J. Phys. Chem. 1986, 90, 993-995
993
Different Energy Gap Laws for the Three Types of Electron-Transfer Reactions in Polar Solvents Toshiaki Kakitani* Department of Physics, Nagoya University, Chikusaku, Nagoya 464, Japan
and Noboru Mataga* Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan (Received: October 28, 1985)
Taking into account the fact that the motion of polar solvent molecules around the charged solute molecule is much more restricted than that around the neutral solute molecule, we have obtained theoretically three different energy gap laws for the photoinduced charge separation (CS), A*.-D or A-D* A--D+, charge shift (CSH), A-DA--D or A+-D A-D', and charge recombination (CR), A--D+ A-D, reactions. The three energy gap laws differ from each other especially in the strongly downhill region; the rates of CS, CSH, and CR decrease scarcely, moderately, and drastically, respectively. This theoretical prediction is consistent with the experimental results obtained so far for CS, CSH, and CR reactions.
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Introduction Electron transfer (ET) is the most fundamental reaction process, and the elucidation of the molecular mechanism which regulates the E T rate is one of the most important problems in physical chemistry and biology. Especially detailed investigations have been made for the luminescence quenching reactions due to charge A--D+, in polar solvents.' separation (CS), A*-D or A-D* The charge recombination (CR) reactions of radical ion pairs A-D, have been inproduced by photoinduced CS, A--D+ vestigated by means of picosecond and nanosecond laser photolysis methods, in the case of various exciplex systems and excited electron donof-acceptor complexes, etc., in polar solvent^.^*^ Furthermore, D--A D-A- type of charge shift (CSH) reaction in solution was studied by means of the pulse radiolysis m e t h ~ d . ~ On the other hand, many theories5-l' have been proposed to explain various aspects of the ET reaction. One of the most important problems in the interpretation of experimental results on the basis of these theories is the energy gap dependence of the ET rate. In this respect, a controversial problem has been the energy gap dependence of the ET rate in the inverted r e g i ~ n , ~ , ~ where the free energy gap is larger than the reorganization energy. Namely, according to the systematic experimental studies of fluorescence quenching reactins of many aromatic fluorescerquencher pairs due to the CS type of ET in acetonitrile solution, the quenching rate constant k , shows a steep rise around zero energy gap toward the downhill side and it shows a constant, diffusion-controlled value even in the strongly downhill region.I2 It is difficult to explain these experimental results in terms of the above theories.
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( I ) See for example: (a) Mataga, N.; Ottolenghi, M. In 'Molecular Interactions"; Foster, R., Ed.; Academic Press: New York, 1979; Vol. 2, p I . (b) Mataga, N. In "Molecular Interactions"; Ratajczak, H., OrvilleThomas, W. J., Eds.; Wiley: Chichester, 1981; Vol. 2, p 509. (2) (a) Mataga, N . Pure Appl. Chem. 1984, 56, 1255. (b) Mataga, N.; Karen, A,; Okada, T.; Nishitani, S.; Kurata, N.; Sakata, Y.; Misumi, S. J . A m . Chem. SOC.1984, 106, 2442. (c) Mataga, N.; Karen, A.; Okada, T.; Nishitani, S.; Sakata, Y.; Misumi, S. J . Phys. Chem. 1984, 88, 4650. (d) Mataga, N.; Okada, T.; Kanda, Y.;Shioyama, H . Tetrahedron, in press. ( e ) Uemiya, T.; Miyasaka, H.; Masuhara, H.; Mataga, N., to be submitted for publication. (f) Shioyama, H.; Mataga, N., to be submitted for publication. (8) Kanda, Y.; Okada, T.; Mataga, N., to be submitted for publication. ( 3 ) Wasielewski, M. R.; Niemczyk, M. P.; Svec, W. A,; Pewitt, E. B. J . A m . Chem. SOC.1985, 107, 1080. (4) Miller, J. R.; Calcaterra, L. T.; Closs, G. L. J . A m . Chem. SOC.1984, 106, 3047. (5) Marcus, R. A. J . Chem. Phys. 1956, 24, 966. (6) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155. ( 7 ) Dogonadze, R. R. Dokl. Chem. (Engl. Transl.) 1959, 124, 9 . (8) Levich, V . G.; Dogonadze, R. R. Collect. Czech. Chem. Commun. 1961, 26, 193. (9) Levich, V . G. Adu. E1ectrochem. Electrochem. Eng. 1966, 4, 249. (10) Kestner, N . R.; Logan, J.; Jortner, J. J . Phys. Chem. 1974, 13, 161. ( 1 1 ) Ulstrup, J.; Jortner, J. J . Chem. Phys. 1975, 63, 4358. (12) Rehm, D.; Weller, A. Isr. J . Chem. 1970, 8, 259.
0022-3654186 , ,12090-0993$01.50/0 I
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Contrary to the above CS type ET reaction from the fluorescent state, it has been demonstrated that the CR reaction of photochemically produced ion pairs in polar solvents shows a strong energy gap dependence.* For example, the CR rate constant W 2 3 X 10" s-' in the case of ethyletioporphyrin (EEP) (D)toluyuinone (TQ) (A) system in acetone soIution2a-Cwith AG (between the ion pair and ground state) -1.46 eV, while W 3 X lo7 s-' for pyrene (A)-N,N-dimethylaniline (DMA) (D) in acetone solution2a*d,g with AG -2.88 eV. Pyromellitic dianhydride (PMDA) (A)-pyrene (D)2a,d.fand 1,2,4,5-tetracyanobenzene (TCNB) (A)-toluene (D)2a,esystems in various solvents with AG values between the above two extremes show W values also between the above two extremes, as it will be discussed in detail later. Moreover, a large decrease of Win the inverted region was also observed by picosecond laser photolysis investigations upon the C R reactions of a series of tetraphenylporphyrin (TPP) or Zn-TPP-quinone combined systems in solutio~~ Although .~ the study of the photoinduced CS reactions was also made for the same system, the observation of the C S rate constant in the inverted region was not possible.3 Picosecond pulse radiolysis studies have been made upon a series of biphenyl (D) and various hydrocarbons or quinones (A) combined with a rigid saturated hydrocarbon spacer: The rate D-A-, increases constant W of intramolecular CSH, D--A at first with an increase of the -AG value and shows a marked decrease in the region of large -AG values.J Any theoretical treatment of the ET reaction should be able to explain all aspects of the above experimental results. In this respect, we pointed out recently an important role of the following property of polar solvent molecules in the ET r e a c t i ~ n . l ~ -The '~ motion of polar solvent molecules around a charged solute molecule undergoing ET reaction is much more restricted than that around a neutral one. Therefore, the force constant of the phonon mode of polar solvent coordinated around a charged solute molecule is much larger than that surrounding a neutral solute molecule. By using this property, we have theoretically obtained the result that the photoinduced CS rate in strongly polar solvent does not appreciably decrease even in the strongly downhill region,l3.l4in agreement with experimental results.I2 We have also obtained the result that the CR rate drastically decreases in the strongly downhill r e g i ~ n , ' ~in, 'agreement ~ with experimental r e ~ u l t sat ~.~ least qualitatively. In the present report, we first formulate the energy gap law with a refined model of the solvent mode, where we put special emphasis on the CSH reaction which was not treated by us before. Then, we show that the experimental data so far obtained on the
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(13) Kakitani, T.; Mataga, N. Chem. Phys. 1985, 93, 381. (14) Kakitani, T.; Mataga, N. J . Phys. Chem. 1985, 89, 8 . (15) Kakitani, T.; Mataga, N. J . Phys. Chem. 1985, 89, 4752
D 1986 American Chemical Societv -
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The Journal of Physical Chemistry, Vol. 90, No. 6 , I986
Letters
three types of the ET reaction can be reproduced by our theoretical treatment using reasonable and consistent values of parameters.
Theoretical Method The large difference between the force constant of the solvent mode around the charged molecule and that around the neutral molecule is realized only when the dielectric saturation of the polar solvent molecule surrounding the charged molecule takes place to some extent.I3 In this respect, we have solved recently the problem of the dielectric polarization field around a charged-sphere molecule self-consistently, on the basis of the dielectric continuum model.16 Results of this calculation show that the polarization in the region corresponding to the first solvent shell is in weak dielectric saturation. Accordingly, we may call this strongly solvated layer of solvent the coordination shell, although the solvent-orientating force in this layer will not be so strong as that around a small metal ion. The orientation polarizations of other solvent molecules outside the coordination shell are not saturated but can still contribute to the solvation energy in some systems. We shall call the contributions to the phonon mode from the above two kinds of solvent layers c mode (coordination mode) and s mode (solvent mode), respectively. In the following, we consider these two solvent modes together with the q mode (intramolecular quantum mode) and investigate the three types of ET reactions: CS, CSH, and CR. The ET rate constant W can be written by a convolution forrnaIisml3 W ( A E ) = {mdel -m W q ( t I ) -m l m d t S,(AE 2 - t2)8,(t2 - el)
(1)
where AE is the energy gap and W,(e,) is the rate constant for the virtual energy gap t l by considering only the q mode. W, involves parameters of the average angular frequency ( w ) , the displacement parameter S, and the maximum ET rate constant A I 7 8, and 8,are the thermally averaged Franck-Condon factors for the c and s modes, respectively. Since frequencies of the solvent modes are expected to be smaller than the thermal energy k T , we treat them classically. The force constant of the s mode is nearly equal between the initial and final states in all the above ET reactions. Therefore, 8, is given
" 0
'O'1.0 -1.5
0:5
I:O
1.5
2:O
2:5
;!O
-AG(eV)
Figure 1. (A) Calculated curves of &',(e) in the CSH reaction for p = 0.1, 0.3, 1.0, and m. A, was taken to be 1.0 eV. (B) comparison of the experimental data4 (0)for the CSH reaction with theoretical curves: I , p = 0 . 1 , & = l . 0 e V , A s = 0 . 3 e V , d = 1.4X1010s~';2,p=0.3,1,= 1.0 eV, As = 0.3 eV, d = 3.0 X l o i 0 s-I; 3, A, = 0.9 eV, d = 3.0 X 10'' s-l, with neglect of the c mode. The parameter values for the q mode are given in the text.
c
where A, is the reorganization energy of the s mode. The approximate analytical formulas of 8, for the C S and CR reactions have been given p r e v i ~ u s l y . l ~For - ~ ~the CSH reaction, however, a similar analytical formula of 8, is valid only in much limited cases." Accordingly, we make a numerical integration of the following formula19which is obtained without approximation
In the above equations, k , and k , are force constants of the c mode around the charged and neutral solute molecules, respectively, and a large difference between k , and k , ( k , >> k,) leads to a much smaller value of p than unity. 6x is the origin difference of coordinates between the charged and neutral states, respectively. The definition of the reorganization energy A, of eq 7 in the CSH reaction is in accordance with that of CS and CR reactions. The free energy gap -AG is given byJ3 -AG = AE - (YRTIn
S,(AE - e*) = CJmdy exp(-E cosh y ) X cosh
(
2Il27 cosh
4)
cosh
(
2II27 sinh
4)
(3)
C = ['('
where (Y
where + ')'I/'
rkT
exp[(AE - t 2 - p(l
p = -. k ,
k , - k,'
1
+ 2@)AC]/2kT]
1
-Ac = -k,6x2 2 2
(16) Kakitani, T.;Mataga, N. Chem. Phys. Lett., in press. (17) Jortner, J . J . Chem. Phys. 1976, 64, 4860. (18) Hopfield, J. J. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 3640 (19) Kakitani, T.;Mataga, N., to be submitted for publication.
(4)
(7)
( l i b )
= +1
for C S reaction
=0
for CSH reaction
= -1
for CR reaction
(9)
Numerical Calculations and Discussions In Figure lA, the calculated curves of $,(e) for the CSH reaction are shown for some values of p, with a fixed value of A, = 1.O eV. The temperature is taken to be 300 K, and parameters for the q mode are assumed as h ( w ) = 0.10 eV and S = 3 throughout this work. All these parameter values are similar to those used in the previous ~ o r k s . ~ $,(e) ~ - ' ~diverges at c = 0 for a finite value of p. It is remarkable that, as the value of /3 decreases, the CY,(€) values are increased greatly around t = 0 and in the inverted region. The $,(e) curve in Figure 1A appears t o
The Journal of Physical Chemistry, Vol. 90, No. 6 , 1986 995
Letters
TABLE I: Molecular Species Used in the Studies of the Charge Recombination (CR) Reaction, A--..D+ (-AG),and Transition Rate Constants ( W ) symbol A D solvent -AG, eV a TO EEP acetone 1.46
PMDA
b
pyrene
C
d e
TCNB
toluene
f g
pyrene .. pyrene
h 1
p-DCNB DMA
acetonitrile acetone acetonitrile acetone 2-butanone hexanenitrile octanenitrile acetonitrile acetone
1.70 1.74 2.66
2.70 2.72 2.72
2.76 2.80 2.88
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A...D, Solvents, Free Energy Gaps
w,s-I 2 3 x 10" 1 x 10" 6.7 X 10" 1.8 x 109 1.7 x 109 1.3 x 109 1.2 x 109 8.2 X IO8 -3 x 107 -3 x 107
ref
2a-c 2a,f
2a,e
2aAn 2a,d,g
1011
'gl.0
-0;;
"0'
(
I
I
015
IlO
1.5
2.0"2.5
3.0
- A G (eV)
Figure 2. Comparison of the experimental data3 for CS 0 ) and CR (m) reactions of porphyrin-quinone combined systems with theoretical curves: 1 and 3, (J' = 0.1, d = 1.0 X l O I 3 s-'; 2 and 4, (J' = 0.3, d = 3.0 X l O I 3 s-l. Ac and As are fixed to 1 .O and 0.3 eV, respectively, and parameters for the q mode are given in the text.
be intermediate between that of the CS reaction which is like a step functionT3and that of the C R reaction which is like a 6 function.I5 Figure 1B shows the comparison of theoretically calculated CSH rates with the experimental result^.^ In this calculation, we have not attempted to get the best fit to the observed values, but we have just showed how the general tendency of the experimental results can be explained theoretically. One can clearly see that a moderate decrease of the CSH rate in the inverted region is well explained by use of the smaller value of (3. In Figure 2, calculated results are compared with the observed results on the photoinduced C S and CR reactions of porphyrinquinone combined system^,^ where we have employed only those experimental data in the strongly polar solvent butyronitrile. We have made this choice partly because our present theory is useful for the ET in considerably polar solvent and partly because it is not very clear how large the solvation energy in the very weakly polar solvent is and also whether the correction term, -t2/eR, which is taken to be quite large in such a solvent, is reasonable or not. The results in Figure 2 show that the experimental energy gap dependences can be explained by a combination of separate curves for CS and CR reactions by using the smaller value of (3. It should be noted that we have used the same set of parameter values for the CS and CR reactions. This is reasonable because the donor and acceptor parts are fixed by the chemical bond. In Figure 3, the results2 of picosecond and nanosecond laser photolysis studies on the CR rate of uncombined donor-acceptor systems in strongly polar solvents are compared with the calculation, and the experimental data are summarized in Table I. The negative slope in the inverted region is a little smaller, and a larger solvation energy L&must be used in Figure 3 than in Figure 2 (the large value of A, shifts the curve to the more downhill side making the bell shape broader). Until now, we have not yet resolved, from the molecular point of view, the problem in what kind of systems
-AG
(eV)
Figure 3. Comparison of the experimental data of CR reaction given in Table I with theoretical curves: 1, (J' = 0.1, d = 3.0 X lo1*s-l; 2, p = 0.3, d = 4.0 X l o r 2s-l. Ac and A, are fixed to 1.0 and 0.9 eV, respectively, and parameters for the q mode are given in the text.
and reactions A, should become large. However, it should be pointed out that the systems in Figure 3 differ from those in Figure 2 in that the former are the pairs of strongly solvated radical ions, while the donor and acceptor are combined with a bulky hydrocarbon bridge in the latter. In general, the CR reaction from the radical ion pair state in strongly polar solvents shows a strong dependence not only upon the energy gap but also upon the molecular nature of the individual donor and a ~ c e p t o r . l In ~ ~the ~ ~framework ~ * ~ ~ of the present theory, the latter dependence means that Wq plays an important role in the CR reaction, which is just what our theory is predicting.I4,I5 Although we have selected in Figure 3 the experimental data for some T donors and K acceptors, excluding n donors for instance, it appears that they still show some deviations characteristic to each kind of molecule from the calculated curve. It should also be noted that the A value used in Figure 3 is considerably smaller than that in Figure 2. Although the reason for this difference is not clear, some "through bond" interaction might be contributing to enhancing the electronic interaction in the case of the combined system in Figure 2 . It should be noted here that our calculated results almost quantitatively agree with experimental ones so far. In this respect, we would like to emphasize here that, intrinsically, the inverted effect exists not only in the CSH and CR reactions but also in the CS reaction, but its observation becomes difficult when p is small due to the strong solvation of ions produced by CS from neutral reactants in strongly polar solvents. When p cannot become so small because both reactants and products are charged as in the reaction Fe3+ + Fe2+ Fe2+ + Fe3+,for instance, our theory is almost reduced to the previous theoretical treatments, and the inverted effect might be observed.
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Registry No. TQ, 553-97-9; EEP, 89378-33-6; PMDA, 89-32-7;
TCNB, 712-74-3; p-DCNB, 623-26-7; DMA, 121-69-7; pyrene, 12900-0; toluene, 108-88-3;acetone, 67-64-1;acetonitrile, 75-OS-8; 2-butanone, 78-93-3; hexanenitrile, 628-73-9; octanenitrile, 124- 12-9.
