Calculation of the exciton band structure of the lowest energy singlet state of fluorene

C@m@alPhysiCs 30.(19?8) 353r359 ? Nort&Hcilland Publishing Company

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CALCIiiTIO~ OF TRE EXCLTON BAND STRUCTURE OF THE LOWEST ENERGY SINGLET STATE OF FLUORENE .A_-BRBEand M. EDELSON Chemistry

Department, University Of British Columbia, Vancouver, British Columbia, Canada V6T J WS

Received 17 Cktober 1977 The exciton band structure of the 33000 cm-! transition of fluorene has been calculated in the dipole approximation. It is shown that the interactions between the static dipole moment of one molecule and the transition dipole moment between two e&ted states on other molecules make a significant contribution to the offdiagonal elements of the energy matris, and c&se a contraction in the energy spread of the four exciton branches. The calculated intervals in the bag viironic stack do not agree with the~experimental values; this difficulty is not removed by the inclusion into the calcu!atron of a transition quadrupole moment, which is shown to be small.

1. Introduction A two-photon fluorescence excitation study together with some one-photon absorption and fluorescence studies [1,2] have provided information abodt the energies of the factor-group components of the low-en-

frames on II, III and IV are generated from I by applying two-fold screw axis rotations about the axes a,b and c, respectively. Each free-molecule state gives rise to four zero wavevector crystal levels, and the correla-

ergy singlet state of fluorene. The aim of the present work is to, find if it is possible, in the dipole approximation, to account for some of the features observed in the spectra. In particular, it wilt be important to calculate the magnitudes and sense of the Davydov splittings of the .origin band (the most intense band in the onephoton spectrum), and to show that the vibrational intervals in the (ac) and (bc) two-photon spectra should be disturbed from their values in the isolated molecule in the manner observed.

2. Calculations The free fluorene molecule is essentially planar with symmetry C,,. The symmetries of the singlet states below about 50000 cm-t have been assigned in earlier studies [2-41. The molecule and an appropriate molecular axis frame are shown in fig. 1. The fluorene crystal [5] is orthorhombic (pnam). .Tlre four molecules in the unit cell occupy sites which are labelled I; Ii, III and IV in fig. 1. The molecular axis

-X

Fig. 1. The orthorhombic unit cell of the tluorene crystal defming the molecular orientations and the four available sites.

A. Bree, M. EdeIson/Exciton baridstructure of the singlet state offi’uorene

354

tion between the two sets of states may be found from table 1 of ref. [I] where appropriate selection rules are also given. , The crystal energy levels were calculated using the theory as outlined by Craig and Wahnsley [6] ; to avoid the necessity of re-deriving the equations, we have adopted their symbolism in what follows. 2.1. The dipole approximation Lattice sums were evaluated for unit point rnultipoles placed at the centre of mass of the fluorene mol&ules. Dipole-dipole sums were calculated with a complete neglect of retardation corrections using the Ewald-Komfeld method [7] and a planewise summation method [S] . The relevant data required include the unit cell dimensions and the direction cosines between the molecular and crystal axes which are available from the crystal structure determination [51. The cell edges are a = 8.49 8, b = 5.721 aand c = 18.97 Ii. A unit dipole $ directed along the short in-plane axis of molecule I has components (0.57119, -0.82082, 0.0) along a, b and c, while 8, lies parallel to c (see fig. 1). The exciton levels which are optically accessible are those for which k = 0. Dipoie sums fork = 0 evaluated using the Ewald-Komfeld method [7] are listed in table 1. The non-analytic behaviour [7,9] of the dipole sums implies that the sums may change for different directions of propagation of the incident light in the crystal, we have calculated the sums for light falling normal to the three principal planes of the orthorhom-

bit crystal. The planar sums did_not converge to the infinite plane result (the Ewald-Komfeld sum) for square planes of half-edge less than 1000 8, and, while the sums were close to the Ewald-Korifeld values, the computing time required was too extensive to make fhe convergence complete. The calculated crystal spectrum reported.here is mode&d on a solution spectrum that has the following / features. There are three long-axis transitions at 33250, 38200 and 48500 cm-l with transition dipole lengths of 0.391, 1.Ol and ! .OO8, respectively. The intensity of the high energy system is a rounded-off estimate from the reflection spectrum reported by Tanaka [3], and the other data were taken from a spectrum in n-heptane. Only the low-energy system shows resolved vibrational sublevels, although the structure in the room-temperature solution spectrum is not sufficie&y resolved for a complete comparison with the low-temperature crystal spectrmm. For our model calculation, the intensity of the 33250 cm-l band in the n-heptane spectrum was split in the ratio 7 : 2 : 1 (the rough ratio observed in the 4.2 K ;r-heptane spectrum [2]) amongst transitions from the vibrationless ground state to upper states that have no vibrations, one quantum of a 210 cm-l and one quantum of a 400 cm-l vibration, respectively. The activity of all higher vibronic states of the low-energy system were lumped together in the observed room-temperature solution bands at 34000 and 34550 cm-l with dipole lengths 0.18 and 0.24 A. The molecular constants used in calculating the crystal spectrum are collected in table 2. The integrated intensities were measured from a room temperature

Table 1 Dipole-dipole

interaction

sums (in CC’ A’)

for unit transition I-I

illa

a)

YYb, zz

illb

ilIe

YY

1761.2 -1608.5

I-III

I-IV

-373 1.5

1603.8

-1563.6

-199.2

-525.4

I-U

-510.7

-373;s

1603.8

-1563.6

n

-1058.1

-1783.3

1058.6

-1061.2

YY

3345.2 -2125.3

-5315.6 -116.0

-8.6

7.z

1761.2

dipoles evaluated by the Ewald-Kornfeld

19.8

a) i is a unit vector, and is used here to show the direction of propagation of the tight beam. b, y and z are the molecular axes shown in fii. 1 and label the diection of the transition dipole.

20.4 6.1

method fork = 0

A. Bree. iK EdelsonjExciton band stmclwe of the sin$et state of fluorene Table 2

transitions and transition dipole lengths (d ) derived from a room temperature Spectrumof fluorene in n-Keptane

hfolecular

E (cni’)

d (A)

33250 33460 33650 34000 34550 38200 48500

0.21 a) 0.11 a) 0.08 a) 0.18 0.24 1.01 1.00a)

spectrum of fluorene in n-heptane, and were essentially the same when measured from a published spectrum in methanol [lo] _Under these circumstances no Lorentz correction was made for the presence of the solvent [I l] Retaining dipole-dipole resonance interaction sums only, the dispersion of the exciton branches with k were calculated for first order and the result for the origin band is shown in fig_2a. Other results fork = 0 are collected in table 3, these results refer only to the opticalFor convenience,

we

have set all the crystal shift termsw to zero for all excited electronic states r. While it is true that an estimate

-05 -a4

-03

-0.2 -0.1

!

0

bk&h

L

0.1

has beenmade for the crystal shift term of the first excited state [l] , the correeonding data for the other electronic states is not available. As the results in table 3 show, the Davydov splittings calculated fo first order are very much larger than the experimental values [l] . Indeed, the b2, component of the origin band is displaced so high in energy that a reversal of the two low energy levels is calculated to occur. The curve labelled b2g in fig. 2 suggests that the origin

a)Seetest.

ly accessible transverse branches.

355

I

02

I1

03

04

I

0.5

Fig. 2. The energies (given with respect to Aw +D) of the four exciton branches of the origin band fork normal to (010) [and (loo)] in the dipole approximation.For simplicity,the branches are labelledby their symmetriesat k = 0. (a) Calculationto fiit order of perturbation, and @) more highly excited molecular states included.

band in the (ac) two-photon spectrum has a shape that is largely determined by indirect transitions. Fluorene molecules do not occupy inversion sites so that phonons are mixed motions of translations and librations about the equilibrium position in the lattice. In this case, the restriction of downward intraband scattering by librons having a wavevectorg near zero as has been proposed for anthracene [12] need not apply and all k states of the exciton branch may become optically accessible. The effect of allowing different electronic states of the molecule to mix through the (dipole) interaction potential of the crystal is now examined. Because of the symmetv of the sites occupied.by the molecules in the lattice, only other B2 molecular states may m& with the lowest B2 state under consideration [l] . Energy matrices (of dimension seven, corresponding to the restricted basis indicated by the data in table 2) were set up and diagonahsed using‘an IBM 370/168 computer. Some results from these calculations are shown in fig. 2b and table 3. Some important consequences that follow from these calculations are: (i) The Davydov splittings are very much reduced for the lowest energy system by allowing different electronic states of the molecule to interact in the crystal. Only the b2g and b3g components of the origin band have been identified defmitely by experiment [l] and so only this splitting of 105 cm-1 may be compared with the calculated results in table 3. The calculated splitting is 387 cm-l and is clearly still too large. (ii) There is considerable mixing of the vibronic states of the low-energy system; this is especially true for the states with zero and one quantum of the 210 cm-1 vibration. (iii) The b3, branch is predicted to be lowest in energy, very close to the au branch; this conflicts with experiment [l] . The fact that the b3, branch is incorrectly placed in energy by the calculation is emphasized by the vibrational

356

A. Bree. M. Edekm/Exciton

band stmct&

Table3 -. : ~kAated exciton energies (in m-i’) for fluorene modelled on the parameters listed-i%table 2. The crystal shift term CD’, is setto zero; the data are for f = 0 with &[6 and so refer to transve?se branches only \ First order All states ?) au

blu

33165

of the singlet stnte of fluorene

because the appropriate linear cor&mation of the dipole resonance interaction sums given in table 1 is negative. The upshot of this is&at the AA,vibratioiial intervals in the (bc) two-photon spe&-um:are predicted by the calculations to expand iather than-to contract as observed. The inclusion of more highly excited states only makes the situation worse (see table 3). The mixing together of different free-molecule states

33437

32969 33426

33638 33938 34439 36231 46570

33626 33874 34375 36194 46953

the oriented-gas value. The oriented-gas ratio A(~B~~)] A(bzg) for the two-photon spectrum is 2.1, while the observed ratio for the origin band is 3.3 [l] . The calculated eigenvectors tell us how intensity transfers may occur, and show that intensity is shifted into the b3g component of the origin band of the low-encirgy syitem

33353 33488 33565 34075 34684 40572 50825

33300 33478 33659 34042 34634 40143 51406

from all transitions at higher energy and out of the b2g component of the origin band into the higher transitions. The calculation willagreewith observationprovided the two-photon cross sections to the higher excited ‘B2 free-molecule states are greater than to the lowest lB2

accounts for changes in the pola&ation

r&o away from

state. The two-photon spectrum of a fluorene solution at room temperature [13] shows that transitions to ‘B, states are weak and tend to be hidden under stronger transitions to lAl states. Because of the lack of quan33324 bjg 33632 33492 titative data for the relative two-photon absorption 33565 cross sections accurate values for the polarisation ratios 33666 33705 cannot be made. 34281 34070 The calculations to this point have placed the transi34699 35049 tion point dipoles at the centres of mass of the fluorene 47034 42193 molecules. The position of the transition dipole in the 62982 57160 molecule cau be found from the wavefunctions of the states involved, and these are not avaSabIe. Thus, an32937 bsg 33161 33424 other dimension to the problem is made available by 33436 33624 33637 treating the location of the transition dipole as a vari33866 33935 able. Clearly, the dipole must be placed on the molecular 34365 symmetry axis and remain within the framework pro34434 36122 vided by the bonded carbon atoms. In table 4, we show 36149 46904 the result of placing the dipole at the two extreme po46490 sitions, and compare these interaction sums with-those a) AU ‘Bj molecular states shown in table 2 are included in calculated with the transition dipole at the centre of this calculation. mass. Table 4 shows that the variation in the dipole sums spacings observed in the (bc) two-photon spectrum .[I]. with the positioning of the dipole in the fluoreue moThe frequencies of a21the totally-symmetric frequencies lecule is small. The most significant change occurs for close by about the same amount (24 cm-‘), and this the I-IX interactions, and it is readily seen that the enhas been interpreted to imply that the bSp origin band is shiftedfunher to higher energy than the other (weaker) ergy of the b3, exciton branch is at a maximum with the transition dipole near the centre of mass. All further vibronic bands by 24 cm-l [l] . The fust-order calcucalculations ari based on interaction sums with transilation shows that the b3g levels are depressed in energy tion tiultipoles placed at the centre of mass.

.A. Bree, M. Edekon/Exciton

band shuchue of the singlet state of fluorene

357

Table 4 gwald~KornfekI dipole-dipole interaction sums (in cni’ a;‘) for kb = 0 with unit dipoles placed (a) on the carbon-carbon bond joining the two benzene rings, (b) at the centre of mass, and (c) at the carbon atom of the CHa group

t

I-1

I-II

I-III

I-IV

-3808.0

1590.5

-1576.6

-2116.3

1063.1

-1065.4

caseb yy 1761.2 zz -1058.1

-3731.5 -1783.3

1603.8 1058.6

-1563.6 -1061.2

casec yy

-4226.8

1614.9

-1549.3

-4040.2

1054.9

-1055.4

casea yy

1761,2 zz -1058.1

1761.2

zz -1058.1

I

0

02

0.4

06

,

02

-

03

0.4

Fig. 3. The variation in the energies of the four exciton branches at k = 0 with the oscillator strength of the transition between the fist two excited t Ba states of the molecule. (a) and (b) refer to two different choices for the vibrational overlap factors (see text for details). The D shifttermis set to zero.

21.1. The inclusionof K integrals The integral Km [6] makes a contribution

sets the overlap factors to be the same as for the transi-

tion between a static m&pole

to the off-diagonal element ffr that connects electronic states r and s in the energy matrix, and describes the fnterac-

moment of the ground

tion from the ground to the fast excited state; this corresponds to the (unlikely) assumption that the second excited 1B, state has the same geometry as the ground

state on one molecule with a transition multipole moment that links the statesr and s on another molecule. In the approximation considered here, both of these

state. This set of overlap factors leads to the curves in fig. 3a. The second choice is to put aiI the dipole intensity into the transition between the vibrationless states.

multipole moments are taken to be dipole moments. The static dipole moment of fluorene has been reported to be 0.58 D [12] and 0.53 D [13] _Earlier de-

This corresponds

terminations were 0.82 D [14] and 0.28 D [15]. In what follows we accept the mean of the two more recent values, which corresponds to the displacement unit electronic charge through 0.12 A.

of

The energy of the lowest excited electronic state in the crystal is influenced most by interactions with the B, state next highest in energy, and so we consider K integrals that connect only these two states. While in principle the transition dipole moment between these states may be found by experiment, in practice the transition energy falls in a technically difficult region of the infrared. We show how the energies of the four exciton levels (corresponding to the fnst excited vibrationless electronic state of the free molecule) change as this transition moment is varied as a parameter (see fig. 3). The overlap factors involved in the transition, however, represent another source of unknowns in the calculation. For the sake of deftiteness, two different limiting sets of overlap factors are treated. One choice

roughly to supposing that the first two excited *B2 states have the same geometry, and

leads to the curves in fig. 3b. The effect of adding the extra excited states into the initial calculation is to lower the energy of the b3a component of the origin band for the low-energy absorption system and so to cause the vibrational spacingsto ex-

pand (see table 3). To counter this effect, the static and transition dipole moments used in evaluating the integrals KTSwere chosen with opposing senses. The results of the calculation are shown in fig. 3,

which indicates that the overall effect is independent of the details of the model. As the oscillator strength v) of the transition dipole is increased, the energies of the ar, and b3z components gradually increase. This occurs because the algebraic-sign of the K integral was chosen to oppose the contribution (J integral, see ref. [6]) already included in the off-diagonal elements of the energy matrix. As these off-diagonal elements pass through zero and their absolute magnitude again increases (asfincreases), the exciton energy levels are again depressed. No cancellation occurs for the b,, and

358

.-

A_Bree, M: EdelsonjExcitonbandst?uctureof the singletstateoffuoiene

b,, branches whiih show a steady energy depression. T&e Curves in fiiig.3 cover the range off that might reasonably be expected for the intensity of the transition, and the effect is a dramatic contraction in the calculated Davydov splittings. .‘. The energy difference between the bzg and b3, components (the only measured splitting available for comparison with the calculations) is 10.5 cm-l. Values for the parameterfrequired to fit this observed splitting are 0.46 from fig. 3a and 0.14 froth fig. 3b. Thus, the assumption that it is the inclusion of the R integrals that causes the calculated splittings to contract to the observed ones is equivalent to supposing that the transition between the excited states of fluorene has a medium intensity. -The vibrational intervals in the bSg spectrum remain larger than the experimental values. Even in the more favourable circumstances defied in fig. 3b the fast vibrational intervals are 280 ar.d 515 cm-l, in between the values reported in table 3. A possible way out of this difficulty is to assume that a transition multipole of higher order than a dipole plays a significant role in determining the exciton energies. Since the origin band has by far the largest overlap factor, it will be shifted in energy more than the other vibronic bands. 2.2. The inclusion of a quadrwole transition moment Another way to make the calculated energies of the factor-group components fit the experimental pattern may be to include as parameters values for the various transition muhipole moments, of which the next most important is the transition quadrupole moment. There Table 5 Dipole-quadrupole (ii ~6’ X3) and quadrupole-quadrupole intelaction sums (in cni’ a”] evaluated for unit transition multipoIes and fork = 0 d-Q a)

Q-Q 9

I-I

0.000

I-II I-III I-IV

9.688 9.230 9.348

-138.72 58.48 -252 -1.20

a) The fust written symbol refers to the reference molecule and the second to the surrounding molecules;d represents a dipole and Q a quadrupole moment.

:

is only one allowed component of the transition I&:

tipcle moment.for t& transition .1J5i + ‘A1 and We will r&f&rtq this c?rned+t s&npli as Q. interactions between a. tiriit point dipole 6ti.a reference molecule and unit point quadtipoles on the surrounding molecules are given in table 5, where carresponding data for quadnipole-quadrupole inieractions are also collected..The interactions tiere.evaiuated by the planewise summation method (for square planes of half-edge 300 A). The sums are the same for IICand bc, the planes of experimental iriterest. We now seek the value of Q which.&ill make the energy of the b3, exciton branch as large as possible. To the first order of perturbation, we fmd E(b,,) - (Aw +O) = (-307.61 - 6.952 Q --76.52 Q2)E2,, where .$is the Franck-Condon overlap factor for the origin band of the low-energy system. The energy is maximised when Q is -0.045 eA2, but the energy change when the quadrupole is included is 0.05 cm-1 and is negligible. Clearly, the effect of a-quadrupole transition moment may be neglected.

3. Conclusion Calculations carried out to the fust order of.perturbation in the dipole app.roximation overestimate the extent of the Davydov splitting for the low-energy system of fluorene (see fig. 2a). This overestimate is reduced by including the effect of interactions with more highly excited molecular states (see fig. 2b) and may be further reduced to equal theobserved value by the inclusion of K integrals in the energy matrix of the crystal. The quadrupole transition moment for the lowenergy system is shown to-be small, making a negligible correction to the energies of *he exciton levels. None of these corrections change t$e ordering in energy of the four exciton levels of the origin band from the initial fust order calculation. Within the limits of sophisticatidn adopted in this treatment, our model calculations predict that the energy of the a, level is not below the b3g level but is slightly above it. This conclusion suggests that the weak one- and two-photon absorption centred near 32900 cm-l [l ] arises fr0m.a #her highbensit$ (of-the order of 1%) of X-traps in the nielt-grown samples used, and preeludes the possibility that the weak absorption marks

A. Bree. M. Edekon/Exc.?on &andstructure of the singlet state offruorene

the location of the a,, DaGdov component. No attempt-has be& made to in&de interactions which show an even more rapid convergence on the intermolecular separation (R) than R”. We note that all collective states, such.as charge transfer states, have been omitted from these calculations. Retardation and exchange effects are also unaccounted for in the belief that they are small; however, it may be that the sbortrange-exchange interactions may contribute significantly and should be included in a later calculation. While it should be emphasized that only model calculations are reported here, it must be remarked that the results are in error in that the intervals in the b,, vibronic stack do not agree with tbe experimental values [I]. It may prove more satisfactory to take another approach to the problem, and that is to include the effects of all high-order transition multipoles by making use of cakulated wavefunctions.

Acknowledgement We are pleased to acknowledge a grant from the National Research Council of Canada which made this research possible, and the authors would like to thank Drs. G.J. Small and C. Taliani for helpful discussions regarding this work.

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359

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[l] A. Bree, M. Edelson and C- TaIiani, Chem. Phys.30 (1978) 343. [2] A. Bree and R. Zwarich, J. Chem. Phys. 51 (1969) 903. [3) M. Tanaka, Bull. Chem. Sot. Japan 49 (1976) 3382. 141 T. Hoshi, H. Inoue, J. Shiraishi and Y. Tanizaki, Bull. Chem. Sot. Japan 44 (1971) 1743. [S 1 D.M. Burns and J. Ibarl, Proc. Roy. Sot. A227 (1955) 200. [6] D.P. Craig and S.H. Wahnsley, Escitons in molecular crystals (Benjamin, New York, 1968). [7] M.R. Philpott and J.W. Lee, J. Chem. Phys. 58 (1973) 595. [8] M.R. Philpott, J. Chem. Phys. 58 (1973) 588. [9] A.S. Davydov and E-F. Sheka, Phys. Stat. Sol. 11 (196.5) 877. [IO1 The SadtIer standard spectra, Sadtler Research Laboratories, Philadelphia, 1968. [ 111 L.A. D&ado, J. Phys A3 (1970) 608. [12] L.A. Dissado, Chem. Phys. 8 (1975) 289. [13] R.P. Drucker and WM. McCIain, J. Chem. Phys. 61 (1974) 2616. [14] H. Lumbroso, C.R. And. Sci. 228 (1949) 1425. [lS] J. Sirkm and E. Schott-Lova, Acta Physiochem. USSR 19 (1944) 379. 1161 E.D. Hughes, CG. le F&e and RJ.W. le F&e, J. Chem. sot. (1937) 202. [ 171 E. Bergmann, L. Engel and M. Hoffmann, Z. Physik. Chem. B17 (1932) 92..