Representative tunneling effects

The Jahn-Teller (JT) effect represents a classical example of vibronic admixture and, in the present review, we shall mainly deal with such an aspect, while the aspects connected with the wide field of tunnel effects will be treated only marginally. 

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. 

Fig. 2.6 Tunnelling effect for a free particle in a one-dimensional box with wall of finite height. The broken line represents the corresponding energy level should both potential barriers be infinite (see, for example, ref. 13).

A reaction involving hydrogen atom transfer is usually characterized by a significant tunneling effect, which is represented in the rate constant calculation by the tunneling correction factor kw as... 

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... 

They used Eq. (11.10), using k to represent a rate constant without a tunneling correction, to conclude that tunneling should not cause Swain-Schaad exponents to deviate strongly from a value of 1.44, unless the exponent for the tunnel effect, Shdj is very different from 1.44 and most of the kinetic isotope effect arises from Qh/Qd-... 

FIGURE 3.4 The fitting of the quantum tunneling effect based computed universal half-life times for nuclei emitting alpha particles with their observed values for representative radioactive series adapted from (Rohlf, 1994 HyperPhysics, 2010). 

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