Temperature dependent exciton–exciton annihilation in the LH2 antenna complex

Chemical Physics 357 (2009) 140–143

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Temperature dependent exciton–exciton annihilation in the LH2 antenna complex Ben Brüggemann a,*, Niklas Christensson b, Tõnu Pullerits b a b

Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany Department of Chemical Physics, Lund University, P.O. Box 124, SE-22100 Lund, Sweden

a r t i c l e

i n f o

Article history: Received 19 June 2008 Accepted 2 December 2008 Available online 10 December 2008 Keywords: Exciton–exciton annihilation Photosynthetic antenna system LH2 Pump-probe spectroscopy Internal conversion

a b s t r a c t Two-color pump-probe measurements of the peripheral light harvesting complex LH2 of Rb.sphaeroides reveal strong temperature dependence of the annihilation rate. The experimental results were modeled via multi-exciton density matrix theory. Based on available literature data we can set an upper limit for the feasible intramolecular internal conversion rate. We show that this also restricts the possible values of the still ill-determined energy of the doubly-excited molecular level of the bacteriochlorophyll, which is responsible for the annihilation process. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Basic research of ultrafast primary stages of photosynthesis gives fundamental understanding of one of the most important natural processes on the Earth [1]. It also provides a versatile and well controlled test field for theories of various events in complex molecular systems [2,3]. For example the temperature dependence of primary electron transfer in photosynthetic reaction centre [4] was well described by the inverted region of Marcus Theory [5]. Spectral band-shifts of carotenoid molecules upon excitation of other nearby pigments in light harvesting complexes were explained by Stark shift [6,7]. Energy transfer after light absorption in different antenna complexes has been described using many approaches ranging from weak electronic coupling Förster theory [8] to various flavors of strong coupling exciton description [9,10]. Polaron formation has been identified as laying behind some low temperature observations [11,12] and recent coherent multidimensional electronic spectroscopy results have been discussed in terms of coherent excitation dynamics in the antenna complexes [13]. Here, we are interested in a process called exciton–exciton annihilation. It takes place when two excitations are simultaneously generated within a so called domain – the part of an antenna within migration radius of an excitation [14,15]. If two excitations come into close proximity, they may fuse forming a doubly-excited state. From this state fast internal conversion to the first excited state follows reducing the number of excitations. The process has little significance for the natural photosynthesis

* Corresponding author. Tel.: +49 30 2093 4996; fax: +49 30 2093 4725. E-mail address: [email protected] (B. Brüggemann). 0301-0104/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2008.12.001

since the ambient sun intensities seldom generate two excitations simultaneously in a domain. However, annihilation conditions are easy to create by pulsed laser excitation and it can be utilized to study size, connectivity and electronic structure of photosynthetic antenna systems [15–17] and other molecular complexes [18–21]. It was shown recently that the annihilation-related fifth order terms may interfere with the common third order terms in coherent multidimensional spectroscopy experiments [22]. The effect can substantially distort the experimental outcome since excitation intensities in such experiments are usually quite high. In the present work we study temperature dependence of the excitation annihilation in the peripheral light harvesting antenna (LH2) of photosynthetic purple bacterium Rhodobacter (Rb.) sphaeroides. Multi-exciton density matrix formalism is used to describe the experimental kinetics. The outcome is formulated in terms of temperature dependence of the intramolecular internal conversion rate in bacteriochlorophyll (BChl) molecules. 2. Experimental section The experimental set-up has been described in detail previously [19]. In brief: 100 fs pulses from a regenerative Ti:Sa amplifier are used to pump an optical parametric amplifier (TOPAS). The idler beam is frequency doubled in a thin BBO crystal to generate the pump wavelength of 850 nm. Pulse duration was 100–130 fs. Part of the amplifier output is split off and focused in a sapphire plate to generate continuum which serves as a probe. A monochromator and a set of diodes were used to detect the transient absorption at 865 nm (stimulated emission) with a bandwidth of 3 nm. The pump beam was polarized at magic angle relative to the probe.

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The intensity of the pump pulses was controlled by neutral density filters and corresponded to approximately 1013 —1014 photons per pulse cm2. A continuous flow cryostat was used for the low temperature studies using either LN2 or LHe as the cryogenic medium. LH2 complexes of Rb.sphaeroides were obtained from prof. R. Cogdell (University of Glasgow, UK). The concentrated complexes were dissolved in Tris-buffer with a pH of 8.3. LDAO (lauryldimethylamine oxide) (0.1%) was added to the solution to prevent aggregation of the complexes in the stock solution. Glycerol was added in 2:1 ratio (by volume) to the buffer solution to obtain optical quality glass. Additional LDAO (total 1%) was added to the glycerol– buffer mixture to avoid aggregation in the sample. The mixture was gently stirred on a mechanic stirrer until the two fluids had merged. The sample was then quickly transferred to a 3 mm plastic cuvette and inserted into the cryostat. Not more than 5 min elapsed from adding the glycerol until start of the cool down. The samples in all experiments had an optical density of 0.4 at the peak of absorption at 850 nm. 3. Theory of exciton–exciton annihilation Since the annihilation process interrogates the higher lying states of the molecules it is necessary to go beyond a two level approximation. Each chromophore is modeled as a three level system using the ground state S0 , the first excited state S1 and an higher excited state Sn which is responsible for the excited state absorption from the S1 state. The transition energies are given by 1 for the S0 ! S1 transition and 2 ¼ 1 þ D for the S1 ! Sn transition introducing the energy shift D  1 . The lowest local states of the whole LH2 antenna system states are besides the ground state j0i the states where e.g. only the mth chromophore is in the S1 state j mi. States with two excitations follow: the states where e.g. chromophores m and n are in the S1 state j m; ni and the states with a Sn excitation of e.g. the mth chromophore j m; mi. The later states have approximately twice the S1 energy and can be seen as molecular doubly-excited states. All other chromophores are in the S0 ground state. The diagonalization of the excitonic Hamiltonian gives rise to the delocalized exciton states. For the one-exciton manifold we P get j a1 i ¼ n C a1 ðmÞ j mi, and for the two-exciton manifold P j a2 i ¼ m;n C a2 ðm; nÞ j m; ni using the respective exciton expansion coefficients C a1 ðmÞ and C a2 ðm; nÞ. It should be noted that the twoexciton states include monomeric Sn as well as two separate S1 excitations on two chromophores in a single complex. The laser pulse induced time evolution of the exciton states is performed using a multi-exciton density matrix theory. The details of the approach are presented in Ref. [23–25]. The electric field of the laser pulse is included explicitly in the time evolution scheme. Here, we will focus on relaxation processes within the two- and the one-exciton states as well as between the two-exciton states and the one-exciton states. The intra-manifold relaxation rate between the N-exciton state aN and the N-exciton state bN is given by ðN ¼ 1; 2Þ

for the one-exciton states, and by

X

Jða2 b2 ; b2 a2 ; xÞ ¼

C a2 ðk; mÞC b2 ðk; mÞ

k–m;k–n

 C b2 ðk; nÞC a2 ðk; nÞjðxÞ

ð3Þ

for the two-exciton states [26]. Here we have made the assumption of a local uncorrelated spectral density. Therefore both Eqs. (2) and (3) include the following single chromophore spectral density

jðxÞ ¼

  1 x2 x=x1 x2 x=x2 e þ e ; 3 3 2 x1 2x2

ð4Þ

with x1 ¼ 7 cm1 and x2 ¼ 30 cm1 . The single molecular spectral density used here is a simplified version of the one used previously [23–25]. It is derived from hole-burning [27] and fluorescence line narrowing [28] experiments on LH2. The exciton–exciton annihilation rate ka2 !b1 from the two-exciton state a2 to the one-exciton state b1 is proportional to the square of the overlap of an one-exciton wavefunction at a molecule and the doubly-excited molecular part of the two-exciton wavefunction at the same molecule. Contributions from all molecules need to be added multiplied by the internal conversion rate kIC between the twofold excited state and the first excited state of the molecule. We obtain

ka2 !b1 ¼

X

j C a2 ðn; nÞC b1 ðnÞj2 kIC :

ð5Þ

n

As discussed in Ref. [25] in detail the annihilation rate is given here by the local overlap of the two delocalized wavefunctions. The internal conversion rate kIC from the Sn state to the S1 state of a BChl has the following form [3,29]

kIC ¼

2p j Hna j2 Dð2 Þ h 

ð6Þ

with the nonadiabatic coupling term Hna . The temperature dependent density of states Dð2 Þ is given as

  1 EA Dð2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp   ; 4kB T 4pkkB T

ð7Þ

where the activation energy EA ¼ ð2  2kÞ2 =4k can be expressed in terms of the reorganization energy k and the transition energy 2 (see Fig. 1). Note that an effective temperature has been introduced to describe the appropriate low temperature behavior according to Ref. [29]

kB T  ¼

1  =2kB TÞ  cothðhx hx 2

ð8Þ

. using the mean vibrational energy h x

S

n

2

kaN !bN ¼ 2pX ðaN ; bN Þð1 þ nðXðaN ; bN ÞÞÞ

λ

 ðJðaN bN ; bN aN ; XðaN ; bN ÞÞ  JðaN bN ; bN aN ; XðaN ; bN ÞÞÞ; where nðXÞ denotes the Bose–Einstein distribution, and  hXðaN ; bN Þ the energy difference between the two states. The spectral density JðaN bN ; bN aN ; XaN ;bN Þ describes how effective the energy difference between the states j aN i and j bN i can be deposited into vibrational modes both of the chromophore and the protein matrix. It is given by

Jða1 b1 ; b1 a1 ; xÞ ¼

X m

j C a1 ðmÞC b1 ðmÞj2 jðxÞ

ð2Þ

E

A

ð1Þ

ε2 S1 Fig. 1. Sketch of the S1 and Sn potential energy surfaces with the transition energy 2 , the reorganisation energy k, the activation energy EA and the mean vibrational . energy  hx

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4. Results and discussion The two color pump-probe kinetics at different temperatures are shown in Fig. 2 (pump pulse at 850 nm, probe at 865 nm). The room temperature kinetics have a clear fast component due to the annihilation. This component slows down with decreasing temperature. A simple mono-exponential fit gives an annihilation time constant of 0.55 ps at room temperature. This time constant slows down by a factor of 1.5 at 77 K, by 5 at 40 K and by 8 at 5 K. Using the above described theoretical model we calculate the transient absorption signal in single LH2 complexes at times up to 5 ps and compare with the experiment. The BChl energy is set equal to 810 nm, diagonal (energetic) disorder is applied using a Gaussian distribution with a FWHM of 450 cm1 . The S0 ! S1 or Q y transition dipole moment orientations were obtained from the LH2 structure [30], the nearest neighbor couplings are set to 288 cm1 and 320 cm1 [31], for the two different pairs of BChl molecules. As no further experimental data are available, the dipole orientations and couplings are assumed to be similar for the Sn state. Due to fast relaxation and disorder averaging, coherent effects on the transient absorption spectrum can be neglected here. Therefore, we assume that the transient absorption DA in the main absorption band can be expressed as DA  P1 þ 2P2 , where P 1 and P2 denote the total population of one and two-exciton states, respectively.

A first result is that the faster relaxation at higher temperatures within the pump pulse duration reduces the maximal number of excitons in the LH2 complex (Fig. 2). For 5 K just after the pump pulse we get P 0 ¼ 0:21, P 1 ¼ 0:34 and P 2 ¼ 0:45 resulting in an average number of 1.24 excitons per LH2 complex, where the fraction of LH2 complexes which stay in the ground state is given by P0 . After the annihilation we have P1 ¼ 0:79 and P 2 ¼ 0 whereas P0 stays unchanged, resulting in an average population of 0.79. As a higher pump pulse intensity has been used for the room temperature measurement, a higher average number of excitations is reached here (Fig. 2). The temperature influences the kinetics in two ways: (i) the intra-band dynamics leading to equilibrium depends on the temperature (Eq. 1); (ii) the internal conversion rate is temperature dependent after Eq. 6. The most ill-determined model parameter is the D. The excited state absorption spectrum of BChl shows a broad band with a maximum slightly on the blue side of the ground state bleach [32]. Since it is difficult to extract the exact value for the D, we have carried out a set of calculations allowing D to vary within limits which do not contradict experimental observation. For each D and T we obtain a certain value for kIC when the result of the simulation fits the measured kinetics. The simulations are shown in Fig. 2 together with the experimental data for D ¼ 0 cm1 and D ¼ 500 cm1 . For D ¼ 500 cm1 the low temperature simulations show deviations from a mono-exponential behavior, but give a similar good agreement with the experiment as the simulations for D ¼ 0 cm1 . The dependence of the internal

1.4

1000

1.2

4K

1.1 1

-1 kIC (fs)

Excitons/LH2

1.3

295 K

77 K

100

40 K

295 K

0.9

10 Excluded region

0.8 0

1

2

3

4

5

t (ps)

1

-400

-200

1.3 1.2

Excitons/LH2

0

200

400

Δ (cm-1) Fig. 3. Dependence of the internal conversion rate on the S2 energy shift D in the simulation.

5K

1.1 40K

1

1

77K

Δ = 0 cm -1

0.9 0.8 0

1

2

t (ps)

3

4

5

Fig. 2. Simulation of the average number of excitons in a LH2 complex after laser pulse excitation at different temperatures, convoluted with a Gaussian probe pulse envelope of full width half maximum 130 fs (100 fs for 295 K). As comparison the (scaled) transient absorption kinetics measured at 295 K (upper panel), 77 K (*), 40 K () and 5 K (+), lower panel is shown. The simulation has been averaged over 100 LH2 complexes including static disorder, and random orientations. The simulation uses an S2 energy shift of D ¼ 500 cm1 (solid) and D ¼ 0 cm1 (dashed). Note that for 295 K a higher intensity of the pump-beam has been used both in the experiment (no cryostat) and in the simulation (three times higher). The behavior between 2 ps and 3 ps in the upper panel and the rise at later times is most likely an experimental artifact.

k IC (norm.)

0.8

0.6

Δ = -500 cm -1

0.4

0.2

0 0

50

100

150

200

250

300

T (K) Fig. 4. Temperature dependence of the internal conversion rate kIC for D ¼ 500 cm1 (dashed) and D ¼ 0 cm1 (solid, both normalized at T = 295 K). The lines give the best fits according to Eqs. (6)–(8).

B. Brüggemann et al. / Chemical Physics 357 (2009) 140–143

conversion time 1=kIC on the value of D is shown in Fig. 3. For each temperature an exponential dependence results, with some deviations for low temperature and D ¼ 500 cm1 . The temperature dependence of kIC is shown in Fig. 4, it is in good agreement with Eq. 6. From the comparison we get the values  ¼ 9 cm1 for the mean vibrational energy. For D ¼ 0 cm1 and of x D ¼ 500 cm1 the activation energy is EA ¼ 16 cm1 and EA ¼ 20 cm1 , respectively. The temperature behavior for D ¼ 100 cm1 and D ¼ 500 cm1 is similar to the one shown for D ¼ 0 cm1 . The reorganisation energy k changes from 0:482 for D ¼ 0 cm1 to 0:472 for D ¼ 500 cm1 . Now we can address the question how the twofold temperature dependence discussed before enters the temperature behavior of the exciton–exciton annihilation process. For D ¼ 0 cm1 and D ¼ 100 cm1 the temperature dependence of the intra-manifold relaxation rate (Eq. 1) does not contribute much and the whole temperature dependence is given by kIC (Eq. 6, Fig. 4). For D ¼ 500 cm1 the intra-manifold relaxation after Eq. 1 slightly hinders the annihilation process thus resulting in a larger internal conversion rate kIC needed when going to lower temperatures (not shown). On the other hand, for D ¼ 500 cm1 the intramanifold relaxation facilitates the annihilation process thus a smaller internal conversion rate kIC is needed for low temperatures (Fig. 4). This behavior is related with our exciton model: a chromophore with low S1 energy which could trap an excitation at low temperatures has also low Sn energy, it is also attracting a second excitation. Thus the trapping does not hinder the annihilation process. Though it is not known how the energetic disorder in S1 and Sn is correlated, our assumption seems to be plausible. Depending on the D, the simulation leads to very different values of the internal conversion rate. The broad and featureless excited state absorption spectrum of BChl in Ref. [32] possibly supports a very fast internal conversion, though internal conversion times below 10 fs can still be excluded as the corresponding (lifetime) broadening of the spectrum would exceed the measured one. Lowering the Sn energy results in a slower internal conversion time constant needed to fit the experiment. This restricts the upper boundary of D to 100 cm1, the value used in a number of previous publications [22–25,33]. 5. Conclusion Pump–probe measurements of the temperature dependence of the intracomplex excitation annihilation in LH2 were modeled via multi-exciton density matrix theory. The calculations enable us to set an upper limit for the energy of the doubly-excited BChl state which is mixed into the two-exciton manifold and is responsible for the annihilation.

143

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