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).Contrary to Marcus' approach the inner sphere electron transfer theory of Noel S. Hush refers to a continuous change of the electron density during transfer along a geometrical coordinate (adiabatic case), and takes also into account the solvent influence as did Marcus. Hush's formulation is known as Marcus–Hush theory.Hush, N.S. Trans. Faraday Soc. 1961, 57,557
For electron transfer reactions without making or breaking bonds Marcus theory takes the place of Eyring's transition state theoryP. W. Atkins: Physical Chemistry, 6. Ed., Oxford University Press, Oxford 1998 p.830R.S. Berry, S. A. Rice, J. Ross: Physical Chemistry, Wiley, New York 1980, S. 1147 ff, 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 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 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.Miller J.R., Calcaterra L.T., Closs G.L.: "Intramolecular long-distance electron transfer in radical anions. The effects of free energy and solvent on the reaction rates", J.Am.Chem.Soc. 1984, 106, 3047,
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.
Outer vs inner 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 (ET) to give successor complex. The pair then dissociates. For a one electron transfer the reaction is
{D}+A <=>[k_{12}][k_{21}] [D{\dotsm}A] <=>[k_{23}][k_{32}] [D+{\dotsm}A^-] ->[k_{30}] {D+} + {A^-}
(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.
Problem 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 .
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.
For outer sphere redox reactions there cannot be such a reaction path, but nevertheless one does observe an activation energy. The rate constant for activation-controlled reactions has the same exponential form as the Eyring equation,
is the Gibbs free energy of the formation of the transition state, the exponential term represents the probability of its formation, A contains the probability of crossing from precursor to successor complex.
Marcus model The consequence of an electron transfer is the rearrangement of charges, and this greatly influences the solvent environment. For the dipolar solvent molecules rearrange in the direction of the field of the charges (this is called orientation polarisation), and also the atoms and electrons in the solvent molecules are slightly displaced (atomic and electron polarization, respectively). It is this solvent polarization which determines the free energy of activation and thus the reaction rate.
Substitution, elimination and isomerization reactions differ from the outer sphere redox reaction not only in the structural changes outlined above, but also in the fact that the movements of the nuclei and the shift of charges (charge transfer, CT) on the reactions path take place in a continuous and concerted way: nuclear configurations and charge distribution are always "in equilibrium". This is illustrated by the SN2 substitution of the alkaline hydrolysis of an alkyl halide where the rear side attack of the OH− ion pushes out a halide ion and where a transition state with a five-coordinated carbon atom must be visualized. The system of the reactants becomes coupled so tightly during the reaction that they form the activated complex as an integral entity. The solvent here has a minor effect.
By contrast, in outer sphere redox reactions the displacement of nuclei in the reactants are small, here the solvent has the dominant role. Donor-acceptor coupling is weak, both keep their identity during the reaction. Therefore, the electron, being an elementary particle, can only "jump" as a whole (electron transfer, ET). If the electron jumps, the transfer is much faster than the movement of the large solvent molecules, with the consequence that the nuclear positions of the reaction partners and the solvent molecules are the same before and after the electron jump (Franck–Condon principle).W.F. Libby, "Theory of Electron Exchange Reactions in Aqueous Solution" J.Phys.Chem. 1952, 56, 863 The jump of the electron is governed by quantum mechanical rules, it is only possible if also the energy of the ET system does not change "during" the jump.
The arrangement of solvent molecules depends on the charge distribution on the reactants. If the solvent configuration must be the same before and after the jump and the energy may not change, then the solvent cannot be in the solvation state of the precursor nor in that of the successor complex as they are different, it has to be somewhere in between. 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. Also, in this state the energy of precursor and successor in their solvent environment would be the same.
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. This means that real charge distribution and required solvent polarization are not in an "equilibrium". 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. This, however, requires energy. This energy may be provided by the thermal energy of the solvent and thermal fluctuations can produce the correct polarization state. Once this has been reached the electron can jump. The creation of the correct solvent arrangement and the electron jump are decoupled and do not happen in a synchronous process. Thus the energy of the transition state is mostly polarization energy of the solvent.
Theory description Macroscopic system: two conducting spheres 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 Pe and the slow atom and orientation polarisation Pu 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).