J. Phys. Chem. 1988, 92, 5059-5068
5059
Electron-Transfer Reaction in Frozen Solution. 1. Theory Toshiaki Kakitani* Department of Physics, Nagoya University, Furo-cho, Chikusa- ku, Nagoya 464, Japan
and Noboru Mataga* Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan (Received: September 1 1 , 1987; In Final Form: February 23, 1988)
Survival probability of an electron situated at a donor in a frozen solution is theoretically formulated by taking into account both distributions of the donor-acceptor distance and the orientation of solvent molecules around the donor and acceptor. When all the possible electron-transfer pathways from a donor to any acceptor are incorporated in the theory, the resultant formula has become a generalization of that of Inokuchi and Hirayama, which was derived for the triplet-triplet energy transfer in a rigid solution. It has been shown that the time course of the survival probability is considerably altered by considering the distribution of the solvent orientation, which gives rise to a distribution of the energy gap for the electron-transfer reaction. This effect of solvent orientation differs considerably,depending on the frozen state of solvent, which can be classified into two extremes, type I and type 11: the reorientation time of the solvent around the anion donor is much shorter than the average electron-transfer time in the former case, while the opposite holds in the latter case. The effect of the difference between the frequency of the solvent motion around a neutral reactant and that around a charged reactant is also examined. The apparent energy gap law that is obtained from the time course of the survival probability in frozen solution of type I or type I1 is compared with the energy gap law of the electron-transfer rate in fluid solution. Many of the formulas that were so far developed for explaining the survival probability curve of the electron transfer in rigid solution are systematically derived and classified within our generalized formulas.
Introduction In the nonadiabatic mechanism of electron transfer (ET’) in polar solution, its rate W is characterized by a factor k due to an electron-tunneling matrix element and the thermally averaged Franck-Condon factor F due to the intramolecular and intermolecular vibrations as follows:
solvent orientation plays an essential role in activating the system to the transition state Of the electron-transfer reaction. This activation energy and, accordingly, the thermally averaged Franck-Condon factor are neatly correlated with the energy gap AE in the electron-transfer reaction. As we have shown in a previous paper: AE is composed of the two parts; a partial energy gap AE - t seen by the intramolecular quantum mode (q mode) and the other partial energy gap t seen by the solvent mode. When the solvent motion is fluctuting as it is at room temperature, the partial energy gap E rapidly changes with time. Owing to this property, we obtain a unique formula for the ET rate as expressed by a convolution between the thermally averaged Franck-Condon factor due to the q mode and that due to the solvent mode. In contrast to this, when all the solvent motions are frozen, the partial energy gap t is fmed at each reaction site, depending on the frozen state of the orientation of solvent molecules. The thermally averaged Franck-Condon factor in such a frozen state is that due to only the q mode, Fq. The value of Fq has a strong dependence on the partial energy gap AE - t, which varies considerably among the donor and acceptor pairs, depending on the different environment. Namely, we obtain a different electron-transfer rate for each donor and acceptor pair, depending on the frozen solvent orientation. Under such a situation, it must happen that the ET rate between the donor and acceptor pair that is not in the nearest neighborhood but is surrounded by solvent molecules frozen in favorable orientation is larger than that between the donor and acceptor pair that is in the nearest neighborhood but is surrounded by solvent molecules frozen in unfavorable orientation. In this regard, treatment neglecting the effect of the solvent orientation will be insufficient. On the basis of these considerations, it will be important to take into account all the possible ET pathways realized by the combination of the two kinds of distributions of solvent orientation and donor-acceptor distance. As we have pointed out e l ~ e w h e r e ,the ~ ? ~fluctuation of the solvent orientation around a neutral reactant will be much easier than that around a charged reactant in solution. In other words, the force constant of the orientational motion of solvent molecules around a neutral reactant is expected to be much smaller than
*
W = AF (1) The energy gap dependence of the rate W ( A E ) originates from the factor F. The rate is also dependent upon the distance R between a donor and an acceptor. Its R dependence is mainly due to the factor A . Although the factor F may also have a considerable R dependence through the reorientation energy of solvent when the donor-acceptor distance is small,’ it is not the case in the dilute frozen solution as treated in the present paper. The R dependence of W was extensively investigated by Miller et a1.24 using frozen solution at low temperature. Their analysis of the experimental data of the survival probability P ( t ) that an electron stays at a donor site is based on the following assumption: The effect of freezing the solvent is to fix donors and acceptors at random positions. They have employed following procedures:” (a) Only the electron-transfer process to the acceptor that is at the nearest neighbor from each donor is taken into account. (b) The R dependence of A is represented in the form A = A. exp(-a(R - R o ) J (2) where Ao, Ro, and (Y are constants. Taking the average over R on the basis of the above assumption and procedures, they obtained a plausible value of a 1.2 A-’ by simulating the experimental curves of P(t).394 The result of this investigation has disclosed that the electron transfer in frozen solution is considerably long range. Now, we notice here that the role of the frozen solvent assumed above will be insufficient in the following point: The effect of freezing the solvent orientation is not taken into account. To make the problem more sensible, we first consider a nonfrozen solution (hereafter simply called a solution), where fluctuation of the
-
(1) Marcus, R. A. J . Chem. Phys. 1956, 24, 966. (2) Miller, J. R. J . Phys. Chem. 1978, 82, 767. (3) Beitz, J. V.; Miller, J. R. J . Chem. Phys. 1979, 71, 4579. (4) Miller, J. R.; Beitz, J. V.; Huddleston, R. K. J. Am. Chem. Soc. 1984,
( 5 ) Kakitani, T.; Mataga, N. Chem. Phys. 1985, 93, 381. (6) Kakitani, T.; Mataga, N. J . Phys. Chem. 1985, 89, 8 .
106. 5057.
0022-3654188 , ,12092-5059$01.50/0 I
0 1988 American Chemical Societv -
5060 The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
that around a charged reactant. Indeed, recently we have carried out a Monte Carlo simulation of a polar solution including a charged or neutral solute molecule, based on a spherical hard-core model of solvent and solute m o l e c ~ l e s and , ~ it was shown that a thin dielectric saturation shell of solvents exists just around a charged solute molecule and that the curvature of the free energy curve of the saturation shell as a function of the radial polarization is much larger than that of the free energy curve of the nonsaturation shell. This result of the Monte Carlo simulation is consistent with our expectation. By taking into account the considerable difference between those two force constants, we have theoretically obtained three different energy gap corA---B+), responding to the charge-separation reaction (A-B A-B), and charge-shift charge-recombination reaction (A-.-B+ A--.B). The tendency of the electron-transfer reaction (A-Brate decreasing at the large energy gap is called the inverted effect, and the corresponding region of the energy gap is called the inverted region. According to the energy gap laws predicted by our theory for the electron transfer in polar solution, the inverted region will be observed scarcely in the C S reaction, moderately in the CSH reaction, and remarkably in the C R reaction. Those theoretical predictions that are obtained only by assuming the large difference of the two force constants are consistent with the ' available experimental result^."^ In this context, it is interesting to see how much the survival probability P ( t ) of the electron at the donor in frozen solution is affected by the introduction of the difference between the two kinds of force constants of the solvent mode. The objective of this paper is, therefore, 2-fold: (1) to formulate theoretically the time dependence of the survival probability P ( t ) by taking into account the distribution of the frozen solvent orientation as well as the random distribution of the donor-acceptor distance and to examine the effect of the frozen solvent orientation by neglecting a possible difference of the force constants and (2) to investigate how much the effect of the difference between the two force constants of solvent orientation is. The comparison between theoretical calculations and experimental results will be made in a forthcoming paper. To avoid unnecessary complexity, we confine ourselves, in this paper, to the following electron-transfer reaction in frozen polar solvent:
-
-
D-+ A
D
-+
+ A-
-
(3)
where D- and A denote singly negatively charged donor and neutral acceptor, respectively. Using this scheme, we can also treat the solvated electron, where D- represents the electron trapped in the solvent. In this case, D is solely a vacancy in the solvent. Foundation of the Theoretical Model At the beginning, we settle the theoretical model, in accordance with experimental conditions. We prepare a polar solution involving two kinds of neutral solute molecules, one of which becomes the acceptor and the other of which becomes the donor after accepting an electron produced by pulse radiolysis. In the case where the solvated electron is the donor, we prepare a solution inchding only one kind of solute molecule, which becomes the acceptor. The solution before radiolysis is cooled from room temperatue to low temperature, TI, where solute molecules are frozen at random positions. The solvent orientation around a neutral solute is fixed in a manner such that its energy follows
Kakitani and Mataga
Figure 1. Potential energy surfaces of reactants A and D in the neutral and charged states as functions of solvent orientations. x and y are solvent orientational coordinates around reactants A and D, respectively. k is the curvature of the potential surface.
the Boltzmann distribution at a certain temperature Th. When the sample is cooled very rapidly, Th will be near room temperature. When the sample is cooled slowly, Th will be near the solidification point of the solvent. In the next step, the anion radical of the donor or trapped electron is produced by irradiation of the frozen solution with a rapid electron pulse. In the frozen state of solvent, it takes a certain time T~ for the donor to deepen the potential well due to solvation. This time becomes longer when the temperature TI is very low. The vibration of the q mode will not be substantially affected by the freezing of solvent and will be in thermal equilibrium at low temperature, TI. We assume that the frozen solution is very dilute, satisfying the following condition: N,>> N , >> Nd (4) where N,, N, and Nd are the number of solvent molecules, acceptor molecules, and donor molecules or trapped electrons in the sample volume V, respectively. With regard to the R dependence of the electron tunneling, some detailed models were p r ~ p o s e d ' in ~ , addition ~~ to a simple form of eq 2. However, so far as one remains concerned with a dilute frozen solution where many of the donor-acceptor pairs are at long distance, a simple form of eq 2 may be acceptable. Furthermore, since the exponential form is convenient in our formulation, we use it. Formulation of the Survival Probability Including the Role of the Solvent Orientation Let us go on to formulate the survival probability P(t) in frozen state. The kth acceptor Ak locates at the distance RJkfrom the ;. The solvent orientational coordinates around Ak j t h donor D and D ; are denoted by xk and y]. The energy surfaces along the solvent coordinates are shown in Figure 1. The partial energy gap seen by the solvent mode in the ET reaction is denoted by c(xk,yJ).Then, the residual energy gap seen by the q mode of this donoracceptor pair is AE - c(xk,y,), and the ET rate in this case is written as
where
Z = Cexp(-E,/kT,) U (7) Hatano, Y . ;Saito, M.; Kakitani, T.; Mataga, N. J . Phys. Chem. 1988, 92, 1008. (8) Kakitani, T.; Mataga, N . J . Phys. Chem. 1985, 89, 4752. (9) Kakitani, T.; Mataga, N . J . Phys. Chem. 1986, 90, 993. (10) Rehm, D.; Weller, A. Isr. J . Chem. 1970, 8, 259. (1 1) Miller, J. R.; Calcaterra, L. T.; Closs, G . L. J . Am. Chem. Soc. 1984, 106, 3047. (12) Wasielewski, M. R.; Niemczyk, M. P.; Svec, W. A.; Pewitt, E. B. J . Am. Chem. SOC.1985, 107, 1080. (13) (a) Mataga, N.; Kanda, Y.; Okada, T. J . Phys. Chem. 1986,90,3880. (b) Mataga, N.; Shioyama, H.; Kanda, Y . J. Phys. Chem. 1987,91, 314. (c) Mataga, N. Acta Phys. Pol. 1987, A71, 767. (d) Mataga, N.; Asahi, T.; Kanda, Y . , manuscript in preparation.
In the above equations, E , and E, denote vibrational energies of the q mode in the initial and final states, respectively, I(uqluq)12 the Franck-Condon factor of the q mode, and (iIH'(R.k)V) the electron-tunneling matrix element. The average angular #requency ( u )of the q mode is assumed to be the same between the initial (14) Dainton, F. S.; Pilling, M. J.; Rice, S. A. J . Chem. SOC.,Faraday Tram. 2 1975, 71, 1311. (15) Doktorov, A. B.; Khairutdinov, R. F.; Zamaraev, K. L. Chem. Phys. 1981, 61, 351.
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 5061
Electron-Transfer Reaction in Frozen Solution and final states. Then, Wq is quantum-mechanically formulated as follows:’6 Wq@E
- 4%Y,)’R,k) = A(R,k) Fq(AE - 4X&,Y,)) (8)
Defining new parameters as
where A(R,k) = A ( R , k ) / h W
(9)
Fq(AE - 4%Y,)) = exp[-S(f~ 1)]IbI(2S(a(~ 1 ) ) 1 / 2 ) [ (+~ l ) / ~ ] p /(10) ~
+ 0
+
= [exp(h(w)/kTJ - 11-l
p=
(11)
w - 4Xk>Y,)I/hb)
(12)
s = &/2
(13)
In the above equations, Fq is the thermally averaged FranckCondon factor due to the q mode, Zbl the modified Bessel function, and d the origin shift of the normal coordinate of the q mode. The function A(Rjk)corresponds to A in eq 2. The probability Ql(t&,xk,y,) that the electron is transferred from D/- to Ak until time t is written as Q I ( ~ , R , ~ J ~ , Y=J 1 - exP[-tWq(m
- ~ x ~ , Y , ) & ) I (14)
The next step is to take the ensemble average over the coordinates xk and y!. As already stated in the preceding section, the solvent orientation around the neutral reactant, the acceptor in our reaction 3, is frozen by keeping the Boltzmann distribution of energy at temperature Th. Then, the frozen distribution function for xk is written as fa(x) =
(
kan
)I/’
exp(
-2kBTh
kdc
)I”
&(x,r) = -(2Aa)’/2X- (2Ad)l12Y
exp( --kd.’) 2kBTl
+ Aa +Ad
(26)
Similarly, replacing x and y with X and Y in eq 15-17, we obtain
(15)
where k, and kBare the solvent force constant around the acceptor in the neutral state and the Boltzmann constant, respectively. The suffix k for x is dropped for simplicity. The distribution function for yj changes considerably, depending on the solvation time 7,of the D; and the average electron-transfer time T,. When T , 0 and yd > 0. It should be noticed that when the force constants of the solvent orientation around the charged and neutral reactants are the same, Pa and Pb in the above equations are infinite and c(X,Y) is expressed by a linear function of X and Y as follows:
fa(x) dx = f’a(X) dx
(27)
fib) dJJ =fdI(r) dY
(28)
h ” b ) dy =fh”(r) d Y
(29)
where
(16)
where kdc is the solvent force constant around the DJ-. When 7 , >> T , holds (called type 11), the following frozen distribution function at temperature Th is usable: fJ1b) =
(
2~k~Th
exp[ - kdnb
- Yd)2
2kBTh
]
(17)
where kdnand yd are the solvent force constant around the donor in the neutral state and the displacement of its solvent coordinate accompanying the electron release (see Figure l ) , respectively. Next, we go on to obtain t(x,y) in analogy to the resonance theory of the excitation transfer following the treatment by Hopfield” and our previous works.69 It should be noted that the quantity c(x,y) is independent of the distribution function of x and y. From Figure 1, the resonance-transferring energy E is expressed as
E = Ed - j/2kdrY2 + j/2kdnb - Yd)’
(18)
+
(19) E = E, - )/2kac(x- x,)’ l/ZkanXZ where Ea and Ed are the energy gaps at the acceptor and donor sites, respectively, and are related to the energy gap e(x,y) by the equation c(X,Y) 2 Ea - E d (20) Substituting eq 18 and 19 into eq 20, we obtain (16) Jortner, J. J. Chem. Phys. 1976, 64, 4860. (17) Hopfield, J. J. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 3640.
In the following, we treat the two extreme cases separately for the sake of convenience. Type I ( ? , > 1 holds
g(z) = (In z ) + ~ hl(ln z ) + ~ h2 In z
+ h3 + 0[eT2(lnz ) ~ z - ~ ] (55
where h , = -3I”(l) = 1.731 64699, h2 = 31‘”(1) = 5.93433597 h3 = - r y i ) = 5.44487446 (56) Equation 53 can be regarded as a kind of generalization of the Inokuchi and Hirayama formula.I8 Type ZZ ( T >> ~ ?e). Next we consider case 11, where the solvent relaxation is so slow that the solvent orientation is virtually frozen in the form of the Boltzmann distribution determined at the temperature Th. In such a case, the ET rate between the jth donor and the kth acceptor is given by Wq(& - e‘(X,Y),Rjk). Therefore, the probability SII1(t,Rjk)that the electron on the j t h donor is transferred to the kth acceptor is given by S,”(t,Rjk) = 1 S_Iuf’a(X)
J mdYf’& Y)exp [-t Wq(AE - e’(X Y),RjA 1 -m
(57) Adopting the same procedure for the ensemble average over RJkas before and taking into account all the possible ET pathways from a donor, we obtain the survival probability P”(t) of an electron at a certain donor as follows:
we can rewrite P ( t ) as y)’y-’ d y p (46)
Comparing eq 58 with eq 53, we find that the ensemble average over the coordinate Y is shifted from the inside of the g function to the outside of it.
Y” = exp(-yRv/Ro) (47) Since, y , is much smaller than 1, we obtain the relation
Distribution of the Energy Gap Due to the Frozen Solvent Orientation So far, we have derived formulas of the survival probability P ( t ) using the distribution functions f’,(X) and fh( r). However, it will be helpful for a more thorough understanding of the effect of the frozen solvent orientation in comparison with the nonfrozen solution if we can transform the distribution of solvent orientations into the distribution of energy gaps. For this purpose, we calculate
P’(t) = (JmdXf’,(X)J’exp(-zy)(1n -m Y”
where
Y J 3 - XdZ) + Ob, ln Y,I3
(48) (18) Inokuchi, M.; Hirayama,
F. J . Chem. Phys. 1965, 43,
1978.
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 5063
Electron-Transfer Reaction in Frozen Solution the distribution function Z in terms of the effective energy gap AE'seen by the quantum mode. For the type I frozen state, only the variable X participates in the energy gap distribution. Then, we obtain Z'(AE9 = l : d X f ' , ( X ) 6(AE'- AE
+ c'(X,Y))
(59)
Since no coupling term between the variables X and Y appear in d(X,Y), the shape of the distribution function Z'(AE) is uniquely determined by eq 59, and its distribution can shift along the energy gap AE', depending on the values of AE and Y. Actually, the fluctuation of Y in the type I frozen state is sharply centered at Y = 0 as seen in eq 3 1. Therefore, virtually we can put Y = 0 in eq 59. For the type I1 frozen state, both variables X and Y participate in the energy gap distribution. Then, we obtain Z"(AE')
6(AE'- A E
+ t'(X,Y))
P ( t ) = lJRvdR w ( R ) e l ( t , R ) P = ~ X [-( P
1/ (.Ra))3g(tA~eYFq(AE))I (67)
(60)
The shape of the distribution Z"(AE?is uniquely determined by eq 60, and its distribution can shift along the energy gap AE'in proportion to the value of AE. When 0,= Ob = a,c'(X,Y) is a linear function of X and Y as given by eq 26. In this case, we can analytically integrate eq 59 and 60 as follows:
The formula of eq 67 is the same as obtained by Tachiya and M o z ~ m d e r . ' ~Since F,(AE) is a strongly varying function of AE, P ( t ) given by eq 67 can greatly differ from P'(t) or P"(t). Next, we consider the further approximate treatment where only the ET between the donor and acceptor in the ne_arest neighbor (nn) is taken into account. The survival probability P,(t) in this case is obtained as
Pnn(t)=
S R v d Rw,,(R) exp[-tW,(AE,R)]
(69)
>> R,
as follows:
0
where wn,(R) is given in the extreme of R,
Equation 61 represents Z1(AE') as a Gaussian distribution function, centered a t A E - A, - A d and with a variance (4AakBTh)'lZ. Similarly, Z"(AE') is a Gaussian distribution function, centered at A E - A, + Ad and with a variance (4(A, Ad)keTh)'/'. Both distribution functions are normalized as
+
l:Z'."(AE')
dAE' = 1
(66)
Q ~ ( t , R j d= 1 - exp[-tW,(AE,R,JI
With the procedure similar to that used in the preceding section, the survival probability P(t) of an electron a t a donor with consideration of ET processes to all the acceptors is obtained as
Equation 67 can be also obtained by simply setting Fq(AE c'(X,Y)) equal to Fq(AE) in eq 53 or 58 and using the property
=
S_:dxf',(x)SmdYf'&Y) -m
ceptor might affect the energy level of those reactants randomly from site to site but that such an effect on the electron-transfer rate might be smeared out on the average. This idea will be critically examined in the following. The probability that the electron is tr2nsferred to the kth acceptor from t h e j t h donor, denoted by Ql(t,Rjk) in this case, is given by
(63)
Since the magnitude of the Franck-Condon factor Fqis definitely determined by the effective energy gap AE' by means of eq 10, we can obtain the distribution of the Franck-Condon factor using eq 61 and 62. Here, it should be noted that F, changes by many orders of magnitude in accordance with the variation of a few electronvolts of AE'. So, it is reasonable to represent distribution functions in terms of log F . (Throughout this paper, log represents loglo.) Setting z = log F,, we can obtain the distribution function of z as follows:
wnn(R) = (3Rz/Ra3) e x ~ [ - ( R / R a ) ~ l (70) The formula derived by MillerZocan be obtained by introducing a further approximation to eq 69 as follows: For this purpose, we define a certain critical distance R, satisfying the condition tWq(AE,R,) = 1
(71)
Equation 71 can be rewritten as R, = ( l / a ) In (tAoeYFq(AE))
(72)
Now, we use a property where the function exp[-tW,(AE,R)] is a rapidly decreasing function of R for R IR,. Then, we replace exp[-tWq(AE,R)] by a step function as is done by Doktorov et a1.*5
exp[-tW,(AE,R)]
=1
for R
=0
> R,
-
for R IR, (73) Substituting eq 73 icto eq 69 and setting R, m, we obtain the survival probability PnnMunder such approximations as follows:
PnnM(t) =
S m w n n ( R dR ) = Rc
D',"(z) = l:Z'."(AE')
6(z - log F,(AE')) dAE'
(64)
This distribution function is also normalized as
exp[-( 1/ ~ i R , ) ~ ( l(tAoe'Fq(AE)))3] n (74) or PnnM(t) = e x ~ [ - ( 4 ~ / 3 ) C a I ( l / a )In (tAoFq(AE))
After calculating D'(z) and D"(z) with use of eq 64, z should be replaced with log Fq. Survival Probability Neglecting the Role of the Solvent Orientation Before going into numerical calculations, let us derive other forms of survival probability based on the simpler model of frozen solution, where the role of the frozen orientational distribution of solvent molecules is completely neglected although the role of the random distribution of the positions of donor and acceptor is taken into account. This model is based on the assumption that the randomly oriented solvent molecules around donor and ac-
+ ROI~I
(75)
Equation 75 and eq 72 are the same formulas as obtained by Miller et al.334920by a more intuitive method. From the condition R, > 0 in eq 72, we obtain a time region that is appropriate to be used in eq 74 and 75 as t > eY/Agq(AE) (76) Comparing eq 74 with eq 68, we find that by neglecting the ET pathways between the donor and the acceptor more distant than the nearest neighbor and introducing the approximation of eq 73, the function g(tAoe'Fq(AE)) has been replaced by the (19) Tachiya, M.; Mozumder, A. Chem. Phys. Lett. 1974, 28, 87.
(20) Miller, J. R. J . Chem. Phys. 1972, 56, 5173.
5064
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
Kakitani and Mataga
L
0
la 0.40.2-
-10
log t Figure 2. Calculated curves of the survival probability PI(?) with fi = m, for various values of the energy gap AE as denoted in the figure (in units of electronvolts) on each curve. Throughout this paper, log t denotes log,, t.
log t Figure 3. Calculated curves of the survival probability P"(t) with p = a,for various values of the energy gap AE.
function (In (tAoeYFq(AE)))3.The condition of eq 76 corresponds to z > 1 in the function g(z). Therefore, formula 74 or 75 corresponds to taking only the first term in the logarithmic expansion of g(z) in eq 55. We also see that the condition t >> e Y / @ (AI?) always holds except at a very early time region where P,,%(t) is nearly 1. This condition is equivalent to z >> 1. Therefcre, the first term of g(z) should be always dominant and thus PnnM(t) should be close to P ( t ) . This expectation will be confirmed by the numerical calculation in the next section. Numerical Calculations. 1. Effect of the Distribution of Solvent Orientations In this section, numerical calculations _are made for elucidating qualitative properties of P'(t), PI1(t),P ( t ) , and PnnM(t).Our purpose is to analyze the effect of the distribution of solvent orientations. To attain this object most effectively, we assume that the force constants of the solvent orientation are the same between the initial and final states. We adopt the following typical values of the parameters: A~ = ioi3 s-l, h ( w ) = 0.10 eV, S = 3, R,, = 7 A, Th = 300 K, TI = 77 K, A, = 0.50 eV, C, = 0.025 M (77) Ad = 0.50 eV, a = 1.0 .&-I,
The calculated results of P1(t)and PII(t)are shown in Figures 2 and 3, respectively, with v_ariation_sof the energy gap. The calcula_ted results of P ( t ) or PnnM are shown in Figure 4 (solid curve, P ( t ) ;broken curve, PnnM(t)) with variations of the energy gap. From tkese numerical results, the following are found: (a) The graph of P ( t ) is similar to that of PnnM(t) as seen from Figure 4. In relation to this result we have also calculated P,,(t) using
-8
-6
-4
-2
0
2
4
log t Figure 4. C_alculated curves of the survival probabilities P ( t ) (solid curves) and PmM(t)(broken curves), for various values of the energy gap AE.
eq 69. The calculated graph of F,,,,(r) is found to coincide with that of P(t) completely. This fact indicates that the nearestneighbor approximation between the donor and acceptor is very good under the low acceptor concentration of C, = 0.025 M if we neglect the distribution of the solv_entorientation. (b) Then, a little difference between P ( t ) and PnnM(t) must be-due to the can step function approximation of eq 73. The curve of PnnM(t) be fitted to the curve of P ( t ) by shifting laterally by about -log 2. This result is in accordance with the correction g i n the modified version of Huddleston and Miller's formula.21 (c) Each of the for the different value of the energy curves of P ( t ) or PnnM(t)) gap can be completely fitted to one another by shifting laterally. Therefore, those curves with different energy gaps never cross each other. (d) The graph of P1(t)in Figure 2 changes slightly in shape with variation of the energy gap, and crossing of the curve occurs a little. A remarkable thing is that the energy gap dependence of each curve of P1(t)is considerably different from that of P(t) or FmM(r). (e) The graph of PII(t)in Figure 3 changes drastically with variation of the energy gap. Crossing of the curves occurs evidently. (f) The curve representing the most rapid decay of the survival probability (curve 1.2 in Figure 2, curve 0.3 in Figure 3, or curve 0.3 in Figure 4), which is called the "critical curve", hereafter, nearly coincides among Figures 2-4 with a little lateral shift. These energy gaps are called the "critical energy gap" AE,, hereafter. (8) The energy g a j corresponding to the critical curve is nearly the same between P(t) and P"(t) but is greatly shifted to the large value in P'(t). (h) The curve P ( t ) decreases considerably even whzn the energy gap is negative, ca. -0.3 eV, while curves P ( t ) and P(t) do not. We have seen that the shape of the critical curve remains the same, irrespective of consideration of the distribution of solvent orientation or of the type of the frozen state although the corresponding energy gap differs considerably. Furthermore, it was demonstrated that the survival probability curves cross each other when the distribution of the solvent orientation is taken into account and its tendency is strengthened in the type I1 frozen state as compared with the type I frozen state. In the following, we analyze the mechanism by which the above properties are brought about, with use of the distribution functions of the effective energy gap AE' and the Franck-Condon factor F Figure 5 shows the calculated curves of Zp' (AE?and Z"(A,!?') for the value of A,!? = 0.6 eV. The distribution function Z'(AE9 is a Gaussian function, with a peak at AE'= -0.40 eV and a width of 0.45 eV. The distribution function ZI'(A,!?') is also a Gaussian function, with a peak at AE' = 0.60 eV and a width of 0.63 eV. That is, Z"(AE9 is shifted to a larger energy gap and has a broader distribution than Z'(AE'). The distribution functions D1(log F,) and D"(1og F,) are calculated by using Figure 5 and are plotted for each value of the energy gap A,!? in Figure 6, parts A and B. In this calculation, we have used the property where (21) Huddleston, R. K.; Miller, J. R. J . Phys. G e m . 1982, 86, 200
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 5065
Electron-Transfer Reaction in Frozen Solution
1 .o
B=Cc 3 h
0.8
2
2 -
v
N
-N.
-a
1 -
0 -1.5
-1.0
0
-0.5
1.0
0.5
0.6 0.4
1.5
A E' (eV)
Figure 5. Calculated curves of the distribution functions Z'(hE') and Z"(AE') with (3 = m. The energy gap AE is chosen as 0.6 eV.
0'7
1
jl0
A
-6
-4
-2
0
2
4
Figure 7. Calculated curves of the survival probability P'(t) with (3 = 0.3, for various values of the energy gap AE.
I
the critical curve of P'(t) or P"(t) is obtained. Indeed, the distribution of the solvent orientation works least effectively at the critical energy gap due to the very narrow distribution of log Fq. This fact is reflected in the result that the shape of the crsical curve of P1(t)or P"(t) is nearly the same as that of P ( t ) or PnnM, which is obtained by neglecting the effect of the solvent orientation. We have seen that the distribution of log F, becomes gradually broader by keeping its shape in nearly Gaussian form as AE becomes larger than AE,. In such cases, a certain mean ET rate might be apparently realized, and then the curves of P1(t)or P"(t) can have nearly the same shape with a lateral shift among various energy gaps satisfying AE > AE,. In contrast to this, as A E becomes smaller than AE,,the distribution of log Fq comes to have a very broad tail at the smaller side of log FT In such cases, it is impossible to replace the overall reactions by a certain averaged ET rate even approximately. What happens in such cases is a flattening of the survival curve. Such flattening effect is more evidently seen in the type I1 frozen state. The asymmetric behavior of the survival curves between the two cases AE > AEc and A E < AE, causes the crossing among the survival curves. The reason such an asymmetry appears can be ascribed to the asymmetric energy gap dependence of F,(AE) due to the quantum effect of the intramolecular vibration, i.e., the curve of Fq is much sharper for A E < S h ( w ) than for AE > S h (a).
07-
B
B =m
06-
0 3-
0 5 -
0.2 0. I
0 -12
-8
log t
0.6 0.5
0.2
-11
-10
-9
-8
-7
-6 log
-5
-4
-3
-2
-I
0
F,
Figure 6. Calculated curves of the distribution function D'(log F,) (A) and D"(log F,) (B) with (3 = for the various values of the energy gap AE as denoted in the figure.
Z'(AE') and Z"(AE') are just shifted but keep the same shape when the value of the energy gap AE is changed. From those curves it is clearly seen that the distribution of log Fqchanges drastically when the energy gap AE is varied. The distribution D1 has the sharpest peak at a large value of log F, when AE =, 1.2 eV. As the energy gap becomes larger than 1.2 eV, the distribution becomes gradually broader and its peak is shifted to a smaller value of log F,, by keeping the whole shape nearly Gaussian in form. In contrast to this, as the energy gap becomes smaller than 1.2 eV, the distribution becomes dramatically broader with a large tail to the small-value side of log Fg. The distribution D" has the sharpest peak a t a large value of log F, when AE = 0.3 eV. As the energy gap becomes larger than 0.3 eV, the distribution becomes gradually broader and its peak is shifted to a smaller value of log F,, in a way similar to @(log F,). However, when the energy gap becomes smaller than 0.3 eV, the peak of the distribution is weakened and a very long tail at the smaller side of log F, appears, and finally no trace of the peak is seen within our calculated range of log F,. The energy gap that gives rise to the distribution of the sharpest peak at the largest value of log F corresponds to the critical energy gap (AE, = 1.2 eV for type I an%AE, = 0.3 eV for type 11) where
Numerical Calculations. 2. Effect of the Frequency Change of Solvent Mode So far, we have assumed that the force constant of the orientational vibration of solvent surrounding a charged reactant is the same as that surrounding a neutral reactant. In this section, we extend our calculations to the case where those force constants differ considerably from each other. That is, we choose a typical value /3, = Pb= /3 = 0.3, which was often used in the analysis of the energy gap dependence of the ET rate in s o l u t i ~ n . ~ - ~ The calculated curves of P1(t)and P"(t) are plotted in Figures 7 and 8, respectively. The properties of these curves are summarized as follows: (a) In comparison with the curves of P'(t) for /3 = m, the curves of P'(t) for /3 = 0.3 cross more clearly each other. The shape of the critical curve for /3 = 0.3 is nearly the same as that for /3 = m, with a certain lateral shift (about 0.6 log unit). (b) Comparing with the curves of P"(t) for /3 = 03, we find that the curves of PI1(t)for /3 = 0.3 cross each other very often. Indeed, we see a great deal of flattening in the curves for a small energy gap or a negative energy gap. The shape of the critical curve for /3 = 0.3 is quite the same as that for /3 = except for a considerable lateral shift (about 1.0 log unit). To analyze the mechanism by which the above properties are brought about, we have calculated the distribution functions Z'(AE') and ZII(AE') using eq 59 and 60, as shown in Figure 9. In this case, 2' and Z" are no longer Gaussian functions. The distribution Z' is broad (with ca. 1 .O-eV width) and has a long tail to the smaller side of the effective energy gap AE'. The distribution Z" is broader (with ca. 1.3-eV width) and has long
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
5066
Kakitani and Mataga
,@=0.3
-
a
3=0 3 0 2
0 -10
-8
-6
-2
-4
0
2
4
-12
-11
-10
-9
-7
-8
-6 log
log t
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
-1
0
Fq
Figure 8. Calculated curves of the survival probability P " ( f ) with @ = 0.3, for various values of the energy gap A E . -"
=%
N
-N.
l!yk 1
R=O3
21
0 -1
5
-1
0
-0 5
0
0 5
10
031
1 5
I
A E ' (eV)
0.2;
Figure 9. Calculated curves of the distribution functions Z ' ( A E ' ) and Z " ( A E ' ) with @ = 0.3. The energy gap AE is chosen as 0.6 eV.
tails to both sides of AE'. Using these curves, we have obtained the distribution functions @(log Fq)and D"(1og F,) for each value of the energy gap A E as shown in Figure 10, parts A and B. The distribution D' has the sharpest peak at the large value of log Fq when AE = 1.2 eV, which is the critical energy gap of P'(t) as seen in Figure 7 . It is remarkable that some amount of the distribution remains as a long tail extending to the very small value of log Fq. As the energy gap AE becomes larger than 1.2 eV, the distribution becomes much broader by keeping the whole shape in a nearly Gaussian form. On the other hand, as the energy gap becomes smaller than 1.2 eV, the distribution becomes very broad and has no appreciable peak for AE I0.6 eV. This kind of change in the distribution D' for /3 = 0.3 is somewhat similar to that in the distribution D" for /3 = m in Figure 6B, although the energy gap is shifted to some extent. In the case of the distribution D" for (3 = 0.3, it has the sharpest peak when AE = 0.3 eV, which is the critical energy gap as seen from Figure 8. As the energy gap AE becomes larger than 0.3 eV, the distribution is deformed very much to a broad one, far from the Gaussian form. This property is quite different from the case of /3 = a. As the energy gap AE becomes smaller than 0.3 eV, the distribution becomes a very flat one, but its distribution is quite different from the one for AE > 0.3 eV. Summarizing the above results, it can be concluded in general that the distribution of log Fq for fi = 0.3 is much broader than for (3 = m. Such a broadening effect due to the force constant change of the solvent mode is more exaggerated in the type I1 frozen case. As a result, the flattening of the survival curves of P'(t) is enhanced for AE < AE, in the case of fi = 0.3 compared with the case (3 = a. The survival curve of P"(t) is slightly flattened even for AE > AE, and is greatly flattened for AE C AE,. Owing to these properties, crossing among the survival curves occurs more frequently in the case (3 = 0.3 than in the case fi = m.
Discussion In the preceding sections, we have shown that the shape of the critical curve of the survival probability remains nearly the same, irrespective of the nature of the frozen state, type I or type 11, and almost independent of the analytical method of considering or neglecting the solvent orientation effect. The shape of the survival probability curve is rather strongly correlated with the value of cy. Therefore, the value of cy can be determined unam-
-12
-11
-10
-8
-9
-6
-7
log FQ
Figure 10. Calculated curves of the distribution functions D'(1og F,) (A) and d'(log F,) (B) with @ = 0.3 for the various values of the energy gap AE as denoted in the figure. I
0.3 .\,W , J-=; ,':-.. . ' ,w.,3== . \ ,.,'
,Wi '---
1
,'\
7t
I
~
1i ll2l 2
,
AE (eV)
Figure 11. Apparent energy gap laws (solid curves) obtained from relations between -log f , * and AE for PI(?)with @ = m, PI(?)with @ = 0.3, P"(f) with /3 = m , P Ii ( t ) with 0 = 0.3, and &t). The broken curves represent the energy gap laws of W with fl = m and W with @ = 0.3, which are obtained in solution. The energy gap dependence of W, is also plotted for comparison.
biguously by simulating the observed critical curve. This will be done in forthcoming paper. Using the survival probability curves that were obtained in this paper, we can derive an apparent ET rate for each curve as follows: We define t l , 2 as the time for the survival probability being 0.5. In Figure 11, we have drawn the relation between -log t 1 / 2and the energy gap AE by the solid curve. Since ( t l 2)-1 roughly corresponds to the apparent ET rate, the above reiation can be regarded as an apparent energy gap law of the ET rate in the frozen solution. From this figure, we can see that the peak of the apparent energy gap law locates at the large energy gap in the type I frozen state but at the small energy gap in the type I1 frozen
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 5067
Electron-Transfer Reaction in Frozen Solution
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state. The energy gap law obtained from P where the effect of the solvent orientation is neglected is rather similar to that of the type I1 frozen state with 0 = a. It will be interesting to compare these apparent energy gap laws in the frozen solution with the usual energy gap law in solution. If we follow the notation in this paper, the ET rate in solution is given by
0.8 0.6 -c
W(AE) = s m -m d X f ’ a ( a s m-md Y f ’ J ( Y ) W,(AE -t’(X,Y,Ro))
p‘
0.4
(78) with Th = T, = 300 K. In eq 78, it is assumed that the ET reaction dominantly occurs when an encounter complex is formed with a donor-acceptor distance of Ro. Then, the survival probability in solution is given by
Po’(t)= exp[-tW(AE)]
(79)
The energy gap laws in solution calculated from eq 78 are shown in Figure 11 by the broken curves. The energy gap dependence Wq(AE,Ro)due to the q mode given by eq 8 is also drawn in the same figure for comparison. It is seen that overall shapes of the apparent energy gap laws obtained from PI with 0 = m and 0 = 0.3 agree well with those of the energy gap laws of W(AE)with 0= m and = 0.3, respectively. It should be noted that the energy gap laws with 0 = 0.3 is shifted a little to the low energy gap side from those with 0 = a. With regard to the type I1 frozen state, overall shapes of the apparent energy gap laws with 0 = m and 0 = 0.3 are rather similar to that of the energy gap law of W,(AE). The scme thing applies to the apparent energy gap law obtained from P. We can summarize the above results as follows: (a) The apparent energy gap law obtained with use of the survival curve of the type I frozen state roughly corresponds to the energy gap law W(AE)in solution. (b) The apparent energy gap law obtained with use of the survival curve of the type I1 frozen states roughly corresponds to the energy gap law Wq(hE)due to only the q mode. This result indicates that mostly independent information is obtained from the different types of the frozen state. If one designs an adequate experiment based on the above results, it may become possible to derive much detailed information on the intramolecular and intermolecular parameters accompanying the E T reaction, which was not obtained from the expeJiment in the fluid solution. In Figure 4, we have seen that P ( t ) almost coincides with PnnM(t).This fact indicates that the nearest-neighbor approximation is good when the effect of frozen solvent orientation is not considered. Then, there will arise the question of how the situation changes if the effect of frozen solvent orientation is taken into account. To answer this question, we calculate the survival probability in the type I frozen state with use of the nearestneighbor approximation. The formula is obtained by substituting wnn(R) of eq 70 into w(R) of eq 40 as follows:
The calculated result with 0= is shown in Figure 12. We can see that the curve of P,I(t) is considerably flattened for the energy gap smaller than 1.2 eV and that the crossing occurs clearly among the curves. Its tendency is even more remarkable than in P’(t) with fl = m in Figure 2. The above properties have also been found in the type I1 frozen solution. These properties can be understood as follows: For the nearest-neighboring donor-acceptor pairs that are energetically under favorable circumstance with respect to the surrounding solvent orientation, the ET occurs promptly. However, there will be considerable amount of the nearestneighboring donor-acceptor pairs that are energetically under the unfavorable circumstance. Those pairs greatly contribute to the flattening of the survival curve as shown in Figure 12. Under such circumstances, if all the neighboring acceptors are allowed to take part in the ET reaction, the donor looks for the other acceptors where energetically favorable situations are realized. The ET rate by this pathway can be more rapid than that between the nearest-neighboring donor-acceptor pair with unfavorable frozen Q,
0.2 0 -10
-8
-6
-4
-2 log t
0
2
4
Figure 12. Calculated curves of the survival probability P,,’(t) with 0 = m, for various values of the energy gap AE.
solvent orientation. This pathway can play a role of enhancing the ET rate at longer time, in agreement with the result of numerical calculations. On the basis of these results, we may conclude that the nearest-neighbor approximation is not appropriate once we take into account the effect of frozen solvent orientation. In this paper, we have considered only one solvent mode by allowing the variation of the value of 0.However, when we try to compare the present theory with the experimental data, some care is necessary. According to our recent simulation study,’ a thin dielectric saturation shell of solvent molecules exists just around a charged molecule, and the outside of this saturation shell forms a nonsaturated region. The solvent motion inside the saturation shell contributes to the coordinated solvent mode (c mode) with a 0 value of 0.1-0.3. The solvent motion in the nonsaturation region corresponds to the ordinary polaron solvent mode (s mode) with a 0value of infinity. We have already shown that both of the c mode and s mode as well as the q mode are indispensable to reproduce the experimental data of the various kinds of the ET reaction in polar solution. Therefore, it seems to be necessary to take into account both the c mode and s mode in the analysis of the survival curve in frozen states. The extention of our theory to such cases is straightforward, although the numerical calculation would be very time-consuming. It is guessed that by taking into account both solvent modes, the distribution functions Z1(hE’)and Z”(AE3 would become broader than given in Figure 5 and that crossing among the survival curves of P’(t) or P”(t) would occur more clearly than in Figure 7 or 8. The experimental data obtained by Beitz and Miller3 using the trapped electron as a donor have shown that some considerably flattened curves exist in addition to many “normal” curves. This fact suggests a significant contribution of the c mode in the modification of the survival curves. The detailed comparison of the theory with the experiment will be made in a forthcoming paper. It will be valuable to consider here in what system the condition of type I or type I1 frozen state is realized. The type I1 frozen state will be attained by cooling the polar solution rapidly to a very low temperature such as liquid helium temperature and producing the donor by means of the radiolysis. When the donor is an anion radical, the situation will be ideal. However, the preparation of an optically transparent matrix at liquid helium temperature will not be easy. On the other hand, the experiment by Beitz and Miller3 may correspond mostly to the case of type I frozen state, where the polar solution (the solvent being methyltetrahydrofuran (MTHF)) was cooled to the liquid nitrogen temperature and the trapped electron was produced by radiolysis. From the time-dependent shift of the absorption spectra of trapped electron in M T H F at 77 K, it has been estimated that the reorientation of the surrounding solvent molecules to form the stable trapped electron takes place within ca. 10 ~ s . Under ~ * the strong field of the electron, the motion of the coordinated solvent molecules is considered to be in thermal equilibrium because only a small amplitude of the orientational fluctuation of the solvent molecules is required to be in the equilibrium state. In contrast
5068
The Journal of Physical Chemistry, Vol. 92, No. 17, I988
to this, the solvent molecules surrounding the neutral acceptor molecules are exerted by no appreciable field from the reactant. In such a case, a large amplitude of reorientational motion is required for solvent molecules that were in thermal equilibrium at high temperatue, Th,to relax to the thermal equilibrium state at low temperature, T,. Indeed, it will be very hard to occur for solvent molecules in the glassy media at low temperature. Then, the orientation of solvent molecules surrounding the neutral acceptor molecules is considered to be virtually frozen in the configuration at high temperature, Th. Here, the problem arises whether the solvent molecules outside the coordinated solvent molecules surrounding the trapped electron will be in thermal equilibrium or not. In our opinion, the motion of this s-mode solvent would be also in thermal equilibrium because the saturation shell width is estimated to be very thin, ca.1 A,’ and so the solvent molecules that mostly contribute to the reorientation energy are the ones located close to the trapped electron and feel a strong electric field. So far, a Poisson distribution formula4 was sometimes used for the expression of Wq(AE) instead of eq 8. Indeed, the FranckCondon factor of the q mode consists of many progressions of intramolecular vibrational modes. Quite often Wq(AE)reserves some vibrational structure even if it is not as remarkable as in the Poisson distribution. In this sense, the expression given by eq 8, which has no vibrational structure in its energy gap dependence, is much simplified. However, the solvent mode surrounding the reactant has a distribution of the energy gap as shown in Figure 5 or 9. The width of the distribution of ca. 0.5-0.7 eV is much larger than a possible separation of the vibrational progression in Wq(AE). That is, even if we take into account the vibrational structure of W,(AE),its effect is completely smeared out so far as the observable quantity is the ensemble-averaged one, and the calculated result becomes equivalent to that obtained by using eq 8. Recently, Fiksel et a1.2zinvestigated the effect of the scatter in the parameter value of the frequency factor u ( F , in our sense) as well as that of a due to the random distribution of the solvent molecules. That is, assuming arbitrarily six orders of scatter in the value of v, they have also obtained a flattening of the survival curve. Then, it will be valuable to make clear the relation between their theoretical model and ours. For the purpose of reproducing their formula from our formalism, we start to assume that the ET occurs only between the nearest-neighboring donor and acceptor pairs and that the randomly orienting solvent molecules cause the distribution of the energy gap for the ET reaction, in the type I frozen state. Then, the survival probability curve becomes Pn,’(t) in eq 80. Next, we adopt the step function approximation to this as in eq 7 3 . Then, we finally obtain the survival curve as
Equation 81 corresponds to the combined formula of eq 3 and eq 6 in ref 22, although the distribution functionsf’,(X) andfh’(fl are interchanged in their formula. As we have discussed already, the step function approximation is very good, but the nearestneighbor approximation for the E T is not appropriate so far as the distribution of the solvent orientation is taken into account, giving much flattening effect to the survival curve at the small energy gap. The apparent small flattening in the result of Fiksel is due to a rather small scatter in the value of v, compared with the very large distribution of Fqgiven in Figure 6A or 10A for the small value of AE. So far, we have considered only the charge-shift reaction involving the anion as shown in eq 3. The following reaction involving the cation in the frozen state was experimentally studied by Kira et al.:23,24 (22) Fiksel, A. I.; Zhdanov, V. P.; Parmon, V. N. Chem. Phys. Lett. 1985, 113, 467.
Kakitani and Mataga
+
D A + + D + +A (82) It is obvious that all the present theoretical results hold true for this reaction as well. Furthermore, it is possible to extend the present method to the case of the charge-separation as well as charge-recombination type of reactions in frozen or rigid state. In the case of the charge-recombination reaction, a possible influence of electrostatic interactionsz5 between the donor and acceptor on the energy gap distribution may be significant. It should be noted here that only the ET reactions in the low-temperature frozen solutions but also many other ET reactions in more or less fixed or confined state, where positions of donor and acceptor molecules are fixed and molecular motions in the environment surrounding the donor and acceptor are very restricted compared with the case of fluid solutions, are interesting problems in relation to the present theoretical treatment. For example, electron donor and acceptor chromophores adsorbed on various membranes and crystalline surfaces as well as those fixed in synthetic or biological polymers, especially in proteins, will provide interesting systems to be investigated on the basis of a method similar to the one developed here. It seems quite probable, on the basis of such energy gap dependence of the ET rate as given in Figure 11, that the peak positions of the bell-shaped curve of photosynthetic model systems of donor and acceptor will be at a much smaller energy gap side when fixed in protein than in solutions. On the other hand, concerning the problem of the coupling between the relaxation in the solvation process of charged species and the ET process, we have considered in the present paper only extreme cases (type I and type 11). In most cases of the ET by weak interaction in nonviscous (fluid) polar solutions the solvation relaxation (with the longitudinal relaxation time T L ) will be considerably faster than the ET process itself. We had treated such cases in our previous theoretical studies of the energy gap dependences of various types of ET in polar s o l ~ t i o n s . ~ , ~ , * , ~ - ~ ~ When the solution becomes more viscous or the interchromophore electronic interaction responsible for the ET becomes stronger, however, the TL-’ will become comparable to or smaller than the ET rate. Such a case is a subject of current interest and theoretical s t u d i e ~ . ~ ’It should be noted also that even in the case of the systems fixed on solid or membrane surfaces or in polymer matrices, restricted motions of polar groups in the environment around the donor or acceptor chromophores may be possible when the temperature is not very low. In such a case, similar problem as discussed above for the more or less viscous solutions will arise.29 More detailed discussions on these problems will be given in forthcoming paper. In this paper, we have theoretically shown that the various curves of the survival probability under the different frozen states involve much valuable information for the detailed mechanism of the ET reaction that will not be obtained from the study of the ET reaction in fluid solution. In analyzing those experimental data, however, it is essentially important to take into account both distributions of the donor-acceptor distance and of the solvent orientation around the reactants. Although we have treated rather ideal cases of the frozen state in this paper, a skillful design of the experiment and analysis of its result based on the present theory will be very fruitful. Acknowledgment. The numerical calculations were performed by using the FACOM M-382 in the Computation Center of Nagoya University. T.K. was supported by a Grant-in-Aid (62580218) from the Japanese Ministry of Education, Science and Culture, and N.M. was partly supported by a Grant-in-Aid (62065006) from the Japanese Ministry of Education, Science and Culture. (23) Kira, A. J . Phys. Chem. 1981, 85, 3047. (24) Kira, A,; Imamura, M. 1984, 88, 1865. (25) Leeuwen, J. W. V.; Levine, Y. K. Chem. Phys. 1982, 69, 89. (26) Kakitani, T.; Mataga, N. J . Phys. Chem. 1987, 91, 6277. (27) See, for example: (a) Calef, D. F.; Wolynes, P. G J . Phys. Chem. 1983,87, 3387. (b) Sumi, H.; Marcus, R. A. J . Chem. Phys. 1986,84,4894. ( 2 8 ) Kevan, L. Advances in Radiation Chemisfry;Burton, M., Magee, J. L., Eds.; Wiley: New York, 1974; Vol. 4, pp 182-305. (29) Nakatani, K.; Okada, T.; Mataga, N., manuscript in preparation
