摘要:

Date 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 1981 1981 1982 1982 FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Ion-Solvent Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Photoelectrochemistry High Resolution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Electron and Proton Transfer Oxidation 413 Volume 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 * 66 67 68 69 70 71 72 73 74 * Not available; for current information on prices, etc., of available voiumes, please contact the Marketing Officer, Royal Society of Chemistry, Burlington House, London WI V OBN stating whether or not you are a member of the Society.Date 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 1981 1981 1982 1982 FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Ion-Solvent Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Photoelectrochemistry High Resolution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Electron and Proton Transfer Oxidation 413 Volume 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 * 66 67 68 69 70 71 72 73 74 * Not available; for current information on prices, etc., of available voiumes, please contact the Marketing Officer, Royal Society of Chemistry, Burlington House, London WI V OBN stating whether or not you are a member of the Society.

ISSN:0301-7249    

DOI:10.1039/DC98274FX001

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982,74, 7- 15 THE SECOND R. A . ROBINSON MEMORIAL LECTURE Electron, Proton and Related Transfers* BY RUDOLPH A. MARCUS Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125, U.S.A. Received 14th September, 1982 Past and current developments in electron and proton transfer and in related fields are described. Broad classes of reactions have been considered from a unified viewpoint which offers a variety of experimental predictions. This introductory lecture considers various aspects of this many-faceted field. A simple equation is given for a highly exothermic electron-transfer reaction. 1. INTRODUCTION In the early days of electron transfer one text which I found particularly helpful was Robinson and Stokes, Electrolyte Solutions.' It provided an overview, as well as a detailed and current picture of electrolyte solutions, to one newly arrived in the field.It is a real pleasure to acknowledge in this Memorial Lecture my debt to Professor Robinson. In the intervening years a number of Faraday Society Discussions related to the subject matter of the present Discussion have been held, including those on Oxidation- Reduction Reactions (1960), Proton Transfer Processes (1 965) and Electrode Reactions (1 968). The present Discussion embraces all three and so emphasizes a trend whereby a formalism has been developed which attempts to unify the three different fields and which has now been extended to an increasingly broad class of reactions in chemistry. We survey some of the developments in this area in the present lecture.2. WEAK-OVERLAP ELECTRON TRANSFERS One of the virtues of studying simple weak-overlap electron transfers has been the absence of bond-breaking and bond-forming processes, with all their attendant un- certainties regarding the potential-energy surface in the transition-state region. In this way it was possible, with approximate models for the coordination shell and for the solvent outside it, to allow for the reorganization prior to and following the elec- tron-transfer Libby, in his pioneering and stimulating suggestion on the application of the Franck-Condon principle to electron tran~fer,~ thought of the co- ordination shell and other changes as arising from a vertical transition, as in spec- troscopy, rather than of a prior re~rganization.~ We introduced, instead, a prior and post reorganization.The ensuing history of the field has been nicely summarized in several recent articles, for example ref. (5)-(11). The apparent simplicity of the weak-overlap electron transfer reaction, and certainly of the model, permitted a detailed a n a l ~ s i s , ~ ~ - ' ~ of topics such as the effect of driving force (AGO) on the reaction rate, effect of molecular parameters and of * Contribution no. 6728 from Caltech.8 ELECTRON AND PROTON TRANSFER solvents (when not specifically interacting with the reactants) on the rate,* relation between cross-reaction rates and those of isotopic exchange reactions, relation of homogeneous reaction rates to charge-transfer spectra, l8 chemiluminescent electron tran~fers,'~ relation to electrochemical electron transfers, 7 effect of driving force (activation overpotential) on the electrochemical rate constant and effect of amount of charge transferred.The interaction between experiment and theory in these fields has provided an exciting experience, a source of pleasure and occasionally of dismay. In the late 1950s and early 1960s I visited Brookhaven National Laboratory and spoke often with Dick Dodson and Norman Sutin, who were doing pioneering experiments in the field. Those visits were particularly stimulating. Taube was of course making giant strides, but a t that time I did not have much contact with him. Chemistry, of course, embraces much more than electron-transfer reactions, and it became natural to think about the relation of the formalism developed for weak- overlap electron transfers to other more complicated rea~tions.~O-*~ The main in- gredients of the formalism include work terms, w' and - wp, not necessarily coulombic, for bringing the reactants together and for separating the products, the intrinsic barrier i2/4 (additively related for the cross-reaction to those of isotopic exchange reactions), and the standard free energy of reaction of the elementary electron-transfer step, AGO 12-17 9 k z K Z exp (- AGz/kT) (1) where, in a classical treatment of the nuclear motion, with AG"' = AGO + wp - w'.IC describes the non-adiabaticity of the reaction ( K x 1 for an adiabatic reaction) and 2 is the bimolecular collision frequency.Analogous equations are obtained for unimolecular reactions, for electrochemical reactions, and for reactions at an interface, in which each reactant is in a different phase. Z is replaced by the appropriate analogue in each case, and in the electrochemical case AGO is replaced by the activation overpotential. 3. OTHER CLASSES OF REACTIONS Any extension of the concepts of electron transfer to other classes of reactions must be aware of the differences. In atom-transfer reactions, for example, simultan- eous bond breaking and forming occur and cannot be treated by a pair of intersecting harmonic-oscillator potential-energy or quadratic free-energy surfaces. For this reason a rather different simple model was con~idered,~~ one which originated with Harold Johnston (BEB0).24 When further simplified (with potential energies replaced in an intuitive way by free energies-forward and reverse rate constants obey microscopic reversibility) 23 and with work terms added, this yielded eqn (l), with AGZ now given by ;1 AGO' 1 AG"' A@ = wr + .-- + - + - - In cosh y , y = 2AGo'(ln2)/A 4 2 Y and with i2 having the additivity property as before.Provided that lAGo'1/12 is less than and not too close to unity, this expression is well represented by the slightly simpler quadratic expression, eqn (2).23 Eqn (3) or, * A useful recent summary is given by M-S. Chan and A. C. Wahl in J. Phys. Chem., 1982,86,126. 7 An up-to-date account is given by M. J. Weaver in J. Phys. Chem., 1980,84, 568.R . A . MARCUS 9 more usually eqn (2), has now been applied to atom transfers, proton transfers, methyl-radical transfers, hydride-ion transfers 25 and concerted proton transfers.26 An excellent review of the field has been given by Alber~.~' In the electrochemical proton-transfer case two alternative opinions, discussed here by Krishtalik,28 have arisen, and an effort at a unifying theory which included both as limiting cases was made.29 Several central questions are those such as the following: how far does the proton jump, and hence how much solvent rearrangement has to occur? (In electron transfers, the centre-to-centre jump distance is always quite large, even when the reactants are in van der Waals' contact.) When is the reaction coordinate in the transition-state region the protonic coordinate and when, as expected for sufficiently highly exothermic and thereby barrierless reactions, is it the intermolecular separation coordinate for the two reactants ? 4.QUANTUM EFFECTS Quantum corrections to eqn (1) and (2) are relatively minor at room temperature for typical reactions in the ' normal region ' (1AG"'I < A), for example see ref. (30) and (31). The corrections become larger in the inverted region and at low temper- atures. The classical eqn (1) and (2) have a simplicity which facilitates their applic- ation to and testing by experiment. In some cases, for example in the cross-relation between the rate constants of cross-reactions and those of isotopic exchanges, it has been possible as a result to eliminate by cancellation the individual molecular proper- ties, and so relate the rate constants to each other and to the equilibrium constant (the ' cross-relation To illustrate some of the features of the quantum-mechanical rate expression we consider for simplicity the case of a very highly exothermic non-adiabatic reaction.In the quantum theory of non-adiabatic electron transfer reactions Levich and Dogonadze 32 adapted to the problem, as they pointed out, an earlier result of Kubo and Toyozawa 33 developed for other processes. The theory and its ensuing develop- ment employs what is now known as the theory of radiationless tran~itions.~-~l The rate constant for electron transfer between fixed sites is given by k = (F.C.) (4) where we use the notation of Bixon and Jortner in this discussion. (F.C.) is the Franck Condon factor and Y the matrix element for the electron transfer transition. One problem in electron transfers in polar media, as compared with radiationless transitions involving only intramolecular vibrations, is that the former can have huge entropies of reaction.The latter are not adequately modelled by quadratic potential- energy functions, even though a suitable and applicable free-energy function may be fairly quadratic as a function of some charging parameter. As a result, an approach has been adopted in which the classical expression eqn (2), which does allow for large possible entropy changes, is introduced into the solvent contribution in (F.C.), to replace a quadratic potential-energy expression there.34 For the case in which the intramolecular vibration frequencies are high enough, and the reaction sufficiently exothermic, (F.C.) is given by o3 S" exp[ - (AGO + 3Lo + ~kco)~/43L~kT] (47d0kT)"~ (F.C.) = 2 e-' v = o where, for notational brevity, the oscillators have been taken to have a common10 ELECTRON A N D PROTON TRANSFER angular frequency co. S is the contribution A, of these vibrations to the A of section 2, in units of ha, and A, is the contribution to 3, of the solution outside the coordin- ation shell.One sees that the effect of the high-frequency vibrations in this highly exothermic case is, like lo, to absorb large amounts uho of the excess energy. In effect, it reduces this exothermicity and makes the reaction faster than would be the case if none were absorbed (v = 0). Eqn (5) is, of course, still a little cumbersome, although if each of the molecular parameters is known or guessed at it is rapidly and painlessly computed.There are approximations which we can introduce, which reduce the sum in eqn (5) to a single term: We replace u! by a continuous function, the gamma function, T(u + l), replace the sum by an integral ovei- u, and treat the integrand as a Gaussian about some maximum, which for convenience we will denote by u itself. If I'(u + 1) is then replaced by Stirling's formula, one obtains * Svexp[ - (AGO + lo + ~ h c o ) ~ / 4 l ~ k T ] r ( v + i)tiw (F.C.) GS e-s where v is the solution of a transcendental equation, which when approximated by one iteration simplifies to A comparison of eqn (5) and (6) is given in table 1, where the approximate eqn (6) is seen to yield reasonable agreement over the range of parameters studied. When the TABLE 1 .-COMPARISON OF ' EXACT ' AND APPROXIMATE FRANCK-CONDON FACTORS -AGi * 2 0 exact approx.b approx.c /cm-' /crn-' S /10l2 cm-' cm-' cm- 14 500 500 0.655 1 .o 0.9 0.8 lo00 0.600 1.6 1.8 1.7 1500 0.545 3.3 3.7 3.3 2000 0.490 6.4 8.0 6.4 2500 0.435 12.4 18.3 12.4 a AG E AGO + lo; ha = 1350 cm-', T = 298 K; exact : eqn (5) and approx.3 eqn (6), and (7); see text. transcendental equation for v is solved exactly and when Stirling's formula is avoided, the last column of table 1 is obtained. The linear dependence of In k on AGO for these highly exothermic reactions, the well known energy-gap law,35 is also seen from eqn (6) and (7) (cf. steepest descent type of derivation of that law): replacement of AGO + vhw + A.in the exponent in eqn (6) by the logarithmic term in eqn (7) largely removes that AGO, while linear dependence on v of the exponent of S"/T(u + l), i.e. in (S/v)", and roughly of u on AGO via eqn (7) yields the gap law. An equation similar to but, as the authors note, different from our eqn (6) has been obtained in ref. (36). Eqn (6) and (7) are applicable, incidentally, to recent results relating the rate of radiationless transitions to the energy gap and to solvent effects.36 Apart from an * More precisely, the ko in the denominator in eqn (6) has a cofactor (1 + 2&kT/~(hw>~)+, which is close to unity for the systems in table 1. (u was typically between 6 and 10.)R. A . MARCUS 11 entropic term -(AGO + A,) is the energy of the O+O transition, and so the equations relate the rate constant to the frequency of that transition.We turn next to other experimental results in the inverted region. 5. THE INVERTED REGION Outside the range ]AGO'] < 3, there is a considerable difference between eqn (2) and (3). Whereas eqn (2) shows a decrease of rate with increasing driving force in the inverted region (i.e. in the region where IAG"'l/A > I), eqn (3) displays no such phenomenon. This difference is readily understood when one considers how the potential surface leading to eqn (2) differs from that leading to eqn (3). While the quantum corrections to eqn (1) in the inverted region significantly reduce the inverted effect, unless the relevant vibration frequencies are sufficiently low, they do not eliminate it.The inverted effect was predicted in 1960. Its analogue in radiationless transitions, the energy-gap law of Siebrand,35 is well known. The search for examples of the inverted effect in electron-transfer reactions during the ensuing twenty-odd years has, perhaps because of its novelty, been a very active one. Examples where the effect has been reported have sometimes involved a two-phase reaction, e.g. where one reactant is in a micelle 6p37*38 or is a semiconductor electrode 39 and the other is outside, or a reaction in frozen rnedi~m.~' (To see the second of these4' amid a lot of scatter it is necessary to divide the reactants into The effect has also been invoked to explain the apparent slowness of the back reaction in bacterial photo~ynthesis.~~ In this reaction the bacterial chlorophyll dimer cation and bacterial pheophytin anion are fixed, presumably, rather than mobile, in the membrane.The inverted effect has also been invoked to explain the relatively larger rate constant estimated for forming a triplet state in this back reaction, compared with that estimated for the back reaction forming the ground-state singlet.42 In the case of bimolecular reactions in solution Bard has ascribed to the inverted effect the nearly 100% yield of electronically excited products he observed in some reactions, rather than of ground-state Creutz and Sutin earlier reported vestiges of the inverted region.44 In the reaction of electrons with solutes in hydro- carbons, AGO was varied by changing the solvent, and the results provide some evi- dence for an inverted e f f e ~ t .~ ~ ' ~ ~ Apart from these examples the search for the effect in bimolecular reactions in solution has yielded, instead, a constant diffusion-controlled value at very negative AGO, as exemplified in ref. (48)-(54). Effects which can thwart the observation of an inverted region in bimolecular solution reactions are several-fold : masking by diffusion control, the existence of alternative mechanisms, such as reaction via exciplexes,s5 atom transfer, or formation of electronically excited products, which reduces the magnitude of AGO for the elementary step. Picosecond studies have been proposed to reduce the diffusion-control effect .56 The atom-transfer alternative can be reduced in attractiveness by keeping the reactants physically separated in different phases or in a frozen medium, by holding them apart by rigid chemical bonds or by using a suitable choice of reactants with atoms which cannot undergo atom transfer.There is a predicted relationship between the inverted region and the high- frequency tail of the related charge-transfer spectrum, when the weak-overlap and Condon approximations can be made for both.s6 When that high-frequency tail is not obscured by a new absorption band it should be quite revealing of what to expect for the related thermal electron-transfer rate constant in the inverted region in this weak-overlap case.12 ELECTRON A N D PROTON TRANSFER 6. EFFECT OF SEPARATION DISTANCE One of the newer areas of interest has been the effect of separation distance r on the rate of electron transfer.This effect, which provides a connection between geo- metry and rates, has been of considerable interest in biological electron transfers, e.g. ref. (57). Here, reactants, more or less fixed in a membrane may not have the close contact that they do in solution, and their electron transfer rate may be dominated by this factor. Efforts are being made, by building rigid bridges, for example, to study the effect of r on the rate. Another approach involves the study of reactions in frozen media: The nearest reactants react first, and the kinetics have a peculiar dependence on time: when the rate constant at an edge-to-edge separation distance Y, k(r), behaves as the unreacted fraction of one reactant varies with time roughly as (In k 3 ) 3 / ~ 3 .[A simple derivation is given in ref. (%).I k; = k,exp(d,), where R, is the distance of closest approach. In a reaction between an aromatic molecule and an aromatic anion 0: has been estimated 59 to be of the order of 1.1 The application of this result to a biological electron transfer between cytochrome c and cytochrome c peroxidase provides an interesting example of a current and early connection between the two fields. From a knowledge of the structure of each com- ponent and estimates based on bringing the opposite charges near each other and aligning the hydrogen bonds, the haem-haem edge-to-edge distance r has been esti- mated to be ca. 16.5 A.6o (The latter may be compared with the value of 14.3 8, based on a quite different type of estimate, fluorescence quenching.)61 The minimum rate of electron transfer between the two haems (minimum because this step may not be the rate-limiting one) is ca.lo4 s-'. (T. Yonetani and J. Yandell, personal communi- cation.) The maximum rate of electron transfer is, at close contact, ca. 1013 s-'. Multiplying this by exp(-Crr) yields a maximum calculated rate constant of 1013 x or lo5 s-l, which is to be compared with the experimental lower bound of lo4 s-I.* However, a not much higher ct, or a not much larger r, would not fit in. The point of this exercise is not to compare the current extent of quantitative agreement but to indicate that when better esti- mates of a, of haem-haem edge-to-edge separation distance, of orientation effects, and of rate constants become available, there will be interesting and useful comparisons to be drawn.A word about the assumed 1013 s-' for the maximum value of the k, is perhaps in order : J. R. Miller (personal communication) estimated from his experimental data that the maximum ko for reactions between molecules and/or ions was ca. 10I4 s-', or somewhere between 1013 and lOI4 s-l. (He used data obtained near the maximum of the In k against AGO plot. He extrapolated yield against time data some six orders of magnitude from data obtained over nine orders of magnitude.) I have chosen a value of loi3 s-' so that eqn (4) will yield at Y = 0 the maximum it can be, which is the frequency of nuclear motion, 1013 s-l. If, experimentally, k , proves to be 1014 s-l, eqn (4) can still be used but will break down for r values below that given by exp (-w) = 1013/1014 = 0.1, the reaction then becoming adiabatic at that r (ca.2 A). k(r) = koe-ar (4) Thus these two rate constants are consistent. 7. THIS DISCUSSION The papers in this Discussion describe many facets of this field of electron and Many of the points touched upon in the previous sections, and more, * Also included is a nuclear tunnelling factor which typically varies from 1 to ca. 5 at room tem- proton transfer. perature. The classical factor is of order lOI3 s-'.R . A . MARCUS 13 are well illustrated by the papers of this symposium. Kuznetsov, Kuznetsov and Ulstrup, Bixon and Jortner, and Friedman and Newton consider various quantum- mechanical and other theoretical aspects of the problem.The calculations focus on the polarized solvent, the vibrations and the electronic structure, with different emphases. The comparison of the theory described earlier with experiments on electron exchange reactions is discussed by Brunschwig et al. Experimental work on homo- geneous electron transfers is presented by Bruhn et al. on the effect of added salts, by Amouyal et al. on electron transfer from various photoexcited organic molecules, and by Huppert et al. on intramolecular electron transfer. Electron transfer at interfaces is discussed by SavCant and Tessier, who describe the relation between the observed dependence of the electrochemical transfer co- efficient on overpotential and the theory of section 2, and by Willig and CharlC, who treat electron transfers between ions and molecules adsorbed on organic electrodes.Proton transfers are treated at electrodes by Krishtalik (the hydrogen evolution reaction), who discusses some of the controversy referred to earlier. In solution they are treated by Limbach et al. (isotope effects and double proton transfers) and by Albery, who analyses the concerted proton-transfer problem. Hydride transfers are also discussed in relation to the theory of section 2 by Roberts et al., while Caldin et al. examine the effect of polar solvents on hydrogen-atom transfers. The complexity of proteins and of biological molecules generally requires an in- creasingly detailed knowledge of the structure, to make the interpretation of the electron-transfer rates as meaningful as possible.Structural and other aspects are described for cytochrome c by Roberts et al. The electron transfer reaction of cyto- chrome c on a modified metal electrode (adsorbed organic layer), and its relation to physiological redox reactions is discussed by Eddowes and Hill. Homogeneous electron transfers of cytochrome c, with emphasis on entropy and volume of activ- ation, are treated by Heremans et al. Proton transfers in biological systems are equally important, and are discussed in the paper of Rich (together with electron transfers), Gavach et al., and Kell and Hitchens. The coupling between electron transfers, proton transfers and ATP synthesis represents, of course, a particularly important problem, and one which has been the subject of different views. These are touched upon in this part of the Discussion.The general area reflects the trend towards an increasing knowledge of structure, kinetics and thermodynamics, and increasingly fast and accurate experi- mental methods. Perhaps guided by results obtained in the simpler systems described in earlier parts of this Discussion, we can look forward to striking developments. The Organizing Committee is to be congratulated for having arranged such a broad and interesting programme. The posters, contributions which 1 have not had a chance to see beforehand but whose content is hinted at by the titles, add their strength to this broad Discussion of Electron and Proton Transfers. The Organizing Committee has done its part. It remains for us to begin.Various aspects of my research in this field have been supported by the National It is a pleasure to acknow- I am indebted to Paul Siders for his helpful suggestions and Science Foundation and by the Office of Naval Research. ledge their support here. for providing the results in table 1. R. A. Robinson and R. H. Stokes, Electrolyte Solutions (Butterworths, London, 1955). ’ R. A. Marcus, J. Chem. Phys., 1956,24, 966. R. A. Marcus, Discuss. Faraday Soc., 1960, 29, 21. W. F. Libby, J . Phys. Chem., 1952, 56, 63.14 ELECTRON AND PROTON TRANSFER P. P. Schmidt, in Electrochemistry (Specialist Periodical Report, The Chemical Society, London, 1975), vol. 5. J. Ulstrup, Charge Transfer Processes in Condensed Media, in Lecture Notes in Chemistry (Springer-Verlag, Berlin, 1979).D. Devault, Quart. Rev. Biophys., 1980, 13, 387. N. Sutin, Prog. Inorg. Chem., 1983,30,441. R. Dogonadze, T. Marsagishvili and G. Chonishvili, J. Chem. SOC., Furuday Trans. I , to be pub1 ished. A. M. Kuznetsov, Furaduy Discuss. Chem. SOC., 1982, 74, 49. M. Bixon and J. Jortner, Furuduy Discuss. Chem. SOC., 1982, 74, 17. l2 R. A. Marcus, J. Phys. Chem., 1963, 67, 853, 2889. l 3 R. A. Marcus, Annu. Rev. Phys. Chem., 1964, 15, 155. l4 R. A. Marcus, J. Chem. Phys., 1965, 43, 679. l6 R. A. Marcus, Phys. Chem. Sci. Res. Rep., 1975, 1, 477. l7 R. A. Marcus, ONR Technical Report No. 12, Nonr 839 (09) (1957), reprinted in Special Topics R. A. Marcus, Electrochim. Actu, 1968, 13, 995. in Electrochemistry, ed. P. A. Rock (Elsevier, New York, 1977), p.180. N. S. Hush, Prog. Inorg. Chem., 1967, 8, 391. l9 R. A. Marcus, J . Chem. Phys., 1965, 43, 2654; 1970, 52,4803. ’* N. Sutin, Proc. Exchange Reactions Symp., (Upton, N.Y., 1965), p. 7. 21 A. Haim and N. Sutin, J. Am. Chem. SOC., 1966, 88, 434. 22 R. R. Dogonadze, A. M. Kuznetsov, and V. G. Levich, Sou. Electrochem. (Engl. Trunsl.), 1967, 23 R. A. Marcus, J. Phys. Chem., ;968, 72, 891. 24 H. S, Johnston and C. Parr, J. Am. Chem. SOC., 1963, 85, 2544. 25 R. M. G. Roberts, D. Ostovic and M. M. Kreevoy, Furuduy Discuhs. Chem. SOC., 1982,74,257. 26 W. J. Albery, Furuduy Discuss. Chem. SOC., 1982, 74, 245. 27 W. J. Albery, Annu. Rev. Phys. Chem., 1980,31, 227. z8 L. I. Krishtalik, Furuduy Discuss. Chem. SOC., 1982, 74, 205. 29 R. A. Marcus, Proc. Third Symp.Electrode Processes, ed. S. Bruckenstein, J. D. E. McIntyre, 30 P. Siders and R. A. Marcus, J. Am. Chem. Soc., 1981, 103, 741. 31 E. Buhks, M. Bixon, J. Jortner and G. Navon, J. Phys. Chem., 1981, 85, 3759. 32 V. G. Levich and R. R. Dogonadze, Coll. Czech. Chem. Commun., 1961, 26, 193; Trans]., 0. 33 R. Kubo and Y. Toyozawa, Prog. Theor. Phys., 1955, 13, 160. 34 N. Kestner, J. Logan and J. Jortner, J. Phys. Chem., 1974,78, 2148. 35 W. Siebrand, in The Triplet State, ed. A. B. Zahlan (Cambridge University Press, London, 1967) 36 J. V. Caspar, B. P. Sullivan, E. M. Kober and T. J. Meyer, Chem. Phys. Lett., 1982, 91, 91. 3, 648. B. Miller ed. E. Yeager (Electrochemical Society, Princeton, New Jersey, 1980), p. 1. Boshko, University of Ottawa, Ontario, Canada. 31.37 A. J. Frank, M. Gratzel, A. Henglein and E. Janata, Ber. Bunsenges. Phys. Chem., 1976, 80, 294. 38 A. J. Frank, M. Gratzel, A. Henglein and E. Janata, Ber. Bunsenges. Phys. Chem., 1967,80,547, 39 F. Willig and K.-P. Charle, Furuduy Discuss. Chem. Soc., 1982, 74, 141. 40 J. V. Beitz and J. R. Miller, J. Chem. Phys., 1979, 71, 4579. 41 J. R. Miller, personal communication. 42 R. Haberkorn, M. E. Michel-Beyerle and R. A. Marcus, Proc. Nut1 Acud. Sci. USA, 1979, 76, 43 W. L. Wallace and A. J. Bard, J. Phys. Chem., 1979, 83, 1350. 44 C. Creutz and N. Sutin, J. Am. Chem. SOC., 1977, 99, 241. 45 A. 0. Allen, T. E. Gangwer and R. A. Holyrod, J. Phys. Chem., 1975, 79, 25. 46 A. Henglein, Can. J . Chem., 1977, 55, 2112. 47 S. Lipsky, J. Chem. Educ., 1981, 58, 93. 48 D. Rehm and A. Weller, Isr. J. Chem., 1970, 8, 259. 49 L. V. Romashov, Yu. I. Kiryukhin and Kh. S. Bagdasar’yan, Dokl. Phys. Chem. (Engl. Trunsl.), 4185. 1976,230,961. R. Scheerer and M. Gratzel, J. Am. Chem. Soc., 1977, 99, 865. R. Ballardini, G. Varani, M. T. Indelli, F. Scandola and V. Balzani, J. Am. Chem. SOC., 1978, 100,7219. 52 V. Balzani, F. Bolletta, M. T. Gandolfi and M. Maestri, Top. Curr. Chem., 1978,75, 1 . 53 J. Eriksen and C. S . Foote, J. Phys. Chem., 1978, 82, 2659. 54 E. Vogelmann, S. Schreiner, W. Rauscher, and H. E. A. Kramer, 2. Phys. Chem. (Frunkfurr am Main), 1976, 101, 321.R. A . MARCUS 15 55 A. Weller and K. Zachariasse, Chem. Phys. Lett., 1971, 10, 590. 56 R. A. Marcus and P. Siders, J. Phys. Chem., 1982, 86, 622. 57 J. J. Hopfield, Proc. Nail Acad. Sci. USA, 1974, 71, 3640. 59 I. V. Alexandrov, R. F. Khairutdinov and K. I. Zamaraev, Chem. Phys., 1978, 32, 123. 6o T. L. Poulos and J. Kraut, J. Biol. Chem., 1980,255, 10322. 61 J. J. Leonard and T. Yonetari, Biochemistry, 1974, 13. 1465, M. Tachiya and A. Mozumder, Chem. Phys. Lett., 1974, 28, 87.

ISSN:0301-7249    

DOI:10.1039/DC9827400007

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 74, 17-29 Quantum Effects on Electron-transfer Processes B Y MORDECHAI BIXON AND JOSHUA JORTNER Department of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel Received 2nd June, 1982 The non-adiabatic multiphonon theory of electron transfer provides a complete description of nuclear tunnelling and final-state excitation phenomena, which modify the classical Marcus theory. Ultrafast electron-transfer processes, where the electronic process competes with medium-induced vibrational relaxation, cannot be handled by the conventional theory, being amenable to description in terms of quantum-mechanical models. 1. INTRODUCTION During the last decade it has been recognized that a unified theoretical framework exists for the description of non-radiative, intermolecular, electronic, relaxation phenomena in condensed phases, which span a broad spectrum of processes in the areas of solid-state physics, physical chemistry and bi0physics.l All these diverse non-radiative phenomena can be described in terms of a relaxation process between nuclear states which correspond to two distinct zero-order electronic configurations and whose energy conservation is ensured by absorption and emission of phonons.For example, electron-transfer (e.t.) processes between ions in ~ o l u t i o n ~ - ~ and in biophysical ~ y s t e m s , ~ ’ ~ which are the subject matter of the present paper, can be envisaged as a non-radiative process within a “ supermolecule ” consisting of the donor (D) and the acceptor (A) pair, together with the entire medium, the relevant electronic states being DA and D+A-.7-13 The following ingredients of the relaxation process are central : (1) The electronic coupling V, which involves the two-centre, transfer integral between D and A.(2) The modification of the nuclear vibrational states of the “ supermolecule ” by the change in the electronic states. These involve the changes in the intramolecular vibrational states specified by the coordinates qc of the D and A centres, as well as the response of the exterior polar medium characterized by the coordinates 4m to the change in the charge distribution. The microscopic rate constant, W,, for e.t. from a single vibronic state xI(Q), where Q = qc,qm, of vibronic states of D+A- characterized by the energies Ej, can usually be described in terms of a non-adiabatic f~rmalism,~-’~ being given in terms of a product of an electronic term 1VI2 and a nuclear Franck- Condon factor The basic implicit assumption is that medium-induced vibrational relaxation and18 ELECTRON TRANSFER vibrational excitation processes in the initial ( / I > } manifold are fast on the timescale of the electronic-vibrational process,14J5 i.e.wZ @ YVR (1 -3) where y V R z 1011-1012 s-l is a typical vibrational relaxation rate.' One can then perform a separation of the timescales for " fast '' vibrational relaxation and the " slow " electronic-vibrational process. The conventional transition probability, W, for e.t. is given by the thermal average of the microscopic rate where F = Zrl 2 exp( -Ez/kT) Fz (1 Sa) I is the thermally averaged Franck-Condon factor and 21 = 2 exp(-Ezlkr) I (1.5b) is the partition function for the initial manifold.Two limiting cases of the non-adiabatic multiphonon rate can be distinguished. (a) The low-iemperature limit is exhibited when the thermal energy kT is low relative to all the characteristic relevant vibrational energies kc^)^). The low- temperature rate is Wz = ,,. This low-temperature limit corresponds to nuclear tunnelling from the vibrationless nuclear state of the initial-state potential surface to the final vibronic states, which are quasidegenerate with it.7 The temperature range where the rate is temperature independent can be specified by the ~ o n d i t i o n ~ ~ J ~ kT < fiW0,/5 for all the relevant vibrational frequencies mK. (b) The high-temperature limit: when the thermal energy kT exceeds all the characteristic frequencies, a classical treatment of P can be utilized which amounts to the replacement of the discrete sums in eqn (1.4) by multidimensional configura- tional integralsI8 where U,(Q) and U,(Q) are the nuclear potential surfaces of the initial and final states, respectively.The high-temperature rate [eqn (1 A)] may be written as The Marcus t h e ~ r y ~ - ~ provides a complete description of the classical nuclear contribu- tion FC to the high-temperature e.t. rate within the framework of the harmonic approximation2 -4*7 Fc(AE;Er) = (4nErkT)-) exp[-(AE + Q2/4E,kT] (1.8) where AE is the electronic energy gap,' while E, is the sum of the classical solvent (Es) and intramolecular inner-sphere (E,) reorganization energies.The only quantum phenomenon surviving in the high-temperature non-adiabatic rate [eqn (1.7)] is the pre-exponential electronic coupling. In order to assess the effects of nuclear contributions to the e.t. rate one has to focus attention on the relevant frequencies { f i ~ ~ ) = (kw,}, {k~,,,}, which incorporate both molecular modes {kcoc> and medium modes {kw,>. For e.t. in solution some ofM. BIXON AND J . JORTNER 19 the medium modes are low ( h m , , , ~ 1 cm-l),19 while for ice hmmz 180 cm-1.20 Molecular-type frequencies are kmC z 200-500 cm-l for the first coordination layer of solvated ionss-13 and hmc z 250-1000 cm-l for intramolecular vibrational modes of molecules D and/or A, e.g.porphyrin rings involved in photosynthe~is.~~~ It is apparent that the low-temperature limit cannot be accomplished for e.t. in solutions, where exterior reorganization provides a dominant contribution. On the other hand, for e.t. in low-temperature aqueous glasses 21 and in low-temperature photo- synthetic reaction centres 5*6 the low-temperature limit is realized. Between these two extremes diverse cases can be encountered. In examining quantum effects on e.t. rates two distinct classes of phenomena will be considered: (A) Quantum effects on conventional e.t. rates occurring from a thermally equilibrated initial manifold. These involve: (1) electron coup1ir.g: for non- adiabatic e.t., W K I YI2, (2) nuclear tunnelling effects: the breakdown of the classical limit for the F factor [eqn (1.8)] provides a signature of nuclear tunnelling effects on the e.t.rate. (B) Quantum effects originating from the breakdown of the separability of time- scales for the slow electronic e.t. process and the fast vibrational relaxation. The conventional e.t. rates, eqn (1.4) and (1.5), are applicable only under some restrictions. For ultrafast e.t. processes competing with vibrational relaxation, the entire conceptual framework of the theory has to be modified. 2. QUANTUM EFFECTS ON CONVENTIONAL ELECTRON The unimolecular reaction probability W(R), eqn (1.4), for a spatially fixed DA pair at a centre-to-centre distance R can be utilized to provide a simple expression for the bimolecular e.t. rate, k. We shall now consider the manifestations of quantum tunnelling on e.t.in solution when the thermal energy kT is sufficiently high to warrant a classical treatment of the solvent, while kT is sufficiently low so that a quantum treatment of the molecular modes is required. Within the framework of the harmonic approximation for all nuclear modes, the Franck-Condon factor is TRANSFER I N SOLUTIONS The parameters involved are: (a) the electronic energy gap AE, (b) the Marcus reorganization energy E, of the exterior solvent and (c) the parameters specifying the first coordination layer, which involve the vibrational frequencies (mK‘> and (mK”) of the first coordination layer in the initial and in the final states, respectively. (d) The changes in the equilibrium configuration {ArK) in the first coordination layer originate from totally symmetric modes, which are expressed for the case of octahedral complexes in reciprocal frequency (cm) units (AdK)‘ = 6(Ar,)’m/h.The calculation of multidimensional Franck-Condon factors can be accomplished by various techniques, such as the direct methods,’* the saddle-point method,8v10-22 the convolution technique,13 the series expansion 23 or the semiclassical m e t h ~ d . ~ Rather than dwell on cumber- some technical details we shall examine the effects of nuclear tunnelling on these e.t. rates in solution, which are manifested by the following phenomena. ENHANCEMENT OF ABSOLUTE ELECTRON-TRANSFER RATES Nuclear tunnelling effects in the first coordination layer are expected to increase the absolute rate of electron transfer (e.t.) relative to the classical In fig.1 we present the results of model calculations for a symmetric e.t. process (AE = 0)20 ELECTRON TRANSFER I 1 I I I 1 : - - - - loo : 200 600 I000 intramolecular frequencylcm- FIG. 1 .-Ratio of the quantum-mechanical Franck-Condon factors to their high-temperature classical limit plotted against intramolecular frequency in symmetric electron-exchange reactions at room temperatures (kT = 200 cm-') (I) (Ad)' = 0.1 cm, (2) (Ad)2 = 0.025 cm and (3) (Ad)' = 0.0025 cm. at room temperature characterized by a single mode, co, which suffers a configurational change, (Ad)2, while all the other frequencies are not modified. We have calculated the ratio F/FC, where FC(AE; E, + E,) with E, = (Ad)2tico2 is the classical vibrational overlap, eqn (1.8).From these results it is apparent that the role of nuclear tunnelling is pronounced for high frequencies and for large configurational changes. For the R u ( N H ~ ) ~ ~ + - R u ( N H ~ ) ~ ~ i- exchange, where inner-shell reorganization is very 10w,22*25 i.e. ( A q 2 z 5 x cm and tzw z 400 cm-', the quantum enhancement effect of the rate is only lo%, being exceedingly small. For the CO(NH,)~~+-CO(NH~)~~+ exchange reaction, which is characterized by large inner-shell r e ~ r g a n i z a t i o n , ~ ~ ~ ~ ~ i.e. (Ad)2 z 0.1 cm and tzw z 400 cm-l, the quantum effects enhance the e.t. rate by about one order of magnitude. Detailed calculations 22*25 for this system indicate a huge (lo-*) Franck-Condon reduction of the e.t. rate. However, most of this dramatic nuclear effect is accounted for by the classical result, the quantum effect being just F/FC = 5-7.22*24 Nuclear tunnelling effects on symmetric e.t.reactions at room temperature are modest. TEMPERATURE-DEPENDENT ACTIVATION ENERGY The contribution of high-frequency modes results in the increase of the activation For symmetrical e.t. the activation energy is 8*25 encrgy with increasing temperature. 8lnF E, (Ad,J2tzwL2(cosech2v: + cosech2vz) E = - - a( l / k T ) = 4 -k ? 2(coicothvL -t o.&cothv:)2M. BIXON AND J . JORTNER 21 For the symmetric CO(NH3)62+-cO(NH3)63+ exchange reaction in solution EJ4 = 7.0 kcal mol-' and E,/4 = 11.5 kcal rnol-', so that the " high-temperature '' limit of the activation energy is E,(T + a) = 18.5 kcal mol-I.The activation energy is calculated to vary in the range E, = 15.7 kcal mo1-1 at 273 K to E, = 16.6 kcal mo1-I at 343 K. KINETIC ISOTOPE EFFECTS Isotopic substitution in the first coordination layer of ions, or in a molecular donor or acceptor, is expected to modify the e.t. rate. This kinetic isotope effect originates from frequency changes and from distortion of the equilibrium configurations of totally symmetric stretching modes.26 On the basis of detailed calculations 26 for the C O ( N H ~ ) ~ ~ + / ~ + and C O ( N D ~ ) ~ ~ + / ~ + exchange reactions, we have concluded that the effects of frequency changes are minor and the major contribution to the isotope effect arises from the configurational changes of the totally symmetric mode. The isotope effect for symmetric e.t.is then given by26 where kH/kD = exp(YD - yH) (2.4) Y, = (E,/hco,) tanh - a = H or D (2) ; where E,, the internal reorganization energy, is invariant to isotope substitution. From the temperature dependence of kH/kD for symmetric e.t. (fig. 2), it is apparent that (i) the highest low-temperature value is reached at hco/kT >, 10, (ii) at high temperatures (kT/hw w 1) one can expand eqn (2.5) to get h(kH/kD) cc T3, (iii) in the high-temperature classical limit (kH/kD + 1) quantum effects erode, and (iv) the isotope effect is modest at room temperature, e.g. for the CO(NH&~+/~+ exchange kH/kD varies from 1.26 at 25 "C to 1.12 at 70 "C. The isotope effect is normal, i.e. kH/kD >, 1. FIG. 2.-Temperature dependence of deuterium kinetic isotope effect for symmetric electron- exchange reactions (AE = 0) between hexa-ammine complexes.22 ELECTRON TRANSFER The effect of the change of the electronic energy gap AE on k,/kD (fig.3) reveals the following features. (a) The maximum value of kH/kD is exhibited for symmetric reactions. (b) The AE dependence of h(kH/kD) in the normal region is described by a bell-shaped curve. (c) kH/kD reaches its minimum value for activationless and barrier- less reactions at AE = F(Es + Ec). (d) For highly exothermic and highly endo- W E , FIG. 3-Deuterium isotope effect for e.t. between ammine complexes as a function of the reduced electronic energy gap AEIE,. E, is the total reorganization energy E, = E, + E,. The parameters are as follows: AoH/kT = 2.0 and EJE, = 0 (- .-), E,/E, = 1 (-), E,/E,= 2 (- - -). thermic e.t., i.e. IAEl > E, + E,, the isotope effect increases with increasing the electronic energy gap. It is important to emphasize that the deuterium isotope effects on outer-sphere e.t. reactions originate, in our opinion, from the quantum motion of the entire ligand with respect to the ion. Thus the isotope effect originating from the replacement of an H2160 molecule by a D2160 molecule in the first coordination layer will practically be identical with the effect induced by substituting H2l60 by H2180, extending the scope of isotope effects. FINAL-STATE VIBRATIONAL A N D ELECTRONIC EXCITATION Marcus predicted a decline of the rate in the " inverted region " for highly endo- ergic reaction^.^,^ Excitation of internal quantum states of the donor-acceptor system results in a substantial moderation of the decline of the rate in the inverted region, exhibiting large quantum corrections on the rates of exoergic reactions.For e.t. processes involving only the ground electronic states of DA and of D+A-, the e.t. process results in the vibrational excitation of the internal vibrational modes, which have to be treated quantum me~hanically.~'-~~ The e.t. process involves contributions from parallel decay channels (2.6) D($) A(&$) + D+(E;) A-(E$) where EEL?) are the energies of the vibrational quantum states of the donor (acceptor) in its initial (final) state. The total e.t. rate can be expressed as a sum over allM. BIXON AND J . JORTNER 23 vibrational excitation channels.EP = EP = 0, the rate isl2?l3 For an idially internally " frozen " system, i.e. where SD(O,&F) and SA(O,&) are the Franck-Condon factors for the donor and the acceptor, respectively, each initially being in its ground vibrational state. The effective energy gag is FC(AE - &; - &$), so that a reduction of the electronic gap by the vibrational excitation is exhibited. The contribution of vibrational excitations adds new channels effective at high IAE], thereby reducing the inversion effect. It has been recognized that excitation of final quantum states can also involve electronic excitation when the exoergic e.t. process results in an electronic excitation of the A- and/or the D+ centres.27 The e.t. rate now incorporates additional additive decay channels to final states, characterized by electronic energy gaps A&, AE2, etc.which are lower than AE. The e.t. rate incorporating both electronic and vibrational excitations can be expressed in terms of the sum where Wis given by eqn (2.7), and W(AEj) is given by the same equation with AEj replacing AE. The contribution of the electronic excitations yields a substantial contribution at large IAEl resulting in the moderation of the decrease of k. From the point of view of the experimentalist, there is a qualitative difference between the role of electronic and vibrational excitations in the sense that Vibrational excitations will exhibit a unimodal " distribution " of Ink against AE, while the presence of an additional channel involving an electronic-excited final state will result in a bimodal distribution.An interesting analogy can be drawn in this context with line shapes in optical emis~i0n.l~ The interrogation of free-energy e.t. relations can be considered as an experiment in " chemical type " spectroscopy. The general appearance of Ink plotted against AE bears a close analogy to the line shape L ( h ) in emission of a photon hv, where the dependence cr; r(hv) on (AE - hv) for the optical process is analogous to the dependence of W(AE) on AE for the thermal e.t. process. When emission occurs to a single final electronic state, the line shape will be unimodal, while when emission involves two lower-lying electronic configurations, a bimodal intensity distribution will be exhibited. The examination of nuclear tunnelling phenomena and the effects of excitation of internal quantum states results in some quantitative modification of classical relations and correlations.The following conceptual and technical extensions of the theory will be of interest. (1) Electronic energy gap and the free-energy change. The multiphonon rate expressions incorporate the electronic energy gap AE as the relevant energy parameter, while the Marcus classical r e l a t i ~ n , ~ - ~ eqn (1.8), with AG replacing AE, contains the free-energy change. In this context, one should note that the general multiphonon rate W, eqn (1.4), for the DA -+ D+A- reaction, which will be denoted by W(DA + D+A-), and the rate W(D+A- -+ DA) for the inverse process, satisfy the condition of detailed balance W(DA -+ D+A-)/W(D+A+ --f DA) = exp(-AG/kT) AG = AE - TAS A S = kln(Z&) (2.9) with the entropy change being determined by the ratio of the partition functions for24 ELECTRON TRANSFER the initial DA, eqn (1.5), and for the final D+A- states.Practical calculations, which rest on the harmonic model for nuclear potential surfaces, result only in a minor contribution to A S from frequency changes,19 while the major contribution to A S due to rotational reorganization of the solvent and to electrostriction is missing. (2) The electronic coupling, Y. Three general coupling schemes can be dis- tinguished : (i) spin-allowed direct e x ~ h a n g e , ~ * ~ ~ ' ~ * ~ ~ (ii) spin-forbidden direct exchange,25 and (iii) indirect spin-allowed s~perexchange.~~ We are currently ignorant concerning several basic points.First, the effects of dielectric screening on direct exchange have not been treated in a self-consistent manner. Secondly, for e.t. between molecular D and A, e.g. for e.t. between large aromatic molecules, V has to be calcul- ated using many-electron molecular wavefunctions, as utilized for N a ~ - N a p . ~ * The dependence of Yon distance R provides a grossly over-simplified description and one has to consider orientational dependence. Thirdly, a realistic estimate of super- exchange interactions, which may be important for biophysical processes, will be desirable. The criteria for the validity of the non-adiabatic formalism, which is based on the Landau-Zener are applicable at best to the classical limit of e.t.dominated by the contribution of solvent modes. No satisfactory non-adiabaticity criteria for the multidimensional e.t. problem involving both classical and quantum modes have been provided. In the low-temperature limit only the non-adiabatic kinetic situation is meaningful, while for the limit of large V the zero-order states are heavily mixed (see section 3). The adiabatic rate provides the highest value of the pre-exponential factor for the high-temperature unimolecular rate, which for a single-mode system with a frequency co is given by3' (3) Applicability of the non-adiabatic formalism. (4) An upper limit for a thermal e.t. rate. W = LL) exp[-(A€ + Es)2/4EskT]. (2.10) Accordingly, the upper limit for the rate of an activationless adiabatic process is W M co z 1012-1013 s-'. The corresponding ultrafast bimolecular rate calculated according to eqn (2.10) is k M 1 O I 2 dm3 moI-l s-'.This rate is considerably higher than that of the diffusion controlled rate, kDiff = 109-1010 dm3 mol-' s-l. The observed rate of ultrafast bimolecular e.t. processes is limited by the diffusion process and therefore reveals nothing about the e.t. mechanism. On the other hand, fast unimolecular e.t. is not masked by diffusion and may occur on the picosecond time- scale. However, for such ultrafast e.t. the conventional theory should be modified due to competition between the electronic process and vibrational relaxation. 3. COMPETITION BETWEEN ELECTRON TRANSFER A N D VIBRATIONAL RELAXATION The advent of mode-locked lasers made ultrafast e.t.processes, occurring on the picosecond (and subpicosecond) timescale, amenable to experimental interrogation. Several experiments on ultrafast e.t. which come to mind are: (1) e.t. from electronically excited states in solution and in glasses; (2) e.t. to the ground state of a cation produced by photoionization in a dense medium; (3) geminate recombination of ions and of solvated (or partially solvated) electrons ; (4) intramolecular e.t. from excited states of DBA chemical and biophysical model compounds, where B is a bridge; (5) the ultrafast primary charge-separation process in bacterial photo~ynthesis.'~ For such ultrafast processes, the microscopic rates W,, eqn (1.1), may be compar- able to, or even exceed, the medium-induced vibrational relaxation (v.r.) rates within the initial { / I ) } vibronic manifold.Thus, the basic assumption, eqn (1.3), pertainingM. BIXON AND J . JORTNER 25 to the separation of the timescales for fast v.r. and slow e.t. may no longer be applicable, and the conventional e.t. theory breaks down. We are then concerned with a new and intriguing physical situation, when e.t. occurs during v.r., or even prior to it. One can then assert that for ultrafast e.t. the thermally averaged rate is meaningless, and the microscopic rates, W, [eqn (1. l)] have to be considered explicitly. Obviously, an ultrafast e.t. will be exhibited provided that two conditions are satisfied. This requires: (1) spatial proximity of D and A resulting in large V (ca. 10-100 cm-l) ; (2) large vibrational overlap is exhibited in the vicinity of the crossing point of the potential surfaces.(B) Accessibility of the states with large W,. The system is excited so that the vicinity of the crossing point is accessible during the relaxation process. The relevant v.r. processes during ultrafast e.t. span a broad spectrum of medium- induced phenomena. Some examples are : (a) Rotational relaxation of the solvent during ultrafast e.t. Such an e.t. process occurs on the timescale of the dielectric relaxation time z,, or rather the transverse constant-field dielectric relaxation time 32 7:: = (Dm/Ds) z,. An activationless e.t. characterized by a large Y (10-100 cm-l), where the major nuclear contribution originates from solvent reorganization and which takes place in a system characterized by a long z, (or T:), is expected to occur from a non-equilibrium solvent configuration, which relaxes during the e.t.(b) Con- (A) Some of the W, are large. FIG. 4.-Schematic description of initial excitation followed by vibrational relaxation, which is designated by arrows. (2) When E, > Ex the system can undergo efficient e.t. at the crossing point x before equilibration has been accomplished. (1) When E, < Ex the system usually equilibrated thermally.26 ELECTRON TRANSFER ventional intramolecular v.r. induced by the medium in D and in A, which consist of large polyatomic ions or molecules. At present the treatment of the interesting implications of solvent relaxation and of configurational medium relaxation will be avoided, and we shall consider an ultra- fast e.t.between molecular D and A, where the coupling with exterior-medium modes is small and the major nuclear contribution to each W, originates from intramolecular reorganization. Such a situation of e.t. accompanied by intramolecular reorganiza- tion (e.t.a.i.r.) may prevail in glasses with a low dielectric constant or in biophysical “ rigid ” systems, such as photosynthetic reaction centres, where we expect conven- tional medium relaxation effects of type (b) to prevail. Considering explicitly the conditions (A) and (B) for ultrafast e.t.a.i.r., two relevant physical situations are of interest. First, activationless e.t.a.i.r. involves an exoergic process with A23 = -&, so that the potential surfaces cross at the minimum of the initial-state potential.Second, e.t.a.i.r. during v.r. When W, is large at the crossing point of two potential surfaces and the system is initially excited at an energy E, above the crossing point (Ex), efficient e.t.a.i.r. is expected when the system relaxes vibrationally through the crossing point 1 4 9 1 5 (fig. 4). Let us consider first the problem of ultrafast e.t.a.i.r. occurring from the vibration- less 11 = 0) level, as may be the case for some activationless e.t.a.i.r. The 11 = 0) = 11) level interacts significantly with a small number of final intramolecular states, which will be represented by a single final level IF). The IF) level undergoes medium- induced vibrational relaxation, which may be envisioned as weak interaction with a continuum of medium states superimposed on lower lying intramolecular levels in the final manifold*(fig.5). The residual v.r. coupling can be accounted for by assigning a vibrational relaxation width, I?, to the final state (fig. 5), whose energy is EF. On the other hand, the 11) state, whose energy is E,, is not subjected to v.r. The quantum- ( b ) FIG. 5.-Energy-level scheme for ultrafast e.t. from the vibrationless II> level. (a) Coupling scheme, ( 6 ) model system. The 11) level is coupled by V to IF), which in turn is subjected to v.r. to medium- phonon states superimposed on the IF’) state.M. BIXON AND J . JORTNER 27 mechanical model for the e.t.a.i.r. (fig. 5 ) corresponds to interstate coupling 73 = ITIcF), where FI(Fl is the ]I) - IF) Franck-Condon overlap, with the IF) state being characterized by the width r.This relaxation problem can be handled by solving the secular equation hich has the solutions E = +[EI + EF - iT] &- +[(EI - EF + ir)2 + 4P2]*. (3 -2) Several limiting cases are of interest. AEIF = EI - EF, one obtains to second order the solutions For weak p, i.e. IPI < IAE(IF + i r l , when The energy shifts are of no interest. state has acquired, according to eqn (3.3), a width It is important to note that the coupled initial which is proportional to P2. yI can be, identified with a decay rate, which for a large AEIF, i.e. IAEIFI $ r, is given by yI = V2r/(AE)2, while for near resonance IAEIFI $ r the highest value of yI = P2/r is obtained. These weak coupling situations are analogous to the non-adiabatic limit. The results are drastically different for the strong coupling situation, i.e. P 9 IAEIFI.Strong scrambling between the two zero-order states is exhibited. The resulting states are with the corresponding energies which are split by 2P and have equal decay widths. As the initial state is prepared by optical excitation, the dynamics depends on the coherence properties of the excitation. If Il} and 12) can be excited coherently, which means that the initial state is \I), then the electron is transferred in a decaying oscil- latory manner with an oscillation period of h/2 P and a decay rate of T/h. In real life it would be very difficult to achieve a coherent excitation and more probably the excitation would either be to 11) or to 12). In such a case, the electron is already “ half transferred ” in the preparation act of the initial state, which involves excitation of a charge-transfer state.The e.t. is continued due to v.r. at a rate of r/h, which is then independent of the magnitude of V. This simple quantum mechanical model for the28 ELECTRON TRANSFER activationless ultrafast e.t.a.i.r. is illuminating as it retains all information about coherence effects. In view of inhomogeneous broadening effects and dynamic phase- erosion processes, which prevail even in low-temperature solids, it will be extremely difficult to observe the manifestation of such coherence phenomena in ultrafast e.t. At the other extreme, when coherence effects are insignificant, the stochastic model l5 is applicable. The stochastic model for e.t.a.i.r.I5 is very useful as it adheres to chemical intuition.This model (fig. 6) treats the populations pau(t) of the states in the initial manifold STATE a ”p; a 4 wa 3 a 3 STATE b FIG. 6.-Stochastic model for competition between e.t. and v.r. The microscopic e.t. rates are W,,, while yUju’ denotes the v.r. rates. 11) = lav) of DA in terms of kinetic equations involving medium-induced vibrational relaxation with the rates yu+”*, and e.t. is characterized by the microscopic rates Wa,, which are given by eqn (3.5): . dpau(t) - 2 yu-+u’ pau(t> + 2 yu’-+u pao,(t> - ~ a c p a u ( t > (3.8) dt u‘#u U ’ Z V with appropriate initial conditions. The time-dependent e.t. rate under non- equilibrium conditions is while the population NDA(t) of the D-A pair is NDA(t) = N D A ( 0 ) exp (3.10) exhibiting a non-exponential decay.between e.t. and electronic energy transfer, one can assert that, while the time depen- Pursuing the analogy advanced by HopfieldM . BIXON AND J . JORTNER 29 dence of an ultrafast energy transfer which competes with v.r. is determined by a time-dependent donor-emission/acceptor-absorption spectral overlap the time-dependent e.t.a.i.r. rate [eqn (3.9)] is expressed in terms of a time-dependent transition probability, which in the non-adiabatic limit reduces to a time-dependent Franck-Condon factor. J. Jortner and R. D. Levine, Adu. Chem. Phys., 1981,47, 1 . R. A. Marcus, J. Chem. Phys., 1956,24, 966; 1965, 24, 979; 1957,26,867, 872; Trans. N. Y. Acad. Sci., 1957, 19, 423. R. A. Marcus, Discuss. Faraday Soc., 1960, 29, 21. R.A. Marcus, Annu. Rev. Phys. Chem., 1964, 15, 155. J. J. Hopfield, Proc. Natl Acad. Sci. USA, 1974,71, 3640. J. Jortner, J . Chem. Phys., 1976, 64, 4860. W. Jost (Academic Press, New York, 1970), vol. 9B. N. R. Kestner, J. Logan and J. Jortner, J . Phys. Chem., 1974, 78, 2148. R. R, Dogonadze and A. M. Kuznetsov, Elektrokhimiya, 1967, 3, 1324. 125; 425. ’ V. G. Levich, Physical Chemistry: An Aduanced Treatise, ed. H. Eyring, D. Henderson and lo R. R. Dogonadze, A. M. Kuznetsov and M. A. Vorotyntsev, Phys. Status Sofidi B, 1972, 54, I1 R. Van Dyne and S. Fischer, Chem. Phys., 1974,5, 183. I2 S. Efrima and M. Bixon, Chem. Phys. Lett., 1974,25, 34; Chem. Phys., 1976, 13, 447. l3 J. Ulstrup and J. Jortner, J. Chem. Phys., 1975, 63, 4358. l4 J. Jortner, Philos. Mag., Ser. B, 1979, 14, 317. J. Jortner, J . Am. Chem. SOC., 1980, 102, 6676. l6 V. I. Goldanskii, Dokl. Acad. Nauk USSR, 1959, 124, 1261 ; 1959, 127, 1037. E. Buhks and J. Jortner, J. Phys. Chem., 1980,84, 3370. l8 R. Kubo and Y. Toyozawa, Prog. Theor. Phys., 1955,13, 160. l9 J. Ulstrup, Charge Transfer Processes in Condensed Media (Springer-Verlag, New York, 2o E. Buhks, M. Bixon and J. Jortner, Chem. Phys., 1981,55, 41. 21 J. B. Beitz and J. R. Miller, J. Chem. Phys., 1979, 71, 4579. 22 P. Siders and R. A. Marcus, J . Am. Chem. Soc., 1981, 103, 741. 23 M. H. L. Pryce, Phonons (Oliver and Boyd, Edinburgh, 1966), p. 403. 24 E. Buhks, M. Bixon, G. Navon and J. Jortner, J . Phys. Chem., 1981, 85, 3759. 25 E. Buhks, M. Bixon, J. Jortner and G. Navon, Inorg. Chem., 1979, 18, 2014. 26 E. Buhks, M. Bixon and J. Jortner, J. Phys. Chem., 1981, 85, 3763. 27 P. Siders and R. A. Marcus, J. Am. Chem. Soc., 1981, 103, 748. 28 M. D. Newton, Int. J. Quantum Chem., 1980, 14, 363. 29 H. M. McConnell, J. Chem. Phys., 1961, 35, 508. 30 S. A. Rice and J. Jortner, Physics of Solids under High Pressure, ed. T. Tonizuka and R. M. Emrick (Academic Press, New York, 1965), p. 65. 31 T. Holstein, Ann. Phys. (N. Y.), 1959, 8, 343. 32 A. Mozumder, Electron-Solvent and Anion-Solvent Interaction, ed. L. Kevan and B. Webster 33 I. Kaplan and J. Jortner, Chem. Phys. Lett., 1977, 52, 202; Chem. Phys., 1978, 32, 381. 1979). (Elsevier, Amsterdam, 1976), p. 139.

ISSN:0301-7249    

DOI:10.1039/DC9827400017

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1982, 74, 3 1-47 Quantum Theory of Chemical Reactions of the Solvated Electron BY ALEKSANDER M. KUZNETSOV Institute of Electrochemistry of the Academy of Sciences of the U.S.S.R., Leninsky Prospect 31, 117071 Moscow V-71, U.S.S.R. AND JENS ULSTRUP Chemistry Department A, Building 207, The Technical University of Denmark, 2800 Lyngby, Denmark Received 17th May, 1982 The solvated electron represents a more diffuse charge distribution than do electrons in molecules. The electron-exchange integral in rate expressions for reactions of the solvated electron is therefore also more sensitive to a variation in external parameters such as the free energy of the reaction. We have investigated this effect and have applied the Condon approximation, but we have determined the orbital exponent of the electronic wavefunction (of assumed spherical symmetry) by minimizing the total potential energy with respect to the actual polarization, for which we have also incorporated frequency dispersion, rather than only the equilibrium polarization.For low- frequency nuclear modes this causes a more diffuse charge than at equilibrium in the " normal " free-energy region for the electron-transfer reaction, and a more compact charge in the strongly exothermic region. For high-frequency modes the effect is much larger in the strongly exothermic region and of equal importance to other effects which cause a drop in the rate constant in this region. 1. INTRODUCTION A great deal of attention is presently being given to solvated electrons in frozen Much of this interest refers to the nature of the trapping aqueous and organic glasses.sites as revealed by the absorption spectrum in the near-infrared and visible and to long-range electron transfer of the trapped electron to molecular The kinetics of electron transfer display several features which differ from those of " conventional " electron transfer between molecules in liquid solution. First, the decay of trapped electrons follows a logarithmic time dependence, and the rate depends exponentially on the scavenger concentration.8-11 Both these effects are commonly interpreted as the direct tunnelling of the electron to the scavenger molecule, and the rate laws reflect the strong distance dependence of the tunnelling prob- ability.8 -12 Secondly, reactions of the trapped electron are strongly exothermic, the free energy of reaction exceeding the total nuclear reorganization energy.13-15 The reactions are furthermore often followed at low temperatures (< 100 K) where a substantial part of the nuclear modes are quantum mechanically ' frozen.' Both these conditions are favourable for the observation of nuclear tunnelling.It is therefore of considerable interest to study solvated electrons in terms of electron- transfer theory, but in comparison with molecular reactants in liquids the solvated electron exhibits at least the following differences : (1) The disordered medium cannot adequately be represented by a single-mode appro~irnation'~*~~ as is the case for high-temperature processes.18-20 The electrons32 QUANTUM THEORY OF THE SOLVATED ELECTRON are coupled to a broad continuum of nuclear modes, which requires a significant reinterpretation of the Brransted and Arrhenius relations at low temperatures.l8 (2) The perturbation which induces the process cannot solely be identified with the interaction between the electron and the scavenger, as is the case for molecular reactants, since this would invoke conceptual difficulties for the reverse reaction, i.e.the formation of the solvated electron by thermal ionization of the reduced scavenger. (3) The electron represents a more diffuse charge distribution than for molecular reactants. This may be reflected both in the formal kinetics, leading to the importance of diffusion to sites at varying distances from the scavenger, and in the ' elementary ' rate expressions, where the electron-exchange integral may depend on the solvent nuclear coordinates.We have previously analysed several of these effect^.'^^'^^^^'^^ In the present work we investigate the second and third effects in particular and derive expressions for the electron transfer (at a fixed distance) which incorporate these effects. These rate expressions can subsequently be combined with kinetic rate laws for direct tunnellinglo or variable-range diffusion.21-24 2. HAMILTONIANS AND REACTION CHANNELS We shall now introduce zero-order Hamiltonians for the initial and final states corresponding to localization of the electron in a solvent ' trap ' and on the scavenger molecule, respectively. Subsequent application of perturbation theory is then justified in view of the large (20-40 A)"-" electron-transfer distance. The total Hamiltonian of the trapped electron and the molecular scavenger in the medium is fi= Te -+ TN + p e , t r + pe m + p z , t r + p e z + pzm + ptr,m + f i s c + om (2.1) where T, and TN are the kinetic energies of the trapped electron and the nuclei of the medium, V e , t r and Ve, the energies of interaction of the localized electron with the trap (or the nearest solvent molecules) and the scavenger, screened by the inertia-less polarization of the medium, and Ve, the interaction between the electron and the inertial polarization of the medium outside the trap.We have here assumed that the inertia-less part of the polarization is separated adiabatically from the motion of the trapped electron.V z , t r and Vzm are the corresponding interaction energies for the scavenger, Vtr,m the energy of interaction between the trap and the medium, H,, the (vacuum) Hamiltonian of the scavenger, and Urn finally the potential energy of the unperturbed medium. It is now convenient to introduce the reaction channels, corresponding to basis functions of the electron in the trap and on the scavenger m o l e c ~ l e . ~ ~ * ~ ~ However, in contrast to reactions between molecules, the electronic wavefunction of the trapped electron is not determined by its interaction with a molecular core. It is therefore necessary to introduce the reaction channels in the following way, which differs from the procedure for molecular reactants.We assume that the interaction of the electron with the external medium is linear. The total polarization of the medium in either state is then a superposition of the contributions from the scavenger and the trapped electron. For neutral and negatively charged scavengers the spatial distribution of the polarization field furthermore creates two potential wells for the electron, centred in the trap region and at the acceptor. For positively charged scavengers a potential well at the acceptor site is created by superimposed polarization and molecular potentials. This distribution is represented by the equationA. M. KUZNETSOV AND J . ULSTRUP 33 where the total inertial polarization, P(r), is given as a sum of the inertial polarizations in the trap region when the scavenger is absent, P1(r), and in the scavenger region, P2(r), when the solvated electron is absent.This division refers to the polarization field and not to dynamic variables which describe the solvent. Polarization ffuctua- tions of sufficient depth to trap the electron at a given site are thus incorporated in the channel Hamiltonian for localization of the electron at this site, and we assume that possible distortions of electronic wavefunctions at one reactant by the polarization field or molecular potential from the other reactant are sufficiently small to be ignored or handled by perturbation theory. These points and their relation to a formulation in terms of adiabatic basis functions, which include the electronic interactions between the centres, have recently been analy~ed.~’’~’ We are then in a position to introduce the Hamiltonians of the ingoing (Hi) and outgoing (Hf) channels (initial and final states).The total Hamiltonian can be written I? = rii + Pi = Ei, + Vf f i i = p e +,f+N +Ape,tr + “2 + ptr,m + fz,tr + pzm (2.2) where the channel Hamiltonians are + [Vez + V%]dia + urn + LVe,tr + V%ldia + (2.3) (2.4) (2.5) H f = p e +- F N + Fez+ v% f Qtr,m + pz tr + pzm The perturbations which induce the forward and reverse reactions are f i = ivez + p%]c~ff-d; pf = [Ve,tr + v!z]off-d. VLZ and VL2 are the energies of interaction of the electron with the polarization components Pl(r) and P2(r), respectively, and the perturbations Vi and Vf contain only the ‘ off-diagonal ’ matrix elements (off-d) with respect to the electronic wave- functions at the two different centres, whereas the diagonal parts (dia) are included in the reaction channels.The matrix elements have the form = <ilPez + PL2jf) - <ilPez + P!zli><ilf> (2.6) with an analogous expression for (Vi)fi, where (i) and If) are the electronic wave- functions in the initial and final states. Eqn (2.3)-(2.6) define the zero-order channel Hamiltonians and the perturbations. In particular, we see that if there are no pre-existing traps so that V,,,, = 0, the perturbation for the reverse reaction is the interaction with the polarization field in the absence of the scavenger, i.e. Pl(r), and an analogous contribution appears in the perturbation for the forward reaction. We conclude this section by introducing the Born-Oppenheimer channel wave- functions.The total electronic-nuclear wavefunctions are written Ysb; !z) = lys(x; 4)xs((l) (s = i,f> (2.7) where lys(x; q ) are the electronic wavefunctions and solutions to the stationary electronic Schrodinger equation ( H s - TN)Vs(x; q)l= & S ( ~ ) W S ( ~ ; q)- (2 - 8) x is a set of electronic coordinates, q the total set of all local and “ collective” medium nuclear coordinates, and ~ ~ ( 4 ) the electronic energy in the state s. The nuclear wavefunctions, xs(q), are subsequently found from the equation [ T N + &s(dlxs(!?) = Esxs(4). (2.9)34 QUANTUM THEORY OF THE SOLVATED ELECTRON The polarization contributions to the perturbations are then v g = - J P ( r)&(x; r) dr (2.10) v g = - SP( r)&(x; r) dr (2.11) where &(x; r) is the vacuum field in the point r caused by an electron at x.3. RATE PROBABILITY 3.1. THE CONDON APPROXIMATION REVIEWED We shall assume that the transition probability per unit time, Wif, can be ade- quately calculated by first-order perturbation theory. For large electron-transfer distances a second-order mechanism involving high-energy intermediate electronic states might compete. Such a mechanism can be described by second-order perturba- tion t h e ~ r y , ~ ~ . ~ ~ but we shall not discuss it further in the present work. A quite general form of Wif, averaged over the initial and summed over the final vibrational states, is then 26-31 c + im wif = ( ~ / i h ) do expvFi)Tr(Lipi(l - o)~;pf(9)) (3.1) c - icn where p = (kT)-’, k is Boltzmann’s constant and T is the absolute temperature.pi(l - 8) and pf(8) are the nuclear parts of the statistical operators32 in the initial and final state, at the temperatures T/(1 - 0) and T/8, i.e. pr(l - 0) = exp[-P(l - €J)Z?~]; pf(9) = exp(-P8Ei,) (3.2) where RS = TN + E,(q) (s = i,f). Fi is the free energy of the initial state, i.e. exp@Fi) = {Tr[exp(-pEJ])-l Li an abbreviated notation for the two-centre electronic resonance integral, and L’: its hermitian conjugate. As an operator Li can formally be written where [ Vi(q)lif is given by eqn (2.5)-(2.8). Finally, c is a real number which determines the integration contour in the complex plane. We shall further assume that Li depends sufficiently weakly on the nuclear co- ordinates that it can be replaced by its value at q = q*, for which the trace in the coordinate representation takes its maximum value.We thus formally invoke the Condon approximation, but we do account for the variation of [Vi(g*)IIP with q*. q* generally differs strongly from the equilibrium values of 4 in the initial (4io) and final (&o) states and is subject to significant changes when parameters such as the free energy of reaction vary. For this reason [Vi(*q))lif also differs from both [Vi(qi0)lif and [Vi(qf0)lif and varies through such a series of reactions, and the effect is more important for the trapped electron than for molecular reactants. The transition probability thus takes the form c i - i m wif = (p/ih) [vi(q*)lif12 exp(pFi)Tr{pi(l - e)Pf(e>> (3.4) c - i m and we can proceed to calculate the trace using effective Hamiltonians 26*31933 or the correlation-function Here it is convenient to consider the low-frequency classical and high-frequency ‘ quantum ’ nuclear modes separately.A .M. KUZNETSOV A N D J . ULSTRUP 35 3.2. LIMIT OF CLASSICAL NUCLEAR MODES In this limit the trace in eqn (3.4) can be written as an integral of a functional of the polarization P(r). P(r) represents the dynamic variables of the medium, and the trace could be recast as an integral with respect to a set of harmonic normal c ~ o r d i n a t e s . ~ ~ * ~ ~ * ~ ~ Using the first formulation we can write Wif in the form - W J f l . The potential energies in the initial ( Ui) and final ( Uf) states are (3.5) = um[p(Y)l+ EP - / dr[~i(r) + ~2(r)1/ dxIV/i(x\,12&(x; r) (3 4 Uf = Um[P(r)] + efO - / P(r)Bf(x; r)dr = um[~(r)l+ E ~ o - \dr[~,(*) + ~,(r)l/dxlV4x; r)12&(X; r).U,,,[P(r)] is the potential energy of the unperturbed dielectric medium, determined by the polarizatio'n components P(r) of the different kinds of molecular motion (orienta- tional, atomic etc.) by the relation33 Um[P(r)] = 2 2 / [PV(r)l2dr CV where cV = EL; 1 - 8;' and e, and cV+ 1 are the dielectric constants on each side of the absorption band for the vth polarization mode. For a single (relaxational) mode c = e0-l - E ; ~ , where E, is the optical and E, the static dielectric constant. ey and &Of are finally the electronic energies excluding the interaction with the polarization field. The integrals with respect to 6 and P(r) can be calculated by the saddle-point method for rather general potential surfaces.The saddle points 6" and P(r) are determined by the equations 3 1 9 3 7 9 3 8 (1 - e)(mi/6p) + e(suf/sp) = o (3.7) where the variations must be extended to any discrete molecular or short-range solvent modes, when the appropriate potential-energy contributions have been added to eqn (3.6). We shall assume that the explicit variation of the electric field b(x; r ) with changes in the electronic charge distribution caused by the polarization fluctuations is small compared with the dependence of the electronic resonance integral and Ui and Uf on P(r). This assumption is a reasonable first approximation, but the occasional rather notable dependence of the electronic wavefunctions on P*(r) (section 3.3) does require explicit consideration of the variation of &(x; Y) in these cases.28 Following previous procedures 26*38 P*(r) is then found to be Pv*(r) = (1 - 6*)P,lb(r) + 6*P,o(r); P*(r) = 2 P"*(r) (3.8) V where Pr0(r) and P,'o(r) are the equilibrium inertial polarizations of the initial and final state, respectively.For spherically symmetric charge distributions or for36 QUANTUM THEORY OF THE SOLVATED ELECTRON infinite homogeneous media39 P:o(r) and P,"o(r) are related to the induction fields, D, by the equations c C Pro(r) = 2 Dio(r); Pfo(r) = Dfo(r - R); Pso(r) = 2 P(r,Yo), (r) s = i,f (3.9) which we shall exploit in the following. We thus assume that the electron-transfer distance is sufficiently large, and Ve,tr z 0. Insertion of eqn (3.8) in eqn (3.7) then gives an equation for the saddle point of the Lagrange multiplier O*.Further application of the semiclassical procedure 38 gives the rate probability. If the electron is only coupled to the medium, Wif is (3.10) where E," is the solvent reorganization energy. At high temperatures this quantity is well approximated by the limiting formula for a structureless medium 4.n 47E V Wif = (n/h2kTE;)*1 [ Vi(AFO)]ifl2 exp[-D(Es + Af',-J2/4E:] E: =- [DiO(r) - Dfo(r - R)I2dr (3.1 1) 87~ "I where R is the distance between the trap and the scavenger. At low temperatures (e.g. at 77 K) and for strongly exothermic processes frequency dispersion of the nuclear modes is, however, important,18*20 and E," takes a more general form which we shall discuss in section (3.4).Finally, AFO = (&; - EP) - 1 dr[p,(r) + ~2(r)1\ dxg(x; r)[lvf(x; pfo)12 - Ivi(x; pio)121 is the free energy of reaction. electronic factor on AFo explicitly expressed through P*(r) Eqn (3.10) differs from the usual form of this equation by the dependence of the [~i(p*)lif = / v / ~ ( x ; p*)[vez - / ~2*(r)g(x; r)drlv;(x; ~ ) d x (3.12) The rate probability takes a different form if frequency shift or anharmonicity in discrete modes is important. Such modes could represent the motion of the nearest solvent molecules around the localized electron, for which strong anharmonicity in a low-frequency (50- 100 cm-l) breathing mode has been ~uggested.~O*~~ In such cases the activation energy is still obtained by solution of eqn (3.7), but the pre-exponential factor depends on the detailed topology of the potential surfaces.38 3.3.VARIATION OF THE ELECTRONIC FACTOR We shall now proceed to the effect of the transition configuration on the electronic factor. We shall analyse the effect for the isolated solvated electron, i.e. for sufficiently large electron-transfer distances, and represent the electronic wavefunction by a Mike wavefunction of the form vls(x> = (A3/7E>+ exp(-Alxl) (3.13) where we determine the orbital exponent by the variational calculus introduced byA . M . KUZNETSOV AND J . ULSTRUP 37 Pekar.38 The electronic wavefunction at the initial-state equilibrium polarization is found by minimizing the functional where p is the mass of the electron. Pio(r), Dio(r) and yls(x) are related by the equations where e is the electronic charge, and v/is(x) is obtained from yls(x) by taking the orbital exponent, Lo, corresponding to the equilibrium polarization Pl0(r).Fbo can thus be written (3.16) In comparison, in the transition configuration where P(r) = P*(r) where y f S is the wavefunction for the polarization PT(r) P*(r) = -2 [(l - 6*)D:o(r) + O*D&(r - R) + Dfzo(r - R)]. (3.18) and D the induction fields created by the electron (e) and the scavenger (z) in the initial (i) and final (f) states. By inserting eqn (3.18) in eqn (3.16) and ignoring the field contributions from the acceptor site, the following simplified equation is obtained Fb,(ryl*,) = -\ IVyTs(x)I2dx - & (1 - 6*) ,/ D:o(r)Dr(r)dr. (3.19) The functionals P'! and F i thus only coincide for activationless processes where 8* -+ 0 and Plo(r) = P:(r). The equation for 3, is (3.20) h2 - 1 61Vyys(x)12dx - & D:o(r)dD;o(r)dr = 0 2P whereas the equation takes the following form in the ' normal ' (0 > 6* > 1) and strongly exothermic (6* < 0) freeAenergy regions k2 - ,/ 61Vy,*,(x)12dx - f (1 - O*)/ Dieo(r)dDf(r)dr = 0.2P 4x h2 2P (3.21) With the particular form of yls given by eqn (3.13) this equation becomes (3.22) h2 - 3, - (1 - 6*)I(A) = 0 2P where38 QUANTUM THEORY OF THE SOLVATED ELECTRON This quantity is calculated in the appendix. The result is (3.24) and in combination with eqn (3.22) and the known value of A. for 8" = 0 this equation provides (numerical) values for A. Before doing so we shall, however, derive a simplified expression for A valid for small 8*.For IS*] < 1 the solution of eqn (3.22) can be written A(8*) M A,{ 1 - 8*/[1 - p(1 - 8*)(dZ/dA)lA = ~~/h']> = &(l - @*) < = [l - p(dl/dA)IA = ~Jh~1-l. (3.25) Since dI/dA < 0, it follows that 0 < 5 < 1. From eqn (3.24) its value is found to be 5/11, and A thus decreases with increasing 8". Eqn (3.25) has several implications. Since the main contribution to the two- centre resonance integral in eqn (3.12) is provided by a small region, Ax*, around a point x*, where the integrand is maximum, [Vi(P*)lir is approximately where the superscripts refer to an electron at the trap or at the acceptor. If the explicit interactions for large distances are introduced, eqn (3.26) can be written [ Vi(P*)lif M Ax*&(x* ; P*)ylfs(x*; '*){ [- ~~1.r: RI where ze is the charge of the acceptor.For small 8* the perturbation is dominated by the screened electrostatic interaction with the acceptor. On the other hand, for finite lo* I or large electrostatic screening the perturbation is dominated by the second term, i.e. the interaction of the electron with the inertial polarization field. The major effect is, however, on the overlap integral, and the resonance integral has the approximate form [Vi(P*)Iif = [ ~ i < ~ i o > J i ~ e x ~ ( ~ o ~ ~ ' I x * l ) . (3.28) The electronic transmission coefficient, Kif(P*), would then be 'Cif(P*) = 'C*r(Pio)exP(2Ao@* Ix* I). (3.29) An important effect is thus that in the " normal " free-energy region the radius of the solvated electron is larger than at equilibrium.For Ix*J M 5 A (an electron-transfer distance of ca. 10 A), A x 0.5-0.8 A-';l and O* = 0.1 5, i.e. a fairly exothermic process, the effect amounts to a factor of 1.2-1.3 in the radius (1-l) and 1.4-1.7 in the trans- mission coefficient, but may arise to a factor of 20-30 for larger distance (Ix*l = 15 A) and larger l8*I (0.3), where A must, however, be calculated numerically. The effect vanishes for activationless processes, while the electron is more strongly localized in the transition region than at equilibrium for strongly exothermic processes, where 8 isA . M. KUZNETSOV A N D J . ULSTRUP 39 negative. The effect is here enlarged by the fact that the first two terms on the right- hand side of eqn (3.27) partly cancel each other.These effects should be reflected in the activation volumes being significant and positive in the normal, but negative in the strongly exothermic region. 3.4. EFFECT OF HIGH-FREQUENCY NUCLEAR MODES In this section we consider the variation of the electronic factor when a continuum of high-frequency modes is present. Such modcs are important at low temperatures and even at room temperature for processes in aqueous solution where the vibrational frequencies of a notable fraction of the solvent modes are higher than kT/h.41943 At low temperatures the number of such modes which fluctuate by nuclear tunnelling rather than by thermal activation further depends much more sensitively on the temperature than at high temperatures.18-20 In the following we consider a continuous set of " linear," and either classical or quantum-mechanical solvent modes.We could incorporate local modes of rather general form as well. If they have " low" vibrational frequencies such as the an- harmonic breathing modes around the solvated electron 40741 we can follow a modified version (see below) of the procedure represented by eqn (3.7) and (3.8). If they are of a quantum-mechanical nature their Franck-Condon overlap factors and vibrational energies appear in the expression for Wif such as shown e l ~ e w h e r e . ~ ~ ~ ' * * ~ ~ It is convenient to follow the procedure based on effective Hamiltonians 26931*33 but slightly modified from previous formulations. The Hamiltonian of the solvent corresponds to a set of harmonic oscillators (3.30) The vibrational spectrum is dztermined by the spectrum for dielectric absorption, and the coordinates related to the Fourier components of the polarization vector, P k , by a linear transformation (3.31) where &v are coefficients determined by the dielectric permittivity function.between the field and the solvent polarization is In the presence of electric charges giving the field cf(r), the energy of interaction (3.32) where 8 k are the Fourier components of the field. The corresponding equilibrium co- ordinate shifts, qkv0, are (3.33) The transition probability per unit time is40 QUANTUM THEORY OF THE SOLVATED ELECTRON The integrals can be calculated by the saddle-point method and the saddle points {q&) and 6* determined by the equations This corresponds to the polarization components Pz in the saddle point phmkv(1 - e*) 2 tanh P ; = z & ( V phUkv(1 - 0*) phmkv0* &’ + tanh- 2 tanh phmkv6* tanh- 2 phWkv@* .;i.+ tanh- phmkv(1 - 0.) + 2 2 tanh Using the sum rules for the polarization we can write eqn (3.38) as phm(1 - 0*) 2 tanh phmS* & + tanh - phm(1 - 0*) 2 2 tanh pmo* tanhT (3.38) L + tanh- + ptim(1 - 8”) 2 2 tanh where G(k,m) are the space and time Fourier components of the retarded Green’s function for the solvent polarization, and Im denotes its imaginary part, If spatial dispersion of the medium can be ignored, the k-dependence of G vanishes and eqn (3.39) has the form pm(1 - 0”) 2 + tanh- tanh phcug * ‘i ( r ) p*(r) =2[*mG(m)( ?c tanh pkm(1 - 2 - S*) 2 PhC06* tanh If we can also ignore spherical asymmetry in the charge distribution caused by a non- spherical trap or by the field of the scavenger, G(m) is related to the frequency-depen- dent dielectric permittivity, &(a), by the equation G(m) = L( 1 - &&).4n (3.41)A . M. KUZNETSOV AND J . ULSTRUP 41 The equation for the saddle point @* is then pkw(28" - 1) (3.42) 2 sinh - 2 sinh J and the rate probability [ Y,(P,,)]~~~~ exp[-p@*A& - @(@*)I (3.43) ptiw(1 - 0) pti~08 Pha sinh - . (3.44) 2 2 sinh @(O) = /dr[gi0(r) - gfo(r)l2 :/% f ImG(w) sinh - 2 We shall now modify this formalism by including the variation of the electronic resonance integral with 8". The procedure is again to minimize the function given by eqn (3.17) but now the polarization must be replaced by eqn (3.40), which incorporates the frequency dispersion.P is approximately determined solely by the field of the trapped electron, Bio(r). phw(1 - @*) tanh Q&). (3.45) phm(1 - $8) + tanh- ptiwo* P*(r) z ImG(co) K c i , 2 2 tanh Further calculation of P*(r) now requires specification of ImG(w). ImG(w) is determined by the absorption spectrum of the medium. For liquid water this is well represented by a broad " dissipative " band with maximum at 1-10 cm-I and several narrower but still relatively wide infrared resonance^,^^ while the spectrum is likely to be dominated by infrared absorption bands for amorphous solids. We can, however, obtain some insight into the behaviour of P*(r) without specification of ImG(w) by dividing all the modes into two broad classes separated by a particular frequency 6.43 For w < G the hyperbolic tangent functions are replaced by their arguments (the classical limit) and for w > i3 by unity (the quantum limit).i3 is therefore ca. kT/h, but more precise values can be determined from a fit of the approximate expressions to eqn (3.43). Note that since@* is typically numerically small for reactions of the solvated electron, validity of the replacement of tanh@hwO/*2) by unity does not follow solely from the presence of high frequencies. We can then write P*(r) in the form p*(r) z x w z p E I m G ( w ) ( l - 8*)B:o(r) roo 1 c?:o(r). (3.46) +iJ $Im(Gw) 1 + tanh pfic00* - 2 w If O* is sufficiently small that ptiw8*/2 < 1 in spite of large frequencies, eqn (3.46) takes the simple form P*(r) z (1 - o*)P;;(r) + Py;(r) = Pc*I(r) + P&(r) (3.47)42 QUANTUM THEORY OF THE SOLVATED ELECTRON where the classical [P$(r)] and " quantum " [P$(r)] components refer to parts of P* (r) where the integration with respect to w is from zero to 5, and from 6 to infinity, respectively.If the inverse inequality, phw8*/2 + 1, is valid, then P*(r) FS (1 - t?*)P;;(r) + +Pyl(r) = PC*1(r) + P&(r). (3.48) For the 1s-like wavefunctions 3, is now determined by the equation (3.49) h2 P - a -f(e*)~(;l) = o where I@) is given by eqn (3.24), and the functionf(6)* is phw(i - e*) (3.50) 2 + tanh- tanh ptiwe* * f(S*) = f $Ie w ImG(w) pti@(i - e*) 2 2 tanh This equation is numerically tractable, but for small 18* I the following simple solution emerges. Expansion off(8*) to first order gives h2 lu f(8*) NN f(8) + O*df(B*)/dO*lp = 0 = 1 + df(8*)/d8*le* = 0 = -a/I(a).(3.51) By means of eqn (3.50) and by following the same procedure as for the classical limit we obtain From eqn (3.52) it is seen that p = 1 if ImG(w) # 0 only for classical modes. Other- wise p > 0 since for all values of Btiw/2. Insertion of eqn (3.53) in eqn (3.43) gives Eqn (3.54) can be further converted to the following form valid close to the activation- less region where is small. In this region 's 8* E (EP' + AFo)/pA2 (3.55) where the total nuclear reorganization energy, E,!Ot, is (3.56) E:O' = z/[&'io(r) 1 - d'fo(r)]2dr/$ ImG(w) and A2 = ,/dr[Bio(r) - tiTfo(r)l2 $paImG(w)coth P a -. 2 (3.57)A . M. KUZNETSOV AND J . ULSTRUP 43 The rate probability then takes a Gaussian dependence of AF,, + AFo)2/A2].(3.58) If EPt and A are unaffected by the variation of the electronic wavefunction, Wit takes the following form for small l8*1 Since p > 1, eqn (3.59) shows that the variation of the electronic wavefunction with O* has a stronger effect for high-frequency modes than for the classical limit, caused by a stronger dependence of the high-frequency part of the polarization on O*. The effect is therefore expected to increase with decreasing temperature. For larger 8 we must use numerical solutions of eqn (3.42), (3.45) and (3.50). An analytical rate expression can, however, still be obtained for the particular case where ImG(co) vanishes in a frequency region around kT/h, so that ImG(co) consists of separated low- and high-frequency parts, provided that the latter constitutes a narrow resonance centred at a frequency co,,.For large negative AFo where O* < 0 the equation for O* is -AFo = E;'(l - 20*) + E$"exp(-Phw,,B*) (3.60) where E;' and E$" are the reorganization energies of the low- and high-frequency modes, respectively. O* is then approximated by 8" ---@hc~~,,)-~ ln(/AF,J/E:") (3.61) and the transition probability x exP[-2~ol~*lp(p~co,.)-' ~n(lAFol/E,9") (3.62) where y = ln(lAFoI/EI,Qu) - 1. Eqn (3.62) is the energy-gap law known from the theory of radiationless p r o c e s ~ e s , ~ ~ ~ ~ ~ but it incorporates here the variation of the electronic wavefunction with the energy gap. 4. NUMERICAL CALCULATION A N D CONCLUDING REMARKS Eqn (3.42) is the basis for numerical calculation of 8*. Taking E:"' as a parameter this equation can be rewritten as (AFo/EPt) = 2 Phm(20* - 1) ImG(co)/[@ co2 ImG(co).sinh - 2 Similarly, eqn (3.44) can be given the form (4.1) Phco(1 - 0) phcoo sinh - 2 Phw ImG(co)/[*ImG(co). co (4.2) sinh - 244 QUANTUM THEORY OF THE SOLVATED ELECTRON I I I - - e* -0.5- \. - \. I 1 1. 'a,. - 1 - 2 - 3 -0.4 When 8* is inserted in eqn (3.43) and (3.44) the rate probability can be calculated numerically for the continuous distribution of nuclear modes represented by ImG(co). These equations must subsequently be combined with eqn (3.49) and (3.50) in order to incorporate the variation of the electronic factor. For disordered media representative forms of ImG(m) are combinations of Debye and resonance absorption bands ImG(co) = where RDi and QRj are the *frequencies of maximum absorption for the Debye and resonance bands, respectively, rj the widths of the latter, and Ggi and Gij constants determined by the dielectric constants in the same way as for the c constants above. For solid glasses the Debye functions must be considered as empirical distribution func- tions since relaxational motion is here of minor importance.Fig. 1 shows plots of 8* against AFo for two temperatures representative of experi- mentally investigated reactions of the solvated electron. We have represented the medium by a single Debye band of maximum frequency 10 cm-l for T = 298 K (corresponding to a liquid soiution) and 150 cm-l for T = 77 K where the medium is a solid. Fig. 2 shows the corresponding dependence of the nuclear part of Wi, where against AF,.These curves exhibit characteristic asymmetric maxima which reflect tunnelling of the nuclear modes in the high-frequency branches of the distribution. For large values of E:O* and high temperatures the numerical results are only insig- nificantly modified if, for the same ImG(w) is represented by one Debye band with the same maximum frequency and one narrow resonance at 50-100 cm-l corres- ponding to the " breathing " mode of the solvent molecules nearest to the electron.A . M. KUZNETSOV A N D J . ULSTRUP 45 - 1 - 2 -3 AFoleV FIG. 2.-lnA plotted against AFo, where A is the " nuclear part " of the rate probability. The symbols The difference is, however, more pronounced at lower temperatures and values of E:"'.Fig. 3 shows the variation of 1 with O* calculated from eqn (3.22)-(3.25) for the classical limit and from eqn (3.49) and (3.50) for the general case covering also high- frequency modes. In the classical limit there is no significant numerical difference from the limiting form for small lO*l [eqn (3.25)] up to values of 1O*I z 0.3. For larger 16.1, are the same as in fig. 1. Energy quantities in eV. 1.5 2 1 - --- 0 - 5 I 1 -0.5 0 0.5 8* FIG. 3.-A/A0 calculated from eqn (3.22), (3.24), (3.49) and (3.50), plotted against 8*. (-) The classical limit [eqn (3.22) and (3.24)]. (- - - -) The limiting classical case for small lo* I [eqn (3.25)]. (. - . - .) A/A, calculated from eqn (3.49) and (3.50). The ordinate scale for the latter curve should be multiplied by two.46 QUANTUM THEORY OF THE SOLVATED ELECTRON A varies more slowly in the strongly exothermic region but more strongly in the " normal " region.For high-frequency modes A also depends rather slowly on 8" in the normal region. However, for 8" < 0 there is a very significant rise in A/& to values which imply that the rate constant could drop by several orders of magnitude as a consequence of this effect. The cause of this difference is the much stronger dependence of P*(r) on 8* in this region as reflected in the high- and low-frequency parts of this quantity [eqn (3.45) and (3.46)]. The effect of the variation of the electronic factor on the drop in the rate constant with increasingly negative AF' for strongly exothermic processes is thus expected to be no less important than the drop caused by increasing vibrational excitation of the nuclear modes such as revealed by fig. 2.This effect should therefore be incorporated in analysis of experimental data for electron transfer in this free-energy region. APPENDIX We derive here the integral in eqn (3.24) from eqn (3.21) and (3.23). Since I(1) arises from the second term in eqn (3.21) we calculate at first the integral 9 = Dfo (Y) D; (Y) dv. (Al) 9 can be rewritten in terms of the Fourier components of Dfo ( r ) and Df(r), i.e. Dfo(k) and Dt(k), respectively I where Df* is the complex conjugate of D:. The explicit form of Dfo(k) is DX(r) exp(-ik r) = --e dxl~/?~(x)l' exp(-ik x) D(k) (A3) I where exp[ik (r - x)]. An analogous equation is valid for Df(k).form Since k and D(k) are parallel D(k) takes a simple The integral 9 then becomes After insertion of eqn (3.13) 9 can be transformed Differentiation of this equation with respect to 1 finally gives eqn (3.24). We thank Prof. R. R. Dogonadze, Institute of Inorganic Chemistry and Electro- chemistry of the Academy of Sciences, Tbilisi, and Dr M. A. Vorotyntsev, Institute ofA . M. KUZNETSOV AND J . ULSTRUP 47 Electrochemistry of the Academy of Sciences, Moscow, for helpful comments. We thank the Danish Natural Science Research Council for financial support. L. Kevan, Adv. Rad. Chem., 1974, 4, 181. N. V. Klassen, H. A. Gilles and G. G. Teather, J. Phys. Chem., 1972, 76, 3847. G. V. Buxton, H. A. Gilles and N. V. Klassen, Chem. Phys. Lett., 1975, 32, 533.G. Dolivo and L. Kevan, J. Chem. Phys., 1979, 70,2599. J. W. van Leeuwen, L. H. Strover and H. Nauta, J. Phys. Chem., 1979, 83, 3008. Cz. Stradowski, M. Wolszczak and J. Kroh, Rad. Phys. Chem., 1980, 16,465. J. R. Miller, J. Phys. Chem., (a) 1972,56, 5173; (6) 1975, 79, 1070. F. S. Dainton, M. J. Pilling and S, A. Rice, J. Chem. Soc., Faraday Trans. 2, 1975,71, 131. ' Cz. Stradowski and J. Kroh, Rad. Phys. Chem., 1980,15, 349. lo S. A. Rice and M. J. Pilling, Prog. React. Kinet., 1978, 9, 93. l1 J. V. Beitz and J. R. Miller, J. Chem. Phys., 1979, 71, 4579. l2 M. Tachiya and A. Mozumder, Chem. Phys. Lett., 1974,28, 87. l3 R. P. Van Duyne and S. F. Fischer, Chem. Phys., 1974, 5, 183. l4 J. Ulstrup and J. Jortner, J. Chem. Phys., 1975, 63, 4358. l5 R. R. Dogonadze, A. M.Kuznetsov and M. A. Vorotyntsev, 2. Phys. Chem. (NF), 1976,100, l6 I. Webman and N. R. Kestner, J. Phys. Chem., 1979, 83, 431. lB R. R. Dogonadze, A. M. Kuznetsov, M. A. Vorotyntsev and M. G. Zakaraya, J. Electroanal. l9 A. M. Kuznetsov, N. C. Sonderglrd and J. Ulstrup, Chem. Phys., 1978,29, 383. 2o E. M. Itskovitch, A. M. Kuznetsov and J. Ulstrup, Chem. Phys., 1981, 58, 335. 21 A. M. Kuznetsov and J. Ulstrup, in preparation. 2L W. P. Helman and K. Funabashi, J. Chem. Phys., 1977,66,5790. 23 W. H. HamiH and K. Funabashi, Phys. Rev. B, 1977, 16, 5523. 24 S. A. Rice, J. Phys. Chem., 1980,84, 1280. 25 M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964). 26 (a) R. R. Dogonadze and A. M. Kuznetsov, Physical Chemistry: Kinetics (VINITI, Moscow, 27 A. M. Kuznetsov, Nouv. J. Chim., 1981, 5, 427. 28 A. M. Kuznetsov and J. Ulstrup, Phys. Sfat. Sol., in press. 29 M. V. Vol'kenshtein, R. R. Dogonadze, A. K. Madumarov and Yu. I. Kharkats, Dokl. Akad. 30 A. M. Kuznetsov and J. Ulstrup, J. Chem. Phys., 1981, 75, 2047. 31 R. R. Dogonadze, A. M. Kuznetsov and T. A. Marsagishvili, Electrochim. Acta, 1980,25, 1. 32 D. N. Zubarev, Nonequilibriurn Statistical Thermodynamics (Plenum, New York, 1974). 33 R. R. Dogonadze, A. A. Kornyshev and A. M. Kuznetsov, Teor. Mat. Fiz., 1973, 15, 127. 34 S. F. Fischer, J. Chem. Phys., 1970, 53, 3195. 35 J. Ulstrup, Charge Transfer Processes in Condensed Media, Lecture Notes in Chemistry (Springer- 36 S. I. Pekar, Untemuchungen iiber die Elektronentheorie der Kristalle (Akademie-Verlag, Berlin, 37 R. R. Dogonadze, A. M. Kuznetsov and M. A. Vorotyntsev, Phys. Sfat. Sol., 1972, 54, 125; 1. N. R. Kestner, J. Phys. Chem., 1980, 84, 1270. Chem., 1977, 75, 315. 1973); (b) Prog. Surf. Sci., 1975, 6, 1. Nauk SSSR, Ser. Fiz. Khim., 1971, 199, 124. Verlag, Berlin, 1979), vol. 10. 1954). 425. R. R. Dogonadze and Z. D. Urushadze, J. Electroanal. Chem., 1971,32, 235. 1976, 72, 361. 39 Yu. I. Kharkats, A. A. Kornyshev and M. A. Vorotyntsev, J. Chem. Soc.: Faraday Trans. 2, 40 D. A. Copeland, N. R. Kestner and J. Jortner, J. Chem. Phys., 1970,53, 1189. 41 K. Fueki, D. F. Feng and L. Kevan, J. Am. Chem. Soc., 1973,95, 1398. 42 A. A. Ovchinnikov and M. Ya. Ovchinnikova, Zh. Eksp. Teor. Fiz., 1969, 56, 1278. 43 M. A. Vorotyntsev, R. R. Dogonadze and A. M. Kuznetsov, Dokl. Akad. Nauk SSSR, Ser. Fiz. Khim., 1970, 195, 1135. 44 (a) J . B. Hasted, Specialist Periodical Report. Dielectric and Related Molecular Processes (The Chemical Society, London, 19721, vol. 1, p. 1 ; (6) M. N. Afsar and J. B. Hasted, Infrared Phys., 1979, 18, 835. 45 F. K. Fong, Theory of Molecular Relaxation (Wiley, New York, 1975). 46 R. Englman, Non-Radiative Decay of Ions and Molecules in Solids (North-Holland, Amsterdam, 1979).

ISSN:0301-7249    

DOI:10.1039/DC9827400031

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982,74, 49-56 Quantum Theory of the Processes of Charge Transfer Recent Advances BY ALEXANDER M. KUZNETSOV Institute of Electrochemistry of the Academy of Sciences of the U.S.S.R., Leninsky Prospect 31,117071 Moscow V-71, U.S.S.R. Received 10th May, 1982 A new aspect of the interaction of an electron with the polarization of the surrounding medium is discussed. Effects caused by the modulation of the electron wavefunctions by fluctuations in the polarization are considered. New methods of calculating the transition probability are presented. They permit one to take into account the effects mentioned above and to go beyond the Condon approximation. Many new results have been obtained in the quantum-mechanical theory of elementary charge-transfer processes in polar media during recent years.Some of these are related to the fundamentals of the theory itself, others concern the application of the theory to various phenomena, including chemical and electrochemical reactions of the electron and proton transfer, adsorption, processes involving solvated and trapped electrons in solids and in liquids, biochemical reactions etc. In this paper we focus attention on those results which are of principal importance in developing the fundamentals of the theory. They will be presented for homo- geneous electron-transfer reactions as an example. A review of earlier results may be found in ref. (1)-(5). INTERACTION OF THE ELECTRON WITH THE POLARIZATION OF THE MEDIUM (A NEW ASPECT) It is understood that fluctuations in the polarization of the medium play an important role in the processes of charge transfer.The microscopic mechanism of a non-adiabatic electron transfer between two centres A"1 and BZ2 located at a distance R from each other in a polar medium may be described briefly as Owing to the fluctuation of the inertial polarization of the medium from the initial equilibrium value Poi to the transitional configuration P* the equalization of the electron energies E , and e, required by the Franck-Condon principle occurs. In this configuration the electron transfer from the centre A'I to the centre Bz2 takes place followed by the relaxation of the polarization to the final equilibrium value. The behaviour of the degrees of freedom describing local vibrations in the medium or intramolecular vibrations in A and B depends on the values of their vibrational frequencies and on the potential-energy profiles, and may be either classical or quantum-mechanical in nature.It is of no major importance in the context of the problems discussed below. The work required to change the polarization from Poi to P* determines the50 A QUANTUM THEORY OF CHARGE TRANSFER activation free energy, Fa. The transmission coefficient K is largely determined by the value of the electron-resonance integral where qA and tpB are the zeroth-order electron wavefunctions describing the initial and final electron states, Prd the off-diagonal parts of the interaction operator pi leading to the transition of the electron from A to B. To calculate Fa we must final states Ui and U,. In reference model (RM), the approximation are described where c = I/&, - I/cS and is REFERENCE MODEL know the free-energy surfaces (FES) of the initial and the simplest model, which will be referred to as the free-energy surfaces of the solvent in the continuum by the formula U = (27~1~) P2(r)d3r the polaron-theory parameter.The electrostatic interaction of a charge ez with the solvent is chosen in the RM in the form where cfz(t) is the vacuum electric field created by the charge ez. Due to this interaction a polarization Po(r), which depends on the charge distribu- tion in the system, arises in the medium. It is assumed in the RM that the fluctuations of the polarization do not change the charge distribution (i.e. the electron-density distribution) in the initial and final states.Then the free-energy surfaces Ui and U, are quadratic in the deviation of the polarization from the initial (Poi) and final (Po,) equilibrium values. This results in the known expression for the activation free energy F," = (Er + AF)2/4E, where E, is the reorganization energy of the medium and AF is the free-energy of the transition. To calculate the electron-resonance integral we must know the zeroth-order wave- functions and the interaction Pfd. In the RM the interaction of the electron with the second centre (B), Vl;, is usually chosen as the interaction causing the electron transfer from A to B. The superscript od denotes the off-diagonal part of this interaction. The interaction of the electron with the polarization, yep, which is usually strong, is involved in the zeroth-order Hamiltonians determining the electron wavefunctions VIA and VBZ where T, is the kinetic energy of the electron, and the superscript d denotes a diagonal part of the interaction which does not lead to the electron transition and only distorts the electronic states.Note that in the RM the entire interaction of the electron with the polarizationA . M. KUZNETSOV 51 Ve, is involved in eqn (4) in determining the electronic states. Furthermore, the dependence of qA and pB on the polarization is neglected when calculating the transition probability, as mentioned above. A NEW APPROACH Recently the role of the interaction of the electron with the polarization has been analysed in detai1.7-13 It was suggested that the zeroth-order states should be determined using the equations of the type iTe + VeA + Ve$ + VgB + (V&>dlpA = &AvA [Te + VeB + vz + v,dA + ( V & ) d b B = EBvB ( 5 ) where VeAp is that part of the interaction of the electron with the polarization which creates (together with VeA) the potential well for the electron near centre A.I/: is defined in a similar way. Then the perturbation operators pyd and PFd have the form Ppd = (veB)od + (v&)od = VeB - (VeB)d + V$ - <V$>d (6) = ( VeA)od + (v$)od = VeA - ( veA)d + - (V&)d* The above definitions of the zeroth-order states and of the perturbation operators are more reasonable from a physical point of view than those used in the RM. The zeroth-order states determined in the RM, taking into account all interactions with the polarization at various values of the polarization, may not describe the electron localized near a given centre.In our new approach eqn (5) involve only that part of the interaction of the electron with the polarization which retains its location near a given centre at any value of the polarization. This new approach enables us to consider all the physical effects which are due to the interaction of the electron with the polarization of the medium and to take them into account in calculating the transition probability. These effects are: (i) The eflect of diagonal dynamic disorder. The fluctuations of the polarization change the position of the electronic energy levels E~(P) and cB(P), enabling us to satisfy the requirements of the Franck-Condon principle.(ii) The eflect of of-diagonal dynamic disorder. The interaction of the electron with the polarization fluctuations near the other centre leads to the appearance of additional terms V$ and V$ in the perturbation operators for forward (pi) and backward ( vf) transitions in eqn ( 5 ) . These terms depend on the polarization of the medium. In some cases they may considerably exceed the terms describing a direct interaction with the centres B (VeB) and A (VeA). The interaction of the electron with the polarization fluctuations plays an important role in processes involving the solvated, trapped or weakly bound electrons, and in particular in the electrochemical generation of solvated electrons. (iii) The eflect of diagonal-ofl-diagonal dynamic disorder.The fluctuations of the polarization produce a variation in the distributions of the electron densities near the donor A and the acceptor B (i.e. they cause a modulation of the electronic wave- functions qA and vB). This leads to a modulation of the overlapping of the electron clouds of the donor and the acceptor and hence to a change in the transmission coefficient as compared to the value calculated in the approximation of constant electron density (ACED) (table 1). The variation in the (iv) An additional efect of the diagonal dynamic disorder.52 A QUANTUM THEORY OF CHARGE TRANSFER TABLE 1 .-KINETIC PARAMETERS OF ELECTRON TRANSFER FOR THE SYMMETRICAL SYSTEM IN THE CONDON APPROXIMATION TAKING DUE ACCOUNT OF THE MODULATION OF THE ELECTRON WAVEFUNCTIONS BY FLUCTUATIONS IN THE POLARIZATIONu PISb FJF; 2a,R = 4" 2a,R = 8" KlKo WlWo KIKO WlWO 0 0.811 4.70 56.99 22.11 268.0 0.5 0.864 5.21 76.92 27.11 400.5 1 0.893 5.54 93.39 30.69 517.4 2 0.925 6.05 117.92 36.60 713.4 -0.1 0.795 5.11 58.34 26.09 298.0 a Calculated neglecting the mutual influence of the donor and the acceptor on the zeroth-order distribution of the electron densities near the centres A and B due to fluctuations in the polarization leads to a change in the interaction of the electron with the polariza- tion of the medium and hence a change in the shape of the free-energy surfaces iYi and U,.This effect leads to a change in the activation free energy Fa with respect to the value Fi calculated in the ACED (table 1). A detailed discussion of these effects and numerical calculations are given in ref.(7)-(13). It is shown that the electron densities in the transitional configuration and the activation free energy calculated with due account for these effects and beyond the Condon approximation may differ significantly from those calculated in ACED.7-13 characteristics; PIS = 16z/5c~,; up = 5mce2/16h2. DEVELOPMENT OF THE METHODS OF CALCULATION Below we shall discuss the methods of calculating the probability per unit time, W, of the elementary act of a non-adiabatic charge transfer and the approximations used. One of the main methods of calculation for the non-adiabatic reaction is quantum-mechanical perturbation theory. It is shown that the approximate expression for W may be written as follows:i4 = pe[q*, nA(q*), nB(q*)l pn(Pi,Pf> (7) (8) (9) where Pe and P, are the electronic and nuclear factors, given by pe [q,nA(x * 4) ,nB (x * 4) 1 = A2 1 (x * 4) I nA(x * q>nB (x * 4) Pn(Pi,pf> = (l/ihkT) S do exp(PFi - PoAJ)Tr[pi(l - e)p,(e)l where pi is the off-diagonal part of the interaction leading to the transition, n A and nB are the electron densities of the transferable electron near the donor A and the acceptor B, respectively, Fi is the free energy of the initial state, AJ is the free energy of transition, p = l/kT, pi and p f are the statistical operators of heavy particles in the initial and final states calculated at the temperatures T/(1 - 0) and 778, q is the set of coordinates of the heavy particles, x* the point of the maximum overlapping of the electron " clouds " of the donor and the acceptor, A the effective region giving the major contribution to the electron resonance integral, and q* the transitional configura- tion.If the dependence of pi, n A and nB on q is neglected, we obtain the approximation uf constant eZectron densities (ACED). The results obtained in this approximation are those obtained in the reference model. The usual Condon approximation (CA) is obtained if the dependence of the electronic factor P , on q in eqn (7) is taken into account but the nuclear factor P, is calculated in the same manner as in the ACED.A . M. KUZNETSOV 53 tf in calculating pi and pf involved in P, we take into account the modulation of the electron densities nA and n B by the vibrations of heavy particles, we obtain an improved Condon approximation (ICA).If a classical approximation is used for the calculation of pi and pf we may consider all the cases of Condon approximation (ACED, CA, ICA) and in certain models we may go beyond the Condon approximation (BCA).l0-l4 ZEROTH-ORDER ELECTRON STATES The calculation of the zeroth-order electron states at arbitrary values of the nuclear coordinates is a very complicated problem. It becomes a little simpler if the vibrational subsystem is a classical one. To calculate the transition probability in the CA we have to find the electron wavefunctions only in the equilibrium configura- tion and in the transitional configuration. In the classical limit the transitional configuration is determined as follows 4" = (1 - Q*)qoi + 6"qof (1 1) where 8" is a symmetry factor.If the interaction of the electron with the vibrational subsystem is linear, the terms V& and V$ involved in eqn (5) in the transitional configuration will differ from those in the equilibrium configuration only by the factors (1 - 8*) and 1 9 * . ~ This fact simplifies significantly the calculation and enables us to use the methods developed for the equilibrium configurations. In particular a direct variational method has been The wavefunctions of the initial and final states are chosen in the form qi z exp(-air), qf z exp(-EfIr - RI). (12) The dependence of the orbital exponents ai and orf on the symmetry factor were calculated. It was shown that for 0 < 8* < 1 the radii of the localization of the electron near the donor and the acceptor in the transitional configuration are larger than those in corresponding equilibrium configurations.More complicated functions may also be used to approximate qi and qf. The explicit dependence of qi and pf on the nuclear coordinates is necessary for the calculations in the ICA and BCA. The model considered assumes, in particular, that the major contribution to the transition probability comes from symmetrical polarization fluctuations near the donor and the acceptor of the type &PA = (c/4z)t D?(r,Poi), SPB = ( c / ~ z ) v DeB(r,Pof> (13) where t and 7 are independent variables, and D2(r,Poi) and D:(r,Pof) are the electro- static inductions due to the electron at the equilibrium values of the polarization. This model enables us to take into account the dependence of pi and qf on the polarization vibrations and the change in shape of the free-energy surfaces Ui and U,, and to go beyond the 'Condon a p p r o x i m a t i ~ n .l ~ * ~ ~ ~ ~ ~ Calculations beyond the Condon approximation have shown that the effects of deviation from the CA are important only for long-range tran~fer.'~.'~,'~ At short transfer distances the ICA is satisfactory. More exact results may be obtained by considering polarization fluctuations of a more complicated shape. PATH-INTEGRAL METHOD The above analysis uses the assumption that the major contribution to the transi- tion probability arises from polarization fluctuations of a certain type [see eqn (1 3)].54 A QUANTUM THEORY OF CHARGE TRANSFER A more general and more consistent method has been This starts from the Fermi golden rule, and may be expressed in terms of Feynman path integrals of the initial and final states for the interacting electron and nucleus subsystems.The calculation gives the following expression for the transition probability14*15 W = const x p?(x*,q*)Zi’ pr dxdx’ q;”(x,O)qi(x’,O)qf(x,O)pf*(x’,O) c-ica where pi is the off-diagonal part of the interaction leading to the transition, pi(x,z) and pf(x,z) are the electron wavefunctions of the donor and the acceptor, respectively, zi = 1 - 8, zf = 6, and Zi is the statistical sum of the initial state. Rif(m,z,z’,zi,zf) = ([sinhphco(zi - ~)~inh~hwz’~inh~hmz~sinh~hwz~cos(~hw/2) + sinhphmzsinhphm(zf - z’)sinh~hwzisinh~htwzfcosh(~hw/2) + sinh~hwzsinh~h~r’sinh~hmzisinh~hwzfcosh(~w/2)] /2sin h(Phw/2)sin h/lhwz, sin hphwz,) (20) where qi0(z) and qio(z) are the equilibrium values of the coordinates of the nuclear subsystem in the initial and final states corresponding to the electron wavefunctions pi(x,z) and p,(x,z). wK is the frequency spectrum of the nuclear subsystem, and Hi and I€! are the Hamiltonians of the electron in the initial and final states.In the simplest case they take the form Hi = -(h2/2m)A + VeA Hof = -(h2/2m)A + V,,A. M. KUZNETSOV 55 where VeA and VeB are the interaction energies of the electron with the donor and the acceptor. In a more rigorous consideration Hi involves in addition the diagonal part of the interaction of the electron with the acceptor and Hi involves the diagonal part of the interaction of the electron with the donor.Eqn (14)-(22) are valid for the case when the nuclear subsystem can be described by the harmonic approximation and for the case of the electron interaction with the fluctuations of the polarization or of the density of the medium in the long-wave approximation. In the case when the vibrations of the medium polarization play the major role in the transition, the quantities involved in eqn (16)-(22) may be expressed in terms of the complex dielectric function of the medium, ~ ( k , m ) , using the summation where Dak(7) is the Fourier component of the electrostatic induction Da(r7Z) due to an electron in the state qa(x,z). In the classical limit similar expressions for the transition probability are obtained for the case of a linear interaction of the electron with any fluctuating field, provided the fluctuations of the field are Gaussian. In this limit more general expressions may be obtained for the free-energy surfaces of arbitrary shape.The electron wavefunctions qi(x,z) and qf(x,z) are determined by the condition of the maximum value of the transition pr0babi1ity.l~ In particular in the improved Condon approximation the equations for the calculation of qi and qf have the form & i S f [ ~ i ~ i l + &iSp[~i,qf,zi,rf] = 0 & S [ ~ f , ~ f l + ~ & j ' p [ ~ i ~ ~ f , ~ i , ~ f l = 0 (25) (26) where Bpa denotes the variation with respect to qa. Note that the equations for pi and pf are coupled and describe a self-consistent state of the electrons in the donor and in the acceptor.This reflects the fact that in the transitional configuration the electron interacts with the polarization fluctuation near its own location centre and with that near the other centre. In the classical limit the expression for the transition probability takes the form14*15 T f * - 5 k l . r {[ T&Di(r,z)]' + [ 1 dzDf(r,z)12 + 2 lo:z ,/~z'Di(r,z)Of(r,zl)) 871 0 0 1 c 1 - /odr(CiiHi]f+) + E-/ d3r{\ 0 d~D~(r,r)}~]}; a = i, f where Ti* = 1 - 8*, T ~ * = O* [8* is the symmetry factor determined as a saddle point in the integral over 0 in eqn (14)], pi is the electron wavefunction of the donor in the initial equilibrium configuration, and Di the corresponding electrostatic induction. In particular for the symmetrical transition 8* = zi* = zf* = 1/2.We may use a direct variational method rather than have to solve eqn (23) and (24) for the wave- functions in the transitional configuration and similar equations for the wavefunctions56 A QUANTUM THEORY OF CHARGE TRANSFER in the equilibrium configurations. Using the exponential functions of the type in eqn (12) as probe functions for pi and pi and varying the expressions for the transition probability and for the statistical sum of the initial states over a and a. we may find these parameters and thus calculate the free energy of activation. The calculation in this case, making the assumption that the mutual influence of centres A and B on the zeroth-order wavefunctions qi and pf may be neglected, gives for the free energy of activation14J5 where F," = (5/32)ce2a0 is the free energy of activation calculated at R + co neglecting the modulation of the electron wavefunctions by the phonon field; a.is the inverse localization radius of the electron in the equilibrium configuration. As one can see from eqn (28) Fa may differ significantly from F,", especially for the transfer of weakly bound electrons between neutral or negatively charged centres. CONCLUSIONS The above analysis shows that the interaction of the electron with the vibrational subsystem leads to various effects which should be taken into account when calculating the probability of electron transfer. Some of these effects (viz. the effect of diagonal dynamic disorder, respresenting the dependence of the electron energies on the nuclear coordinates) have been considered earlier. Two new methods of calculation of the transition probability presented in this paper enable us to incorporate into the theory other effects which are due to the modulation of the electronic wavefunctions caused by the change in the nuclear configuration. These effects are of especial importance for processes involving weakly bound, trapped and solvated electrons and for long- range electron transfer. The results obtained provide deeper understanding of the mechanism of elementary charge-transfer processes in polar media. ' R. R, Dogonadze and A. M. Kuznetsov, in Itogi Nauki i Techniki, ser. Kinetika i Katalis (VINITI, Moscow, 1978), vol. 5 . R. R. Dogonadze and A. M. Kuznetsov, Prog. Surf: Sci., 1975,6, 1. J. Ulstrup, Charge Transfer Processes in Condensed Media, in Lecture Notes in Chemistry (Springer-Verlag, Berlin, 1979). P. P. Schmidt, in Electrochemistry (Specialist Periodical Report, The Chemical Society, London, R. A. Marcus, Annu. Rev. Phys. Chem., 1964,15, 155. R. A. Marcus, J . Chem. Phys., 1956, 24, 979. A. M. Kuznetsov, Nouv. J. Chim. , 1981, 5 , 427. a A. M. Kuznetsov, Elektrokhimiya, 1982, 18, 594. A. M. Kuznetsov, Elektrohhimiya, 1982, 18, 598. lo A. M. Kuznetsov, Elektrokhimiya, 1982, 18, 736. l1 A. M. Kuznetsov, Poverchnost, 1982, 1, 119. l2 A. M. Kuznetsov and J. Ulstrup, Phys. Stat. Sol. B, 1982, 114, in press. l3 A. M. Kuznetsov and J. Ulstrup, Elektrokhimiya, in press. l4 A. M. Kuznetsov, Chem. Phys. Lett., 1982, 91, 34. l5 A. M. Kuznetsov, Khim. Fiz., 1982, 11, 1496. 1975), VOI. 5 , pp. 21-131.

ISSN:0301-7249    

DOI:10.1039/DC9827400049

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 74, 57-72 Variation of the Electrochemical Transfer Coefficient with Potential BY JEAN-MICHEL SAV~ANT AND DIDIER TESSIER Laboratoire d’Electrochimie de 1’UniversitC de Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France Received 14th May, 1982 The electrochemical electron-transfer rate constant has been determined as a function of the electrode potential for a series of simple electron-transfer processes to organic molecules in media containing acetonitrile or dimethylformamide and a quaternary ammonium salt as supporting electrolyte and using mercury as the electrode material. The reactions and the experimental condi- tions were selected so as to deal with outer-sphere processes and to minimize the magnitude of double-layer corrections. Convolution potential-sweep voltammetry and the impedance method were used for obtaining the kinetic data.Under these conditions, the electrochemical transfer coefficient was observed, in all cases, to vary, beyond experimental error, with the electrode potential. The magnitude of the variation is of the same order of magnitude as that predicted by the Marcus theory of outer-sphere electron transfer. A more complex reaction, the reduction of benzaldehyde in ethanol, involving dimerization steps following the initial electron transfer was also investigated. A definite variation of the transfer coefficient was again observed. This behaviour, observed for various solvents and functional groups, appears as a general phenomenon in the reduction of organic molecules in the case where charge transfer is fast and mainly governed by solvent reorganization. The present theories of electron transfer at electrodes,’ such as the Hush-Marcus theory,2 -4 predict that the electrochemical transfer coefficient should vary with the electrode potential.Being based on an harmonic approximation they imply a quadratic dependence of the activation energy and therefore a linear dependence of the electrochemical transfer coefficient upon the electrode potential. They also predict the magnitude of the variation of the transfer coefficient as a function of the reorganization factor. The smaller the reorganization factor, i.e. the faster the electron transfer, the larger the variation of the transfer coefficient with potential. Over the last 15 years there have been several attempts to detect such variations experimentally and to compare their magnitude with that anticipated from the Earlier work5-11 in this area has not provided a clear answer to the question thus raised.The same systems that were first regarded as exhibiting such a potential dependence were later, after more accurate analysis, shown not to give rise to a definite variation that was clearly greater than experimental error.7 In order to obtain acceptable evidence that the transfer coefficient does or does not vary with the potential, the system under study must fulfil several requirements. The first of these is that the electrochemical reaction should follow a simple mechanism, preferably involving a single one-electron step giving rise to a chemically stable species, at least in the time range of the experiments.This does not mean that it is impossible to investigate the potential dependence of the transfer coefficient in more complex processes, involving for example follow-up chemical steps. However, it appears safer to start with elementary electron-transfer processes. When going to systems involving associated chemical reactions, the nature of these chemical steps should be ascertained and their occurrence be under proper experimental control. It is also required that the electron transfer be an outer-sphere process with no concomitant bond breaking58 ELECTROCHEMICAL TRANSFER COEFFICIENT or bond formation. Similarly, adsorption of reactants on the electrode surface should be avoided.Since double-layer effects can interfere with the apparent varia- tion of the transfer coefficient with potential, it is important that the structure of the electrode-electrolyte interface be properly defined. In this connection, it is preferable to use mercury as the electrode material rather than solid electrodes. The influence of double-layer effects increases with the charge of the reactants. Since corrections of the double-layer effects are sources of uncertainty, + 1/0 or O/- 1 couples are expected to be the most useful. In order to obtain the maximum accuracy in the detection of variations of the transfer coefficient, the potential range where the kinetic determinations are carried out should be as large as possible. Within a given time-scale the available potential range increases as the rate of electron transfer decreases.However, the variations of the transfer coefficient decrease as the rate of electron transfer decreases, The investigation of very fast electron transfers is, however, limited by the necessity of going to very short time-scales in order for the system to depart from electrochemical reversibility. The kinetic determinations then become less and less accurate. Of the reactions investigated with the aim of detecting the anticipated potential dependence of the transfer coefficient, the reduction of organic molecules giving rise to stable anion radicals in non-aqueous organic media, in the presence of quaternary ammonium salts on a mercury electrode, appears a good candidate for fulfilling the requirements discussed above.Quaternary ammonium cations are not specifically adsorbed on the mercury electrode surface in non-aqueous solvents such as aceto- nitrile 2o and DMF.21 The effect of the double layer on the electron-transfer kinetics can thus be estimated with reasonable accuracy using the Gouy-Chapman theory of the double layer in the absence of specific adsorption of the ions of the supporting electrolyte. Since the considered redox couple involves a neutral molecule and a mononegative species the double-layer effects are minimized. The use of an aprotic organic solvent also favours the chemical stability of anion radicals since attack by acids or electrophiles are minimized. There are a number of aromatic molecules that give rise to stable anion radicals under such conditions, provided the presence of good leaving groups such as halogens is avoided.The negative charge is then generally delocalized over a rather large volume leading to small solvation reorganization factors and hence to fast electron transfer.22 Mononitro derivatives such as nitro- durene and nitromesitylene appear to be good choices for investigating slower electron- transfer processes with the ensuing advantage of more accuracy in rate determinations. A large portion of the negative charge in the anion radical is located on the nitro group, giving rise to larger solvation reorganization factors and thus slower electron trans- Aliphatic nitro compounds can also be investigated since the nitro group both facilitates the reduction and chemically stabilizes the anion radical.23 In this paper we describe and discuss the results obtained from the reduction of organic molecules, including nitro compounds, on mercury in acetonitrile (ACN) and dimethylformamide (DMF) containing a tetra-alkylammonium salt as supporting electrolyte. These compounds give rise to a stable anion radical which allowed the investigation of an electrochemical reaction consisting of an outer-sphere electron transfer.The reduction of benzaldehyde in buffered ethanol will provide an example of a more complex electrochemical reaction where an initial outer-sphere electron- transfer step is followed by a dimerization reaction. Two different electrochemical techniques were employed for the rate determinations : convolution potential-sweep voltammetry (c.P.s.v.) and impedance measurements.J-M. SAVEANT AND D.TESSIER 59 THE PREDICTED POTENTIAL DEPENDENCE OF THE ELECTROCHEMICAL TRANSFER COEFFICIENT Considering the electron transfer A + e - - - L B z being the charge of A and z - 1 that of B, the rate law can be written as I> = k(E){(CA)o - (G)o exp[&E F - E") FS where i is the current, S the electrode surface area, (C,J0 and (CB)o the reactant con- centrations just outside the diffuse double layer, E the electrode potential, E* the standard potential of the A/B couple and k(E) the potential-dependent rate constant of the forward electron transfer. According to the Marcus t h e ~ r y , ~ . ~ k(E) depends quadratically upon the electrode potential according to where ;lo is the reorganization factor, Zel the heterogeneous collision frequency and qr the potential difference between the reaction site and the solution.The apparent transfer coefficient as defined by is thus given by a , , = z s + BE (I - There are two ways of defining the E = 0.5 a = 0.5 E and a are related by: 20: = 0.5 + F (E - Ee - d). BE transfer coefficient : - a. is readily derived from map, which is obtained from the experimental plots of lnk(E) against E, while a is conveniently used for obtaining the reorganization factor Lo from the same plots, according to with60 ELECTROCHEMICAL TRANSFER COEFFICIENT Comparison between the predicted and experimental variations of the transfer co- efficient can be made using CC or a SIMPLE ELECTRON-TRANSFER REACTIONS CONVOLUTION POTENTIAL-SWEEP VOLTAMMETRY Convolution potential-sweep voltammetry (c.P.s.v.) 24*2s is a procedure for treating the current-potential curves obtained from cyclic voltammetry (c.v.).The experi- mental current, i, is transformed by convolution with the linear diffusion characteristic function (nt)-* into a “ convoluted ” current, Z: The convoluted current is then used jointly with the current itself to determine the forward rate constant k(E) as a function of the electrode potential according to: where DA is the diffusion coefficient of the reactant, Z, the plateau value of Z, assuming that the reversible half-wave potential and the standard potential of the A/B couple are practically the same. Fig. 1 gives an example of a cyclic voltammetric curve and of the corresponding convolution potential-sweep curve for t-nitrobutane in DMF + 0.1 mol dm-3 Bu4NI.This also illustrates a convenient way of determining the standard potential of the A/B couple, based on the determination of the potential, FIG. 1 .-Cyclic voltammetry and convolution potential-sweep voltammetry of t-nitrobutane in DMF + 0.1 mol Bu4NI. Concentration, 1.5 mmol dm-3; sweep rate, 17.9 V s-’, E is referred to the Ag, AgI electrode.J - M . SAVEANT AND D. TESSIER 61 Ei ship : where the backward C.V. current intersects the potential axis, using the relation- C.P.S.V. presents two advantages over the conventional use of cyclic voltammetric peak separation. The first of these is that the form of the rate law, k(E), need not be a priori specified for treating the data. The second is that all the information contained in the voltammetric curve is used instead of only that provided by the C.V.peaks. Fig. 2(a) shows plots of k(E) against E obtained by these procedures for t-nitro- butane in ACN and DMF, and for nitrodurene and nitromesitylene in ACN. The results of an experiment carried out with nitrodurene in ACN in the presence of 2% H,O are also given. Electron transfer is then slower than for ACN without added water, emphasising the specific solvation of the anion radical by water molecules. In all cases, the plots of log k(E) against E appear as bent toward the potential axis, indicating a dependence of the apparent transfer coefficient on the potential. This is better seen on the plots of a,* against E [fig. 2(b)] obtained by differentiation of the log k(E) against E curves [eqn (l)].The apparent transfer coefficient thus appears as an approximately linear function of the electrode potential. However, we should ask whether this reflects an actual variation of the true transfer coefficient or a double-layer effect deriving from the variation of qr with the electrode potential. Differentiation of eqn (2) leads to: Fig. 3 shows the variations of p2, the potential difference between the outer Helmoltz plane and the solution, and of ap2/aE and a2q2/aE2 with the electrode potential. If we assume that the reaction site is located at the 0.H.p. it is seen that in the pertinent potential ranges a2p2/aE2 is so small that the second term in eqn (4) is negligible in comparison with the first.Even if the reaction site is closer to the electrode surface as it indeed appears to be (by ca. 20% in the case of the considered molecules with NBuZ as supporting cation),26 the above estimation will not be significantly altered. A first conclusion is that the electrochemical transfer coefficient does vary with potential beyond experimental error for the reactions considered. It is then of interest to compare the observed variation with that predicted by the Marcus theory. This is shown in table 1. k, and hence Lo were estimated on the basis of two different assumptions. In the first of these qr is taken as equal to zero and in the second pr = q2, i.e. the reaction site is regarded as located at the 0.H.p. With both assumptions it is observed that the experimental and predicted variations of the transfer coefficient with potential are of the same order of magnitude. Location of the reaction site closer to the electrode surface 26 would obviously not significantly alter this con- clusion.Note that electron transfer to the same compound, t-nitrobutane, is faster in DMF than in ACN. This reflects the stronger solvating power of the latter toward anions or, alternatively, the greater availability of residual water, a specific solvation agent, in ACN than in DMF. In both cases this is in agreement with ACN being a stronger acid and a weaker base (in a broad sense) than DMF. On the other hand, note that the value of a(E* + qz) is smaller than predicted by the Marcus theory ( O S ) , the difference being more important in ACN than in DMF.62 ELECTROCHEMICAL TRANSFER COEFFICIENT n 2 M d 0.5 0 I M -.0.5 0 0 n 3 -Y M - 4 - 0.5 - - 0 ,+*' ,f I I I I 0 5 0 I 0 n 2 en - -4 0.5 d 0.2 t- 0 I 0-5 (b2 1 0.51 \ . *.fa . - , . . .?;'a& -.. d .l .,' . I 1 0 ( b 3 ) I 0-5 rr" 0.51 \ ... ....... ........ .: .... I I 0 I Oa5 I (64) 0.2 O . 1 ", .- . a d I i 1 I 0 I 0-5 0.5 0 E - Ee/V 0.5 0 E - E ~ VJ-M. SAVEANT AND D. TESSIER 63 A possible explanation of this phenomenon is that the solvation mode is different in the starting molecule and in the anion radical especially as far as preferential solvation by water is concerned. In the framework of an harmonic approximation the parabola representing the potential energy of the product as a function of the reaction coordin- ate would then be tighter than that corresponding to the reactant.IMPEDANCE MEASUREMENTS The reduction of nitromesitylene in ACN [fig. 2(a), table 11 illustrates the limit- ation of the C.P.S.V. technique. For this relatively fast electron transfer, ks,ap = 0.2 cm s-l, reasonably accurate kinetic data could only be obtained in a narrow range of sweep rate and thus a narrow range of potentials. In order to overcome these difficulties the impedance method was used with the aim of investigating this even faster electron-transfer processes. The measurement technique we used was basically the same as already described 22 involving the use of a two-electrode configuration and of a lock-in amplifier providing the in-phase and quadrature component of the first harmonic current response to a small ( 5 mV amplitude) sinusoidal input voltage.Since we wished to investigate fast electron transfers, frequencies up to 20 kHz were used. Special care is then required to extract the faradaic resistance and capacitance from the overall output signal. On the other hand, the measurements have to be carried out for a large number of values of the d.c. potential since we are looking for small variations of a. These are the two reasons why the in-phase and quadrature component were digitized and then stored and treated in an on-line computer. A detailed description of the instrumentation and procedures used is given el~ewhere.~' The potential-dependent charge-transfer rate constant was derived from the faradaic resistance, Rf, and capacitance, C,, according to : 27 where OJ is the pulsation of the input signal, E the d.c.potential and E+ the reversible half-wave potential. Di and E+ were derived from the height and location, respec- FIG. 2.-(a) Forward charge-transfer rate constant in ACN and DMF with 0.1 mol dm-3 Bu4NI as a function of the electrode potential as obtained from c P.S.V. (b) Variation of the apparent transfer coefficient with potential. (1) 3.0 mmol dmP3 t-nitrobutane in ACN; (2) 2.0 mmol dm-3 nitrodurene in ACN + 2% H20; (3) 2.1 mmol dm-3 nitrodurene in ACN; (4) 1.64 mmol dm-3 nitromesitylene in ACN; (5) 2.5 mmol dm-3 t-nitrobutane in DMF. The various points were obtained with the following sweep rate (V s-l) system cathodic scan ~~ ~ 1 2166 724 216 72.2 23'6 6.60 2.40 0.75 0.15 2 2194 707 228 66.1 22.6 6.55 2.42 0.65 0.25 3 2322 651 221 69.4 23.5 6.96 2.31 4 659 238 65.1 23.6 6.56 5 2165 750 218 71.3 22.9 6.95 2.39 0.70 0.24 system anodic scan 1 2161 669 212 67.7 22.1 6.96 2.34 0.77 0.14 2 2076 596 205 56.1 20.4 6.41 2.10 0.64 0.21 3 2227 722 218 69.4 4 5 2171 729 208 68.8 21.9 6.70 2.3064 ELECTROCHEMICAL TRANSFER COEFFICIENT I I -1.0 -1.5 EIV us.Ag, AgI ( C ) . - 0.1 -0 --.-.- .-.-. - -.-. 1 I -1.5 -2.0 E/V us. Ag, Ag+ -0.5 -1.0 E/V us. SCE FIG. 3.-Potential difference between the 0.H.p. (--) and the solution and its first (- - -) and second (-. - .) derivatives. (a) ACN + 0.1 mol dm-3 Bu4NI; (b) DMF + 0.1 mol dmP3 Bu,NI; (c) EtOH + 0.4% H 2 0 + 1 mol dm-3 Bu,NI + 0.012 mol dm-3 Bu4NOH. tively, of the minimum of the faradaic capacitance as a function of the d.c.potential: where C" is the initial concentration of A. Fig. 4(a) shows plots of logk(E) against E obtained for nitromesitylene, nitro- durene, terephthalonitrile, phthalonitrile and p-diacetylbenzene in DMF. Differen- tiation of these curves gives the variation of the apparent transfer coefficient with the potential. While this is clearly the case for the first three compounds, the results obtained with phthalo- nitrile and p-diacetylbenzene show less accuracy, corresponding to the fact that these An approximate linear dependence is again observed [fig. 4(b)].TABLE 1 .-PREDICTED AND OBSERVED VARIATION OF ELECTROCHEMICAL TRANSFER COEFFICIENT * I Cr = A 4'2 7 Cr = 0 c (aa/aE)/V-' (au/aE)/V-' solvent reactant -Eoa DA Z,, {aa,,/a;E) k, (A,/F) u(E* r-~-, k, (llo/F) a(E* ,------ \ /V /cm2sWi /ems- ~ /V- /ems-' /V +pr) exptl Marcus /cm s-' /V +p,) exptl Marcus ACN t-nitrobutane 1.998 3.0 x lo-' 6200 0.33 2.0 x 10- DMF t-nitrobutane 1.277 1.3 x lod5 6200 0.48 4.8 (f0.03) (& 1.0) ACN + nitrodurene 1.619 1.9 x 4700 0.45 1.6 2 %H20 (h0.04) (3ZO.1) x 10-2 ACN nitrodurene 1.714 1.9 x 4700 0.55 6.0 (3~0.06) (50.5) x 10-2 ACN nitromesitylene 1.667 1.9 x 4900 0.75 2.0 (3ZO.10) (f0.1) x 10-1 (f0.01) kto.2; x 10-3 1.28 ( f 0.01) 0.44 0.165 0.165 1.9 (& 0.006) ( f 0.002) ( f 0.2) x 10-2 0.51 0.240 0.175 5.0 (f0.015) (A0.003) (&l.l) x 10-2 0.44 0.225 0.195 1.2 (f0.018) (f0.002) (fO.l) x 10-1 0.45 0.275 0.218 4.7 (f0.031) (f0.002) (f0.4) x 10-1 0.48 0.37 0.243 1.64 (f0.05) (fO.001) (f0.06) ~ ~ 1.29 0.41 0.174 0.194 (& 0.006) (* 0.002) (10.01) 1.19 0.46 0.249 0.210 (f 0.03) ( f 0.0 16) ( k 0.004) 1.08 0.40 0.245 0.23 ( f 0.01) ( f 0.020) (f 0.002 0.94 0.40 0.294 0.267 ( f 0.01) (h0.033) (f0.003) 0.81 0.40 0.40 0.307 (-CO.Ol) (f0.08) (f0.002) E is referred to the Ag/Ag+ electrode in ACN, and to the Ag/AgI electrode in DMF.66 0- G 5 - - 1 - ELECTROCHEMICAL TRANSFER COEFFICIENT d 0.5 0.2 1 I I I 1 0.1 0 0.1 0 0.2 c .. .. 0.1 I 0 (bla) L .. - . .. . . : ... . - 3 0.5 0.2 t- 1 1 I I 0.1 0 E - E*/V I 1 I I 0.1 0 0.1 0 .. 1 0.1 ... I .:" 0 (a101 1 1 1 I 0.1 0 E - Ee/VJ-M. SAVBANT AND D . TESSIER 67 compounds give rise to very fast charge transfer (table 2). The variations of q2 with potential [fig. 3(b)] are again too small to be responsible for the variation of the apparent transfer coefficient with potential.In fact, they can again be neglected when passing from the variation of the apparent transfer coefficient to the variation of the intrinsic transfer coefficient [eqn (4)]. Comparison of the experimental and predicted potential dependence of the transfer coefficient was made in the same way as in the preceding section. The results (table 2) show that the experimental and predicted variations are again of the same order of magnitude. REACTIONS INVOLVING CHEMICAL STEPS COUPLED WITH THE ELECTRON-TRANSFER PROCESS REDUCTION OF BENZALDEHYDE IN ALKALINE ETHANOL The reduction of benzaldehyde in alkaline ethanol or water + ethanol mixtures has been shown to involve dimerization reactions following the initial charge-transfer ~tep.~'-~' Since the apparent rate of dimerization increases with decreasing pH, the reduction mechanism is likely to involve the following steps: C6H5CH0 + e- + C6H5CHO*- C6H5CHO*- + H+ === C6H5&OH (KA) ko 2 C6H5CHO'- C6H5-c7C-c6H5 I I 0- 0- OHOH Since the mono- and di-pinacolate are rapidly protonated in the considered pH range (15.5-17) and since the protonation of the anion radical is fast and reversible, the overall reaction scheme is equivalent to : A + e- B (E", ks, CL) kd 2B -+ products with kd the apparent dimerization rate constant. The kinetic control of the overall reaction depends upon two parameters featuring the rate of the electron transfer and the rate of the dimerization reaction relative to the rate of diffusion.In cyclic voltammetry and convolution potential-sweep voltam- FIG.4.-(a) Forward charge-transfer rate constant as a function of the electrode potential as obtained for impedance measurements. (b) Variation of the apparent transfer coefficient with potential. Solvent: DMF; supporting electrolyte: 0.5 mmol dm-3 NBuJ. (6) 0.76 mmol dm-3 nitromesi- tylene; (7) 1.0 mmol dmw3 nitrodurene; (8) 0.95 mmol dmW3 terephthalonitde; (9) 0.94 mmol dm-' phthalonitrile; (10) 1.0 mmol dm-3 p-diacetylbenzene. The various points were obtained with the following frequencies (kHz): 1, 2.5, 5, 10, 20 for (6) and (7); 2.5, 5, 10, 20 for @)-(lo).TABLE 2.-cHARACTERISTICS OF THE CHARGE-TRANSFER KINETICS AND COMPARISON WITH THE MARCUS THEORY reactant -Ee DA /v us. /10-6 SCE cm2 S - l nit r odurene 1.390 6.2 nitromesitylene 1.351 9.5 terephthalonitrile 1.482 9.2 phthaloni trile 1.560 9.0 p-diacetylbenzene 1.389 6.9 Ze 1 /cm S - l 4700 4900 5500 5500 4900 0.98 f 0.09 0 P2 1.09 f 0.10 0 P2 0.56 0.09 0 PZ 0.88 f 0.18 0 P2 1.0 & 0.2 0 P2 k,/cm s-’ 0.15 f 0.01 0.82 i 0.04 0.29 0.01 1.85 & 0.05 0.82 f 0.03 5.55 f 0.20 2.00 f 0.05 23.5 + 0.6 2.30 & 0.12 22.3 + 1.2 1.05 & 0.01 0.88 f 0.01 0.99 0.01 0.80 -+ 0.01 0.90 i- 0.01 0.70 i 0.01 0.81 & 0.02 0.55 f 0.02 0.78 & 0.02 0.55 & 0.02 a(Ee + PA 0.45 0.36 0.51 0.41 0.48 0.43 0.62 0.54 0.60 0.51 Marcus exptl 0.238 5 0.002 0.284 f 0.002 0.253 f 0.002 0.312 f 0.002 0.279 & 0.002 0.357 i- 0.002 0.311 f 0.003 0.451 & 0.003 0.321 f 0.003 0.456 f 0.003 0.49 & 0.05 0.54 f 0.05 0.28 f 0.05 0.44 f 0.09 0.50 f: 0.10 0 mJ - M .SAVEANT AND D. TESSIER 69 1 1 metry, which is the technique we used for investigating this reaction, the two para- meters are conveniently expressed as: V where C" is the initial concentration of A. A semi-quantitative representation of how the kinetic control varies with A and A is shown in fig. 5.31 In this representation the potential dependence of the transfer coefficient is neglected and a is taken as equal to 0.5. This does not correspond to the actual characteristics of the reaction we are investigating. However, it is sufficient for defining the various kinetic zones and the KP I IR - 4 4 FIG. 5.--Kinetic zone diagram for an irreversible dimerization following the charge-transfer process. The two oblique segments (1) and (2) show the shift of the system when varying the sweep rate from 0.07 to 2330 V s-' for pH 17 and 15.5, respectively. nature of the kinetic control.DO corresponds to diffusion-controlled Nernstian behaviours, QR and IR to quasi-reversible and totally irreversible kinetic control by charge transfer, KP to " pure kinetic " control by the dimerization reaction and KO to mixed diffusion-kinetic control involving the dimerization reaction with no inter- ference from the charge-transfer kinetics in the two latter cases. KI still corresponds to " pure kinetic " conditions with regard to the dimerization reaction, i.e. to mutual compensation between diffusion and the chemical reaction. Kinetic control by charge transfer, however, interferes concomitantly.KG represents the general case. Information about the kinetics of the electron-transfer steps can be obtained as long as the system is in a kinetic situation corresponding to the QR, IR or KI zones. For QR, the potential-dependent forward rate constant can be derived from the kinetic data in the same manner as in the absence of follow-up reaction, i.e. according to eqn (3). In the IR case, this relationship simplifies into: Il - I log(k)E = log(D%) - log 7. In the KI zone, k ( E ) can still be obtained using a different expression: Il - I - i+ exp[ (&) (E - E,)] logk(E) = log(D2) - log----- I (5)70 with 0 - n 3 3 - 1 - ELECTROCHEMICAL TRANSFER COEFFICIENT Ek can be obtained as soon as the system enters the KP zone. Then: RT I , - I E=Ek+---ln- F i+ ' (7) At pH 17 (0.012 mol dm-3 Bu,NOH + 1 mol dmV3 Bu,NI), the system lies in the KP zone at low sweep rates (0.07, 0.22 and 0.68 V s-l).The corresponding log analysis [eqn (7)] gives rise to a straight line with the correct slope (59 & 1 mV). The potential location of the straight line provides the value of Ek, - 1.451 V us. SCE. At high sweep rates (69, 227,700 and 2330 V s-l) the system is shifted into the QR zone. The standard potential can then be determined as E d = -1,615 V. From eqn (6) it follows that kd = (3.3 1.5) lo5 dm3 mol-' s-l. On the other hand, application of eqn (3) provides the potential-dependent forward electron-transfer rate constant k(E) in the high-sweep-rate region [fig. 6(a)] and, by differentiation, the variations of the apparent transfer coefficient with potential [fig.6(b)]. .- .: I 1 I -1.6 - 1.7 -1.6 - 1.7 E/V FIG. 6.-Reduction of benzaldehyde in ethanol. I- . . L I I - 1.6 - 1.7 I I I - 1.6 - 1.7 EIV (a) Forward electron-transfer rate constant as a function of the electrode potential (from C.P.S.V. data); (b) variation of the apparent transfer coefficient with potential. (1) pH = 17, u = 69, 227, 700 and 2330 V s-'; (2) pH = 15.5, u = 0.22, 0.69, 2.27, 6.8, 22.4, 69, 222, 689 and 2200 V s-'. At pH 15.5 (phenol buffer), the system lies in the KI zone in almost all the available sweep-rate range. The KP zone is reached only for the slowest sweep rate (0.07 V s-l) which gives access to Ek, Ek = - 1.41 3 V. We thus found that kd = (5 & 3) x 10' dm3 mol-l s-l, taking for E* the same value as at pH 17.This confirms the increase of the apparent dimerization rate constant upon decreasing the pH. In the KI zone, the treatment of data through eqn (5) taking into account the above value of & pro- vides k(E) [fig. 6(a)] and, by differentiation, aaP [fig. 6(b)]. At both pH zap appears as an approximately linear function of potential. AgainJ-M. SAVBANT AND D. TESSIER 71 the variations of the 0.H.p. potential q2 with the electrode potential [fig. 3(c)] are too small to be responsible for the variation of asp. Comparison with the Marcus theory is shown in table 3, along the same lines as in the preceding sections. TABLE 3 .-COMPARISON OF PREDICTED AND OBSERVED ELECTROCHEMICAL TRANSFER COEFFICIENT DA k S (8alaE)IV-I pH SCE s-' s-' /V-' qr s-' /V + pr) Marcus exptl -E* Z,, (da,, /V us.cm2 /cm /aE) /cm a w e r--7 17 1.615 6.3 6070 0.85 0 0.17 1.07 0.52 0.24 0.42 15.5 1.615 6.3 6070 0.94 0 0.12 1.10 0.54 0.23 0.47 ~2 1.5 0.84 0.43 0.30 0.44 ~ ) 2 1.2 0.87 0.43 0.29 0.48 CONCLUDING REMARKS For all the organic systems investigated in this study there is a definite tendency for the electrochemical transfer coefficient to vary with the electrode potential whether the investigation was carried out by convolution potential-sweep voltammetry or by the impedance method. This was shown to occur for a series of simple electron- transfer reactions but could also be detected for a more complex reaction involving a chemical reaction following the initial electron-transfer process. In all cases, the variations of the potential at the reaction site with the electrode potential were too small to be considered as responsible for the observed variation.It is thus concluded that the actual transfer coefficient does vary with the electrode potential. Correlating the observed potential dependence with the reorganization factor according to the Marcus theory, it was observed that the experimental and predicted variations are of the same order of magnitude. In most cases the agreement is not quantitative. The concordance between theory and experimental data can, however, be considered as satisfactory taking into account the crudeness of the Marcus model. Rather than an unexpected quantitative agreement, the important point is that the variations of the electrochemical transfer coefficient are experimentally significant, indicating a definite curvature of the potential-energy surfaces of the same order of magnitude as predicted by the theory.It is interesting to note that, although the rate data depend upon electrode pre-treatment, the variations of tc found for t-nitrobutane on platinum are of the same order of magnitude as those found on mercury.18 These results contrast with those found for the reduction of chromium complexes in water, for which the possible variations of tc have been very thoroughly and critically in~estigated.'~ In these cases, no potential dependence of the transfer coefficient was detected, at least in potential ranges cathodic to the standard potential. Experimental precision and extension of the explored potential range were sufficient for leaving little doubt that if a variation of tc of the order of magnitude predicted by the Marcus theory had occurred it would have been detected.More recent investigations l9 revealed that for the aquo C13+/Cr2+ system variations of tc with potential exist on the anodic side while they do not on the cathodic side, but are much larger than predicted for an outer-sphere electron transfer. In the case of the above organic systems it is worth emphasizing that the variations of a are the same in the potential regions either negative or positive to the standard potential. The outer-sphere character of the electron-transfer process in the case of the chromium complexes is not evident.72 ELECTROCHEMICAL TRANSFER COEFFICIENT Electron transfer is very slow (k, is of the order of cm s-l), implying considerable re-organization in the inner coordination sphere.The above organic reactions involving fast electron transfer meet the requirements of an outer-sphere process. They most probably involve small changes in bond distances and angles, solvation being the main reorganization factor. We thank Dr D. Garreau for his helpful collaboration in the impedance determin- ations. This work was supported in part by the C.N.R.S. (ERA 309 " Electrochimie Molkculaire "). P. P. Schmidt, Electrochemistry (Specialist Periodical Report, The Chemical Society, London, 1974), vol. 5, p. 21. R. A. Marcus, J. Chem. Phys., 1965,43, 679. R. A. Marcus, in Dalhem Workshop on the Nature of Sea Water, Dahlem Konferenzen, ed. E. D. Goldberg (Abakon Verlagsgesellschaft, West Berlin, 1973, pp. 447-504. N. S. Hush, J. Chem. Phys., 1958,28, 962. R. Parsons and E. Passeron, J. Electroanal. Chem., 1966,12, 525. D. M. Mohilner, J. Phys. Chem., 1969, 73, 2652. D. H. Angel1 and T. Dickinson, J. Electroanal. Chem., 1972, 35, 55. K, Suga, H. Mizota, Y. Kanzaki and S. Aoyagni, J. Electroanal. Chem., 1973,41, 313. ' F. C. Anson, N. Rathjen and R. D. Frisbee, J. Electrochem. Soc., 1970,117, 477. lo E. Momot and G. Bronoel, C.R. Acad. Sci., 1974,278, 319. l1 P. Bindra, A. P. Brown, M. Fleischmann and D. Pletcher, J. Electroanal. Chem., 1975, 58, 39. l2 J-M. SavCant and D. Tessier, J. Electroanal. Chem., 1975, 65, 57. l3 M. J. Weaver and F. C. Anson, J. Phys. Chem., 1976, 80, 1861. l4 Z. Samec and J. Weber, J. Electroanal. Chem., 1977, 77, 163. l5 J-M. SavCant and D. Tessier, J. Phys. Chem., 1977, 81, 2192. l6 J-M. Savkant and D. Tessier, J. Phys. Chem., 1978, 82, 1723. l8 D. A. Corrigan and D. H. Evans, J. Electroanal. Chem., 1980, 106, 287. l9 P. D. Tyma and M. J. Weaver, J. Electroanal. Chem., 1980,111, 195. 2o W. R. Fawcett and R. 0. Loufty, Can. J. Chem., 1973,51, 230. 21 W. R. Fawcett, B. M. Ikeda and J. B. Sellan, Can. J. Chem., 1979, 57, 2268. 22 H. Kojima and A. J. Bard, J. Am. Chem. Soc., 1975, 97, 6317. 23 M. E. Peover and J. S. Powell, J. Electroanal. Chem., 1969, 20, 427. 24 J. C. Imbeaux and J-M. Savkant, J. Electroanal. Chem., 1973, 44, 169. 25 J-M. Saveant and D. Tessier, J. Electroanal. Chem., 1975, 62, 57. 26 J-M. Saveant and D. Tessier, to be submitted. 27 D. Garreau, J-M. Saveant and D. Tessier, J. Electroanal. Chem., 1979, 103, 321. 28 L. Nadjo and J-M. Saveant, J. Electroanal. Chem., 1971, 33, 419. 29 F. Ammar, L. Nadjo and J-M. SavCant, J. Electroanal. Chem., 1973,47, 146. 30 J. W. Hayes, 1. Ruzic and D. E. Smith, J. Electroanal. Chem., 1974,51, 269. 31 L. Nadjo and J-M. SavCant, J. Electroanal. Chem., 1973, 48, 113. D. Garreau, J-M. Saveant and D. Tessier, J. Phys. Chem., 1979, 83, 3003.

ISSN:0301-7249    

DOI:10.1039/DC9827400057

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 74, 73-81 Theory of Fe2+-Fe3+ Electron Exchange in Water BY HAROLD L. FRIEDMAN Department of Chemistry, State University of New York, Stony Brook, New York 11794, U.S.A. AND MARSHALL D. NEWTON Department of Chemistry, Brookhaven National Laboratory, Upton, New York 11973, U.S.A. Received 14th May, 1982 The detailed theory reported earlier by Tembe, Friedman and Newton is supplemented by calculations of the electronic overlap of the initial and final states for reaction geometries different from those treated earlier. A dynamical theory of the outer-sphere reorganization process provides an explicit model for deviations from the transition-state theory. These results support the earlier calculation of the rate constant. Tembe, Friedman and Newton recently presented a comprehensive theory for the aqueous Fe2 +-Fe3 -t electron-exchange rate constant formulated as (1) a form which is also used to analyse the data for the nuclear spin relaxation in 27A13+ induced by collisions with Ni2+.It is assumed that the equilibrium pair-correlation function g23(r) is the same function of ionic composition and tempFrature in the two cases and that in the spin relaxation process the local rate constant kZ3(r) has the form that may be deduced from the Splomon-Bloembergen equations. In the case of the exchange reaction the theory of k23(r) was developed with respect to the contributions from slow inner-shell or outer-shell reorganization (activation) dynamics. It was concluded that these complications are not important for Fe2+-Fe3+ and that the controlling dynamics is the crossing from the reactant to the product diabatic Born- Oppenheimer surface.While there are some uncertain and even surprising features in the model, the calculated rate constants are in good agreement with the experimental data for both electron-exchange and spin-relaxation processes as functions of temperature and ionic strength, as far as comparison is allowed by the limited data. For the electron-exchange reaction the crucial factor in the calculation of k23(r) is the r-dependence of the electronic matrix element that connects the zero-order initial state (reactant) and final state (product). Ref. (1) used values of HAB derived from ab initio molecular-orbital calculations that were based on a structural model in which optimal electronic overlap was achieved by having the octahedral hexa-aquo reactant ions approach in a staggered face-to-face configuration [fig. l(a)].This approach permits values of r as small as 4.5 A without74 Fe2+-Fe3+ ELECTRON EXCHANGE H \ \ 0-H i H-O \ H H 2+ / Fe 0 H \ 3+ \ 0 Fe H / (6) FIG. 1 .-Approach of octahedral hexa-aquo complexes. Only the water molecules between the Fe atoms are shown. (a) Face-to-face approach, S, point group. (6) Apex-to-apex approach, DZh point group. disrupting the octahedral geometry of either Fe(H20)E+ ion.' The calculated maximum in the integrand in eqn (1) at ca. 5.3 8, is nearer to 4.5 A than to 6.9 A, the value of r at which the spherical envelopes of the hexa-aquo ions touch. Note that recent ab initio calculations for water dimers appropriate to the fig.l(a) approach geometry yield a minimum energy for rH . . . = 2.36 A, corresponding to a 5.3 A metal-metal distance. On the other hand, extended Hiickel m.0. calculations reported by Larsson suggest a mechanism by which H A B may be relatively large even at r 2 781 owing to strong participation of the water ligands. This report has led us to consider approach geometries such as fig. l(b) at the ab initio level, to be described below. As noted in ref. (l), calculation of k23(r) for electron exchange rests on the assumed validity of time-dependent perturbation theory (the golden rule) implemented with a Boltzmann distribution of internal states of the reactants, although for sufficiently large HAB one would have a different distribution that reflected the non- zero relaxation times associated with both inner-sphere and outer-sphere reorganiz- ation processes.It was concluded tentatively, on the basis of a chemical kinetics model for the relaxation processes, that the characteristic time Z~ for the electron to jump, once the required reorganization is complete, is long enough for all values of rH . L. FRIEDMAN AND M. D. NEWTON 75 of interest so that the Boltzmann distribution of reactant states is a good approxim- ation; i.e. transition-state theory is applicable. This use of a chemical kinetics model to represent the effects of molecular-relaxation processes is supported by an earlier report by Northrup and Hynes of a dynamical analysis of a rather general reaction model.They identified deviations from transition-state theory owing to relaxation processes in what they term " stable states." Moreover they expressed their results in terms of a chemical kinetics model, their eqn (4.6), which is essentially equivalent to eqn (2.13) of ref. (1). Further developments of the theory by Hynes and coworkers extend to the particular case of chemical reactions that are coupled to polarization processes in the solvent, as reported by van der Zwan and Hynes.6 A different analysis of the same problem, but with many points of contact, is offered below. REACTIVE ENCOUNTER GEOMETRIES FACE-TO-FACE APPROACH The staggered face-to-face approach geometry [fig. l(a)] gives a good account of the kinetic data,' but some aspects seem to require further attention.Thus the electronic matrix elements HAB [eqn (2)] were based on high-spin loA, coupling of Fe2+ and Fe3+ in which the transferring t2g electron begins in a " sigma " 3d-like orbital on the Fe2+, an orbital that is symmetric with respect to the Fe-Fe axis, and ends up in a similar 3d-like orbital on the other Fe ion. Note that sigma orbital occupation is energetically favoured for the " excess " electron because of the crystal- field stabilization of this orbital by the adjacent Fe3+ reactant. The preference for the ''A state (i.e. sigma occupation) relative to the doubly degenerate 'OE state of the reactant complex is estimated to be ca. 2.5/Oeff kcal rnol-l, where Deff is the effective dielectric constant for the electronic system.Model calculations of Deff using an ellipsoidal dielectric model, as in fig. 4 of ref. (l), indicate that Deff < 5 for shielding fields between the Fe atoms. Then the difference in energy between the I0A and 'OE states is great enough so the latter can be neglected at room temperature, as in the earlier calculations;2 if the 'OE states were populated their contribution to HAB would probably be less than for the ''A state, at least to the extent that direct Fe-Fe overlap is involved. TABLE 1 .-ELECTRONIC MATRIX ELEMENTS' geometry 4%. - Hke/cm - fig. l(a) 5.25 121bpd 137" fig. 1(4 7.3 gb** 2.7" fig. l(b) 7.3 27b fig. l(b) 7.8 8b a The atoms explicitly represented in the calculations are those shown in fig. l(a) or l(6). See Full treatment of ligand valence Extrapolated using results calculated for ref.(2) for discussion of the relationship between HAB and HAB electrons. 5.5 < r / A < 6.75, ref (7) and (8). Point-charge model for ligands, ref. (2). It may be recalled that the values of HAB employed in ref. (1) were based on a model which only allowed direct Fe-Fe overlap because the H,O ligands were represented solely by their crystal fields.2 New calculations including all of the valence electrons of the six waters between the two Fe atoms in the fig. l(a) configuration give similar results 'p8 (see table 1). Although the new results cannot be uniquely partitioned76 Fe2+-Fe3+ ELECTRON EXCHANGE between direct Fe-Fe and ligand-mediated contributions, the agreement with the ligand-field model suggests that Fe-Fe overlap is the dominant coupling mechanism for Fe2+-Fe3+ electron exchange in water.The characteristic time Z~ has been re- calculated with the new HAB coefficients as shown in fig. 2; again we see that the cal- 0.11 I I I I I I 4 5 6 7 r1.H FIG. 2.-Computed te as a function of r. The dashed horizontal lines indicate the estimates in ref. (1) for T : , ~ and the lower bound for ~i,.,. The solid curve is te calculated in ref. (1) based on a point-charge model for the ligands [ref. (2)], while the dotted curve is the result of a calculation that is similar except that the ligand valence electrons are included as specified in table 1 [ref. (7) and (811. culations are quite stable with respect to the change in the way the water molecules are represented. In all of these cases the HA, coefficients were calculated for " super-molecules " consisting of the two metal-ion hydrates, but without contributions from the surrounding polarized dielectric.Additional calculations based on an ellipsoidal cavity in the dielectric, as in fig. 4 of ref. (l), indicate that in the spatial region that is important for determining HAB, namely between the metal atoms, where Deff is < 5, the electrostatic field owing to the polarized dielectric is small compared with the fields from the Fe ions. APEX-TO-APEX APPROACH This geometry, more completely specified in fig. l(b), also merits attention. Al- though the minimum r is necessarily much greater than for face-to-face contact, theH . L . FRIEDMAN A N D M . D . NEWTON 77 possibility of relatively strong overlap between adjacent water molecules makes it necessary to estimate the contribution to H A B from this structure.Even allowing H H contacts as short as 2.OA permits values of Y no less than 7 . 3 A. Calculations for this structure, using basis sets analogous to those described e l ~ e h h e r e , ~ * ~ * ~ lead to the results in table 1. The suppression of the water molecules that are not between the Fe atoms is unlikely to cause serious error inasmuch as it is the region near the Fe-Fe axis that is most important for calculating HAB. Also this procedure is con- sistent with the face-to-face calculations 7 9 8 which included only the water molecules between the Fe atoms. The ‘ excess ’ electron in these calculations was placed in a d, orbital, where d, denotes functions which are primarily “ tzs ” 3d Fe atomic orbitals that are symmetric in the DZh plane (i.e.3 4 , , where xz is the DZh plane and z is the Fe-Fe axis). In view of the fact that ze depends on HAB2 the comparison in table 1 indicates that the fig. l(6) geometry is less important for the collision complex than that in fig. l(a). This conclusion was supported by the results of additional calculations for the fig l(6) geometry at r = 7.8 A (table 1). Furthermore, although all orientations of the two hexa-aquo ion complexes become accessible for Y > 7.3 A, the matrix elements NAB seem to be small enough so that the conclusions in ref. (1) about the importance of the close contact of the fig. 1 geometry will not be greatly modified when the full co- ordinate space of the reacting complexes can be taken into account.THE CAVITY FIELD EXISTENCE OF THE CAVITY FIELD In the simplest possible terms the transition-state theory for a process gives the rate constant kZ3(r) as knr = [exp( -A:/RT)]/.r (3) where At is the free energy of the transition state minus that of the ground state and l/z, often written kBT/h, is the frequency with which systems in the immediate neighbourhood of the transition state pass from the reactant manifold of states to the product manifold, all at given r. Eqn (3) is accurate only if the dynamical processes associated with A= are sufficiently fast so that the process associated with z does not significantly deplete the concentration of transition-state ~ y s t e m s . ~ * ~ * ~ Here the effect of the dynamics of the outer-shell activation process [the ‘‘ out ” process in ref.(l)] is studied. For simplicity we consider the question raised above only in the limiting case in which the “ in ” activation process is negligible compared with the “ out ” process. For the important configuration [fig. l(a)] in the Fe2+-Fe3+ electron exchange (or for a typical mixed-valence complex) the reacting system can be rep- resented approximately as an ellipsoidal “ cavity ” of low dielectric constant &,,,, in a medium of high, frequency-dependent dielectric constant &(COO). For such a model the electron transfer that changes the configuration from Fe2+, Fe3+ to Fe3+, Fe2+ is associated with a change of electric dipole moment from - 1/2, + 1/2 to + 1/2, - 1/2 [fig. 3(a) and (b)] superimposed on fixed charges +24, +23 at the metal centres.This picture has been described by Cannon9 who showed how to combine it with the Kirkwood-Westheimer theory of the electrostatics of a system of charges in an ellip- soidal cavity lo to get an estimate of the Marcus outer-sphere reorganization free energy.” In terms of the cavity field we interpret Cannon’s calculation as follows: a fixed dipole in a cavity in a dielectric medium elicits an electric field from the medium, Onsanger’s reaction field,12 which lowers the energy of the dipole, the effect used in Cannon’s calculation.78 Fe2+-Fe3+ ELECTRON EXCHANGE The dynamical effect that corresponds to Onsager’s reaction field was described by Nee and Zwanzig in their generalization of Onsager’s theory of the dielectric ons st ant.'^ Thus a rotating dipole in a cavity elicits from the medium a lagging time- FIG.3.-(a) Fez+, Fe3+ hexa-aquo ions in an ellipsoidal cavity in the dielectric. The net dipole moment is shown. (b) Fe3+, Fez+ hexa-aquo ions in an ellipsoidal cavity. The net dipole moment is shown. (c) Fluctuating cavity field G(t). In some units the norm of G ( t ) is -1 with the electronic configuration in (a) and +1 with the electronic configuration in (b). dependent reaction field which tends to slow the rotating dipole. In applications in dielectric theory,13*14 the dipole in a cavity represents a real molecule with thermally driven rotational velocity whose r.m.s. value may not be small enough for the Nee and Zwanzig theory to be accurate.Hubbard and Wolynes l4 improved this aspect of the theory and, in the process, introduced the dynamical cavity field concept which we now develop. The general basis for the cavity field may be described as follows. The rotational motion of a dipolar molecule in a dense fluid medium can be represented by the generalized Langevin equation [eqn (29) and (B2) of Nee and Zwanzig] l3 1 b(t) = -fi t 1 r D ( t - 4) * + 4 0 (4) 0 where n(t) is the instaneous rotational velocity of the dipolar molecule, k? its time derivative, I the inertial tensor of the molecule, c,(t) is the time-dependent dynamical friction tensor [in the absence of memory effects it reduces to <,d(t) and one recovers the strict Langevin equation, an accurate approximation only if the dipolar molecule is very large compared with the molecules of the medium] and A ( t ) is the random torque that is conjugate to the frictional torque, these being complementary projections of the torque G(t) x p(t) acting on the dipole p(t), the random torque providing heat by as much as the friction cools.Here we have introduced our notation G(t) for the dynamical cavity field, the electric field “ felt ” by the dipole in the cavity. If we notice that A ( t ) = G(t) x ~ ( t ) is the random torque acting on a fixed dipolar body and assume the Debye form for the dielectric function of the mediumH . L . FRIEDMAN A N D M. D . NEWTON 79 we find l4 the following form for the time correlation function of the cavity field where a is the radius of the cavity (assumed spherical) and p = l/kT.The most important feature of eqn (6) is that it provides an explicit example, albeit for a continuum model of the solvent, of a dynamical cavity field G(t) whose variance (G G) - (G) (G) and correlation time 50 rG = fit( G(t) G(O))/( G G ) 0 (7) do not depend upon the probe (i.e. the dipole in the cavity) used to measure this field. Thus the cavity field is a manifestation of the polarization fluctuations of the medium that are associated by linear response theory with the measurable ~(co). RATE THEORY WITH THE CAVITY FIELD Now, returning to the electron-transfer problem, we propose that the question as to whether the outer-sphere reorganization process is fast enough so that eqn (3) is valid can be approached in the following way.Fig. 3(c) shows (schematically) G(t) over a period of time long enough so that at least one electron-transfer process is likely to occur [where G(t) is the projection of G(t) on the Fe2+-Fe3+ vector]. The norm (average) of G(t) when the electron is on the left [cf. fig. 3(a)] or right is Osanger's static reaction field l2 in either case. For the electron to jump in a radiationless way we have to wait for a fluctuation in which G = 0. In the neighbourhood of G = 0 the mean time for the electron to jump is z~, as defined in ref. (1); it is a particular case of the z parameter in eqn (3). So now we calculate the mean first-passage time rmef p . , the time we have to wait, on average, for a fluctuation of G(t) from a normative value to a G = 0 state in which the electron does jump.We make this calculation under the linear response theory, having noticed that linear response of the dielectric medium to the changing electric dipole in the cavity has already been assumed in estimating the barrier EiUt associated with the outer-sphere reorgani~ation.'*~*'~ To calculate the mean first-passage time we begin with a model in which a ball of mass m rolls on a potential surface U(x) while governed by a Langevin equation where x is the location of the ball of mass m, m/z; is the friction coefficient and R(t) the random force conjugate to it. The diffusion coefficient is D = kTt;/m. Using a convenient version of first-passage time theory l5*I6 which allows us to specify bU(x) = A2x2, x > a and a < 0 (9) with V(x) = 00 for x < a < 0 and a rate constant rc, for passage from reactant to product states at x = a (" radiation boundary condition "), together with Chan- drasekhar's analysis l7 of the motion under eqn (8) and (9) when a = -00, giving z, f /)x(t)x(O)> dt/ { x2 ) = mp/A2z;80 Fe2+-Fe3 + ELECTRON EXCHANGE we find where z, defined in eqn (10) for the a = -a limit is expected to be very nearly the same when a is finite as long as PET = PU(a) > 9. While the theory of zrnqf.,.is applicable to low barrier heights l59I6 the bound 9 < PET has been assumed to get the simple result in eqn (1 1). Finally we identify x with the fluctuation G(t) - (G) of the cavity field when the electron is on the left (or right), we identify l / ~ , with the characteristic time ze for the electron to jump to the right (or left) when G(t) = 0, we identify z, with zG, the charac- teristic time for fluctuations in the cavity field, and we identify 1 / ~ ~ .~ . ~ . with the rate constant Iz23(~) to get This equation reduces to the transition state form if ze is big enough. The difference between Er and A : [as in eqn (3)] is not important for the process of interest. The factor n1/2, which is not in eqn (3), is a consequence of the mean first-passage time approach together with the radiation boundary condition. ESTIMATES OF TG Parameterizing eqn (6) for water at 25 "C (z, = 8 ps, E, = 78.35, E,, = 2) leads to zG = 0.3 ps. Since it was found in ref. (1) that the most important value of ze is near 1.0 ps (see fig. 2) it follows that the continuum model calculation giving eqn (6) leads to the expectation that the outer-sphere reorganization process is fast enough so that it gives no deviation from transition-state theory.This conclusion is not sensitive to the shape of the cavity.6 It is desirable to have a more empirical basis for estimating zG. Fortunately there are data for zG', the correlation time for the gradient G' = vG of the cavity field. First we recall that Hynes and Wolynes l8 modified the Nee-Zwanzig theory l3 to calculate the dynamical friction for a slowly rotating electric quadrupole. Their results give l8 zD = 0.3 ps for water at 25 "C. (1 3) p' - . 5 EOP - 3 E s + 2E,p Now it may be recalled that just as diamagnetic metal ions having nuclei with nuclear spin quantum number I > 3 have their nuclear spin relaxation dominated by the coupling GI : vG(1 + y), where 6a is the nuclear quadrupole and 1 + y the anti- shielding factor, so paramagnetic metal ions with electron spin quantum number S > have their electron spin relaxation dominated by the coupling Q : vG, where now Q is the electronic q u a d r u p ~ l e .' ~ * ~ ~ (The term Q : VG is most often written as the zero-field splitting term S * D S, another way to describe the same physics.)20 Because the electron Larmor frequency II), is a thousand-fold larger at a given magnetic field than typical nuclear Larmor frequencies ol, the dispersion association with z 'coLarmor can be measured for the e.p.r. cases although not for the n.m.r.*OV2l One finds that P' 2: 2 ps is quite typical, implying that the continuum model calculation for zG' is not very realistic.However, if we assume that the continuum model result that zG N zG' is reliable then we have an estimate that zG 21 2 ps in water at 25 "C. Even with zG = 2 ps the deviation of eqn (12) from the limiting transition-state theory form is quite small if ze 2 1 .O ps, as estimated in ref. (1). However, note that GH . L . FRIEDMAN AND M. D . NEWTON 81 for an electron-transfer process in the adiabatic limit one has Z~ - ph - 0.16 ps, in which case the T~ term in eqn (1 0) is quite important, even if PET is as large as 9. The work at Brookhaven National Laboratory was carried out under contract with the U.S. Department of Energy and supported by its Department of Basic Energy Sciences.The work at Stony Brook was supported by the National Science Foundation. We thank Dr J. Logan for permission to include some of the results in table 1 and fig. 2 prior to publication [ref. (S)] and Prof. J. T. Hynes for communic- ating ref. (6) prior to publication. We also thank Prof. N. Kestner for permission to present some of the results of ref. (3) prior to publication. B. L. Tembe, H. L. Friedman and M. D. Newton, J . Chem. Phys., 1982,76, 1490. N. Kestner and M. D. Newton, J . Chem. Phys., in press. S. Larsson, J. Am. Chem. Soc., 1981, 103, 4034. S. H. Northrup and J. T. Hynes, J . Chem. Phys., 1980, 73, 2700. G. van der Zwan and J. T. Hynes, J . Chem. Phys., 1982, 76, 2993. M. D. Newton, Adv. Chem. Ser., 1982, 198, 255. R. D. Cannon, Chem. Phys. Lett., 1977, 49, 299. R. A. Marcus, J. Chem. PhyJ., 1979, 24, 979. L. Onsager, J. Am. Chem. SOC., 1936,58, 1486. ' M. D. Newton, Int. J . Quantum Chem. Symp., 1980, 14, 363. * J. Logan and M. D. Newton, J. Chem. Phys., in press. lo J. G. Kirkwood and F. H. Westheimer, J . Chem. Phys., 1938, 6, 506 and 513. l 3 T. W. Nee and R. Zwanzig, J. Chem. Phys., 1970, 52, 6353. l4 J . B. Hubbard and P. G. Wolynes, J. Chem. Phys., 1978, 69, 998. If, A. Szabo, K. Schulten and Z . Schulten, J. Chem. Phys., 1980, 72, 4350. l6 J. M. Deutch, J. Chem. Phys., 1980, 73, 4700. l7 S. Chandrasekhar, Rev. Mod. Phys., 1943, 15, I . l9 C. F. Anderson, L. P. Hwang and H. L. Friedman, J . Chem. Phys., 1976, 64, 2806. 2o H. L. Friedman, in Protons and Ions Involved in Fast Dynamic Phenomena, ed. P. Laszlo J. T. Hynes and P. G. Wolynes, J. Chem. Phys., 1981, 75, 395. (Elsevier, Amsterdam, 1978), p. 27. H. L. Friedman, M. Holz and H. C. Hertz, J . Chem. Phys., 1979, 70, 3369.

ISSN:0301-7249    

DOI:10.1039/DC9827400073

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. C. A. G. 0. Varma (University of Leiden) said: May I ask Prof. Marcus to comment on the following remarks? There are serious difficulties in understanding why solvent reorientation should be an essential factor determining the rate of electron-transfer processes. In correlating experimental rate constants with theoretical expressions many authors obtain good correlation when AG contains terms arising from solvent reorientation. I think that as far as solvent repolarization is concerned only the readjustment of the induc- tion polarization should be significant for the rate constant. The following consider- ations may perhaps support my opinion. A polarization fluctuation brings a popu- lated sublevel of the initial electronic state into resonance with one or more sublevels of the final electronic state at time t = 0.Then the electron has to be transferred from donor to acceptor in a time Atl which is short compared with the decay of the polarization, i.e. At, < zo, where zo is the solvent orientational relaxation time. The vibrational sublevels of the final electronic state, which become populated as a result of the electron transfer, are coupled to a dense manifold of levels. As a consequence the recurrence time T, for reverse transfer is infinitely long compared with the vi- brational relaxation time T,. Unless a new suitable polarization fluctuation occurs at the scene within a period Atl + z, reverse electron transfer in the product pair is unlikely. If the energy gap between initial and final electronic state is larger than a few times kT, vibrational relaxation rescues the new-born product pair from reverse transfer in a time shorter than that required for solvent reorientation.Starting from the initiating polarization fluctuation and terminating in a state of thermal equilibrium, formation of products consists of a sequence of experimentally distinguishable steps. Therefore the rate of disappearance of reactants is not the same as the rate at which product-solvent equilibration is achieved. The former should not depend on solvent reorientation and the latter may be determined completely by the orientational relax- ation time. In the event that the solvent reorientation is really important for the rate of disappearance of reactants, the electronic motion should be kept coupled to the intramolecular vibrations and the solvent motions throughout the quantum mech- anical calculation of the rate constant.Prof. R. A. Marcus (California Institute of Technology, Pasadena) said : To undergo electron transfer the system needs to attain a state of ‘ resonance,’ e.g. in classical terms, reach an intersection of the reactants’ and the products’ potential-energy sur- surfaces. [This intersection is an ( N - 1)-dimensional surface if there are N co- ordinates.] It can do so by undergoing a change in vibrational configuration, in solvation configuration (reorientation) or both. (Typically the equilibrium vi- brational and solvation properties both differ for reactants and products.) Not surprisingly, and confirmed by calculation, the minimum-energy path for reaching the ‘ intersection ’ region involves changes in both sets of coordinates, not just one.In quite highly exothermic reactants the intersection may occur near the minimum of the reactants’ potential-energy surface, and then little or no fluctuations in vibrational and solvation reorientation coordinates are needed for the forward reaction, but large fluctuation in both would be needed for the reverse reaction.84 GENERAL DISCUSSION In summary, the need for solvent and vibrational reorganization is a matter of favourable energetics for reaching the lowest part of the intersection surface. Dr. C. A. G. 0. Varma (University ofLeiden) said: 1 want to add the following to the review given by Prof. Marcus in his introductory lecture.The need for experimental investigations concerning the kinetics of electron- transfer processes, which are not limited in their rate by mass diffusion of reactants, has become evident here after the review given by Prof. Marcus. I want to draw attention to an experimental result which we obtained recently and which offers interesting opportunities to study electron-transfer kinetics in the picosecond time domain.' We excited a solution of of ca. 0.1 mol dm-3 in a non-polar liquid freon with a laser pulse of 265 nm, f.w.h.m. of 8 ps and total energy content of 0.5 mJ deposited in a volume of 1.0 mm3. The vibrationally unrelaxed primary excited state of C6F6 yields a biradical, which is an intermediate in the isomerization to the Dewar-type valence isomer, with a rise-time zR(l) < 10 ps.This time constant zR(l) is determined by the competition between biradical formation and relaxation to the bottom of S1, from which fluorescence with a lifetime of 1.7 ns is emitted. In addition to these species the ions C6F6' and C6F6- can be observed by means of absorption and emission spectroscopy. The rise-time for the formation of ions is z ~ ( ~ ) = 40 ps and the neutralization reaction has a time constant zD = 700 ps. The ion concentration varies linearly with the total energy of the laser pulse. These facts reveal that pairs of oppositely charged ions are formed predominantly by transfer of an electron, over a distance of not more than a few molecular diameters, from a vibrationally relaxed C6F6 molecule in the state s1 to a C6F6 molecule in the ground state. The C6F6' fluorescence observed simultaneously with and after the ion formation rules out the possibility that the absorptions attributed to the ions arise from an excimer.The ions in the initial pair are free in the sense that the motions of electrons in the positive ion are not, or only very weakly, correlated with those of electrons in the negative ion. The decay constant of 700 ps for the ions seems to indicate that cation and anion in an ion pair are able to escape from their mutual Coulomb attraction, although the solvent has a dielectric constant of only 2.4. An estimate of AH for the formation of ions may give an impression of the range of distances R over which the electron may be transferred. As usual, the estimate of AH is not quite satisfactory.Molecules and ions are treated as hard spheres embedded in a dielectric continuum. The radii of the spheres cannot be assigned values un- ambiguously. Entropy terms are neglected because they are believed to be small enough, but essentially because the temperature dependence of the dielectric constant is not known. Specific solute-solvent and ion-solvent interactions are considered to be absent. When donor and acceptor are separated by a single or a just a few solvent molecules still nothing significant can be said about AH. If R is less than the diameter of a solvent molecule, i.e. R < 6 A, the solvation energy of the ion pair is found by treating the pair as an electric dipole of magnitude p = eR in a spherical cavity of radius R .Then AH is given by where I. and A , are the ionization energy and electron affinity of a ground-state molecule in vacuum, respectively, E is the 0-0 energy separation between So and S1 in solution, F is Onsager's reaction field and Co is the Coulomb attraction in vacuum. When R 6 A Born's expression for ion solvation AHB is used and AH is then A H = I0 - A0 - E - 2HB - CGENERAL DISCUSSION 85 with C the Coulomb term in solution. Using the experimental values I, = 9.9 eV,2 A, = 1.2 eV 3 9 4 and E = 4.4 eV the value of AH in the first case is found to be AH = 0 at R = 4.6 A and AH = - 1.3 eV at R = 3.5 8, with R measured along the coincid- ing two sixfold axes of reactants. The second case, i.e. R 9 6 8, yields, with both ion radii equal to 3.0 A, a value AH > 2.4 eV.These estimated values of AH may be helpful in making a quick connection with the expressions for the transfer rate con- stant given in the theoretical contributions at this meeting. An adequate description of the rates of growth and decay of the ion concentration will have to describe the distribution of initial distances R carefully and will have to account for the time dependence of rotational and translational diffusion in the short-time regime. To me it appears impossible to treat the variation of the orientational- and distance-dependent mutual interaction of donor and acceptor properly in an analytic fashion. Molecular- dynamics-type calculations are probably the only means to treat the problem properly. J. L. G. Suijker and C.A. G. 0. Varma, Chem. Phys. Lett., submitted for publication. G. Dujardin, Chem. Phys., 1980,46, 407. F. M. Page and G. C. Goode, Negative Ions and the Magnetron (Wiley-Interscience, New York, 1969). U. Sowada and R. A. Holroyd, J. Phys. Chem., 1980, 84, 1150. Prof. R. A. Marcus (California Institute of Technology, Pasadena) said: It is very interesting to hear of these results. It should be particularly revealing to examine the nature of the 700 ps decay curve to determine whether it be exponential or non- exponential. ~ Any appreciable distribution of initial formation distances of the ion pair, and of final distances just prior to neutralization, would contribute to a non- exponential behaviour. Dr. M. R. V. Sahyun (3M Center, St. Paul) said: The expression for the rate of electron transfer over large distances, r, k = k,exp(-ur) may be derived from quantum-mechanical tunnelling theory, assuming transmission through a vertical potential barrier from low-lying vibronic states.' Continuity, but not necessarily coupling, of the wavefunctions appropriate to each side of the barrier is also assumed ; this set of assumptions is inconsistent with adiabatic electron-transfer models.Recently reported data on electron-transfer rates in long-distance redox quenching of excited states encourage this view.2 These data indicate pre-exponential factors of 10'4-10'7, too large for molecular vibration, but consistent with the kinetic energies of excited state electrons, inferred from the virial theorem. Accordingly, electron transfers may occur with a broad spectrum of mechanisms lying between limiting cases corresponding to adiabatic processes approximately describable by the Marcus equation, and quantum-mechanical tunnelling from low-lying vibronic states ; this is analogous to the proposal offered by Krishtalik in this Discussion for the inter- pretation of proton transfers.The possibility of electron tunnelling directly from low-lying vibronic states should be considered for the case of the cytochrome c/cytochrome c peroxidase system, for which the model based on the Marcus equation predicts electron-transfer rates close enough to the observed, before inclusion of reorganization, to preclude inclusion of the requisite reorganisation for reaction from higher vibronic states. Interpre- tation in terms of tunnelling obviates this difficulty; it allows for a larger pre-exponen- tial factor and thus leaves room in the prediction for the expected work term in the86 GENERAL DISCUSSION free energy of activation.The implication of this picture is that whatever reorganis- ation occurs with electron transfer in the cytochrome c/cytochrome c peroxidase system does not appreciably affect the separation of the metal centres, r, as borne out by the observation reported by Prof. Williams in this Discussion, namely that the Fe-S distance in cytochrome c is essentially independent of the Fe oxidation state. This derivation is reproduced, e.g., in W. F. Leonard and T. L. Martin, EEectronic Structure and Transport Properties of Crystals (Kreger, New York, 1980), pp.630 ff. J. R. Miller et af., J. Am. Chem. SOC., 1982, 104, 4296. Prof. R. A. Marcus (California Institute of Technology, Pasadena) said: The ko appearing in the formula k = ko exp (- ar), where r is now the edge-to-edge separation distance, is not be identified with a frequency of electronic motion. As discussed else- where the electronic frequency enters into k itself in a quite different fashion, e.g. via Landau-Zener type theory. The derivation in the book Dr. Sahyun mentions is applicable only when, unlike the case of the present reactions, no nuclear motion is needed to bring the reactants into resonance. It is not clear from ref. (2) cited by Dr. Sahyun or ref. (9)-(11) cited therein, what error is involved in the kh determination. Usually one would not expect this ko to exceed the value for an adiabatic reaction which is about 1013 s-l.Nuclear tunnelling would not increase this value for the rate constant. Indeed, any additional reaction barrier such as nuclear tunnelling would make it < 1013 s-l. The point raised is certainly interesting, and error estimates on the experimental ko would therefore be very useful. I have discussed with Dr. Miller his pre-exponential factor, ko. The current best value for its maximum for reactions between molecules and ions (excluding solvated electrons), he feels, is between 1013 and 1014 s-l (cf. discussion in my paper). My first remark applies also to the cytochrome c/cytochrome c peroxidase reaction -one can’t equate the pre-exponential factor, either in ko exp( - ar) or in the Arrhenius expression for k, with the frequency for electronic motion.R. A. Marcus, Annu. Rev. Phys. Chem., 1964, 15, 155. Dr. J. E. Baggott (Oxford University) said: In his lecture Prof. Marcus discussed the early experimental work of Rehm and Weller on the quenching of aromatic hydrocarbon fluorescence as an example of a system in agreement with Marcus theory in the “normal” region (endoergic or slightly exoergic electron transfer) but in apparent disagreement in the “ inverted ” region (very exoergic electron transfer). It has since been recognised * that the mechanism of quenching becomes dominated by exciplex formation and decay in the “ inverted ” region, reflecting the fact that both the overall free-energy changes for electron transfer and the binding energies of the exciplexes are related to the redox potentials of the reactants in a similar f a ~ h i o n .~ However, recent experiment work on these systems conducted by Dr. M. J. Pilling and myself indicates that exciplex-type interactions may be important in the “ nor- mal ’’ region also. Our data cannot be interpreted in terms of a simple outer-sphere electron-transfer mechanism, and we have suggested that the correlation observed between quenching rate constant and overall free-energy change for reactions reflects the fact that charge-transfer interactions play an important part in the quenching mechanism. D. Rehm and A. Weller, Zsr. J. Chem., 1970, 8, 259. J. Joussot-Dubien, A. C. Albrecht, H. Gerischer, R. S. Knox, R. A. Marcus, M. Schott, A. Weller and F.Willig, Light-induced Charge Separation in Biology and Chemistry, ed. H. Gerischer and J. J. Katz (Verlag Chemie, New York, 1979).GENERAL DISCUSSION 87 A. Weller, in The Exciplex, ed. M. Gordon and W. R. Ware (Academic Press, New York, 1975). * J. E. Baggott and M. J. Pilling, J. Chem. SOC., Faraday Trans. I , in press. Prof. J. M. SavCant (University of Paris) (communicated): There are few investi- gations of the inverted region in organic chemistry. An indirect approach to this question was developed by Forno et a2.l in the case of the reduction of anthracene and naphthalene in dimethylformamide in the presence of phenol. The reaction mechanism : A + l e s A - - (1) (2) AH. + le AH- (3) and/or: AH- + A*- 2 AH- + A- (Kd) (3') A*- + PhOHZAH. + PhO- involves the competition between the heterogeneous [reaction (3)] and the homo- geneous [reaction (3')] electron transfer to AH-.AH. is easier to reduce than A, i.e. (RT/F) In Kd = E ( i H - A H - 1 - E * ( A / A - ) > 0. In Kd = 15 for anthracene and 25 for naphthalene. In the context of the Marcus parabolic relationship between In kd and In &, In kd is predicted to be 9.5 for an- thracene and 2.5 for naphthalene, as shown in fig. 1. Jt is then anticipated that the 10 9 h 7 7.5 v) I rl - 2 3 a --- O 5 s c ..I 2.5 5 10 20 In Kd \ FIG. 1.-Plot of In kd against In Kd for anthracene and naphthalene.88 GENERAL DISCUSSION homogeneous reaction (3') will prevail over the heterogeneous reaction (3) in the case of anthracene and vice versa for naphthalene.This is what was indeed found by the above-cited authors using single-potential-step chronoamperometry. However, it has been shown more recently that the single-step technique is not powerful enough to actually discriminate between the two pathways. Double-potential-step chrono- amperometry is much more efficient in this respect. Using the latter technique it was found that the homogeneous pathway is followed also for naphthalene. It ensues that In kd is at least 6 for the case of naphthalene. This shows that the magnitude of the inverted effect, if any, is much smaller than predicted by the parabolic relationship. A. E. J. Forno, M. E. Peover and R. Wilson, Trans. Faraday SOC., 1970,66, 1322. C. Amatore and J. M. Saveant, J. Electroanal. Chem., 1980, 107, 353.Dr. W. Schmickler (University of DiisseZdorf )said : In connection with the effect of nuclear tunnelling on the reaction rates of electron-transfer reactions I would like to mention electrochemical reactions at low temperatures which Dr. Stimming and I are studying. We have measured the reaction of the Fe2+/3+ couple at a platinum electrode in solutions of HC104 5.5H20; below ca. -40 "C this system forms a homogeneous frozen solution. We have measured the exchange-current density in the temperature range -40 to -120 "C, and observed a decrease of the energy of activation with decreasing temperature. While these experiments are still in an early stage, and a definite interpretation is not yet possible, this may well be an indication of the occurrence of nuclear tunnelling.If the typical frequencies of the frozen solu- tion are of the order of 1014 s-l, as they are for ice, one would indeed expect to see this effect in the temperature region which we investigate. Prof. H. L. Friedman (State University of New York, Stony Brook) said: In view of Prof. Jortner's analysis of the role of slow nuclear motions in the theory of the rate constant for electron-transfer reactions, it may be helpful to recall that an analysis by the ordinary methods of chemical kinetics leads to the expression where we assume that the effect of nuclear motion is accounted for by the usual inner (IN) and outer shell (OUT) reorganization terms, with relaxation time zILN and z&, respectively, and where T~ is the lifetime of a diabatic state at the transition-state con- figuration.The (. - .)-l term in the denominator expresses a deviation from tran- sition-state theory; it is the consequence of slowness in the " in " and " out " re- organization processes. Only when the (- * .)-' term is negligible do we have that equilibrium between initial states and transition state which is assumed in transition- state theory. This equation seems to be useful for estimating the importance of deviations from transition-state theory in electron transfer, although at this time there is some difficulty in estimating z:N and Z&.~'' A mechanical (as opposed to chemical kinetic) deriv- ation in a more general case has recently been given by Northrup and Hynes and in the electron-transfer case by us as well as by van der Zwan and H y n e ~ .~ B. L. Tembe, H. L. Friedman and M. D. Newton, J. Chem. Phys., 1982,76, 1490. S. H. Northrup and J. T. Hynes, J. Chem. Phys., 1980,73, 2700. G. van der Zwan and J. T. Hynes, J. Chem. Phys., 1982,76, 2993. ' H. L. Friedman and M. D. Newton, Faraday Disc. Chem. SOC., 1982,74,73. Dr. M. D. Newton (Brookhaven National Laboratory) said: The preceding com- ment by Prof. Friedman concerning the role of relaxation times is based on a simpleGENERAL DISCUSSION 89 transition-state model which does not include nuclear tunnelling. However, I wish to emphasize that Prof. Friedman, for the purposes of illustration, is employing a special limiting case of a general kinetic expression derived by Tembe et aZ.l which in- cludes both nuclear and electronic tunnelling: ’ The effective activation energy is defined as A$ = -alnk(r)/a(l/kT) where k(r) is the local rate constant calculated by whatever model is desired (classical, semiclassical, or quantum-mechanical).For the case of non-adiabatic charge transfer we obtain A$(r) = Eii(r) + E&(r) - (kT)/2 - A n where --A, is the effective reduction of the classical barrier arising from nuclear tunnelling. B. L. Tembe, H. L. Friedman and M. D. Newton, J. Chem. Phys., 1982, 76, 1490. Prof. A. M. Kuznetsov (Academy of Sciences of the U.S.S.R., Moscow) (communi- cated): A difference between the Marcus approach and the approach based on the theory of multiphonon transitions using the model of harmonic oscillators has been mentioned in Prof. Jortner’s paper. In the former approach the quantity [E, + AFI2/4Er represents the free energy of activation and involves the free energy of the reaction AF.In the latter model, as noted by the authors, the quantity [E, + AEI2/ 4E, is the energy and it does not involve the entropy which comes from the orientational modes of the medium molecules. I would like to mention that, in the approach based on the theory of multiphonon transitions, methods of calculation exist which take into account orientational entropy. It has been rigorously shown ’ that the expression for the transition proba- bility involves the factor W - exp[- F(B*)/kT] where F(O*) is the difference of the free energies of the system in the transition con- figuration and in the initial state. The method of calculation given in this paper does not use the model of harmonic vibrations and to some extent takes into account the effects of the dielectric saturation.In particular it has been shown that F(e*) has the form F(O*) = (E, + AF)’/4Er and a concept of the energy of the reorganization of the medium E,(B*) which is necessary to reach the transitional configuration has been introduced. This has the significance of free energy. We may mention also the classical calculation in ref. (2) and (3) using the Hamiltonian of the effective harmonic oscillators, which para- meters depend on the temperature through the dependence of the dielectric properties of the medium on T. The expression for W obtained with the use of this method is similar to that obtained in the long-wavelength approximation without using the model of harmonic o~cillators.~ Recent results presented in our paper also take into account the change of the orient ational entropy.90 GENERAL DISCUSSION A.M. Kuznetsov, Elektrokhimiya, 1981, 17, 84. W. Schmickler, Ber. Bunsenges. Phys. Chem., 1976,80, 834. R. R. Dogonadze and A. M. Kuznetsov, Elektrokhimiya, 1971, 7 , 763. A. A. Ovchinnikov and M. Ya. Ovchinnikova, Zh. Eksp. Tear. Fiz., 1969,56, 1278. Prof. J . M . Savkant (University of Paris) (communicated): The following remark is intended to illustrate the use of Brarnsted-Marcus free-energy relationships in the analysis of the mechanism of organometallic reactions. The problem is to decide whether or not the reaction of alkyl halides on Fe' and Co' porphyrins yielding the corresponding alkyl-metal compounds proceeds via an outer-sphere electron-transfer mechanism or via an SN2 mechanism.The former would involve the electron trans- fer from the Fe' or Cot to the alkyl halide leading to an alkyl radical and an Fe" or Co" complex which would then couple leading to the final organometallic complex. To this end we have determined the rate constants of the reaction of several n-butyl halides with several Fe' and Cot porphyrins. The results were plotted against the E" values of the corresponding Fe"/Fe' and CO~~/CO' couples, as shown in fig. 2. The -1.0 -1.5 -2.0 E"IV us. SCE FIG. 2.-Rate constant of the reaction of Fe' and Co' porphyrins and of aromatic or heteroaromatic anion radicals with n-butyl halides as a function of the standard potential of the reactant.TPP, tetraphenylporphyrin; C12TPP, basket-handle tetraphenylporphyrin bearing two 12 aliphatic carbon chains; OEP, octaethylporphyrin. (a) n-BuI, (b) n-BuBr, (c) n-BuC1. a, Aromatic or hetero- aromatic anion radicals; A, n-BuI + Fe-OEP; x , n-BuBr + Fe'OEP; +, n-BuC1 4- Fe'OEP; A, n-BuBr + Co'TPP; B, n-BuBr + Fe'CI2TPP; 0, n-BuBr + Fe'TPP.GENERAL DISCUSSION 91 same figure shows the comparison with similar results obtained for the reaction of aromatic or heteroaromatic anion radicals with the same alkyl halides. In this case we showed from a stereochemical investigation that the reaction proceeds through a prior a outer-sphere electron transfer step. There is then (fig. 2) a fair linear correlation between the In K and the E" values.It is seen that the reactions with the Fe' and CO' complexes lie far above the straight line corresponding to the aromatic anion radicals. It is therefore concluded that the reaction of alkyl halides with Fe' and Co' por- phyrins does not involve an outer-sphere rate-determining electron transfer, but rather follows a classical SN2 mechanism. Dr. T. Markvart (University of Southampton) said: I would like to make a comment which concerns some of our recent results on the non-radiative transition rate between two potential curves. In the semiclassical limit we have obtained the result for the transition rate W(T) from curve U2 to U, (fig. 3):l where Qc is the crossing point (in general, a complex number) of the potential curves, V(Q) is the electronic matrix element, ( * - - )T denotes the thermal average at tempera- ture T, Qi (i = 1,2) lie in the classical region of motion in potential Ui( Q), f varies A \ E _I H I \ i i / i Re Q ( 2 ) WMAvUv branch cuts 8 Qc FIG.3.-Configuration diagram of the accepting mode (top) and the corresponding complex Q plane (bottom) for general potential curves. weakly with E, and the remaining symbols are clear from fig. 3. With the arrange- ment of branch cuts as in fig. 3, and assuming the square roots are positive immediately above the real Q axis for @+a, only the crossing point Q;l) contributes to the integral in eqn (1). Eqn (1) provides a connection between the tunnelling (low T ) and92 GENERAL DISCUSSION activated (high T ) regimes, and makes it possible to evaluate the transition rate for a wide range of potential curves.Although a similar result was suggested recently by Holstein for small polarons, eqn (1) (which was obtained by calculating the overlap integral and using the first-order perturbation theory) contains a number of new and interesting features. These will be discussed in detail in a forthcoming publication. Here I would like to focus attention on the following points. (1) If the crossing point Qc has a non-zero imaginary part (i.e. in the usual con- figuration diagram the potential curves do not cross), eqn (1) describes a novel type of tunnelling between the potential curves. A similar situation was considered by Form~sinho,~ who approximated the non-radiative rate by tunnelling probability through an idealized potential barrier.Tunnelling between “ non-intersecting ” potential curves was also recently used to explain the enhancement of defect reaction rates in semic~nductors.~ In this situation, the usual high-temperature Arrhenius regime may not be realized. (2) If the crossing point Qc lies on the real Q axis, we assume that it has a small negative imaginary part q, and let q+O at the end of the calculation. The low- temperature tunnelling then occurs through a well defined potential barrier. It is 0 t FJG. 4.-Configuration diagram (top) and the complex Q plane (bottom) for potential curves that cross near the real axis. inteiesting to note that in this case, there is a one-to-one relationship between the function W(T) and the barrier width A(E). This can be inverted to obtain an expres- sion for A(E) in terms of W(T):5 where p = l/kT, q(p) = In@‘@), the prime denotes differentiation with respect to p, and P(E) is the solution of p’(P) = --E.Note that q‘(p) (GO) is a monotonically increasing function of p so that the argument of the square root in the integrand in eqn (2) is always positive. Eqn (2) can be used to obtain directly the shape of the potential curves from the temperature dependence of the non-radiative transition rate. (3) Eqn (1) contains only the matrix element V(Q) evaluated at the crossing pointGENERAL DISCUSSION 93 Qc. This justifies a form of Condon approximation, where the crossing point is used, rather than the equilibrium positions. I am grateful to Prof. Marcus and Dr. Sahyun for bringing my attention to ref.(2) and (3). T. Markvart, J . Phys. C, 1981, 14, L895. T. Holstein, Philos. Mag., Sect. B, 1978, 37, 49. S. J. Formosinho, J. Chem. Soc., Faraday Trans. 2, 1974, 70, 605. T. Markvart, Physica, Sect. B, to be published. T. Markvart, J. Phys. C, 1981, 14, L435. Prof. R. A. Marcus (California Institute of Technology, Pasadena) said: This interesting semiclassical model has recently been described, at least in part, by Scher and Holstein.' H. Scher and T. Holstein, Philos. Mag., Sect. B, 1981, 44, 343. Prof. R. A. Marcus (California Institute of Technology, Pasadena) said : Would Dr. Ulstrup care to comment on the relation of solvation effects on the orbitals (and hence on the electron-transfer matrix elements) to the vibrational effects on those quantities, treated by Redi and Hopfield for large separation distances? M.Redi and J. J. Hopfield, J. Chern. Phys., 1980,72, 6651. Dr. J. Ulstrup (Technical University of Denmark, Lyngby) said: The physical basis for the process is that the solvent polarization in the trap region fluctuates around the qquilibrium value corresponding to the charge distribution given by the electronic wavefunction. The instantaneous polarization can therefore be either smaller or larger than the equilibrium polarization, in both cases corresponding to a higher energy for the electron and the surrounding medium than at equilibrium. In the " normal '' free-energy region the absolute value of the non-equilibrium polariz- ation P*(r) is smaller than the equilibrium value. The electron is therefore trapped in a weaker polarization field, it has a smaller potential energy, and as a consequence it billows out in space.This is partly compensated by a lowering of the kinetic energy of the electron and the polarization energy of the solvent, but the overall effect is a destabilization of the electron. These results therefore bear a certain conceptual resemblance to the considerations of Redi and Hopfield, even though the nature of the electronic-vibrational coupling differs in the two cases. However, in the strongly exothermic region the wavefunction of the solvated electron " shrinks " and its kinetic energy increases, as a response to polarization fluctuations towards larger absolute values. This represents in fact a stabilization of the electron but the total potential energy of the electron-solvent system is higher than at equilibrium, since the potential energy required to create the increased polarization is now larger.This energy quantity depends quadratically on the absolute value of P*(r) and is therefore more important the larger is P*(r). In the strongly exothermic region the solvation effects on the electronic wavefunctions treated here are therefore different from the concepts introduced by Redi and Hopfield. The solvation effect, in terms of the dependence of the orbital exponent A on the parameter 8*, is apparently much more pronounced in the strongly exothermic region when the electronic-vibrational coupling is dominated by high-frequency modes. However, this reflects the nuclear tunnelling of these modes and the fact that \@*I remains small even for very large changes in the external parameters A& and T.P*(r) therefore stays at values which are not widely different from the initial-state equilibrium value.94 GENERAL DISCUSSION Prof. J. Jortner (Tel-Aviv University) said: The Condon approximation for non- radiative processes, which assumes that the electronic matrix elements are weakly dependent on the nuclear coordinates, has been widely utilized in the theory of non- radiative electron-transfer (e.t.) processes. Dr. Kuznetsov considered the extension of the theory of e.t. to include contributions beyond the Condon approximation. Such " non-Condon " contributions are well known in the theory of non-radiative electronic transitions in solids and in large molecules.2 For such spin-allowed solid-state and molecular-relaxation processes, the electronic matrix element involves the nuclear momentum and exhibits a marked dependence on the nuclear coordinates. On the other hand, for e.t.between ordinary ions or/and molecules in solution and in solids the matrix element involves a two-centre electronic transfer integral, which is expected to exhibit a weak dependence on the nuclear coordinates, whereupon non- Condon corrections will be of minor importance for most e.t. processes. A notable exception is the case of e.t. involving solvated electrons, whose wavefunction exhibits a marked dependence on the nuclear coordinate^.^ For this class of e.t. processes, non-Condon corrections will be appreciable. The work of Dr.Kuznetsov and Dr. Ulstrup on the quantum theory of chemical reactions of solvated electrons, which rests on the Condon approximation, should be extended to incorporate non-Condon correc- tions. V. A. Kovarskii, Sou. Phys. Solid State, 1963, 4, 1200; 1963, 4, 2345. A. Nitzan and J. Jortner, J. Chem. Phys., 1972,56, 3360. D. A. Copeland, N. R. Kestner and J. Jortner, J. Chem. Phys., 1970,53, 1189. Dr. J. Ulstrup (Technical University of Denmark, Lyngby) said: The reason why non-Condon effects are expected to be important for the solvated electron in particular is the strong sensitivity of the wavefunction of this entity to fluctuations in the nearest surroundings, i.e. the solvent polarization fluctuations including the first solvation shell. This is physically based on the trapping of the electron by the polarization field rather than by an atomic or molecular field.Our approach certainly incorporates this sensitivity by the explicit dependence of the electronic factor on the instantaneous polarization rather than the constant equilibrium polarization. The dependence is furthermore subsequently calculated by the variational procedure and conditions, where the variation of the electronic factor with AF,, and T is comparable to the variation of the nuclear factor have been found. Even though we have applied the Condon approximation, our approach has therefore modified previous procedures significantly. The effect of the variation of the electronic factor is in fact large in some cases. Eqn (3.28) and (3.29), and fig.3 of our paper show that it is generally small in the " normal " free-energy region (0 < 6* < 1). In the strongly exothermic region (6* < 0) [eqn (3.46) and fig. 31; numerical calculations reveal, however, that the effect may amount to several orders of magnitude and is comparable to the drop in the rate constant caused by the variation of the unclear Franck-Condon factor. We are in fact presently trying to calculate the electron-transfer probability by means of a non-Condon approach and incorporating the explicit dependence of the electronic wavefunctions on external parameters [ref. (28) of our paper]. This dependence is introduced in the form [cf. eqn (3.18), where 5 and q represent the external parameters]. This procedure provides a prescription for the calculation of A(t,q) and shows not unexpectedly that non-Condon effects contribute significantly.GENERAL DISCUSSION 95 Dr.M. D. Newton (Brookhauen National Laboratory) said: As Prof. Jortner has just emphasized, the Condon approximation is expected to be quite reliable for processes involving tightly bound electrons, as in the case of electron transfer between transition-metal complexes. I should like to give some specific, quantitative results on the validity of the Condon approximation for the aqueous Fe2+/Fe3+ exchange. I will consider only the effect of inner-sphere water molecules, since the work of Friedman and Newton has shown that the screening provided by the outer-sphere medium is very small in the region important for electronic overlap (i.e. between the two Fe complexes).Applying ab initio electronic structure techniques to the apex- to-apex inner-shell Fe2+/Fe3+ encounter complex (with rFe . . . Fe = 7.3 A), one finds that the magnitude of the electronic matrix element, HAB,1,2 at the “ crossing- point ” (i.e. rg:o = rg:o = rieo = 2.06 A; H,& = 27.5 cm-l) changes by < 1 cm-’ or 4% as the inner-shell “ coordinate,” r&!o - r:Lo, is varied from zero (at rS) to as much as 0.08 A (or ca. 60% of the distance from the transition state to equilibrium reactants or products). It should also be noted that in order to ensure proper detailed balance, the Condon approximation should in general be implemented only at the crossing point. This is because the matrix elements for forward and reverse processes in the case of non- orthogonal ( S A B # 0) initial and final states ( y A , y B ) are, respectively, (forward) H A = H B A - S B A H B B .(reverse) At the crossing, H A A and H B B are equal and H A B = H A A , as r e q ~ i r e d . ~ H. L. Friedman and M. D. Newton, Faruday Discuss. Chem. Soc., 1982, 74, 73. J. Logan and M. D. Newton, J . Chem. Phys., in press. M. D. Newton, Adv. Chem. Ser., 1982, 198, 255. Prof. L. I. Krishtalik (Academy of Sciences of the U.S.S.R., Moscow) (communi- cated): Prof. Savtant has undertaken a systematic investigation of the dependence of the true value of a on the electrode potential and has given some interesting examples of such a dependence. The problem is really very important and hence every case of the kind deserves detailed investigation. I think that some features of the systems under consideration require further study.The first problem arises in connection with the yl-effect. The conclusion on the negligibly small influence of Vl-potential was drawn on the basis of identification of the yl-potential with the potential of the outer Helmholtz plane, yo, the latter being estimated according to Gouy-Chapman theory. However, this assessment is not precise enough, for in the systems studied the iodide anion of the background electrolyte possesses a significant specific adsorb- ability [see, e.g., ref. (l)]. The other side of this problem is connected with the difference of effective radii of the background cation Bu,N+ (ca. 5 A) and of the reacting groups, e.g. nitrogroups (ca.1.5-2 A). Due to this difference the charge centre of the reactant lies deep inside the dense double layer, and hence a substantial fraction of the whole potential drop does not act on the charge-transfer process. This means that the y1 potential is far from yo and the corresponding correction may be rather large and strongly potential-dependent. Both these problems might possibly be solved by using various non-surface-active background electrolytes with different and relatively small cations, for example (CH3)4NC104 and NaC10,. The other problem to be clarified experimentally is the potential dependence of the adsorption of an organic depolarizer. For many systems this dependence is quadratic, and this might also contribute to the distortion of the true current- potential relationship.96 GENERAL DISCUSSION All the foregoing does not mean the denial in principle of a smooth dependence of a on potential.Moreover, I believe such a dependence of the atomic hydrogen ioniz- ation found recently by Babenko et aL2 reflects the true dependence, which is not influenced substantially by the I,V~ potential. In their experiments the ionization of hydrogen atoms formed via trapping of photoemitted electrons by H,O+ ions was studied. In my opinion3 hydrogen atoms under these conditions react in a non- adsorbed state. In this case a substantial difference in behaviour of this “ photo- hydrogen ” and phenomena during cathodic hydrogen evolution seems to be remark- able. It may favour an explanation of the constancy of 01 based on Kuznetsov’s idea about the change of the effective charge of adsorbed hydr~gen.~ M.D. Levi and I. A. Bagotskaya, Elektrokhimimiya, 1982, 18, 1103. S. D. Babenko, V. A. Benderskii and A. G. Krivenko, Dokl. Akad. Nauk SSSR, 1982,262,360. L. 1. Krishtalik, J. Electroanal. Chem. Interfacial Electrochem., 1981, 130,9. A. M. Kuznetsov, J . Electroanal. Chem. Interfacial Electrochem., in press. Prof. J. M. SavCant (University of Paris) said: We did not find any significant variations of the rate constants when changing the anion of the supporting electrolyte from I- to C10, and BF;. This is not surprising since adsorption of I- on mercury in DMF or CH3CN is not expected to play an important role in the rather negative potential range where the reduction of the investigated molecules occurs.The question of the location of the reaction site and of the potential profile in the compact double layer is certainly interesting. The potential profile in the compact double layer can be expressed on the rational scale as q r = YOHP + I ( q e l - VoHp) with R = K/(K + K’), where K is the integral capacity of the portion of the compact double layer comprised between the electrode and the considered plane and K‘ the integral capacity between the latter plane and the 0.H.p. (outer Helmholz plane). In the (negative) potential range of interest the integral capacities are practically not dependent upon the charge of the electrode and hence upon the electrode potentiaL2 R does not therefore depend appreciably upon the potential. It follows that a2qr/aE2 is of the same order of magnitude as a2qOHp/aE2, i-e.very small. The second term in the right-hand side of eqn (4) of our paper does not consequently contribute sig- nificantly to the relationship between the variations of the apparent transfer co- efficient and those of the “ true ” transfer coefficient. Thus since 6qoHP/aE is small (see tables 1 and 2 of our paper). A first conclusion is there- fore that a variation of the apparent transfer coefficient with the potential can only reflect a corresponding variation of the “true ” transfer coefficient. The former can, however, be smaller than the latter if A is not negligible. If the difference between the observed and predicted variations were to be attributed to the above discussed effect, it follows that R would at most be 0.5 and much less in a number of cases.That this effect is not anticipated to be very large for the investigated systems derives from the following reasons. The investigated molecules are rather bulky, having a radius 9 1.5-2.0 8, since they are not simply composed of an NO, group (NO, radius is itself ca. 2.5 A)., A large part of the potential drop in the compact double layer occurs within a thin layer of low permittivity adjacent to the electrode, where the solvent dipoles are strongly oriented. The reaction site is likely to be located farther from the electrode surface. Deep penetration of the reactant into the compact layer leads to a substantial decrease of the electron-transfer rate in the present case of un-GENERAL DISCUSSION 97 charged reactants.Consequently, electron transfer may well mostly occur at larger distances from the electrode in the context of a distance-dependent expression of the rate c o n ~ t a n t . ~ We are currently investigating the effect of the size of the counter- cations on the electrode kinetics with similar reactants. It is interesting to note that t-nitrobutane exhibits variations of a with potential of the same ordqr of magnitude on platinum as on mercury in spite of the fact that the rates are lower and much dependent on electrode pre-treatment [see ref. (18) of our paper]. The rather complex interfacial structure which exists in the case of platinum does not seem to have a pronounced effect on a variations. Solvent reorganization appears as the main factor with the investigated systems.It responds harmonically. It is not therefore surprising to find a definite variation of the transfer coefficient with potential in these cases. What would be actually sur- prising would be the opposite. We have not found experimental evidence for the adsorption of the considered reactants. This is expected for organic molecules in organic solvents. W. R. Fawcett, J. Chem. Phys., 1974, 61, 3842. W. R. Fawcett, B. M. Ikeda and J. B. Sellan, Can. J. Chem., 1979, 57, 2268. M. E. Peover and J. S. Powell, J. Electroanal. Chem., 1969, 20, 427. R. R. Dogonadze and A. M. Kuznetsov, J. Electroanal. Chem., 1975, 65, 545. Dr. A. M. Kuznetsov (Academy of Sciences of the U.S.S.R., Moscow) (communi- cated): This comment concerns the variation of the transfer coefficient with the electrode potential.All theories of redox reactions lead to the conclusion that a should vary with the potential and it is surprising that this has not been definitely confirmed experimentally until now. In this respect the data presented in Prof. Saveant’s paper are of great interest. We have a paradoxical situation. The possibility of the variation of u may be derived from rather general physical consider- ations and may be easily explained, but it is difficult to find experimental data showing this variation. Indeed, in the well investigated reaction of the discharge of hydrogen ions a is constant over a very wide range of potential variation (ca. 1.5 V) but it is very difficult to find a physical mechanism which could explain this fact.Recently we have suggested a new physical mechanism leading to a constant value of a. It is based on the fact that the charge of the adsorbed hydrogen atom should in principle vary due to fluctuations in configuration of the solvent molecules. This variation leads to a change in the interaction of Hads with the medium and to a change in the electron energy in the system Hads + metal, and hence leads to a distortion of the shape of the potential and free-energy surfaces of the final state. The calculation of this effect using the Anderson model for the adsorbed hydrogen atom resulted in a linear dependence of the activation free energy on the free energy of transition for some values of the parameters. The dependence of the transfer co- efficient a on the electrode potential is different depending on the relationship between the reorganization energy of the medium E, and the parameter U characterizing the repulsion of the electrons in Hads.If U >> E, the charge of the adsorbed hydrogen atom depends weakly on the con- figuration of the solvent molecules and the dependence of a on ~1 is similar to that for ordinary redox reactions. If U z E,, the dependence of u on p is much weaker and, for some relationships between the parameters, a is practically constant throughout the span of the variation of the electrode potential of order E,. This effect may be also seen from an approximate expression for o! valid for small variations of the charge of Hads: a = +[198 GENERAL DISCUSSION The above dependence of cc on AF differs from the normal one cc = 3[1 + AF/E,] by the presence of the factor (1 - 2E,/U), which may, however, be rather small.Dr. B. R. Eggins (Ulster Polytechnic) said: It is pertinent to consider the impli- cations of the variation of cc with potential on other techniques such as polarisation studies and cyclic voltammetry for irreversible waves. The aa/aE factor should be observable as a deviation from the assumed linearity of Tafel plots, taken over a suitable potential range. For irreversible voltammetric waves, ccn, is commonly determined from Ep - Ep/2 = 0.0477/~n,.~ Now the maximum value of &/aE from the results of SavCant and Tessier is ca. 0.5 V-l, which over the usual range of Ep - EPj2 of ca. 0.1 V would give a variation in 0: of h0.05, equivalent to a variation in ED - Ep/2 of &lo mV.Experimental data for cyclic voltammetry are usually better than that (say &2 mV), so theoretical considerations involving irreversible cyclic voltammograms should take account of &/aE. R. S. Nicholson and 1. Shain, Anal. Chem., 1964, 46, 706. Prof. J. M. SavCant (University of Paris) said: The results in table 1 of our paper were obtained using convolution potential-sweep voltammetry (c.P.s.v.). This is just a more convenient reading of the cyclic voltammetric data using together with the current itself its convolution integral with the linear diffusion characteristic function (nt)-1/2. In other words, it is a way of expressing quantitatively the shape of the voltammograms as Ep - Ep,2 also does, although with less accuracy. There is thus certainly an effect of the cc variations on Ep - E,,, and more generally on the shape of the cyclic voltammogram, C.P.S.V.being merely a method for revealing it. The effect on the variations of OL with potential on the Ep - Ep,2 of irreversible cyclic voltammogram will, however, be less than that indicated by Dr. Eggins. &/aE = 0.5 V-l corresponds to quite rapid electron transfer, with ktp N" 2 cm s-l. In such a case the cyclic voltammetric pattern will be quasi-reversible over the whole available sweep rate range (complete irreversibility would be reached at a sweep rate of ca. lo6 V s-l, clearly inaccessible in practice). The variation of Ep - Ep/2 will then be less that predicted for a purely irreversible voltammogram.On the other hand, complete irreversibility would be reached at a reasonable sweep rate, say 100 V s-', for ka&' = 2 x cm s-l. Then 8a/aE would be of the order of 0.25 V s-l, leading to a variation of Ep/Epjz of &5 mV. It follows that the variations of 0: with potential can be neglected in a number of practical applications of cyclic voltammetry. For example, it will not affect too seriously the determination of the standard rate constants if only a moderate accuracy is looked for. The counterpart of this is that the experi- mental demonstration that a does vary with potential is not an easy task. It requires the investigation of an extended range of sweep rates in cyclic voltammetry when dealing with moderately fast electron transfer and the recourse to high-frequency a.c, techniques for very fast electron transfer.Prof. B. E. Conway (University of Ottawa) said: I should like to make two points. (1) The paper of Savkant and Tessier addresses the important question whether the transfer coefficient, cc, for electron transfer in electrochemical processes is dependent on potential and hence whether anharmonicity effects are relatively insignificant in the activation process involving solvent reorganization. The common difficulties, e.g. double-layer effects, associated with unambiguous experimental proof of thisGENERAL DISCUSSION 99 matter are recognized by the authors but the observed variation of cc is based on data obtained over quite a wide range of potential, ca. 0.7 V. Over this range it is expec- ted l v 2 that there would be an appreciable change of solvent dipole orientation in the inner region of the double layer and hence of local polarization environment of the reactant species in the electron-transfer process.It would be surprising if this chang- ing solvent polarization with potential in the double layer did not affect the reorganiz- ation of solvent in the electron-transfer process and thus lead to a possible potential- dependent component of variation of u. For example, double-layer solvent coplanar and ion cosphere solvent interaction effects in the electrode interphase can be com- ~ a r a b l e . ~ (2) In comparison with the results found by SavCant and Tessier for t-nitrobutane it is striking that 01 for the H2 evolution reaction at Hg, where it is agreed that H30 + discharge is rate-controlling, is remarkably independent of potential over a wide range of ca.0.8-0.9 V, taking into account various authors’ results. This suggests that a much more important role is played by an inner-sphere activation mechanism involving bond stretching in H,O+ in this case, leading to greater importance of anharmonic effects. A related question is the temperature dependence of Tafel slopes and hence cc. For H2 evolution at Hg and some other metals, e.g. Ni, the Tafel slope b = dq/d In i is not simply of the form RT/ccF with 01 = RT/b but rather b = RT/rF + K, where K is a temperature-independent constant and a + 0.5. This led us to suggest that an important component of the potential dependence of the free energy of activation lies in an effect on the entropy of activation, ASt, as well as on the energy term.This effect may be less significant in unstructured solvents such as ACN and DMF used in the work of SavCant and Tessier. J. O’M. Bockris, M. A. V. Devanathan and K. Miiller, Proc. R. SOC. London, Ser. A , 1963, 274, 55. B. E. Conway and H. P. Dhar, Croat. Chem. Acta, 1973, 45, 173. B. E. Conway, J . Efectroanal. Chem., 1981, 123, 81. B. E. Conway, D. J. MacKinnon and B. V. Tilak, Trans. Furuday Suc., 1970, 66, 1203. Prof. J. M. SavCant (University of Paris) said : Dealing with Prof. Conway’s remark (l), I would comment that if such an effect occurs to a significant extent, it is antici- pated that, in the potential range of interest, the dielectric constant of the solvent at the reaction site, Ds, would decrease as the electrode potential becomes more and more negative.This would result in a decrease of the medium reorganization factor according to Marcus theory. The potential-dependent rate constant, k(E), would then increase when going to negative potentials more rapidly than expected from the Marcus kinetic law. In other words, this would lead to a variation of the transfer coefficient opposite to that predicted by the harmonicity of solvent reorganization according to Marcus theory and also opposite to the net variation found experi- mentally. The two types of 01 variation cannot therefore be mistaken. What might happen is that the effect of this potential-dependent polarization of the solvent would result in a net variation of u with potential lower than that predicted from Marcus theory.That such an effect is anticipated to be small results from the following reasons. The decrease in Ds due to solvent polarization should be quite large in order to induce a noticeable variation of A,,, since the prevailing term in A. comes from l/Dop. The increased solvent polarization is anticipated mostly to affect the first monolayer of solvent molecules while the reaction site is most probably located farther from the electrode surface. Change of solvent dipole orientation in the potential range of interest appears as small in dimethylformamide and most probably also in acetonitrile.2 There is no such systematic trend in the experimental results.1 00 GENERAL DISCUSSION W. R. Fawcett, B.M. Ikeda and J. B. Sellan, Can. J. Chem., 1979,57, 2268. W. R. Fawcett and R. 0. Loufty, Can. J. Chem., 1973, 51, 230. Dr. J. Ulstrup (Technical University of Denmark, Lyngby) said: In response to informal comments by Prof. Conway and Prof. Savkant, I would like to remark that we have calculated electron and proton transfer rate constants using anharmonic potentials (Morse potentials and Rosen-Morse potentials)* for the intramolecular modes. The anharmonicity effects can be very pronounced for electrochemical pro- cesses. Thus if a given nuclear mode exists in a highly excited vibrational state, corresponding to stretching, the current-voltage relationship is much " flatter " than if the intramolecular potential is harmonic. Such a result is expected for a classical asymmetric potential such as a Morse potential, since this potential is in fact almost linear over quite wide energy ranges.For a high-frequency mode the physical reason is that even for fairly low overpotentials a significantly larger number of vibrational states contribute than for harmonic potentials. This is again due to the smaller barrier for nuclear tunnelling for the " stretched " part of a Morse potential. For analogous reasons nuclear tunnelling becomes more difficult by " compression " of a bond represented by an asymmetric potential. If the final state corresponds to com- pression of a set of intramolecular anharmonic modes, a larger curvature is therefore expected than for a harmonic potential. All these expectations were substantiated by numerical calculation by means of the Franck-Condon overlap factors for the appropriate potentials. Furthermore, for homogeneous processes where the donor is generally subject to bond compression and the acceptor to bond stretching, the anharmonicity effects cancel to a significant extent.For the two electrochemical reactions in question anharmonicity effects would have the following implications. The bond of the adsorbed hydrogen atom in the electrochemical discharge of H30+ is highly stretched immediately after electron transfer. This means that many vibrational states of the stretching mode of the adsorbed hydrogen atom contribute to the rate, and the Tafel plot becomes close to a linear relationship when a Morse potential is used, in contrast to the results for a harmonic potential.For outer-sphere electron transfer involving transition-metal complexes " flatter " Tafel relationships are expected for the cathodic process and a stronger curvature for the anodic process as compared with harmonic potentials, as long as only symmetric " breathing " modes are considered. This would be in line with the observations for the reduction of CrI'I complexes. On the other hand, if the electron is transferred by an inner-sphere mechanism, the reaction is analogous to the H,O+ discharge, and a " flatter" Tafel relationship would also be obtained for such a mechanism if the intramolecular restructuration is dominated by reorganization of the bridge ligand. Other recent calculations of ours based on the same model for the solvent incorpor- ating various kinds of molecular motion as in my paper with Dr.Kuznetsov at the present Discussion, illustrate the temperature dependence of the transfer coefficient. This quantity increases rather significantly with increasing temperature T when T varies over wide ranges. The reason is that the electronic charges are coupled to solvent modes of many different vibrational frequencies. Some of these (in particular hindered rotation with maximum vibrational frequency around 700 ern-' and hindered translation with a maximum frequency of ca. 200 cm-l) have such values that they become significantly " quantized " in the lower range of the temperature interval investigated for the H30+ discharge. This effect contributes to a decrease of the transfer coefficient cc with decreasing T, and as a result the Tafel " b-coefficient" depends significantly less strongly on T than implied by the linear dependence b = RT/aI;, where R is the gas constant and F is Faraday's number.GENERAL DISCUSSION 101 N.C . SmdergHrd, J. Ulstrup and J. Jortner, Chem. Phys., 1976, 17, 417. Yu. I. Kharkats and J. Ulstrup, J. Electrounul. Chem., 1975, 65, 555. ’ N. Bruniche-Olsen and J. Ulstrup, J . Chem. SOC., Faruday Trans. 2, 1978, 74, 1690. Dr. M. D. Newton (Brookhaven National Laboratory) said: In the case of homo- geneous electron transfer the effects of anharmonicity on the inner-sphere reorganiz- ation energy of the two reactants tend to cancel even though the individual magnitudes can be appreciable. This was demonstrated in terms of model calculations reported by Slzrnderggrd et al.’ The ab initio molecular-orbital calculations of Jafri et a1.2 give realistic quantitative estimates for the case of Fe2+ and Fe3+ hydrate^:^ the anharmonic corrections increase the Fe2+ reorganization energy by 10.1% and decrease that for Fe3+ by 8.6%, yielding a net increase (1.5%) which is smaller than the separate terms by an order of magnitude.In the case of electrode kinetics, anharmonicity would be expected to play a greater role. N. Sanderghd, J. Ulstrup and J . Jortner, Chem. Phys., 1976, 17, 417. J . A. Jafri, J. Logan and M. D. Newton, Zsr. J . Chem., 1980, 19, 340. The data reported here are based on the calculations of Jafri et ul.’ but were not included in their paper. Prof. M. J. Weaver (Purdue University, West Lafayette) said: Savkant and Tessier have provided convincing evidence that demonstrates the approximate validity of the harmonic-oscillator model for electrode reactions involving organic- molecule-radical-anion redox couples.They note that these results contrast those obtained for the Cr:,+/2+ redox couple at the mercury-aqueous-solution interface (where “ aq ” denotes the aquo ligand).lV2 For the latter system, the variation of the transfer coefficient with electrode potential has been found to be negligibly small at cathodic overpotentials and yet markedly larger than predicted at anodic over- potentials.’Y2 They suggest that these results may be due to the involvement of an inner-sphere mechanism, at least for Cri; electro-oxidation. Although reasonable, the available evidence does not support this assertion. Essentially identical results have also been obtained for Eu:,+/~+ and V2,+/2+ redox couples;* the former couple seems especially unlikely to engage in inner-sphere reaction pathways.Moreover, we have recently shown that the marked deviations noted between the observed driving-force dependence of the rate constant for the electro-oxidation of Cri,+, Eui,+, and Vi,+ and the corresponding theoretical predictions are paralleled by comparable deviations seen for the homogeneous oxidation of these and other aquo couples by a variety of outer-sphere ~ x i d a n t s . ~ . ~ Although such discrepancies may be due to the presence of unfavourable work term^,^ they appear to arise at least in part from the presence of electron-transfer barriers that are strongly anharm~nic.~ We have specu- lated that the source of asymmetry in the free-energy barriers for these couples lies in the differences in the short-range solvent structure surrounding the reduced and oxidized com~lexes.~*~ These structural changes create a substantial asymmetry in the potential-energy surfaces even at small free-energy driving f o ~ c e s .~ Such specific solvation effects are likely to be much milder for the systems studied by Savkant and Tessier in view of the smaller and more delocalized charges carried by the redox couples and the use of aprotic media. M. J. Weaver and F. C . Anson, J. Phys. Chem., 1976, 80, 1861. P. D. Tyma and M. J. Weaver, J . Electroanal. Chem., 1980, 111, 195. M. J .Weaver and E. L. Yee, Znorg. Chem., 1980, 19, 1936. M. J. Weaver and J. T. Hupp, A.C.S. Symp. Ser., 1982, 198, 181. Prof. J. M. SavCant (University of Paris) said : I thank Drs. Weaver, Ulstrup and Newton for their comments and turn also to Prof. Conway’s second point.102 GENERAL DISCUSSION When contrasting our results concerning the validity of the harmonic model for electron transfer involving organic-molecule radical-anion couples with those ob- tained by Weaver et al. with chromium complexes we noted that inner-sphere re- organization plays a much more important role in the second case than in the first. We did not suggest the involvement of an inner-sphere mechanism as the only possible reason for the observed difference. Rather we raised the question whether this difference reflects the involvement of an inner-sphere mechanism or the anharmonicity of inner sphere reorganization in the framework of an outer-sphere mechanism.The first of these possibilities should be taken into account since the water ligands can exchange quite rapidly, as noted by Aoyagui et aZ.l The Cr-OH, bonds'stretch upon reduction of Cr"' to Cr", the increase in bond length being particularly large for the two axial H20 molecules (see table 1 in the paper by Sutin et al.). Taking into account the anharmonicity effects in the framework of an outer-sphere mechanism it is thus expected that the Tafel plots will be flatter than predicted for the harmonic model for the reduction and more curved for the oxidation. As noted by Prof. Con- way and Dr.Ulstrup this also should be the case for an inner-sphere mechanism for the reduction process. It does not thus appear possible to distinguish between the two possibilities on these bases. In the absence of specific interaction between the chromium and the electrode material (which appears to be the case) the inner-sphere mechanism differs from the outer sphere mechanism by the involvement, as ligands, of water molecules strongly oriented by the proximity to the electrode surface. This is more likely to be the case on the cathodic side than on the anodic side. K. Suga, H. Mizota, Y . Kanzaki and S. Aoyagui, J. Electroanal. Chem., 1975,41, 313. N . C. Sondergiird, J. Ulstrup and J. Jortner, Chem. Phys., 1976, 17, 417. Prof. W. J. Albery (Imperial College, London) said: Prof. SavCant has used two different solvents in his work.Besides the variation of c( with potential, I would be interested to know if he has any further results on the systematic variation of the reorganisation energy Ro with the nature of the solvent. Prof. J. M. Savkant (University of Paris) said: We used acetonitrile (AN) and dimethylformamide (DMF). Electron transfer is systematically slower in AN than in DMF. The difference is, however, not very large. Ro values are 6-8% larger for AN than for DMF (see the data for t-nitrobutane and nitrodurene in tables 1 and 2 of our paper). Estimation of the solvent reorganization factor, Ro, according to the Marcus model predicts that it should be larger by ca. 1304 for AN than for DMF, assuming that the radius of the reacting particle and the distance between the electrode and the reaction site are the same in both solvents.This is not a bad agreement between theory and experimental results. The investigation of the solvent effect on 2, was, however, far from systematic. A much larger number of systems should be tested in this respect before reaching sound conclusions. On the other hand, the possible effect of residual water, which may not be the same when passing from one solvent to the other, should be taken into account. It was indeed observed that pur- posely added water slows down the electron-transfer reaction (see table l in our paper). There is ample evidence that water may complex anion radicals in aprotic so1vents.l The effect of water on the electron-transfer rate may then be viewed as that of a complexation of the initially formed anion radicals shifting anodically the standard potential or, more probably, as interfering directly in the medium reorganiz- ation accompanying the electron-transfer reaction.' C. Amatore, J. Pinson and J. M. SavCant, J, Electroanal. Chem., 1982, 139, 183.GENERAL DISCUSSION 103 Prof. R. A. Marcus (California Institute of Technology, Pasadena) said : Prof. Gerischer asked me informally about the slope of 0.5. Even for harmonic potentials a deviation from a Brarnsted or a Tafel slope of 0.5 at AG* = 0 (or E - E" = 0) can occur when there is a change in vibration frequencies. For example, we estimated a deviation of ca. 0.04 from the 0.5 value when the force constants in one reactant change by a factor of two as a result of reaction.' In addition there can, of course, be anharmonic contributions. Regarding the an- harmonic possibility in solvation free energies, these are certainly possible (although I am impressed with how well the simple harmonic solvation free-energy model seems to work), particularly with the more highly charged ions.Prof. Savkant's result of a Tafel slope of 0.5 indicates little anharmonic solvation effect for the organic molecules studied (small charges). R. A. Marcus, J. Chem. Phys., 1965, 43, 679, appendix IV. Prof. M. Bixon (Tel-Aviv University) said: It is interesting to note that the auto- correlation function for the random electric field in the cavity, derived by Friedman and Newton, can be used as a basic ingredient in the theory of non-adiabatic electron transfer.In fact, their correlation function leads to a rate constant which is practically identical to the classical result of Marcus. The transition probability in the case of a stochastic solvent model, as derived by Efrima and Bixon,' is given by W = u,/:dtr2cos( AZft' T ) exp (- /: dz(t - z)ls(z)) h2 where mil is the energy difference between the final and initial electronic states, with the solvent at the equilibrium initial state. &(t) is defined as where AU,(t) is the instantaneous difference in the interaction energy with the solvent between the initial and final electronic states. On the assumption that t,(t) decays exponentially with a reasonable decay time (not too short), one may perform the integrations in the above expression to obtain = q== &)I+ exp [ - 2h2&(0) AEi2f 1 ' Substituting the value 2EskT/h2 for &(O) results in Marcus's classical expression for the transition probability. The work of Friedman and Newton (FN) provides an alternative, independent derivation of the transition probability.Let us begin by writing AU, = eR * G where e is the electron charge, R is the vector distance between the two centres and G is the instantaneous electric field in the cavity. h2cs(0) = {(AUJ2) = e2R2(G2cos28) == $e2R2(G2) The equal time correlation is given by which according to F N [eqn ( 6 ) ] , and under the assumption that the cavity radius is also R, gives 2e2kT E, - E,, FN 2 E s + &op'104 GENERAL DISCUSSION Marcus' result for the case of two spherical reactants of equal diameters, R, at contact, is obtained if one uses the following expression.R The ratio between the two expressions is For water at 25 "C this ratio is practically one (158/156). Corrections due to the different shapes of the cavity would be of order unity. S. Efrima and M. Bixon, J. Chem. Phys., 1979,70, 3531. Dr. J. F. Holzwarth (Fritz-Haber-Institut, Berlin) said : The system Prof. Friedman presented, Fe(H,O)~+/Fe(H,O)~+, seems to contain a problem in so far as Fe(H,O);+ is unstable in aqueous solutions at pH > 1 . If one compares his theoretical consider- ations concerning the hexaquo complex of Fellr with experimental results, one always has high concentrations of anions (from the acid used) present. Those anions are very likely to be involved in the transition state because of the five positive charges.We know from our e.t. experiments using positively charged substitution-inert transition-metal complexes like Fe(bi~y)$+/~ + that the anions do not change the e.t. rate constants very much if one uses a series like F-,Cl-,Br-,ClOT ; however, we believe that this is mainly due to effects which are compensating each other. This does not mean that the anions at concentration of 0.1 mol dm-3 or more are not involved in the transition state. What I would like to know is what Prof. Friedman thinks about the influence of the anions? These are certainly involved in the tran- sition state and in the case of the series mentioned above are strongly complexing Fell1 except for ClO;, but this is very difficult to keep free of C1-.Prof. H. L. Friedman (State University of New York, Stony Brook) said : The anions and their interactions with the Fe2+ and Fe3+ ions are explicitly accounted for in our theory. We proceed from a model in which the energy of any configuration of ions is approximated as a sum of solvent averaged ion-ion pair interactions, but we do not neglect any species pairs. Thus we are able to see what the model gives for the three- point Fe2+, Fe3+, C10; correlations, as exemplified in fig. 9 of the report by Tembe et al.' The model calculations do not imply that a close Fe2+, Fe3+ pair is imbedded in a cluster of many C10; ions. While pairwise additivity of the configurational energy at the McMillan-Mayer (i.e. solvent-averaged) level is only an approximation, we are not aware of any ionic solution data that have been shown to be outside of the scope of theories based on such models. B.L. Tembe, H. L. Friedman and M. D. Newton, J. Chem. Phys., 1982,76, 1490. Prof. R. A. Marcus (California Institute of Technology, Pasadena) said : How large was the steric factor caused by the interlocking of the Fe(H20)z+ and Fe(H20)2+ at an r where r2g23(r)k23(r) was a maximum? What was the net quantitative effect on the bimolecular rate constant k ? Prof. H. L. Friedman (State University of New York, Stony Brook) said: In the range 4.6 < r / A < 6.9, where r is the distance between the centres of the two hexa-aquo com-GENERAL DISCUSSION 105 plexes, the complexes lose some rotational freedom.This effect is represented in our models for the solvent-averaged G23(~) potential by the term -kT In 323, where the " switching function " s23 varies from 0 at 4.6 A to 1 at 6.9 A (and greater).l With a cubic switching function one can see from our ' fig. 6(b) that at r = 5 A this term in the potential is ca. 2kT, corresponding to a reduction in g23(r) by a factor e2. The next part of Prof. Marcus' question, about the quantitative effect of the steric factor upon the rate constant, cannot be answered without extending the computations already made; an expensive proposition. What we do know ' is that a rather drastic change in the switching function ~23(~) has little effect upon the computed rate con- stant k (i.e. our k23). This statement is based on studies ' in which we " tune " the models (by adjusting Ai and Rij for Gurney and Granular models, respectively) to fit the relevant thermodynamic and spin relaxation data before calculating k23.Evi- dently this tuning process tends to compensate changes in the model solvent-averaged pair potentials gij. €3. L. Tembe, H. L. Friedman and M. D. Newton, J. Chem. Phys., 1982, 76, 1490. Dr. D. R. Rosseinsky (Exeter University) said: Electron transfer in solids is closely relevant to questions about reactant orientation as posed in the paper by Friedman and Newton. Dielectric relaxometry is shown to give the MnOp + MnOi- electron-transfer rate v in the mixed-valence solid K,(MnO,),. Internal con- sistency is established by the accord demonstrated with the independently measured d.c.conductivity 0 via the phenomenological relationship,2 o = (ne2a2/6kT)v, where n is the density of donors (manganate), a the transfer distance. The first- order rate constant vAQ for the aqueous reaction of the anions is obtained from the observed bimolecular rate constant kAQ by writing kAQ = KvAQ where K the juxtaposition constant is closely enough 0.1 dm3 mol-I, from 4m26rexp (- W/kT) with W the contact Coulomb interaction (symbols otherwise as Sutin's). The values of v at 298 K are compared in table 1, with the corresponding activation energies (that for AQ being corrected for the temperature dependence of W). The striking corres- pondence of parameters strongly suggests that the same process is being monitored in all three measurements. This means that the solid-state configuration of transfer centres is strongly indicative of what pertains in solution. To summarise, the MnOa and MnOZ- dispositions allow line of sight between the Mn centres through confront- ing staggered tetrahedral faces of the anions. Satisfactory prediction of o can thus be achieved using the outer-sphere Marcus formulation of inner (bond-length) and outer (polarisation) reorganisation energies. D.R. Rosseinsky and J. S. Tonge, J. Chem. Soc., Faraday Trans. I , 1982, 78, 3595. D. R. Rosseinsky, J. A. Stephan and J. S. Tonge, J. Chem. SOC., Faraduy Trans. I , 1981,77, 1719. J. C. Sheppard and A. C. Wahl, J. Am. Chem. SOC., 1957,79, 1020. D. R. Rosseinsky, J . Chem. SOC., Dalton Trans., 1979, 731. TABLE 1 .-VALUES OF 25 "C ELECTRON TRANSFER FREQUENCY (V/HZ) FROM AQUEOUS-SOLUTION RATE CONSTANT, FROM D.C.CONDUCTIVITY, AND FROM DIELECTRIC RELAXOMETRY, AND THE CORRESPONDING ACTIVATION ENERGIES (E/kJ mol-I), FOR MnOh/MnOz- VAQ 32 000 I'COND 6600 VDIEL 6785 EAQ 46 i 4 ECOND 46 f 1 ; 49f7" EDIEL 51 & 5 " 'Same sample.106 GENERAL DISCUSSION Dr. D. R. Rosseinsky and Dr. P. Tucker (Exeter University) (communicated): The K3(Mn04), structure shows the tetrahedral disposition of one type of anion about the other, the positions being equivalent. There are two types of counter-cation. The stereogram (fig. 5) shows alternative Mn-0-0-Mn or Mn-0-K-0-Mn routes for electron transfer, which represent longer, indirect, alternative or accompany- ing possibilities for the process. The stereo diagram is that of isomorphous ' Ba3(P04)* but it represents fact in contrast with theory, conjecture, or speculation; only the notable exceptions in solution chemistry, of demonstrable bridging, estab- lish reactant dispositions so clearly.V. G. Erenburg, V. V. Boldyrev, L. D. Anikina and Yu. I. Mikhailov, J. Struct. Chem., (USSR), 1968, 9, 461. H. Taube, H. Myers and R. L. Rich, J. Am. Chem. SOC., 1953, 75, 4118. H. Taube and H. Myers, J. Am. Chem. SOC., 1954,76, 2103. / I \ 0 y o FIG. 5.-Stereo diagram of the K3(Mn0& isomorph structure. Prof. J. Jortner (Tel Aviu University) said : The interesting results of Dr. Rosseinsky on electron transfer in the mixed-valence solid K3(Mn04), provide compelling evidence that electron transfer in solution and in the solid state can be described within a uni- fied theoretical framework. The close correspondence between the activation energies for MnOT/MnOi- electron transfer in the two systems indicates that a main contri- bution originates essentially from configurational changes within the polyatomic ions, as the contribution of the outer-sphere polarization energy is expected to be somewhat different in the solid and in aqueous solution.Charge transport in K3(Mn04), provides an example for small-polaron motion in an inorganic solid.' At high tem-GENERAL DISCUSSION 107 peratures, studied by Rosseinsky, small-polaron motion is an activated process, involving thermally assisted electron hopping from one site to another, with the elec- tron mean free path being equal to the lattice spacing.' At low temperatures, the small-polaron motion is expected to be " coherent," with the mean free path consider- ably exceeding the lattice spacing.' Such "coherent " motion is expected to be characterized by a negative activation energy.' It will be interesting to explore low- temperature conductivity in this system.T. Holstein, Ann. N. Y. Acad. Sci., 1959, 8, 343. Dr. D. R. Rosseinsky (Exeter University) said : Prof. Jortner's comment, that the similarity of aqueous and solid-state rates for MnOr/MnO;- seems to preclude application of Marcus's polarisation treatment, can be countered by the observation that the high-frequency (electronic-only) permittivities of the two phases are not very disparate, whereas the terms involving the static permittivities are 1/78 and ca.1/170 for solution and solid, respectively, again differing only slightly in effect. While K3(Mn0,), can be deemed, as by Jortner, to provide an interesting inorganic example of the small-polaron mechanism, because this terminology might evoke implications of unnecessary subtlety, we ' have preferred to coin the term " site trans- fer conductivity " for this particular phenomenon in view of the clear identifiability of the transfer sites involved, to replace both " polaron " and " hopping '' as being not specific enough. The suggestion that we should at low enough temperature see negative activation energies will depend on obtaining a better crystalline quality than is at present acces- sible. D. R. Rosseinsky, J. A. Stephan and J. S. Tonge, J.Chem. Soc., Faraday Trans. I , 1981,77, 1719. Dr. R. J. P. Williams (Oxford University) said: Some years ago ' p 2 we did a great deal of work on the electron conductivity of inorganic mixed-valence solids. We observed that although hop conductivity was not very effective it could be readily studied. The great advantages of these solid-state systems are that unlike solution reactions the reactants have a known geometric relationship both of distance and orientation. We found that (i) complex hexacyanides gave very poor electron-transfer systems although the metal-metal distances are very short in comparison with bio- logical systems; (ii) some complex chlorides give good conductors ; (iii) oxide lattices with outer-sphere complexes were very poor conductors but blue asbestos provided a very interesting high-temperature system; (iv) the salt [Fe(phenan),12+ IrClZ- was a very poor conductor.Given the considerable advances in understanding of mixed- valence systems and the great improvement in apparatus I believe that mixed-valence crystals will greatly help in the understanding of solution and especially biological electron transfer. In effect they provide the rigid bridges often discussed in solution chemistry . Has Prof. Friedman looked at the effect of D20 on the Fe"/Fe'" exchange? I would like to ask a quite separate question. D. Culpin, P. Day, P. R. Edwards and R. J. P. Williams, J. Chem. Soc. A , 1968, 1155. R. J. P. Williams, in Current Topics in Bio-energetics, ed. A. Sanadi (Academic Press, New York, 1969), VOI. 3, pp.80-156. Dr. D. R. Rosseinsky, Mr. T. E. Booty and Dr. J. S. Tonge (Exeter University) (communicated): In reply to the comment of Dr. Williams that electron transfer in solids was long ago studied by himself and Day, and their coworkers, indeed we have often used their studies as the starting point for our novel application of dielectric108 GENERAL DISCUSSION relaxometry to the direct measurement of electron-transfer rates, inter alia in Prussian B1ue.l We have furthermore sought to study the calculation of thermal electron- transfer rate from the charge-transfer spectrum (a subject curiously missing among others in this Discussion) having some success with Prussian Blue,3 but none for the other mixed-valence iron system just referred to by Dr. Williams, blue asbestos or crocidolite.Our conductivity (ca. lo-' IR-' cm-') yields a frequency some lo4 fold higher than is predictable from the s p e c t ~ u m ; ~ the two observations were made on different samples of material. The high conductivity (accompanied by slight deviation from ohmicity) is perhaps ascribable to an accompanying dominant protonic con- ductivity, but the material invites further study, as do many of the systems so profitably pioneered by the Oxford school. J. S. Tonge, Ph.D. Thesis (Exeter University, 1982). N. S. Hush, Electrochim. Acta, 1968, 13, 1005. S. J. England, P. Kathirgamanathan and D. R. Rosseinsky, J. Chem. SOC., Chem. Commun., 1980,840. G. C. Allen, Transition Metal Chem., 1979, 1, 143. Dr. M. D. Newton (Brookhaven NationaI Laboratory) said: The kinetic isotope effect for the aqueous Fe2+/Fe3+ exchange reaction (kH/kD) is ca. 2.' We have ex- plotted possible sources of this effect, employing the standard non-adiabatic form- alism 23 in conjuction with appropriate ab initio molecular-orbital calculations.TABLE 2.-vARIATION OF YOH WITH CHARGE AND HYDROGEN-BONDING ENVIRONMENT, AND RELATED ISOTOPE EFFECTS a rodA model 3 + 2 1 AriA k FXlk iix (A) frozen a 0.962 0.962 0. OOO 1.1 (B) relaxed rOH (1) no second-shell waters 0.974 0.997 0.023 1.4 (2) second-shell waters 1.003 1.076 0.073 2.8 ' (fig. 6 ) a Based on free water molecule; the corresponding relative values of krX are 1.0, 0.6, and 0.04, respectively. Models in which the inner-sphere reorganization is limited to the FeO symmetric breathing modes yield an isotope effect of only ca.1.1 Since it seemed likely that the OH bond length and stretching frequency for inner-shell water molecules might depend appreciably on the Fe ionic charge (+2 or +3),5 we carried out model ab initio calculations for Mfl+(H20)(H20), clusters, where Mfl+ was rep- resented as a oint charge, separated from the inner-shell oxygen by 2.13 8, (for n = 2) and by 1.99 if (for n = 3),4 and where hydrogen-bonded second-shell water molecules were either absent (m = 0) or present (m = 2), as illustrated in fig. 6. Optimization of the inner-sphere OH distances led to the values summarized in table 2, along with the corresponding calculated values of rate constant and kinetic isotope effect. The kinetic results are based on inner-sphere OH stretching frequencies inferred from the frequency-bond length correlations reported in ref.(6). From table 2 we see that inclusion of the OH stretching mode in the inner-sphere reorganization is potentially capable of accounting for the observed isotope effect ' provided that hydrogen bonding to the second shell of water molecules is included: i.e. the calculated Ar value of 0.023 A in the absence of additional solvent, which yields an isotope effect of 1.4, is (see also table 2).GENERAL DISCUSSION 109 greatly enhanced (Ar = 0.073) when hydrogen-bonded second shell waters are in- cluded, with a correspondingly greater isotope effect of ca. 2.8. The sensitivity of Ar to charge and hydrogen bonding is similar to the case of H,O and H,O+ : the calculated OH bond lengths in H,O+ are only 0.017 A greater than for H,O when the isolated species are considered; if the OH bonds are hydrogen-bonded /' / \ \ FIG.6.-Structural model for inner-sphere water molecule hydrogen-bonded to two second-shell waters. The cluster has C,, symmetry, with M-0 coinciding with the two-fold axis and all atoms but the four peripheral hydrogen atoms lying in a reflection plane. The second shell waters were placed in a plane perpendicular to the M-0 vector; OH - - - - 0 bonds are linear, with the 0 * * . 0 separation fixed at 2.80 A. to additional waters, the H,O bond lengths increase by (0.01 A, whereas the H20+ bonds stretch out an additional 0.029 A.6 It should be emphasized that the data of table 2 constitute only preliminary results based on very crude models.Furthermore, the Ar values and isotope effects should be taken as upper limits since the use of point charges confined to the site of the Fe atom and optimal hydrogen-bonded geometries tends to exaggerate the degree of differential perturbation of the inner-sphere OH bonds with respect to charge state. Never- theless, the role of OH bonds in activated electron transfer would seem to deserve careful attention in future studies. It should, of course, be borne in mind that the non-zero ArOH values which may account for the isotope effect also reduce the mag- nitudes of the individual rate constants (kH and kD) relative to those based on Ar = 0, as noted in table 2 (footnote 6). I would also like to note another result of our ab initio calculations, namely the distance dependence of the electronic matrix element for the aqueous Fe2+/Fe3+ exchange.2*' While an exponential falloff of the form exp (-M.Y) is undoubtedly the major factor, the complete expression can in general be expected to include an addi- tional factor in the form of a polynomial in r, the precise details of which depend on the nature of the atomic orbitals involved in the primary electron exchange process. We have obtained values of 5c and estimated standard deviations (a) corresponding to least-squares fits of the calculated data to various assumed radial forms over the range, rFe... Fe % 5-8 A (the data quoted below refer to calculations for the face-to-face approach including ligand valence electrons). For the aqueous Fe2+I3+ case, where 3d-3d overlap is expected to be of major the simple exponential form which yields M.= 1.2 A-' is found to be less satisfactory (a z 3 cm-') than, for example, a fifth-order polynomial of the form (a + br)r4, which yields the same value of cc but with CT only 0.3 cm-'. Because of uncertainty as to the most appropriate polynomial form, there is in general corresponding uncertainty in the " best ''110 GENERAL DISCUSSION exponential parameter, a. However, the lowest 0 values (-: 1 cm-’) are associated with a values near 1 A-1. J. Hudis and R. W. Dodson, J. Am. Chem. SOC., 1956,75,911. M. D. Newton, Int. J. Quantum Chem. Symp., 1980, 14, 363. E. Buhks, M. Bixon, J. Jortner and G. Navon, J. Phys. Chern., 1981, 85, 3759. M. D. Newton, in Tunneling in Biological Systems, ed.B. Chance, D. C. Devault, H. Fruen- felder, J. R. Schrieffer and N. Sutin (Academic Press, New York, 1979), p. 226. J. Bigeleisen, J . Chem. Phys., 1960, 32, 1583. M. D. Newton, J. Chem. Phys., 1977,67, 12. M. D. Newton, Adv. Chem. Ser., 1982, 198, 255. ’ J. Logan and M. D. Newton, J . Chem. Phys., in press, Prof. J. Jortner (Tel Aviv University) said: The calculations of the electronic coupling matrix element, V, play a central role in the theory of non-adiabatic electron transfer. Authoritative many-electron calculations were conducted by Dr. Newton for the inorganic Fe(H,O)~’/Fe(H,O)~+ system. For electron transfer between large aromatic molecules, Katz et al., have conducted a calculation of V using many- electron wavefunctions, including both (one-electron) Coulomb and (two-electron) exchange contributions. The distance, R, dependence of V for a parallel pair of aromatic molecules, i.e.naphthalene--naphthalene and anthracene--anthracene Y can be expressed in the exponential form Aexp( - aR), where a FS 1.1 A-1.3 The value of the exponent a, originating from these many-electron calculations, is in excellent agreement with the value recommended by Prof. Marcus for biophysical systems. M. D. Newton, Int. J. Quantum Chem., 1980, 14, 363. S. A. Rice and J. Jortner, in Physics of Solids at High Pressure, ed. C . T. Tonizuka and R. M. Emrich (Academic Press, New York, 1975). E. Buhks and J. Jortner, FEBS Lett., 1980, 109, 117. Prof. R. A. Marcus (California Institute of Technology, Pasadena) said: The dependence of rate constant on distance is of particular interest in biological electron transfer, e.g.between haems at large separation distances (r E 15 A), where there is intervening material rather than a vacuum. What results does Dr. Newton find for the effect of intervening material on dependence of electron-transfer rate constant k(r) on separation distance r? Dr, M. D. Newton (Brookhaven National Laboratory) said: First, I should em- phasize that for the range of Fe-Fe separations relevant to the aqueous electron exchange reaction there is no room for an “ intervening medium ” aside from the inner-shell water molecules, and thus one has a situation quite different from that relevant to larger-range interactions typical of biological systems. The only way in which we can estimate a “ medium effect’’’ for the range of distance which we included in our ab initio matrix element calculations (ca.5-8 A) is to compare the results of crystal-field (inner-shell ligands included as point charges only) and full ligand valence- electron calculations.lg2 In this sense [with HAB assumed proportional to exp( - ar)] we find that a larger value of a is obtained for the “ vacuum ” case (i.e. our crystal- field model) as compared with the case where ligand electrons are present: 2.1 and 1.2 A-’ respectively. These results, which pertain to the preferred face-to-face orient- ation of reactants,’ are qualitatively in accord with the medium effect discussed by Marcus and S i d e r ~ . ~ It should also be noted that the apex-to-apex orientation of the reactants, which places two ligands directly between the Fe atoms, yields a larger value of a (ca. 2.4 A-l, based on calculations including ligand electrons) than the face-to-face orientation.GENERAL DISCUSSION 111 H. L. Friedman and M. D. Newton, Faraday Discuss. Chem. Soc., 1982, 74, 73. J. Logan and M. D. Newton, J. Chem. Phys., in press. R. A. Marcus and P. Siders, J. Phys. Chem., 1982, 86, 622. Prof. J. Jortner (Tel Aviv University) said: It has been demonstrated by Buhks et al.’ that the deuteration of the ligands in the first coordination layer is expected to provide a small, normal, isotope effect on electron transfer, which originates from the changes in the ion-ligand frequencies. It has been pointed out that an additional contribution to the deuterium isotope effect may originate from small changes in the band distances within the ligand, which are induced by the change of the charge of the cation. This effect was addressed by Dr. Newton. To assess the contribution of bond-length changes within the ligands on the isotope effect it would be interesting to compare the isotope effects originating from the replacement of the H2160 molecule by D2I60 with the isotope effect due to the replacement of H2160 molecules by H,180 molecules in the first coordination layer. In the latter case only ion-ligand frequencies will be modified, while in the former case both modification of ion-ligand frequencies and changes in intra-ligand bond length will contribute to the isotope effect. E. Buhks, M. Bixon and J. Jortner, J . Phys. Chem., 1981, 85, 3763. Prof. W. J. Albery (Imperial College, London) said : The use of the solvent isotope effect to probe the details of electron-transfer reactions is to be recommended. However, I am a little surprised at Dr. Newton’s calculations of the fractionation in the secondary solvation shell of cations. The fractionation on L30 +, where each site has a factor I = 0.69, is well estab1ished;I there then appears to be little fractionation in the next shell of water molecules. Furthermore from Salomaa’s results fraction- ation around cations such as Li+, Na+ or K+ appears to be negligible.’ So although there may be fractionation on water ligands, attached to transition-metal ions, I would not expect to find significant fractionation in the secondary solvation shell of such cations. V. Gold, Adv. Phys. Org. Chem., 1969, 7 , 259. W. J. Albery, in Proton Transfer Reactions, ed. E. F. Caldin and V. Gold (Chapman and Hall, London, 1975), p. 282. Dr. M. D. Newton (Brookhaven National Laboratory) (communicated) : Let me emphasize that while my results relevant to the kinetic isotope effect for aqueous Fe2+t3+ exchange appear to indicate the importance of second shell water molecules, the primary role of the latter is to serve as proton acceptors for OH - - 0 hydrogen bonds with the inner-sphere waters; my tentative model does not require apprecizble perturbation of the second-shell OH-bonds relative to those of the bulk liquid and thus does not imply any significant H/D fractionation in the second shell. Analogous considerations seem to apply to H30+ and H20, since the relatively small differences between the OH bonds of the isolated species are strongly enhanced when the monomers are hydrogen-bonded to additional water molecules [see my pre- ceding comment and ref. (l)], in qualitative accord with the significant fractionation observed for L,O+ in aqueous solution ( l z 0.69).2 M. D. Newton, J . Chem. Phys., 1977, 67, 12. ’ V. Gold, Adv. Phys. Org. Chem., 1969, 7, 259.

ISSN:0301-7249    

DOI:10.1039/DC9827400083

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 74, 113-127 The Role of Inner-sphere Configuration Changes in Electron-exchange Reactions of Metal Complexes BY BRUCE S. BRUNSCHWIG, CAROL CREUTZ, DONAL H. MACARTNEY, T-K. SHAM AND NORMAN SUTIN Department of Chemistry, Brookhaven National Laboratory, Upton, New York 11973, U.S.A. Receiued 19th May, 1982 Extended X-ray absorption fine structure (EXAFS) techniques have been used to determine the differences in the metal-ligand bond distances Ado for the following couples (couple, Ado/& Fe(H20)i +13 +, 0.13 ; Fe(CN)d-I3-, 0.01 ; Fe(phen): + I 3 + , 0.00; Co(NH& + I 3 +, 0.22; Co(bpy)3 +D+, -0.02; C0(phen)5+'~+, 0.19; Ru(H20)%+13+, 0.08; Ru(NH3)%+13+, 0.04. The Cr(H,O)$+ ion has four short and two long Cr-0 bonds (2.07 and 2.30 A, respectively) and the average Ad, for the Cr(H20)g+/'+ couple is 0.20 A.The rate constant for the Fe(H,O)~+/Fe(H,O)~+ exchange reaction is 1.1 dm' mol-' s-' at 25 "C and 0.1 mol dm-3 ionic strength. The exchange rate is independent of the cation (Li+/Na+) used to maintain constant ionic strength. Exchange rate constants for twelve reactions that span fifteen orders of magnitude are shown to be in excellent agreement with the semi-classical model. The simplest bimolecular electron-transfer reaction, the exchange reaction, i.e. MLg+ + ML:+ -+ML$+ + MLg+ has been extensively studied both experimentally ' v 2 and theoretically.2-11 In aqueous media the bimolecular rate constants for the electron-exchange reactions of metal complexes span at least 15 orders of magnitude.These rate variations are under- stood in a general way in terms of a semi-classical model 2*9~11 in which the observed bimolecular rate constant kobs is the product of a pre-equilibrium constant K A , an effective nuclear frequency vn, and electronic and nuclear factors and K,, respectively kobs = KAVnKeIKn* (1) For electron transfer to occur, the two reactants must be brought into close proximity; K A is the equilibrium constant for the formation of the precursor complex from the separated reactants MLg+ + ML$+-MLg+ .-*ML:+ (2) and its value may be obtained from where the reaction thickness dr is typically ca. 0.8 A 2*9 and w(r) is the work required to bring the two reactants to the separation distance r. If this work is predominantly electrostatic and most of the reaction occurs at r = (a, + a,), where a, and a, are the radii of the two reactants, then w(r) is given by the Debye-Huckel expression114 ELECTRON-EXCHANGE REACTIONS where z2 and z, are the charges on the two reactants, D, is the static dielectric constant of the medium and /? = (8dVe2/1000 D,kT)”’.Although more complex work ex- pressions are available, eqn (4) is adequate for the present purpose. For exchange reactions the nuclear factor contains both solvent and inner-sphere contributions Kn = exp[-(AGL + AGi*,)/RT] ( 5 ) where r, is the inner-sphere nuclear tunnelling factor (frequently set equal to unity),ll Do, is the optical dielectric constant of the medium (equal to the square of the refrac- tive index), fi = 2fif3/cf2 +A) is a reduced force constant for the ith inner-sphere vibration defined in terms of the normal-mode force constants of the two oxidation states, (dz - d:)i = Ad,, is the corresponding difference in the equilibrium bond distances in the two oxidation states, and the summation is over all the intramolecular vibrations.The nuclear frequency appearing in eqn (1) is related to the solvent and inner-sphere reorganization energies by For water as solvent, VOut z 30 cm-’ or vout = 0.9 x 10” s-’ while metal-ligand stretching frequencies are typically 300-500 cm-’ [(9-15) x 10’’ s-’1 and intraligand frequencies 1000-3000 cm-’ [(3-9) x 1013 s-l].’l Within the Landau-Zener framework the electronic factor for an exchange reac- tion is given by ’ where vel is an electronic frequency that is related to the electronic coupling matrix element HAB by It has been recognized ’ that differences in K , arising from inner-shell changes may give rise to exchange rate variations of 9 orders of magnitude and that those arising from changes in the outer-shell barrier span at least 5 orders of magnitude.Systematic data have been absent for many systems, however, so the magnitudes of these factors have remained unknown for many couples. Since little progress can be made without knowledge of the metal-ligand bond lengths for both partners in a redox couple we have begun a systematic investigation of metal-ligand bond lengths in the crystalline state and in solution. Here we report bond-length data obtained by the technique of extended X-ray absorption fine structure (EXAFS).l2-I4 Rates determined for the Fe(H,O); +/Fe(H2O);+ exchange in aqueous LiClO, + HC104 at low ionic strength are also reported.These rates, as well as the rate constants for a number of other exchange reactions between metal complexes in water, are interpreted in terms of the semi-classical model outlined above and it is concluded that, with few exceptions, excellent agreement between theory and experiment is observed.B . s. BRUNSCHWIG et al. 115 EXPERIMENTAL MATERIALS For the EXAFS studies, the complexes [C~(bpy),]Cl,~' [C~(bpy),]Cl~,'~ [Co(~hen)~]- (C104),,17 Ru(H20):+,l8 Ru(H,O);+,~* [CO(NH,)~]C~,,'~ [RU(NH~)~]B~~," [Ru(NH3)6]C12 2o and [Fe(phen)3](C104)321 were prepared by literature methods and characterized spectro- photometrically (bpy = bipyridine, phen = phenanthroline).Commercial samples of (NH4)2Fe(S04)2.6H20, Fe(N03)3-9H20, k[Fe(CN),], K3[Fe(CN)d, [Fe(phen)~ICl2, [Ru- (NH3),JClJ, [Co(NH3),]CI3, Cr(C104)3 and [Co(en),]CI, were used as purchased (en = ethylenediamine). The [C~(phen)~]Cl, was prepared by mixing C0C12 and phen in a 1 : 3.3 ratio in ethanol under argon. The Cr(C1O4), was dissolved in 1 mol dm-3 HC104. The Cr(H,O);+ was prepared by almalgamated zinc reduction of the Cr(H,O)a+ solution. [The presence of Zn2+ in the Cr(H,O);+ solution does not interfere with the EXAFS spectroscopy since the threshold energy for zinc is 3.7 keV higher than that for chromium.] The K4Fe- (CN), was dissolved in 0.1 mol dmV3 NaOH, the K3Fe(CN), in H20, the [Ru(NH3),]Cl3 in 0.1 rnol dmV3 &Sod, the [Ru(NH3),]C12 in 0.2 mol dm-, aqueous ammonia, the [Co(phen),]- (C104), in concentrated HC104, and the remaining complexes in water. The concentrations obtained were in the range 0.3-1 rnol dm-,, except those for the ruthenium compounds which were ca.(3-10) x For the exchange measurements iron(irr) perchlorate (G. F. Smith) was purified by recrystallization from HC104. Stock solutions of ferrous perchlorate were prepared by electrolytic reduction of iron(m) in perchloric acid. The labelled iron(1rr) perchlorate used in the exchange studies was prepared by fuming 59FeC13 (New England Nuclear) in HClO, until a negative test for chloride was obtained. Lithium perchlorate was prepared by neutralization of HC104 with Li2C03, followed by several recrystallizations from distilled water.Sodium perchlorate solutions were made from the monohydrate salt (Baker). Analysis of the metal and acid solutions is described in the l i t e ~ a t u r e . ~ ~ . ~ ~ Sodium and lithium perchlorate solutions were analysed by titration with sodium hydroxide of an aliquot which had been passed through a Dowex 50W-X8 (50-100 mesh) ion-exchange column in the H+ form. mol dm-, All solutions gave negative tests for sulphate and chloride. KINETIC MEASUREMENTS The iron(n)/iron(m) electron-exchange measurements were made according to procedures described previo~sly.~~ A circulating water bath was used to thermostat solutions between 0.3 and 25 "C. The rate constant for the exchange reaction, followed by monitoring the increase in the specific activity ( X ) of *Fe", is given by Plots of ln(X, - X ) against time were linear for at least three half-lives.Experimental rate constants at higher acidities were more reproducible (& <5%) and weighted more heavily than those at lower acidities in the least-squares determination of the specific rate parameters. EXAFS MEASUREMENTS X-ray absorption spectra were obtained at Stanford Synchrotron Radiation Laboratory (SSRL) on focused beam lines XI-3 and 11-2. For higher energies ( > 9 keV) the focusing mirror on beam line 11-2 was removed. The spectra were obtained in the absorption mode. Solid samples were ground with Vaseline and held in a Mylar cell; solution samples were prepared in Mylar bags which were then heat sealed. The path lengths of the solutions were adjusted to give the best absorption spectrum.Samples that were oxygen sensitive were loaded under argon. For the ruthenium solutions a cylindrical absorption cell with Mylar windows and a path length of 1-2 cm was used.116 ELECTRON-EXCHANGE REACTIONS EXAFS ANALYSIS The extended X-ray absorption fine structure is the oscillations in the absorption co- eficients at photon energies higher than the absorption edge.24-27 These oscillations are due to the scattering of the outgoing photoelectron wave by the atoms surrounding the absorbing atom. The EXAFS spectrum function is defined as 2 4 9 2 5 with k = 2/2rn(E - Eo) (2n/h) where k is the wave vector of the photoelectron, m is the mass of the electron, E is the energy of the incident photon, Eo is the threshold energy for the photoelectron emission, I.and 1 are the intensities of the incident and transmitted beams, respectively, and po is the structureless background absorption (the absorption due to a " free " atom). The EXAFS function ~ ( k ) was extracted from the measured p values by using a poly- nomial spline fit (usually of order 3 over an interval of ca. 3 A-') to remove the back- g r ~ u n d . ~ ~ , ~ ~ Only the data in the region ca. 4-14 A-' were fitted. The threshold energy Eo was arbitrarily taken as half-way up the edge jump. Short-range single-electron single-scattering theories of EXAFS relate ~ ( k ) to the local structure around the central metal (absorber) atom 24*25*28*29 ~ ( k ) = CAj(k) sin Qj(k) (1 5 ) where Aj(k) and Q,(k) are the amplitude and phase functions, respectively, for scattering by nj identical atoms at a given distance d j from the central atom.The summation extends over all the ligand atoms and the orientations of the scattering systems have been spherically averaged. The amplitude function is defined by A ,(k) = {njSjFj(k) exp( -2k2aj2) exp[ -2dj/dJ(k)]}/kd: where Sj is an experimental scaling factor, F,(k) is the backscattering amplitude from each of the n, neighbouring atoms, oj is the Debye-Waller-type term accounting for thermal vibrations (and static disorder), and exp[- 2d,/Aj(k)] is a damping factor that approximately accounts for inelastic scattering with dj(k) the electron mean-free-path (this last factor is usually set equal to ~ n i t y ) . ~ ~ . ~ ' The phase function for K edge absorption Qj is defined as 2kdj plus twice the central atom phase shift (pea) and the backscatterer phase shift (q0JZ9 (17) Two methods were used to determine n j , dj and oj.In the first 24*25 a Fourier transform of the x(k)k" data (usually n = 3) was taken. A window was then chosen so as to isolate the radial distribution peak of interest in distance space. The data were then back trans- formed into k space to yield filtered ~(k)k" values and trial phase Qj(k) and amplitude Aj(k) functions. The phase functions of two data sets were then analysed to extract the difference in metal-ligand bond lengths. In computing Adj the values of the central atom and back- scatterer phase shifts were assumed to be the same in the two data sets and the threshold energy of one data set was varied until a plot of the difference in the phase functions against k yielded a least-squares straight line with zero intercept; 2Ad, was obtained from the slope of this line.The shift in threshold energy was then used to adjust the appropriate amplitude function. The natural logarithm of the ratio of the two amplitude functions was then plotted as a function of k2 to yield a least-squares straight line with intercept the natural logarithm of the ratio of nj/djz for the two data sets and slope 2A(aj)'. The two data sets used were either for two different oxidation states of the same complex or solution and solid-state data for the same complex. The sensitivity of Adj and of the coordination Qj(k) = 2kdj + 2~)ca(k) + Vj(k) - TC-B . s. BRUNSCHWIG et al.117 number ratio to the Fourier window and to the type of spline used to remove the background was tested. In general the Adj values were not sensitive to these changes. The second method24*25$30 was to fit directly the x(k)kn data using a non-linear least-squares fitting routine and eqn (12)-(17). Teo and Lee 29 have published values for these functions calculated from theory. These can be used to extract dj and o j ; if the scaling factor S j in eqn (16) is known, n, can also be obtained. Data for one oxidation state or sample were used to determine the scaling factor (assuming a value of nj) and then n, was fitted for the other oxidation state or sample. (Thus only ratios of the coordination numbers were obtained in this method also.) The threshold energy Eo was also used as an adjustable para mete^.^^ For the bipyridine and phenanthroline complexes it was necessary to include the u and p carbon atoms as well as the nitrogen atom of each pyridine ring in order to fit the data satisfactorily.In these cases, the scaling factors of all the scattering atoms were set equal and the Debye-Waller factors for the carbon atoms were also assumed equal. In order to do this pca(k), pj(k) and Fj(k) are needed. RESULTS EXAFS MEASUREMENTS Bond lengths and bond-length differences obtained in this study by the EXAFS technique are presented in table 1. The bond-length differences presented in table 1 were calculated using the first method described above. The two methods for analys- ing the EXAFS data yielded bond-length differences that agreed to within kO.01 A; however, the direct comparison of the experimental phase and amplitude functions for two complexes as done in the first method seems preferable since it should result in some cancellation of errors.The coordination numbers were generally within 10% of the expected values except for Cr(H,O)i+. If only one coordination shell is used to analyse the data for this complex, the coordination number is found to be 4.0 0.5. Consequently a second coordination shell was included. The values of d, n and o were poorly defined for this shell; with a value of 0 of 0.09 A for this shell it was only possible to determine d to within 0.05 A and n as <2. The Debye-Waller factors for all of the other compounds were in the range 0.065-0.085 8, with the values for the 2+ ions ca.10% higher than those for the 3+ ions. No significant difference in the bond lengths was found for complexes determined in the solid state and in solution. Bond-length data obtained by X-ray diffraction are also included in table 1. The values obtained by the diffraction and EXAFS techniques are generally in good agree- ment (&0.02 A in Ado). As expected for an analysis based on a single scattering description the largest disagreement is found for ferricyanide ; for this complex EXAFS gives a bond length that is 0.046 8, shorter than that determined by X-ray diffraction. The linear arrangement of iron, carbon and nitrogen atoms in ferri- cyanide (and ferricyanide) requires the use of EXAFS expressions incorporating multiple ~ c a t t e r i n g .~ ~ Since this extension is not well understood it has not been included. However, it is unlikely that neglect of this effect will significantly affect the determination of Ado. The value of Ado determined for the RU(H,O)~+/~+ couple is 0.08 A (EXAFS) compared with 0.09 A (X-ray diffraction). Although the two values are in reasonable agreement the X-ray diffraction value is preferred because the EXAFS measurements were made on relatively dilute solutions resulting in less precise ~ ( k ) values. In addition the Ru(H,O)g+ solutions could have contained small amounts of Ru(H20)i+ due to oxidation by air. Except for this system, the Ado values determined by the EXAFS technique are used in the subsequent discussion.118 ELECTRON-EXCHANGE REACTIONS TABLE 1 .-BOND LENGTHS AND BOND-LENGTH DIFFERENCES OBTAINED BY EXAFS complex AEoa dO(lit.)*/A ref.' dolA AdOd/A lev Cr( H,O)% + Fe(CN)i - Fe(CN)% - Fe(phen)i + Fe(phen),j + Co(NH3); + Co(NH3): + Co(en): + WbPY )3' CO(bPY)i+ Co(phen)$+ Co(phen): + Ru(H,O):+ Ru(H2O)i + Ru(NH~)~+ Ru(N H3): + 1.959(3) 2.123(6) 1.995(4) 1.900(7) 1.926(3) 1.97( 1) 1.973(8) 2.16 1.965(1) 1.955(5) (2.12) (2.14) - - 2.1 22( 1 6) 2.029( 7) 2.144(4) 2.104(4) - 31 32 31 33 33 34 35 36 37 38 39 39 - 40 40 41 41 2.10( 1)e.f 1 .97(1),' 1.98(l)f 1.88( l)f 1.88(1)e-f 1.97( l)f 1.97( l)f 2.19( 3)' 1.97(1)'*f 1.97( 1)'~f 2.09( 1)" 2.12( 1)' 2.1 1 (2)f 1.91(l)f 2.1 l(2)f 2.11(1),' 2.12(l)f 2.03(2)f 2.14(1)'gf 0.09( l)f 0.32(5)f v h 0.13(l)'*f O.Ol(l)f O.OO( 1)f 0.22(2)" O.OO(l)"~f*' - O.o2(l)e O.l9(2)f 0.08(3)f 0.04( 1 ), ' 0.03( l)f a AE, is the difference in the threshold energy of the reduced and oxidized forms of the redox pair as determined from the absorption data.Metal-ligand distances determined by X-ray diffraction measurements on crystals. Literature reference for the X-ray diffraction measurements, * Metal-ligand distance differences determined by the direct comparison of EXAFS phase functions for the two oxidation states (the first method in the text). Distances determined for solids, f Distances determined for solutions. Equatorial distance. Axial distance. Difference between the cobalt-nitrogen distances in the hexa-ammine and trisethylenediamine cobalt(1n) complexes. EXCHANGE MEASUREMENTS The Fe(H,0)i+/3+ electron-exchange rate constants were measured at several acidities over the temperature range 0.3-25.0 "C at an ionic strength of 0.10 mol dm-3 (LiC10.J.The observed second-order rate constants display an inverse dependence on the acid concentration. The acid dependence was reported previously for this system 22 and was attributed to hydrolysis of the Fe(H20)z+ ion Fe(H20):+ K", Fe(H20)50H2+ + H+ Fe(H,O):+ + Fe(H,O);+ kt_ Fe(H2O);+ + Fe(H20)i+ (19) Fe(H20),0H2+ + Fe(H,O)i+ k2_ Fe(H,O);+ + Fe(H20)50H2+. (20) The kinetic data are consistent with a rate law of the form The rate constants k, and k, were derived from a plot of k,, (1 + Kh/[H+]) against l/[H+] using Kh = 2.45 x mol dm-3 at 25 "C and p = 0.1 mol dm-3 and theB. s. BRUNSCHWIC et al. 119 corresponding thermodynamic parameters AHh = 11.5 kcal mol-', ASh = 27 cal K-I m01-l.~~ The following rate constants were obtained at p = 0.1 mol dm-3 (T/"C, k1/dm3 mo1-l s-l, k2/dm3 mo1-l s-'): 0,3,0.20 5 0.02, 380 & 10; 4.7,0.26 If 0.03, 510 & 10; 13.0, 0.59 & 0.08, 806 & 15; 25.0, 1.12 5 0.20, 1360 & 30.At p = 0.55 mol dm-3 and 7.3 "C the results were kl = 1.4 If 0.1, k2 = 1450 & 30 dm3 mo1-I s-l with NaC10, supporting electrolyte [in very good agreement with the results in ref. (22), kl = 1.4, k, = 1430 dm3 mo1-I s-l, obtained under the same conditions] and kl = 1.3 & 0.1, k2 = 1420 & 30 dm3 mold' s-I with LiClO, supporting electro- lyte. An Eyring plot of the p = 0.1 mol dm-3 data yields AH! = 11.1 kcal rnol-', ASi = -21 &- 3 cal K-I mol-1 and AH$ = 6.7 & 0.5 kcal mol-l, AS2$ = -22 & 2 cal K-' mo1-l (obtained by subtracting AHh and ASh from the respective composite Khk2 activation parameters).The agreement of the Fe(H,O)g+/Fe(H,O):+ exchange rate constants derived from experiments in LiClO, and NaClO, at 7.3 "C and p = 0.55 mol dm-3 indicate that negligible medium effects arise from the nature of the cation of the supporting electrolyte. Similar results have been obtained for the Co(H20)~+/Co(H20)~+ exchange rea~tion,,~ with good agreement in the rate and activation parameters at p = 0.50 mol dm-3 using LiC10, or NaC10,. The lack of Na+/Li+ medium effects in the present system and the demonstration that varying the perchlorate concentration at constant ionic strength by substituting lanthanum perchlorate for sodium perchlorate has no significant effect on the Fe(H,0)~+/Fe(H20)~+ exchange rate 22 allow for com- parisons of the kinetic and thermodynamic parameters at different ionic strengths.If a Debye-Huckel model for the electrostatic work terms is used and it is assumed that the entropy of activation for the exchange arises solely from the temperature dependence of the work terms then the ionic strength dependence of A S is given by 44 where alnDs/alnT = -1.368 for water at 25 "C. [The last term in eqn (22) is to correct for the different prefactors used in transition-state theory and in the semi- classical model used here.] For an M(H20)g+/3+ reaction with r = 6.5 A, eqn (22) predicts that AS$ should increase from -24.3 cal K-I mol-I at p = 0 to - 11.4 cal K-I mo1-l at p = 1.0 mol dm-3.The measured entropies of activation for the present system and for M(H,O); + / 3 + self-exchange and cross-reactions involving vanadium and cobalt ions are generally in the range from -25 to -20 cal K-I mol-I, as shown in fig. 1 43-47 (in the case of the cross-reactions, no correction for AS9 has been made). Within this range the AS$ values appear to exhibit a slight decrease with ionic strength between p = 0.1 and 2.0 mol dmW3. An ionic strength of 0.10 mol dmw3 is a practical lower limit for M(H2O)g+l3+ exchange measurements due to the hydrolysis of the M( H20)36 + ion and the increased uncertainties in extrapolations of k,, to [H+]-' = 0 as p is reduced. Electron-exchange studies with the outer-sphere Ru(NH,),phen2+/Ru(NH3),bpy3+ system 49 have extended the ionic-strength depen- dence to below p = 0.01 mol dm-3 and a similar behaviour of AS$ was observed.At zero ionic strength the calculated and extrapolated experimental entropies are in good agreement. At higher ionic strengths, however, the Debye-Huckel model (with r = 6.5 A) fails to predict adequately the behaviour of AS$ for these systems. Although theoretical models incorporating non-adiabaticity ( K , ~ > 1) and a more elaborate treatment of the ionic interactions yield more negative entropies of acti- ~ a t i o n , ' ~ a satisfactory interpretation of the ionic-strength dependence of the activation entropy has yet to be presented.I20 ELECTRON-EXCHANGE REACTIONS r( I k.4 d -10 - -25 t -30 0.0 0.6 1.2 (ionic strength/mol dm-3)+ 3 FIG.1 .-Ionic-strength dependence of the entropy of activation for electron-transfer reactions involving metal aquo ions; (1) Fe(HzO)i+/Fe(H20)2+ [this work, ref. (22) and (45)]; (2) V(H20)2+/ V(H20)t+ [ref. (46)]; (3) Co(H20)2+/Co(H20)2+ [ref. (43)]; (4) V(H20)i+/Fe(H20)2+ [ref. (47)]; ( 5 ) Fe(H20);+/Co(H2O)g+ [ref. (48)]. Solid line corresponds to A S calculated from the Debye- Huckel equation with an internuclear distance of 6.5 A, DISCUSSION The results presented in table 1 are in accord with the expectation that the addition of electrons to the a*d (e, in Oh symmetry) orbitals perturbs the metal-ligand bonding more than does addition to the ~ d ( t ~ ~ ) orbitals. Compare reduction of Cr(H20)i+ with reduction of Fe(H20)i+ : formation of Cr(H,O)i+ is accompanied by a large axial distortion leading to an average increase in bond length of 0.2 A.For Fe(H20);+ Ad, is smaller, 0.13 A. The effect is even more pronounced when Co(NH3);+j3+ [Ado = 0.22 A, two a*d orbitals populated because Co(NH$+ is high-spin d7 while the Co"' complex is low-spin d6] and Ru(NH3);+13+ (Ado = 0.04 A, a low spin d6/d5 couple) are considered. Simple notions do not, however, account for the contrast between Ru(NH3);+13+ and Ru(H20)g+I3+ (Ado = 0.04 and 0.09 A, respectively). Since the ammine and aquo ruthenium complexes are all low spin, with both ligands the electron addition is to a xd level. Note, however, that the RulI1- OH2 bond is much shorter than the Rur1'-NH3 bond while the bond lengths in the Ru" state are within 0.03 A of each other.The most marked electronic effects occur for bpy, phen or cyanide complexes in which the ligand may act as a zd electron acceptor with concomitant relative shortening of the metal-ligand bond lengths in the low oxidation states. When an electron is added to the nd set in such complexes, e.g. one-electron reduction of Fe(phen);+ or Co(bpy)$+, lAdol is G0.02 A,* but for reduction of low-spin d6 Co(phen)$+ to high-spin d7 Co(phen)$+, Ad, is 0.19 A. Note that, although Ad, for the metal-nitrogen bonds in Co(bpy)$/*+ is very small, the preliminary results of X-ray diffraction studies of the the chloride salts indicate significant Ad, values for the intrabipyridine These changes are neglected in the following discussion. * Co(bpy)3+ is high-spin d8 [see ref.(Sl)].TABLE 2.-PARAMETERS FOR ELECTRON-EXCHANGE REACTIONS couplea elect r onicb Ado/A ,u/mol kc/dm3 r / A KAd/lO-' AGO*,, ref. configuration dm-3 rnol-'s-' dm3 mo1-I /kcal mol-1 Cr(H20)62+/3 + Fe(H20)62+/3+ Fe(phen): +13 + Ru(HZO)z+ 13+ Ru(NH3);+l3 + R ~ ( e n ) ; + / ~ + Ru( bpy); + l3 + Co(H20)$ + I 3 + Co(en): + /3 + Co( bpy); + /3 + Co(sep)2+/3+ Co( bpy): + l 2 + CO( NH3): + I3 + (0.20)= 1.0 0.13 0.1 0.00 5.5 0.09 1 .o 0.04 0.1 0.02 0.75 0.00 0.1 0.21 0.5 0.22 I .o 0.21 1 .o 0.19 0.1 0.17 0.2 - 0.02 0.5 ~2 x 1 0 - 5 6.5 1.1 6.5 3 x lo8 13.6 (lo2) 6.5 3.2 x 103 6.7 2.8 x 104 8.4 4.2 x lo8 13.6 3.3 6.5 > lo-' 6.6 18 13.6 5.1 9.0 (>los) 13.6 7.7 x 1 0 - 5 8.4 0.33 0.05 8.3 0.33 0.07 1 .o 3.3 0.17 0.33 1.2 3.3 0.66 8.3 6.9 6.9 3.3 6.9 6.7 5.3 3.3 6.8 6.8 5.3 3.3 5.0 3.3 f, 52 f , f f, 53 40,45 f 7 55949 56, 57 R, 58 31, 59, 43 f, 61, 60 67, 63 h, 64 65, 65 f, i F m W 36 c Z m Abbreviations used : en, ethylenediamine; bpy, 2,2'-bipyridine; phen, 1 ,lo-phenanthroline; sep, 1,3,6,8,10,13,16,19-octaazabicyclo[6.6.6]eicosane Electron- '' Aver- hAs- (" sepulchrate ").exchange rate constant measured at ionic strength given and 25 "C. age '' value over all six Cr-0 bonds taken equal to {[4(0.09)2 -+ 2(0.32)2J/6)'/2. sumed to be the same as for Co(phen)5+ '+. The notation used refers to the occupation of the 71 and o* d-orbitals (t2, and e,, respectively, in octahedral symmetry). The value of KA was obtained from eqn (3) using the r value tabulated. 'This work. , Assumed to be the same as for Fe(phen)5+l3+. C. Creutz and N Sutin, unpublished work.+ c!122 ELECTRON-EXCHANGE REACTIONS Not only are electronic effects on metal-ligand bond lengths of interest in their own right but they play an important role in determining the rates of electron exchange between metal complexes. To consider this role we present table 2, in which the elec- tronic configurations, Ado values and measured exchange rates for a number of metal complexes are given. It is apparent that the smallest exchange rates [Cr(H20)2+/3+, C ~ ( e n ) $ + / ~ + , etc.] are accompanied by large (ca. 0.2 A) Ado values while the largest rates [Fe(~hen)$+/~+, R~(bpy);+/~+, Co(bpy)3+/2+] are accompanied by Ado G0.02 A. To pursue this qualitative trend at a more quantitative level other factors must, of course, be taken into consideration.Two such factors, the differences in outer-sphere reorganization energy [eqn (6)] and in the precursor complexes’ stability constants [eqn (3)], are taken into account when the observed exchange rate constant kobs is corrected as in Values of KA obtained from eqn (3) and (4) and AGzut are given in table 2. If, for the present purpose, the ln(K,,v,) term in eqn (23) and the metal-ligand force constant in eqn (7) are assumed constant, eqn (7) and (23) suggest a plot of -[ln(kob,/KA) + AG&,/RT] against (AdJ2. The plot obtained for the couples in table 2 is shown in fig. 2. 10 -20 i I I I I 6 3 2O 40 H t9 010 $ 1 I I 0 l20 1-1 08 - 300 E 20 40 60 ( ~ d , , ) ~ / i o ~ A ~ FIG. 2.-Natural logarithm of the observed exchange rate constant corrected for the stability of the precursor complex plus the outer-sphere barrier divided by RT plotted against the square of the difference of the metal-ligand bond distances in the two oxidation states ( 1 ) Cr(H,0)a+/3+ ; ( 2 ) Fe(H,0)i+’3+; (3) Fe(~hen):+/~+; (4) RU(H~O);+/~+; ( 5 ) RU(NH~):+~’+; (6) R~(en):+’~+; (7) R~(bpy):+’~+; (8) C O ( H ~ O ) ~ + / ~ + ; (9) C O ( N H ~ ) ~ + ’ ~ + ; (10) C~(en):+’~+; (11) C ~ ( b p y ) i + / ~ + ; (12) C o ( ~ e p ) ~ + / ~ + ; (13) Co(bpy)Zlz+.B .s. BRUNSCHWIG et al. 123 With the exception of C O ( H ~ O ) ~ + / ~ + , a remarkably good correlation of the left- hand side of eqn (23) with (Ad,)' is seen. The behaviour of the hexa-aquocobalt couple has long been recognized to be anomalous and both inner-sphere 66 and spin- pre-equilibrium 67,68 mechanisms have been proposed for the direct exchange process. An estimate of the effective barrier to outer-sphere electron transfer in cross-reactions is similar to that for the hexa-ammine couple66 and thus quite consistent with the pattern seen in fig.1. The possibility that Fe(H20):+l3+ exchange proceeds by an inner-sphere mechanism should also be considered. That this is not the case is strongly suggested by the fact that the behaviour of this couple is consistent with eqn (23), an outer-sphere model, and by the activation parameters for the exchange process. The entropies of activation and their dependence on ionic strength are very similar for the Fe(H20)i+/3+ and V(H,O)g+/Fe(H,O)~+ reactions. The differences in the rate constants (kFe2+ = 1.1 dm3 mo1-I s-l and kV2+ = 7.9 x lo3 dm3 mo1-I s-',~' 25 "C, An outer-sphere mechanism has been assigned 62 to the V(H,O)i */Fe(HzO)2+ reaction because the redox rate constant exceeds the rate of water exchange on both V(H,O);+ and Fe(H20)i+.70 Thus, despite the fact that electron exchange is slower than water exchange on the participating ~ p e c i e s , ~ ~ - ~ ' there is no reason to invoke an inner-sphere mechanism for the Fe(H,O):+I3 + exchange reaction.Returning to fig. 2, we examine the validity of the assumptions made in construct- ing the plot. The use of (Ad,)' as ordinate is equivalent to the assumption that f i n eqn (7) is invariant with the nature of the couple; however,f varies from ca. 1.7 x los dyn cm-I for CO(NH~);+/~+ to ca.2.2 x los dyn cm-l for RU(NH~);+/~+.' In addition variations in v, are not included in fig. 2; v, ranges from ca. 1 x 10" s-' for Ru(bpy),2+/3+ (AGzut 9 AG;,) to ca. 12 x 1OI2 s-I for the Co"/Co"' couples (AGzut < AGT,). Furthermore, nuclear tunnelling [neglected in eqn (23) and fig. 21 contributes significantly to the exchange rates of couples with large Ad,. For Fe(H,0)gT/3+ (Ad, = 0.13 .$) the rate is enhanced a factor of 3 ; for the CO"/CO~~' couples the enhancement is larger [a factor of 6 for Co(en)i+l3+ with Ad, = 0.21 A]." These variations inf, v, and the inner-sphere nuclear tunnelling factor r, are taken into account in fig. 3 where * the function = 0.10 mol dm-3) are predominantly reflected in the enthalpies of activation. is plotted against [2.15 x 10-3f(Ad,)2]/RT.Except for point (6) [R~(en)$+/~+], the points lie fairly close to the dashed line which has the theoretical slope 1 .O, but an inter- cept of 2.2 (a zero intercept is predicted for K , ~ = 1 .O). The behaviour found suggests that the present semiclassical treatment generally gives results in excellent agreement with experiment although the measured values tend to be systematically smaller (by about a factor of ten) than those calculated. A systematic deviation would imply that one of the factors KA, v,, etc. is consistently underestimated. This possibility cannot be ruled out: a very simple model has been used. Possible oversimplifications include the fact that only metal-ligand (and no intraligand) stretches have been incorporated; inclusion of other modes has ramifications for Ad,,fand v,.Alternatively (or in addition) the value of r (which was taken as a hard-sphere contact distance) may have been systematically overestimated owing to the interpenetration of the inner-coordin- ation spheres of the reactants. This would affect both KA and AGZUt (but in opposing * The following parameters were used in fig. 3 [(data point),.fj105 dyn cm-', In v,, In m]; (1) 1.7, 30.1, 1.79; (2) 2.0, 29.9, 0.7; (3) -, 27.6, 0 ; (4) 2.2, 29.7, 0.4; (5) 2.2, 28.9, 0 ; (6) 2.2, 28.7, 0; (7) -, 27.6, 0 ; (9) 1.7, 30.1, 1.95; (10) 1.7, 30.1, 1.79; (11) 1.7, 30.2, 1.61; (12) 1.7, 30.1, 1.61; (13) 1.7, 29.2, 0.124 ELECTRON-EXCHANGE REACTIONS fashions). Furthermore, the electrostatic work, calculated here from Debye-Huckel theory, has a substantial effect on the magnitude of KA.Since, as shown in fig. 1, Debye-Huckel theory fails in describing cation-cation interactions at finite ionic strength, there is some reason to question the KA values used. The remaining possi- bility that K , ~ is a constant, but not unity, is the least likely source of the trend. For xel < 1, 7ce1 is the ratio of electronic and nuclear frequencies. Since v, increases from 1 X. 10l2 to 12 x 1OI2 s-l as Ado changes from zero to >0.1 A, a constant value of 7ce1 would require that vel increase as v, increased and that vel be substantially greater for (2.15 x ~ o - ~ ) ~ - ( A ~ J z / R T FIG. 3.-Natural logarithm of the observed exchange rate constant corrected for the stability of the precursor complex, the effective nuclear frequency and the (inner-sphere) nuclear tunnelling factor, plus the outer-sphere barrier divided by RT plotted against the inner-sphere barrier divided by RT.(I) Cr(H20)g+/3+; (2) Fe(HzO)k+/3+; (3) Fe(~hen):+\~+; (4) Ru(Hz0):+I3+; ( 5 ) Ru- (NH3)g+/3+; (6) Ru(en)$+l3+; (7) R~(bpy):+'~+; (9) CO(NH~):+/~+; (10) Co(en)$+l3+; (11) Co(b~y>:+/~+ ; (12) C o ( ~ e p ) ~ + / ~ + ; (13) Co(bpy);l2+. the CO~~/CO"' couples (right-hand side of fig. 3) than for the Ru"/Ru"' couples (far left-hand side). There is no reason why Ve1 [which is proportional to the square of the electronic coupling energj, eqn (lo)] should be greater for Co"/Co"'. Indeed there is every reason to expect the opposite since these cobalt exchanges involve a formally forbidden electronic process.Thus, the possibility that the finite intercept in fig. 3 is a consequence of a deficiency in the model cannot be excluded, but it is almost certainly not due to constant 7ce1 < 1. Beyond the excellent fit obtained in fig. 3, the most striking feature is that mentioned above: the CO"/CO~~' exchange reactions are not particularly slower than those involving other electronic configurations. Had no correction for nuclear tunnelling been made, very little evidence that 7ce1 < 1 for these exchanges would be seen. The value I C , ~ = has been calculated by permitting spin-orbit mixing of ground and excited states for C O ( N H ~ ) ~ + / ~ + , ~ ~ but the data in fig. 3 suggest K , ~ > 0.01 for all the cobalt exchanges. This magnitude for K , ~ requires that HAB, the electronic coupling energy, be 230 cm-I.Values of 100-200 cm-I have been estimated from the intensities of metal-to-metal charge-transfer absorption spectra in weakly coupled binuclear mixed-valence complexes 73 and in ion pairs of the type M(CN)4,-/B . s. BRUNSCHWIG et al. 125 Ru(NH,);+ (M = Fe, Ru, 0s) 74 where the interacting metal centres have nd6-nd5 configurations. An even larger value (ca. 800 cm-'?) is obtained for the Co(NH,);+/ Ru(CN);- ion pair (note that this charge transfer produces 2E excited-state rather than 4Tground-state C O " ) . ~ ~ Values of 180,76 30 77 and 70 77 cm-' have been obtained from ab initio calculations for the Cr(H20)2+/3+, Fe(H20)g+/3+ and Ru(NH3);+I3+ exchanges, respectively. Thus the HAB value of 330 cm-l, consistent with 7ce1 >, 0.01, is not significantly smaller than the HAB values implied for couples in which the electron exchange is not formally forbidden.Possibly the K , ~ value of 0.01 is overestimated because the role of nuclear tunnelling has been underestimated. Interestingly, the magnitudes of AS$ for Co' ' / C O ~ ~ " amine exchanges are significantly more negative than those for the other systems considered (AS$ z -30 against -20 cal K-l mol-I). Since both nuclear tunnelling and non-adiabaticity decrease AS$ the magnitude of ASS does help in refining the estimate of 7ce1 for the cobalt sys- tems. and r k = 6 used in the plot, but in view of the errors intrinsic to AS$ measurements values such as 7ce1 z loA2, r, x 20 could also be accommodated [A(bS$) z 15 entropy units].(Of course, the assumption that 7cel = 1 for the reactions for which A S = - 20 cal K-' mol-' is implicit in this argument.) Obviously the treatment of the electronic factor for these cobalt(II)/cobalt(m) exchanges remains an important problem. Possibly the extension of the theoretical treatment to include the mixing of excited states with ground states of non .equilibrium geometry will shed additional light on these systems. To summarize, the semi-classical model predicts the rates of more than ten exchange reactions to within a factor of ca. 30. This should be regarded as excellent agreement since the rates for the couples considered span a range of 10'' with the slow extreme dominated by inner-shell rearrangement barriers and non-classical effects and the rapid extreme by the lower frequency solvent reorganization. We have shown that the rate differences in the systems considered are dominated by the magnitude of the inner-shell barrier.Subtler questions, the calculation of electronic factors and a deeper understanding of the frequency factors, among others, remain to challenge ex- perimentalists and theorists alike. The A(ASt) value of ca. 10 entropy units is consistent with 7ce1 z 4 x We thank Ms M-H. Chou and Dr D. Mahajan for the donation of samples and Dr R. Carr for helping with some of the EXAFS measurements. EXAFS spectra were measured at the Stanford Synchrotron Radiation Laboratory which is supported by the U.S. National Science Foundation through the Division of Materials Research, the U.S.National Institutes of Health through the Biotechnology Resource programme in the Division of Research Resources and the U.S. Department of Energy. 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ISSN:0301-7249    

DOI:10.1039/DC9827400113

出版商:RSC    

年代:1982

数据来源: RSC

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Faraday Discuss. Chem. Sac., 1982, 74, 129-140 Catalytic Influence of the Environment on Outer-sphere Electron-transfer Reactions in Aqueous Solutions BY H. BRUHN, S . NIGAM AND J. F. HOLZWARTH Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-1000 Berlin 33, West Germany Received 28th June, 1982 The continuous-flow method with integrating observation (CFMIO) has been used to investigate irreversible electron-transfer (ET) reactions between negatively charged, substitution-inert transition- metal complexes. Special attention has been paid in order to distinguish between the different contributions to the energy of activation such as size of reactants, long-range charge interactions, influence of the free energy of reactions (difference in redox potential) and the composition (electro- lytic content) of the solutions.We selected the ET reaction between Fe(CN)6Hx"-4 and IrCli- to demonstrate the catalytic rate enhancement caused by the addition of mono-, di- and tri-valent cations. Increasing protonation of Fe(CN),H:-4 decreases the rate of ET; strong association with M2+ and M3+ (where M indicates the metal) has no catalytic effect. All alkali-metal ions show an increas- ing catalytic effect with increasing size; the four tetra-alkylammonium ions show the opposite. The Arrhenius plot of the above-mentioned ET reactions in the presence of cations is strongly curved ; the decreasing slope at higher temperatures indicates a complex reaction mechanism. In the ET reaction between silvertetraphenylporphyrin tetrasulphonate and IrCli- , ET occurs at the axial position of the complex, far away from the negatively charged sulphonate groups.A catalytic effect similar to that found in the reaction with Fe(CN)z- is observed ; this result precludes the cations having a bridge-like function during ET. Monovalent cations of varying size show a maximum rate enhancement when their ionic radius is ca. 0.23 nm. If long-range Coulomb inter- actions are shielded, and a situation in which the free-energy change of reaction is zero is simulated, we extrapolate a maximum for the ET rate constant of 10" dm3 mol-' s-'. Progress in the understanding of electron-transfer (ET) reactions is mainly due to theoretical work by Marcus and Levich and Dogonadze and experimental work by Sutin and T a ~ b e .~ A recent summary of the present state can be found in Ulstrup's In this paper we concentrate on outer-sphere electron-transfer reactions between ligand-substitution-inert anionic transition-metal complexes in the presence of different concentrations of electrolyte, to separate the primary salt effect (the electro- static influence on encounter rates) from other effects the electrolytic content of the solution might have. The unusual kinetic behaviour of such anionic complexes in ET has been observed by several groups before and is summarized for electrodes by Peter et aZ.,6 and Scherer et aZ.,7 and by Holzwarth et aZ.,8 Indelli 9n and Wahl 9b for homo- geneous solution. The transition state of these ET reactions carries a high negative charge, causing a high cation concentration in the surrounding solution.If we apply the formalism of Marcus we can expect that any change in the so-called reorganization energy of the transition state caused by different cations can be detected. The measurements described in the following sections give clear evidence that cations strongly influence the activation energy for ET.130 CATALYTIC EFFECTS O N ELECTRON TRANSFER EXPERIMENTAL All rate constants reported here were measured using the continuous-flow method with integrating observation (CFMIO). The time resolution of this method has been demons- trated by measuring half-lives of first- and second-order reactions as low as 5 x s.l0 The complexes used were prepared according to the literature." HC1O4, NaClO,, HCI, all metal chlorides and Na21rC16 were of Pro Analysis or Suprapure grade and purchased from Merck A.G.; Serva (Heidelberg) supplied silver mesotetraphenylporphyrin tetrasulphonate (AgTPPTS4-), and Kodak (Rochester) the tetra-alkylammonium chlorides. All measure- ments were performed in triply distilled water using reactant concentrations between lo-' and mol dmd3. Stock solutions were never kept for more than 10 h, and special care was taken to avoid photochemical decomposition by the light used to monitor the progress of reaction inside the CFMIO apparatus. RESULTS AND DISCUSSION The electron-transfer (ET) rate constants between IrCIg- as the oxidizing species and Fe(CN);- as the reducing compound were measured by adding different con- centrations of metal chlorides to the solutions.In fig. 1 the results are summarized by plotting the rate constants kET against the root of the ionic strength I. Above the dashed line the solution pH was 5 : below the line 1 rnol dm-3 HC1 (not included in I ! ) was always present, making the pH zero. Only the open symbols show measurements at varying pH values; here fne HCl concentration ranged from to 1 mol dme3. All rate constants at pH 5 show a marked increase with increasing ionic strength of added metal chlorides. Measurements with the same cations (Na+ or K+) and differ- ent anions, such as NO,, PF, or SO:-, instead of C1- never varied by more than 40% at an ionic strength of 0.1 rnol dme3; therefore we have only reported results using metal chlorides. At low ionic strength the rate enhancement increased in the series M+ < M2+ < M3+.At ionic strengths of 0.1 M2+ and M3+ have already reached a I 01 5 10 20 30 40 50' 75 100 102(I&+/mol dm-3)* FIG. 1.-Influence of cations on the rate constants for electron transfer between IrCli- and Fe(CN):- at different ionic strengths ( I ) . Above the dashed line the pH is 5, below the dashed line the pH is 0 (1 mol dm-3 HCl present). A, Increasing concentration of HCl from lo-, to 1 mol dm-3. Tem- perature = 298 K.H . BRUHN, S . NIGAM AND J . F . HOLZWARTH 131 limit for kET, unlike the alkali ions, which still showed an increase in kET at higher ionic strength up to 1 mol dm-3; no limit in kET could be observed for M+. The pronounced decrease in slope for M2+, and at even lower ionic strength Z for M3+, occurs in a concentration range where ion-pair formation with Fe(CN)%- is observed.12 In the series Mg2+ < Ca2+ < Sr2+ < Ba2+ the increase in kET is only small; almost no difference was measured between La3+ and Ce3+.Unexpected behaviour was observed if we increased the concentration of H + from to 1 rnol dm-3. After a small increase in kET around pH 4 we found a decrease in kET which levels to a con- stant value at pH 2. Experiments with a constant concentration of 1 mol dm'3 HCl and increasing amounts of K+ showed an increase in kET similar to the rate at high K+ concentrations without the addition of HCl; 0.1 mol dmW3 Rb+ was slightly more effective than 0.1 mol dm-3 K+ ; Ce3+ and Ca2+ are less effective. To explain the results in fig. 1 we have to take into account that both reactants are negatively charged.This means that the electrostatic repulsion influences the encounter rate. Only at an ionic strength above 0.5 mol dm-3 can we expect that this ET rate-decreasing effect is no longer important l1 (see also fig. 6). The most sur- prising result was measured with an increasing concentration of H+. To gain more insight into the reason for this unusual behaviour we have measured the rate of ET between Fe(CN)z- and IrCli- at a constant ionic strength of 0.93 mol dm-3 and changing electrolytic content from 0.93 mol dm-3 HCIO, to 0.93 mol dm-3 NaC10,. In this way we have shielded the electrostatic repulsion completely but the pH of solutions was gradually shifted from 0.03 to 5. The results are reported in fig.2. FIG. 2.--l)ependence of the rate constants for electron transfer between H,Fe(CN)f-4 and IrCIi- on the pH at a constant ionic strength of 0.93 mol dm-3 and temperature of 295 K. AkET/kET = f5 %. The kET value starts at 2.6 x lo5 dm3 mol-' s-' (as in fig. 1 at 1 rnol dm-3 HC1) and levels into a constant value of 2.1 x lo8 dm3 mol-' s-l above pH 3. In fig. 3 we have plotted the redox potentials of the two reactant couples IrCl;2'-3 and Fe(CN)c3jm4 under the same experimental conditions. Furthermore, we have included the con- centration of the protonated forms of H,Fe(CN)z-4 using the equilibrium constants given in the figure, which were extrapolated from measurements of Jordan and Ewing.13 As we learn from fig. 3, three different forms of H,Fe(CN)Z-, are present,132 CATALYTIC EFFECTS ON ELECTRON TRANSFER PH FIG. 3.-Relative concentrations of the three ions (a) H,Fe(CN);-, (6) HFe(CN);- and (c) Fe(CN)i-, and the redox potentials of 11Cl;-'~- and H,Fe(CN)f-4/3- in the pH range 0-5.Temperature = 295 K. but only the form IrClg- is an electron acceptor in the pH range investigated. The small increase in the redox potential of IrClg-'3- below pH 1 is due to a catalytic influence of Na+ replacing H+. With the results from fig. 3 we can now set up a mechanism for the oxidation of H,Fe(CN),"-4 by IrC1:- (see scheme 1). The three forms of the iron complex, connected by fast protonation/deprotonation equilibria, react with IrClg- to give the same products Fe(CN)g- and IrClg-. In table 1 we have calculated the reorganization energy AI2 using the formalism of Marcus: AG,*, == A2/4 + AG,",/2 + (AG1"2)2/4&2 + AG& The electrostatic interaction term AGX2 (free enthalpy) can be neglected, because the constant ionic strength of 0.93 mol dmq3 shields all long-range Coulomb interactions; SCHEME 1.-Mechanism of the oxidation of H,Fe(CN)f-4 by IrCIi-,H .BRUHN, S . NIGAM A N D J . F . HOLZWARTH 133 this was demonstrated before8*" and can be seen from fig. 6 later. AG,", was cal- culated from the measured differences between the redox potentials of the reactants given in fig. 3. AGf2 is connected with the experimental kET value by the transition- state equation k = 7cZexp(-AGF2/RT). IC was taken as 1, assuming an adiabatic outer-sphere electron transfer with an inter- action energy of 1-2 kJ between the redox orbitals.2, the highest possible ET rate constant in aqueous solution, was taken equal to 10" dm3 mo1-I s-l, and is governed TABLE DEPENDENCE OF THE OXIDATION OF H,Fe(CN):-, BY IrClg- ON pH AT 298 K AND CONSTANT IONIC STRENGTH (0.93 mol dm-3 NaClO, + HCIO,) ~~ ~~ with AGY2 = 0 (Marcus) * I \ PH k E T l AGT2I AG21 AG,*,"l A121 -log[H&,,] dm3 mol-ls-' kJ mol-1 kJ mol-' kJ mol-' kJ mol-' 0.03 0.16 0.33 0.49 0.64 1.03 1.15 1.33 1.64 2.03 2.16 2.33 2.64 3.03 3.33 4.03 5.0 2.6 x 105 4.4 x 105 8.5 x 105 1.6 x lo6 2.3 x lo6 6.2 x lo6 8.8 x lo6 1.3 x 107 2.7 x 107 4.7 x 107 6.1 x 107 8.7 x lo8 1.3 x 10' 1.6 x lo8 1.8 x lo8 1.9 x 10' 2.1 x lo8 31.5 30.3 28.6 27.0 26.2 23.8 22.9 21.9 20.4 18.8 18.2 17.3 16.3 15.8 15.5 15.4 15.1 - 20.9 -23.4 - 26.4 -28.9 -31.5 -35.1 - 36.4 - 37.4 - 40.1 -42.8 -43.4 -44.1 -45.3 - 46.3 -46.6 - 46.7 -46.8 41.3 41.1 40.8 40.3 40.1 39.4 38.9 38.4 37.8 37.1 36.7 35.9 35.3 35.1 34.9 34.8 34.6 165.2 164.4 163.1 161.0 160.4 157.5 155.8 153.5 151.3 148.5 146.6 143.8 141.1 140.5 139.6 139.2 138.2 by the concerted rotations of water dipoles in the dynamic structure of liquid water caused by short-lived hydrogen bridges between the water molecules.The recom- bination reaction of Hkq, and OH,,, with a rate of 1.2 x lo1' dm3 mol-' s-' at 298 K l4 is a good estimate for 2, although there could be some differences because of the charge neutralization. During the reorganization of the reactants in the encounter complex before the electron will actually be transferred (Franck-Condon restriction) ion-dipole and dipole-dipole interactions dominate in the surrounding solution. AG&O is the calculated free enthalpy of activation if the reduction of the measured AGT2 caused by the thermodynamic differences in the free energy between reactants and products AGf2 is taken into account.In this way we simulate conditions under which there is no thermodynamic differ- ence in the energy content of reactants and products. That the formula of Marcus given above is suitable to calculate the influence of the differences in the free energy AG,", was shown by Sutin et al.15 and Holzwarth et a1.I6 From the A12 term in table 1 we can now obtain the free energy of activation AGTF under the conditions AG& = 0. With the relation from the transition-state theory mentioned previously k& can easily be calculated.In fig. 4 the calculated rate constants for ET using AGT; as the activ-134 CATALYTIC EFFECTS ON ELECTRON TRANSFER ation energy are plotted against the pH of solution. We learn from this plot that the kgT value now varies from 5 x lo3 to 8 x lo4 dm3 mol-' s-l, which is markedly less than the change of lo3 in fig. 2. This is due to the effect of the thermodynamic differ- ence in the free energy of the reaction AGi2. These results can be explained by assum- ing that as the pH is shifted below 3 the protonation of Fe(CN)t- becomes important and at a pH of 0 the dominating species is H,Fe(CN):-. If we accept the mechanism given we conclude that ET becomes more difficult as soon as Fe(CN):- is protonated.We can calculate kiT for the three different reaction couples as k&(Fe) = 8 x lo4 dm3 1 0 ~ ~ 0 0 4 I v) I I I I 1 0 1 2 3 4 5 lo3 PH FIG. 4.-Calculated electron-transfer rate constants of the reaction H,Fe(CN):-4 + IrClz- using the activation energy in table 1, where the influence of the difference in redox potential is taken into account (AG:z = 0). For all three lines k&(H2Fe) = 3.3 x lo3 dm3 mol-' s-' and kgT(Fe) = 8 x 104 dm3 mol-'s-'; k&(Hk) varies from 2 x lo4 dm3 mol-'s-' in (a) to 8 x lo3 dm3 mol-'s-' in T = 295 K. Ionic strength = 0.93 mol dm-3. MiT = f 11 %. (6). mo1-l s-l, kgT(HFe) = 1.4 x lo4 dm3 mo1-l s-l and kiT(H2Fe) = 3.3 x lo3 dm3 mol-l s-l. The solid line in fig. 4 was calculated using these rate constants. The accuracy of the computation can be seen from the dashed lines; while calculating (a) kgT(HFe) was taken as 2 x lo4 dm3 rno1-I s-l; for (b) k&(HFe) = 8 x lo3 dm3 mol-1 s-' with both kET(H2Fe) and kgT(Fe) were kept constant.The reason for the slower ET rates of the protonated forms H,Fe(CN)Zm4 is certainly the dissociation of one or two protons connected with ET. Fe(CN)z- is not protonated in the pH range above zero. We cannot decide from these experiments whether the protons have to be completely dissociated before the electron can be transferred (consecutive reaction) or the tran- sition state is better described by a weakening of the proton-ferrocyanide interaction (concerted reaction). However, knowing the ET rates of H,Fe(CN)g- with the very fast Fe-phenanthroline complexes measured before l6 we can exclude the consecutive reaction path because these rate constants are three to four orders higher than the value of kET with IrClg-, and the differences in the redox potentials of the Fe- phenanthroline complexes are reflected in the kET values as predicted by Marcus theory; this would not be the case if the proton dissociation occurs before the electron transfer governing the rate.We therefore believe that we are justified in assuming three parallel ET pathways connected by two proton equilibria which control theH . BRUHN, S . NIGAM AND J . F . HOLZWARTH 135 concentration of the three forms of H,Fe(CN)gm4, as shown in scheme 1. The un- expected behaviour of the ET rate from Fe(CN)z' to IrC1;- in fig. 1 for increasing HCl concentrations between and 1 mol dm-3 (open symbols) can now be explained by the superposition of the increasing electrostatic shielding of the negative charges of the reactants (increasing kET) and the lowering of the difference in redox potential (decreasing kET) as well as the increasing reorganization energy A with increasing degree of protonation (decreasing kET).These three opposing effects result in a kET value which does not change between proton concentrations from to 1 mol dm-3 if no other cations are added. No attempt was made to separate the influence of IrCli' on the reorganization energy L12 from the effect of H,Fe(CN);- because the homo- nuclear electron exchange rate used by other authors of 2 x lo5 dm3 rno1-l s-l 17*18 seems rather low. If we use the Fe(CN):-/4' exchange rate given by Shporer et aZ.19 for a 1.7 mol dm-3 Na+ solution of 5.8 x lo4 dm3 mol-1 s-I we would calculate an almost diffusion-controlled electron exchange for IrCli-'3 - .Nevertheless we can draw the conclusion that the three electron exchange rate constants of H,Fe(CN)g-4/ Fe(CN)i- differ approximately by a factor of 15. The influence on IrCl:-l3- should be very small if H+ is replaced by Na+, as was shown by Holzwarth et To gain more insight into the catalytic effect of cations on the ET rate of anionic transition- metal complexes we have measured the temperature dependence of the reaction IrC1;- + Fe(CN):- on adding 0.1 mol dm-3 of the cations Li+, Na+ and K+. We found no linear relation between 283 and 323 K. This proves that a complex reaction mechanism is acting.For the initial slope we found the activation parameters given in table 2. A- decrease in the activation enthalpy AH* is accompanied by a strong TABLE 2.-MOLAR ACTIVATION PARAMETERS AT 0.1 m01 dm-3 Me+ FOR THE ET REACTION IrC1;- + Fe(CN)%- M+ AH*/kJ mo1-l AS*/J K-' mol-' A/dm3 rn01-'s-~ ~~ ~~~ Li+ ' 3 0 5 3 -3.8 6.3 x 10'' Na+ 25.5 -& 3 - 10.6 2.8 x 10l2 K+ 10.3 =k 2 -55.3 1.3 x lo1* increase in the entropy AS* from Li+ to K+ . Our results show the same trend as the data given by Lemire and Lister 2o for the reaction Fe(CN):- + W(CN):- at a con- centration of M+ of (1.05. The strong increase in entropy from Li+ to K+ shown in fig. 5 is another indication of the participation of the highly catalytic cation K+ in the transition state, as is also expected from the curvature in the Arrhenius plot.The lower value of AH* for K+ might be explained by assuming that this cation does not act in a simple Coulomb type of interaction, which should give higher AH* values, but as a catalyst for the reorientation of the outer sphere in the transition state. In fig. 6 we have included the measurements using silver(I1)tetraphenylporphyrin tetra- sulphonate (AgTPPTS4-) as the electron donor instead of Fe(CN);- in the reaction with IrCli-. We know from experiments with micelles 21 and silverporphyrin com- plexes that AgTPPTS4- transfers electrons via an axial type of reaction rather than through the negatively charged sulphonic acid groups attached to the phenyl rings. If the cations acted as a bridge for ET we should not see a catalytic effect of M+ as in the reactions of fig.1. The results included in fig. 6 show a catalytic ET rate enhancement similar to that in fig. 1. Only at ionic strengths >0.05 mol dm-3 in M+ do the differences between the cations seem to disappear. This can be explained by an aggregation of AgTPPTS at high ionic strength (shown by spectroscopic and other136 CATALYTIC EFFECTS ON ELECTRON TRANSFER evidence) which shields the axial positions of the porphyrin complex in such a way that ET is hindered.21 Besides the reaction of AgTPPTS4- with IrCli- we have also included in fig. 6 results using Os(dipy)g+ as the electron acceptor and W(CN)i- as electron donor to establish very accurate values for completely diffusion controlled ET reactions of this type.These rate constants are necessary to distinguish between the influence of the rate of encounter and the real rate of ET at different ionic strengths 10' r( lo! I fA I 0 3 - E % 'CI . - E: " 2 10' 10' T K 3 20 310 300 290 I r I 3.1 3.2 3 . 3 3 . 4 3.5 103K/T FIG. 5-Temperature dependence of the rate constant for electron transfer between IrCIg- + Fe- (CN)t- at constant ionic strength in the presence of different cations: @, 0.1 mol dm-3 KCI; w, 0.1 mol dm-3 NaCl; A, 0.1 mol dm-' LiCl; a, without additives. for experimental rate constants where both effects are superimposed. The steady state equation k;: = kF$ + kDilff allows one to separate both influences if kDiff is known. The purely electrostatic effect on the encounter rate of AgTPPTS4- (fig. 6) or Fe(CN)z- (fig.1) with IrCIi- can be calculated if we take the sum of the ionic radii as the encounter distance and use the static dielectric constant of water. Under these conditions we expect a maximum for the rate enhancement of 2 x lo2, which agrees well with the measurements in the presence of high concentrations of Li +. All other monovalent cations than H+ show a higher increase in kET than expected from purely Coulomb interactions. In table 3 we have summarized the results of the three electron-transfer reaction couples Fe(CN)i-, W(CN)i- and AgTPPTS4-, with IrCli-. Only kET values with monovalent cations at an ionic strength of 0.1 or 1 mol dm-3 are included in the table. The influence of diffusion (the transport term) was taken into account if this was necessary.In this way we achieved the real rate constants of ET which were then used to calculate the reorganization energy RI2 by applying the for- mula of Marcus together with the differences in redox potential either measured orH . BRUHN, S . NIGAM AND J . F . HOLZWARTH 137 taken from the l i t e r a t ~ r e . ~ ~ ' ~ ~ In the last column of table 3 we have included the now- corrected kET values by subtracting the influence of diffusion and the thermodynamic free energy (AGf2 = zFAE) of these reactions. These kET rate constants are the only ones which are suitable for a comparison between different reactions to show the Zimol dm-3 410'2 0.1 0.25 0.5 1 I 1 I Na' 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 I+/rnoI* dm-3/2 FIG. 6.-Influence of cations on the rate constant of electron transfer between silver(1r)meso-tetra- phenylprophyrin tetrasulphonate and IrClE- at different ionic strengths together with completely diffusion-controlled electron-transfer reactions of Os(bipy)3 + and W(CN):-. a, Os(bipy)$ + + W(CN):- ; ., Os(bipy)!+ + Ag"TPPTS4- ; 0, 0, A, x , IrCla- + Ag"TPPTS4- ; v, IrClb- + W(CN): - .catalytic influence of the cations. In fig. 7 these quasi-equilibrium electron-transfer rate constants for the three electron donors W(CN);f-, Fe(CN);- and AgTPPTS4- in reactions with the electron acceptor IrC12- are plotted against the crystallographic radii of several monovalent cations. The first important result from fig. 7 is that there seems to be an optimum value for the radius of monovalent cations around 0.23 & 0.02 nm to cause the maximum rate enhancement possible.All three reactions show the same relative differences between the cations, although their absolute k,, values reflect the nature of the electron donor, giving the highest rate constants for the largest anion (AgTPPTS4-) and the lowest rate constant for the smallest [Fe(CN);-]. The tetra-alkylammonium cations complete the set of data on the side of the larger radii. They could not be used with AgTPPTS4- because of a specific interaction with the porphyrin ring, but experiments with W(CN)i- and N(CH,)Z not included in138 CATALYTIC EFFECTS ON ELECTRON TRANSFER TABLE 3. --CORRECTED ELECTRON-TRANSFER RATE CONSTANTS FOR MONOVALENT CATIONS AT 298 K ionic strength/ experimental results calculated using Marcus mol dm-3 kET(AGi2 # o)/ W 2 l &z/ ~ET(WZ = 0)l A- C+ dm3 mol-'s-' kJ mol-' kJ mol-I dm3 mol-l s-I kE-r C+ IrCIi- + Fe(CN):- - I = 0.1 PH 5 I = 1 PH 5 I = 0.1 PH 5 I = 1 PH 5 1 = 0.1 P H 5 Li + Na+ K+ C1- NH: Rb+ c s + N(CH3): Br- N(C2H5)2 N(C4H9)2 3.5 x 107 4.7 x 107 2.0 x 10% 1.9 x lo8 5.6 x lo8 1.9 x 10% 3.1 x 109 5.1 x 107 1.9 x 107 - 53.4 - 53.4 - 52.5 - 52.5 - 51.6 - 50.4 -55.4 - 57.9 - 59.3 Li + 1.8 x lo8 -47.5 C1- Na+ 2.9 x 10% - 47.1 K+ 1.7 x 109 - 46.9 NHZ 2.0 x 109 - 46.9 kET C + IrClg- + W(CN)i- - H+ 2.8 x 107 - 42.9 Li + 2.8 x 107 -42.9 Na+ 4.3 x 107 - 42.9 C1- K+ 2.3 x 10% - 42.3 Rb+ 5.6 x 10% - 42.1 cs+ 3.1 x 109 -41.7 C1- Na+ 2.4 x 10% - 39.2 R b+ 5.3 x 109 - 39.0 kEr IrCIg- + AgTPPTS4- 7 Li + 7.3 x lo8 -45.3 C1- Na+ 9.8 x 10% -45.3 K+ 1.6 x 109 - 45.3 c s + 2.7 x 109 -45.3 168.8 165.5 148.0 148.6 134.8 112.7 152.9 171.3 184.4 141.5 135.5 114.9 113.1 155 155 150.1 131.1 120.8 100.5 126.0 90.3 122.6 119.2 113.5 76.9 4.0 x 103 5.6 x 103 3.2 x 104 1.2 x 105 2.0 x 104 3.1 x 103 8.2 x loz 6.0 x 104 1.1 x 105 3.1 x 10' 1.2 x lo6 8.8 x 105 1.1 x 106 1.6 x 104 1.6 x 104 2.6 x 104 1.8 x 105 5.0 x 105 3.1 x 105 1.1 x 107 3.9 x lo6 4.2 x 105 5.9 x 105 4.3 x 107 1.1 x lo6H .BRUHN, S . NIGAM AND J . F. HOLZWARTH 139 fig. 7 showed similar results as Fe(CN)i-. In fig. 7 only measurements at an ionic strength of 0.1 mol dm-3 in M + are included to avoid unwanted complications caused by aggregation of reactants (AgTPPTS4-) or precipitation of reactants [Fe(CN)t- + 1 mol dm-3 Cs+].At an ionic strength of 0.1 mol dm-3 electrostatic repulsion between the reactants is still important; therefore we have included some measurements at 1 mol dm-3 M+ in table 3. A comparison with the same reactions at 0.1 mol dm-3 of similar cations show that the rate constant can still be accelerated by a factor of 20 in i lo9 I I I I I I I I I ;, .ti Is+ I \ \ I I I \ I 1 I I I I \ \ \ I I \ \ I \ ! u Na. Li+ Rcrystlnm FIG. 7.-Calcutated eIectron-transfer rate constants for the reactions between IrCIz- and W(CN)g- ( A), Fe(CN);- (0) or AgTPFTS4- (0) using the results from table 3 (AG,", = 0). Temperature = 298 K, pH 5 and I = 0.1 mol dm-3. the case of Na+, where only a small catalytic effect could be observed. The maximum rate constant for electron transfer extrapolated from fig.7 to 109-1010 dm3 mol-' s-' is in good agreement with the maximum rate of ET possible in aqueous solutions of 10" dm3 mol-l s-' if we add the electrostatic term of 20. The equal results for H+ and Li+ in table 3 for IrClg- + W(CN)4,- prove that the proton behaves like Li+ if it is not strongly bound to the reactants, as in the case of Fe(CN)i- at low pH values. C O NCLU S I ON Our measurements of the rate of electron transfer between anionic substitution-inert transition-metal complexes such as Fe(CN)i-, W(CN);- and AgTPPTS4- with TrCIi- is convincing evidence that cations can accelerate ET. This cannot be explained by a simple increase in the encounter rate due to shielding of the electrostatic repulsion140 CATALYTIC EFFECTS ON ELECTRON TRANSFER between the negatively charged complexes.Strong ion associates, like those between H+ and Fe(CN)%- or between di- and tri-valent cations and the reactants at high con- centrations, show no catalytic activity. A simple bridging mechanism like the one found by Taube for inner-sphere ET reactions is unlikely because identical catalytic effects were observed in the ET reactions of AgTPPTS4- + IrCIi- and Fe(CN)i- + IrC1;- as well as W(CN)i- + IrCli-, although the ion-association behaviour of these complexes is different. The differences between the monovalent cations are almost independent of their concentration between loF3 and lo-' mol dm-3. The temper- ature dependence of the reactions in the presence of cations shows a strong curvature, and the entropy of activation for the catalytic active ion K+ is very high and accom- panied by a low AH* in comparison to Li+, which does not act as a catalyst.This proves that the cations are participating in the transition state but they have to be still mobile to cause the catalytic rate enhancement. We believe that the catalytic influence of cations can be described by three effects acting in concert. (1) They influence the water dipoles in the surrounding solution of the activated complex in such a way that their orientation is faster; (2) they allow for a better adjustment of the ligand central metal bonds before electron transfer; (3) they promote the interaction of the redox orbitals which are involved in the reaction, so that the transfer of charge is facilitated. R. A.Marcus, Annu. Rev. Phys. Chem., 1964,15, 155. Phys. Chem. Sect. (Engl. Transl.), 1959, 124, 9. N. Sutin, Acc. Chem. Res., 1968, 1, 225. H. Taube, H. Myers and R. L. Rich, J. Am. Chem. Soc.. 1953,75, 4118; 1955, 77, 4481. J. Ulstrup, Lecture Notes in Chemistry, vol. 10, Charge Transfer Processes in Condensed Medig (Springer-Verlag, Berlin, 1979). G. Scherer and F. Willig, J. Electroanal. Chem., 1977, 85, 77. J. F. Holzwarth and L. Strohmaier, Ber. Bunsenges. Phys. Chem., 1973, 77, 1145. (a) A. Indelli, Gazz. Chim. Ital., 1962, 92, 365; Zsr. J . Chem., 1971, 9, 301 ; (b) R. J. Campion, C. F. Deck, P. King and A. Wahl, Inorg. Chem., 1967, 6, 672. lo J. F. Holzwarth, in Techniques and Applications of Fast Reactions in Solution, ed. W. J. Gettins and E. Wyn-Jones (D. Reidel, Dordrecht, 1979), pp. 13-24. l1 J. F. Holzwarth and H. Jurgensen, Ber. Bunsenges. Phys. Chem., 1974, 78, 526. l2 G. I. H. Hanania and S. A. Israelian, J. Solution Chem., 1974, 3, 57. l3 J. Jordan and G. J. Ewing, Inorg. Chem., 1962, 1, 587. l4 J. F. Holzwarth, in Techniques and Applications of Fast Reactions in Solution, ed. 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ISSN:0301-7249    

DOI:10.1039/DC9827400129

出版商:RSC    

年代:1982

数据来源: RSC