text: 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. From thermodynamics it is well known that the energy of such a state can be determined if a reversible path to that state is found. Marcus was successful in finding such a path via two reversible charging steps for the preparation of the "transition state" from the precursor complex.
  Four elements are essential for the model on which the theory is based: 
  # Marcus employs a classical, purely electrostatic model. The charge (many elementary charges) may be transferred in any portion from one body to another. 
  # Marcus separates the fast electron polarisation P and the slow atom and orientation polarisation P of the solvent on grounds of their time constants differing several orders of magnitude. 
  # Marcus separates the inner sphere (reactant + tightly bound solvent molecules, in complexes + ligands) and the outer sphere (free solvent ) 
  # In this model Marcus confines himself to calculating the outer sphere energy of the non-equilibrium polarization of the "transition state". The outer sphere energy is often much larger than the inner sphere contribution because of the far reaching electrostatic forces (compare the Debye–Hückel theory of electrochemistry).
  Marcus' tool is the theory of dielectric polarization in solvents. He solved the problem in a general way for a transfer of charge between two bodies of arbitrary shape with arbitrary surface and volume charge. For the self-exchange reaction, the redox pair (e.g. Fe(HO) / Fe(HO)) is substituted by two macroscopic conducting spheres at a defined distance carrying specified charges. Between these spheres a certain amount of charge is reversibly exchanged.
  In the first step the energy W of the transfer of a specific amount of charge is calculated, e.g. for the system in a state when both spheres carry half of the amount of charge which is to be transferred. This state of the system can be reached by transferring the respective charge from the donor sphere to the vacuum and then back to the acceptor sphere. Then the spheres in this state of charge give rise to a defined electric field in the solvent which creates the total solvent polarization P + P. By the same token this polarization of the solvent interacts with the charges.
  In a second step the energy W of the reversible (back) transfer of the charge to the first sphere, again via the vacuum, is calculated. However, the atom and orientation polarization P is kept fixed, only the electron polarization P may adjust to the field of the new charge distribution and the fixed P. After this second step the system is in the desired state with an electron polarization corresponding to the starting point of the redox reaction and an atom and orientation polarization corresponding to the "transition state". The energy W + W of this state is, thermodynamically speaking, a Gibbs free energy G.
  Of course, in this classical model the transfer of any arbitrary amount of charge &Delta;e is possible. So the energy of the non-equilibrium state, and consequently of the polarization energy of the solvent, can be probed as a function of &Delta;e. Thus Marcus has lumped together, in a very elegant way, the coordinates of all solvent molecules into a single coordinate of solvent polarization &Delta;p which is determined by the amount of transferred charge &Delta;e. So he reached a simplification of the energy representation to only two dimensions: G = f(&Delta;e). The result for two conducting spheres in a solvent is the formula of Marcus
  Where r and r are the radii of the spheres and R is their separation, &epsilon; and &epsilon; are the static and high frequency (optical) dielectric constants of the solvent, &Delta;e the amount of charge transferred. The graph of G vs. &Delta;e is a parabola (Fig. 1). In Marcus theory the energy belonging to the transfer of a unit charge (&Delta;e = 1) is called the (outer sphere) reorganization energy &lambda;, i.e. the energy of a state where the polarization would correspond to the transfer of a unit amount of charge, but the real charge distribution is that before the transfer. In terms of exchange direction the system is symmetric.