Marcus nonadiabatic

R.A. Marcus, Nonadiabatic processes involving quantum-like and classical-like coordinates with applications to nonadiabatic electron transfers, J. Chem. Phys., 81, 4494—4500 (1984). 

Beginning in the 1950s, Marcus developed and refined a classical theoretical description of redox reactions occurring in homogeneous solution [79-84]. He also extended the treatment to include electrode processes [81-86]. Related treatments are the quantum models of Levich and Doganadze and the works of Gerisher. More recently, Chidsey [29] presented a derivation of Marcus nonadiabatic (MNA) behavior, characterized by the quadratic free energy relationship for electrode processes ... 

The ZN formulas can also be utihzed to formulate a theory for the direct evaluation of thermal rate constant of electronically nonadiabatic chemical reactions based on the idea of transition state theory [27]. This formulation can be further utilized to formulate a theory of electron transfer and an improvement of the celebrated Marcus formula can be done [28]. 

The factor k takes into acount the effects of nonadiabatic transition and tunneling properly. Also note that the electronic coupling //ad is assumed to be constant in the Marcus formula, but this is not necessary in the present formulation. The coupling Had cancels out in k of Eq. (126) and the ZN probability can be calculated from the information of adiabatic potentials. 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

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... 

Figure 5. Energy diagram for charge separation resolved into reactant-like and product-like diabatic surfaces. The two diabatic curves do not intersect, but interact, to give an avoided crossing, whose energy gap is twice the electronic coupling, Vei, for the interaction. Also depicted is the Marcus-Hush classical rate expression for nonadiabatic ET.

When electron transfer is forced to take place at a large distance from the electrode by means of an appropriate spacer, the reaction quickly falls within the nonadiabatic limit. H is then a strongly decreasing function of distance. Several models predict an exponential decrease of H with distance with a coefficient on the order of 1 A-1.39 The version of the Marcus-Hush model presented so far is simplified in the sense that it assumed that only the electronic states of the electrode of energy close or equal to the Fermi level are involved in the reaction.31 What are the changes in the model predictions brought about by taking into account that all electrode electronic states are actually involved is the question that is examined now. The kinetics... 

The breakdown of the Condon approximation can actually lead to gating phenomena if particular angles are very much favored. It can also lead to situations in which the Marcus-Hush-Jortner formula must be generahzed to deal with the Condon breakdown, and the results can be quantitatively important. DNA seems to be a situation in which Condon breakdown is striking, and this should be kept in mind when comparing adiabatic and nonadiabatic transport mechanisms. 

By measuring the temperature dependence of kex, activation parameters (Aff and AS ) could be calculated and were reported. However, I am not sure how to physically interpret these numbers. The temperature dependence of rate can be fit to other expressions, and here it is fit to the Marcus equation for nonadiabatic electron transfer in the case of degenerate electron transfer (e.g., AG° = 0)... 

For x 1, the process is nonadiabatic and is expected to follow the Marcus-Hush theory with the rate constant of e.t. given as... 

In general, for the types of linked donor-acceptor systems to be discussed in this review, electron transfer is assumed to occur in the nonadiabatic regime. That is, the mixing between the electronic state of the donor and acceptor before electron transfer occurs and the corresponding state after electron transfer is weak (< kBT) [9], The actual electron transfer event is assumed to be fast compared to the time scale of nuclear motions. Marcus has proposed [11, 15] that the electron transfer rate constant ke, is given by Eq. 1. 

To sum up the survey of the past work on oxidation-reduction reactions the only experimental results obtained thus far which strongly indicate nonadiabatic effects are some obtained by Matteson and Bailey (22) and by Chan and Wahl (23) for self-exchange reactions. In addition, it is very likely that such effects are significant also for reactions of f electron redox agents, particularly on reaction with one another. Marcus has advocated consistently the position that nonadiabatic effects are relatively unimportant for the ordinary oxidation—reduction reactions which have been studied. But many experimentalists, including myself, have been much slower to arrive at it. Further work may show that a nonadiabatic factor is significant in many other processes, but at the present level of development of the subject, there are not many cases where it needs to be invoked. 

See - nonadiabatic (diabatic) process, -> Marcus theory, - Randles, and - Gurney, - adiabatic process (thermodynamics). 

According to the Marcus theory of ET (32), the driving force dependence of the rate constant of nonadiabatic intramolecular ET is given by Eq. 1, where V is the... 

These theoretical considerations also gave a basis for the consideration of the optimal distance of discharge, which is a result of competition between the activation energy AG and the overlap of electronic wave functions of the initial and final states. The reaction site for outer-sphere electrochemical reactions is presumed to be separated from the electrode surface by a layer of solvent molecules (see, for instance, [129]). In consequence, the influence of imaging interactions on AGJ predicted by the Marcus equation is small, which explains why such interactions are neglected in many calculations. However, considerations of metal field penetration show that the reaction sites close to the electrode are not favored [128], though contributions to ks from more distant reaction sites will be diminished by a smaller transmission coefficient. If the reaction is strongly nonadiabatic, then the closest approach to the electrode is favorable. 

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