Quantum theory of electron-transfer reactions

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. 

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. 

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 transport of a proton is involved in many electrochemical reactions like, e.g., the hydrogen evolution reaction. Existing theories have mostly been extensions of the theory of electron transfer reactions, in which the proton tunnels from its initial to its final state. So far, quantum simulation techniques have only been applied to proton transfer in the bulk aqueous phase and clusters (e.g., Ref. 240-246), but not near interfaces. 

The first main idea of this work is to refuse the assumption of possible one-step transfer of several (more than one) electrons in one elementary electrochemical act and to consider any real many-electron process as a sequence of one-electron steps. Although this idea is not new (it follows immediately from quantum theories of electron transfer [4]), it is not followed consistently in research practice. The reason is that a number of significant problems ought to be overcome in such an approach description of the accompanying intervalence chemical reactions, general scheme of the mechanism, estimation of stability of low-valence intermediate species and... 

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... 

The theoretical description of the kinetics of electron transfer reactions starts fi om the pioneering work of Marcus [1] in his work the convenient expression for the free energy of activation was defined. However, the pre-exponential factor in the expression for the reaction rate constant was left undetermined in the framework of that classical (activate-complex formalism) and macroscopic theory. The more sophisticated, semiclassical or quantum-mechanical, approaches [37-41] avoid this inadequacy. Typically, they are based on the Franck-Condon principle, i.e., assuming the separation of the electronic and nuclear motions. The Franck-Condon principle... 

A. Dynamical Test of the Centroid Quantum Transition-State Theory for Electron Transfer Reactions... 

The quantum mechanical nature of the proton transfer reaction has been dealt with in analytic theory and simulations by Warshel and Chu and by Borgis and Hynes.i s-i o xhe methods they describe explicitly take into account the solvent effect on the proton tunneling process. In that sense, they are perhaps more closely related to simulations of electron transfer reactions than to the simulations we have described so far. For that reason, we shall refer the reader to the original papers for further description of these techniques. 

One of the currently most promising approaches to a quantitative theory of electron transfer at an electrode is that of Marcus, whose fundamental assumption is that only a weak electronic interaction of the two reactants is required for a simple electron transfer process to occur. Interesting and significant deductions have been made quantum mechanically for simple electrode reaction in which no rupture or formation of chemical bonds occurs in the electron transfer step. The elaboration of the theory to include bond rupture is of obvious importance for the treatment of organic electrode processes. 

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

---