Electron transfer nuclear tunnelling

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. 

The last 50 years have witnessed the establishment of a truly molecular-level description of electron transfer chemistry. From the Marcus description of how solvent polarization defines the ET reaction coordinate, to fully quantum treatments that describe electron and nuclear tunneling contributions to the kinetics, to atomistic simulations of reaction coordinate motion, a comprehensive view of biological ET is emerging (1-5). 

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. 

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... 

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. 

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. 

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,... 

From the expressions given for example in Refs. [4,9,29], it can be seen that the nuclear factor, and consequently the electron transfer rate, becomes temperature independent when the temperature is low enough for only the ground level of each oscillator to be populated (nuclear tunneling effect). In the opposite limit where IcgT is greater than all the vibrational quanta hco , the nuclear factor takes an activated form similar to that of Eq. 1 with AG replaced by AU [4,9,29]. The model has been refined to take into account the frequency shifts that may accompany the change of redox state [22]. 

In the electron transfer literature it has become common to describe electron transfer reactions that occur through vibrational distributions below the intersection as having occurred by nuclear tunneling and the actual electron hopping event as electron tunneling . 

The present chapter comprises a review of tunneling phenomena in nuclear, atomic, and solid-state physics, tunneling effects during the transfer of atoms in chemical reactions in gases, liquids, and solids, as well as of tunneling phenomena in electron transfer reactions in gases. 

Manifestations of nuclei tunneling in chemical reactions in gaseous, liquid, and solid phases are consecutively considered in Sects. 4.2-4.5. Also discussed in this chapter are (1) manifestations of nuclear tunneling in the vibrational spectra of ammonia-type molecules (Sect. 4.6), (2) electron tunneling in gas-phase chemical reactions of atom transfer (the so-called "harpoon reactions, Sect. 4.2), and (3) tunneling of hydrated electrons in the reactions of their recombination with some inorganic anions in aqueous solutions (Sect. 4.4). 

Secondly, the model of thermal diffusion does not allow one to explain the independence of the reaction rate on temperature observed for many low-temperature electron transfer processes. Indeed, the thermal diffusion of molecules in liquids and solids is known to be an activated process and its rate must be dependent on temperature. True, at low temperatures when activated processes are very slow, diffusion itself can be assumed to become a non-activated process going on via a mechanism of nuclear tunneling, i.e. by tunneling transitions of atoms over very short (less than 1 A) distances. A sequence of such transitions can, in principle, result in a diffusional approach of reagents in the matrix. Direct tunneling of the electron, whose mass is less than that of an atom by a factor of 10 or 104, can, however, be expected to proceed much faster. 

Nuclear tunnelling in the aqueous iron (2+)-iron (3+) electron transfer has been investigated467 and the rate enhancement for H2O has been assessed to be 65 times the classical rate, and that for D2O 25 times the classical rate, yielding a H/D isotope effect of 2.6. The occurrence of the general base catalysis and sizable primary D KIEs indicated that the isomerization of l//-indene-l-carboxylic acid to l//-indene-3-carboxylic acid in aqueous solution takes place through an enolization-reketonization sequence468. Kinetic HH/HD/DD isotope and solvent effects have been used in a dynamic NMR study469 of the tautomerization of 15N-and 2H -labelled bicyclic oxalamidines. 

So far the attention has been on the nuclear reorganization barrier. Nevertheless, other important factors previously hidden in the pre-exponential factor (and ultimately in the standard rate constant) have to be considered, namely, the fundamental question of the magnitude of the electronic interaction between electroactive molecules and energy levels in the electrode (i.e., the degree of adiabaticity) and its variation with the tunneling medium (electrode-solution interface), the tunneling distance, and the electrode material. Thus, within the transition-state formalism, the rate constant for electron transfer can be expressed as the product of three factors [39—42] ... 

Thus the studies of the QA-involved electron tunneling reactions have shown that analysis of the reaction rate vs — AG° dependences provides information about the character of nuclear motions coupled to electron transfer in RC of photosynthetic bacteria. In particular, as kT becomes smaller than ha the dependence of electron transfer rate on — AG° becomes sensitive to the magnitude of ha. In other words, it is possible to find out from these experiments what aspects of protein and cofactor dynamics are important for the reaction. 

The temperature dependence of the rate constant of electron transfer over large distance from the first triplet state of Zn porphyrin to Rum(NH3)5 covalently attached to histidine-33 in Zn-substituted cytc was studied in Ref. [318]. A temperature independent triplet quenching process with the rate constant 3.6 s-1, was observed at 10-100 K and tentatively attributed to electron transfer facilitated by nuclear tunneling. 

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