Reaction Rates from Dynamics Simulations

In general, it seems reasonable to believe that we should be able to quantitatively account for any large deviations that may occur between the kinetics of MD simulations (i.e., from numerical experiments ) and the kinetics predicted by simple theoretical models of reaction rates (such as transition state theory). We usually should be able to obtain numerically the asymptotic reaction rate from MD simulations at a particular E by integrating Hamilton s equations of motion for an ensemble and counting the number of these trajectories that correspond to reactants at any particular time. 

Let us consider the MD simulation of an elementary gas-phase unimo-lecular isomerization reaction of the form of Eq. [1]. We first define a suitable dividing surface between reactants and products. If we next initiate an ensemble of trajectories all with the same energy (we might do this uniformly on the constant-energy surface of possible initial positions and momenta, forming a molecular microcanonical ensemble ) and all initially on the reactant side of the dividing surface, we can monitor the decay of this initially nonequilibrium distribution of trajectories as a function of time.  

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Summary. Rate constants of chemical reactions can be calculated directly from dynamical simulations. Employing flux correlation functions, no scattering calculations are required. These calculations provide a rigorous quantum description of the reaction process based on first principles. In addition, flux correlation functions are the conceptual basis of important approximate theories. Changing from quantum to classical mechanics and employing a short time approximation, one can derive transition state theory and variational transition state theory. This article reviews the theory of flux correlation functions and discusses their relation to transition state theory. Basic concepts which facilitate the calculation and interpretation of accurate rate constants are introduced and efficient methods for the description of larger systems are described. Applications are presented for several systems highlighting different aspects of reaction rate calculations. For these examples, different types of approximations are described and discussed. 

Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

As described above, silicon crystals can be grown from a variety of gas sources. Because the rate of growth can be modulated using these techniques, dopants can be efficiently incorporated into a growing crystal. This results in control of the atomic structure of the crystal, and allows the production of samples which have specific electronic properties. The mechanisms by which gas-phase silicon species are incorporated into the crystal, however, are still unclear, and so molecular dynamics simulations have been used to help understand these microscopic reaction events. 

Buehler et al. presented a preliminary study on formation of water from molecular oxygen and hydrogen using a series of atomistic simulations based on ReaxFF MD method.111 They described the dynamics of water formation at a Pt catalyst. By performing this series of studies, we obtain statistically meaningful trajectories that permit to derive the reaction rate constants of water formation. However, the method requires calibrations with either ab initio simulation results in order to correctly evaluate the energetics of OER on Pt. Thus, this method is system specific and less reliable than the ab initio methods and will not replace ab initio methods. Nevertheless, this work demonstrates that atomistic simulation to continuum description can be linked with the ReaxFF MD in a hierarchical multiscale model. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

The computer simulation is one of the essential means to investigate dynamic and steady-state behavior as well as control of metabolic pathways. A metabolic simulator is a computer program that performs one or several of the tasks including solving the steady state of a metabolic pathway, dynamically simulating a metabolic pathway, or calculating the control coefficient of a metabolic pathway. Its mathematical model generally consists of a set of differential equations derived from rate equations of the enzymatic reactions of the pathway. 

Another illustration of the power of molecular dynamics simulation can be drawn from the sphere of enzyme catalysis. Many enzyme-catalyzed reactions proceed at a rate that depends on the diffusion-limited association of the substrate with the active site. Sharp et al. [28] have carried out Brownian dynamics simulations of the association of superoxide anions with superoxide dismutase (SOD). The active center in SOD is a positively charged copper atom. The distribution of charge over the enzyme is not uniform, and so an electric field is produced. Using their model, Sharp et al. [28] have shown that the electric field enhances the association of the substrate with the enzyme by a factor of 30 or more. Their calculations also predict correctly the response of the association rate to changes in ionic strength and amino... 

Laboratory simulations of aqueous-phase chemical systems are necessary to 1) verify reaction mechanisms and 2) assign a value and an uncertainty to transformation rates. A dynamic cloud chemistry simulation chamber has been characterized to obtain these rates and their uncertainties. Initial experimental results exhibited large uncertainties, with a 26% variability in cloud liquid water as the major contributor to measurement uncertainty. Uncertainties in transformation rates were as high as factor of ten. Standard operating procedures and computer control of the simulation chamber decreased the variability in the observed liquid water content, experiment duration and final temperature from 0.65 to 0.10 g nr3, 180 to 5.3 s and 1.73 to 0.27°C respectively. The consequences of this improved control over the experimental variables with respect to cloud chemistry were tested for the aqueous transformation of SO2 using a cloud-physics and chemistry model of this system. These results were compared to measurements made prior to the institution of standard operating procedures and computer control to quantify the reduction in reaction rate uncertainty resulting from those controls. 

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