Determination of the reaction path

In order to determine the reaction path one starts at the saddle point, determined by the Newton-Raphson technique described in the previous sections, and follows the steepest descent path to the reactant and product valleys. Along the steepest descent path one takes steps in the direction which leads to a maximal decrease in potential energy. Thus we wish to maximize 

By following the steepest descent path we change the coordinates of all the N atoms of the system. If the atoms were moving infinitely slowly the path would be mapped out by the classical equations of motion. However, the displacement of the atoms should not be mass dependent. This can be avoided by introducing mass-weighted coordinates in a similar manner to what is done in the normal mode treatment of vibrational motion. We therefore introduce the socalled intrinsic reaction path (IRP) [19]. The IRP is defined as the steepest descent path in mass-weighted coordinates. Thus we have the mass weighted cartesian position coordinates and notation, such that 

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Therefore, an a posteriori approach seems to be an attractive alternative, in which the finite temperature MD simulation is performed along the pre-determined reaction paths.33 Such an approach seems to cost more computational time, as it requires determination of the reaction path prior to MD simulation. However, it can be often less expensive than repeating a simulation due to unexpected problems, e.g. a pronounced hysteresis. 

Hsu K, Buenker RJ, Peyerimhoff SD, Theoretical determination of the reaction path in the prototype electrocyclic transformation between cyclobutene and cis-butadiene Thermochemical process J Am Chem Soc 93, 2117-2127 (1971)... 

K. Hsu, R. J. Buenker, S. D. Peyerimholf, Theoretical Determination of the Reaction Path in the Prototype Electrocyclic Transformation between Cyclobutenebndci. -Butadicne. Thcrrnochcrnical Process. ./. Am. Chem. Soc. 1971,93,2117-2127. 

The motion of the composition point is no longer always in a direction normal to the equipotential line, but at an angle determined by the matrix A. This angle changes with position but, since the matrix A is diagonal, the components of the vector 3 in each direction are completely independent. Hence, the determination of the reaction path may be made easily from the equipotential lines if A is known. 

For the monomolecular system the determination of the proper Liapounov function for a proper choice of coordinates reduces the complexity of the problem and leads to the immediate determination of the reaction paths. This formulation is, of course, merely the solution given in Section II rewritten in a different form. Its importance hes in the suggestion that the complexity of some classes of nonhnear systems may be reduced by such an approach even if the complete solution is not obtained. 

In Table I, the various stages in the determination of the mechanism of an electrocatalytic reaction and the methods used to make such determinations are outlined. The techniques used to determine the over-all reaction are simple except in the case of some reactions, as for example anodic oxidation of certain organic compounds, where several reactions may occur in parallel. The analysis of the over-all reaction also gives the number of electrons transferred per molecule. As seen from Table I, a determination of the reaction path and rate-determining step involves ... 

DETERMINATION OF THE REACTION-PATH HAMILTONIAN FROM AB INITIO CALCULATIONS... 

The differences in rate for the two positions of naphthalene show clearly that an additional-elimination mechanism may be ruled out. On the other hand, the magnitude of the above isotope effect is smaller than would be expected for a reaction involving rate-determining abstraction of hydrogen, so a mechanism involving significant internal return had been proposed, equilibria (239) and (240), p. 266. In this base-catalysed (B-SE2) reaction both k and k 2 must be fast in view of the reaction path symmetry. If diffusion away of the labelled solvent molecule BH is not rapid compared with the return reaction kLt a considerable fraction of ArLi reacts with BH rather than BH, the former possibility leading to no nett isotope effect. Since the diffusion process is unlikely to have an isotope effect then the overall observed effect will be less than that for the step k. ... 

The procedure and methods for the MEP determination by the NEB and parallel path optimizer methods have been explained in detail elsewhere [25, 27], Briefly, these methods are types of chain of states methods [20, 21, 25, 26, 30, 31]. In these methods the path is represented by a discrete number of images which are optimized to the MEP simultaneously. This parallel optimization is possible since any point on the MEP is a minimum in all directions except for the reaction coordinate, and thus the energy gradient for any point is parallel to the local tangent of the reaction path. 

Through a procedure such as umbrella sampling we can calculate the correlation function C(t) for a particular time t. For a determination of the reaction rate constants, however, we need the derivative of C(t). Of course, the time derivative of C(t) could be determined by calculating C(t) at different path lengths t and taking the derivative numerically. Fortunately, such a computationally expensive procedure is not necessary. One can derive expressions for the reaction rate constant that... 

To invoke our geothermometer, we need to recombine the vapor and fluid phases and then heat the mixture to determine saturation indices as functions of temperature. We could do this in two steps, first titrating the vapor phase into the liquid and then picking up the results as the starting point for a polythermal path. We will employ a small trick, however, to accomplish these steps in a single reaction path. The trick is to add the vapor phase quickly during the first part of the reaction path but use the cutoff option to prevent mass transfer over the remainder of the path. The commands to set the mass transfer are... 

Thus, there are four possible mechanisms for the hydrogen evolution in an acid solution (1) CT is RDS, CD fast (2) CD is RDS, CT is fast (3) CT is RDS, ED is fast and (4) ED is RDS, CT is fast. Different paths and different mechanisms have different Tafel slopes. Readers are referred to Refs. 11, 15, 21-23, and 26 for determination of the reaction mechanisms. 

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