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Chemical Physics 336 (2007) 91–98

Chain length dependence of singlet and triplet excited states ofoligofluorenes: A density functional study

Emil Jansson *, Prakash Chandra Jha, Hans Agren

Department of Theoretical Chemistry, Royal Institute of Technology, S-106 91 Stockholm, Sweden

Received 13 March 2007; accepted 10 May 2007Available online 2 June 2007

Abstract

Using time dependent density functional theory, we investigate the chain length dependence of the energies of excited states of a seriesof conjugated 9,9-dihexylfluorene-2,7-diyl oligomers. Excited state optimization reveals that upon excitation the dihedral angle betweentwo adjacent monomer units moves towards zero, forming a planar structure within the oligomer. The calculated energies of the opticaltransitions in absorption, fluorescence, phosphorescence and triplet–triplet absorption are compared with recently reported experimentaldata. The calculated as well as experimentally reported energies involved seem to saturate very fast as the chain length increases. Theenergy dispersion and saturation indicates that the triplet ground state is somewhat more confined than the first singlet excited state.Our calculated energies agree well with the experimental findings where available, showing small but systematic deviations.� 2007 Elsevier B.V. All rights reserved.

Keywords: Polyfluorene; Excited states; Chain length dependence

1. Introduction

An area that has attracted wide interest of the researchcommunity during the last ten years is the fabricationand design of electronic devices based on organic p-conju-gated systems such as light emitting diodes [1] (OLEDs)and photovoltaic systems [2,3]. This area of interest hasbeen studied by both experimentalists and theoreticians,and thanks to the combined efforts some of these systemshave reached the stage of manufacturing for commerciali-zation. Even so, a need for deeper understanding of thematerials being used is crucial in order to increase the effi-ciency of the devices and for the technology to reach newlevels.

In addition to physical and chemical properties requiredfor device fabrication, the luminescence properties are themost important for applications in light emitting diodes.Luminescence in OLEDs originates from the radiative

0301-0104/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2007.05.021

* Corresponding author.E-mail address: emil@theochem.kth.se (E. Jansson).

decay of singlet excitons that are generated when electronsand holes, injected from the electrodes to the polymerlayer, form pairs. Since the charge pairs are nongeminatethey have random spin orientation and the singlet and trip-let colliding pairs are equally probable. Hence, based onspin statistics it has been assumed that the ratio of gener-ated singlet and triplet excitons has a maximum value ofrS/rT = 25%, the so-called statistical limit. In recent timesit has been suggested from both theoretical [4] and experi-mental [5] grounds that this ratio may even exceed 1 andthus, the efficiency can go beyond this statistical upperlimit. In order to breach the statistical limit there must bea difference in the transition dipole moments within the sin-glet and triplet subspaces. If the singlet and triplet chargetransfer excitons decay equally fast, the limit will indeedbe 25% but if the triplet charge transfer states decay signif-icantly slower there is a chance that triplet excitons canpopulate the singlet subspace in two ways. Either, throughspin–orbit coupling induced intersystem crossing (ISC),which works as a link between the singlet and triplet sub-spaces transferring triplet excitons to singlet excitons, or,the triplet excitons may dissociate and form new charge

Table 1Expectation value of S2 calculated by UB3LYP by Gaussian before spinannihilation

Oligomer (F)n S2

n = 1 2.0392n = 2 2.0338n = 3 2.0274n = 4 2.0286n = 5 2.0282n = 6 2.0282n = 7 2.0277

92 E. Jansson et al. / Chemical Physics 336 (2007) 91–98

transfer excitons. Knowledge of the excited states is there-fore of paramount importance in order to enhance the per-formance of OLEDs.

Among the wide range of polymers studied in this con-text during, last two decades, polyfluorene-based polymersrepresent a special class due to their unique characteristics,in particular exceptional luminescence properties suited forapplications in OLEDs [6–8]. They provide high fluores-cence quantum yields, excellent solubility and film formingcapability together with chemical and thermal stability [9].Moreover, their structures can flexibly be varied to obtainthe desired electro-optical properties. In fact, this particu-lar class of polymers is the only known polymer till datewhich can emit colors spanning the entire visible range withhigh efficiency and low operating voltage [10].

It follows from the above discussion that a good under-standing of chain length dependence of photophysical andelectrochemical properties of the oligofluorenes is essential.By analyzing the results from the oligomer systems, knowl-edge about the corresponding polymer system can beobtained if the properties saturate within the oligomerapproximation. Using well-defined oligofluorenes [11–13],Wasserberg et al. [14] reported the first systematic experi-mental studies of energies of the lowest singlet and tripletstates of a series of 9,9-dihexylfluorene-2,7-diyl oligomers((F)n, n = 1,3,5,7, . . . , 50) and addressed the evolution ofthese energies from monomer to polymer. Fluorescenceand phosphorescence spectroscopy were used in order todetermine the emission properties, whereas the absorptionproperties were determined using UV/vis and photo-induced absorption spectroscopy. Motivated by their work,we have performed theoretical calculations to explore theorigin of the chain length dependence of the excited statesand to compare with experimental findings. We have usedtime dependent density functional theory (TDDFT) as ithas previously been shown to be applicable for calculationsof both absorption and emission properties for a largenumber of molecules [15–17]. The aim of our study is two-fold: (i) To verify how close TDDFT can reproduce theexperimental excitation energies of oligofluorenes; (ii) Toshed some light on their photophysical properties.

2. Computational details

In this work we have considered the oligofluorenes frommonomer to heptamer [(F)n, n = 1–7], the structure of oneof the oligomer, n = 3, is shown in Fig. 1. Ground state (S0)and first excited triplet state (T1) molecular geometries of

(F)n=3

12

34

5

6 7

8

9

1011

12

13 14 15

16

1718

19

2021

0

θ1 θ2

Fig. 1. Structure of oligofluorene (F)n = 3, bond numbering and labelingof dihedral angles.

these oligomers were optimized at density functional theory(DFT) using the hybrid B3LYP exchange-correlation func-tional [18] and the all-electron 6-31G* double-f plus polar-ization basis set [19]. The first excited singlet state (S1) wasoptimized using configuration interaction singles (CIS)[20]. Due to the increasing polymer size (monomer to hep-tamer), these optimizations have been performed using the3-21G basis set [21], as the number of determinantsincreases tremendously. The absorption and emission spec-tra were calculated by the means of TDDFT using 6-31G*

basis set on the optimized structures. As always, when per-forming open shell calculations the question of spin con-tamination comes into the picture. In Table 1 we havecollected the expectation value of S2 of the Kohn–Shamdeterminants, and we conclude that the spin contaminationon the T1 structures is small and hence the contaminationshould not affect any of the conclusion reached in thisstudy. Although we have not considered any environmen-tal effects in our calculations, the conclusions derived fromthis study are expected to be unchanged qualitatively. Sinceall corresponding experiments are done either in inert sol-vent or in frozen solution. It has been shown time andagain that there is not much drastic change as a result ofsolvent [22,23]. All calculations have been performed usingthe quantum chemistry program Gaussian 03 [24].

3. Results and discussion

Before any conclusions based on the calculations arebeing made, it is important to validate the methods used.The applied electronic structure calculation method isalways a compromise between accuracy and computationalcost. This is of importance since we are performing ourstudy for quite large polymeric systems. When it comesto the accuracy versus computational cost ratio DFT hasbeen shown to be a stronger candidate than the ab initio

methods such as Hartree–Fock or electron correlatedmethods. For systems as large as the ones in this paperDFT is the preferred method, especially if we want to per-form calculations using an adequate basis set. Based onprevious studies [17,15] on similar systems, we believe6-31G* to be adequate for optimization of the singletground state and the first triplet excited state. In the caseof the first excited singlet state optimizations, we haveused CIS and hence the basis set used, 3-21G, was smaller.

0102030

0102030

0102030

0102030

Δ A

ngle

Num

ber

(Deg

rees

)

0102030

-1 0 1 2 3 4Angle Number

0102030

n=3

n=2

n=4

n=5

n=7

n=6

Fig. 2. B3LYP/6-31G*: Changes in dihedral bond angles betweenadjacent monomer units upon excitation from S0 to T1 states, numberingas given in Fig. 1.

E. Jansson et al. / Chemical Physics 336 (2007) 91–98 93

On the one hand, we are not aware of any work in the lit-erature that has experienced such extensive calculations; onthe other hand, Wang et al. [25] have reported TDDFT cal-culations based on CIS/3-21G optimized structure andbased on their agreement with experiment we believe thatthis level of theory is good enough for our purpose. Whenusing both DFT and CIS, two factors need to be kept inmind when comparing the structures; the amount of elec-tron correlation included is different, since by using DFTwe are able to capture some of the dynamic correlationswhereas in CIS only static correlation is taken intoaccount. Secondly, the use of different basis sets may causediscrepancies. In order to determine error associated withthe smaller basis set, we performed CIS/6-31G* optimiza-tion for some of the smaller systems and on the basis of thiscalculation we conclude that for interesting dihedral angles(such as hmax discussed in the next section) the differencesare less than 1�. Also, the corresponding bond length differ-ences are negligible between the smaller and larger basissets. Selecting the proper functional is always of impor-tance and considering that our goal is to obtain emissionand absorption spectra, Becke’s three-parameter hybridfunctional was chosen since recent works have shown thatit gives reliable excitation energies [15,17,26]. Also, it hasshown to give reliable description of both ground and firstexcited triplet state in a variety of different conjugated sys-tems [27]. When it comes to interpretation of Kohn–Sham(KS) orbitals, it has been debated on whatever KS orbitalscan be interpreted as molecular orbitals in the same senseas in Hartree–Fock theory. This question comes into thepicture since the KS orbital energies do not carry any phys-ical meaning. However, it has been shown that KS orbitalsare good basis for qualitative interpretation of molecularorbitals [28,29].

In the coming subsections we will compare our results tothe ones reported by Wasserberg et al. [14]. All discussionsregarding experiment should be referred to this study ifnothing else is mentioned. In connection to this, it is impor-tant to clarify that our calculations correspond to verticaltransitions which do not always correspond to a maximumof spectra in an electronic excitation, since there is an over-lap between neighboring electronic and vibronic bands.

3.1. Geometrical relaxation of the T1 and S1 states

All our calculations have been performed for oligomersby approximating 9,9-dihexylfluorene-2,7-diyl with fluo-rene (i.e. polyfluorene with hydrogens instead of hexylgroup at the 9-position), see Fig. 1. We rely this approxi-mation on two things: (i) It is well known that the purposeof the alkyl chains in position 9 is mainly to increase thesolubility of the systems. (ii) Comparison of the T1 energylevel of 9,9-dihexylfluorene-2,7-diyl [14] and fluorene [30]agrees reasonably well, which implies that the hexyl groupsare of less importance when it comes to the absorption andemission properties of interest in this paper. Looking at theoptimized structures for S0 oligomers where n > 2, a twist

in the dihedral angle (hn) between adjacent monomer unit(see Fig. 1 for labeling of hn) which rotates the monomerunits out of the plane is experienced. The chain lengthdependence on the oligomers structure at the S0 state hasalready been calculated by Wang et al. [25] and we confirmthe alternating character of hn. Compared to the previousstudy we have incorporated additional polarization func-tions in the basis set which gives somewhat smaller hn,roughly 0.8� smaller for dimer and trimer and 0.5� smallerfor n P 4, which can be considered as negligible. As far aschange in the bond lengths is concerned, there is no appre-ciable change.

Coming to the structure of the T1 state we observechanges in both hn and in bond lengths. A more detailedanalysis of hn reveals that two adjacent monomer units inthe oligomer goes towards planarity during the excitationwhich is seen clearly from Fig. 2, in which we have plottedthe change of hn upon excitation from S0 to T1. We havecollected the calculated values of the dihedral angle whichis experiencing the largest deviation (hmax) in Table 2. Wesee that the hmax almost goes to zero, i.e. the two adjacentmonomers are only a few degrees from being planar.

It is interesting to see that the structural change in dihe-dral angles becomes more and more localized as we con-sider longer oligomers. This gives us valuable informationabout the excited state distribution and we expect to seea more localized excited state relative to size, as we gotowards longer chains. A closer look at the calculated bondlengths indicates that the perfect aromatic structure of S0

has an ‘impurification’ present at hmax for the T1 state.Bonds parallel to the elongation direction become longer

Table 2Calculated hmax (�) for S0 (B3LYP/6-31G*), T1 (B3LYP/6-31G*) and S1

(CIS/3-21G)

Oligomer (F)n S0 S1 T1

n = 2 37.5 6.0 1.4n = 3 36.9 23.6 18.6n = 4 37.5 13.2 1.2n = 5 37.4 14.2 1.9n = 6 36.7 14.7 2.7n = 7 37.1 14.6 6.6

-0.040

0.04

-0.040

0.04

-0.040

0.04

-0.040

0.04

Δ B

ond

Len

gth

(Å

)

-0.040

0.04

-0.040

0.04

-20 -15 -10 -5 0 5 10 15 20Bond Number

-0.040

0.04

n=1

n=2

n=3

n=4

n=5

n=6

n=7

Fig. 3. B3LYP/6-31G*: Changes in bond lengths upon excitation from S0

to T1, numbering as given in Fig. 1.

Table 3B3LYP/6-31G* calculated transition energies (eV) of oligofluorenes,experimental numbers (in parentheses) from Ref. [14]

Oligomer (F)n S1 S0 S1! S0 T1! S0 Tn T1

n = 1 4.68 (4.08) 4.21 (4.08) 3.03 (2.86) 3.48 (2.98)n = 2 3.80 3.21 2.53 2.30n = 3 3.45 (3.22) 2.97 (3.13) 2.47 (2.25) 1.67 (1.85)n = 4 3.30 2.89 2.42 1.50n = 5 3.21 (3.04) 2.86 (2.97) 2.42 (2.18) 1.36 (1.58)n = 6 3.16 2.85 2.42 1.27n = 7 3.13 (2.99) 2.85 (2.88) 2.42 (2.11) 1.24 (1.46)

94 E. Jansson et al. / Chemical Physics 336 (2007) 91–98

and other bonds become shorter forming a quinoid struc-ture. The changes in bond length are shown in Fig. 3 wherethe numbering given in Fig. 1 is used. The behavior is sim-ilar for all oligomers but the center of the quinoid struc-tures is different for n = 1 and n = 3 where the center islocated at the linking bond within a monomer unit ratherthan at hmax which is the case for other oligomers. InFig. 3 we have not plotted the changes in bond length forthe entire structure simply because as one goes furtheraway from hmax the changes are negligible. The geometricaldeformation takes place over no more than two benzenerings, and a cutoff of j0.01j A gives an indication of theextension of the T1 state.

Having performed S1 optimization at another level oftheory we do not make any quantitative comparison ofthe structure with the S0 and T1. Nevertheless, a glanceat the structure reveals the same trend for S1 as for T1

but the changes in hmax are less pronounced, see Table 2.It is hence expected that the S1 state is less localized com-pared to its triplet counterpart T1, while both have a morelocalized character than S0. This is also confirmed by

experimental observations [14,31] based on the chainlength dependence of the T1 and S1 excitation energies.

Based on these observations we conclude that as the oli-gomer goes from the S0 state to either the S1 or T1 excitedstate, a more ordered segment is formed on the chain. Thiscan be of interest for determination of the origin of the so-called b-phase, which was first observed by Bradley et al.[32–34] as a small shoulder peak at 437 nm. Dias et al.[35] have recently concluded that the formation of the b-phase is a two step mechanism: first an intramolecular pro-cess in which a relatively disordered segment goes to anordered conformation (b) on a single chain followed byaggregation of chains containing this ordered b-conforma-tion. Based on the observed planarization of the excitedstates, we argue that the first intramolecular step suggestedby Dias et al. [35] may be caused by molecular excitation.The formation of b-conformation may hence be aggrega-tion of excited oligomers. We intend to confirm or dismissthis hypothesis in a forthcoming paper.

3.2. Absorption and emission spectra

The UV/vis absorption and steady-state fluorescencespectra of polyfluorenes have been reported in frozenMeTHF at 80 K by Wasserberg et al. According to theseresearchers, both (S1! S0) emission and the (S0! S1)absorption exhibit a clear 0–0 transition and a vibronicprogression with 0.16 ± 0.01 eV energy spacing, character-istic for modes related to C@C and CAC stretch vibrationsof conjugated polymers. However, at room temperature theabsorption spectra of polyfluorenes show no vibrationalresolution [11,12]. This difference at room temperatureand at 80 K has been attributed to the conformational dis-order of the inter-ring dihedral angles between repeat units[14]. Furthermore, Chi et al. [31] have show that the shapeand position of fluorescence spectra of the heptamer do notchange with concentration, which of course rules out theimportance of aggregates and excimers. Before we startcomparing our calculated results with the experimentallyreported ones, it is important to mention that the correctway of calculating these excitation energies would be totake the difference in the zero-point vibration correctedenergy (ZPVE) of the optimized S0 and S1 energy levels.In order to obtain the S1! S0 excitation energy, we haveperformed TDDFT calculation of the S0! S1 process at

E. Jansson et al. / Chemical Physics 336 (2007) 91–98 95

the S1 optimized geometry. Hence, the calculated fluores-cence energies are expected to differ somewhat from theexperimental 0–0 energies. As seen in Table 3, the agree-ment between theoretically calculated UV/vis absorptionand experiment is quite good. In fact, our calculated spec-tra are also in agreement with some previously reportedtheoretical energies [25]. The difference between computa-tional and experimental absorption spectra (S0! S1) forthe monomer case has been found to be relatively high(0.60 eV). For larger systems (n P 3) the errors is systemat-ically overestimated but with differences less than 0.23 eV,which can be considered within the experimental range oferror. As we increase the chain length there is a continuouschange in the inter dihedral angle, which drags the poly-mers from the planar position to a non-planar one. Thismeans that as we move from the monomer to the heptamer,the conformational freedom of the polymer unit changes,which leads to significant changes in the spectra.

Moving on to the steady state fluorescence spectra, thecomparison between experiment and theory is reported inTable 3. Our calculated fluorescence energies (S1! S0)are in better agreement than the absorption energies(S0! S1). The excitation energy becomes smaller for largeroligomers. This observation concords with the continuousshift of experimental spectra in the excited states. It is alsoin agreement with the fact that there is an increased delo-calization of the excited state wave function for longer olig-omers. Wang et al. reported steady state fluorescenceemission energies using TD-DFT/B3LYP calculations ofthe S1! S0 transition for n = 2,4 with a 6-31G basis set.By comparing with their results we come to the conclusionthat there is only a minor role of the extra polarizationfunction since the maximum deviation of the excitationenergies obtained by the two different basis sets is less than0.07 eV for S0! S1 and less than 0.1 eV for S1! S0. Thesaturation of energy occurs almost for n = 5 in the caseof the S0! S1 absorption as well as for the S1! S0

emission.Switching to the triplet absorption spectra, experimental

peaks together with our calculated absorption energies arereported in Table 3. We could reproduce the peaks in theT1! Tn absorption spectra. The largest deviation fromexperiment is 0.5 eV for the monomer which is quite consis-tent with the singlet absorption spectra. The differencebetween our calculated results and experiment is roughlyaround 0.21 eV, which to some degree could be due tothe negligence of the ZPVE correction. It has also beenobserved experimentally that for n > 5, the T1! Tn spec-tral features are very similar and there is appearance ofone distinct peak, which seems to be related to the satura-tion in energy. In general, the T1! Tn energies for thepolyfluorene chain could reproduce the trend in energiesquite well compared to all the reported experimental results[14].

Considering the fact that the relaxation between the S0

and T1 state geometries becomes less with increasing chainlength, it is expected that the relative importance of some

vibronic states increases in the phosphorescence spectra.This has also been observed by Wasserberg et al. [14]. Theyobserved that the 0–0 transition intensifies, as chain lengthincreases and by looking at their phosphorescence spectra(Fig. 2, Ref. [14]) one can also see that the energy of the0–0 transition saturates.

Hence, we have calculated T1! S0 energies by takingthe differences of the energies of the ground states and trip-let states at their optimized geometries, neglecting ZPVE tobe consistent. Our calculated energies are consistently over-estimated (see Table 3) which can be due to the fact that wedo not consider vibrational effects. The excitation energyshifts towards lower energies as chain length increasesand it starts to saturate at n � 3. Our calculation confirmsthis observation showing a definite saturation for n = 4.The overall agreement seems to be satisfactory in this casetoo. It is important to notice that in all cases the deviationsbetween observations and calculations are small but sys-tematic. Of course, our calculations can suffer from cancel-lation of errors between chosen functional and basis set,which cannot be ruled out.

3.3. Effective conjugation length of polyfluorenes

In order to determine the limiting values of the energiesEn with the increasing chain length of polymers, Meieret al. [36] proposed an empirical relationship. Accordingto them the length dependence on excitation energy ofthe lowest excited states can be expressed by the followingempirical relation [36]:

En ¼ E1 þ ½E1 � E1� � e�aðn�1Þ; ð1Þwhere n is the number of repeat monomer units, E1, E1 arethe excitation energies for the monomer and the infinitelylong polymer and a is the fitting parameter, respectively.This could be directly related to how fast the saturationhas been reached, which in turn means that the higherthe value of a the faster is the saturation. In Figs. 4 and5, we show the energies of various calculated optical tran-sitions with respect to the reciprocal number of repeat unitsobtained as a result of fitting of Eq. (1). In the same plot,we have also compared our calculated results with theexperimental ones. We have reported the fitted parametersin Table 5 and compared them with the experimentally ob-tained ones. The trends seen for parameter a are the sameas seen in the previous study performed by Wasserberget al. [14]. This parameter is larger for phosphorescencethan for fluorescence, which indicates that the T1 state ismore confined than the S1 state. We note that the disper-sion in energy is much lower for phosphorescence (0.89)than for fluorescence (1.33).

Cornil et al. [37] have reported S1 and T1 excitationenergies for oligofluorene based on semi-empirical INDO/SCI calculations and concluded that the excitation energieshave a linear dependence towards the inverse chain length.They only considered oligomers of size n = 1– 4. On thebasis of these calculations the authors suggested a linear

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

3.5

4

4.5

5

1/n

Ene

rgy

(eV

)

S0

S1

T1

Tn

S0

S1(exp)

T1

Tn(exp)

→

→

→

→

Fig. 4. B3LYP/6-31G*: Calculated and experimentally obtained absorp-tion energies for the S0! S1 and T1! Tn transitions for Fn as a functionof the inverse system size (1/n). The solid lines are the fitted functions toEq. (1).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

1/n

Ene

rgy

(eV

)

T1

S0

S1

S0

T1

S0(exp)

S1

S0(exp)

→

→

→

→

Fig. 5. B3LYP/6-31G*: Calculated and experimentally obtained emissionenergies for the S1! S0 and T1! S0 transitions for Fn as a function of theinverse system size (1/n). The solid lines are the fitted functions to Eq. (1).

Table 4B3LYP/6-31G* Calculated fluorescence lifetimes s (ps) and electric dipoletransition moments l (Debye)

Oligomer (F)n scalc.a scalc.

b sexp.c lcalc.

a lcalc.b lexp.

c

n = 2 1179 1384 789 9.67 11.5 6.8n = 3 891 1063 663 12.85 14.7 9.3n = 4 726 905 578 15.28 16.7 11.5n = 5 611 808 549 17.31 17.9 13.0n = 6 523 746 520 19.15 18.8 14.4n = 7 460 701 397 20.72 19.4 15.0

a Calculation performed at the ground state optimized geometry.b Calculation performed at the first excited singlet state optimized

geometry.c From Ref. [39].

Table 5Fitting parameter a and the excitation energies for the infinite polymerchain as obtained by Eq. (1), numbers based on experimental observationfrom Ref. [14] (in parentheses)

Transition a E1 E(PF)

S1 S0 0.81 (0.74) 3.14 (2.97) 2.96S1! S0 1.31 (0.82) 2.86 (2.91) 2.88T1! S0 3.06 (0.91) 2.42 (2.14) 2.11Tn T1 0.72 (0.67) 1.22 (1.47) 1.46

Excitation energies, from Ref. [14], for polyfluorene (PF) are given for

96 E. Jansson et al. / Chemical Physics 336 (2007) 91–98

extrapolation to get the excitation energies for the infiniteoligomer. Comparing our results with the one reportedby Cornil et al. [37], we also observe the linear trend. Butaccording to our observation the linear extrapolationscheme does not hold if one goes beyond n = 4, which isquite consistent with earlier reported results [36,38]. In fact,in order to get more accurate excitation energy for the infi-nite oligomer, one needs to use higher order polynomials.The effective conjugation length (ECL) at which saturationis reached for the optical properties, absorption and emis-sion, is found to be n = 5 for Fn. This nonlinear behavior isnot surprising. We have already seen that the optimizedstructures saturate at n = 5, where there is no significant

change in dihedral angles between adjacent monomer unitsand even the bonding behavior is saturated (see Fig. 3).

3.4. Fluorescence lifetimes

In order to make a conclusive comparison of the com-puted data with the experimental observations, we havealso analyzed the fluorescence lifetimes. The fluorescencelifetime from the first excited singlet state S1 is given by

1

s¼ 4

3t0

a30ðDEÞ3

X

a2fx;y;zgjMaj2; ð2Þ

where t0 = (4p�0)2�h3/mee4, a0 is the fine-structure constant,

DE is the transition energy, and Ma is the a-axis projectionof the electric dipole transition moment between theground state and the first singlet excited state. Here itshould be mentioned that we do not calculate theFranck–Condon factors, which indeed are an approxima-tion. In order to get the experimental lifetimes one needsto calculate the oscillator strength, which is obtained byintegrating the molar extinction coefficient (�) over theabsorption band. Given both the calculated and the ob-served lifetimes, we have compared the results in Table 4in which both calculations performed at the S0 and S1 opti-mized geometries have been included for comparison. Theexperimental numbers have been obtained by Chi et al. [39]and similar lifetimes are reported by Wasserberg et al. [14]but for a smaller number of oligomers. The calculated fluo-rescence lifetimes follow observed trend of a shorter life-

comparison.

Fig. 6. B3LYP/6-31G*: Molecular orbitals of the S0 state of oligofluorene (n = 7). (a) LUMO + 1, (b) LUMO, (c) HOMO and (d) HOMO � 1.

Table 6Character of the S1, T1 and Tn excited states in the heptamer oligofluorene

Excitation Determinants

S0! S1 0.66[HOMO! LUMO] �0.16[HOMO � 1! LUMO + 1]S0! T1 0.74[HOMO! LUMO]T1! Tn (a) 0.60[HOMO! LUMO] �0.19[HOMO! LUMO + 1]

(b) 0.65[HOMO! LUMO] �0.25[HOMO � 2! LUMO]

E. Jansson et al. / Chemical Physics 336 (2007) 91–98 97

time as the chain length increases, even the absolute num-bers are not too far off. Having Eq. (2) in mind and analyz-ing the transition dipole moments it is clear that the goodagreement, in terms of absolute numbers, is an artifactdue to cancellation of errors. The transition moments(see Table 4) are overestimated, whereas the excitationenergies (see Table 3) are underestimated. The errors donot cancel completely resulting in a somewhat overesti-mated lifetime. Experimental lifetimes are based on electricdipole transition moments obtained from the ground stateto the first excited state. In order to get a correct picture,one should instead use the electric dipole transition mo-ments from the first excited state to the ground state. Based

Fig. 7. B3LYP/6-31G*: a-Molecular orbitals of the T1 state of olig

Fig. 8. B3LYP/6-31G*: b–Molecular orbitals of the T1 state of olig

on our CIS optimized structure and the fact that the trendof our calculated lifetimes is in good agreement, we forcean increase of the electric dipole transition moment at theexcited state (see Table 4). This will of course alter the ob-served lifetimes towards shorter timescales.

3.5. Nature of the excited states

In order to determine the nature of the excited states inthe infinite polymer, we have analyzed the KS orbitals bytaking the heptamer as an example. The KS orbitals(addressed as the molecular orbitals (MO)) which are ofinterest are of course the ones which are involved in theS1 S0, T1 S0 Tn T1 transitions. As seen in Fig. 6the HOMO and LUMO of the S0 state are predominantlylocalized on the five inner monomers compared toHOMO � 1 and LUMO + 1, which are localized on allmonomers except the central one. Upon excitation fromS0 to either S1 or T1, it is clear that electrons are removedfrom HOMO to LUMO (see Table 6). The S0! S1 tran-sition has also non-negligible contribution from

ofluorene (n = 7). (a) LUMO + 1, (b) LUMO and (c) HOMO.

ofluorene (n = 7). (a) LUMO, (b) HOMO and (c) HOMO � 2.

98 E. Jansson et al. / Chemical Physics 336 (2007) 91–98

HOMO � 1 LUMO + 1, which can explain the more delo-calized character of the S1 state compared to T1. Also, theformation of the quinoid structure can be seen as the bond-ing character of the ‘0’ bond (see Fig. 1) goes from antibonding in the HOMO to bonding character in theLUMO. Combining the orbital plots in Fig. 6 together withTable 6, it is clear that all transitions are, as expected, of p–p* character.

Turning our attention to the triplet subspace, we firstnotice that the excited a-electron is strongly localized athmax (see Fig. 7b). The T1 state seems to be a single electronexcitation compared to Tn, which have strong contribu-tions from excitation in both a and b subspaces. By lookingat Figs. 7 and 8, one could try to characterize the Tn state.On the one hand, it seems that the a electron becomes moredelocalized upon excitation and on the other hand, the belectron seems to become more localized. This makes it dif-ficult to draw any conclusions concerning the character ofthe Tn state.

4. Conclusions

By studying the monomer to heptamer oligomer of oli-gofluorene, this work gives an idea about the conjugationlength dependence of the excitation energies of differentoptical transitions. It is observed that the energies of allthe optical transitions irrespective of their nature decreaseslinearly with the reciprocal number of repeat units beforesaturation. The saturation of S0! S1, S1! S0 andT1! Tn occurs around n > 5, which corresponds to morethan 10 aromatic units. For the T1! S0 the saturationstarts around n = 4. We have also considered the excitedstates structures in details. It seems that the structuralchange occurring as the system undergoes excitation toeither S1 or T1 is strongly localized on the chain. More spe-cifically, the dihedral angle between two adjacent monomerunits goes towards zero. It is important to notice that evenwith common density functionals and limited basis sets, theagreement between theory and experiment is quite encour-aging. At this point one can say that density functional the-ory level of theory with decent basis set can give quite aclear picture about the optical properties of conjugatedorganic polymers in general and polyfluorene in particular.This may be of practical importance for the use of thesetheoretical methods to predict or guide the experimentalistin designing their experiments.

Acknowledgments

The authors are grateful to Professor Hugh D. Burrows(University of Coimbra) for helpful advice and fruitful dis-cussions. One of the author, P.C.J., acknowledges financialsupport from Wenner-Gren Foundations for supportingthrough a Postdoc Grant. We acknowledge the use of com-putational resources at the National Supercomputer Centre(NSC) in Linkoping, Sweden.

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