Curvature Transmission Coefficient

Large-Curvature Tunneling Without Vibrational Excitations 

As stated, the large-curvature tunneling (LCT) methods use the ground-state vibrationally adiabatic potential to define classical reaction-coordinate turning points for a total energy E by inverting the equation 

The end points of the tunneling paths in the reactant and product valleys are defined as So and Si, and they obey the resonance condition 

This expression provides a relationship between so and si so that either one or the other is an independent variable. Unless stated otherwise, we use sq as the independent variable, and when si appears, its dependence on sq is implicit. The tunneling path is a straight-line path in mass-scaled Cartesian coordinates defined by 

The total tunneling amplitude along the incoming trajectory at energy E includes contributions from all tunneling paths initiated in the reactant valley 

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The use of Eq. (5-10) to evaluate the reaction rate is characterised by the calculation of Hessians for a large number of points along the MEP which are required to locate the free energy maximum and also to evaluate the curvature required for evaluation of the transmission coefficient. In view of the associated computational expense, high-level electronic structure calculations are not feasible and alternative strategies, one of which is to use a semi-empirical method, are usually employed [81]. 

In practice, we approximate the exact transmission coefficient by a mean-field-type of approximation that is we replace the ratio of averages by the ratio for an average or effective potential. For gas-phase reactions with small reaction-path curvature, this effective potential would just be the vibrationally adiabatic ground-state potential. In the liquid phase and enzymes we generalize this with the canonical mean-shape approximation. In any event, though, the transmission coefficient should not be thought of as a perturbation. The method used here may be thought of as an approximate full-dimensional quantum treatment of the reaction rate. 

As before, the energy in the reaction coordinate, Ct is measured from the top of the barrier. Solutions to equation (7.67) for the barrier of figure 7.28, are plotted in figure 7.29 for three values of hv, 3000, 2122, and 1000 cm , but for the same curvature at the top of the barrier. These are chosen to correspond approximately to a reaction in which the critical coordinate involves an H atom transfer, a D atom transfer, and perhaps a C—C bond break. The marked differences show the well known effeet of the reduced mass on the transmission probabilities. Note that even at the top of the barrier, the transmission coefficient equals 0.5 exactly. Note also that while the Eckart equation is a function of the curvature, k, which depends upon the critical frequency and the reduced mass, the transmission coefficient depends only upon the critical frequency. 

Figure 7.29 Tunneling probability for the indicated Eckart barrier with three different critical vibrational frequencies. The latter are related to the curvature at the top of the barrier. Note that the transmission coefficient is equal to 0.5 at the saddle point.

The conditions (133.HI) mean that the tunneling through the adiabatic potential barriers (99.HI) is neglected however, the non-adiabatic transitions from a lower to a higher electronic state are not excluded. The reflexion effects due to the reaction path curvature may be simply included in the dynamical definition (121.11) of the reaction coordinate. An adiabatic separation of that coordinate may be also used as an approximation at these conditions. Then, the "transmission coefficient — 1 takes into account only sudden changes... 

Assuming a parabolic fission barrier with a height of Vq and curvature of hto, then one can get the well-known Hill-Wheeler transmission coefficient... 

There we show that the dependence of the transmission coefficients on the local geometry of a circular fiber involves only the radius of curvature p in the plane of incidence of the ray, i.e. the radius of curvature of the interface or turning-point caustic in the plane defined by the ray path or the tangent to the ray path, respectively, and the normal. We then claim that the locally valid transmission coefficients for interfaces or turning-point caustics of arbitrary curvature have the same functional dependence on p. In these cases p depends on two prindpal radii of curvature instead of the single radius of curvature of the circular fiber. 

A turning-point caustic of arbitrary curvature is illustrated in Fig. 7-5(b), and is defined by the surface n = (r,p), where r,p is the position of the turning point in the radial direction. We assume that the turning point is not too far from the intersection of the radial axis with the interface at r = p, beyond which the profile is uniform, i.e. n — n p) = n i- The tangent to the ray path at P lies in the tangent plane at P and makes angles n/2 — 0 and 0, with the y- and z-axes, respectively. Otherwise the notation is identical with Fig. 7-5(a). In this situation, the local transmission coefficient is given by Eq. (35-55) [19]... 

The interface and turning-point caustic are curved surfaces. They are defined by two principal radii of curvature which depend on both the core radius p and the bend radius R. Under these conditions we use the localized transmission coefficients of Section 7-14 each time a ray loses power by tunneling. When power is lost by refraction, we employ the Fresnel coefficient of Eq. (35-50) for the step profile, and assume complete power loss for the clad parabolic profile, i.e. T= 1. 

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