all the way to rigorous path integral methods that are essentially exact. Various applications are discussed to illustrate the variety of dynamical treatments. 

Path Integral Methods Reaction Path Hamiltonian and its Use for Investigating Reaction Mechanisms Reactive Scattering of Polyatomic Molecules State to State Reactive Scattering Statistical Adiabatic Channel Models Time Correlation Functions Transition State Theory Unimolecular Reaction Dynamics. 

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