text: 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. However, the polarization still is determined by the total ensemble of the solvent molecules and therefore can still be treated classically, i.e. the polarization energy is not subject to quantum limitations. Therefore, the energy of solvent reorganization can be calculated as being due to a hypothetical transfer and back transfer of a partial elementary charge according to the Marcus formula. Thus the reorganization energy for chemical redox reactions, which is a Gibbs free energy, is also a parabolic function of &Delta;e of this hypothetical transfer, For the self exchange reaction, where for symmetry reasons &Delta;e = 0.5, the Gibbs free energy of activation is &Delta;G(0) = &lambda;/4 (see Fig. 1 and Fig. 2 intersection of the parabolas I and f, f(0), respectively).
  Up to now all was physics, now some chemistry enters. The self exchange reaction is a very specific redox reaction, most of the redox reactions are between different partners e.g.
  and they have positive (endergonic) or negative (exergonic) Gibbs free energies of reaction .
  As Marcus calculations refer exclusively to the electrostatic properties in the solvent (outer sphere)  and  are independent of one another and therefore can just be added up. This means that the Marcus parabolas in systems with different  are shifted just up or down in the  vs.  diagram (Fig. 2). Variation of  can be affected in experiments by offering different acceptors to the same donor.
  Simple calculation of the intersection point between the parabolas i  and   give the Gibbs free energy of activation
  where  =  and = c. The intersection of those parabolas represents an activation energy and not the energy of a transition state of fixed configuration of all nuclei in the system as is the case in the substitution and other reactions mentioned. The transition state of the latter reactions has to meet structural and energetic conditions, redox reactions have only to comply to the energy requirement.  Whereas the geometry of the transition state in the other reactions is the same for all pairs of reactants, for redox pairs many polarization environments may meet the energetic conditions.
  Marcus' formula shows a quadratic dependence of the Gibbs free energy of activation on the Gibbs free energy of reaction. It is general knowledge from the host of chemical experience that reactions usually are the faster the more negative is . In many cases even a linear free energy relation is found. According to the Marcus formula the rates increase also when the reactions are more exergonic, however only as long as  is positive or slightly negative. It is surprising that for redox reactions according to the Marcus formula the activation energy should increase for very exergonic reaction, i.e. in the cases when  is negative and its absolute value is greater than that of . This realm of Gibbs free energy of reaction is called "Marcus inverted region". In Fig. 2 it becomes obvious that the intersection of the parabolas i and f moves upwards in the left part of the graph when  continues to become more negative, and this means increasing activation energy. Thus the total graph of  vs.  should have a maximum.
  The maximum of the ET rate is expected at  Here  and  (Fig. 2) which means that the electron may jump in the precursor complex at its equilibrium polarization. No thermal activation is necessary: the reaction is barrierless. In the inverted region the polarization corresponds to the difficult-to-imagine notion of a charge distribution where the donor has received and the acceptor given off charge. Of course, in real world this does not happen, it is not a real charge distribution which creates this critical polarization, but the thermal fluctuation in the solvent. This polarization necessary for transfer in the inverted region can be created – with some probability – as well as any other one. The electron is just waiting for it for jumping.