Frequency-dependent friction dynamics

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]. 

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...

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 ... 

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 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],... 

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. 

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

In Hgl, possibility C is the best description [8]. The dephasing time constant is -150 fs and the overall time for vibrational cooling is -200 fs. Thus coherence is seen in the vibrational excited states, and in the ground state as well. A molecular dynamics simulation of rigid Hgl in ethanol was used to imderstand the VER mechanism [90]. The computed frequency-dependent friction is shown in figure C3.5 8 [90]. Notice this function is much more complicated than in liquid O2 (figure C3.5.6). and an exponential gap law is not observed. The simulation results... 

A similar conclusion has been reached by Ciccotti et al. s-iao jj, their studies of the model ion association reaction. Their system consisted of two equally massive ions, modeled as Lennard-Jones spheres with a positive or negative charge, in a solvent of dipolar molecules. Each solvent molecule was modeled as a Lennard-Jones sphere with a dipole moment of either 2.4 or 3.0 D and with a mass equal to that of the ionic mass. As with the simulations of Karim and McCammon, Ciccotti et al. started the dynamics at the transition state, as determined from the free energy calculations, and ran 104-144 trajectories to determine the transmission coefficient. The values of the transmission coefficient they found were 0.18 in the 2.4 D solvent and 0.16 in the 3.0 D solvent (which are surprisingly, and perhaps coincidentally, close to the results of Karim and McCammon e). Ciccotti et al. also calculated the frequency-dependent friction that the solvent exerted on the reaction coordinate in order to compare the simulation results with Grote-Hynes theory for the rates. The comparison with Grote-Hynes theory was quite close, although within the outer reaches of the calculated uncertainties in the molecular dynamics transmission coefficients. 

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. 

Bagchi, B., and Oxtoby, D. W., 1983. The effect of frequency dependent friction on isomerization dynamics in solution, J.Chem.Phvs>. 78 2735. 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

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... 

An attempt has been made to answer the following questions. What is the relation between r)s(t) and (r) at short times Does the ratio between the two retain a Stokes-like value at all times And how does the relation behave as a function of frequency The analysis seems to suggest that if one includes only the binary interaction in the calculation of the time scale of the short-time dynamics, both viscosity and friction exhibits nearly the same time scale. When the triplet dynamics is included, both the responses become slower with the viscosity being affected more than the friction. The time scale of both the responses axe of the order of 100 fs. It is shown that both the frequency-dependent viscosity and the friction exhibit a clear bimodal dynamics. 

When friction is present, there will be solvent-induced recrossings of the barrier and non-equilibrium solvation, and the frequency Ar by which products are formed depends both on the equilibrium curvature of the barrier and on the dynamics of the solvent as expressed by the time-dependent friction kernel. The following two limiting cases should be noticed. 

Thus, one may conclude that, in the region of comparatively low frequencies, the schematic representation of the macromolecule by a subchain, taking into account intramolecular friction, the volume effects, and the hydrodynamic interaction, make it possible to explain the dependence of the viscoelastic behaviour of dilute polymer solutions on the molecular weight, temperature, and frequency. At low frequencies, the description becomes universal. In order to describe the frequency dependence of the dynamic modulus at higher frequencies, internal relaxation process has to be considered as was shown in Section 6.2.4. 

The deviation of nitrobenzene from the solid line (the slope = 1) in Figure 2 is probably attributed to the frequency dependent dielectric friction for the reaction dynamics around the barrier top, i.e., the much slower dielectric fluctuation of nitrobenzene (tl - 6 ps at 298K) compared with the ET rate hardly works as friction for the barrier crossing. In such case, the friction is shows tl (a[Pg.400]

Collision-induced vibrational excitation and relaxation by the bath molecules are the fundamental processes that characterize dissociation and recombination at low bath densities. The close relationship between the frequency-dep>endent friction and vibrational relaxation is discussed in Section V A. The frequency-dependent collisional friction of Section III C is used to estimate the average energy transfer jjer collision, and this is compared with the results from one-dimensional simulations for the Morse potential in Section V B. A comparison with molecular dynamics simulations of iodine in thermal equilibrium with a bath of argon atoms is carried out in Section V C. The nonequilibrium situation of a diatomic poised near the dissociation limit is studied in Section VD where comparisons of the stochastic model with molecular dynamics simulations of bromine in argon are made. The role of solvent packing and hydrodynamic contributions to vibrational relaxation are also studied in this section. 

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