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Exciton transfer in disordered molecular aggregates: Computer modeling of optical line shapes in ring bacteria antenna systems
a __
__ fif!B
JOURNAL
OF
LUMINESCENCE
ELSEVIER
Journal
of Luminescence
76&77
(1998) 33
I 324
Exciton transfer in disordered molecular aggregates: Computer modeling of optical line shapes in ring bacteria antenna systems
Abstract The structure of the LH2 subunit Rps. acidoplzi/a consists of a ring of 9 $ heterodimers with 1X 8850 BChI molecules sandwiched between the r and p subunits. For Frenkel excitons moving on the ring we investigate the influence of dynamic and static disorder on their optical line shapes. The dynamic and static disorders are taken into account bq fluctuations of the local energies of the BChl molecules. The fluctuations are represented by dichotomic Markov processes with coloured noise. Comparison of our theoretical results with experimental observations indicates. that the strength of the local energy fluctuation d should be smaller than J ‘2. (’ 1998 Elsevier Science B.V. All rights reserved. Kq,~o&:
Coherent
transfer:
Optical
line shapes: Antenna
systems:
1. Introduction
The structure of the LH2 subunit Rps.acidopkiln consists of a ring of 9 rwBheterodimers [ 11. 18 B850 BChl molecules are sandwiched between the s( and p subunits. The regime of the exciton transfer in the ring of LH2 antenna system remains still an open question. The transition dipole moments in the B850 ring are oriented head-head, the basic unit is a dimer. The ring and whole LH2 has 9-fold symmetry. Short distances between BChl850 molecules lead to a strong mutual interaction J. Estimates reach from 50 to 750 cm ‘. Absorption spectrum of the homogeneous ring of N molecules, with their transition dipole moments in the plane [l]. would be characterized [2] by peaks at the energy E0 in the z-polarization and *Corresponding author. barvik’lr’karlo~.mll;cunl.c7.
Fax:
+ 420 2 296764:
e-mall:
0022-3313 98,$19.00 (‘ 1998 Elsevicr Science B.V. All rights reserved PI/ soo22-23 13(97)00220-2
Photosynthesis
at the double degenerate energy E, in the .Y,Ypolarization (E, = f: + 2 Jcos((2xn);‘N)). We shall take IJI = 1. The explanation of the optical absorption and the hole-burning spectroscopy which relates the energy difference between two lowest states to the experimental value 2OOcm -‘[-?I would lead to the estimation of the transfer integral greater than 700 cm- I. To bring the theoretical values closer to the experimental one, we have tried 14.51 to take into account the nonhomogeneity of the local energies I:, and I:~ and of the transfer integrals J12 and J13 and to change in such a way [6] the energy difference between the two lowest states. Small recently suggested [S]. as Crondelle did before [7], to include the influence of the static disorder. Estimates of the static local energy fluctuations d between 150 and 500 cm ’ can be found in literature. Our aim is an estimation of the strength of the local static disorder d from the optical properties
1. Bun%
332
et (11.
Jou~~ul
o#‘Luminrscrncr
of the ring structures [S]. Ratio A/J determines whether we are allowed to treat the exciton transfer inside of the ring LH2 as a quasicoherent one [9].
2. Method
77 IIUW)
331-334
Due to a lack of knowledge of the microscopical mechanisms we shall use a stochastic approach which describes the influence of the static and dynamic disorder by a stochastic process with prescribed properties. A stochastically time-dependent part of the Hamiltonian H,(t) = c C,(t)n;U,, m
The Hamiltonian H,, = 2 Eaia, m
76&
+ c
Jmna~an
(1)
m+n
describes the purely coherent exciton transfer in the ring of 18 BChl molecules with the same energy F. ai and a, are creation and annihilation operators for an excitation at site m and J,, the excitonic transfer matrix element between sites m and II.
(2)
models the influence of the disorder via fluctuations of the local exciton energies c,(t). Mean values and correlation functions are given by (3)
(&l(r)) = 0, = d,,A’exp(
(~,(t)~,(r))
- R(r -
7)).
160
140
120
100
80
60
-2
-1 Fig. 1. Optical
0
1
2
3
hne shapes I,Jw,) and I,(w) for d = 0.5 and i = 0.05.
4
(4)
1. Burcik et
al.
: Journal
of Luminescence 76& 77 (IWX)
Especially for dichotomic stochastic processes, A is the amplitude of the local energy fluctuations, and i, describes the average switchover rate between the two values.
3. Results We have calculated in a series of papers [lo-121 (where also a thorough description of the method can be found) the optical absorption line shapes I((II) for Frenkel excitons on cyclic models of antenna systems. The optical line shape has been determined from an exact closed system of coupled linear equations for correlation functions. The parameters in these equations are the transfer matrix element J between nearest neighbours, the strength A of the fluctuations of the local excitation energies and the decay constant /. of the correlation functions of these fluctuations. We calculated [lo-121 the optical line shapes for various arrangement of the transition dipole moments and different polarizations of the light. The arrangement of dipoles changes not only the rules for allowed optical transitions but changes also due to the dipole-dipole interaction the transfer integrals [2]. For small values of i., i.e. slow fluctuations, the line shape is strongly structured; it could be related to the energy structure in the static case. With increasing values of r., the lines become broader and the structure of the spectra is smeared out. Furthermore, due to the disorder. transitions are allowed which in the completely ordered case are forbidden. For large values of j. motional narrowing of the optical lines takes place. In this paper we are dealing with the exciton optical line shapes of the ring with 18 BChl 850 molecules. We investigate the influence of the local energ!, ,fluc.tuations within the so-called RPA3 decoupling [lO,ll]. Exciton optical line shapes [lo-1211,(w), for the transition dipole moments and the light polarization parallel with z-axis of the ring, and I,((IJ), for the transition dipole moments and the light polarization perpendicular to the z-axis are displayed on Fig. 1 for i, = 0.05 and A = 0.5.
331-334
73.7
4. Conclusions Raising the strength of the local energy fluctuations A the optical transition to other states than El also gain oscillator strength in I,((II). Results of 4.2 absorption and hole-burning experiments on the B850 absorption band of isolated LH2 complexes revealed [3] direct observation of the lowest exciton level of the B850 ring as a weak but distinct shoulder at the red edge of the B850 band. Comparison between our theoretical (Fig. I) and experimental results indicates. that the strength of the local energy fluctuation d should be weaker than 52. For A = 59 the optical absorption to lowest state Et1 (which is originally. without disorder, not allowed) is comparable with that to the state E, An increase of the local energy fluctuation strength A changes aho the energy difference between E,) and El. Relative weakness of the static local energy disorder allows us to treat the exciton transfer inside of the LH2 ring as a quasicoherent one [9].
Acknowledgements The support of the Deutsche Forschungsgemeinschaft (SFB 239) are gratefully acknowledged. This work has been also funded (I.B.) by contract No. 105!95 of the Charles University and No. 1235197 of the Development Foundation.
References [I]
A.A.
Freer.
S.M.
Prince,
Hawthornthwalte-Lawiess. [I’]
N.W.
Issacs. Structure
R.M.
Pearlstein.
Sauer.
M.Z.
G. McDermott.
Paper.
A.M.
R..l. Cogdell.
4 (1996) 449.
H. Zuber.
tennas and Reaction Springer
K.
in: Michel-Beyerle
(Ed.). An-
Centers of Photosynthetic
Series in Chemical
Bactcrla.
PhysIca. vol. 12. Springer.
Berlin. 1985. [3]
H.-M.
Wu. N.R.S. Reddy. G.J. Sma1L.l. Phys Chem. B IO1
(19971651. [4]
I. Barlik.
In: ESE workshop:
Light-Harvecting
Physics.
BirStonas. 21. 1996. [S]
I. Barvik. 35, 1996.
in: ERC:
Biophysics of Photosynthesix
Sitges.
I. Barr% ei ul.
334
[6]
I. Barvik.
[7]
R. van Grondelle.
P. Herman.
[8]
1. Barvik.
[9]
V. Szocs. 1. Barvik,
BBA 1187 (1994) ICL’96.
Czech J. Phys.. submitted.
J.P. Dekker.
T. Gillbro.
[IO] [I I]
Ch.
Warns.
Th.
Neidlinger,
1996. pp. 7-94. Czech. J. Phys. B 33 (1983) 674.
P. Reineker. Ch. Warns, Th. Neidlinger.
I. Barvik. Chem.
Phys. 177 (1993) 715.
V. Sundstrom,
I.
P. Reineker.
Prague.
1Journul of Luminescerwe 76& 77 IIYYH) 331-334
Ch. Warns, I. Barvik.
P. Reineker. Th. Neidhnger.
Chem.
Phys. 194 (1995) 117.
In: [II]
P. Reineker.
Ch.
Warns.
Lumin. 65 (1995) 169.
Th.
Neidlinger.
I. Barvik.
J.
__ fif!B
JOURNAL
OF
LUMINESCENCE
ELSEVIER
Journal
of Luminescence
76&77
(1998) 33
I 324
Exciton transfer in disordered molecular aggregates: Computer modeling of optical line shapes in ring bacteria antenna systems
Abstract The structure of the LH2 subunit Rps. acidoplzi/a consists of a ring of 9 $ heterodimers with 1X 8850 BChI molecules sandwiched between the r and p subunits. For Frenkel excitons moving on the ring we investigate the influence of dynamic and static disorder on their optical line shapes. The dynamic and static disorders are taken into account bq fluctuations of the local energies of the BChl molecules. The fluctuations are represented by dichotomic Markov processes with coloured noise. Comparison of our theoretical results with experimental observations indicates. that the strength of the local energy fluctuation d should be smaller than J ‘2. (’ 1998 Elsevier Science B.V. All rights reserved. Kq,~o&:
Coherent
transfer:
Optical
line shapes: Antenna
systems:
1. Introduction
The structure of the LH2 subunit Rps.acidopkiln consists of a ring of 9 rwBheterodimers [ 11. 18 B850 BChl molecules are sandwiched between the s( and p subunits. The regime of the exciton transfer in the ring of LH2 antenna system remains still an open question. The transition dipole moments in the B850 ring are oriented head-head, the basic unit is a dimer. The ring and whole LH2 has 9-fold symmetry. Short distances between BChl850 molecules lead to a strong mutual interaction J. Estimates reach from 50 to 750 cm ‘. Absorption spectrum of the homogeneous ring of N molecules, with their transition dipole moments in the plane [l]. would be characterized [2] by peaks at the energy E0 in the z-polarization and *Corresponding author. barvik’lr’karlo~.mll;cunl.c7.
Fax:
+ 420 2 296764:
e-mall:
0022-3313 98,$19.00 (‘ 1998 Elsevicr Science B.V. All rights reserved PI/ soo22-23 13(97)00220-2
Photosynthesis
at the double degenerate energy E, in the .Y,Ypolarization (E, = f: + 2 Jcos((2xn);‘N)). We shall take IJI = 1. The explanation of the optical absorption and the hole-burning spectroscopy which relates the energy difference between two lowest states to the experimental value 2OOcm -‘[-?I would lead to the estimation of the transfer integral greater than 700 cm- I. To bring the theoretical values closer to the experimental one, we have tried 14.51 to take into account the nonhomogeneity of the local energies I:, and I:~ and of the transfer integrals J12 and J13 and to change in such a way [6] the energy difference between the two lowest states. Small recently suggested [S]. as Crondelle did before [7], to include the influence of the static disorder. Estimates of the static local energy fluctuations d between 150 and 500 cm ’ can be found in literature. Our aim is an estimation of the strength of the local static disorder d from the optical properties
1. Bun%
332
et (11.
Jou~~ul
o#‘Luminrscrncr
of the ring structures [S]. Ratio A/J determines whether we are allowed to treat the exciton transfer inside of the ring LH2 as a quasicoherent one [9].
2. Method
77 IIUW)
331-334
Due to a lack of knowledge of the microscopical mechanisms we shall use a stochastic approach which describes the influence of the static and dynamic disorder by a stochastic process with prescribed properties. A stochastically time-dependent part of the Hamiltonian H,(t) = c C,(t)n;U,, m
The Hamiltonian H,, = 2 Eaia, m
76&
+ c
Jmna~an
(1)
m+n
describes the purely coherent exciton transfer in the ring of 18 BChl molecules with the same energy F. ai and a, are creation and annihilation operators for an excitation at site m and J,, the excitonic transfer matrix element between sites m and II.
(2)
models the influence of the disorder via fluctuations of the local exciton energies c,(t). Mean values and correlation functions are given by (3)
(&l(r)) = 0, = d,,A’exp(
(~,(t)~,(r))
- R(r -
7)).
160
140
120
100
80
60
-2
-1 Fig. 1. Optical
0
1
2
3
hne shapes I,Jw,) and I,(w) for d = 0.5 and i = 0.05.
4
(4)
1. Burcik et
al.
: Journal
of Luminescence 76& 77 (IWX)
Especially for dichotomic stochastic processes, A is the amplitude of the local energy fluctuations, and i, describes the average switchover rate between the two values.
3. Results We have calculated in a series of papers [lo-121 (where also a thorough description of the method can be found) the optical absorption line shapes I((II) for Frenkel excitons on cyclic models of antenna systems. The optical line shape has been determined from an exact closed system of coupled linear equations for correlation functions. The parameters in these equations are the transfer matrix element J between nearest neighbours, the strength A of the fluctuations of the local excitation energies and the decay constant /. of the correlation functions of these fluctuations. We calculated [lo-121 the optical line shapes for various arrangement of the transition dipole moments and different polarizations of the light. The arrangement of dipoles changes not only the rules for allowed optical transitions but changes also due to the dipole-dipole interaction the transfer integrals [2]. For small values of i., i.e. slow fluctuations, the line shape is strongly structured; it could be related to the energy structure in the static case. With increasing values of r., the lines become broader and the structure of the spectra is smeared out. Furthermore, due to the disorder. transitions are allowed which in the completely ordered case are forbidden. For large values of j. motional narrowing of the optical lines takes place. In this paper we are dealing with the exciton optical line shapes of the ring with 18 BChl 850 molecules. We investigate the influence of the local energ!, ,fluc.tuations within the so-called RPA3 decoupling [lO,ll]. Exciton optical line shapes [lo-1211,(w), for the transition dipole moments and the light polarization parallel with z-axis of the ring, and I,((IJ), for the transition dipole moments and the light polarization perpendicular to the z-axis are displayed on Fig. 1 for i, = 0.05 and A = 0.5.
331-334
73.7
4. Conclusions Raising the strength of the local energy fluctuations A the optical transition to other states than El also gain oscillator strength in I,((II). Results of 4.2 absorption and hole-burning experiments on the B850 absorption band of isolated LH2 complexes revealed [3] direct observation of the lowest exciton level of the B850 ring as a weak but distinct shoulder at the red edge of the B850 band. Comparison between our theoretical (Fig. I) and experimental results indicates. that the strength of the local energy fluctuation d should be weaker than 52. For A = 59 the optical absorption to lowest state Et1 (which is originally. without disorder, not allowed) is comparable with that to the state E, An increase of the local energy fluctuation strength A changes aho the energy difference between E,) and El. Relative weakness of the static local energy disorder allows us to treat the exciton transfer inside of the LH2 ring as a quasicoherent one [9].
Acknowledgements The support of the Deutsche Forschungsgemeinschaft (SFB 239) are gratefully acknowledged. This work has been also funded (I.B.) by contract No. 105!95 of the Charles University and No. 1235197 of the Development Foundation.
References [I]
A.A.
Freer.
S.M.
Prince,
Hawthornthwalte-Lawiess. [I’]
N.W.
Issacs. Structure
R.M.
Pearlstein.
Sauer.
M.Z.
G. McDermott.
Paper.
A.M.
R..l. Cogdell.
4 (1996) 449.
H. Zuber.
tennas and Reaction Springer
K.
in: Michel-Beyerle
(Ed.). An-
Centers of Photosynthetic
Series in Chemical
Bactcrla.
PhysIca. vol. 12. Springer.
Berlin. 1985. [3]
H.-M.
Wu. N.R.S. Reddy. G.J. Sma1L.l. Phys Chem. B IO1
(19971651. [4]
I. Barlik.
In: ESE workshop:
Light-Harvecting
Physics.
BirStonas. 21. 1996. [S]
I. Barvik. 35, 1996.
in: ERC:
Biophysics of Photosynthesix
Sitges.
I. Barr% ei ul.
334
[6]
I. Barvik.
[7]
R. van Grondelle.
P. Herman.
[8]
1. Barvik.
[9]
V. Szocs. 1. Barvik,
BBA 1187 (1994) ICL’96.
Czech J. Phys.. submitted.
J.P. Dekker.
T. Gillbro.
[IO] [I I]
Ch.
Warns.
Th.
Neidlinger,
1996. pp. 7-94. Czech. J. Phys. B 33 (1983) 674.
P. Reineker. Ch. Warns, Th. Neidlinger.
I. Barvik. Chem.
Phys. 177 (1993) 715.
V. Sundstrom,
I.
P. Reineker.
Prague.
1Journul of Luminescerwe 76& 77 IIYYH) 331-334
Ch. Warns, I. Barvik.
P. Reineker. Th. Neidhnger.
Chem.
Phys. 194 (1995) 117.
In: [II]
P. Reineker.
Ch.
Warns.
Lumin. 65 (1995) 169.
Th.
Neidlinger.
I. Barvik.
J.