Reorganization outer sphere

Here X is tire reorganization energy associated witli the curvature of tire reactant and product free energy wells and tlieir displacement witli respect to one another. Assuming a stmctureless polarizable medium, Marcus computed the solvent or outer-sphere component of tire reorganization energy to be... 

The elementary electrochemical reactions differ by the degree of their complexity. The simplest class of reactions is represented by the outer-sphere electron transfer reactions. An example of this type is the electron transfer reactions of complex ions. The electron transfer here does not result in a change of the composition of the reactants. Even a change in the intramolecular structure (inner-sphere reorganization) may be neglected in many cases. The only result of the electron transfer is then the change in the outer-sphere solvation of the reactants. The microscopic mechanism of this type of reaction is very close to that for the outer-sphere electron transfer in the bulk solution. Therefore, the latter is worth considering first. 

In typical outer sphere electron transfer on metal electrodes, A is in the weakly adiabatic region and thus sufficiently large to ensure adiabaticity, but too small to lead to a noticeable reduction of the activation energy. In this case, the rate is determined by solvent reorganization, and is independent of the nature of the metal [Iwasita et al., 1985 Santos et al., 1986]. 

In order to fully evaluate Eq. (17), expressions involving the outer-sphere reorganization term A,q, as well as the work terms, were also established [5],... 

An expression of the type in Eq. (29) has been rederived recently in Ref. 13 for outer-sphere electron transfer reactions with unchanged intramolecular structure of the complexes where essentially the following expression for the effective outer-sphere reorganization energy Ers was used ... 

As with the Marcus-Hush model of outer-sphere electron transfers, the activation free energy, AG, is a quadratic function of the free energy of the reaction, AG°, as depicted by equation (7), where the intrinsic barrier free energy (equation 8) is the sum of two contributions. One involves the solvent reorganization free energy, 2q, as in the Marcus-Hush model of outer-sphere electron transfer. The other, which represents the contribution of bond breaking, is one-fourth of the bond dissociation energy (BDE). This approach is... 

It has been shown so far that internal and external factors can be combined in the control of the electron-transfer rate. Although in most cases a simple theoretical treatment, e.g. by the Marcus approach, is prevented by the coincidence of these factors, it is clear that the observed features for the isoenergetic self-exchange differ by the electronic coupling and the free energy of activation. Then it is also difficult to separate the inner- and outer-sphere reorganization energies. 

The value of E° was hence determined by the reaction of R4M with Fe3+ complexes as outer-sphere SET oxidizers. Using five complexes with a range of different E° values, from 1.15 to 1.42 V, the rate constants were determined193. This was followed up by Eberson who, by application of the Marcus theory, was able to determine from the E° values (shown in Table 18) standard potentials and reorganization energies. Most compounds... 

Two types of electron transfer mechanisms are defined for transition metal species. Outer-sphere electron transfer occurs when the outer, or solvent, coordination spheres of the metal centers is involved in transferring electrons. No reorganization of the inner coordination sphere of either reactant takes place during electron transfer. A reaction example is depicted in equation 1.27 ... 

A natural question is In which temporal order do the reorganization processes and the proper electron transfer take place The answer is given by the Frank-Condon principle, which in this context states First the heavy particles of the inner and outer sphere must assume a suitable intermediate configuration, then the electron is exchanged isoenergetically, and finally the system relaxes to its new equilibrium... 

Figure 6.1 Reorganization of inner and outer sphere during an electron-transfer reaction.

To develop these ideas into a quantitative theory, we require models for the inner and outer sphere and their reorganization. The problem is similar to that encountered in infrared and Raman spectroscopy, where... 

A is a measure for the energy required to reorganize the inner and outer sphere during the reaction. The energy of activation for the oxidation is the saddle point energy minus the initial energy ered, which gives ... 

To obtain an estimate for the energy of reorganization of the outer sphere, we start from the Born model, in which the solvation of an ion is viewed as resulting from the Coulomb interaction of the ionic charge with the polarization of the solvent. This polarization contains two contributions one is from the electronic polarizability of the solvent molecules the other is caused by the orientation and distortion of the... 

The reorganization of the solvent molecules can be expressed through the change in the slow polarization. Consider a small volume element AC of the solvent in the vicinity of the reactant it has a dipole moment m = Ps AC caused by the slow polarization, and its energy of interaction with the external field Eex caused by the reacting ion is —Ps Eex AC = —Ps D AC/eo, since Eex = D/eo- We take the polarization Ps as the relevant outer-sphere coordinate, and require an expression for the contribution AU of the volume element to the potential energy of the system. In the harmonic approximation this must be a second-order polynomial in Ps, and the linear term is the interaction with the external field, so that the equilibrium values of Ps in the absence of a field vanishes ... 

During the reaction the dielectric displacement changes from Dox to Dred (or vice versa), and the equilibrium value from Dox/2aeo to Drec[/2a eo. From Eq. (6.5) the contribution of the volume element AV to the energy of reorganization of the outer sphere is ... 

The total energy of reorganization of the outer sphere is obtained by integrating over the volume of the solution surrounding the reactant ... 

When a reaction is adiabatic, the electron is transferred every time the system crosses the reaction hypersurface. In this case the preexponential factor is determined solely by the dynamics of the inner-and outer-sphere reorganization. Consequently the reaction rate is independent of the strength of the electronic interaction between the reactant and the metal. In particular, the reaction rate should be independent of the nature of the metal, which acts simply as an electron donor and acceptor. Almost by definition adiabatic electron-transfer reactions are expected to be fast. 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

We first consider outer sphere transfer (ET) reactions, e.g. D" + A -> D + A, a donor-acceptor electron transfer without significant coupled internal reorganization of the D and A species [27,29,30]. A hallmark of such reactions, which has been long appreciated [27], is that the reactive coordinate is itself a many-body collective solvent variable (and is not the coordinate of the electron itself)- In particular, if R and P stand for the reactant and product, then the reactive coordinate is... 

The coordinate pertaining to solvent reorganization, z, is the same fictitious charge number as already considered in the Hush-Marcus model of outer-sphere electron transfer (Section 1.4.2), and so is the definition of 2q [equation (1.27)] and the difference between the Hush and Marcus estimation of this parameter. The coordinated describing the cleavage of the bond is the bond length, y, referred to its equilibrium value in the reactant, yRX. Db is the bond dissociation energy and the shape factor ft is defined as... 

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