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Journal ArticleDOI

Unified statistical model for ’’complex’’ and ’’direct’’ reaction mechanisms

15 Sep 1976-Journal of Chemical Physics (American Institute of Physics)-Vol. 65, Iss: 6, pp 2216-2223
TL;DR: In this article, a unified statistical theory for bimolecular chemical reactions is developed, which is correct for this situation, and if the reaction proceeds via a long-lived collision complex, it reduces to the statistical model of Light and Nikitin.
Abstract: A unified statistical theory for bimolecular chemical reactions is developed. In the limit of a ’’direct’’ mechanism it becomes the usual transition state theory, which is correct for this situation, and if the reaction proceeds via a long‐lived collision complex it reduces to the statistical model of Light and Nikitin. A general criterion for locating the ’’dividing surfaces’’ that are central to statistical theory is also discussed. This prescription (Keck’s variational principle) is shown not only to locate the usual dividing surfaces that pass through saddle points and minima of the potential energy surface, but it also selects the critical surfaces relevant to the ’’orbiting’’ and ’’nonadiabatic trapping’’ models of complex formation.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors present an overview of the current status of transition-state theory and its generalizations, including recent improvements in available methodology for calculations on complex systems, including the interface with electronic structure theory, progress in the theory and application of transitionstate theory to condensed-phase reactions, and insight into the relation of transition state theory to accurate quantum dynamics.
Abstract: We present an overview of the current status of transition-state theory and its generalizations. We emphasize (i) recent improvements in available methodology for calculations on complex systems, including the interface with electronic structure theory, (ii) progress in the theory and application of transition-state theory to condensed-phase reactions, and (iii) insight into the relation of transition-state theory to accurate quantum dynamics and tests of its accuracy via comparisons with both experimental and other theoretical dynamical approximations.

1,919 citations

Journal ArticleDOI
TL;DR: Several formally exact expressions for quantum mechanical rate constants (i.e., bimolecular reactive cross sections suitably averaged and summed over initial and final states) are derived and their relation to one another analyzed in this paper.
Abstract: Several formally exact expressions for quantum mechanical rate constants (i.e., bimolecular reactive cross sections suitably averaged and summed over initial and final states) are derived and their relation to one another analyzed. It is suggested that they may provide a useful means for calculating quantum mechanical rate constants accurately without having to solve the complete state‐to‐state quantum mechanical reactive scattering problem. Several ways are discussed for evaluating the quantum mechanical traces involved in these expressions, including a path integral evaluation of the Boltzmann operator/time propagator and a discrete basis set approximation. Both these methods are applied to a one‐dimensional test problem (the Eckart barrier).

791 citations

Journal ArticleDOI
TL;DR: In this article, a review concentrates on recent developments in the study of elementary reaction kinetics in relation to the modeling and prediction of low-temperature combustion and autoignition, with specific focus placed on the critical alkylperoxy and hydroperoxyalkyl reactions.

555 citations

Journal ArticleDOI
TL;DR: This review is concerned with the theoretical and computational modeling of bimolecular reactions, especially with generally applicable methods for kinetics (i.e., overall rates as opposed to detailed dynamics), and includes a basic theoretical framework that can be used for gas-phase thermal reactions, gas- phase microcanonical and state-selected reactions, and condensed-phase chemical reactions.
Abstract: A review of the theoretical and computational modeling of bimolecular reactions is given. The review is divided into several sections which are as follows: gas-phase thermal reactions; gas-phase state-selected reactions and product state distributions; and condensed-phase bimolecular reactions. The section on gas-phase thermal reactions covers the enthalpies and free energies of reaction, kinetics, saddle points and potential energy surfaces, rate theory for simple barrier reactions and bimolecular reactions over potential wells. The section on gas-phase state-selected reactions focuses on electronically adiabatic reactions and electronically nonadiabatic reactions. Finally, the section on condensed-phase bimolecular reactions covers reactions in liquids, reactions on surfaces and in solids and tunneling at low temperature.

534 citations

Journal ArticleDOI
TL;DR: In this paper, the authors discuss two minimum density-of-states criteria for the location of generalized transition states for chemical reactions and prove that both provide upper bounds on the exact classical equilibrium rate constant.
Abstract: We discuss two minimum‐density‐of‐states criteria for the location of generalized transition states for chemical reactions. One is due to Bunker and Pattengill; the other is due to Wong and Marcus. We prove that both provide upper bounds on the exact classical equilibrium rate constant. In addition, we show that for several‐dimensional systems both methods are exact at threshold, and in the limit of an infinite number of dimensions they agree with the variational theory of reactions of Wigner, Horiuti, and Keck. However, it is also shown that for a finite number of degrees of freedom both methods yield rate constants which are only as accurate as or less accurate than rate constants given by the variational theory of reactions. We note that, where tested by others for actual systems, the differences of the results obtained with the variational and Bunker–Pattengill criteria have been minor.

481 citations

References
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Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of estimating the reaction rate of elementary reactions in terms of the energy and velocity distribution of the molecules in the system, and showed that the results can be obtained by the application of quantum mechanics to molecular systems.
Abstract: According to our present notions, the theory of reaction rates involves three steps. First, one should know the behaviour of all molecules present in the system during the reaction, how they will move, and which products they will yield when colliding with definite velocities, etc. Practically, this amounts in most cases to the construction of the energy surface for the reacting system. Professor Eyring told us about the results which can be obtained by the application of quantum mechanics to molecular systems for this part of the theory. The second step in the theory I would call the statistical part. It endeavours to solve the problem of the rate of elementary reactions. Assuming only the material on the left side of a chemical equation to be present in a vessel, and the molecules of these to have the Maxwell-Boltzman energy and velocity distribution, one wants to know how many molecules corresponding to the right side of the equation will be formed in unit time. The elementary properties of the molecules are supposed to be known in this second step and one wants to express the reaction rate of elementary reactions in terms of these. The present paper will be devoted entirely to this second step. The third step is the consideration of the co-operation of the various elementary reactions, which may occur beside and must occur after each other in order to complete a real reaction. In especially favourable cases there is only one important chain of reactions leading to the final products and this has one link which is so much slower than all the others, that it is made responsible for the observed rate. The others are then assumed to be so much faster that one has practically equilibrium between the two sides of their chemical equations.

768 citations

Journal ArticleDOI
TL;DR: In this paper, an exact quantum mechanical transition state theory is defined, i.e., a model which invokes the basic transition state idea to calculate the rate of a chemical reaction but which is free of any auxiliary approximations.
Abstract: An ``exact'' quantum mechanical transition state theory is defined, i.e., a model which invokes the basic transition state idea to calculate the rate of a chemical reaction but which is free of any auxiliary approximations. Most importantly, for example, it is not necessary to assume that the Hamiltonian is separable about the saddle point. It is argued that this model should provide an accurate description of the threshold region of the reaction where quantum effects are most significant. Finally, an even more general model, a new kind of semiclassical approximation, is presented which is essentially a synthesis of this quantum mechanical transition state model and the completely classical trajectory procedure for determining the rate constant; at sufficiently high temperatures, quantum effects become negligible, so that the correct rate constant is obtained; while at low temperature, the correct result is obtained because the transition state model becomes valid.

629 citations

Journal ArticleDOI
TL;DR: In this article, the authors derived the semiclassical limit of quantum mechanical transition state theory by invoking the classical path approximation for the Boltzmann density operator and making use of the stationary phase approximation; separability of motion along a reaction coordinate is not assumed.
Abstract: The semiclassical limit of quantum mechanical transition state theory is derived by invoking the classical path approximation for the Boltzmann density operator and making use of the stationary phase approximation; separability of motion along a reaction coordinate is not assumed. The resulting expression for the rate constant bears an interesting similarity to that of conventional transition state theory, although all quantities in it refer to the full classical dynamics on the potential energy surface. In place of the vibrational frequencies of the ’’activated complex’’ which appear in the conventional theory, for example, the semiclassical expression contains characteristic frequencies related to the stability properties of a periodic classical trajectory. Conservation of total angular momentum is easily accounted for in a rigorous manner so that the semiclassical model can be applied to three−dimensional dynamical systems.

606 citations

Journal ArticleDOI
TL;DR: In this paper, the classical mechanics of chemically reactive linear collisions are investigated for vibrationally near-adiabatic reactions, and the reaction coordinate for the adiabatic system is shown to be that curve on which local vibrational and internal centrifugal forces balance pointwise.
Abstract: The classical mechanics of chemically reactive linear collisions is investigated for vibrationally near‐adiabatic reactions. A coordinate system which passes smoothly from one suited to the reactants to one suited to the products is used. The Hamilton—Jacobi equation is then solved in the adiabatic approximation by introduction of an ``adiabatic‐separable'' method. Nonadiabatic corrections, which describe the probability of vibrational transitions, are also calculated. They involve the Fourier component of local internal centrifugal and vibration frequency‐change terms. The reaction coordinate for the adiabatic system is shown to be that curve on which local vibrational and internal centrifugal forces balance pointwise. Applications can be made to the role of translational—vibrational energy interchange in reactions, reaction‐cross‐section theory, bobsled effect, and other topics. The results may be compared with electronic computer calculations as they become available.

422 citations

Journal ArticleDOI
TL;DR: In this article, the authors used exact solutions of the classical equations of motion (numerically obtained trajectories) to construct the corresponding classical approximation to the time-independent S-matrix elements for use in quantum mechanical expressions for cross sections; it is argued that this should accurately describe many quantum effects in heavy particle collisions.
Abstract: The aim of this work is to show how one can use exact solutions of the classical equations of motion (numerically obtained trajectories) to construct the corresponding classical approximation to the time‐independent S‐matrix elements for use in quantum mechanical expressions for cross sections; it is argued that this should accurately describe many quantum effects in heavy particle collisions. The expression for the S matrix in terms of the classical trajectory is given for systems of any number of degrees of freedom, and the matter is pursued in detail for the A + BC collision system. It is shown that within this classical limit the magnitude of an S‐matrix element is explicitly determined by its phase. Constancy of total angular momentum is used throughout to reduce the 12 first‐order differential equations of the A + BC system in its center of mass to eight equations. A practical method is also given (in Appendix B) for further reducing the number of coupled equations to six, the minimum number possible.

331 citations