Marcus model rate theory

We now turn to the electronically adiabatic ET reaction problem (cf. Sec. 2.2). There has been a spate oftheoretical papers [8,11 28,33,35,36,50] dealing with the possible role of solvent dynamics in causing departures from the standard Marcus TST rate theory [27,28] (although many of these deal with nonadiabatic reactions). The ET reaction considered is a simplified symmetric model, A1 2 A1/2 A1/2 A1/2, in a model solvent similar to CH3C1. The technical and computational... 

The rationalization of PER by Marcus is based on a simple model of the reaction profile, that of two intersecting parabolas (Pigure 5.10) [35]. In most applications of the Marcus equilibrium-rate theory, the reaction coordinate is a normalized quantity between 0 and 1, measuring in a generalized way the progress of reaction it is usually poorly defined from a geometrical, structural point of view. Indeed, when the word structure is used in works on PER, it refers mainly to the connectivity... 

Since the Marcus model was initially a classical or later a semi-classical theory, the introduction of quantum effects was considered to account for these observations. In particular, tunnelling pathways of e.t. would increase the rates of some reactions. A closer analysis has, however, led to the conclusion that this could not be a general explanation [80]. 

Weller behaviour, so that in intermolecular reactions the rate constant remains at the diffusion controlled limit in the most exergonie processes in apparent contradiction with the results of Marcus theory [84], On the other hand, observations of the M.I.R. have been made essentially in the case of thermal charge recombination reactions, with a few examples of charge shifts. Why should some e.t. systems follow the Marcus model, and others not This is the fundamental theoretical question which has been addressed to by different authors in recent years and several interesting models have been suggested to account for the different behaviours of the Rehm-Weller and Marcus systems. 

There are however essential differences between the Marcus model and the theory of radiationless transitions. In the former, the decrease of the rate constant in the inverted region results from an activation barrier which must be overcome by thermal energy, whereas the rates of radiationless transitions are in principle temperature independent. As implied in [14], there is no normal region in the case of nonradiative transitions, a no bell-shaped curve is expected from the plot of the rate constant against the energy gap. 

The temperature dependence of the rates of e.t. processes is therefore the obvious criterion for a choice between the Marcus model and a model of e.t. as a type of radiationless transition. In this respect, it must, however, be noted that an apparent temperature effect may appear in the theories of radiationless transitions, if the... 

Application of Marcus rate theory to proton transfer in enzyme-catalyzed reactions was discussed by Kresge and Silverman, 1999. Relationships of log KIE and kinetics of the enzyme catalysis (kcat) and parameters of the reaction driving force were found to be in agreement with the Marcus model. 

The results obtained clearly demonstrate that the Marcus model for ECL processes may be used for qualitative as well as for quantitative descriptions of this kind of electron transfer reactions. The more sophisticated approach, taking into account the vibronic excitation in the reaction products (important in the inverted Marcus region), solvent molecular dynamics (important in the case of large values of the electronic coupling elements), as well as the changes in the electron transfer distance, should be used. The results indicate that the Marcus theory may also be used for predicting the ECL efficiency, provided that some conditions are fulfilled. Especially, during the ECL process, only the annihilation of ions should occur, without any competitive reactions. The necessary rate constants can be evaluated from pertinent electrochemical and spectroscopic data. 

As already mentioned in the introduction to this chapter, kinetic models describing electron transfer processes at metal electrodes, had been used for a long time before Marcus developed his theory. At a fairly early stage a transition state model was applied the rate constants were described in terms of an activation energy so that we have... 

Marcus12 and others13 extended this model to include reactions in which electron transfer occurred during collisions between the donor and acceptor species, that is, between the short-lived Dn—Am complexes. In this context, electron transfer within transient precursor complexes ([Dn — A" in Scheme 1.1) resulted in the formation of short-lived successor complexes ([D(, + — A(m 1)] in Scheme 1.1). The Debye-Smoluchowski description of the diffusion-controlled collision frequency between D" and A " was retained. This has important implications for application of the Marcus model, particularly where—as is common in inorganic electron transfer reactions—charged donors or acceptors are involved. In these cases, use of the Marcus model to evaluate such reactions is only defensible if the collision rates between the reactants vary with ionic strength as required by the Debye-Smoluchowski model. The requirements of that model, and how electrolyte theory can be used to verify whether a reaction is a defensible candidate for evaluation using the Marcus model, are presented at the end of this section. 

Background and Useful Models The Marcus equation is an extension of earlier models from collision rate theory. As such, compliance with collision rate models is a prerequisite to defensible use of the Marcus equation. This is particularly important for reactions of charged species, and therefore, for reactions of many inorganic complexes. In these cases, the key question is whether electron transfer rate constants vary with ionic strength as dictated by electrolyte theory, on which the collision rate models are based. When they do not, differences between calculated and experimental values can differ by many orders of magnitude. 

Debye-Hiickel theory that has not been criticized. From this perspective, electrolyte models are simply the best tools available to assess whether the dependence of electron transfer rate constants on ionic strength is sufficiently well behaved to justify use of the Marcus model. For this, and despite their shortcomings, they are indispensable. 

The recent theoretical approaches include a theory of barrierless electronic relaxation which draws on the model of nonradiative excited state decay, and a general treatment of the effect of solvent dielectric relaxation based on the theory of optical line shapes, as well as treatments based on classical and quantum rate theories. Equation(5) does not hold for all solvents and, more generally, may be frequency-dependent. Papers by Hynes, Rips and Jortner, Sumi and Marcus, and Warshel and Hwang " contain good overviews of the theoretical developments. 

Fig. 8.2 Oxidation potential of model electrolyte clusters vs. the rate estimated from the Marcus electron transfer theory. EC(—H) denotes a neutnil EC radical with one proton abstracted from it in the process of the oxidation-induced reaction. Oxidation potentials are from M05-2X/6-31+G calculations with PCM(e=20) [16]. Rates were ceilculated in this work...

---