The Classical Marcus Theory

To fully understand the relaxation pathways for photoinduced charge-transfer reactions in solutions we need to take solvent effects into account. For that reason it is necessary to recall some basic principles of the classical Marcus Theory for electron-transfer reactions in solution. 

Electron transfer profoundly affects chemical reactivity by inverting normal electron densities in electron donor/acceptor pairs and therefore activating previously inaccessible reaction modes. The basic principles have been widely discussed in several reviews [69-72]. 

Employing the Franck-Condon principle, i.e. preservation of the nuclear configuration of reactant and product at the point of transition, we can assume a horizontal transition between the donor (D) and acceptor (A). In terms of the 

With Eq. 4.6 we can formulate the free-activation-energy dependence of the rate constant for electron-transfer reactions 

The exponential term of 4.7 in conjunction with 4.6 contain an important prediction, namely that three distinct kinetic regimes exist, depending on the driving force of the electron transfer process. The three kinetic regimes are also shown schematically in Fig. 4.2 (lower part) in terms of the classical Marcus parabolas  

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In the early 1980s, the classical Marcus theory was reanalyzed to consider the influence of quantum effects, notably electron tunneling [23]. Qualitatively, this gives the right direction, since it increases the observed rate constants when the thermally activated process becomes slow but it was concluded that it could not account for the quantitative discrepancies of observed Rehm-Weller type plots from Marcus behaviour. For this reason the hypothesis was retained that the... 

Figure 4.2 illustrates the parabolic free-energy surfaces as a function of reaction coordinate. In particular, three different kinetic regimes are shown in accordance with the classical Marcus theory. The reorganization energy, X, represents the change in free energy upon transformation of the equilibrium conformation of the reactants to the equilibrium conformation of the products when no electron is... 

The classical Marcus theory is thus of paramount importance since it defines the free energy barrier in terms of two quantities only the free energy of reaction AG and the reorganization energy A. Consequently, when obeying Marcus s law, ET processes... 

V. Like the classical Marcus theory, equation 1 predicts an inverted region, although the decrease of rates for highly exoergic reactions may be less pronounced than in the Marcus theory. The quantum mechanical theory also predicts modifications of the effects of temperature and polarity. Some principal features of these predictions have been verified by experiments using both pulse radiolysis and laser photoexcitation. 

The shape of the rate vs. free-energy curve departed somewhat from that predicted by the classical Marcus theory because of quantum mechanical effects... 

Qualitative predictions of the classical Marcus theory of ET have been confirmed for a wide variety of systems. Given this success, theory is now challenged to calculate absolute ET rates on the basis of separately obtainable parameters. Regarding the solvent, the input is the dielectric relaxation spectrum. It can be measured or even calculated independently to a high degree of accuracy. Intramolecular modes participating in the ET... 

One of the most exciting developments in the field of electron transfer has been the experimental verification of the classical Marcus theory [326,327], which predicts a decrease in the rate constant for the electron transfer as the exothermicity of the electron fransferring process increases. As it turns out, this so-called Marcus inverted region occurs frequently for photo-... 

According to (37), is actually temperature independent. Hence, one can rewrite (35) into a form which agrees with the rate predicted by the classical Marcus theory... 

Most of the redox proteins have their active centre deeply buried inside the protein matrix which slows down or sometimes insulates the transfer of electrons from protein to the electrode. As per the classical Marcus theory for the monolayer redox couple [Eq. (1)] [16, 43, 44], A et decreases exponentially with the distance of electron transfer, d, where is the electron transfer rate constant at the distance of closest contact do. ... 

Table 1 summarizes the behavior, in the form of activation enthalpies (AH ), for each of 18 reactions. The values listed are somewhat larger than published values [36], reflecting corrections for unrecognized thermal control errors in the original investigation. As expected from classical Marcus theory, decreases in rate are accompanied by increases in AH. Curiously, however, as the reaction is pushed progressively further into the inverted region, AH increases by... 

The higher rate of the reverse electron transfer in the dimer radical cations than in the monomer radicals was explained by classical Marcus theory as follows [52,53]. The -AG° values for the reverse electron transfer from NS and DCS to TPB were estimated as 1.69 and 1.61 eV from the redox potentials, respectively. The reduction potential of the dimer radical cation should be less negative by 0.65 and 0.59 V than that of the radical monomer radicals due to the stabilization energy. The -AG° values for the reverse electron transfer from the dimer radical cation to TPB were thus estimated to be 1.04 and 1.02 eV for (NS+ -NS ) and (DCS+ -DCS ), respectively [68],... 

The total reorganization energy >. in classical Marcus theory was estimated by the method described below. A plot of the maximum of the IPCT band in solution (T op) against the driving force of electron transfer within the contact... 

Fig. 3 A schematic showing how, within the context of classical Marcus theory, the ET rate varies with the ergonicity - or, equivalently, the driving force (= — AG°) - of the reaction.

Thus, the semi-classical Marcus theory of non-adiabatic ET expresses the ET rate constant in terms of three important quantities, namely Vel, A, and AG°. It therefore follows that an understanding of ET reactions entails an understanding of how these three variables are dependent on factors such as the electronic properties of the donor and acceptor chromophores, the nature of the intervening medium and the inter-chromophore separation and orientation. 

In the classical Marcus-Hush theory, the initial nuclear geometry of the reactant state undergoes reorganization to the transition state prior to electron transfer [30, 31, 39]. The energy of the transition state, AGe, is gained by intermolecular collisions, in order to satisfy conservation of energy and momentum. The nuclear factor is related to the activation energy, i.e.,... 

Thus, classical Marcus theory predicts an electron transfer rate that has a Gaussian dependence on the free energy of the reaction (Marcus, 1956 Marcus and Sutin, 1985). 

The self-exchange electron-transfer (SEET) process, in which a radical is trapped by the parent molecule, has been studied using the intersecting-state model (ISM). Absolute rate constants of SEET for a number organic molecules from ISM show a significant improvement over classical Marcus theory in the ability to predict experimental SEET values. A combination of Marcus theory and the Rips and Jortner approach was apphed to the estimation of the amount of charge transferred in the intramolecular ET reactions of isodisubstituted aromatic compounds. ... 

More recent work has concentrated on quantitative evaluations of the dependence of Os on Fred- Two studies in particular have examined the dependence of spectral sensitization efficiency on the redox potentials of the sensitizers using classical Marcus theory (Eqs. (105), (106)) [188]. 

We now turn to the hierarchy of electron-transfer rate theories that have developed since the 1950s, starting with classical Marcus theory of homogeneous reactions and the development of eq. 4.4. In later sections we shall consider theories of nonadiabatic ET, which allow the identification and evaluation of the prefactor A in eq. 4.4, and also electrochemical ET, which differs from homogeneous reactions in that an electronic conductor is one of the reactants . 

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