J . Phys. Chem. 1987, 91, 6277-6285
Acknowledgment. This work was supported by the U S . Department of Energy, Office of Basic Energy Sciences, under Grant DE-FG02-84ER 13269. This support does not constitute an endorsement by DOE of the views expressed in this article. Preliminary experimental work by J. A. Silver, laboratory technical support by P. Saccone and S. Kallelis, and useful discussions with F. Kaufman, D. Herschbach, M. Zahniser, and K. McCurdy contributed to the results presented here. Helpful comments by reviewers were also appreciated. Registry No. CHBr3, 75-25-2; Na, 7440-23-5; K, 7440-09-7; CH':, 3315-37-5; N20, 10024-97-2;CHI, 74-82-8; H,,1333-74-0; 0 2 , 778244-7; C12, 7782-50-5.
molecules such as HF, HCl and DF, DC1, as well as more precise determinations of the CH Clz rate constant. The C H D2 system should also be studied in the present apparatus to avoid any systematic differences between this and previous work.
+
6277
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5. Conclusion
A new, continuous chemical source of methylidine radicals designed for use in fast-flow reactors is described and has been used to make rate measurements for five C H reactions near room temperature. The CH ClZreaction rate constant is measured for the first time. Other results are in good agreement with those of previous studies.
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Comprehensive Study on the Role of Coordinated Solvent Mode Played in Electron-Transfer Reactions In Polar Solutions Toshiaki Kakitani* Department of Physics, Nagoya University, Furo-cho, Chikusaku, Nagoya 464, Japan
and Noboru Mataga* Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan (Received: March 18, 1987; In Final Form: June 1 1 , 1987)
The role of the coordinated solvent mode in the three kinds of electron-transfer reactions-charge separation, charge recombination, and charge-shift reactions in polar solutions-is extensively investigated. That is, taking into account the fact that the vibrational frequency of the possibly coordinated solvent molecules around the reactant in the neutral state is considerably smaller than that in the charged state, we have derived formulas for the electron-transfer rate as a function of the energy gap which are quite different among the three kinds of electron-transferreactions. In such theoretical treatments, we have derived both formulas in an approximate analytical form and those in an exact integral form for the Franck-Condon factor of this solvent mode. By means of the extensive numerical calculations using the analytical and integral formulas, we have revealed the criterion within which the analytical formulas are used safely. By use of those formulas, the property of three different energy gap laws corresponding to the three kinds of electron-transfer reactions is extensively analyzed. The energy gap law of electron-transfer reactions including the transition-metal complex is discussed from the standpoint of a generalized charge-shift reaction. On the basis of the above elaborate calculations, we have introduced an index denoting the ability of annihilation of the inverted region. This index is found to be useful for the judgment of the inverted region of all the electron-transfer reactions considered.
Introduction Electron transfer (ET) is one of the most important reactions in chemistry and biology. The E T rate could be as fast as 1 ps-' and as slow as 1 yr-', depending on reactants and environmental conditions. It is quite valuable if one can discriminate various mechanisms which regulate the ET reaction rate. From a theoretical point of view based on Fermi's golden rule for the transition rate, which applies to the nonadiabatic process, the E T rate is governed by the electron-tunneling matrix element and the Franck-Condon factor. The former and the latter contribute to the reaction rate by the form of the preexponential factor and the exponential factor related to the thermal activation, respectively. The tunneling matrix element has a strong distance dependence between donor and acceptor molecules, because the overlap of electronic wave functions of those reactants is roughly proportional to the tunneling matrix element. Some experimental studies have shown that its distance dependence could be e~ponential.'-~ If the electronic wave function of a donor or acceptor molecule is mixed with that of environmental molecules or groups, the transferring electron can travel via those environmental species
and the overlap of electronic wave functions between donor and acceptor molecules can be effectively much larger than in the vacuum state. Then, the ET can be a long-range reaction. A study on this mechanism is still in p r o g r e s ~ . ~ .In ~ this respect, the following fact should be pointed out. When the electron-tunneling matrix element exceeds a certain value, the character of the ET process changes to adiabatic and the preexponential factor is replaced by a frequency factor, as expressed by the transition-state theory. Recently, it has been elucidated that when the preexponential factor somewhat exceeds 1 / ~ the ~ , inverse of the longitudinal dielectric relaxation time of solvent, the solvent dynamics can be effective to the ET reaction, the preexponential factor of the ET rate being limited to a much smaller factor than the frequency On the other hand, the role of the Franck-Condon factor has been extensively investigated. This factor is concerned with the thermal activation factor mostly represented by the exponential term in the ET rate, irrespective of the transition being adiabatic or nonadiabatic. If the decay of the survival probability of the transferring electron is expressed by a single-exponential function of time, the activation factor is not affected by the dynamical
(1) Beitz, J. V.;Miller, J. R. J . Chem. Phys. 1979, 71, 4579. (2) Miller, J. R.; Beitz, J. V.;Huddleston, R. K. J. Am. Chem. SOC.1984, 106, 5057. ( 3 ) Mataga, N.; Karen, A.; Okada, T.; Nishitani, S.; Kurata, N.; Sakata, Y.;Misumi, S. J. Phys. Chem. 1984, 88, 5138. (4) Leland, B. A.; Joran, A. D.; Felker, P.M.; Hopfield, J. J.; Zewail, A. H.; Dervan, P. B. J . Phys. Chem. 1985,89, 5571. ,
#
I
( 5 ) Larsson, S. J . Chem. Soc., Faraday Trans. 2 1983, 79, 1375. (6) Larsson, S. J . Phys. Chem. 1984, 88, 1321. (7) Calef, D. F.; Wolynes, P. G . J . Phys. Chem. 1983, 87, 3387. (8) Sumi, H.; Marcus, R.A. J . Chem. Phys. 1986, 84, 4894. (9) Gennett, T.; Milner, D. F.; Weaver, M. J. J . Phys. Chem. 1985, 89,
2787.
0 1987 American Chemical Societv -
6278 The Journal of Physical Chemistry, Vol. 91, No. 24, 1987
property of the solvent. A sufficient condition for this is that the E T rate is smaller than 1 / ~ ~ .Such ' a condition is believed to be satisfied in a nonviscous dielectric solvent such as acetonitrile. We treat such systems in this paper. The Franck-Condon factor is strongly correlated with the magnitude of energy gap between the initial and final states of reaction. Such a correlation is called the energy gap law. The first experimental study of the energy gap law for the photoinduced charge-separation (CS) reaction has been done by Rehm and Weller'O in the case of the fluorescence quenching reaction in polar solvents, A* + B A[ + B,'. The results of measurements showed a rapid increase of the bimolecular rate constant k, of the ET reaction around the zeroenergy gap toward the downhill side and no appreciable decay of the E T rate at the large energy gap (strongly exothermic) region. The behavior at the small energy gap region was consistent with the current theory such as the Marcus theory.I1J2 However, the observed behavior at the large energy gap region contradicted the above theory which predicted a rapid decrease of the ET rate at the large energy gap region, usually called the inverted region. In this sense, the large energy gap region in the above experimental resultlo was called the anomalous region. This problem has not been quantitatively resolved even by means of a more advanced theory incorporating the intramolecular quantum mode.I3 Several other possibilities were suggested many years ago to interpret this problem for the lack of an "inverted region" in the observed relation between k, and the free energy gap -AG between B,') states. One interpretation assumes (A* B) and (A; the formation of a nonfluorescent charge-transfer (CT) complex at the diffusion-controlled e n ~ 0 u n t e r . l ~However, recent picosecond laser photolysis studies by Mataga et al. upon aromatic hydrocarbon (AH)-tetracyanoethylene (TCNE) systems in acetonitrile, which undergo strongly exothermic photoinduced CS, have rejected this inter~retati0n.l~Namely, the ion pairs, AH,+-.TCNE;, produced in the bimolecular fluorescence quenching reaction seem to have a loose structure with a lifetime 1 ns, while the excited state of the C T complex of ca. 100 ps in acetonitrile undergoes ultrafast nonradiative deactivation to the ground state within a few picosecond^.'^ Another possibility is the participation of excited electronic states of the ion pair (A>-B,') when -AG becomes too large, keeping the FranckCondon factor of the ET process large.1~10~16 However, the picosecond studies on the above AH-TCNE systems in acetonitrile have so far never showed any indication of the formation of the excited ion pair state immediately after ET.I5J7 Therefore, those interpretations proposed previously do not seem probable. We need more reasonable new interpretations, as we proposed recently.'' On the other hand, Wasielewski et al.19 synthesized a complex which involves tetraphenylprphine (donor) and quinone (acceptor) molecules linked by nonconjugated bonds (spacer) and showed that the charge-recombination (CR) rate of the intramolecular ion pair produced by photoinduced CS dropped rapidly at the large energy gap, although no data were obtained for the normal region, Le., for the region where the rate increases with the increase of the energy gap. Miller et a1.20 studied the intramolecular A--Sp-B, where chargeshift (CSH) type. of reaction, A-SpBSp is a rigid saturated hydrocarbon spacer, by means of the pulse radiolysis method and showed the bell-shaped energy gap dependence of the ET rate; Le., the rate increased with increase of the energy gap and then decreased considerably with further
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(10) Rehm, D.; Weller, A. Isr. J . Chem. 1970, 8, 259. (11) Marcus, R. A. J . Chem. Phys. 1956, 24, 966. (12) Marcus, R. A. Annu. Reo. Phys. Chem. 1964, 15, 155. (13) Ulstrup, J.; Jortner, J. J. Chem. Phys. 1975, 63, 4358. (14) Masuhara, H.; Mataga, N. Arc. Chem. Res. 1981, 14, 312. (15) Mataga, N.; Kanda, Y.: Okada, T.J . Phys. Chem. 1986, 90, 3880. (16) Mataga, N. Bull. Chem. SOC.Jpn. 1970, 43, 3523. (17) Mataga, N.; Kanda, Y., to be submitted for publication. (18) Kakitani, T.:Mataga, N. Chem. Phys. 1985, 93, 381. (19) Wasielewski, M. R.; Niemczyk, M. P. J . Am. Chem. SOC.1984,106, 5043. (20) Miller, J. R.: Calcaterra, L. T.; Closs, G . L. J . Am. Chem.SOC.1984, 106, 3047.
Kakitani and Mataga increase of the energy gap. The advantage of using such a linked complex is to eliminate the structural ambiguity in the collision complex which is thought to be formed in the bimolecular reaction. At this point, one may consider that the anomalous region is characteristic of the reaction between uncombined donor and acceptor. However, this is not the case. The inverted region has been o b ~ e r v e dfor ' ~ the ~ ~CR ~ ~reaction ~~ of the ion pairs of uncombined donor and acceptor produced by the fluorescence quenching reaction where, however, the observation of the inverted region was not possible. Moreover, for this CR reaction of the ion pairs, the observation has been made not only for the inverted region but also for the normal region, completing the bell-shaped energy gap dependence of the ET rate c o n ~ t a n t , although ' ~ ~ ~ ~ the data for the normal region were not available in the case of the CR reaction of the porphyrin-quinone intramolecular ion pair of Wasielewski et aI.l9 It should be noted here that, in the abovediscussed CR reaction of the ion pairs of uncombined donora c c e p t ~ r , ' ~and * ~also ~ - ~the ~ CS, C R as well as CSH reactions of combined donor-acceptor systems of Wasielewski et a l l 9 and Miller et al.,20the condition that the rate of ET is considerably smaller than is satisfied. In such a situation as described above, we have proposed a new aspect of the role of solvent molecules played in the ET reaction.18,22-24That is, the solvent motion surrounding a charged reactant is considered to be more restricted than that surrounding a neutral reactant. Indeed, on the basis of the dielectric continuum model and the Lorentz model of the internal field, we have theoretically shown that the force constant for solvent vibration surrounding a charged reactant can be much larger than that surrounding a neutral reactant.25 We have discriminated solvent molecules in the first layer surrounding a reactant in proximity from those outside the first layer. The vibrations of the former ones are called the c mode because those solvent molecules are considered to be weakly coordinated to a charged reactant. The vibrations of the latter ones are called the s mode. The vibrational frequency of the c mode is much affected by the charged state of the reactant. By introducing the c mode of the solvent, we have succeeded in reproducing theoretically the anomalous region in the C S reaction.I6J8 On the other hand, our theory predicts a strong energy gap dependence of the ET rate and its large decrease with increase of the energy gap in the inverted region for the CR and also a moderate decrease of the ET rate at large energy gap for the CSH reaction.22 Therefore, the predictions by our theory are consistent with all of the experimental results given for the photoinduced CS,l0 CSH,20and CR15J9s21322 reactions, which indicates that whether or not the inverted region appears depends upon the type of the E T reaction. In view of the qualitative success of our theory, the FranckCondon factor of the c mode plays an important role in regulating the energy gap law, depending on the type of ET reaction. Accordingly, it will be very important to investigate the property of the Franck-Condon factor due to the c mode in more detail. Indeed, in the above theoretical study of the C S and C R reactions18,22-24 we have used approximate analytical formulas which are correct when the ratio of vibrational frequencies in the charged and neutral states is infinity. For the CSH reaction, we have just showed a final result of the exact integral formula.22 In this paper, we complete the theoretical framework of the approximate analytical formula and exact integral formula for the CS, CR, and CSH reactions. Then, we examine to what extent the analytical formulas are valid by comparing numerical results obtained by the analytical and integral formulas. Since we did not present any derivation of the formula for the CSH reaction so far, the main course of our formalism in this paper will be along the CSH reaction. After completing the formalism and numerical calculations, we shall introduce an index denoting the ability of annihilating the inverted region. It will be shown that a qualitative (21) (22) (23) (24) (25)
Mataga, N . Acta Phys. Pol. 1987, A71, 767. Kakitani, T.;Mataga, N . J . Phys. Chem. 1986, 90, 993. Kakitani, T.; Mataga, N . J . Phys. Chem. 1985, 89, 8. Kakitani, T.; Mataga, N . J . Phys. Chem. 1985, 89, 4752. Kakitani, T.;Mataga, N . Chem. Phys. Lett. 1986, 124, 437.
Electron-Transfer Reactions in Polar Solutions
The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 6279
discussion of the inverted region becomes very easy for all the kinds of electron-transfer reactions by evaluating this index. Theoretical Formulation of the CSH Reaction We consider the following CSH reaction: A...B- -+ A-...B
(1)
Among the many vibrations, we take into account only the intramolecular quantum mode (q mode) and the coordinated solvent mode (c mode). The other modes such as the solvent mode due to the solvent molecules surrounding the coordinated solvent molecules (s mode) and the intermolecular vibrational mode of reactants (r mode) might play a substantial role in some systems. However, we focus upon the role of the coordinated solvent mode in this paper. Let us start from the following general formulals for the charge-shift reaction rate V S H ( - A G ) fYCSH(-AG)= x:8,CSH(t) Wq(AE-e) de
0
The energy E emitted by reactant A or absorbed by reactant B- is respectively
(2)
where
0
Solvent coordinate x Solvent coordinate y around molecule A around molecule B Figure 1. Potential surfaces along the solvent coordinate around the neutral and charged states of reactants A and B.
E = E,
+ 1/2kanX2- 1/2kac(X- x,)'
(12)
E = Eb
- !hkd2 + y 2 k b n b - Yb)'
(13)
and
W,(AE-t) = A e x p ( ~ (+ 2 ~I ) ) Z , I ( ~ S [ O ( B
+ I)]'/~)[(o + I)/o]P/~
B = [exp(h(u)/kT) -
11-l
(3)
(4)
p = (AE - t)/h(u)
(5)
s = 62/2
(6) (7)
where kac and kbnare force constants of the c mode of solvent surrounding A- and B, respectively. x , and yb are origin shifts of the solvent coordinate x and y accompanying the ET reaction, respectively. From eq 12 and 13, we obtain 1 x=(kacxa f [kankag? + 2(kac - kan)(Ea - E)1'/2) kac - kan (14) 1 Y = kkbnYb [kbckb13.'b2 - 2(kbn - kbc)(Eb kbc - kbn
*
and ScCSH(e)= J I D a ( E ) &'(E) d E
In the above equations, W is the ET rate when only the q mode is taken into a c c o ~ n t . ~ ~ScCSH . ~ ' represents the energy density of emitting the energy e of the c mode in the CSH reaction. -AG and AE are the free energy gap and energy gap, respectively. The relation between -AG and A E is given later. In eq 3-7, Zb1, ( u ) , 6, I(ilHlf)l, and T denote the modified Bessel function of the first kind, average angular frequency and origin shift of the normal coordinate of the q mode, electronic tunneling matrix element, and temperature, respectively. In eq 8, Da(E)and %(E) represent the electron insertion spectrum into the reactant A and the electron removal spectrum from the reactant B-, respectively. We derive a formula of ScCSH(e), using a theoretical method in analogy with the excitation transfer by the Forster mechanismZs as before.18 The potential energy surfaces of solvent motion are shown in Figure 1 . For simplicity, we assume that the solvent motions surrounding reactants A and B are independent although, strictly speaking, it is not the case.27 The energy parameter e is related to the energy changes E, and Eb in Figure 1 through the equation t
= E, - Eb
(15)
(8)
(9)
We define here new parameters as follows
(10)
P b b ) = ( k k / 2 a k T ) ' / * exp(-kd2/2kT)
(11)
where k,, and kbc are force constants of the c mode of solvent surrounding A and B-, respectively. (26) Kestner, N. R.;Logan, J.; Jortner, J. J . Phys. Chem. 1974, 78, 2148. (27) Jortner, J. J . Chem. Phys. 1976, 64, 4860. (28) Forster, Th.Ann. Phys. 1948, 2, 55.
= kan/(kac - kan)
(16)
Pb
=
(17)
kbn/(kbc
- kbn)
Aac = kacX?/2 = kdb2/2
E,' = Ea + PaAac
(18)
(19) (20)
E< = Eb (21) In the above parameters, Aacand Ab represent the reorientation energies of the coordinated solvents around A and B molecules when the reactants change from a neutral state to a charged state. The spectra D,(E) and D
