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JOHNF. ENDICOTT
It has been suggested that mixing may be incomplete even after the stream reaches the esr cavity4 and that reaction 1 cannot necessarily be assumed to be complete within the mixing chamber.13 A reasonable value8 for k~ is -lo3 M-’ sec-I. Thus, reaction 1 could very well be a minor source for -OH up to several hundred milliseconds after mixing. The pure Ti(II1)H z 0 2 system without substrate is complicated,8 arid additional sources for .OH could easily be present. The possible contribution of Ti(1V) or its complexes to the decay of substrate radicals has been mentioned p r e v i ~ u s l y . ~ Substrate ~J~ complexes in the abstraction process were considered by Turkevich;5 (4)and (5) may therefore involve complexed CH30CHSOCH3. Similarly, the substrate radicals in (11)-(14) may be serving as ligands in metal complexes, and this possibility is suggested by the low values of the velocity constants
for over-all decay relative to those expected for diffusion-controlled reactions of neutral molecules. Arrhenius plots (Figure 4) of log k vs. T-I gave apparent activation energies of 14.8 and 13.0 kcal/mol and preexponential factors of 4 X l O I 4 and 10“ M-’’2 sec-l for RI and RP, respectively. The apparent activation energies are high for purely diffusioncontrolled reactions. Because of the low activation energies for radical combination reactions, the major parts of the E A values could presumably be associated with reactions 13 and 14,respectively. Acknowledgments. This research was supported by Robert A. Welch Foundation, Grant A-177. The esr spectrometer was made available by National Science Foundation Grant GP-3767. (13) P. Smith and P. B. Wood, Can. J . Chem., 45, 649 (1966). (14) J. K. Kochi, Science, 155, 415 (1967).
The Effects of Magnetic Exchange Interactions on the Rates of Electron-Transfer Reactions
by John F. Endicott Department of Chemistry, Boston University, Boston, Massachusetts OR216 (Received November 21, 1968)
Magnetic exchange interactions between adjacent paramagnetic metals in solids and in known binuclear complexes are often very large even at room temperature. Similar interactions would be expected to occur in the “activated complexes” of at least some electron-transfer reactions. Since such interactions would define the spin alignment of donor and acceptor orbitals (at the reductant and oxidant, respectively), some magnetic restrictions on the probability of electron transfer are to be expected. A simplified application of the theory proposed for magnetic interactions in solids leads to the conclusion that at least some very slow reactions may involve a magnetic restriction on the electron-transfer probability.
Introduction Electron-transfer reactions between metal ions in solution are simple enough that very detailed mechanistic information may be obtained and that reasonably sophisticated theoretical treatments of the reaction rates have been developed. Many of the theoretical discussions have found qualitative and semiquantitative experimental justification (for pertinent reviews and discussion see ref 1-4). Despite the great deal of analytical thought and discussion, very large reactivity differences in some of the seemingly simplest systems have not yet been satisfactorily accounted for. Probably the most remarkable instance of this frustration is the -I06-fold variation in The Journal of Physical Chemistry
reactivity observed for the simple isotope-exchange rates between aquo ions in solution4 M3+ + *M2+ 142+ + *M3+ A common starting point in most theoretical discussions is the formation of a binuclear ‘‘intermediate’’in which the reactant metal centers are near enough that the interaction of donor and acceptor orbitals can lead (1) R.A. Marcus, Ann. Rev. Phys. Chem., 15, 155 (1964). (2) W. L. Reynolds and R. W. Lumry, “Mechanisms of Electron Transfer,” Ronald Press, Inc., New York, N. Y., 1966. (3) J. Halpern and L. E. Orgel, Discussions Faraday Soc., 29, 32 (1960). (4) A. G. Sykes, Advan. Inorg. Chem. Radiochem., 10, 163 (1967).
EFFECTS OF MAGNETIC EXCHANGE INTERACTIONS to electron transfer. Frequently these electron-transfer reactions involve more than one paramagnetic species. If both metal centers of the binuclear “intermediate” involved in the electron-transfer step are paramagnetic, then there may be magnetic exchange and thus spin coupling in this species just as there is in many known, stable binuclear complexes.6 Such spin coupling can, in principle, result in an antiferromagnetic alignment of the magnetic orbitals of adjacent nuclei and thus inhibit electron transfer. The present paper uses the theory of magnetic exchange in insulating solidsa to classify electron-exchange reactions according to the probable spin alignment of paramagnetic nuclei in the reaction intermediate.
General and Kinetic Considerations To simplify the development of a model for magnetic exchange interactions in the activated complex for electron-transfer reactions, most of the discussion in this paper will be confined to one-electron exchange reactions (e.g., Fe2+ *Fea+)so that AG N 0. This discussion will be further simplified by examining interactions only in those reactions in which one ligand (e.g., H20) acts as a bridging group between the reactant metal centers in the activated complex for electron exchange ( i e . , reactions of the inner-sphere type4) There are two principal advantages of this latter restriction: (1) the Anderson treatment of magnetic exchange interactions is directly applicable and (2) known binuclear complexes can be regarded as direct analogs. The possibility of removing these several restrictions is considered briefly in a later section of this paper. A general reaction scheme (similar to that used by Marcus*) provides a convenient means of organizing the discussion. Thus the over-all reaction may be represented by a series of steps.
+
.’
M-X
+ *M+
M-X- *M + M +-X-*M
M-X-*M+
(1)
h!t +-X- *M
(2)
+ X-*M
(3)
M+
For purposes of this discussion it is not assumed that the intermediates, n!t-X-*M + and M +-X-*M, have the ligands so arranged that the donor and acceptor orbitals at the reactant metal centers are matched in energy. Thus, in the present treatment (1) and (3) represent simply the formation and dissociation of the respective dimeric species, and (2) simply represents the redistribution of charge within the binuclear intermediate. Following Halpern and Orgel,3 it is convenient to define an average lifetime, ri, of the binuclear intermediate and a characteristic lifetime, re, for the redistribution of charge in the binuclear intermediate.g There are then two important limiting cases: (1) T i >> re, in which case the over-all rate of electron exchange is proportional to the concentrattion of binuclear intermediate; (2) T i > r e. It should be noted that super exchange and direct exchange, defined in this mannerj6 do not provide (5) See E. Konig, “LandoltrBornstein. New Series,” Vol. 2, K. H. Hellwege and X. M. Hellwege, Ed., Springer-Verlag, New York, N. Y., 1966. (6) P. W. Anderson, “Magnetism,” Vol. 1, G. T. Rad0 and H. Suhl, Ed., Academic Press, Inc., New York, N. Y., 1963, Chapter 2. (7) This is not meant to imply that the aquo ion exchange reactions all have inner-sphere mechanisms or that the mechanism of these reactions is unequivocal. The argument which follows only examines interactions which must occur for inner-sphere reactions. The aquo ion reactions are used for examples and comparison (Table 111) because of their simplicity and because some relevant experimental information exists. (8) R. A. Marcus, J. Chem. P h y s . , 24, 966 (1956). (9) Note that in the present formulation KZ = ka/k-a ‘u 1 and that if Y is the frequency of charge transfer in the intermediate, then re may be defined as re = l / v . (10) See especially ref 3 and 11 and reviews in ref 1-4. (11) P. George and J. S. Griffiths, “The Enzymes,” Academic Press, Inc., New York, N. Y., 1959, pp 1, 289.
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JOHN F. ENDICOTT
mechanisms for the net transfer of electronic charge. Furthermore, the interaction energies seem to run in the order:6,12double exchange super exchange > direct exchange. Thus in the present context, ie., for r i < re, the antiferromagnetic interactions are expected to be larger in magnitude. Although super-exchange interactions are generally larger in magnitude than direct-exchange interactions, the net spin exchange interaction can be ferromagnetic if the number of the latter terms is sufficiently greater than the number of super-exchange terms.6 The most relevant of the d-orbital interactions are presented in Table I. For a linear AI-X-*M+ geometry, the classification of orbital interactions (Table I) follows Anderson's prescriptions. With nonlinear geometries there is a much greater multiplicity of interactioq6 and they probably vary greatly in their magnitudes. For example a strong dZY-pY-d*ZY ( T ) interaction in a linear geometry might be expected to be much weaker when the lV-X-*Al angle is less than 180"; furthermore, as the R!I-X-*RI angle becomes small (-90") the direct drz-d*zz overlap should begin to make an antiferromagnetic contribution.6 Since the magnitudes of these interactions are difficult to assess, two limiting cases of bent geometries, the doubly bridged and triply bridged dimers, are also considered in Table I. I n the case of the triply bridged (of face shared) dimer the antiferromagnetic interactions are probably either e,-s-e*, ( u ) or due to the direct overlap of tzg wave functions. The spirit of Anderson's argument seems to be that the u interactions would be much the largest and that the tzg-t*ze interaction mould not be important except for cases where the tzg wave functions have considerable spatial extension or where the larger terms are not present. For purposes of the enumeration of terms in Table I1 no allowance has been made for difference in the different magnitudes of the various interactions. From a comparison of terms enumerated in Table I with data in Table I1 it can be concluded that direct, antiferromagnetic tz,-t *zg interactions are relatively small for the relevant first row transition metals. Thus these terms (included in parentheses in Table 11) should be omitted in the comparison of ferromagnetic and antiferromagnetic term^.'^^'^ Anderson has shown that for a linear geometry the antiferromagnetic interactions are larger in magnitude than the ferromagnetic interactiow6 Thus a binuclear complex is expected to have two paramagnetic metals with their net spins parallel only if the number of ferromagnetic interactions significantly exceeds the number of antiferromagnetic interactions. Examination of Tables I and I1 suggests that (1) for a linear complex, the net spins of paramagnetic metals are paired if the number of antiferromagnetic terms, N A ,is greater than one half the number of ferromagnetic terms, NF; (2) that for a doubly bridged complex, the antiferromagnetic interactions are weaker and the
>
The Journal of Physical Chemistry
Table I : Interactions between Magnetic Orbitals for Different Dimer Geometries" Antiferromagnetic
Ferromagnetic
A. Linear geometry M-X-*M d,z-d * E z d,,-d*z, d,rd*,,
da2-(d*ii, d*yz) dzE-(d*azj d*yz) dyi-(d*s2, d*zs)
B. Bent geometry (singly bridged) M-X
I
M* d,z-d *zz d,z-d *zy dyz-d*yz dzs-(d*r2, d*sc)
d,rd*,, dyz-(d*zS, d*zz) dzd*yz
C. Bent geometry (doubly bridged or edge-shared) X
/ \
M
M*
(triply bridged or face-shared) X
/ \ * \ /
M-x--M
X
d,z-d * z z dzz-y2--drzz-y2 dzy-d * z y d d *z. dye-d*yc
ds*(d*zs, d*yz, d*zy) dzZ-y*(d*zy, d*ss, d*yz) dzc-(d*z$, d*zZ-y2, d*zy, d*gz) dsy-(d*ye,d*,,, d*,%,d*Zz-y2) dys-(d*zz, d*z2-yZt d*zz, d*zy)
To simplify enumeration, the following conventions are used for naming orbitals. (1) For singly bridged dimers the M-X bond defines the z axis and the X-M* bond the z* axis. T h e other coordinates of the M-X-*M plane are the x axis (perpendicular to M-X) and the x* axis (perpendicular to X-*M). Thus the z, z*, x, and x* axes are coplanar. (2) For the doubly bridged dimer, the bridging atoms are placed at the intersection of the z and z* axes and the x and x* axes. Again z, z * , x, and x* are coplanar. ( 3 ) For the triply bridged dimer, the bridging atoms are placed at the intersection of the z and z* axes, the x and x* axes, and the y and y * axes.
-
complexes can only be antiferromagnetic when N A NF; (3) that the bent, singly bridged dimer should be intermediate between the above two cases. (12) P. G. de Gennes, Phys. Rev., 118, 141 (1960). (13) It should be noted that if only the e, orbitals are magnetic, the
faceshared dimer would have spins coupled antiferromagnetically. (14) The face-shared and edge-shared dimers are not considered as useful models for reaction intermediates in the context of this discussion. The formation and dissociation of such an intermediate would have to occur in two or more steps, and this would tend to make the condition T , J B ~>J I-. Since a decrease in antiferromagnetic coupling corresponds to an increase in the probability20 of electron transfer, electron-exchange rate constants for the Cr2+J+ and V2+t3+reactions should increase in the same order; L e . , k ~ J B r - > J I - for the coupling. For (VV-X-V)~+and (CI-X-C~)~+ dimers, the antiXhe Journal of Physical Chemistry
Outer-Sphere Electron-Exchange Reactions Electron-exchange reactions for which the activated complex does not have at least one ligand corrdinated simultaneously to both reacting metal centers are more difficult to analyze for spin-exchange interactions. The absence of a bridging ligand suggests that the dimer formalism developed above may not be directly applicable. There is an additional problem in that the geometry of the activated complex ( L e . , the disposition of ligands around the RIII-*h!I+ axis) is not well defined. Furthermore, the magnetic exchange interactions considered above are largest in magnitude when mediated by a ligand.6 And finally the interaction lifetimes are
+
EFFECTS OF MAGNETIC EXCHANGE INTERACTIONS no doubt very short (probably of the order of “solvent cage’’ lifetimes) for metal centers which react by means of an outer-sphere mechanism. For these reasons a specific model for magnetic exchange interactions in outer-sphere reactions is not considered here. One point that should be noted is that there are probably also different spin-multiplicity restrictions for outer-sphere reactions: the rule of net spin conservation (Condition 1: AST = 0) formulated above for inner-sphere reactions should be replaced by the more restrictive prohibitions on spin-multiplicity changes at each metal center ( A S = 1, AS* = 1). This spin restriction has been suggested as a major reason for slowness of the Co(NHs)2+ *CO(NH&~+reaction.4,28-30
+
Extension to Reactions for Which
CYGO # 0 . The above discussion of spin-correlation effects is, in principle, as applicable to reactions between metal ions of different elements as it is to reactions between ions of the same element. Unfortunately, even a qualitative discussion of the “mixed” electron-transfer reactions is far more complex. As noted above, several factors seem to make contributions to reactivity. I n a consideration of exchange reactions some of these factors can be assumed similar through a series of comparisons or can be neglected altogether. A notable example of the latter is the variation of reactivity with the free energy of reaction. Although correlations between AGO and reactivity seem to be common for electrontransfer reactions, there is little agreement on the generality of such correlations or the form which they should take (see discussion in ref 1, 2, 4, and 31-33). Despite these difficulties some general observations concerning this application are of interest. If the formalism of eq 1-3 is used to describe the “mixed” reaction, then in general the equilibrium constant for step 2, .& > 1, and RiI-X-*M+ and M+-X-*M are different chemical species. The magnetic exchange model discussed above should apply to both M-X-*M+ and M+-X-*M. Electron transfer is magnetically allowed only if the transfer from donor to acceptor orbital does not require the electron to invert its spin moment.34
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Summary and Critique Anderson’s analysis of magnetic exchange interactions in solids and the magnetic properties of relevant binuclear complexes of transition metals have been used to develop a model for analyzing the effects of magnetic exchange on the rates of inner-sphere electron-transfer reactions. It has been shown that magnetic exchange effects can result in antiferromagnetic coupling between reactant metals and such coupling can account for the relative slowness of some electron-exchange reactions. Systematic, predictable variations in the kind of magnetic exchange terms also account for some of the specific ligand effects observed in these reactions. The treatment presented here has necessarily been qualitative. The theoretical estimation of coupling parameters is still very crude.6 However as more reliable information becomes available about the magnetic properties of dimers formed from paramagnetic metals, it should be possible to obtain reasonably appropriate estimates of the magnitudes of the various interaction terms. Such information would permit a nearly quantitative evaluation of the effects of magnetic exchange on electron-transfer reactions.
Acknowledgment. The author gratefully acknowledges support of this research by the National Science Foundation through research grants GP 3467 and GP 7849.
(28) (a) D. R. Stranks, Discussions Faraday Soc., 29, 73 (1960); (b) N. S. Biradar and D. R. Stranks, Trans. Faraday Soc., 58, 2421 (1963). (29) The exchange rate constant (extrapolated to 25O) of -10-12 M-1 sec-lz8 for this reaction can be contrasted to an exchange rate M-1 sec-1 for the Ru(NHa)$+ *Ru(NHs)&S+ constant of ~ 1 0 3 reacti0n.m (30) T. J. Myer and H. Taube, Inorg. Chem., 7, 2369 (1968). (31) N. Sutin, Ann. Rev. Phys. Chem., 17, 119 (1966). (32) R. C. Patel and J. F. Endicott, J. Amer. Chem. Soc., 90, 6364 (1968). (33) N. Sutin, Acct. Chem. Res., 1, 225 (1968). (34) This may be stated symbolically as follows: if S and 8’ are the total spin momenta of iM-X-*M and M +-X-*M, respectively, then the transfer of an electron is spin allowed if S - 8’ = 0.
+
+
Volume 78, Number 8 August 1989
