Dynamic reaction path Hamiltonian

A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-71]. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. 

A quantum dynamical study of the Cl- + CH3 Br 5k2 reaction has been made.78 The calculations are described in detail and the resulting value of the rate constant is in much better agreement with experiment than is that derived from statistical theory, hi related work on the same reaction, a reaction path Hamiltonian analysis of the dynamics is presented.79 The same research group has used statistical theory to calculate the rate constant for the 5n2 reaction... 

Of course, there is more to a chemical reaction than its rate constant the reaction path or mechanism is also of central interest. Once again, nonequilibrium solvation is crucial in describing this path. In an equilibrium solvation picture, the solvent polarization would remain equilibrated throughout the reaction course, but this assumption is rarely satisfied for an actual reaction path, because of the same considerations noted above for the rate constant. Indeed these nonequilibrium solvation effects can qualitatively change the character of the reaction path as compared with an equilibrium solvation image. Dielectric continuum dynamic descriptions thus have an important role to play here as well. Indeed, we will employ in this contribution the reaction path Hamiltonian formulation previously developed [48,49], which can be used to generate a reaction path which is the analog in solution of the well-known Fukui reaction path in the gas phase [50], The reaction path will be discussed for both reaction topics in this contribution. 

In a study of the rate of isomerization of HCN to CNH, Rice and co-workers [19] suggested exploiting a reaction path Hamiltonian as a device to permit extension of classical statistical reaction rate theory from few-dimensional to many-dimensional systems. In that approach the dynamics of the reacting molecule is reduced to that of a system with a complicated but one-dimensional reactive DOF coupled with other effective DOFs. Although their calculations based on this approach yield an accurate description of the isomerization rate as... 

The most common assumption is one of a reaction path in hyperspace (Miller et al. 1980). A saddle point on the PES is found and the steepest descent path (in mass-weighted coordinates) from this saddle point to reactants and products is defined as the reaction path. The information needed, except for the path and the energies along it, is the local quadratic PES for motion perpendicular to the path. The reaction-path Hamiltonian is only a weakly local method since it can be viewed as an approximation to the full PES and since it is possible to use any of the previously defined global-dynamical methods with this potential. However, it is local because the approximate PES restricts motion to lie around the reaction path. The utility of a reaction-path formalism involves convenient approximations to the dynamics which can be made with the formalism as a starting point. 

Clearly, there are many ways in which the ideas we have proposed must be extended. Among the more important extensions we cite variational optimization of the shape, duration, and separation of the pulses used to generate the selectivity of reactivity, and analysis of the changes induced by the inclusion of all degrees of freedom of the molecule (say in the sense of a reaction path Hamiltonian, or a dynamical path Hamiltonian). For studies involving more degrees of freedom, a swarm of classical trajectories should be a very useful tool. 

In the context of reaction path studies. Billing has employed classical trajectories directly to simulate the dynamics of an approximation to the reaction path Hamiltonian (including rotational motion) [89,93]. Classi-... 

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. 

With a Hamiltonian one can begin to describe dynamics, and this section considers some of the dynamical models that have been based on the reaction-path Hamiltonian, beginning with the simplest approaches and proceeding to more rigorous ones. 

Fig. 2 shows an example of the SCP-IOS model, i.e., Eq. (3.9), applied to vibrational excitation of H2 by collision with He atoms. ° One sees that this simple dynamical model based on the reaction path Hamiltonian does an extremely good job of describing vibrational inelasticity. 

I have attempted in this paper to illustrate the wide variety of dynamical treatments that can be usefully based on the reaction path Hamiltonian model, from simple "back of the envelope" statistical approximations (TST, RRKM, etc.) all the way to rigorous computational methods that can be practically applied to polyatomic systems. Given the necessary "input" which characterizes the model — i.e., the quantum chemistry calculations of the reaction path, and the energy and force constant matrix along it — the example applications that have been discussed show that it provides a quantitative ab initio approach to reaction dynamics in polyatomic molecular systems. 

For the Li + HF reaction Dunning, Kraka Fades (7) showed that the features of the reaction valley can be readily understood in terms of the changes in the electronic structure of the system as it evolves during the reaction. For the OH + H2 reaction they showed that the terms in the reaction path Hamiltonian provide a rationale for many of the qualitative features of reaction dynamics, including such fine effects as the deposition of reactant vibrational excitation into product vibrational modes. The reaction valley approach thus provides a direct connection between the electronic structure of the system, the potential energy surface and the reaction dynamics. 

Moreover, even in view of dynamical calculations, the IRP has a rather severe drawback (a similar situation prevails for the genuine steepest-descent path) according to the Reaction Path Hamiltonian philosophy, initially developed by Miller, Handy and Adams [33] and extended by others [15,18,31,36,37], the curvilinear displacement along the IRP (i.e. the parameters t or T above) should be taken as one of the internal coordinates describing the system This is undoubtedly a clever idea, since the value of that very coordinate would, throughout the dynamical calculations, be a measure of the advancement of the reactive process. The complications stem from the fact that (i) it can be fairly complicated to define the complementary 3N-7 internal coordinates (except for few isomerizations in which they can be considered small-amplitude vibrational coordinates) and (ii) in many cases, multivaluation difficulties cannot be avoided, especially when large-amplitude motions drive the representative point of the system in... 

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