Marcus formulation

In the context of the Marcus formulation, the lowering of the activation barrier in an inner-sphere process could arise from the reduction of the work term wp as a result of the strong interaction in the ionic products, e.g., [RitSn+ IrCU3 ] and [RitSn+TCNE ]. The electrostatic potential in such an ion pair is attractive and may cause the tetraalkyltin to achieve a quasi five-coordinate configuration in the precursor complex, reminiscent of a variety of trigonal bipyramidal structures already well-known for tin(IV) derivatives, i.e.,... 

One difficulty that arises within the Marcus formulation is that the intrinsic barrier term is ideally treated as a constant. Earlier applications of Marcus theory were based on this assumption (Cohen and Marcus, 1968 Kreevoy and Konasewich, 1970 Kreevoy and Oh, 1973 Albery et al., 1972). However, as regards methyl transfer, this assumption is clearly invalid. 

Therefore, unlike the empirical Butler-Volmer theory, in the Marcus formulation the heterogeneous electron transfer rate constant is sensitive to both the structure of the redox center and the solvent. 

In the view of many, the theoretical contributions of Marcus have earned a special place in the electron transfer field [132,133]. Over three decades ago, Marcus formulated the rate of an electron transfer reaction as a function of two parameters a) its driving force, i.e. the free energy, AG°, of the reaction, and b) a solvent... 

The Marcus formulation relates the new well depth (AE,) to the depth in the unperturbed, (that is, symmetric) system, AE, and the change in basicity of B, 5PA ... 

It should be stated parenthetically that the Marcus formulation is not the only one that could in principle reproduce the patterns of H-bond energy changes precipitated by proton affinity changes. There are other extant theories - that differ from the Marcus equation chiefly in the last term, a rather unimportant one in many cases as the relationship is very nearly linear anyway. 

One can rationalize a simple relationship between the strength of an ionic H-bond (A—H B) on one hand and the difference in proton affinity between the two partners A and E on the other. The Marcus formulation provides a convenient framework for predicting the El-bond energy of an arbitrary system based on knowledge of the interaction energy in a symmetric system (A=B), and the difference in proton affinity between the two partners. This model has proven successful in a series of interoxygen and internitrogen H-bonds. 

Here we have treated the nuclear degrees of freedom classically as in the Marcus formulation [1]. 

Numerically, it is now a common practice to calculate within the dielectric continuum formulation but employing cavities of realistic molecular shape determined by the van der Waals surface of the solute. The method is based upon finite-difference solution of the Poisson-Boltzmann equation for the electrostatic potential with the appropriate boundary conditions [214, 238, 239]. An important outcome of such studies is that even in complex systems there exists a strong linear correlation between the calculated outer-sphere reorganization energy and the inverse donor-acceptor distance, as anticipated by the Marcus formulation (see Fig. 9.6). More... 

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 quantum mechanical formulation of electron transfer (Atkins, 1984 Closs et al, 1986) as a radiationless transition, the rate of ET is described as the product of the electronic coupling term J2 and the Frank-Condon factor FC, which is weighted with the Boltzmann population of the vibrational energy levels. But Marcus and Sutin (1985) have pointed out that, in the high-temperature limit, this treatment yields the semiclassical expression (9). 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

Marcus developed a quantum mechanical formulation of Kassel-Rice-Ramsperger theories in which zero point energies have been taken into account (see flow chart). However, due to lack of data for individual molecules it is difficult to apply the theory of Rice-Ramsperger-Kassel-Marcus (RRKM)... 

In Marcus original formulation of ET theory, the free energy curves Gj and Gj are assumed to be quadratic in x (linear response approximation). Using this assumption, Marcus derives the relationship between the activation free energy and the reaction free energy... 

Occasionally, the successful application of the Marcus expressions (5.35) and (5.37) to a reaction can support its designation as outer-sphere. The reduction of a series of substituted benzenediazonium salts by Fe(CN)5 and (Me5cp)2pe conforms to the simple Marcus expression and represents supporting evidence for the formulation of these reactions as outer sphere (or non-bonded electron transfer in organic systems)... 

In the 2-level limit a perturbative approach has been used in two famous problems the Marcus model in chemistry and the small polaron model in physics. Both models describe hopping of an electron that drags the polarization cloud that it is formed because of its electrostatic coupling to the enviromnent. This enviromnent is the solvent in the Marcus model and the crystal vibrations (phonons) in the small polaron problem. The details of the coupling and of the polarization are different in these problems, but the Hamiltonian formulation is very similar. ... 

In the absence of dynamic and static disorder, all partially filled band systems would exhibit coherent transport over long distances. With static and dynamic disorder, the modulation of the simple molecular orbital or band structure by nuclear effects entirely dominates transport. This is clear both in the Kubo linear response formulation of conductivity and in the Marcus-Hush-Jortner formulation of ET rates. The DNA systems are remarkable for the different kinds of disorder they exhibit in addition to the ordinary static and dynamic disorder expected in any soft material, DNA has the covalent disorder arising from the choice of A, T, G, or C at each substitution base site along the backbone. Additionally, DNA has the characteristic orientational and metric (helicoidal) disorder parameters arising from the fundamental motif of electron motion along the r-stack. 

The electrical contact of redox proteins is one of the most fundamental concepts of bioelectronics. Redox proteins usually lack direct electrical communication with electrodes. This can be explained by the Marcus theory16 that formulates the electron transfer (ET) rate, ket, between a donor-acceptor pair (Eq. 12.1), where d0 and d are the van der Waals and actual distances separating the donor-acceptor pair, respectively, and AG° and X correspond to the free energy change and the reorganization enery accompanying the electron transfer process, respectively. 

Marcus stressed that only harmonic modes U = were involved in the ion-solvent interactions and went further than Weiss in formulating a simple equation for the rate of adiabatic electron transfer, taking the case of an isotopic reaction so that the AG° term was eliminated. Under this condition and using Eq. (9.32), the current density (or electrochemical reaction rate) at a given overpotential t], in the cathodic direction (T] is negative) is... 

V. Levich and R. R. Dogonadze, Dokl. Akad. Nauk. SSSR 124 123 (1959). Hamiltonian formulation for electron transfer dielectric polarization approach. Quantum aspects of Weiss-Marcus model developed. 

To place experimental studies in context, it is helpful to recall Marcus semiclassi-cal formulation of ET reaction rate theory. Briefly, for an ET reaction involving weak electronic interactions, the first-order rate constant can be written as [19,20]... 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

The second article deals with probably the most fascinating predictions of modem electron transfer theories i.e. the Marcus Inverted Region (M.I.R.) . It was shown only one decade ago, nearly 20 years after the first formulation of the Marcus theory, that the M.I.R. does indeed exist First for thermal charge shifts and later for charge recombination. Even a charge separation reaction was recently found to behave according to the Marcus theory. Nevertheless, many reactions do not follow the Marcus model and therefore the second contribution of this issue is mainly concerned with this question. 

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