Free energy curves, for electron transfer

The environmental (i.e., solvent and/or protein) free energy curves for electron transfer reactions can be generated from histograms of the polarization energies, as in the works of Warshel and coworkers [79,80]. 

T Ichiye. Solvent free energy curves for electron transfer A non-lmear solvent response model. J Chem Phys 104 7561-7571, 1996. 

Following the early studies on the pure interface, chemical and electrochemical processes at the interface between two immiscible liquids have been studied using the molecular dynamics method. The most important processes for electrochemical research involve charge transfer reactions. Molecular dynamics computer simulations have been used to study the rate and the mechanism of ion transfer across the water/1,2-dichloroethane interface and of ion transfer across a simple model of a liquid/liquid interface, where a direct comparison of the rate with the prediction of simple diffusion models has been made. ° ° Charge transfer of several types has also been studied, including the calculations of free energy curves for electron transfer reactions at a model liquid/liquid... 

T. Ichiye,/. Chem. Phys., 104,7561 (1996). Solvent Free Energy Curves for Electron Transfer... 

Studies[71-73] of the free energy curves for electron transfer at liquid/liquid interfaces have been concerned with several issues. First, to what degree is the linear response assumption which leads to parabolic free energy curves accurate Second, what qualitatively new features does the interface region introduce into the solvent free energy curves Finaly, how do continuum electrostatic models for the free energy curves compare with the molecular dynamics results Here we consider the first two points. For a recent study of continuum models see reference [73]. 

Fig. 12.2. Free energy data for electron transfer between the protein cytochrome c and the small acceptor microperoxidase-8 (MP8), from recent simulations [47]. Top Gibbs free energy derivative versus the coupling parameter A. The data correspond to solvated cytochrome c the MP8 contribution is not shown (adapted from [47]) Bottom the Marcus diabatic free energy curves. The simulation data correspond to cyt c and MP8, infinitely separated in aqueous solution. The curves intersect at 77 = 0, as they should. The reaction free energy is decomposed into a static and relaxation component, using the two steps shown by arrows a static, vertical step, then relaxation into the product state. All free energies in kcalmol-1. Adapted with permission from reference [88]...

Fig. 2 Free energy diagram for electron transfer. Solid curve represents free energy of the equilibrium state, E = Eoq. Dashed curve represents free energy of for E = Eoq + r. ... 

The results of the free energy calculations for electron transfer in bulk water show that the full free energy curves are well approximated by paraboli. The calculations for electron transfer at the solution/metal interface are also, in general, in agreement with the linear response assumption. 

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. 

FIgurt 12 (a) Potential energy curve for electron transfer from P to Q. (b) Bell-shaped curve of rate constant of electron transfer vs. free energy change (AG) predicted by Marcus theory, (c) Saturation curve of rate constant of bimolecular electron transfer observed in solution. 

RB Yelle, T Ichiye. Solvation free energy reaction curves for electron transfer Theory and simulation. J Phys Chem B 101 4127-4135, 1997. 

Our problem now is to determine the functional form of this experimental free energy curve for the intrinsic rate constant ki for electron transfer. In addition to the Marcus eq 4, two other relationships are currently in use to relate the activation free energy to the free energy change in electron transfer reactions (15, JL6). 

Figure 11. The solid line depicts the quantum adiabatic free energy curve for the Fe /Fe electron transfer at the water/Pt(lll) interface (obtained by using the Anderson-Newns model, path integral quantum transition state theory, and the umbrella sampling of molecular dynamics. The dashed line shows the curve from the classical calculation as given in Fig. 5. (Reprinted from Ref 14.)...

Figure 13. Adiabatic free energy curves for the electron transfer reaction Fe + e" for...

Figure 17.6 Free energy curves for reactant and product states of an electron transfer process in the kinetic regimes of the Marcus model.

Fig. 9.16. Adiabatic free energy curves for the electron-transfer reaction for Fe +e-Fe2+ for an overpotential q and electronic coupling coefficient, r. (Reprinted from I. Benjamin and D. A. Rose, J. Chem. Phys. 100 3545, 1994 with permission of the American Institute of Physics.)...

Fig. 9. Plots of log(k[M-1 s 1] for fluorescence quenching of excited states [21, 40]. The solid curve is a Rehm-WeBer plot and the broken one a Marcus plot, both with X = 9.6 kcal mol-1 = 40 kJ mol-1. The dotted curve corresponds to a Marcus plot with X = 38 kcal mol-1 = 159 kJ mol-1 (X = reorganization energy, AG° = corrected standard free energy change of electron transfer) — taken from Ref. [lb]...

Figure 2.10 Parabolic free energy curves for heterogeneous electron transfer processes... 

The free-energy curves depict the zero-order or diabatic states of the system. Figure 1 shows the diabatic free-energy curves for a self-exchange reaction (Eq. la, AG° = 0) and Figure 2 the curves for an electron transfer reaction accompanied by a net chemical change (Eq. lb, AG° < 0 for an exergonic reaction). 

Figure 8 shows the free energy curves for a model electron transfer reaction at the water/1,2-dichloroethane (DCE) interface. The model[72] represents the... 

Figure 9 Free-energy curves for simple outer-sphere electron-transfer reactions. Xj4 is the activation energy at equilibrium, X being the solvent reorganization energy.

Figure 19. PI-QTST activation free-energy curves for a model A-H-A proton transfer system which demonstrate the effect of solvent electronic polarizability (see Ref. 77). The solid line depicts the classical free-energy curve for the solute in isolation with a rigid A-A distance. The short-dashed line is for the rigid solute in an electronically nonpolarizable solvent. The long-dashed line depicts the quantum free-energy curve for the rigid solute solvated in a quantum polarizable solvent, while the dot-dashed line depicts the quantum free energy for the same solute but in a classically polarizable solvent.

This brief report outlines a new method for studying electron transfer reactions. The electric field effect on the emission lineshape combined with the effect on the amplitude and induced polarization are found to be sensitive to parameters which are quite difficult to obtain by other approaches. For well-defined model systems, many of the parameters characterizing dipoles are fixed by the molecular structure. For the RC, these parameters are unknowns of considerable interest in understanding the mechanism. It is also possible to obtain information on the rate vs. free energy curve for the molecule of interest, without resorting to a synthetic series of molecules. Details of the application to native RCs [12] and to RC mutants [13] can be found in the primary literature. 

Use die activated complex theory for explaining clearly how the applied potential affects the rate constant of an electron-transfer reaction. Draw free energy curves and use proper equations for your explanation. 

FIGURE 35.3 Free-energy functions for reactant (AE) and product Ag (AE) of an electron transfer reaction as calculated using umbrella sampling within a simple dipolar diatomic solvent. AG° is the reaction free energy. Solid lines are polynomial fittings to the simulated points. Dashed lines are parabolic extrapolations from the minimum of the curves. (From King and Warshel, 1990, with permission from the American Institute of Physics.)... 

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