The Journal of
Physical Chemistry
0 Copyright, 1988, by the American Chemical Society
VOLUME 92, NUMBER 17 AUGUST 25, 1988
LETTERS Adlabaticity Criteria for Outer-Sphere Bimolecular Electron-Transfer Reactions Jose Nelson Onuchict instituto de F h c a e Qdmica de Siio Carlos, Uniuersidade de Siio Paulo, 13560, Szio Carlos, SP, Brazil
and David N. Beratan* Jet Propulsion Laboratory, California institute of Technology, Pasadena, California 91 109 (Received: March 23, 1988)
A model is presented for outer-sphere bimolecular electron-transfer reactions which is correct in the adiabatic, nonadiabatic, and intermediate dynamical regimes for an overdamped solvent coordinate. From this model we deduce the conditions for the transfer to be “adiabatic” or “nonadiabatic”. Evidence for the two regimes exists in the experimental literature, so this work will provide guidance for mapping out the transition between these regimes. The time-scaleseparationsneeded to adequately describe the process as an average over (distant dependent) unimolecular rates are described.
1. Introduction
Bimolecular outer-sphere electron-transfer reactions are a source of continuing theoretical and experimental interest.’*2 New sets of experiments are providing detailed tests of the models for these reaction^.^" Under particular scrutiny is the dynamics of these reactions. Most theoretical models analyze the reaction rates by assuming a single reaction mechanism (adiabatic or nonadiabatic) and averaging the rate expression over the appropriate volume. Such an analysis is not general and does not provide an understanding of the transition from adiabatic to nonadiabatic behavior. The goals of this paper are to describe the time-scale separations needed to define a bimolecular electron-transfer rate and to develop the theory of bimolecular electron-transfer reactions (in the overdamped reaction coordinate limit) without making assumptions about the reaction mechanism. We will, in this way, describe the nature of the solvent dependence in the two limits as well as in the intermediate regime. ‘Work performed in part during a visit to the Jet Propulsion Laboratory.
0022-3654/88/2092-4817$01.50/0
We first consider the relevant time scales for the bimolecular charge-transfer problem. Because the donor and acceptor are not bound together, the motion along the coordinate separating them, as well as along the reaction coordinate, must be considered. The separation coordinate is particularly crucial because it determines the tunneling matrix element. We focus on how motion along two coordinates affects the overall electron-transfer rate. Then, we perform the distance averaging of the general distance dependent “unimolecular” rate to arrive at the exact bimolecular ~~~~
~
(1) (a) Taube, H. Electron Transfer Reactions of Complex Ions in Solution; Academic: New York, 1970. (b) Sutin, N. Prog. Inorg. Chem. 1983, 30,441. (c) Marcus, R. A,, Sutin, N. Bimhim. Biophys. Acta 1985,811,265. (2) Eberson, L. Electron Transfer in Organic Chemistry; Springer Verlag: New York, 1987. (3) Nielson, R. M.; Golvin, M. N.; McManis, G. E.; Weaver, M. J. J. Am. Chem. SOL.1988, 110, 1745. (4) Grampp, G . ;Harrer, W.; Jaenicke, W. J . Chem. SOC.,Faraday Trans. I 1987.83, i61. (5) (a) Kosower, E. M.; Huppert, D. Annu. Reu. Phys. Chem. 1986, 37, 127. (b) Kosower, E. M. J . Am. Chem. SOC.1985, 107, 1114. ( 6 ) McGuire, M.; McLendon, G. J. Phys. Chem. 1986, 90, 2549.
0 1988 American Chemical Society
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The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
rate as a function of the molecular and solvent parameters.
2. Time-Scale Separations in the Two-Mode Problem When an electron-transfer reaction is coupled to two diffusive reaction coordinate^,'^ the unimolecular rate is given by the following equation providing certain conditions are met.’-”
where
Letters As in the two reaction coordinate case, once k ( z ) is written we must still consider whether the decay of the donor population is exponential in time or not. To address this question, we introduce the concept of a “reaction region”. The value of the tunneling matrix element at donor-acceptor contact (TDA(R0)) determines the maximum value of g (eq IC),g(Ro). If this value is less than 1, every electron-transfer reaction is basically nonadiabatic. The final rate is dominated by acceptor molecules within a shell of width /3 around the donor. However, if g(Ro) is greater than 1, there is a shell around the donor of width u (see section 3) within which the transfer is basically adiabatic. If u > P most of the transfer will occur in this “adiabatic shell”. To assure that the donor decay is exponential in time we require that the flux of acceptors into the reaction region defined by this shell be much faster than the reaction time. The volume of the reaction region is donor, acceptor, and solvent dependent. The observed rate of electron-transfer reactions is generally written
- =1 - + - 1 (similarly for gz). The reorganization energy’ of the reaction is ER = ERY ER’. TDA is the electron tunneling matrix element. The rate may be written as an integral over the z coordinate as in eq 2 where the driving force (t) is now z dependent provided that gy Ro, kdiffis of the form14-16 kdiif
4ADRRreacNA
(4)
where N A is the number density of acceptors. R,,, is the outer radius of the reactive region. It is always larger than Ro and is determined by the adiabaticity parameter, g, through its influence on the reactive region volume (discussed in section 3). DR is the sum of the donor and acceptor diffusion constants.
3. Bimolecular Rate Calculations If the initial distribution of acceptors around each donor is the same as the equilibrium distribution, the short-time behavior is given by k,, and as long as “diffusion” is fast enough, exponential decay occurs with rate k,,,. We now focus on the calculation of k,,, using eq 2 with a single overdamped reaction coordinate and a rate expression valid for all g’s and transfer distances. We write k,,, = 47rjmR2k,(R) 0 P(R) dR
(5a)
where P(R) is the pair distribution function for acceptors a distance R from the donor. In the case of a uniform distribution of acceptors (expected to be a good first approximation when at least one reactant is neutral) where the sum of the donor and acceptor radii is equal to Ro, P(R) NA for R > Ro so k,,, = 47rNALIR2k,(R) dR Taking TDA2= A exp[-(R written
(5b)
- Ro)/@],the unimolecular rate is
where
J. N.; Wolynes, P. G., manuscript in preparation. (8) (a) Zusman, L. D. Chem. Phys. 1980, 49, 259. (b) Zusman, L. D. Chem. Phys. 1983, 80, 29. (9) (a) Rips, I.: Jortner, J. J . Chem. Phys. 1987,87, 2090. (b) Ibid. 1987, 87, 6513. (10) Onuchic, J. N.; Beratan, D. N.; Hopfield, J. J. J . Phys. Chem. 1986, 90, 3707. (11) Wolynes, P. G. J . Chem. Phys. 1987, 86, 1957. (12) Rips, I.; Jortner, J. Chem. Phys. Lett. 1987, 133, 411. (13) Heitele, H.; Michel-Beyerle, M. E.; Finckh, P. Chem. Phys. Lett. 1987, 138, 237. (14) Marcus, R. A.; Siders, P. J . Phys. Chem. 1982, 86, 622.
(15) Marcus, R. A. Discuss. Faraday SOC.1960, 29, 129. (16) Collins, F. C.; Kimball, G. J . Colloid Sci. 1949, 4, 425. (17) Calef, D.; Wolynes, P. G. J . Phys. Chem. 1983, 87, 3387. (18) (a) Sumi, H.; Marcus, R. A. J . Chem. Phys. 1986,84,4272. (b) Ibid. 1986, 84, 4894. (19) Alexandrov, I. V . Chem. Phys. 1980, 51, 449.
Letters 2T A exp(-Ea/kT)
z = -h
(4nERkT)lJ2
(6c)
Q!!
102-
is the solvent longitudinal relaxation time. If we consider the solvent polarization to be the diffusive reaction coordinate, then wc = 1/ T L . ~ - ~ ’Assuming ~ ~ - ~ ~ a uniform distribution of acceptors (see ref 14 and 20 for the proper approaches in more complicated cases), and spherical donors and acceptors undergoing free diffusion,
P
=lo
_-________
\
-
T~
10-2 lO-1
This can be rewritten 103
10-2
io-’
io0
io1
io2
103
104
105
106
g’ Figure 1. k,,,/4rZNAP3 is plotted vs g’ with Ro/P = 10. The upper dashed line corresponds to the nonadiabatic limit and the lower dashed curve to the adiabatic limit. The solid curve is the exact result computed using eq 3.8-3.10.
where
Direct integration (for ul) or integration by parts gives the rapidly converging series, for g’ I1,
I
loot
t
(9)
lo-’
4
10-2
For 0 Ig’ 5 1, u, varies from zero to the Riemann 7 function ~ ( n ) .Analytically continuing these functions for g ’ 1 1 gives u,(g? = In g’+ u , ( l / g ?
(loa)
In2 g’ u2(g? = -j- + 27(2) - u2(1/g?
(lob)
For g’> 1 we use these equations with the series in eq 9 to calculate the rate in eq 8a. Details of this calculation will be reported shortly.2’
10-2
k r = Z exp[-(R
- RO)/@]
(1la)
and
The bimolecular rate in this limit is basically the rate at R = Ro times the reactive volume, a shell of thickness p around the acceptor. When g’is larger than 1, there is a shell of thickness u (from Roto Ro + u ) within which the unimolecular rates are essentially adiabatic. u is defined so that g’exp(-u/P) = 1. If u is much larger than p, most of the bimolecular rate arises from the adi~~~~
(20) Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1985, 35, 437. (21) Beratan, D. N.; Onuchic, J. N., submitted for publication.
io0
io’
io2
103
104
105
106
g’ Figure 2. g‘ka,,/4~ZNAP3 is plotted vs g’with Ro/P = 10. The upper dashed line is the nonadiabatic limit and the lower dashed curve is the adiabatic limit. The solid curve is the exact result.
abatic region rather than from the region where R the rate can be written approximately as
> Ro + u and
or e
Adiabatic and Nonadiabatic Limits When g’
