Curve crossing electron transfer reactions

Figure C3.2.1. A slice tlirough tlie intersecting potential energy curves associated witli tlie K-l-Br2 electron transfer reaction. At tlie crossing point between tlie curves (Afy, electron transfer occurs, tlius Tiarjiooning tlie species,...

The use of the terms adiabatic and non-adiabatic in this way leads to a source of confusion. Normally, in describing surface-crossing processes, a process which remains on the same potential curve is called adiabatic and in that sense every net electron transfer reaction is an adiabatic process. Processes which involve a transition between different states as between the two different potential curves in Figure lb are usually called non-adiabatic. Such processes have some special features and will be returned to in a later section dealing with the inverted region and excited state decay. 

Figure 5 Potential energy curves for an electron-transfer reaction p(A+- -B) — s A- -B+), showing vibrational quantization, assuming the same vibrational frequency v in precursor and successor states. Some of the vibrational wavefunctions are indicated. The dotted arrows refer to electron transfer below the energy of the crossing of the two curves...

Note that assumptions (2) and (3) are about timescales. Denoting by x, and tlz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities Tr A Ts tlz hold. The validity of this assumption has to be addressed, but for now let us consider its consequences. When assumptions (1)—(3) are satisfied we can invoke the extended transition-state theory of Section 14.3.5 that leads to an expression for the electron transfer rate coefficient of the form (cf. Eq. 14.32)... 

In this review, almost all of the simulations we have described use only classical mechanics to describe the nuclear motion of the reaction system. However, a more accurate analysis of many reactions, including some of the ones that have already been simulated via purely classical mechanics, will ultimately require some infusion of quantum mechanical methods. This infusion has already taken place in several different types of reaction dynamics electron transfer in solution, > i> 2 HI photodissociation in rare gas clusters and solids,i i 22 >2 ° I2 photodissociation in Ar fluid,and the dynamics of electron solvation.22-24 Since calculation of the quantum dynamics of a full solvent is at present too time-consuming, all of these calculations involve a quantum solute in a classical solvent. (For a system where the solvent is treated quantum mechanically, see the quantum Monte Carlo treatment of an electron transfer reaction in water by Bader et al. O) As more complex reaaions are investigated, the techniques used in these studies will need to be extended to take into account effects involving electron dynamics such as curve crossing, the interaction of multiple electronic surfaces and other breakdowns of the Born-Oppenheimer approximation, the effect of solvent and solute polarization, and ultimately the actual detailed dynamics of the time evolution of the electronic degrees of freedom. 

A quite different response is found for couples O/R where Iq is large (in fact, where Iq > 10" /l or k > 10" cm s ). Then the electron transfer reaction at the surface is rapid enough that under most mass transport conditions obtainable experimentally, the electron transfer couple at the surface appears to be in equilibrium. Then the surface concentrations may, at each potential, be calculated from the Nernst equation, a purely thermodynamic equation, and the current may be calculated, for example, from equation (1.57). The I-E curve has the form shown in Fig. 1.16 the I-E curve crosses the zero current axis steeply and there is no overpotential for oxidation or reduction. Systems with these characteristics are often termed reversible . On the other hand, the limiting current densities do not depend on the kinetics of electron transfer closer to E. Hence the limiting current densities for reversible and irreversible reactions are the same. 

Note that assumptions (2) and (3) are about timescales. Denoting by Xx, Xg, and xiz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities... 

Figures 8c and 8d represent the projection of the reaction pathways on the same plane, namely the front plane in which the dissociative electron transfer step is represented. This two-dimensional representation is easier to decipher than the 3D representation for determining the preferred pathway. They may however be misleading if it is not borne in mind that, in the 2D representation, the crossings between the three curves should not be considered as actual crossings of reaction pathways.

Fig. 1. The Marcus parabolic free energy surfaces corresponding to the reactant electronic state of the system (DA) and to the product electronic state of the system (D A ) cross (become resonant) at the transition state. The curves which cross are computed with zero electronic tunneling interaction and are known as the diabatic curves, and include the Born-Oppenheimer potential energy of the molecular system plus the environmental polarization free energy as a function of the reaction coordinate. Due to the finite electronic coupling between the reactant and charge separated states, a fraction k l of the molecular systems passing through the transition state region will cross over onto the product surface this electronically controlled fraction k l thus enters directly as a factor into the electron transfer rate constant...

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