Reorganization energy inner sphere

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 ... 

Table 5-6. Comparison of ADEs and VDEs, and inner sphere reorganization energies (A.oxi) (eV) for iron-sulfur protein, Rd...

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. 

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 ... 

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. 

The Marcus classical free energy of activation is AG , the adiabatic preexponential factor A may be taken from Eyring s Transition State Theory as (kg T /h), and Kel is a dimensionless transmission coefficient (0 < k l < 1) which includes the entire efiFect of electronic interactions between the donor and acceptor, and which becomes crucial at long range. With Kel set to unity the rate expression has only nuclear factors and in particular the inner sphere and outer sphere reorganization energies mentioned in the introduction are dominant parameters controlling AG and hence the rate. It is assumed here that the rate constant may be taken as a unimolecular rate constant, and if needed the associated bimolecular rate constant may be constructed by incorporation of diffusional processes as ... 

The exothermicity dependence is in p, (p = — AG°/hco) whereas the reorganization energy is expressed in s, (s = X,/ho)). This limit can be appropriate for the inner sphere reorganization whereas for the outer sphere reorganization one can assume very low frequencies and take the high temperature (classical) limit, obtaining... 

The essentials of quantum kinetics were in place by 1954, Weiss having added to the Gurney theory a comprehensive theory of redox reactions. By this date, tunneling, adiabatic and non-adiabatic electron transfer, the simplicity introduced by considering redox reactions between isotopes, the separate contribution from outer sphere and inner sphere, and in particular the equation for the reorganization energy involving and stat had all been published. 

This treatment is oversimplified because, in addition to neglecting inner sphere contributions to the reorganization energy it approximates the dielectric frequency spectrum to a single frequency, 0jo — 1011 s 1, corresponding to the Debye dielectric relaxation which probably varies in the vicinity of the ions. The cathodic current is given by... 

E, depends on the inner-sphere reorganization energy, E., the outer-sphere reorganization energy, E, on the enthalpyr4nanges,... 

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