3728
SA = [ l
J. Phys. Chem. 1992, 96,3728-3736
+ ksT/kl]-l[l + 3k4(k3 + k4)/k3(k4 + k 4 ) l - l
(17)
and for k one can use the same equations as in (12)-( 15) for the samefimited cases. (a) If k3, k4 < k+, corresponding to kqT< 6 X lo9 M-’ s-I, approximating gives
SA
1/[1 + 3(k4/WI[l + (ksT/kl)l
(d) If k3 < k4 and k4 > k4, corresponding to 4.5 X lo9 < kqT 1O1OM-I sd, approximating gives
< 1.1 X
+
= [1 + (kS~/k1)]-’[1 (3k-,j/k3)]-’
(18)
+ (ksT/kl)]-l = 0 4 - 2 5
(21)
Registry No. 02,7782-44-7;tetramethyl-p-benzoquinone,527-17-3; tetrafluoro-p-benzoquinone,527-2 1-9; tetrachloro-p-benzoquinone,1 1875-2; tetrabromo-p-benzoquinone, 488-48-2; anthracene, 120-12-7; 9-
which, with the assumption that k3 = k4, gives SA= 0-0.25. (b) If k3, k4 > k4, corresponding to kqT> 6 X lo9 M-] s-I, approximating gives S A = !/4[1
< 0.25
bromoanthracene, 1564-64-3;9,1O-dibromoanthracene,523-27-3;acetophenone, 98-86-2; 2’-fluoroacetophenone,445-27-2; 2’,6’-difluoroacetophenone, 13670-99-0;2-chloroacetophenone, 532-27-4; 4’-chloroacetophenone, 99-91-2; 2-bromoacetophenone, 70-11-1; 3’-bromoacetophenone, 2142-63-4; 4’-bromoacetophenone, 99-90-1; 2,4’-dibromoacetophenone, 99-73-0.
(19)
(c) If k3 > k4 and k4 < k4, corresponding to 1.5 X lo9 < kqT X lo9 M-I s-l , a pproximating gives
< 7.5
Interplay of Solvent Motion and Vibrational Excitation in Electron-Transfer Kinetics: Experiment and Theory Gilbert C. Walker,+ Eva ikesson,t Alan E. Johnson, Nancy E. Levinger, and Paul F. Barbara* Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: October 2, 1991; In Final Form: January 9, 1992)
This paper reports new kinetic data on the electron-transfer kinetics of betaine-30 and the related compound tert-butylbetaine. The experimental data are in excellent agreement with a new theoretical model which is an extension of the approach given by Sumi and Marcus (Sumi, H; Marcus, R. A. J. Chem. Phys. 1986,84,4894). Most of the parameters required for the kinetic predictions can be obtained in a straightforward fashion by fitting the static absorption spectra of the charge-transfer band. The combined theoretical and experimental results demonstrate that an accurate model for electron transfer in the inverted regime in solution minimally requires the following three nuclear degrees of freedom: (i) a solvent mode with a frictional response, (ii) an intramolecular low-frequency (classical) mode, and (iii) a high-frequency (quantum mechanical) intramolecular mode. Furthermore, the analysis allows for a detailed understanding of the combined effects of solvation dynamics and vibrational excitations in ultrafast electron-transfer kinetics.
Introduction Ultrafast experiments on intramolecular charge transfer are leading to important insight into the mechanism of electrontransfer reactions in In particular, the strong agreement of electron-transfer rates with estimated solvation rates for certain barrierless reactions has been taken as experimental support of the theoretical prediction that solvent coordinate motion should be rate limiting for barrierless electron-transfer reactions. The excited-state electron transfer of bianthryl is an example of a barrierless reaction that falls in this category.6 Other molecules, including ADMA,’J DMAPS,4v9 and betaine-30,I0J I exhibit electron-transfer rates that can substantially exceed solvation rates, indicating that intramolecular modes are promoting the electron transfer in these molecules. The direct photoinduced electron-transfer reaction of betaine30 (Figure 1) is particularly promising for unraveling the complex interaction of solvent motion and vibrational promotion in electron-transfer kinetics, as the static electronic spectroscopy of this compound can be used to estimate the parameters needed to predict the electron-transfer rate. The parameters include vibrational frequencies, vibrational reorganization energies, and solvent reorganization energies.]* Ultrafast spectroscopy on betaine-30 has shown that k,, for this compound has previously unobserved behavior: k, is approximately equal to the estimated rate of solvation dynamics in rapidly relaxing solvents (e.g., Present address: Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104. *Presentaddress: Department of Physical Chemistry, University of U m d , Umei, Sweden.
acetone, acetonitrile), but tends toward a constant value which greatly exceeds the rate of solvation dynamics in slowly relaxing solvents (e.g., glycerol triacetate (GTA)). Qualitatively, the results suggest that there is a shift from kinetic control by solvation dynamics in rapidly relaxing solvents to kinetic control by intramolecular vibrational motion in the slower solvents. The apparent shift of ket dependence from solvation dynamics to vibrational dynamics qualitatively agrees with the predictions of the model of Sumi and Marcus.I3 Indeed, this model leads to a clear picture of the origin of this effect. In order to further test the suitability of the Suhi/Marcus model, we recently made quantitative predictions using this model employing parameters that were determined from static absorption spectroscopy.lOJ1 Surprisingly, the predicted values were more than lo6 times too slow. ( I ) Barbara, P. F.; Walker, G.C.; Smith, T. P.Science, in press. (2) Barbara, P. F.; Jarzeba, W. Adu. Photochem. 1990, 15, 1.
(3) Maroncelli, M.; MacInnis, J.; Fleming, G.R. Science 1989,243,1674. (4) Simon, J. Acc. Chem. Res. 1988, 21, 128. (5) Kosower, E. M.; Huppert, D. Annu. Reu. Phys. Chem. 1986,37, 127. (6) Kang, T. J.; Jarzeba. W.; Barbara, P. F.;Fonseca, T. Chem. Phys. 1990. 149. 81. (7) Tominaga, K.; Walker, G.C.; Kang, T. J.; Barbara, P. F.; Fonseca, T. J. Phys. Chem. 1991, 95, 10485. (8) Tominaga, K.; Walker, G.C.; Jarzeba, W.; Barbara, P. F. J . Phys. Chem. 1991, 95, 10475. (9) Si.mon, J.; Su,S.-G. Chem. Phys. 1991, 152, 143. (10) Akesson, E.: Walker, G.C.; Barbara, P. F. J. Chem. Phys. 1991,95, 4188. (1 1 ) A k m n , E.; Johnson, A. E.; Levinger, N. E.; Walker, G.C.; DuBruil, T. P.; Barbara, P. F. J. Chem. Phys., in press. (12) Kjaer, A. M.; Ulstrup, J. J. Am. Chem. Sac. 1987, 109, 1934. (13) Sumi, H.; Marcus, R. A. J. Chem. Phys. 1986, 84, 4894. ~
.
0022-365419212096-3728$03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3129
Electron-Transfer Kinetics
D+A-
DA
R
TABLE I: Multiexponential Fit Parameters for the Transient Pump-Probe Signal of Betaine30 and tert-Butylbetaine in Various
Solvents"
solvent
b. R:
0
0
Betaine -30
propylene carbonate acetonitrile acetone benzonitrile triacetin (GTA) methyl acetate ethyl acetate
m-dichlorobenzene benzene toluene
Aib 72, PS Betaine-30
71, PS
1.1 5 0.7 3.6 3.4 1.3 1.5 4.3 2.7 3.2
5.8 2.0 2.3 4.7 >30 8.0 7.0
-0.35 -0.45 -0.08 -0.16 -0.17
tert-Butylbetaine 0;81 7.0
-0.19
acetone GTA
0.9
m-dichlorobenzene
7.9 4.2 4.2
benzene toluene
5.3
0.69 0.67 0.65 0.55 0.92 0.84 0.83 1.0 1.0 1.0
A*b
1 1 1 1
-0.3 1
-0.33
621 630 687 690 697 725 761 784 840 855 689 832 844
"Temperature 293 K. b A l and A2 are the amplitudes of each exponential in the model decay function.
solvent coordinate Figure 1. Free energy surfaces which represent localization of the transferring electron on the donor (DA) and acceptor (D'A-) in the betaines along the solvent coordinate. Optical excitation occurs from the bottom of the (D+A-) well to the nonequilibrium region of the (DA) well. The Sumi/Marcus model13 predicts back electron transfer at the curve crossing a. The Jortner/Bixon model16 and our model both predict electron transfer to excited vibrational levels of the ground state at the curve crossings marked b. The poor agreement with experimentl0JI stems from the classical approximation employed to model the intramolecular vibrational modes in the published implementati~n'~ of the Sumi/Marcus model. It is well-known that electron-transfer theories which employ purely classical descriptions of vibrational modes can grossly underestimate rates for inverted regime reaction~.'~-'~The origin of this effect can be seen in Figure 1. Classical theories predict that electron transfer should occur exclusively at the free energy curve crossing for the reactant (DA) and product (D+A),specifically, circle a in Figure 1. In the inverted regime, as the reaction becomes more exothermic, a crossing at region a occurs at larger activation energies, thereby producing a slower rate. However, if quantum mechanical effects are included, the electron transfer can occur via vibrationally excited product states, the crossings marked b in Figure 1. Some of these reaction channels will have small activation energies and can tremendously accelerate the rate in favorable cases. In this paper, we report an extension of the Sumi/Marcus model that includes two intramolecular modes. The first is a classical mode, in analogy to the Swni and Marcus approach.I3 The second is a quantum mechanical (high-frequency) mode which we treat by the usual a p p r o a ~ h . ' ~We J ~ also make a second modification to the basic Swni/Marcus model by changing the initial conditions along the solvent coordinate. This second modification is required to correctly describe the initial probability distribution for a direct photoinduced electron-transfer reaction. (14) In the Sumi/Marcusd e l , the electron-transfer process occurs along vibrational coordinates. In the published implementation (ref 12) of this model, a classical electron-transfer expression is invoked (see eq 4.1 of ref 13). However, Sumi and Marcus indicated that quantum mechanical expressions could also be used. This is the approach taken in the present paper. (15) Newton, M. D.Chem. Reu. 1991,91,767. Weaver, M.J.; McManis, G. E. 111 Acc. Chem. Res. 1990, 23, 294. (16) Jortner, J.; Bixon, M. J . Chem. Phys. 1988.88, 167. (17) Marcus, R. A.; Sutin, N. Eioehim. Eiophys. Acra 1985, 811, 265.
Electron-transfer simulations using the new model are in unexpectedly good agreement with experiment over a large range of temperatures and solvent environments. Indeed, the observed rapid rate in slow solvents is predicted. Additionally, the predicted rates no longer differ from experiment by orders of magnitude. An in-depth analysis of the simulation results verifies that the rapid electron-transfer rate in the slower solvents is due to vibrational promotion and is not controlled by solvation dynamics. In addition to the new model, this paper also includes some new ultrafast data on betaine-30 and a related molecule, tert-butylbetaine.
Experimental Methods Most of the transient pumpprobe measurements were made using the same color for the excitation and analyzing light. The measurements were made with two different laser systems, one based on a copper vapor laser (CVL) for amplification (resulting in 70-fs pulses at 792 nm, 1 pJ at 8.2 kHz) and the other based on a regenerative Nd:YAG amplifier (resulting in 150-fs pulses at 820 nm, 10 rJ at 500 Hz).The pulses are split into pump (WO) and probe (10%) pulses, which follow different optical paths of variable relative length. Both pulses are focused by a lens ( t y p ically 100-mm focal length) and overlapped in the sample. The change in transmission of the probe pulse is monitored by a photodiode whose signal is input into a boxcar averager. A reference pulse is used for normalizing each laser pulse by subtracting the reference intensity from the probe pulse intensity in order to reduce shot-to-shot noise and to minimize the influence of drift in laser intensity on the measurements. The instrument response function, obtained by measuring the optical Kerr effect in water, is 100 fs (fwhm) for the CVL system and 200 fs (fwhm) for the regenerative Nd:YAG amplifier system. Betaine-30 was purchased from Aldrich Chemicals and used without further purification. tert-butylbetaine was a generous gift from Prof. C. Reichardt. All solvents used were HPLC or spectroscopic grade. Polymer films were made by casting from solutions of methylene chloride or benzonitrile. Absorption spectra were collected using a Cary 17 absorption spectrometer. Experimental Results We have made femtosecond transient absorption measurements on betaine-30 and rert-butylbetaine in various solvents and in polystyrene polymer films. Additionally, we have studied the temperature dependence. The measurements made at room temperature with identical pump and probe colors are summarized in Table I. All the transient absorption spectra were analyzed with a convolute-and-comparealgorithm using a multiexponential model for the decay and the instrument response function. Any coherent component of the transient was ignored when the elec-
Walker et al.
3130 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
-A0
0
Figure 2. Transient absorption at 792 nm of betaine-30 in (a) toluene, (b) benzene, and (c) acetone. Points are data and the solid lines are fits to the data.
tron-transfer rate, kct,was being determined. All transients for the both betaine dyes displayed an initial bleach. The subsequent behavior depends on the relative position of the electron-transfer absorption spectrum peak and the wavelength of the analyzing light; due to the solvent-dependent shifts of the absorption spectrum, the probe pulse interrogated different regions of the absorption band depending on the polarity of the solvent used. Some of these measurements have been previously reported.lO~ll In nonpolar solvents, e.g., benzene or toluene, both the betaine dyes have an absorption maximum close to the color of the laser and show an initial bleach followed by a single-exponential decay as shown in Figures 2a,b and 3a,b. The transient absorption signal shows a different pattern for the more polar solvents. See, for example, Figures 2c and 3c, which show betaine-30 and tertbutylbetaine in acetone. We still observe an initial bleach but a t later times the signal changes sign; there is an increased absorption. In intermediate-polarity solvents, (e.g., GTA, methyl acetate, ethyl acetate) the amplitude of the increased absorption is very small when 792-nm light is used. However, when the regenerative Nd:YAG laser system is used, whose wavelength is further to the red (820 nm), the increased absorption signal is more pronounced. For the quickly relaxing solvents, we tentatively interpret this effect as a transient shift in the betaine spectrum due to local solvent heating subsequent to ground-state return. The static absorption spectra of betaine show a very pronounced red shift with increasing temperature. In the slower solvents, studies using continuum as a probe indicate that inhomogeneous electron-transfer kinetics may also lead to a probe wavelength variation in the observed dynamics. In general, the same trends for the electron-transfer rates in different solvents are observed for betaine30 and tert-butylbetaine. Both display a delayed sign reversal of the signal in polar solvents like acetone, acetonitrile, and propylene carbonate that is absent in nonpolar solvents. However, the rates for tert-butylbetaine are slower compared with betaine-30 in the same environment. Figure 4 shows transients of betaine-30 in benzonitrile with a peak power of 2.5 and 0.2 pJ/pulse. The transients displayed no dependence on the laser intensity. This excludes the possibility of a twephoton absorption influencing the kinetics. Additionally, measurements with different relative polarization for the pump and probe pulses yielded no difference in the signal except a
(wc)
20
Figure 3. Transient absortion at 792 nm of fert-butylbetaine in (a) toluene, (b) benzene, and (c) acetone. Points are data and the solid lines are fits to the data.
-AOC
0 (psec) 20 Figure 4. Power dependence of transient absorption at 792 nm of betaine-30 in benzonitrile. Pump energies were 2.5 pJ/pulse (solid line) and 0.2 pJ/pulse (points).
decreased amplitude. This indicates that there are no complications due to rotational reorientation or overlapping electronic states. We have studied the temperature dependence of the electron transfer of betaine-30" and tert-butylbetaine in toluene and GTA (seeTable 11). The results axhibit approximately linear Arrhenius plots, as shown in Figure 5 for terr-butylbetaine in GTA and toluene. Table I11 shows that there is little variation in the activation energies obtained from the slopes of the Arrhenius plots. The same value, E, = 500 cm-I, was obtained for both dyes in both solvents. It is worth noting that for GTA the Arrhenius plots for betaine-30 and tert-butylbetaine are linear throughout the entire temperature interval studied, despite the fact that the solvent goes through a phase transition. We recently reported, in a preliminary paper," the influence of temperature variation in polystyrene films doped with betaine-30. The polymer fims data (Table 11) show lower activation energies, E, = 20 cm-I (films cast from benzonitrile solution) and E, = 27 cm-I (films cast from methylene chloride solution), than the activation energies found in liquids. The films cast from benzonitrile show a smaller rate than those cast from methylene chloride. The absorption spectra are also slightly different, indicating that there may be some residual benzonitrile in the films. The results mentioned thus far have utilized the same wavelength for the pump and the probe light. In order to investigate the dependence of the kinetics on the analyzing wavelength, we
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3731
Electron-Transfer Kinetics TABLE [I: Temperature Dependence of k , of Betaine-30 and tert-Butylbetaine in Various Solvents T, K k.,. 1012s-l T. K k.,. 1012s-1 Betaine-30 Solvent: GTA 347 0.47 f 0.06 303 0.35 f 0.04 333 0.43 f 0.03 295 0.29 f 0.02 0.41 f 0.02 0.29 f 0.01 328 293 27 1 0.40 f 0.02 0.22 f 0.03 323 318 0.39 f 0.02 263 0.22 f 0.02 314 0.34 f 0.02 259 0.26 f 0.01 228 0.39 f 0.04 0.18 f 0.02 313 0.36 f 0.01 308 331 295 293 282
Solvent: Toluene 269 0.55 i 0.02 249 0.34 i 0.02 223 0.31 f 0.02 0.32 f 0.04
331 293
Solvent: Benzene 0.50 f 0.05 286 0.37 f 0.03 271
29 1 63
Solvent: Polystyrene" 0.066 f 0.003 34 0.043 f 0.004
0.031 f 0.004
293 144
Solvent: Polystyreneb 0.11 f 0.012 76 0.091 f 0.02
0.077 f 0.01
0.29 f 0.02 0.20 f 0.01 0.15 f 0.01
0.42 f 0.08 0.33 f 0.07
Figure 5. Arrhenius plots of tert-butylbetaine in (a, top) toluene and (b, bottom) GTA.
tert-Butylbetaine 333 327 295 291 280
Solvent: Toluene 260 0.28 f 0.02 237 0.36 f 0.03 205 0.20 f 0.01 188 0.24 f 0.02 0.18 0.02
295 282 267
Solvent: GTA 0.18 f 0.01 247 223 0.16 i 0.01 198 0.12 i 0.01
0.17 f 0.02 0.12 f 0.01 0.080 0.005 0.062 f 0.01
0.1 1 f 0.06 0.077 f 0.01 0.057 f 0.008
" Cast from benzonitrile solution. *Cast from methylene chloride solution. TABLE III: Summary of Parameters Obtained from Arrhenius Plots of Betaine-30 and tert-Butvlbetaine in Various Solvents molecule solvent A E., cm-I betaine-30 toluene 29.52 595 betaine-30 GTA 28.77 472 terf-butylbetaine toluene 28.62 506 terf-butylbetaine GTA 28.28 492 betaine-30 polystyrene" 25.5 27 betaine-30 polystyreneb 25.0 20 *Film cast from methylene chloride solution. *Film cast from benzonitrile solution.
TABLE I V Bleach Recovery (Electron-Transfer)Times Measured by Transient Absorption with A,,- # L compound betaine-30 betaine-30 tert-butylbetaine tert-butylbetaine
solvent acetone benzonitrile benzene benzene
Xpump
hprok
792 792 792 792
700 700 800 820
T,
PS
1.2 3.8 3.5 3.6
are currently using continuum generation to vary the wavelength of the probe light. Some preliminary results are summarized in Table IV. The trends observed at different wavelengths of probe = Aprok. light are in agreement with trends observed for ,A, Some discrepancies exist which may be due to the effect on the betaine spectrum of local heating, solvation in the ground state, and inhomogeneous kinetics in slowly relaxing solvents. We are currently studying these issues and will report on them in a future publication. We would like to stress that these discrepancies are
small and the data still support the same conclusions when compared to those of the various models described below.
Theoretical Results and Analysis Basic Models. We have employed three models to describe our experimental observations of the electron-transfer rate. The models include both solvation dynamics and vibrational modes but differ in their treatment of the vibrational degrees of freedom. Model A, a treatment by Sumi and Marcus,13 includes a low-frequency (classical) vibrational degree of freedom and a classical solvent degree of freedom. Model B, a treatment of Jortner and Bixon,16 includes a high-frequency (quantal, hv >> kET)vibrational mode in addition to a classical solvent degree of freedom. Thus, model A involves a classical treatment of nuclear degrees of freedom, and model B involves a mixed classical/quantum description. Our model, model C, is a hybrid of models A and B, which we developed recently by extending model A to include an additional high-frequency quantum vibrational mode. The results of the analyses using the three models are found in Table V. In this table, one can see that model A is orders of magnitude too small, model B is in good agreement with experimental values for rapidly relaxing solvents but fails for the slow solvents, and model C can predict the kinetics for all environments studied. The Sumi/Marcus model, model A, is a purely classical model which uses a diffusive mechanism to describe motion along the solvent coordinate. The electron-transfer event occurs along the classical vibrational coordinate. Because the barrier in the vibrational direction is a function of the solvent configuration, X , the rate is also a function of solvent coordinate. The classical probability distribution function, P(X,t), is assumed to evolve according to a diffusion equation
(1)
with a diffusion constant D = ( k B T ) / ( 2 & ~ s ) . Vis the reaction potential and k ( X ) is the solvent coordinate dependent probability rate constant. The physical significance of P(X,t) can be seen in Figure 6. Figure 6a shows the free energy surfaces of the reactant and product states as a function of the solvent coordinate. Figure 6b shows how P(X,t) varies as a function of delay after
3732 The Journal of Physical Chemistry, Vol. 96, No. 9,1992
Walker et al.
TABLE V Rate Estimates Using Models A, B, and C“ and Parameter Values of Table VI rates, 10l2 s-l
T,K
solvent acetonitrile acetone methyl acetate ethyl acetate benzonitrile GTA
a
k,,(C,) is the rate estimate from
I
7g
kc1,ote 2.0 1.4 0.77 0.67 0.27 0.38 0.40 0.36 0.34 0.29 0.22 0.18
7s-l
298 298 298 298 298 318 313 308 303 293 283 228
2.0 1.2 0.60 0.38 0.21 8X 6x 4 x 10-2 3 x 10-2 8X 2 x 10-3 10-7
ke,(A) 10-4 2 x 10-5 3 x 10-5 5 x 10” 2 x 10-5 7 x 10-5 1 x 10-4 2 x 10-4 1 x 10-4 i x 10-4 2 x 10-5 4 x 10-9
of model C. Similarly, kc,(Cb) is the rate estimate from
I
I
I
I
1
0
1
2
kem 0.28 0.22 0.17 0.12 7 x 10-2 6X 4 x 10-2 3 x 10-2 2 x 10-2 1 x 10-2 4 x 10-3 1 x 10-7 ib of
ket(C,) 1.1 0.78 0.72 0.44 0.30 0.36 0.34 0.33 0.34 0.35 0.41 1.2
kel(Cb) 1.8 1.2 1.05 1.63 0.44 0.40 0.35 0.30 0.29 0.25 0.25 0.71
model C.
are the solvent reorganization energy, Lk,the driving force, AGO, and the average solvent relaxation time, 7,. If the solvent is frozen, is = a,then the reaction still occurs, but with a distribution of rates reflecting a distribution of solvent environments, inhomogeneous kinetics. In the opposite limit, as T , approaches zero, fluctuations in the solvent polarization can accelerate the reaction by allowing the distribution to sample regions of X where k(X) is larger. As T , is varied from zero to infinity, the rate predicted by model A will tend to decrease, but the trend, under certain circumstances, will be more mild than a simple 1/T,dependence. If solvation occurs too slowly to cause a large increase in k(X), the reaction will proceed along the vibrational coordinates at the initially excited probability distribution. Using this model and parameters obtained from the static spectra, we were able to reproduce qualitative trends in the data including the shift from solvation dynamics control to intramolecular vibrational control. However, the calculated rates were orders of magnitude too slow. The Jortner/Bixon model, model B, differs from model A by treating the vibrational coordinates quantum mechanically. In actual practice, a single vibrational mode is used and the total electron-transfer rate is given as a sum of rates from the u = 0 level of the DA state to all of the n vibrational levels of the D+Astate.
I
Solvent Coordinate
(4)
Figure 6. (a, top) Free energy surfaces for the charge-transfer reaction and (b, bottom) diffusive evolution of population along the upper surface with Vel = 0. The distribution is created at X = 1 and evolves to the minimum at X = 0 with a characteristic time, T ~ .
laser excitation if the reaction rate is strictly zero; this is achieved by fixing Vel = 0, Le., no coupling between DA and D’A-. P(X,t=O) reflects the equilibrium solvent distribution in the ground state, which is “frozen” during the ultrashort excitation pulse. P(X,t) evolves in time due to the solvation process. The average value of X relaxes toward zero exponentially with a relaxation time T ~ .The integral under P(X,t) is called the survival probability S(t). It is related to fraction of reactant molecules surviving at time t . Of course, if Vel = 0, then no reaction occurs and S(t) does not decrease with increasing time. However, when Velis nonzero, the chemical reaction proceeds, P(X,r) and S(r) reflecting the evolving reaction. In model A, the reaction occurs thermally via the vibrational modes of the reactant with a rate k(X).
k(X) = Y eXp(-AG*(X)/kBT)
(2)
Here, A G ’ Q is the X-dependent activation energy and Y, in the nonadiabatic limit, is the effective frequency factor defined by
(3) where Xi is the vibrational reorganization energy and Vel is the electronic matrix element. The remaining parameters required
In this expression for the rate, n is the number of quanta accepted g is the nonadiabatic rate constant by the high-frequency mode, k for the transition to the nth level, and %f) is the adiabaticity parameter of the nth level. The adiabaticity parameter can be expressed as
%!$ = 4rq(TS)/&,,iv
(5)
where V,Z is the electroniC matrix element multiplied by the appropriate Franck-Condon factor for the nth vibrational state. The nonadiabatic rate constant in this model is
-
Here, AG&,, is the activation energy for the 0 n channel. Using this model and the parameters from the spectroscopic fits, we predicted rates much faster than we did using model A, but we were unable to reproduce the switch from solvation dynamical control to vibrational control. The model we present, model C, is based model A, but differs from the treatment of model A in two ways. First, in our calculations we define the initial distribution P(X,t=O) displaced along the solvent coordinate, in analogy to preparation by laser excitation from the ground state. Sumi and Marcus used an equilibrated distribution at t = 0 in their simulations. This difference can lead to substantially different kinetics, i.e., evolution along the solvent coordinate before electron transfer. The second difference is the expression we use for the solvent-dependent rate constant, k(X),
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3733
Electron-Transfer Kinetics
TABLE VI: Best-Fit Parameters of Betaine-30 Absorption Spectra and Potential Parameters Used in Electron-Transfer Simulations
parameters, cm-’ solvent acetonitrile acetone ethyl acetate
methyl acetate benzonitrile GTA
toluene benzene cyclohexane“ a
T, K
AGO
b M
298 298 298 298 298 318 313 308 303 293 283 228 298 298 298 298
-1 1645 -10954 -10 363 -10465 -11 196 -10532 -10557 -10583 -10609 -10660 -10711 -10993 -9 993 -9 993 -10025 -9 097
1276 924 952 1016 1172 1085 1143 1200 1258 1372 1487 2119 526 526 639 474
V~~
1554 1736 1600 1697 1631 1800 1800 1800 1800 1800 1800 1800 1566 1566 1639 1703
&I
3644 3104 2628 2812 3006 2776 2812 2851 2889 2964 3039 345 1 1695 1695 1901 1669
hl,vib
bl,l
1223 1223 1223 1223 1223 1223 1223 1223 1223 1223 1223 1223 1223 1223 1223 1669
2221 1881 1405 1589 1783 1553 1591 1628 1666 1741 1816 2228 472 472 678 0
fa,
PS
0.5 0.83 2.63 1.66 4.76 12 18 28 40 125 410 7 x 106 5.7 0.5 0.5
Parameters for electron-transfer simulations in cyclohexane are determined from the absorption spectrum of rert-butylbetaine in cyclohexane.
which includes multiple vibronic channels (Figure lb). Sumi and Marcus used the Levich” expression for k(X), which is a single vibronic channel model (Figure la); the vibrational degrees of freedom were assumed to be purely classical. We divide the classical reorganization energy obtained from the spectral fit into two parts: first, a term corresponding to classical vibrations of the solute and, second, a term corresponding to the classical solvent reorganization. The X-dependent rate constant can be expressed as k(X) =
CGT(X) n
(7)
-
where
I 8000
I
energy (cm-l)
30,000
Figure 7. Static absorption spectra of betaine-30 in GTA: experimental (dashed line) and simulated (solid line).
The nonadiabatic frequency factor, A N A , is ANA =
2
4
h (4*Xcl,vibkBT)”2
(9)
a l , v i b is the reorganization energy of the low-frequency, classical vibrations of the solute and is neither X nor T, dependent. On the other hand, A G k n is solvent coordinate dependent as follows:
AG& =
Llv+ AGO - 2Xx,,, + nhu
(10) Note that the classical reorganization energy has been partitioned into the two previously mentioned parts: one due to the solvent, Xsolv, and one due to low-frequency, classical vibrations of the solute, Xcl,vib. An assumption of model C is that vibrational relaxation of the internal degrees of freedom in the reactant and product is fast on the electron-transfer time scale. Determination of Parameters. In order to obtain the spectroscopic potential parameters used in Table V, we have fit the static absorption spectra to a line shape model which includes one classical degree of freedom and one high-frequency quantal degree of freedom. The form of the line shape is therefore a sum of Gaussians in a Franck-Condon distribution (vibronic shape factor) as follows:
Here, AGO is the reaction free energy (driving force), A,, is the is the intramolecular total classical reorganization energy, hM reorganization energy due to the quantum, high-frequency vibration, and uQM is the frequency of the quantum mode. We in a partition XcI into and Llvby assuming XcI = nonpolar solvent. More specifically, the value for was obtained by fitting absorption spectra of tert-butylbetaine in cy-
clohexane and GTA using a model with one quantum and one classical mode. The difference in the classical reorganization energies from these two fits, 1741 cm-I, was assumed to represent the additional classical reorganization energy provided by the solvent nuclear polarization of GTA. When the absorption spectrum of betaine-30 in GTA was fit in the same manner, the classical reorganization energy was 2964 cm-I. We assumed that the difference between 2964 and 1741 cm-l represented the classical reorganization energy of betaine-30 due to the classical intramolecular degree of freedom. A least-squares fitting procedure was used, and except in the case of GTA, free fits yielded the values in Table VI. For GTA the frequency of the quantal degree of freedom was held fmed for purposes of fit stability. The static absorption spectrum of betaine30 in GTA is shown in Figure 7 along with the fit. We note trends in the parameters for betaine-30 which are similar to those observed by Kjaer and Ulstrup for betaine-26.12 For instance, more polar solvents exhibit more solvent reorganization energy. As expected, for a system with a larger dipole moment in the ground state than in the excited.state, there is a smaller free energy change for less polar solvents. However, the difference is smaller than would be expected from continuum approximations. Another pattern observed is the increase in reorganization energy, XQM, in the high-frequency mode, uQM, with increasing solvent polarity. This may reflect a non-Franck-Condon effect in the vibronic profile. Simulation Results. Model C combines the important features of both models A and B to accurately predict the electron-transfer kinetics of betaine-30 and tert-butylbetaine in a wide range of environments. A closer examination of the model can lead to insight into the interplay of solvent motion and intramolecular motion in electron-transfer kinetics. Figure 8 shows one of the roles that classical vibrations play in determining k(X). Figure 8a is a plot of k(X) with XcIvib = 50 cm-’, while Figure 8b shows k(X) with &l,db = 1223 cm-’; the
3134 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
Walker et al. A
3
-
'1 10
'1
2
1
0
Solvent Coordinate
Solvent Coordinate Figure 8. Solvent-dependent electron-transfer rate constant, k(X), predicted by model C. In panel a (top), hcl,vib = 50 cm-' and in panel b (bottom), &l,db = 1223 cm-l. The parameters for (b) correspond to those for betaine-30 in acetone.
remaining parameters are the same. One effect of the classical vibrational reorganization energy is the reduction of the "coarse graining" brought about by modeling the vibronic levels, which act as accepting modes with a single mode, rather than a multimode model, which would presumably be a more realistic approach. As in model A, k(X) tends to be largest at values of X which correspond to small energy gap. Figure 8b depicts log k(X) for betaine-30 in acetone as a function of solvent coordinate. In the region of X prepared by laser excitation, (no solvent displacement from the ground-state equilibrium solvent configuration), k(X) is much slower than near the minimum of the excited-state free energy surface. In a quickly relaxing solvents like acetone, solvation occurs faster than k(X) at the initially prepared distribution, P(X,t=O). Therefore, solvation dynamics cause an evolution of P(X,t) along X toward the free energy minimum where k(X) is faster than solvation dynamics (Figure 9B). This type of inductive behavior can be seen in Figure 10. In contrast, the simulations reveal that in the slowly relaxing solvents, T, > 10 ps, the evolution along X due solvation is so slow that P(X,t)decreases due to the reaction faster than it shifts toward lower energy, as shown in Figure 9A. The reaction is nearly at the frozen solvent limit where the predicted k,, is no longer dependent on 7,. In this case, the model predicts inhomogeneous kinetics, the reaction occurring at different rates for different solvent configurations. The complex interaction of solvation dynamics and electrontransfer kinetics causes the time dependence of the survival probability to be nonexponential. Therefore, following the lead of Sumi and Marcus," we report two characteristic survival times, 7 , and Tb defined by T,
= l S ( t ) dt
7b
= l f S ( f )dt
and
The greater the difference between
T,
and
(13) Tb, the
more nonex-
Figure 9. (A) Movement of population in a slow solvent (betaine-30 in GTA), demonstrating the inhomogeneous kinetics that result in the "frozen" solvent limit. (B) Movement of population in a fast solvent (betaine-30 in acetone). Notice that, even in acetone, a significant portion of the reaction occurs before the population reaches X = 0.
0
1
2
3
4
5
Time (psec)
Figure 10. Survival, S(r),for a fast solvent. Note the induction period caused by the interplay of solvent motion and the solvent-dependent rate constant, k(X).
ponential are the dynamics. It is apparent from both k(X) and the values of r a and Tb reported in Table V that the kinetics are sometimes highly nonexponential. Because T, is more sensitive to the early time behavior of S ( t ) , it tends to demonstrate the inductive behavior of S ( t ) in solvents whose relaxation time is faster than k(X) in the region of laser excitation (see Figure 10). Thus, the calculations from model C strongly suggest that the observed dependence of T,, on rScan be assigned to qualitatively different regimes: the first is controlled by solvation dynamics, and the second, in a frozen solvent limit, is controlled by thermal electron transfer along vibrational coordinates. These two regimes are a natural consequence of the treatment by Sumi and Marcus.13 However, it should be emphasized that in order to treat ultrafast electron transfer in the inverted regime, the inclusion of highfrequency vibrational accepting modes and correct initialization of the population distribution is necessary. Comperisoa with Experimeat. Figure 11 shows the remarkable agreement between the observed electron-transfer times and the simulated times. These results are summarized in Table V,
Electron-Transfer Kinetics
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3135 TABLE VII: Temperature Dependence of k, of Betaine-30. Observed Rates and Rates Predicted by Model C rates, lo1*s-I
4-
E
c‘
3-
357 331 293 286 27 1 237 203
G
c
E
;” 2 c e al
3 I-
u.8
t 0
20
40 Solvation Time
60
80
Figure 11. Experimentally observed electron-transfer times, 0,and survival times, 7, and ‘ T ~ . predicted using model C.
1o2 lo41
L - - - - ’ - - . . ’ . - . . L 3.5 4.0 4.5
Temperature”
IC1)
Figure 12. Arrhenius plot of experimental data and theoretical predictions for betaine-30 in GTA.
calculated using the parameters found in Table VI. In these cases, the matrix element, Vel,has been fixed a t 2500 cm-I to optimize agreement of the last plotted data point, GTA at 298 K. The value of V, used lies within the range of estimates made using the Hushls approximation (1400-2800 cm-’ for electron transfer using either a valence bond description or the ring center-to-center distance, respectively). The small temperature dependence we have observed for betaine30 and tert-butylbetaine is similar to the observations of Closs and co-workers on a related inverted regime e~amp1e.I~A summary of the Arrhenius behavior of betaine30 in GTA is shown in Figure 12, along with the predictions of models A-C. The potential parameters used in generating the figure are based on direct, spectroscopic measurement, except for the lowest temperature point, 228 K, which is made by extrapolation from the potential parameters a t higher temperatures. Obviously, model C predicts the experimentally observed values much better than model A or B. Again, the failure of model A may be traced to the large barrier along the classical vibrational coordinate for the electron transfer at lower temperatures, and the failure of model B may be traced to the model’s dependence on solvent diffusion. The activation energies predicted by model C in toluene and benzene are listed in Table VII. In this case an additional (18) Hush, N. S. Prog. Inorg. Chem. 1967, 8, 391. Liang, N.; Miller, J. R.; Closs, G. L. J . Am. Chem. SOC.1990, 11.2,
(19)
5355.
Solvent: Benzene 0.61 0.58 0.54 0.53 0.50 0.47 0.42
0.50 f 0.05 0.37 f 0.03 0.42 f 0.08 0.33 f 0.07
33 1 295 293 282 269 249 223 202
Solvent: Toluene (7,= 0.5 ps)” 0.158 0.173 0.55 f 0.02 0.143 0.156 0.34 f 0.02 0.31 d~ 0.02 0.136 0.149 0.32 f 0.04 0.130 0.143 0.29 f 0.02 0.122 0.134 0.20 f 0.01 0.113 0.125 0.15 f 0.01 0.107 0.118
331 295 293 282 269 249 223
Solvent: Toluene (7,= 5.7 p ~ ) ~ 0.087 0.1 10 0.55 0.02 0.082 0.106 0.34 f 0.02 0.31 f 0.02 0.080 0.104 0.32 f 0.04 0.29 f 0.02 0.073 0.096 0.20 f 0.01 0.15 f 0.01
“is is set at 0.5 ps in a somewhat arbitrary fashion, since little information is available. This might correspond to the fluctuation frequency of some type of *-stacking excimer-like complex. 7,is set at 5.7 ps, again in an arbitrary fashion.
‘..
I
0.49 0.47 0.44 0.43 0.41 0.38 0.35
approximation has been made: that the potential parameters are not temperature dependent. This is less critical for nonpolar solvents. The predicted activation energies, listed in Table VII, are small and on the order of the observed activation energies. All three models, A-C, are flawed by their treatment of solvent dynamics as overdamped and characterized by a single relaxation time. Various experimental and theoretical results have demonstrated that solvent dynamics are nonexponentialZ4 and also suggest that inertial effects may be i m p ~ r t a n t . ~ *The ~ ~ nonexponential dynamics could be treated by using a generalized diffusion equation. In a crude way, we could treat inertial effects by adding an additional quantum mechanical degree of freedom to represent the inertial properties of the solvent.24 Also, we have only considered dielectric friction. If betaine undergoes significant large-amplitude displacements during the reaction, collisional friction could be important; we assumed that the internal classical bath had an infinite relaxation rate. This leads to another obvious problem: the use of a single average high-frequency vibrational mode and a single low-frequency mode. A more appropriate treatment would use all modes found in a resonance Raman spectrum of betaine, including vibrational excitations of the electronic excited state. Other approximations are more difficult to discern, but result from employing the Fermi Golden Rule expression for the nonadiabatic rate. This neglects change in curvature of the vibronic surfaces due to the strong electronic coupling. Furthermore, in certain situations state/state coherences are expected to play an important role, as demonstrated in the work of Jean et alez5for multimode quantum rate processes which are analogous to electron (20) Rosenthal, S. J.; Xie, X.;Du, M.; Fleming, G. R. J. Chem. Phys.
1991, 95, 4715.
(21) Maroncelli, M.; Fleming, G. R. J . Chem. Phys. 1988, 89, 5044. (22) Maroncelli, M. J . Chem. Phys. 1991, 94, 2084. (23) Fonseca, T.; Ladanyi, B. J . Phys. Chem. 1991, 95, 2116. (24) Walker, G. C.; Barbara, P. F.; Doorn, S. K.; Dong, Y.; Hupp, J. T. J . Phys. Chem. 1991, 95, 5712. (25) Jean, J. M.;Fleming, G. R.; Friesner, R. A. Ber. Bunsen-Ges. Phys. Chem., in press.
J. Phys. Chem. 1992, 96, 3736-3741
3736
transfer. Our simpler model neglects any possible coherent effects. Summary
We have measured the electron-transfer rate of betaine-30 and tert-butylbetaine in various solvents, both polar and nonpolar. We find the electron-transfer rate depends on solvation dynamics for fast solvents, but does not scale with solvation dynamics. Rather, the rate remains fast even in very slowly relaxing solvents. Also, the temperature dependence of the electron-transfer rate in these very slowly relaxing solvents is small. Because the treatments of Sumi and MarcusI3 and of Jortner and Bixon16 are unable to reproduce all of the observed trends, we have introduced a hybrid model. The model includes two classical degrees of freedom: (i) an intramolecular vibrational degree of freedom of the solute with an arbitrarily fast correlation time and (ii) a solvent degree of freedom characterized by a single
relaxation time. The model also includes a single quantal degree of freedom corresponding to a high-frequency intramolecular vibrational mode of the solute. This vibrational mode serves to reduce the effective activation energy of the reaction, compared to the prediction of a purely classical model. Partitioning the classical energy into a low-frequency intramolecular degree of freedom and the solvent provides a mechanism for thermally activated electron transfer in rigid media. Acknowledgment. Support of this research by the National Science Foundation and the Office of Naval Research is gratefully acknowledged. N.E.L. was supported by ,a National Science Foundation Postdoctoral Fellowship and E.A. was supported by a Swedish Natural Science Research Council Postdoctoral Fellowship. Registry No. Betaine-30, 10081-39-7.
Electron Transfer from Ground-State Triethylamine to the Second and Lowest Excited Triplet States of Haloanthraquinones (1-Chloro, 2-Chloro, 1,5-DIchloro, 1,8-Dlchloro, and 1,8-DIbromo Compounds) in Acetonitrile at Room Temperature Studied by Picosecond and Nanosecond Laser Spectroscopy Kumao Hamanoue,* Toshihiro Nakayama, Satoshi Asada, and Kazuyasu Ibuki Department of Chemistry, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku. Kyoto 606, Japan (Received: October 9, 1991; In Final Form: January 2, 1992)
-
-
Picosecond laser photolysis of haloanthraquinones (XAQ) at room temperature in acetonitrile-triethylamine (TEA) reveals the existence of two new absorptions (bands X and Y) which are different from the triplet-triplet (T” T2 and T’ TI) absorptions due to the second [3XAQ*(T2)]and lowest [3XAQ*(Tl)] excited triplet states of XAQ. The rise of bands X is faster than that of bands Y, which increase with accompanying decrease of T’ T I absorptions. Moreover, the intensity ratios of bands X to bands Y at 2-11sdelay increase with increasing T2 TI internal conversion times. Since nanosecond laser photolysis reveals that bands Y shift to bands X in the submicrosecond time regime with a rate matching increase of transient photocurrents and that bands X decay in the millisecond time regime following second-order reaction kinetics which yields haloanthrasemiquinoneradicals and triethylamine radical, it is concluded that bands X are the ahrptions of the free-radical anions of XAQ produced via the direct electron transfer from ground-state TEA to 3XAQ*(T2)and that bands Y are the absorptions of the exciplexes between 3XAQ*(Tl)and ground-state TEA or the ion pairs (or the contact ion pairs) between the free-radical anions of XAQ and the free-radical cation of TEA.
-
-+
Introduction In previous papers, we have reported that picosecond laser photolysis of 1,8-dichloroanthraquinone1 and 1,8-dibromoanthraquin~ne~*~ in solutions at room temperature give rise to the appearance of not only the triplet-triplet (T’ T I ) absorption bands of the lowest excited triplet states [3XAQ*(Tl)] but also the T” T2absorption bands of the second excited triplet states [3XAQ*(T,)] located below the lowest excited singlet (SI)states and that the T2 T1 internal conversion times are -700-750 ps for 1,8-dichloroanthraquinone and -70-1 10 ps for 1,8-dibromoanthraquinone. Although no clear T” T2 absorption bands have been observed for anthraquinone, l-chloroanthra-
quinone, 2-chloroanthraquinone, and 1,5-dichloroanthraquinone, analysis of the change of transient absorptions with time has led us to the conclusion that the T2states also exist below the SIstates and the T2 T1 internal conversion times are less than 70 P S . ~ In spite of these circumstances, an electron transfer from ground-state triethylamine to triplet anthraquinone and haloanthraquinones (XAQ) forming the exciplexes has been found to occur via 3XAQ*(TI)in both toluene and ethanol, and these exciplexes change to the contact ion pairs between the radical anions of XAQ and the radical cation of triethylamine, followed by the first-order proton transfer generating the anthrasemiquinone radicals and the triethylamine r a d i ~ a l ; ~neither - ~ * the free-radical
(1) Hamanoue, K.; Nakajima, K.; Kajiwara, Y.; Nakayama, T.; Teranishi, H. Chem. Phys. Lett. 1984,110, 178. Hamanoue, K.; Nakayama, T.; Shiczaki, M.; Funasaki, Y.; Nakajima, K.; Teranishi, H. J. Chem. Phys. 1986, 85, 5698. Hamanoue, K.; Yamamoto, Y.; Nakayama, T.; Teranishi, H. Physical Organic Chemistry 1986 Kobayashi, M.,Eds.; Studies in Organic Chemistry; Elsevier: Amsterdam, 1987; Vol. 31, p 235. (2) Nakayama, T.; Ito, M.; Yuhara, Y.; Ushida, K.; Hamanoue, K. UI-
(4) Nakajima, K. Master Thesis, Faculty of Engineering and Design, Kyoto Institute of Technology, 1983. ( 5 ) Hamanoue, K.; Yokoyama, K.; Kajiwara, Y.; Kimoto, M.; Nakayama, T.; Teranishi, H. Chem. Phys. Lett. 1985, 113, 207. (6) Ha,ma,noue, K.;Nakayama, T.; Yamamoto, Y.; Sawada, K.; Yuhara, Y . ;Teranishi, H. Bull. Chem. SOC.Jpn. 1988, 61, 1121. (7) Hamanoue, K.; Kimoto, M.; Kajiwara, Y.; Nakayama, T.; Teranishi, H. J. Photochem. 1985, 31, 143. (8) Hamanoue, K.; Nakayama, T.; Ibuki, K.; Otani, A. J. Chem. SOC.,
+
+
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+
trafast Phenomena VI, Springer Ser. Chem. Phys. 1988, 48, 489. (3) Hamanoue, K.; Nakayama, T.; Ito, M. J. Chem. Soc., Faraday Trans. 1991, 87, 3487.
0022-3654/92/2096-3736$03.00/0
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Faraday Trans. 1991, 87, 3731.
0 1992 American Chemical Society
