Reaction path coordinate

The close-coupling approach works readily and simply if the reaction is purely melastic . The method can also be made to work very simply for a single product arrangement (as in collinear reactions), by using a twisted coordinate system, most conveniently reaction path coordinates [37, 38 and 39] as shown in figure B3.4.3. 

The iGLE also presents a novel approach for studying the reaction dynamics of polymers in which the chemistry is driven by a macroscopic force that is representative of the macroscopic polymerization process itself The model relies on a redefined potential of mean force depending on a coordinate R which corresponds locally to the reaction-path coordinate between an n-mer and an (n -t 1 )-mer for R = nl. The reaction is quenched not by a kinetic termination step, but through an (R(t))-dependent friction kernel which effects a turnover from energy-diffusion-limited to spatial-diffusion-Iimited dynamics. The iGLE model for polymerization has been shown to exhibit the anticipated qualitative dynamical behavior It is an activated process, it is autocatalytic, and it quenches... 

Suppose the reactive polyatomic molecule of interest can undergo uni-molecular reaction to form several products, and we imagine carrying out a constrained reaction path analysis for each of the product channels. To carry out the analysis of a particular constrained reaction path, Zhao and Rice adopted a system-bath model [74] in which the reaction path coordinate delines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the optimal control field for motion along the reaction coordinate. 

Figure 4-9. Scheme of saddle point configurations, that is, energy vs. reaction path coordinate r. rp conforms to different states of atomic particles in a crystal, for example, sublattices in homogeneous crystals or lattice planes in inhomogeneous crystals. 

FIGURE 3. AMl-calculated reaction path coordinate for the aldol reaction of 150b with CF3CHO. Reproduced from Y. Makino, K. Iseki, K. Fujii, S. Oishi, T. Fhrano and Y. Kobayashi, Tetrahedron Lett., 36, 6527. Copyright 1995, courtesy of Elsevier... 

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

Several approximate models for computing resonance energies and widths have been compared, with numerical Illustrations for blmolecular reactive resonances for which an adiabatic separation in reaction-path coordinates Is reasonable. For systems In which the reaction-path curvature Is not too great, the adiabatic model... 

To describe these approximations, we provide the treatment of dynamics within a correlation function formalism (Yamamoto 1960 Miller et al. 1983). Let the reaction-path coordinate be denoted by u and the other coordinates by U. Define the saddle point on the path as u = 0. Then the thermally averaged rate constant for transition from m < 0 to u > 0 is given by... 

Curvilinear coordinates may be defined in terms of (x, 2Q and potential parameters. We sketch their use here, leaving detailed definitions and references for the following sections. Reaction path coordinates are usually designated s, along the path, and p, perpendicular to it. The Schrodinger equation becomes... 

Reaction-path coordinates were first described in detail by Marcus (1966). Choosing a curve 4 in the two-dimensional configuration space (x, X) for the reaction AB + C -> A + BC, he introduced two new variables the distance s along ( , and r, the shortest distance of nearby points in the plane to < . He then proposed an adiabatic-separable method that included curvilinear motion effects. Writing for the potential V, without loss of generality,... 

The great majority of reactive collinear collisions have been studied computationally, with physical insight being extracted from the final numerical results. An alternative to this approach is the development of models that may be interpreted in terms of approximate analytical solutions. This has been done by Hofacker and collaborators. A non-adiabatic model was introduced (Hofacker and Levine, 1971) to describe population inversion in reactions. It makes use of reaction path coordinates and assumes harmonic vibrational motion perpendicular to the path. 

To obtain the Hamiltonian function for this reaction path potential, it is necessary to express the kinetic energy in terms of the momenta congugate to the reaction path coordinates. The result, for the J=0 case, is [12]... 

If the reaction path coordinate were truly separable from the other coordinates, then classical recrossing of the barrier would not occur and our k[E) would give the exact quantum mechanical recrossing rate. Unfortunately, this situation is impossible. For example, if our chemical reaction is... 

A natural compromise is to characterize just the reaction valley leading from reactants through the transition state to products, i.e., the (3N-7)-dimensional valley centered upon the reaction path. The idea of characterizing a reacting system by a coordinate set based on such a concept has frequently been used in chemistry see, e.g., the papers listed in Ref. 5. Miller, Handy Adams (6) placed this approach on a firm theoretical foundation by deriving the nuclear Hamiltonian in a set of (3N-6) reaction path coordinates, namely, the IRC and the (3N-7) vibrational modes transverse to the reaction path. In this approach the reaction valley is... 

The next value of the reaction path coordinate s +i = s + 8s is given by choosing the value of to satisfy the following integral equation ... 

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