Effects of the Donor-Acceptor Distance Distribution on the Energy Gap

---

J. Phys. Chem. 1992, 96, 5385-5392 (7) Kuznesof, P. M.; Shriver, D. F.; Stafford, F. E. J . Am. Chem. SOC. 1 9 4 , 90,2551.

(8) Frisch, M. J.; Head-Gordon, M,; Schlegel, H, B.;Raghavachari, K,; Binkley, J, S,;Gonzales, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.;Melius, C. F.;Baker, J.; Martin, R.L.; Kahn, L. R.;Stewart, J. J. P.; Fluder. E. M.; Topiol, S.; Poplc, J. A. GAUSSIAN88; Gaussian, Inc.: Pittsburgh, PA, 1988. (9) Amos, R.D.; Rice, J. E.CADPAC The Cambridge Analytic Deriuatiues Package, Issue 4.0 Cambridge, 1987, CRAY Version. (10) For a description of basis sets see: Hehre, W. J.; Radom, L.: Schleyer, P. V. R.; Pople, J. A. Ab Initio Molecular Orbital Theory, Wiley: New York, 1986. (11) DeFrees, D.

J.; McLcan, A. D. J . Chem. Phys. 1985, 82, 333. (12) Sugie, M.; Takea, H.; Matsumura, C . Chem. Phys. Lett. 1979, 64, 573. (13) Suendram, R. D.; Thorne, L. R. Chem. Phys. Lett. 1981, 78, 157. (14) For a reference to expcrimental work and rcccnt ab initio calculation of vibrational frquencies, see: Brint, P.; Sangchakr, B.; Fowler, P. W. J . Chem. Chem., Faraday Trans. 2 1989, 85, 29.

5385

(15) Hess, Jr., B. A,; Schaad, L.

T. Chem. Reu. 1986,86, 709. (16) SEt: Ekaudet. R. A. In Aduunces in Boron and fhe Boranes, Licbman,

J, F., Greenberg, A,, Williams, R, E,, Eds,;VCH: New York, 1988; p 417, (17) Dewar, M. J. S.;McKee, M. L. J. Mol. Srrucr. 1980, 68, 105. (18) Buehl, M.; Schleyer, P.v. R.In Electron Deficient Boron and Carhn Clusters; Olah, G . A,, Wade, K.,Williams, R. E.. Eds.; Wiley: New York. 1991; p 113. (19) Raghavachari, K.; Schleyer, P.v. R.;Spitmagel, G. W. J. Am. Chem. SOC.1983. 105. 5917. (20) Shore, S.G.; Hall, C. L. J . Am. Chem. Soc. 1966,88, 5346. (2!) Purcell, K. F.;Kotz, J. C. Inorgonic Chemistry; Saunders: Phiadelphla, 1977; pp 399-401. (22) Purcell, K. F.; Devore, D. D. Inorg. Chem. 1987, 26, 43. (23) Ha, T. THEOCHEM 1986, 136, 165. (24) Budzelaar, P. H.M.; Kos, A. J.; Clark, T.; Schleyer, P. v. R. Organometallics 1985, 4, 429. (25) Sana, M.; Leroy, G.; Henrict, Ch. J. Chim. Phys. 1990, 87, 1. (26) Carnetre-Mellon Quantum Chemistry Archive, 3rd 4.; CarnegicMellon University, Pittsburgh, PA, 1983.

Effects of the Donor-Acceptor Distance Distribution on the Energy Gap Laws of Charge Separation and Charge Recombination Reactions in Polar Solutions Toshiaki Kakitani,* Akira Yoshimori, Department of Physics, Faculty of Science, Nagoya University, Furo-cho, Chikusa- ku, Nagoya 464-01, Japan

and Noboru Mataga* Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan (Received: October 9, 1991; In Final Form: February 21, 1992)

We have investigated energy gap laws of the intermolecular charge separation (CS) and charge recombination (CR) reactions by considering the distance distribution between donor and acceptor molecules. The experimentally obtained energy gap dependence of the CS reaction rate constant, which shows a large width and flat shape, has been theoretically explained as follows. The sharp increase in the normal region is due to the donor-acceptor molecules in close contact which give the energy gap laws with small widths and are located at small energy gaps. The flat shapes of the top region and the inverted region are due to the donoracceptor molecules at large mutual distances which give the energy gap laws with large widths and are located at large energy gaps. The experimentallyobtained energy gap law of the CR reaction, which is nearly bell-shaped and located at the rather large energy gap, cannot be explained by assuming the same distance distribution as in the CS reaction. A narrow distance distribution of the geminate radical ion pair was necessary for the satisfactory interpretation of the energy gap law of the CR reaction. It was also shown that the energy gap law of the CR reaction can be reproduced rather well by considering a dynamical process of relaxation of the initial distance distribution of the geminate radical ion pair which is produced by the photoinduced CS reaction.

Introduction Various types of electron-transfer (ET) reactions play important roles in the energy conversion in chemical and biological systems. For a better understanding of those functions, the mechanism which regulates the E T rate must be elucidated first of all. Theoretically, it was predicted that the ET rate constant depends on the free energy gap between the initial and final states.lJ This relation is called the energy gap law. According to the classic Marcus theory,',* the energy gap law is given by a Gauss function of the energy gap. Recent experiments using the donoracceptor linked systems demonstrated that the obtained energy gap law for the charge shift (CSH) reaction is consistent with the Marcus theorye3 On the other hand, the intermolecular E T reactions in solution are more complicated. The energy gap law of the photoinduced charge separation (CS) reaction as demonstrated by the fluorescence quenching rate constant in the stationary state for various fluorescer-quencher pairs is that the quenching rate constant increases rapidly in the normal region at a small energy gap (ca. -AG < 0.3 eV) and it becomes nearly constant in the larger energy gap region (0.3 eV < -AG < 2.5 eV)! The latter fact indicates that the intrinsic E T rate is masked by the slower 0022-3654/92/2096-5385S03.00/0

diffusion rate of reactants. Recently, the intrinsic CS rate constants in this diffusion limit region have been experimentally evaluated by analyzing the transient effect in the fluorescence decay curvea5 The obtained CS rate constants are larger than the diffusion rate constant and they do not show a typical bellshaped energy gap dependence but show a rather flat shape. In contrast to this, the energy gap law of the intermolecular charge recombination (CR) reaction was nearly bell-shaped and the normal region is considerably (0.6-0.7eV) shifted to the larger energy gap side6as compared with that of the CS reaction. Under these situations, it is of crucial importance to clarify the mechanism why the energy gap laws of intermolecular CS and CR reactions are so much different. In our former theoretical studies using harmonic free energy curves,'-I* we proposed a possible role of nonlinear response of solvent polarizations. That is, we showed that, when the nonlinearity is increased by changing the free energy curvature of the ion pair state and keeping the reorganization energy of the linear response case constant, the energy gap law of the CS reaction becomes broader than that of the C R reaction and the inverted region of the former energy gap law becomes smeared out. This theoretical proposal caused much discussion as to 0 1992 American Chemical Society

5586 The Journal of Physical Chemistry. Vol, 96, No. 13, 1992

whether a large nonlinearity exists in molecular systems and whether the nonlinearity actually affects the electron-transfer processes. With regard to the first point, the calculations by means of Monte Carlo and molecular dynamics simulations revealed that the nonlinear solvent effect may not be SO large in water and organic molecular solution^,^^-^^ although the problem is not solved yet when the motion of polar groups is restricted by the environment." With regard to the second point, it was theoretically shown that the free energy curvatures at the free energy minimum can be different between the ion pair and neutral pair states when the nonlinear response of solvent exists but that the free energy curves cannot be extrapolated by the quadratic form to the large value of the free energy.20 If the free energy curves for the both states are taken consistently, the overall free energy curves (and so the energy gap laws of the CS and C R r e a c t i o n ~ ~ become ~J~) rather similar so far as the nonlinearity is not so large.20,21 Furthermore, it was theoretically proved that, if proper anharmonic free energy curves are taken as a function of the exact reaction coordinate, the energy gap law of the CS reaction remains always sy"etric.l5 It was also shown that, when the nonlinearity is large, the inverted region of the C R reaction is much shifted to the small energy gap side as compared with that of the CS reaction, and so the energy gap law of the CS reaction is much broader than that of the C R r e a ~ t i o n . ' ~ , ' ~ Judging from the above results, we may say that the nonlinear response effect cannot be the main reason of the significant difference in the energy gap laws between the CS and CR reactions in polar solution. As an alternative mechanism to affect the energy gap law, we consider the donor-acceptor distance distribution effect. In general, when the intermolecular ET reaction takes place in solution, the donor-acceptor distance distributes inevitably. Similar considerations on this problem were made previously for the luminescence quenching reaction in s o l ~ t i o n . ~However, ~-~~ it will be important to notice that the manner of the distance distribution would be considerably different between the CS and C R reactions. In the CS reaction, the donor-acceptor distance distribution would be uniform, because the reactants are neutral and the thermal equilibrium would be attained in the initial state. Contrary to this, in the CR reaction, the distance distribution of geminate radical ion pairs produced by the fluorescence quenching will be a more localized one. In the present study, we concentrate on the three points: (1) theoretical elucidation of by what mechanism the distance distribution affects the energy gap laws, (2) examination of whether the theoretically calculated energy gap laws consistently fit the experimental data of the C S and C R reactions, and (3) demonstration that the dynamical process of relaxation of the distance distribution of the geminate radical ion pair inevitably takes part in the course of CR reaction. Although the nonlinearity is expected to be small in the present system and would not play a major role in the energy gap law, we theoretically treat the nonlinearity as variable so that our theoretical analysis may be applied to any molecular systems.

Theory Most of the detailed theoretical formalisms of the energy gap law in the presence of the steady-state distance distribution were given elsewhere.24 We do not repeat the derivation of them, but we give here some basic equations which are necessary for the discussion, with some modification of notations. We present ample discussion on the physical background of the theoretical model adopted. In the following theoretical treatment, the reactant molecules are assumed to be spherical. The actual molecule is not spherical but planar in most cases. But, when the two reactant molecules are apart, there are many mutual configurations. Since we must finally average the ET rate over those configurations, the spherical shape approximation becomes eventually valid. On the other hand, when the two reactant molecules are proximate, the specific shape of the reactant molecule is effective to determine the distribution of the distance and the mutual orientation. In the planar reactant molecules, the face to face configuration will be produced at the

Kakitani et al. very close distance. In such a way, the spherical shape approximation becomes inappropriate at very close distances. However, theoretical formulations using spherical shape of molecules are much simpler and easy to see through the distance distribution effect. So, we adopt the spherical shape approximation in the present theoretical analysis all the way. It appears that a similar kind of approximation is adopted implicitly in the experimental study of the energy gap law using many species of donor and acceptor molecules with different energy gaps. In the last section, we shall reexamine the effect of the molecular shape a t close distance of reactant molecules. (It will be shown that a proper account of the molecular shape is necessary to satisfactorily explain the experimental data of the normal region in the CS reaction.) Similarly, we treat the solvent molecules by the spherical shape approximation. This assumption may be more satisfactory, because their shapes are not planar and many solvent molecules with various mutual configurations take part in the E T reaction, There are three factors which depend on the donor-acceptor distance R . The first is the frequency factor v(R) pertinent to the ET process. For convenience, we write it as

with where va,J and A ( R ) are the frequency factors in the adiabatic and nonadiabatic mechanisms, respectively. A. and (Y are constants. R 1 is the solvent-separated donor-acceptor distance. The second factor is the local energy gap -AGJ(R) which can be modified by the mean ion pair potential Ulp(R) as follows: -AGCS(R)= -AGocs - OP(R)

(3)

+ OP(R)

(4)

-AGCR(R)= -AG°CR

where -AGoj (J = CS, CR) is the standard free energy gap. It should be noticed that the mean ion pair potential works to increase the local energy gap of the CS reaction and to decrease the energy gap of the C R reaction, and that its effect is appreciable only for small R. In the following treatments, we adopt spherical hard-core models for reactant and solvent molecules. On the basis of Monte Carlo simulation results using such model systems,25we assume Op(R) = - t / R

for R

> Ro

(5)

where t is a constant and Ro is the contact ion pair distance. The third factor is the reorganization energy which is an increasing function of R . When the nonlinear response works, the reorganization energy can be different between the CS and C R r e a c t i o r ~ s . ' ~ JAccording ~J~ to our recent Monte Carlo simulation res~lts,l*-'~ Xa(R) is considerably (about 20%)larger than XCR(R) and they are much smaller (about a half) than the one calculated by the Marcus formula using the dielectric continuum model.2 It was also found by King and Warshel that Xa(R) is about 13% larger than XCR(R)by reading the curves in Figure 4 of ref 16. Taking into account the simulation result, we assume the reorganization energy for the CS reaction as Xcs(R) = c - ( d / R )

for R

> Ro

(6)

where c and d are constants. After determining free energy curves of the initial and final states, we determine XCR(R). Under these preparations, the ET rate kJ where only the contribution from solvent motions is considered can be written as ~ J ( - A G o ~ ,= R )v(R)e-BAG'(-AG'(R).h'(R)) (J = CS, CR) (7) where j3 = l / k B T . The activation energy AG* can be related to the free energy curve of the initial state Gr(x;R)as a function of the reaction coordinate x as follows:'s*20 A@(-AGJ(R),XJ(R))= Gr(AGJ(R)+ G,(R) - G,(R);R)- Cr(R) ( 8 )

The Journal of Physical Chemistry. Vol. 96, NO. 13, 1992 5387

Charge Separation and Recombination Reactions

In eq 8, G, and G, are constants which are defined by

by Strw and makes the inverted region mild,

Energy Cap Law of CS Reaction We assume that the distance distribution of donor and acceptor molecules is uniform. Then, the intrinsic distance-averaged CS rate constant denoted by kfi is written as22923 where Gp(x;R)is the free energy curve of the final state and it is related to Gf(x;R)in the Gp(x;R) = G'(x;R) - x

(11)

We express the free energy curve of the neutral pair and ion pair states as

+

G"P(x;R)= a(x-xo(R))" b(x-x0(R))*

G'p(x;R) G"P(x;R)- x

(12)

where CY, b, and xo(R)are constants and n is an even number larger than 2. The Erst term in the right-hand side of eq 12 represents the nonlinear effect of solvent polarization. We have included the nth nonlinear term in order to cover the broad range of molecular systems by adjusting the coefficients u and b. Here, we define the nonlinear parameter p as

?

Cn/cc - cn

(13)

where C, and C, are the free energy curvatures of the neutral pair and ion pair states at their minima, respectively. This B value may change somewhat depending on the environment (such as water or organic polar molecules or other systems). So, we treat $ as a changeable parameter in this paper in order to see what happens in the various systems. Then, u and b are expressed as

a=

(n - 1)"'(2 2"'n"(n/3

+ nB)"' - B + 1)"

1

[ACS(R)]"I

(14)

where NA is Avogadro's number. This is the bimolecular reaction rate constant. In the very early stage after the photoexcitation, the observed reaction rate would be close to k:$ This value might be obtained by measurement of the transient effect in the fluorescence decay process. On the contrary, when the stationary state is realized at long time after excitation, the observed CS rate constant would be given by 1

kFG(-AGOcs) =

1 k5Z(-AGocs)

+- 1

kJ,(-AG",,R) = l : k J ( - A G o , - e,R)Fq(e) de

(J = CS, CR) (16)

where F,(e) is the Franck-Condon factor of the quantum mode which is written as2'

Fq(c) = 1 exp[-S(2a

["'p'

+ l)]Ip[Sa(a+ I ) ] -

(17)

with B'

1 exp(@hw)- 1 p = e/hw

(19)

s = 62/2

(20)

where I, is the modified Bessel function, w is the effective angular frequency, and 6 is the shift of the effective normal coordinate between the initial and final states. This quantum mode contributes to shift the energy gap law to the larger energy gap side

kdiff

where kdinis the rate constant of the diffusional encounter of the donor and acceptor. This case would correspond to the fluorescence quenching experiment in the stationary state. For numerical calculations, we adopt the optimum parameter values which were estimated p r e v i o u ~ l yexcept , ~ ~ that we choose n = 6 in this paper. The chosen parameter values are as follows: v,d

= 5 x 1O"s-',

'40

= 5 x lo1*s-'

Ro = 4.4 A, R , = 8.8 A, a = 1.2 A-I Shw = 0.30 eV, hw = 0.10 eV kdiff

= 2 x lolos-l,

t = 1.3 eV*A

c = 2.7 eV, d = 9.7 eV.A

where ACS(R) is the reorganization energy given in eq 6. We treat ACS(R)as independent fmction of 8. So, we change the parameters u and b simultaneously, according to eqs 14 and 15 when the nonlinear parameter $ is varied. The reason for doing so is discussed in the last section. It should be noticed that G"p(x;R)becomes Gf(x;R) in the CS reaction and that C;nP(x;R)- x becomes O(x;R)in the CR reaction using eq 11. Combining eqs 14,15,12, and 8, we obtain an explicit form of AG* as functions of -AGJ(R) and AJ(R). Taking into account the contribution from the intramolecular vibration in addition to the solvent motion, we can write the total ET rate k; as

(22)

(23)

The temperature is put always 300 K. In obtaining the above values for Ro and R,,effective radii of reactants and solvent molecules were assumed to be 2.2 A. The radius for the solvent molecule is referred to that of acetonitrile. The radius for the reactant molecule was determined by referring to the planar aromatic molecule. The validity of the values of v,d and A. is reconfirmed later. Under these parameter values, we find from eq 1 that the adiabatic mechanism applies for R < 8.8 A and the nonadiabatic mechanism for R > 8.8 A. The reorganization energies so determined are a little smaller than those of our Monte Carlo simulation calculations,l* and this is reasonable because the electronic polarizability of solvent which reduces the reorganization energy16J9 was not taken into account in this Monte Carlo simulation. The ion pair potential P ( R ) chosen above is of considerable magnitude (0.30 eV at R = 4.4 A). This large ion pair potential is necessary in order to reproduce the experimental data that the normal region in the energy gap law of the CS reaction shifts considerably (0.6 eV - 0.7 eV) to the negative energy gap side as compared with that of the C R reaction. This amount of magnitude for OP(R) was also obtained by the recent Monte Carlo simulation calculations using the spherical hard core model of m o l e ~ u l e s . ~Although ~ ~ ~ ~ it may happen that some electronic mixing may takes place at the very proximity of the ion pair, we neglected its effect in our spherical shape approximation in this paper. In Fi ure 1, the calculated energy gap laws of k,C,s(-AGocs) and kZ&-AGocs) are drawn by solid and broken curves, respectively, and the are compared with the experimental data (filled circles for kFi5 and crosses for kF24). In the present system with fluorescer-quencher distance distribution, the shorter distance pair undergoes faster quenching due to C S leading to the timedependent C S process (the transient effectz9). By measuring accurately the fluorescence decay curves with picosecond dye laser as excitation source and single-photon-counting detection and analyzing the transient effect in the quenching process with equations given in the l i t e r a t ~ r ewe , ~ ~have obtained the::k values

5388 The Journal of Physical Chemistry, Vol. 96, NO.13, I992

Kakitani et al. 0

D

- A G & = -08 A G & = 0.0

-

0

-0.8

ll

1

-0.4

1

0.0

1

1

0.4

1

I

0.8

I

I

I

1.2

- AG&

I I 1.6

I

I

2.0

I

2.4

I

1

2.8

I

t

3.2

44

ev 1

64

74

94

64

104

114

124

134

144

R(A)

Fi re 1. Calculated results of the distance-averagedenergy gap laws kaa( AGOcs) and k:i in the CS reaction, in com arison with the ex-

8-

perimental data. Theoretical curves for k s and &Bare solid curves and

broken curves, respectively. Experimental data for k s and kg are filled circles taken from ref 5 and crosses taken from ref 4. 6 denotes the degrcc of nonlinearity of the solvent polarization. 0

I

,R = 7 4

I

- AG&( e V ) Figure 2. Calculated results of the R2-weighted energy gap law R 2 k ~ S ( - A G o c s , Rfor ) some values of R in the CS reaction (solid curves). The unit of R is angstrom. The broken curve represents the distance-

averaged energy gap law k$(-AGocs) for comparison. of the filled circles in Figure 1.s These values are somewhat larger than the previously reported onesz4which were obtained by using an approximate formula (long time a roximation). It has been found that the calculated curve of k,,, is broad and flat around the maximum, in agreement with the experimental data. The calculated normal region locates at a little uphill side of the energy gap than the experimental points, and this tendency is more remarkable in the case of the larger nonlinearity (small value of 8). However, the nonlinear effect appears to be small, as compared with the case where the distance distribution was not considered.15 A plausible 6 value might be between 1.0 and -. In the last section, we discuss the reason why the theoretical curves becomes higher than the experimental data in the normal region. In order to elucidate the mechanism by which the distancebeaveraged energy gap law of the CS reaction kf:(-AGocs) comes broad, we have drawn the R2-weighted energy gap law R2kfS(-AGo,R) in the case of 8 = 1 for some fured values of the distance R by solid curves in Figure 2. (The same discussion in the following applies to the case 8 = -). The R2-weighted energy gap law for R = 4.4 A is narrow and its maximum locates at a very small energy gap with large amplitude. This small width is due to the small Acs at small R. The location of the maximum at a very small energy gap is due to a large uphill shift of the

l?

54

Figure 3. Calculated results of the initial distribution function go(R)of the ion pair for some values of -AGO, with which the geminate ion pair is produced by the fluorescence quenching. The unit of -AGOcs is electronvolt

energy gap by W ( R ) in addition to the small value of Acs. On the other hand, the R2-weighted energy gap law for R = 1.4 A is broad and its maximum locates a t a rather large energy gap. Those properties are due to the larger value of Aa and the small value of W ( R ) a t large R. The height of the maximum for R = 1.4 A is even lar er than that for R = 4.4 A. This is due to the fact that R2kf is larger in the former case and that the adiabatic mechanism still applies. The R2-weighted energy gap law for R = 10.4 A is much broad and its maximum locates at a very large energy gap due to the very large Aa. However, the height of the maximum becomes smaller than that for R = 7.4 A. This is because A(R) for R = 10.4A is considerably smaller than v,d and this ET process proceeds by the nonadiabatic mechanism. The above analysis shows clearly that the very broad distance-averaged energy gap law of the CS reaction is firstly due to the overlap of the different kinds of the R2-weighted energy gap law with considerable amplitude and secondly due to a large uphill shift of the R2-weighted energy gap law for small R. The R2-weighted CS rate R2kFs as a function of the distance R for a fixed value of -AGOcs represents the relative magnitude of the contribution from the donor-acceptor pair with distance R to the CS reaction. Then, it represents the initial distribution function for the CR reaction of the geminate radical ion pair produced by the CS reaction. We define the normalized initial distribution function go(R) as

8

J;I:R2k~s(-AGoC,R) d R The calculated results of go(R) in the case 8 = 1 are plotted in Figure 3. (The same discussion in the following applies to the case of 8 = m.) We find that, when -AGO is small (ca. -0.8 or 0.0 eV), go(R) has a sharp distribution function with a maximum at R = 4.4 A. When -AGO is increased, go(R) becomes broader and the position of the maximum shifts to the larger value of R. (At the very large energy gap of 2.4 eV, the maximum position is at 10.0A and the full width at half-maximum is about 2.5 A.) When the distance distribution function is sharp, the distribution of Acs(R) is rather limited. But, when the distance distribution function is broad, the distribution of Acs(R) becomes considerable, giving rise to a possibility of choosing optimum value of XCs (Le., Acs + Shw -AG). Energy Gap Law of CR Reaction

We examine the energy gap law of the C R reaction taking into account two cases. In the first case, the steady state is attained for the geminate ion pair distance distribution before the CR reaction. In the second case, the distance distribution is on the

The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5309

Charge Separation and Recombination Reactions

-0

CR

.i-% \\ g

Y b

e B e

=0.1 =0.3 =1.0

S = w

5Y

b

-0.8

-0.4

0.0

0.8

0.4

1.2

1.6

2.0

2.4

2.8

3.2

- AG&( eV ) Figure 4. Calculated results of the distance-averaged energy gap law kCR(-AGoCR)in the CR reaction (solid curves), in comparison with the experimental data (triangles taken from ref 6 ) .

way to the steady state at the time of the CR reaction. The reality will be close to the sccond case. However, the first case is valuable to investigate theoretically because we can compare the distance distribution effect on the energy gap law between the CS and CR reactions both under the stationary condition. We examine the first case with the distribution function g(R) given in the form

J%

where u and R, are constants. Since this steady state is not equivalent to the thermal equilibrium, the relation -kBT In g(R) = UlP(R) does not necessarily holds. The distanceaveraged energy gap law of the CR reaction kCR(-AGo) is written as kCR(-AGoCR)=

J;

kFR(-AGocR,R)g(R)R2dR

- AG:,(

the Monte Carlo simulations does not have a deep minimum at the solvent-separated ion pair distance but has at the contact ion pair d i s t a n ~ e . ~This ~ ~ ~fact * indicates that the narrow distance distribution g(R) centered at the solvent-separatedion pair distance is only possible in a steady state, and it will not be realized in the thermal equilibrium state. Now, we go to the second case where the distance distribution can change in the course of CR process. In this case, we treat the problem that the CR reaction proceeds concomitant with the relaxation of the distance distribution of the geminate ion pair produced by the fluorescence quenching. Nevertheless, in this paper, we do not solve this dynamical problem exactly, but qualitatively show that the experimental data can be reasonably well explained by this mechanism. We adopt the following Gaussian distribution function for the distance

(26)

This is the monomolecular reaction rate. We choose the optimum values for u and R, so that the calculated energy gap law may fit the experimental data.24 The result is u = 0.25 A-* and R, = 8.8 A which is equivalent to the solvent-separated ion pair distance. In Figure 4, we draw the calculated curve of kCR(-AGOCR)for some values of 8. The experimental data are shown by trianglesa6 We have dropped the experimental points of three donoracceptor pairs from the data in ref 6, because their electronic or chemical properties differ greatly from the others. We see that the calculated curve is nearly bell-shaped and it differs from each other considerably, depending on the magnitude of the nonlinearity The best fit to the experimental data is obtained for 6 = 1.O in our parametrization. In Figure 5, we draw the R*g(R)-weighted energy gap law in the case of 8 =: 1.0. (The same discussion in the following applies to the case of B = ma) We find that the curve with considerable amplitude is very limited due to the localized distance distribution of the geminate ion pair. The other reason why the distanceaveraged energy gap law does not deviate greatly from the bell-shape is the shift of the RZg(R)-weighted energy gap law for small R to the larger energy gap side owing to the mean ion pair potential. This fact indicates that even when the distance distribution is uniform, the distance-averaged energy gap law of the CR reaction is narrower than that of the CS reaction. Here, we should note the following facts. Although the distance distribution function used in the above analysis has a maximum at R = 8.8 A which is the solvent-separated ion pair distance, the mean ion pair potential of eq 5 does not have a minimum at this distance. Furthermore, the mean ion pair potential obtained by

B.

eV )

Figure 5. Calculated results of the R2-weighted energy gap law R2k:R(-AGoCR,R)for some values of R in the CR reaction (solid curves). The unit of R is angstrom. The broken curve represents the distance-averaged energy gap law kCR(-AGo) for comparison.

e-(R-Rn)’/(d4’R2 dR

where R,,and d are constants which depend on the condition when the geminate ion pair is produced. The factor u corresponds to a normalized time. In this treatment, it is assumed that the width of the distribution increases with time but the distance corresponding to the maximum distribution remains the same. We set f(R,u) for u = 1 as the distribution function just after the formation of geminate radical ion pairs. Therefore, f(R,l) must correspond to gO(R)in eq 24. Although go(R) dose not look like Gaussian for small -AGO, we approximate f(R,l) to a Gauss function having the same width and the maximum position as go(R). We choose 13 donor-acceptor pairs with various values of -AGOcs, as shown in Table I. Those are taken from ref 6. For each value of -AGcs, we calculate go(R), read R, and d from its initial distribution function and obtainf(R,u) by means of eq 27. Using this f(R,u), we calculate the CR rate with kCR(-AG0cR,u) ~ ~ k F R ( - A G o ~ R , R ) ~ RdR , ~ ) R z (28) The calculated values of kCR(-AGoCR,~) for u = 1, 2, and 3 are listed in Table I. Those values for u = 2 and 3 correspond to the CR rate when the distance distribution functions are expanded successively, in the course of time. In the last column, the experimental data of the CR rate are listed. It is seen that the calculated CR rate increases with increase of u when the rate is small (no. 1-4 and 8-13) but decreases when the rate is large (no. 5-7). We plot those calculated results as well as the ex-

5390 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992

Kakitani et al.

TABLE I: Calculated Values of the CR Rate k m ( - A C o a , u ) for Some Values of the Distance Distribution Expansion Parameter u, in Comparison with the Experimental Values" for CS no. donor acceptor -AGOcs, eV 2.30 TCNE 1 Per* 2.46 TCNE 2 Bpcr' TCNE 2.21 3 DPA*

TCNE PMDA MA Per* PA

Py* Per* Per* TMPD Per* BPer*

4

5 6

7 8 9 10 11

o-DMT Py*

12

R,,

A

9.5 9.8 9.3

9.8

2.44 1.50 1.21 1,09 0,74 0,91 0.47 0.41 0.88 0.51

PA Per* Per* PA

DMA

kCR(-~c0, a), s-I

8.0 7.1 6,7 5.6 6.1 4.9 4.8 6.0 5.0

d, A 1.8 1.8 1.7 1.8 1.6 1.5 1.4 1.0 1.2 0.66 0.60 1.2 0.69

-AGOcK,

0.55 0.79 0.89 0.90 1.35

1.64 1.76 2.1 1 2.34 2.38 2.44 2.46 2.83

eV

=1 1.0 x 2.7 x 6.7 x 8.0 x 7.0 X 1,3 X 9.6 X 8.6 X 1.2 X 2.2 x 9.6 X 3.5 x 2.7 x

107 io8 108

108 1O1O 10" lo9 loio lo9 109 lo6 108 104

u = 2

u = 3

kZPtIts1 s-I

8.0 x 107

1.2 x 108 1.5 x 109 2.8 x 109

6.1 x io* 1.9 x 109 1.3 x 109 2.6 x 109 >4 X IOio > 1 X 10" > 6 X lolo 1 X loi" 5.2 X loq 2.1 x io* 6.8 X lo* 1.4 x 109