Marcus formula theory

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]. 

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. 

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. 

To use the master equation, one needs a general formula for the rate constant, kj, out of minimum j through transition state f. In the micro-canonical ensemble this relation is provided by Rice-Ramsperger-Kassel-Marcus (RRKM) theory [166] ... 

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. 

LAUNAY - Not exactly. The use of 1/n in the Marcus formula comes from the theory of non equilibrium polarization of the solvent considered as a dielectric continuum. One may think that the contribution of the solvent to the activation energy is broken in two terms a term due to orientational polarization and a term due to electronic polarization. Only the first term is kept, because it is slow and the corresponding rearrangement must occur before electron transfer (cf the case of the first coordination sphere). The second term is deleted because it is fast and thus can occur during electron transfer. Since at optical frequencies, only the electronic polarization can respond, this term 1/n = op separation between the fast... 

Let us now consider the results of theoretical calculation of the reorganization energy. Marcus formulas (3.14) and (3.16) represent the simplest version of the theory. For a reaction involving two ions of the same radius a, which are in direct contact (R = 2a), as well as for the electrode reaction of an ion which approaches the electrode until they are in direct contact (R = a), these formulas can be simplified and lead to the following relations ... 

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 simplest model of a single-channel transfer (without excitation of reaction products), the rate obtained with the perturbation theory of R. Marcus is given by the following formula [32,79] ... 

Marcus inverted region, 284. 286 Marcus theory. 284-86 Markovnikov and anti-Markovnikov, 468 Mataga-Nishimoto formula, 53, 55 MCD spectrum, acenaphthylene, 157-58, 169... 

From these calculations, under application of the Marcus theory, which leads to the formula... 

For route 2, charge transport mechanism is similar to the hopping mechanism of small molecules. According to the charge transport theory developed hy Bredas et al. [22], the charge carrier transport in organic materials can be described by Marcus electron transfer theory (Eq. 1.3). In organic ciystals, the AG is 0 because the electron transfer happens in a same kind of molecules. Thus the Eq. 1.3 formula can be simplified into Eq. 1.4 as follows ... 

ET rate via electronic coupling for a multi-dimensional system in the Marcus inverted regime, (a) pEa—6.7, (b) pSa = 10.0, and (c) pEa = 20.0. Ea represents the minimum energy on the seam surface. Solid line present result dashed line the results predicted from the LZ formula dotted line results from perturbation theory. 

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