Nuclear reorganization, Marcus theory

In the classical theory of Marcus, the rate determining factors involve nuclear reorganization. We write the first-order rate constant, ke, as [37]... 

Marcus theory predicts that the nuclear factor in the electron transfer rate expression will be maximal when - AG° = X. Under these conditions, the electron transfer process will be temperature independent. The first two electron transfer steps in the RC approximately exhibit this behavior. The rate of the D 4>a step actually increases slightly (by a factor of 2-4) as the temperature decreases from 300 to 8 K (180). Consequently, X may be estimated to be in the range 0.3 to 0.5 V (—7-10 kcal/mol) from the A ° values for these two steps (Table IV). These values are approximately 4 times smaller than those observed for ruthenated proteins discussed previously. Sequestration of the redox groups in a membrane-bound protein complex, away from aqueous solution, may serve to decrease the value of X by minimizing the reorganization energy of a highly polar solvent. 

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 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. 

In the classical Marcus-Hush theory, the initial nuclear geometry of the reactant state undergoes reorganization to the transition state prior to electron transfer [30, 31, 39]. The energy of the transition state, AGe, is gained by intermolecular collisions, in order to satisfy conservation of energy and momentum. The nuclear factor is related to the activation energy, i.e.,... 

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