Tunneling probabilities

In the case of a strongly exothermic reaction the final term turns into an absorbing wall, and the transition is completed whenever the distance AB reaches a certain value and the A-B bond is broken. The intra- and intermolecular coordinates Q and q are harmonic and have frequencies (Oo and oji, and reduced masses mo and mi. At fixed intermolecular displacement the tunneling probability equals... 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

In the opposite case of slow flip limit, cojp co, the exponential kernel can be approximated by the delta function, exp( —cUj t ) ii 2S(r)/coj, thus renormalizing the kinetic energy and, consequently, multiplying the particle s effective mass by the factor M = 1 + X The rate constant equals the tunneling probability in the adiabatic barrier I d(Q) with the renormalized mass M, ... 

Sato and Iwata [1988] solved straightforwardly the Shrddinger equation in two dimensions and found the increase in tunneling probability in a three-atom fragment ABA, resulting from an increase in the quantum number of the low-frequency A-A vibration, n. For a barrier height equal to 1300 cm (3.72 kcal/mol), a vibration frequency of 450 cm and = 100, the tunneling... 

Figure 2.3. Tunnelling of a wave with kinetic energy E through a rectangular potential energy barrier, height V. The narrower the barrier, the smaller the mass of the particle and the smaller the difference between V and E, the greater the tunnelling probability. If the amplitude of the wave has not reached zero at the far side of the barrier, it will stop decaying and resume the oscillation it had on entering the barrier (but with smaller amplitude).

Figure 2.4. Reaction coordinate diagram for a simple chemical reaction. The reactant A is converted to product B. The R curve represents the potential energy surface of the reactant and the P curve the potential energy surface of the product. Thermal activation leads to an over-the-barrier process at transition state X. The vibrational states have been shown for the reactant A. As temperature increases, the higher energy vibrational states are occupied leading to increased penetration of the P curve below the classical transition state, and therefore increased tunnelling probability.

Based on C-H versus C-D zero point vibrational differences, the authors estimated maximum classical kinetic isotope effects of 17, 53, and 260 for h/ d at -30, -100, and -150°C, respectively. In contrast, ratios of 80,1400, and 13,000 were measured experimentally at those temperatures. Based on the temperature dependence of the atom transfers, the difference in activation energies for H- versus D-abstraction was found to be significantly greater than the theoretical difference of 1.3kcal/mol. These results clearly reflected the smaller tunneling probability of the heavier deuterium atom. 

One has to keep in mind that STM images show contours of constant tunnel probability rather than height contours directly. Nevertheless, for simple cases such as a metal surface, both quantities are closely related to each other. The dependence of the tunnel current IT on the tunnel voltage UT and on other parameters is given for the ideal case in Eq. (5.1) ... 

One has to keep in mind, that STM images show contours of constant tunnel probability rather than height contours directly. Nevertheless, for simple cases such... 

Ness H, Fisher AJ (1997) Nonperturbative evaluation of STM tunneling probability from ab initio calculations. Phys Rev B56 12469... 

An extension of this QC model, including tunneling probabilities between the confined crystallites and the bulk, has been developed [Fr6]. The QC model for microporous silicon formation, however, is still qualitative in character, and a quantitative correlation between anodization parameters and the morphology and properties of the porous structure is at yet beyond the capability of the model. 

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