Representative tunnelling energy

Figure 9 Plot of the first integrand of Eq. [327] versus x = E — at three different energies for the H + CH4 reaction. The maximum of the curves indicates the representative tunneling energy. The top of the barrier is located at x = 0.

Fig. 2 Electronic conduction of a benzene ring between two conducting electrodes. These calculations are performed by the time-dependent method presented here (solid line) and by the ESQC method (dashed line). The electrodes are connected either in ortho (left column) or meta (right column) position. Two regimes are investigated tunneling with v = —0.25 ev (upper row), pseudoballistic with v = —2 ev (lower row). The vertical dashed lines represent the energy of the benzene s molecular orbitals...

Leforestier et al [53] have compared their experimental results for vibrational-rotational-tunneling energy levels of the water dimer for all six intermolecular vibrations with the corresponding data obtained from the MCY, RWK2 and ASP-w (I) and ASP-w (II). It turns out that while the structure of the dimer, quantified by the rotational constants, is well represented by all these four potentials, none of them is able to describe the tunneling dynamics of intermolecular vibrations even at a qualitatively correct level of accuracy. 

Fig. 10. Models proposed for proton translocation (Model A) single proton translocation with carbonium ion formation. (Model B) concerted double proton translocation with retroretinal formation, (c) Tunneling potential energy barriers for formation of prelumirho-dopsin. represents the energy barrier and cIq the width or translocation distance. 

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.

Figure 2.4 Carboxylic acid dimer in the potential energy minima with local vibrational states. OV represents the correlation time for a thermally activated proton transfer, and TU, the correlation time for tunneling transfer. (Reproduced with permission from ref. 29.)...

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 ... 

Figure 4.27. (a) Schematic of a STM z- Ft injection spectrum (solid curve). The dashed curves represent typical STM tip displacements observed at a clean metal surface, (b) Energy band diagrams for STM tunnelling through a vacuum barrier into the organic thin film and (c) through a Schottky-like barrier with the tip in contact. In both cases, Ft < 0 relative to Ep is shown. Adapted from Muller et al, 2001. 

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