Theory Marcus

The contemporary theory of the electron transfer reaction was proposed by Rudolph Marcus. The theory is based to a large extent on the harmonic approximation for the diabatic potentials involved, i.e. the diabatic curves represent parabolas. One of the parabolas corresponds to the reactants VR(q), the other to the products Vp(q) of the electron transfer reaction (Fig. 14.22). Now, lei us assume that both parabolas have the same curvature (force constant f). The reactants correspond to the parabola with the minimum at qp (without loosing generality we adopt a convention that at = 0 the energy is equal zero) 

Rudolph Arthur Marcus (b. 1923), American chemist, professor at the University of iiii-nois in Urbana and at Caii-fornia institute of Technoiogy in Pasadena, in 1992 Marcus received the Nobei Prize for his contribution to the theory of electron transfer reactions in chemical systems . 

So far we just treat the quantity AG as a potential energy difjerence Vp(qp) — Vplqn) of the model system under cotisideration (H + H2 or the pendulum HF), even though the symbol suggests that this interpretation will be generalized in the future. 

Such parabolas represent a simple situation. The parabolas intersection point qc satisfies by definition Vg(qc) = Vp(qc)- This gives 

Of course on the parabola diagram, the two minima are the most important, the intersection point qc and the corresponding energy, which represents the reaction barrier reactants products. 

From a kinetic viewpoint, electron transfer processes involving excited states, as well as those involving ground state molecules, can be dealt with in the frame of the Marcus theory [4] and of the successive, more sophisticated theoretical models [5]-The only difference between electron transfer processes involving excited state instead of ground state molecules is that in the first case, in the calculation of the free energy change, the redox potential of the excited state couple has to be used (2.6 and 2.7). 

In an absolute rate formalism (Marcus model [4]), potential energy curves of an electron transfer reaction for the initial (i) and final (f) states of the system are represented by parabolic functions (Fig. 2.4). The rate constant for an electron transfer process can be expressed as 

Equations 2.19 and 2.20 predict that for a homogeneous series of reactions (i.e. for reactions having the same X and feei values) a In fcei versus AG plot is a bellshaped curve (Fig. 2.5, solid line) involving  

The reorganizational energy X can be expressed as the sum of two independent contributions corresponding to the reorganization of the inner (bond lengths and angles within the two reaction partners) and outer (solvent reorientation around the reacting pair) nuclear modes  

The outer reorganizational energy, which is often the predominant term in electron transfer processes, can be estimated, to a first approximation, by the 

A special case of a non-adiabatic reaction is electron transfer. The dynamics of electron-transfer processes have been studied extensively, and the most robust model used to describe 

For this simple case, Marcus theory predicts the rate constant for electron transfer to be 

The reorganization energy term derives from the solvent being unable to reorient on the same timescale as the electron transfer takes place. Thus, at the instant of transfer, the bulk dielectric portion of the solvent reaction field is oriented to solvate charge on species A, and not B, and over the course of the electron transfer only the optical part of the solvent reaction field can relax to the change in tire position of the charge (see Section 14.6). If the Bom formula (Eq. (11.12)) is used to compute the solvation free energies of the various equilibrium and non-equilibrium species involved, one finds that 

A key point that must be made is diat quantum mechanical tunneling through the Marcus-theory barrier when it is non-zero can increase the rate for electron transfer just as is true for any other activated process. Because the electron is so light a particle, tunneling can be a major contributor to die overall rate. Models for electron tunneling will not, however, be presented here. 

The intersection of two parabolas representing Morse potentials defines the shape of the reaction coordinate. 

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Klippenstein S J 1992 Variational optimizations in the Rice-Ramsperger-Kassel-Marcus theory calculations for unimolecular dissociations with no reverse barrier J. Chem. Rhys. 96 367-71... 

Eig. 2. Electron-transfer reaction rate, vs exoergicity of reaction the dashed line is according to simple Marcus theory the soUd line and data poiats are... 

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

This discussion of sources of curvature in Br insted-type plots should suggest caution in the interpretation of observed curvature. There is a related matter, concerning particularly item 5 in this list, namely, the effect of a change in transition state structure. Br nsted-type plots are sometimes linear over quite remarkable ranges, of the order 10 pK units, and this linearity has evoked interest because it seems to be incompatible with Marcus theory, which we reviewed in Section 5.3. The Marcus equation (Eq. 5-69) for the plot of log k against log K of the same reaction series requires curvature, the slope of the plot being the coefficient a. given by Eq. (5-67). A Brjinsted plot, however, is not a Marcus plot, because it correlates rates and equilibria of different reactions. The slope p of a Br nsted plot is defined p = d log kobs/d pK, which we can expand as... 

As a consequence, it appears to be valid to apply Marcus theory (Section 5.3) to Sn2 reactions. Note that we may expect structure-reactivity relationships in Sn2 reactions to be functions of both the bond formation and bond cleavage processes, just as in acyl transfers. 

This is the reverse of the first step in the SnI mechanism. As written here, this reaction is called cation-anion recombination, or an electrophile-nucleophile reaction. This type of reaction lacks the symmetry of a group transfer reaction, and we should therefore not expect Marcus theory to be applicable, as Ritchie et al. have emphasized. Nevertheless, the electrophile-nucleophile reaction possesses the simplifying feature that bond formation occurs in the absence of bond cleavage. 

Other measures of nucleophilicity have been proposed. Brauman et al. studied Sn2 reactions in the gas phase and applied Marcus theory to obtain the intrinsic barriers of identity reactions. These quantities were interpreted as intrinsic nucleo-philicities. Streitwieser has shown that the reactivity of anionic nucleophiles toward methyl iodide in dimethylformamide (DMF) is correlated with the overall heat of reaction in the gas phase he concludes that bond strength and electron affinity are the important factors controlling nucleophilicity. The dominant role of the solvent in controlling nucleophilicity was shown by Parker, who found solvent effects on nucleophilic reactivity of many orders of magnitude. For example, most anions are more nucleophilic in DMF than in methanol by factors as large as 10, because they are less effectively shielded by solvation in the aprotic solvent. Liotta et al. have measured rates of substitution by anionic nucleophiles in acetonitrile solution containing a crown ether, which forms an inclusion complex with the cation (K ) of the nucleophile. These rates correlate with gas phase rates of the same nucleophiles, which, in this crown ether-acetonitrile system, are considered to be naked anions. The solvation of anionic nucleophiles is treated in Section 8.3. 

THE BELL-EVANS-POLANYIPRINCIPLE/HAMMOND POSTULATE/MARCUS THEORY... 

According to the Marcus theory [64] for outer-sphere reactions, there is good correlation between the heterogeneous (electrode) and homogeneous (solution) rate constants. This is the theoretical basis for the proposed use of hydrated-electron rate constants (ke) as a criterion for the reactivity of an electrolyte component towards lithium or any electrode at lithium potential. Table 1 shows rate-constant values for selected materials that are relevant to SE1 formation and to lithium batteries. Although many important materials are missing (such as PC, EC, diethyl carbonate (DEC), LiPF6, etc.), much can be learned from a careful study of this table (and its sources). 

Magnesium, arylation 273 Marcus theory, application to dediazonia-tion 197... 

Applications of the Marcus theory are by no means limited to electron transfer. The generality of this approach has been cited.38... 

Marcus theory. Show that a key result is that the reaction rate constant is... 

Marcus theory. Consider that the reorganization energy for the ET reaction, AAb, can be approximated as the mean of the reorganization energies for the EE reactions Aab = (Aaa + ABb)/2. Show that substitution of this expression into Eq. (10-63) gives the usual form of the Marcus cross relation. 

Marcus theory. Prove the point that A = 4AGJ by making use of the analytic expressions for the equation of a parabola. The two equations should be those that describe the curves on the left side of Fig. 10-11. 

Marcus cross relation, 243-246 Marcus theory, 239-248 Mean lifetime (see Lifetime)... 

It is often useful to compare the reactivity of one compound with that of similar compounds. What we would like to do is to find out how a reaction coordinate (and in particular the transition state) changes when one reactant molecule is replaced by a similar molecule. Marcus theory is a method for doing this. ... 

Marcus theory can be applied to any single-step process where something is transferred from one particle to another. It was originally derived for electron... 

In order to find the relation between Eq. (122) and the Marcus theory, we employ the linear response approximation. In this case, the free energies Fj )(i =1)2) for the donor and acceptor become a parabolic function of as... 

Samec Z, Weber J (1973) The influence of chemisorbed sulfur on the kinetic parameters of the reduction of Fe " ions on a platinum electrode on the basis of the Marcus theory of electron transfer. J Electroanal Chem Interfacial Electrochem 44 229-238... 

More recently, a value of 3x 10 l.mole . sec has been calculated for the exchange rate coefficient at 10 °C and zero ionic strength by Campion et al. using the Marcus theory and rate coefficients for the reactions... 

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