Theories of Electron-Transfer Reactions

Electron transfer reactions, treated by continuum theory, suggested that the Franck-Condon barrier (the barrier for the vertical transition of electrons), which is about four times the activation barrier for the isotopic electron transfer in solution, is due to Bom continuum solvation processes. Specific contributions for the activation of ions come from the solvent continuum far from the ion the important contribution from the solvent molecules oriented toward the central ion in the first and second solvation shells is neglected.  

The theory of homogeneous electron transfer reactions in solution has been formulated in terms of models in which the transferring electron is localized at a donor site in the reactant and at an acceptor site in the 

However, for electrochemical electron transfer reactions at a metal electrode, one gets  

The continuum theory expression of the free energy of activation AG (continuum) can be expressed as 

The free energy of activation in the continuum theory is based on the prequantal Born continuum solvation equation for ions in solution, 

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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 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 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... 

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. 

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... 

Sutin N (1983) Theory of electron transfer reactions Insights and hindsights. In Prog Inorg Chem, Lippard SJ (Ed) 30 441-448, John Wiley Sons, New York... 

Rudolph A. Marcus Chemistry Theory of electron transfer reactions... 

MARCUS , RUDOLPH A. 11923-1. Prolessor Marcus from the California Institute of Technology. Pasadena. California, won the Nobel prize lor chemistry in 1992 for his contributions in the theory of electron transfer reactions in chemical systems. 

The assumptions, equations and several applications of a recently formulated theory of electron transfer reactions of solvated electrons are outlined. The relationship of the reorganization terms to those of ordinary electron exchange and electrochemical reactions is described, together with the role played by an effective standard free energy of reaction. Applications include prediction of conditions under which chemiluminescence might be found and description of conditions under which reactions might not be diffusion-controlled. 

K. F. Purcell and B. Blaive, Theory of Electron Transfer Reactions, in Photoinduced Electron Transfer, Part A, edited by M. A. Fox and M. Chanon, Chapter 1.3, pp. 123-160, Elsevier,... 

He received the 1992 Nobel Prize in Chemistry for his contributions to the theory of electron transfer reactions in chemical systems. Dr. Marcus is a Member of the National Academy of Sciences of the U.S.A. (1970) and a Foreign Member of the Royal Society (London, 1987). His other distinctions include the Wolf Prize in Chemistry (Israel, 1985) and the National Medal of Science (1989). Our conversation was recorded in Dr. Marcus s office at the California Institute of Technology on February 19, 1996. ... 

As mentioned above, the formation of excited states in chemical reactions may be understood in the context of an electron transfer model for chemiluminescence, first proposed by Marcus [2]. According to this model the formation of excited states is competitive with the formation of the ground state, even though the latter is strongly favored thermodynamically. Thus, understanding the factors that determine the electron transfer rate is of considerable importance. The theory of electron transfer reactions in solution has been summarized and reviewed in many reviews (e.g., [30-36]). Therefore, in this chapter the relevant ideas and equations are only briefly summarized, to serve as a basis for description of the ECL experiments. 

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