Reaction coordinate problem

Fig. 8. Generic free energy landscapes can illustrate the reaction coordinate problem in complex systems. If both q and q are important ingredients of the reaction coordinate top left panel), leaving out one of them, e.g. q, will lead to a wrong prediction of the transition state region bottom panel), and hence a wrong mechanism and a statistically inaccurate rate constant. See text for more details...

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The cyclopentadienyl radical and the cyclopentadienyl cation are two well-known Jahn-Teller problems The traditional Jahn-Teller heatment starts at the D k symmetry, and looks for the normal modes that reduce the symmetry by first-01 second-order vibronic coupling. A Longuet-Higgins treatment will search for anchors that may be used to form the proper loop. The coordinates relevant to this approach are reaction coordinates. 

Click Coached Problems for a self-study module on reaction coordinate diagrams. 

There have been a number of investigations of the formulation of the problem of electron transfer accompanied by atom transfer particularly with regard to the simultaneous movement of the proton (which, in view of its small mass, may in fact be an atypical case). A possible model for such processes would assume a conservation of bond order along the reaction coordinates (Johnston, 1960). It is of interest that the results of such calculations are similar to those for electron transfer for weak coupling, although the interpretation of the process and parameters (such as a) are different. 

In the general case of a piston flow reactor, one must solve a fairly small set of simultaneous, ordinary differential equations. The minimum set (of one) arises for a single, isothermal reaction. In principle, one extra equation must be added for each additional reaction. In practice, numerical solutions are somewhat easier to implement if a separate equation is written for each reactive component. This ensures that the stoichiometry is correct and keeps the physics and chemistry of the problem rather more transparent than when the reaction coordinate method is used to obtain the smallest possible set of differential... 

Example 7.17 illustrates the utility of the reaction coordinate method for solving equilibrium problems. There are no more equations than there are independent chemical reactions. However, in practical problems such as atmospheric chemistry and combustion, the number of reactions is very large. A relatively complete description of high-temperature equilibria between oxygen and... 

The maximum set will consist of Equations (14.1) and (14.3) and N versions of Equation (14.2), where N is the number of components in the system. The maximum dimensionality is thus 2- -A. It can always be reduced to 2 plus the number of independent reactions by using the reaction coordinate method of Section 2.8. However, such reductions are unnecessary from a computational viewpoint and they disguise the physics of the problem. 

The limitation of the problem to a single energy profile is depicted in Fig. 5.5 as a symbolic dependence of the potential energy of the reacting system on the reaction coordinate. 

The motion of activated complexes within the transition state may be analyzed in terms of classical or quantum mechanics. In terms of classical physics, motion along the reaction coordinate may be analyzed in terms of a onedimensional velocity distribution function. In terms of quantum mechanics, motion along the reaction coordinate within the limits of the transition state corresponds to the traditional quantum mechanical problem involving a particle in a box. 

Tx and Tx are the kinetic energies of the atomic coordinates and X variables, respectively. The As are treated as volumeless particles with mass mx. Since the X variables are associated with the chemical reaction coordinates , the A-dynamics method can utilize the power of specific biasing potentials in the umbrella sampling method to overcome sampling problems that require conventional FEP calculations to be performed in multiple steps. 

Enyne metathesis using ruthenium catalyst 52a was developed by Mori and Kinoshita [18]. When enyne 62a was treated with Grubbs s ruthenium catalyst 52a in benzene at room temperature for 22 h, the cyclized product 63a was obtained in only 13% yield (Eq. 29). It seems that the catalyst was coordinated by the diene generated in this reaction. This problem was overcome by the study of... 

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