Classical Marcus-Hush theory

(3.2)

Following charge separation, charge recombination commonly acts as an excited-state quencher restoring the ground-state of the reactants by thermal charge recombination. However, charge recombination to excited neutral species is also possible. Charge recombination occurring within a geminate ion pair is termed geminate recombination while recombination between well-separated ions is termed homogeneous recombination.

(3.3) (3.4) (3.5) (3.6)

One distinguishes between intramolecular ET where the acceptor and the donor are covalently linked, and intermolecular (also called bimolecular) ET where A and D are two separate molecules.

3.1 Classical Marcus-Hush theory

The theory developed by Marcus and Hush [166, 167, 168, 169, 170] provides a framework to describe the rate of ET reactions according to its thermodynamics. The main postulates of the theory are that both precursor and successor states can be described by quadratic functions of the reaction coordinate with equal force constants, nuclear reorganization proceeds classically through low frequency vibrational and rotational modes, and that nuclear reorganization is kinetically symmetrical with , where represents the rate of nuclear reorganization. The theory uses transition state theory (TST) where one generalized nuclear coordinate Q, is introduced in order to simplify the complex potential energy surface of each molecule within the system (including the surrounding solvent) by a one-dimensional profile. The reaction coordinate Q represents the average configuration of all molecules of the system.

Figure 3.2: Parabolic Gibbs free energy (G) surfaces of the reactants (R) and products (P) according to the classical Marcus-Hush theory.

Figure 3.2 gives a representation of the parabolic free energy surfaces of the reactants and products. The system of equations describing the free energy surfaces can be solved at the point of intersection of the two parabolas enabling the activation energy to be expressed in terms of the free energy of the reaction and the total reorganization energy, :

(3.7)

The total reorganization energy is comprised of both the solvent reorganization energy and the intramolecular reorganization energy, :

(3.8)

More specifically, the solvent reorganization energy represents the energy needed by the solvent molecules in order to reach an equilibrium orientation with the products, while the intramolecular reorganization energy represents the change in free energy required to deform

50 the reactant from the reactant equilibrium position to the product equilibrium position without transferring an electron.

Within the TST, the overall rate of ET, depends on the probability of reaching the transition state, the time required to cross the transition state and the probability of the electronic transition. It is given by the Arrhenius equation:

(3.9)

where A is the pre-exponential factor, T is the temperature and is the Boltzmann constant. The pre-exponential factor gives the probability for a system in the transition state region to cross over to the reactant free energy surface and is thus highly dependent on the electronic coupling between the two states. One distinguishes between non-adiabatic coupling where is small and adiabatic coupling where is large. In the first instance, R and P can be represented by unperturbed potential energy surfaces, while in the second case the large coupling implies an avoided crossing between the potential energy surfaces with an energy splitting of . These two cases are schematized in figure 3.3.

Figure 3.3: Schematic representation of the free energy surfaces for a) non-adiabatic and b) adiabatic ET where P represents the product surface and R the reactant surface.

(3.10)

where is the probability and is the oscillation frequency of the system in the reactant well. In the case of adiabatic coupling, using either time-dependent perturbation theory or the classical Landau-Zener theory, Levich and Dogonadze [171] have shown that the pre- exponential factor is proportional to :

(3.11)

Using equation 3.7 and replacing equation 3.11 into equation 3.9, it is possible to express the rate as a function of the free energy of the reaction :

(3.12)

The plot of as a function of shown in figure 3.4 has a bell-shape where one distinguishes three regions characterized by their exergonicity:

1. Normal regime:

The rate of electron transfer increases with increasing exergonicity 2. Barrierless regime:

There is no activation energy for the ET reaction and the ET rate is minimal 3. Inverted regime:

52 Figure 3.4: Schematic representation of the normal, barrierless and inverted Marcus regimes.

Unlike other excited-state processes like electron exchange or intersystem crossing, electron transfer can be predicted using the redox potentials of the donor and acceptor. Indeed, the thermodynamic efficiency of photoinduced ET can be determined using the equation proposed by Rehm-Weller [172, 173, 174] which gives the free energy of the reaction,

in terms of the oxidation and reduction potential of the acceptor (A) and donor (D): (3.13)

where is a coulomb term representing the electrostatic interaction between the ions which accounts for the fact that ions are formed at close distance and not infinite separation, represents a corrective factor for the solvent when a solvent other than acetonitrile is used,

is the energy of the 0-0 transition of either or , the reduction potential represents the following reaction:

and the oxidation potential describes the following reaction:

(3.15)

The coulomb term can be estimated based on a point charge model in which the charges are separated by a solvent which is treated as a dielectric continuum:

(3.16)

where is the static dielectric constant of the solvent and is the interionic centre-to-centre distance.