Friction frequency dependent

Poliak E, Grabert H and Hanggi P 1989 Theory of activated rate processes for arbitrary frequency dependent friction solution of the turnover problem J. Chem. Phys. 91 4073... 

VER occurs as a result of fluctuating forces exerted by the bath on the system at the system s oscillation frequency O [5]. Fluctuating dynamical forces are characterized by a force-force correlation function. The Fourier transfonn of this force correlation function at Q, denoted n(n), characterizes the quantum mechanical frequency-dependent friction exerted on the system by the bath [5, 8]. 

Equation (C3.5.2 ) is a function of batli coordinates only. The VER rate constant is proportional to tire Fourier transfonn, at tire oscillator frequency Q, of tire batli force-correlation function. This Fourier transfonn is proportional as well to tire frequency-dependent friction q(n) mentioned previously. For example, tire rate constant for VER of tire Emdamental (v = 1) to tire ground (v = 0) state of an oscillator witli frequency D is [54]... 

Chemical reaction dynamics is an attempt to understand chemical reactions at tire level of individual quantum states. Much work has been done on isolated molecules in molecular beams, but it is unlikely tliat tliis infonnation can be used to understand condensed phase chemistry at tire same level [8]. In a batli, tire reacting solute s potential energy surface is altered by botli dynamic and static effects. The static effect is characterized by a potential of mean force. The dynamical effects are characterized by tire force-correlation fimction or tire frequency-dependent friction [8]. 

Figure C3.5.8. Computed frequency-dependent friction (inverseiy proportionai to tire VER iifetime T ) from a ciassicai moiecuiar dynamics simuiation of rigid Hgl moiecuies in etiianoi soiution, from [90]. The Hgl vibrationai...

At present the body of data on reactions in clusters is insufficient to test the above two microcanonical approaches. For electron transfers in solution it seems clear that the vibrational assistance approach, stemming from Eq. (1.2), with its extensions mentioned earlier, is the one that has been the most successful [27-30]. For slow isomerizations Sumi and Asano have pointed out that an analysis based on Eq. (1.2) was again needed [40]. An approach based on Eq. (1.1) or on its extension to include a frequency-dependent friction, they noted, led to unphysical correlation times [40]. In investigations of fast isomerizations the most commonly studied system has been the photoex-cited trans-stilbene [5, 41-43,46]. Difficulties encountered by a one-coordinate treatment for that system have been reported [4, 8]. Indeed, coherence results for photoexcited cw-stilbene have shown a coupling of a phenyl torsional mode to the torsional mode about the C=C bond [42, 47]. 

Many questions in the analysis of solvent dynamics effects for isomer-izations in solution have arisen, such as (1) when is a frequency-dependent friction needed (2) when does a change of solvent, of pressure, or of temperature change the barrier height (i.e., the threshold energy), and (3) when is the vibrational assistance model needed, instead of one based on Eq. (1.1) or its extensions ... 

D. Derivation of the Final Expression for the Frequency-Dependent Friction... 

The relation between friction and viscosity goes beyond the Stokes relation. The Navier-Stokes hydrodynamics has been generalized by Zwanzig and Bixon [23] to include the viscoelastic response of the medium. This generalization provides an elegant expression for the frequency-dependent friction which depends among other things on the frequency-dependent bulk and shear viscosities and sound velocity. 

Note that the above expression is known as the generalized Einstein equation and that the memory function, ((z), is the frequency-dependent friction. 

The time-dependent VACF is obtained by numerically Laplace inverting the frequency-dependent VACF, which is related to the frequency-dependent friction through the following generalized Einstein relation given by Eq. (81) ... 

Thus in this scheme the frequency-dependent friction has been calculated self-consistently with the MSD. 

The self-consistency is implemented through the following iterative scheme. First, the VACF is obtained from Eq. (153) by replacing the total frequency-dependent friction, C(z) by its binary part, (B(z)- The VACF thus obtained is used to calculate the MSD through Eq. (152). Now this MSD is used to calculate Rpp(t) and Rrr(t) and thus (z). This total friction is used to calculate the new VACF, which again is used to determine MSD and thus (z). This iterative process is continued until the VACF obtained from two consecutive steps overlap. 

In the previous section we have discussed the relation between the time- and frequency-dependent friction and viscosity in the normal liquid regime. The study in this section is motivated by the recent experimental (see Refs. 80-87) and computer simulation studies [13,14, 88] of diffusion of a tagged particle in the supercooled liquid where the tagged particle has nearly the same size as the solvent molecules. These studies often find that although the fric-... 

The mode coupling expression for the frequency-dependent friction as presented in Section IX is given by... 

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. 

The situation is far more complex for reactions in high viscous liquids. The frequency-dependent friction, (z) [in the case of Fourier frequency-dependent friction C(cu)], is clearly bimodal in nature. The high-frequency response describes the short time, primarily binary dynamics, while the low-frequency part comes from the collective that is, the long-time dynamics. There are some activated reactions, where the barrier is very sharp (i.e., the barrier frequency co is > 100 cm-1). In these reactions, the dynamics is governed only through the ultrafast component of the total solvent response and the reaction rate is completely decoupled from the solvent viscosity. This gives rise to the well-known TST result. On the other hand, soft barriers... 

In order to apply the Grote-Hynes formula to realistic cases, the frequency-dependent friction is required which is calculated from the mode coupling theory (MCT) presented in Section IX. 

The total frequency-dependent friction calculated from the MCT, (z), is plotted against the Laplace frequency (z) in Fig. 14. In the same figure the Enskog friction (e and the binary contribution (z) are also shown. Note here that in the high-frequency regime the frequency-dependent total friction is much less than the Enskog friction and is dominated entirely by the binary... 

Figure 14. The frequency-dependent total friction Ctoui(z) (solid line) and the binary friction (B(z) (dashed-dot line) plotted as a function of Laplace frequency (z). For comparison, the calculated Bnskog friction is also shown (dashed line). The calculation has been performed for p = 0.85 and T = 0.85. The frequency-dependent friction, the Enskog friction, and the frequency are scaled by xsc l. This figure has been taken from Ref. 170.

The isomerization rate is calculated using the Grote-Hynes formula, given by Eqs. (320) and (322). The frequency-dependent friction ( (z)) and viscosity (rj) has been obtained from the mode coupling theory presented in Section IX. For convenience the rate is expressed in terms of the dimensionless quantity k in the following form ... 

There have been several other theoretical studies by different authors [144, 145,156,163] where the frequency-dependent friction was modeled by using the modified version of the generalized hydrodynamic expression [23, 165]. These theories failed to reproduce the experimental results at certain limits. Barbara and coworkers attributed this failure of the G-H theory to the nonavailability of a reliable frequency-dependent friction and called for the use of a friction better than the hydrodynamic friction. 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

The frequency (z)-dependent velocity correlation function Cv(z) is related to the frequency dependent friction by the well-known generalized Einstein relation,... 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

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