3424
J. Phys. Chem. 1994,98, 3424-3431
Analytical Treatment of Nonexponential Electron Transfer Coupled to Intramolecular and Medium Modes A. A. Zharikovt Institute of Chemical Kinetics and Combustions, Novosibirsk 630090, Russia
P. 0. J. Scherer and S. F. Fischer' Technische Universitat Miinchen, Physikdepartment T- 38, 0-85748 Garching, Germany Received: December 20, 1993Q
A many-mode model for electron transfer is developed. The electron transfer is coupled to the classical solvent dynamics and to a molecular subsystem, consisting of many internal modes which form a dense manifold of states for electron transfer in the inverted region. The fluctuations of the solvent coordinate are treated as an uncorrelated stochastic process. For the limit of slow fluctuations it is shown that a transition from a solventcontrolled transfer to a solvent-independent one occurs as a function of the free energy of reaction. The model predicts a nonexponential electron-transfer kinetics of the intermediate region. Introduction The electron transfer between molecules or molecular fragments in polar media is accompanied by a reorganization of many medium and molecular modes. Experimental data on electron transfer are usually treated in the framework of a simplifiedmodel, where it is assumed that the electron transfer is assisted by the reorganization of one classical medium coordinate and possibly a few quantum molecular vibrational coordinates. This model predicts for slow solvent relaxation the so-called solvent-controlled transfer, whereby the electron transfer is limited by the rate of fluctuations of the classical coordinate.l-4 In this model the necessary rearrangement of the system to energy resonance conditions can be achieved only for a discrete set of values of the solvent coordinate describing the activated transfer complex. Experimentally it is found now for several systems that the electron-transfer rate can exceed the solvent-controlledlimitaS9 Moreover, the study of the back electron transfer by charge recombination in nonpolar solutions cannot be interpreted in the frame of such a model. In particular, the observed weak energy gap dependence of the rate constant in the inverted requires a model which includes many internal molecular vibrational degrees of freedom such that they form a dense manifold of states for the electron transfer in the inverted regime.'>-" The transfer becomes an intramolecularradiationless process, and the medium modes might freeze out on the fast time scale. This situation is important for the initial charge separation in photosynthesis. In the present paper we consider kinetics of the electron transfer in a many-mode model. The fluctuationof the solvent coordinate is modeled by an uncorrelated stochastic process. In the framework of this model, analytical expressions for electrontransfer kinetics are derived. The transition from the solventcontrolled rate constant in the normal region to the solventindependent one in the inverted region with an increase of free energy has a nonexponential character. In the case of photoinduced charge separation, the delayed kinetics and the quantum yield of fluorescence are analyzed in the limit of slow solvent fluctuations. General Formalism We consider for the electron-transfer transitions between two electronic states with potential surfaces VI(* and &(X) which
Figure 1. Potential energies Ul and Uz for the excited state P* and the charge-transferstate, CT, respectively,as functions of the solvent reaction coordinate X with the internal mode in their ground states. AG is the total free energy difference for the electron transfer, A, is the solvent reorganization energy, and E. is the solvent activation energy. AU(X) plays the role of the free energy for the internal degree of freedom for fixed solvent coordinates.
are functions of a classical solvent coordinate X (see Figure 1) and introduce rate constants K(X) and K-l(X) for the forward and back transfer, respectively. If the solvent motion is approximated by a Markovian stochastic process, the following equations holdl,2,4,l8.19
where P(X,t) AX and C(X,t) AX are the probabilities of finding the system in the electronic state P* or P+A-, respectively, within the interval between X and X + AX. -Cl and L2 are operators describing the motion in the two electronic states, more explicitly given in (4), and the rates r1-land 72-2 account for additional depopulation by radiative or radiationless transitions to the electronic ground state or to another excited electronic state. The total populations of P* and F A - at time t are given by
To whom correspondence should be addressed. Humboldt Fellow. On leave from Novosibirsk. Abstract published in Adwnce ACS Abstructs, March 1, 1994.
QQ22-3654/94/2Q98-3424104.50/0 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3425
Nonexponential Electron Transfer (at)) =
J dX C(X,t)
If the dynamics is only considered along the classical coordinate
X,the transfer can occur only at Ul(X)= Uz(X). However, transitions for AU(X) = U l ( X )- U2(X) # 0 are possible once coupling of the electronic states to intramolecular vibrational states is considered. In thecaseof fast electronicphase relaxation, the results can be expressed by an overall rate and the back rate follows from the detailed balance. In general, K ( X ) is a function of the energy detuning AU(X) and is determined by the character of the vibrational subsystem and its coupling to the electronic states. For example, in the case of coupling to one vibrational mode with energy hw and a reorganization energy Xi = Shw, the function K(AU) has the form20-22
x, xln X Figure 2. Rate constant K(X)as function of the solvent coordinate X. The values X. and X,,, are defined implicity via AV(X,) = 0 and AV(X,) = Xi, respectively. Xi is the reorganization energy of the internal molecular
vibrational modes. we have
S sinh -
2kT
which after Fourier transformation
Here I,,(z) is the modified Bessel function. This functional dependence shows resonance behavior due to the discretestructure of thevibrational energiesand a nonactivated character for AU > 0 in the case of a quantum mode. For Xi > hw the largest resonance is found at AU Xi. If many intramolecular vibrational states couple to the initial electronic state, providing a large manifold of dense states, then K ( X ) approximates a smooth function (see Figure 2) and transitions occur at any value of AU, The rateconstant &(X) of the backtransfer will bedetermined from the principle of detailed balance as
The ET dynamics described by eq 1 is often studied in the framework of a diffusion model for the solvent motion. This model provides a realistic description of the stochastic dynamics of the solvent polarization which can assist the electron transfer. Due to the complexity of eq 1 in this model, ET dynamics has been studied in the case of parabolic potentials only for some special forms of K(X)(K(X)= a 6(X - XI) + b 6(X - XZ),K(X) = con~tant).I,2,4,12,23,2~ The diffusion model is a limiting case of a more general class of discontinuous Markovian processes. Here the fluctuations X(t) are modeled by sudden jumps between different values of X. The conditional probability density of changing thevalue X’toXby a singlejump is denoted byflX,X’). In the time interval between two jumps, the value X remains constant.2k28In caseof a Poisson distribution of the time between two jumps, the stochastic process X(t) is Markovian and the operator f has the form p(x,f) = -T(X)-’ p(X,t)
+ J f(X$?
p(X’,t)
T(x1-I
dx’ (4)
where 7 ( X )is the mean time betweenjumps, which may in general depend on the value of X. We shall further assume that r(X) = 7 is constant and that f(X,X’j has the functional form AX$’) = AX - 7x9. In this case the functionfcan be derived as follows. From the stationary condition of the equilibrium distribution qq(X),29
gives
The parameter y with JyII 1 characterizes the degree of correlation. For y = -1 the process is anticorrelatedor dichotomic. In the limit y 1 the time T between jumps approaches zero such that (1 - Y ) / T 1 / and ~ AX$’) ~ becomes a 6 function -6(X - X’). This is the correlated or diffusion process. For y = 0, the function
--
f(x-rxl)= CpCqQ
(6)
does not depend on X’. This characterizes the uncorrelated process. The correlated as well as the uncorrelated model are well-known from spectroscopy as models for weak and strong collisions, respectively. Independent of y, the autocorrelation function of X(t) is exponential ( ~ ( t~(0)) ) =
X2e+C
with a correlation time
The degree of correlation (i.e. the value of y) shows up in the dependence of higher order correlation functions on their order number. In this paper we use the uncorrelated Markovian process to model the stochastic motion. This way we are able to investigate arbitrary forms of K ( X ) and U(X). For this model, eq 1 takes the form
i i ( X , t ) = -;i(x) ii(X,t) where
+ 7-1 B(x) ( i i ( f ) )
(7)
Zharikov et al.
3426 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
[
K(X)
+ 7' + rl-l
= -K(X)
-K-,(X) K-,(X)
+ + 72-l T-'
1
(i) For
71
< 7 2 we have
and where Here ' p 1 , 2 ( X ) are the equilibriumdistributions in the two electronic states which are normalized according to ( ~ 1 . 2 ) = 1. The kinetics of the total populations ( R ( t ) )obeys the integral equation
(ii(t)) = (e-'(m'
R,(x)) + 7-l
Jo'dt' (e-'(*('-''
are the Laplace represent3tions of the kinetics for 72-l = 0 and a rescaled time constant rl. (ii) In the opposite case, 7 2 < 71, we have
B ( X ) ) ( & ( t l ) > (8)
which can be solved with the help of a Laplace transformation
Electron-Transfer Dynamics For initial conditions, we assume that only the state P* is populated at t = 0 and that the motion within the corresponding potential is already equilibrated, i.e.
Then from eq 9 one can obtain the Laplace representation of the population dynamics of the two states as
where
are the Laplac: transformed kinetics for = 0 and a rescaled time constant 72. Equations 13 and 14 result from the equation of motion (eq 1) and hold for any arbitrary form of the stochastic motion. Thus in order to study the features of the electrontransfer dynamics, it is sufficient to investigate the kinetics of P1,zand C1.2. In the following we concentrateon thedepopulation of the state P*. The character of the kinetics depends on the values of the change in the free enthalpy AG and the solvent reorganization energy A,. In the normal region (AG < A&, the transfer is thermally (aK-1)2 h),the kinetic approach (eq 17) may be applied only in the oase of fast solvent motion(~+O,K1.2 = (KI,z)E").At slow motion theETdynamics shows static stages and a motion accelerated or slowed down one.
-
Since p1,2(X)
-
e-%(*, it can be easily shown that
K, = e-AG/kTK
p 1
- e-AG/kTCcp cp-
-
if K(X) and K-l(X)satisfy the condition of detailed balance. For fast motion T 0 the rate constant approaches the kinetic rate
Nonactivated Transfer at Slow Solvent Motion At AG 1 A,, ET can occur without activation of the classical subsystem Let us consider pure static ET dynamics with frozen solvent motion ( T - ~ 0). Further, we assume that the lifetimes of the two electronic states are long
(a.
(K(X) For the last approximation we assumed that K(X) varies slowly. With increasing T the rate constant decreases, and for K(X)T >> 1 the transfer becomes solvent controlled
-
+ K-,(X))q,z >> 1
(29)
In this case one easily obtains from eqs 12-1 4 and 29 the following form of the static kinetics
plst(t) =
J d~ c p l ( ~[ c ~ ( xe+(* )
+
~ m )+' e(-PtfX)l+l)']
which may be simplified by making use of the detailed balance condition to give
p i t ( t ) = J' d~ cpl(m
+
+
K~Q)t
P'(X)e(*(a'r2)r] where
After the redistribution process has relaxed, a delayed decay remains
e+(m
(30)
3428 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
For a fixed value of X the kinetics is similar to eqs 20 and 28, which represent the results of a simple kinetic scheme. For each X there are a redistribution stage and a decay due to the finite lifetimes of the two states. Let us now consider the static redistribution stage
Zharikov et al.
and hence
where with
P' = J
cp,(X) dX
(32)
In the case where the function K(X) + K-I(X) varies slowly as compared to the functions cpl(X), P ( X ) a e(X- X,), and P ( X ) a e(& - X ) , the kinetics of eq 32 can be approximated by
-
Since IUl'( kT
c a t
where X+ is the position of the maximum off'(*. equation
It obeys the
Av(x'(t)) = - arcsinh(t/2r)
7
< 7zeAU(xl)/kTfor A, < AG
where XI is determined by the condition
kT
The second term in eq 47 plays the main role in the activated regime (AG < A,, E, > kT). As AG increases, its amplitude decreases, and for AG > A,, it becomes insignificant.
Nonactivated Transfer-Motion Accelerated Stage The solvent motion changes the character of the ET process in the nonactivated regime significantly. After the static redistribution stage, the populations of the two states are not in thermal equilibrium. The motion along the terms leads to equilibration. First, it accelerates the decay of states which remain in the P state, and second, it slows down the delayed decay stage (eq 46) in the case of an unstable P state. Thecharacteristic time of the kinetics (eq 46) for x* = 0 is 71 exp(AU(O)/kT) (see eq 43). This time is shorter than the time of equilibrium delayed decay, 71 exp(AG/kT) (see q 25). Let us consider the case of long lifetimes 71,~.In the limit 71.2 we have the pure redistribution kinetics
-
The delayed kinetics of q 47 is realized if the conditions (eq 50) are not satisfied, but due to the exponential factors, 7 has to be very long. In the case of an unstable P state, the kinetics Z(t) (eq 45) has two regimes. The pure redistribution stage is realized when
If eq 51 is not satisfied, the solvent motion influences only the decay of systems in the state C:
pl(t)
pe-Ki.r
+ pte+i + p
The static decay Z(t) is realized if 7
>~
7
e-+fi'rrt
eq
St
(52)
is very long
~ e ( ~ AG ~ --A, ~> kT ) / ~ ~ (53)
or The second term in eq 49 describesthe motion accelerated stage. The rate of this stage coincides with that of the monoexponential redistribution stage (eq 21) for the solvent-controlled transfer (eq 24). But the kinetics of eq 49 may be approximated by a single exponential only in the activated regime AG < A,, U(X,) > kT, b A,, Ul(X,) > kT,the kinetics is monoexponential again but the rate constant Kin is now determined by the rate of static redistribution. At long times, P d approaches the q u i librium value Po9 with a rate constant PqT1-l or Cq~2-l for a finite lifetime of the P or C state, respectively. The kinetics eq 49 is realized when the redistribution is faster than the decay due to the instability of the electronic states. For an unstable state C this is the case if
where XI is determined by eq 50.
Quantum Yield Usually the initial state P* decays by fluorescence as well as by a nonradiative process. Its lifetime is given by T ~ - *= rn-l + fn1-1 where 7n-1 and 7,;l are the rates of the two decay channels. Electron transfer to the state C reduces the fluorescence quantum yield, which is defined as t#~
1
I K P ( t ) dt = -&O) 711
=e 711
From eqs 11, 12, and 54 one easily obtains
(54)
Zharikov et al.
3430 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 1
d = do 1+-
+
1 - T K ~ 72K2
d=dF
where do = 71/7flis the quantum yield in the absence of electron transfer and the rates K1,2are given by 16. The influence of the ET process upon the quantum yield is usually treated in the frameworkofthesimple~net~cscheme (eq 17). isalsoassumed that therateconstantsK1,2areindependentof ~~,2andcharacterize the rate of the electronic transitions. In particular, in the of fast solvent motion ( 7 0). ea 55 reproduces the result of a simple kinetic scheme (eq 17)'
-
where (K1,2)hnare the kinetic rate constants (eqs 22 and 23). For AG > kT, in view of eq 22, the back transfer can be neglected and the quantum yield becomes (574
Using eq 57b the result (eq 5 5 ) can be cast into the form of eq 57a: 1
where
The result, eq 58, is different from eq 57, which describes an irreversibletransfer. It does not describe pure irreversibletransfer as it involves 7 2 and K-l(X). For 72-1 0 and in the limit of frozen motion T a,we have 4 = 40.This shows that we have neglected only the back reaction from the bottom of the potential well of the C state. The effective rate constant K does not depend on 71.2 as long as 7 h, according to bt = 1. Kdepends significantly on 71,2 if 7 > 71.2. If 71 = 72, then the result (eq 59) is applicable with 7 replaced by V I / ( ? + 71). For 7 >> 71 this gives
(61)
With increasing AG the quantum yield decreases. This reflects the fact that only thosepartsofsystemscontributetothequantum yield which remain in the P state after the static redistribution stage* The systems which have undergone ET to the state decay via destabilization of this state. Only if 7 2 is longer than fluorescence (q53) does the characteristictime Of the thecontributionof systems in the C state become essential, leading to 4 4 0 as 7*
- -.
Discussion
The most important parameter, whichcharacterizesthe electron transfer, is the energy, A = AG - &, of the "vertical transition" measured from the bottom of the initial state potential to the charge-transfer state at a fixed configuration of the solvent. For A < 0, the electron transfer is in the normal region. For A > 0, solvent nonactivated transfer takes place. For charge-separation processes from neutral molecular complexes, A is practically independent of the solvent polarity. For the slow motion limit (Kin7> l), the model predicts a solvent-controlled rate constant in the normal region and a solvent-independentone in the "solvent nonactivated" region. In the intermediate region, where the activation energy is of the order of the kTregion, the kinetics has a nonexponential character with two characteristic times. A transition between these limits might be achieved by varying the molecular system or by applying an external electric field to the system prepared in low-polarity solution. The model is applicable also to charge-recombination processes. But for such processes the initial electronic state has a dipole moment and A becomes dependent on the solvent polarity (decreases with increasing polarity). In this case it may be possible to observe the change of the kinetics by varying the solvent polarity. Experimentally, it is found for some systems that charge-recombination kinetics reveals a two-exponential behavior.30 The question about the nature of this "nonexponentiality" requires more detailed study but might be related to our model. It should be noted that the condition of the "solvent nonactivated" region, A > 0, does not coincide with the condition of the so-called inverted region. The last is realized at A > Xi when the rate constant decreases with increasing AG. In the "solvent nonactivated" region, the kinetics is determined by electron coupling with the vibrational subsystem. As a mechanism of coupling, it can serve the direct coupling of an electron with vibrational modes, due to charge redistribution during electron transfer, and indirect coupling due to anharmonic coupling between one strong interaction with an electron vibrational mode and other vibrational modes of the molecule. The description of transfer in the "solvent nonactivated" region requires a more detailed study. In the case of electron coupling induced via a low-frequency mode, the latter plays a role similar to that of the solvent coordinate. In the limit of slow relaxation of this mode, the "solvent nonactivated" region can be subdivided into an activated region and a nonactivated region with regard to this coordinate, leading also to nonexponential kinetics. Acknowledgment. A.A.Zh. gratefully acknowledges a fellowship from Alexander von Humboldt Stiftung. References and Notes (1) Zusman, L. D.Chem. Phys. 1980, 49, 295. ( 2 ) Yakobson, B. I.; Burshtein, A. I. Chem. Phys. 1980, 49, 385.
Nonexponential Electron Transfer (3) Calef, D.F.; Wolynes, P. G. J. Phys. Chem. 1983, 87, 3387. (4) Rips, I.; Jortner, I. J . Phys. Chem. 1987,87,6513; Chem. Phys. Le??. 1987, 133, 41 1. ( 5 ) Kobayshi, T.; Takagi, Y.; Kandori, H.; Kemnitz, K.; Yoshihara, K. Chem. Phys. Le??.1991, 180,416. (6) Jortner, J.; Bixon, M.; Heitele, H.; Michel-Beyerle, M. E. Chem. Phys. Let?. 1992, 197, 131. (7) Akesson, E.; Walker, G. C.; Barbara, P. F. J . Chem. Phys. 1991,95, 6. ( 8 ) P6llinger, F.; Heitele,H.; Michel-Beyerle, M. E.; Anders,C.; Futscher, M.; Staab, H.A. Chem. Phys. Le??.1992, 198,645. (9) Tominaga, K.; Kliner, D. A. V.; Hupp, J. T.; Barbara, P. F. Uhafas? Phenomena VIIk Springer-Seriesin Chemical Physics, Vol. 55; Martin, J. L., Migus, A., Mouron, G. A., Zewail, A. H.; Springer-Verlag: Berlin/Heidelberg, 1993; p 582. (10) Levin, P. P.; Pluzhnikov, P. F.; Kuzmin, V. A. Chem. Phys. Le??. 1988, 147, 135. (11) Levin, P. P.; Pluzhnikov, P. F.; Kuzmin, V. A. Chem. Phys. 1989, 137, 331. (12) Penfield, K. W.; Miller, J. R.; Paddon-Row, M. N.; Cotsaris, E.; Oliver, A. M.; Hush,N. S . J . Am. Chem. Soc. 1987, 109, 5061. (13) Van Dayne, R. P.; Fischer, S . F. Chem. Phys. 1974, 5, 183. (14) Ulstrup, J.; Jortner, J. J . Chem. Phys. 1975, 63, 4358.
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3431 (15) Frey, W.; Klann, R.; Laermer, F.; Elsaesser, T.; Baumann, E.; Futscher, M.; Staab, H. A. Chem. Phys. Le??.1992, 190, 567. (16) Scherer, P. 0. J.; Fischer, S . F. Chem. Phys. Le??.1992, 190, 574. (17) Scherer, P. 0. J. Chem. Phys. Le??.1993,214, 149. (18) Garg, A.; Onuchic, J. N.; Ambegaokar, V. J. Chem. Phys. 1985,83, 4491. (19) Sumi, H.; Marcus, R. A. J . Chem. Phys. 1986,84,4894. (20) Jortner, J. J . Chem. Phys. 1976, 64, 4860. (21) Webman, I.; Kestner, N. R. J. Chem. Phys. 1982, 77, 2387. (22) Bogdanchikov, G. A.; Burshtein, A. I.; Zharikov, A. A. Chem. Phys. 1984,86,9. (23) Burshtein, A. I.; Frantsuzov, P. A.; Zharikov, A. A. J. Chem. Phys. 1992, 96, 4261. (24) Burshtein, A. I.; Kofman, A. G. Chem. Phys. 1979, 40, 289. (25) Feller, W. Anlntroduction roProbabilify Theory a n d h Applications; Wiley: New York, 1966; Vol. 2. (26) Montroll, E. W.; Weiss, C. A. J . Math. Phys. 1965, 6, 167. (27) Burshtein, A. I. Lectures on Quantum Kinerics (in Russian); Novosibirsk State University: Novosibirsk, 1968. (28) Rautian, S. G.; Eksp, 2. H. Theor. Fiz. 1966, 51, 1176. (29) Doktorov, A. B.; Burshtein, A. I. Sou. Phys.-JETP 1973,36,411. (30) Asahi, T.; Mataga, N. J. Phys. Chem. 1989, 93, 6575; 1991, 95, 1956.
