Free energy profile, electron transfer

Outer sphere electron transfer (e.g., [11-19,107,160-162]), ion transfer [10,109,163,164] and proton transfer [165] are among the reactions near electrodes and the hquid/liquid interface which have been studied by computer simulation. Much of this work has been reviewed recently [64,111,125,126] and will not be repeated here. All studies involve the calculation of a free energy profile as a function of a spatial or a collective solvent coordinate. 

FIGURE 1.13. Free-energy profiles in outer-sphere electron transfer according to the Butler-Volmer approximation (a) and to the Marcus-Hush model (b). 

Calhoun and Voth also utilized molecular dynamic simulations using the Anderson-Newns Hamiltonian to determine the free energy profile for an adiabatic electron transfer involving an Fe /Fe redox couple at an electrolyte/Pt(lll) metal interface. This treatment expands upon their earlier simulation by including, in particular, the influence of the motion of the redox ions and the counterions at the interface. 

Figure 14.8 Simplified scheme for the transfer of the first electron from a reductant R to a NAC (adapted from Eberson, 1987). Panels (a) and (b) show free energy profiles of reactions where the actual electron transfer (a) or other steps such as precursor formation (b) are rate determining. Note that the subscript 1 is used to denote transfer of one electron to the NAC.

In Fig. 7 we have taken a symmetrical reaction where, apart from the isotopic mixing, AG ° = 0. One of the first successes of the Marcus theory was the correlation of rates for such homogeneous reactions with the rates found for the same electron transfer taking place on an electrode (Marcus, 1963). The theory then went on to predict the rates of cross reactions between two different redox couples in terms of the kinetic and thermodynamic properties of the two redox couples. The free energy profile for an unsymmetrical cross reaction such as (17) is shown in Fig. 8. The free energy of activation depends... 

The schematic free-energy profiles in Fig. 1 also illustrate the relationships between the various free-energy barriers of fundamental significance in electron-transfer kinetics. The activation free energy for the overall... 

The dynamical theory also provides a framework for the study of the diabatic free energy profiles as functions of the reaction coordinate required in the theory of non-adiabatic electron transfer reactions. We illustrate this new application by calculating the free energy profiles in solvents covering a wide range of polarity. 

Non-Linear Effects in the Free Energy Profile of Electron Transfer Reaction ... 

The free energy profile for the electron transfer reaction in a polar solvent is examined based on the extended reference interaction site method (ex-RISM) applying it to a simple model of a charge separation reaction which was previously studied by Carter and Hynes with molecular dynamics simulations. Due to the non-linear nature of the hypemetted chain (HNC) closure to solve the RISM equation, our method can shed light on the non-linearity of the free energy profiles, and we discuss these problems based on the obtained free energy profile. 

Electron transfer (ET) reaction in a polar solvent has been one of the central issues in physical chemistry and biophysics. In the presence of an immersed solute, a polar solvent around the solute continuously fluctuates due to the translational and orientational motion of its constituent molecules. The concept of a free energy profile governing these fluctuations as a function of solvent coordinates plays a central role in the theory of ET. Recently, we developed a molecular theory for obtaining the free energy profiles of ET reactions based on ex-RISM, and obtained quantitatively good results in terms of agreement with the simulation data. 

Here we briefly outline the methrxl for obtaining the free energy profiles of ET reactions based on ex-RISM, an integral equation theory developed for molecular liquids. (For more detailed description of the method, see Ref. I.) We denote the charge distribution of the solute ions, the donor (A) and the acceptor (B) ions, at the reactant state as (ca, Cb). The distance between A and B is fixed. Let z denote a fraction of the electronic charge transferred from A to B. We define a reaction coordinate as. 

Figure 1. Free-energy profile for simple electrochemical reaction + e - Y plotted against the nuclear-reaction coordinate. Fig. lA shows the overall electrochemical free-energy profile I, bulk reactant P, precursor state S, successor state 11, bulk product. Figs. IB and C show the components of the free-energy profile arising from the solution species (Y, Y ), and transferring electron, respectively.

As already mentioned, Schmickler considered the adiabatic electron transfer between a redox center and a metal electrode and derived free energy profiles of the system as follows. 

This behavior has important implications for the free energy profiles of the most rapid electron transfer reactions (51). The fastest rates should occur when the free energy change for electron transfer in the complex approaches zero. Equivalently, the reduction potential difference between donor and acceptor... 

Fig. 3. Free energy profile for electron transfer reactions, (a) Inefficient profile, with large activation energies and stable intermediates along the pathway, (b) Efficient profile, with low activation barriers separating intermediates. Values of E° and A , the reduction potential difference between free and bound A and B species, respectively, are indicated.

Non-equilibrium Free Energy Profile for the Electron Transfer Reactions... 

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