Marcus Theory - The Probability of The Electron Jump

The Probability of The Electron Jump

The strength of the electronic coupling of the donor and acceptor decides whether the electron transfer reaction is adiabatic or non-adiabatic. In the non-adiabatic case the coupling is weak, i.e. H_AB in Fig. 3 is small compared to the reorganization energy and donor and acceptor retain their idendity. The system has a certain probability to jump from the initial to the final potential energy curves. In the adiabatic case the coupling is considerable, the gap of 2 H_AB is larger and the system stays on the lower potential energy curve.

Marcus theory as laid out above, represents the non-adiabatic case. Consequently the semi-classical Landau-Zener theory can be applied, which gives the probability of interconversion of donor and acceptor for a single passage of the system through the region of the intersection of the potential energy curves

where H_if is the interaction energy at the intersection, v the velocity of the system through the intersection region, s_i and s_f the slopes there.

Fig. 3 Energy diagram for Electron Transfer including inner and outer sphere reorganization and electronic coupling: 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 (inclusive solvent reorganization)

Working this out one arrives at the basic equation of Marcus theory

where is the rate constant for electron transfer, is the electronic coupling between the initial and final states, is the reorganization energy (both inner and outer-sphere), and is the total Gibbs free energy change for the electron transfer reaction ( is the Boltzmann constant and is the absolute temperature).

Thus 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 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 “reorganize” the system structure from initial to final coordinates, without making the charge transfer. 2. 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).