J. Phys. Chem. 1982, 86,200-203
200
Toriyama et al.3 From their ENDOR study, they suggested that the radical is formed by one-electron loss or oneelectron gain by the acetate group and that the radical is displaced along the C-C bond. Our simulations clearly show that the methyl radical has moved along the b axis closer to the acetate group, which indicates that the radical formation is by an electron loss. Otherwise, the movement of the methyl radical toward the acetate group would result in chemical bonding if the acetate group still had an unpaired electron.
This study demonstrates the power of electron spin echo modulation spectrometry for probing subtle structural changes in powder samples. In this regard it seems superior to matrix ENDOR analysis of similar systems.18 Acknowledgment. We thank the Army Research Office for support of this research. (18) Kevan, L.; Narayana, P. A.; Toriyama, K.; Iwasaki, M. J. Chem. Phys. 1979, 70,5006.
Determination of Electron-Transfer Rate Constants from Data on Tunneling to Randomly Distributed Acceptors in a Rigid Medium R. Kurt Huddleston' and John R. Mlller Chemistry Dlvlsbn, Argonne Natbml Laboratory, Argonne, Illlnols 60439 (Received: September 9, 198 1)
The problem of analyzing data for an electron-transfer reaction in rigid solution to obtain the rate constant as a function of distance is considered in detail. Effects due to the distribution of donor-acceptor separations and the finite volumes of the individual molecules are explicitly included. Calculations are presented which show that the assumptions of a previously presented tunneling model are essentially correct; reactions with non-nearest-neighboracceptors are, in general, unimportant, and the concept of a mean-time-dependentreaction radius for tunneling is valid, in spite of the fact that the electron transfer is occurring simultaneously over a range of distances. The earlier tunneling model is modified by making a more careful choice of the reaction radius, and it is shown that the modified model yields accurate rate constants. Convenient procedures for correcting for the volume occupied by acceptor molecules and for the fact that a donor cannot be arbitrarily close to an acceptor are presented.
I. Introduction The study of electron-transfer (ET) reactions in rigid matrices1-13has the unique feature that the reactants are held apart at fixed distances. Experimental electrontransfer distances larger than 35 A have been rep0rted.3,~JJ' The experiments confirm that the rates decrease exponentially with distance 2 3 7 11,12 8
9
and is usually 4-7 A. A goal of the experiments is to obtain k(r) from measurements on rigid solutions of randomly distributed reactants. This was first done13by defining an effective reaction radius R(t) at time t by
9
k ( r ) = v exp[-(r - R,)/a]
(1)
where k(r) is the rate constant for a donor-acceptor pair at (center-to-center) distance r. The frequency factor v and attenuation length a are interpreted in terms of a Franck-Condon-weighted density of states and half the attenuation length of the one-electron exchange matrix element. Ro accounts for the finite size of the reactants (1) J. W. van Leeuwen, M. G. J. Heijman, H. Nauta, and G. Casteleijn,
J. Chem. Plays., 73,
1483 (1980). (2) A. Kira, Y. Nosaka, and M. Imamura, J. Phys. Chem., 84, 1882 (1980). (3) J. V. Beitz and J. R. Miller, J. Chem. Phys., 71, 4579 (1979). (4) A. Barkatt, C. A. Angell, and J. R. Miller, J. Phys. Chem., 82,2143 (1978). (5) J. R. Miller, J. Phys. Chem., 82, 767 (1978). (6) H. A. Gillis, G. G. Teather, and G. V. Buxton, Can. J. Chem., 56, 1889 (1978).
(7)'K. L'Zamaraev, R. F. Khairutdinov, and J. R. Miller, Chem. Phys. Lett., 57, 311 (1978). (8) J. Kroh and Cz. Stradowski, Radiat. Phys. Chem., 12,175 (1978). (9) G. V. Burton and K. G. Kemsley, J. Chem. SOC.,Faraday Trans.
where P(t)is the experimental observable, the surviving fraction of electron donors, and c is the (number) concentration of acceptors. In equation 2 R ( t ) is interpreted as the average radius of a reaction volume V ( t )at time t. In principle V ( t )could be nonsperical. In deriving this capture volume model, Miller somewhat arbitrarily equated the average reaction radius to the distance at which k(R) = t-l. Equation 1 then relates R to the rate parameters v and a for an isolated donor-acceptor pair: R ( t ) = Ro + a In ut (3) We show below that this approximation in the capture volume model yields accurate measurements of a, but it consistently overestimates Y by almost a factor of two. An exact model for energy transfer by exchange was derived earlier by Inokuti and Hirayama.14 Tachiya and Mozumder applied similar methods to the electron-tunneling problem, giving the exact survival pr~bability'~ as
1, 72, 1333 (1976).
(10)G. V. Buxton and K. G. Kensley, J. Chem. SOC.,Faraday Trans. I , 72, 466 (1976). (11) J. R. Miller, J.Phys. Chem., 79, 1070 (1975). (12) J. R. Miller, Chem. Phys. Lett., 22, 180 (1973). (13) J. R. Miller, J. Chem. Phys., 56, 5173 (1972). 0022-3654/82/2086-0200$01.25/0
(14) M. Inokuti and F. Hirayama, J. Chem. Phys., 43, 1978 (1965).
0 1982 American Chemical Society
Electron-Transfer Rate Constant Determinatlon
The Journal of Physical Cbmlstry, Vol. 86, No. 2, 1982 201
f ( v t ) = (In vtI3 + 1.7316 (In ~ t+ 5.9343 ) ~ In vt
+ 5.4449
(4b) This differs slightly from the capture volume model for which f ( v t ) is13 f ( v t ) = (In ~ t= )[ R (~t ) / a I 3 (44 Here we have assumed point particles (Ro= 0) in order to make a direct comparison with the exact model,15which does not allow for the finite sizes of the reactants.le Dainton, Pilling, and Rice17J8obtained results similar to the exact model by numerical integration using a slightly modified form for k(r). Still the importance of reactions with non-nearestneighbor acceptors has not been determined.15J7J8We will show that only a small error is introduced by considering reactions between nearest neighbors only. Section I1 presents a revised form of the capture volume model. Section I11 gives numerical results that compare the “revised capture volume model” and the exact solution, and also clarifies the meaning of the capture radius and the origin of the error in the simple model. Section IV discusses the assumptions of the model presented here. Section V allows the generalization from point particles to include effects due to finite molecular volumes. 11. The Revised Capture Volume Model A highly accurate approximation to the exact model, which retains the convenient form of the capture volume model, can be obtained by replacing equation 3 for the reaction radius with R ( t ) = Ro + a In gvt (5) where g is simply a constant. The survival probability is still given by (2) or (4a) with
f ( v t ) = (In g ~ t ) ~
P(t)
t, s
exactb
capture volumeC
nearest neighbord
revised capture volumee
10.’’ 0.8910 0.9082 0.8918 0.8929 lo-‘] 0.7417 0.7679 0.7437 0.7431 0.5399 0.5705 0.5435 0.5406 0.3313 0.3589 0.3359 0.3313 0.1651 0.1842 0.1694 0.1649 1 0.0644 0.0744 0.0672 0.0642 l o z 0.0190 0.0228 0.0203 0.0188 104 4.06 x 10-3 5.09 x 10-3 4.49 x 10-3 4.02 x 10-3 lo6 6.09 X 8.00X 7.03 X 6.00 X 108 6.16 x 10-5 8.54 x 10-5 7.51 x 10-5 6.05 x 10-5 a Using a = 1 A, u = lOI5 s-’, c = 0.025 M,R , = 0. Using eq 4a and 4b. Using eq 2 and 3. Using eq 8-10. e Using eq 7 with g = 1.9.
transfer when two solutes are present, the “exact model” is impractical, as Tachiya pointed 0 ~ t . lThis ~ is also true for correlated electron acceptors, e.g., two acceptor groups on the same molecule.20 111. Numerical Results and the Role of Nonnearest Neighbors The survival probability P ( t ) calculated by the exact model, the revised capture volume model (g = L9), and the original capture volume model (g = 1.0) are given in Table I. Also included for comparison is a numerical calculation of the exact survival probability Punwhen electron (or hole) donors are considered to react only with nearest neighbors.
(64
or = (Ro/a + In g ~ t ) ~
(6b)
so that
+ a In g
TABLE I: Comparison of Approximations for P(t)”
1
~ t ) ~
Here p ( r ) is the survivial probability for a donor having its nearest acceptor at r p ( r ) = exp[-vt exp(-r/a)]
(9)
and w(r) is the nearest-neighbor distribution function21 (7)
Again we suppressed Roin (sa) for direct comparison with
w(r) = exp(-4~cI.3/3)(4~cr~)
(10)
(16)M.Tachiya and A. Mozumder, Chem. Phys. Lett., 28,87 (1974). (16)The assumption of point particles is readily relaxed to allow for the effect of finite molecular radii on the rate vs. distance relation. This is done in the capture volume model by including R, (e.g., eq 3),and in
The contribution from nonnearest neighbors is given by the difference between P, and the exact model. Table I compares the results of these four methods over a wide range of times. The dependence of the survival probability on a and c is identical in the capture volume models and the exact result, since either may be written in the form of equation 4a. It is clear that the revised capture volume model with g = 1.9 gives excellent agreement with the exact result. Because the use of g = 1.9 optimizes the results for intermediate times, the error increases at latter times. At lo2s the error in P(t) is 1% , but the absolute error is only 2 X loa. The error does not reach 2% of P(t) until lo9 s where P = 3.0 X The optimal value of g varies weakly with vt, and is rigorously independent of a and c. The value g = 1.9 should be more than adequate for most practical purposes. The errors of the original capture volume model (g = 1)are much larger, reaching 20% of P(t) at lo2s, although the absolute error in P(t) is still small (0.004). Only a fraction of the error of the simple model is attributable to neglect of reactions with nonnearest neighbors. Com-
aday Tram. 2,71, 1311 (1975). (18)S.A. Rice and M. J. Pilling, h o g . React. Kinet., 9,93 (1978).
(19)M.Tachiya, J. Chem. SOC., Faraday 1 1 2 0 ~ 2, . 75, 271 (1979). (20)R. K. Huddleston and J. R. Miller, J. Phys. Chem., 85, 2292 (1981). (21)S. Chandrasekhar, Reu. Mod. Phys., 15, 1 (1943).
the exact model in (4). The form of (6a) is the same as the multiterm form (4b), except that the coefficients of the powers of In vt differ. But for g = 1.9, equation 7 gives results nearly identical with those of the exact model over the time range to lo2 s, as we show in the following section. Retaining the form of the capture volume model has several advantages: The “exact model” is only exact for randomly distributed point particles; by retaining the simple form and the concept of a capture radius, it is possible to conveniently correct for effects of the finite volumes of the acceptor molecules (see Section V). Although small, the errors from neglecting the effects of finite volumes can be larger than errors from the original capture volume model. For nonrandom distributions of donoracceptor distances, which apply to intermolecular electron
the exact model by using a fictive Y, larger by exp(Ro/a). This effect is distinct from the finite volume effects discussed in section IV,which alter the distribution of donor-acceptor distances. (17)F. 5.Dainton, M. J. Pilling, and S. A. Rice, J. Chem. SOC.,Far-
202
Huddleston and Miller
The Journal of Physical Chemistry, Vol. 86, No. 2, 1982
parison with the nearest neighbor only calculation shows that the contribution to the error from neglect of reaction with other than nearest neighbors is only one third of the total error of the simple model at lo2 s. At earlier times, it is smaller. Thus, most of the error in the original capture volume model is due to the careless choice of the average reaction radius. That choice k ( r ) = t-' gives a value for R (equation 3) which is very close to the most probable distance of electron transfer at time t. However, the distribution of electron jump distance occurring at time t is skewed to long distances, giving a larger average value (equation 5).
IV. Discussion of the Validity of the Assumptions of the Model The model presented here, like earlier treatments,13J5J8 assumes a random distribution of noninteracting, point particles and neglects any effects due to a distribution of molecular orientations. Here we discuss the validity and applicability of the neglect of orientation effects and the assumption of noninteracting particles. The next section discusses generalization from point particles. The model presented here neglects the fact that the ET rate between a given donor and acceptor will depend on their relative angular orientation as well as on the distance between them. From simple considerations of the symmetry properties of electronic wave functions, it is clear that electron exchange matrix elements will be typically positive for some relative orientations and negative for others (except in cases where both have spherical symmetry). There must then be a locus of points in angle space for which the E T rate is zero. Orientation effects have been previously invoked for reactions of trapped electrons but were not thought to have a major efect on observed decay curves.l' A theory of orientation effects is desirable but would' be somewhat complex and cumbersome to apply. It would also be of limited usefulness, because orientation effects are unlikely to significantly alter the observed decay in rigid solutions of randomly distributed reactants. This is so because the decay curves are insensitive to a dispersion in rates. For example, consider the simplest angle-dependent case: that of a totally symmetric orbital interacting with an orbital having a single nodal plane at angle 6 relative to the intermolecular axis. If we assume that v = vo cos 6,we can estimate the angle averaged survival probability as
between D and A will cause fluctuations in electron exchange matrix elements leading to finite v's, even at the most unfavorable orientations. We, therefore, neglect orientation effects, not because they are uninteresting, but because experiments in random solutions are likely to be insensitive to them, since they measure only the angle-averaged rate at distance R. We also neglected departures'from a random distribution due to long-range forces between the reactants. This omission seems entirely justified for cases in which the added acceptors are neutral molecules or, for ionic acceptors, in solvents having very high dielectric constants. The present treatment would require modification for ions in weak dielectrics.
V. Effect Due to The Finite Volumes Occupied By Donors and Acceptors The fact that donor and acceptor molecules are not point particles has two immediate consequences: (1) They occupy a finite fraction of the total sample volume, and (2) they cannot be closer than the sum of their radii. In what follows, the molecules are assumed to be spherical.23 A treatment of nonspherical molecules would be considerably more complex, but would not be expected to significantly alter the results. Two distinct electron-transfer processes need to be considered: (1) trapped electron e; (or hole) capture from the matrix and (2) intermolecular electron transfer. The fact that different acceptors cannot be arbitrarily close results in an increased reaction efficiency over point particles. Consider the process of adding acceptors to random locations in the sample one at a time. Each subsequent acceptor sees a slightly reduced free volume remaining. Therefore, the added acceptor is likely to go into regions of the sample such that its capture volume overlaps less with the capture volumes of other acceptors, which leads to an increased effective concentration of acceptors. This effective concentration can be estimated from the average free volume seen by an acceptor, or, equivalently, the free volume left when half the acceptors have been added. Thus, the effective concentrations of acceptors is given approximately by
where CAis the actual concentration of acceptors of radius RA. It is clear that equation 12 applies to both e; (hole) capture and intermolecular electron transfer. The fact that donor and acceptor centers cannot be where P(6,t) is calculated by equation 7. We found ( H t )) to differ by less than 0.01 from P(6(~),t), where 3 = 0 . 6 3 6 ~ ~ closer than the s u m of their radii leads to larger separations on the average, which decreases the probability of reaction. is the mean (angle averaged) value of v. This result was A sphere of radius Ro = RA + RD is not included in the found by using equation 11 and 7 over the range lo* to reacton volume, where RD is the radius of the donor. Thus, lo3 s where ( P ( t ) )decreased from 0.951 to 0.044 with a the reaction volume becomes = 1A, vo = 1013s-l, and Ro = 0 at a concentration of 0.025 M. V ( t )= 4 / 3 ~ [ R (-t )Ro3] ~ (13) The angular dependence of the rates has such a small effect on the decay curves because only a small fraction leading to an increased survival probability of the D-A pairs would be a t angles giving v's reduced by P ( t ) = exp[-CAV(t)] (14) more than an order of magnitude from the average. The effect could be larger for interactions between orbitals of particularly at short times when Ro is relatively larger other symmetries. compared to R(t). Equation 13 only applies for intermoThe impact of the small fraction of D-A pairs at unfalecular electron transfer since an excess electron (or hole) vorable angles would be further reduced by reactions with may be initially localized arbitrary close to (or actually on) second and farther nearest neighbors and by any vibraan acceptor. tional motions which alter 6. We also speculate that, in the superexchange model,n vibrations of solvent molecules
( W=)JP(e,t) d s / S d o
(11)
(22) J. R. Miller and J. V. Beitz, J. Chem. Phys., 74, 6746 (1981).
(23) For practical calculations, the molecular radius can be approximated aa the geometric mean of the extent of the molecule in three perpendicular directions.
The Journal of Physical Chemistty, Vol. 86, No. 2, 1982 203
Electron-Transfer Rate Constant Determinatlon
TABLE 11: Effect of Finite Volume Correction%
P-
P-
6.0 0.6133 0.6113 0.6217 0.6197 6.0 0.0532 0.0470 0.0578 0.0511 4.0 0.6133 0.6127 0.6158 0.6152 5.0 0.6133 0.6121 0.6182 0.6170 8.0 0.6133 0.6084 0.6334 0.6287 10.0 0.6133 0.6036 0.6532 0.6442 a For v = 1 x lOI5 s-l, a = 0.7 A . Taking R*= RD. Taking the effective concentration from Using eq 7. eq 12. e Using eq 13. f Using eq 13 and taking CA' from eq 12. 0.025 0.15 0.025 0.025 0.025 0.025
It is not necessary to take into account that two potential donor molecules cannot be closer than 2RD The concentration of available donor sites does not affect the distribution of acceptors relative to an occupied donor. The actual concentration of occupied donors could conceivably affect the electron-transfer kinetics if it were high enough to substantially deplete the number of available acceptors. Such a situation can be avoided by proper choice of the experimental conditions. Finally, for an intermolecular electron-transfer process studied by pulse radiolysis, it is necessary to account for the fact that some of the charge will be initially captured by the acceptor. The effect is really of a somewhat different nature than the others discussed here and will be dealt with in a separate publication." Briefly, a certain fraction of the charge will be captured by acceptors that would have been captured by donors if the acceptors were not present. However, if a donor and an acceptor are close enough to compete for charge from the matrix, intermolecular electron transfer is likely to occur at short times in any case. In short, competition and reaction are correlated. The magnitude of the correction depends on the detailed kinetics of charge capture and intermolecular electron transfer but may be made with some effort24or neglected in many cases, particularly if CD >> Ck A few results of calculations of the effects represented by equation 12 and 13 are presented in Table 11. Values are given for the survival probability calculated by (1) neglecting both finite volume effects, (2)including only the effective concentration of acceptors, (3)including only the mininum distance of approach between donor and acceptor, and (4) including both the finite volume effects. It is concluded that the corrections due to the two finite volume effects are small except when (1)CAis large, (2) Ro is large, or (3) the reaction is slow (or at early times) such that R(t)is relatively small compared to R,,. The two corrections have opposite effects so that some cancellation is obtained. It can be seen that for the somewhat typical (24)R. K. Huddleston and J. R. Miller, J. Phys. Chem., in press.
TABLE 111: Summary of Applicability of Modifications to the Tunneling Model et' (or h') capture increase in rate constant by a factor of g, eq 7 excluded volume of acceptors, eq 8 approach distance of donor-acceptor pairs, eq 9 competition correction
intermolecular transfer
yes
Yes
Yes
Yes
no correction
yes
no correction
yes
case of CA= 0.025 M and Ro= 6.0 A the corrections (
