J. Phys. Chem. B 1998, 102, 1641-1648
1641
Detailed Balance in Fo1 rster-Dexter Excitation Transfer and Its Application to Photosynthesis Philip D. Laible,*,† Robert S. Knox,‡ and Thomas G. Owens† Section of Plant Biology, Cornell UniVersity, Ithaca, New York 14853-5908, and Department of Physics and Astronomy and Rochester Theory Center for Optical Science and Engineering, UniVersity of Rochester, Rochester, New York 14627-0171 ReceiVed: September 15, 1997; In Final Form: December 16, 1997
Under the conditions assumed both by Fo¨rster in his resonance-transfer theory and by Kennard and Stepanov in their theory of a universal relation between fluorescence and absorption, the rates of transfer between two chromophores k and l are related by Flfk/Fkfl ) exp[-(∆Gk - ∆Gl)/kBT)], where the energies ∆Gj are the free energies of excitation relating to the vibrational and other bath states interacting with the chromophore. They are dominated by, but not identical with, the 0-0 transition energies of the two chromophores. We show that ∆Gk - ∆Gl can be computed reliably with Fo¨rster’s overlap integral approximation only when the Kennard-Stepanov relation is obeyed for both chromophores. We examine the extent to which approximations to Fo¨rster-Dexter transfer agree with this detailed balance result and the implications of the result for multichromophore systems.
1. Introduction Fo¨rster’s excitation-transfer theory1,2 was applied immediately to the problem of photosynthesis3 and is well established in its application to a huge variety of systems. However, it still attracts much attention at the fundamental level. One of the main questions about the theory deals with its limits of validity in the interpretation of ultrafast rate processes: when the rates are large, is Fo¨rster’s approach relevant or must one use an exciton representation? It has long been known that excitons are the preferred representation when the intermolecular coupling strength exceeds the optical widths of the transition involved.4 Despite this, ultrashort optical pulses can localize excitation even under strong-coupling conditions,5 and the decay of coherence between the exciton states, which results in delocalization, may be related to the rate of Fo¨rster transfer.6 Another broad question, which will be the primary concern of this article, has been raised by recent studies on the photosynthetic antenna. The question is whether the theory predicts consistent uphill/downhill rates of pairwise transfer between pairs of inequivalent chromophores. Fo¨rster’s original theory was applied to fluorescence depolarization in a homogeneous solution and was accurate at low concentrations, where it could be assumed that two identical chromophores alone were interchanging excitation. While Fo¨rster later extended his theory to the case of nonidentical chromophores,7 no explicit consideration of the uphill/downhill ratio of rates appears to have been published. Because of its occurrence in a variety of excitation energy transfer situations, the uphill/downhill rate question deserves careful attention. Before beginning any formulation, however, it is important to distinguish three levels of approximation within what is generally called “Fo¨rster theory”. All of them refer to * Corresponding author. Present address: Chemistry Division, Argonne National Laboratory, Argonne, IL 60439-4831. † Cornell University. ‡ University of Rochester.
a situation in which an excitation is initially localized at a site with a certain probability, and this probability flows to a new distribution over all sites during the lifetime of the excitation. At the first level, one admits no approximation, and the probability flow just described will be called a Fo¨rster process. At the second level, a specific method of calculating the rate of flow of the probability away from the initial site is adopted, using the principles of first-order time-dependent perturbation theory (“golden rule”). This will be referred to as Fo¨rster theory. An important aspect of Fo¨rster theory is an assumption of intramolecular excited-state thermal equilibrium, which will be discussed in some detail in this article. Finally, at the third level is the well-known Fo¨rster dipole-dipole approximation, which results from a particular choice of interaction matrix element within Fo¨rster theory. Through a well-known equation, it expresses a rate of flow in terms of a spectral overlap integral and an R-6 dependence on the molecular separation R. We introduce the concept “Fo¨rster process” in order to distinguish it from the process occurring if the initially excited state is not localized and is more nearly an eigenstate of the entire electronic system. Such a state, an exciton state, is wellknown in systems with narrow spectral lines and strong interactions, especially at low temperatures. It is intimately related to the original Frenkel8 exciton, whose properties in organic crystals were studied extensively by Davydov.9 When excitons are valid initial states, it is necessary to consider their relaxation behavior instead of the Fo¨rster process, and one may call this a “Davydov process”.10 Recent experiments on excitation transfer in photosynthetic systems have shown the need to consider excitons in aggregates of large molecules that have rather broad spectra.11 Because in many cases it is not possible to determine the nature of the initial excitation from steady-state spectra themselves, the consequences of both Fo¨rster and Davydov processes must be considered and compared. A choice between these processes is not at issue. They coexist, and both are present and accounted for in a density-matrix formulation of the transfer problem.6,12 In any given experiment,
S1089-5647(97)03010-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/07/1998
1642 J. Phys. Chem. B, Vol. 102, No. 9, 1998 one of the two processes may be the dominant relaxation mechanism. The present work is concerned only with Fo¨rster processes. Fo¨rster theory predicts an approximate rate for a Fo¨rster process, and an interaction matrix element must be chosen to quantify this prediction. The choice depends heavily on the nature of the excited states involved, and when the transitions are optically forbidden, Dexter’s extension of the theory13 can be used. The square of the interaction matrix element may have an R-6 dependence or some other dependence such as exponential in the case of Dexter’s exchange theory. It may require numerical evaluation, as in the case of “monopole interactions” for chromophore pairs whose spatial separation is of the same order of magnitude as their dimensions.14,15 The principal result of the present article will be seen to be independent of these details and will rely only on the principal assumptions of Fo¨rster theory (golden rule, excited-state thermal equilibration). Should these assumptions not be valid, as in the case of “hot” transfer (to be discussed below), Forster theory is not necessarily useful for estimating the rate of a Forster process. However, this does not mean that a Forster process, as defined here (the spatial relaxation of excess probability density), cannot exist. When the Fo¨rster dipole-dipole (d-d) approximation is appropriate, the nature of the matrix element discussed above leads to an elegant separation of the rate into factors related to the emission and absorption of the donor and acceptor chromophores, respectively. At this level of approximation, a direct connection may be made with the Kennard-Stepanov (KS) relation,16,17 which also deals with absorption and emission by thermally equilibrated molecules in their ground and excited states, respectively. It has been suggested18 that the KS theory might be used to supply emission spectra for use in Fo¨rster d-d computations when a donor emission spectrum is not known. The question of whether the KS relation holds for an inhomogeneous aggregate of chromophores when these chromophores are exchanging energy19-24 has usually been treated within the context of the Fo¨rster dipole-dipole approximation. There is a more restrictive question, which may be cast into the following form: is there a consistent definition of a characteristic energy for each excited chromophore such that a Boltzmann factor for the ratio of uphill to downhill pairwise rates exists? Our answer is yes, provided that the thermal equilibration assumption is good, and we show furthermore that this is true for Fo¨rster theory in general, not merely within the Fo¨rster dipole-dipole approximation. Whether the KS relation holds for the entire complex is a more difficult problem, to which we will return in section 4. Section 2 reviews the original Fo¨rster theory in the d-d approximation and brings it into a form that enables direct comparison with the KS relation. The KS relation is derived in the Fo¨rster context. Section 3 first demonstrates our detailed balance result in the d-d approximation and then extends the argument to the generalized Fo¨rster theory. Section 4 discusses the relevance of our result to calculations of transfer rates and to the theory of diffusing excitations in multichromophore systems. 2. Fundamentals of Fo1 rster and Kennard-Stepanov Theory For two reasons we use Fo¨rster’s original theoretical structure and notation,2 including energy normalization of certain final states. First, the derivation is rendered model-independent, which serves to focus attention on the basic principles. Second,
Laible et al.
Figure 1. Energy levels involved in the two-chromophore transfer process and in the luminescence and absorption process. For the transfer process, W is the energy of a virtual near-zone photon exchanged between the chromophores. For the optical processes, the left diagram represents a transition in which a photon of energy W is emitted, and the right diagram, one in which a photon of energy W is absorbed.
the theory in this form is most easily combined with the Stepanov17 treatment of the relation between absorption and emission. We begin with Fo¨rster’s expression for the rate of transfer of excitation from an originally excited chromophore k to an originally unexcited chromophore l at a distance Rkl. We rely heavily on his formal paper, of which an English translation is available:2
∫
∫
∫
∞ ∞ 2π ∞ dW w )0dwl w ′)0dwk′ × l k p W)0 gk′(wk′) exp(-wk′/kBT) gl(wl) exp(-wl /kBT) |ulk|2 (1) Zk′ Zl
Fkfl )
[
][
]
Energies w are those of the vibrational and bath states. Primes indicate quantities associated with the excited electronic state (Figure 1). W is the energy transferred between the chromophores, and the integral over W sums the individual probabilities of occurrence of that transfer. Because of Fo¨rster’s energy normalization of final-state wave functions, the square of the interaction matrix element, |ulk|2, is dimensionless. Our g’s, in contrast with the originals (see below), have dimensions 1/(energy). The principal differences one sees on comparing eq 1 with Fo¨rster’s eq 9 are the terms in square brackets, which replace Fo¨rster’s g’s. His symbol g referred to the normalized probability that the chromophore, in thermal equilibrium, is in a group of states at the energy concerned. In eq 1, we have made the Boltzmann factor and the normalization explicit so that our g’s are simply the densities of vibrational and bath states. This adjustment is essential for our development of the theory. The partition functions that normalize the distributions are
Zk′ )
∫w∞′)0dwk′ gk′(wk′) exp(-wk′/kBT)
(2)
∫w∞)0dwl gl(wl) exp(-wl/kBT)
(3)
k
and
Zl )
l
In the Fo¨rster d-d approximation, the square of the interaction matrix element in eq 1 is given by
|ulk|2angular average )
2 Mk(wk, wk′)2 Ml(wl, wl′)2 6 3n Rkl 4
(4)
In the original theory,2 both chromophores have the same 0-0 energy, Wk ) Wl ) W0. When the original treatment is generalized to the case of different electronic transition energies on the two chromophores (as was done later by Fo¨rster7), the
Detailed Balance in Excitation Transfer
J. Phys. Chem. B, Vol. 102, No. 9, 1998 1643
various energies are related by (see Figure 1)
Wk + wk′ - W ) wk
(5a)
Wl + wl′ - W ) wl
(5b)
In eq 4, n is the index of refraction of the medium and the M’s are matrix elements of the electric dipole operator between states in the ground and excited manifolds at the energies indicated. An approximation that is implicit in this “separable” case is that the vibrational and bath states associated with the two chromophores are uncorrelated. Cases involving such correlations have been treated by Soules and Duke.25 Using an angular average for the square of the dipole-dipole interaction does not reduce the generality of any results based on eq 4, since a factor accounting for the explicit angular dependence is easily restored. We return shortly to the full theory in which ulk is much more general. Introducing eq 4 into eq 1 and rearranging, we find
Fkfl )
4π 3pn4Rkl6
(
∞ 1 ∞ dW ∫w ′)0dwk′ gk′(wk′) × ∫W)0 Zk′ k
)( ∫
M(Wk - W + wk′, wk′)2 exp(-wk′/kBT)
1 Zl
∞
wl)0
dwl gl(wl) ×
)
M(wl, W - Wl + wl)2 exp(-wl/kBT) (6) The two bracketed quantities were recognized by Fo¨rster to be proportional to the emission spectrum Ak(W) of chromophore k (in quanta s-1 J-1) and the absorption cross section σl(W) of chromophore l, respectively: 4 3
(c/n)2I(ν) 8πhν3σ(ν) Z Z′
)
∞ dw′ g′(w′) M(hν0 - hν + w′, w′)2 exp(-w′/kBT) ∫w′)0 ∞ dw g(w) M(w, hν - hν0 + w)2 exp(-w/kBT) ∫w)0
(11) Kennard and Stepanov observed that g′(w′) M(hν0 - hν + w′, w′)2 dw′ in the integral in the numerator is equal to g(w) M(w, hν - hν0 +w)2 dw in the integral in the denominator. This is due to the fundamental microscopic reversibility of transition rates, or is a consequence of the Einstein A/B relation, which is the way the authors actually made the argument. (A precise application of the A/B relation would include a stimulated emission term. However, for the photon frequencies and temperatures of interest, this is safely ignored, since it is exp(-72) times as large as the spontaneous term.) With this substitution and using eq 5a in the exponential in the numerator, we see that the entire integral cancels out, leaving
(c/n)2I(ν) 8πhν3σ(ν)
∫w∞′)0dwk′ gk′(wk′) ×
3p c 1 A (W) ) 3 k Z 4nW k′
We switch to frequency units, in which the standard KS relation is usually presented. Thus W ) hν and W0 ) hν0. The absorption cross section is unchanged because it is a point function. The emission spectrum is a density, so we set A(W) dW ) F(ν) dν; hence hA(W) ) F(ν), where F is the spectrum in quanta s-1 Hz-1. To convert to a power spectrum, we set I(ν) ) hνF(ν) so that I measures the spectrum in W Hz-1. As a result, eq 10 becomes
or
k
M(Wk - W + wk′, wk′) exp(-wk′/kBT) (7) 2
ln and
3n2pc 1 σl(W) ) 2 Z 4π nW l
∫w∞)0dwl gl(wl) × l
M(wl, W - Wl + wl)2 exp(-wl/kBT) (8) In eq 8, σl(W), the absorption cross section, has replaced Fo¨rster’s Einstein B coefficient, using B(W) ) (c/nW)σ(W). Introducing eqs 7 and 8 into eq 6, we obtain the standard Fo¨rster expression that contains a spectral overlap integral:
Fkfl )
3p4c4 4πn4Rkl6
Ak(W) σl(W)
∞ dW ∫W)0
W4
(9)
Equations 7 and 8 may also be used to establish the so-called universal relation discovered by Kennard16 and Stepanov17 for homogeneous systems, in the present case ensembles of single chromophores. Omitting chromophore indices and calling the 0-0 energy W0, we divide eq 7 by eq 8 and rearrange to obtain
A(W) W2 × ) 3 2 σ(W) p π (c/n)2 Z Z′
∞ dw′ g′(w′) M(W0 - W + w′, w′)2 exp(-w′/kBT) ∫w′)0 ∞ dw g(w) M(w, W - W0 + w)2 exp(-w/kBT) ∫w)0
(10)
[
)
(c/n)2I(ν) 3
]
8πhν σ(ν)
Z exp[-(hν - W0)/kBT] Z′
)-
(
)
hν0 hν Z + + ln kBT kBT Z′
(12)
(13)
This form of the KS relation has been subjected to many analyses over the years (see, for example, refs 26-29). It is not universally obeyed because the assumptions on which it is based are violated to various degrees. In particular, the excited electronic state vibrational manifold can only approximate a thermal equilibration because of its various intrinsic loss mechanisms, and many systems for which it has been tested are probably not homogeneous. Nonetheless, it provides a familiar reference frame for theoretical investigations involving the excited electronic manifold. 3. Uphill and Downhill Rates of Transfer If the chromophores k and l discussed in part 2 have spectra that conform to the KS relation, which, as we have seen, will be the case under the assumptions made in the Fo¨rster theory of transfer, it is possible to find a simple relation between the downhill (k to l) transfer rate and the uphill (l to k) transfer rate. To this end we use eq 9 to compute the ratio
Flfk ) Fkfl
∞ dW W -4Al(W) σk(W) ∫W)0 ∞ dW W -4Ak(W) σl(W) ∫W)0
(14)
and then use eq 12, with chromophore labels restored, to substitute the absorption spectrum for the emission spectrum in each case:
1644 J. Phys. Chem. B, Vol. 102, No. 9, 1998
Laible et al.
Flfk ) Fkfl
mophores concerned. Hence the central result is
∫ ∞ dW W-4{σk(W)(Zk/Zk′) exp[-(W - Wk)/kBT]}σl(W) ∫W)0 ∞
dW W-4{σl(W)(Zl/Zl′) exp[-(W - Wl)/kBT]}σk(W) W)0
(15)
Zl Zk′ ) exp[-(Wk - Wl)/kBT] Zl′ Zk
(16)
Equation 16 may be put into the form of a single Boltzmann factor by introducing the free energies
∆Gk ) Wk - kBT ln(Zk′/Zk), ∆Gl ) Wl - kBT ln(Zl′/Zl) (17) These may be called the free energy differences between the excited and ground states, taking account of the fact that the distribution of vibrational frequencies and bath modes may have changed during electronic excitation. Indeed, under conditions of very rapid dissipation of excess excitation energy (e20 ps), they are the free energy differences. The excited-state partition functions Z′ are referred to the 0-0 energies of the chro-
FlFk/FkFl ) exp[-(∆Gk - ∆Gl)/kBT]
(18)
Using a harmonic oscillator model of the vibrational states in which the sets of ground- and excited-state vibrational angular frequencies are {ωk} and {ωk′}, respectively, and ignoring other types of bath states, we find that
(
∆Gk ) Wk - kBT ln
sinh(pωk/2kBT) ∏ k
)
sinh(pωk′/2kBT) ∏ k
(19)
where the products run over all modes. Those modes whose frequencies are unaffected by the electronic transition will contribute factors of 1 to the quotient. It appears likely that affected modes will thus contribute on the order of kBT to the quantity we are calculating. This could be significant, but if both chromophores are of the same type, the difference in the free energies will be fairly close to the difference of the 0-0 transition energies.
In the foregoing we have used the case of the separable dipole-dipole interaction to argue the validity of the Fo¨rster transfer formula, the KS relation, and the uphill/downhill Boltzmann factor. It is clear that the matrix elements M2 need not have referred to the dipole approximation; any multipole interaction would have led to the same result. We now argue that the detailed-balance argument can be extended to cover even the Dexter13 exchange interaction and its generalizations30 and therefore any process describable by first-order time-dependent perturbation theory in the Fo¨rster-Dexter sense. We find the ratio Flfk/Fkfl by applying eq 1: ∞ ∞ ∞ dW∫w )0dwk∫w ′)0dwl′ gl′(wl′) exp(-wl′/kBT) gk(wk) exp(-wk/kBT)|ukl|2 ∫W)0 ∞ ∞ ∞ dW∫w )0dwl∫w ′)0dwk′ gk′(wk′) exp(-wk′/kBT) gl(wl) exp(-wl/kBT)|ulk|2 ∫W)0
-1 Flfk (Zl′Zk) ) Fkfl (Z ′Z )-1 k
l
k
l
l
k
Since the interaction u now does not factor into parts dealing with the individual chromophores, we cannot use the same microscopic detailed-balance argument (Einstein relation) as before. We may however rely on a more general statement of the principle, that each group of transitions contributing to transfer of an amount of energy between W and W + dW must be a reversible process, requiring
dwk dwl′ gl′(wl′) gk(wk)|ukl|2 ) dwl dwk′ gk′(wk′) gl(wl)|ulk|2 (21) Although this equation does not appear to be symmetrical, recall that Fo¨rster’s energy normalization for the final-state wave function is still being used, so that each side of eq 21 has effectively yet another factor gg′, in the square of the matrix element, which we have continued to suppress for brevity. The microscopic states involved in eq 21 are related by the restrictions imposed by eqs 5, in this case,
Wk + wk′ - wk ) Wl + wl′ - wl
(22)
wk + wl′ ) Wk - Wl + wk′ + wl
(23)
or
During the integration over W, the restriction ∆W ) 0 selects only processes meeting these energy criteria. When eqs 22 and 23 are used in eqs 20 and 21, again the integrals cancel and again we obtain the result (eq 18)
(20)
Flfk/Fkfl ) exp[-(∆Gk - ∆Gl)/kBT] This relation is therefore generally true for transition probabilities based on eq 1 without reliance on the d-d approximation represented by eq 4. It is still restricted to a system whose internal and external environments immediately thermalize upon electronic excitation. 4. Discussion and Applications a. Free Energies in the Boltzmann Factor. The appearance of free energies in our result may not be surprising to chemists who deal with reactions, but a word may be in order for those acquainted with the more austere detailed-balance proscriptions of statistical physics. The connection is this: for all the microscopic transfer processes, strict detailed balance31a holds; for processes within the subsystem comprising the chromophores’ electronic states only, the thermal-activation form of detailed balance31b holds, providing the factor exp[-(Wk Wl)/kBT]. Finally, one has to consider the asymmetry of the golden rule: there is an aVerage over initial states but a sum over final states. The average involves not only sums but also normalization factors (Z, Z′), and it is these that emerge as the noncanceling terms in the rate ratio when the chromophores are nonidentical. These partition functions are brought into Fo¨rster theory through the optical spectra when the d-d approximation is used, and they are wholly implicit in that case. To our knowledge they have not previously been made explicit in Fo¨rster theory. The equivalent of eq 18 has been employed in the literature, notably in the analysis of triplet-triplet transfer.32,33 In that
Detailed Balance in Excitation Transfer case the introduction of free energies was based on analogy with electron-transfer theory, for which extensive model calculations are available (e.g., Ulstrup and Jortner34). Free energies of excitation also appear in the Fo¨rster cycle theory of protolytic reactions,35,36 but are again introduced without formal derivation. Because of the complexity of most chromophores to which the theory is intended to apply, accomplishing a satisfactory formal evaluation of the entropic part of the free energy will be unlikely in most cases. The vibrational contribution, shown in eq 19, an approximation, is only one participating factor. Solvent rearrangement energies and entropies will also be important. The brighter side of this picture is that the generality of the result (within the golden rule context) produces some flexibility in the assignment of the energies ∆Gk and ∆Gl, releasing them from a strict interpretation as absorption peak energies or 0-0 transition energies. If such a strict but erroneous interpretation is made, there is danger of an inconsistent set of rates developing in a multichromophore system. b. Approximations to the Uphill/Downhill Rate Ratio Predicted by Fo1 rster Theory. Fo¨rster theory has been applied most successfully to energy transfer between identical chromophores (self-depolarization of fluorescence) and very different chromophores (the case of negligible uphill transfer). In neither of these cases does the question of the ratio of uphill and downhill rates arise quantitatively. Recently this question has become prominent in connection with photosynthetic antenna systems consisting of very similar or identical chromophores whose transition energies are distributed, the typical transition energy differences between transferring chromophores being on the order of kBT.23,24,36-45 We now consider several ways of determining the theoretical uphill/downhill ratio of transfer rates. 1. Deduction of a Boltzmann factor from calculated rates. The ratio F1f2/F2f1 relating to transfers between two manifolds based on particular electronic excited states of chromophores 1 and 2 might be computed in any approximation, and an energy difference ∆E21 ) E2 - E1 is defined implicitly by F1f2/F2f1 ) exp[-(E2 - E1)/kBT]. One must then ask, having defined a similar ∆E31 for the state 1 and another state 3, whether a consistent rate ratio is obtained between levels 2 and 3. Such consistency is guaranteed, according to eq 18, only if F3f2/ F2f3 ) F1f2F3f1/F1f3F2f1. This relation is a necessary but not sufficient condition that the rates connecting the three states have been determined correctly. 2. Flfk/Fkfl ) exp[-(Wk - Wl)/kBT], where Wk - Wl is the difference between the 0-0 energies of the two states. This case is the easiest to relate to our formal result, because they are completely consistent if the ratios of the vibrational-bath partition functions in the excited and ground states are the same for the two chromophores. This is a good assumption if the two are identical, have identical environments, and suffer only a small environmental shift. In any event, one must be certain that the 0-0 energy differences have been ascertained correctly. Frequently they are approximated as the absorption-fluorescence crossing point for “normalized” spectra or, nearly equivalently, by using the point halfway between the absorption and fluorescence peaks. 3. Flfk/Fkfl ) exp[-(∆Eklabs)/kBT], where ∆Eklabs is the separation between the absorption peaks of the two states. This method is less reliable than the preceding method. It has all of its approximations and effectively adds the assumption that the Stokes shift is identical for the two chromophores. 4. Flfk/Fkfl is obtained from Fo¨ rster’s equation inVolVing the oVerlaps of donor emission and acceptor absorption, in
J. Phys. Chem. B, Vol. 102, No. 9, 1998 1645 cases where all four spectra are known experimentally to good accuracy. Here the focus shifts from how well we are approximating the partition functions to how well we are approximating the excitation interaction u. The spectra already contain partition function information, as we have seen. If the d-d approximation is a good one for the chromophores in question, a firm statement can be made. This is the most accurate method of obtaining the uphill/downhill ratio as well as the values of the rates themselves. When this method is used, the Kennard-Stepanov relation should be tested on the spectra of the participating chromophores, measured ideally in the environment in which the transfer is taking place. If the KS relation does not hold, the extent to which it does not hold can be taken as an indication of the reliability of the calculations of the transfer rates, because of their sensitivity to all of the approximations common to the theory of the two effects. 5. Flfk/Fkfl is obtained from the oVerlaps of donor emission and acceptor absorption spectra, but the fluorescence spectrum of one or both chromophore(s) exist(s) by Virtue of using the Kennard-StepanoV relation. This method is formally equivalent to method 4, assuming that the KS relation holds for each of the chromophores. Whereas in all the previously described methods the thermal equilibrium approximation is left implicit, using the KS relation to find the fluorescence is an explicit application of that approximation. If either or both of the fluorescence spectra are not available for use in method 4, the cost of using method 5 is to lose a potential test for thermal equilibration. 6. Flfk/Fkfl is obtained from the oVerlaps of donor emission and acceptor absorption spectra, but the mirror relation is used for the fluorescence of one or both chromophore(s). This method is also formally equivalent to method 4, but using the mirror law2 makes it unsatisfactory in two respects. First, the mirror law itself should always be a last resort because it is a very crude approximation based on normally unwarranted assumptions about excited-state vibrational frequencies. Second, it bears no relation to thermodynamics and therefore cannot represent the equilibrium assumption in any rational way. c. Limitations on the Usefulness of Spectral Information. It is well to take stock of the approximations that are incurred on the way to the Fo¨rster d-d approximation. There are three reasons one might go wrong in using a spectrally determined uphill/downhill ratio in a Fo¨rster process: (1) Given exceptionally accurate spectra for both directions of transfer, there may yet be corrections to the basic Fo¨rster d-d approximation that have different Franck-Condon factors, so that the ratios implied by the uphill/downhill spectral overlap integrals are not faithful. (2) Using inaccurate or approximate spectra for either or both directions of transfer will not represent the correct FranckCondon factors (see below); thus clearly Flfk/Fkfl will be incorrect even when there is no need to correct the d-d approximation. Numerous examples are given in the thesis of one of the present authors.18 (3) There may not be thermal equilibrium in the donor excited state, either during fluorescence or during transfer or during both. Keeping in mind that our transferring pair are a fortiori different molecules, or are in different environments, or are both, one cannot hope that the effects of nonequilibrium will be the same for both directions of transfer (in the sense of canceling out in the ratio). The possibility of appreciable departure from excited-state thermal equilibrium is, of course, a major concern
1646 J. Phys. Chem. B, Vol. 102, No. 9, 1998
Laible et al.
for the calculation of the rate of a Forster process itself. We turn briefly to this subject. Recent time-resolved studies have provided ample evidence of measurably low rates of excited-state relaxation compared with radiative and nonradiative rates in chromophores bound in a light-harvesting proteins46 with relaxation dynamics similar to isolated pigments in polar organic solvents.47 These rates can be identified with parameters in the Brownian oscillator model of optical processes, specifically including absorption and fluorescence.48 Application of this model to the rate of excitation transfer has been initiated by Mukamel and Rupasov,49 who showed that a time-dependent Forster rate exists in the high-temperature and slow-dynamics limits. In these limits Gaussian absorption and emission bands are predicted by the theory, and the time-dependent transfer rate is found to be proportional to a Forster overlap of a time-dependent donor fluorescence band with the acceptor’s absorption band. The donor’s fluorescence undergoes its Stokes shift during the course of the relaxation from the donor’s “hot” state. We do not know of any applications of this method at present, but it appears to provide one of the most natural ways to link the usual Forster theory with “hot” transfer. We return to a discussion of contingencies that must be faced in cases where excited-state equilibrium is a valid assumption. Although the uphill/downhill ratio is known from eq 18, it is correct only if the transition rates have been calculated exactly at the golden rule level. Recall that in the d-d case one approximation is made and one assumption is made at the first stage of the calculation. The approximation is the BornOppenheimer or adiabatic approximation, and the assumption is that the donor and acceptor electronic states are nondegenerate. (The latter is largely for convenience and may in some cases be circumvented.) With these in place, the fundamental matrix element of the transition rate takes the form
completely different from those that couple to the singlet transitions. Reference 46 shows that the rate of triplet-triplet transfer can be either overestimated or underestimated by using singlet-triplet spectra. The preceding paragraph applies to the Hcorr contributions to excitation transfer in varying degrees. It applies directly if there are triplet admixtures to the transferring states, and if the configuration mixing of singlets is due to intermolecular coupling rather than intramolecular. (If it were the latter, one could say that the same configuration interaction would be involved in the spectrum.) It is less clear that the cautionary argument applies in the monopole and singlet exchange cases. These corrections still involve the same intramolecular electronic states, so they introduce closely related or identical FranckCondon factors. There is no guarantee that intermolecular interactions will not change things, because of the proximity of the donor and acceptor when these terms are in fact important. The problem is compounded by the fact that the quantity we deal with ultimately is |Hdd + Hcorr|2, which has cross-terms as well as terms coming from Hdd and Hcorr, a fact that ensures hybrid Franck-Condon factors if different vibrational modes are brought in by Hcorr. d. Application to Photosynthetic Antenna Systems. Photosynthetic antenna systems are large aggregates (30-10 000 pigments), bound in specific protein environments, that serve to absorb light energy and to transfer that energy with high efficiency to the photosynthetic reaction center. The free energy of excitation appears in many forms in the analysis of excitedstate dynamics in these antenna systems. For example, consider a small light-harvesting array consisting of a single low-lying chromophore (“trap”) that is in contact with a pool of N antenna chromophores, each having a free energy that is higher by ∆Gat ) ∆Ga - ∆Gt and the same rate of energy transfer Faft to the trap. The effective rate of transfer out of the trap is51
ukl ) 〈f, νf|H′|i, νi〉 ) Hdd + Hcorr (24)
NFaft exp(-∆Gat/kBT) ) Faft exp[(-∆Gat - kBT ln N)/kBT] (25)
The first term Hdd represents that part of the matrix element that is given by the dipole-dipole approximation to the electronic interactions and which, in the standard derivations of the Fo¨rster d-d equation, is the origin of spectral contributions. It involves a set of vibrational overlap factors, denoted schematically by 〈Vf|Vi〉, that when squared result in the FranckCondon factors associated with the spectra. The term Hcorr represents a correcting electronic matrix element that might be due to any combination of the following things: monopole corrections,14,15 higher multipole and exchange and overlap terms,13 spin-orbit admixtures of triplet states, configurational mixing of higher singlet states, and configurational mixing of charge-transfer states.30 It would be very convenient if indeed the associated vibrational overlap factors were the same as those appearing in the Hdd term, because we could then argue that all electronic factors in the matrix elements drop out of the ratio Flfk/Fkfl. But the vibrational overlaps in Hcorr are not necessarily the same as those appearing in Hdd except under very special circumstances.50 Consider triplet states, for example, to which optical transitions are not allowed. How is any spectrum relevant to transfer? It happens through the triplets’ coupling to singlets, which will bring into the transfer matrix element some vestiges of a singlet-triplet spectrum, but if the transfer matrix element is dominated by the triplet states themselves, it is impossible on the basis of spectra to say what the Franck-Condon factors are. They can involve modes
This expression is in fact more generally valid, since it can readily be derived regardless of the number of closely interacting antenna-trap molecules, as long as the intra-antenna transfer rate is high compared with all other rates in the system.43 While antenna inhomogeneity will alter this result in detail, it is fundamentally sound and illuminates a picture of antenna activity that is quite well removed from the traditional “funnel” picture. Should there be any funnel effect, in this context it would be to bias the antenna excitation population to produce a similar result with N replaced by some effective antenna size that could be smaller or larger than the actual size.40 The term -kBT ln N can be identified as a T∆S contribution to the overall excited-state free energy. Such an interpretation involves an assumption of thermal equilibration on a much broader scale than the intramolecular considerations of KS or the pairwise interactions of the Fo¨rster process. The origin of this assumption is in part based on the increased density of final states caused by the multiplicity of chromophores, and it is complicated by the spectral diversity of the chromophores in a typical antenna system (different 0-0 energies in otherwise identical or nearly identical chromophores induced by static differences in the protein “solvent” environment). There are considerable experimental data to suggest that such a quasi-thermal equilibrium is established in the excited states of many photosynthetic antenna systems. The strongest evidence for this comes from the extension of the intramolecular
) 〈f|H′|i〉〈Vf|Vi〉 + 〈f|H′|i〉′〈Vf|Vi〉′
Detailed Balance in Excitation Transfer
J. Phys. Chem. B, Vol. 102, No. 9, 1998 1647
formalism of KS to the photosynthetic antenna complexes and photosystems. In this intermolecular case, the assumption is now that there is an equilibrium among all of the excited electronic and vibrational states of the coupled chromophore system. This necessitates that rates of energy transfer among the coupled chromophores are rapid compared to that of any competing processes that result in loss of the electronic excited state from the complex. Ross and Calvin19b were the first to apply the KS formalism to photosystems (PS) I and II of higher plants. Szalay et al.20 and Zankel and Clayton21 extended the application to algae and photosynthetic bacteria, respectively. More recently, Dau and Sauer23 demonstrated nearly perfect thermal equilibration in PS II by the use of a modified KS relation, while Croce et al.22 showed only small deviations from thermal equilibration in PS I. In this context, it is also important to consider the dynamics of intramolecular vibrational relaxation and rapid energy transfer among pairs of tightly coupled chromophores in these antenna systems. In both PS I and PS II, the average time constants for pairwise energy transfer are on the order of a few hundred fs or less,52,53 the same time scale as vibrational relaxation in these chromophores.54 Theoretical treatments of this “hot transfer” case have long been available55 but, as with the more recent treatment discussed above,49 have not yet been applied in photosynthetic excitation transfer studies. Thus it is possible that for each individual transfer the requirement for thermal equilibrium may not be met. Nevertheless, the observation of near-KS behavior in these antenna systems indicates thermal equilibration within the entire antenna-reaction center system on the time scale of the excitedstate lifetime. Dau24 has formally extended the KS theory to aggregates of tightly coupled chromophores. Near-thermal equilibration in photosynthetic antenna systems is also supported by theoretical studies. Pearlstein,56 using analytical solutions to excited-state dynamics in two- and threedimensional arrays of chlorophylls, showed that the lowest moment of the excited-state decay could dominate the overall dynamics (so-called “zero-mode dominance”). Laible et al.43 used the exact solution to the Pauli master equation to predict the occurrence of thermalized states in structural models of PS I and PS II. Interestingly, the deviation from quasi-thermal equilibrium seen by Croce et al.22 in their PS I samples (a depletion of the excited-state density in the vicinity of the photochemical trap P700) was predicted in the simulations of Laible et al.43 This near-thermal equilibrium of the excited state in photosynthetic systems has been termed transfer equilibrium by Laible et al.43 and excitation equilibrium by Dau and Sauer.23 The utility of the KS formulation in analysis of multichromophore systems is strictly tied to the requirement of detailed balance and microscopic reversibility:
| | | | | | N
pi(∞) )
Fifj
∑ j)1 F
jfi
-1
N
)
FCifj
∑ j)1 FC
jfi
-1
N
)
Iifj
∑ j)1 I
-1
(26)
jfi
where pi(∞) is the probability of excitation of chromophore i at infinite time after the creation of an excited state in the system, and FCifj and Iifj are the Franck-Condon factor and spectral overlap integral (the integral appearing in eq 9) for the corresponding Fo¨rster rate Fifj. In the absence of any loss of excited state from the system, the first equality is simply a statement of detailed balance. The second equality assumes only the application of the Born-Oppenheimer approximation. In the Fo¨rster d-d limit, and in the absence of the complications detailed in section 4c, the Franck-Condon factors are completely described by the spectral overlap integral, leading to the
third equality. These same spectral features underlie the KS formalism as outlined above. Combining the first equality in eq 26 with eq 18 gives
pi(∞) )
|
|
N
exp[-(∆Gj - ∆Gi)/kBT] ∑ j)1
-1
)
1
exp(-∆Gi /kBT) (27) Zant where N
Zant )
exp(-∆Gj /kBT) ∑ j)1
(28)
Equation 28 has frequently appeared, with ∆G replaced by some approximate ∆E, in the kinetics of the energy transfer within aggregates such as the photosystems of photosynthetic organisms (see, for example, refs 43 and 57-59). Its origin is a broad assumption of thermal equilibration and a consideration of increased density of final states caused by the multiplicity of chromophores. 5. Summary The central result of this paper is a clear statement of the relationship between the rates of uphill and downhill excitation transfer between two chromophores that is directly based on, and therefore consistent with, Fo¨rster’s theory of excitation transfer. The result, eq 18, has a familiar and simple form resulting from the approximation that excited manifolds are in thermal equilibrium. A second result is that the exponential factor in eq 18, while completely general within the context of the theory and normally impossible to compute exactly, can actually be obtained exactly by using Fo¨rster’s dipole-dipole approximation if the d-d interaction dominates and the spectra of the participating chromophores obey the Kennard-Stepanov relation. We have discussed the implications of failure of these two criteria and commented on the application to multichromophore arrays. Acknowledgment. The authors wish to thank Profs. H. Dau, K. Sauer, A. Albrecht, and R. Loring for stimulating and helpful discussions in the initial stages of the work. The research was supported in part by NIH Grant 08-T26M08267A (P.L.), U.S. Department of Agriculture NRICGO Grants 95-37306-2014 (R.K.) and 93-37306-9042 (T.O.), and National Science Foundation Grants MCB-92-05165 (T.O.) and PHY-94-15583 (R.K.). References and Notes (1) Fo¨rster, Th. Naturwissenschaften 1946, 33, 166. (2) Fo¨rster, Th. Ann. Physik (Leipzig) 1948, 2, 55. English translation: In Mielczarek, E. V.; Greenbaum, E.; Knox, R. S. Biological Physics; American Institute of Physics: New York, 1993; p 148. (3) Fo¨rster, Th. Z. Naturforsch. 1947, 2b, 174. (4) Simpson, W. T.; Peterson, D. L. J. Chem. Phys. 1957, 26, 588. (5) Knox, R. S.; Gu¨len, D. Photochem. Photobiol. 1993, 57, 40. (6) Rahman, T. S.; Knox, R. S.; Kenkre, V. M. Chem. Phys. 1979, 44, 197. Erratum: Chem. Phys. 1980, 47, 416. (7) Fo¨rster, Th. Z. Naturforsch. 1949, 4a, 321; Fluoreszenz Organischer Verbindungen; Vandenhoeck u. Ruprecht: Go¨ttingen, 1951. It is difficult to pinpoint Fo¨rster’s generalization in the literature, but this paper and this book clearly reflect it. (8) Frenkel, J. Phys. ReV. 1931, 37, 17, 1276. (9) Davydov, A. S. Zh. Eksp. Teor. Fiz. 1948, 18, 210. (10) Reddy, N. R. S.; Picorel, R.; Small, G. J. J. Phys. Chem. 1992, 96, 6458. (11) Edington, M. D.; Riter, R. E.; Beck, W. F. J. Phys. Chem. 1996, 100, 14206.
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