Marcus Jortner theory

Figure 24. Electron-transfer rate versus electronic coupling strength. The temperature is T = 500 K. Solid line with circle-present results from Eq. (126) with the transition probability averaged over the seam surface. Solid line with square-present results with the transition probability taken at the minimum energy crossing point (MECP). Dashed line-Bixon-Jortner theory Ref. [109]. Dotted line-Marcus s high temperature theory. Taken from Ref. [28].

The resulting theory, named as the Marcus-Hush theory [17], has been the widest and most accepted theory for kinetics overviews since then. However, the theory is based basically on classical kinetics for electron transfer, and the quantum nature of the process is almost shielded by using other related concepts. This is rather strange since, between 1960 and 1970, electron quantum mechanics by Jortner and Kuznetsov [18-20] was well accepted in the specialized literature for non-radiant transitions. 

Excited State Charge Transfer. Our goal here is to discuss aspects of ET theory that are most relevant to the charge transfer processes of excited molecules. One important point is that often the solvent relaxation is not well modeled with a single t, but rather a distribution of times apply. This subject has been treated by Hynes [63], Nadler and Marcus [65], Rips and Jortner [66], Mukamel [67], Newton and Friedman [68], Zusman [62], Warshel [71], and Fonseca [139], We also would like to study ET in the strongly adiabatic regime since experimental results on BA indicate this is the correct limit. Finally, we would like to treat the special case of three-well ET, which is the case for BA. 

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).91 Absolute rate constants of SEET for a number organic molecules from ISM show a significant improvement over classical Marcus theory92-94 in the ability to predict experimental SEET values. A combination of Marcus theory and the Rips and Jortner approach was applied to the estimation of the amount of charge transferred in the intramolecular ET reactions of isodisubstituted aromatic compounds.95... 

In contrast to the experimentally based work discussed above, in the most recent comprehensive theoretical discussion [21d], Bixon and Jortner state that the question of whether non-adiabatic or adiabatic algorithms describe electron-transfer reactions was settled in the 1960s, and that the majority of outer-sphere electron-transfer reactions are non-adiabatic. This is certainly true for the reactions that occur in the Marcus inverted region in which these authors are interested, but we think the question of whether reactions in the normal region are best treated by adiabatic theory that includes an electronic transmission coefficient or by non-adiabatic equations remains to be established. 

Despite its utility at room temperature, simple Marcus theory cannot explain the DeVault and Chance experiment. All Marcus reactions have a conspicuous temperature dependence except in the region close to where AG = —A. Marcus theory does not predict that a temperature-dependent reaction will shift to a temperature-independent reaction as the temperature is lowered. Hopfield proposed a quantum enhancement of Marcus theory that would permit the behavior seen in the experiment [11]. He introduced a characteristic frequency of vibration hco) that is coupled to electron transfer, in other words, a vibration that distorts the nuclei of the reactant to resemble the product state. This quantum expression includes a hyperbolic cotangent (Coth) term that resembles the Marcus expression at higher temperatures, but becomes essentially temperature independent at lower temperatures. Other quantized expressions, such as a full quantum mechanical simple harmonic oscillator behavior [12] and that of Jortner [13], give analogous temperature behavior. 

Excellent reviews on the theory of ET are available in the literature. Among them, the Nobel lecture of Marcus [77], the 1996 reviews of Barbara et al. [78] and of Balzani et al. [79], both the 1999 Jortner and Bixon issues of Advances in Chemical Physics [80], and the 2000 monograph of May and Kiihn [81] are worthy of note. 

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. 

The adiabatic inner-sphere redox reactions were first treated by MARCUS /145/, who made use of the classical and semiclassical statistical theory A quantum-mechanical treatment of the two-frequency oscillator model by DOGONADZE and KUSNETSOV /147/ provides tractable rate expressions for non-adiabatic processes in both high and low temperature ranges. Similar results were obtained by KESTNER, LOGAN and JORTNER /148/. 

The main differences between Hopfield s theory and the earlier theory of Marcus are the explicit use of the spectral shape functions D E) (though this difference is removed when Gaussian functions are used), the lack of reference to a transition state, and the extension to lower temperatures where equation (16) is replaced by a non-Arrhenius form, not quoted here. Jortner, however, later argued that at low temperatures the Hopfield theory is inapplicable, since... 

The subscrips in and out introduced here refer to inner- and outer-sphere by analogy with the two reorganization energies of the Marcus theory [equation (4)J. Actually the theory does not assume that high frequencies are associated exclusively with inner-sphere bond vibrations, though this is probably a good approximation. In Jortner s notation vjn and vout are written cos and [Pg.8]

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