1.4. Excited-State Dynamics with Interchromophoric Interactions Interchromophoric Interactions
1.4.4. Electron Transfer
The simplest chemical reaction one can think of is the transfer of one electron from one molecule to another. Electron-transfer (ET) processes in solution are among the most important reactions in chemistry and biology. Photosynthesis, metabolism, photography, corrosion, oxidations and reductions of organic molecules or of metal ions in protein cores are only a few examples in which electron transfer processes play a key role.
Although the transfer of an electron is an elementary reaction which has been studied for many decades and for the modelling of which the Nobel Prise was awarded to R. A. Marcus in 1992, the understanding and the control of ET reactions still forms a very broad and active research area, especially since the ultrafast time scale has become experimentally avail-able [87-91]. Other topics all dominated by electron transfer and electron transmission in molecular systems, such as photoelectrochemistry, solar energy conversion, organic light emitting diodes, or molecular electronic devices, are also presently subjects of intense research at the interface of science and technology.
Electron transfer reactions can be either thermally activated, or photoinduced, in which case one of the partners reacts in an electroni-cally excited state. Although most of the discussion hereafter applies to both types of ET, it will be restricted to photoinduced electron transfer
(PET). A PET reaction between two neutral molecules can be
D and A are the electron donor and acceptor species, respectively. Equa-tion (1.105) describes a &'?*." )"/?*?$(%9 (CS) process in which an ion pair is formed. Very often, the two ions do not have time do diffuse away and form free ions and the two ions recombine to form neutral products, which may be in an excited state:
D A D A
Equation (1.106) describes &'?*." *"&%#,(9?$(%9 (CR) reactions which are usually not photoinduced. PET can also occur between an excited neutral molecule and an ion, in which case it is called a &'?*.")'(!$ reac-tion. In addition, ET reactions can be either (9$*?#%-"&0-?* or (9$"*#%-"&0-?*, depending whether the donor and acceptor groups be-long to the same molecule or not.
A general scheme for bimolecular PET in solution is represented in Figure 1.11. An electron donor (D) or acceptor (A) is excited by light.
This leads to an increase of its electronic affinity and simultaneously to a decrease of its ionisation potential. In order for ET to occur, the mole-cules must be at reacting distance, which, in the case of a bimolecular reaction, requires first diffusion (with a rate constant 5diff) in solution. The
Figure 1.11. General reaction scheme for a photoinduced electron transfer reac-tion in polar solvents.
CS process may then occur with a rate constant 5ET. The formed ions can either diffuse apart with a rate constant 5sep or recombine to the neutral ground state of the reactants with a rate constant 5CR.
a) Energetics
A PET reaction is only efficient if it is also thermodynamically favourable, that is, exergonic. The Gibbs free energy, IOET, of a PET reaction can be estimated from the oxidation potential of the donor, 2ox
# $
D , the reduc-tion potential of the acceptor, 2red# $
A , the energy of the excited state involved, 200, and a Coulombic term, E, representing the energy gained when the two product ions are brought from infinite separation (as as-sumed in the oxidation-reduction potential difference) to the actual reac-tion distance [92]:# $ # $
ET ox D red A 00
O " 2. 2 / 2 E
I " 1 ! 2! 0 . (1.107)
All terms in this equation proposed by Rehm and Weller are expressed in joules except the oxidation-reduction (redox) potentials which are in volts and the elementary charge, ", which is in coulombs.
Figure 1.12. Energetics of a photoinduced electron transfer reaction. The differ-ent contributions to the exergonicity of the electron transfer reaction, IOET, are represented, as well as the exergonicity of the charge recombination reaction,
OCR
I .
The Rehm-Weller equation shows that the redox potential of a mole-cule in an excited state is more favourable by an energy difference 200
(Figure 1.12). The Coulombic term is difficult to estimate with precision.
In a point charge model, it is simply E"I I1 2/*, where I1 and I2 are the charges of the two ions which are separated by a distance * in a solvent with dielectric constant , [40]. In a polar solvent, E is usually on the order of 0.1 to 0.2 eV or lower and therefore this term is often neglected.
b) The Classical Marcus-Hush Theory
The theoretical background establishing a link between the thermody-namics and the kinetics of electron transfer was developed 50 years ago independently by Marcus and Hush [93-99]. Although it was initially in-tended to address thermally activated ET in electrochemistry, it is still the most popular model which accounts for ET reactions in many areas of chemistry. This model is however based on transition state theory, which limits its range of applicability. The main assumption is that the system is at quasi-equilibrium with its environment along the whole pathway from the reactants to the products, which implies that the electron transfer process must be much slower than the solvent relaxation.
The present discussion is restricted to non-adiabatic ET, that is, ET which is described as a transition between two distinct PESs, implying that the coupling between the donor and the acceptor is weak. The PESs
Figure 1.13. Free energy surfaces of the reactants (R) and products (P) when considered as harmonic oscillators in the frame of Marcus-Hush theory. The crossing point determines the activation energy, IOT. The reorganisation energy, (, and the driving force for electron transfer, IOET, are also indicated.
commonly represent the potential energy of the system as a function of a generalised solvent coordinate P. The reactants are most of the time near the bottom of their PES, that is, at or near equilibrium. Random solvent fluctuations may however drive the system up to a point, the transition state, where the two PESs cross and where the system can switch to the product surface with a certain probability before relaxing to the equilib-rium configuration. The classical Marucs-Hush theory excludes any nu-clear tunnelling between the PESs.
Because of the underlying assumptions to the description of the sol-vent coordinate, the PESs have to be reinterpreted as free energy surfaces (FESs) [64, 100]. Also, as a linear response of the solvent to any electric field change is assumed, the FESs of reactants and products can be taken to depend quadratically on P (Figure 1.13). In other words, they are pa-rabolas with equal curvature (equal force constant). This greatly simplifies the mathematical treatment.
where U is a frequency factor. Because the geometry of the FESs is para-bolic, the equations describing the FESs can be solved at the crossing point to give the height of the activation barrier, IOT. This quantity is directly related to the free energy of the reaction and to (, the so-called
*"%*.?9()?$(%9"9"*.+:
( represents the energy needed by the system to move from the reactants to the products without any change in configuration neither of the reac-tants (bond lengths, angles) nor of the surrounding solvent (reorientation of the dipoles). It accounts for the horizontal displacement of the reactant and product parabolas. IOET, on the other hand, describes their vertical
displacement. Combining equations (1.108) and (1.109) finally yields the classical Marcus-Hush expression:
#
ET$
2ET exp
4 5 O
56 ( (
7 I 0 8
9 :
" U !
9 :
; <. (1.110)
The most straightforward consequence of equation (1.110) is that, at constant reorganisation energy, the dependence of the ET rate constant of the driving force is not linear but Gaussian. Three regimes are com-monly distinguished (Figure 1.14). In the so-called 9%*#?-*".(%9 where
OET (
I V , 5ET increases with the exergonicity of the reaction. At some point, a maximum is reached when IOET C(. This is the ,?**("*-"))
*".(%9. If the exergonicity keeps on increasing, IOET F( and the rate constant starts decreasing. This very non-intuitive behaviour is known as the Marcus inverted region and was very controversial until it could be univocally proved experimentally in the mid eighties [101]. The effect of larger solvent reorganisation energy would be to shift the maximum of the bell-shaped curve to a more exergonic IOET value.
Figure 1.14. Relative position of the free energy curves of reactants (R) and products (P) according to the classical Marcus-Hush model in (from left to right) the normal region, the barrierless region, and the inverted region. The Gaussian dependence of the ET rate constant on the exergonicity of the reaction is shown in the lower left part of the figure.
The preexponential factor U in equation (1.110) describes the prob-ability for the system to switch from the reactant to the product FES whenever the system crosses the transition state region. It is related to the square of the coupling between reactants and products (see [64, 99, 100]
for more details).
c) The Semi-Classical Marcus-Hush Theory
Experimentally, the Marcus inverted region has been univocally observed for many types of electron transfer reactions except bimolecular charge separation in solution [87]. In all these observations, however, the de-crease of the rate constant in the highly exergonic regime was found to be lower than predicted by classical Hush theory. Classical Marcus-Hush theory cannot either account for the observed temperature de-pendence of the ET rate constant in the Marcus inverted region. The
Figure 1.15. Inclusion of high-frequency intramolecular vibrational modes into the non-adiabatic electron transfer theory. The products can be formed in an excited vibrational level, which lowers the activation barrier in the inverted re-gion.
idea behind the semi-classical Marcus-Hush approach is to keep on con-sidering the solvent classically while high frequency intramolecular vibra-tional quantum modes of the reactants and of the products are included into the treatment (Figure 1.15). This allows the ET to occur from ther-mally populated vibrational levels of the reactants to high-frequency vibra-tional levels, 8, of the product, that is, nuclear tunnelling is considered.
The effect is to lower the classical barrier in the inverted region, but not in the normal or barrierless regions where the crossing point with vibra-tionally excited levels of the product surface would be even higher in energy on the reactant FES. With this correction, the observed faster rate than predicted by classical theory can be accounted for.
The semi-classical approach requires several rate constants for the electron transfer from the reactant ground vibrational level to each of the accessible product excited vibrational levels, 50O8, to be considered:
ET 0 8
where the semi-classical preexponential factor, Usc, takes into account that the coupling between reactants and products is no more purely electronic but also vibrational, and where (S is the reorganisation energy due only to configurational changes of the solvent.
The semi-classical Marcus-Hush description of electron transfer is valid as long as the energy of the solvent low-frequency modes is smaller than thermal energy. If this requirement is not fulfilled, a complete quan-tum mechanical description of the system must be implemented [99, 100]. It would however be beyond the scope of the present work to treat this aspect here. Diffusion-controlled ET will neither be discussed, al-though the aforementioned discussion on ET implicitly implied that the partners undergoing ET would stay at a fixed distance, which is certainly not true when the two partners can diffuse freely in solution (see Figure
1.11). It should simply be mentioned that, in the case of bimolecular PET in dilute solution in the absence of ground-state complexes, the rate limit-ing step is diffusion, so that no rate constant larger than the diffusion rate constant of the solute in the given solvent (equation (1.62)) can in princi-ple be measured.