Marcus Theory of Electron Transfer

Abstract

In 1992, Rudolph Marcus received the Nobel Prize for chemistry. His theory is currently the dominant theory of electron transfer in chemistry. Originally Marcus explained outer sphere electron transfer, introducing reorganization energy and electronic coupling to relate the thermodynamic transition state to nonequilibrium uctuations of the medium polarization. We begin with a phenomenological description including diusional motion of the reactands. Then, we apply a simplied model with one reaction coordinate to calculate the reaction rate as a function of reorganization energy and reaction free enthalpy. Next, we apply a continuum model for the dielectric medium and derive the free energy contribution of the non equilibrium polarization quite generally. Reorganization energy and activation energy are calculated and transition state theory is applied to calculate the rate constant. We consider a model system consisting of two spherical reactants to calculate the reorganization energy explicitly and discuss charge separation and charge shift processes. We introduce the energy gap as a reaction coordinate and include inner shell reorganization. Finally, the mutual dependency of the electronic wavefunction and the polarization are discussed within a simple model for charge delocalization and self-trapping.

Problems

16.1

Marcus Cross Relation

• (a) Calculate the activation energy for the self-exchange reaction

  $$ \text {A}^{-}+\text {A}\quad \left. \begin{array}{c} k_{AA}\\ \rightarrow \\ \, \end{array}\right. \quad \text {A}+\text {A}^{-} $$

  in the harmonic model

  $$ G_{R}(Q)=\frac{a}{2}Q^{2}\quad \quad G_{P}(Q)=\frac{a}{2}(Q-Q_{1})^{2}. $$

• (b) Show that for the cross reaction

  $$ \text {A+D}\quad \left. \begin{array}{c} k\\ \rightarrow \\ \, \end{array}\right. \quad \text {A}^{-}+\text {D}^{+} $$

  the reorganization energy is given by the average of the reorganization energies for the two self-exchange reactions

  $$ \lambda =\frac{\lambda _{AA}+\lambda _{DD}}{2} $$

  and the rate k can be expressed as

  $$ k=\sqrt{k_{AA}k_{DD}K_{eq}f} $$

  where $k_{AA}$ and $k_{DD}$ are the rate constants of the self-exchange reactions, $K_{eq}$ is the equilibrium constant of the cross reaction and f is a factor, which is usually close to unity.