Computer simulation of fluorescence spectra for molecular ring:exciton dynamics

DAVID ZAPLETALUniversity of Pardubice

Faculty of Econ. and Adm., Inst. of MathematicsStudentska 95, 532 10 Pardubice

CZECH REPUBLICdavid.zapletal@upce.cz

PAVEL HERMANUniversity of Hradec Kralove

Faculty of Science, Department of PhysicsRokitanskeho 62, 300 03 Hradec Kralove

CZECH REPUBLICpavel.herman@uhk.cz

Abstract: Software package Mathematica is used for computer simulation of steady state fluorescence spectraof ring molecular system. The cyclic antenna unit LH2 of the bacterial photosystem from purple bacteriumRhodopseudomonas acidophila can be modeled by such system. Dynamic disorder, interaction with a bath, isincluded in Markovian approximation. Also three different models of uncorrelated static disorder are taking intoaccount in our simulations: Gaussian disorder in local excitation energies, Gaussian disorder in nearest neigh-bour transfer integrals and Gaussian disorder in radial positions of molecules in the ring. The cumulant-expansionmethod of Mukamel et al. is used for the calculation of spectral responses of the system with exciton-phonon cou-pling. The peak positions of single ring spectra and localizations of exciton states depend on realization of staticdisorder. Dynamic disorder also influences the peak position. We discuss different types of exciton dynamics, thatare coupled to above mentioned effects.

Key–Words: LH2; fluorescence spectrum, static and dynamic disorder, exciton states and dynamics, Mathematica

1 IntroductionIn 1995 the crystal structure of peripheral light-harvesting complex LH2 from purple bacteriumRhodopseudomonas acidophila was determined withhigh resolution [1, 2]. That is why ultrafast initialphases of photosynthesis in purple bacteria could beenthoroughly studied in last two decades. LH2 is ahighly symmetric ring of nine pigment-protein sub-units, each containing two transmembrane polypep-tide helixes and three bacteriochlorophylls (BChl).

Due to the strong interaction between BChlmolecules, an extended Frenkel exciton states modelis considered in our theoretical approach. In spite ofextensive investigation, the role of the protein moietyin governing the dynamics of the excited states hasnot been totally clear yet. At room temperature thesolvent and protein environment fluctuate with char-acteristic time scales ranging from femtoseconds tonanoseconds. The simplest approach is to substitutefast fluctuations by dynamic disorder and slow fluctu-ations by static disorder.

Kumble, Hochstrasser [3] and Nagarajan et al.[4–6] studied static disorder effect on the anisotropyof fluorescence for LH2 rings. We have extendedthese investigations by consideration of dynamic dis-order. After studying the influence of dynamic dis-order for simple systems (dimer, trimer) [7–9] we

added this effect to our model of LH2 ring by usinga quantum master equation in the Markovian [10] andnon-Markovian limits [11]. Influence of four typesof uncorrelated static disorder [12, 13] and correlatedstatic disorder (elliptical deformation) [11] was inves-tigated. We also compared models of rings with dif-ferent arrangement of optical dipole moments (radialarrangement) [14–16].

Recently we have focused on the modeling of thesteady state fluorescence and absorption spectra forabove mentioned model of LH2 ring [17]. Main goalof the present paper is the investigation of differenttypes of exciton dynamics, that are coupled to exci-ton states with different degree of localization. Thedynamic disorder - interaction with the phonon bathin Markovian approximation is taken into account si-multaneously with uncorrelated static disorder.

2 Physical modelThe Hamiltonian of an exciton in the ideal ring cou-pled to a bath of harmonic oscillators reads

H0 =∑

m,n(m6=n)Jmna

†man +

∑q

hωqb†qbq +

+1√N

∑m

∑q

Gmq hωqa†mam(b†q + b−q). (1)

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Here the first term corresponds to an exciton, e.g. thesystem without any disorder (the operator a†m (am)creates (annihilates) an exciton at site m), Jmn (form 6= n) is the so-called transfer integral betweensites m and n, the second term represents phononbath in the harmonic approximation (the phonon cre-ation and annihilation operators are denoted by b†qand b−q, respectively). Last term in (1), describesexciton-phonon interaction which is assumed to besite-diagonal and linear in the bath coordinates (theterm Gmq denotes the exciton-phonon coupling con-stant).

Inside one ring the pure exciton Hamiltonian canbe diagonalized using the wave vector representationwith corresponding delocalized ”Bloch” states α andenergies Eα. Considering homogeneous case withonly nearest neighbour transfer matrix elements

Jmn = J0(δm,n+1 + δm,n−1) (2)

and using Fourier transformed excitonic operators(Bloch representation)

aα =∑n

eiαkn, α =2π

Nl, l = 0,±1, . . . ,±N

2, (3)

the simplest exciton Hamiltonian in α - representationreads

H0ex =

∑α

Eαa†αaα, Eα = −2J0 cosα. (4)

Influence of static disorder is modeled by:(I) the local excitation energy fluctuations δεn (uncor-related, Gaussian distribution, standard deviation ∆,HI

s =∑n δεna

†nan),

(II) the transfer integral fluctuations δJmn (un-correlated, nearest neighbour approximation, Gaus-sian distribution, standard deviation ∆J , HII

s =∑mn(m6=n) δJmna

†man),

(III) fluctuations of the radial positions of moleculeson the ring δrn (uncorrelated, nearest neighbour ap-proximation, Gaussian distribution, standard devia-tion ∆r).Hamiltonian of the uncorrelated static disorder Hx

sadds to the Hamiltonian H0

ex.The cumulant-expansion method of Mukamel et

al. [19, 20] is used for the calculation of spectralresponses of the system with exciton-phonon cou-pling. Absorption OD(ω) and steady-state fluores-cence FL(ω) spectrum can be expressed as

OD(ω) = ω∑α

d2α×

×Re

∫ ∞0

dtei(ω−ωα)t−gαααα(t)−Rααααt, (5)

FL(ω) = ω∑α

Pαd2α×

×Re

∫ ∞0

dtei(ω−ωα)t+iλααααt−g∗αααα(t)−Rααααt. (6)

Here ~dα =∑n c

αn~dn is the dipole strength of eigen-

state α, cαn are the expansion coefficients of the eigen-state α in site representation and Pα is steady statepopulation of the eigenstate α. The inverse lifetime ofexciton state Rαααα [18] is given by the elements ofRedfield tensor [21] (a sum of the relaxation rates be-tween exciton states) Rαααα = −

∑β 6=αRββαα. The

g-function and λ-values in (6) are given by

gαβγδ = −∫ ∞−∞

dω

2πω2Cαβγδ(ω)×

×[coth

ω

2kBT(cosωt− 1)− i(sinωt− ωt)

], (7)

λαβγδ = − limt→∞

d

dtIm{gαβγδ(t)} =

=

∫ ∞−∞

dω

2πωCαβγδ(ω). (8)

The matrix of the spectral densities Cαβγδ(ω) inthe eigenstate (exciton) representation reflects one-exciton states coupling to the manifold of nuclearmodes. In what follows only a diagonal excitonphonon interaction in site representation is used (see(1)), i.e., only fluctuations of the pigment site ener-gies are assumed and the restriction to the completelyuncorrelated dynamical disorder is applied. In suchcase each site (i.e. each chromophore) has its ownbath completely uncoupled from the baths of the othersites. Furthermore it is assumed that these baths haveidentical properties [12, 22, 23]

Cmnm′n′(ω) = δmnδmm′δnn′C(ω). (9)

After transformation to exciton representation wehave

Cαβγδ(ω) =∑n

cαncβncγncδnC(ω). (10)

Several models of spectral density of the bath are usedin literature [18, 24, 25]. In our present investigationwe have used the model of Kuhn and May [24]

C(ω) = Θ(ω)j0ω2

2ω3c

e−ω/ωc (11)

which has its maximum at 2ωc.Delocalization of the exciton states contributing

to the steady state FL spectrum can be characterized

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Figure 1: Resulting absorption OD(ω) (dashed lines) and steady-state fluorescence FL(ω) spectra (solid lines) ofLH2 at room temperature kT = 0.5 J0 averaged over 2000 realizations of Gaussian uncorrelated static disorder:in local excitation energies δε (left column), in nearest neighbour transfer integrals δJ (middle column), in radialpositions of molecules δr (right column). Experimental fluorescence profile averaged in time and over the particlesfor room temperature [18] is also displayed (points).

by the thermally averaged participation ratio 〈PR〉,which is given by

〈PR〉 =

∑α PRα e

− EαkBT∑

α e− EαkBT

, PRα =N∑n=1

|cαn|4. (12)

Time evolution of exciton density matrix ραβ isgoverned by Redfield equation

∂ραβ(t)

∂t= −iωαβραβ(t) +

∑γδ

Rαβγδργδ(t), (13)

exciton density matrix in site representation is givenby

ρmn =∑αβ

cαncβnραβ. (14)

3 Computational point of viewTo have steady state fluorescence spectrum it is neces-sary to calculate single ring spectrum FL(ω) for largenumber of different static disorder realizations createdby random number generator. Finally these resultshave to be averaged over all realizations of static dis-order. Time evolution of exciton density matrix wascalculated for each realization of static disorder. Thatis why it was necessary to put through numerical in-tegrations for each realization of static disorder (see(6)).

For all calculation the software package Mathe-matica [26] was used. This package is very conve-nient not only for symbolic calculations [27] whichare needed for expression of all required quantities,but it can be used also for numerical ones [28]. Thatis why Mathematica was used by us as for symboliccalculations as for numerical integrations and also for

final averaging of results over all realizations of staticdisorder.

4 ResultsThree above mentioned types of uncorrelated staticdisorder have been taken into account in our simula-

Figure 2: The distribution of 〈PR〉 values as a func-tion of FL spectrum peak position at room temper-ature kT = 0.5 J0 calculated for 2000 realizationsof Gaussian uncorrelated static disorder: in local ex-citation energies δε (first row), in nearest neighbourtransfer integrals δJ (second row), in radial positionsof molecules δr (third row). Columns: three differentstrengths of static disorder.

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Figure 3: The diagonal exciton density matrix ele-ments in the site representation ρnn at room temper-ature (kT = 0.5J0) are shown as a function of timeτ for Gaussian uncorrelated static disorder type (I) -fluctuations of local excitation energies δεn. First andsecond row shows the coherent dynamics (without dy-namic disorder) for the realizations of static disorderwith lowest (first row) and highest (second row) 〈PR〉values. The third and fourth row shows the same butwith the dynamic disorder effect taking into account.Columns: three different strengths of static disorder.

tions simultaneously with dynamic disorder in Marko-vian approximation. Dimensionless energies normal-ized to the transfer integral J12 = J0 and dimension-less time τ have been used. Estimation of J0 variesin literature between 250 cm−1 and 400 cm−1. Forthese extreme values of J0 our time unit (τ = 1) cor-responds to 21.2 fs or 13.3 fs.

Contrary to Novoderezhkin et al. [18] differentmodel of spectral density (the model of Kuhn andMay [12]) has been used. In agreement with our pre-vious results [29] we have used j0 = 0.4 J0 andωc = 0.212 J0 (see (11)). The strengths of individualtypes of uncorrelated static disorder have been taken

Figure 4: The same as in Figure 3 but for the Gaussianuncorrelated static disorder type (II) - fluctuations ofnearest neighbour transfer integrals δJmn.

in agreement with [30]. For each type of static disor-der the simulations have been done for three values ofthe static disorder strength:(I): ∆ = 0.1, 0.3, 0.6 J0,(II): ∆J = 0.05, 0.15, 0.30 J0,(III): ∆r = 0.02, 0.06, 0.12 r0, where r0 is the radiusof unperturbed ring.

Resulting absorption OD(ω) and steady state flu-orescence spectra FL(ω) for room temperature kT =0.5 J0 averaged over 2000 realizations of each staticdisorder type and strength can be seen in Fig. 1. Com-parison of calculated FL spectra with experimental FLprofile averaged in time and over the particles [18] isalso displayed in Fig. 1.

Fig. 2 shows the distribution of 〈PR〉 values as afunction of FL spectrum peak position at room tem-perature kT = 0.5 J0 calculated for 2000 realizationsof the disorder.

Dynamics of the diagonal exciton density ma-trix elements in site representation ρnn is shown inFig. 3-5 for three above mentioned static disordertypes. Initial density matrix ραβ(t = 0) is cho-

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sen by us corresponding to coherent wavepacket withsteady state populations Pα and arbitrary fixed phasesϕα [18],

ραβ =√PαPβ ei(ϕα−ϕβ), ραα = Pα ∼ e−

εαkT . (15)

Timescale τ ∈ 〈0; 70〉 corresponds to t ∈ (0; 1.4 ps)or (0; 1 ps) for limit values of J0 mentioned above.Initial exciton density matrix is defined as follows:initial populations in eigenstate representation (ραα)correspond to a thermally equilibrated wavepacket atroom temperature (kT = 0.5 J0), initial coherences(ραβ , α 6= β) have arbitrary fixed phases. Contraryto [18] the dynamic disorder have been taken intoaccount in our simulations. First and second rowsof Fig. 3-5 show the coherent dynamics (without dy-namic disorder) for the realizations of static disorderwith lowest and highest 〈PR〉 values. The third andfourth rows show the same but with the dynamic dis-order effect taking into account.

5 ConclusionSoftware package Mathematica has been found by usvery useful for the simulations of the spectra of molec-ular rings. From the comparison of our simulatedFL spectra with experimental data (Fig. 1) the valuesof the interpigment interaction energy J0 and unper-turbed transition energy from the ground state ∆E0

were obtained [31].From Figure 2 it can be seen growing of 〈PR〉

values in case of higher strength of static disorder. Itcorresponds to more localized exciton states. Shiftto lower wavelengths is evident for the static disor-der type (III) in comparison with other static disordertypes.

From Figure 3 - Figure 5 no significant differencein exciton density matrix dynamics for different typesof static disorder is visible. Different degree of lo-calization 〈PR〉 produce different types of coherentexcitation dynamics. The population distribution forsmall 〈PR〉 is more or less uniform, i.e., excitationcan be found on any part of the ring. The wavepacketmoves around the ring in this case. On the other hand,higher values of 〈PR〉 lead to higher localization ofexcitation on a smaller group of pigments or even ona single pigment. Excitation can be even totally lo-calized on a single molecule without any migration tothe other sites in case of highest 〈PR〉. If the dynamicdisorder is taken into account, oscillations in excitondynamics are suppressed. The dynamics can be char-acterized by relaxation.

Acknowledgements: Support from the Faculty ofScience, University of Hradec Kralove (project of spe-

Figure 5: The same as in Fig. 3 but for the Gaussianuncorrelated static disorder type (III) - fluctuations ofradial positions of molecules δrn.

cific research No. 2116/2011 - P. Herman) is grate-fully acknowledged.

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