Solvent fluctuations in electron transfer processes

ELSEVIER Journal of Molecular Liquids, 63 (1995) 77-88

journal of MOLECULAR LIQUIDS

Solvent Fluctuations in Electron Transfer Processes

Victor P6rez, Jos6 M. Lluch and Juan Bertr~n Departament de Qufmica, Universitat Aut6noma de Barcelona

08 193 Bellaterra, Barcelona, Catalonia, Spain

S u m m a r y

In this brief review the fundamental role of solvent fluctuations in electron transfer reactions in solution is stated. The mechanism of outer-sphere bimolecular electron transfers is outlined and the usefulness of computer simulations for analyzing solvent fluctuations is discussed. Finally, the importance of dynamic solvent effects is mentioned.

Introduction Electron transfer reactions are among the most fundamental chemical processes.

Because of their ubiquity the understanding of the factors which determine electron transfer rates is of importance in a wide variety of physical, chemical and biological systems (refs. 1-6).

Until about 1950 it was commonly thought that simple electron transfer reactions, such as the Fe 3÷ + Fe 2÷ exchange in aqueous solution, could proceed very rapidly, because the electron would only have to jump from the ferrous ion to the ferric ion, with no change in energy (refs. 4 and 7-9). This would be true for an exchange reaction in gas phase since a resonant electron transition between similar particles occurs with large probability. However no such resonance phenomena take place in solution because the ion energy levels are displaced by interaction with the solvent, which is different for ions which are similar in nature but differ in charge (ref. 10). As a consequence most electron transfers in solution are not diffusion controlled and require significant thermal activation. Nowadays there is common agreement that the core of the electron transfer problem is the change (fluctuation) in equilibrium nuclear configuration that takes place when a species gains or loses an electron (refs. 11 and 12). In general these fluctuations involve vibrational coordinates (i.e., metal-ligand and intraligand bond lengths and angles) as well as the coordinates of the surrounding solvent molecules. For reaction to occur the reactants change their nuclear coordinates until a less stable configuration which allows a resonant electron transfer is reached. The electron is transferred and then the products relax to their most stable configuration.

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Mechanism of Outer-Sphere Bimolecular Electron Transfer Reactions in Solution

We shall be concerned hereafter primarily with outer-sphere bimolecular electron transfer reactions in solution, in which no bond are made or broken during the course of the process. This kind of reactions between a donor D and an acceptor A follow a tree-step mechanism (refs. 13-21):

D + A - ~ D [ A

DI A - 4 D + I A - (1)

D + A--->D + + A -

Since electron transfer is favored by small separation of the two reactants, the first step is the diffusion together of the separated reactants D and A to form the precursor complex D I A. The actual electron transfer takes place within the precursor complex D I A to form a successor complex D ÷ I A-. The dissociation of the successor complex leads to the separated products in the third step.

Opposing the close approach of the reactants is the coulombic work required to bring together similarly charged reactants and ultimately the electron-electron repul- sions (refs. 22-24). Due to the compromise between these opposing factors a bimolecular electron transfer will occur over a range of separation distances.

Then the rate of electron transfer reactions is given by:

I~ = I P(R-~k(R-->)d 3R-> (2)

where d3R->is the volume element of the system, P(~) is the probability that D and A will be at a given distance and orientation, ~ and k (~ ) is the rate of electron transfer at a fixed ~ If the function P(R-~)k(R ->) has a sharp maximum at some value of ~ say R'~m, and accordingly the rate constant can be approximated by P(KOm)k(R-~m), the study of electron transfer processes can be carried out by holding the two redox sites separated a fixed distance.

We will focus our attention on the factors that determine k(R ->) for electron transfer at fixed ~ This problem is frequently formulated in terms of a suitable diabatic (one that does not diagonalize the Born-Oppenheimer electronic Hamiltonian 101 of the system) two-state basis. It consists in the electronic wave functions q~p and hos, which can be directly identified in a valence-bond structure sense with the precursor and successor complexes, respectively (refs. 21 and 25). With the hop and hos functions specified, the integrals Hpp = < hop I l~l I hop > and Hss = < hos 1 151 1 hos > can be calculated.

Then Hop (~-~) and Hss(q---~), where ~~stands for the nuclear coordinates, are the corresponding diabatic potential energy hypersurfaces. Both are actually N-dimensional surfaces that should be represented in a N+I dimensional space where N is the

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number of independent variables necessary to define the vector q-~, that is, the vibrational coordinates of the solutes and the coordinates of the surrounding solvent molecules. Schematic one-dimensional profiles of such many-dimensional surfaces are given in figure 1. Both Hpp(~ ->) and Hss(~ ->) hypersurfaces have a minimum that corresponds to the equilibrium configuration of the precursor and successor complexes, respectively, and which are denoted as q->~ and ~->0. These hypersurfaces intersect, being q~ the nuclear coordinates corresponding to the intersection hypersurface S*. According to this diabatic representation a thermal electron transfer reaction involves a nonradiative jump from the q~p function in the ~-->p0 region to the qJs function in the

~-->0 region.

Eot/// A H~('~~ l'Iss(q~ =

-m- 0 -m- -m- 0 qp q qs q Figure ] : Diabatic potential encrgy hypcrsurfaccs as a function of nuclear coordinatcs q->

A thermal electron transfer is governed by the Franck-Condon principle (ref. 26). According to this principle during the actual (instantaneous) electron transfer the nuclei do not have time to change either their positions or their momenta. This Franck-Condon principle is taken into account in a different manner in the classical, semiclassical and quantum-mechanical electron transfer theories.

Quantum-mechanical models (refs. 19, 21, 22, 25, and 27-50) are based on the Golden rule of the time-dependent perturbation theory, which gives the transition probability per unit time for passing from the vibrational states of the precursor complex to the vibrational states of the successor one. In most applications the Condon approximation is applied in such a way that the transition probability is a function of nuclear overlap integrals (Franck-Condon factors) and the electronic coupling integral Hps = <h°p I 1~I I tlJs>.

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On the other hand classical models (refs. 13-15, 17, 19, 21 and 50-60) generally use the formalism of the transition state theory (ref. 61). To satisfy the Franck-Condon principle, the electron transfer must occur, within this classical frame, at the intersection region of the diabatic potential energy hypersurfaces Hpp(q --~) and Hss(q-3). It is important to remark that all the configurations belonging to the S* region, i.e., with coordinate q--i that fulfil the relation H0p(q --z/) = Hss(q-i), define the transition state of the electron transfer. This region S* is reached by a suitable thermal fluctuation in the nuclear configurations of the precursor complex. The electronic coupling integral between both diabatic states is assumed to be large enough so that the reactants are converted into products with unit probability in the intersection region (an adiabatic process), but small enough so that it can be neglected in calculating the amount of energy required to reach S*. Obviously, these classical models ignore quantum-mechanical effects such as nonadiabaticity and nuclear tunnelling. Other approximate procedures, called semiclassical (refs. 19, 21, 25 and 50), have tried to include such quantum-mechanical effects. Anyway it seems that the classical model adequately runs well for many electron transfers in solution. It has to be remarked that nuclear tunnelling correction can be important at low temperature and high exothermicities. Nonadiabatic electron transfer should be considered when the reorganization barrier is reduced by the exothermicity of the reaction or when reactants are kept far apart, for example, in biological systems or in frozen media (ref. 21 ). So assuming the transition state theory within the classical formalism the rate constant is given by (refs. 19 and 50):

k = agn exp(-AF~/RT) (3)

where D n is the frequency of passage through the intersection region (refs. 19 and 62-64), and AF ¢ is the activation free energy for the electron transfer, that is, the reversible work required to establish the solvent and ligand configurations for which the two redox sites are degenerate. These nuclear configurations are those belonging to S* region.

To evaluate AF t a two-mode model has commonly been employed (refs. 14, 15, 19, 21, 33, 45, 47, 54 and 59) in such a way that:

AF ~ = AF~n + AFo~ut (4)

The system is divided in two parts: the inner shell and the outer shell. The internal reorganization term features the changes in bond lengths and angles accompanying electron transfer. On the other hand the external reorganization term concerns the fluctuational reorganization of the solvent molecules. Assuming that the ligand vibra- tions are harmonic, the contribution of the inner shell to the free energy barrier (A ~n) can be estimated from the equilibrium bond lengths and symmetrical stretching (breathing) frequencies of the reactants (ref. 55).

To obtain the free energy barrier AFout, required to reorganize the solvent outside

the inner coordination shell of the reactants, the solvent can be treated as a dielectric continuum with a polarization made up of two parts, a relatively rapid electronic and a slower vibrational-orientational polarization (refs. 13-15, 20, 22, 25, 33, 53, 54, 60, 63 and 65-69). One of the most important results of this classical theory, that was originally deduced for structureless electron donor and electron acceptor, is the Marcus' relationship (refs. 5, 13, 23 and 70):

AF ~ (cz+AFo) 2 - 4o~ ( 5 )

In this equation ~ is the reorganization free energy, that is, the free energy released when the system evolves from the equilibrium configurations corresponding to the reactants to those corresponding to the products, while an electronic wave function, that can be directly related to products in a valence-bond structure sense, is maintained for describing the solute. In turn AF0 is the free energy difference between the products

and the reactants.

Analysis of Solvent Fluctuations by Computer Simulation

The rapid development of modem computers and the methods of numerical simulation (Monte Carlo (refs. 71-77) and Molecular Dynamics (refs. 76 and 78-82) methods) in the last years has provided a new microscopic way of acquiring very interesting information about the solvent fluctuations that allow the electron transfer occurs. In this section we focus on systems in which the nuclear coordinates only correspond to solvent coordinates.

Two problems can be analyzed, first the energetic and structural nature of those configurations that belong to the intersection region S* of diabatic hypersurfaces (refs. 83-86), and secondly the calculation of the probability of reaching S*, what is equivalent to obtain the free energy barrier of the reaction (refs. 84 and 87-98).

Regarding the first point it has been shown that the configurations belonging to S* region appear as a compensation between solvation contributions from the different solvation shells. Any attempt to reduce the effect of solvent fluctuations to the first hydration shell thus leads to an oversimplified model. In addition it is interesting to remark that the transition state does not correspond to an only one well-defined structure, but an important geometry and energy dispersion in the configurations corresponding to S* region exists. This dispersion is due to the great complexity of the solvent fluctuations which lead to electron transfer itself.

With respect to the second point two diabatic free energy curves corresponding to reactants and products, respectively, and depending on an unique one-dimensional

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reaction coordinate can be obtained. The intersection of both curves determines the free energy barrier of the reaction A1 ~. The parameter AE = Hss - Hpp has shown to

behave as a good reaction coordinate for electron transfer reactions (ref. 90). This AE value has to be calculated for each solvent configuration generated during the numerical simulation. Then the configuration space is partitioned in different subsets S, each one being associated with a particular value AE s of the reaction coordinate AE. The reactants' region (SR) is identified with the most populated interval when the Hpp potential is used. The intersection region S* corresponds to the interval centred at the value of AEs = 0. Then, this value of the reaction coordinate explicitly characterizes

the transition structures for the electron transfer. The activation free energy AF ~ corresponds to the transition from SR to S*, in such a way that the factor exp(-AF~/kT) expresses the probability that the reaction system will be in the transition state region S* relative to the probability of being in SR.

The complete sampling of the configuration space in order to obtain AF as a function of the reaction coordinate would generally require an extremely long simulation. To circumvent this problem two useful approaches have been used: the umbrella sampling (refs. 84 and 91) and specially the statistical perturbation theory (refs. 90 and 97-106).

Within the perturbation procedure a mapping potential energy hypersurface of the form H m = (1 - ~m)Hpp + ~mHss, is defined (refs. 89, 90, 97). The parameter ~m

changes fom 0 to 1 upon motion from the precursor to the successor states. With the H m potential corresponding to Lm = 0 the most populated subspace is SR. As ~'m

increases, the system is forced to evolve towards the intersection region S*. Now the diabatic free energy along the reaction coordinate corresponding to reactants is obtained by using the following expression as a function of AEs.

exp(-AF(AEs)/kT) = exp(-AF0 ._, m/kT)<exp[Hm(q-~ S) -

Hpp(q-~E S)]/kT>Hm(q(rnS)/Qrn)(Qpp/q(p SR)) (6)

An analogous expression for products can also be deduced.

The magnitude AF0~ m of the first factor gives the free energy associated with changing the potential corresponding to )~m = 0 to the sampling potential Hm, according to the expressions:

m-1

AFo_~ m = AF(Xo = 0 ---) ~m) = ~ a AF(~j ----> ) . j+ l ) (7) j=0

• t - - •

AF0~j ----) )~j' ) = - kT ln<exp - (HJkTHJ)>n j (8)

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where the average is for sampling based on the potential Hj, so the Hi, potential corresponds to a perturbed system. In the second factor, Hm(q-)~ S) and Hpp(~-~e S), represent the sampling potential and the diabatic potential corresponding to the precursor complex, respectively, evaluated for any configuration belonging to the subset S centred at the AE s value. The average is extended over the subspace S using the potential H m. The factor q(mS)/Qm is the probability that using the potential H m the

configurations generated belong to the subset S, while the factor,~(SR)/n represents "app "'<pp

the probability that using the potential Hop the configurations generated belong to the reactants' subset SR. The convergence of the numerical calculation is fast to evaluate AF(AEs) when in each case the mapping potential H m, for which the most populated subset S is one centred at AEs, is used.

Dynamic Solvent Effects on Electron Transfer Reactions

In deducing expression (3) for the electron transfer rate constant it was assumed that the states from which reaction occurs have a thermal equilibrium population unperturbed by the reaction. According to this transition state theory description the rate depends on the polar solvent free energies but not on the solvent dynamics. In recent years, the possible role of the solvent dynamics in influencing the rates of electron transfers in polar solvents has begun to receive considerable attention (refs. 89 and 107-133). When the free energy barrier is small and the electronic factor is favorable, the dynamics of the solvent reorganization may become rate determining. During the course of the reaction, the charge is rapidly redistributed among reactants. How the reaction couples to its solvent environment depends critically on how fast the solvent can evolve in comparison with changes in reactant charge distribution. The influence of solvent dynamics leads to the modification of preexponential factor Un in the classical equation (3).

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Solvent fluctuations in electron transfer processes

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