- Email: [email protected]
The anisotropy of fluorescence in ring units
ARTICLE IN PRESS
Journal of Luminescence 110 (2004) 258–263 www.elsevier.com/locate/jlumin
The anisotropy of fluorescence in ring units Michal Reitera, Pavel Herˇ manb,, Ivan Barvı´ ka a
Institute of Physics of Charles University, Faculty of Mathematics and Physics, CZ-12116 Prague, Czech Republic Department of Physics, University of Hradec Kra´love´, V. Nejedle´ho 573, CZ-50003 Hradec Kra´love´, Czech Republic
b
Available online 21 September 2004
Abstract The time dependence of the anisotropy of fluorescence after an impulsive excitation in the ring unit (resembling the B850 ring of the purple bacterium Rhodopseudomonas acidophila) is calculated. Fast fluctuations of the environment are simulated by dynamic disorder and slow fluctuations by static disorder. Both types of disorder are taken into account simultaneously. Without dynamic disorder modest degrees of static disorder are sufficient to cause the experimentally found initial drop of the anisotropy on a sub-100 fs time scale. Two different models of the spectral density JðoÞ of phonons have been often used in the literature to describe influence of the bath. The spectral density JðoÞ enters the non-Markovian dynamical equations for the one exciton density matrix (entering the time dependence of the anisotropy of fluorescence). The spectral density JðoÞ with high-energy tail leads to faster anisotropy decay. r 2004 Elsevier B.V. All rights reserved. PACS: 82.39.k; 82.53.Ps; 87.15.Aa Keywords: Exciton transfer; Density matrix theory; Fluorescence
1. Introduction Highly efficient light collection and excitation transfer towards the reaction center takes place in the so-called light-harvesting systems (LHs). We are dealing with the ring-shaped units resembling those from antenna complex LH2 from the purple bacterium Rhodopseudomonas acidophila. Our theoretical treatment considers an extended Frenkel Corresponding author. Tel.: +420-493-331-186; fax: +420
495-513-890. E-mail address: [email protected] (P. Herˇ man).
exciton states model but invokes (moderate) static disorder in the excitation energies of the individual BChls. Time-dependent experiments of the femtosecond dynamics of the energy transfer and relaxation [1,2] led for the B850 ring in LH2 complexes to the conclusion that the elementary dynamics occurs on a time scale of about 100 fs [3–5]. For example, depolarization of fluorescence was studied already quite some time ago for a model of electronically coupled molecules [6,7]. Rahman et al. [6] were the first who recognized the importance of the off-diagonal density matrix
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.08.018
ARTICLE IN PRESS M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
elements (coherences) [8] which can lead to an initial anisotropy larger than the incoherent theoretical limit of 0.4. Already some time ago substantial relaxation on the time scale of 10–100 fs and an anomalously large initial anisotropy of 0.7 was observed by Nagarjan et al. [3]. The high initial anisotropy was ascribed to a coherent excitation of a degenerate pair of states with allowed optical transitions and then relaxation to states at lower energies which have forbidden transitions. Nagarjan et al. [4] concluded, that the main features of the spectral relaxation and the decay of anisotropy are reproduced well by a model that considers decay processes of electronic coherences within the manifold of the exciton states and thermal equilibration among the excitonic states. In that contribution the exciton dynamics was not calculated explicitly. In several steps [9–12], we have recently extended the former investigations of the timedependent optical anisotropy by Kumble and Hochstrasser [13] and Nagarjan et al. [4]. For a Gaussian distribution of local energies in the ring units we added the effect of dynamical disorder by using a quantum master equation in the Markovian and non-Markovian limits. Two different models of the spectral density JðoÞ of phonons have been often used in the literature to describe influence of the bath. The spectral density JðoÞ enters the non-Markovian dynamical equations for the one exciton density matrix (entering the time dependence of the anisotropy of fluorescence). In the present investigation, we are comparing results for the spectral density JðoÞ of the bath with and without high-energy tail. Up to now there is only one result of molecular dynamics simulation [14] which shows that the real spectral density contains resonance peaks at higher frequencies in addition to a low-energy peak.
2. Model In the excitation excitation the ideal
following, we assume that only one is present on the ring after an impulsive [13]. The Hamiltonian of an exciton in ring coupled to a bath of harmonic
259
oscillators reads X X H0 ¼ J mn aym an þ _oq byq bq q
m;nðmanÞ
1 XX m þ pffiffiffiffiffi G q _oq aym am ðbyq þ bq Þ N m q ¼ H 0ex þ H ph þ H exph :
ð1Þ
H 0ex
represents the single exciton, i.e. the system. The operator aym ðam Þ creates (annihilates) an exciton at site m. J mn (for man) is the so-called transfer integral between sites m and n. H ph describes the bath of phonons in the harmonic approximation. The phonon creation and annihilation operators are denoted by byq and bq ; respectively. The last term in Eq. (1), H exph ; represents the exciton–bath interaction which is assumed to be site–diagonal and linear in the bath coordinates. The term G m q denotes the exciton-phonon coupling constant. Inside one ring, the pure exciton Hamiltonian H 0ex (Eq. (1)) can be diagonalized using the wave vector representation with corresponding delocalized ‘‘Bloch’’ states and energies. Considering homogeneous case with only nearest-neighbor transfer matrix elements J mn ¼ J 12 ðdm;nþ1 þ dm;n1 Þ and using Fourier transformed excitonic operators (Bloch representation) X 2p l; l ¼ 0; 1; . . . N=2; ak ¼ an eikn ; k ¼ N n (2) ~ the simplest exciton Hamiltonian in k representation reads X H 0ex ¼ E k ayk ak with E k ¼ 2J 12 cos k: k
(3) Influence of static disorder is modelled by a Gaussian distribution for the uncorrelated local energy fluctuations n with a standard deviation D which adds to the Hamiltonian of the ideal ring X H ¼ H0 þ Hs ¼ H0 þ n ayn an : (4) n
All of the Qy transition dipole moments of the chromophores (bacteriochlorophylls (BChls) B850) in a ring without static and dynamic
ARTICLE IN PRESS M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
260
disorder lie approximately in the plane of the ring and the entire dipole strength of the B850 band comes from a degenerate pair of orthogonally polarized transitions at an energy slightly higher than the transition energy of the lowest exciton state. The dipole strength ~ ma of eigenstate jai of the ring with static disorder and the dipole strength ~ ma of eigenstate jai of the ring without static disorder read ~ ma ¼
N X
can~ mn ;
~ ma ¼
n¼1
N X
can~ mn ;
(5)
n¼1
where can and can are the expansion coefficients of the eigenstates of the unperturbed ring and the disordered one in site representation, respectively. In the case of impulsive excitation the dipole strength is simply redistributed among the exciton levels due to disorder [13]. Thus, the impulsive excitation with a pulse of sufficiently wide spectral bandwidth will always prepare the same initial state, irrespective of the actual eigenstates of the real ring. After impulsive excitation with polarization ~ ex the excitonic density matrix r [10] is given by [4] 1 ex ~ rab ðt ¼ 0;~ ex Þ ¼ ð~ ma Þð~ mb ~ ex Þ; A X A¼ ð~ ex ~ ma Þð~ ma ~ ex Þ:
ð6Þ
a
The usual time-dependent anisotropy of fluorescence hS xx ðtÞi hS xy ðtÞi ; hS xx ðtÞi þ 2hS xy ðtÞi Z Sxy ðtÞ ¼ Pxy ðo; tÞ do
rðtÞ ¼
ð7Þ
a
In long time approximation coefficient dOðt ! 1Þ becomes time independent. All details of a calculation leading to the timeconvolutionless dynamical equations for the exciton density matrix are given elsewhere [12] and we shall not repeat them here. The full time dependence of dOðtÞ is given through time dependent parameters [15] Z t X X i_ n Apmn ðtÞ ¼ o2k ðG m G rk k G k Þ N 0 r k X hbjrihrjaihajmihpjbi a;b
eði=_ÞðE a E b Þt f½1 þ nB ð_ok Þ eiok t þ nB ð_ok Þ eiok t g dt:
ð10Þ
Obtaining the full time dependence of dOðtÞ is not a simple task. We have succeeded to calculate microscopically full time dependence of dOðtÞ only for the simplest molecular model namely dimer [16]. In case of molecular ring we should resort to some simplification [12]. In what follows, we use a simple model for correlation functions C mn assuming that each site (i.e. each chromophore) has its own bath completely uncoupled from the baths of the other sites. Furthermore, it is assumed that these baths have identical properties [1,17]. Then only one correlation function CðoÞ is needed C mn ðoÞ ¼ dmn CðoÞ ¼ dmn 2p½1 þ nB ðoÞ½JðoÞ JðoÞ:
ð11Þ
Here JðoÞ is the spectral density [17] and nB ðoÞ the Bose–Einstein distribution of phonons. Two models of the spectral density ðJðoÞ often used in the literature are
is determined from XX Pxy ðo; tÞ ¼ A raa0 ðtÞð~ ma 0 ~ ey Þð~ ey ~ ma Þ a0
½dðo oa0 0 Þ þ dðo oa0 Þ:
exciton density matrix obtained in Ref. [15] read X d rmn ðtÞ ¼ iðOmn;pq þ dOmn;pq ðtÞÞrpq ðtÞ: (9) dt pq
ð8Þ
The brackets h i denote the ensemble average and the orientational average over the sample. The crucial quantity entering the time dependence of the anisotropy in Eq. (7) is the exciton density matrix r: The dynamical equations for the
o2 o=oc e ; 2o3c o ðBÞ JðoÞ ¼ YðoÞj 1 2 : o þ o2D ðAÞ JðoÞ ¼ YðoÞj 0
ð12Þ
Spectral density (A) has its maximum at 2oc : We shall use (in agreement with [1]) j 0 ¼ 0:2 or j 0 ¼
ARTICLE IN PRESS M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
0:4 and oc ¼ 0:2: Parameters j 1 and oD are chosen in such a way that heights and positions of maxima of (A) and (B) coincide. Difference between models (A) and (B) is the presence of high-energy tail in model (B).
261
cm1 : Estimation of the transfer integral J12 varies between 250 and 400 cm1 : For these extreme values of J12 our time unit (t ¼ 1) corresponds to 21.2 or 13.3 fs.
1.5
The anisotropy of fluorescence (Eq. (7)) has been calculated using dynamical equations for the exciton density matrix r to express the time dependence of the optical properties of the ring units in the femtosecond time range. Details are the same as in Ref. [12]. In Ref. [13], which does not take the bath into account, the anisotropy of fluorescence of the LH2 ring decreases from 0.7 to 0.3–0.35 and subsequently reaches a final value of 0:4: One needs a strength of static disorder of D 0:4–0.8 to reach a decay time below 100 fs. Our numerical results are presented graphically. We use dimensionless energies normalized to the transfer integral J12 and the renormalized time t: To convert t into seconds one has to divide t by 2pcJ 12 with c the speed of light in cm s1 and J12 in
J(ω) (arb. units)
3. Results
(A) (B) 1.0
0.5
0.0 0
1
2
3
4
5
hω/ (2πJ12) Fig. 1. The difference between two models of the spectral density used in our calculations. Parameters j 1 and oD are chosen in such a way that heights and positions of maxima of (A) and (B) coincide.
Fig. 2. The time and static disorder D dependence of the anisotropy depolarization for the non-Markovian treatment of the dynamic disorder, the spectral density (A) (left) and (B) (right) and the strength of dynamic disorder j 0 ¼ 0:2: In rows there are the results for the temperature kT ¼ 0:5 (upper row) and kT ¼ 1:0 (lower row).
ARTICLE IN PRESS 262
M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
Fig. 3. The same as on Fig. 2 but for j 0 ¼ 0:4:
In Fig. 1, we show the difference between two models of the spectral density (Eq. (7)) used in our calculations. Parameters j1 and oD are chosen in such a way that heights and positions of maxima of (A) and (B) coincide. In Figs. 2 and 3, the time and D dependences of the anisotropy depolarization are given for the non-Markovian treatment of the dynamic disorder with the strength j 0 ¼ 0:2 and j 0 ¼ 0:4; respectively. In rows there are the results for the temperature kT ¼ 0:5 (upper row) and kT ¼ 1:0 (lower row) and in columns the results with the spectral density (A) (left) and (B) (right). The decay of the anisotropy depolarization with the spectral density (B) is always somewhat faster than with (A) one. The model with the spectral density (B) supports more of those transitions between states with higher energy difference. Rahman et al. [6] were the first who recognized the importance of the off-diagonal density matrix elements (coherences) [8] which can lead to an initial anisotropy rð0Þ larger than the incoherent theoretical limit of 0.4. The difference between the (A) and (B) calculations measured by the time at which the anisotropy depolarization reaches r ¼ 0:4 is maximum (as
much as 25%) for the small static disorder. Increasing the static disorder D the difference between Model A and Model B calculations becomes smaller.
Acknowledgements This work has been funded by the project GACˇR 202-03-0817.
References [1] V. Sundstro¨m, T. Pullertis, R. van Grondelle, J. Phys. Chem. B 103 (1999) 2327. [2] V. Novoderezhkin, R. van Grondelle, J. Phys. Chem. B 106 (2002) 6025. [3] V. Nagarjan, R.G. Alden, J.C. Williams, W.W. Parson, Proc. Natl. Acad. Sci. USA 93 (1996) 13774. [4] V. Nagarjan, E.T. Johnson, J.C. Williams, W.W. Parson, J. Phys. Chem. B 103 (1999) 2297. [5] V. Nagarjan, W.W. Parson, J. Phys. Chem. B 104 (2000) 4010. [6] T.S. Rahman, R.S. Knox, V.M. Kenkre, Chem. Phys. 44 (1979) 197. [7] K. Wynne, R.M. Hochstrasser, Chem. Phys. 171 (1993) 179.
ARTICLE IN PRESS M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263 [8] O. Ku¨hn, V. Sundstro¨m, T. Pullertis, Chem. Phys. 275 (2002) 15. [9] P. Herˇ man, U. Kleinekatho¨fer, I. Barvı´ k, M. Schreiber, J. Lumin. 94&95 (2001) 447. [10] P. Herˇ man, U. Kleinekatho¨fer, I. Barvı´ k, M. Schreiber, Chem. Phys. 275 (2002) 1. [11] I. Barvı´ k, I. Kondov, P. Herˇ man, M. Schreiber, U. Kleinekatho¨fer, Nonlin. Opt. 29 (2002) 167. [12] P. Herˇ man, I. Barvı´ k, Czech. J. Phys. 53 (2003) 579.
263
[13] R. Kumble, R. Hochstrasser, J. Chem. Phys. 109 (1998) 855. [14] A. Damjanovic, I. Kosztin, U. Kleinekatho¨fer, K. Schulten, Phys. Rev. E 65 (2002) 031919. [15] V. Cˇa´pek, Z. Phys. B 99 (1996) 261. [16] I. Barvı´ k, J. Macek, J. Chin. Chem. Soc. 47 (2000) 647. [17] V. May, O. Ku¨hn, Charge and Energy Transfer in Molecular Systems, Wiley-WCH, Berlin, 2000.
Journal of Luminescence 110 (2004) 258–263 www.elsevier.com/locate/jlumin
The anisotropy of fluorescence in ring units Michal Reitera, Pavel Herˇ manb,, Ivan Barvı´ ka a
Institute of Physics of Charles University, Faculty of Mathematics and Physics, CZ-12116 Prague, Czech Republic Department of Physics, University of Hradec Kra´love´, V. Nejedle´ho 573, CZ-50003 Hradec Kra´love´, Czech Republic
b
Available online 21 September 2004
Abstract The time dependence of the anisotropy of fluorescence after an impulsive excitation in the ring unit (resembling the B850 ring of the purple bacterium Rhodopseudomonas acidophila) is calculated. Fast fluctuations of the environment are simulated by dynamic disorder and slow fluctuations by static disorder. Both types of disorder are taken into account simultaneously. Without dynamic disorder modest degrees of static disorder are sufficient to cause the experimentally found initial drop of the anisotropy on a sub-100 fs time scale. Two different models of the spectral density JðoÞ of phonons have been often used in the literature to describe influence of the bath. The spectral density JðoÞ enters the non-Markovian dynamical equations for the one exciton density matrix (entering the time dependence of the anisotropy of fluorescence). The spectral density JðoÞ with high-energy tail leads to faster anisotropy decay. r 2004 Elsevier B.V. All rights reserved. PACS: 82.39.k; 82.53.Ps; 87.15.Aa Keywords: Exciton transfer; Density matrix theory; Fluorescence
1. Introduction Highly efficient light collection and excitation transfer towards the reaction center takes place in the so-called light-harvesting systems (LHs). We are dealing with the ring-shaped units resembling those from antenna complex LH2 from the purple bacterium Rhodopseudomonas acidophila. Our theoretical treatment considers an extended Frenkel Corresponding author. Tel.: +420-493-331-186; fax: +420
495-513-890. E-mail address: [email protected] (P. Herˇ man).
exciton states model but invokes (moderate) static disorder in the excitation energies of the individual BChls. Time-dependent experiments of the femtosecond dynamics of the energy transfer and relaxation [1,2] led for the B850 ring in LH2 complexes to the conclusion that the elementary dynamics occurs on a time scale of about 100 fs [3–5]. For example, depolarization of fluorescence was studied already quite some time ago for a model of electronically coupled molecules [6,7]. Rahman et al. [6] were the first who recognized the importance of the off-diagonal density matrix
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.08.018
ARTICLE IN PRESS M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
elements (coherences) [8] which can lead to an initial anisotropy larger than the incoherent theoretical limit of 0.4. Already some time ago substantial relaxation on the time scale of 10–100 fs and an anomalously large initial anisotropy of 0.7 was observed by Nagarjan et al. [3]. The high initial anisotropy was ascribed to a coherent excitation of a degenerate pair of states with allowed optical transitions and then relaxation to states at lower energies which have forbidden transitions. Nagarjan et al. [4] concluded, that the main features of the spectral relaxation and the decay of anisotropy are reproduced well by a model that considers decay processes of electronic coherences within the manifold of the exciton states and thermal equilibration among the excitonic states. In that contribution the exciton dynamics was not calculated explicitly. In several steps [9–12], we have recently extended the former investigations of the timedependent optical anisotropy by Kumble and Hochstrasser [13] and Nagarjan et al. [4]. For a Gaussian distribution of local energies in the ring units we added the effect of dynamical disorder by using a quantum master equation in the Markovian and non-Markovian limits. Two different models of the spectral density JðoÞ of phonons have been often used in the literature to describe influence of the bath. The spectral density JðoÞ enters the non-Markovian dynamical equations for the one exciton density matrix (entering the time dependence of the anisotropy of fluorescence). In the present investigation, we are comparing results for the spectral density JðoÞ of the bath with and without high-energy tail. Up to now there is only one result of molecular dynamics simulation [14] which shows that the real spectral density contains resonance peaks at higher frequencies in addition to a low-energy peak.
2. Model In the excitation excitation the ideal
following, we assume that only one is present on the ring after an impulsive [13]. The Hamiltonian of an exciton in ring coupled to a bath of harmonic
259
oscillators reads X X H0 ¼ J mn aym an þ _oq byq bq q
m;nðmanÞ
1 XX m þ pffiffiffiffiffi G q _oq aym am ðbyq þ bq Þ N m q ¼ H 0ex þ H ph þ H exph :
ð1Þ
H 0ex
represents the single exciton, i.e. the system. The operator aym ðam Þ creates (annihilates) an exciton at site m. J mn (for man) is the so-called transfer integral between sites m and n. H ph describes the bath of phonons in the harmonic approximation. The phonon creation and annihilation operators are denoted by byq and bq ; respectively. The last term in Eq. (1), H exph ; represents the exciton–bath interaction which is assumed to be site–diagonal and linear in the bath coordinates. The term G m q denotes the exciton-phonon coupling constant. Inside one ring, the pure exciton Hamiltonian H 0ex (Eq. (1)) can be diagonalized using the wave vector representation with corresponding delocalized ‘‘Bloch’’ states and energies. Considering homogeneous case with only nearest-neighbor transfer matrix elements J mn ¼ J 12 ðdm;nþ1 þ dm;n1 Þ and using Fourier transformed excitonic operators (Bloch representation) X 2p l; l ¼ 0; 1; . . . N=2; ak ¼ an eikn ; k ¼ N n (2) ~ the simplest exciton Hamiltonian in k representation reads X H 0ex ¼ E k ayk ak with E k ¼ 2J 12 cos k: k
(3) Influence of static disorder is modelled by a Gaussian distribution for the uncorrelated local energy fluctuations n with a standard deviation D which adds to the Hamiltonian of the ideal ring X H ¼ H0 þ Hs ¼ H0 þ n ayn an : (4) n
All of the Qy transition dipole moments of the chromophores (bacteriochlorophylls (BChls) B850) in a ring without static and dynamic
ARTICLE IN PRESS M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
260
disorder lie approximately in the plane of the ring and the entire dipole strength of the B850 band comes from a degenerate pair of orthogonally polarized transitions at an energy slightly higher than the transition energy of the lowest exciton state. The dipole strength ~ ma of eigenstate jai of the ring with static disorder and the dipole strength ~ ma of eigenstate jai of the ring without static disorder read ~ ma ¼
N X
can~ mn ;
~ ma ¼
n¼1
N X
can~ mn ;
(5)
n¼1
where can and can are the expansion coefficients of the eigenstates of the unperturbed ring and the disordered one in site representation, respectively. In the case of impulsive excitation the dipole strength is simply redistributed among the exciton levels due to disorder [13]. Thus, the impulsive excitation with a pulse of sufficiently wide spectral bandwidth will always prepare the same initial state, irrespective of the actual eigenstates of the real ring. After impulsive excitation with polarization ~ ex the excitonic density matrix r [10] is given by [4] 1 ex ~ rab ðt ¼ 0;~ ex Þ ¼ ð~ ma Þð~ mb ~ ex Þ; A X A¼ ð~ ex ~ ma Þð~ ma ~ ex Þ:
ð6Þ
a
The usual time-dependent anisotropy of fluorescence hS xx ðtÞi hS xy ðtÞi ; hS xx ðtÞi þ 2hS xy ðtÞi Z Sxy ðtÞ ¼ Pxy ðo; tÞ do
rðtÞ ¼
ð7Þ
a
In long time approximation coefficient dOðt ! 1Þ becomes time independent. All details of a calculation leading to the timeconvolutionless dynamical equations for the exciton density matrix are given elsewhere [12] and we shall not repeat them here. The full time dependence of dOðtÞ is given through time dependent parameters [15] Z t X X i_ n Apmn ðtÞ ¼ o2k ðG m G rk k G k Þ N 0 r k X hbjrihrjaihajmihpjbi a;b
eði=_ÞðE a E b Þt f½1 þ nB ð_ok Þ eiok t þ nB ð_ok Þ eiok t g dt:
ð10Þ
Obtaining the full time dependence of dOðtÞ is not a simple task. We have succeeded to calculate microscopically full time dependence of dOðtÞ only for the simplest molecular model namely dimer [16]. In case of molecular ring we should resort to some simplification [12]. In what follows, we use a simple model for correlation functions C mn assuming that each site (i.e. each chromophore) has its own bath completely uncoupled from the baths of the other sites. Furthermore, it is assumed that these baths have identical properties [1,17]. Then only one correlation function CðoÞ is needed C mn ðoÞ ¼ dmn CðoÞ ¼ dmn 2p½1 þ nB ðoÞ½JðoÞ JðoÞ:
ð11Þ
Here JðoÞ is the spectral density [17] and nB ðoÞ the Bose–Einstein distribution of phonons. Two models of the spectral density ðJðoÞ often used in the literature are
is determined from XX Pxy ðo; tÞ ¼ A raa0 ðtÞð~ ma 0 ~ ey Þð~ ey ~ ma Þ a0
½dðo oa0 0 Þ þ dðo oa0 Þ:
exciton density matrix obtained in Ref. [15] read X d rmn ðtÞ ¼ iðOmn;pq þ dOmn;pq ðtÞÞrpq ðtÞ: (9) dt pq
ð8Þ
The brackets h i denote the ensemble average and the orientational average over the sample. The crucial quantity entering the time dependence of the anisotropy in Eq. (7) is the exciton density matrix r: The dynamical equations for the
o2 o=oc e ; 2o3c o ðBÞ JðoÞ ¼ YðoÞj 1 2 : o þ o2D ðAÞ JðoÞ ¼ YðoÞj 0
ð12Þ
Spectral density (A) has its maximum at 2oc : We shall use (in agreement with [1]) j 0 ¼ 0:2 or j 0 ¼
ARTICLE IN PRESS M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
0:4 and oc ¼ 0:2: Parameters j 1 and oD are chosen in such a way that heights and positions of maxima of (A) and (B) coincide. Difference between models (A) and (B) is the presence of high-energy tail in model (B).
261
cm1 : Estimation of the transfer integral J12 varies between 250 and 400 cm1 : For these extreme values of J12 our time unit (t ¼ 1) corresponds to 21.2 or 13.3 fs.
1.5
The anisotropy of fluorescence (Eq. (7)) has been calculated using dynamical equations for the exciton density matrix r to express the time dependence of the optical properties of the ring units in the femtosecond time range. Details are the same as in Ref. [12]. In Ref. [13], which does not take the bath into account, the anisotropy of fluorescence of the LH2 ring decreases from 0.7 to 0.3–0.35 and subsequently reaches a final value of 0:4: One needs a strength of static disorder of D 0:4–0.8 to reach a decay time below 100 fs. Our numerical results are presented graphically. We use dimensionless energies normalized to the transfer integral J12 and the renormalized time t: To convert t into seconds one has to divide t by 2pcJ 12 with c the speed of light in cm s1 and J12 in
J(ω) (arb. units)
3. Results
(A) (B) 1.0
0.5
0.0 0
1
2
3
4
5
hω/ (2πJ12) Fig. 1. The difference between two models of the spectral density used in our calculations. Parameters j 1 and oD are chosen in such a way that heights and positions of maxima of (A) and (B) coincide.
Fig. 2. The time and static disorder D dependence of the anisotropy depolarization for the non-Markovian treatment of the dynamic disorder, the spectral density (A) (left) and (B) (right) and the strength of dynamic disorder j 0 ¼ 0:2: In rows there are the results for the temperature kT ¼ 0:5 (upper row) and kT ¼ 1:0 (lower row).
ARTICLE IN PRESS 262
M. Reiter et al. / Journal of Luminescence 110 (2004) 258–263
Fig. 3. The same as on Fig. 2 but for j 0 ¼ 0:4:
In Fig. 1, we show the difference between two models of the spectral density (Eq. (7)) used in our calculations. Parameters j1 and oD are chosen in such a way that heights and positions of maxima of (A) and (B) coincide. In Figs. 2 and 3, the time and D dependences of the anisotropy depolarization are given for the non-Markovian treatment of the dynamic disorder with the strength j 0 ¼ 0:2 and j 0 ¼ 0:4; respectively. In rows there are the results for the temperature kT ¼ 0:5 (upper row) and kT ¼ 1:0 (lower row) and in columns the results with the spectral density (A) (left) and (B) (right). The decay of the anisotropy depolarization with the spectral density (B) is always somewhat faster than with (A) one. The model with the spectral density (B) supports more of those transitions between states with higher energy difference. Rahman et al. [6] were the first who recognized the importance of the off-diagonal density matrix elements (coherences) [8] which can lead to an initial anisotropy rð0Þ larger than the incoherent theoretical limit of 0.4. The difference between the (A) and (B) calculations measured by the time at which the anisotropy depolarization reaches r ¼ 0:4 is maximum (as
much as 25%) for the small static disorder. Increasing the static disorder D the difference between Model A and Model B calculations becomes smaller.
Acknowledgements This work has been funded by the project GACˇR 202-03-0817.
References [1] V. Sundstro¨m, T. Pullertis, R. van Grondelle, J. Phys. Chem. B 103 (1999) 2327. [2] V. Novoderezhkin, R. van Grondelle, J. Phys. Chem. B 106 (2002) 6025. [3] V. Nagarjan, R.G. Alden, J.C. Williams, W.W. Parson, Proc. Natl. Acad. Sci. USA 93 (1996) 13774. [4] V. Nagarjan, E.T. Johnson, J.C. Williams, W.W. Parson, J. Phys. Chem. B 103 (1999) 2297. [5] V. Nagarjan, W.W. Parson, J. Phys. Chem. B 104 (2000) 4010. [6] T.S. Rahman, R.S. Knox, V.M. Kenkre, Chem. Phys. 44 (1979) 197. [7] K. Wynne, R.M. Hochstrasser, Chem. Phys. 171 (1993) 179.
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