Eckart potential energy

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

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. 

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 ...

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 potential energy surface for the proton motion is not of a simple shape in our model calculation. Hence, the Eckart barrier formula is used for proton tunneling by adjusting the two variables in it [41]. Thus, the barrier for the proton transfer was fitted to the following Eckart formula ... 

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.

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 ... 

There are some reservations about the strict applicability of the treatment outlined above to actual reactions. In the first place a parabolic energy barrier becomes unrealistic for configurations far removed from the transition state, and a more appropriate type of barrier is shown by the full curve in Figure 22(a), in which the broken curve represents a parabola. There is one potential energy function of this kind for which an explicit expression for the permeability can be obtained, commonly known as the Eckart barrier. For a symmetrical barrier (AH = 0) the equations are... 

A slightly more realistic description of the change in potential energy along the minimum energy path is provided by the following Eckart function (76) ... 

ABSTRACT. The results of multireference singles and doubles Cl calculations of potential energy surfaces for hydrogen atom addition to O2, N2, and NO and recombination of OH -h O are discussed. The errors due to the use of externally contracted Cl and due to the neglect of correlation of O 2s and N 2s electrons are analyzed. Similarities and differences between the surfaces for the addition reactions are discussed. The calculated HN2 addition surface is used in a simple dynamical treatment (one-dimensional tunneling through an Eckart barrier) to estimate the lifetime of the HN2 species. The OH -h O recombination potential is found to exhibit complex features which require that electrostatic forces (dipole-quadrupole) and chemical forces be treated consistently. 

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]).

Figure 6.14. Potential energy barriers for the transition state with a total energy c of an approaching particle the lower curve represents an Eckart barrier.

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