Parabolic barrier mechanisms

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 ... 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

D. ChruScMski, Quantum mechanics of damped systems II. Damping and parabolic barrier, J. Math. Phys. 45 (2003) 841-854. 

In the second section the calculation of the rate constant was discussed from the classical mechanics viewpoint. Voth, Chandler, and Miller derived a quantum mechanical expression for the rate constant based on a path integral formalism. Using this expression as a starting point, Voth and O Gormani derived an effective barrier model to allow the calculation of the barrier tunneling contribution to the quantum mechanical rate constant for reactions in dissipative baths. The spirit of their derivation is quite similar to that which treats Grote-Hynes theory o as transition state theory for a parabolic barrier in a harmonic bath. 

Fig. 6.2 (a) Bell (parabolic) and Eckart barriers, both widely used in approximate TST calculations of quantum mechanical tunneling, (b) Transmission probability (Bell tunneling) as a function of energy for two values of the reduced barrier width, a... 

A similar study by Schoofs et al. [43] of methane dissociation on the Pt(l 11) surface produced qualitatively similar results An exponential increase in the value of S0 with increasing normal energy and a weak dependence of S0 on surface temperature, Ts. Further, like Rettner et al. [40], Schoofs and coworkers find these trends consistent with a hydrogen tunneling mechanism through a one-dimensional parabolic-shaped activation barrier. 

Abstract This contribution deals with the modeling of coupled thermal (T), hydraulic (H) and mechanical (M) processes in subsurface structures or barrier systems. We assume a system of three phases a deformable fractured porous medium fully or partially saturated with liquid and a gas which remains at atmospheric pressure. Consideration of the thermal flow problem leads to an extensively coupled problem consisting of an elliptic and parabolic-hyperbolic set of partial differential equations. The resulting initial boundary value problems are outlined. Their finite element representation and the required solving algorithms and control options for the coupled processes are implemented using object-oriented programming in the finite element code RockFlow/RockMech. 

As was mentioned above, the value of AE is determined by the barrier shape. The characteristic values of AE determining the behavior of the system near the top and at the bottom of the barrier are generally different so that in the general case we get two parameters which can be described in terms of the energy levels of a particle in the potential well formed by inverting the barrier (as shown in Figure 3.6). These parameters are AE, the distance from the bottom of the well to the lowest level in it, and AEj, the distance from the upper level in the well to the level of a free particle outside the well. The criterion for the classical behavior is AEq < kT, and for the quantum-mechanical behavior, AEj > 8.6kT. Naturally, the relation between AE and AEj depends on the barrier shape. Finally, we shall formulate the criterion for the classical and quantum-mechanical behavior for a frequently encountered symmetrical barrier formed by the intersection of two parabolic terms with the same natural frequency o) the system behaves quantum. mechanically for ho) > 2.8 kT, and classically for "ho) < kT(l ia)/E ) (here E is the barrier height). 

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