J. Phys. Chem. 1993,97, 6793-6800
6793
Effect of Temperature, Energy Gap, and Distortion of Potential Surfaces on Photoinduced Intramolecular Electron Transfer R. Islampour,+R. G. Alden, George Y. C.Wu,#and S. H . Lin' Department of Chemistry and Center for the Study of Early Events in Photosynthesis, Arizona State University, Tempe, Arizona 85287- 1604 Received: September 22. 1992
We report the quantitative examination of the effect of distortion of potential energy surfaces and the temperature effect on photoinduced intramolecular electron transfer (PIET) as a function of electronic energy gap. The results demonstrate the importance of distorted oscillators in determining the dependence of the rate of PIET on the energy gap. This phenomenon may in some cases lead to misinterpretations of experimental data, when undistorted oscillators are assumed to be involved in the PIET process. The condition for observing the linear (rather than parabolic) dependence of the ET rate will be determined using the multimode model. The inclusion of multivibrational modes in the nuclear factors results in energy gap behavior, which is markedly different from the single mode case. Finally, a comparison between PIET and internal conversion is discussed.
1. Introduction
Photoinduced electron transfer (ET) and excited-state properties of covalently linked porphyrindimers have been extensively studied in order to elucidate the functions of the organized chromophores in the photosynthetic reaction centers (RC).l-S Although the postulated initial excited state in the wild-type RCs is not a completecharge-transfer (CT) state, the preparation and investigation of the complete CT state of porphyrin aggregates are useful for an understandingof the intradimer CT character.6.7 Recently, Segawa et a1.8 have reported the direct measurement of intradimer photoinduced ET and energy gap dependence of the ET kinetics in metalloporphyrin heteroaggregates. By the photoexcitation of the heterodimer, the contact radical ion pair (CIP) of the heterodimer is formed through an extremely fast process within a few picoseconds. The CIP does not dissociate readily to solvated radicals but tends to decay through a nonradiative charge recombination process. The rate constant of charge recombination km of the CIP decreases as the exothermicity of the ET increases; the plot of log k ~ ~the v senergy gap of ET exhibits a linear relation. The nonadiabatic (time domain) expression for PIET can be written so as to include changes of curvature as well as shifts of equilibrium position of the potential energy surfaces. However, the integration over time becomes complicated,and most authors have chosen to neglect the effects of frequency changes on the basis that they are smallcompared to thoseof equilibriumposition shifts. Ulstrupand Jortnergpointedout theimportanceof effects of frequency changes on the overall transition probability. Kakitani and Kakitani'o investigated the possibility that the remarkable temperature dependence of the photosynthetic electron-transfer rate might emerge from the vibrational frequency changebetween theinitial and f i i states. Kakitaniand Matagall pointed out that the anomalous energy gap dependence of the photochemical charge-separation reaction rate in polar solvents can be explained by the fact that the frequency of the solvent mode surrounding the initial neutral state, Aq-B, would be much smaller than that surrounding the A--W state. They used the classical method of Hopfield. In this paper, we shall study the effect of energy gap, temperature, and distortion of potential surfaceson photoinduced intramolecular ET (PIET). For this purpose, both the single-
mode case and multimode case will be investigated. We shall demonstratethe condition under which the linear relation between log km and energy gap of ET can be observed. We shall also compare the photoinduced intramolecular ET with radiationless transition (i.e., electronic relaxation). As can be expected, for the case where the donor group and acceptor group interact strongly, the PIET reduces to ordinary radiationless transition. The ET process in betaine-3012may be interpreted as electronic relaxation rather than PIET. 2. General Consideration
It has been shown that the PIET rate can be expressed as13
where Pfudenotes the Boltzmann factor, I( 0,$3b)12represents the Franck-Condon factor and TBis the electronic matrix element for ET:
Here the Condon and Placzek approximations have been wed. Using the integral representation for the delta function 6 ( E p - Etu), WBcan be written as
WB= -$"I2~dt
exp(itw,JG(t)
where h w denotes ~ the electronic energy gap and if 0, and eb can be written as a product of vibrational wave functions, i.e.
then G(t) = &Gl(t) and Gl(t) is given by
(01 + '/z)w,H (2.5) Here for simplicityvibrations are assumed to be harmonic. The anharmonic effect has been examined in a previous paper.13 Notice that Gj(r) can be expressed a d 4
t Permanent address: Department of Chemistry, Teacher Training University, 49 Mofateh Ave., Tehran,Iran. t Permanent address: Department of Chemistry, Chinese Culture University, Taipei, Taiwan, ROC.
0022-3654/93/2091-6793104.00/0
(2.3)
Q 1993 American Chemical Society
6794 The Journal of Physical Chemistry, Vol. 97, No. 26, 1993
Islampour et al. where pi
< 1. Carrying out the integration in eq 2.17 yields
[
4(2!pj)2]6(
In particular if 1-Pj)uJ
of = w;,
"rot+
9
2np;-u;--ppj 2
then eq 2.18 becomes
I
(2.9) We shall let (wj} and {wj'} refer to the vibrational frequencies of the initial and the final electronic states, respectively, and {dj} be the displacement of the equilibrium configuration in the two electronic states. Here we have fij
= (e"4"-
11-1,
pi
= (w, - w;)/wj
(2.10)
In particular, for the displaced oscillator case, Cj(t) reduces to ~ , ( t )= exp(Sj(l
(2.18)
(2.19) where N = ( w ~ + f 1/2p,q)/wI. Applying the saddle-point approximation to eq 2.1 7 yields
+ D,) + Sj[(l + fij)e"@j+ fi/eirwj])
(2.1 1) where S, = w#,2/2h, the coupling constant (or the Huang-Rhys factor). On the other hand, for the distorted oscillator case we have
(2.20) fiI)* e
fi?e+]
[ej'(*-pj)w]- e~'(1-p,)~,]~/2 (2,121
where
The Fourier integral in eq 2.3 is often evaluated by using the saddle-point method;15 the result is given by
where t* denotes the saddle-point value o f t and is determined by iwIf = G'(t*)/G(t*) (2.14) For the displaced oscillator case, the results are well-known:15
where
It has been shown that for the case in which the distortion is large, the singular point exists in the integral of W, for T # 0 due to the curve crossing. This phenomenon generates a very pronounced temperature effect in' W Thegeneral WJ(Le., with bothnormalcoordinatedisplacements and normal frequency displacements) can be expressed as
at T = 0. Here S,l = w,ldj/2h. wV=
cSpj[(Z+ Aj)eirrwj-fip-"*wJ] (2.16)
For the single mode case, if
pj
= 0 then eq 2.22 reduces to
I
Now we consider a two mode system in which the ith mode is displaced, while the jth mode is distorted. Furthermore, we assume that the molecule is initially cold (i.e., ii, = AI = 0).Then we have
where Nj = w1/Iwj, and if Sj = 0, it reduces to
Wp= *T/?12Jmdt exp[ it(wp - ZjPIoI) 1 -OD
][
S,(1 - eifwi) 1 + A P2 ( 1 - e2""~) 4(1- Pj)
(2.23B) where Nj = (wif
+ 1/2pjq)/w[.
Photoinduced Intramolecular Electron Transfer
The Journal of Physical Chemistry, Vol. 97, No. 26, 1993 6195
Applying the saddle-point method to eq 2.22 yields
end we may write
W" =
G(t) = (exp(ifi!/h)
=
exp(-ifi,r/h))
(exp-[(i/ h x d i '0091 ) (3.1) where
0(f)= exp(ifiit/h)(fif - hi)exp(-ifiir/h)
(3.2)
exp-[ ...I is the negative time ordering operator, and (...) means an equilibriumTnsembleaverage over the initial vibrational states. Here Hi and Hfare the vibrational Hamiltonians of the initial and the final states, respectively. To the second-order approximation we 0btain13.~0 (2.24)
G(t) = exp[(i/h)(o)r
+ (i/h)2Jdr'(r
-~ 9 ( ~ l ( 0 ) ~ l ( r 9 ) l (3.3)
where
01=0-(0), O = H f - H i
where
(3.4)
In the short-time approximation, eq 3.3 immediately leads to the following expression for the ET rate constant:
+
wfl = (l/h2)I~f112(2?rh2/a~)1/2 exp[-(Ef-
(0))2/2a,z1 (3.5) where
to
where Nj = W ~ / Wand J , for SI = 0 it reduces to
= (02)- (0)2
(3.6) It can be seen that the thermal average of the vibrational gap between two electronic states can be interpreted as the reorganization energy. Applying the saddle-point method the rate constantexpression, eq 2.3, can be written in terms of cumulants as follows: 0, 2
Again, for the single mode case and for p j = 0 eq 2.24 reduces
Wfl = (l/hz)ITf112[2?rli2/(~l(0) fr,(t*)))]1/2exp[(i/h) X (E,-E, + (i?))t* + (i/h)21'dt'(t* - t')(Ol(0)0](t?) (3.7) where
-
where Nj = (wr+ 1/2p*j)/w/. Using the Stirling approximation, n! 6 n n e - " , which is in error by less than 2% for n > 4, it is seen that eqs 2.23A and 2.23B reduce to eqs 2.26A and 2.26B, respectively. Thus eqs 2.26A and 2.26B are good approximations for eqs 2.23A and 2.23B. In eq 2.3, the ET rate constant has been expressed in terms of the Fourier transform of the time-correlation function G(t). The rate constants of numerous electronic processes and optical processes can be expressed in a similar form. This time-correlation function G(t) has been evaluated exactly and analytically by numerous workers for linear coupling case (i.e., displaced oscillator case), quadratic coupling case (i.e., distorted oscillator case), and linear plus quadratic coupling case (Le., displaced-distorted oscillator case).1"19 The Fourier integral in eq 2.3 for the ET rate constant can be carried out numerically. In this paper, the saddle-point method is used to obtain the expressions of ET rate constants. 3. Cumulant Expansion of
at)
For comparison, we shall consider the evaluation of PIET rate constants by the cumulant expansion method. To investigate the effects of frequency change on PIET, we shall expand the time domain correlation function G(t) in terms of cumulants. To that
(l/h)(E,-E,+
( 0 ) )= -i(l/h2)fdt'(01(0)
O,(t?)
(3.8) In the harmonic oscillator basis sets (of the electronic state li)), we have13J0 I
2a;[(1
+ fij)eifwJ+ rt,e-ifwJ]2)(hwj)2(3.10)
where
= -'/4~j(2 - P,), Sj = (1 -pi)' (3.11) Now G(t) in eq 2.3 can be written as G ( t ) = nic,(t)with aj
Gj(t) = exp[-j3;Sj(l
exp[bj(itoj)
+ 2fij) - '/p;(l + 2fij)']
X
+ 2a;fij(1 + rt,)(itw,)' + /3;Sj[(1 + fij>ei'"l + fiJeAfwJ] + '/2a;[(1 + fij)eifwl+ fip-ifwj])(3.12)
where bj=
-
[bj(l-/3,)Sj+aj(l-aj)(l
If the molecule is initially cold (Le., fij
+2fij)]
(3.13)
0), eq 3.12 reduces to
6796 The Journal of Physical Chemistry, Vol. 97, No. 26, 1993
Islampour et al. with eqs 3.16A and 3.16B, respectively. These equations are related to each other by the Stirling approximation. Finally we should mention that for the displaced harmonic potential surfaces and in the high-temperature range, we can write eq 3.5 as follows:13
wfi = (i/hZ)~~,12(.rrh2/k~(0)) exp[-(hw, + (0))'/4kT( e ) ] (3.22) where (0)= &Sjhwk Equation 3.22 is the conventional form of the ET rate constant.
In this case Wp is given by
4. Electron Transfer vs Electronic Relaxation
(a:/2)"'/m,!(Ni where Ni = (wuif pj = 0, then
- 2mi)! (3.15)
[pi( 1 - &)SI + a,( 1 - a j ) ] ) / w , , In particular,
W, = (2?r/h2wi)lT/l12e"ls,"l/N,!
(3.16A)
and if SI = 0
It should be noted that the photoinduced intramolecular ET is a nonadiabatic process. For the case in which the interaction between the donor group D and acceptor group A is very strong, the PIET becomes an ordinary electronic relaxation. This is particularly true when there are no spacer groups between D and A. For spin-allowed electronic relaxation processes (i.e., internal conversion), the rate constant is given by15J6
Introducing&s3.9and3.lOintoeq3.7, weobtainthe following expression for the rate constant within the saddle-point approximation: Im
I2
(4.1) where R i v ) represents the electronic matrix element involved in IC:
Ri(f?)= -h2(
@Hi)
and Qi denotes the promoting mode: Here for the purpose of comparison, we consider only the displaced oscillator case. At T = 0, eq 4.1 reduces to
-Ca:[(l 1 2,
+ fi,)ei'*wI + fi,e4'*wJ]2) (3.17)
where
For the single mode and the initially cold molecules eq 3.17 becomes [27r/(b:Si W, = (ITA2/h2wi) exp(-j3;Si'/+;
+ a:xj)] ' ~ z ~ , - ( N p l X/ z ) + j3;S,xi+ ' / p ; x ; ) (3.19)
Comparing eqs 4.1 and 4.3 with the results given in section 2 for PIET, we can see that in PIET the electronic matrix element is mainly due to electrostatic interaction, while in IC the electronic matrix element is mainly due to the vibronic coupling. Furthermore, in IC due to the vibronic coupling the electronic energy gap is modified because of the terms like eitwfand e-"or due to the promoting mode and the temperature effect is also modified because of the terms like 1 + fii and fii. That is
w,
= W,(1)
+ W,(2)
(4.4)
where
where
= {-@:Si + 4 (s: + 4 N , ~ ~ ~ ) ' / ~ ) / 2(3.20) a~ = 0 and S, = 0, eq 3.19 reduces to x,
For pi
w'
e+/2(a:/ 2)WZ 2* (3.2 1B) =~ T ' 2 ~ ~ ( N i / 2 ) N J / 2 e - N , / 2
respectively. Equations 3.21A and 2.21B should be compared
CS,((l i
+ 2 9 ) + C S , [ ( l + fij)ei"' + fi,e-i'Uj])]
(4.6)
i
In other words, the ET rate constant'W consists of two parts, Wfi(1) and W'(2). In Wp(l), the electronic energy gap is ai/-
The Journal of Physical Chemistry, Vol. 97, No. 26, 1993 6797
Photoinduced Intramolecular Electron Transfer
TABLE I P
Wp, w 2.1 Wp, 2.24 w,,t9 2.1, x 103 Wp, cq 2.24, X lk3
-0.28 0.155 0.149
-0.18 0.141 0.137 0.608 0.599
0.980 0.962
-0.12 0.128 0.124 0.339 0.335
0 0.0983 0.0964 0.0579 0.0573
-0.06 0.1 13 0.111 0.161 0.159
parameters used S’= 1, o,*/w’ = 4, A = 0
+0.06 0.0838 0.0816
Sf=1,~~/w‘=8,A=0
TABLE II P
Wp, e~ 2.24 Wp, 3.21 Wp, w 2.24 Wp, w 3.21 Wp, w 2.24, X 10 Wp, e~ 3.21, X 10
-0.24 1.27 1.13 0.945 1.oo 1.34 1.27
-0.18 1.40 1.30 0.909 0.968 0.834 0.729
-0.12 1.54 1.48 0.837 0.883 0.441 0.364
-0.06 1.66 1.85 0.726 0.748 0.185 0.154
wi and the temperature effect due to the promoting mode i is through the factor Ri + 1 . Similarly in Wfi(2),the electronic energy gap is w u + wi and the temperature effect due to the promoting mode is through the factor Ri.
0 1.77 1.77 0.579 0.579 0.0546 0.0546
+0.06 1.84 1.82 0.407 0.402
0.12 1.84 1.74 0.237 0.246
0.18 1.73 1.54
parameters used S = 2, wu/w 2, A S = 2,w1//w
0
4, A = 0
crossing point. It follows that
t c = E i / h+ C Si j X 2 ( 1 - p j ) 4 / [ ( l + X ) - X ( ~ - P ~ ) ~ ] ’ (5.6) For the multimode displaced oscillator, p, = 0, we easily obtain
5. “ d o n
Electron transfer (ET) is modulated by a number of parameters, including temperature, energy gap, and nuclear factors (encompassing the displacements of both vibrational frequencies and coordinates). The energy gap and temperature are experimentally adjustable parameters. The energy gap is typically varied by altering electron-withdrawing or -donating groups around the periphery of a macrocycle, such as porphyrin?’ or by using the solvent in condensed-phasesystems to stabilize or destabilize the charge-separatedstate relative to theinitialdonor state. “Solvent” effects are the impetus for site-specific mutagenesis studies in protein systems where the protein acts as a structured “solvent” environment. The temperature dependence provides important information on the dependence of the ET rate on the nuclear factors, which are commonly referred to as the reorganization energy associated with the ET event. To find a relationship among the various parameters involved in the ET processes we shall calculate the minimum energy crossing for the multidimensional harmonic potential energy surfaces. At the crossing point
or
(5.2)
where qj = (w/h)l/ZQj.We search for a minimum point on the potential energy surface of the initial electronic state such that the quality in eq 5.2 holds. To that end we employ the Lagrange multiplier method and minimize the following expression:
where Xis the Lagrange multiplier. Setting deldq, = 0, we obtain
Inserting this expression into eq 5.2 yields
X(l
- Pj)’]’ ( 5 . 5 )
From eq 5.5 we can determine h and the position of the minimum
and for a single displaced-distorted mode t,
= EJ h
+
Uj[(
1 - p j ) 4 / p ; ( 2 - pj)2] {*sjllz7
[ s j / ( l - pj)’
- ( w i f / w j ) ~ j (-2 p j ) / ( l - P~)~I’/’I’( 5 . 8 )
which gives the following inequality: (5.9)
inorder that c,bearealquantity. It isseen that for largeelectronic energy gap, pi is mainly negative and for large coupling constant Sj it can be positive as well. Also, the respective criteria for the so-called normal, activationless, and inverted crossing are
The cumulant expansion method is often applied to calculate the band-shape function^^^^^^^^^ and rate constants of relaxation processes.13 It has been shown that for the displaced oscillator systems, the second-order approximation of the cumulant expansion method yields the same expressions as those of the exact ones. However, the performance of the cumulant expansion method applied to other systems has not been carefully examined. This will be done in the following. First we shall compare the various approximations that have been considered in this paper. For the single-modecase and for the low-temperature limit, we have numerically compared the exact rate constant expression, eq 2.1 with the exact-saddle pointapproximated eq 2.24, for the various values of p and for two different values of oijlo. (Notice that Wfi= (IT/i12/h2u,)W/if). The numerical results are collected in Table I. In Tables I1 and I11 we have compared the exact-saddle-point-approximatedrate expression,eq 2.24 with the cumulant-saddle-point-approximated eq 3.21 at low temperatures. These tables are prepared for various values of the electronic energy gap, wif/w, keeping S constant, and for various values of the coupling constant S,keeping the electronic energy gap constant. It is seen that the agreement between the two approximations is quite satisfactory. Figures 1 and 2 show the dependence of the rate of ET on the distortion, p, for a series of energy gaps. The curves are representativeof log(W’’), where, Wfi’ = [ ; A C ~ - ~ ) / ( I T M - Wfi, ~)~] which provides a means for broader application of the data. The effects of distortion become increasingly more pronounced as the energy gap, wi//wj increases from 2 to 10. The overall magnitude
6798
Islampour et al.
The Journal of Physical Chemistry, Vol. 97, No. 26, 1993
TABLE I11 -0.18 0.957 0.728 0.352 0.368 0.909 0.968
-0.24 1.42 1.18 0.438 0.482 0.945 1.oo
P
Wp, w 2.24 Wp, e~ 3.21 Wp, w 2.24 Wp,w 3.21 Wp, w 2.24 Wp, e~ 3.21
-0.12 0.569 0.412 0.262 0.260 0.837 0.883
-0.06 0.280 0.213 0.174 0.168 0.726 0.748
0 0.101 0.101 0.0983 0.0983 0.579 0.579
+0.06
parameters used
0.12
S 0.5, w r / w = 4, = 0
s = 1,wu/w=4,R=O
0.0430 0.0522 0.407 0.402
s = 2, w,/w = 4, A = 0
0.237 0.246
11 I 11
10
10 n
9
=I
8
9
. .c h
. z W
8
E=
W
M
ho
0
0
3
I
7
3
7
1
6
'1%
- ,1 '
, -0.24 .-0.12 & ; L
--
,E 0.00
0.12
1
8
0.24
P
r.
Figure 3. Dependence of ET rate on distortion for various temperatures: (A) Rj = 5; (B)Rj = 4; (C) Rj = 3; (D) Rj = 2; (E) A] 1; (F) Rj = 0. Sj = 0.5; .I///./ = 8.
- Saddle-Point -- Energy Gap Law
-
11
10 h
. E
B
W
9
on 0
3
3
2000
4000
6000
8000
10000
Energy Gap(cm- 1) Figure 4. Comparison of energy gap law and saddle-point method using data in eq 5-1 1. -0.24 -0.12
0.00
0.12
0.24
P Figure 2. Dependence of ET rate on distortion for various energy gaps (the same parameters as those in Figure 1 except R j = 3).
of the energy gap dependence on the ET rate can be reduced by as much as a few orders of magnitude relative to the undistorted displaced oscillator case. This dependence clearly indicates that the effects of distortion in some cases may result in a misrepresentation of the nuclear factors coupling the donor and acceptor states, if the data are modeled using simple displaced oscillators. The magnitude of temperature dependence of the ET rate as a consequence of distortion is shown in Figure 3. In general, the rate slows down consistently over the entire range of distortions calculated as the temperature is decreased and the relative change in magnitude is roughly constant.
The saddle-point method has been shown to be an excellent technique for calculating rates of ET in complicated systems involving multimode participation and large temperature ranges, which cannot be accomplished using conventional approximations, such as the strong coupling limitl5 and the high-temperature approximation.20 In a previous paper, Lin and c0-workers2~ utilized the saddle-point method to determine the multimode effects on ET with direct applications to bacterial photosynthetic reaction centers. In Figure 4 the saddle-point method, which has been shown to be quite accurate,24is used to calculate the ET rate (in the multimode case using six modes) as a function of energy gap over a large range from 100 to 14 000 cm-l. In addition, the energy-gap dependence is also calculated using the energy-gap law (see eq 5.13), which uses an averaged frequency and coupling constants. The results indicate that for large energy gaps there is good agreement between the two methods.
Photoinduced Intramolecular Electron Transfer
The Journal of Physical Chemistry, Vol. 97, No. 26, 1993 6799
Mataga and co-workers8 have recently reported the energygap dependence of a series of heteroaggregate porphyrin dimers in which peripheral substituent groups have been altered to shift the redox potentials of the donor and acceptor. They found that the energy gap dependence of the ET rate is linear in a 3000-cm-' range. This phenomenon is also observed at large energy gap theoretically, as shown in Figure 4. For numerical calculationsof the multimode case, we consider the following displaced oscillator model: 12.25
w1 = 100 cm-', S , = 2.61; wg = 750 cm-', S, = 0.04; w, = 1400 cm-',
w2 = 224 cm-', S, = 0.06 w4 = 1200 cm-I, S4 = 0.08
S, = 0.04;
06
= 1520 cm-', s6 = 0.08
12.00
t'
Y
I
500
1000
1500
Energy Gap(cm-' ) Figure 5. Energy gap dependence of ET rate at T = 300 K.
(5.1 1)
ITA = 200 cm-'
(5.12)
In Figure 4, we show the effect of energy gap on the ET rate at T = 0. When the energy gap is large, we may use the so-called energy gap law expression: Q = (l/hZ)ITJ(2./w@)'/Z
x
e x p { S - (w,,/G)[ln(w,,/%)
- 1)) (5.13)
Here S = E$,, the total coupling constant (in this case, S = 2.79), whiles denotes the sum of the coupling constants for highfrequency modes (in this case, = 0.12). Equation 5.13 is obtained by noticing that the saddle-point equation
s
w,, =
ESPjexp(it*wj)
(5.14)
i
can be solved for the high-energy-gap case where only highfrequencymodesmake significant contributionsin eq 5.14. That is w,, = Si exp(it*G)
(5.15)
it* = (I/;)ln(w,//SG)
(5.16)
or
where is the average frequency for high-frequency modes and represents the sum of coupling constants of high frequency modes. Substituting eq 5.16 into the saddle-point expression for Wfi we obtain the expression for Wr given by eq 5.13. We have emphasized the multimode effect on ET. So far we have shown that the weak couplings of high-frequency modes can cause the inverse and weak temperature dependence of ET24 and exhibit the energy-gap law dependence (see Figure 4). Next we shall show that the multimode effect of ET can slow down the decrease of ET rate with increasing electronic energy gap after the ET reaches the maximum. For the purpose of demonstration, we still use the same six modes as described by eq 5.1 1 with some changes in the coupling constants:
w1 = 100 cm-', S,= 4.8; w2 = 224 cm-', w3 = 750 cm-', S3= 0.1; S2= 0.1; w4 = 1200 cm-', S4 = 0.1; w5 = 1400 cm-', s.j = 0.1; w6 = 1520 cm-', s6 = 0.1 (5.17) Here ITd = 200 cm-l is used. For comparison, the single-mode ET case with w = 100 cm-l and S = 5.3 is also shown in Figure 5.
From Figure 5 we can see that the initial increase of ET with energy gap is mainly due to 01 and SI, and after the ET rate
reaches the maximum, depending on the strengths of the coupling constants of high frequency modes the ET rate can remain relatively constant for a certain energy gap range. This effect cannot be observed in the single-mode case (see Figure 5 ) . The purpose of Figures 4 and 5 is to compare the single-mode ET and multimode ET, and to show the importance of highfrequency modes in ET. Figures 4 and 5 also show that the multimode ET usually does not exhibit the parabolic energy-gap dependence. The choice of the six modes shown in eq 5.17 is motivated by the resonance Raman spectroscopic studies of the porphyrin systems. Due to the fact that there exist possible singularities in the time-correlation function G(t) for the quadratic coupling case, the ET rate constant will exhibit a dramatic temperature effect near the condition where a singular point exists. This effect has been observed for radiationless transitions of an isolated system (i.e., in a collision-free condition); in this case the rate constant shows a dramatic dependence of excess vibrational energy.2sFor a system in dense media, this effect may be observed for ultrafast photoinduced intramolecular ET by femtosecond pump-probe experiments,that is, by systematicallyvarying the excitation (or pumping) wavelengths and measuring (or probing) the ET (or other relaxation) rate. For the case of multimode ET with a total S >> 1, but with individual S, I 1, the short-time approximation can be used to evaluate the Fourier integral of the ET rate constant. In this case we obtain the Marcus-Levich type expression of the ET rate constants. However, if in additionto the displacement of potential surfaces there exists the distortion of potential surfaces, then the free-energy change of ET will depend on temperatures and the activation energy of ET in the Arrhenius form of the ET rate constant will vary with temperatures.13 In concluding the paper, we would like to point out that in this paper and previous papers,l3~2~ we have been emphasizing the importance of multimode effects on ET. We have shown that it can modify the behaviors of the temperature dependence and energy-gap dependence of ET rates. In this paper, we have examined the effect of distortion of potential surfaces on ET and we have also demonstratedhow powerful the saddle-point method is in calculating ET rates. Acknowledgment. Publication 150 is from Arizona State University Center for the Study of Early Events in Photosynthesis. The Center is funded by U.S.Department of Energy Grant DEFG02-MER13969 as part of the USDA/DOE/NSF Plant Science Center program. R.G.A. is funded by a NSF postdoctoral fellowship in Plant Biology. This work was supported in part by NSF and NATO. The authors wish to thank the referees for helpful suggestions and discussions.
6800 The Journal of Physical Chemistry, Vol. 97, No. 26, I993
References and Notes (1) Bergkamp, M. A.; Chang, C. K.;Netzel, T. L. J. Phys. Chem. 1984, 87,4441. (2) Wasielewski, M. R.; Johnson, D. G.; Niemczyk, M. P.; Gaines 111, G. L.; ONeil, M. P.; Svec, W. A. J. Am. Chem. Soc. 1990,112,6482. (3) Regev, A.; Galili, T.; Levanon, H.; Harriman, A. Chem. Phys. Lett. 1986,131,140. (4) Helms, A.; Heiler, D.; McLenden, G. J. Am. Chem. Soc. 1991,113, 4325. ( 5 ) Osuka,A,; Maruyama, K.; Yamazaki, I.; Tamai, N. Chem. Phys. Lett. 1990,165,392. (6) Boxer, S . G.;Goldstein, R. A.; Lockhart, D. J.; Mittendorf, T. R.; Takiff, L. J. Phys. Chem. 1989,93,8280. (7) McDowell, L. M.; Kirmaier, C.; Holten, D. J. Phys. Chem. 1991,95, 3379. ( 8 ) Segawa, H.; Takehara, C.; Honda, K.; Shimidzu, T.; Asahi, T.; Mataga, N. J. Phys. Chem. 1992,96,503. (9) Ulstrup, J.; Jortner, J. J. Chem. Phys. 1975,63,4358. (10) Kakitani, T.; Kakitani, H. Eiochim. Biophys. Acta 1981,635,498. (11) Kakitani, T.; Mataga, N. Chem. Phys. 1985,93,381. (1 2) Akesson, E.;Johnson, A. E.;Levinger, N. E.; Walker, G. C.; DuBruil, T. P.; Barbara, P. F. J . Chem. Phys. 1992,96,1862.
Islampour et al. (13) Islampour, R.; Lin, S.H. J. Phys. Chem. 1991,95,10261. Lin, S. H. J. Chem. Phys. 1989,90,7103.Islampour, R.; Lin, S . H. Chem. Phys. Lerr. 1991,179, 147. (14) Lin, S. H.J. Chem. Phys. 1973,59,4458. (15) See, for example: Eyring, H.; Lin, S.H.; Lin, S.M. Baric Chemicul Kinetics; Wilcy-Interscience: New York, 1980. (16) Lin, S.H.J. Chem. Phys. 1966,44,3759. (17) Heller, E.J. Annu. Reu. Phys. Chem. 1984,35, 563. (18) Lagos,R.; Friesncr, R. A. J. Chem. Phys. 1984,81,5899. Won,Y.; Lagos, R.;Friesner, R. A. Ibid. 1986,84,6561. (19) Chan, C. K.;Pagc, J. B. J. Chcm. Phys. 1983,79,5234;Chem. Phys. Lett. 1984,104,609. (20) Islampour, R. Chem. Phys. 1989,133,425. (21) Land, E.J.; Lexa, D.; Bensasson, R. V.;Gust, J. b.;Moore, T. A,; Moore, A. L.; Liddell. P. A.; Memeth, G.A. J. Phys. Chem. 1987,91,4831. (22) Faid, K.; Fox, R. F. Phys. Rev. 1986,A34,4286. (23) Islampour, R.; Lin, S.H. Trends Chem. Phys. 1991, 1, 249. (24) Alden, R.G.;Chang, W.D.;Lin,S. H.Chem. Phys.L.err. 1992,194, 318. (25) Kono, H.; Lin, S.H.; Schlag, E. W. Chem. Phys. Lerr. 1988,145, 280.
