Quantum mechanical treatments of electron transfer processes

The essential term is the exponential function of (here a gaussian distribution function of the empty energy states in the redox system), which is mathematically the same as in the original Marcus theory. Accordingly, the rate equation is almost identical for both theories although the basic models are conceptually different. The reason is that a harmonic oscillator type of fluctuation of the solvent molecules is assumed in both cases. 

3 Quantum Mechanical Treatments of Electron Transfer Processes 

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Since electrochemical surface reactions involve electron transfer to or from the surface, a quantum mechanical approach becomes necessary to account for electron tunneling in such processes. Quantum mechanical treatments of electron transfer and adsorption have been reviewed recently (67-77). The Gurney treatment (68, 72, 73) assumes the transfer of an electron at the Fermi level of the metal to an H3O at its ground state at the outer Helmholtz plane. The electrode potential changes the minimum vibrational energy of the bond necessary to induce tunneling. Levich (67) has... 

A decrease of the theoretical value of kq in this exothermic region is due to a decrease of the k23-value which is ascribed to small values of Franck-Condon factors in this process. A similar theoretical consideration was given by Levich and Dogonadze, using the polaron model (5). A quantum mechanical treatment of electron transfer was developed by Kestner et al. (6). These results indicated a similar bell-shaped curve for the relation between log kq and AG23. None of the treatments can interpret the experimental results. 

The basic idea of the theory of electron-transfer concerning the dynamic role of the solvent, first suggested by LIBBI /150/, has been developed by MARCUS /40a/ for outer-sphere redox processes on the basis of a classical continuum model for solvent polarization. A quantum-mechanical treatment of the same model was done by LEVICH and DO-GONADZE /40b,143/, making use of the theory of non-adiabatic radiationless electron transfer in polar crystals. 

Beyond a general recognition of their importance there is no consistent theory to model the participation of electrons in chemical processes. The well-known empirical procedures to describe charge transfer within and between molecules are commonly introduced with the disclaimer that a full quantum-mechanical treatment would be required to describe the mechanisms more rigorously. However, attempts to formulate such a treatment are rare. 

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. 

Marcus[195] gave a quantitative interpretation of this idea and above all, the role of solvent rearrangement within the framework of the absolute rate theory. Later, he also extended these concepts to electrochemical processes[196]. Similar concepts were also developed by Hush[197,198]. An important result of this work was the establishment of the relation between the transfer coefficient for adiabatic reactions and the charge distribution in the transient state. Gerischer[93,199] proposed a very useful and lucid treatment of the process of electron transfer in reactions with metallic as well as semiconductor electrodes. While the works mentioned above were mainly based on transition state theory, a systematic quantum-mechanical analysis of the problem was started by Levich, Dogonadze, and Chizmadzhev[200-202] and continued in a series of investigations by the same group. They extensively used the results and methods of solid state physics, and above all the Landau-Pekar polaron theory[203]. 

To make QM studies of chemical reactions in the condensed phase computationally more feasible combined quantum me-chanical/molecular mechanical (QM/MM) methods have been developed. The idea of combined QM/MM methods, introduced first by Levitt and Warshell [17] in 1976, is to divide the system into a part which is treated accurately by means of quantum mechanics and a part whose properties are approximated by use of QM methods (Fig. 5.1). Typically, QM methods are used to describe chemical processes in which bonds are broken and formed, or electron-transfer and excitation processes, which cannot be treated with MM methods. Combined QM and MM methods have been extensively used to study chemical reactions in solution and the mechanisms of enzyme-catalyzed reactions. When the system is partitioned into the QM and MM parts it is assumed that the process requiring QM treatment is localized in that region. The MM methods are then used to approximate the effects of the environment on the QM part of the system, which, via steric and electrostatic interactions, can be substantial. The... 

In recent years, electrochemical charge transfer processes have received considerable theoretical attention at the quantum mechanical level. These quantal treatments are pivotal in understanding underlying processes of technological importance, such as electrode kinetics, electrocatalysis, corrosion, energy transduction, solar energy conversion, and electron transfer in biological systems. 

Thermodynamic treatments in physical chemistry were effectively identical with the theory of the subjectin the nineteenth century. No oneunderstoodelectron transfer at interfaces at that time (J. J. Thompson did not discover the electron until 1897). But whereas the molecular kinetic approach gradually seeped into many parts of chemistry by the 1930s, the chemistry of electrode processes remained reluctantly bound up with the older thermodynamic viewpoint. The Faraday Society meeting in Manchester, U.K. in 1947 was a turning point in the application of a molecular-level concepts and even of quantum mechanics. By the mid-1950s, research papers in electrode process chemistry (except for those dealing with electroanalytica] themes)10 were fully kinetic. 

The ET processes under discussion here correspond by definition to pure ET, in which molecular or medium coordinates may shift (the polaron response) [17], but no overall bonding rearrangements occur. More complex ET processes accompanied by such rearrangements (e.g., coupled electron/proton transfer and dissociative ET) are of great current interest, and many theoretical approaches have been formulated to deal with them, including quantum mechanical methods based on DC treatment of solvent [31,32],... 

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