Electron transfer nonadiabatic limit

The electron transfer rates in biological systems differ from those between small transition metal complexes in solution because the electron transfer is generally long-range, often greater than 10 A [1]. For long-range transfer (the nonadiabatic limit), the rate constant is... 

In this section, we switch gears slightly to address another contemporary topic, solvation dynamics coupled into the ESPT reaction. One relevant, important issue of current interest is the ESPT coupled excited-state charge transfer (ESCT) reaction. Seminal theoretical approaches applied by Hynes and coworkers revealed the key features, with descriptions of dynamics and electronic structures of non-adiabatic [119, 120] and adiabatic [121-123] proton transfer reactions. The most recent theoretical advancement has incorporated both solvent reorganization and proton tunneling and made the framework similar to electron transfer reaction, [119-126] such that the proton transfer rate kpt can be categorized into two regimes (a) For nonadiabatic limit [120] ... 

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. 

When electron transfer is forced to take place at a large distance from the electrode by means of an appropriate spacer, the reaction quickly falls within the nonadiabatic limit. H is then a strongly decreasing function of distance. Several models predict an exponential decrease of H with distance with a coefficient on the order of 1 A-1.39 The version of the Marcus-Hush model presented so far is simplified in the sense that it assumed that only the electronic states of the electrode of energy close or equal to the Fermi level are involved in the reaction.31 What are the changes in the model predictions brought about by taking into account that all electrode electronic states are actually involved is the question that is examined now. The kinetics... 

Figure 12. The rate of the electron transfer reaction Fe + e Fe in the nonadiabatic limit. The dotted line is the result of molecular dynamics computer simulations [Eqs. (27) and (28)] and the solid line corresponds to Eqs. (29)-(31). (Adapted from Ref 163.)...

The Fermi Golden rule describes the first-order rate constant for the electron transfer process, according to equation (11), where the summation is over all the vibrational substates of the initial state i, weighted according to their probability Pi, times the square of the electron transfer matrix element in brackets. The delta function ensures conservation of energy, in that only initial and final states of the same energy contribute to the observed rate. This treatment assumes a weak coupling between D and A, also known as the nonadiabatic limit. 

The observation of excited products in metal-molecule reactions is mostly limited to the simplest molecules. The formation of excited species is probably much more general and should be probed since the electron transfer can correlate excited-state reagents and excited-state products. In fact, the nascent excited-state products from reactions yielding complex polyatomic systems are quenched by nonadiabatic... 

In the weak-interaction limit, the exchange mechanism can be described in terms of thermodynamic quantities, according to a classical approach which parallels that for nonadiabatic electron transfer (see Eqs. (1) and (2)) [59-63]. 

Charge injection is fast compared with nuclear relaxation of the excited state (k k,). In this case, interfacial charge transfer would take place from the prepared hot vibronic level (Eq. (34)) and the quantum yield for the primary injection process would be close to unity = 1). For both limiting cases, k[ kr and k[ kr, relation (30) would be relevant, provided electron transfer is nonadiabatic. 

Several classes of precursor complexes can be distinguished (see Table I), depending on the relative magnitudes of A and H g. The properties of class I systems are predominantly those of the separate components, and the electron transfer is nonadiabatic. Limiting class I behavior corresponds to the zero-interaction case discussed earlier. Class II systems possess new optical and electronic properties in addi-... 

Ill) Rate Determined by Conversion of Precursor to Successor Compiex. This is the case for the ordinary electron transfer discussed in earlier sections. However, even though all of the preceding steps may be rapid and no unfavorable preequilibria are involved, the rate constants for ordinary electron transfer saturate below the diffusion-controlled limit when the reaction is nonadiabatic (k, << 1). For example, when AG = 0 (the normal condition for a diffusion-controlled reaction), k, for a nonadia-... 

In a few systems, quenching rate constants tend toward saturation below the diffusion-controlled limit . This may be observed when the quenching reaction does not involve an electron-transfer process (e.g. , [ RuL3] -[Cr )] ), when the electron transfer is highly nonadiabatic, when the reactive form of the quencher is not the dominant form in solution (e.g. , [ RuL3] -[CU( )] , or when a substitutional or conformational change becomes rate determining. ... 

The Marcus theory, as described above, is a transition state theory (TST, see Section 14.3) by which the rate of an electron transfer process (in both the adiabatic and nonadiabatic limits) is assumed to be determined by the probability to reach a subset of solvent configurations defined by a certain value of the reaction coordinate. The rate expressions (16.50) for adiabatic, and (16.59) or (16.51) for nonadiabatic electron transfer were obtained by making the TST assumptions that (1) the probability to reach transition state configuration(s) is thermal, and (2) once the reaction coordinate reaches its transition state value, the electron transfer reaction proceeds to completion. Both assumptions rely on the supposition that the overall reaction is slow relative to the thermal relaxation of the nuclear environment. We have seen in Sections 14.4.2 and 14.4.4 that the breakdown of this picture leads to dynamic solvent effects, that in the Markovian limit can be characterized by a friction coefficient y The rate is proportional to y in the low friction, y 0, limit where assumption (1) breaks down, and varies like y when y oo and assumption (2) does. What stands in common to these situations is that in these opposing limits the solvent affects dynamically the reaction rate. Solvent effects in TST appear only through its effect on the free energy surface of the reactant subspace. 

When the system passes the intersection at a high velocity, that is, the above condition is not met even approximately, it will usually jump from the lower R surface (before S along the reaction coordinate) to the upper R surface (after S). That is, the system behaves in a nonadiabatic (or diabatic) fashion, and the probability per passage of electron transfer occurring is small (i.e., k< C1). The nuclear coordinates of the system change so rapidly that it cannot remain at equilibrium. At the nonadiabatic limit, the time interval for passage between the two states at point S approaches zero, that is, (tf — t,) —> 0 (infinitely rapid), and the probability density distribution functions that describe the initial and final states remain unchanged ... 

The electron-transfer reactions that occur within and between proteins typically involve prosthetic groups separated by distances that are often greater than 10 A. When we consider these distant electron transfers, an explicit expression for the electronic factor is required. In the nonadiabatic limit, the rate constant for reaction between a donor and acceptor held at fixed distance and orientation is ... 

We note that Equations 1.1.5 and 1.1.6 were obtained in the framework of perturbation theory and are therefore strictly applicable only in the limiting case of weak electronic couphng (nonadiabatic electron-transfer regime) [14,15]. In a more general case (any Vy < 0.5A,), the thermal activation barrier also depends on electronic coupling while the prefactor is a function of the attempting nuclear frequency v (frequency for nuclear motion along the reaction coordinate) and the electronic frequency that is equal to the prefactor in Equations 1.1.5 and 1.1.6... 

This equation can also be further modified to accommodate other effects, such as site energy fluctuations, external fields, or disorder. In this chapter, however, we will limit ourselves to the nonadiabatic electron transfer regime. 

Applications are then presented in Section IV. These examples should served as a guide as to what kinds of problems can be studied with these techniques and the limitations and possibilities for these methods. We present three examples (1) a dynamical test of the centroid quantum transition-state theory for electron transfer (ET) reactions in the crossover regime between adiabatic and nonadiabatic electron transfer, (2) the primary electron transfer reaction in bacterial photosynthesis, and (3) the diffusion kinetics of a Brownian particle in a periodic potential. Finally, Section V offers an outlook and a perspective of the current status of the field from our vantage point. 

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