Marcus Theory - Inner Sphere Electron Transfer

Inner Sphere Electron Transfer

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 a configuration which is an identical one of as well the precursor as the successor complexes, of course highly distorted. In this case the energy requirement is fulfilled automatically.

In this inner sphere case the Arrhenius concept holds, the transition state of definite geometric structure is reached along a geometrical reaction coordinate determined by nuclear motions. 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.

For the self-exchange system Fe2+/Fe3+ only the symmetrical breathing vibration of the six water molecules around the iron ions is considered. Assuming harmonic conditions this vibration has frequencies and, the force constants f_D and f_A are and the energies are

where q₀ is the equilibrium normal coordinate and the displacement along the normal coordinate, the factor 3 stems from 6 (H₂O) ½. Like for the outer-sphere reorganization energy potential energy curve is quadratic, here, however, as a consequence of vibrations.

The equilibrium normal coordinates differ in Fe(H₂O)₆2+ and Fe(H₂O)₆3+. By thermal excitation of the breathing vibration a geometry can be reached which is common to both donor and acceptor, i.e. the potential energy curves of the breathing vibrations of D and A intersect here. This is the situation where the electron may jump. The energy of this transition state is the inner sphere reorganization energy _in.

For the self-exchange reaction the metal-water distance in the transition state can be calculated

This gives the inner sphere reorganisation energy

It is fortunate that the expressions for the energies for outer and inner reorganization have the same quadratic form. Inner sphere and outer sphere reorganization energies are independent, so they can be added to give and inserted in the Arrhenius equation

Here, A can be seen to represent the probability of electron jump, exp that of reaching the transition state of the inner sphere and exp that of outer sphere adjustment . For unsymmetrical (cross) reactions like

the expression for can also be derived, but it is more complicated. These reactions have a free reaction enthalpy G0 which is independent of the reorganization energy and determined by the different redox potentials of the iron and cobalt couple. Consequently the quadratic Marcus equation holds also for the inner sphere reorganization energy, including the prediction of an inverted region. One may visualizing this by (a) in the normal region both the initial state and the final state have to have stretched bonds, (b) In the G = 0 case the equilibrium configuration of the initial state is the stretched configuration of the final state, and (c) in the inverted region the initial state has compressed bonds whereas the final state has largely stretched bonds. Similar considerations hold for metal complexes where the ligands are larger than solvent molecules and also for ligand bridged polynuclear complexes.