J. Chern. Soc., Furaday Trans. I, 1989, 85(6), 1207-1215 Effect of Solvent Fluctuations in the Electron-transfer Process between two Fe+ Ions Angels Gonz6lez-Lafont, Jo& M. Lluch, Antonio Oliva and Juan Bertran* Dept. de Quimica, Universitat A d n o m a de Barcelona, Bellaterra (Barcelona), Spain A Monte Carlo simulation of the solvent intervention in the process Fe++Fe+ 4 Fe2++ Fe" in aqueous solution has been carried out within a classical model. The configurations whose solvation energy does not change when the electron- transfer process occurs, have been determined. The results obtained show that the effect of the outer hydration shells can be very important in order to reach these isoenergetic configurations. Furthermore, an important energy dispersion in the configurations that makes possible the electron-transfer process has been found, this being due to the great complexity of the solvent fluctuations which leads to isoenergetic con- figurations.The elucidation of the molecular mechanism which regulates the rate of electron-transfer processes is one of the most important problems in physics, chemistry and biology. The homogeneous outer-sphere electron-transfer reactions in solution are especially interesting since they occur at a rate that is noticeably slower than the diffusion rate. This peculiar behaviour has been e~plainedl-~ through a three-step mechanism : formation of a precursor complex from the separated reactants, actual electron transfer within this complex to form a successor complex and dissociation of the latter complex into separated products.The reaction rate is usually controlled by the electron-transfer step, this step being governed by the Franck-Condon prin~iple.~ According to this principle, internuclear distances and nuclear velocities do not change during an electronic transition. This principle is embodied in classical electron-transfer theories5-' using an activated-complex formalism in which the electron transfer occurs at the intersection of two potential-energy surfaces, one for the reactants and the other for the products. This implies that the second step necessarily involves the reorganization of the solvent before and after the electron transfer itself is produced. Thus it is obvious that the solvent must play an essential role in the rate of electron transfer reactions in solution.As Levich' mentioned in 1967 a consistent statistical theory of the liquid state did not exist at that time, this fact being particularly true in the case of such a complex liquid as a polar solvent. For this reason, very simple solvent models have been generally used in order to study electron-transfer processes in solution. The original applications by Levich and co-workersg~ lo included only a single solvent harmonic vibrational mode (or equivalently, many solvent modes, all with a common frequency). A two-mode model has commonly been employed in subsequent work, 1,11-13 this model including the contributions of a low-frequency solvent mode and of a high-frequency inner-shell mode associated with the symmetrical stretching.Thus these models imply a drastic reduction of the degrees of freedom of the system and in them the movement of the solvent is represented by harmonic oscillations. Moreover, the two-mode model considers the inner shell in a discrete way, the rest of the solvent molecules being taken as a continuum characterized by its longitudinal optical frequency. In recent years, statistical methods based on numerical simulations, such as the Monte Carlo m e t h ~ d ' ~ - ' ~ and molecular d y n a m i c ~ ' ~ - ~ ~ techniques, have been revealed as very powerful tools in the treatment of chemical systems in solution. Such studies of electron- 12071208 Electron Transfer between Two Fe+ Ions transfer reactions would permit one to treat explicitly many solvent molecules in a discrete representation, without the usual reduction in the number of the degrees of freedom or the common adoption of a harmonic oscillator model.These statistical methods open a very hopeful perspective for the characterization of the solvent nuclear reorganization which has to be produced before the electron transfer itself takes place. However, their actual application is limited because of the enormous number of configurations which ought to be generated in order to obtain configurations appropriate to electron transfer in the intersection region of both potential-energy surfaces. The object of the present work is to carry out a simple Monte Carlo simulation of the process Fe+ + Fe+ + Fe2+ + Feo in aqueous solution within a classical model. We have chosen this system because it takes place at a very fast rate.Thus, we expect to find some configurations which make electron transfer possible in this system in spite of the large number of degrees of freedom that are involved when a reasonable number of water molecules are explicitly considered. Method of Calculation In this paper we have studied the disproportionation process of Fe+ leading to Fe2+ and Feo which takes place in clusters of 12 and 50 water molecules ground the system composed by two Fe+ ions separated by a reasonable distance of 5 A. We have chosen a number of 12 water molecules, since in this way the first hydration shell of six water molecules around each Fe+ ion is represented. On the other hand, calculations with 50 water molecules permit us to obtain a first estimate of the effect of the next solvation shells.As discussed above since internuclear distances and nuclear velocities do not change during an electronic tran~ition,~ the actual electron transfer on the precursor complex occurs at essentially constant nuclear configuration and momentum. This requirement is incorporated into classical electron-transfer theories by postulating that the electron transfer occurs at the intersection of the reactants' (precursor complex) and products' (successor complex) potential-energy surfaces. ' 9 2 * 5-7 This intersection region is reached by a suitable fluctuation in the nuclear configurations of the reactants. To determine which fluctuations favour the electron-transfer process, we have performed the following scheme of calculations in both clusters : (1) minimum-energy structures have been obtained using the Metropolis2' Monte Car10'~-'~ method as a minimization technique.21 In each case the only minimum found represents the solvent configuration of minimum energy in the potential-energy surface of this precursor complex at 0 K.(2) Statistical simulations of clusters at 298 K have been carried out by the Monte Carlo method using the Metropolis algorithm.20 Statistical equilibration has been achieved after 3 x lo5 configurations and statistical analysis has been done over 3 x lo5 or 8 x lo5 additional configurations for the systems with 12 and 50 water molecules, respectively. (3) Pairwise additive potential functions have been used to evaluate the solvation energy of each configuration generated.The MCY 22 potential for the water-water interaction and ab initio analytical potentials generated by us for the Fe2+-water,23 F e + - ~ a t e r ~ ~ and FeO-~ater~~ interactions have been employed. (4) For each one of the solvent configurations generated around the system composed by two Fe+ ions, the solvation energy of this system, and the solvation energy of the system obtained by keeping unchanged the water molecules' coordinates but replacing the two Fe+ ions by Fe2+ and FeO, have been calculated. The difference between these two solvation energies gives the change (AE,,,,) in solvation energy between the potential-energy surface of the successor complex and the potential-energy surface of the precursor complex, for the frozen- solvent configuration.AEsolv could be obtained merely by subtracting the solute-solvent interaction energies after and before the electron transfer, given that the solvent-solvent interactions are kept constant.A . Gonzalez-Lafont et al. 1209 Fig. 1. Minimum-energy structure for the cluster containing 12 water molecules around the Fe+-Fe' system. The values of AEsolv permit us to determine which of the configurations generated are isoenergetic when the electron transfer takes place, and therefore correspond to the intersection region of both potential-energy surfaces. Note, however, that in this work only the change of the solvation energy has been taken into account to define isoenergetic configurations, the interaction between both metal atoms and the energy associated with the electron transfer in vacuum not having been considered.( 5 ) Micro- scopical information of the configurations generated has been obtained through a procedure proposed by one of us.25 In this procedure, a geometrical criterion permits one to classify the configurations into different classes in such a way that each class corresponds to a significant structure of the solvent. In this way, the most significant structures of the system can be identified. Results and Discussion We will first present the results corresponding to the minimum-energy structures at 0 K for the clusters, containing 12 or 50 water molecules, around the Fe'-Fe' system. In both cases, the configuration of the first solvation shell that we have found is very similar, six water molecules being octahedrally disposed around each Fe' ion (see fig.1). Since in this paper we intend to study the electron-transfer process Fe' + Fe' + Fez' + FeO, it is convenient to distinguish between both Fe' ions. This has been done in fig. 1 and thereafter by calling 'reductor' the Fe' ion which loses an electron and transforms into Fez+ and 'oxidant' the Fe+ ion which gains and electron and transforms into Fe". Table 1 presents for each cluster the mean distances Rred and R,, between both Fe' ions and the oxygen atoms of water molecules of the first solvation shell. As it might be expected both mean distances are identical in each cluster. On the other hand, the comparison of the results obtained for the two clusters shows that the first hydration shell is more expanded in the cluster containing 50 water molecules, this fact being due to the effect of the solvent molecules which form the other hydration shells.Following the strategy described in the method of calculation, we have calculated the energy difference, AEsolv, associated with the electron-transfer process which would take place keeping unchanged the geometry structure of the precursor complex at 0 K. The calculated values of AEsolv for the clusters with 12 and 50 water molecules are presented in the third column of table 1. It is interesting to remark that the two values are very different, AESolv being negative for the cluster containing 12 water molecules and positive for the other cluster. This different behaviour can easily be rationalized in the activated-complex formalism, in which the electron transfer occurs at the intersection region between the N-dimensional hypersurfaces of reactants and products, N being the number of independent variables which are necessary to define the nuclear configuration1210 Electron Transfer between Two Fe+ Ions Table 1.Mean Fe+-0 distances and solvation energy difference associated with the electron- transfer process at 0 K for the clusters containing 12 and 50 water molecules n Rred/A Rox/A AE,,,,/kJ mol-' 12 2.30 2.30 - 67.2 50 2.3 1 2.3 1 57.8 of the solvent. According to the relative position of the intersection region with respect to the equilibrium configurations of the precursor and of the successor complexes, the two situations shown schematically in fig. 2 can be c o n ~ i d e r e d .' ~ ~ ~ ~ In the normal-energy region [fig. 2 (a)] the crossing q* is situated between the equilibrium configurations of the precursor (4;) and of the successor (4:) complexes, and this leads to a positive value of AEsolv. On the contrary, in the abnormal or inverted energy region [fig. 2 (b)] the crossing q* is situated to the side of the equilibrium configuration of the precursor complex and ALEsolv is negative. Thus, the results obtained in this work indicate that the clusters containing 12 and 50 water molecules around the Fe+-Fe+ system provide examples of the abnormal and of the normal energy regions, respectively. This different behaviour between the two clusters studied seems to indicate that the water molecules of the first hydration shell and the rest of the water molecules act in opposite senses.To confirm this hypothesis we have decomposed the value of ALEsolv for the cluster containing 50 water molecules into two components : the one corresponding to the 12 water molecules of the first hydration shell and the one which is due to the rest of the water molecules. It is interesting that this decomposition needs only the evaluation of the interaction energy of the solvent molecules with the metal atoms, since the disposition of the water molecules is the same before and after the electron transfer. The values obtained are - 60.6 and 118.4 kJ mol-', respectively. These values show clearly that the water molecules of the first hydration shell afford an exothermic component to ALEsolv, while the rest of water molecules contribute an endothermic component.It is also interesting that the contribution of the water molecules of the first hydration shell is similar to the value of ALEsolv for the cluster containing 12 water molecules (see table 1). This similarity indicates that the presence of outer hydration shells does not essentially change the interaction of the two metal atoms with the first hydration shell. The calculated values of ALEsolv imply that the electron-transfer process between the two Fe+ ions in the clusters containing 12 and 50 water molecules is not possible at 0 K, since, according to the Franck-Condon principle, the electron-transfer process can only be produced in the intersection region between the hypersurfaces of reactants and products.We will now investigate the possibility that solvent fluctuations due to the effect of thermal agitation lead to the existence of some configurations which possess nearly the same energy before and after the electron transfer. To this aim, statistical calculations have been done at 298 K. Table 2 presents, for the clusters containing 12 and 50 water molecules, the mean value of the solvation energy over all the configurations generated at 298 K, along with the solvation energy corresponding to the minimum-energy structure at 0 K, in order to compare both values. The increase in temperature is accompanied by a decrease in the solvation energy owing to the fact that thermal agitation makes possible the existence of higher-energy configurations.To study the effect of such configurations on the electron-transfer process, we have calculated AEsOlv for each of the configurations generated of the clusters containing 12 and 50 water molecules. According to the value of ALEsolv, we have selected three kinds of configurations that we have identified as isoenergetic, highly exoenergetic andA . Gonzalez-Lafont et al. 121 1 I I I I I I I I I I I I I I 4* 4; 4s” (6) Fig. 2. Schematic representation of the normal (a) and abnormal (6) energy regions in the crossing of the potential-energy surfaces of the precursor and successor complexes. Table 2. Solvation energy of clusters containing 12 and 50 water molecules at 0 and 298 K 12 I o 50 f 0 1 298 1 298 - 1261.2 - 1163.4 - 3032.8 -2513.61212 Electron Transfer between Two Fe+ Ions Table 3. Percentage of the total number of configurations belonging to the three groups defined in the text and mean Fe+-0 distances (A) for the first solvation shell in each group n = 12 n = 50 ” Rred ” Rred isoenergetic 5.2 2.34 2.30 3.8 2.28 2.33 highly 44.5 2.26 2.33 4.1 2.28 2.36 highly 6.3 2.36 2.26 62.8 2.32 2.28 exoenergetic endoenergetic highly endoenergetic.The first are those for which IAEsolvl < 6 kJ mol-’, i.e. those for which the solvation energy hardly changes when the electron transfer is produced, the geometry of the precursor complex being kept frozen. A configuration is identified as highly exoenergetic when AEsOlv < -67.2 kJ mol-’ in the case of the cluster containing 12 water molecules or AEsol, < -28.9 kJ mol-’ in the case of the cluster containing 50 water molecules.Finally, the highly endoenergetic configurations are those for which AEsolv > 33.6 or 57.8 kJ mo1-l for the clusters with 12 or 50 water molecules, respec- tively. The choice of these boundary values is rather arbitrary; the indicated values have been selected from the values of AEsolv at 0 K (table 1). Once the configurations belonging to the three abovementioned groups have been identified, we have determined the most significant structure of each group. Table 3 presents the percentage of the total number of configurations belonging to each one of the three groups for the two clusters studied along with the two mean Fe+-0 distances between the ‘reductor’ and the ‘oxidant’ Fe+ ion and the oxygen atoms of the water molecules of the first hydration shell, these mean distances being calculated for the most significant structure of each group.The first thing to observe in table 3 is that the number of isoenergetic structures in the clusters containing 12 or 50 water molecules is not negligible. This fact indicates that the increase in temperature to 298 K strongly enhances the probability of the electron-transfer process to be produced. In each cluster, the greatest number of configurations belong to the group for which AEsolv has the same sign as the one presented at 0 K. Let us now analyse the two mean Fe+-0 distances. This analysis will permit us to understand the way in which the first hydration shell has to be varied starting from the minimum-energy structure for the system to reach an isoenergetic configuration. As has been shown previously, the cluster containing 12 water molecules corresponds to the energy-inverted region [fig.2 (b)]. In this case, the intersection region, q*, is approached when solvent fluctuations displace the minimum-energy structure of the precursor complex, qi, in the opposite direction to the one which would lead to the minimum- energy structure of the successor complex, q:. In good agreement with this prediction, table 3 shows that isoenergetic configurations are reached when the first hydration shell of the ‘reductor’ Fe+ ion is expanded and the first hydration shell of the ‘oxidant’ Fe+ ion is contracted. Obviously, expansion and contraction are more important when highly endoenergetic configurations have to be reached.On the contrary, highly exoenergetic configurations are obtained when the fluctuation of the solvent leads to a contraction and to an expansion of the first hydration shells of the ‘reductor’ and of the ‘oxidant’ Fe+ ions, respectively. On the other hand, the cluster containing 50 water molecules corresponds to the normal energy region [fig. 2 (a)], and the intersection region q* is approached when the minimum-energy structure of the precursor complex, qi, is displaced to that of the successor complex, 4:. Again, the results of table 3 confirm thisA . Gonzalez-Lafont et al. 1213 2( l! 11 % 10 % 5 -1150 -2625 - 2575 -1100 1 -2525 Esolv 1 -2475 -2 -c, L 25 Fig. 3. Histograms of the distribution (YO) of isoenergetic configurations at 298 K against their total solvation energy for the clusters containing (a) 12 and (b) 50 water molecules.prediction, since isoenergetic configurations correspond to a contraction of the first hydration shell of the ‘reductor’ Fe’ ion and to an expansion of the first hydration shell of the ‘oxidant’ Fe’ ion. When a comparison is made between the clusters, one can observe that the variation of the two mean distances from one group to another is appreciably smaller in the cluster containing 50 water molecules. This fact seems to indicate that the fluctuations of the outer hydration shells are not negligible. Any attempt to reduce the effect of solvent fluctuations in the electron-transfer process to the expansion or contraction of the first hydration shell is thus an oversimplification.1214 Electron Transfer between Two Fe+ Ions Now the geometrical aspects of the solvent fluctuations which lead to isoenergetic configurations have been analysed, let us consider the energy dispersion in this group of configurations.Fig. 3 shows two histograms in which the percentage distribution of isoenergetic configurations at 298 K is plotted against their total solvation energy for the clusters containing 12 [fig. 3(a)] or 50 [fig. 3(b)] water molecules. In each case, one can observe that the energy dispersion is very important, this fact being a direct consequence of the large number of degrees of freedom of the system. The same reason explains why the energy dispersion is much more important in the case of the cluster containing 50 water molecules.This increase in the energy dispersion when the number of water molecules is increased again confirms that the outer hydration shells play a non- negligible role for the system to reach an isoenergetic configuration. To obtain a more thorough understanding of the role played by the outer hydration shells in the electron- transfer process we have decomposed the value of AEsolv for each of the isoenergetic configurations of the cluster with 50 water molecules in the same way that we did in the case of its minimum-energy structure at 0 K, i.e. we have separated the effect of the first hydration shell from that of the other hydration shells. An analysis of this decomposition clearly shows that the existence of isoenergetic configurations is usually due to a compensation between both contributions.As an example, let us take the case of an isoenergetic configuration whose total solvation energy is - 25 15.44 kJ mol-'. The decomposition of AEsolv into the two components leads to values of - 116.98 kJ mol-' for the contribution of the first hydration shell and 116.92 kJ mol-' for the remaining hydration shells. The fact that this configuration belongs to the isoenergetic group arises from a compensation of both contributions, in such a way that the total value of AEsolv is only -0.06 kJ mol-'. Finally, some words have to be said about the limitations which are present in this work. First, the accuracy of the pairwise approximation can perhaps be questioned for transition-metal atoms in solution, where the effect of the d-orbital splitting owing to the symmetry of the environment should be considered through the use of an anisotropic pair potential.Secondly, only the solvation energy in the determination of isoenergetic configurations has been considered. However, we have to emphasize that the purpose of this paper has not been to afford quantitative results for the reaction studied, but to show that the models developed up to now, which imply a drastic reduction of the degrees of freedom of the solvent system, do not allow us to give a complete description of the nuclear reorganization of the solvent, prior to the electron transfer. In this sense, we believe that, in spite of the abovementioned limitations, the results presented here open very interesting perspectives for the treatment of electron- transfer processes in solution.Conclusions In this work we have carried out a theoretical study of the solvent intervention in the electron-transfer process between two Fe+ ions. The calculations have been done with a classical model, no quantum effects having been introduced. However, it must be emphasized that we have taken into account a factor not usually considered in this kind of study, since we have explicitly considered a large number of water molecules with all their degrees of freedom. Our statistical calculations show that the usual separation between inner and outer solvent shells does not appear. As a matter of fact, the existence of isoenergetic configurations is generally due to a compensation between solvation contributions from both shells.Any attempt to reduce the effect of solvent fluctuations to the first hydration shell thus leads to an oversimplified model. We have also shown that there exists an important energy dispersion in the configurations which makes possible the electron- transfer process, this dispersion being due to the great complexity of the solvent fluctuations which lead to isoenergetic configurations. Thus we believe that the usualA . Gonzalez-Lafont et al. 1215 treatment in which the degrees of freedom of the solvent are reduced to only two harmonic oscillations is an oversimple representation of the actual movement of the solvent. This work was supported by the Spanish ‘Comision Asesora de Investigacion Cientifica y T h i c a ’ under contract no.3344/83. References 1 B. S. Brunschwig, J. Logan, M. D. Newton and N. Sutin, J. Am. Chem. SOC., 1980, 102, 5798. 2 B. L. Tembe, H. L. Friedman and M. D. Newton, J. Chem. Phys., 1982, 76, 1490. 3 N. Sutin, in Inorganic Reactions and Methods, ed. J. J. Zuckerman (V.C.H., Deerfield Beach, Florida, 4 W. F. Libby, J. Phys. Chem., 1952, 56, 863. 5 R. A. Marcus, J. Chem. Phys., 1965, 43, 679. 6 N. S. Hush, Trans. Faraday SOC., 1961, 57, 155. 7 N. Sutin, Annu. Rev. Nucl. Sci., 1962, 12, 285. 8 V. G. Levich, in Physical Chemistry, ed. H. Eyring, D. Henderson and W. Jost (Academic Press, New 9 V. G. Levich, Adv. Electrochem. Electrochem. Eng., 1966, 4, 249. 1986), vol. 15. York, 1970), vol. 9B, p. 1004. 10 R. R. Dogonadze, A. M. Kuznetsov and V. G. Levich, Electrochim. Acta, 1968, 13, 1025. 11 N. R. Kestner, J. Logan and J. Jortner, J. Phys. Chem., 1974, 78, 2148. 12 I. Webman and N. R. Kestner, J. Chem. Phys., 1982, 77, 2387. 13 E. Buhks, M. Bixon, J. Jortner and G. Navon, J. Phys. Chem., 1981, 85, 3759. 14 J. P. Valleau and S. G. Whittington, in Statistical Mechanics, ed. B. J. Berne (Plenum Press, New York, 15 J. P. Valleau and G. M. Torrie, in Statistical Mechanics, ed. B. J. Berne (Plenum Press, New York, 16 W. W. Wood, in Physics of Simple Liquids, ed. H. N. V. Temperley, J. S. Rowlinson and 17 J. Kushick and B. J. Berne, in Modern Theoretical Chemistry: Part B, ed. B. J. Berne (Plenum Press, 18 A. Warshel, J. Phys. Chem., 1982, 86, 2218. 19 A. Warshel and J. K. Hwang, J. Chem. Phys., 1986, 84, 4938. 20 N. A. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. Teller and E. Teller, J. Chem. Phys., 1953, 21 K. S. Kim, M. Dupuis, G. C. Lie and E. Clementi, Chem. Phys. Lett., 1986, 131,451. 22 0. Matsuoka, E. Clementi and M. Yoshimine, J. Chem. Phys., 1976, 64, 1351. 23 A. Gonzalez-Lafont, J. M. Lluch, A. Oliva and J. Bertran, Znt. J. Quantum Chem., 1986, 30, 663. 24 A. Gonzalez-Lafont, J. M. Lluch, A. Oliva and J. Bertran, Znt. J. Quantum Chem., 1988, 33, 77. 25 0. Tapia and J. M. Lluch, J. Chem. Phys., 1985, 83, 3970. 26 M. D. Newton and N. Sutin, Annu. Rev. Phys. Chem., 1984, 35, 437. 1977), vol. A, chap. 4. 1977), vol. A, chap. 5. G. S. Rushbrooke (North-Holland, Amsterdam, 1968), chap. 5. New York, 1977), vol. 6. 21, 1087. Paper 7/00045F; Received 22nd June, 1987