Time-dependent transmission coefficient

Fig. 2. (Left) A schematic illustration of the free energy W along a reaction coordinate for weak coupling to the reaction coordinate. The dotted hue at indicate the position of the barrier top. (Right) Comparison of the time-dependent transmission coefficients kAB/k B for quantum (QRB) and classical (CRB) sam-...

Figure 2 (left panel) shows the energy profile for a two-level system weakly coupled to the reaction coordinate. Both the ground and excited state surfaces have two minima separated by a high barrier at (Rq) = The right panel of this figure compares the time dependent rate coefficients for quantum (QRB) and classical (CRB) treatments of the reaction coordinate for a moderately low temperature (/3 = 2). At t = 0, the CRB result for the time-dependent transmission coefficient, (t) = where is determined from a... 

Figure 5 Time-dependent transmission coefficients K t) for (a) cyclohexane isomerization in CS2 (adapted from ref. 62) (b) the Cl + CH3CI 5 2 reaction in water (adapted from ref. 35). Note the more complicated structure in the cyclohexane K(r) at longer times.

Here, q is the time derivative of the reaction coordinate q and 0 x) is the Heaviside step function. The estimate k-j-si is accurate to the extent that reactive trajectories cross the surface q = q only once during a transition. In the second step of the procedure, corrections to transition state theory are computed by initiating many fleeting trajectories from the q = q surface. The fates of these trajectories determine the time-dependent transmission coefficient. 

There is an additional advantage to the reactive flux method. In the limit of very short times, the coordinate q t) will be in the reactants region if its velocity at time t = 0 is in the direction of the reactants. Therefore the zero time limit of the reactive flux expression is just the one-dimensional TST estimate for the rate. This means that if one wants to study corrections to TST all one needs to do numerically is compute the transmission coefficient k defined as the plateau value of the time-dependent transmission... 

The transmission coefficient Cl (Qj,t), considering transient (broadband) sources, is time-dependent and therefore accounts for the possible pulse deformation in the refraction process. It also takes account of the quantity actually computed in the solid (displacement, velocity potential,...) and the possible mode-conversion into shear waves and is given by... 

However, often the minimum in Si or Ti which is reached at first is shallow and thermal energy will allow escape into other areas on the Si or Ti surface before return to So occurs (Fig. 3, path e). This is particularly true in the Ti state which has longer lifetimes due to the spin-forbidden nature of both its radiative and non-radiative modes of return to So-The rate of the escape should depend on temperature and is determined in the simplest case by the height and shape of the wall around the minimum, similarly as in ground state reactions (concepts such as activation energy and entropy should be applicable). In cases of intermediate complexity, non-unity transmission coefficients may become important, as discussed above. Finally, in unfavorable cases, vibronic coupling between two or more states has to be considered at all times and simple concepts familiar from ground-state chemistry are not applicable. Pres-... 

For many chemical reactions with high sharp barriers, the required time dependent friction on the reactive coordinate can be usefully approximated as the tcf of the force with the reacting solute fixed at the transition state. That is to say, no motion of the reactive solute is permitted in the evaluation of (2.3). This restriction has its rationale in the physical idea [1,2] that recrossing trajectories which influence the rate and the transmission coefficient occur on a quite short time scale. The results of many MD simulations for a very wide variety of different reaction types [3-12] show that this condition is satisfied it can be valid even where it is most suspect, i.e., for low barrier reactions of the ion pair interconversion class [6],... 

This method (transmission coefficient ratios) does not suffer from sensitivity to time referencing errors, although the available frequency range is somewhat lower than the direct rdlection method—the decrease in the upper cut-off frequency bdng dependent upon the length of dielectric-filled line. Loeb et al. have demonstrated how the two methods can be used in a complementary manner to yidd the optimum predsion and bandwidth. 

This theory highlights the importance of the time dependence of the friction and demonstrates how it may be taken into account at the leading order. In the overdamped regime this is done so through the insightful and compact Grote-Hynes (subscript GH) formula for the transmission coefficient [4]. 

In Eq. [7], the frequency-dependent friction is the Laplace transform of the time-dependent friction The presence of the Laplace transform means that the time-dependence of the friction must be known in order to determine the Laplace transform. This friction can be readily determined from molecular dynamics simulations in the approximation where the motion along the reaction coordinate is fixed at x = 0. (A discussion of some subtle, but important, aspects of this approximation is given by Carter et al. ) In that case, the random force R(t) can be calculated from equilibrium dynamics in the presence of this one constraint. From R(t), the time-dependent friction (t) can be calculated and the implicit Eq. [7] solved. The result gives the Grote—Hynes value of the transmission coefficient for that system. 

Diffusion coefficients and solubility coefficients are the fimdamental parameters that control mass transfer in packaging systems. In practice, however, the permeability P is the most important of the permeation properties. Characteristic values for the permeability are the transmission rates (oxygen transmission rate, OTR water vapor transmission rate, WVTR). They indicate the amount of a given substance that passes a unit surface area of film per unit time, depending on the partial pressure gradient for a given temperature, the relative humidity and the given thickness of the film. 

Dogonadze et al. have commented on the possibility of obtaining experimental information about the transmission coefficient x [equation (1)]. Referring again to equation (2), if the reactants A+ and B can be generated suddenly, and if the subsequent electron-transfer reaction is rapid compared with the rate of diffusion (which requires that electron transfer can occur over long distances), then the dependence of reaction rate upon time can be shown to be related to the dependence of transition... 

During the last decade we have studied a number of enzymatic reactions, not only H-transfer processes, and compared the transmission coefficient of the catalyzed reaction with that of the counterpart process in aqueous solution [77. 98, 99]. A useful approach for the calculation of the transmission coefficient is provided by Grote-Hynes theory, where this coefficient can be obtained from a generalized Langevin equation where a time-dependent friction acts on the advance of the system along the reaction coordinate [100] ... 

The interface and turning-point caustic are curved surfaces. They are defined by two principal radii of curvature which depend on both the core radius p and the bend radius R. Under these conditions we use the localized transmission coefficients of Section 7-14 each time a ray loses power by tunneling. When power is lost by refraction, we employ the Fresnel coefficient of Eq. (35-50) for the step profile, and assume complete power loss for the clad parabolic profile, i.e. T= 1. 

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