Eckart potential barrier

The analytical solutions for the following three exactly solvable cases are presented (1) delta-function barrier, (2) parabolic potential barrier, and (3) Eckart potential barrier. These solutions may be useful for some analyses. 

Unfortunately, this method cannot be used in the case of an Eckart potential barrier, discussed in Section 2.1.3. The corresponding appropriate comparison equation method has not yet been developed in this case. 

Figure 2 Illustration of nonequilibrium solvation for the simple reaction model of a Eckart potential barrier representing the solute coupled linearly to a single harmonic oscillator representing the solvent. The thin curves are equipotential contours as a function of solute coordinate and solvent coordinate. The dashed line is the equilibrium solvation path for this model. The thick lines are the conventional transition slate dividing surfaces for the gas-phase reaction (vertical line that is defined in terms of the solute coordinate only) and for the solution-phase (line that makes a 28° angle with the abscissa)...

Figure 4 demonstrates that in order to variationally describe a realistic barrier shape (Eckart potential) by an effective parabolic one, the frequency of the latter, should drop with decreasing temperature. At high temperatures, T > T, transitions near the barrier top dominate, and the parabolic approximation with roeff = is accurate. 

Such calculations have been performed by Takayanagi et al. [1987] and Hancock et al. [1989]. The minimum energy of the linear H3 complex is only 0.055 kcal/mol lower than that of the isolated H and H2. The intermolecular vibration frequency is smaller than 50cm L The height of the vibrational-adiabatic barrier is 9.4 kcal/mol, the H-H distance 0.82 A. The barrier was approximated by an Eckart potential with width 1.5-1.8 A. The rate constant has been calculated from eq. (2.1), using the barrier height as an adjustable parameter. This led to a value of Vq similar to that of the gas-phase reaction H -I- H2. 

A second widely used approximation uses the more smoothly shaped Eckart barrier (Fig. 6.1), which for a symmetric barrier may be expressed as V = V sech2(x) = V [2/(ex + e x)]2 where x = jts/a with s a variable dimension proportional to the displacement along MEP, and a a characteristic length. Like the Bell barrier the Eckart potential is amenable to exact solution. The solutions are similar and tunnel corrections can be substantial. In both the Bell and Eckart cases one is implicitly assuming separability of the reaction coordinate (MEP) from all other modes over the total extent of the barrier, and this assumption will carry through to more sophisticated approaches. 

The tunnelling correction P is the transmission probability through the potential barrier averaged over all possible crossing points and potential energies . An asymmetrical banier of the Eckart type l is assumed in the present model. 

Figure 8. Example of an Eckart potential with an energy change of 5000 cm and a barrier height of 10208 cm ...

From the normal mode analysis at the classical hairier, and computed from the curvature along the Eckart potential at the adiabatic barrier. 

In the case of not having a potential barrier independent of the distance, like in the Eckart potential, some approximations can be proposed. The Wentzel, Kramer, and Brillouin (WKB) approach is a clear example to overcome the problem. If the energy equation inside the potential barrier is... 

It has been shovm 764,129,130/ that the formula (177.Ill) is fairly accurate for a variety of potential barriers if S < 3/2 or T (2/3). Such is,for instance,the generalized Eckart barrier (77.11). In general, this holds true for any barrier shape which can be approximated by a parabola in the energy range... 

To answer this. Stern and I (22) tued the same model systems previously Investigated by Schneider and Stern, but Includ the tunneling correction In the Isotope effect. Values of v were those calculated In their work barrier heights V were not needed In their work, and we have simply assumed values of 1, 5, 10, 20, and 30 kcal/mole for Vg. A symmetrical Eckart potential was i ised. For reasons discussed in our paper (22), two sets of calculations were made one with Vg = Vg and one with Vg = Vg +... 

If information on the reaction path is available, as, for instance, in variational transition state theory, this can be used to calculate k [69,70]. In transition state theory, only the knowledge of the energy and its first and second derivatives at the reactant and transition state locations is needed and the barrier is typically approximated by a simple functional form. One possibility is to describe the reaction barrier by an Eckart potential [75] (also called a sech potential, depending on the literature source), k in Eq. (7.19) is defined as the ratio of transmitted quantum particles to classical particles and the resulting integral for the Eckart potential can be solved numerically. An approximate solution is the Wigner tunneling correction ... 

Analytical expressions for the contributions of tunneling to the transmission coefficient have been obtained for some model barriers, V(x), notably the Eckart potential, for which the solution is exact [46], and the truncated parabola, for which an approximate, but accurate, solution has been found [47]. These references should be consulted for details of the calculations and tabulations of tunneling corrections. A hrst approximation to tunneling through a barrier of arbitrary shape is given by the following equation [48] ... 

Figure 4. Classical potential energy, Vj p (lower curve and left scale), and ground-state vibrationally adiabatic potential energy, (upper solid curve and right scale), as functions of the reaction coordinate s along the MEP for the CH4 + system. The long-dashed portions are extrapolations (see text). The short-dashed curve is an Eckart potential fit to the adiabatic barrier (reproduced with permission from [147]).

We can complete the calculation of k for the Eckart potential. Taking jB = 0 reduces the barrier to a simple step of energy A with no maximum in V. For this special case the reflection coefficient is... 

Finally, a potential barrier that closely approaches the classical reaction path of an atom-transfer reaction and that also has an analytical solution for the transmission probability, is the Eckart barrier [14]... 

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