ELECTRON TRANSFER REACTIONS Some Basic Principles

1. ELECTRON TRANSFER REACTIONS Some Basic Principles Processes involving the transfer of electrons occur widely in the sciences. They range from simple exchange reactions in chemistry to processes that drive energy storage and respiration in biological systems e.g. the cytochrome-c/cytochrome oxidase couple Both are heme proteins– oxido-reductases Cytochrome oxidase Cytochrome-c e- Fe Fe Fe Fe

2. Photoscientists are keenly interested in electron transfer reactions because: • they occur as primary events in many photoprocesses. • they can be conveniently studied using photophysical techniques. • photophysical methods offer excellent ways of testing the theories. Kinetic Aspects of Bimolecular Reactions An electron transfer reaction between individual molecules freely diffusing in a mobile liquid has characteristics of all such bimolecular reactions: The reactants diffuse together, react, and two different entities diffuse apart. The process proceeds via a collision complex, one or more reaction intermediates, or a transition state. Micro-reversibility applies, and the process occurs on a continuous potential energy surface. No excited states have yet been invoked, so we can imagine the process to be adiabatic (no crossings to other PE surfaces)—not always true.

3. We can define rate constants for both forward and reverse processes, kf and kr, These need to be measured and the factors that influence them understood. The overall reaction, like all chemical reactions, will have characteristic Keq, DG0, DH0, and DS0 parameters. In addition, we can relate DG0 to reduction potentials, viz., Thermodynamic constants are useful for describing equilibrium states-- no information on mechanistic details. Experiment shows that for exoergic electron transfer processes, rate constant values occur over a wide range, up to the limit imposed by diffusion. This needs to be understood. Consider a detailed scheme for the overall bimolecular process shown above.

4. D + A D+ + A- kd k-d k-d kd kn ne k’-n k-n ne k’n [DA] Precursor complex [DA]# Reorganized precursor complex [D+A-]# Successor complex [D+A-] Reorganized successor complex

5. A steady state treatment leads to • Where • if • When ka >> kd, then kobs kd • Let us examine the expression for ka • Under conditions when i.e., the electron moves to A more • rapidly than the reorganized complex relaxes, then • Here k-n is the rate of relaxation of vibrationally excited precursor, and DG#n is the energy barrier to nuclear reorganization. • Under these conditions ka is independent of ne ( the electron-hopping rate.)

6. (ii) On the other hand, when In both cases (and all others), the nuclear reorganization process is a barrier to electron transfer and imposes an activation step. Thus the overall (measured) rate constant is a combination of diffusion-dependent (kd) and activation-dependent (ka) terms: Therefore when ka >> kd, then kobs kd. and our kinetic measurements can provide no information on the activation-dependent process because everything is limited by diffusion.

7. However, the central segment of the sequence occurs independently of how the sequence is initiated To investigate the role of activation in the sequence we need to circumvent the diffusion limiting problem. Later we will see how this can be done. For now, we assume that it can be done and proceed to examine ka. We see that the parameters k-n, ne, and DGn# are important in determining the magnitude of ka. Theoreticians have examined these using classical mechanics Marcus, Sutin, Hush And semi-classical/quantum methods Jortner, Levich.

8. THE TRANSITION STATE APPROACH The scheme is The steps prior to (D/A) and those after electron transfer are ignored. D and A may be polyatomic molecules, aquated metal ions, etc., and the reaction above proceeds with changes in bonding coordinates in D and A, and solvation around the complex (D/A)# and (D+/A-)# have identical nuclear configurations they differ only in that a single electron has moved. The situation resembles a Franck-Condon type event, or even a radiationless transition between two states. Using the radiationless transition approach, we write where r is an average density of states in the acceptor.

9. In electron transfer theory, r usually appears as a Franck-Condon weighted density of states (FCWD). The electronic matrix element term be2 contains the operator driving the process The classical picture due to Marcus (1956) is less rigorous and simpler, but provides useful physical insights. It provides a comparison to the transition state theory (TST) of kinetics. Marcus chose to represent the complex multidimensional PE surfaces of polyatomic reactant pairs as a parabolic energy curve in "nuclear configuration space"

10. REACTANT STATE PRODUCT STATE DG l DGn# Nuclear configuration • is the energy required to move the electron in the (D/A) to (D+/A-) without prior nuclear reorganization. Resembles a Franck-Condon event.

11. DGn# - the energy required to reconfigure the precursor complex to a non-equilibrium nuclear configuration in which the electron transfer can occur The system switches from the reactant state curve to the product state surface. Note thatDGn# < l. At the curve crossing, the electron can hop from one curve to another with some probability (rate). The situation shown in the schematic is for DG0= 0 an isoergic process. Analytical geometry of intersecting parabolas showed Marcus that in this case WhenDG0=\= 0 Thus the energy barrier (DGn#) to electron transfer depends on DG0 and l in a quadratic manner.

12. DG0 the effect on DG#n as the value of DG0 becomes increasingly negative. As the product parabola is lowered wrt the reactant curve, the nuclear reorganization barrier first becomes less and then increases again the only change is in the overall driving force no shape changes, no shifts in curve minimum.

13. inverted region normal region ka l Earlier we saw that or Thus, Marcus theory predicts that for weakly exoergic reactions log ka increases as -DG0 increases It maximizes at it decreases again as -DG0 increases beyond l (inverted region) This remarkable result flies in the face of intuition — WHY? it led to the Nobel Prize in Chemistry for Rudy Marcus in the early 1990s

14. we have assumed that the product state is formed in its zeroth vibrational state. this is a simplification and in fact the formation of product species in vibrational states above the zero point is very possible. DG Nuclear coordinate the dashed curves represent four vibrational energies of the product state.

15. The red dashed curve (v = 0) crosses the reactant curve in the inverted region, The higher vibrational modes do not. The rate constant observed will be a weighted sum of the contributions from all the modes, and it will be larger than that if the v = 0 mode was the only contributor. Thus the inverted region will be less pronounced than otherwise; the parabola will depart from the symmetrical form The Marcus approach, being geometrical, assumes symmetrical sets of pure parabolas there is only very weak interaction between surfaces at the crossing point. Thus the reaction is by necessity non-adiabatic, since curve crossing must occur.

16. at the crossing point the two curves are degenerate then time dependent perturbation theory instructs us that the probability of transferring to the product curve will be proportional to The vibrational motion of the nuclei in the reactant state will be such that it spends only a short time in the crossing region. Thus for states where is small (< kBT), the probability of reaction will be small system will continue on the reactant surface for many passages through the crossing region. when the perturbation is large (> kBT) – adiabatic case the sin2 term is large the oscillatory frequency will be sufficient to ensure effective passage into product space.

17. The reorganization energy The barrier to electron transfer, per Marcus, is manifest as a free energy term composed of DG0 and l components. The former is a thermodynamic state property, defining the overall free energy changes in going from reactants to products. The quantity l is an energy term that is identified in Marcus theory as being the energy of a FC transition from the relaxed precursor state into the corresponding nuclear configuration space of the product state. It is assumed that l has two additive parts: (i) -- inner shell, derived from required changes in the internal nuclear geometry of the donor and acceptor molecules. (ii) – outer shell, arising from the required readjustment of solvent dipoles to accommodate the shift in the electronic charge. Marcus (1959), assuming that a solvent behaves as dielectric continuum (no local structure; hard sphere molecules) derived the expression

18. For polar organic liquids (CH3CN) ls ~ 0.75 V For non-polar organic liquids (cyclohexane) ls ~ 0.15 V The value of lo is not easily calculated usually estimated from considerations of the force constants of normal mode vibrations in the reactant and product species. A typical value for lo is ~ 0.4 V Some exchange reactions such as (H2O)6Fe2+/3+ and (NH3)6Co2+/3+ the redox change necessitates large M—Ligand bond length changes (140 and 220 pm respectively). These lead to large lo values of 8.4 and 17.6 kcal mole-1 respectively.

19. The ne parameter: ne has been used to represent the frequency (or rate) at which an electron can shift between (D/A)# and (D+/A-)# . This can be expanded as nn is the frequency for nuclear reorganization and Kel is the probability of the electron transferring from one PE curve to another in the reorganized configuration. In TST terms, nn can be regarded as being similar to an entropic term, viz., The maximum value (DS# = 0) of nn is kBT/h, approximately 1013 s-1 Kel is similar to the transmission coefficient of TST It takes values between zero and one. It represents the probability of the system moving from reactant to product surface it depends on the interaction energy between the two surfaces at the crossing region Thus V is the electronic coupling matrix element: equivalent to beused earlier.

20. D RDA A D RDA A Molecular wave functions decrease exponentially with distance from their maximum amplitude. Overlap increases as the distance between the reacting species decreases

21. It is found that where br (not to be confused with be ) is a multiplier of RDA having dimensions of m-1 It is equivalent to a molecularresistance. If br is small for a given RDA , then is large and electron transfer occurs effectively if br is large for a given RDA , then is less and the transfer efficiency is reduced Thus the overall rate constant for electron transfer in the activated case (ka) depends on distance between the participants the frequency with which nuclei reorganize a reorganization barrier

22. Overall the activated rate constant can be described by The maximum value of nn is approximately 1013 s-1. The electronic coupling factor, Kel, depends on RDAand on the relative orientation between the dipoles in the donor and acceptor. The l parameter is the reorganization energy, which depends on solvent reorientation and on nuclear changes internal to the molecules. Our development has been independent of the electronic state of the reacting species.

23. Electron Transfer in the Photosciences Advantages of photoexcitation 1No transfer occurs until the photons are absorbed. Thus systems can be manipulated (synthesized, mixed, etc.) in their ground states. 2Wavelength specificity allows preferential excitation of one component. 3Photon absorption is virtually instantaneous and thus experimental "dead" time is non-existent. 4Two rate constants can usually be determined for each D/A couple (Figure) 5Electronically excited states have > 1 eV more energy than the parent ground states and are therefore better oxidants and reductants.

24. D* + A kET D+ + A- k-ET D + A

25. log k diffusion limit increasing driving force DG0 Experimental Observations: Measurements of bimolecular electron transfer rate constants permeate the literature, but they are of little use in testing the theoretical picture since (1 Distance and orientation effects are averaged over the population of reactants. Diffusion effects tend to obscure the details of the log kvs. DG0 curve at high DG0 As the driving force increases, so does the bimolecular rate constant for the reaction (the normal region). However at some point, even though the driving force continues to increase, the rate constant levels off because the rate-limiting step is now the diffusion of the reactants together. Rehm-Weller plots 0

26. D A SPACER Progress made when experimentalists confined D and A in close proximity No need for diffusion to bring reactants together Three basic approaches to this: 1: Retain D and A as discrete molecules, but disperse them in rigid glasses (J.R. Miller, 1975; G. McLendon, 1983) A relatively crude approach since a distribution of RDA values generated (1However, analysis showed that a distance relationship existedb ~ 0.01 pm-1. (22: Link D to A via a rigid spacer entity, using covalent bonding, e.g. Variations in spacer length allow RDA changes to be effected (keep DG0 constant) Change the reduction potentials of D and A at constant RDA allows DG0 effects to be investigated. (Closs and Miller seminal paper in 1983).