Friction time dependent

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

In the Smoluchowski limit the reaction is by definition the slow coordinate, such that y(kr) y(0) = dL yW Lrand Agii fO). Though the time-dependent friction... 

Straub J E, Berne B J and Roux B 1990 Spatial dependence of time-dependent friction for pair diffusion in a simple fluid J. Chem. Phys. 93 6804... 

Haynes G R, Voth G A and Poliak E 1994 A theory for the activated barrier crossing rate constant in systems influenced by space and time dependent friction J. Chem. Phys. 101 7811... 

From Eq. (70) we see that the time-dependent friction coefficient is given in terms of the force correlation function with projected dynamics. Instead, in MD simulations the time-dependent friction coefficient is computed using ordinary dynamics. 

Equation (75) shows that (u(t) is an exponentially decaying function for long times with a decay constant /p. For very massive B particles M N mN with M/mN = q = const, the decay rate should vary as 1 /N since p = mNq/ (q + 1). The time-dependent friction coefficient (u(t) for a B particle interacting with the mesoscopic solvent molecules through repulsive LJ potentials... 

Figure 9. (a) Semilogarithmic plot of time-dependent friction coefficient as function of time... 

The time dependent friction coefficient, per solute mass p, is related to the fluctuating forces exerted by the solvent on the solute coordinate x through their time correlation function ... 

This self-consistent equation has a simple message the relevant friction for the reaction is determined by the (Laplace) frequency component of the time dependent friction at the reactive frequency X. This frequency sets the basic time scale for the microscopic events affecting k. 

Here m ", and ClXO refer to the equilibrium barrier frequency and the time dependent friction for the solvent coordinate 8AE note the contrast with (2.1), which refers to the corresponding quantities for a solute reactive coordinate. 

So far, the solvent coordinate has not been defined. As noted at the beginning of this Section, the time dependent friction is to be found for the reacting solute fixed at the transition state value x of x. By (3.14), its dynamics were related to those of an (unspecified) solvent coordinate. v. One strategy to identify the solvent coordinate, its frequency, friction, etc., would be to derive an equation of motion for the relevant fluctuating force SF there. To this end, one can use a double-membered projection technique in terms of 8F and 8F. In particular, we define the projection operator... 

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],... 

What does this time dependent friction look like To answer this, we describe the MD calculated results [5c] for (t) for the reactive asymmetric stretch coordinate for the Cl" + CH3C1 SN2 system in H20 solvent, and its associated Fourier spectrum. The latter is particularly illuminating, since it displays peaks clearly identifiable from the spectrum of the same pure H20 liquid. Thus contributions from the H20 bends and... 

Of related interest are results for water response to an instantaneous change in the dipole of a solute [44a], for the time scale of the solvent response for several charge-transfer reactions in water, including the SN2 reaction [49], and for a similar response for Fc21 - Fe3+ in water [44b]. The time scales found in those studies for the water solvent relaxation - and that originally found in [5] for time-dependent friction on the Sn2 transition state - are similar to those observed for the prior reorganization of the solvent H20. 

Actually, all of the above results are in contradiction to the currently conventional view [32-35] that solvent dynamical effects for electronically adiabatic ET reactions are determined by solvent dynamics in the R and P wells, and not the barrier top region. This misses the correct picture, even for fairly cusped barrier. Instead, it is the solvent dynamics occurring near the barrier top, and the associated time dependent friction, that are the crucial aspects. It could however be thought possible that, for cusped barrier adiabatic ET reactions in much more slowly relaxing solvents, the well dynamics could begin to play a significant role. However, MD simulations have now been carried out for the same ET solute in a solvent where the... 

The possibility even exists of including dynamical effects with time-dependent friction terms (plus random forces at finite temperatures).77-80 Flowever, it may not be advisable to take advantage of this possibility, as the simulation would become increasingly slow with increasing number of time steps. Moreover, the simulation will slow down considerably in higher dimensions because of the nonorthogonality of the dynamical coupling in reciprocal space. 

The normal mode transformation imphes that q = uqoP + 2 ujoyj and that p = uooq + UojXj. One can show, that the matrix element uqo may be expressed in terms of the Laplaee transform of the time dependent friction and the barrier frequency A ... 

Optimal planar dividing smface VTST has been used to study the effects of exponential time dependent friction in Ref 93. The major interesting result was the prediction of a memory suppression of the rate of reaction which occms when the memory time and the inverse damping time (f) are of the same order. When... 

PGH theory has been extended. It can be used in conjunction with VTST and optimized planar dividing surfaces,in which case, the energy loss is to be computed along the coordinate perpendicular to the optimal planar dividing surface. In the same vein it has been generalized to include the case of space and time dependent friction. ... 

The expressions for the depopulation factor as given in Eqs. 29 and 30 for the single and double well potential cases respectively, remain unchanged. This version of the turnover theory for space and time dependent friction has been tested successfully against numerical simulation data, in Refs. 68,137. 

The quantity 17(f) is the time-dependent friction kernel. It characterizes the dissipation effects of the solvent motion along the reaction coordinate. The dynamic solute-solvent interactions in the case of charge transfer are analogous to the transient solvation effects manifested in C(t) (see Section II). We assume that the underlying dynamics of the dielectric function for BA and other molecules are similar to the dynamics for the coumarins. Thus we quantify t](t) from the experimental C(t) values using the relationship discussed elsewhere [139], The solution to the GLE is in the form of p(z, t), the probability distribution function. 

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

The comparative study between the time-dependent friction and the viscosity at p = 0.844 and T = 0.728 is depicted in Fig. 3. In this figure both the viscosity and the friction have been normalized to unity at t = 0 by their respective initial values. This figure has several interesting features. Both friction and viscosity exhibit a pronounced ultrafast Gaussian decay which accounts for almost 90% of the total relaxation. The Gaussian time constants... 

In order to investigate the quantum number dependence of vibrational dephasing, an analysis was done on two systems C-I stretching mode in neat-CH3I and C-H mode in neat-CHCl3 systems. The C-I and C-H frequencies are widely different (525 cm-1 and 3020 cm-1, respectively) and so also their anharmonic constants. Yet, they both lead to a subquadratic quantum number dependence. The time-dependent friction on the normal coordinate is found to have the universal nonexponential characteristics in both systems—a distinct inertial Gaussian part followed by a slower almost-exponential part. 

The liquid is characterized by the reduced atomic density per ., and the reduced temperature is given by T = kBT/ ,-, . The time-dependent friction is calculated following the procedure presented in Section IX. 

Figure 10. The calculated total friction (C(0) as a function of time, along with the relative contributions to it from the binary ( and the density relaxation Rpp t) terms for the system CH3 in CH3I. The reduced temperature T (= kaT/e) is 1.158 and the reduced density p for CH3I is 0.918. The time-dependent frictions are scaled by t 2, where = [mirTj/fcgT]1/2 1.1 ps. i and j represent the solute atom and the solvent atom, respectively. The plot shows a clear Gaussian component in the initial time scale for the binary part (r) and slower damped oscillatory behavior for the Rpf t) part.

The same arguments should hold well for C and H solutes in the case of CHCI3, because their individual masses are considerably different (C = 12 g/ mol and H = 1.008 g/mol), which means that H is more free to move than C and therefore the solvent influence on H is likely to have a dominant role in the determination of the C-H bond friction. Furthermore, C is actually shielded by the presence of three Cl atoms, a factor that has not been considered here. However, this is not expected to be serious because the dephasing is more sensitive to the friction dynamics of the H atom. The time-dependent friction profiles for H and C show similar strong bimodal behavior as in the case of CH3 and I systems. 

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