Marcus theory

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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 hop 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 aren't directly bonded to each other, but it was also extended to inner sphere electron transfer reactions, in which the two chemical species are attached by a chemical bridge, by Noel Hush (Hush's formulation is known as Marcus-Hush theory). 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 cell and more. Besides the inner and outer sphere applications, Marcus theory has been extended to address heterogeneous electron transfer.

Marcus's theory builds on the traditional Arrhenius equation for the rates of chemical reactions in two ways:

  1. It provides a formula for the pre-exponential factor in the Arrhenius equation, based on the electronic coupling between the initial and final state of the electron transfer reaction (i.e., the overlap of the electronic wave functions of the two states).
  2. It provides a formula for the activation energy, based on a parameter called the reorganization energy, as well as the Gibbs free energy. The reorganization energy is defined as the energy required to move an electron from the electron donor to the electron acceptor, without moving any of the atomic nuclei, but it can be thought of as describing the distance the atomic nuclei need to move as the electron hops from the donor to the acceptor.

Marcus theory is currently the dominant theory of electron transfer in chemistry. Marcus theory is so widely accepted because it makes surprising predictions about electron transfer rates that have been nonetheless proven true experimentally over the last several decades. The most significant prediction is the fact that the rate of electron transfer will increase as the electron transfer reaction becomes more exothermic – but only to a point. Past that point, the electron transfer rate will actually decrease as the reaction becomes more exothermic, 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 \lambda 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:

Image:Parameters of the Marcus Equation.JPG


Here the vertical axis is the free energy, and the horizontal axis is the "reaction coordinate" – a simplified axis representing the motion of all the atomic nuclei. 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.


[edit] Additional information

[edit] 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.

[edit] References

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