Marcus theory

In theoretical chemistry, Marcus theory is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron transfer reactions – the rate at which an electron can move or jump from one chemical species (called the electron donor) to another (called the electron acceptor). It was originally formulated to address outer sphere electron transfer reactions, in which the two chemical species only change in their charge with an electron jumping (e.g. the oxidation of an ion like Fe²⁺/Fe³⁺), but do not undergo large structural changes. It was extended to include inner sphere electron transfer contributions, in which a change of distances or geometry in the solvation or coordination shells of the two chemical species is taken into account (the Fe-O distances in Fe(H₂O)²⁺ and Fe(H₂O)³⁺ are different).

For electron transfer reactions without making or breaking bonds Marcus theory takes the place of Eyring's transition state theory which has been derived for reactions with structural changes. Both theories lead to rate equations of the same exponential form. However, whereas in Eyring theory the reaction partners become strongly coupled in the course of the reaction to form a structurally defined activated complex, in Marcus theory they are weakly coupled and retain their individuality. It is the thermally induced reorganization of the surroundings, the solvent (outer sphere) and the solvent sheath or the ligands (inner sphere) which create the geometrically favourable situation prior to and independent of the electron jump.

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 ${\displaystyle \Delta G^{\circ }}$ of the redox reaction. The most startling result of Marcus' theory was the "inverted region": whereas the reaction rates usually become higher with increasing exergonicity of the reaction, electron transfer should, according to Marcus theory, become slower in the very negative ${\displaystyle \Delta G^{\circ }}$ domain. Scientists searched the inverted region for proof of a slower electron transfer rate for 30 years until it was unequivocally verified experimentally in 1984.

R. A. Marcus received the Nobel Prize in Chemistry in 1992 for this theory. Marcus theory is used to describe a number of important processes in chemistry and biology, including photosynthesis, corrosion, certain types of chemiluminescence, charge separation in some types of solar cells and more. Besides the inner and outer sphere applications, Marcus theory has been extended to address heterogeneous electron transfer.

In a redox reaction an electron donor D must diffuse to the acceptor A, forming a precursor complex, which is labile but allows electron transfer to give successor complex. The pair then dissociates. For a one electron transfer the reaction is

(D and A may already carry charges). Here k₁₂, k₂₁ and k₃₀ are diffusion constants, k₂₃ and k₃₂ are rate constants of activated reactions. 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). The ligand shells around A and D are retained. This process is called outer sphere electron transfer. Outer sphere ET is the main focus of traditional Marcus Theory. The other kind or redox reactions is inner sphere where A and D are covalently linked by a bridging ligand. Rates for such ET reactions depend on ligand exchange rates.

In outer sphere redox reactions no bonds are formed or broken; only an electron transfer (ET) takes place. A quite simple example is the Fe²⁺/Fe³⁺ redox reaction, the self exchange reaction which is known to be always occurring in an aqueous solution containing the aquo complexes [Fe(H₂O)₆]²⁺ and [Fe(H₂O)6]³⁺. Redox occurs with Gibbs free reaction energy ${\displaystyle \Delta G^{\circ }=0}$.

From the reaction rate's temperature dependence an activation energy is determined, and this activation energy is interpreted as the energy of the transition state in a reaction diagram. The latter is drawn, according to Arrhenius and Eyring, as an energy diagram with the reaction coordinate as the abscissa. The reaction coordinate describes the minimum energy path from the reactants to the products, and the points of this coordinate are combinations of distances and angles between and in the reactants in the course of the formation and/or cleavage of bonds. The maximum of the energy diagram, the transition state, is characterized by a specific configuration of the atoms. Moreover, in Eyring's TST 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.