Classical Marcus theory

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 . 

ET reactions of transition metal complexes, where ligand-field effects can coerce achange from the low-spin to the high-spin configuration on change of redox state, are an exception. 

In the classical description of this section, we ignore both die coupling of die R and P states (which causes the R and P surfaces to split where they would odierwise cross) and die possibility of nuclear tunnelling from one surface to the other through the intervening energy barrier radier dian passage over the barrier. These effects are considered in the next section. 

The thermal fluctuations that take R° to the transition-state configuration are dominated by the internal vibrations of D and A and sympathetic adjustments in the state of polarisation of the surrounding medium. These both have a roughly harmonic effect on the energy of the system, that is, the energy in each mode varies roughly as the square of its displacement from equilibrium. Thus the R° and P° minima in Fig. 4.7 are depicted 

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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... 

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... 

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. 

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... 

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. 

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]. 

Figure 4.8 Classical Marcus theory section across the reaction coordinate X through the free energy hypersurface of the reaction complex R and product complex P for an ET reaction, showing the activation barrier AG, the reorganisation energy A and the free energy of reaction AG°.

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... 

To conclude this section, it is important to note that, in its classical form, Marcus s theory only implies two parameters the activation energy to the free energy of the reaction and the reorganization energy. It does not explicitly depend on the importance of the energetic coupling between the initial (reactant) and final (product) state. This effect is explored in the semi-classical Marcus theory [88] and will not be detailed here. 

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... 

For inverted-region ET, the rate was almost completely independent of temperature. This result was in strong contrast to the predictions of classical Marcus theory, but in excellent agreement with quantum mechanical modifications that include the effects of high-frequency molecular vibrations in the donor and acceptor groups. 

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