Tunneling factor

The quantity / is just a further combination of constants already in Eq. (10-70). The value of Z is taken to be the collision frequency between reaction partners and is often set at the gas-phase collision frequency, 1011 L mol-1 s-1. This choice is not particularly critical, however, since / is nearly unity unless is very large. Other authors29-30 give expressions for Z in terms of the nuclear tunneling factors and the molecular dimensions. 

In the case where they represent quantum vibrational modes, this leads to the appearance of a small tunnel factor in the transmission coefficient k. ... 

The quantity Ef is the energy of the reorganization of all the classical degrees of freedom of the local vibrations and of the classical part of the medium polarization, and crc is the tunneling factor for quantum degrees of freedom 1) which do not... 

A recently proposed semiclassical model, in which an electronic transmission coefficient and a nuclear tunneling factor are introduced as corrections to the classical activated-complex expression, is described. The nuclear tunneling corrections are shown to be important only at low temperatures or when the electron transfer is very exothermic. By contrast, corrections for nonadiabaticity may be significant for most outer-sphere reactions of metal complexes. The rate constants for the Fe(H20)6 +-Fe(H20)6 +> Ru(NH3)62+-Ru(NH3)63+ and Ru(bpy)32+-Ru(bpy)33+ electron exchange reactions predicted by the semiclassical model are in very good agreement with the observed values. The implications of the model for optically-induced electron transfer in mixed-valence systems are noted. 

Classically, the rate of electron transfer is determined by the rate of passage of the system over the barrier defined by the surfaces. In the semiclassical model (13) a nuclear tunneling factor that measures the increase in rate arising from... 

According to a recent model (13) nuclear tunneling factors for the inner-sphere modes can be defined by... 

The value of log rn for the Fe(H20) 2+ - Fe(H20)6 + exchange (which features a relatively large inner-sphere barrier) is plotted as a function of 1/T in Figure 5. The nuclear tunneling factors are close to unity at room temperature but become very large at low temperatures. As a consequence of nuclear tunneling, the electron transfer rates at low temperatures will be much faster than those calculated from the classical model. 

Figure 5. Plot of the logarithm of the nuclear tunneling factor vs. 1/T for the Fe(H20)62 -Fe(H20)63 exchange reaction. The slope of the linear portion below 150 K is equal to Ein/4R (13).

To summarize, in this article we have discussed some aspects of a semiclassical electron-transfer model (13) in which quantum-mechanical effects associated with the inner-sphere are allowed for through a nuclear tunneling factor, and electronic factors are incorporated through an electronic transmission coefficient or adiabaticity factor. We focussed on the various time scales that characterize the electron transfer process and we presented one example to indicate how considerations of the time scales can be used in understanding nonequilibrium phenomena. 

The nuclear tunnelling factor can be accurately estimated from a 1-mode model based on the high frequency inner-sphere breathing mode (10, Tl)... 

Electronic non-adiabaticity can give rise to a factor K which is less than unity the nuclear tunnelling factor, T, on the other hand, is always greater than or equal to unity. 

Non-resonant Golden Rule Nuclear Tunnelling Factor 1 (298) for Self-exchange. 

It is interesting to calculate the tunnelling factor, using this analytical expression, for typical outer-sphere exchanges. We select the self-exchanges in water Fe(II)(H O) /... 

Non-resonant Nuclear Tunnelling Factor T(298) for Self-exchange in Aqueous Solution... 

Here it was found that the tunneling factor k is very close to unity. However this result is uncertain because the magnitude of k is sensitive to the choice of the potential energy surface which is not as well established for reaction 6.20 as it is for 6.6. For that matter, learning whether a given reaction rate is significantly influenced by tunneling either on the basis of theory or experiment is not a trivial problem as will be pointed out in further discussion. 

An improved and direct correlation between the experimental rate constant and [obtained using Eq. (49)] is observed if v = /zd is used instead of v = 1/Tt, the solvent-dependent tunneling factor is utilized, and only AG (het) of Eq. (8) is used in Eq. (49) (see triangles in Fig. 18). Furthermore, the inverse of the longitudinal solvent relaxation time Xi is not necessarily the relevant one to use as the frequency factor v (see empty circles in Fig. 18). Similar conclusions were reached by Barbara and Jerzeba for the electron transfer reaction in homogeneous solutions. Barbara and Jerzeba measured the electron transfer time... 

Dr. Berkowitz I must question the validity of Dr. Teichmiiller s rather definite conclusions about the relative roles of time, temperature, and pressure in the coalification process. From an examination of Ruhr coals, Dr. Teich-miiller said that only temperature plays a significant role. I suggest that conclusions drawn from data for coals in other areas (e.g., Alberta and Pennsylvania) would lead to the conclusion that pressure rather than temperature was the determining variable therefore, I doubt whether Dr. Teichmiiller s quite unqualified statements could have general validity. Indeed, from first principles one would deduce a rather complex and variable situation. Thermodynamically, one could perhaps rule out time as an important parameter since, unless one accepted the concept of a "tunnelling factor, time alone will not... 

The right side in equation (23) has a clear physical meaning. The probability flux ji is proportional to C, the probability of finding the system in the left well. Also the probability flux is proportional to w/(2tt), the frequency with which the particle hits the barrier wall, and to the exponential tunneling factor, which is the probability of tunneling through the barrier at each hit. 

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