- Email: [email protected]
Exciton scattering in light-harvesting systems of purple bacteria
Journal of Luminescence 94–95 (2001) 447–450
Exciton scattering in light-harvesting systems of purple bacteria . b, Ivan Barv!ıkc, Michael Schreiberb Pavel Her$mana,*, Ulrich Kleinekathofer a
! e, ! V. Nejedleho ! 573, CZ-50003 Hradec Kralov ! e, ! Czech Republic Department of Physics, University of Hradec Kralov b Institut fur . D-09107 Chemnitz, Germany . Physik, Technische Universitat, c Institute of Physics of Charles University, Faculty of Mathematics and Physics, CZ-12116 Prague, Czech Republic
Abstract Using the reduced density matrix formalism the exciton scattering in light-harvesting systems of purple bacteria is calculated. Static disorder (fluctuations of the site energies) as well as dynamic disorder (dissipation) is taken into account in this work. Circular aggregates with 18 pigments are studied to model the B850 ring of bacteriochlorophylls within LH2 complexes. It can be shown that the influence of dissipation may not be neglected in the simulation of the time-dependent anisotropy of fluorescence. Also an elliptical deformation of the ring could be essential. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Exciton transfer; Density matrix theory; Fluorescence
1. Motivation Highly efficient light collection and excitation transfer to the reaction center initiates the energy conversion in photosynthesis. This process takes place in the so-called light-harvesting antenna network. Particularly, the ring subunit of the peripheral light-harvesting antenna (LH2) [1] of purple bacteria has been extensively studied [2]. The very symmetric arrangement with short distances between the pigments gave new impulses to the discussion about coherence in the exciton transfer and the exciton delocalization in LH2. Both can be reduced by dynamic and static disorder. Time-dependent experiments made it possible to study the femtosecond dynamics of energy transfer
and relaxation [3]. Kumble and Hochstrasser [4] presented a time-domain analysis of the effects of static disorder on the dynamics of optical excitations. But the interpretation of time-dependent experiments on the femtosecond time scale requires a theory which incorporates static and dynamic disorder. The aim of the present paper is to extend the investigation by Kumble and Hochstrasser taking into account the simultaneous influence of static and dynamic disorder after impulse excitation. In addition, we calculate the time-dependent anisotropy of fluorescence not only for the symmetric but also for elliptically distorted rings.
2. Model The Hamiltonian
*Corresponding author. Fax: +420-49-551-3890. E-mail address: [email protected] (P. He$rman).
0 H ¼ Hex þ Hph þ Hex2ph þ Hs þ Hel
0022-2313/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 0 1 ) 0 0 3 3 4 - 9
ð1Þ
448
P. He$rman et al. / Journal of Luminescence 94–95 (2001) 447–450
describes the transfer of a single exciton 0with a ; the transfer integral J along the ideal ring Hex independent heat baths for each chromophore ðHph Þ; the site-diagonal and linear interaction between the exciton and the bath ðHex2ph Þ; the static disorder ðHs Þ with Gaussian distribution (standard deviation D) and elliptical distortion of the ideal ring ðHel Þ: 0 Diagonalization of the Hamiltonian Hex of the ideal ring leads to eigenstates jki and eigenenergies En ¼ 2J cos kn ; where kn ¼ 2pn=N; n¼ 0; 71; 72; y; N=2: For a symmetric coplanar arrangement of site transition moments ~ mn dipole-allowed transitions populate only the degenerate n ¼ 71 levels of the ideal ring. If the ring is elliptically distorted the eigenenergies of the Hamiltonian 0 Hex þ Hel are no longer degenerate as is the case for the ideal ring. When static disorder of the site energies is taken into account ðDa0Þ; the stationary states, i.e. the eigenstates jai of the 0 Hamiltonian Hex þ Hs ; correspond to mixtures of jki and an excitation will prepare a superposition of the jki states. The dipole strength ~ ma of state jai of the ring with static disorder and the dipole strength ~ ma of state jai of the ideal one read ~ ma ¼
N X n¼1
can~ mn ;
~ ma ¼
N X
can~ mn ;
ð2Þ
n¼1
where can and can are the expansion coefficients of the eigenstates of the ideal and disordered rings in site representation. Kumble and Hochstrasser [4] concluded, that in the case of pump-pulse excitation the dipole strength is simply redistributed among the exciton levels due to disorder. So the amplitudes of site excitations and the phase relationships in the initial state are necessarily identical to that of an equal superposition of n ¼ 71 excitons of the ideal ring. Thus, generally, the excitation with a pump pulse of sufficiently wide spectral bandwidth will always prepare the same initial state, irrespective of the actual eigenstates of the real ring. The nature of this initial state is entirely determined by the selection rules of the ring without static disorder. The initial condition for the density matrix after impulse excitation with polarization
~ ex is given by (Eq. (1a) in Ref. [5]) 1 ~ rab ðt ¼ 0; ~ ex Þ ¼ ma ~ mb ; ex ~ ex ~ A P where A ¼ a ~ ma ~ ma : ex ~ ex ~
ð3Þ
3. Anisotropy of fluorescence Kumble and Hochstrasser [4] calculated the usual time-dependent anisotropy of fluorescence: hSxx ðtÞi Sxy ðtÞ rð t Þ ¼ ð4Þ hSxx ðtÞi þ 2 Sxy ðtÞ where, e.g., 2 + * X a * a ði=_ÞE t a ~ e ~ e ~ Sxy ðtÞ ¼ m ~ m cn cn e : a;l;n x a y a ð5Þ The indices a and a label the eigenstates of the ring without and with static disorder and Ea is the eigenenergy of the state jai: The brackets /S denote the ensemble average and the average over the direction of the laser pulses with fixed relative directions ~ ex and ~ ey : To include dynamic disorder which contributes to dephasing of the initial wave packet and promotes thermalization of the dephased populations one has to work within the exciton density-matrix formalism [6] instead of using only the exciton wave functions. Then we have to determine Z Sxy ðtÞ ¼ Pxy ðo; tÞ do; ð6Þ where Pxy ðo; tÞ ¼ A
XX a
raa0 ðtÞ ~ ey ~ ma 0 ~ ma ey ~
a0
½dðo oa0 0 Þ þ dðo oa0 Þ :
ð7Þ
4. Density matrix formalism for exciton transfer and relaxation Provided that the exciton dynamics is not very fast, its coupling to the bath weak, and excepting the initial time interval tttd (td =dephasing time
P. He$rman et al. / Journal of Luminescence 94–95 (2001) 447–450
of the bath), the Redfield equation [7,8]: d 1 ð8Þ i rðtÞ ¼ ½H; rðtÞ þ RrðtÞ dt _ is adequate for a factorized initial state. The relaxation superoperator R describes the influence of the thermal bath on the exciton dynamics. $ a! pek applied several different ways of obtainC ing of convolutional and convolutionless dynamical equations for the exciton density matrix in the site basis [9]. After Markovian approximation they have the form d 1 i rðtÞ ¼ ½H; rðtÞ dOrðtÞ: ð9Þ dt _ We proved [6] the equivalence of the Redfield theory without secular approximation with $ a! pek’s equations after Markovian approximaC tion. Redfield secular approximation can distort true time dependence of physical quantities calculated by means of r [10].
5. Results In Kumble and Hochstrasser’s modeling, the anisotropy of fluorescence of the ring LH2 subunit decreases from 0.7 to 0.3–0.35 and subsequently reaches a final value 0.4. Kumble and Hochstrasser concluded that one needs static disrorder of strength DE0.4–0.8J to reach a decay time below 100 fs. Our results for the dependence of the anisotropy of fluorescence on time t ¼ tJ=_ in the symmetric ring are shown in Fig. 1. In addition to static disorder we consider two strengths of dynamic disorder the spectral density J ðoÞ ¼ j0 entering YðoÞj0 o2 =2o3c eo=oc with cut-off frequency oc =0.2J [11]. Inclusion of dynamic disorder leads to faster decay during the initial stage. Smaller strength of the static disorder than predicted in Ref. [4] would be necessary to guarantee the decrease of the anisotropy of fluorescence during the first 100 fs. Based on measurements by Chachisvilis et al. [12] it was concluded [4] that the time decay of the anisotropy of fluorescence during the first tens of fs is temperature independent in the case of LH2 subunits. Our calculation for the symmetric
449
Fig. 1. Time dependence of the anisotropy of fluorescence rðtÞ for the symmetric ring with static disorder D=0.4J. The influence of dynamic disorder is displayed for low (ii, iv) and room temperature (iii, v) for j0=0.2J (ii, iii), and j0=0.4J (iv, v) compared to j0=0 (i).
ring shows that such result can be obtained only for D > 0:8J: But because the time resolution of the experiments [12] was not very high this restrictive statement about the strength of the static disorder can be made with caution. We expect that some temperature dependence as in Fig. 1 can be seen in experiments with shorter laser pulses. Recent results obtained by single molecule spectroscopy [13] can be interpreted [14] admitting a C2 distortion of the LH2 ring. It has, up to now, not been concluded whether such C2 distortion of the LH2 ring is present also in samples in vivo. We have taken into account the possible C2 distortion of the ring using the model C by Matsushita et al. [14] of elliptical distortion of the ring in which the transfer integral J is cosine modulated. Local transition dipole moments lie tangentially to the ellipse. Our results for the anisotropy of fluorescence (Eq. (4)) are shown in Fig. 2 for three values of the elliptical deformation.
450
P. He$rman et al. / Journal of Luminescence 94–95 (2001) 447–450
Acknowledgements This work has been partially funded by the $ $ MSMT CR (Kontakt CZE0014), BMBF and DFG. While preparing this work, I.B. and P.H. experienced the kind hospitality of the Chemnitz University of Technology and U.K. experienced the kind hospitality of the Charles University in Prague.
References
Fig. 2. Time dependence of the anisotropy of fluorescence rðtÞ for the elliptically distorted ring with static disorder D=0.4J and dynamic disorder j0=0.2J. The influence of the elliptical deformation is displayed by dotted lines for V2=0.05J, dashed lines for V2=0.1J and long dashed lines for V2=0.2J compared to V2=0 (solid lines) for low and room temperature (upper and lower set of lines, respectively).
It is seen, that the inclusion of an elliptical deformation leads to faster decay of the anisotropy of fluorescence and diminishes the influence of the dynamic disorder. Our calculation for the elliptically distorted ring (V2=0.2J [14]) shows that the temperature independent time decay of the anisotropy of fluorescence during the first tens of fs can be obtained even for DE0.4J.
[1] G. McDermott, S.M. Prince, A.A. Freer, A.M. Hawthornthwaite-Lawiess, M.Z. Papiz, R.J. Cogdell, N.W. Issacs, Nature 374 (1995) 517. . [2] V. Sundstr . om, T. Pullerits, R. van Grondelle, J. Phys. Chem. B 103 (1999) 2327. [3] V. Nagarjan, R.G. Alden, J.C. Williams, W.W. Parson, Proc. Natl. Acad. Sci. USA 93 (1996) 13774. [4] R. Kumble, R. Hochstrasser, J. Chem. Phys. 109 (1998) 855. [5] V. Nagarjan, E.T. Johnson, J.C. Williams, W.W. Parson, J. Phys. Chem. B 103 (1999) 2297. . [6] P. He$rman, U. Kleinekathofer, I. Barv!ık, M. Schreiber, Chem. Phys., accepted for publication. [7] A.G. Redfield, IBM J. Res. Dev. 1 (1957) 19. [8] A.G. Redfield, Adv. Magn. Reson. 1 (1965) 1. $ apek, Z. Phys. B 99 (1996) 261. [9] V. C! $ apek, I. Barv!ık, P. He$rman, Chem. Phys. 270 (2001) [10] V. C! 141. [11] V. May, O. Kuhn, . Charge and Energy Transfer Dynamics in Molecular Systems, Wiley-VCH, Berlin, 2000. [12] M. Chachisvilis, O. Kuhn, . T. Pullerits, V. Sundstrom, . J. Phys. Chem. B 101 (1997) 7275. . [13] A.M. van Oijen, M. Ketelaars, J. Kohler, T.J. Aartsma, J. Schmidt, Science 285 (1999) 400. [14] M. Matsushita, M. Ketelaars, A.M. van Oijen, . J. Kohler, T.J. Aartsma, J. Schmidt, Biophys. J. 80 (2001) 1604.
Exciton scattering in light-harvesting systems of purple bacteria . b, Ivan Barv!ıkc, Michael Schreiberb Pavel Her$mana,*, Ulrich Kleinekathofer a
! e, ! V. Nejedleho ! 573, CZ-50003 Hradec Kralov ! e, ! Czech Republic Department of Physics, University of Hradec Kralov b Institut fur . D-09107 Chemnitz, Germany . Physik, Technische Universitat, c Institute of Physics of Charles University, Faculty of Mathematics and Physics, CZ-12116 Prague, Czech Republic
Abstract Using the reduced density matrix formalism the exciton scattering in light-harvesting systems of purple bacteria is calculated. Static disorder (fluctuations of the site energies) as well as dynamic disorder (dissipation) is taken into account in this work. Circular aggregates with 18 pigments are studied to model the B850 ring of bacteriochlorophylls within LH2 complexes. It can be shown that the influence of dissipation may not be neglected in the simulation of the time-dependent anisotropy of fluorescence. Also an elliptical deformation of the ring could be essential. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Exciton transfer; Density matrix theory; Fluorescence
1. Motivation Highly efficient light collection and excitation transfer to the reaction center initiates the energy conversion in photosynthesis. This process takes place in the so-called light-harvesting antenna network. Particularly, the ring subunit of the peripheral light-harvesting antenna (LH2) [1] of purple bacteria has been extensively studied [2]. The very symmetric arrangement with short distances between the pigments gave new impulses to the discussion about coherence in the exciton transfer and the exciton delocalization in LH2. Both can be reduced by dynamic and static disorder. Time-dependent experiments made it possible to study the femtosecond dynamics of energy transfer
and relaxation [3]. Kumble and Hochstrasser [4] presented a time-domain analysis of the effects of static disorder on the dynamics of optical excitations. But the interpretation of time-dependent experiments on the femtosecond time scale requires a theory which incorporates static and dynamic disorder. The aim of the present paper is to extend the investigation by Kumble and Hochstrasser taking into account the simultaneous influence of static and dynamic disorder after impulse excitation. In addition, we calculate the time-dependent anisotropy of fluorescence not only for the symmetric but also for elliptically distorted rings.
2. Model The Hamiltonian
*Corresponding author. Fax: +420-49-551-3890. E-mail address: [email protected] (P. He$rman).
0 H ¼ Hex þ Hph þ Hex2ph þ Hs þ Hel
0022-2313/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 0 1 ) 0 0 3 3 4 - 9
ð1Þ
448
P. He$rman et al. / Journal of Luminescence 94–95 (2001) 447–450
describes the transfer of a single exciton 0with a ; the transfer integral J along the ideal ring Hex independent heat baths for each chromophore ðHph Þ; the site-diagonal and linear interaction between the exciton and the bath ðHex2ph Þ; the static disorder ðHs Þ with Gaussian distribution (standard deviation D) and elliptical distortion of the ideal ring ðHel Þ: 0 Diagonalization of the Hamiltonian Hex of the ideal ring leads to eigenstates jki and eigenenergies En ¼ 2J cos kn ; where kn ¼ 2pn=N; n¼ 0; 71; 72; y; N=2: For a symmetric coplanar arrangement of site transition moments ~ mn dipole-allowed transitions populate only the degenerate n ¼ 71 levels of the ideal ring. If the ring is elliptically distorted the eigenenergies of the Hamiltonian 0 Hex þ Hel are no longer degenerate as is the case for the ideal ring. When static disorder of the site energies is taken into account ðDa0Þ; the stationary states, i.e. the eigenstates jai of the 0 Hamiltonian Hex þ Hs ; correspond to mixtures of jki and an excitation will prepare a superposition of the jki states. The dipole strength ~ ma of state jai of the ring with static disorder and the dipole strength ~ ma of state jai of the ideal one read ~ ma ¼
N X n¼1
can~ mn ;
~ ma ¼
N X
can~ mn ;
ð2Þ
n¼1
where can and can are the expansion coefficients of the eigenstates of the ideal and disordered rings in site representation. Kumble and Hochstrasser [4] concluded, that in the case of pump-pulse excitation the dipole strength is simply redistributed among the exciton levels due to disorder. So the amplitudes of site excitations and the phase relationships in the initial state are necessarily identical to that of an equal superposition of n ¼ 71 excitons of the ideal ring. Thus, generally, the excitation with a pump pulse of sufficiently wide spectral bandwidth will always prepare the same initial state, irrespective of the actual eigenstates of the real ring. The nature of this initial state is entirely determined by the selection rules of the ring without static disorder. The initial condition for the density matrix after impulse excitation with polarization
~ ex is given by (Eq. (1a) in Ref. [5]) 1 ~ rab ðt ¼ 0; ~ ex Þ ¼ ma ~ mb ; ex ~ ex ~ A P where A ¼ a ~ ma ~ ma : ex ~ ex ~
ð3Þ
3. Anisotropy of fluorescence Kumble and Hochstrasser [4] calculated the usual time-dependent anisotropy of fluorescence: hSxx ðtÞi Sxy ðtÞ rð t Þ ¼ ð4Þ hSxx ðtÞi þ 2 Sxy ðtÞ where, e.g., 2 + * X a * a ði=_ÞE t a ~ e ~ e ~ Sxy ðtÞ ¼ m ~ m cn cn e : a;l;n x a y a ð5Þ The indices a and a label the eigenstates of the ring without and with static disorder and Ea is the eigenenergy of the state jai: The brackets /S denote the ensemble average and the average over the direction of the laser pulses with fixed relative directions ~ ex and ~ ey : To include dynamic disorder which contributes to dephasing of the initial wave packet and promotes thermalization of the dephased populations one has to work within the exciton density-matrix formalism [6] instead of using only the exciton wave functions. Then we have to determine Z Sxy ðtÞ ¼ Pxy ðo; tÞ do; ð6Þ where Pxy ðo; tÞ ¼ A
XX a
raa0 ðtÞ ~ ey ~ ma 0 ~ ma ey ~
a0
½dðo oa0 0 Þ þ dðo oa0 Þ :
ð7Þ
4. Density matrix formalism for exciton transfer and relaxation Provided that the exciton dynamics is not very fast, its coupling to the bath weak, and excepting the initial time interval tttd (td =dephasing time
P. He$rman et al. / Journal of Luminescence 94–95 (2001) 447–450
of the bath), the Redfield equation [7,8]: d 1 ð8Þ i rðtÞ ¼ ½H; rðtÞ þ RrðtÞ dt _ is adequate for a factorized initial state. The relaxation superoperator R describes the influence of the thermal bath on the exciton dynamics. $ a! pek applied several different ways of obtainC ing of convolutional and convolutionless dynamical equations for the exciton density matrix in the site basis [9]. After Markovian approximation they have the form d 1 i rðtÞ ¼ ½H; rðtÞ dOrðtÞ: ð9Þ dt _ We proved [6] the equivalence of the Redfield theory without secular approximation with $ a! pek’s equations after Markovian approximaC tion. Redfield secular approximation can distort true time dependence of physical quantities calculated by means of r [10].
5. Results In Kumble and Hochstrasser’s modeling, the anisotropy of fluorescence of the ring LH2 subunit decreases from 0.7 to 0.3–0.35 and subsequently reaches a final value 0.4. Kumble and Hochstrasser concluded that one needs static disrorder of strength DE0.4–0.8J to reach a decay time below 100 fs. Our results for the dependence of the anisotropy of fluorescence on time t ¼ tJ=_ in the symmetric ring are shown in Fig. 1. In addition to static disorder we consider two strengths of dynamic disorder the spectral density J ðoÞ ¼ j0 entering YðoÞj0 o2 =2o3c eo=oc with cut-off frequency oc =0.2J [11]. Inclusion of dynamic disorder leads to faster decay during the initial stage. Smaller strength of the static disorder than predicted in Ref. [4] would be necessary to guarantee the decrease of the anisotropy of fluorescence during the first 100 fs. Based on measurements by Chachisvilis et al. [12] it was concluded [4] that the time decay of the anisotropy of fluorescence during the first tens of fs is temperature independent in the case of LH2 subunits. Our calculation for the symmetric
449
Fig. 1. Time dependence of the anisotropy of fluorescence rðtÞ for the symmetric ring with static disorder D=0.4J. The influence of dynamic disorder is displayed for low (ii, iv) and room temperature (iii, v) for j0=0.2J (ii, iii), and j0=0.4J (iv, v) compared to j0=0 (i).
ring shows that such result can be obtained only for D > 0:8J: But because the time resolution of the experiments [12] was not very high this restrictive statement about the strength of the static disorder can be made with caution. We expect that some temperature dependence as in Fig. 1 can be seen in experiments with shorter laser pulses. Recent results obtained by single molecule spectroscopy [13] can be interpreted [14] admitting a C2 distortion of the LH2 ring. It has, up to now, not been concluded whether such C2 distortion of the LH2 ring is present also in samples in vivo. We have taken into account the possible C2 distortion of the ring using the model C by Matsushita et al. [14] of elliptical distortion of the ring in which the transfer integral J is cosine modulated. Local transition dipole moments lie tangentially to the ellipse. Our results for the anisotropy of fluorescence (Eq. (4)) are shown in Fig. 2 for three values of the elliptical deformation.
450
P. He$rman et al. / Journal of Luminescence 94–95 (2001) 447–450
Acknowledgements This work has been partially funded by the $ $ MSMT CR (Kontakt CZE0014), BMBF and DFG. While preparing this work, I.B. and P.H. experienced the kind hospitality of the Chemnitz University of Technology and U.K. experienced the kind hospitality of the Charles University in Prague.
References
Fig. 2. Time dependence of the anisotropy of fluorescence rðtÞ for the elliptically distorted ring with static disorder D=0.4J and dynamic disorder j0=0.2J. The influence of the elliptical deformation is displayed by dotted lines for V2=0.05J, dashed lines for V2=0.1J and long dashed lines for V2=0.2J compared to V2=0 (solid lines) for low and room temperature (upper and lower set of lines, respectively).
It is seen, that the inclusion of an elliptical deformation leads to faster decay of the anisotropy of fluorescence and diminishes the influence of the dynamic disorder. Our calculation for the elliptically distorted ring (V2=0.2J [14]) shows that the temperature independent time decay of the anisotropy of fluorescence during the first tens of fs can be obtained even for DE0.4J.
[1] G. McDermott, S.M. Prince, A.A. Freer, A.M. Hawthornthwaite-Lawiess, M.Z. Papiz, R.J. Cogdell, N.W. Issacs, Nature 374 (1995) 517. . [2] V. Sundstr . om, T. Pullerits, R. van Grondelle, J. Phys. Chem. B 103 (1999) 2327. [3] V. Nagarjan, R.G. Alden, J.C. Williams, W.W. Parson, Proc. Natl. Acad. Sci. USA 93 (1996) 13774. [4] R. Kumble, R. Hochstrasser, J. Chem. Phys. 109 (1998) 855. [5] V. Nagarjan, E.T. Johnson, J.C. Williams, W.W. Parson, J. Phys. Chem. B 103 (1999) 2297. . [6] P. He$rman, U. Kleinekathofer, I. Barv!ık, M. Schreiber, Chem. Phys., accepted for publication. [7] A.G. Redfield, IBM J. Res. Dev. 1 (1957) 19. [8] A.G. Redfield, Adv. Magn. Reson. 1 (1965) 1. $ apek, Z. Phys. B 99 (1996) 261. [9] V. C! $ apek, I. Barv!ık, P. He$rman, Chem. Phys. 270 (2001) [10] V. C! 141. [11] V. May, O. Kuhn, . Charge and Energy Transfer Dynamics in Molecular Systems, Wiley-VCH, Berlin, 2000. [12] M. Chachisvilis, O. Kuhn, . T. Pullerits, V. Sundstrom, . J. Phys. Chem. B 101 (1997) 7275. . [13] A.M. van Oijen, M. Ketelaars, J. Kohler, T.J. Aartsma, J. Schmidt, Science 285 (1999) 400. [14] M. Matsushita, M. Ketelaars, A.M. van Oijen, . J. Kohler, T.J. Aartsma, J. Schmidt, Biophys. J. 80 (2001) 1604.