Theories of electron transfer

The experiment which triggered much of the theoretical interest in electron transfer in biological systems was that of DeVault and Chance (125), who studied the temperature dependence of the cytochrome c (Cyt c) reduction of the primary donor (BchljJ (bacteriochlorophyll) in the photosynthetic bacterium Chromatium vinosum. The reaction is 

The rest of this section is organized as follows a fairly straightforward account of modern Marcus theory is given in Section III,A which emphasizes the distinction between electron and nuclear tunneling. The superexchange mechanism and other QE theories are considered in Section III,B, and the analogy between them and certain solid state problems are considered. It is worth reemphasizing that. 

Consider the situation in which D and A are sufficiently well separated so that their mutual electronic interaction is small. The nuclear potential energy surfaces of reactants and products for the reaction 

The overall rate constant, however, will be determined not so much by the rate of electron tunneling, but by the probability of the nuclei reaching the crossing point owing to thermal fluctuations. This rate constant has been obtained by Marcus and is given by (1) 

Inclusion of quantum effects on the nuclear dynamics can be accomplished by using Fermi s Golden Rule (134), which is really a manifestation of first-order time-dependent perturbation theory and conservation of energy during a transition. In this level of refinement, and formally allowing for the inclusion of solvent effects, the rate is given by 

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Although not discussed in detail here, the normal mode analysis method has been used to calculate the electron transfer reorganization spectrum in / M-modified cytochrome c [65,66]. In this application the normal mode analysis fits comfortably into the theory of electron transfer. 

R. A. Marcus (California Institute of Technology) contributions to the theory of electron transfer reactions in chemical systems. 

Theory of electron transfer reactions insights and hindsights. N. Sutin. Prog. Inorg. Chem., 1983, 30, 441-498 (148). 

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]. 

Samec Z, Weber J (1973) The influence of chemisorbed sulfur on the kinetic parameters of the reduction of Fe " ions on a platinum electrode on the basis of the Marcus theory of electron transfer. J Electroanal Chem Interfacial Electrochem 44 229-238... 

The theoretical method employed is based on and largely similar to the theory of electron-transfer reactions in solution [123,124,125]. Thus the intramolecular spin conversion may be described as a transition between an initial manifold of states [f((r, qc)ZK,( c)[Pg.94]

In the last decade, an intense and successful investigation of this phenomenon has focused on its mechanism. The experimental facts discovered and the debate of their interpretation form large portions of these volumes. The views expressed come both from experimentalists, who have devised clever tests of each new hypothesis, and from theorists, who have applied these findings and refined the powerful theories of electron transfer reactions. Indeed, from a purely scientific view, the cooperative marriage of theory and experiment in this pursuit is a powerful outcome likely to oudast the recent intense interest in this field. 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

Marcus RA (1965) On the theory of electron-transfer reactions. VI. Unified treatment for homogeneous and electrode reactions. J Chem Phys 43 679... 

FIGURE 6.6 Potential energy diagram for the theory of electron transfer reactions. The activated complex is at S. For reasonably fast reactions, the reactant adheres to the lower curve and slithers into the product curve through the activated complex—that is, an adiabatic electron transfer occurs. 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

In our description of the Marcus theory of electron-transfer reactions we have found it helpful to plot the free energy change in the three dimensional picture shown in Fig. 10 (Albery, 1975c, 1980). This picture emphasizes that... 

Electron Transfer Far From Equilibrium. We have shown how the Marcus Theory of electron transfer provides a quantitative means of analysis of outer-sphere mechanisms in both homogeneous and heterogeneous systems. It is particularly useful for predicting electron transfer rates near the equilibrium potential,... 

A quantitative description must account for the band structure of the electrode, and can be formulated within the theory of electron-transfer reactions presented in Chapter 6. We start from Eq. (6.12) for the rate of electron transfer from a reduced state in the solution to a state of energy e on the electrode, and rewrite it in the form ... 

The first term is the intrinsic electronic energy of the adsorbate eo is the energy required to take away an electron from the atom. The second term is the potential energy part of the ensemble of harmonic oscillators we do not need the kinetic part since we are interested in static properties only. The last term denotes the interaction of the adsorbate with the solvent the are the coupling constants. By a transformation of coordinates the last two terms can be combined into the same form that was used in Chapter 6 in the theory of electron-transfer reactions. 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

A few references on the quantum theory of electron-transfer reactions follow. The article by P. P. Schmidt [1] is particularly useful for learning the... 

The last part covers a few theoretical issues. I expect that theory will play an increasingly role in electrochemistry, so every student should be introduced into the basic ideas behind current models and theories. I have tried to keep this section simple and in several cases have provided simplified versions of more complex theories. Only the last chapter, which covers the quantum theory of electron transfer reactions, requires some knowledge of quantum mechanics and of more advanced mathematical techniques, but no more than is covered in a course on quantum chemistry. 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

Marcus, R. A. and Siddarth, P. Theory of electron transfer reactions and comparison with experiments, NATO ASISer., Ser. C, 376(Photoprocesses in Transition Metal Complexes, Biosystems and Other Molecules) (1992), 49-88... 

Sutin, Norman, Theory of Electron Transfer Reactions Insights and Hindsights 30 441... 

The basic assumption of the Marcus theory of electron transfer is that only a weak interaction between the reactants is needed in order for a simple electron-transfer process to occur. Marcus theory considers... 

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