Zero-point vibrations tunneling models

If zero-point vibration amplitudes of the dot are comparable with the Fermi length of the electrons, the shuttling takes place at small bias voltage. This is the case for cold dots. The constructive interference of electron waves in the tunnel gap center effectively charges the dot. In the quantum limit, this charging requires a justification of the tunnel-term concept based on the Schrodinger equation. In next section we address a more rigorous quantum mechanical picture based on the "ab-initio" SET model. 

Fig. 2 Schematic representation of the so-called semiclassical treatment of kinetic isotope effects for hydrogen transfer. All vibrational motions of the reactant state are quantized and all vibrational motions of the transition state except for the reaction coordinate are quantized the reaction coordinate is taken as classical. In the simplest version, only the zero-point levels are considered as occupied and the isotope effect and temperature dependence shown at the bottom are expected. Because the quantization of all stable degrees of freedom is taken into account (thus the zero-point energies and the isotope effects) but the reaction-coordinate degree of freedom for the transition state is considered as classical (thus omitting tunneling), the model is ealled semielassieal.

The one-frequency model represented by Eqs. (11.3)-(11.8) shows single isotopic frequency expressions for the MMI (mass/moment of inertia), ZPE (vibrational zero-point energy), and EXC (excited vibrations) terms of the usual Bigeleisen equation [21]. The extra term tun is the truncated Bell tunnel correction [22], used here to provide a simple way to express a tunneling effect in terms of a reaction-coordinate frequency, vh... 

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