Primary Processes and Structure of the Photosystem II Reaction

J. Phys. Chem. B 2000, 104, 11563-11578

11563

Primary Processes and Structure of the Photosystem II Reaction Center: A Photon Echo Study†,‡ Valentin I. Prokhorenko and Alfred R. Holzwarth* Max-Planck-Institut fu¨ r Strahlenchemie, Stiftstrasse 34-36, D-45413 Mu¨ lheim a.d. Ruhr, Germany ReceiVed: June 30, 2000; In Final Form: September 19, 2000

An experimental and theoretical photon echo (PE) study of the primary charge separation process in the photosystem II reaction center (PS II RC) at low temperature (T ) 1.33 ( 0.01 K) is reported. Experiments were carried out at low excitation intensities of 5 × 1012 photons/cm2 with time and spectral resolution of about 0.5 ps and 1 nm, respectively, using the two-pulse photon echo technique (2PE). The data were interpreted in the framework of the exciton model. For that purpose the theory of the PE formation and energy transfer in an excitonically coupled system, including explicitly the electron-bath interaction, is developed. By comparing the measured and the simulated PE kinetics, we draw the conclusion that the accessory chlorophyll in the active branch of the RC core is the primary electron donor. The charge separation occurs with an intrinsic time constant of ≈1.5 ps, in good agreement with previously published data (Wasielewski, M. R.; Johnson, D. G.; Seibert, M.; Govindjee Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 524; Jankowiak, R.; Tang, D.; Small, G. J.; Seibert, M. J. Phys. Chem. 1989, 93, 1649; Tang, D.; Jankowiak, R.; Seibert, M.; Small, G. J. Photosynth. Res. 1991, 27, 19). However, the dipole-dipole interaction between pigments leads to a very wide distribution of the effective charge separation kinetics ranging from 1.5 ps up to a few nanoseconds. Thus, the experimentally observable effective distributive charge separation rate differs strongly from the intrinsic one. In this work the effect of the charge separation process in an excitonically coupled system is described for the first time. Energy transfer rates, calculated on the basis of developed theory, show that the energy transfer occurs in the 100-200 fs time domain in agreement with our own experimental observations, and previously published data. This fast energy transfer contributes to the intense and narrow peak at early delay times in the 2PE kinetics. In contrast, the slow dephasing observed in the 2PE kinetics at time delay above ca. 1 ps reflects mainly the primary charge separation process.

I. Introduction The fundamental primary reaction of photosynthesis is charge separation in a specialized pigment-protein complex, the photosynthetic reaction center (RC). The protein complex performing that function in photosystem II of higher plants, the so-called D1-D2-cyt-b559 complex, contains 6 chlorophyll a (Chl a) and 2 pheophytin a (Pheo a) molecules,1,2 i.e., two Chls more than the bacterial RC. The two additional Chls are weakly coupled to the RC core and are considered more like antenna Chls. The PS II RC displays a substantial structural similarity to the bacterial RC complex.3 Charge separation within the RC core occurs from the primary electron donor, generally believed to be a special pair of Chls, called P680, to the first electron acceptor - presumably one of the two Pheo a’s. In the isolated D1-D2-RC the following electron transfer step is blocked and the primary radical pair, which generally is believed to be P680+Pheo-, loses its energy by singlet-triplet conversion and * To whom all correspondence should be addressed. E-mail: [email protected]. Fax: (+49)2083063951. † A preliminary account of this work has been given at the XIth International Congress on Photosynthesis, Budapest, Hungary (1998). ‡ Abbreviations: reaction center, RC; photosystem II, PS II; chlorophyll, Chl; pheophytin, Pheo; bacteriochlorophyll, Bchl; energy transfer, ET; transient absorption spectroscopy, TA; time-resolved fluorescence spectroscopy, FS; hole-burning spectroscopy, HB; photon echo, PE; accumulated photon echo, APE; two-pulse photon echo, 2PE; three-pulse photon echo, 3PE; photomultiplier, PMT; cross-correlation function, CCF; Huang-Rhys factor, HRF; phonon wing, PW.

eventual charge recombination to the triplet state of P680, with a relatively high efficiency especially at low temperature.4-6 Excitation of the primary electron donor occurs either directly by light or via energy transfer (ET) from one of the two extra “antenna” Chls. Since the first isolation of the PS II RC complex some 10 years ago,7 the processes of energy transfer and charge separation (CS) have been studied by a number of groups using a variety of methods. It is now well established that, in contrast to the bacterial RC, in the PS II RC the excited states of the various chromophores are nearly isoenergetic; i.e., their absorption bands in the Qy region are almost totally overlapped at room temperature and the Stokes shift of the emission is small. Therefore, the interpretation of spectroscopic data such as transient absorption spectroscopy (TA), time-resolved fluorescence spectroscopy (FS) or hole-burning spectroscopy (HB) is very difficult and often ambiguous. Using HB spectroscopy at liquid helium temperature a charge separation time of ≈1.9 ps was found by Small and co-workers.8,9 In addition two longer components of 12 and 50 ps were observed as well,10 which were assigned to the ET from Pheo a and the accessory Chls to P680. In a TA study by Wasielewski’s group a monoexponential charge separation kinetics of 1.4 ps lifetime was observed at 15 K and a 3 ps lifetime at 4 °C.11,12 However in more recent studies13,14 they reported a charge separation time of 5 and 8 ps at 7 K and 5 °C, respectively. Low-intensity TA spectroscopy with high resolution by Mu¨ller et al.15 revealed an intrinsic charge separation time of ≈2.4 ps (4 °C) and two components

10.1021/jp002323n CCC: $19.00 © 2000 American Chemical Society Published on Web 11/08/2000

11564 J. Phys. Chem. B, Vol. 104, No. 48, 2000 of ∼9 and 20 ps, assigned to slow ET from the external “antenna” chlorophylls. FS experiments at 277 and 77 K by Holzwarth et al.16 indicated also that the lifetime of the P680* excited state is a few picoseconds. Somewhat different data were reported by Groot et al.17 from a TA study at several temperatures. An ultrafast component that accelerated from ∼2.6 ps at 20 K to ∼0.4 ps at 240 K was observed and this lifetime was attributed to the CS to an intermediate state with should be located near the singlet-excited state of P680. Note that a ∼100 fs component at room temperature was observed earlier by Durrant and co-workers18,19 and by Holzwarth and coworkers15,20 which was thought to derive from energy equilibration among pigments in the RC core due to fast ET processes.18 In a study of Leegwater et al.,21 based on the “multimer” model22 of the RC core, such an equilibration process was in agreement with theoretical calculations. From a HB study23 a red “trap” state at 682 nm was characterized along with two “blue” pigments with decay times of ∼200 and 12 ps at 1.2 K. Despite the many studies carried out so far there is a great deal of uncertainty and controversy as to the ET and CS kinetics and the nature of the various states involved. Photon echo spectroscopy is free from many disadvantages inherent to conventional spectroscopy. In simple terms, it measures directly the decay of the coherence of the excited state(s) created by a coherent light pulse (for details see, e.g., the monograph in ref 24). The processes of ET and CS both destroy the coherence of the excited state. The measurement of the PE decay thus allows determining the overall dephasing time25 T2, from which under certain conditions the population relaxation lifetime T1 and the charge separation time can be extracted. A great advantage of the PE spectroscopy compared to TA spectroscopy is insensitivity to reversible processes in the population dynamics like, e.g., reversible ET. So far only two PE studies of PS II RCs were reported, and both were performed using the accumulated photon echo technique (APE) which has some drawbacks due to the use of a long-lived intermediate state, in that case the triplet state of P680. In the first study,26 two slow components of ∼200 and ∼500 ps in the APE decay were observed at 1.5 K, i.e., no fast charge separation process was observed. In the study of Aartsma and co-workers27 time constants of the APE decays of 1.5, 35 and 250-400 ps in the wavelength range 673-685 nm were found. The fast component was assigned to the primary charge transfer from directly excited P680* to the active Pheo a, the middle component was interpreted to an “energy-transfer-limited” CS upon excitation of a pigment roughly isoenergetic with P680 (the “red trap” state) and the slow component was attributed to ET from the long-wavelength pigments. It is not quite clear at present whether these kinetic components found by APE spectroscopy are the same as found by other (TA and FS) spectroscopies. Some of the reasons for these uncertainties are related to the particular problems of interpreting APE measurements. We present here an extended photon echo study of the primary processes in the D1-D2-cyt-b559 complex at low temperatures using the simple 2PE technique. While the general theory for optical PE phenomena for individual molecules is quite well developed,24 the applicability of this theory to molecular complexes is much less clear and meets some so far unsolved difficulties. An important aspect of the present work - besides the experimental results - is thus the development of an adequate theory of photon echo formation in excitonically coupled systems interacting with an environment (bath). In particular this theory include also a proper description of charge separation under the restrictions of our particular experiment.

Prokhorenko and Holzwarth The present paper is organized as follows. In section II we present a theory of the PE formation in excitonically coupled systems, including charge separation effects. This theory is developed in the so-called Markov limit, i.e., under assumption of a short bath memory time. The expression for the energy transfer rates (and for the corresponding dephasing rates) are derived in a stringent form and connected to the experimentally measurable Huang-Rhys factor (HRF) and the phonon profile. In Sec. III we discuss the main experimental results of the 2PE experiment in the D1-D2 RC. Then in section IV we compare the measured and the modeled echo signals and derive an intrinsic primary charge separation rate and information about the primary electron donor in the D1-D2 RC. All modeling calculations are based on the structural model of Svensson et al.28 Recent electron microscopy studies of photosystem II performed with 8 Å resolution29,30 show that this model fits very well the experimentally measured spatial structure of the RC core. II. Theory Photon Echo Formation. As mentioned above, the theory of photon echo formation in simple systems such as single molecules is very well developed24 and properly describes the main features in the observable PE kinetics. This theory is based on the cumulant expansion approach, but it is not generalized so far for excitonically coupled systems (so-called many-body systems) such as photosynthetic subunits like antenna complexes or reaction centers. A key element of this theory is consideration of the system-bath interaction in the non-Markov limit, i.e., by taking into account a finite system-bath interaction memory. This allows us to correctly describe the system behavior on the femtosecond time scale, where this effect plays a major role. However, the measured 2PE kinetics in RCs shows us that the main features, mostly reflecting the charge separation process, are concentrated in the picosecond time scale. Thus, we can consider here the system-bath interaction in the Markov limit and develop the appropriate theory of the PE formation using the reduced density operator approach. Without a loss of generality we have neglected here the excited state absorption which is insignificantly contributing to the kinetics in our case (long wavelength range). A photon echo signal, measured by homodyne detection, can be simply described as24

SPE(τ) )

∫-∞∞|PB(t,τ)|2 dt

(1)

where B P is the induced polarization in the medium. It can be calculated as an expectation value of the system transition dipole Bˆ operator m

Bˆ 〉 ≡ 〈Ψ|m Bˆ |Ψ〉 ≡ Tr(m Bˆ ,Fˆ s) B P ) 〈m

(2)

where ψ is the time-dependent total wave function, and Fˆ s a system density operator. The total Hamiltonian of the excitonically coupled system, which contains N molecules, is

H ˆ tot )

∑i (Hˆ mol)i + Hˆ int + Hˆ E

(3)

where the summation runs over i ∈ [1, ..., N], H ˆ int ) ∑i,k;i*kJik|i〉〈k| describes the dipole-dipole interaction between molecules, and

H ˆE ) -

∑i bµi(|0〉〈i| + |i〉〈0|)EB

(4)

Photon echo in D1D2 RC

J. Phys. Chem. B, Vol. 104, No. 48, 2000 11565

describes the interaction with the electrical field B E (assuming that the molecular dipole moments µ for the S0 f S1 and S1 f S0 transitions are equal). The Hamiltonian of the particular molecule can be generally given as

H ˆ mol ) |1〉ν〈1| + H ˆb + H ˆ e-b

(5)

where |1〉 is an electronic wave function in the S1 excited state (we consider the molecules within the two-level approximation), ν the energy of the S0 f S1 electronic transition, H ˆ b describes the bath, and for the electron-bath interaction we take a linear coupling approximation, neglecting the interaction in ground ˆ b and Fˆ are Hermitians state: H ˆ e-b ) |1〉〈1|Fˆ . The operators H acting only on the bath variables, which we assume to be sitedependent. They can be expressed in an explicit form for specific bath models (i.e., discrete or continuous distributed sets of oscillators, the Brownian oscillator model, etc.), but in this theory we do not need a specific expression that makes the obtained solutions quite general. The primary charge separation can be incorporated by using the complex transition energies νi ≡ ν˜ i - i/2δixΓCS (δix is the Kroenecker δ-function), assuming that the electron transfer occurs only from one of the six core pigments which is in principle unknown and is labeled by index x, and ΓCS represents the intrinsic charge separation rate from this pigment. Because of charge separation, the line width of the primary electron donor broadens. For the total Hamiltonian we thus get

H ˆ tot )

∑i |i〉νi〈i| + i,k:i*k ∑ Jik|i〉〈k| + ∑i (Hˆ b)i + ∑i |i〉〈i|Fˆ i ∑i bµi(|0〉〈i| + |i〉〉0|)EB (6)

where the first two terms on the right-hand side correspond to the excitonically coupled system, the third term corresponds to the bath, and the fourth term describes the system-bath interaction. For applying the reduced density matrix formalism, the system-bath interaction must appear as a perturbation. Thus, an appropriate unitary transform of the total Hamiltonian should be performed. Choice of the basis set for such a transform depends on the electron-phonon coupling strength. According to31 the relation between the reorganization energy νr and the interaction energy, J can be used as a criterion. The case νr < |J| corresponds to the so-called weak electron-phonon coupling, and for νr > |J| the system and bath should be considered strongly coupled. The reorganization energy is related to the Stokes shift SS ) 2νr, which can be given as a product of the mean phonon frequency ωm and the HRF S0 (at T ) 0): SS ) 2ωmS0. For the D1-D2 RC core at low temperatures S0(RC core) ≈ 1.6-1.9, ωm ≈ 20 cm-1,10,32,33 and |J| ≈ 120-180 cm-1,32,34-37 thus νr < |J| and electron-phonon coupling is essentially weak. In this case for the Hamiltonian in eq 6 the excitonic transform can be performed which results in

∑k ∑k 〈φi|k〉〈k|φj〉|φi〉〈φj|)‚Fˆ k - ∑m b kB E(|0〉〈φk| + |φk〉〈0|) ∑k (∑ i,j k

ˆs + H ˆb + H ˆ s-b + H ˆE ) H ˆ tot ) H

k|φk〉〈φk| +

term describes an interaction of the system with the electric b ‚E B, field and can be written in analogous form as H ˆ E ) -m where the operator m b ) ∑km b k(|0〉〈φk| + |φk〉〈0|) is the dipole operator of the entire system. The energies of the excitonic states k are obtained by diagonalization of the corresponding Frenkel Hamiltonian matrix (first two terms in eq 6), but unlike for the ν-set (see above), where only νx has an imaginary part due to charge separation, now all energies of excitonic transitions get a complex character k ) ˜ k - i/2Γ′k. The charge separation rates Γ′k for the various excitonic states will be significantly different from the intrinsic charge separation rate ΓCS. The origin of this difference is a redistribution of the charge separation process over the whole excitonically coupled ensemble of pigments due to their interaction (dipole-dipole in our case) which leads to line broadening of all excitonic states. The magnitude of this broadening directly depends on the probability of occupation of the primary electron donor in a given exciton state: Γ′k ) |Uxk|2ΓCS. Thus, for a system with strong excitonic coupling in which an electron donor is participating in all excitonic states, such an effect can by very pronounced. Besides the line broadening redistribution, the energies of the exciton states suffer small shift as compared to those calculated from the Frenkel Hamiltonian matrix alone. This shift is related to the Lamb shift, which is negligibly small for the D1-D2 RC (for an intrinsic charge separation time of a few picoseconds it is around ∼1 cm-1). The total wave function of the system is the sum of all stationary excitonic wave functions φn with time-dependent coefficients c(t): Ψs ) ∑nc(t)nφn. By plugging this expression and m ˆ into eq 2 we immediately get

∑k mbk[(F0k)s + (Fk0)s]

B P)

which means that the induced polarization and the PE signal are determined only by the off-diagonal elements of the density matrix. For describing the time evolution of the density operator the Liouville equation is usually used, but if the Hamiltonian is not hermitian, the general expression should be applied:

∂Fˆ )H ˆ totFˆ - Fˆ H ˆ /tot ∂t

ip

where |φk〉 ) ∑iUik|i〉 is the exciton wave function, and m bk ) ∑ib µiUik is the dipole moment of the kth excitonic transition. The first term on the right-hand side in eq 7 corresponds to the system, the second to the bath, and the third describes their interaction, which is small compared to the first term. The last

(9)

where * denotes the complex conjugate. For the reduced density matrix in the Markov limit we now get

(F˘m′m)s ) -iωm′m(Fm′m)s - (Γ′′m′ + Γ′′m)(Fm′m)s + i Rm′mkn(Fkn)s - 〈φm′|[H ˆ E,Fˆ ]|φm〉 (10) p k,n

∑

where ωm′m ) (˜ m′ - ˜ m)/p, Γ′′m ) Γ′m/2p, and R is the Redfield relaxation tensor,38 which can be written as + + Γnmm′k - δnm Rm′mkn ) Γnmm′k

(H ˆ b)k +

(7)

(8)

+ - δm′k∑Γnssm′ ∑s Γm′ssk s

(11)

and its components can be expressed in a general form over the system-bath correlation functions as39

∫0∞dτ e-iω

+ ) Γnmm′k

1 p2

Γnmm′k )

1 p2

m′kτ

∫0∞dτ e-iω

〈H ˆ s-b(τ)nm‚H ˆ s-b(0)m′k〉bath

(12.1)

〈H ˆ s-b(0)nm‚H ˆ s-b(τ)m′k〉bath

(12.2)

nmτ

where H ˆ s-b(τ) ) eiHˆ bτH ˆ s-be-iHˆ bτ. The equation of motion for

11566 J. Phys. Chem. B, Vol. 104, No. 48, 2000

Prokhorenko and Holzwarth

the reduced density matrix (eq 10) differs from the one obtained from the Liouville equation38 by a term describing the decay of the system due to charge separation (so-called “open system”). The working system of density matrix equations40 can be obtained from eq 10 by expressing H ˆ E:

i F˘n0 ) -iωn0Fn0 - (Γ′′n + γn0)Fn0 - m E(Fnn - F00) b nB p i m b kB EFnk pk*n

∑

i F˘0n ) iωn0F0n - (Γ′′n + γ0n)F0n + m E(Fnn - F00) + b nB p i

∑mbkBEFkn pk*n

disorder plays a dominant role, and the 2PE signal must be synthesized in the way mentioned above. For this we need to calculate the energy transfer and dephasing rates of each individual RC of the ensemble. As can be seen from eq 13, only the sets of γnn, Wmn, γn0, γ0n need to be determined which is describing in the following. Transfer Rates. The system-bath interaction appears in the Hamiltonian (eq 7) as a product of two operators, acting on the independent variables. Thus, the contributions of the system and bath in the general expression (eq 12) can be separated,41 and they are simplified to + Γnmm′k )

/ U/anUamUbm′ UbkΘ+ ∑ abnk a,b

(16.1)

) Γnmm′k

/ U/bnUbmUam′ UakΘabm′k ∑ a,b

(16.2)

∫0∞dτe-iω

i b F -m F˘kn ) -iωknFkn - (Γ′′k + Γ′′n + γkn)Fkn + (m b nFk0)E B p k 0n

+ ) Θabm′k

1 p2

i b F -m b kFn0)E B F˘nk ) -iωnkFnk - (Γ′′n + Γ′′k + γnk)Fnk + (m p n 0k

Θabnm )

1 p2

F˘nn ) -(2Γ′′n + γnn)Fnn + i

F00 )

i

∑ WnNFNN + pmbnBE(F0n - Fn0) N*n

∑N mbN(FN0 - F0N)EB

(13)

p

γm′m )

+ + + Γ∑k (Γm′kkm′ mkkm) - Γmmm′m′ - Γmmm′m′

(14.1)

Wmn ) Γ+ nmmn + Γnmmn

(14.2)

Two important features can be seen from eq 13. First, the system is not conservative anymore (i.e., the trace of the density matrix is not a constant), which reflects a chemical reaction (in our case charge separation) that occurs after excitation and the initial system decays. Second, the dephasing due to charge separation is two times slower as compared to the corresponding population relaxation. The system of coupled equations (eq 13) can be solved analytically by using a rotating wave approximation (RWA). All details of this solution up to the third order of the applied field are derived in Appendix A. Here we take up only the final expression for the induced polarization in the 2k B2 - B k1 direction:

B P)2

∑k mbkRe

{

i

eiωLt[(Ck + C0)I2,k]Xe-(iδk+Γ2k)t

2

}

(15)

(for meaning of terms see Appendix A). This polarization should be averaged over a disordered ensemble of RCs. Because of disorder all RCs are different with their individual set of the excitonic transitions, energy transfer, charge separation, and dephasing rates. After that averaging the 2PE signal can be numerically calculated using eq 1. Such a situation strongly differs from the PE formation in simple systems like, e.g., an ensemble of noninteracting molecules in a solvent, where their individuality can be described in the framework of inhomogeneous broadening, and averaging over the ensemble can be performed analytically. If disorder is weak and does not lead to a strong individuality, the inhomogeneous broadening approach can be also used. However, in the PS II RC the diagonal

(16.3)

nmτ

〈Fˆ b(0) Fˆ a(τ)〉bath

(16.4)

where Θ+, Θ- is the Fourier-Laplace transform of the bath correlation functions. In eqs 16.1 and 16.2 the summation runs over a,b ∈ [1, ..., Nexc], where Nexc ≡ N is a total number of excited states in an excitonically coupled system. For dephasing rates γn0, γ0n we get from eq 14.1

γn0 )

The transfer rates γ and W are41

∫0∞dτe-iω

〈Fˆ a(τ) Fˆ b(0)〉bath

m′kτ

∑k Γ+nkkn,

γ0n )

∑k Γ-nkkn

(17)

On the other hand, from eq 14.2 it follows for the energy transfer rate γnn (which describe the population relaxation of the nth exciton state)

γnn )

∑k (Γ+nkkn + Γ-nkkn) ≡ ∑k Wkn

(18)

and by comparing with eq 17 we see that

γnn ) γn0 + γ0n

(19)

Taking into account a general property of the transfer coef+ 41 which reflects the stationary charficients (Γmnkl)* ) Γlknm, acter of the time correlation functions, we can easily see that the transfer rates γnn are real numbers, i.e., γ/nn ) γnn. This gives the following set of relations for the dephasing rates:

γ/n0 + γ/0n ) γn0 + γ0n γ/n0 ) γ0n γ/0n ) γn0

(20)

which can be satisfied only if

Re(γn0) ) Re(γ0n) Im(γn0) ) -Im(γ0n)

(21)

The real part of the dephasing rates describes the dephasing of excitation due to energy transfer, and the imaginary part describes a spectral shift of the excitonic transition due to dephasing. Its value can be strong enough (depending on the energy transfer rate, see below) that it should be taken into account also for the calculation on the steady-state spectra.



Henceforth we will denote γ′n ) Re(γn0) ≡ Re(γ0n) as the dephasing rate proper and γ′′n ) lm(γn0) ≡ -lm(γ0n) as the dephasing spectral shift of the nth exciton level. From eq 19 we immediately get a relation between the population relaxation and the dephasing rates:

γnn ) 2γ′n

1 p

∑a |Uan|2|Uam|2(Θ+aamn + Θ-aanm)

2

g(t) )

C ˜ (ω) -iωt e + ω2 C ˜ (ω) C ˜ (ω) 1 ∞ 1 ∞ (27) dω 2 - it dω -∞ 2π -∞ 2π ω ω

1 ∞ ∫0tdt′∫0t′dt′′ C(t′′) ≡ 2π ∫-∞dω

∫

(22)

which is generalizing the well-known relation 1/T2 ) 1/2T1 + 1/T/2 to the excitonically coupled system case. That is, in the excitonically coupled system the dephasing time of the excited exciton level is two times slower than its population relaxation time. In our analysis we have omitted from consideration the T/2 process which is usually very slow at low temperature as compared to the excitation dynamics. It can be included by using the quadratic approximation for description of the electron-bath interaction.42,43 Above we have assumed the site dependence of the bath variables, which means that each molecule behaves independently. In other words, the two-site correlation functions 〈Fˆ a(τ) Fˆ b(0)〉bath ) 〈Fˆ b(0) Fˆ a(τ)〉bath ) 0, if a * b. By this assumption it follows from eqs 14, 18, and 16 that the Fourier-Laplace transformed bath correlation functions are contributing to the energy transfer rates as the sum

Wmn )

the temperature. The line shape function can be expressed over the system-bath correlation function:24

where the tilde denotes the Fourier image of the time correlation ∞ dt eiωt〈Fˆ (t) Fˆ (0)〉bath. By introducing function C(ω) ) 1/p2∫-∞ the spectral density of the bath states

F(ω,T) )

∫-∞∞dτ eiω

nmτ

g(t,T) ) -

1 2π

FIωnm(〈Fˆ a(τ) Fˆ a(0)〉bath) (24)

where FIωnm denotes the Fourier image of the time correlation + function at ω ) ωnm. From (Γmnkl)* ) Γlknm it follows that + + Θaamn + Θaanm ) 2Re(Θaamn) ≡ 2Re(Θaanm). The imaginary part of Θ, which is necessary for calculation of the dephasing spectral shift γ′′n, can be obtained from its real part due to the causality principle (i.e., the integral in the Fourier-Laplace transforms runs from a finite initial time point), and by assuming that the Fourier image of the time correlation functions decays faster (or comparable to) than ω-1.44 At these conditions the Kramers-Kronig relation is applicable, and we get

Im(Θ+ aamn)

1 ) PP 2π

FIΩ(〈Fˆ a(τ) Fˆ a(0)〉bath) -∞ Ω - ωmn

∫

∞

(25)

F˜ (0,T) - F˜ (t,T) - itΛ(T) (29) where

F˜ (t,T) )

σa(ω,T) )

1 2π

∫-∞∞dτ exp[i(ω - ωge)τ - g(τ,T)]

(26)

where ωge is the frequency of the electron transition and T is

∫-∞∞dω F(ω,T)e-iωt

1 2π

F˜ (0,T) ) Λ(T) )

1 2π

1 2π

∫-∞∞dω F(ω,T)

∫-∞∞dω ωF(ω,T)

(30)

Here F˜ (0,T) ≡ S(T) is the temperature-dependent HRF and Λ is the reorganization energy. The spectral density of states can be derived in straightforward manner from the measured absorption spectrum. Inserting eq 29 into eq 26 gives a wellknown relation between the absorption spectrum and the spectral density of states:45

σa(ω) ) 1 2π

∫-∞∞dτ exp[i(ω - ω00)τ + F˜ (τ,T) - F˜ (0,T)]

(31)

where ω00 ) ωge - Λ denotes the 00-transition. Following ref 46, we expand exp [F˜ (τ,T)] into a Tailor series:

σa(ω,T) ) eF˜ (0,Τ)

[

1

∞ dτ exp[i(ω - ω00)τ] + ∫ -∞ 2π

1

where PP stands for the principal part of the integral. Thus, for calculations of the dephasing spectral shift, energy transfer and dephasing rates, we need only to establish a Fourier image of the time correlation function. We can link this Fourier image to the steady-state absorption spectrum. Let us consider a single molecule, immersed into an environment (protein), whose Hamiltonian is given by eq 5. Its absorption spectrum can be properly described only beyond the Markov limit. A cumulant expansion technique gives an expression for the absorption spectrum in terms of the so-called line shape function g(τ,T):24,42,45

(28)

ω2

∫-∞∞dω F(ω,T)[exp (-iωt) + iωt - 1] ≡

(23)

〈Fˆ a(τ) Fˆ a(0)〉bath ≡

C ˜ (ω,T)

the line shape function (eq 27) can be rewritten as

By taking into account a stationary character of the correlation functions we see that Θ+ aamn + Θaanm )

∫

∞

1

∑ ∫ dτ [F˜ (τ,T)]m exp[i(ω - ω00)τ] 2πm)1m! -∞ ∞

]

Then, by inserting into this equation the Fourier images of [F˜ (t,T)]m

[F˜ (t,T)]m )

1 (2π)

∞ ∞ ∞ dω1 ∫-∞dω2 ‚‚‚ ∫-∞dωm ∫ -∞ m m

m

ωi)t]∏F(ωi,T) ∑ i)1 i)1

exp[-i(

we obtain a very familiar expression for the impurity center absorption line shape:45,46

σa(ω,T) ) Φ0(ω,T) + Φ(ω,T)

(32)



TABLE 1: Expressions for Transfer Rates and Dephasing Spectral Shifts for an Excitonically Coupled Systema,b decay of population in exciton level n

γnn ) 1.18

∑|U

2 2 2 an| |Uak| ωnk Sa(T)f′′ a(ωnk,T)

k,a

energy transfer from level n to level m

Wmn ) 1.18

∑|U

an|

|Uam|2ωnm2Sa(T)f′′a(ωnm,T)

2

a

dephasing of level n due to energy transfer

1 γ′n ) γnn 2

dephasing spectral shiftc of the level n

γ′′n )

∑|U k,a

an|

|Uak| Sa(T)PP

2

2

∫

∞

-∞

Ω2f′′a(Ω,T) Ω - ωkn

dΩ

a f ′′(ω,T) ) the phonon function (see text) with its area normalized to 1; S(T) ) Huang-Rhys factor; k, n, m ) indexes for exciton levels; a ) index for molecule in coupled system; Uan ) amplitude of the nth excitonic wave function (Uan ) eigenvecs(HFrenkel)an); ωnm ) ωn0 - ωm0, where ωk0 is the energy of the exciton transition (from ground state to the kth exciton state), ωk0 ) eigenvals(HFrenkel)k; T ) temperature; HFrenkel ) Σaνa + Σa,b:a*bJab ) is the interaction matrix whose diagonal terms νa correspond to the location of electron transitions for different molecules, labeled by a,b. b Units: [ωnm, γ′′n, νa, Jab] ) cm-1; [f ′′] ) cm; [γnn, γ′n, Wmn] ) ps-1, [U] ) dimensionless; [S] ) dimensionless. c Shift of exciton transition ωn0.

Here Φ0(ω,T) ) e-S(T)δ(ω - ω00) is the zero-phonon line (ZPL), which is a δ-function due to the use of the linear approximation ∞ for the electron-bath coupling, and Φ(ω,T) ) ∑m)1 Φm(ω,T) is the phonon wing (PW), given over the phonon function:45,47

f(ω,T) ) Φm(ω,T) )

{

[nj(ω,T) + 1]f0(ω), [nj(-ω,T) f0(-ω),

if ω g 0 if ω e 0

(33)

1

∞ ∞ ∞ dω1 ∫-∞dω2 ‚‚‚ ∫-∞dωm Φ0(ω - ω00 ∫ -∞ m! m

m

∑ i)1

ωi)*

f(ωi,T) ∏ i)1

(34)

and nj(ω,T) ) [exp(pω/kT) - 1]-1 is the Bose-Einstein occupation number. The function f0(ω) is usually called a weighted density of phonon states, which is nonzero only for ω > 0. Thus, from comparison of these expressions to eqs 24 and 28 we see that

Θ+ aamn

+

Θaanm

)

2πωnm2f(ωnm,T)

(35)

The area of the phonon function is simply equal to the HRF:45

S(T) )

∫-∞∞dω f(ω,T)

(36)

which can be measured in HB experiments or can be obtained from site-selective fluorescence spectroscopy. The shape of the phonon function f(ω,T) can be derived from the measured PW by numerical solution of an integral equation:48

f(ω,T) )

[

eS(T) Φ(ω,T) -

1 ω

∫0ωdΩ f(Ω,T) ΩΦ(ω - Ω)]

(37)

On the other hand, the phonon function can be also derived from HB experiments, 3PE-peak-shift measurements of single pigments in the same environment (as its Fourier image), or calculated in the framework of the appropriate bath model. We have now obtained a completely closed set of working expressions for calculating all energy transfer rates, dephasing rates, and dephasing spectral shifts, given in terms of an experimentally measurable HRF and phonon function. All expressions are listed in Table 1. In addition, due to charge separation which occurs in one of the pigments with intrinsic rate Γx, the

Figure 1. Absorption spectrum of the PS II RC at 4.2 K used in this study. The inset shows the second derivative of the spectrum (reversed).

population decay and dephasing rates should be corrected to 1 2Γ′′n ) s|Uxn|2ΓCS and Γ′′n, respectively (see Appendix A). p III. Experiment Material and Methods. The PS II RC was prepared from spinach leaves according to the method of van Leeuwen et al.49 with slight modifications as described in ref 50. The sample was diluted 2-fold with glycerol to obtain a high optical quality at liquid helium temperature. The 0.5 mm thick cell of 15 µL volume was filled in the dark under a nitrogen atmosphere at 4 °C and was slowly cooled to 4.2 K by immersing it in the bath cryostat with liquid helium. The absorption spectrum (Figure 1) was measured at 4.2 K in situ directly before the PE measurements on a rapid-recording spectrophotometer “Omega10” (Bruins Instruments, Mu¨nchen, Germany). The second derivative of the absorption spectrum of the PS II RC shows the presence of “red” shoulder at 684.7 nm. The laser system included a mode-locked model 3800 Nd: YAG pump laser, an optical compressor model 3695 (both Spectra Physics, Mountain View, CA, USA) with frequency doubler, and a home-built single-jet chirp-compensated tunable synchronously pumped dye laser with intracavity dumper. The laser allowed to producing transform limited pulses in the range 615-720 nm (DCM dye was used). Analysis of the shapes of the ACF and the average of τp*∆ν for ≈103 pulses showed that the pulse shape corresponded to an asymmetric two-sided



exponential function.51 The pulse energy was 20-30 nJ with stability from pulse to pulse of (2%, while the typical duration and spectral width were τp ) 550 fs (fwhm) and ∆λ ) 0.8 nm, respectively. A helium-bath cryostat model B29 (Institute of Physics, National Academy of Sciences of Ukraine, Kiev) with pumping system permits measurements in the range of 1.3-4 K for ca. 10-12 h. The temperature was controlled by the vapor pressure of the helium atmosphere and was measured by a semiconductor transducer with an accuracy of (0.01 K. The cryostat contained six depolarization-free windows and a cell holder with the temperature transducer. The 2PE study was carried out in the typical one-color pump-probe scheme by using of the homodyne detection. The photon echo signal was registered in the k ) 2k1 - k2 direction by a photomultiplier (PMT) R374 (Hamamatsu, Hamamatsu City, Japan). The optical part of the setup had two equivalent arms, one for the 2PE measurement and the other for obtaining the cross-correlation function (CCF) of the pulses. The probe pulse was delayed by a delay line with a minimal step of 1 µm, the energy ratio of pump/probe pulses was 1.1/1. The beams were focused into the sample by a lens with focal length of 200 mm to a diameter in the sample of approximately 180 µm. The angle between the beams was 20 mrad. For optical noise suppression a spatial filter was used after the sample and a double lock-in-technique was applied. The absolute sensitivity of the 2PE setup was approximately 200 photons per shot in the echo pulse. The 2PE measurements were generally carried out at 1.33 ( 0.01 K with a 8 kHz repetition rate, an 81/46.3 Hz chopper frequency ratio and pump pulses with excitation energy density of 5 × 1012 photons/cm2. The accumulation time of the PE signal was typically 1 s per measured point in the decay trace with 0.2 ps step delay over a total scan range of 200 ps. The full radiation dose per measured trace was 14 J/cm2. Using the relevant data from the literature6 under our conditions a maximal triplet state population 10% was estimated, which is low enough to not significantly disturb the measurements. For comparison of measured and modeled decay traces one needs to remove the particular absorption dependence of the 2PE, because the generation and absorption of photon echo occurs simultaneously in the same medium. Therefore, the amplitudes of the measured 2PE traces were absorption corrected by using of the expression from ref 52

SPE(λ,∆t) ∝

[

A(λ)‚10A(λ)

100.5A(λ) - 10-0.5A(λ)

]

2

SP(λ,∆t)

(38)

where SP(λ,∆t) is the measured 2PE signal at wavelength λ at a delay time ∆t between the pump and the probe pulses, and A(λ) is the absorbance. The analysis of the obtained decay curves was started only 1.5 ps after the “zero point” (defined where pump and probe pulses are fully overlapping), since the dynamics of the 2PE signal around the “zero point” is very complicated (see below). Thus, the single and global data analysis on the data was performed without deconvolution procedure in a multiexponential fashion: n

SPE(∆t) )

Figure 2. 2PE traces, measured at 1.33 K. The inset shows the measured and absorption-corrected 2PE spectra at 1.5 ps delay.

(Ampl)i exp(-∆t/(τPE)i) ∑ i)1

(39)

where (τPE)i corresponds to the decay constants of the component with amplitude Ampli and n is the number of decay components. In the single analysis case (τPE)i ≡ f(λ) while for global analysis (τPE)i are independent of the wavelength λ.

Figure 3. (a) Full view of the 2PE decay trace at 680 nm. (b) 2PE trace around “zero delay”, recorded with resolution of 6.7 fs. (c) Comparison of the CCF (solid line) and maxima of the interference fingers in the 2PE signal (symbols).

Results. Figure 2 shows the 2PE traces of the PS II RC at various wavelengths. Long-lived decay components in the 2PE signals were observed in the region 674-688 nm. The experimental and absorption-corrected 2PE spectra are presented in Figure 2 (inset) at a 1.5 ps delay. At short wavelengths the 2PE signal showed only a very narrow peak, whose duration is comparable to the CCF. Figure 3 displays in detail a 2PE trace recorded at 680 nm. The signal, recorded around zero delay with a minimal delay step of 6.7 fs (Figure 3b), shows an interference structure of this peak. The maxima of the interference spikes correspond well to the CCF1.5(τ) of the laser pulses (Figure 3c). All kinetic traces contain such an intense and narrow



Figure 4. (a) Dephasing time distributions obtained from the single decay analysis. (b) Amplitudes of the decay components (symbols) and Gaussian fit (solid lines) with parameters listed in Table 3.

TABLE 2: Dephasing Decay Times τPE, Spectral Position of the Maxima λmax of Amplitude Distributions, and Their Widths ∆νFWHM, Recovered from the Measured 2PE Traces by Three-Exponential Fitting dephasing component fast middle slowa

τPE (ps)

λmax (nm)

∆νfwhm (cm-1) amplitudes (au)

2-6 680 116 ( 9 10-50 680.6 104 ( 7 150-500 680.9 81 ( 5 (684.8b) (20 ( 55)

1.6 ( 12% 1.7 ( 10% 1.3 ( 7% (0.8 ( 20%)

a By fitting with 2 Gaussian peaks. b Corresponds to the “red” peak in the absorption spectrum.

peak at zero delay, which could not be resolved well under our experimental conditions. Therefore, the analysis of the 2PE decays was carried out only starting 1.5 ps after zero delay where the contribution of this peak is negligibly small. The global analysis resulted in decay components τPE with 3, 30, and 184 ps. From Figure 3a (inset) and from measured traces with smaller delay step of 6.7 fs (not shown) it follows that a strong component with lifetime shorter than the pulse width (ca. 100-200 fs) is present in addition. This component cannot be determined exactly in this analysis under our experimental conditions. Figure 4a shows the wavelength dependence of the dephasing times, obtained from the threeexponential decay analysis of the single 2PE traces. One recognizes basically three groups of dephasing times that strongly depend on the wavelength. The fast component changes from 2 to 5 ps in the measured spectral range, the middle component from 10 to 50 ps, and the slow component from 150 to 500 ps. The corresponding absorption-corrected amplitude distributions for these components are plotted in Figure 4b. Here the solid curves are Gaussian fits with the parameters listed in Table 2. The slow component shows an additional peak at 684.8 ( 0.7 nm, whose location exactly corresponds to the red shoulder in the absorption spectrum (see inset in Figure 1). There were also some additional reproducible features in the 2PE decay traces observed. Around 682 nm the traces show oscillations with a relatively long period of ≈67 ps. This can also be seen in Figure 5a where the trace at 683 nm is shown. A Fourier analysis of the residuals displays a series of low frequency modes, equally spaced by ≈0.5 cm-1 (Figure 5b).

Figure 5. 2PE decay trace at 683 nm (a) after 1.5 ps delay and it residuals (difference signal to the exponent decays). (b) Fourier transform of residuals.

The oscillations have equal frequencies and phase at different wavelengths. We can clearly exclude an experimental artifact, which might occur, e.g., by multireflection of light pulses between the cryostat windows. In an analogous 2PE study on the antenna complex LHC II53 and in other measurements using the same setup under similar conditions no such oscillations were observed. The nature and origin of these oscillation modes is not clear at present. The 2PE-power dependence was measured at 680 nm at ∆t ) 1.5 ps by attenuation of the laser pulses energy and is shown in Figure 6a. In a simple fashion this dependence for temporally nonoverlapping excitation pulses with equal energies can be given as54,55



Figure 7. Arrangement of pigments in the RC core according to the structural model of Svensson et al.28

TABLE 3: Exciton Interaction Energies Jik (cm-1) for the PS II RC Core Model of Svensson et al.28 Chlacc2 Chl1P680 Chl2P680

Figure 6. Power dependence of the 2PE signal at 680 nm and delay of 1.5 ps, plotted vs excitation intensity (a) and vs pulse area (b). Solid line: fit with eq 41. For details see section III.

SPE ∝ Const‚µ [sin(µθ)(1 - cos(µθ))] 2

2

(40)

where µ is the relevant transition dipole moment, the constant is related to the particular measuring conditions and θ ) 1/p ∫-∞ ∞ E(t) dt is the electrical field pulse area, related to the measured energy density p of the laser light pulses as θ = x2τppω/p0nc (0 is the permittivity of vacuum, c is the light speed, ω is the laser field frequency, and n is the refractive index). For quantitative analysis this dependence should be averaged over the randomly distributed in space ensemble of the transition dipole moment vectors:

SPE ∝

1 2π

∫0πdφ ∫0πdΘ |µb(Θ,φ) sin(µb(Θ,φ)θB)[1 -

Chlacc2 Chl1P680 Chl2P680 Chlacc1 Pheo2inact Pheo1act

-74.3

IV. Modeling and Comparison with Experiment General Aspects. As can be seen from section II, for the modeling of the 2PE kinetics one needs the structural arrangement of pigments in the RC core and their spectral location. Unfortunately, the spatial structure of D1-D2 RC is not resolved with atomic resolution, and the exact pigment coordinates are still unknown. Our theoretical analysis is based on the structural model of the PS II RC core developed by Svensson et al.28 As proposed already by Tetenkin et al.,34 in this model the distance

Chlacc1 +17.4 -78.4 -101.6

Pheo2inact Pheo1act +86.7 +15 -3 -5.7

-5 -5.3 +13 +77.8 +2.1

between chlorophylls in the special pair P680 is increased relative to the one in the reaction center of the purple bacterium R. Viridis,3 which leads to a decrease of the interaction energy in P680. Thus, the similar neighbor interaction requires the excitonic calculations for the whole pigment assembly in the RC core. The first analysis of the PS II RC core spectral properties in the framework of the exciton model (using only the Frenkel Hamiltonian, without taking into account the electron-phonon interaction) has been carried out in ref 22, where the static diagonal disorder was already included. The information about the structure of the PS II RC core has been taken from the Brookhaven Protein Data bank, file 1DOP, and the corresponding arrangement of the pigments in the RC core is shown in Figure 7. The dipole-dipole interaction energy Ji,k between the ith and kth pigments was calculated in the pointdipole approximation:56

cos(µ b(Θ,φ)θ B)]|2 sin(Θ) (41) The experimental 2PE dependence, plotted vs the excitation pulse area, is shown in Figure 6b. As can be seen, it fits very well with eq 41 for µ ) 15 D, which is approximately 3 times higher than for a single Chl a molecule. The absolute accuracy of the measured energy density under our experimental conditions we estimate to be (30%, which gives an error of the recovered dipole moment of (3 D. This experimental result clearly indicates that the excitation energy is delocalized over ∼3 pigments in the Qy absorption region and supports the “multimer” model of the RC core,22 and also the structural model of Svensson et al.,28 which gives the same degree of delocalization (see below).

-16.4 +144.3

Ji,k )

b µ i‚µ bk |R Bik|

3

-3

(µ bi‚R Bik)(µ bk‚R Bik) |R Bik|5

(42)

Here b µi corresponds to the transition dipole moment of the ith pigment and B Rik is the distance vector between the ith and kth pigment, defined with reference to the Mg-atoms of the Chl’s, or to the geometrical center of the pheophytin chlorin rings. The transition dipole strengths µ2 for Chl a and Pheo a were assumed to be 23 and 13.8 D2, respectively.57 For comparing with previous calculations22 and experimental data,32,34-37 the obtained values for Jik in the Qy absorption region are given in Table 3. A result of the simplest excitonic calculation, performed for the nondisordered RC core with isoenergetic spectral position for all pigments and for the ensemble of disordered RCs, is shown in Figure 8. Here the probability of occupation for each pigment that corresponds to the square of the wave function amplitudes |Uia|2 is plotted vs the wavelength for different exciton states. As can be seen, the exciton interaction between pigments leads to the appearance of a quasi-multidimer structure. The upper and lower exciton transitions of the P680 dimer could be clearly resolved. They are located around 660 and 680 nm respectively, and are split by ∼20 nm. It allows us to conclude that the role of the exciton interaction among all pigments is



Figure 9. Calculated absorption stick-spectrum of the RC core (ensemble of 2000 RC’s with diagonal disorder of 210 cm-1). Solid line: excitonic calculation by using the Frenkel Hamiltonian only; dashed line: with dephasing spectral shift (see text).

Figure 8. Calculated probability of the pigment occupation in an undisordered RC core (a) and for the diagonally disordered ensemble (b) of 2000 RCs (probability is proportional to the map density).

very pronounced and for kinetic modeling the exciton model for the whole system is indeed required, at least for low temperatures. In our modeling we have also introduced a diagonal disorder of the pigment spectral location: νa ) 〈νa〉 + ∆νa, where 〈νa〉 denotes the average transient excitation energy of the pigments and ∆νa are the inhomogeneous offset energies, for which we take a Gaussian distribution with fwhm of 210 cm-1. The dipole strengths of the exciton transitions were calculated by using the formula from ref 58. The spectral positions of the exciton states were corrected by introducing the calculated dephasing shifts, for which the weighted density of phonon states f0(ω), derived from the measured and published PW profile33,59 was used, and the same HRF of 1.8 was applied for all pigments. Figure 9 demonstrates the significance of the dephasing spectral shift. There the absorption stick spectra, calculated for an ensemble of 2000 disordered RCs with nearly isoenergetic location of the individual pigment transitions at 〈νa〉 ) 14 850 cm-1 (see below), are plotted with and without the dephasing shift. It can be seen that the dephasing leads to a blue shift of the overall spectrum by approximately 2 nm (however, the individual transitions can be shifted either to the red and blue). Around 680 nm the model also predicts ∼3 times higher dipole strength with respect to a single Chl a molecule, thus supporting our experimental data. The degree of delocalization, which can be estimated in its simplest fashion by using the expression for the inverse participation ratio from ref 60 is also ≈3 pigments and agrees with the calculated one in the framework of the

“multimer” model.22 The calculated density of exciton states in the Qy absorption band is ∼0.4 per nm, thus under our experimental condition only one single exciton state will typically be excited. Energy Transfer Contribution. Our estimates show that the energy transfer in the D1-D2 RC core should be very faststhe population relaxation time, calculated in framework of this structural model, is ∼100 fs at 1.3 K in the Qy absorption band and also has a dispersive character. Because of disorder the ET rates are distributed, but not strongly enough to explain the presence of the long-lived components in the 2PE kinetics. These distributions are plotted in Figure 10 for several wavelengths. For comparison the distributions of charge separation rates are also shown. The population relaxation times are in agreement with the ones observed experimentally in the TA kinetics at room temperature (see Introduction). It should be pointed out, however, that our ET theory cannot be straightforwardly extended up to room temperature due to the used excitonic transform of the total Hamiltonian, which is applicable only for a sufficiently low pigment-bath interaction as compared to the pigment-pigment interaction. This fast energy transfer of the initial excitation, i.e., the equilibration process,19 could not be resolved in our experiments and contributes to high and narrow peak at zero delay times in 2PE kinetics (Figure 3). Thus, the observed slow decay components in the 2PE kinetics of D1D2 RC, stretched from a few ps up to 1 ns, reflect only the dephasing of the lowest exciton state in an individual RC complex from which uphill ET cannot occur at 1.3 K. Of course the nature of the lowest exciton state of each individual RC complex depends on the particular site energy position of individual pigments within the inhomogeneous distribution. 2PE Kinetics. By modeling of the 2PE kinetics we can separate the fast dephasing due to ET (equilibrium of excitation energy), and the relatively slow dephasing, which reflects almost exclusively the charge separation process. Because of disorder we observe both contributions in the 2PE signal at each wavelength. The wide separation of the time scales for the ET and CS processes allows us to calculate the 2PE signal in a delay window of 1.5-200 ps taking into account only the lowest exciton states of individual RC averaged over disordered ensemble. For analysis within a delay window starting from 1.5 ps we can use the so-called impulsive limit (i.e., the pump and probe pulses are well separated in time and sufficiently shorter than the dephasing times). Taking into account a



Figure 11. Measured (a) and calculated (b, c) 2PE traces at different wavelengths after 1.5 ps delay. (b) corresponds to the assumption that Chlacc1 is the primary electron donor, and (c), that one of the P680 Chls is the primary electron donor.

Figure 12. Comparison of measured (symbols) and calculated 2PE spectra (solid lines) at 1.6, 5, 15, 64, and 178 ps delay times. Figure 10. Calculated distributions of the population decay times (white bars) and charge separation times (black bars) for disordered ensemble of RC’s at different wavelengths.

relatively low density of exciton states, eq 15 is greatly simplified and we get for the 2PE signal, generated from the directly excited lowest state:

S2PE(∆t,λ) ∝ 〈(

∑i bµiUilow)4 exp[-2∆t/(T2)low]〉disorder+orient2

(43)

where 1/(T2)low ) Γ′′low + 1/T/2 and index “low” denotes the lowest exciton transition of each individual particle, located at the excitation wavelength λ within a spectral window of 0.8 nm (corresponding to the fwhm of the excitation light). The brackets denote an averaging over the disordered ensemble and over the space orientation of the randomly distributed dipole moment vectors. We have assumed T/2 ≈ 4-8 ns for all pigments61 (however, this is not a critical parameter). The calculations of the 2PE decays were carried out assuming that charge separation occurs only from one of the chromophores. The intrinsic charge separation time τCS was a fit

parameter. By calculating the 2PE traces it was found that the photon echo kinetics is highly sensitive to the assumption which particular pigment is the putative primary electron donor. The best agreement with the measured data was obtained by assuming that the charge separation occurs with a ∼1.5 ps lifetime and the primary electron donor is Chlacc1. Figure 11 shows the measured and calculated 2PE decays as a function of wavelength starting at a time delay of 1.5 ps. For better comparison, the measured and calculated 2PE spectra, taken at 1.6, 5, 15, 64, and 178 ps delay times, are plotted in Figure 12. As can be seen from these figures, agreement between the measured and simulated spectra is very good in this case. Figure 11c shows for comparison the 2PE kinetics simulated assuming that the primary electron donor is one of the Chls in the “special pair” P680. In this case the kinetics shows only very fast components (with ∼2ps dephasing time) in vast disagreement with experimental observations. Variation of the charge separation time in the 1-100 ps range in that case does not give any agreement between simulated and measured 2PE kinetics. For all the other assumed electron donor pigments the overall agreement between simulated and experimental data is also very poor and unsatisfactory. The most drastic deviations were in

11574 J. Phys. Chem. B, Vol. 104, No. 48, 2000 fact obtained if any of the two “special pair” chlorophylls, i.e., the traditionally assumed P680 electron donor, was assumed to be involved in primary electron transfer. The simulations of the 2PE kinetics were performed assuming the identical pigment spectra, except for the primary electron donor, Chlacc1, for which the best agreement between measured and modeled 2PE kinetics was obtained when 2 nm blue-shifted in respect to the other maxima. Any small shift (1 nm of zeroenergy pigment spectral positions did not perturb markedly the 2PE spectrally temporal behavior, however. From eq 43 it follows that the 2PE kinetics have a pronounced multiexponential character due to the diagonal disorder with a wide continuous distribution of dephasing times. However, if we analyze the simulated 2PE kinetics with a small number of decay components, as performed in section III for analysis of the experimental data, we can also well reproduce the apparent wavelength dependency of the dephasing times. V. Discussion and Conclusions The main result of this work is the assignment of the nature of the primary electron donor in the PS II RC to be the accessory chlorophyll. Within our model calculations the case when one of the accessory chlorophylls is assumed as the primary donor is drastically differ from all the others (see above), and is the only assignment that yields an excellent agreement with the experiment. It is now important however to discuss the limits within which this assignment is correct. There are two important parameters to consider: the structural arrangement of the pigments, or more specifically the distance between the “special pair” pigments “P680”, and the degree of static disorder. We used the Svensson et al.28 structural model for our calculations. In this model as well as in the “multidimer” model of Durrant et al.22 the distance between chlorophylls in the so-called special pair “P680” is increased substantially as compared to the bacterial RC. This is a decisive point since it leads to a strong decrease of the dipole-dipole interaction in the “P680 dimer”. Thus, the interaction energy becomes comparable to those between P680-chlorophylls and accessory chlorophylls. At these conditions the delocalization of excitation over most of the RC will be pronounced, and thus the effect of redistribution of the CS process among excitonically coupled pigments, which leads to dispersive CS kinetics, will be also significant. Small changes of other structural parameters, e.g. orientation of transition dipole moments or exact pigment locations relative to the P680chlorophylls, will not change the conclusion. Indeed the recent structural data30 as well as spectroscopic data show convincingly that the distance between chlorophylls in P680 of the PS II RC is increased as compared to the one in the bacterial reaction center (approximately 11 Å compared with 7.6 Å). Moreover, the relatively weak interaction energy between these chlorophylls (140-180 cm-1) is confirmed by a large number of experimental data.32,34-37 Thus, the structural model used for our calculations reflects properly the main structural features of the native PS II RC irrespective of any future more precise structural information. The second important requirement that must be fulfilled in order for our conclusion to be correct is the degree of static disorder. Because of disorder, whose degree (∼200 cm-1) is comparable to the interaction energy (as follows from HB experiments), the CS process will be distributed over a wide range of lifetimes running from the intrinsic CS time up to ∼1 ns. Thus, this requirement is also fulfilled. It is important to note that within these principal limits the exact numbers for these parameters (i.e., distance of special pair pigments and disorder) will not change the assignment of the primary electron

Prokhorenko and Holzwarth donor. In the other extreme case for the RC (i.e., a strongly coupled special pair, and a relatively small disorder), the excitation would be delocalized only over the special pair, and the CS kinetics should have a nondispersive behavior without significant distribution of the CS times. This means, that the effective (observable) CS time would be much closer to the intrinsic one. This latter situation is clearly not realized in the PS II RC. Energy Transfer. Our experimental and theoretical results clearly demonstrate that the energy transfer in the D1-D2 RC core is very fast and appears in the 2PE kinetics at early delay times as a high and narrow peak. Under our limited time resolution we could not separate the fast energy transfer on one hand, and the initial fast dephasing due to pigment-protein interaction (non-Markov process), which also contributes to the 2PE kinetics at zero delay times. As can be seen in Figure 2, the narrow peak is present also at 686 nm, were we cannot expect any energy transfer at low temperature. However, our model calculations for the energy transfer rates explain convincingly the presence of the fast subpicosecond equilibration component as observed in the TA kinetics around room temperature.15,18-20 Fast but not very well resolved processes with characteristic lifetimes of ∼40-80 fs was also seen in the TA kinetics over a wide temperature range from 20 K up to 240 K in ref 17 and in ref 62 at 77 K (470 fs component in DAS spectra for excitation at 670 nm). The presence of the fast equilibration process was supported by the theoretical modeling of Leegwater et al.,21 where a simple exciton model, including electron-phonon interaction with an arbitrarily chosen density of states, was employed for analysis of anisotropy in the TA kinetics. In contrast to this work and some other publications63,64 we have here connected the energy transfer rates to the experimental properties of single molecules, immersed into their environment, such as e.g. the HRF and PW. However, our theoretical analysis of the energy transfer rates was performed assuming a linear electron-bath coupling. As it was mentioned in section II, including a quadratic approximation leads to the appearance of the so-called pure dephasing process (which is completely absent in the linear approximation, see, e.g., ref 65) but does not change the energy transfer rates. Our theory presented here is evaluated in the Markov limit, thus the expressions for the ET rates are applicable only if the calculated relaxation times are longer than the characteristic bath memory time. The latter can be estimated as a characteristic decay time of the system-bath correlation function C(t). Recent experimental 3PE-peak-shift measurements66-68 and theoretical calculations69,70 of these functions for different systems show that at room temperature the molecules loose their coherence within the first few femtosecond, and the lower limit for the application of the derived transfer rate expressions can be estimated to be ∼50 fs at low temperatures. Charge Separation Kinetics. In numerous studies of the primary charge separation process in the PS II RC by means of time-resolved spectroscopy11,12,15-17,20,27,62 the shortest decay components typically were assigned to the charge separation lifetime in a more or less arbitrary manner. In the present study we show convincingly that the fastest stage in the pump-probe kinetics corresponds to the fast energy transfer in the RC core. We show furthermore that the exciton interaction between pigments leads to a wide distribution of the CS rates over a time scale from 1.5 ps up to several nanoseconds. For these reasons the specific interpretation of Groot et al.17 of a ∼400 fs component at 240 K as a charge separation lifetime must be questioned. Similarly a kinetics of ∼2.6 ps at 20 K, that was



recovered by using the global analysis procedure, was also arbitrarily assigned to charge separation. Likewise, the assignment of a discrete charge separation time by HB investigations8-10 is also inadequate within the scope of our results. In view of the distribution of the effective charge separation lifetimes and their wavelength dependence global analysis cannot be applied. Thus, the assertion that the charge separation process should accelerate with temperature17 also appears doubtful and requires further investigations. The Role of “P680”. The Chl a dimer, traditionally called the “special pair” P680 and assumed to be primary electron donor, consists of two relatively weakly coupled Chls (Figure 7) as proposed originally by Tetenkin et al.34 In this study we demonstrate that the two Chls in the “P680” dimer do not carry the primary electron donor function. Our data provide clear evidence that at least at low temperature, the primary donor is the monomeric Chlacc1 pigment. The monomeric nature of the primary donor was also proposed already by Tetenkin et al.34 though they did not imply the Chlacc1 pigment with that. This is the first time for PS II RC where the accessory monomer Chl is shown to be the primary donor. We note that for bacterial RCs the monomeric Bchl can in principle act efficiently as a primary donor71 although in the wild-type RC of bacteria this process does not give a high yield because of the high energy of the Bchl/monomer state and the rapid competing energy transfer to the special pair. However in the PS II RC all pigments excited state are nearly degenerate and thus such competing depopulation processes are not pronounced. Thus, for low temperatures the Chlacc1 pigment seems to be the regular electron donor. Although at room temperature the same mechanism might still be valid, we cannot finally conclude this from the present data. Proposing Chlacc1 as the primary electron donor poses the question of the secondary electron transfer step(s). Because of loss of coherence PE spectroscopy is sensitive only to the first step in the charge separation; i.e., we cannot observe the system after that. In the TA kinetics the entire dynamics is observed. This makes it difficult to separate the various processes, however combining both time-resolved techniques allows to separate these steps. We have investigated the primary steps in charge separation by means of one-color TA spectroscopy at 1.3 K.72 From analysis of the long-lived component in the TA kinetics and comparison of the decay times to the 2PE data we conclude that the charge separation occurs in two steps, where one of the Chls in “P680” is the secondary electron donor, which reduces Chl+acc1 with a time constant of ∼25 ps. This can be schematically shown as

(RC core)*′

w

100÷500 fs (equilibrium)

(RC core)*

number of pigments this effect is not dominant. However, in the CP47 antenna complex, which is not as strongly coupled as the D1-D2 RC core and is characterized by an approximately 2-3 times smaller HRF, we can expect a strong dispersion of the energy transfer rates which leads to dispersive behavior of dephasing. A similar effect we also observed in the dephasing wavelength dependence of the LHC II antenna complex.53 The theoretical result obtained in this study applies not only to the primary charge separation in the PS II RC but has a much wide relevance. The effect of the lifetime distribution in an excitonic system is directly applicable to other ultrafast photochemical reactions. An example that is close to photosynthesis is a quenching problem that occurs in the engineering of artificial antenna complexes as aggregates with a large number of excitonically coupled molecules.74 As follows from the lifetime distribution effect, incorporating a quencher molecule with fast intramolecular nonradiative decay, does not necessarily lead to equivalent lifetime shortening of the whole antenna. The theory of the FID and energy transfer in excitonically coupled system, developed in the present work, is evaluated specifically for describing a chemical reaction. However, this theory is also fully applicable for a nonreactive stable system (so-called “closed system”) by setting the reaction probability to zero. Appendix A Assuming that the 2PE signal is very weak and does not affect the medium polarization, an electrical field in eq 13 can be given as the sum of two pump (1) and probe (2) fields, separated in a wave-vector space:

1 r r B E(t,r) ) B + e-i(ωLt-kB1b) ]+ E ˜ (t)[ei(ωLt-kB1b) 2 1 1B r r + e-i(ωLt-kB2b) ] (A1) E ˜ (t)[ei(ωLt-kB2b) 2 2 where ωL is a mean frequency of laser light, E ˜ 1,2(t) are the pulse envelopes and B k1,2 is the wavevectors of the propagating fields. In the RWA the off-diagonal density matrix elements can be given as

(A2)

By plugging eqs A1 and A2 into the system eq 13, and assuming that the exciton states are well separated as compared to the laser spectrum width, we can neglect of the contribution of Fkn (k * n * 0) which givs the fast oscillatory terms, and get

σ˘ 0N ) -(iδN + Γ2N)σ0N +

w

∼1.5 ps (intrinsic)

i B ˜ 1e-ikB1br + m b (F - F00)(E 2p N NN B E ˜ 2e-ikB2br )

/ σN0 ) σ0N

(“P680” Chl+acc1Pheo-1) w ("P680+′′ Chlacc1 Pheo-1) ∼25 ps

where “P680” denotes one from the two Chls in the “special pair”. Wavelength Dependency. The dispersive behavior of dephasing was observed earlier in disordered molecular aggregates73 and in the CP47 antenna complex of the PS II.37 Unlike the RC core, the J-aggregates with a large number of coupled molecules have a much higher density of exciton states. In ref73 the observable dispersive dephasing was assigned to dispersion of the pure dephasing T/2, determined by the total density of vibrational molecular modes, like in the molecular crystals.31 Our investigation shows that in the coupled system with a small

FN0(t) ) σN0(t)e-iωLt

F0N(t) ) σ0N(t)eiωLt

F˘NN ) -Γ1ΝFΝΝ +

i

B ˜ 1eikB1br + E ˜ 2eikB2br ) m b N[σ0N(E 2p B ˜ 1e-ikB1br + B E ˜ 2e-ikB2br )] + σN0(E

∑ WNnFnn

n*N

i F˘00 ) 2p

∑N mbN[σ0N(EB˜ 1eikB br + BE˜ 2eikB br ) - σN0(EB˜ 1e-ikB br + 1

2

1

B E ˜ 2e-ikB2br )] (A3) Here δN ) ωL - (ωN0 + γ′′N) denotes a tuning of the laser field from the transition of the Nth exciton state, shifted by γ′′N due to dephasing, and Γ2N ) Γ′′N + γ′N, Γ1N ) Γ′′N + γNN are the



dephasing and relaxation population rates, respectively. The solution of the system eqn. A3 can be obtained analytically to third order in the applied field: F(E3). Let us introduce an inversion ∆N ) FNN - F00. At initial conditions t f -∞, F00 f1 in zero order for the inversion we have ∆(0) N ) -1. Thus, the solution of eq A3 for the off-diagonal elements in first order is

i (1) σ0N )2

∑q e-ikB br [Iq,N(t) X e-(iδ +Γ q

N

2N)t

]

(A4)

B where Iq,N ) mN/pE ˜ q(t) cos(m b N,E ˜ q), the index q ∈ [1,2] denotes the laser pulse, and the symbol X denotes a time-dependent t convolution integral: f1 X f2 ) ∫-∞ dt′ f1(t′) f2(t - t′). By plugging eq A4 (and its complex conjugate) into system eq A3 we get in second order for the diagonal elements of F: (2) F˘(2) NN ) -Γ1NFNN +

F3(2) 00 ) -

∑

0 WNnF(2) nn + SN + n*N ei(kB2-kB1)br SNBk 2-kB1 + ei(kB1-kB2)br SNBk 1-kB2

∑n [S0n + ei(kB -kB )br SBnk -kB 2

1

2

1

(A5.1)

+ ei(kB1-kB2)br SNBk 1-kB2] (A5.2)

Here the term S0N describes a scalar contribution to the population and the last term describes an amplitude grating due to interference of two electrical laser fields:

S0N )

1

Iq,N[Iq,N X e-(iδ +Γ ∑ 4 q N

2N)t

1 SKBk 2-kB1 ) [I2,N(I1,N X e-(iδN+Γ2N)t) + I1,N(I2,N X e(iδN-Γ2N)t)] 4 1 SNBk 1-kB2 ) [I1,N(I2,N X e-(iδN+Γ2N)t) + I2,N(I1,N X e(iδN-Γ2N)t)] 4 (A6) The solution of eq A5.1 can be expressed as i(k B2-k B1 ) b r BN + ei(kB1-kB2)br CN F(2) NN ) AN + e

where for coefficients AN, BN, CN the corresponding expressions can be given in a vector form:

(A7)

sN ) {S0N, SBkN2-kB1,SBkN1-kB2}T. with vectors b aN ) {AN, BN, CN}T and b The decay matrix is

{

WNn, W ˜ ) -Γ 1N,

n*N n≡N

(A8)

The system eq A7 has a trivial solution at initial conditions t f -∞, b a f 0:

b aN )

∑n ∑k Ln,kN(bs n X eλ t) k

a0 ) where vector b a0 ) {A0, B0, C0}T is b for inversion in second-order we get

t ∑n∫-∞ dt′b s n(t′).

(A10) Thus,

i(k B2-k B1)b r (2) (2) (BN + B0) + ∆(2) N ) FNN - F00 ≡ AN + A0 - 1 + e

ei(kB1-kB2)br (CN + C0) (A11) Inserting result into eq A3 for the off-diagonal elements get an expression in the third order of the applied field:

i (3) (3) ) -(iδN + Γ2N)σ0N + ∆(2) σ˘ 0N N 2

∑q Iq,Ne-ikB br q

(A12)

As can be seen from this expression and from eq A11, the offdiagonal elements of the density matrix can be also expressed as the sum of different components in a wave-vector space: (3) (3) (3) (3) (3) σ0N ≡ σ0N (k B1) + σ0N (k B2) + σ0N (2k B1 - B k2) + σ0N (2k B2 - B k1). With respect to the 2PE formation we should take only components that correspond to the direction of the PE propagation, that is, the last two terms. They are mirror symmetric in the time domain, thus we need to analyze only one of them. For the 2k B2 -B k1 direction we get from eq A12: (3) (3) (2k B2 - B k 1) ) -(iδN + Γ2N)σ0N (2k B2 - B k 1) + σ˘ 0N i (C + C0)I2,N (A13) 2 N (3) with a trivial solution at initial conditions t f -∞, σ0N f 0:

i PE+ (3) σ0N ≡ σ0N (2k B2 - B k 1) ) [I2,N(CN + C0)] X e-(iδN+Γ2N)t 2 (A14)

+ Iq,N X e(iδN-Γ2N)t]

b aN ) W ˜ ‚a bN + b sN

i(k B2-k B1)b r F(2) B0 - ei(kB1-kB2)br C0 00 ) 1 - A0 - e

(A9)

where LNn,k ) TNk(T-1)kn, the vector B λ corresponds to the eigenvalues of W ˜ , and the matrix T denotes its eigenvectors. Solution of eq A5.2 follows immediately after integration with initial conditions t f -∞, F(2) 00 f 1:

k1 direction. Finally, where the index PE+ denotes the 2k B2 - B for the off-diagonal density matrix elements we obtain

i F0N ) eiωLt[I2,N(CN + C0)] X e-(iδN+Γ2N)t 2

(A15)

and FN0 is complex conjugate to F0N. With respect to the analysis of the present experimental data, we need to consider two important cases: (1) the laser pulse duration is much longer than the dephasing time and (2) it is much shorter than that. For both cases, we assume that the spectrum of laser light pulses can overlap only with one of the exciton states, labeled by index Q (due to a low exciton density of states). (1) In this case the system eq A3 can be directly solved in the so-called stationary approach which gives for the offdiagonal elements of the density matrix:

F0Q )

2(iΓ2Q - 2δQ)

I 2I eiωLt 2 2 1,Q 2,Q

(Γ2Q + 4δQ ) 2

(A16)

After integrating over the laser pulse spectrum and using eq 1 we obtain for the 2PE signal:

S2PE(τ) ∝

1 Γ2N

2

mQ8

∫-∞∞dt AL2(t) AL(t - τ)

(A17)

where AL(t) is the laser pulse intensity envelope and τ the separation between pulses. As can be seen, the integral in eq A17 is simply proportional to the cross-correlation function of laser pulses and can be approximated as S2PE(τ) ∝ [CCF(τ)]1.5. (2) Here we can pick out from the convolution integrals the decay terms. For pulses, well separated in time by delay τ, the upper limit in the convolution integrals can be replaced by ∞



which simplifies coefficient SBkN1-kB2 in eq A6 to

1 SNBk 1-kB2 = I2,Ne(iδN-Γ2N)t 4

∫-∞∞dt′

1 I1,N(t′)e-iδNt′ ≡ I2,Ne(iδN-Γ2N)tSp(ω′′N0 - ωL) (A18) 4 where Sp(ω) denotes the spectral shape of the electrical laser pulse and ω′′N0 ) ωN0 + γ′′N is the transition frequency of Nth exciton state, shifted due to dephasing. The first term in SBkN1-kB2 vanishes because of time separation of pulses. A narrow spectrum of laser light separates only the Q-transition from the whole manifold of exciton states, and for coefficient C0 we get

∫-∞t dt′ I2,Qeiδ t′

1 C0 ) Sp(ω′′Q0 - ωL)e-Γ2Qτ 4

Q

(A19)

For coefficient CQ in eq A15 we have

1 CQ ) Sp(ω′′Q0 - ωL)e-Γ2Qτ[ 4

Q λ (t-τ) e ]∫-∞dt′ I2,Qeiδ t′ ∑k LQ,k k

t

Q

(A20)

By plugging these expressions into eq A15 and replacing the upper limit in the convolution integral by ∞ (the PE pulse is delayed relative to the second laser pulse) we have

i F0Q ) eiωLtSp3(ωL - ω′′Q0)eiδQ(2τ-t)-Γ2Qt 8

(A21)

where we assume that the laser pulses have identical and symmetrical time envelopes, i.e., Sp(ωL - ω′′Q0) ) Sp(ω′′Q0 ωL). From this expression one can see that the PE pulse is located at 2τ delay, and the laser line shape selected only a small portion of the whole inhomogeneously broadened ensemble. This means that the duration of the PE pulse determined by the duration of laser light pulses. Acknowledgment. We thank Mr. M. Reus for preparing the samples, Mr. N. Dickmann, and Mr. H-V. Seeling for able technical assistance. This work was supported in part by the Deutsche Forschungsgemeinschaft, through Sonderforschungsbereich 189, Henrich-Heine-Universita¨t Du¨sseldorf and MaxPlanck-Institut fu¨r Strahlenchemie, Mu¨lheim a.d. Ruhr. References and Notes (1) Kobayashi, M.; Maeda, H.; Watanabe, T.; Nakane, H.; Satoh, K. FEBS Lett. 1990, 260, 138. (2) Eijckelhoff, C.; Dekker, J. P. Biochim. Biophys. Acta 1995, 1231, 21. (3) Deisenhofer, J.; Epp, O.; Miki, K.; Huber, R.; Michel, H. Nature 1985, 318, 618. (4) Takahashi, Y.; Hansson, O.; Mathis, P.; Satoh, K. Biochim. Biophys. Acta 1987, 893, 49. (5) Durrant, J. R.; Giorgi, L. B.; Barber, J.; Klug, D. R.; Porter, G. Biochim. Biophys. Acta 1990, 1017, 167. (6) Groot, M.-L.; Peterman, E. J.; van Kan, P. J. M.; van Stokkum, I. H. M.; Dekker, J. P.; van Grondelle, R. Biophys. J. 1994, 67, 318. (7) Nanba, O.; Satoh, K. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 109. (8) Jankowiak, R.; Tang, D.; Small, G. J.; Seibert, M. J. Phys. Chem. 1989, 93, 1649. (9) Tang, D.; Jankowiak, R.; Seibert, M.; Small, G. J. Photosynth. Res. 1991, 27, 19. (10) Tang, D.; Jankowiak, R.; Seibert, M.; Yocum, C. F.; Small, G. J. J. Phys. Chem. 1990, 94, 6519. (11) Wasielewski, M. R.; Johnson, D. G.; Govindjee; Preston, C.; Seibert, M. Photosynth. Res. 1989, 22, 89. (12) Wasielewski, M. R.; Johnson, D. G.; Seibert, M.; Govindjee Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 524. (13) Greenfield, S. R.; Seibert, M.; Govindjee; Wasielewski, M. R. J. Phys. Chem. B 1997, 101, 2251.

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