Barrier frequency

Fig. 3. One-dimensional barrier along the coordinate of an exoergic reaction. Qi(E), Q i(E), QiiE), Q liE) are the turning points, coo and CO initial well and upside-down barrier frequencies, Vo the barrier height, — AE the reaction heat. Classically accessible regions are 1, 3, tunneling region 2.

Fig. 4. Variationally determined effective parabolic barrier frequency co ff for the Eckart barrier in units of 2n/hfi [Voth et al. 1989b], The dotted line is the high-temperature limit co = co. ...

The dimensionless upside-down barrier frequency equals = 2(1 — and the transverse frequency Qf = Q. The instanton action at = oo in the one-dimensional potential (4.41) equals [cf. eq. (3.68)]... 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

If the potential is parabolic, it seems credible that the inverted barrier frequency A should be substituted for the parabolic barrier transparency to give the dissipative tunneling rate as... 

Figure 5. Reaction probabilities for a given instance of the noise as a function of the total integration time Tint for different values of the anharmonic coupling constant k. The solid lines represent the forward and backward reaction probabilities calculated using the moving dividing surface and the dashed lines correspond to the results obtained from the standard fixed dividing surface. In the top panel the dotted lines display the analytic estimates provided by Eq. (52). The results were obtained from 15,000 barrier ensemble trajectories subject to the same noise sequence evolved on the reactive potential (48) with barrier frequency to, = 0.75, transverse frequency co-y = 1.5, a damping constant y = 0.2, and temperature k%T = 1. (From Ref. 39.)...

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. 

We consider the reactive solute system with coordinate x and its associated mass p, in the neighborhood of the barrier top, located at x=xi=0, and in the presence of the solvent. We characterize the latter by the single coordinate. v, with an associated mass ps. If the solvent were equilibrated to x in the barrier passage, so that there is equilibrium solvation and s = seq(x), the potential for x is just -1/2 pcc X2, where (, , is the equilibrium barrier frequency [cf. (2.2)]. To this potential we add a locally harmonic restoring potential for the solvent coordinate to account for deviations from this equilibrium state of affairs ... 

In this nonadiabatic limit, the transmission coefficient is determined, via (2.8) by the ratio of the nonadiabatic and equilibrium barrier frequencies, and is in full agreement with the MD results [5a-5c]. (By contrast, the Kramers theory prediction based on the zero frequency friction constant is far too low. Recall that we emphasized for example the importance of the tail to the full time area of the SN2 (t). In the language of (3.14), the solvation time xs is not directly relevant in determining... 

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 ... 

A useful trial variational function is the eigenfunction of the operator L for the parabolic barrier which has the form of an error function. The variational parameters are the location of the barrier top and the barrier frequency. The parabolic barrierpotential corresponds to an infinite barrier height. The derivation of finite barrier corrections for cubic and quartic potentials may be found in Refs. 44,45,100. Finite barrier corrections for two dimensional systems have been derived with the aid of the Rayleigh quotient in Ref 101. Thus far though, the... 

The free energy iv[f] must now be varied with respect to the location f as well as with respect to the transformation coefficients ao, aj j = 1,.. . , N. The details are given in Ref 107 and have been reviewed in Ref 49. The final result is that the frequency A and collective coupling parameter C are expressed in the continumn limit as functions of a generalized barrier frequency A, One then remains with a minimization problem for the free energy as a function of two variables - the location f and A, Details on the mmierical minimization may be found in Refs. 68,93. For a parabolic barrier one readily finds that the minimum is such that f = 0 and that X = In other words, in the parabolic barrier limit, optimal planar VTST reduces to the well known Kramers-Grote-Hynes expression for the rate. 

In this equation J(cd) is the spectral density of the bath, C° is the correlation function for the nncoupled 1-dimensional problem, Bi and B2 are functions that depend on the characteristics of the bath and on the barrier frequency COb (the detailed forms of these functions are given elsewhere ) and... 

Many experiments (see Refs. 154-160) have shown that Kramers theory fails to describe the viscosity dependence of rate in isomerization reactions. This is especially the case where the barrier frequency (< , ) giving the curvature at the barrier top is large. In this case both experimental (see Refs. 154-160) and simulation studies [147, 153] find a rate that decreases with viscosity at a rate much slower than that predicted by Kramers theory. In fact, at high viscosities, it is often found that the rate could be fitted to a form given by... 

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... 

A detailed study of the viscosity dependence of the rate has been carried out for a large number of thermodynamic state points. A subsequent analysis reveals that over a large variation of viscosity, the rates can indeed be fitted well to Eq. (316) and the exponent a is found to depend strongly on the barrier frequency (ft) ). 

Here, coR is the frequency of motion in the reactant well, and Eb is the height of the transition-state barrier. Xr is the effective barrier frequency with which the reactant molecule passes, by diffusive Brownian motions through the barrier region and is given by the following self-consistent relation... 

Equation (320) predicts the TST result for very weak friction (Ar to ) and predicts the Kramers result for low barrier frequency (i.e., (ob —> 0) so that (2r) can be replaced by (0) in Eq. (322). If die barrier frequency is large (ia>b > 1013 s 1) and the friction is not negligible ( (0)/fi — cob), then the situation is not so straightforward. In this regime, which often turns out to be the relevant one experimentally, the effective friction (2r) can be quite small even if the zero frequency (i.e., the macroscopic) friction (proportional to viscosity) is very large. The non-Markovian effects can play a very important role in this regime. 

Figure 15. Calculated values of the transmission coefficient k plotted as a function of the solvent viscosity rf for four barrier frequencies a>b at 7 = 0.85. The squares denote the calculated results for to = 3 x 1012 s I, the asterisks denote results for to = 5 x 1012 s-1, the triangles denote results for to = 1013 s I and the circles denote results for a>b = 2 x 1013 s l. The solid lines are the best-fit curves with exponents a 0.72 for wb = 3 x 1012s l, a 0.58 for wb = 5 x 1012 s-1,a 0.22 for wb = 1013 s l, and a 0.045 for cob = 2 x 10I3s-. Note here that the barrier crossing rate becomes completely decoupled from the viscosity of the solvent at wb = 2 x 10l3s-1. The transmission coefficient k is obtained by using Eq. (326). Note here that the viscosity is calculated using the procedure given in Section X and is scaled by a2/ /mkBT, and a>b is scaled by t -1. For discussion, see the text. This figure has been taken from Ref. 170.

The Value of the Exponent a Obtained from the Best-Fit Curves (Fig. 15) for Four Different Barrier Frequencies a>b, at T =0.85... 

As can be seen from the numbers, the exponent a is clearly a function of barrier frequency (cob) and its value is decreasing with increase in a>b- For cob — 2 x 1013 s-1, its value almost goes to zero (a < 0.05), which clearly indicates that beyond this frequency the barrier crossing rate is entirely decoupled from solvent viscosity so that one recovers the well-known TST result that neglects the dynamic solvent effects. 

In Fig. 16, the values of the exponent a are plotted against the barrier frequency this is clearly a kind of master curve that summarizes the essence of much of the work reported here. When the rate is calculated from Grote-Hynes theory, this curve depends only weakly on temperature. 

Figure 16. Calculated values of the exponent a plotted as a function of the barrier frequency cOj,. The solid circles are the values of a obtained for different barrier frequencies. The solid line is simply an aid to the human eye. The calculations have been performed by varying the reduced density p at a constant reduced temperature T = 0.85 for a particular value of a>b- For further details, see the text. This figure has been taken from Ref. 170.

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