J. Phys. Chem. 1992, 96, 8441-8444 (12) Herbich, J.; Sepid, J.; Waluk, J. J . Mol. Strut. 1984, 114, 329. (1 3) McMorrow, D.; Aartsma, T. J. Chem. Phys. Lett. 1986, 125, 58 1. (14) Konijnenberg, J.; Huizer, H. H.; Varma, C. A. G. 0.J . Chem. Soc., Faraday Trans. 2 1988,84, 1163. (15) Moog, R. S.;Bovino, S.C.; Simon, J. D. J . Phys. Chem. 1988, 92, 6545. (16) Negrerie, M.; Gai, F.; Bellefeuille, S. M.; Petrich, J. W. J . Phys. Chem. 1991, 95, 8663. (17) Moog, R. S.;Maroncelli, M. J . Phys. Chem. 1991, 95, 10359. (18) Collins, S.T. Doctoral Thesis, 1981, Florida State University. (19) Chou, P.-T.; Martinez, M. L.; Cooper, W. C.; Collins, S. T.; McMorrow, D. P.; Kasha, M. J. Phys. Chem. 1992,96, 5203. (20) Under the assumption that the extinction coefficient of the groundstate tautomer of 7AI is the same as that of 7MPP, absorption measurements indicate less than 0.5% could be presmt in the solvents we have examined here. (21) N6grerie, M.; Bellefeuille, S. M.; Witham. S.;Petrich, J. W.; Thomburg, R. W. J. Am. Chem. Soc. 1990, 112,7419. (22) Robison, M. M.; Robison, B. L. J. Am. Chem. Soc. 1955,77,6554. (23) Demas, J. N.; Crosby, G. A. J . Phys. Chem. 1971, 75, 991. (24) Siano, D. B.; Metzler, D. E. J. Chem. Phys. 1969, 51, 1856. (25) Chapman, C. F.; Fee, R. S.; Maroncelli, M. J. Phys. Chem. 1990,94, 4929. (26) Maroncelli, M.; Fleming, G. R. J. Chem. Phys. 1987, 86, 6221. (27) Evidence for the presence of overlapping transitions comes from recent polarization anisotropy studies by Petrich and co-workers (unpublished results). (28) For related indole data, see, for example: Meech, S.R.; Phillips, D.; Lee, A. G. Chem. Phys. 1983,80, 317. And the more recent papers on the 'L, and 'Lbstates of indole: Callis, P. R. J . Chem. Phys. 1991, 95, 4230. Tubergen, M. J.; Levy, D. H. J . Phys. Chem. 1991,95,2175. Ruggiero, A. J.; Todd, D.C.; Fleming, G. R. J. Am. Chem. Soc. 1990,112,2175. Demmer, D. R.; Leach, G. W.; Outhouse, E. A.; Hager, J. W.; Wallace, S.C. J . Phys. Chem. 1990, 94, 582. (29) Reichardt, C.; Eschner, M.; Schafer, G. Leibigs Ann. Chem. 1990, 75. (30) We also examined these frequency shifts relative to the T* polarity scale of Taft and co-workers (Kamlet, M. J.; Abboud, J. L. M.; Abraham, M. J.; Taft, R. W. J. Org. Chem. 1983,48,2877). Here the aprotic solvents and alcohols fell on two distinct (and typically parallel) correlations, which indicates the importance of hydrogen bond donation by solvents in determining the solvation energies in these molecules. (31) Waluk, J:; PakuL, B.; Balakier, G. Chem. Phys. Left. 1983,94, 58. (32) Caminat], W.; Di Bernardo, S.; Trombetti, A. J . Mol. Sirucr. 1990, 223, 415.
8441
(33) Calculations performed by using the MOPAC (Stewart, J. J. P. MOPAC 6.0, QCPE #455, Indiana University, Bloomington, IN 47405) and Gaussian 88 (Friscb, M. J.; Head-Gordon, M.; Schlegel, H. B.;Raghavachari, K.; Binkley, J. S.;Gonzalez, C.; Defrees, D. J.; Fox, D. J.; Whiteaide, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R.; Kahn, L. R.; Stewart, J. J. P.; Fluder, E. M.; Tiopol, S.;Pople, J. A. Gaussian, Inc., Pittsburgh, PA,1988) program packages. The ab initio calculations used the MNDO optimized geometries and the 6-31G' basis set. (34) Marrone, T. J.; Maroncelli, M. To be published. (35) See the review: Maroncelli, M. J . Mol. Liquids, in press. (36) Suppan, P. Faraday Discuss. Chem. Soc. 1988,85. 173. (37) If one assumes the usual Stokes-Einstein result with equal diffusion and reaction radii, the diffusion-limited reaction rate constant is given by k = (2.2 X 107)(T/g) M-' s-I, where T i s the temperature in kelvin and is the solvent viscosity in centipoise (Birks, J. B. The Photophysics of Aromatic Molecules; Wiley: New York, 1970; Chapter 10). Using this expression the expected times are 180 and 50 ps in the ether and acetonitrile solutions, respectively. We note that the observed times for these types of "diffusive solvation" shifts always seem to be 50-100% slower than the times predicted as above. (See also. Chapman, C. F.; Maroncelli, M. J . Phys. Chem. 1991, 95, 9095.) (38) From fits to data collected in: Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents; Wiley: New York, 1986. (39) We observe similar behavior in dilute alcohol mixtures as well. Our data indicate that isolated 1:l complexes undergo reaction in a time less than our instrumental resolution of -20 ps. (40) If one can assume that near the peak of the N emission there is no contribution from the tautomer then irreversibility im ies eqs 3-6) that the decay should be single exponential with rate constant k? i h e n the neat H 2 0 data are globally fit to a double-exponential function across all wavelengths there is a 5% component appearing with a different time constant. This would suggest that some reversibility was detectable. However, the decays at these wavelengths can be equally well fit by a single-exponential function and therefore the present data are inconclusive. Improved data might be able to decide the matter. Simulationsshow that if +2 does appeer in the water decays with a 5% amplitude it should be definitively observed in data collected with our instrumentation if 150000 counts are collected in the maximum channel ( x 2 1.5). (41) Since the radiative rates of the tautomer are negligible compared to the fluorescencedecay rates, these k, values are all simply the inverse of the observed decay time. (42) Waluk, J.; Pakda, B.; Komorowski, S . J. J . Phorochem. 1987,39,49. (43) Chapman, C. F.; Maroncelli, M. Unpublished results. N
New Explanation for the Lack of the Inverted Region in Charge Separation Reactions M. Tachiya* and S. Murata National Chemical Loborarory for Industry, Tsukuba, Ibaraki 305, Japan (Received: May 27, 1992; In Final Form: July 21, 1992)
According to the Marcus theory, the first-order rate constant of electron transfer between donor and acceptor depends on the donoracceptor distance through the transfer integral and the reorganization energy. The second-order diffusion-mediated rate constant for charge separation reactions was calculated by using a recently developed theory of diffusion-mediatedreactions and by properly taking into account the donoracceptor distance dependence of the fmt-order rate constant. A new explanation for the lack of the inverted region is given.
Introduction The rate of electron-transfer reactions is controlled by the donoracceptor distance, the free energy change of the reaction, the polarity of the solvent, the dynamical property of the solvent, and the temperature. MarcusIJ was the first to study theoretically how the rate depends on these factors. According to his theory, the rate constant should increase with decreasing free energy change AG for relatively high values of AG,while it should decrease with decreasing AG for low values of AG. Throughout this paper we mean by 'low AGn that AG is negative and has a large magnitude. Accordingly, the dependence of the rate constant on AG should be bell-shaped. The region where the rate constant decreases with decreasing AG is called the inverted region. The bell-shaped dependence of the rate constant on AG has in fact recently been observed for charge shift reactions3 (A- + B A + B-)and charge recombination reactions4 (A+ + B- A
-
-
-
+ B). However, for charge separation reactions (A + B A+ + B-),Rehm and WellerSobserved experimentally that the rate
constant remains constant even for very low values of AG. In other words, the inverted region has not been observed for charge separation reactions. This discrepancy between theory and experiment has bothered people in this field for more than 20 years. In the Marcus model the solvent is described by using the dielectric continuum model which is essentially a linear response theory and does not take into account any nonlinear effect. For some time, the reason for the lack of the inverted region in charge separation reactions was attributed to the dielectric saturation effect of the solvent? However, this explanation was later shown to be The purpose of the present paper is to give a new explanation for the lack of the inverted region in charge separation reactions. It is important to notedhat the Marcus equation gives the first-
0022-3654/92/2096-8441$03.00/0@ 1992 American Chemical Society
8442 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
Tachiya and Murata
order rate constant for a donoracceptor pair with a fixed separation, while the Rehm-Weller result is concerned with the second-order rate constant. We first discuss the dependence of the Marcus first-order rate constant on the donoracceptor distance. We then calculate the second-order diffusion-mediated rate constant by using a recently developed theory of diffusion-mediated reactions and by properly taking into account the donoracceptor distance dependence of the first-order rate constant. We finally compare the calculated results with the experimental results of Rehm and Weller.
lo”
ll----l
‘10‘0 5 2
First-Order Electron-Transfer Rate According to Marcus,’V2 the rate constant of electron transfer between donor and acceptor which are separated by distance R is given by
10’
108 5
10
15
20
R I A
where h is the Planck constant divided by 2u. J is the transfer integral which is determined by the overlap between the initialstate and the final-state electronic wave functions, kB is the Boltzmann constant, and Tis the temperature. X is the quantity called the reorganization energy. Within the framework of the dielectric continuum model X is expressed asl,2 a
b
R
where e is the electronic charge, % and 4 are the optical and static dielectric constants of the solvent, respectively, and a and b are the radii of the donor and the acceptor, respectively. AG is the free energy change of the reaction. It is important to note that eq 1 gives the first-order rate constant for a donoracceptor pair with a fixed separation. The bell-shaped dependence of the rate constant on AG is derived from eq 1 on the assumption that the donoracceptor separation is fmed. In fact, the bell-shaped dependence has so far been observed experimentally only for either first-order reactions with a fixed donor-acceptor separation3 or geminate rec~mbination.~ On the contrary, charge separation reactions studied by Rehm and WellerS are bulk second-order reactions. In this example there are a number of acceptors around one donor. In addition, both the donor and the acceptors move randomly, and electron transfer occurs between the donor and either one of the acceptors. In this situation the distance between the donor and the acceptor between which electron transfer occurs is not fixed but variable. The electron-transfer rate constant given by eq 1 depends on the donor-acceptor distance R through the transfer integral J, the reorganization energy A, and the free energy change AG. The transfer integral depends on the distance approximately exponentially and may be expressed as9
J2 = Jo2exp[-j3(R - (a
+ b))]
(3)
The value of j3 is roughly 1 A-l. The dependence of the reorganization energy on the donor-acceptor distance is given by eq 2. The free energy change of the reaction is given by AG = IP - EA - Ags - e2/R
(4)
where IP is the ionization potential of the donor, EA is the electron affinity of the acceptor, Agsis the solvation energy of the produced ion pair and is given by e2
Ags = -(1 2 - (;I)(!
+ 1-
i)
and d / R is the Coulomb attractive energy between the produced ion pair. With the aid of eq 5, the free energy change is rewritten as A G = I P - E A - - ( l e2 -,,I)(I+ e2 (6) 2 a €8
j)--
Rigorously speaking, the free energy change depends on the do-
Figure 1. First-order electron-transfer rate constant k in acetonitrile as a function of the donoracceptor distance R for a variety of values of the free energy change AG. The figures attached to the curves stand for the values of AG in units of electronvolts.
noracceptor distance through the term. However, in polar solvents the magnitude of this term is small. For example, in the case of acetonitrile solvent the magnitude of this term is at most 0.064 eV, if the distance at the closest approach is 6 A. So, in polar solvents one can as a first approximation neglect this term and assume that the free energy change is independent of the distance. Figure 1 shows the distance dependence of the Marcus firstorder rate constant for a variety of values of the free energy change AG in the case of acetonitrile solvent. The radii of the donor and the acceptor are both taken as 3 A. The values of the parameters for the transfer integral are chosen as Jo= 100 cm-l and j3 = 1 A-’ on the basis of the results of recent ab initio cal~ulations.~ It is often assumed that the electron-transfer rate constant depends on the donor-acceptor distance However, according to Figure 1, the functional dependence of the rate constant on the distance changes with the value of AG. If the solvent is the same, the value of AG is determined by the ionization potential of the donor and the electron affinity of the acceptor, as seen from eq 6. For a donoracceptor combination which has a relatively high value of AG the rate anstant decreases monotonously with increasing distance. (See, for example, the curve for AG = 0 eV.) However, for a donor-acceptor combination which has a low AG the rate constant has a maximum at a certain distance. (See, for example, the curve for AG = -3 eV.) Such complicated distance dependence of the rate constant results from the interplay of the distance dependences of the transfer integral and the reorganization energy in eq 1.
Second-Order Diffusion-Mediated Rate In charge separation reactions studied by Rehm and Weller,5 the distance between the donor and any acceptor changes with time as they independently move randomly. Electron transfer occurs between the donor and either one of the acceptors at a rate which depends on their separation, as illustrated by the curves in Figure 1. The second-order diffusion-mediated rate constant for this situation can be calculated from the following diffusion equation DVz c(R) - k(R) c(R) = 0
(7)
where
D is the sum of the diffusion coefficients of the donor and the acceptor, c(R) is the concentration of the acceptor at a distance R from the donor, and k(R) is the first-order rate constant illustrated by curves in Figure l. The first term in eq 7 represents
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8443
Inverted Region in Charge Separation Reactions the effect of diffusion of acceptors relative to the donor, and the second term represents the loss of acceptors by reaction. The boundary conditions for c(R) are given by
[$I
lo’*
=o
9 t 1
R=a+b
lim c(R) = c,
R--
(9)
Equation 8 shows that acceptors cannot approach the donor closer than the encounter distance Q b. c, is the bulk concentration of the acceptor. In terms of c(R) which satisfies eqs 7-9, the second-order rate constant is expressed as 1 k, = c(R) 4*R2 dR (10)
+
b/
I /
2
10’
\
O
\i
c_-+f(R)
Fayer et a1.,I0 Fleming et al.,” and Marcus and Sided2calculated the rate amstant by numerically solving diffusion equations similar to eq 7, while other calculated it by using empirical equations. Recently, Wilemski and Fixman’s-16have developed an a p proximate method for calculating the second-order rate on the basis of a diffusion equation which has a nonlocal sink term. We mean by *nonlocal’’ that the sink is not a delta function. This method is called the closure approximation. According to the closure approximation, the second-order rate constant is expressed as1’
where k , = $ k ( R ) d3R
[
kD = $ k ( R ) d3R]’/$
(12)
$ k ( R ) G(R,R’) k ( R ’) d3R d 3 R (13)
In eq 13 G(R,R’) is the Green function for the diffusion operator of eq 7 , namely DV’G = -6(R - R’) (14)
[g]
=o
R=a+b
It is convenient to introduce the following quantity which is obtained by integrating the Green function over the angles 0’ and
4‘.
G = LrX2‘G(R,R’) sin 0‘ dV d4’ This quantity satisfies DV‘C = -b(R - R ’)
[g]
=o
R=a+b
The solution of eqs 17 and 18 is given by G=- 1 RCR‘ DR ’ =- I R>R‘ DR With the aid of eq 19, eq 13 is rewritten as ko- = 4*0[
a+b
k(R)R* dR]’
(16)
10’
I
-3
I 1
I
-2
-1
1
0
dG / s V Figure 2. Second-order diffusion-mediated electron-transfer rate constant k, in acetonitrile as a function of the free energy change AG. The prcscnt theoretical result is shown by the full line, while the experimental results of Rehm and Weller5 are shown by the circles.
The second-order diffusion-mediated rate constant was calculated by using eqs 1 1 , 12, and 20 together with eq 1 . The sum of the diffusion coefficients is assumed as D = 3 X cm2/s. The values of the parameters used for the first-order rate constant (eq 1 ) are the same as those used in Figure 1 . In the simplest theory in which electron transfer is assumed to occur only at the encounter distance, kD is given by kD = ~ * D ( Q + b) (21)
This equation yields kD= 1.4 X 10” M-I P1.On the other hand, kDcalculated from eq 20 depends on the value of AG and varies from kD = 1.5 X 10” M-’ S-l for AG = 0 eV to kD = 3.1 M-’ s-’ for AG = -3 eV.
X
10”
Discussion Figure 2 shows the calculated rate constant k, as a function of the free energy change AG. The measured rate constants of Rehm and WellerSare also included for comparison. Some remarks are in order in this comparison. It is important to note that the measured rate constants actually represent the fluorescence quenching rates (see ref 5 for details). Therefore, if fluorescence is quenched not by electron transfer but by other mechanisms, the measured rate constants do not represent the electron-transfer rate which is of our concem. Recently, Kikuchi et al.l8 have shown that in the region around AG = 0 eV fluorescence is actually quenched by exciplex formation. This may partly explain the discrepancy between the calculated and the measured rate constants in the region around AG = 0 eV. As seen from Figure 2, the present treatment also predicts the existence of the inverted region. However, it predicts the inverted region at much lower values of AG compared with previous treatments. In other words, as AG is decreased, the rate constant starts to decrease at lower values of AG in the present result than in previous results. This is the right direction for the explanation of the results of Rehm and Weller. Although the inverted region is not seen in their results, it is generally believed that the inverted region itself should exist and would be observed at still lower values of AG than so far examined. By using Figure 1 , one can intuitively explain the reason the inverted region appears at lower values of AG in the present result than in previous results. In the simplest theory of diffusionmediated reactions, it is assumed that the reaction occurs only at one specified distance between a donor and an acceptor. This distance is called the reaction radius, and the rate of reaction at this specific distance is called the intrinsic rate. In previous treatments of diffusion-mediated electron-transfer reactions, it was assumed that electron transfer occurs only at the encounter distance. According to Figure 1, this means that it m u r s at R = 6 A with the intrinsic rate proportional to the ordinate at R
8444 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
= 6 A. For relatively high values of AG (AG> -1 eV) the above assumption seems allowable. However, for low values of AG (AG < -1 eV) it is not. In this region the rate constant has a maximum as seen from Figure 1. It is more reasonable to assume that electron transfer occurs at this maximum. In the case where electron transfer cccurs at this maximum, both the reaction radius and the intrinsic rate are greater than in the case where it is assumed to occur at the encounter distance. Therefore, the resultant second-order rate constant should also be greater. This is the reason the inverted region appears at lower values of AG in the present result than in previous results. In the present treatment the complicated distance dependence of the first-order rate constant as illustrated in Figure 1 is properly taken into account. According to Figure 1, the effective reaction radius should increase with decreasing AG. In other words, electron transfer occurs at longer donoracceptor distances as AG is decreased. This has been experimentally confirmed by Angel and Peters.” They evaluated the effective reaction radius by analyzing the transient effect of fluorescence decay curves in the presence of quenchers. Their results indicate that the effective reaction radius increases with decreasing AG. The present treatment includes five more or less uncertain parameters, namely, the radii a and b of the donor and the acceptor, the transfer integral Jo at the encounter distance, the decay parameter i3 of the transfer integral, and the sum D of the diffusion coefficients of the donor and the acceptor. We have chosen the following values for the parameters as reasonable values: a = b = 3 A, Jo = 100 cm-l, 0 = 1 A, D = 3 X cm2/s
Agreement between theory and experiment in Figure 2 can be improved if one chooses smaller values for a and b or a larger value for Jo or a smaller value for @ or a smaller value for D. However, we believe that the above values are reasonable ones. In the present treatment the contribution of the intramolecular vibrations of the donor and the acceptor to the Franck-Condon factor is neglected. If the contribution of the intramolecular vibrations is taken into account, agreement between theory and experiment in Figure 2 should be further improved. Concluding Rema& In previous treatments diffusion-mediated electron-transfer rate constants were calculated on the assumption that electron transfer from donor to acceptor occurs only at the encounter separation. For relatively high values of AG the first-order electron transfer rate constant decreases rapidly with increasing donor-acceptor distance, so this assumption seems allowable. However, for low
Tachiya and Murata values of AG the first-order rate constant has a maximum at a certain donor-acceptor distance which is different from the encounter distance, and electron transfer occurs mainly at this distance. In the present treatment the distance dependence of the fust-order rate constant has been properly taken into account by using a recently developed theory of diffusion-mediated reactions. Agreement between the calculated results and the experimental results of Rehm and Weller has been greatly improved by so doing. The present treatment differs from that of Kakitani et aI.l4 in two points. First, the present treatment is based on the Marcus equation for the first-order rate constant, while their treatment is not. One purpose of the present paper is to show to what extent the results of Rehm and Weller are theoretically reproduced within the framework of the Marcus theory. Second, in their treatment the diffusion-mediated rate constant was calculated by using an empirical equation. On the other hand, in the present treatment it has been more rigorously calculated by using a recently developed theory of diffusion-mediated reactions. Burshtein et aLzo have also considered the effect of the distance dependence of the first-order rate constant on the second-order diffusion-mediated rate constant.
References and Not@ (1) Marcus, R.A. J . Chem. Phys. ,1956,24,966. (2)Marcus, R.A. Annu. Rev. Phys. Chem. 1964,15, 155.
(3) Miller, J. R.;Calcaterra, L. T.; Closs, G. L. J . Am. Chem. Soc. 1984, 106,3047. (4)Mataga, N.; Asahi, T.; Kanda, Y.;Okada, T.; Kakitani, T. Chem. Phys. 1988,127,249. ( 5 ) Rehm, D.; Weller, A. Isr. J . Chem. 1970,8, 259. (6)Kakitani, T.;Mataga, N. J . Phys. Chem. 1985,89,8. (7) Tachiya, M. Chem. Phys. Left. 1989,159,505. (8) Tachiya, M. J. Phys. Chem. 1989,93,7050. (9)Logan, J.; Newton, M. D. J . Chem. Phys. 1983,78,4086. (10)Song, L.;Dorfman, R. C.; Swallen, S. F.; Fayer, M. D. J . Phys. Chem. 1991,95,3454. (11) Eads, D.D.; Dimer, B.G.; Fleming, G. R. J . Chem. Phys. 1990,93, 1136. (12)Marcus, R. A.; Siders, P. J . Phys. Chem. 1982,86, 622. (13) Brunschwig, B. S.;Ehrenson, S.; Sutin, N. J . Am. Chem. Soc. 1984, 106,6858. (14)Kakitani, T.;Yoshimori, A.; Mataga, N. Electron Transfer in Inorganic, Organic and Biological System; Advances in Chemistry Series; Bolton, J., et al., Eds.;American Chemical Society: Washington, DC, 1991;Chapter 4. (15) Wilemski, G.;Fixman. M. J . Chem. Phys. 1973. 58, 4009. (161 Wilemski. G.:Fixman. M.J. Chem. Phvs. 1974. 60.866. 878. (i7j m i , M. Cheh. Phys. i975,11,107, 115. (18) Kikuchi, K.; Tahhashi, Y.;Hoshi, M.; Niwa, T.; Katagiri, T.; Miyashi, T. J. Pbys. Chem. 1991,95,2378. (19)Angel, S.A,; Peters, K.S. J . Phys. Chem. 1991, 95,3606. (20)Burshtein, A. I.; Frantsuzov, P. A,; Zharikov, A. A. Chem. Phys. 1991, 155,91.
