Reactive motion

An elegant explanation for the unusual viscosity dependence was provided by the non-Markovian rate theory (NMRT) of Grote and Hynes [149] which incorporates the idea of frequency dependence of the friction. According to this theory the friction experienced by the reactive motion is not the zero frequency macroscopic friction (related to viscosity) but the friction at a finite frequency which itself depends on the barrier curvature. The rate is obtained by a self-consistent calculation involving the frequency-dependent friction. 

Figure 27 illustrates reactive motion through cylindrical manifolds and construction of the manifolds. It shows that a trajectory initially trapped in conformer A eventually enters the interior of the cylinder W. By going through W, it reacts, and it goes to conformer B by entering manifold Wg. The cylinder Wj mediates all pre-reactive motion A B, and the cylinder Wg mediates all... 

Phase space curves, associated with a one-dimensional attractive potential such as that for He-I, are illustrated in figure 8.12. The separatrix (i.e., dashed curve) is called a reaction separatrix, because it is a boundary between bound unreactive motion and unbound reactive motion. Regardless of the length of time the trajectory is evaluated it will remain on this curve and, therefore, the separatrix is called an invariant curve. It is plotted in figure 8.13(a). 

These plots are shown in Figure 5, with trapped motion indicated by a T and reactive motion indicated by an R. Note that the curve labeled T corresponds to an ellipse that does not span both isomers (there is another trapped curve that corresponds to being trapped in isomer B with the same torsional energy),... 

Next considering the case of reactive motion at the same energy E, we realize that the situation at hand is not very different. The phase space of qi is still elliptical. The phase space of is not elliptical, but it is a simple closed curve, and it still therefore has the same topology as a one-dimensional sphere (every point on a closed curve can be uniquely mapped onto a sphere). Thus, the phase space of reactive motion consists of foliated tori that span both sides of the potential barrier. These reactive tori will be skinny when sliced along the ( 2 Pz) compared to the trapped tori, because they have less energy in the vibrational coordinate and more in the reaction coordinate. In Figure 8 these are labeled Qab j... 

R. Q. Topper, Ph.D. Dissertation, Yale University, New Haven, CT, 1990. The Dynamics and Kinetics of Reactive Motion Between Multiple Geometric Conformers. 

Let us summarize the steps quickly. First, we use the Marcus theory to obtain the reaction free-energy surface. Second, we adopt the Grote-Hynes theory to obtain the reaction rate. The latter needs frequency-dependent friction on the reactive motion, which is the solvent polarization. Third, we use the solvation time correlation function to obtain the frequency-dependent friction. 

The quasiclassical description of the motion along the reaction coordinate x is also possible when the initial state of the system is assumed to be a bound-state, in which the potential energy V(x,y ) has a minimum in direction x and other directions y corresponding to the non-reactive motions. In the vicinity of the point (x = x, yj = y ) of this minimum, a series expansion yields... 

HYNES - You raise an important point that concerns us. Electronic solvent polarization would presumably adjust "instantly to the reactive motion and exert no force to induce recrossing, however we use the "bare" H2O dipole moment in the calculations and not the solvent polarizability enhanced value. At some stage, polarizability should be included in such calculations, but we do not know how to do it. 

Haddadin, S., Urbanek, H., Parusel, S., Burschka, D., Rofimann, J., Albu-Schaffer, A., Hirzinger, G. Realtime reactive motion generation based on variable attractor dynamics and shaped velocities. In lEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS 2010), Taipeh, Taiwan, pp. 3109-3116 (2010)... 

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