The Theory of Electron Transfer

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. 

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]

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. 

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. 

The Golden Rule approach has been used for many years by Levich, Dogonadze, and co-workers (39, 40), who have stressed the difference between the roles of "quantum" (high-frequency) and "classical" coupling modes in discussing the theory of electron transfer, and by a number of subsequent workers (16, 41). 

The theory of electron transfer to electrodes has been worked out by various... 

Marcus attempted to calculate the minimum energy reaction coordinate or reaction trajectory needed for electron transfer to occur. The reaction coordinate includes contributions from all of the trapping vibrations of the system including the solvent and is not simply the normal coordinate illustrated in Figure 1. In general, the reaction coordinate is a complex function of the coordinates of the series of normal modes that are involved in electron trapping. In this approach to the theory of electron transfer the rate constant for outer-sphere electron transfer is given by equation (18). 

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. 

Catalysis is used to control many kinds of chemical reactions, including natural enzymatic reactions [1,2] as well as most industrial chemical processes [3,4]. The electron transfer reaction is the most fundamental, since the electron is the minimal unit of the change in chemical reactions. Interest in electron transfer reactions in many areas of chemistry has developed rapidly in the last several decades since Marcus established the theory of electron transfer [5-7],... 

The theory of electron transfer in chemical and biological systems has been discussed by Marcus and many other workers 74 84). Recently, Larson 8l) has discussed the theory of electron transfer in protein and polymer-metal complex structures on the basis of a model first proposed by Marcus. In biological systems, electrons are mediated between redox centers over large distances (1.5 to 3.0 nm). Under non-adiabatic conditions, as the two energy surfaces have little interaction (Fig. 5), the electron transfer reaction does not occur. If there is weak interaction between the two surfaces, a, and a2, the system tends to split into two continuous energy surfaces, A3 and A2, with a small gap A which corresponds to the electronic coupling matrix element. Under such conditions, electron transfer from reductant to oxidant may occur, with the probability (x) given by Eq. (10),... 

The theory of electron transfer in polar media is considered in detail in Kuznetsov s book [13] where numerous references on the different problems are cited. It should be noted that the tunneling transfer of protons accompanied by the reorganization of the vibration degrees of freedom may be examined in the same way as the electron transfer. The theory of this phenomenon is investigated in details in the monograph of Gol danskii, Trakhtenberg and Flerov [28], and also in Ref. [13]. 

At the classical description of the nuclear degrees of freedom (it is the classical approach that is used in the theory of electron transfer in polar media (see Chapter 2, Section 5)), the fulfillment of criterion (73) results in the following. Instead of one effective oscillator (see Chapter 2) where the equilibrium position is shifted at the transition, it should now introduce two... 

The basic principles describing the efFects of CT complexes on the energy profile along the reaction coordinate stem from the theory of electron transfer. Redox processes may occur (i) as ground-state thermal reactions, (ii) by direct irradiation of the CT band, and (iii) upon photoexcitation of one of the redox partners followed by diffusional complex formation [4, 24], as depicted in Chart 3. 

Information about further energy levels in a redox system can be derived from the theory of electron transfer between a redox system and an electrode, which has been derived by various authors [2-5]. In all these theories it is assumed that the vibration of redox molecules and their surrounding solvation shell is slow, compared to the actual electron transfer, i.e. it is assumed that the Frank-Condon principle is valid. As shown by Gerischer [44], this assumption leads to the consequence that the energy levels involved in the charge transfer differ from the thermodynamic value Ep, jox- This model leads to a distribution of empty and occupied states versus electron energy, as illustrated in Fig. 6. These electron states are not discrete energy levels, but are distributed over... 

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. 

Andrzej Kapturkiewicz gives a thorough account on the theory of electron transfer reactions that lead to electrochemiluminescence (ECL). He discusses in detail the conditions under which the Marcus theory can give a more quantitative description of ECL processes. 

Marcus, R. A. (1965) On the Theory of Electron-Transfer Reactions. VI. Unified Treatment for Homogeneous and Electrode Reactions, J. Chem. Phys. 43, 679-701. 

Halpem J. and Oigel L. E. (1960), The theory of electron transfer between metal ions in bridged systems , Z)Acm55. Faraday Soc. 29, 32 1. 

Gao Y. Q., Georgievskii Y. and Marcus R. A. (2000), On the theory of electron transfer reactions at semiconductor electrode/liquid interfaces , J. Chem. Phys. 112, 3358-3369. 

It is pointed out above that metal deposition is different from outer-sphere charge-transfer reactions in that charge is carried across the metal/solution interface by the ions, not by electrons. Although this has been acknowledged by several noted electrochemists, a theory of charge transfer by ions, comparable in detail and depth to the theories of electron transfer, has yet to be developed. So far,... 

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