Kubo and Toyozawa

Another approach to ET reactions originated in the work of Weiss and Libby, who suggested that activation energy for ET reactions in solution does not arise from the collisional-vibrational or electrostatic interaction of the solvent in the first and second solvation sphere, but rather from continuum solvent polarization fluctuation far out in solution. This approach, called continuum theory, was further developed by Kubo and Toyozawa, " Marcus, Platzmann and Erank, and by Levich and Dogo-nadze. ... 

Using the harmonic and the Franck-Condon approximations, it is possible to advance in calculating the sums of the (18) type using the method of generating functions developed by Kubo and Toyozawa [8, 9] for the calculation of the probabilities of optical and non-radiative transitions in the impurity centres in crystals. According to this method we can rewrite eqn. (18) as... 

But if we examine the localized near the donor or the acceptor crystal vibrations or intra-molecular vibrations, the electron transition may induce much larger changes in such modes. It may be the substantial shifts of the equilibrium positions, the frequencies, or at last, the change of the set of normal modes due to violation of the space structure of the centers. The local vibrations at electron transitions between the atomic centers in the polar medium are the oscillations of the rigid solvation spheres near the centers. Such vibrations are denoted by the inner-sphere vibrations in contrast to the outer-sphere vibrations of the medium. The expressions for the rate constant cited above are based on the smallness of the shift of the equilibrium position or the frequency in each mode (see Eqs. (11) and (13)). They may be useless for the case of local vibrations that are, as a rule, high-frequency ones. The general formal approach to the description of the electron transitions in such systems based on the method of density function was developed by Kubo and Toyozawa [7] within the bounds of the conception of the harmonic vibrations in the initial and final states. 

An important achievement of the early theories was the derivation of the exact quantum mechanical expression for the ET rate in the Fermi Golden Rule limit in the linear response regime by Kubo and Toyozawa [4b], Levich and co-workers [20a] and by Ovchinnikov and Ovchinnikova [21], in terms of the dielectric spectral density of the solvent and intramolecular vibrational modes of donor and acceptor complexes. The solvent model was improved to take into account time and space correlation of the polarization fluctuations [20,21]. The importance of high-frequency intramolecular vibrations was fully recognized by Dogonadze and Kuznetsov [22], Efrima and Bixon [23], and by Jortner and co-workers [24,25] and Ulstrup [26]. It was shown that the main role of quantum modes is to effectively reduce the activation energy and thus to increase the reaction rate in the inverted... 

In 1954, Platzmann and Frank indicated the possibility of using the so-called radiationless theory of transitions developed by Lax for polyatomic molecules, and by Pekar for polar crystals, to the process involving charge transfer in liquids. The most general method in the theory of the radiationless transitions was suggested by Kubo and Toyozawa in 1955. Subsequently it was used in many other works. The first calculations for processes in polar liquids in the framework of the polaron theory were performed by Davydov and Deygen, " who investigated the properties of metal-ammonia solutions. 

Concluding this brief survey, we may note that most of the papers on this aspect of chemical kinetics derive from the further development of the theory of radiational and radiationless transitions. The most general approach is the density matrix method developed in recent years which is a generalization of the method of Kubo and Toyozawa. ... 

Assumptions (a), (c), and (d) are standard in the literature on multiphonon processes (Englman and Jortner, 1970 Kubo and Toyozawa, 1955 Lin, 1966). The usual criterion for the validity of the adiabatic approximation is that the electronic energy difference is large compared with the energy of the phonons which mediate the transition, i.e., (e - ) >, and this same criterion also... 

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

R. Kubo and Y. Toyozawa, Prog. Theorel. Phys. 130 411 (1955). Early formation of reorganization energy in electron transfer. 

Y. Toyozawa, Dynamical Processes in Solid State Optics, R. Kubo and H. Kamimura, Eds., Syokabo Tokyo, 1967, p. 90. 

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

Y. Toyozawa, in Solid State Sciences. Vol. 18 Relaxation of Elementary Excitations, R. Kubo and E. Hanomura, Eds., Springer, Berlin, 1980, p. 3. 

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