Marcus theory

The original classical Marcus theory for outer sphere electron transfer reactions demonstrates the importance of the solvent and leads the way to the calculation of the Gibbs free energy of activation, using the polarization properties of the solvent, the size of the reactants, the transfer distance and the Gibbs free energy

Scientists searched the inverted region for proof of a slower electron transfer rate for 30 years until it was unequivocally verified experimentally in 1984.

Besides the inner and outer sphere applications, Marcus theory has been extended to address heterogeneous electron transfer.

The total reaction may be diffusion controlled (the electron transfer step is faster than diffusion, every encounter leads to reaction) or activation controlled (the "equilibrium of association" is reached, the electron transfer step is slow, the separation of the successor complex is fast).

In outer sphere redox reactions no bonds are formed or broken; only an electron transfer (ET) takes place.

Moreover, in Eyring's TST[4][5] a quite specific change of the nuclear coordinates is responsible for crossing the maximum point, a vibration in this direction is consequently treated as a translation.

The system of the reactants becomes coupled so tightly during the reaction that they form the activated complex as an integral entity.

By contrast, in outer sphere redox reactions the displacement of nuclei in the reactants are small, here the solvent has the dominant role.

For the self-exchange reaction for symmetry reasons an arrangement of the solvent molecules exactly in the middle of those of precursor and successor complex would meet the conditions.

This means that the solvent arrangement with half of the electron on both donor and acceptor would be the correct environment for jumping.

However, the electron as an elementary particle cannot be divided, it resides either on the donor or the acceptor and arranges the solvent molecules accordingly in an equilibrium.

The "transition state", on the other hand, requires a solvent configuration which would result from the transfer of half an electron, which is impossible.

Yet it is possible that the solvent takes a configuration corresponding to the "transition state", even if the electron sits on the donor or acceptor.

The creation of the correct solvent arrangement and the electron jump are decoupled and do not happen in a synchronous process.

On the basis of his reasoning R.A. Marcus developed a classical theory with the aim of calculating the polarization energy of the said non-equilibrium state.

Marcus was successful in finding such a path via two reversible charging steps for the preparation of the "transition state" from the precursor complex.

In a second step the energy WII of the reversible (back) transfer of the charge to the first sphere, again via the vacuum, is calculated.

Shrinking the two-sphere model to the molecular level creates the problem that in the self-exchange reaction the charge can no longer be transferred in arbitrary amounts, but only as a single electron.

In the outer sphere model the donor or acceptor and the tightly bound solvation shells or the complex' ligands were considered to form rigid structures which do not change in the course of electron transfer.

However, the distances in the inner sphere are dependent on the charge of donor and acceptor, e.g. the central ion-ligand distances are different in complexes carrying different charges and again the Franck–Condon principle must be obeyed: for the electron to jump to occur, the nuclei have to have an identical configuration to both the precursor and the successor complexes, of course highly distorted.

No further nuclear motion is necessary to form the successor complex, just the electron jumps, which makes a difference to the TST theory.

The reaction coordinate for inner sphere energy is governed by vibrations and they differ in the oxidized and reduces species.

[12] For the self-exchange system Fe2+/Fe3+ only the symmetrical breathing vibration of the six water molecules around the iron ions is considered.

Consequently, the quadratic Marcus equation holds also for the inner sphere reorganization energy, including the prediction of an inverted region.

In the adiabatic case the coupling is considerable, the gap of 2 HAB is larger and the system stays on the lower potential energy curve.

Thus Marcus's theory builds on the traditional Arrhenius equation for the rates of chemical reactions in two ways: 1.

But all experiments with series of reactions of more and more negative ΔG0 revealed only an increase of the reaction rate up to the diffusion limit, i.e. to a value indicating that every encounter lead to electron transfer, and that limit held also for very negative ΔG0 values (Rehm-Weller behaviour).

[15] It took about 30 years until the inverted region was unequivocally substantiated by Miller, Calcaterra and Closs for an intramolecular electron transfer in a molecule where donor and acceptor are kept at a constant distance by means of a stiff spacer (Fig.4).

There are, however, other concepts for the phenomenon,[1] e.g. the participation of excited states or that the decrease of the rate constants would be so far in the inverted region that it escapes measurement.

They have included inter alia statistical aspects and quantum effects,[17] they have applied the theory to chemiluminescence[18] and electrode reactions.

Fig. 1. The parabolas of outer-sphere reorganisation energy of the system two spheres in a solvent. Parabola i: the charge on the first, transfer to the second, parabola f: the charge on the second, transfer to the first. The abscissa is the transferred amount of charge Δe or the induced polarization P, the ordinate the Gibbs free energy. ΔG(0) ‡ = λ o /4 is the reorganization energy at Δe = 0.5, it corresponds to the activation energy of the self-exchange reaction.
Fig. 2 Marcus-Parabolas for different redox reactions: f 1 for positive , for the self-exchange reaction with (broken line), for moderately negative with and for strongly negative . The free energy of activation decreases from ( ) via (a) to (zero) and increases again for ("Marcus inverted region").
Energy diagram
Fig. 3 Energy diagram for Electron Transfer including inner and outer sphere reorganization and electronic coupling: The vertical axis is the free energy, and the horizontal axis is the "reaction coordinate" – a simplified axis representing the motion of all the atomic nuclei (including solvent reorganization)
Fig.4. Marcus behaviour in a molecule, which is composed of a biphenyl entity, whose anion (produced by means of pulse radiolysis) acts as a donor, a steroid entity, which is a rigid spacer and different aromatic hydrocarbons and quinones, which are the acceptors (A).