J. Phys. Chem. 199t96.7187-7193
7187
ARTICLES Solvent Dynamicai Effects in Electron Transfer: Some Predicted Modifications in the Presence of Reactant Vibrational Activation and Comparisons with Experlment Donald K. Phelps and Michael J. Weaver* Department of Chemistry, Purdue University, West Lqfayette, Indiana 47907-1393 (Received: March 4, 1992; In Final Form: April 27, 1992)
The predicted alterations in the solvent-dependent reaction dynamics of electron-transfer processes brought about by the presence of reactant vibrtional (inner-shell) activation in addition to overdamped solvent motion are examined by using a of nuclear modified form of the theoretical treatment due to Sumi, Nadler, and Marcus. Allowance is made for the tunneling involving the high-frequency vibrational modes and for barrier crossing driven by solvent inertial polarization as well as vibrational activation. The latter enables the emergence of the zero-friction (transition-statetheory, TST) limit to be included along with providing a unified description in the absence and presence of vibrational activation. Attention is focusad on the calculated dependenciesof the reaction rate constant,ku,and barriercrossing frequency, v,, upon the overdamped solvent relaxation time, 7L, over a range of parameters and in a format which can be compared directly with experimental ranging &, from 0 to 1.5 are solvent-dependent kinetic data. Ratios of inner- and outer-shell reorganization energies, &I explored, corresponding to commonly encountered values of the vibrational barrier height. The addition of small or moderate in kU as well as v,,. Unlike barrier components arisiig from high-frequency vibrations is predicted to yield substantial inthe TST limit, however, ket and v, can still exhibit marked dependencies on rL under these conditions, especially when the vibrational frequencies are much higher than TL-'. The log v,-log T ~ - 'slopes, however, are commonly predicted to lie well below unity even for small (ca. 0.5-1 kcal mol-') inner-shell barriers, as a result of the coupled influence of vibrations and solvent inertia, and decrease markedly for larger barriers. Comparisons with solvent-dependent kinetic data indicate only semiquantitative agreement: the attenuation of the rate-TL-l dependencies by reactant vibrational activation as deduced from some experiments is less marked than predicted by the theory.
The mqnition that the solvent can exert important influences on the barrier-crossing dynamics of electron-transfer reactions, as for other types of chemical processes, has attracted considerable reccnt attention. (For ovaviews, see refs 1 and 2.) In many cases, the frequency of collective solvent motion required for electron transfer is expected to be markedly slower than the limit prescribed by the inertia of the molecular solvent dipoles. The resulting reaction kinetics are described in terms of overdamped solvent motion, with the s ~ ~ ~ l"solvent l e d friction" determining to what extent the barrier-crossing dynamics fall below that anticipated from the conventional transition-state theory (TST).'" The majority of quantitative experimental examinations of such solvent dynamical effects have involved ultrafast photoinduced processes where the activation barriers are small or negligible.' While complicated by the need to separate the dynamical and energetic components of the observed solvent effects, some reliable information along these lines has also been obtained for thermal electron-exchange reactions.2 The latter class of reactions, featuring moderate or large activation barriers (AG*1 5-10 kBT), have also been the focus of a plethora of theoretical approaches. (Refmnces 3-10 arc only a rqnemtative selection.) They predict close links between the nuclear barrier-crossing frequency, v,, and the relaxation times for overdamped solvent dynamics, in accord with some experimental data for adiabatic electron-transfer reactions as extracted from solvent-dependent rate data.2 Most of these theoretical treatments assume that the entire activation barrier, AG*,is associated with solvent reorganization. A very commonfeature of experimental systems, however, is the presence of a significant and even substantial additional component of AG* from intramolecular reactant (Le., inner-shell) distortions, especially associated with bond vibrations. This arises from the well-known tendency of electron addition or subtraction to yield significant changes in reactant bond lengths and angles, thereby forming part of the activation process required for reaction.11As
a consequence, it is of particular practical interest to explore the manner and extent to which the reaction dynamics arc influenced additionally by such bond distortions, thereby diminishingthe role of solvent dynamics upon the barrier-crossing frequency. Direct experimental information on this question is spar~e.l2-~5 This is due in part to the paucity of reactions having welldefmed (and preferably variable) inner-shell barriers together with the other properties required for quantitatively interpretable solvent-dependent studies. In addition, it is usually difficult to distinguish experimentally between the attenuation of solvent dynamical effects caused by reaction nonadiabaticity (i.e., from insufficient donor-acceptor electronic coupling) as opposed to inner-shell activation. As a consequence, theoretical guidance as to the anticipated experimental manifestation of these effects would be invaluable. Probably the most w f u l theoretical treatment of the coupled effects of reactant vibrations and overdamped solvent dynamics in electron-transfer processes is due to Sumi, Nadler, and Marcus (referred hereafter to as the "SNM"theory)? Basically, their model is a modified version of the conventional picture involving solvent diffusional motion along a polarization coordinate, X. Rather than such dipolar-solvent fluctuations consummating the reaction, however, barrier crossing is perceived to occur instead by motion along a (separate) vibrational reaction coordinate, q.&ls The nature and extent to which the latter motion influenas the overall reaction dynamics are determined in part by the relative inner- and outer-shell reorganization enegies, & and A,,. When & > k,) where activation proceeds largely via the thendominant q coordinate.6J6 A commonly anticipated experimentalsituation is where & is sisnificant relative to k, yet & S X,(X,is often in the range 15-25 kcal mol-'). Even though the vibrational motion is usually characterized by higher effective frequencies, the overdamped solvent dynamics can still largely determine the reaction rate since the system must diffuse (along X ) up to the barrier-top region where consequent motion along q is effective to consummating e l a w n transfer without too mere an additional energy cost. The situation is illustrated pictorially in Figure 1; the ellipsoids r e p resent equipotential contours, with their centers representing the energy minima associated with reactants and products. Unlike the TST treatment where the system is in quasi-equilibrium at all points along the reaction coordinate, the influence of the solvent motion upon the reaction rate may therefore still be felt when these overdamped dynamics are much slower than the reactant vibrations. Consequently, then, the nature of the solvent dependence of the adiabatic reaction kinetics is expected to be determined by an intttesting interplay between thew factors, ranging from total control by overdamped solvent motion to an emerging dominance by reactant vibrational (Le., solvent-independent)dynamics. The purpose of this communication is to provide a numerical assessment of the adiabatic solvent-dependent dynamics for e l e c t r o n - e x ~ ereaction$ under these circumstanca, for a range of experimentallyrelevant amditions, in a fashion most transparent to, the therefore usable by, experimentalists (such as ourselves). Two modifications are made to the SNM model for thip purpose. The first involves the explicit inclusion of solvent inertial effects in the calculations. In addition to yielding the appropriate TST (i.e., zero-friction) l i t , " inclusion of this factor provides a self-consistent treatment of the barrier-crossing dynamics in the absence as well as presence of the vibrational reaction coordiante (vide infra). The sccond modification involves the incorporation of quantum (nuclear-tunneling) effects in the vibrational dynamics. Perhaps not unexpectedly, this factor turns aut to yield siignificant modifications to the predicted form of the solvent-dependent dynamics, especially in the presence of high-frequency vibrations. We a h consider briefly some pertinent experimental data in comparison with these theoretical predictions. l h m t i c a i Framework
Prior to presenting and discussing the numerical results, it is mathematical features appropriate to describe briefly the -tial of the SNM model along with the modifications employed here. Sumi and Marcus model the electron-transfer reaction with the reaction-diffusion equation:"
The distribution function P [=P(X,t)] describes the probability of finding the reactant system at potarization X at time t , and Yequals S / 2 , the solvent polarization potential. The diffusion
Q(t) = s m -m P ( X , t )dx
(3)
The resulting average survival time, T*, can be used to estimate the electron-transfer rate constant simply from ka = T [ I . This numerical solution is employed in the model calculations below. Since Nadler and Marcus provided a detailed description, we merely summarize the necessary mathematical detail required for its implementation in Appendix A. The solvent polarization coordinate, X , and the vibrational coordinate, q, are of course only simpWkations to the real physical situation. The solvent polarization of the system is a function of numerous nuclear and electronic coordinates, and the vibrational motion is determined by the quantum mechanical state of the reactant. Sumi and Marcus were able to simplify this to a single polarization coordinate, X,as a result of the earlier seminal work of Marcus.'* The free energy of the system is assumed to be quadratic in X; i.e., v(x)= 3 / 2 . The reactant vibrational modes are treated classically in ref 6 and have been reduced to a single harmonic coordinate, q, with a vibrational frequency, vq, given by a weighted average of all the relevant normal-mode frequencia. (Note that, unlike ref 6, v as used here refers only to an adiabatic process; nonadiabatic effects are not considered in the present study.) The vibrational coordinate is assumed to be fast relative to the solvent polarization coordinate and to always be in q u i librium with the surrounding bath. Four limiting cases of the SNM model are usefully disting u i s h a P (1) the "slow diffusion" limit, (2) the "fast diffusion" limit, (3) the "narrow reaction window" limit, and (4) the "wide reaction window" limit. The last two c88t8 are noted above. Case 2 is also worth mentioning in the present context: this refers to the situation where the diffusion along Xis rapid relative to the consequent passage over the barrier via the q coordinate. In this circumstance, T,,-I >> k,, where k, is the "thermally equilibrated" rate constant: yielding prctquilibrium at all points along the reaction hypersurface (Le. the TST limit). More generally, however, solvent friction impedes progress along the X coordinate so that the actual rate constant k, falls below k,. This situation can be described by eqs 1-3 above together with an appropriate description of the dynamics along the q coordinate as contained in &(a. An expression for k(X) is6 k(X) = vq ~ x P [ - A G ( X ) / ~ B ~ ' ~ (4) where AG(X) is the barrier height along the q ooordinate starting from a given X polarization coordinate. While clearly useful in the presence of significant inner-shell activation, eq 4 is strictly inapplicable in the "narrow reaction window" limit, when the q coordinate eventually vanishes. A more broad-based treatment involves replacing uq with a generalized "reactive" frequency, v,, defined byI9 v, = [(wo/2.)2(X0/X,) + vg2]1/*
-
(5)
where wo is the solvent inertial frequency." Since usually vQ > w o / 2 u , when A,, A, then v, = vq. I n the ("narrow reaction
,
Solvent Dynamical Effects in Electron Transfer window") limit when &/Xi a,k(X) = (wo/*/2*)6(X - Xc), yielding a delta function reaction term at the transition state (see Appendix B). Physically, in the absence (or near absence) of the vibrational coordinate solvent inertial fluctuations are considered to send the system over the barrier once the transition state has been approached. As already noted, in addition to providing a formally self-consistent treatment of bamer cmsing in the absence and presence of inner-shell activation, this approach yields a TST-like result in the absence of solvent friction. The SNM model presumes that vibrational motion along the q coordinate is classical. For many experimental systems, however, the inner-shell frequencies are sufficiently high so that nuclear tunneling becomes significant, whereby barrier crossing is accelerated by tunneling from points energetically below the classical transition state. Such nuclear-tunneling effects are most rigorously accounted for by a quantum-mechanical calculation of transition rates between all the available reactant and product states. More commonly, however, "semiclassical" treatments are employed that entail calculating a correction factor, r,, to the classical rate. Approximate closed-form expressions are available for this purpose.20 The approach employed here involves evaluating r, for each k(X) vibrational contribution along the solvent polarization coordinate. Since r, depends on the reaction exothermicity,AEo, as well as the reorganization energy Xi and the vibrational frequency Y,, the nuclear-tunneling correction will generally be a function of X,so that it is necessary to calculate r, for each k(X). This involved utilizing the expression20s21
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7189
+
The required X-dependent driving f o r k along the q coordinate, AE, was obtained as outlined in Appendix C. One limitation of eq 6 is that it applies to a cusplike barrier, i.e., does not allow for barrier-top roundedness resulting from donor-acceptor electronic coupling. The assumption of a cusp bamer, however, is contained within the SNM model (vide infra). One other, more minor, modification to the SNM model made here is prompted by our desire to calculate the reaction dynamics for elcctron-exchange processes (either homogeneous self-exchange or electrochemical exchange), since most experimental data gathered so far are for such reactions. In this case, the free energy driving force is necessarily zero, whereas the SNM model was formulated (for simplicity) for highly exoergic reactions. The correction required to transpose the latter results to the former case, however, is straightforward, involving merely a 2-fold decrease in the barrier-crossing frequency.22
Numerical Results and Discussion The usual tactic for exploring solvent dynamical effects in activated electron-exchange reactions involves evaluating the observed rate constant, k,,, in various media and extracting the dynamid component of the k,--sohrent dependence by correcting for the solvent-dependent bamer height, AG*,either from theory or from near-infrared optical measurements2 This approach is most straightforward (and reliable) when polar Debye (or near-Debye) solvents are used, i.e., those exhibiting a single (Debye) relaxation in the dielectric-loss spectra, enabling values of T L to be extracted. Fortunately, polar Debye solvents are available that span a sizable range of rLvalues, from ca. 0.2 to 10 ps at ambient temperatures,23so that the anticipated variations in the overdamped dynamics can be markedly larger than the uncertainties in the solvent-dependent AG* values.2 Reflecting this experimentalsituation, Figure 2 displays plots of log k, versus log rL?, over the TL range 0.15-1 5 p, calculated
1
I
v. = 4.5 x 10" s-'
V I
11.3
I
11.8
I
12.3
12.8
Figure 2. Logarithmic plot of unimolecular rate constant versus inverse solvent longitudinal relaxation time for a sequence of inner/outer shell barrier ratios, Xi/& = 0, 0.1, 0.2, 0.5, and 1.0. X, = 16 kcal mol-', vP. = 4.5 X IOt3 s-I (Le., 1500 cm-l), and 4 2 % = 1.8 X 10l2 s-l.
from the present modified SNM model for five &/A,, ratios as indicated, from 0 to 1.O. The A,, value is fixed at 16 kcal mol-' (equivalent to an outer-shell bamer, AG,*, of 4 kcal mol-'), and u, IS taken to be 1500 cm-'(r4.5 X 1013s-I). This relatively high bond frequency was chosen to accentuate the influence of vibrational dynamics, although nuclear-tunneling corrections are not applied in Figure 2 for simplicity. The solvent inertial frequency, wo, was fixed at 10 ps-', appropriate for acetonitrile and other small polar solvents." Since the barrier height is constant for each curve in Figure 2, the rL-induced variations in log k,, reflect purely the dependence of the barrier-crossing dynamics upon solvent dynamics. F v 2 contains several instructive features. All the log k,-lOg T ~ traces - ~ are curved, their slopes decreasing toward higher rL-' values. This curvature arises chiefly from the influence of solvent inertia. Thus, the log ket-log 7L-I slope for &/A,, = 0 diminishes increasingly below the value of unity, corresponding to complete control by overdamped solvent dynamics, as the rate becomes increasingly limited by solvent inertia toward high 7L-l values. The addition of increasing amounts of inner-shell activation (Le., increasing &/&)exerts two immediately noticeable effects. Firstly, for &/A,, S 0.2 the rate at a given TL-' increases as Xi/&, is enlarged, even though the activation barrier is becoming higher under these conditions. This is because the increasingly unfavorable activation energetics are being more than offset by the more rapid reaction dynamics provided by the vibrational coordinate. For &/A,, 2 0.2, however, the former factor increasingly dominates, yielding sharply diminished reaction rates. The second clearly discernible consequence of increasing &/A,, is to depress the log k,-log 7L-l slopes. This reflects an inmasing control of the bamtr-crossing dynamics wrought by the reactant vibrational rather than the overdamped solvent dynamics as the barrier associated with the q coordinate is enlarged. Nevertheleas, w 2 that the log k-log 7L-l slopes it is interesting to note from F remain quite substantial, ca. 0.4-0.6,even in the presence of significant or even large vibrational bamers, &/A,, 5 0.5. Indeed, 0.1-0.2) even the addition of a small inner-shell bamer (&/A,, increases the log &,-log 7L-I dependence ab high 7L-I values. This surprising effect results from an attenuation of the rate-limiting influence of solvent inertia afforded by the presence of a more rapid (vibrational) barrier-crossing pathway. The numerical examination of a wider range of system parametersupon the solventdependentdynamics is clearly of interest. To this end, a more systematic set of numerical results is given
-
7190 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 1
I
Phelps and Weaver 1
I
I
"'"1 4
I
I
I
0.5
w*
1.o
I
I
I
I
0.5
1 .o
I
1.5
1.5
Figure 3. Plot of log k, versus xi/& with & = 16 kcal mole1,u = 4.5 X lOI3s-' (is., 1500 cm-I), and w,,/2w = 1.8 X IOL2s-l. The solid/%ashed
Figure 5. As in Figure 3, but with X, = 16 kcal molp1and u9 = 1.5 X 1013 s-1.
curvca are calculated with/without nuclear tunneling.
. 1 .o
0.5
0.5
1.o
1.5
Ai/h
WO Figare 4. As in Figure 3, but with & = 24 kcal mol-I and
Figure 6. As in Figure 3, but with X, = 16 kcal mol-I and vP = 4.5 u9
= 4.5
X
1013s-1.
in graphical form in Figures 3-6. Each of these figures consists of plots of log k, versus &/A,,, again for a fmed A,, value as indicated. The sequence of traces shown in each figure refers to the log T'-I values as indicated, spanning from 10.8 to 12.8 (Le., the same TL range as on the x axisof F w 2). The corresponding dashed and solid lines refer to calculations performed in the ab#nce and p"x,respsctivey, of nuclear tunneling comctions. (The four cro89e8 marked on they axes of Figures 3,5, and 6 are the log ka values for each of these curves in the limit where &/X, 0, from F w 2.) In F v 3,5, and 6, A, is fmed at 16 kcal mol-' (Le., AGO,,, 4 kcal mol-'); a higher value, 24 kcal mol-', is chosen in Figure 4. The vibrational frequency is set at 4.5 X 10" s-' in Figures 3 and 4, while progressively lower values, 1.5 X lOI3 and 4.5 X 1012s-l (i.e., 500 and 150 cm-I), are chosen for Figures 5 and 6, respectively. The highest frequency is roughly appropriate for the sesquibicyclic hydrazine redox couples that we have recently examined in connection with reactant vibrational effects in electron transfer,Is while the lower two frequencies are representative of the metal-ligand vibrations commonly encoun-
-
-
1.5
X
10'2 s-1.
tered in inner-shell activation. A limitation of the calculations for the lowest vibrational frequency is that this value can be insufficiently faster than the overdamped solvent dynamics to satisfy the time scale separation requirement of the SNM model noted above. Inspection of these figus, both separately and together, reveals several significant points. As expected, nuclear tunneling exerts an increasingly important rate-acceleratingeffect toward larger vibrational frequencies. More notably, the inclusion of nuclear-tunneling corrections under these conditions can nevertheless enhance markedly the extent of solvent dynamical effects remaining in the presence of reactant vibrational activation. Thus, Figures 3 and 4 show that the vertical displacement between the various traces, corresponding to the sensitivity of log k, to variations in log TL-', can be substantially larger with consideration of q-coordinate nuclear tunneling. While the effect of nuclear tunneling is relatively small for (&/A,,) 5 0.2, markedly greater - ~ as well as larger k, values are seen log k,-log T ~ dependencies to arise for larger inner-shell bamers. This effect can be perceived most simply as arising from a diminution in the effective vibra-
Solvent Dynamical Effects in Electron Transfer
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7191 14.0
14.0
13.5
M
2
12.0-/
*
M
2
11.5
12.0j.
I
'
D
e
4
11.04
11.5
12.0
111.5
12.5
a reaction with selected vibrational barriers versus T ~ - ' ,with parameters appropriate for a series of commonly used solvents. (Note that a 45O line has a slope of unity.) The outer-shell barriers were calculated using the usual dielectric continuum model'*b assuming a pair of 4-A spherical reactants in contact, yielding selected AG*, = 4.5-5.5 kcal mol-'. The inner-shell (vibrational)activation energies selected are 0.0,0.5, 1.0,2.5,
5.0, and 10.0 kcal mol-', shown as crossed circles, circles, squares, triangles, inverted triangles, and diamonds, respectively. Filled/open symbols denote calculations performed with/without nuclear tunneling. The solvents are, from left to right, hexamethylphosphoramde,nitrobenzene, benmnitrile, dimethyl sulfoxide, Nfldimethylfmmidc, nitromethane, acetone, and acetonitrile, with inertial limit frequencics from ref 25 and T~ values from refs 23 and 24.
tional activation energy caused by nuclear tunneling "through" the barrier. Larger inner-shell barriers are therefore required to bring about a given attenuation of the role of solvent dynamics upon ka in the presence of nuclear tunneling than in its absence. The magnitudes of these differences, however, are sharply diminished as the vibrational frequency is lowered, so that nuclear tunneling exerts little influence on the predicted reaction dynamics for u = 4.5 X lo'* s-l, as seen by the near identity of the solid and lashed traces in Figure 6. Comparing Figures 3, 5, and 6 (for which uq = 4.5 X lOI3, 1.5 X lOI3, and 0.45 X lOI3 s-l, respectively) enables the effects of altering the vibrational frequency to be diwmed more generally. As expected, increasing uq enhances the degree to which ka is accelerated for a given & value. Nevertheless, the residual degree of rLdependentdynamics, Le., the log &,,-log 7L-I dependencies at a given &/X, value, also tends to be enhanced as uq is increased, especially after inclusion of nuclear tunneling. The latter, perhaps surprising, trend can be rationalized from the greater extent to which the rate should be limited by diffusion along the X coordinate as the ensuing vibrational motion over the bamer becomes more rapid. Such a result, whereby the solvent motion influences importantly the overall reaction dynamics when the overdamped frequencies are much lower than those of the coupled vibrations, is qualitatively different from the usual TST situation. Consequently, then, although the barrier-crossing frequency and even the reaction rate can be accelerated substantially by the presence of a vibrational component of the activation barrier, the reaction dynamics can still maintain a marked sensitivity to variations in the slower overdamped motion associated with the solvent polarization coordinate. The predicted effects of altering the outer-shell (solvent) bamer can be gleaned by comparing Figures 3 and 4, for which h, = 16 and 24 kcal mol-', respectively. While the former shows clearly the survival of sL-'-dependent reaction dynamics to larger Xi/& values, similar solvent dependencies are obtained when the com-
li.5
10dT;')
log(T;')
Flow 7. Logarithmicplot of the effective barriertrotsing frequency for
12.0
Figure 8. As in Figure 7,except with vq = 1.5
X 10"
s-I.
parison is made for a given absolute Xi value. While such "model-solvent" results provide a useful means of assessing general trends, it is also instructive to undertake related calculations for a sequence of actual solvents. Some illustrative results of the latter type are displayed in Figures 7 and 8. These are essentially logarithmic plots of the bamer-crossing frequency, extracted from rate constants, versus T~-I;this and related analym are commonly applied to experimental electron-transfer data with the aim of exploring solvent dynamical effects.2 The plots were generated for a series of eight polar solvents, commonly utilized for this purpose, as noted in the caption to Figure 7. They axis, log r,v,,, is essentially a logarithmic barrier-crossing frequency, extracted from the calculated rate constants by using log m u ,
log k,, + (AG*,
+ AG*is)/2.3kBT
(7)
The outer-shell barrier, A G * , was calculated in each solvent using the usual Marcus continuum model'8bfor a pair of 4-A reacting spheres in contact. (These values fall in the range ca. 4.5-5.5 kcal mol-'.) The (classical) inner-shell barriers, AG*h, chosen are 0 (crossed circles), 0.5 (circles), 1.0 (squares), 2.5 (triangles), 5.0 (inverted triangles), and 10 kcal mol-' (diamonds). The filled and open symbols refer to calculation with and without vibrational nuclear-tunneling corrections, respectively. The latter situation therefore constrains r,,to be unity. (The two highest sets of data points with inclusion of nuclear tunneling are omitted from Figure 7 for clarity; they are off scale on the plot shown.) The vibrational frequency is taken to be 4.5 X l O I 3 and 1.5 X 10" s-I in Figures 7 and 8, respectively. The log r,v, values for each AG*h chosen are plotted, as before, versus log T ~ - 'in Figures 7 and 8. The TL values are taken from compilations in refs 23 and 24; the solvent inertial frequencies,also required for the calculations, are as given in refs 17 and 25. The log I',v,,-log 7L-I plots in the absence of vibrational activation (crossed circles, Figures 7 and 8) display the expected near-unit slope (ca. 0.8-0.9) at low TL-' values, with deviations in the low-friction (i.e., high 7L-I) solvents arising from the emergence of partial rate control by solvent inertia, similarly to Figure 2. (Note that the "scatter" in the points for these media arises from variations in wo; cf. ref 23.) Adding small (0.5 or 1.0) vibrational barriers (circles, squares) yields, as expected, marked (ca. 5-10-fold) increases in m u , and some diminution in the log I',,v,-log 7L-I slopes (to ca. 0.4-0.65), especially for the smaller vibrational frequency (Figure 8 versus Figure 7). The presence of larger vibrational barriers, AG*i, 1 2.5 kcal mol-', attenuates severely the degree of solvent dynamical control, although some moderate TL-' dependence is observed even for AG*is = 2.5 kcal
7192 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992
mol-' in Figure 7 with inclusion of nuclear tunneling (filled triangles). Consequences and Comparisons with Experiment While the SNM model predicts the Occurrence of partial rate control by overdamped solvent dynamics for electron-transfer reactions featuring substantial inner-shell reorganization, the presence of even mild degrees of vibrational activation is clearly anticipated to attenuate significantly the 7Ldependence of the reaction dynamia. Two other related factors should be mentioned in this context that are expected to diminish further the role of solvent dynamics. First, in most cases the degree of donor-acceptor electronic coupling is predicted to be insufficient to yield complete reaction adiabati~ity.'~.~~ In the nonadiabatic limit, the barrier-crossing frequency should become essentially independent of the nuclear dynamics. Since this limit can be approached eventually by accelerating the nuclear dynamics, log kei-log 7L-1 plots are often expected to be nonlinear and exhibit fractional sl0pes.I~9~~ Direct experimental evidence for such behavior has been observed for several metallocene self-exchange reactions, featuring varying degrees of electronic coupling.23 Furthermore, even weaker rate-rL-' dependencies are predicted for bimolecular homogeneous and electrochemical processes as a result of systematic rL-'-dependent variations in the reactant internuclear geometries contributing to electron transfer.26 Taken together, then, there are good reasons to expect the occurrtnceof only mild log r a w o r log vn-) log 7L-1dependencies, especially in the presence of sisnificant vibrational activation. The metallocene self-exchange reactions are worthy of further consideration in this regard since the Xi/& values are known a p proximately, lying in the range ca. 0.1-0.25.27 An unusually quantitative solvent dynamical analysis could be undertaken for these systems, in part because of the availability of solvent-dependent activation barriers from optical data.z8 While the ferrocenium-ferrocene couples display virtually solvent-independent reaction dynamics resulting from weak electronic coupling, the fastest cobaltocenium-cobaltocene systems (particularly the decamethyl derivative Cp;Co+/O) yield bamercorrected log ratelog TL-' s l o p that approach unity at smaller log T ~ - Given ~ . ~that ~ Xi/&, 0.2 for the latter systemz7and uq = 1 X 1013s - ' , ~such behavior is somewhat unexpected on the basis of the SNM model. Some other experimental evidence also suggests that the SNM model may overestimate the attenuation of solvent dynamical effects in the presence of vibrational activation. A marked dependence of log v, upon log TL-' (albeit with subunit slopes) was observad for cobalt clathrochalate electrochemical exchanges, even though Xi/X, 1 and uq 1 X 1013 s-l.12 In addition, the electrochemical exchange of sequibicyclic hydrazines exhibits 1, near-unit log v,-log 7L-' slopes; for these systems Xi/& although the v values are suffciently large, -5 X 10" s-I, to yield a prediction o? significant rate-.sL-I dependencies from the SNM model (see Figure 3).15 Lastly, the self-exchangeof aromatic redox couples based on tetracyanoquinodimethane,tetrathiafulvalene, and tetracyanoethylene also yielded markedly dynamical solvent-dependent rate behavior, even though apparently & 2 X,for these systems. Together, then, these experimental observations may be considered to cast some doubt on at least the qualitative validity of the SNM model. As mentioned above, however, a difficulty with most solvent-dependent rate studies is that the separation of the dynamical and energetic factors usually relies on theoretical models " l ' ties in the parametrization far experimental systems. with l Consequently, the nature as well as degree of solvent dynamical effects present in most outer-sphere electron-transfer reactions is open to some doubt.2b To obviate these limitations, it would clearly be desirable to obtain more experimental information on solvent-dependent energy barriers from optical data. Another complication for bimolecular reactions is that the overall reaction energetics and possibly dynamics can contain solvent-dependent contributions associated with the formation of the precursor state from sewvated reactants. Direct evaluation of solvent-dewndent unimolkular reaction kinetics for suitable binuclear or'related
-
-
-
-
Phelps and Weaver redox systems would provide invaluable information since these difficulties would be avoided.2b Nevertheless, it is appropriate to note possible limitations of the SNM model utilized here. As already mentioned, the treatment presumes the presence of a cusp-shaped barrier, yet significant or even substantial barrier-top roundedness usually results from the degree of donor-acceptor electronic coupling necessary for the Occurrence of adiabatic pathways and hence control by nuclear (including solvent) dynamics. Such barrier-top roundedness can alter significantly the overdamped reaction dynamics, as enunciated in particular by Hyne~.~OQualitatively, the net solvent friction tends to increase as the electronic coupling is enlarged. From the foregoing, a greater influence of overdamped solvent motion in the presence of vibrational activation might be anticipated under these conditions. At least qualitatively, then, this trend is in accord with the observed apparent discrepancies between experiment and the SNM predictions. Despite these details, the survival of at least partial rate control by overdamped solvent motion in the face of faster vibrational dynamics is clearly indicated by the theoretical findings as well as by experiment. Given the near-universal presence of inner-shell distortions in activated electron-transfer processes, the further delineation of such effects, especially in a comparative fashion between theory and experiment, would seem to be well worthwhile. Acknowledgment. We are grateful to Prof. R. A. Marcus for helpful comments concerning his theoretical model. This work is supported by the Office of Naval Research. Appendix A As we have noted with respect to eqs 1-3, Nadler and Marcus have provided a numerical means to obtain Q(t) and hence 7c1 [=&,,I. What follows is a summary of their method; for more explicit details the reader is referred to ref 6b. The generalized moments algorithm used here was discussed originally by Nadler and S~hulten.~' They construct an approximation to Q(t) by means of the Laplace transform of Q(t)
e($),
and the use of L+(X) = . i l [
-&-
X I Ea
the adjoint Fokker-Planck operator. The Laplace transform of Q ( t ) is written as
Q(s>= ( l i b - L+(X)+ k(X)I-'ll)o
(A.3)
where ( q V 0denotes S-",P,(X) V(X) V(X) dX. Equation A.1 can be expanded in terms of generalized moments of pnsuch that (A.4)
n=O
and
- E(-s)~~-*, m
Q(s)
n-0
for s
O
('4.5)
The moment of particular interest to us is pn = (II[k(X) c1-I
- L+(X)lnI1)o
= X m Q ( t )dr = 7a
('4.6) (A.7)
In order to evaluate p-' numerically, the Fokker-Planck operator L(X) = D[a/aX + (1/kBn(BV(X)/3X)] is transformed into its Hermitian, symmetric form. This operator LtS)(X)is related to L(X) and L+(X) via Lcs)(X) = Po-l(X) U X ) P O W = P O W L+(X)PO-'(X)
where po(X) = (P0(X))'/*and
(A.8)
The Journal of Physical Chemistry, Vo1.96, No. 18, 1992 7193
Solvent Dynamical Effects in Electron Transfer
-E- [ x - ( 2 & / k B q 1 2 kBT 2 so that p-l becomes p-1
= ( p o Q l [ k B - ~5(~)(X)l-'Ipo(X)) (A.10)
and one evaluates p-l = (po(X))p-l(X)). Higher order p-" functions can be obtained from a generalization of this procedure which is described in detail by Nadler and Marcus.6b Equation A.10 can be evaluated b obtaining the matrix representation of the operator [k(X)- L(J7(X)]-1.The diffusion space in which the reaction occurs is limited to a finite interval, Xnnge, and discretized into N , segments. The necessary portion of polarization space is determined by making the range large enough that all significant paths over the transition state are represented in the calculation. A (N, X NJ matrix is formed With the elements
+
w(i+i-l)]
i = j t l i =j all other i and;
The rate of transition between cells is determined by W(i+jJ; thus, the L,,terms determine the rate at which reactants diffuse in the polarization coordinate, and the time scale for this diffusion is T ) . The matrix representation of the operator just described is used in
= [k(X) - L'"(X)I-'IPo(X))
[q - (2&/kBT)I2 (c,2)
2
where E is the energy. In order to determine AEo, one needs to evaluate the minimum energy in the q coordinate for a given X . For any given X I the minimum of the potential-energy surface is aE/aq = 0. Setting the derivatives of eqs C.l and C.2 equal to zero implies that q = 0 and q = 2Xi/kBTgiven eqs C.1 and C.2, respectively. Substituting these values into eqs C.l and (2.2 and subtracting yields M -
p
+
I--
kBT
2
[ x - (2&/kBT)I2
(C.3)
2
where M = AEo(X);eq C.3 simplifies to eq (2.4. hEo(X)
3
-m0 + 2b2/keT
(C.4)
References and Notes (A.ll)
where W(i+j) = T ~ ~ @ ~ [ X ~ ) ] / ~T ~) =[ X 6 2(~ ~Land ,) ] )6 , = X-/N, The reaction term in this operator becomes the diagonal matnx K where K,, = k[X(i)l6,, (A.12)
P-18
+
(1) (a) Hynes, J. T. In The Theory of Chemical Reactions;Baer, M., Ed.; CRC Ress: Boca Raton, FL, 1985; Vol. 4, p 171. (b) Barbara, P. F.;Jamba, W. Adu. Photochem. 1990,15, 1. (c) Maronelli, M.; MacInnis, J.; Fleming, G. R. Science 1989,243,1674. (d) Bagchi, G. Annu. Rev. Phys. Chem. 1989, 40, 115. (e) Fleming, G. R.; Wolynes,P. G. Phys. Today 1990,43 (May), 36. (2) (a) Weaver, M. J.; McManis, G. E. Acc. Chem. Res. 1990, 23,294. (b) Weaver, M. J. Chem. Rev. 1992, 92,463. (3) Zusman, L. D. Chem. Phys. 1980,19, 295. (4) (a) Calef, D. F.;Wolynes, P. G. J. Phys. Chem. 1983,87, 3387. (b) Wolynes, P. G. J. Chem. Phys. 1987.86, 5133. (5) (a) van der Zwan, G.; Hynes, J. T. J. Chem. Phys. 198q76.2993. (b) van der Zwan, G.; Hynes, J. T . Chem. Phys. Lett. 1983,101,367. (c) Hynes, J. T. J. Phys. Chem. 1986,90, 3701. (6) (a) Sumi, H.; Marcus, R. A. J. Chem. Phys. 1986, 84, 4894. (b) Nadler, W.; Marcus, R. A. J. Chem. Phys. 1987, 86, 3906. (1) Newton. M. D.: Friedman. H. L. J. Chem. Phvs. 1988. 88. 4460. i 8 j Rips, I.'Jortne;, J. J . Chem. Phys. 1987, 87, 2690. (9) . . (a) . . Morillo, M.; Cukier, R. I. J. Chem. Phys. 1988, 89, 6736. (b) Yang, D. Y.; Cukier, R. I. J. Chem. Phys. 1989,91,281. (10) (a) Sparpaglione, M.;Mukamel, S. J. Phys. Chem. 1987,91, 3938. (b) Sparpaglione, M.; Mukamel, S . J. Chem. Phys. 1988, 88, 3263. (c) Mukamel, S.; Yau, Y . T. Acc. Chem. Res. 1989, 22, 301. (11) For example, sct: Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. (12) (a) Nielson, R. M.; Weaver, M. J. J . Electroanal. Chem. 1989,260, 15. (b) Weaver, M. J.; Phelp, D. K.; Nielson, R. M.;Golovin, M. N.; McManis, G. E. J . Phys. Chem. 1990,94, 2949. (13) Grampp, G.; Jaenicke, W. Ber. Bunsen-Ges. Phys. Chem. 1991,95, 904. (14) Su,S.-G.; Simon, J. D. J. Chem. Phys. 1988,89,908. (15) Phelp, D. K.;Ramm, M. T.; Wang, Y.; Nelsen, S. F.;Weaver, M. I
(A. 13)
A Gaussian elimination procedure can be used to solve for P - ~ ( X ) and hence p-l from (A.10).
APpeadlX B The delta function form of k(X) noted in the text with regard to the narrow reaction window limit arises as follows:32 In the limit & 0 cq 5 becomes Y, = ( ~ ~ / 2 7 r ) ( & / A Note J ~ / ~here . that the vibrational coordinate is no longer present, and thus Y, no longer has a vq contribution. Additionally, we know from ref 6b that the activation energy AG(X) in eq 4 is given by A G Q = (&/AJ[(X X c ) 2 / 2 ] .Then, noting that
-
-
and letting a = 2Xi/X0and y = X - X,; inserting these terms in eq 4 yields 2Wb-4
which is the desired delta function reactive term. AppcadixC The solvent-polarization (X-) dependence of the nuclear tunneling factor (eq 6) can be obtained once the X dependence of the q aumlinate driving force, AE,,,is known. The latter is derived from the equations representing the reactant and product potential energy surfaces. These surfaces are given by
,
J. J . Phys. Chem., submitted. (16) For a brief summary, see: Marcus, R. A.; Sumi, H. J. Electroanal. Chem. 1986, 204. 59. (17) McManis, G. E.; Gochev, A.; Weaver, M. J. Chem. Phys. 1991,152, 107. (18) (a) Marcus, R. A. J. Chem. Phys. 1956,24,979. (b) Marcus, R. A. J. Chem. Phys. 1965,43,679. (19) Equation S is similar to the relation suggested in footnote 28 of ref 6a. (20) See for example: Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. (21) Scher, H.; Holstein, T. Philos. Mag. 1981, B44, 343. (22) Zhu, J.; Rosaiah, J. C. J. Chem. Phys. 1991, 95, 3325. (23) McManis, G. E.; Nielson, R. M.; Gochev, A.; Weaver, M. J. J . Am. Chem. Soc. 1989,111,5533. (24) McManis, G. E.; Golovin, M. N.; Weaver, M. J. J. Phys. Chem. 1986, 90, 6563. (25) McManis, G. E.; Weaver, M. J. J . Chem. Phys. 1989, 90,912. (26) Gochev. A,; McManis, G. E.; Weaver, M. J. J . Chem. Phys. 1989, 91, 906. (27) See footnote 47 of ref 23. (28) McManis, G. E.; Gochev, A.; Nielson, R. M.; Weaver, M. J. 1.Phys. Chem. 1989, 93, 7733. (29) Nielson, R. M.; Golovin, M. N.; McManis, G. E.; Weaver, M. J. J . Am. Chem. Soc. 1988, 110, 1745. (30) (a) Van der Zwan, G.; Hynes, J. T. J. Phys. Chem. 1985.89.4181. (b) Hynes, J. T. J. Phys. Chem. 1986, 90, 3701. (31) Nadler, W.; Schulten. K. J . Chem. Phys. 1986. 84, 4015. (32) Marcus, R. A. Private communication.
