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# Theory of Dissepative Ultrafast Exciton Motion in Photosynthetic Antennae

Abstract (Summary)
The dissipative dynamics of excitons in photosynthetic pigment--protein--complexes has been investigated in the framework of the density matrix theory. Two different model pigment--protein--complex Hamiltonians could be developed. In a first so called effective mode model the local coupling of the pigments to one effective protein mode per pigment has been considered. The full quantum motion of these effective modes was taken into account. In this way the model is capable to describe coherent nuclear motion and also memory effects in the exciton--vibrational interaction. The latter usually appear in a perturbation theory with respect to the exciton--vibrational interaction as memory integrals in the equation of motion for the dynamic variables. And often, a Markov approximation is applied, i.e. memory effects are neglected. In a non--pertubative treatment of the coupling between excitons and effective protein modes non-Markovian effects are of course included. The remaining low frequency modes of the protein and high frequency intramolecular modes of the pigments were treated as a heat bath. The coupling of pigments and effective protein modes to the heat bath was described in second order perturbation theory. This coupling includes (i) a damping of the motion of the effective modes, (ii) a modulating of the inter pigment {\it Coulomb} interactions, and (iii) internal conversion transitions between the higher excited singlet $S_n$--states and the first excited $S_1$--states of the pigments. The coupling to external light fields as well as the inter--pigment Coulomb interaction are included non--pertubatively. A microscopic description of exciton--exciton annihilation processes could be offered, which together with the non--pertubative inclusion of external fields allows to simulate the intensity dependence of non--linear optical spectra. In a second so called multi--mode model the whole exciton--vibrational interaction was described in second order perturbation theory, and the standard Redfield theory was applied in the representation of multi--exciton eigenstates of the pigment--protein--complex. Exciton relaxation in this model is characterized by the coupling weighted density of states (so called spectral density) of the low frequency protein vibrations, which form a multi--mode heat bath. The standard multi--level Redfield theory has been applied also in \cite{Kueh97} on another pigment--protein complex (LH-2). However, what is new in the present approach is the formulation of a correlation radius of protein vibrations. It enables one to charaterize how the couplings of different pigments with their local protein environments are correlated. The multi--exciton spectral density of the protein vibrations can be discussed in terms of the correlation radius and molecular spectral densities characterizing the local coupling of pigments and proteins. These spectral densities have been taken equal for all pigments, and the theory was formulated in such a way that exciton relaxation could be used for a global shape analysis of the spectral density of protein vibrations. An efficient propagation scheme for the density matrix could be developed. It allows for an exact inclusion of external fields and for a distinction of the different spatial contributions of the light induced polarization wave. An expansion of the density matrix with respect to the carrier waves of the external light fields has been carried out. The high efficiency of this method is due to the absence of the high frequency part of the external fields. Instead only the envelopes of the external fields enter the equation of motion for the expansion coefficients of the reduced density matrix. This efficient propagation scheme allowed for the treatment of up to 9 electronic states including two effective modes. Up to now in the literature a 3 electronic state system with two effective modes could be investigated \cite{Matr95}. The ultrafast exciton--vibrational dynamic in a Chla/b hetero dimer of the light--harvesting complex LHC-II of green plants has been studied within the effective mode model. From the simulation of the two--color pump--probe spectra of \cite{Bitt94} evidence could be obtained for the geometry of the optical transition dipoles of the Chla and Chlb pigments absorbing at 680 nm and 650 nm, respectively. They are arranged rather in line than like a sandwich. The measured femtosecond component in the pump--probe signal reflects the Coulomb interaction induced redistribution of oscillator strength among the one-- and the two--exciton transitions of the dimer. Changing the dipole geometry resulted in a qualitative change of the signal. The inclusion of higher excited singlet $S_n$--states of the pigments and internal conversion transitions to the first excited singlet $S_1$--states allowed for a microscopic description of exciton--exciton annihilation. Within this model the intensity dependence of the two--color pump--probe signal measured in \cite{Bitt94} could be simulated, revealing an inverse internal conversion rate of $1/R^{(\rm IC)}_{S_n\rightarrow S_1}=2.2$ ps, and a ratio of dipole moments $\mu_{S_1\rightarrow S_n}/\mu_{S_0\rightarrow S_1}=$1.19. The energy of the $S_n$--state has been taken twice the $S_1$--state energy for both pigments. Consistently to the simulation of the two--color pump--probe experiments also the 77 K one--color pump--probe experiment of \cite{Viss96} could be successfully simulated. It could be demonstrated that again the redistribution of oscillator strength by the {\it Coulomb}interaction can explain the observed switch of the sign of the signal after about 2 ps. In this time exciton relaxation between the high-- and the low energetic exciton states of the dimer occurs and a two exciton transition starting from the low energetic one exciton state becomes possible. To look for an influence of intramolecular excited state absorption of the pigments the higher excited $S_n$--state of Chl{\it a} has been shifted to reach resonance with the probe pulse. A further redistribution of oscillator strength among the two exciton states could be found. However, the principle effect is already included if only interacting two--level molecules are considered. Finally it could be demonstrated that if the effective modes are shifted into the heat bath, the simulation of the one-- and two--color pump--probe experiments gives significantly worse results. Such a simplified approach corresponds to the multi--mode model applied afterwards on the FMO--complex. Only the behavior on the femtosecond or the picosecond time scale could be understood within the simple model. For a consistent description over the whole time range memory effects in the exciton--vibrational interaction had to be taken into account. However the interaction of the Chl{\it a/b} dimer with the remaining pigments in the LHC-II monomer have been neglected. Therefore it would be interesting to compare the dimer results within the effective mode model with a multi --mode approach including all 12 pigments of the LHC-II monomer. A major difficulty arises from the fact that the geometry of the optical transition dipoles of the pigments in the LHC-II has not been resolved yet. Therefore it is difficult to estimate the mutual {\it Coulomb} interactions between the pigments. However, a promising result has been published recently in \cite{Guel97} giving estimates for the dipole geometries of the Chl{\it a} molecules. It is tempting to combine these results with the result of this work and arrange the remaining Chl{\it b} transition dipoles in such a way that within the Chl{\it a/b} dimers {\it in li ne} geometry is obtained. However first calculations up to now do not give such a consistent fit of the one-- and two-- color pump--probe experiments as it could be obtained within the effective mode model. Further work is in progress. For the bacterial light harvesting FMO complex a high resolution structural investigation allowed to give the geometry of the $Q_y$ transition dipoles of the 7 BChls \cite{Tron85,Li97}. In the standard approaches \cite{Pear92,Pear93,guelen97,Louw97} of the simulation of the linear response of FMO--complexes the homogeneous line width has been always neglected. In this work, a microscopic model of exciton relaxation formulated in the multi--mode model could be used to calculate realistic homogeneous exciton transition line shapes. Moreover, since temperature enters the density matrix theory, it was possible to simulate the temperature dependends of the spectra measured in \cite{Frei97}. A simultaneous fit of the linear absorption at two different temperatures was essential for getting the right microscopic parameter set which could be verified afterwards in the simulation of non--linear optical experiments in the time domain. These parameters include the site energies of the 7 BChl molecules, the global shape of the spectral density of protein vibrations, and the correlation radius of exciton--vibrational coupling. For the latter a value $R_c=$ 21 \AA{} could be obtained, which lies in the middle between smallest (11\AA) and largest (30\AA) center to center distances of pigments in the FMO monomer. Hence the coupling of the protein vibrations to different pigments can be characterized as partly correlated. The dependence of the exciton dynamics and linear absorption spectra on $R_c$ was investigated, and as a general trend it could be obtained that a small correlation radius enhances exciton relaxation. Using the same approach and the same parameters ultrafast pump--probe experiments of \cite{Frei97} performed at 20 K could be successfully simulated at three different probe wavelengths. The nice agreement with the measured data did allow for an estimation of the intramolecular excited state absorption of the pigments in the FMO protein. A ratio of dipole moments $\mu_{S_1\rightarrow S_n}/\mu_{S_0\rightarrow S_1}=0.5$ was obtained. This was the only parameter, which could not be determined from the linear absorption fit, since it appears only in the non--linear optical response. A closer examination of the pump--probe simulations revealed some deviations from the measured data for small delay times ($<250 fs$). In the light of the results obtained in the effective mode model one possible explanation could be the presence of {\it non-Markovian} effects in the exciton--vibrational interaction, which have not been included in this simulation. Therefore one alternative description would be the introduction of potential energy surfaces for the excitonic {\it eigenstates}, a model which has been theoretically discussed in this work. However, only a small number of effective modes could be included within such a model. Another possible approach would be to apply a multi--mode model, but take into account the memory integrals which appear in a perturbation theory with respect to the exciton--vibrational coupling. Within such an approach linear absorption could easily be calculated taking into account frequency dependent line broadenings. Having fixed all microscopic parameters, in the following also the temperature dependence of the two--color pump--probe spectra of \cite{Frei97} could be successfully simulated. An important point in the simulation of the ultrafast pump--probe spectra was the explicit consideration of the vector character of the external fields. An orientational average over randomly orientated pigment--protein--complexes in the sample has been carried out numerically. A correct simulation of the non--linear optical response demands for such an average. Finally the 19 K transient anisotropy measured in \cite{Savh97} could be also simulated revealing exciton quantum beats between the lowest notwo one--exciton states of the FMO complex. The developed model easily allows to simulate further experiments as, for example, hole burning studies and photon echo experiments. In addition, it would be interesting to study the intensity dependence of the pump--probe spectra. Up to now there is only a picosecond annihilation study on FMO-complexes in the literature \cite{Gulb96}. A femtosecond experiment would be desirable. After having discussed the intricate quantum dynamics of excitons under non--linear excitation conditions and at extreme temperatures in different experiments, one should go back to the central question about the structure--function relation in these antennae. The simulations gave insights into the interplay of exciton and protein dynamics. In the LHC-II the presence of memory effects in the exciton--vibrat ional interaction could be found. For the FMO monomers a correlation radius of the protein vibrations could be determined and the global shape of the spectral density of the protein vibrations could be estimated. The idea to use exciton relaxation for such estimates is new in the literature. Also the microscopic parameters describing the electronic properties of the pigments in their local protein environment were obtained from the simulation. Now all the obtained microscopic information should somehow be used to calculate a quantity which directly relates to the function of the antennae. A simple way should be found to measure the efficiency of light harvesting antennae. Then the different antenna systems could be compared, and it may be possible to follow the evolutionary strategy of nature in the development of these systems. In the following a first step in this direction will be done. The efficiency of a light harvesting antennae may be defined as the ability to absorb sun light in a broad spectral range and to quickly dissipate the excess energy of the excitons on their way to the reaction center. These two features can be quantified in the following way. One may use a weak ultrashort pulse which is energetically broad enough to excite the different one exciton transitions of the pigment--protein--complex with nearly the same intensity. Thus a broad spectral excitation as it occurs under natural conditions can be obtained. Such ultrashort pulses have been used for example in this work to apply the time dependent formulation of the linear absorption coefficient in the simulation of linear absorption spectra of the Chl{\it a/b} dimer. From the propagation of the density matrix the total energy of the pigment system can be calculated as $={\rm Tr}_{\rm S}\{H_{\rm S}\hat{\rho}(t)\}$ and the dissipation of system energy by the protein environment can be followed by looking at the decay of the above defined energy. The efficiency of a light harvesting antenna may be obtained from the following quantity \be U(t)=\frac{-_{\rm eq}}{_{\rm eq}}\,, \label{U} \ee where $_{\rm eq}$ is the energy of the pigment system after exciton relaxation has finished, i.e. when an equilibrium among the excited pigment states has established. A high efficiency should result in a high initial value of $U(t)$ and a fast decay to the equilibrium value. From the initial value the dynamic range over which exciton relaxation occurs in the pigment--protein--complex follows, whereas the decay indicates how fast exciton excess energy can be absorbed by the proteins. To make the calculated signal independent on the applied intensity (of course linear excitation conditions are provided) the difference of the two considered energies in Eq. \ref{U} will be divided by a reference energy. For the latter the equilibrium value has been chosen in the above equation for U(t). In Fig. \ref{efficiency} the signal $U(t)$ has been calculated for the Chl{\it a/b} dimer of the LHC-II (dashed line) and the FMO complex of {\it chlorobium tepidum} (solid line). For the propagation of the density matrix the determined microscopic parameters of this work were used and an ultrashort 10 fs pulse has been applied in the simulations. The bacterial pigment--protein--complex shows a higher efficiency. The initial value of $U(t)$ is larger, and also the decay is faster than in the Chl{\it a/b} dimer. Of course this result is due to the reduced description of the LHC-II, as a single dimer. Such a model could be used to explain a special experimental situation for which the applied laser fields due to their carrier frequencies and spectral widths mainly probed this particular dimer absorbing at 650 nm and at 680 nm. However if a 10 fs pulse is applied also the remaining pigments are excited and will contribute to the signal $U(t)$. The calculated signal for the FMO complex should, however, change only lightly if instead of FMO--monomers, as done here, FMO--trimers, which represent the next larger subunit of this antenna, would be used for the simulation. Realistic absorption spectra could be obtained already if the 7 BChls contained in the FMO--monomer were included. In addition the interaction between the pigments in different FMO monomers is weak. Therefore the FMO--signal $U(t)$ can serve as a starting point, and it will be interesting to compare the obtained signal to those of other antenna systems. Finally it can be concluded that the density matrix theory provides a tool of just the right complexity to study exciton relaxation in photosynthetic antennae. By taking into account three electronic states per pigment and a few global characteristics of the protein vibrations it was possible to simulate a large number of optical experiments, and interesting information on structure--function relationships in photosynthetic pigment--protein--complexes could be obtained.
This document abstract is also available in German.
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