Quantum tunneling Bell model

Bell quantum tunneling model, 33, 34-35, 72 (3-Cyclodextrin/amino acid complexes, 220t Bidentate ligands, 153 Biomacromolecules... 

This means that at low temperature where P is large the HD reaction is ca. twice as fast as the DD reaction. Equation (6.31) has been used in connection with the Bell-Limbach tunneling model to describe the stepwise double proton transfer in porphyrins, azophenine, and oxalamidines, as will be discussed in Section 6.3. Smedarchina et al. [16] used the same equations for their quantum-mechanical treatment of the porphyrin tautomerism. 

Figure 21.5 The Bell tunnel model (a) Quantum mechanical harmonic oscillator with its ground state wavefunctions. (b) Inverted harmonic oscillator potential, (c) A stream of particles with a Boltzmann distribution of energies hits the barrier. Classical only those particles with W>Vq can pass the barrier.

In electrochemical proton transfer, such as may occur as a primary step in the hydrogen evolution reaction (h.e.r.) or as a secondary, followup step in organic electrode reactions or O2 reduction, the possibility exists that nonclassical transfer of the H particle may occur by quantum-mechanical tunneling. In homogeneous proton transfer reactions, the consequences of this possibility were investigated quantitatively by Bernal and Fowler and Bell, while Bawn and Ogden examined the H/D kinetic isotope effect that would arise, albeit on the basis of a primitive model, in electrochemical proton discharge and transfer in the h.e.r. 

As soon as bound states are considered there are only discrete energy levels. Nevertheless it was shown by Bell [77] that it is possible to employ approximately a continuum of energy levels for the calculations of the tunnel rates, which is adequate for the description of many experimental systems. In the simplest form (see Fig. 21.5) of the Bell model, the potential barrier is an inverted parabola. This allows the use of the known solution of the quantum mechanical harmonic oscillator for the calculation of the transition probability through the barrier. The corresponding Schrodinger equation is... 

The tunnel correction is not now a fundamentally defined number rather it is defined by the equation Q = kobJk, where kobs is the observed rate constant for a chemical reaction and k is that calculated on the basis of some model which is as good as possible except that it does not allow tunnelling. In this chapter the definition used for k is that calculated by absolute reaction rate theory [3], i.e., k = KRT/Nh)K where X is the equilibrium constant for the formation of the transition state. The factor k, the transmission coefficient, is also a quantum correction on the barrier passage process, but it is in the other direction, that is k < 1. We shall here follow the customary view (though it is not solidly based) that k is temperature-independent and not markedly less than unity. The term k is used following Bell [1] the s stands for semi-classical, that is quantum mechanics is applied to vibrations and rotations, but translation along the reaction coordinate is treated classically. 

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