Marcus theory cross reactions

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. 

Figure 1.13 Potential energy diagrams describing electron transfer processes according to Marcus theory. (A) Self-exchange (B) Cross Reaction.

One might ask how the Marcus case of crossing in the harmonic region can arise. In a sense, that is the surprising situation. How can significant reaction occur without reaching the anharmonic part of the potential To think of this, it is probably wise to remember that Marcus theory was first applied to electron transfer of the outer-sphere variety. Albery answers by pointing out that a reaction coordinate which is constructed from the intersection of parabolic surfaces for several... 

Within exact C2 symmetry, the two states may cross. The system can, however, deviate from this ideal symmetry to allow coupling between the two states so that the crossing is avoided. If we equate the 81 -> 82 excitation energy in 1 to the reorganization energy. A, of Marcus theory [11], we can use the Marcus-Hush expression [Eq. (2)] to estimate the activation energy, AE, based on the calculated values for A and the heat of reaction, AE —12kcalmol" ) ... 

From this brief description it is quite apparent that the qualitative elements of the Marcus treatment for an electron transfer process are identical to the CM model. In CM terms the reaction involves the avoided crossing of reactant (Fe2+ + Fe3+) with product (Fe3+ + Fe2+) configurations, with the reaction co-ordinate just being the distortion-relaxation motion of the solvation sphere. Thus in CM terms any electron transfer reaction involves the avoided crossing of the DA (donor-acceptor) and D+A" configurations, and for such reactions at least, based on the equivalence with Marcus theory, the CM model has a solid foundation. 

The theoretical results obtained for outer-sphere electron transfer based on self-exchange reactions provide the essential background for discussing the interplay between theory and experiment in a variety of electron transfer processes. The next topic considered is outer-sphere electron transfer for net reactions where AG O and application of the Marcus cross reaction equation for correlating experimental data. A consideration of reactions for which AG is highly favorable leads to some peculiar features and the concept of electron transfer in the inverted region and, also, excited state decay. 

In Fig. 7 we have taken a symmetrical reaction where, apart from the isotopic mixing, AG ° = 0. One of the first successes of the Marcus theory was the correlation of rates for such homogeneous reactions with the rates found for the same electron transfer taking place on an electrode (Marcus, 1963). The theory then went on to predict the rates of cross reactions between two different redox couples in terms of the kinetic and thermodynamic properties of the two redox couples. The free energy profile for an unsymmetrical cross reaction such as (17) is shown in Fig. 8. The free energy of activation depends... 

Fig. 9 Test of the Marcus theory of electron transfer where fcca,c for the cross-reaction O, + R - R, + On is calculated from the thermodynamic free energies and the free energies of activation of the symmetrical reactions. The symbols are as follows O, Ce(IV), x IrCl -, + Mo(CN)j-, Fe(CN) ", R O, Fe(CN)J , A Mo(CN)f, W(CN)<-...

We now explore whether the pattern of reactivity predicted by the Marcus theory is found for methyl transfer reactions in water. We use equation (29) to calculate values of G from the experimental data where, from (27), G = j(JGlx + AG Y). The values of G should then be made up of a contribution from the symmetrical reaction for the nucleophile X and for the leaving group Y. We then examine whether the values of G 29) calculated for the cross reactions from (29) agree with the values of G(27) calculated from (27) using a set of values for the symmetrical reactions. The problem is similar to the proof of Kohlrausch s law of limiting ionic conductances. 

In a previous paper [22], it has been shown that the reaction coordinate is well defined only in the Marcus theory, that is in the classical regime. In this case, the reaction coordinate is along the minimum crossing point in the multi-dimensional potential surfaces (i.e., multi-mode case). In the quantum regime, the reaction coordinate is not well defined especially in the case of the non-relaxed ET like Wm. 

Thus the Marcus theory gives rise to a free energy relationship of a type similar to those commonly used in physical organic chemistry. It can be transformed into other relationships (see below) which can easily be subjected to experimental tests. Foremost among these are the remarkably simple relationships that were developed (Marcus, 1963) for what have been denoted cross reactions. All non-bonded electron-transfer processes between two different species can actually be formulated as cross reactions of two self-exchange reactions. Thus the cross reaction of (59) and (60) is (61), and, neglecting a small electrostatic effect, the relationship between kn, k22 and kl2... 

It was recently shown (Ratner and Levine, 1980) that the Marcus cross-relation (62) can be derived rigorously for the case that / = 1 by a thermodynamic treatment without postulating any microscopic model of the activation process. The only assumptions made were (1) the activation process for each species is independent of its reaction partner, and (2) the activated states of the participating species (A, [A-], B and [B ]+) are the same for the self-exchange reactions and for the cross reaction. Note that the following assumptions need not be made (3) applicability of the Franck-Condon principle, (4) validity of the transition-state theory, (5) parabolic potential energy curves, (6) solvent as a dielectric continuum and (7) electron transfer is... 

Marcus theory showed a good correlation between experimental and calculated rate constants using Eq. (5). The 22 value was set at 10 M sec for this purpose and is considered as an upper limit for selfexchange of the diethyldithiocarbamate radical/anion pair. From the oxidation rates it was also estimated that E (edtc /edtc ) = 0.43(3)V vs SCE. A free-energy analysis for the oxidation of diethyldithiocarbamate (edtc ) by [FeiCNlgT also showed that the initial outer-sphere oxidation of the thiolate anion to its thio radical (Eq. 36) is the main energy barrier to be crossed along the reaction coordinate. 

The electron exchange rate constant of the iron(III) complex in DMSO was estimated from the cross reactions with hydroquinone and catechol, which was compared with the rate constant obtained electrochemically. The mechanism of the ascorbic acid oxidation reaction in DMSO is discussed based on the Marcus theory. 

The Marcus-Hush theory proved to be most advantage in describing of a cross-reactions ... 

Marcus theory has been so successful in predicting cross reaction rates in outer-sphere electron transfer reactions that it is often used to give evidence that an outer-sphere reaction is taking place. If the observed rate constant is within an order of magnitude of the calculated rate constant, then it is likely that an outer-sphere mechanism is occurring. In this way, you will use Marcus theory to give evidence for the reaction mechanism assigned to your rate constant, kos, determined in Experiment 5.5. 

Another widely used result of Marcus theory deals with the extraction of useful kinetic relationships for cross reactions from parameters for self-exchange reactions. Consider the cross reaction, Equation (6.22), for which the rate... 

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