Marcus formula

In the strong-coupling limit at high temperatures the electron transfer rate constant is given by the Marcus formula [Marcus 1964]... 

The symmetry coefficient = —P d nk/dAE is usually close to j, in agreement with the Marcus formula. Turning to the quantum limit, one observes that the barrier transparency increases with increasing AE as a result of barrier lowering, as well as of a decrease of its width. Therefore, k grows faster than the Arrhenius rate constant. At 7 = 0... 

The ZN formulas can also be utihzed to formulate a theory for the direct evaluation of thermal rate constant of electronically nonadiabatic chemical reactions based on the idea of transition state theory [27]. This formulation can be further utilized to formulate a theory of electron transfer and an improvement of the celebrated Marcus formula can be done [28]. 

The factor k takes into acount the effects of nonadiabatic transition and tunneling properly. Also note that the electronic coupling //ad is assumed to be constant in the Marcus formula, but this is not necessary in the present formulation. The coupling Had cancels out in k of Eq. (126) and the ZN probability can be calculated from the information of adiabatic potentials. 

In the classical limit when the thermal energy K T is much higher than the energy ha of the vibrational frequencies that are coupled to the electron transfer reaction, the Franck-Condon factor can be expressed in terms of AG and X and equation 2 converts to the classical Marcus formula for the electron transfer rate ... 

In certain cases, the classical Marcus formula is not sufficient to explain the observed-dependence of the electron transfer rate on temperature or AG, which could indicate that it is necessary to use a Franck-Condon term in which the contribution of the nuclei is treated in quantum mechanical terms. In this treatment, the Franck-Condon term equals the thermally-weighted sum of the contributions from all possible vibrational states of the reactants, each multiplied by their Franck-Condon factor i.e. the square of the overlap integral of a nuclear wave function of the reactant with the nuclear wave function of the product state that has the same total energy. 

Some difficulties in comparing the experimental kinetic data with the outer-sphere reorganization energy calculated from the Marcus formula (28) result from several assumptions made in this theory. The reactant was assumed to have a spherical shape with a symmetric charge distribution. No field penetration into the metal was considered. Also, the spatial dispersion of the dielectric permittivity of the medium was not taken into account. In fact, the positions and orientations of dipoles around a given ion are correlated with each other therefore the reorientation of one dipole, under the influence of the external field, changes to some extent the reorientation of other dipoles within the distance defined by the correlation length. 

As a result, from all three parameters AG°, A and Vad in the Marcus formula (46), only A is strongly dependent on the electrolyte solution. Within this dependence the solvent and ion contributions can be separated,... 

The approach used to obtain the EVB free-energy functionals (the Ag of Equation (7)) has been originally developed in Ref. 25 in order to provide the microscopic equivalent of the Marcus theory for electron transfer (ET) reactions.38 This approach allows one to explore the validity of the Marcus formula and the underlying linear response approximation (LRA) on a microscopic molecular level.39 While this point is now widely accepted by the ET community,40 the validity of the EVB as perhaps the most general tool in microscopic LFER studies is less appreciated. This issue will be addressed below. 

This work addressed the issue of enzyme catalysis focusing on the principle of physical organic chemistry and the power of computer simulation approaches. It was shown that when such concepts as reorganization energy and Marcus parabolas are formulated in a consistent microscopic way, they could be used to explore the origin of enzyme catalysis. It was also clarified that phenomenological applications of the Marcus formula or related expressions can lead to problematic conclusions. 

In the Marcus formula, reorganization energy plays an important role. This energy is the main reason for the electron-transfer reaction barrier. 

Marcus Formula The electron-transfer reaction barrier is calculated as ... 

Nelsen and co-workers measured the ET rates within the 2,7-dinitro-naphthalene anion radical in different solvents and noted that the solvent dynamic effect was not important. They thus tried to use the Marcus formula or the BJ theory in the perturbation limit to explain their experimental results. However, both cannot explain the experimental results correctly because the electronic coupling is not weak enough. Since the solvent dynamics are fast, here we can use the rate expression (eqn (12.17)) with ZN transition probability. Indeed, the predicted rates are in excellent agreement with the experiment. The results are shown in Figure 12.6. In the calculations, the reorganization energy and electronic coupling are estimated by Nelsen s measurement. 

We have presented several approaches to calculate the rate constants of electron transfer occurring in solvent from the weak to strong electronic couplings. In the fast solvent relaxation limit, the approach based on the nonadiabatic transition state theory can be adopted. It is related to the Marcus formula by a prefactor and referred as a modified Marcus formula. When the solvent dynamics begin to play a role, the quantum Kramers-like theory is applicable. For the case where the intramoleeular vibrational motions are much faster than the solvent motion, the extended Sumi-Mareus theory is a better ehoice. As the coherent motion of eleetron is ineorporated, such as in the organic semiconductors, the time-dependent wavepaeket diffusion approach is proposed. Several applications show that the proposed approaches, together with electronic structure calculations for the faetors eontrolling eleetron transfer, can be used to theoretically predict electron transfer rates correctly. 

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