Theoretical Aspects of Electron Transfer

The behavior of a colliding system at the avoided crossing depends on the speed with which the crossing is traversed, as well as the separation and slopes of the adiabatic curves. If the atoms approach slowly enough on the covalent curve, the 

The probability of a diabatic hop, P, is given to good approximation by the Landau-Zener (LZ) relation,  

from Equation (2), low speeds or large k (large or large separation between the curves) gives a small P, and the probability of a diabatic hop between potential curves is low. Under these circumstances, the crossing is adiabatic and the system smoothly changes from a covalent to an ionic description. In this process the electron jumps from the Na atom to the I atom. 

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A number of publications in recent years have demonstrated an active interest in the theoretical aspects of electron transfer (ET) processes in biological systems (1.-9). This interest was stimulated by the extensive experimental information regarding the temperature dependence of ET rates measured over a broad range of temperatures (10-16). The unimolecular rate of cyto-chrome-c oxidation in Chromatium (10-12), for example, exhibits the Arrhenius type dependence and changes by three orders of... 

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). 

Like the 1994-issue of the series "Electron Transfer", the second volume again covers various aspects of this fundamental process. The articles are concerned with the experimental and theoretical aspects of electron transfer in chemistry and biology. In the latter, emphasis is given to energy transfer, which is also part of photosynthesis. 

The discussion in the previous section was helpful in identifying the factors at the molecular level which are involved when electron transfer occurs. Two different theoretical approaches have been developed which incorporate these features and attempt to account for electron transfer rate constants quantitatively. The first, by Marcus34 and Hush,35 is classical in nature, and the second is based on quantum mechanics and time dependent perturbation theory. The theoretical aspects of electron transfer in chemical36-38 and biological systems39 have been discussed in a series of reviews. 

Recent progress in understanding the theoretical basis of electron transfer has been rapid. Theoretical aspects of electron transfer are addressed in detail in other contributions to this series, and authoriative, up-to-date reviews are available [9-14], For our purposes, it will be sufficient to review some very basic electron transfer theory which will serve as a framework for the discussion of artificial photosynthetic systems which follows. 

Theoretical Aspects of Electron Transfer from Solid Proteins to Ions in Solution... 

Prior to the 1970 s, electrochemical kinetic studies were largely directed towards faradaic reactions occurring at metal electrodes. While certain questions remain unanswered, a combination of theoretical and experimental studies has produced a relatively mature picture of electron transfer at the metal-solution interface f1-41. Recent interest in photoelectrochemical processes has extended the interest in electrochemical kinetics to semiconductor electrodes f5-151. Despite the pioneering work of Gerischer (11-141 and Memming (15), many aspects of electron transfer kinetics at the semiconductor-solution interface remain controversial or unexplained. 

The ubiquity of electron transfer processes make them a familiar subject for all chemists and a consequence of that familiarity is a general feeling that they are well understood. There is justification for that feeling in broad, general terms. General or large-scale patterns of electron transfer behavior can usually be predicted with considerable confidence. On the other hand, the continuing flow of basic experimental and theoretical work is a clear indication that there are fundamental aspects of electron transfer behavior that are not well understood. 

The theoretical aspects of charge-transfer complexes have been discussed by Briegleb [33], Andrews and Keefer [34], by Mulliken and Person [35], and by Foster [36]. Formation of the charge-transfer complex is generally regarded as involving an electron transfer from the highest occupied molecular orbital of the donor (D) to the lowest unoccupied molecular orbital of the acceptor (A) and can be represented as shown in Scheme IV. 

Most of the AIMD simulations described in the literature have assumed that Newtonian dynamics was sufficient for the nuclei. While this is often justified, there are important cases where the quantum mechanical nature of the nuclei is crucial for even a qualitative understanding. For example, tunneling is intrinsically quantum mechanical and can be important in chemistry involving proton transfer. A second area where nuclei must be described quantum mechanically is when the BOA breaks down, as is always the case when multiple coupled electronic states participate in chemistry. In particular, photochemical processes are often dominated by conical intersections [14,15], where two electronic states are exactly degenerate and the BOA fails. In this chapter, we discuss our recent development of the ab initio multiple spawning (AIMS) method which solves the elecronic and nuclear Schrodinger equations simultaneously this makes AIMD approaches applicable for problems where quantum mechanical effects of both electrons and nuclei are important. We present an overview of what has been achieved, and make a special effort to point out areas where further improvements can be made. Theoretical aspects of the AIMS method are... 

When we transfer the theoretical aspects of these models to or nometallic chemistry, we are aware of the fact that, e.g. when altering the number of rr-electrons by two (67T-, 4ir- or 2Tr-electrons) in the elementary steps of organometallic reactions, we have, step-by-step to await a change in the type of process (S or A) (in analogy to Fig. 1 in Scheme 2.1-3). On the other hand, we also can learn that the energies of the FMO s change systematically, going from 6n- to 4ir- to 27r-electron stems (from e.g. bis-ir-allyl- to rr-allyl-a-allyl to bis-o-allyl-ntetal-complexes). 

The kineties of eleetron-transfer reactions, which is also affected by the electrode potential and the metal-water interface, is more difficult and complex to treat than the thermodynamic aspects. While the theoretical development for electron transfer kinetics began decades ago, a practical implementation for surface reactions is still unavailable. Popular transition state-searching techniques such as the NEB method are not designed to search for minimum-energy reaction paths subject to a constant potential. Approximations that allow affordable quantum chemistry calculations to get around this limitation have been proposed, ranging from the electron affinity/ionization potential matching method to heuristic arguments based on interpolations. 

In Chap. 2, the analysis of diffusion-limited reaction rates of Smolu-chowski, Collins and Kimball, and that of Noyes is followed. The considerable literature on reaction rates between solute species is also presented. Additional and important other factors which influence the rate of reaction are a coulomb interaction between reactants, long-range energy or electron transfer and an angular dependence of the rate of reaction. These topics are considered in the Chaps. 3—5. The experimental and theoretical work are compared and contrasted. When the reactants are formed in pairs (by bond fission of a precursor), the rate or probability of recombination can be measured and is of considerable interest. Chapters 6 and 7 discuss the theoretical aspects of the recombination of neutral and ionic radical pairs and also appeal to the extensive literature on the experimentally measured rate of recombination. The weaknesses of this theoretical... 

Kinetic and Theoretical Aspects of Outer-Sphere Electron Transfer Reactions... 

A number of excellent reviews on the theoretical aspects of photoinduced electron transfer have appeared in the literature and the interested reader may refer to these original articles. Due to the fundamental importance of the electron transfer step for the title process and in order to facilitate reading this chapter a brief introduction on PET will be given [2,6]. 

Theoretical aspects of the subject are considered and then experimental data on the various types of transfer are reviewed. Mention is also made of applications of energy transfer processes to give information about electronically excited states, which is often difficult to obtain in other ways. 

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