Quantum Mechanics of Electron Transfer

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. 

Electron transfer from the excited states of Fe(II) to the H30 f cation in aqueous solutions of H2S04 which results in the formation of Fe(III) and of H atoms has been studied by Korolev and Bazhin [36, 37]. The quantum yield of the formation of Fe(III) in 5.5 M H2S04 at 77 K has been found to be only two times smaller than at room temperature. Photo-oxidation of Fe(II) is also observed at 4.2 K. The actual very weak dependence of the efficiency of Fe(II) photo-oxidation on temperature points to the tunneling mechanism of this process [36, 37]. Bazhin and Korolev [38], have made a detailed theoretical analysis in terms of the theory of radiationless transitions of the mechanism of electron transfer from the excited ions Fe(II) to H30 1 in solutions. In this work a simple way is suggested for an a priori estimation of the maximum possible distance, RmSiX, of tunneling between a donor and an acceptor in solid matrices. This method is based on taking into account the dependence... 

The exponential dependence of the efficiency of fluorescence quenching on the distance between a donor and an acceptor may be explained by the tunneling mechanism of electron transfer from a singlet-excited molecule of the donor to the acceptor. Indeed, in case of stationary excitation of donor particles, the value of J is determined by the stationary concentration n of the excited donor particles J = An where A is a constant. The value of n is, in its turn, inversely proportional to the rate constant, k, of deactivation of excited particles nft = nJexcexciting light, quantum yield of excited molecules, and n is the concentration of non-excited donor molecules. Thus, J = AnJexc4>lk. Hence, one can easily obtain... 

Some notions of the mechanism of electron transfer were given in Section 4.2. Any theory must be realistic and take into account the reorientation of the ionic atmosphere in mathematical terms. There have been many contributions in this area, especially by Marcus, Hush, Levich, Dog-nadze, and others5-9. The theories have been of a classical or quantum-mechanical nature, the latter being more difficult to develop but more correct. It is fundamental that the theories permit quantitative comparison between rates of electron transfer in electrodes and in homogeneous solution. 

Our interest in quantum dot-sensitized solar cells (QDSSC) is motivated by recent experiments in the Parkinson group (UW), where a two-electron transfer from excitonic states of a QD to a semiconductor was observed [32]. The main goal of this section is to understand a fundamental mechanism of electron transfer in solar cells. An electron transfer scheme in a QDSSC is illustrated in Figure 5.22. As discussed in introduction, quantum correlations play a crucial role in electron transfer. Thus, we briefly describe the theory [99] in which different correlation mechanisms such as e-ph and e-e interactions in a QD and e-ph interactions in a SM are considered. A time-dependent electric field of an arbitrary shape interacting with QD electrons is described in a dipole approximation. The interaction between a SM and a QD is presented in terms of the tunneling Hamiltonian, that is, in... 

While it is possible to develop MM parameters specifically for reactions, but this is highly laborious, and the resulting parameters may not be transferable. Also, the form of the potential function can impose serious limitations, such as the neglect of electronic polarization. Methods that take the fundamental quantum mechanics of electronic structure into account are more generally applicable. Quantum mechanical methods (i.e. methods to calculate molecular electronic structure) are often preferable and can be easier to apply than involved molecular mechanics type approaches. The major problem with electronic structure calculations on enzymes is the large computational resources required, which significantly limits the size of the system that can be treated. Quantum chemical approaches to modelling enzyme reactions are described in the next couple of sections. 

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... 

Since in this case, electrons could only be excited in a single well the photocurrent was small. On the other hand, the quantum yield, that is, the number of transferred electrons per absorbed photons, reached values of up to = 0.63 [80]. This might appear surprisingly high for a relatively thick outer barrier layer. However, calculations and measurements of the temperature dependence of the photocurrent showed that at room temperature the mechanism of electron transfer out of the well was thermionic emission over the barrier [80]. The rate of thermionic emission at lattice temperatures in the range of 200—300 K was sufficient to keep up with the measured rate of interfacial electron transfer. Studies with very thin outer barriers (20 A) have shown that the mechanism of charge transfer was field-assisted tunneling, and the photocurrent was then independent of temperature. 

Discuss the physical origins of quantum mechanical tunneling. Why is tunneling more likely to contribute to the mechanisms of electron transfer and proton transfer processes than to mechanisms of group transfer reactions, such as AB -I- C —> A -I- BC (where A, B, and C are large molecular groups) ... 

Computer simulations of electron transfer proteins often entail a variety of calculation techniques electronic structure calculations, molecular mechanics, and electrostatic calculations. In this section, general considerations for calculations of metalloproteins are outlined in subsequent sections, details for studying specific redox properties are given. Quantum chemistry electronic structure calculations of the redox site are important in the calculation of the energetics of the redox site and in obtaining parameters and are discussed in Sections III.A and III.B. Both molecular mechanics and electrostatic calculations of the protein are important in understanding the outer shell energetics and are discussed in Section III.C, with a focus on molecular mechanics. 

First, we shall discuss reaction (5.7.1), which is more involved than simple electron transfer. While the frequency of polarization vibration of the media where electron transfer occurs lies in the range 3 x 1010 to 3 x 1011 Hz, the frequency of the vibrations of proton-containing groups in proton donors (e.g. in the oxonium ion or in the molecules of weak acids) is of the order of 3 x 1012 to 3 x 1013 Hz. Then for the transfer proper of the proton from the proton donor to the electrode the classical approximation cannot be employed without modification. This step has indeed a quantum mechanical character, but, in simple cases, proton transfer can be described in terms of concepts of reorganization of the medium and thus of the exponential relationship in Eq. (5.3.14). The quantum character of proton transfer occurring through the tunnel mechanism is expressed in terms of the... 

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. 

In the quantum mechanical formulation of electron transfer (Atkins, 1984 Closs et al, 1986) as a radiationless transition, the rate of ET is described as the product of the electronic coupling term J2 and the Frank-Condon factor FC, which is weighted with the Boltzmann population of the vibrational energy levels. But Marcus and Sutin (1985) have pointed out that, in the high-temperature limit, this treatment yields the semiclassical expression (9). 

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 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 aspects of electron transfer mechanisms in aqueous solution have received considerable attention in the last two decades. The early successes of Marcus Q, 2), Hush (3, 4), and Levich (5) have stimulated the development of a wide variety of more detailed models, including those based on simple transition state theory, as well as more elaborate semi-clas-sical and quantum mechanical models (6-12). 

The Quantum Mechanics of Long-Range Electron Transfer Kinetics. 52... 

The general framework of the quantum mechanical rate expression for long-range electron transfer processes in the very weak or non-adiabatic regime will be presented in Sect. 2 with an emphasis on the inclusion of superexchange interactions. The relation between the simplest case of direct donor-acceptor interactions, on the one hand, and long-range electronic interactions important in proteins, on the other, is considered in terms of the elements of electron transfer theory. 

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