- Email: [email protected]
Temperature dependence of the anisotropy of fluorescence in ring molecular systems
ARTICLE IN PRESS
Journal of Luminescence 122–123 (2007) 558–561 www.elsevier.com/locate/jlumin
Temperature dependence of the anisotropy of fluorescence in ring molecular systems Pavel Herˇ mana,, Ivan Barvı´ kb a
Department of Physics, University of Hradec Kra´love´, Rokitanske´ho 62, CZ-50003 Hradec Kra´love´, Czech Republic Institute of Physics of Charles University, Faculty of Mathematics and Physics, CZ-12116 Prague, Czech Republic
b
Available online 14 March 2006
Abstract The time dependence of the anisotropy of fluorescence after an impulsive excitation in the molecular ring (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 uncorrelated static disorder. 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. In the present investigation we are comparing results for the time-dependent optical anisotropy of the molecular ring for four models of the uncorrelated static disorder: Gaussian disorder in the local energies (model A), Gaussian disorder in the transfer integrals (model B), Gaussian disorder in radial positions of molecules (model C) and Gaussian disorder in angular positions of molecules (model D). Both types of disorder—static and dynamic— are taken into account simultaneously. r 2006 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 We are dealing with the ring-shaped units resembling those from antenna complex LH2 of the purple bacterium Rhodopseudomonas acidophila in which a highly efficient light collection and excitation transfer through the LH1 units towards the reaction center takes place. Large interaction between bacteriochlorophylls J (in the range 2502450 cm1 ) has been acquired by ab initio calculations. Our theoretical approach therefore uses the strong coupling limit in J and considers an extended Frenkel exciton states model. But experimental steady and timedependent optical data do not correspond to that of the ideal, isolated BChl’s ring. Time-resolved absorption and transmission spectroscopy has provided some information on the initial relaxation processes occurring after photoexcitation. Femtosecond fluorescence experiments revealed
Corresponding author. Tel.: +420 493 331 186; fax: +420 495 513 890.
E-mail address: [email protected] (P. Herˇ man). 0022-2313/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2006.01.224
an ultrafast relaxation of optical excitations within an inhomogeneously broadened density of states. Nevertheless, despite long time intensive study, the precise role of the protein moiety in governing the dynamics of the excited states of the rings is still under debate [1]. The simplest approach is to decompose the line profile into homogeneous and inhomogeneous contributions of the dynamic and static disorder. Yet, a satisfactory understanding of the nature of the static disorder in lightharvesting systems has not been reached [1]. Working in the local site exciton basis, the static disorder can be present in both diagonal (local energies) and off-diagonal (transfer integrals) matrix elements of the Hamiltonian. Silbey et al. [1], in the extended study, pointed out following questions: It is not clear whether only the consideration of the former is enough or the latter should be included as well. If both are considered, then there remains a question about independence. Rahman et al. [2], dealing with the time-dependent optical properties, were the first who recognized the importance of the off-diagonal density matrix elements
ARTICLE IN PRESS P. Herˇman, I. Barvı´k / Journal of Luminescence 122–123 (2007) 558–561
(coherences) [3] which can lead in the (partially) coherent exciton transfer regime 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. [4]. In several steps [5–8] we have recently extended the former investigations by Kumble and Hochstrasser [9] and Nagarjan et al. [10]. We have investigated not only the diagonal static disorder (fluctuations of the site energies) but also the off-diagonal static disorder (fluctuations of the transfer integrals J) as well as possible elliptical deformation of the ring for several models of inclusion of the dynamic disorder (dissipation). At first, for a noncorrelated 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 also in the non-Markovian limit and later on also the elliptical deformation. The estimates of the static disorder in local excitation energies needed for agreement with experimental data are still quite high not only by us but also in literature. That is why the attention has been taken to the off-diagonal static disorder (disorder in transfer integrals J) in the last few years. Recently, we started to investigate influence of the noncorrelated static disorder in transfer integrals [11]. Later on, we have done extended study of the time dependence of fluorescence depolarization in the ring LH2 scale under the influence of three models of noncorrelated off-diagonal static disorder [1]. The paper is structured as follows: Section 2 introduces the ring model, types of disorder and gives a description of the anisotropy expression. The results for the time development of the anisotropy of fluorescence for four models of the noncorrelated static disorder are given in Section 3. In Section 4 some conclusions are drawn. 2. Model
approximation, Gaussian distribution, standard deviation DJ ). (C) The fluctuations of radial positions of molecules (rn ¼ r0 ð1 þ drn Þ, uncorrelated, Gaussian distribution, standard deviation Dr , hrn i ¼ r0 ) leading to H C s . (D) The fluctuations of angular positions of molecules on the ring without the changing of orientations of transition dipole moments (nn ¼ n0n þ dnn , uncorrelated, Gaussian distribution, standard deviation Dj ) leading to H D s . Hamiltonian of the static disorder H X s adds to the Hamiltonian of the ideal ring. All of the Qy transition dipole moments of the bacteriochlorophylls (BChls) B850 in a ring without 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 n¼1
can~ mn ;
~ ma ¼
N X
can~ mn ,
(2)
n¼1
where can ðcan Þ are the eigenstate expansion coefficients of the unperturbed (disordered) ring in site representation. In the case of impulsive excitation, the dipole strength is simply redistributed among the exciton levels due to disorder [9]. Thus, the 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 [6]. The usual time-dependent anisotropy of fluorescence reads hS xx ðtÞi hSxy ðtÞi , hS xx ðtÞi þ 2hSxy ðtÞi Z Sxy ðtÞ ¼ Pxy ðo; tÞ do.
rðtÞ ¼
In the following we assume that only one excitation is present on the ring after an impulsive excitation [9]. The Hamiltonian of an exciton in the ideal ring coupled to a bath of harmonic oscillators reads H 0 ¼ H 0ex þ H ph þ H ex2ph .
559
(1)
H 0ex
represents the exciton on the ideal ring without an interaction with a bath. H ph describes the bath of phonons in the harmonic approximation. H ex2ph represents the exciton–bath interaction (assumed to be site-diagonal and linear in the bath coordinates). Influence of static disorder is modeled by P y (A) The local energy fluctuations n (H A s ¼ m n am an , uncorrelated, Gaussian distribution, standard deviation D). (B) P The transfer integral fluctuations dJ nm (H Bs ¼ y m;nðmanÞ dJ mn am an , uncorrelated, nearest neighbor
ð3Þ
The brackets h i denote the ensemble average and the orientational average over the sample. The crucial quantity entering Eq. (3) is the exciton density matrix r. The dynamical equations for r obtained by Cˇa´pek [12] read X d rmn ðtÞ ¼ iðOmn;pq þ dOmn;pq ðtÞÞrpq ðtÞ. (4) dt pq In long time approximation coefficient dOðt ! 1Þ becomes time independent. All details of calculations leading to Eq. (4) are given elsewhere [8] and we shall not repeat them here. Obtaining of the full time dependence of dOðtÞ is not a simple task. We have succeeded to calculate it microscopically only for the simplest molecular model namely
ARTICLE IN PRESS P. Herˇman, I. Barvı´k / Journal of Luminescence 122–123 (2007) 558–561
560
dimer [13]. In case of molecular ring we should resort to some simplification [8]. In what follows we use Markovian version of Eq. (4) with a simple model for bath correlation functions C mn assuming that each site has its own bath completely uncoupled from the other site ones. Furthermore, it is assumed that these baths have identical properties [14,15]. Then only one correlation function CðoÞ is needed. We shall use the spectral density of the bath JðoÞ in agreement with [14]. 3. Results The time dependence of the anisotropy of fluorescence (Eq. (3)) has been calculated using dynamical equations for the exciton density matrix rðtÞ to express the optical properties of the ring units in the femtosecond time range. Details are the same as in Refs. [8,16,11]. In Ref. [9], which does not take the bath into account, the anisotropy of fluorescence decreases from 0.7 to 0.3–0.35 and subsequently reaches a final value of 0.4.
∆
∆
∆
∆
∆
τ
∆
τ
τ
τ
∆
∆
One needs a strength of static disorder of D 0:420:8 to reach a decay time below 100 fs. Our results are shown in Figs. 1 and 2. We use dimensionless energies normalized to the transfer integral J and the renormalized time t. To convert t into seconds, one has to divide t by 2pcJ, with c being the speed of light in cm s1 and J in cm1 . Estimation of the transfer integral J varies between 250 and 400 cm1 . For these extreme values of J our time unit ðt ¼ 1Þ corresponds to 21.2 or 13.3 fs. The time and D dependence of the anisotropy depolarization is pictured in Fig. 1 for four models of the static disorder: (A) Gaussian disorder in local excitation energies; (B) Gaussian disorder in transfer integrals J; (C) Gaussian disorder in radial positions of molecules and (D) Gaussian disorder in angular positions of molecules. In columns there are the results without dynamic disorder (left column), with dynamic disorder for the temperature kT ¼ 0:25J (middle column) and for the temperature kT ¼ 0:5J (right column). The strength of dynamic disorder j 0 ¼ 0:4. To convert T into kelvins one has to
∆
τ
∆
τ
τ
τ
∆
∆
τ
τ
τ
τ
Fig. 1. The time and D dependence of the anisotropy depolarization for four models of the static disorder: (A) Gaussian disorder in local excitation energies (first row); (B) Gaussian disorder in transfer integrals J (second row); (C) Gaussian disorder in radial positions of molecules (third row) and (D) Gaussian disorder in angular positions of molecules (fourth row). In columns are the results without dynamic disorder (left column) and with dynamic disorder (middle and right columns) for the temperature kT ¼ 0:25J (middle column) and kT ¼ 0:5J (right column). The strength of dynamic disorder j 0 ¼ 0:4.
ARTICLE IN PRESS P. Herˇman, I. Barvı´k / Journal of Luminescence 122–123 (2007) 558–561
∆
∆
δ
δ
561
∆ν
δν
Fig. 2. Distribution of fluctuations dJ for three models of off-diagonal static disorder: Gaussian disorder in transfer integrals J (left), Gaussian disorder in radial positions of molecules (middle), Gaussian disorder in angular positions of molecules (right).
divide T by k=J with k being the Boltzmann constant in cm1 K1 and J in cm1 . Distributions of fluctuations dJ for three models of offdiagonal static disorder (B)–(D) are shown in Fig. 2.
The distributions of transfer integrals J in the case of r— disorder and n—disorder are not Gaussian. Not only maxima are shifted in the case of r—disorder and n— disorder but also a form is changed. The distributions have different wings (tails).
4. Conclusions According to Rahman et al. [2], the initial anisotropy rð0Þ can be, in presence of the coherent transfer regime of the exciton, larger than the incoherent theoretical limit of 0.4. Without dynamic disorder, the noncorrelated diagonal static disorder 0.4–0.8 is sufficient to cause the experimentally found initial drop of the anisotropy on a sub-100 fs time scale. Taking into account dynamic disorder smaller values of static disorder are sufficient to produce the same time decay rates of the anisotropy of the fluorescence in comparison with the results by Kumble and Hochstrasser [9]. On the other hand, temperature-independent decay can be obtained only for D40:8 [6]. Application of four models of the uncorrelated static disorder shows: difference between time interval in which the anisotropy depolarization reaches r ¼ 0:4 (the incoherent theoretical limit) using Gaussian static disorder in the local energies (model A) or the Gaussian static disorder in the transfer integrals (model B) is almost as much as 100% for the same value of the static disorder D ¼ DJ . It means that the same drop of the anisotropy may be caused even by the diagonal static disorder (model A) with D or by the static disorder in the transfer integrals with DJ 0:5D. This difference between the model A and the model B calculations is still present also in the case when the exciton interaction with the bath is taken into account. Disorder with Dr ¼ 0:1r0 in model (C) and Dn ¼ 0:035 in model (D) has practically the same effect D ¼ 0:6J in model (A). All differences between different models of static disorder are visible also in presence of the dynamic disorder.
Acknowledgement This work has been funded by the project GACˇR 20203-0817. The support from the Ministry of Education, Youth and Sports of the Czech Republic (research plan MSM0021620835) (I.B.) is also gratefully acknowledged.
References [1] S. Jang, S.F. Dempster, R.J. Silbey, J. Phys. Chem. B 105 (2001) 6655. [2] T.S. Rahman, R.S. Knox, V.M. Kenkre, Chem. Phys. 44 (1979) 197. [3] O. Ku¨hn, V. Sundstro¨m, T. Pullerits, Chem. Phys. 275 (2002) 15. [4] V. Nagarjan, R.G. Alden, J.C. Williams, W.W. Parson, Proc. Natl. Acad. Sci. USA 93 (1996) 13774. [5] P. Herˇ man, U. Kleinekatho¨fer, I. Barvı´ k, M. Schreiber, J. Lumin. 94–95 (2001) 447. [6] P. Herˇ man, U. Kleinekatho¨fer, I. Barvı´ k, M. Schreiber, Chem. Phys. 275 (2002) 1. [7] I. Barvı´ k, I. Kondov, P. Herˇ man, M. Schreiber, U. Kleinekatho¨fer, Nonlinear Opt. 29 (2002) 167. [8] P. Herˇ man, I. Barvı´ k, Czech. J. Phys. 53 (2003) 579. [9] R. Kumble, R. Hochstrasser, J. Chem. Phys. 109 (1998) 855. [10] V. Nagarjan, E.T. Johnson, J.C. Williams, W.W. Parson, J. Phys. Chem. B 103 (1999) 2297. [11] P. Herˇ man, I. Barvı´ k, M. Reiter, J. Lumin. 112 (2005) 469. [12] V. Cˇa´pek, Z. Phys. B 99 (1996) 261. [13] I. Barvı´ k, J. Macek, J. Chin. Chem. Soc. 47 (2000) 647. [14] V. Sundstro¨m, T. Pullerits, R. van Grondelle, J. Phys. Chem. B 103 (1999) 2327. [15] V. May, O. Ku¨hn, Charge and Energy Transfer in Molecular Systems, Wiley-WCH, Berlin, 2000. [16] M. Reiter, P. Herˇ man, I. Barvı´ k, J. Lumin. 110 (2004) 258.
Journal of Luminescence 122–123 (2007) 558–561 www.elsevier.com/locate/jlumin
Temperature dependence of the anisotropy of fluorescence in ring molecular systems Pavel Herˇ mana,, Ivan Barvı´ kb a
Department of Physics, University of Hradec Kra´love´, Rokitanske´ho 62, CZ-50003 Hradec Kra´love´, Czech Republic Institute of Physics of Charles University, Faculty of Mathematics and Physics, CZ-12116 Prague, Czech Republic
b
Available online 14 March 2006
Abstract The time dependence of the anisotropy of fluorescence after an impulsive excitation in the molecular ring (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 uncorrelated static disorder. 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. In the present investigation we are comparing results for the time-dependent optical anisotropy of the molecular ring for four models of the uncorrelated static disorder: Gaussian disorder in the local energies (model A), Gaussian disorder in the transfer integrals (model B), Gaussian disorder in radial positions of molecules (model C) and Gaussian disorder in angular positions of molecules (model D). Both types of disorder—static and dynamic— are taken into account simultaneously. r 2006 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 We are dealing with the ring-shaped units resembling those from antenna complex LH2 of the purple bacterium Rhodopseudomonas acidophila in which a highly efficient light collection and excitation transfer through the LH1 units towards the reaction center takes place. Large interaction between bacteriochlorophylls J (in the range 2502450 cm1 ) has been acquired by ab initio calculations. Our theoretical approach therefore uses the strong coupling limit in J and considers an extended Frenkel exciton states model. But experimental steady and timedependent optical data do not correspond to that of the ideal, isolated BChl’s ring. Time-resolved absorption and transmission spectroscopy has provided some information on the initial relaxation processes occurring after photoexcitation. Femtosecond fluorescence experiments revealed
Corresponding author. Tel.: +420 493 331 186; fax: +420 495 513 890.
E-mail address: [email protected] (P. Herˇ man). 0022-2313/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2006.01.224
an ultrafast relaxation of optical excitations within an inhomogeneously broadened density of states. Nevertheless, despite long time intensive study, the precise role of the protein moiety in governing the dynamics of the excited states of the rings is still under debate [1]. The simplest approach is to decompose the line profile into homogeneous and inhomogeneous contributions of the dynamic and static disorder. Yet, a satisfactory understanding of the nature of the static disorder in lightharvesting systems has not been reached [1]. Working in the local site exciton basis, the static disorder can be present in both diagonal (local energies) and off-diagonal (transfer integrals) matrix elements of the Hamiltonian. Silbey et al. [1], in the extended study, pointed out following questions: It is not clear whether only the consideration of the former is enough or the latter should be included as well. If both are considered, then there remains a question about independence. Rahman et al. [2], dealing with the time-dependent optical properties, were the first who recognized the importance of the off-diagonal density matrix elements
ARTICLE IN PRESS P. Herˇman, I. Barvı´k / Journal of Luminescence 122–123 (2007) 558–561
(coherences) [3] which can lead in the (partially) coherent exciton transfer regime 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. [4]. In several steps [5–8] we have recently extended the former investigations by Kumble and Hochstrasser [9] and Nagarjan et al. [10]. We have investigated not only the diagonal static disorder (fluctuations of the site energies) but also the off-diagonal static disorder (fluctuations of the transfer integrals J) as well as possible elliptical deformation of the ring for several models of inclusion of the dynamic disorder (dissipation). At first, for a noncorrelated 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 also in the non-Markovian limit and later on also the elliptical deformation. The estimates of the static disorder in local excitation energies needed for agreement with experimental data are still quite high not only by us but also in literature. That is why the attention has been taken to the off-diagonal static disorder (disorder in transfer integrals J) in the last few years. Recently, we started to investigate influence of the noncorrelated static disorder in transfer integrals [11]. Later on, we have done extended study of the time dependence of fluorescence depolarization in the ring LH2 scale under the influence of three models of noncorrelated off-diagonal static disorder [1]. The paper is structured as follows: Section 2 introduces the ring model, types of disorder and gives a description of the anisotropy expression. The results for the time development of the anisotropy of fluorescence for four models of the noncorrelated static disorder are given in Section 3. In Section 4 some conclusions are drawn. 2. Model
approximation, Gaussian distribution, standard deviation DJ ). (C) The fluctuations of radial positions of molecules (rn ¼ r0 ð1 þ drn Þ, uncorrelated, Gaussian distribution, standard deviation Dr , hrn i ¼ r0 ) leading to H C s . (D) The fluctuations of angular positions of molecules on the ring without the changing of orientations of transition dipole moments (nn ¼ n0n þ dnn , uncorrelated, Gaussian distribution, standard deviation Dj ) leading to H D s . Hamiltonian of the static disorder H X s adds to the Hamiltonian of the ideal ring. All of the Qy transition dipole moments of the bacteriochlorophylls (BChls) B850 in a ring without 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 n¼1
can~ mn ;
~ ma ¼
N X
can~ mn ,
(2)
n¼1
where can ðcan Þ are the eigenstate expansion coefficients of the unperturbed (disordered) ring in site representation. In the case of impulsive excitation, the dipole strength is simply redistributed among the exciton levels due to disorder [9]. Thus, the 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 [6]. The usual time-dependent anisotropy of fluorescence reads hS xx ðtÞi hSxy ðtÞi , hS xx ðtÞi þ 2hSxy ðtÞi Z Sxy ðtÞ ¼ Pxy ðo; tÞ do.
rðtÞ ¼
In the following we assume that only one excitation is present on the ring after an impulsive excitation [9]. The Hamiltonian of an exciton in the ideal ring coupled to a bath of harmonic oscillators reads H 0 ¼ H 0ex þ H ph þ H ex2ph .
559
(1)
H 0ex
represents the exciton on the ideal ring without an interaction with a bath. H ph describes the bath of phonons in the harmonic approximation. H ex2ph represents the exciton–bath interaction (assumed to be site-diagonal and linear in the bath coordinates). Influence of static disorder is modeled by P y (A) The local energy fluctuations n (H A s ¼ m n am an , uncorrelated, Gaussian distribution, standard deviation D). (B) P The transfer integral fluctuations dJ nm (H Bs ¼ y m;nðmanÞ dJ mn am an , uncorrelated, nearest neighbor
ð3Þ
The brackets h i denote the ensemble average and the orientational average over the sample. The crucial quantity entering Eq. (3) is the exciton density matrix r. The dynamical equations for r obtained by Cˇa´pek [12] read X d rmn ðtÞ ¼ iðOmn;pq þ dOmn;pq ðtÞÞrpq ðtÞ. (4) dt pq In long time approximation coefficient dOðt ! 1Þ becomes time independent. All details of calculations leading to Eq. (4) are given elsewhere [8] and we shall not repeat them here. Obtaining of the full time dependence of dOðtÞ is not a simple task. We have succeeded to calculate it microscopically only for the simplest molecular model namely
ARTICLE IN PRESS P. Herˇman, I. Barvı´k / Journal of Luminescence 122–123 (2007) 558–561
560
dimer [13]. In case of molecular ring we should resort to some simplification [8]. In what follows we use Markovian version of Eq. (4) with a simple model for bath correlation functions C mn assuming that each site has its own bath completely uncoupled from the other site ones. Furthermore, it is assumed that these baths have identical properties [14,15]. Then only one correlation function CðoÞ is needed. We shall use the spectral density of the bath JðoÞ in agreement with [14]. 3. Results The time dependence of the anisotropy of fluorescence (Eq. (3)) has been calculated using dynamical equations for the exciton density matrix rðtÞ to express the optical properties of the ring units in the femtosecond time range. Details are the same as in Refs. [8,16,11]. In Ref. [9], which does not take the bath into account, the anisotropy of fluorescence decreases from 0.7 to 0.3–0.35 and subsequently reaches a final value of 0.4.
∆
∆
∆
∆
∆
τ
∆
τ
τ
τ
∆
∆
One needs a strength of static disorder of D 0:420:8 to reach a decay time below 100 fs. Our results are shown in Figs. 1 and 2. We use dimensionless energies normalized to the transfer integral J and the renormalized time t. To convert t into seconds, one has to divide t by 2pcJ, with c being the speed of light in cm s1 and J in cm1 . Estimation of the transfer integral J varies between 250 and 400 cm1 . For these extreme values of J our time unit ðt ¼ 1Þ corresponds to 21.2 or 13.3 fs. The time and D dependence of the anisotropy depolarization is pictured in Fig. 1 for four models of the static disorder: (A) Gaussian disorder in local excitation energies; (B) Gaussian disorder in transfer integrals J; (C) Gaussian disorder in radial positions of molecules and (D) Gaussian disorder in angular positions of molecules. In columns there are the results without dynamic disorder (left column), with dynamic disorder for the temperature kT ¼ 0:25J (middle column) and for the temperature kT ¼ 0:5J (right column). The strength of dynamic disorder j 0 ¼ 0:4. To convert T into kelvins one has to
∆
τ
∆
τ
τ
τ
∆
∆
τ
τ
τ
τ
Fig. 1. The time and D dependence of the anisotropy depolarization for four models of the static disorder: (A) Gaussian disorder in local excitation energies (first row); (B) Gaussian disorder in transfer integrals J (second row); (C) Gaussian disorder in radial positions of molecules (third row) and (D) Gaussian disorder in angular positions of molecules (fourth row). In columns are the results without dynamic disorder (left column) and with dynamic disorder (middle and right columns) for the temperature kT ¼ 0:25J (middle column) and kT ¼ 0:5J (right column). The strength of dynamic disorder j 0 ¼ 0:4.
ARTICLE IN PRESS P. Herˇman, I. Barvı´k / Journal of Luminescence 122–123 (2007) 558–561
∆
∆
δ
δ
561
∆ν
δν
Fig. 2. Distribution of fluctuations dJ for three models of off-diagonal static disorder: Gaussian disorder in transfer integrals J (left), Gaussian disorder in radial positions of molecules (middle), Gaussian disorder in angular positions of molecules (right).
divide T by k=J with k being the Boltzmann constant in cm1 K1 and J in cm1 . Distributions of fluctuations dJ for three models of offdiagonal static disorder (B)–(D) are shown in Fig. 2.
The distributions of transfer integrals J in the case of r— disorder and n—disorder are not Gaussian. Not only maxima are shifted in the case of r—disorder and n— disorder but also a form is changed. The distributions have different wings (tails).
4. Conclusions According to Rahman et al. [2], the initial anisotropy rð0Þ can be, in presence of the coherent transfer regime of the exciton, larger than the incoherent theoretical limit of 0.4. Without dynamic disorder, the noncorrelated diagonal static disorder 0.4–0.8 is sufficient to cause the experimentally found initial drop of the anisotropy on a sub-100 fs time scale. Taking into account dynamic disorder smaller values of static disorder are sufficient to produce the same time decay rates of the anisotropy of the fluorescence in comparison with the results by Kumble and Hochstrasser [9]. On the other hand, temperature-independent decay can be obtained only for D40:8 [6]. Application of four models of the uncorrelated static disorder shows: difference between time interval in which the anisotropy depolarization reaches r ¼ 0:4 (the incoherent theoretical limit) using Gaussian static disorder in the local energies (model A) or the Gaussian static disorder in the transfer integrals (model B) is almost as much as 100% for the same value of the static disorder D ¼ DJ . It means that the same drop of the anisotropy may be caused even by the diagonal static disorder (model A) with D or by the static disorder in the transfer integrals with DJ 0:5D. This difference between the model A and the model B calculations is still present also in the case when the exciton interaction with the bath is taken into account. Disorder with Dr ¼ 0:1r0 in model (C) and Dn ¼ 0:035 in model (D) has practically the same effect D ¼ 0:6J in model (A). All differences between different models of static disorder are visible also in presence of the dynamic disorder.
Acknowledgement This work has been funded by the project GACˇR 20203-0817. The support from the Ministry of Education, Youth and Sports of the Czech Republic (research plan MSM0021620835) (I.B.) is also gratefully acknowledged.
References [1] S. Jang, S.F. Dempster, R.J. Silbey, J. Phys. Chem. B 105 (2001) 6655. [2] T.S. Rahman, R.S. Knox, V.M. Kenkre, Chem. Phys. 44 (1979) 197. [3] O. Ku¨hn, V. Sundstro¨m, T. Pullerits, Chem. Phys. 275 (2002) 15. [4] V. Nagarjan, R.G. Alden, J.C. Williams, W.W. Parson, Proc. Natl. Acad. Sci. USA 93 (1996) 13774. [5] P. Herˇ man, U. Kleinekatho¨fer, I. Barvı´ k, M. Schreiber, J. Lumin. 94–95 (2001) 447. [6] P. Herˇ man, U. Kleinekatho¨fer, I. Barvı´ k, M. Schreiber, Chem. Phys. 275 (2002) 1. [7] I. Barvı´ k, I. Kondov, P. Herˇ man, M. Schreiber, U. Kleinekatho¨fer, Nonlinear Opt. 29 (2002) 167. [8] P. Herˇ man, I. Barvı´ k, Czech. J. Phys. 53 (2003) 579. [9] R. Kumble, R. Hochstrasser, J. Chem. Phys. 109 (1998) 855. [10] V. Nagarjan, E.T. Johnson, J.C. Williams, W.W. Parson, J. Phys. Chem. B 103 (1999) 2297. [11] P. Herˇ man, I. Barvı´ k, M. Reiter, J. Lumin. 112 (2005) 469. [12] V. Cˇa´pek, Z. Phys. B 99 (1996) 261. [13] I. Barvı´ k, J. Macek, J. Chin. Chem. Soc. 47 (2000) 647. [14] V. Sundstro¨m, T. Pullerits, R. van Grondelle, J. Phys. Chem. B 103 (1999) 2327. [15] V. May, O. Ku¨hn, Charge and Energy Transfer in Molecular Systems, Wiley-WCH, Berlin, 2000. [16] M. Reiter, P. Herˇ man, I. Barvı´ k, J. Lumin. 110 (2004) 258.