My watch list
my.chemeurope.com

# Marcus theory

Marcus Theory is a theory originally developed by Rudolph A. Marcus to explain outer sphere electron transfer, the transfer of electrons between two metal spheres, by introducing reorganization energy and electronic coupling to the reaction coordinate diagram typically portraying the Gibbs' Free Energy and thermodynamic transition state. The theory was also later extended to inner sphere electron transfer by Noel Hush. The resultant theory, called Marcus-Hush theory, will also be discussed here. Besides the inner and outersphere applications, Marcus theory has been extended to address heterogeneous electron transfer.

Rudolph Marcus received the Nobel Prize in Chemistry in 1992 for this theory. Marcus theory is used in various aspects of chemistry and biology, including photosynthesis, corrosion, certain types of chemiluminescence and more.

Marcus theory is currently the dominant theory of electron transfer in chemistry. Marcus theory is so widely accepted because it makes several predictions concerning electron transfer that have been proven true over the last several decades. The most salient points of the theory are;

1. The energy required to optically excite an electron from an electron donor to an electron acceptor (the so-called reorganization energy) is intimately related to the activation energy for thermal electron transfer process.
2. The rate of electron transfer from one redox site to another will increase with the thermodynamic driving force for the reaction — to a point. When the driving force becomes too large the electron transfer rate will decrease, in the so-called "Marcus inverted region." This aspect of Marcus theory was controversial from the time the theory was proposed in 1956 until John Miller's group at Argonne National lab found empirical proof of it in 1986.[1]

The basic equation of Marcus theory is:

$k_{et} = \frac{2\pi}{\hbar}\mathbf{H}_{AB}^2 \frac{1}{\sqrt{4\pi k_bT}}\exp \left ( \frac{-(\lambda +\Delta G^\circ)^2}{4\lambda k_bT} \right )$

where ket is the rate of electron transfer, $\mathbf{H}_{AB}$ is the electronic coupling between the initial and final states, λ is the reorganization energy, and $\Delta G^\circ$ is the total Gibbs free energy change for the electron transfer reaction (kb is the Boltzmann constant).

The key parameters are diagrammed here:

The left hand parabola represents the Potential energy surface for the nuclear motion of the reactants in the initial state (where the electron is still on the donor molecule or group}, and the right hand parabola represents the potential energy surface for the nuclear motion of the products in the final state (after the electron has transferred from the donor to the acceptor). The unusual dependence of the electron transfer rate on the free energy change (i.e., the $(\lambda +\Delta G^\circ)^2$ term in the equation), which leads to the Marcus inverted region, follows simply from assuming that the potential energy of the initial and final states varies quadratically with some reaction coordinate (i.e. that both potential energy surfaces are parabolas), solving for the activation energy in terms of $\Delta G^\circ$ and λ, and plugging the result into the Arrhenius equation.

• Marcus' Nobel Lecture

## Marcus's Key Papers

Marcus, R.A. J. Chem. Phys. 1956, 24, 966.

Marcus, R.A. J. Chem. Phys. 1956, 24, 979.

Marcus, R.A. J. Chem. Phys. 1957, 26, 867.

Marcus, R.A. J. Chem. Phys. 1957, 26, 872.

Marcus, R.A. Disc. Faraday Soc. 1960, 29, 21.

Marcus, R.A. J. Phys. Chem. 1963, 67, 853.

Marcus, R.A. Annu. Rev. Phys. Chem. 1964, 15, 155.

Marcus, R.A. J. Chem. Phys. 1965, 43, 679.

Marcus, R.A.; Sutin N. Biochem. Biophys. Acta 1985, 811, 265.

## References

1. ^ Closs, G.L.; Calcaterra, L.T.; Green, N.J.; Penfield, K.W.; Miller, J.R. J. Phys. Chem. 1986, 90, 3673-3683.