J . Phys. Chem. 1988,92, 3049-3056
3049
Electronic Structure Analysis of Electron-Transfer Matrix Elements for Transition-Metal Redox Pairs Marshall D. Newton Chemistry Department, Brookhaven National Laboratory, Upton, New York 1 I973 (Received: July 22, 1987;
In Final Form: October 28, 1987) Calculated electron-transfer matrix elements (Hif)for redox processes of the type ML62++ ML:+ * MLs3+4- ML62+(M = Fe, Co, or Ru; L = H 2 0 or NH3) have been analyzed in terms of various orbital concepts. The matrix elements are based on ab initio wave functions for model supermoleculeclusters of the type (ML,-.L,M)S+, with n = 1 or 3 . The many-electron Hifquantities are analyzed as effective one-electron expressions of the type Hif a (X’)2hC,,Lf,where X’ is the metal-ligand covalency parameter and hi,Lris a local one-electron matrix element for ligand orbitals in contact in the transition state. The data for apex-to-apex approach of reactants yield an excellent linear least-squares fit (r2 = 0.996) and imply a value of -5000 cm-’ for thus showing that significant coupling can occur in the absence of formal bonding between reactants.
Introduction Electronic structure plays a pivotal role in the kinetics of electron-transfer reactions, as one might well expect from the name of this class of reactions.14 If the initial (Qi) and final (qf)states of an electron-transfer process are characterized in terms of diabatic potential (or free) energy surfaces, then one recognizes the distinct roles of electronic structure in determining both the energy surfaces themselves (diagonal matrix elements) and the coupling between them (off-diagonal matrix elements), without which the reaction would not proceed. Much of the current interest in the off-diagonal coupling elements (referred to below simmply as transfer integrals) has been focused on the distance dependence of their magnitude for situations of transfer over relatively large distances (tens of angstroms), where questions of electronic overlap and electronic participation by the intervening medium clearly become ~ r i t i c a l . On ~ ~ the * ~other hand, even for relatively small donor-acceptor systems essentially in contact, kinetically significant electronic structural effects on transfer integrals may be expe~ted.~ In the present paper we develop this notion by analyzing transfer elements for a set of prototype transition-metal complex redox pairs, as modeled by a b initio electronic structure calculations for suitable supermolecule clusters.I0 In particular, we shall consider the following class of reactions involving hexacoordinate complexes in aqueous solutions ML62+
+ MI+j3+
MLL ,~,+
+ ML,52+
(1)
(1) Halpern, J.; Orgel, L. E. Discuss. Faraday SOC.1960, 29, 32. (2) McConnell, H. M. J . Chem. Phys. 1961,35, 508. (3) (a) Newton, M. D. Int. J . Quantum Chem., Quantum Chem. Symp. 1980, No. 14,363. (b) Logan, J.;Newton, M. D. J . Chem. Phys. 1983, 78, 4086. (c) Logan, J.; Newton, M. D.; Noell, J. 0. Int. J. Quantum Chem., Quantum Chem. Symp. 1984,’No. 18, 213. (d) Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1984,35,437. (e) Newton, M. D. J . Phys. Chem. 1986, 90, 3734. (0 Newton, M. D. In Tunneling; Jortner, J., Pullman, E., Eds.; Reidel: Dordrecht, Holland, 1986; p 305. (4) (a) Miller, J. R.; Beitz, J. V. J . Chem. Phys. 1981, 74, 6746. (bJ Closs, G. L.; Calcaterra, L. T.; Green,”. J.; Penfield, K. W.; Miller, J. R. J . Phys. Chem. 1986, 90, 3673. (5) (a) Larsson, S. J . A m . Chem. SOC.1981,103, 4034. (b) Larsson, S. Chem. Phys. Let?. 1982, 90, 136. (c) Larsson, S. J . Chem. SOC.Faraday Trans. 2 1983, 79, r375. (d) Larsson, S. J . Phys. Chem. 1984,88, 1321. ( e ) Larsson, S.; Stahl, K.; Zerner, M. C. Inorg. Chem. 1986, 25, 3033. (6) (a) Hopfield, J. J. Prm. Natl. Acad. Sci. U.S.A.1974, 71, 3640. (b) Beratan, D. N.; Onuchic, J. N.; Hopfield, J. J. J. Chem. Phys. 1985,83, 5325. (c) Beratan, D. N.; Onuchic, J. N.; Hopfield, J. J. J. Chem. Phys. 1987, 86, 4488. (7) (a) Ondrechen, M. J.; Ratner, M. A.; Ellis, D. E. Chem. Phys. Lett. 1984, 109, 50. (b) Mikkelsen, K. V.; Ratner, M. A. Chem. Rev. 1987, 87, 113. (8) (a) Ohta, k.; Close, G. L.; Morokuma, K.; Green, N. J. J . Am. Chem. SOC.1986,108, 1319. (b) Ohuta, K.; Morokuma, K. J . Phys. Chem. 1987, 91, 401. (9) Kuki, A.; Wolynes, P. Science 1987, 236, 1647. (10) Details concerning the ab initio calculations are given in ref !b,c,e. A preliminary discussion of the analysis of the calculated results in terms of orbital concepts is given in ref 3f.
0022-3654/88/2092-3049$01.50/0
where M and L refer to a transition metal and a ligand, respectively. Inevitably, in modeling condensed-phase reactions of this type, one defines localized donor (D) and acceptor .(A) species which, typically, are immersed in a dielectric medium.3d In the present case (eq I), the D and A species are the ML6*+I3+redox partners, whose structural identity is maintained throughout the reaction (i.e., a so-called “outer-sphere” reaction, in which M-L bonds are not disrupted (see below)). Since the D and A species are taken to be in contact in the transition state, we may consider the influence of the outer aqueous medium on the transfer integrals to be relatively minor and concentrate on the role played by the electronic properties of D and A.” The electronic aspects of ligand/metal interactions identified as being important for understanding transfer integrals may be quantitatively assessed both by model calculations, as reported here, and also by various experimental probes. Some examples of the latter are discussed below. Before addressing the specific systems of interest, we will find it useful to consider some general questions about electron-transfer reactions. These questions will be considered again at the end, in the context of the detailed analysis reported below. To what extent may electron-transfer reactions be considered unique? Aside from specific features such as polaron coupling to a dielectric medium, one in general expects relatively “weak” coupling beween initial and final states. Whereas the diabatic model invoked above is a generic one applicable to many types of chemical reactions, including those involving complex rearrangements of chemical bonds, electron-transfer reactions (at least those of the outer-sphere type, to which we confine our attention) do not entail disruption of short-range chemical bonds-hence the description of the initial state/final state interactions as “weak” (Le., a small transfer integral).3d These notions may be discussed in terms of schematic energy profiles along a reaction coordinate, as depicted in Figure 1. In Figure 1a we see the extreme of very weak coupling, in which a transfer integral of very small magnitude yields a very “cuspy” avoided crossing of the diabatic energy surfaces. The surfaces for \k, and \kf are distinguished by the fact that some structural coordinates have equilibrium values which differ in the initial and final states, even though bonds are not “broken” (Le., they are only “perturbed”). The other extreme (Figure lb) displays a “smooth” transition state associated with strong mixing of \kf and \kf in the interaction region, which is seen to be relatively broad compared with the (11) ):( Situations where the role of an intervening or enveloping “medium may be appreciable have been dealt with in ref 1 , 2,4-9, and 1lb-d; ref 1 1 b,c deals with the consequences to transfer through a medium, whereas ref 1 Id emphasizes the long-range influence of the medium on the electronic structure inside a “cavity” containing the donor and acceptor. (b) Kuznetsov, A. M. Faraday Discuss. Chem. SOC.1982, 74, 49; Chem. Phys. Lett. 1982, 91. 34. (c) German, E. D. J . Chem. SOC.,Faraday Trans. I 1985,81, 1153. (d) Mikkelsen, K. V.; Dalgaard, E.; Swanstrom, P. J . Phys. Chem. 1987, 91, 308 1 .
0 1988 American Chemical Society
3050 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
In terms of eq 2-6, the extremes represented by Figure 1 correspond to the limits of nonadiabatic (Figure l a , K , ~
