The Reorganization Energy

The reorganization energy. A, is usually treated as the sum of two contributions, an inner (Aj ) and an outer (Aout) sphere reorganization energy, i.e. 

Marcus has considered the vibration of the ion-solvent bond within the first solvation shell and treated it as an harmonic oscillator. He obtained 

The slow component due to orientation polarization is then given by 

The energy involved upon any change of polarization is given by 

With respect to the electron transfer, we are interested in the rather slow orientation change of the solvent molecules. The corresponding energy involved in such a process can be obtained by performing a thought experiment as follows. In the first step the electric field is switched on very slowly so that all processes can follow. In the second step the field is switched off very rapidly so that only the distortion of electrons can follow. The outer sphere reorientation energy is thus given by 

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Within this framework, by considering the physical situation of the electrode double layer, the free energy of activation of an electron transfer reaction can be identified with the reorganization energy of the solvation sheath around the ion. This idea will be carried through in detail for the simple case of the strongly solvated... 

Figure 1 Temperature dependence of the reorganization energy and effective charges on...

This model permits one to immediately relate the bath frequency spectrum to the rate-constant temperature dependence. For the classical bath (PhoOc < 1) the Franck-Condon factor is proportional to exp( —with the reorganization energy equal to... 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

Note that only Er, which is actually the sum of the reorganization energies for all degrees of freedom, enters into the high-temperature rate constant formula (2.62). At low temperature, however, in order to preserve E, one has to fit an additional parameter co, which has no direct physical sense for a real multiphonon problem. 

Marcus theory. Consider that the reorganization energy for the ET reaction, AAb, can be approximated as the mean of the reorganization energies for the EE reactions Aab = (Aaa + ABb)/2. Show that substitution of this expression into Eq. (10-63) gives the usual form of the Marcus cross relation. 

The change in the inner-sphere structure of the reacting partners usually leads to a decrease in the transition probability. If the intramolecular degrees of freedom behave classically, their reorganization results in an increase in the activation barrier. In the simplest case where the intramolecular vibrations are described as harmonic oscillators with unchanged frequencies, this leads to an increase in the reorganization energy ... 

The reorganization energy of the slow polarization is, roughly speaking, almost one-half of that for the bulk solution. 

The reorganization energy of the slow polarization for the reactions at metal electrodes can be calculated with the use of Eqs. (34.11). For a spherical model of the reacting ion, it is equal approximately to... 

Figure 2.4 Adiabatic potential energy E q) for various values of A at equilibrium. The solvent coordinate q has been normalized so that the initial state corresponds to = 0 and the final state to = 1. The reorganization energy was taken as A = 1 eV.

In subsequent works, Marcus developed his theory further in a series of papers providing expressions for the work terms, the reorganization energy and the macroscopic ET rate constants [3 6]. Assuming a sharp liquid-liquid boundary, the solution of the mean molar volume of reactants yields an expression for of the form... 

The flat interface model employed by Marcus does not seem to be in agreement with the rough picture obtained from molecular dynamics simulations [19,21,64-66]. Benjamin examined the main assumptions of work terms [Eq. (19)] and the reorganization energy [Eq. (18)] by MD simulations of the water-DCE junction [8,19]. It was found that the electric field induced by both liquids underestimates the effect of water molecules and overestimates the effect of DCE molecules in the case of the continuum approach. However, the total field as a function of the charge of the reactants is consistent in both analyses. In conclusion, the continuum model remains as a good approximation despite the crude description of the liquid-liquid boundary. 

Very often the reorganization energy is rather large (the barrier for the process is high), ET F E - X/Redl, and Eq. (5.3.14) attains a simplified form,... 

According to the Marcus theory [9], the electron transfer rate depends upon the reaction enthalpy (AG), the electronic coupling (V) and the reorganization energy (A). By changing the electron donor and the bridge we measured the influence of these parameters on the charge transfer rate. The re-... 

The problem of the physical meaning of the quantity Hx and of the reorganization energy of the medium Es has been analyzed in Ref. 11. Following Ref. 11, we write the expression for the transition probability per unit time in the form3... 

Here (x)t and (x)f denote the mean values of the relative coordinate x over the states of the proton in the first and second potential wells, respectively. Equation (107) shows that the inertia effects lead to a decrease of the activation factor in the transition probability due to an increase of the reorganization energy. The greater the mass, m of the tunneling particle and the frequency of the vibrations of the atom, w0, the greater is this effect. The above result corresponds to the conclusion drawn in Ref. 66. 

These structural factors are relevant for the redox-activity of bis- and oligo-electrophoric systems and for the reorganization energy associated with intramolecular electron hopping. 

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