Eckart potentials

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. 

Figure 11.9 Eckart potentials and wave functions used in the simulation of laser catalyst Potential parameters were A = 0 a.u., B - 6.247 a.u., / = 4.0 a.u., and m = 1060.83 a.iu P denotes E, 1+) state of text, denotes E, 2 ) state of text, and P0 denotes Eq) state" text. (Taken from Fig. 2, Ref. [308].)...

As an illustration, consider laser catalysis with an Eckart potential [376, 377jy the ground state , . ) 1... 

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

A standard set of solvable potentials with critical behavior can be found in many text books on quantum mechanics [49,50], like the usual square-well potentials and other piecewise constant potentials. Also there are many potentials that are solvable only at d - 1 or for three-dimensional, v waves like the Hulthen potential, the Eckart potential, and the Posch-Teller potential. These potentials belong to a class of potentials, called shape-invariant potentials, that are exactly solvable using supersymmetric quantum mechanics [51,52], There are also many approaches to make isospectral deformation of these potentials [51,53] therefore it is possible to construct nonsymmetrical potentials with the same critical behavior as that of the original symmetric problem. 

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

B.1 Solution for the Transmission Probability under the Eckart Potential.93... 

We will consider the Eckart potential energy, U(x), to be larger than the total energy of the electron, E ... 

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... 

The interpretation of the accurate quantal results in terms of variational and supernumerary transition states is consistent with model studies of scattering by unsym-metrical one-dimensional Eckart potentials (84). These studies show that both maxima in the unsymmetrical potentials are associated with poles of the scattering matrix, and some of these poles are associated with an increase in the transmission probability, while others are not. 

We discussed the implications of the O + H2 reaction s multiple bottleneck regions in terms of variational and supernumerary transition states. We related the observed features to the scattering results for asymmetrical Eckart potentials. We emphasized that global control is maintained to very high energy (1.9 eV) and very high levels of v2. We demonstrated the influence of quantized transition states at the level of state-selected reaction probability for this reaction. 

In the molecule, each normal mode, including the one for the reaction coordinate, contains ZPE. However, the Eckart potential is unbound on the reactant and product side so that there is no ZPE energy in the reaction coordinate. In reconciling these two energy references, we need consider only the energy in the one-dimensional reaction coordinate. The energy in the other modes, which are perpendicular to the reaction coordinate, can continue to be referenced to the ZPE level. 

The above treatment ignores any effects resulting from the coupling of the reaction coordinate with the other degrees of freedom. The main feature in the potential energy surface that causes coupling is curvature in the reaction path (Miller et al., 1980). These effects cannot be corrected in the above approach in which an Eckart potential is substituted for the true reaction path since it is a priori a one-dimensional function. In... 

Figure 5.H The top panel shows the cumulative reaction probabilities A/"exact( ) (black oscillatory curve) and A/"weyi( ) (red smooth curve) for the Eckart-Morse-Morse reactive system with the Hamiltonian given by Eq. (66) with e = 0. It also shows the quantum numbers (rii, rii) of the Morse oscillators that contribute to the quantization steps. The bottom panel shows the resonances in the complex energy plane marked by circles for the uncoupled case e = 0 and by crosses for the strongly coupled case e = 0.3. The parameters for the Eckart potential are o = 1, A = 0.5, and 6 = 5. The parameters for the Morse potential are = 1, Dj 3 = 1.5, and Om = = 1. Also, h ff = 0.1.

In Appendix F, we present an application of this approach to the Eckart potential, which is a one-dimensional potential applicable to chemical reaction problems and which yields to analytical solutions. 

Figure F.l Eckart potential for three values of the shape parameter, k 4,1/3,-4.

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