3720
J. Phys. Chem. B 1999, 103, 3720-3726
Theory of Excitation Transfer in the Intermediate Coupling Case T. Kakitani*,† and A. Kimura Department of Physics, Graduate School of Science, Nagoya UniVersity, Chikusa-ku, Nagoya 464-8602, Japan
H. Sumi Institute of Materials Science, UniVersity of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan ReceiVed: July 29, 1998; In Final Form: January 26, 1999
Theory of excitation transfer in the intermediate coupling case is proposed. This theory is appropriate to such molecular systems where the interaction between donor and acceptor is considerably strong so that the excitation transfer takes place before the thermal equilibrium is attained in the excited intermediate state of the donor but is not so strong as the exciton mechanism holds true. The expected remarkable properties of the intermediate coupling excitation transfer are that there exists a concurrent process of the superexchange mechanism and the sequential mechanism in the same way as the electron transfer reaction. We have proposed a new method of evaluating the degree of sequentiality which bases on much physical background, namely counting the fraction of density flow passing through the intermediate state.
1. Introduction Excitation transfer is important in the photosynthesis because it is a fundamental process for light-harvesting in antenna systems.1 Excitation transfer is also called excitation energy transfer. So, we use an abbreviation EET for the excitation energy transfer or excitation transfer in order to discriminate from the electron transfer which is abbreviated as ET. Recently, many of three-dimensional structures of antenna systems have been elucidated by electron crystallography2 and X-ray crystallography.3-6 Those structures disclosed that the antenna chromophores are located in many variety of symmetry. Donor and acceptor for EET in the neighborhood are rather in close distance and those not in the neighborhood are in a variety of distances. Kinetic data for the EETs also suggested that the couplings between donor and acceptor would be considerably strong or substantial, depending on the antenna systems7-9. The current famous theories of EET are applicable to two limiting cases: a strong coupling case where coherent exciton mechanism works and a weak coupling case where incoherent Fo¨rster mechanism works.10 So far, some attempts were made to construct a theory interpolating these two limiting cases. Among them is the theory of Kenkre and Knox using the generalized master equation11. However, it is not enough to incorporate many of the important energetic and structural information which are obtained now. On the other hand, there were considerable efforts at treating coherent and incoherent exciton dynamics systematically.12-15 Some applications to the light-harvesting antenna complexes were also made.16,17 In such histories, many attempts to interpolate directly the strong coupling EET and the weak coupling EET have been made, with some success from a theoretical point of view. However, the theory is not formulated for the experimentalist to use it easily. So, a theory of the EET in the intermediate coupling case expressed in a compact form is absolutely necessary. Here, we mean the intermediate coupling case where †
E-mail:
[email protected]. Fax: +81-052-789-3528.
the interaction between donor and acceptor is not so strong as to form an exciton state but is still considerably strong that the EET takes place before thermal equilibrium is attained in the intermediate state. In this paper, we provide a general framework of our new EET theory in the intermediate coupling case, as well as some results of numerical calculations. We first notify that the intermediate coupling EET can be treated in a similar way as the usual ET for the triad; Here, we call a donor (D) and acceptor (A) system bridged by a mediator (M) as triad. In this, the mediator can be either a compact molecule or a molecular enviroment which intervenes between donor and acceptor. At present we have a considerable accumulation of theoretical works on ET for the triad.18-32 Some of those works were applied to the photoinduced ET of the reaction center of photosysnthetic bacteria. It is a great advantage for developing an EET theory in the intermediate coupling case to make use of those well-known ET theories of triad. 2. Excitation Transfer Schemes Based on the Three-State Model Before going into the problem of EET, we briefly describe the model of ET in the triad. We schematically show potential energy surfaces of the three states in Figure 1a. The initial state, denoted as d state, corresponds to the excited state of donor D*. The intermediate state, denoted as m state, corresponds to the ET from donor to mediator D+M-. The final state, denoted as a state, corresponds to the ET from donor to acceptor D+A-. The coupling of ET between the d state and the m state is denoted by J1, and the coupling between the m state and the a state by J2. In the initial state, the vibrations are assumed to be in thermal equilibrium. In the intermediate coupling case, there are two pathways for the transition from the initial state to the final state. One is the process depicted by the solid lines, representing successive, realistic transitions from the d state to the m state and from the m state to the a state. This mechanism is called the sequential mechanism. This transition takes place
10.1021/jp9832185 CCC: $18.00 © 1999 American Chemical Society Published on Web 04/16/1999
Excitation Transfer in the Intermediate Coupling Case
Figure 1. (a) Schematic diagram of the type I electron transfer (ET) in the three-state model of the triad. The initial state, denoted as d (D*MA), comprises donor (D) in the excited state and mediator (M) and acceptor (A) in the ground state. The intermediate state, denoted as m (D+M-A), comprises D in the cation form, M in the anion form, and A in the neutral form. The final state, denoted as a (D+MA-), comprises A in the cation form, M in the neutral form, and A in the anion form. J1 is the coupling for the ET between D and M. J2 is the coupling for the ET between M and A. The arrows on the solid line denote the sequential mechanism of the ET. The arrow on the broken line denotes the superexchange mechanism of the ET. (b) Schematic diagram of the type I excitation transfer (EET) in three-state model. The initial state, denoted as d (D*MA), comprises D in the excited state and M and A in the ground state. The intermediate state, denoted as m (DM*A), comprises M in the excited state and D and A in the ground state. The final state, denoted as, a (DMA*), comprises D and M in the ground state and A in the excited state. U1 is the coupling for the EET between D and M. U2 is the coupling for the EET between D and A. A sequence of arrows on solid lines denotes the sequential processes. Several arrows indicate that EET is possible on the way of vibrational relaxation. The dotted curve denotes the superexchange process in which the EET takes place directly from the d state to the a state.
only through the crossing points of the potential energy surfaces between the d state and the m state and between the m state and the a state, respectively. It is possible that when the coupling J2 is considerably large, the transition from the m state to the a state can take place from various vibrational states before relaxation as depicted by the many solid lines. We call this process “hot sequential” in analogy with hot luminescence in optical processes. The other is the process depicted by the broken line. That is, the ET takes place from the d state to the a state by virtually passing through the m state. This process is called the superexchange mechanism. The importance of the superexchange process relative to the sequential process increases when the potential energy minimum of the m state is considerably higher than that of the d state20 and/or when the couplings J1 and J2 are so strong that the vibrational relaxation at the m state does not disturb the superexchange process substantially.30 Now, let us consider the problem of EET, in analogy with the ET of the triad. In general, we encounter two cases of the EET. One is the case that thermal equilibrium is attained for vibrations of the excited state of donor after photoabsorption. The other is the case that thermal equilibrium is not attained for vibrations of the excited state of donor after photoabsorption. Let us first consider the former case. As shown in Figure 1b, we find that an intimate parallelism holds true between ET and EET; namely, we take the excited state of donor D* as the initial state (d state), the excited state of mediator M* as the intermediate state (m state), and the excited state of acceptor
J. Phys. Chem. B, Vol. 103, No. 18, 1999 3721
Figure 2. (a) Schematic diagram of the photoinduced ET (type II ET) in the three-state model. The initial state, denoted as d (DA + hν), comprises D and A in the ground state plus a photon to be absorbed. The intermediate state, denoted as m (D*A), comprises the mediator in the excited state M* and A in the ground state. The final state, denoted as a (D+A-), comprises the donor in the cation form D+ and acceptor in the anion form A-. K is the coupling between donor and electromagnetic field. J is the coupling for the ET between D* and A. The others are the same as Figure 1a. (b) Schematic diagram of the photoinduced excitation transfer (type II EET) in the three-state model. The initial and intermediate states are the same as in part a. The final state, denoted as, a (DA*), comprises the donor in the ground state D and acceptor in the excited state A*. U is the coupling for the EET between D and A. The others are the same as in part a.
A* as the final state (a state). The coupling parameters U1 and U2 represent interactions for the EET between donor and mediator, and between mediator and acceptor, respectively. In the intermediate coupling case, two kinds of pathways are possible for the EET from d state to a state. One is the pathway due to the sequential processes as shown by solid lines. The other is the nonsequential pathway as shown by the broken line. We call the latter superexchange process for EET in analogy with that for ET. The importance of the superexchange process must increase according as the couplings U1 and U2 become considerably large and/or the energy level of the m state is considerably higher than the d state, in analogy with the ET reaction.30,31 The arrows of the solid lines represent the actual transition between the two successive states. When the coupling U2 is considerably large, the transition from the m state to the a state can take place from various vibrational states before relaxation, as depicted by many solid lines. We call them “hot sequential”, as in the case of ET. Only the solid line through the vibrationally ralaxed state corresponds to the EET by the Fo¨rster mechanism10 (Figure 1b). We call the EET and ET in Figure 1 the type I EET and type I ET, respectively, hereafter. Let us next consider the latter case of EET where thermal equilibrium is not attained for vibrations of the excited state of donor. For this case, we choose the ground state of the donor plus a photon before it is absorbed D + hν, where the thermal equilibrium is attained, as the initial state (d state). We take the excited state of donor D* as the intermediate state (m state) and the excited state of acceptor A* as the final state (a state), as shown in Figure 2b. K is the coupling between photon field and donor. This idea of the three-state model is the same as the three-state model of Cho and Sillbey27 for the photoinduced ET, although the superexchange mechanism was not taken into account by them. The photoinduced ET corresponding to the EET of Figure 2b is shown in Figure 2a. A remarkable thing in
3722 J. Phys. Chem. B, Vol. 103, No. 18, 1999
Kakitani et al.
the scheme of Figure 2 is that a direct excitation of acceptor (Figure 2b) or a direct charge separation (Figure 2a) by absorption of a photon is possible by means of the superexchange mechanism (broken line). This terminology of the superexchange mechanism is a simple extension of the terminology used in Figure 1. We adopt it also for this case because we notice that a nice parallelism exists between the two kinds of three-state model, namely between Figure 1 and Figure 2. The arrows of solid lines represent the EET or ET from the various vibrational states in the excited state of donor after photoabsorption. We call the EET and ET in Figure 2 the type II EET and type II ET, respectively, hereafter. The coupling parameter U in Figure 2b represents the interaction between donor and acceptor for the EET, and the coupling parameter K represents the interaction between the photon field and donor, which are expressed as
U)
∫∫φ′a*(1)φ/b(2)Vφa(1)φ′b(2) dτ1dτ2
(1)
nk K ) i 2πpωk V
(2)
[
]
1/2
(mmd‚ekλ)
where φ′a (φ′b) and are the wave functions in the ground and excited states of donor (acceptor), respectively, and V is the Coulomb interaction between donor and acceptor. The quantities nk and ωk are the photon number in surrounding media and the angular frequency of photon with wavenumber k, respectively. mmd, ekλ, and V are the transition dipole moment of a donor, vector of the electromagnetic wave with wavenumber k and polarization λ, and volume, respectively. From eq 2, we find that |K|2 is proportional to the light intensity and square of the transition dipole moment of donor.
In this section, we give a concrete formula for the EET in the intermediate coupling case, by making use of the triad ET theory. There have been developed some triad ET theories up to the present time, based on different pictures and theoretical techniques.18-33 Among them, we make use of the triad ET theory which was recently developed by Sumi and Kakitani30 and Sumi.31 In this theory, the renormalization effect due to the ET from the m state to the a and d states which causes the decay factor of the m state is properly taken into account. Originally, this decay factor was introduced phenomenologically30 and later it was derived rigidly by the nonperturbative calculation.33 Owing to this renormalization effect, current formulas of the ET by the sequential mechanism through the vibrationally relaxed m state and the ET by the superexchange mechanism were correctly reproduced in two limiting cases of the vibrational relaxation time.30 The most important advantage of this theory is that compact, analytical formulas of the overall ET rate is obtained.30 The total Hamiltonian for EET is given by
(3)
H0 ) Hd|d〉〈d| + Hm|m〉〈m| + Ha|a〉〈a| V ) Jmd(|m〉〈d| + |d〉〈m|) + Jam(|a〉〈m| + |m〉〈a|)
|
kad ) 2π
JmdJam p
|∫ 2
∞
0
∫0τCm(τ′) dτ′] dτ
f(τ) exp[-
(5)
The function f(τ) in the integrand in eq 5 is given by
Sna R [erf(x + f(τ) ) [(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4]1/2n)0n! ∞
1
{
iy)] exp -Sa - y with
x)
2
1
∑
2 [E(n) am(τ) + Emd]
2 D 2 + D 2 + 2D 2 - 2D (τ)2 a d m m
[
]
4 2 2 2 2 τ (Da + Dm )(Dm + Dd ) - Dm(τ) 2 p Da2 + Dd2 + 2Dm2 - 2Dm(τ)2
}
(6)
1/2
(7)
y) 2 2 2 |[Dd2 + Dm2 - Dm(τ)2]E(n) am(τ) - [Da + Dm - Dm(τ) ]Emd|
{2[(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4][Da2 + Dd2 + 2Dm2 - 2Dm(τ)2]}1/2
(8)
3. Theory of Excitation Transfer by the Three-State Model
H ) H0 + V
intramolecular vibrations and phonons of the surroundings. Jmd and Jam are the couplings between d and m states, and between m and a states, respectively, which are U1 (J1) and U2 (J2) for the type I EET (ET), and K and U (J) for the type II EET (ET), respectively. In accordance with Sumi,31 we treat the molecular vibrations of the environment surrounding donor and acceptor semiclassically. We treat the intramolecular reorganization due to the transition from the m state to the a state quatum mechanically. Then, in analogy with the triad ET theory,30 the formula for the overall rate kad of the transition from the d state to the a state in the EET is written as follows:
(4)
where Hd, Hm, and Ha are Hamiltonians of vibrations of the d, m, and a states, respectively. The vibrarions are composed of
where erf(x + iy) represents the error function for a complex argument x + iy, R represents the real part, and Sa is the dimensionless vibrational coupling constant (called the HuangRhys factor) of the intramolecular vibrations due to the transition from the m state to the a state. Energies Emd and E(n) am(τ) are defined by
Emd ) ∆Gm + λm + λd
(9)
E(n) am(τ) ) ∆Ga + λa + λm - 2λm(τ) +npωa
(10)
where λm(τ) represents the energy of vibrational reorganization at the m state at time τ with λ(0) ) 0 and λ(∞) ) λm. An explicit expression of λm(τ) might be given as
λm(τ) ) λm exp(-τ2/τm2)
(11)
in consideration of dλm(τ)/dτ ) 0 at τ ) 0, where τm is the vibrational relaxational time at the m state. In our theory, the role of τm is incorporated into the EET (ET) reaction only through eq 11. Although this is an approximation, it is still useful expression for the purpose of semiquantitatively treating the dynamical behavior of vibrational relaxation. λd and λa are the total vibrational reorganization energies of the d state and a state, respectively. The variance of fluctuations in the semiclassically vibrational reorganization energy has been represented by Dd, Dm, and Da, respectively, at the d, m, and a states in thermal equilibrium, and Dm(τ) represents its value at time τ with Dm(0) ) 0 and
Excitation Transfer in the Intermediate Coupling Case
J. Phys. Chem. B, Vol. 103, No. 18, 1999 3723
Dm(∞) ) Dm. An explicit expression of Dm(τ) is given as
Dm(τ)2 ) 2λm(τ)kBT′
(12)
j (1/2pω j /kBT) kBT′ ) 1/2pω
(13)
with
j , where T′ in parallel with Dm(τ)2 ) 2λm(τ)kBT for kBT . pω represents an effective temperature for vibrational fluctuations with an average angular frequency ω j at temperature T. ∆Gm(∆Ga) is the energy gap between the m state (a state) and the d state, respectively (∆Gm is negative when the energy of the m state is lower than that of the a state). Here, it should be mentioned that the term [Da2 + Dd2 + 2Dm2 - 2Dm(τ)2] in the denominator of eq 8 was written erroneously as [Da2 + Dd2 + 2Dm2 - Dm(τ)2] in ref 31). The exponent Cm(τ) in eq 5 represents the decay rate of the excitation at the m state due to the transition from the m state to the a state and the d state at time τ expressed as
Cm(τ) ) kdm(τ) + kam(τ)
(14)
where kdm(τ) and kam(τ) are the rates for the transition from the m state to the a and d states, respectively, at time τ as expressed by
[
] [
|Jmd|2 2π kdm(τ) ) p D 2+D 2 d m kam(τ) )
[ ]
|Jam|2
2π
p
Dd2 + Dm2
1/2
∞
∑
Sna
n)0n!
1/2
2 1 Edm(τ) exp 2D 2+D
[
exp - Sa -
d
]
(15)
2 m
(n) 2 1 Eam(τ)
]
2D 2+D 2 a m
Figure 3. Two propagators of an excitation in the state d, shown by thin solid lines, evolve from thermal equilibrium therein at time zero until time t1 or t′1, from which two propagators in the state m, shown by thick solid lines, evolve until time t. It should be noted that Hamiltonians of the propagators represented by thin and thick lines are Hd and H, respectively.
luminescence in second-order optical processes. It basically follows the idea of Sumi and Kakitani and we explicitly count the opportunity of passing through the relaxed m state. Now, let us go to formulate two kinds of sequentiality. We first calculate the density of m state Nm(t) at time t, which is defined as
Nm(t) ) 〈m|F(t)|m〉
where F(t) is the density operator. Applying the nonperturbative method33 for the calculation of the exact propagator at the m state 〈m|e-iH(t-t1)p|m〉 and 〈m|e-iH(t-t′1)p|m〉 in Figure 3, and combining together the case t1 > t′1 and the case t1 < t′1, we obtain
| |∫
Jdm 2 t t1 dt1 0 dµcos[(∆Gm + λm + λd)µ/p] × 0 p t-µ t 1 µ2 exp - (Dm2 + Dd2) 2 - 0 Cm(τ) dτ - 0 Cm(τ) dτ 2 p (18)
Nm(t) ) 2 (16)
4. Degree of Sequentiality In order to characterize the intermediate coupling EET distinctly, we introduce an index, degree of sequentiality, defined as follows. Here, the degree of sequentiality, or more simply sequentiality, is a fraction of the processes of EET by the sequential mechanism in the total processes. It is well-known in the ET reaction that a perfect distinction of the sequential mechanism from the superexchange mechanism in the whole reaction is impossible because both processes are quantummechanically mixed in principle.30 However, to make separation of the sequential processes from the superexchange process even approximately is useful to elucidate the mechanism of ET and/ or EET reactions which proceed ultrahigh speed. In the definition of the degree of sequentiality, there is some variance at present. Bixon and Jortner considered that all the processes passing through the actual intermediate state devote to the sequential mechanism.32 According to their idea, the hot sequential processes as well as the process passing through the vibrationally relaxed m state constitute the sequential processes. In contrast to this, Sumi and Kakitani considered that the hot process cannot be separated from the superexchange process in a rigid sense and so only the process passing through the vibrationally ralaxed m state can be safely defined as the “ordinary sequential” process.30 In the present paper, we try to compromise between these two ideas, by defining two kinds of sequentiality. One is the total sequentiality PSQ which basically follows the idea of Bixon and Jortner and we explicitly count the opportunity of passing through the actual m state. The other is the ordinary sequentiality POS in EET, in parallel to ordinary
(17)
[
∫
∫
]
∫
where
µ ) t1 -t′1
(t1 g t′1)
(19)
The details of the derivation of eq 18 are given elsewhere.33 The time difference t - t1 represents the elapsed time after arriving at the real m state (region of the two bold lines in Figure 3 appearing at the same time). During this time, the vibrational relaxation proceeds in the m state. Therefore, the transition rate from the m state to the a state at time t is represented by kam(t-t1). Therefore the density flow rate of the m state to the a state is obtained by multiplying kam(t-t1) to the integrand of t1 of eq 18. Then, the integration of the density flow rate gives the fraction of the reaction passing through the m state, which we call total sequentiality PSQ, as follows
| |∫
PSQ ) 2
Jdm p
2
∞
0
[
∫0t
dt1 kam(t-t1)
1
dµ cos[(∆Gm + λm +
1 µ2 λd)µ/p] exp - (Dm2 + Dd2) 2 2 p
∫0t-µCm(τ) dτ -
]
∫0tCm(τ) dτ
(20)
In principle, Nm(t) and PSQ should not exceed 1. However, we cannot guarantee it due to the approximation (momentarity ansaz) in deriving Nm(t) of eq 18. Even if there is such a flaw, PSQ defined by eq 20 is a nice index to measure the sequentiality in a much wider region of parameters. In the next paper, we
3724 J. Phys. Chem. B, Vol. 103, No. 18, 1999
Kakitani et al.
Figure 4. Total sequentiality PSQ drawn in the two-dimensional coordinates, ∆Gm and U1 ()U2). τm ) 2 × 10-13 s.
shall give the result of comparison with the index obtained by the other method. The second sequentiality, ordinary sequentiality POS, is obtained as follows. Among the density flow from the m state to the a state, we keep only the density flow visiting the vibrationally relaxed m state. We can select those processes by defining the restricted transition rate kham(t-t1) as follows
k˜am (t-t1) ) kam(t-t1) )0
for t - t1 g τm
for t - t1 < τm
(21)
Then, we obtain the ordinary sequentiality POS as follows:
| |∫
POS ) 2
Jdm p
2
∞
0
dt
∫0t dt1k˜am(t-t1)∫0t dµ cos[(∆Gm + 1
[
1 µ2 λm + λd)µ/p] exp - (Dm2 + Dd2) 2 2 p
]
∫0t-µCm(τ) dτ - ∫0tCm(τ) dτ
-
(22)
The above definition of POS is based on the same concept as the definition with PSQ. Therefore, we use POS in the following, instead of the sequentiality defined previously by Sumi and Kakitani30 which bases on more qualitative method.
to increase as ∆Gm increases for ∆Gm > 0. This is an indication of the shortcoming of our expression of the superexchange process in a rather strong coupling region, which we discuss later. In Figure 5, we show the sequentialities PSQ and POS as a function of τm, for ∆Gm ) 0 and U1 ) U2 ) 20 cm-1. We see that PSQ is nearly constant (about 0.25) for τm > 10-15 s and increases rapidly for τm < 10-15 s. We also see that POS is almost the same as PSQ for τm < 10-14 s and it decreases gradually to nearly zero as τm increases from 10-14 to 10-12 s. This τm dependence of POS just coincides with what was expected by Sumi and Kakitani.30 In Figure 6, we show the overall rate kad as a function of the coupling strength U1(U2) for ∆Gm ) 0 and τm ) 2 × 10-13 s. The value of U2 is taken as the same as U1. As U1(U2) increases, kad increases very rapidly (almost proportional to |U1|4). However, after about 20 cm-1, the rate of increase becomes gradually dull and it approaches a constant value after about 200 cm-1. This tendency can be explained as follows. In Figure 4, we see that PSQ is less than 0.3 in all the regions of U1(U2). Therefore, the superexchange mechansim dominantly contributes to kad. In the pure superexchange mechanism, its rate k(SX) ad is expressed as30
k(SX) ad )
5. Numerical Calculations For the purpose of exemplifying the property of the intermediate coupling EET, we give some numerical results corresponding to the type I EET in Figure 1b. Therefore, Jmd is U1 and Jam is U2. The parameters are chosen as follows:
λd ) λm ) λa ) 40 cm-1 Sa ) 0,
Figure 5. Sequentialities PSQ and POS as a function of log τm, ∆Gm ) 0 and U1 ) U2 ) 20 cm-1.
m
a
- Em + iΓm
|
2
δ(Ea - Ed)
(23)
In Figure 4, we show the total sequentiality PSQ drawn in the two-dimensional space of ∆Gm and U1(U2). For simplicity, the value of U2 is taken the same as U1. τm is 2 × 10-13 s. When U1(U2) is small, PSQ increases continuously as ∆Gm decreases. This is consistent with the idea of Bixon and Jortner.32 As U1(U2) increases, the contour line goes down to the small value of ∆Gm; namely, PSQ decreases for a given value of ∆Gm as U1(U2) increases. This fact indicates that the superexchange mechanism is more important when the coupling is increased. When U1(U2) is very large, the property changes; PSQ becomes
〉
(24)
Ed
where we put U1 ) U2 ) U, and Ed, Em, and Ea are the vibrational energies of the d, m, and a states, respectively. Γm is the decay constant of the m state. In the present notation, Γm may be expressed as
Γm ) Cm(0) ) kdm(0) + kam(0)
∆Ga ) 0
T ) 300 K
〈
〈Ea|Em〉〈Em|Ed〉
∑ |∑ E E
2π 4 |U| p Ea
(25)
where kdm(0) and kam(0) are proportional to |U|2. Therefore, k(SX) is proportional to |U|4 for small |U| and converges to a ad constant value for large |U|. The curve of Figure 6 shows just this tendency. The long-persisting constant rate at the large value of U1(U2) is not realistic and it indicates that the character of the superexchange mechansim becomes wrong for describing the EET at the large value of U1(U2) (probably U1 > 250 cm-1). It also indicates that the character of the exciton mechanism must be incorporated in this large U region. 6. Discussion Now, we shortly discuss the property of the type II EET. We can also calculate the overall rate in this case for the
Excitation Transfer in the Intermediate Coupling Case
Figure 6. Overall rate kad as a function of U1 ()U2). ∆Gm ) 0 and τm ) 2 × 10-13 s.
transition from the d state to the a state using eq 5. Here, the excitation light is a continuous wave. The excitation continues until all the donor molecules are excited. The overall rate is proportional to the light intensity and does not correspond to the ordinary EET rate such as in the Fo¨rster mechanism. We cannot separate the excitation process from the photon absorption process in the intermediate coupling case for the continuous light irradiation. Here, the intermediate coupling EET corresponds to the rather large value of U, and the coupling K between the photon field and the electric dipole need not be large. The superexchange mechanism in the type II EET indicates direct photoabsorption by acceptor. This phenomenon is similar to the absorption of a photon by the exciton where excited states of donor and acceptor are coherently coupled. A property of exciton is already partially reserved in the superexchange mechanism in EET. In the above, we considered the type II EET when the continuous light is used for irradiation. However, there is another mode of light irradiation. Laser pulse irradiation experiment is very often made. In this case, the time dependence of incoming light intensity is not constant but is Gaussian in shape or a rectangle, and so on. We can treat this case in our theory by inserting envelope functions F(t1)F(t′1) corresponding to the time course of the envelop of the incoming light intensity into the integrand by dt1 and dt′1 in the original form of eq 18. For the case of the delta function-type excitation, the excitation of donor molecule to the Franck-Condon state of the intermediate state takes place instantaneously. In this case, the time course of Nm(t) is obtained by putting µ ) 0 in eq 18. We find that this form of Nm(t) just coincides with PD(t) given by Cho and Silbey.27 (However, it should be noted that Cho and Silbey obtained PD(t) by exponentiating the expression obtained by the second-order perturbation method but we obtained Nm(t) directly by the nonperturbative method). Therefore, in such case as the instantaneous excitation by delta function, the time course of Nm(t) represents an EET phenomenon by the sequential mechanism and the superexchange mechanism does not work. That is, we can decouple the EET process from the light absorption process. When the width of the excitation light pluse increases, the coupling between the excitation process and the EET process increases and the superexchange mechanism emerges in the intermediate coupling case. When the superexchange mechanism coexists with the sequential mechanism of the EET, we might detect experimentally its effect in the transient optical properties of mediator and acceptor; namely, the decrease of the population in the m
J. Phys. Chem. B, Vol. 103, No. 18, 1999 3725 state should take place through the sequential process but the increase of the population in the a state should take place through both of the sequential and superexchange processes. Therefore, the decay time of the fluorescence from the mediator must be larger than the rise time of the fluorescence from the acceptor. The antenna systems will provide a nice example to confirm this remarkable property of the intermediate coupling case. Also in aggregates of chromophores in which the interactions among the chromophores are substantial but not so strong, we expect that a remarkable property of the intermediate coupling EET emerges; namely, it is possible in the intermediate coupling case that only a few chromophores are coherently coupled to one another in the excited state and the coupled coherentincoherent migration inside the aggregates may take place. This type of the EET is already expected by the experiment for the light-harvesting complexes LH1 and LH2 of Rhodobacter sphaeroides in which the coherent length in the ring arrangement of the chromophore were evaluated as a distance of a few chromophores.35-37 Therefore, the intermediate coupling EET mechanism might play a significant role in the light harvesting in antenna systems. The detailed analysis based on the molecular structure will be made hereafter. We have decoupled the EET into two mechanisms, superexchange mechanism and sequential mechanism. However, this decoupling is only approximately possible. In deriving the exponential decay factor of the m state in eq 18 and eq 20 as well as in eq 5, we adopted the “momentarity ansaz”. Usually, the momentarity ansaz is appropriate for the weak coupling case. In Figure 4, we observe a region where PSQ is larger than 1 at the negatively very large value of ∆Gm. Although its region is rather limited, it represents a shortcoming of our theory. We need to diminish its region as much as possible. By improving the nonperturbative method, it might be possible to escape from the momentarity ansaz. Its study is on progress. As the electronic coupling between donor and acceptor becomes much larger, the excited states of donor and acceptor become to mix coherently and display the character of an exciton. For the exciton state, we can no more define the transition rate in an ordinary sense, because only the phase of the electronic state is oscillating between donor and acceptor. Based on these facts, it will be reasonable to reclassify the EET into three cases depending on the coupling strength U: strong coupling case where the exciton picture applies as before; intermediate coupling case where the overall rate can be defined but the separation between the super-exchange and sequential mechanisms is impossible in a rigid sense but is approximately possible, and the weak coupling case where predominantly the sequential mechanism works and the EET rate is well-defined. We insist in this paper that the new classification including the intermediate coupling case in the EET is absolutely necessary in order to adequately describe the whole range of EET. The criterion between the intermediate coupling case and the strong coupling case might be that the overall rate can be well-defined or not, depending on the coupling’s strength and the reorganization energy. The criterion between the intermediate coupling case and the weak coupling case might be whether the hot sequential exists or not, depending on the energy gap ∆Gm and the vibrational relaxation time τm. In the present paper, we showed that the EET process can be theoretically described in parallel with the ET process. However, we should point out one thing. The coupling for the ET is due to the exchange interaction and it is short range. The coupling for the EET is due to the Coulomb mechanism or due to the
3726 J. Phys. Chem. B, Vol. 103, No. 18, 1999 exchange mechanism (Dexter mechanism) depending on the spin state. The former is long range and the latter is short range. In the long range case, the situation is a little more complex because the direct transition from the d state to the a state for the type I EET is possible which we neglected in the above theory. This problem of EET is treated in future work. 7. Concluding Remarks In this paper, we provided a theory of EET in the intermediate coupling case, based on the analogy with the current ET theory for triad. Using the triad ET theory of Sumi and Kakitani30 and Sumi,31 we expressed the overall rate for the intermediate coupling EET in compact, analytical formulas. The intermediate coupling EET in the mode of continuous light irradiation includes concurrent processes of superexchange and sequential mechanisms, which cannot be separated from each other in a rigid sense. However, we separated them approximately. We presented a new method of evaluating the degree of sequentiality which bases on much physical background, namely counting the fraction of density flow passing through the m state. The theory of the intermediate coupling EET may work well also in aggregates of chromophores where the interactions among the chromophores are relatively strong and coupled coherentincoherent excitation migration may take place. On the basis of these facts, we should emphasize that the new classification of the intermediate coupling EET, in addition to the strong and weak coupling cases, is reasonable in order to grasp the physical picture of the whole range of coupling strength for EET. It should be also emphasized that our EET theory presented in eqs 5-20 covers the intermediate coupling case and the weak coupling case together. In this paper, we showed only a general framework of the intermediate coupling EET theory as well as some numerical results for the type I EET. The result of more detailed investigation of the properties as well as the criteria for realizing the intermediate coupling EET will be published elsewhere. Acknowledgment. The authors express their sincere thanks to Dr. T. Yamato in Nagoya University for a valuable discussion. This study was partly supported by Grants-in-Aid for Scientific Research on Priority Area of “Electrochemistry of Ordered Interfaces” from Ministry of Education, Science, and Culture. Japan. References and Notes (1) 1 Knox, R. S. In Bioenergetics of Photosynthesis; Govindjee, Ed.; Academic Press: New York, 1975; Chapter 4.
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