Electron transfer free energy surfaces

Such reactions are possible at interaction of transition metal oxide surface in the redox reactant/surface site pairs possessing low electron transfer free energy and lead to formation of radical ions from the reactant molecules [64]... 

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

Only if one takes into account the solvent dynamics, the situation changes. The electron transfer from the metal to the acceptor results in the transition from the initial free energy surface to the final surface and subsequent relaxation of the solvent polarization to the final equilibrium value Pqj,. This brings the energy level (now occupied) to its equilibrium position e red far below the Fermi level, where it remains occupied independent of the position of the acceptor with respect to the electrode surface. 

FIGURE 34.8 Free-energy surfaces for the dissociative electron transfer reaction (a) for the solvent polarization (b) along the coordinate r of the molecnlar chemical bond. corresponds to stable molecule in oxidized form. U" is the decay potential for the rednced foim. AFj and AF are the partial free energies of the transition determining mntnal arrangement of the two sets of the free-energy surfaces. 

Describing the dissociative electron transfer step, RX e R + X involves determining the saddle point on the intersection of the two following free energy surfaces.22,31... 

The classical Morse curve model of intramolecular dissociative electron transfer, leading to equations (3.23) to (3.27), involves the following free energy surfaces for the reactant (Grx-) and product (Gr +x ) systems, respectively ... 

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...

Consider now temperatures below T in the nonequilibrium region of chemisorption, and let us assume that electron transfer over the surface barrier is rate-limiting. We will examine the case in which initially the surface is completely free of the adsorbed species it has been heated to high temperature, well above T, at a low pressure to remove essentially all the adsorbed gas. The energy bands will be straight out to the surface, and no surface barrier will exist. If the sample is quenched to a low temperature, well below T, and the gas pressure is increased, adsorption will commence. Initially it will be very fast, since the surface barrier, Et, is... 

The concept of free energy surfaces has proven its vitality over many years of fruitful applications to electron transfer kinetics. The direct connection... 

Figure 8 Free energy surfaces for the precursor and successor states of intramolecular electron transfer in a model charge-transfer system. " On the plot the dashed fines indicate the Marcus theory, circles are simulations, and solid lines refer to the Q-model. The vertical dashed line marked Xq indicates the hoimdary of the energy gap fluctuation band predicted by the Q-model. (Reprinted with permission from Ref 54, 1989 American Chemical Society)...

The two-dimensional electron transfer diabatic free energy surfaces in Figure 7 have been analyzed with the Golden Rule rate expression given in Eq. 46. This analysis suggests that FT and EPT are possible for both systems, but FT is the dominant path due to significant overlap between the proton vibrational wave... 

Figure 5. Plot of the adiabatic free-energy surfaces against the reaction coordinate for an electron transfer reaction with AG" = 0 and //ab/- varying from 0 to 0.5.

Assuming a simple Marcus-type model for the interaction with the solvent, one can derive an analytical expression for the potential (free) energy surface of the bond-breaking electron-transfer reaction as a function of the collective solvent coordinate q and the distance r between the fragments R and X [71,75]. The activation energy of the reaction can also be calculated explicitly ... 

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. 

Q (r — fb). In this case, and for transfer of one electron, A(R ) = A(R ) is the difference between the electrostatic potentials at the A and B centers that is easily evaluated in numerical simulations. An example of such result, the free energy surfaces for electron transfer within the Fe i /Fe redox pair, is shown in Fig. 16.5. The resulting curves are fitted very well by identical shifted parabolas. Results of such numerical simulations indicate that the origin of the parabolic form of these free energy curves is more fundamental than what is implied by continuum linear dielectric theory. 

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