THE THEORY OF ELECTRON TRANSFER BETWEEN METAL IONS IN BRIDGED SYSTEMS BY J. HALPERN” AND L. E. ORGEL University Chemical Laboratory, Lensfield Road, Cambridge Received 4th February, 1960 The role of bridging groups in promoting electron transfer between metal ions in solution is discussed. Interactions between orbitals of the metal ions through those of the bridging groups give rise to two possible exchange mechanisms-double exchange and superexchange. Expressions for the exchange frequencies associated with these are derived and their dependence on various factors such as orbital symmetry, overlap and redox potentials is discussed. Particular consideration is given to electron transfer through bridging groups containing conjugated n-electron systems. Electron-transfer reactions between metal ions in solution have been shown 1 in many cases to proceed through intermediates in which the two metal ions are coupled by a bridging group which forms part of the inner co-ordination shells of both.Examples are the oxidation of CrII by CrIII and CoIII complexes through the bridged intermediates (H20)5CrIIIXCrII(OH2) 5 and (H3N) 5 CoIIIXCrII( OH2) 5 , respectively. A characteristic feature of these reactions is the very marked sensi- tivity of their rates to the nature of the bridging group. Thus the rate of the first of the above reactions decreases by a factor of 107 along the series X = Br->NY, Cl->OH->F->NCS->H20,2 and that of the second by a similar factor along the series X = OH- >C1- >p-phthalate >fumarate, H20 >acetate, succinate, m- phthalate, o-phthalate.1 The higher rates of electron transfer with p-phthalate and fumarate (/’‘bi-40 and 0-5 M-1 sec--1 respectively) than with other carboxylic acids (kbi4.1-0.2) are of special interest and have been interpreted in terms of bridged intermediates, such as (H,N),co~~~ .. . o- Y O . . . Cr”(OH,), \c-CH=CH-C OY \OH in which the two metal ions are co-ordinated to different carboxyl groups. Transfer of an electron between the separated CrII, and CoIII ions presumably occurs through the conjugated n-electron system of the bridging group. In all these reactions the overall electron-transfer process is accompanied by transfer of the bridging ligand from the oxidant to the reductant and, in some cases, by reaction of the ligand itself (e.g.hydrolysis of methyl fumarate 2 and cis-trans isomerization of maleate 3). All of these observations emphasize the important role of the bridging group in relation to the mechanism of electron transfer in these systems. Bridged inter- mediates are undoubtedly important also in many other electron-transfer reactions, although the substitution lability of most ions makes this difficult to demonstrate unequivocally. Various theoretical aspects of electron transfer processes in solution have been considered by Libby,s Zwolinski, R. J. Marcus and Eyring,6 R. A. Marcus,7 * on leavc from tlic Chemistry Department, University of British Columbia, Vancouver 8, Canada. 32J . HALPERN AND L. E. ORGEL 33 Orgel 8 and George and Grfith.9 Nearly all these treatments have emphasized the dependence of the rate on the following factors.(i) electrostatic interactions between the overall charges of the reactants ; (ii) the reorganization energy of the ligands and of the surrounding medium prior to and during electron transfer, associated with the Franck-Condon restriction, and (iii) the rate of the electron-transfer process itself in the transition complex. Considerable progress has been made, .particularly by Marcus 7 in the quantitative treatment of the first two of these and although the models employed for this purpose approximate more closely to reactions of the " outer-sphere activated complex " type 1 (i.e. those in which electron transfer occurs without disruption of the first co-ordination shells of the ions) many of the results are at least quali- tatively relevant also to bridged systems, The role of the last factor, however, is still not well understood, although some attempts have been made to treat it both from the standpoint of the time-dependent perturbation theory 5 and of electron tunnelling.6910 For many reactions, the system may pass through a con- figuration in which the electron-transfer probability is sufficiently high for the rate to be determined completely by other factors (i.e.the free energy of activation for attainment of this configuration). On the other hand, it seems likely that some of the variations in rate found in bridged systems, such as those for the oxidation of Cr11 by Con1 through conjugated carboxylic acid bridges, are associ- ated with differences in the rate of the electron-transfer process itself.The concept, based on analogy with similar processes in the gas phase, that electron transfer in solution occurs directly between the orbitals of the exchanging ions, has led to emphasis on the overlap of the latter as a critical factor in deter- mining the electron transfer probability. Calculations based on this approach lead to the conclusion that the probability of electron transfer between ions is indeed appreciable even at distances of the order of lOA and larger.5 However, failure to take into account the effect of the intervening medium renders this, and similar results based on electron-tunnelhg models,6 of doubtful significance for most electron-transfer processes in solution and, in particular, for reactions of the bridged type.For such systems it is probable that direct interaction between orbitals of the metal ions is unimportant relative to interactions through orbitals of the bridging groups. Various possible detailed mechanisms of electron transfer involving such mediated interactions have been considered by Taube and Myers 11 and by George and Griffith.9 In this paper we attempt to examine these suggestions in greater detail, with particular emphasis on their application to electron transfer through conjugated systems. KINETIC CONSJDERATIONS We consider the problems of electron transfer within the framework of the general reaction scheme formulated by Marcus.7 Thus, taking account of the Franck-Condon restriction, we assume that electron transfer occurs through an intermediate atomic configuration of the system, achieved through suitable ad- justment of the ligands and surrounding solvent, such that the electron configur- ations corresponding to the reactants and products (I and 11, respectively) have the same energy.The overall reaction is thus represented formally by the series of elementary steps, ki A+X-B++(Ax-B+) (AX-B +)+(A+x- B) k- I k2 I1 I k - 2 B34 ELECTRON TRANSFER IN BRIDGED SYSTEMS (A+x-B)~A+x- +B 11 and the usual steady-state approximation for (I) and (11) yields for the overall bimolecular rate-constant, k b i = k,l[1 +(I + k-zlk3)k- 1/k2]* (4) For simplicity we assume that A and B are the same and that the structure of the bridged intermediate is symmetrical. Its formation from the reactants involves a substitutional step whose rate constant is included in kl.k2 and k-2 refer to the actual electronic transitions, that is the rate constant for the transition during the period in which the configuration of matching energies is maintained, and k3 and k-1 to disorganizing motions of the ligands or solvent which destroy the energy equivalence. In a symmetrical system such as we are considering, k 2 = k-2 and k-1 = k3. Two limiting cases may be, noted. (1) When the probability of electron-transfer during the lifetime of the sym- metrical intermediate is high (k2 >k-l) then k b i z i k l . ( 5 ) Under these conditions the rate does not depend directly on the electron-transfer probability although it may do so indirectly since, as pointed out by Marcus 7 large electronic interactions lead to relaxation of the Franck-Condon restriction.(2) When the probability of electron transfer is small (k-lbkz) then kbix klk,/k,,. (6) Under these conditions the intermediate (I) is essentially in equilibrium with the reactants and its lifetime is of the order of l/kdl. The transition probability during this lifetime then determines the rate. The transition probability may be computed using the time-dependent per- turbation theory which gives,12# 14 P , = sin2 nut, (7) where P I , I I ( ~ ) is the probability of the system, initially in state I, undergoing a transition to 11, in time d. v is given by in the case where @I and @u (the 11) are orthogonal, and otherwise unperturbed wave functions for the states I and (to first order in overlap) by where and Thus, according to (7) if the system is initially in configuration (I) it will “ oscillate ” between the two configurations (i.e.the electron will exchange between A and B) with the frequencyv. For times z, small compared to l/nv, (7) may be expanded to give P , I,(Z) = I nvz 12. (10)J . HALPERN A N D L. E . ORGEL 35 Under these conditions the transition probability is seen to be proportional to the squares of both v and z, the lifetime of the intermediate.* The latter is of the order of 10-13-10-12 sec and thus it is likely that the electron-transfer probability, in the sense implied here, will be an important rate-determining factor only if v < 1013 sec-1 (HI 11<0.5 kcal). Although, as emphasized by Marcus,7 this con- dition is more likely to apply to electron-transfer reactions of the outer-sphere type, it may also be realized in bridged systems, where coupling of the metal ions by the bridging group is very weak.We proceed to examine the nature of the interactions which give rise to such coupling. DIRECT AND DOUBLE EXCHANGE We consider first a simple model system comprising a one-electron atom A and the corresponding ion, Bf, coupled symmetrically through a bridging ion, X- containing a closed shell of two electrons. If we represent the unperturbed states I and 11, corresponding to the electron being located on A and B, respectively, by the determinental wavefunctions (in which 4~ and & refer to electrons with different spins), I then the exchange frequency is given by 2 or, in the approximation which neglects overlap terms relative to 1 and which we also subsequently use, by Expansion of HI I and HI 11 gives, H I I = (4A(1)$X(2)4X(3) I 1 $A(1)4X(2)4X(3)) - (4A(1)4X(2)4X(3) 1 I 4X(1)4A(2)4X(3)) HI I1 = (4*(1)4X(2)4X(3) 1 H I4d1)4x(mx(3)) - (4A(1)4X(2)4X(3) I I 4X(1)4B(2)4X(3)), The forms of the two terms in the expression for I3111 suggest that the first may be loosely identified with direct exchange of an electron between A and B * It should be noted that the earlier use of the ordinary form of the rate-law for the electron-transfer step in the steady-state kinetic treatment is not strictly consistent with (7) and (10).However, provided that encounters are sufficiently frequent and that the probability of transition per encounter is sufficiently small then the macroscopic rate law is still linear in the time, but the rate constant is dependent on the variation of PI II(T) with T, the lifetime of the “energy-matched” intermediate.It should also be noted that while PI 11 is a quadratic function of YT in the model which we have adopted, the nature of the dependence in a more realistic theory is by no means obvious, and may indeed be linear.36 ELECTRON TRANSFER IN BRIDGED SYSTEMS and the second with a " double exchange " mechanism corresponding to con- certed transfer of an electron from A to X and from X to B. This type of inter- action was first invoked by Zener 14 to account for the electrical conductivity and ferromagnetic properties of mixed valence metal oxides (e.g.those containing Mn3f and Mn4f ions) and Taube and Myers 11 have drawn attention to its possible relevance for electron-transfer processes in solution. Eqn. (11) may be compared with the corresponding result 5 for the unmediated electron transfer frequency (i.e. in the absence of X-) Approximate values of v1 and v2, computed using hydrogen 1s orbitals for C)A, C)X and 4~ are listed in table I. As expected, v2 falls off more rapidly than v1 (roughly as SAB and SAXSAB, respectively) with increasing A-B separation. TABLE 1 .-IS ELECTRON-EXCHANGE FREQUENCIES A-B separation V1 v2 (Bohr radii) sec-1 seC-1 5 3 x 1015 3 x 1014 10 7 x 1013 4 x 1012 15 2 x 1012 4 x 1010 20 3 x 1010 3 x 108 A similar result is obtained if the bridging group contains more than one closed shell, Thus, if there are N filled orbitals, 4xl, #xZ, .. . J(2N + l)! (1 - Six,) i = 1 and, again neglecting overlap terms relative to 1, we get Expansion of HI 11, neglecting terms higher than second order in overlap, givesJ . HALPERN AND L . E. ORGEL 37 The terms in this summation correspond to contributions to double exchange through the various occupied orbitals of the bridging group and may either rein- force or oppose each other depending on the symmetries of the orbitals. We now proceed to apply this result to a system in which A and B are linked through a conjugated bridge. We introduce the following notation and simplifying assumptions, have the same product form as previously (eqn. (14)) but &,, &,. . . . 4 x N now correspond to the N occupied molecular orbitals of the conjugated bridge.We approximate these by wave functions of the Huckel type, (i) @I and where Xk is the atomic p-orbital of the kth atom. All the Xks are assumed to be equivalent. The atoms to which A and B are attached are identified by the subscripts r and s respectively. (ii) In the expression for v (eqn. (17)) we neglect terms involving overlap of non-adjacent orbitals. This makes both H a x i , x i ~ and SAXiSXiB pro- portional to CirCis and thus The coefficients, j ; are approximately, but not strictly, independent of i, Y or s, but it is of interest to note that in the approximation where this dependence is neglected, v becomes proportional to the mobile bond order,l5 prs between the rth and sth atoms of the conjugated system (defined by prs = 2 ClrCis). This result is readily reconciled with the concept of electron transfer by n-electron " conduction " through the conjugated system.To examine some consequences of this result we list in table 2 mobile bond order values for a number of conjugated systems computed from the coefficients all electrons TABLE 2.-MOBILE BOND ORDERS IN CONJUGATED SYSTEMS * 1 A -0.50 - 0.02 - 0.42 B 1 --- * 0.88 0 C -0.07 0 -0.58 -0.21 D P * 0.87 0 -0.39 0 0.30 E lk ak 1 G38 ELEC'I'RON TRANSFER I N BKIDGEL) SYS'I'EJMS TABLE 2. -con t in ucd J K L trends. (9 (ii) The M N of the Huckel orbitals of the corresponding hydrocarbon molecules or anions. The value (whose sign is not relevant in this connection) refers in each case to the bond order between the designated atom and the starred one.We note the following A general tendency for the bond order to fall off with increasing length of the conjugated path. In many cases there is superimposed upon this an alternation effect such that the bond order between atoms separated by an odd number of atoms is zero. (This is a general result for alternant molecules but not necessarily for ions.) introduction of hetero-atoms (e.g. 0 or N) which are normally present in the systems of actual interest as bridging ligands will undoubtedly modify this pattern but is unlikely to alter it qualitatively. Thus the alternation of bond orders is probably relevant to the differences noted between m- and p-phthalate as electron transfer mediators (cf. K and L, table 2).Of related interest is the observation 1 that the fumarate and p-phthalate medi- ated oxidations of CrII by CoIII are accelerated by acids. This has been interpreted in terms of the improved conjugation of the path connecting the two metal ions, resulting from protonation of the carboxyl oxygen adjacent to CoIII, e.g., In terms of our model this explanation finds an analogy in the increase in bond order in going from M to N (table 2). The bond order patterns in some of the aromatic molecules listed in table 2 are also of interest in relation to electron transfer reactions of metal complexes of dipyridyl, o-phenanthroline, porphyrins, etc. Such reactions are generally of the outer-sphere type, but it is likely that electron transfer through the ligands (e.g. the relative effectiveness of different positions of the aromatic system) is governed by similar considerations.SUPEREXCHANGE George and GrBth 9 have drawn attention to another possible detailed mechan- ism of mediated electron transfer which may be important in these systems. This is the mechanism of superexchange first suggested by Kramers 16 also to account for magnetic interactions between transition metal ions in oxide crystals. It1. IiALPEKN AND L. E. OKGCL 39 arises from mixing with the ground states, @I and QI, of excited configurations such as A+X"B+ and AXB (111) (IV) corresponding to transfer of an electron in the first case from A to one of the un- occupied orbitals 4xtj of X and in the second from one of the occupied orbitals through (111) leads to additional contributions to v $xi, of X to B.Interaction of @I and in eqn. (15) of the form where the energy of the excited configuration. The principal which extends over all the unoccupied orbitals of X, are Similarly interactions through (IV) give rise to a term, containing contributions from each of the occupied orbitals, Xi, of the form, - (4A(1)4Xi(2)6Xi(3) 1 I $A(1)4X,(2)48(3>)(~A(1)4Xi(2)4B(3) I I 4Xi(1)4Xi(2)4B(3))* EI-EIv, The magnitude of the superexchange interactions, like those of double exchange, are of the order of the overlap product, SAXSXB. However, because of the energy term in the denominator, it is to be expected that contributions from superexchange will arise principally from interactions through the highest occupied and lowest unoccupied orbitals and that both their absolute and relative importance will be related to the ease of oxidation or reduction of the bridging ligand by the metal ions.The magnitude of the transition frequency, v, is determined by the algebraic sum of the contributions from these simultaneous interactions, i.e., Thus the contributions from the various superexchange interactions may either reinforce or interfere with each other as well as with those from double exchange. A factor of importance in relation to both the double exchange and super- exchange mechanisms is the matching of the symmetries of the metal ion and bridging orbitals. Thus electron transfer between t2g d-orbitals will be favoured through n-bridging orbitals and transfer between eg orbitals through a-bridging orbitals.It should be noted that on the basis of this criterion, neither the CrII+CrIn nor Cra+ CoIII systems, both of which involve electron transfer between eg orbitals, represent favourable cases for mediation by n-bridging orbitals. However, this40 ELECTRON TRANSFER IN BRIDGED SYSTEMS rcstriction may be partly removed by a number of factors (e.g. distortion from planar configuration) and its importance is difficult to assess. In this connection the possibility of participation of excited configurations of the metal ions should also be considered, particularly in reactions involving a net free-energy decrease. CHEMICAL MECHANISMS In addition to these quantum-mechanical interactions it is possible for net electron transfer to be effected through chemical mechanisms in which the bridging group is first reduced by one of the metal ions and subsequently reoxidized by the other (or vice versa).In contrast to superexchange this type of mechanism in- volves the participation of configurations such as 111 and IV as actual chemical intermediates in the reaction and depends on the attainment of such configurations through thermal activation. Thus it is likely to be important only when the redox potential for the oxidation or reduction of the bridging group is very favourable. The limiting case of this type of mechanism involves participation of only one of the metal ions in the rate-determining step (e.g., A+X-->A+ X ; X+ B+X-+ B+) and is readily distinguishable kinetically. This is not commonly encountered in redox reactions between metal ions but possible instances are the oxidation of T1+ by CeOH3+ 17 and the TlBrlfTl+ electron exchange.18 In general, however, even this type of mechanism is likely to be favoured by concerted interactions of both metal ions with the ligand and hence to proceed through an activated complex of the bridged type.Redox mechanisms19.20 involving the transfer of an H atom (i.e. the intermediate reduction of a water molecule) between the hydration shells of the metal ions may be included in this class. Conjugation in the bridging group is important also in relation to this type of mechanism since an electron transferred from the reducing ion to a conjugated bridging group is in a delocalized orbital and can pass readily to the oxidant. Thus the process is analogous 9 to the mechanism of ordinary semiconduction in solids involving thermal excitation of electrons into a conduction band.That an electron passes into the maleate bridge in the oxidation of Vn and CrII by maleato- pentammine-cobaltIII has been inferred from the observation that the maleate group undergoes cis-trans isomerization and deuterium exchange with the solvent during the reaction.4 Also relevant to this theme are recent studies21 on the oxidation of the organic ligand in complexes such as oxalatopentammine cobalt111 by Ce*V and other oxidants. CONCLUSIONS In this paper we have attempted to examine various types of interaction which may be of importance in promoting electron transfer across bridged systems. At this stage neither the theoretical nor experimental evidence appears adequate for quantitative assessment of the importance of these relative to each other or to other factors which may influence the overall rates of such reactions.Among the latter are the rate of the substitutional step leading to the formation of the bridged intermediate, the ligand displacements (particularly that of the bridging ligand) associated with the Franck-Condon restriction and the lifetimes of configurations appropriate for electron transfer compared with frequency of the electron transfer process itself. Qualitatively we have noted a number of factors (bond orders, redox potentials, orbital symmetries, etc.) which may serve as distinguishing criteria for some of the interactions we have considered and these suggest further experiments whereby their importance might be tested.The study of conjugated bridging groups offers a particularly promising approach for it permits the various parameters of interest to be varied in a systematic manner. Thus it would be of interest to compare the effectiveness as electron-transfer mediators of a series ofJ . HALPERN AND L. E . ORGEL 41 dicarboxylic acids connected through conjugated paths of varying length, or of a series of diaza-aromatic compounds varying the relative positions of the two N atoms. Comparison of reactions of VII and CrII which have similar redox poten- tials, but involve transfer of electrons from orbitals of different symmetry, should also be of interest in assessing the importance of the latter factor. Such measure- ments appear to lie within the scope of present experimental methods. We are grateful to Prof. H. C. Longuet-Higgins, Prof. S . Golden and Mr. J. S . Griffith for valuable discussions. One of us (J. H.) also thanks the Nuffield Foundation for a Travelling Fellowship and the University of British Columbia for a leave of absence. 1 Taube, Advances in Inorganic Chemistry and Radiochemistry (Academic Press, New 2 Ball and King, J. Amer. Chem. SOC., 1958, 80, 1091. 3 Fraser, Sebera and Taube, J. Amer. Chem. Soc., 1959, 81, 2096, 3000. 4 Fraser and Taube, J. Anrer. Chem. SOC., 1959, 81, 5514. 5 Libby, J. Physic. Chem., 1952, 56, 863. 6 Zwolinski, Marcus, R. J. and Eyring, Chem. Rev., 1955, 55, 157; J. Physic. Chem., 7 Marcus, R. A., J. Chem. Physics, 1956,24,966; 1957,26, 876, 871. 8 Orgel, Report X ConseiZ Chim. SoZvay (Brussels, 1956), p. 289. 9 George and Griffith, The Enzymes (Academic Press, New York), 1959, 1, 347. 10 Laidler, Can. J. Chem., 1959, 37, 138. 11 Taube and Myers, J . Amer. Chem. SOC., 1953, 76, 2103. 12 Slater, Quantum Theory of Matfer (McGraw-Hill, New York, 1951), p. 81. 13 Gurnee and Magee, J. Chem. Physics, 1957, 26, 1237. 14 Zener, Physic. Rev., 1951, 82,403. 15 Coulson and Longuet-Higgins, Proc. Roy. Soc. A, 1947, 191, 39. 16 Kramers, Physica, 1934,1, 182. 17 Armstrong and Halpern, unpublished. 18 Carpenter, Ford-Smith, Bell and Dodson, this Discussion. 19 Reynolds and Lumry, J. Chem. Physics, 1955,23,2460. 20 Hudis and Dodson, J. Amer. Chem. SOC., 1956, 78, 91 1. 21 Taube, Chem. SOC. Spec. Publ., 1959, 13, 57. York), 1959, 1, 1. 1954, 58, 432.