Reaction path Hamiltonian method

Carrington and Miller (235) developed a method called the reaction-surface Hamiltonian for reactions with large amplitudes perpendicular to the reaction path and for some types of reactions with bifurcation of the reaction path. In contrast to the reaction-path Hamiltonian method, in the reaction-surface Hamiltonian method two coordinates are extracted from the complete coordinate set. One coordinate describes motion along the reaction path and the second one describes the large-amplitude motion. Potential energy in space of the remaining 3JV — 8 coordinates perpendicular to the two-dimensional reaction surface is approximated by quadratic functions. It... 

The separation of the PES into a part determined by the reaction coordinate and a part described by a quadratic approximation in a subspace of the remaining coordinates has recently often been used, typically with the WKB approximation (236,237) Yamashita and Miller (238) utilized the reaction-path Hamiltonian method combined with the path-integral method to calculate the rate constant of the reaction of H + H2. 

Finding a coordinate system that minimizes the coupling between the DOFs has always been a natural aspiration in theoretical chemistry. The so-called reaction-path formalism is just such a procedure, as is the use of Normal-Form theory [13], which is our method of choice. Normal-Form theory gives us sufficient conditions for a Hamiltonian to be transformed into the form of Eq. (1) in the neighborhood of an equilibrium point of center (g) center g) saddle type. This result is well known (see, e.g.. Ref. 13). To summarize, first we perform a Taylor expansion of the Hamiltonian [Eq. (1)] ... 

In the empirical valence bond (EVB) model [304, 349, 370] a fairly small number of VB functions is used to fit a VB model of a chemical reaction path the parameterisation of these functions is carried out to reproduce experimental or ab initio MO data. The simple EVB Hamiltonian thus calibrated for a model reaction in solution can subsequently be used in the description of the enzyme-ligand complex. One of the most ingenious attributes of the EVB model is that the reduction of the number of VB resonance structures included in the model does not introduce serious errors, as would happen in an ab initio VB formulation, due to the parameterisation of the VB framework which ensures the reproduction of the experimental or other information used. This computationally efficient approach has been extensively used with remarkable success [305, 306, 371, 379] A similar method presented by Kim and Hymes [380] considers a non-equilibrium coupling between the solute and the solvent, the latter being treated as a dielectric continuum. 

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. 

The simplest way to include solvation effects is to calculate the reaction path and tunneling paths of the solute in the gas phase and then add the free energy of solvation at every point along the reaction path and tunneling paths. This is equivalent to treating the Hamiltonian as separable in solute coordinates and solvent coordinates, and we call it separable equilibrium solvation (SES) [74]. Adding tunneling in this method requires a new approximation, namely the canonical mean shape (CMS) approximation [75]. 

For large systems (systems with more than four atoms) it is necessary to use methods which reduce the coordinate space for which we have to know the potential energy surface. Since chemical reactions are at most three or four-center reactions, the obvious partitioning is to treat the motion of the three or four atoms defining the reaction center by some of the methods described and the remaining motion using a small-amplitude description identical to the one used in the reaction path Hamiltonian method. In the... 

In 1980, Miller, Handy, and Adams further entrenched the reaction path idea with the derivation of the classical Hamiltonian for a simple potential based on the MEP. The motivations for the development of the reaction path Hamiltonian included the emerging ability to calculate derivatives of the Born—Oppenheimer potential energy (i.e., forces and force constants) directly via electronic structure methods and the desire to develop practical methods for... 

Chapter 2, Michael L. McKee and Michael Page address an important issue for bench chemists how to go from reactant to product. They describe how to compute reaction pathways. The chapter begins with an introduction of how to locate stationary points on a potential energy surface. Then they describe methods of computing minimum energy reactions pathways and explain the reaction path Hamiltonian and variational transition state theory. 

In Quantum Mechanical calculations, the energy is computed from the exact hamiltonian. It is then possible to build a Born-Oppenheimer energy surface which can be used later to perform lattice dynamics or to study the reaction path of a displacive phase transition. These methods give access to the electron density, the spin density and the density of states which are useful to predict electric and optical properties as well as to analyze the bonding. Recently, methods combining a quantum mechanical calculation of the potential and the Molecular Dynamics scheme have been developed after the seminal work of R. Car and M. Parrinello. 

Application of the Reaction Path Hamiltonian Method to the Reaction H2 + OH —> H2O + H Acceptor-Donor (AD) Theory of Chemical Reactims (O 4)... 

Reaction Path Hamiltonian Method 14.5.1 Energy Close to IRC... 

The reaction represents one of a few polyatomic systems for which precise calculations were performed. It may be instructive to see bow a practical implementation of the reaction path Hamiltonian method looks. 

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. 

The reaction path Hamiltonian also provides a very useful framework for the rigorous calculation of the Boltzmann (i.e., thermally averaged) rate constant for a chemical reaction using the path integral methods described by Miller, Schwartz, and Tromp. In that paper it is shown that the rate constant can be expressed as the time integral of a flux-flux autocorrelation function... 

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