Tunneling nuclear

two models designed to deal with nuclear tunneling will be discussed the PKS model and the Bixon-Jortner model. The Bixon-Jortner model (1968) provided the conceptual basis for the understanding of radiationless transitions in excited electronic and vibrational states. 

This model was originally used along with the Day-Hush disproportionation model, mentioned in Section 10.7.2, and got its name from the pioneers S. Piepho, E. Krausz, and P. N. Schatz (PKS). The PKS model is a useful and consistent model for the treatment of spectra of mixed valence systans. The model includes the vibrational levels for MV-2 and MV-3 systems. It is assumed that the coupling is small so that the harmonic approximation can be used. 

Various models that take this problem into account exist. Here, we will mention a model by Kestner et al. based on the Fermi golden rule. It is possible to show (see Section 7.3.2) that the probability per unit time that a system in an initial vibronic (electronic and vibrational) state, n, will undergo a transition into a set of vibronic levels mw is given by 

Assuming a Boltzmann distribution over the vibrational energy levels of the initial electronic state n, the thermally averaged probability per unit time of passing from the initial set of vibronic levels nv to a set of vibronic levels mw is given by 

It may be shown that the equations for nuclear tunneling converge to the ordinary rate equation for a large activation barrier and high temperature. In the case of low temperature or a low activation barrier, the ordinary rate equation based on classical nuclear motion should not be used. 

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The reactions of electron transfer and vibronic relaxation are ubiquitous in chemistry and many review papers have dealt with them in detail (see, e.g., Ovchinnikov and Ovchinnikova [1982], Ulstrup [1979]), so we discuss them to the extent that the nuclear tunneling is involved. 

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. 

A very brief introduction to the important topic of bioinorganic electron transfer mechanisms has been included in Section 1.8 (Electron Transfer) of Chapter 1. Discussions of Marcus theory for protein-protein electron transfer and electron or nuclear tunneling are included in the texts mentioned in Chapter 1 (references 3-7). A definitive explanation of the underlying theory is found in the article entitled Electron-Transfer in Chemistry and Biology, written by R. A. Marcus and N. Sutin and published in Biochem. Biophys. Acta, 1985, 811, 265-322. 

DR. EPHRAIM BUHKS (University of Delaware) I would like to ask your opinion about a possible interpretation of proton transfer in terms of nuclear tunneling effects. Might it be possible that as the energy of the vibrational modes becomes very large, the classical rate theory might not work ... 

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. 

In the classical activated-complex formalism nuclear tunneling effects are neglected. In addition, the electron transfer is assumed to be adiabatic. These assumptions are relaxed in the semiclassical model. 

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. 

The Effective Nuclear Frequency. When nuclear tunneling is important we suggest that the individual frequencies should be weighted by their effective barriers, that is, instead of eq 8 it may be more appropriate to use eq 14... 

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).

Also, when nuclear tunneling is important E n in the denominator should be replaced by 2vinEin csch 2 n ) s evident from eq... 

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. 

It is at this stage that we should now begin to look into the details and ask just the sorts of questions that Dr. Sutin is raising — e.g., the importance of nuclear tunneling or of electronic nonadiabaticity. These are, as we might say, the fine structure of the problem. 

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

More insight into the reaction mechanism is given by the breakdowns in Table III. The negative entropy has important contributions from bimolecular work, non-adiabatic effects and nuclear tunnelling. Nuclear tunnelling (r = 3.5 (10, 11)) also... 

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. 

Does T differ significantly from unity in typical electron transfer reactions It is difficult to get direct evidence for nuclear tunnelling from rate measurements except at very low temperatures in certain systems. Nuclear tunnelling is a consequence of the quantum nature of oscillators involved in the process. For the corresponding optical transfer, it is easy to see this property when one measures the temperature dependence of the intervalence band profile in a dynamically-trapped mixed-valence system. The second moment of the band,... 

How important, though, is nuclear tunnelling for thermal outer-sphere reactions at ordinary temperature If we work in the Golden Rule formalism, an approximate answer was given some time ago. In harmonic approximation, one obtains from consideration of the Laplace transform of the transition probability (neglecting maximization of pre-exponential terms) the following expressions for free energy (AG ) and enthalpy (AH ) of... 

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

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

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