Marcus quantum mechanical model

The theoretical aspects of electron transfer mechanisms in aqueous solution have received considerable attention in the last two decades. The early successes of Marcus Q, 2), Hush (3, 4), and Levich (5) have stimulated the development of a wide variety of more detailed models, including those based on simple transition state theory, as well as more elaborate semi-clas-sical and quantum mechanical models (6-12). 

A comprehensive quantum mechanical model for the effect has been developed by Marcus and his colleagues at the California Institute of Technology. The Gao and Marcus (2001, 2002) model accounts for many of the experimental observations and utilizes classical quantum mechanical RRKM theory in its development. Statistical RRKM theory quantitatively describes the energetics of gas phase atom-molecule encounters and the relevant parameters which lead to either stabilization and product formation or re-dissociation to atomic and molecular species. This is a well-developed theory and will not be described in detail here. An important application of this theory is that it determines... 

The wide ranging experimental results on the conductivity of DNA have motivated theoretical efforts in this direction in order to come out with a clearer picture about the remarkable behaviour of the "molecule of life". The common theoretical approaches include ab initio studies and model calculations. The quantum master equations of the reduced density matrix [153] and the single step transfer theory of Marcus [154-157] govern the model calculations. According to Marcus, quantum-mechanical tunneling can occur when vibrational fluctuations bring the donor and acceptor states of the molecule into resonance. 

Recently, Ovchinnikov and Benderskii gave a quantum mechanical model of the hydrogen evolution reaction at a metal electrode. In this model, they have emphasized the importance of Gurney-based model rather than that of Marcus, Levich, and Dogonadze. However, they tried to combine the principal features of the two models. They have pointed out that transition along the reaction coordinate, rather than the solvent coordinate, was important to explain the Tafel behavior and the constancy of the transfer coefficient. 

Calculations of A<, from the electrostatic continuum model and from quantum mechanical models have been reviewed and compared/ The distinction between electrostatic displacement D and field E is emphasized. Values of A are compared for different physical models of the reacting molecules, e.g., conducting spheres (the model usually considered in previous literature) and cavities of various dimensions. In the electrostatic model a formula for A has been given, which applies to any system which has a symmetrical binuclear structure, and from which Marcus two-sphere " and Cannon s ellipsoidal " models can be deduced as special cases. 

A key point that must be made is diat quantum mechanical tunneling through the Marcus-theory barrier when it is non-zero can increase the rate for electron transfer just as is true for any other activated process. Because the electron is so light a particle, tunneling can be a major contributor to die overall rate. Models for electron tunneling will not, however, be presented here. 

The Levich—Dogonadze—Kuznetsov (LDK) treatment [65] considers that the only source of activation is the polarization electrostatic fluctuations (harmonic oscillations) of the solvent around the reacting ion and uses essentially the same model as the Marcus—Hush approach. However, unlike the latter, it provides a quantum mechanical calculation of both the pre-exponential factor and the activation energy but neglects intramolecular (inner sphere) vibrations (1013—1014 s 1). 

The difference between the two results is in the pre-exponential term. In the quantum mechanical treatments either vet= vaKe or vet = 0.5(tu + reyl depending on the model adopted. In the Marcus equation the pre-exponential term is ZKee p( w lRT) and the time dependence is introduced through the collision frequency, Z. In any case, in the non-adiabatic limit, ve < v , and kobs is given by equation (41). In the adiabatic limit, ve > vn and kobs is given by equation (42), with vn in the range 10n —1013 s-1. 

In long distance ET, the presence of a wave function in the medium between donor and acceptor is the only means for donor and acceptor to communicate. The nature of this connection is expected to influence the reaction rates for ET or EET between the subsystems. Partitioning technique together with the Marcus-Hush model [6,7] may be viewed as an adaptation to practical chemistry of a full quantum mechanical treatment [21], where nuclei and electrons are treated as equal partners. In particular the influence on ET from the medium between the redox centres is formalized. 

The theory for this intermolecular electron transfer reaction can be approached on a microscopic quantum mechanical level, as suggested above, based on a molecular orbital (filled and virtual) approach for both donor (solute) and acceptor (solvent) molecules. If the two sets of molecular orbitals can be in resonance and can physically overlap for a given cluster geometry, then the electron transfer is relatively efficient. In the cases discussed above, a barrier to electron transfer clearly exists, but the overall reaction in certainly exothermic. The barrier must be coupled to a nuclear motion and, thus, Franck-Condon factors for the electron transfer process must be small. This interaction should be modeled by Marcus inverted region electron transfer theory and is well described in the literature (Closs and Miller 1988 Kang et al. 1990 Kim and Hynes 1990a,b Marcus and Sutin 1985 McLendon 1988 Minaga et al. 1991 Sutin 1986). 

Although, AE is not generally equal to AG° unless the frequencies of the reactant and product are the same, this assumption is almost universally made. With this assumption classical Marcus ET theory combined with a quantum mechanical (Landau-Zener) treatment of the barrier crossing also yields Eq. (4) [2,36-39]. This derivation of Eq. (4) is called semiclassical ET theory, and therefore in the rest of this paper Eq. (4) will also be referred to as the semiclassical rate expression or the semiclassical model. 

The Sumi-Marcus model treats both the solvent and intramolecular mode classically. However, the actual system should have not only the classical modes but also quantum mechanical high-frequency modes. Jortner and Bixon have developed an ET model which introduces the effect of the quantum mechanical high-frequency modes [4]. In this model, the reactant surface crosses not only with the vibrational ground state of the product but also with vibrationally excited states of the product. The reaction of oxazines in DMA is activationless, which means that the reaction is not in the inverted region but very close to it. We use a hybrid model of Sumi-Marcus and Jortner-Bixon developed by Walker et al. [15]. In this model, the intramolecular vibration is separated into the quantum mechanical high-frequency mode and classical low-frequency mode ... 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

As already mentioned above, the derivation of the Butler-Volmer equation, especially the introduction of the transfer factor a, is mostly based on an empirical approach. On the other hand, the model of a transition state (Figs. 7.1 and 7.2) looks similar to the free energy profile derived for adiabatic reactions, i.e. for processes where a strong interaction between electrode and redox species exists (compare with Section 6.3.3). However, it should also be possible to apply the basic Marcus theory (Section 6.1) or the quantum mechanical theory for weak interactions (see Section 6.3.2) to the derivation of a current-potential. According to these models the activation energy is given by (see Eq. 6.10)... 

It is curious that the striking deviations of electrochemical kinetic behavior from that expected conventionally, which are the subject of this review, have not been recognized or treated in the recent quantum-mechanical approaches, e.g., of Levich et al (e.g., see Refs. 66 and 105) to the interpretation of electrode reaction rates. The reasons for this may be traced to the emphasis which is placed in such treatments on (1) quantal effects in the energy of the system and (2) continuum modeling of the solution with consequent neglect of the specific solvational- and solvent-structure aspects that can lead, in aqueous media, to the important entropic factor in the kinetics and in other interactions in water solutions. However, the work of Hupp and Weaver, referred to on p. 153, showed that the results could be interpreted in terms of Marcus theory, with regard to potential dependence of AS, when there was a substantial net reaction entropy change in the process. 

Stine and Marcus.63 These results are in excellent agreement (as few %) with the accurate quantum mechanical values obtained by Secrest and Johnson24 even for extremely weak transitions with a probability as small as 10" n. It is thus encouraging that the semiclassical model is able to describe such quantum-like phenomena for which ordinary (i.e. real-valued) classical trajectory methods would clearly be inapplicable. 

The difference between the two results is in the pre-exponential term. In the quantum mechanical treatments either = et = 0-5( rn+Tg) depending on the model adopted. In the Marcus... 

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