Parabolic barrier

Non-parabolic barrier tops cause the prefactor to become temperahire dependent [48]. In the Smoluchowski... 

Of special interest is the case of parabolic barrier (1.5) for which the cross-over between the classical and quantum regimes can be studied in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the parabolic barrier transparency [Landau and Lifshitz 1981],... 

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. ...

Fig. 8. Arrhenius plot of dissipative tunneling rate in a cubic potential with Vq = Sficoo and r jlto = 0, 0.25 and 0.5 for curves 1-3, respectively. The cross-over temperatures are indicated by asterisks. The dashed line shows k(T) for the parabolic barrier with the same CO and Va-...

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Although the correlation function formalism provides formally exact expressions for the rate constant, only the parabolic barrier has proven to be analytically tractable in this way. It is difficult to consistently follow up the relationship between the flux-flux correlation function expression and the semiclassical Im F formulae atoo. So far, the correlation function approach has mostly been used for fairly high temperatures in order to accurately study the quantum corrections to CLST, while the behavior of the functions Cf, Cf, and C, far below has not been studied. A number of papers have appeared (see, e.g., Tromp and Miller [1986], Makri [1991]) implementing the correlation function formalism for two-dimensional PES. 

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 S-14. (A) A parabolic potential barrier and a linear perturbation. (B) Sum of the parabolic and linear functions, showing shift in maximum in accord with the Hammond postulate. (C) Two parabolic potential wells aa and bb are equivalent to the parabolic barrier cc .

Thus, these results indicated the involvement of heavy atom tunneling in the localized biradicals. The rates of decay for 19,20, and 9 could be fitted with Bell s simple model of tunneling through a parabolic barrier. Assuming log A (s ) = 8.0, and... 

Interestingly, this equation does not appear in this form in the oft-cited classic text by Bell (Ref. la), and refers to an earlier derivation. Bell derived a simpler expression, described in Ref. la, for tunneling through a parabolic barrier which affords approximately the same results. See Ref 2a for discussion. 

Although Eqs. (4-1) and (4-2) have identical expressions as that of the classical rate constant, there is no variational upper bound in the QTST rate constant because the quantum transmission coefficient Yq may be either greater than or less than one. There is no practical procedure to compute the quantum transmission coefficient Yq- For a model reaction with a parabolic barrier along the reaction coordinate coupled to a bath of harmonic oscillators, the quantum transmission... 

In reality, as the barrier becomes narrower, it deviates from the square shape. One often used model is the parabolic barrier (dashed line in Fig. 1). When the barrier is composed of molecules, not only is the barrier shape difficult to predict, but the effective mass of the electron can deviate significantly from the free-electron mass. In order to take these differences into account, a more sophisticated treatment of the tunneling problem, based on the WKB method, can be used [21, 29-31]. Even if the metals are the same, differences in deposition methods, surface crystallographic orientation, and interaction with the active layer generally result in slightly different work functions on either side of the barrier. 

The temperature dependence of the large isotope effect for the 2,4,6-collidine is just as striking (see Chart 1 and Fig. 2). In place of the expected unit value of Ah/Aq, a value around 0.15 was found accompanied by an enormous isotopic difference in enthalpies of activation, equivalent to an isotope effect of 165. Both of these results had earlier been shown by Bell (as summarized by Caldin ) to be predicted by a onedimensional model for tunneling through a parabolic barrier. The outlines of Bell s treatment of tunneling are given in Chart 2, while Fig. 3 shows that the departure of the isotopic ratios of pre-exponential factors from unity and isotopic activation energy differences from the expected values are both predicted by the Bell approach. 

In Equation 21, T is the absolute temperature, h is Planck s constant, is Boltzmann constant, and AG is the free energy barrier height relative to infinitely-separated reactants. The temperature-dependent factor r(7) represents quantum mechanical tunneling and the Wigner approximation to tunneling through an inverted parabolic barrier ... 

We will denote the positive solution of this equation as f. As shown in Refs. 39,83,84 one may consider the parabolic barrier problem in terms of a Fokker-Planck equation, whose solution is known analytically. One may then obtain... 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

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. 

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