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. 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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.