Photosynthesis Research 49: 269-276, 1996. (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.
Regular paper
Site inhomogeneity and exciton delocalization in the photosynthetic antenna T a t i a n a V. D r a c h e v a 1, V l a d i m i r I. N o v o d e r e z h k i n 1'2 & A n d r e i E R a z j i v i n 1
1A.N. Belozersky Institute of Physico-Chemical Biology, M.V. Lomonosov Moscow State University, Moscow 119899, Russian Federation; 2Scientific Research Center on Technological Lasers, Russian Academy of Science, Troizk, Moscow Region 142092, Russian Federation Received 18 December 1995; accepted in revised form 31 July 1996
Key words: exciton, light-harvesting antenna, primary processes, bacterial photosynthesis, site inhomogeneity
Abstract
The influence of energy disorder on exciton states of molecular aggregates (the dimer and the circular aggregate) was analyzed. The dipole strength and inhomogeneous line shapes of exciton states were calculated by means of numerical diagonalization of Hamiltonian with diagonal energy disorder without intersite correlation. The disorder degree corresponding to destruction of coherent exciton states was estimated. The circular aggregates were treated as a model of light-harvesting antenna structures of photosynthetic bacteria. It was concluded that the site inhomogeneity typical for LH1 and LH2 complexes of purple bacteria cannot significantly influence the exciton delocalization over the whole antenna.
Abbreviations: BChl-bacteriochlorophyll; LH1 and L H 2 - ' c o r e ' and 'peripheral' light-harvesting complexes from purple bacteria; RC-reaction center
Introduction
Nowadays there are two theoretical models describing spectral and kinetic properties of the light-harvesting antenna of purple photosynthetic bacteria. The first one assumes excitation localization at the BChl dimer (or small cluster of strongly coupled Bchls). It treats the energy transfer process as hopping of such localized excitation between weakly coupled dimers (clusters) (Pullerits et al. 1994; Somsen et al. 1994). Another model assumes exciton delocalization over the circular aggregate of strongly coupled BChls with CNsymmetry, where N is the number of BChl monomers or dimers in LH1 or LH2 complex (Dracheva et al. 1995; Novoderezhkin and Razjivin 1993, 1994, 1995a,b). The crystal structure of the antenna complex shows that the light-harvesting BChl molecules aggregate in the form of a perfect ring (McDermott et al. 1995). The distances between BChls are about 9 ,~ and correspond
to strong exciton coupling. This is an argument for the delocalized exciton hypothesis. Exciton-phonon interactions as well as the site inhomogeneity induced by static interactions of pigments with the surroundings can destroy the delocalized exciton states. In this case the radius of exciton delocalization will be significantly reduced. The only method to determine the exciton radius is some kind of nonlinear technique, such as pump and probe (excited state absorption) spectroscopy. It is known that the nonlinear response of a molecular aggregate is proportional to the number of excitonically coupled pigment molecules (Spano and Mukamel 1989). Experiments with different purple bacteriae species showed that the induced absorption at the bleaching peak of the antenna is 4-16 times greater than the bleaching of the BChl dimer of the RC (Danielius et al. 1989; Kennis et al. 1994; Novoderezhkin and Razjivin 1993, 1995b). This is direct evidence for exciton delocalization over many light-harvesting BChls.
270 On the other hand, experimental data shows an inhomogeneous broadening of the antenna spectral lines (Van Mourik et al. 1992, 1993; Visschers et al. 1993; Van Grondelle et al. 1994). This fact is usually treated as an indication for excitation localization at the Bchl dimer (or small Bchl aggregate) of the antenna (Somsen et al. 1994; Van Grondelle et al. 1994). In this paper we demonstrate that there is no contradiction between the fact of inhomogeneous broadening and the exciton delocalization hypothesis. The main problem is the influence of site inhomogeneity on the exciton structure of the antenna. The theory of exciton spectra and dynamics in the antenna of purple bacteria is presented for arbitrary degree of spectral disorder. The disorder degree value corresponding to destruction of coherent exciton states is estimated. These critical disorder values for several bacteria are compared with the experimental ones revealed by holeburning spectroscopy, site selection spectroscopy, and other methods. The exciton structure and lineshape of disordered aggregates were studied for one-dimensional chains with parallel transition dipoles (Knapp 1984), for Jaggregates (Fidder et al. 1991, 1993; Knoester 1993) and for a BChl dimer (Koolhaas et al. 1994). In this paper we analyze the same problem for the ring-like BChl aggregates with circularly degenerated dipoles as a model for the antenna of purple bacteria. To calculate the exciton structure of an aggregate with arbitrary degree of spectral disorder we used a numerical procedure for diagonalization of molecular Hamilton±an and Monte-Carlo integration of energies and dipole strengths of spectral components. The analytical solution for exciton state energies, wave function and dipole moments can be obtained only for the simplest molecular aggregate (dimer). This solution is presented in the first section. A numerical solution for the twelve-fold dodecamer as an example of circular aggregate considered in the next section.
Symmetric dimer Let us consider a dimer of identical molecules with parallel transition dipole moments. Suppose that electronic excitation energies AE of both molecules exhibit a statistical shift due to interaction with the surroundings. The Hamilton±an of such a system can be written as H = (AE + A1)B+B1 + (AE + A2) B+B: + M(B+B2 + B+B1),
(1)
where operators B +, B + and B1, B2 create and destroy the localized states I1>, 12> with the energies El, E2; A1 and A2 describe a diagonal disorder; M is nondiagonal coupling energy (let everywhere below M +IMI
m
~.E+@
m
*lMI
~x
IUl
•"E +>IMI) the difference between oscillator strengths of both components decreases; their widths approach A0. This case is close to the case of two noninteracting molecules with different energies. All these conclusions concerning exciton delocalization (based on studies of spectral features) can be obtained directly by exciton wavefunction calculation. The wavefunction of I-4-> states in localized basis is -
+
(1
-
c2i = (1/v"2)[1 ~ (1 - e2)1/2] 1/2
14- > = cffll > +c2~12 >
(5)
In the case of weak disorder (e is near to unity) the amplitudes of exciton wavefunction, 1cl,21 = 1/V~, are the same for both sites, i.e., the exciton is uniformly delocalized over two molecules. The difference between Icll and Ic21 increases with the increasing of disorder degree; the radius of delocalization reduces.
Twelve-fold symmetry dodecamer Let us consider a circular aggregate with CN-symmetry where N is the total number of identical BChl molecules. As an example we take N = 12. The orientation of the BChl transition dipole moment d,~ is characterized by angles ~band qo, where ~ois the angle between dn and circle plane~0~; 4) is the angle between the projection of vector dn to the ~'0~7plane and the tangent to the circle. The Hamiltonian for the circular aggregate is the nearest neighbors' approximation can be written as: N
N
H = Z ( A E + A#).B+B. + Z M ' n=l
(B+B.+, + Bff+IBn) (6)
n=l
Here &ISis the electronic excitation energy of the isolated Bchl molecule; M is the matrix element of resonant interactions between molecules n and (n + 1); B + and Bn are the excitation creation and destruction operators at the n-th site (B++1 and BN+I are equal to B + and B 1). Statistics of A,, are supposed to be Gaussian without intersite correlation: W(Al...An...) = 7r-N/2A(~-nexp[-(A~ + ... + An2 + ...)/~2)] (7) The inhomogeneously broadened spectra of the dodecamer are simulated by Monte-Carlo method. We supposed that ~o = 15 ° and M = - 7 4 0 c m - l ; zero of energy is taken to be AE. It is convenient to introduce the site inhomogeneity width (FWHM), F = 2Ao lv/i-ff2, and the inhomogeneous width of exciton levels (FWHM), rex. We calculated the dipole moments, d2k, the energies, Ek, and the width, Fex, of k-th exciton level versus F (Figure 2 and 3). The spectral components and inhomogeneous line shapes of the absorption spectrum for various disorder degree are shown in Figures 4 and 5. The homogeneous line broadening of higher exciton levels due to relaxation is not taken into account. In the case of homogeneous aggregate (F = Fex = 0) only two lowest levels, k = 0 and 4-1, are dipole allowed. The energy gap between them is 200 cm -1 . Most of the oscillator strength is concentrated in the two-fold degenerated k = 4-1 level. The anomalously high dipole moment for this level, a~il, is a result
272 2
r~
dK
o
g
-2
i
i
I
I
I
2
i
I
Z
V
+B 0 -3
+2
Figure 3. The relation between the exciton level inhomogeneity width, Fex, and the site inhomogoneity width, F, for the lowest exeiton level of the twelve-fold symmetry dodecamer. Fex and F are normalizedto IMI.Numerical results are shown by circles; the solid line corresponds to Fex-FV'~ Insort: the inhomogeneous line shape for the lowest exeiton level, where E is normalized to Inl.
4 I
I
2
I
I
~
I
F
Figure 2. The dipole moments ~ and energies Ek of k-th exeiton level versus the site inhomogeneity width, F, for the twelve-fold symmetry dodecamer (for the lowest levels k - 0, -4-1, -t-2, 4-3 only; the k - 4- 4, 4-5, and 6 levels are not shown; a~ are normalized to d z, Ek and F are normalized to IMI;the zero of energy is taken to be
e 0
0
0
AE).
o o o
o
of coherent summation of N molecular dipoles in the circle plane. In the case of weak disorder (F
Regular paper
Site inhomogeneity and exciton delocalization in the photosynthetic antenna T a t i a n a V. D r a c h e v a 1, V l a d i m i r I. N o v o d e r e z h k i n 1'2 & A n d r e i E R a z j i v i n 1
1A.N. Belozersky Institute of Physico-Chemical Biology, M.V. Lomonosov Moscow State University, Moscow 119899, Russian Federation; 2Scientific Research Center on Technological Lasers, Russian Academy of Science, Troizk, Moscow Region 142092, Russian Federation Received 18 December 1995; accepted in revised form 31 July 1996
Key words: exciton, light-harvesting antenna, primary processes, bacterial photosynthesis, site inhomogeneity
Abstract
The influence of energy disorder on exciton states of molecular aggregates (the dimer and the circular aggregate) was analyzed. The dipole strength and inhomogeneous line shapes of exciton states were calculated by means of numerical diagonalization of Hamiltonian with diagonal energy disorder without intersite correlation. The disorder degree corresponding to destruction of coherent exciton states was estimated. The circular aggregates were treated as a model of light-harvesting antenna structures of photosynthetic bacteria. It was concluded that the site inhomogeneity typical for LH1 and LH2 complexes of purple bacteria cannot significantly influence the exciton delocalization over the whole antenna.
Abbreviations: BChl-bacteriochlorophyll; LH1 and L H 2 - ' c o r e ' and 'peripheral' light-harvesting complexes from purple bacteria; RC-reaction center
Introduction
Nowadays there are two theoretical models describing spectral and kinetic properties of the light-harvesting antenna of purple photosynthetic bacteria. The first one assumes excitation localization at the BChl dimer (or small cluster of strongly coupled Bchls). It treats the energy transfer process as hopping of such localized excitation between weakly coupled dimers (clusters) (Pullerits et al. 1994; Somsen et al. 1994). Another model assumes exciton delocalization over the circular aggregate of strongly coupled BChls with CNsymmetry, where N is the number of BChl monomers or dimers in LH1 or LH2 complex (Dracheva et al. 1995; Novoderezhkin and Razjivin 1993, 1994, 1995a,b). The crystal structure of the antenna complex shows that the light-harvesting BChl molecules aggregate in the form of a perfect ring (McDermott et al. 1995). The distances between BChls are about 9 ,~ and correspond
to strong exciton coupling. This is an argument for the delocalized exciton hypothesis. Exciton-phonon interactions as well as the site inhomogeneity induced by static interactions of pigments with the surroundings can destroy the delocalized exciton states. In this case the radius of exciton delocalization will be significantly reduced. The only method to determine the exciton radius is some kind of nonlinear technique, such as pump and probe (excited state absorption) spectroscopy. It is known that the nonlinear response of a molecular aggregate is proportional to the number of excitonically coupled pigment molecules (Spano and Mukamel 1989). Experiments with different purple bacteriae species showed that the induced absorption at the bleaching peak of the antenna is 4-16 times greater than the bleaching of the BChl dimer of the RC (Danielius et al. 1989; Kennis et al. 1994; Novoderezhkin and Razjivin 1993, 1995b). This is direct evidence for exciton delocalization over many light-harvesting BChls.
270 On the other hand, experimental data shows an inhomogeneous broadening of the antenna spectral lines (Van Mourik et al. 1992, 1993; Visschers et al. 1993; Van Grondelle et al. 1994). This fact is usually treated as an indication for excitation localization at the Bchl dimer (or small Bchl aggregate) of the antenna (Somsen et al. 1994; Van Grondelle et al. 1994). In this paper we demonstrate that there is no contradiction between the fact of inhomogeneous broadening and the exciton delocalization hypothesis. The main problem is the influence of site inhomogeneity on the exciton structure of the antenna. The theory of exciton spectra and dynamics in the antenna of purple bacteria is presented for arbitrary degree of spectral disorder. The disorder degree value corresponding to destruction of coherent exciton states is estimated. These critical disorder values for several bacteria are compared with the experimental ones revealed by holeburning spectroscopy, site selection spectroscopy, and other methods. The exciton structure and lineshape of disordered aggregates were studied for one-dimensional chains with parallel transition dipoles (Knapp 1984), for Jaggregates (Fidder et al. 1991, 1993; Knoester 1993) and for a BChl dimer (Koolhaas et al. 1994). In this paper we analyze the same problem for the ring-like BChl aggregates with circularly degenerated dipoles as a model for the antenna of purple bacteria. To calculate the exciton structure of an aggregate with arbitrary degree of spectral disorder we used a numerical procedure for diagonalization of molecular Hamilton±an and Monte-Carlo integration of energies and dipole strengths of spectral components. The analytical solution for exciton state energies, wave function and dipole moments can be obtained only for the simplest molecular aggregate (dimer). This solution is presented in the first section. A numerical solution for the twelve-fold dodecamer as an example of circular aggregate considered in the next section.
Symmetric dimer Let us consider a dimer of identical molecules with parallel transition dipole moments. Suppose that electronic excitation energies AE of both molecules exhibit a statistical shift due to interaction with the surroundings. The Hamilton±an of such a system can be written as H = (AE + A1)B+B1 + (AE + A2) B+B: + M(B+B2 + B+B1),
(1)
where operators B +, B + and B1, B2 create and destroy the localized states I1>, 12> with the energies El, E2; A1 and A2 describe a diagonal disorder; M is nondiagonal coupling energy (let everywhere below M +IMI
m
~.E+@
m
*lMI
~x
IUl
•"E +>IMI) the difference between oscillator strengths of both components decreases; their widths approach A0. This case is close to the case of two noninteracting molecules with different energies. All these conclusions concerning exciton delocalization (based on studies of spectral features) can be obtained directly by exciton wavefunction calculation. The wavefunction of I-4-> states in localized basis is -
+
(1
-
c2i = (1/v"2)[1 ~ (1 - e2)1/2] 1/2
14- > = cffll > +c2~12 >
(5)
In the case of weak disorder (e is near to unity) the amplitudes of exciton wavefunction, 1cl,21 = 1/V~, are the same for both sites, i.e., the exciton is uniformly delocalized over two molecules. The difference between Icll and Ic21 increases with the increasing of disorder degree; the radius of delocalization reduces.
Twelve-fold symmetry dodecamer Let us consider a circular aggregate with CN-symmetry where N is the total number of identical BChl molecules. As an example we take N = 12. The orientation of the BChl transition dipole moment d,~ is characterized by angles ~band qo, where ~ois the angle between dn and circle plane~0~; 4) is the angle between the projection of vector dn to the ~'0~7plane and the tangent to the circle. The Hamiltonian for the circular aggregate is the nearest neighbors' approximation can be written as: N
N
H = Z ( A E + A#).B+B. + Z M ' n=l
(B+B.+, + Bff+IBn) (6)
n=l
Here &ISis the electronic excitation energy of the isolated Bchl molecule; M is the matrix element of resonant interactions between molecules n and (n + 1); B + and Bn are the excitation creation and destruction operators at the n-th site (B++1 and BN+I are equal to B + and B 1). Statistics of A,, are supposed to be Gaussian without intersite correlation: W(Al...An...) = 7r-N/2A(~-nexp[-(A~ + ... + An2 + ...)/~2)] (7) The inhomogeneously broadened spectra of the dodecamer are simulated by Monte-Carlo method. We supposed that ~o = 15 ° and M = - 7 4 0 c m - l ; zero of energy is taken to be AE. It is convenient to introduce the site inhomogeneity width (FWHM), F = 2Ao lv/i-ff2, and the inhomogeneous width of exciton levels (FWHM), rex. We calculated the dipole moments, d2k, the energies, Ek, and the width, Fex, of k-th exciton level versus F (Figure 2 and 3). The spectral components and inhomogeneous line shapes of the absorption spectrum for various disorder degree are shown in Figures 4 and 5. The homogeneous line broadening of higher exciton levels due to relaxation is not taken into account. In the case of homogeneous aggregate (F = Fex = 0) only two lowest levels, k = 0 and 4-1, are dipole allowed. The energy gap between them is 200 cm -1 . Most of the oscillator strength is concentrated in the two-fold degenerated k = 4-1 level. The anomalously high dipole moment for this level, a~il, is a result
272 2
r~
dK
o
g
-2
i
i
I
I
I
2
i
I
Z
V
+B 0 -3
+2
Figure 3. The relation between the exciton level inhomogeneity width, Fex, and the site inhomogoneity width, F, for the lowest exeiton level of the twelve-fold symmetry dodecamer. Fex and F are normalizedto IMI.Numerical results are shown by circles; the solid line corresponds to Fex-FV'~ Insort: the inhomogeneous line shape for the lowest exeiton level, where E is normalized to Inl.
4 I
I
2
I
I
~
I
F
Figure 2. The dipole moments ~ and energies Ek of k-th exeiton level versus the site inhomogeneity width, F, for the twelve-fold symmetry dodecamer (for the lowest levels k - 0, -4-1, -t-2, 4-3 only; the k - 4- 4, 4-5, and 6 levels are not shown; a~ are normalized to d z, Ek and F are normalized to IMI;the zero of energy is taken to be
e 0
0
0
AE).
o o o
o
of coherent summation of N molecular dipoles in the circle plane. In the case of weak disorder (F
