On the geometry dependence of molecular dimer spectra with an application to aggregates of perylene bisimide

Chemical Physics 328 (2006) 354–362 www.elsevier.com/locate/chemphys

On the geometry dependence of molecular dimer spectra with an application to aggregates of perylene bisimide J. Seibt a, P. Marquetand a, Volker Engel a

a,*

, Z. Chen b, V. Dehm b, F. Wu¨rthner

b

Institut fu¨r Physikalische Chemie, Am Hubland, 97074 Wu¨rzburg, Germany Institut fu¨r Organische Chemie, Am Hubland, 97074 Wu¨rzburg, Germany

b

Received 18 May 2006; accepted 19 July 2006 Available online 21 July 2006

Abstract We study spectroscopic properties of molecular dimers coupled by dipole–dipole interactions within the framework of timedependent quantum mechanics. A systematic variation of the dimer geometry allows to establish relationships between the latter and structures in the absorption spectrum. The theoretical model is constructed with the purpose to characterize the changes in absorption and emission properties arising upon aggregation of perylene bisimides. Measured and calculated spectra are compared, thereby addressing the question if a simple exciton model is capable to describe excited state properties of nanoaggregates of these molecules.  2006 Elsevier B.V. All rights reserved. Keywords: Optical spectroscopy; Aggregates; Molecular dimers; Time-dependent quantum mechanics

1. Introduction It has been realized only recently, that vibronic interactions in molecules are by far more important in photochemistry than generally assumed [1]. The common origin of these interactions is the break-down of the Born–Oppenheimer approximation [2,3] yielding kinetic couplings which cannot be neglected when the electronic wave functions change rapidly with nuclear geometry. In the so-called ‘‘diabatic’’ representation (see e.g., Ref. [4] and references therein), the adiabatic potential energy surfaces belonging to different electronic states of the molecule, are coupled by off-diagonal potential matrix elements. Regarding the form of the Hamiltonian, this is formally equivalent to the coupling between the electronic and nuclear motion in composite systems which are weakly bound to each other, as is the case for molecular aggregates. A particular simple system is a molecular dimer con*

Corresponding author. E-mail address: [email protected] (V. Engel).

0301-0104/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.07.023

sisting of two monomer units interacting via dipole–dipole interaction. This system was treated within the molecular exciton model [5,6], incorporating a single vibrational degree of freedom for each monomer [7–9]. Within this model, absorption, circular dichroism, and Raman spectra were calculated and compared to experimental data [10– 14]. Introducing additional approximations, the treatment can be extended to more extended aggregates up to an infinite size, see e.g., the early work of Herzenberg and Briggs [15] and also more recent work [16,17]. In this paper, we apply a time-dependent quantum methodology to calculate the geometry dependent absorption and emission spectra for dimer aggregates of perylene bisimide (PBI) chromophores and compare the results to experimental data. In a forthcoming paper this theoretical work will be extended to aggregates of larger size, and well-defined dimer and trimer model systems will be synthesized to validate the theoretical results based on experimental data. In doing so, one of the central points is the question if, and to which extent, geometrical properties of the system can be extracted from linear spectroscopy.

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

There are several reasons why such a study is particularly promising and important for PBI chromophores. First, perylene bisimides are an important class of dyestuffs [18]. In particular, many pigments are produced on a large scale and their coloristic properties vary from red to maroon and black [19]. Most important, these color changes do not arise from different substituents like in many other classes of dyestuffs but are governed by the packing of the molecules in the solid state [20]. Second, the monomeric dyes are strongly photo-luminescent but this photoluminescence is often quenched in the aggregated state. For monomeric dyes, well-defined vibronic structures are observed for the S1 S0 transition (polarized along the long axis of the molecule) in absorption as well as in the mirror image fluorescence spectra [18]. Third, novel applications of these dyes as active materials for organic solar cells [21–23] or light emitting devices are dependent on absorption properties, charge carrier and exciton mobilities [24]. All of these properties are governed by the electronic interactions between the molecular building blocks in the aggregated state that are highly dependent on the spatial arrangement of the chromophores [25]. Whilst the packing geometry in crystalline solid state materials can be easily derived from X-ray diffraction studies, this information is not easily available for self-assembled aggregates in solution and mesoscopic and mesomorphic materials including organogels and liquid crystals [26]. Accordingly, it seems particularly appealing to establish a theoretical description for the relationship between aggregate structure and aggregate absorption spectra that enables us to derive the given supramolecular organization of perylene bisimides in their aggregates from UV/Vis spectroscopic data. Recently, we have synthesized a PBI derivative (named PBI 1, in what follows) bearing two tridodecylphenyl sub-

355

stituents at the imide nitrogen atoms [27], see Fig. 1 for the structure. Aggregates of these dyes in solution and in the columnar liquid crystalline phase show pronounced spectral changes in the absorption and emission bands. Nevertheless, despite the fluorescence lifetime is increased, the strongly bathochromically shifted aggregate fluorescence remains quite intensive and well-resolved spectra could be monitored. Having established the optical properties for these PBI 1 aggregates the question is addressed if they can be understood within the simple dimer model. Therefore, we proceed as follows: in Section 2, the theoretical model is presented and the calculation of the absorption and emission spectrum is described. A systematic variation of the dimer geometry and the such induced influence on the spectra is presented in Section 3, which also contains a comparison between experiment and theory and the conclusions drawn from the comparison. 2. Theory 2.1. Monomer Hamiltonian The monomer (M) unit is described by a single-mode Hamiltonian with an electronic ground state jgi and an excited state jei as: M H M ðqÞ ¼ jgiH M g ðqÞhgj þ jeiH e ðqÞhej:

ð1Þ

Within the harmonic approximation, the Hamiltonian for the nuclear motion in the ground state reads: HM g ðqÞ ¼ 

1 d2 1 þ x 2 q2 ; 2 dq2 2

ð2Þ

whereas in the excited state one has HM e ðqÞ ¼ 

1 d2 1 2 þ x2 ðq  qe Þ þ DEM : 2 dq2 2

ð3Þ

Here, q is a scaled coordinate and x the oscillator frequency which is assumed to be equal in both electronic states. Upon excitation, the equilibrium distance is displaced by the value qe and the potential is shifted in energy by DEM. The vibrational mode taken into account within the model, shows as a progression in the monomer spectrum (see below) and corresponds to a breathing motion of the perylene skeleton [28]. 2.2. Dimer Hamiltonian

Fig. 1. Molecular structure of the PBI chromophore in general (left) and definition of the imide substituent R (right) in the PBI 1 derivative whose aggregates have been studied in Ref. [27]. The lower part of the figure illustrates the geometrical parameters of the dimer system where the single monomers are sketched as rectangulars. Here, b is the angle between the monomer transition dipole-moments ~ d n , and an denotes the angle between the monomer–monomer centre-of-mass vector ~ R and ~ dn.

It is not necessary to outline the steps leading to the approximate Hamiltonian for the molecular dimer, see e.g., Refs. [6,9]. Briefly, within the single exciton model (as adopted here), the possibility to excite one or the other monomer unit is considered, taking a coupling between the respective two excited states into account. Within the model, the dimer is to be regarded as a single structural unit and effects resulting from higher order aggregation phenomena are naturally not taken into account.

356

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

The ground state Hamiltonian is of the form M H Dg ðq1 ; q2 Þ ¼ jgð1Þ; gð2ÞiðH M g ðq1 Þ þ H g ðq2 ÞÞhgð2Þ; gð1Þj;

ð4Þ where the indices n = 1, 2 refer to the two monomer units. In the electronic ground state, the dimer is bound together only by the weak van der Waals interaction which here, as usual, is ignored in the theoretical description. The zeroth-order Hamiltonian for the two electronically excited states is H ðD;0Þ ðq1 ; q2 Þ ¼ jeð1Þ; gð2ÞiðH De1 ðq1 ; q2 Þhgð2Þ; eð1Þj e þ jgð1Þ; eð2ÞiH De2 ðq1 ; q2 ÞÞheð2Þ; gð1Þj;

ð5Þ

where the definition M H De1 ðq1 ; q2 Þ ¼H M e ðq1 Þ þ H g ðq2 Þ þ ðDED  DEM Þ;

ð6Þ

H De2 ðq1 ; q2 Þ

ð7Þ

¼H M g ðq1 Þ

þ

HM e ðq2 Þ

þ ðDED  DEM Þ;

for the nuclear Hamiltonians is employed. In bringing the monomer units together, a perturbation caused by the electrostatic interactions of the monomer electron clouds occurs. Here, we encounter the situation where the coupling arises between two configuration corresponding to a localized electronic excitation on one or the other monomer. The respective coupling element is thus much stronger than in the ‘closed shell’ ground state configuration. Keeping only the leading terms in a multi-pole expansion, this leads to the electronic coupling matrix elements [29] ~ ð~ R~ d 1 Þð~ R~ d 2Þ d 1~ d2 3 : ð8Þ 5 3 R R Here, ~ R denotes the vector connecting the center-of-mass of one monomer with that of the other. The transition dipolemoments are defined as J ð~ RÞ ¼

d~n ¼ hgð2Þeð1Þj~ ln jgð1Þeð2Þi;

ð9Þ

where ~ ln is the dipole operator of monomer (n). Introducing the angle b between the dipole vectors and the angles an, which describe the orientation of the d~n with respect to the vector ~ R (see the lower part of Fig. 1), the coupling element reads J ðR; b; a1 ; a2 Þ ¼

d2 ½cosðbÞ  3 cosða1 Þ cosða2 Þ; R3

ð10Þ

where we have used the fact that j~ d 1 j ¼ j~ d 2 j ¼ d and R is the monomer–monomer separation. Thus, the coupling is strongly geometry dependent and we will explore this aspect in detail below. The absolute value of the transition dipole-moment is estimated from the absorption lineshape as d = 8.5 D. At this point it should be noted that the assumptions leading to Eq. (10) are often not fulfilled, see e.g., the early paper of Murrell and Tanaka [30]. Better approximations to the interaction can be formulated such as the extended

dipole model [31], or methods employing multicentre mono-pole or multi-pole expansions, for a discussion on these models see the book of Stone [29]. Having specified the coupling elements, the total excited state Hamiltonian for the nuclear motion can now be written down in matrix form as ! J ðR; b; a1 ; a2 Þ H De1 ðq1 ; q2 Þ D : ð11Þ He ¼ J ðR; b; a1 ; a2 Þ H De2 ðq1 ; q2 Þ As has been noted in Section 1, this Hamiltonian describes a vibronic coupling problem which involves two coupled electronic states each having two vibrational degree of freedoms; for a discussion regarding the topology of the excited state potential surfaces see Ref. [32]. 2.3. Time-dependent expression for the spectra Absorption spectra are calculated from the time-dependent version of Fermi’s Golden rule expression as [33,34] Z 1  iðEg þEÞt= h rab ðEÞ  2R dt e cab ðtÞwðtÞ ; ð12Þ 0

where R denotes the real part, and the time-correlation function is defined as cab ðtÞ ¼ ~ hwg~ dwg i~ : djU e ðtÞj~

ð13Þ

In the above equation, E is the photon energy and ~  the polarization vector of the electric field. A phenomenological damping function w(t) is introduced to account for line broadening caused by interaction with the surroundings [35]. Within the theoretical description, a single parameter is used to account for line broadening mechanisms. Naturally, the origin of the physical process is not identified. The resolution of the spectrum is determined by the temporal width of the window function w(t) which is used to truncate the time-correlation function. In our calculation, we employed a Gaussian window function. The full width at half maximum of the Gaussian in the time domain DT and in the energy domain DE are related via a Fourier relation as DT = 8 ln2h/DE. The initial ground state of energy Eg is denoted as jwgi, and ~ d is the transition dipole-moment between the electronic states under consideration. Finally, Ue = exp(iHet/h) is the propagator for the nuclear motion in the excited state. In the case of emission from an excited initial state jwei of energy Ee the spectrum is calculated as Z 1  iðEe EÞt= h rem ðEÞ  2R dt e cem ðtÞwðtÞ ; ð14Þ 0

where the correlation function contains the propagator Ug(t) in the electronic ground state: cem ðtÞ ¼ ~ hwe~ dwe i~ : djU g ðtÞj~

ð15Þ

The general expressions of this section will now be specialized for the case of monomer and dimer absorption and emission, respectively.

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

2.4. Monomer spectra Assuming the Condon approximation (coordinate independent transition dipole-functions), the correlation functions in the monomer case are of the simple form cab ðtÞ ¼~ hwg~ dwg i~  ¼ d 2 hwg jU e ðtÞjwg i; djU e ðtÞj~ cem ðtÞ ¼~ hw ~ dw i~  ¼ d 2 hw jU g ðtÞjw i; djU g ðtÞj~ e

e



e

e

ð16Þ ð17Þ

where d is the projection of the transition dipole-moment on the laser polarization vector ~ . Although the correlation functions can be evaluated analytically for the case of shifted harmonic potentials [36], below they are determined numerically. 2.5. Dimer spectra The dimer time-correlation function for the absorption process (Eq. (13)) is evaluated in what follows. Therefore, the two transition moments have to be added yielding the expression ( ) 2 X cab ðtÞ ¼ ~  hwg~ d m wg i ~ : ð18Þ d n jU e ðtÞj~

357

At this point, we emphasize that the angle b not only enters into the electronic coupling element but also in the initial condition of the wave function to be propagated in the coupled excited electronic states. For the emission process, the correlation function is calculated as described above. Starting from the symmetric ground state in the electronically excited manifold jwe i ¼ we;0 j1i þ we;0 j2i;

ð23Þ

one finds cem ðtÞ ¼ 2ð1 þ cosðbÞÞhwe;0 d 1 jU g ðtÞjd 1 we;0 i:

ð24Þ

Note that the expressions Eqs. (22) and (24) differ in its structure because the monomer–monomer coupling is neglected in the ground electronic state. Numerically, the time-propagation is carried out with the split operator method [37], where the propagator containing the off-diagonal potential matrix is approximated as is described in Ref. [38]. 3. Results 3.1. Monomer spectra

ðn;mÞ¼1

Here, one has to keep in mind that the propagator U e ðtÞ ¼ expðiHDe t= hÞ is a matrix operator and the wave functions have two components so that ! ~1 d1m d ~ : ð19Þ d m wg ¼ wg d~2 d2m We now place the dimer in the (x,z)-plane, with the x-axis bisecting the angle b between the dipole vectors ~ d 1 and ~ d 2 . In the case of x-polarized light, where ~ ~ d n ¼ d n cosðb=2Þ one finds, taking advantage of the symmetry of the problem,   2 X 2 b cx ðtÞ ¼ cos hwg d n jU e ðtÞjd m wg i 2 ðn;mÞ¼1   2 b ¼2 cos hwg d 1 jU e ðtÞjd 1 wg þ d 2 wg i: ð20Þ 2 The meaning of dnwg is the same as defined in Eq. (19), with the vector ~ d n replaced by its modulus. Likewise, in the case of an y-polarized field, the correlation function reads   2 X 2 b ðnþmÞ cy ðtÞ ¼ sin ð1Þ hwg d n jU e ðtÞjd m wg i 2 ðn;mÞ¼1   2 b ¼2 sin hwg d 1 jU e ðtÞjd 1 wg  d 2 wg i: ð21Þ 2 Regarding the case of unpolarized light ð~  ¼~ x þ~ y Þ, the correlation function, upon using addition theorems for the trigonometric functions, reads cab ðtÞ ¼ cx ðtÞ þ cy ðtÞ ¼ 2fhwg d 1 jU e ðtÞjd 1 wg þ cosðbÞd 2 wg ig:

ð22Þ

Within the applied model, the dimer properties are constructed from those of the monomer. Here, we start with the absorption data recorded for the PBI 1 molecule. Thus, by comparison to experiment, the parameters x, qe and DEM, entering into the monomer Hamiltonian, are determined. The monomer spectra are measured at a very low concentration of 2.1 · 107 M. Out thermodynamic studies indicate that in such low concentration, any form of aggregated species is negligible [27,39]. The measured spectrum is compared to the calculated one in Fig. 2, upper panel. In the calculation, the characteristic oscillator energy is fixed to a value of Ex = hx = 0.175 eV. For the excited state, a displacement of qe = 2.57 (eV)1/2 and an energy shift of DEM = 2.3975 eV is used. In calculating the spectrum, a Gaussian damping function w(t) is employed yielding a resolution of Dx = 0.11 eV. The absorption spectra agree very well, and the same holds for the emission spectra displayed in the lower panel of the figure. 3.2. Dimer spectra Having established the monomer parameters, we proceed and calculate dimer spectra under the assumption that the point-dipole approximation is valid, i.e., the electronic coupling element is taken from Eq. (10). First, we study the influence of the relative orientation of the transition dipolemoments on the spectrum by varying the angular variable b, see also Ref. [10]. Therefore, the monomer separation is ˚ . Additionally, it is kept fixed at a value of R = 3.2 A ~ ~ assumed that the vectors d n and R are orthogonal, i.e., an = 90. The energy shift DED is set equal to DEM. In what follows, low-resolution spectra are discussed (spectral

358

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

o

β = 90

absorption exp theo

o

emission

absorption [arb. units]

β = 80

monomer

o

β = 60

o

β = 40

1.6

2

2.4

2.8

3.2

E [eV]

1.5

2

2.5 E [eV]

3

3.5

Fig. 2. Upper panel: absorption spectra of the PBI 1 monomer. Experimental and theoretical spectra are compared. The calculated spectrum is smeared out with a resolution of 0.11 eV. The lower panel shows the measured and calculated emission spectra. The measured spectra are recorded at a concentration of 2.1 · 107 M in methylcyclohexane.

Fig. 3. Dimer absorption: dependence of the spectra on the angle b, which describes the relative orientation of the monomer transition dipole moments. Here, low-resolution spectra are calculated which do not exhibit the vibrational progression.

resolution of 0.22 eV), for details on the vibrational structure, see below. Spectra for different values of b are shown in Fig. 3, as indicated. For perpendicular transition dipole moments (upper panel), the coupling vanishes and one obtains the low-resolution monomer spectrum. With decreasing angle, the coupling element increases, leading to the well-known exciton (Davydov) splitting D of the single band into two. For two degenerate levels, this splitting is just D = 2jJj. The two bands originate from the interaction with the x- and y-polarized light leading to the respective correlation functions. Because the Fourier-transform is linear two separate spectra can be calculated which combine to the total spectrum. In fact, these two spectra belong to absorption into states of different permutation symmetry [14]. The orientation angle not only enters into the coupling element but also into the initial wave function. This influences the strengths of the two bands, as can be taken from the figure. The relative band intensity is given by the ratio (cos(b/2)/sin(b/2))2. The trends documented in Fig. 2 have also been discussed in a recent paper by Hager et al. [40]. Note, that one obtains identical spectra if the angle b is replaced by b + 90. In this case, the coupling element becomes negative, but also the second component of the initial wave function is multiplied with a phase factor of (1). Both operations result in a unitary transformation of the two-component wave function, thus leaving the correlation function invariant.

We next turn to the influence of the angles an which describe the orientation of the transition dipole-moments relative to the vector ~ R. There are two distinct cases to be considered [41]. The first one is that of equal angles (a1 = a2) where the second term in Eq. (10) becomes negative. The other possibility is characterized by a2 = 180  a1 which leads to a stronger coupling. Let us illustrate this behavior for a fixed value of b = 60. In Fig. 4, we show curves for various pairs of angles (a1, a2), as indicated. For equal angles at 80, one obtains a spectrum similar to the one depicted in Fig. 3. Decreasing the angle to a1 = a2 = 60 exchanges the position of the low and high intensity bands which stems from a sign change in the coupling matrix element. Increasing the coupling even more (40), results in two well separated bands. On the other hand, regarding the cases with a2 = 180  a1, the coupling is always positive and the energy separation of the two bands increases with decreasing angle a1. In order to illustrate the sensitivity of the dimer spectrum with respect to the monomer separation R, calculations with different values of this parameter are performed. Therefore, we use fixed angles b = 60 and an = 90. The resulting spectra are contained in Fig. 5. As is to be expected, with increasing values of the separation R, the sub-bands merge. The splitting is rather sensitive to the distance because the latter enters to the third power in the denominator of the electronic coupling element. The trends which are documented in Figs. 2–4, will help to find

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

o

o

o

absorption [arb. units]

(80 ,80 )

(80 ,100 )

(60o,60o)

(60o,120o)

(40o,40o)

0

1

3

2

o

4

(40o,140o)

0

E [eV]

1

3

2

4

5

E [eV]

Fig. 4. Dimer absorption: low-resolution spectra are shown for different pairs of angles (a1, a2) which characterize the orientation of the two transition dipole-moments with respect to the vector connecting the center of mass of the two monomers.

359

proper parameters in fitting an experimental spectrum, see Section 3.3. There is an additional issue to be discussed, which is the vibrational fine-structure eventually seen in a measured aggregate spectrum [10]. Within our calculation, the resolution of the absorption spectrum can be influenced arbitrarily (see Eq. (12)): for a fast decaying window function w(t), the spectral features are – according to the Fourierrelation – washed out. On the other hand, for a slowly decaying window function the vibrational structure shows up in the spectrum, for a discussion see, e.g., the book of Schinke [42]. Let us, in what follows, assume that line broadening mechanisms are less effective so that there is a vibrational structure present in the absorption spectrum. In that case, it is clear that, for a large enough coupling where the two dimer bands are well separated, these bands themselves will exhibit the vibrational progression. A more interesting case is present when the bands overlap and the energy gap becomes comparable to the vibrational energy spacing. This situation is illustrated in Fig. 6, which shows spectra calculated with different resolutions, as indicated. ˚ , b = 60, and an = 70 enter Here, parameters of R = 3.2 A into the calculation. This results in a coupling element of about J = 0.1 eV so that the splitting D = 2jJj is comparable to the vibrational spacing of 0.175 eV. Whereas for the lowest resolution (250 meV), two diffuse bands can be recognized, already at 126 meV three main peaks appear, and it is not clear where the structures might arise from. With increasing resolution (63 and 84 meV), the fine structure

84 meV

126 meV

4.5 Å

absorption [arb. units]

absorption [arb. units]

4.0 Å

3.5 Å

3.0 Å

1.8

2.2

2.6

3

3.4

E [eV] Fig. 5. Dimer absorption spectra are displayed for various monomer– monomer separations, as indicated. As in the calculations leading to the results in Figs. 3 and 4, low-resolution spectra are presented.

250 meV

2

2.2

63 meV

2.4

2.6

E [eV]

2.8

2

2.2

2.4

2.6

2.8

3

E [eV]

Fig. 6. Dimer absorption spectra, determined for different spectral resolution, as indicated. In increasing the resolution, the vibrational structure of the two electronic bands becomes apparent.

360

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

of the spectrum becomes detectable, where for infinite resolution a complicated stick spectrum is obtained (not shown). The figure documents that here it is difficult to directly extract the electronic coupling strength because the structures can be caused by an electronic coupling and/or by vibrational motion. 3.3. Comparison to experiment We now turn to a comparison with experiment. Before doing so, it is worth to keep in mind that not only dimerization but also the formation of higher aggregated species was observed in the experiment. It is, nevertheless, useful to compare the dimer spectra with the measured one. The latter is displayed in the upper panel of Fig. 7. It shows a diffuse structure with a main maximum occurring around 2.5 eV, and a smaller peak around 2.3 eV. Within the exciton model, the dimer spectrum is rationalized as stemming from the splitting D of a single band, where D is twice the strength of the electronic coupling element. Furthermore, the relative intensity of the two bands is determined by the angle b defined by the orientation of the monomer transition dipole-moments. From Fig. 3 it is clear that, because of the band intensity ratio, the dipole orientation angle b is in the vicinity of b = 60. However, for the values of ˚ and an = 90 which were employed in the calcuR = 3.2 A lation, the splitting is much too large. In order to reduce the gap one would either have to decrease the angles an

aggregate

absorption [arb. units]

exp

dimer

theo

2

2.4

2.8

3.2

E [eV] Fig. 7. Upper panel: measured absorption spectrum at 2.2 · 104 M in methylcyclohexane. The lower panel contains the calculated dimer spectrum. The resolution of the calculated spectrum was taken to be 196 meV.

in the case of equal orientation (a1 = a2), as can be taken from Fig. 4, or dramatically increase the monomer separa˚ , see Fig. 5 (of course the simultion to values above 4.5 A taneous variation of both parameters is possible as well). ˚ , measured by X-ray diffracTaking the distance of 3.5 A tion on crystals into account (see [27], and references therein), the latter value seems to be too large although deviations are to be expected in solution. Therefore, it is not unreasonable to assume that a distortion from the configuration where the dipole moments are orthogonal to the center of mass vector is present. We determine the dimer ˚ , employing values of spectrum for the distance R = 3.5 A b = 55 and an = 70. Fixing the energy shift DED to a value of 2.315 eV leads to the spectrum displayed in the lower panel of Fig. 7, where a resolution of 196 meV is used. A comparison to the measured spectrum shows an excellent (although not perfect) overall agreement: the width of the spectrum, the band positions and also the relative intensities are correct. The experimental spectrum exhibits some additional fine structure, which is not present in the calculated curve. Note, that the found geometry is of course not unique because the monomer separation is taken as fixed in its determination. It is worth to be mentioned that, even in case the explicit form of the coupling (Eq. (10)) is not taken into account and J is taken as a constant parameter to be adjusted in the fitting procedure, the angle b still determines the band intensity ratio so that the dimer spectrum can be used to gain insight into the geometric properties of the system. The agreement of theory and experiment suggests that the excitonic molecular dimer model is able to describe the absorption properties of the PBI 1 aggregates. Thus, the same should be valid in the case of emission. Therefore, the emission spectrum is calculated originating in the ground vibronic state of the dimer excited states. Fig. 8 compares calculated (middle panel) and measured dimer spectra (upper panel). Also shown is the measured monomer spectrum, for comparison (lower panel). The experimental curve, taken at a concentration of 3 · 105 M, contains contributions from the monomer (at higher energies) but differs remarkably from the monomer emission at lower energies. In particular, a broad band dropping only at energies below 1.8 eV is seen. The calculated emission spectrum does not show this band and thus the dimer exciton model cannot account for emission resulting in lower energy photons. This discrepancy could mean that the dimer unit is not responsible for the characteristics of the spectrum so that aggregates containing more monomer units have to be considered. We performed preliminary calculations on trimer systems with model parameters similar to the here established ones and found no considerable changes in neither absorption nor emission properties. This hints at a more complex excited state electronic structure. In particular, one could imagine another excited state which (because of Franck– Condon arguments) is not seen in the absorption but is responsible for the extraordinary emission band. Here,

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

Preliminary calculations indicate that this band does not belong to aggregates built by a larger number of monomers. An extended study of these size effects will be presented elsewhere [39]. In the future, we are aiming at a quantum chemical characterization of the electronically excited states in order to find a better model to describe the linear spectroscopy of PBI 1 nanoaggregates.

experiment (aggregate)

emission [arb. units]

361

theory

Acknowledgement

(dimer)

Financial support by the Deutsche Forschungsgemeinschaft within the Graduiertenkolleg 1221 is gratefully acknowledged. experiment

References

(monomer)

1.3

1.5

1.7

1.9

2.1

2.3

2.5

E [eV] Fig. 8. Upper panel: measured emission spectrum recorded at a concentration of 3 · 105 M in methylcyclohexane. The calculated dimer spectrum is shown in the middle panel. For comparison, the monomer spectrum is included in the lower panel.

quantum chemical input is required to establish a more realistic model. To conclude, we have theoretically investigated the influence of several geometrical parameters on the absorption spectrum of a molecular dimer treated within a simple excitonic model including a single vibrational coordinate for each monomer. It is shown that, under the assumption of a point-dipole interaction between the monomers, the geometrical properties of the system can be well characterized although the parameterization is not unique. If, however, a reasonable guess for the monomer–monomer separation can be found, the three orientational angles belonging to the transition dipole/transition dipole orientation and to the center of mass vector/transition dipole vector angles can be determined accurately. However, the point-dipole approximation might not be valid. With no detailed information on the coupling elements available, one might then assume a constant coupling parameter J and still obtain information about the relative orientation of the monomer transition dipole-moments via the relative intensity of the two absorption bands. The dimer-model is applied to recently investigated perylene bisimide aggregates. Constructing an effective Hamiltonian from the monomer data, dimer spectra are calculated. Whereas the absorption spectrum is in good agreement with experiment, the identical model yields an emission spectrum, which cannot account for the measured broad band occurring at much smaller photon energies.

[1] W. Domcke, D.R. Yarkony, H. Ko¨ppel (Eds.), Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, World Scientific, Singapore, 2004. [2] M. Born, K. Huang, Theory of Crystal Lattices, Oxford University Press, London, 1954. [3] G.A. Worth, L.S. Cederbaum, Ann. Rev. Phys. Chem. 55 (2004) 127. [4] G. Stock, W. Domcke, Adv. Chem. Phys. 100 (1997) 1. [5] M. Kasha, H.R. Rawls, M.A. El-Bayoumi, Pure App. Chem. 11 (1965) 371. [6] V. May, O. Ku¨hn, Charge and Energy Transfer Dynamics in Molecular Systems, Wiley-VCH, Berlin, 2000. [7] A. Witkowski, W. Moffitt, J. Chem. Phys. 33 (1960) 872. [8] R.E. Merrifield, Radiat. Res. 20 (1960) 154. [9] R.L. Fulton, M. Goutermann, J. Chem. Phys. 41 (1964) 2280. [10] A.R. Gregory, W.H. Henneker, W. Siebrand, M.Z. Zgierski, J. Chem. Phys. 63 (1975) 5475. [11] B. Kopainsky, J.K. Hallermeier, W. Kaiser, Chem. Phys. Lett. 83 (1981) 498. [12] R. Friesner, R. Silbey, J. Chem. Phys. 75 (1981) 5630. [13] W. Leng, F. Wu¨rthner, A.M. Kelley, J. Phys. Chem. B 108 (2004) 10284. [14] A. Eisfeld, L. Braun, W.T. Strunz, J.S. Briggs, J. Beck, V. Engel, J. Chem. Phys. 122 (2005) 134103. [15] J.S. Briggs, A. Herzenberg, Mol. Phys. 21 (1971) 865. [16] F.C. Spano, J. Chem. Phys. 114 (2001) 5376. [17] A. Eisfeld, J.S. Briggs, Chem. Phys., in press. [18] H. Langhals, Heterocycles 40 (1995) 477. [19] W. Herbst, K. Hunger, Industrial Organic Pigments, third ed., WileyVCH, Weinheim, 2004. [20] P.M. Kazmaier, R. Hoffmann, J. Am. Chem. Soc. 116 (1994) 9684. [21] C.W. Tang, Appl. Phys. Lett. 48 (1986) 183. [22] P. Peumans, S. Uchida, S.R. Forrest, Nature 425 (2003) 158. [23] F. Yang, M. Shtein, S.R. Forrest, Nat. Mater. 4 (2005) 37. [24] F. Wu¨rthner, Chem. Commun. (2004) 1564. [25] J. Cornil, D. Beljonne, J.-P. Calber, J.-L. Bredas, Adv. Mater. 13 (2001) 1053. [26] F. Wu¨rthner (Ed.), Supramolecular Dye Chemistry, Topics in Current Chemistry, vol. 258, Springer, New York, 2005. [27] F. Wu¨rthner, Z. Chen, V. Dehm, V. Stepanenko, Chem. Commun. (2006) 1188. [28] M. Sadrai, L. Hadel, R.R. Sauers, S. Husain, K. Krogh-Jespersen, J.D. Westbrook, G.R. Bird, J. Chem. Phys. 96 (1992) 7988. [29] A.J. Stone, The Theory of Intermolecular Forces, Cambridge University Press, Oxford, 1996. [30] J.N. Murrell, J. Tanaka, Mol. Phys. 7 (1964) 634. [31] V. Czikklely, H.D. Fo¨rsterling, H. Kuhn, Chem. Phys. Lett. 6 (1970) 207.

362

J. Seibt et al. / Chemical Physics 328 (2006) 354–362

[32] W.J.D. Beenken, M. Dahlbom, P. Kjellberg, T. Pullerits, J. Chem. Phys. 117 (2002) 5810. [33] R.G. Gordon, Adv. Mag. Reson. 3 (1968) 1. [34] E.J. Heller, Acc. Chem. Res. 14 (1981) 368. [35] V. Engel, R. Schinke, S. Hennig, H. Metiu, J. Chem. Phys. 92 (1990) 1. [36] E.S. Medvedev, V.I. Osherov, Radiationless Transitions in Polyatomic Molecules, Springer, Berlin, 1995. [37] M.D. Feit, J.A. Fleck, A. Steiger, J. Comput. Phys. 47 (1982) 412. [38] J. Alvarellos, H. Metiu, J. Chem. Phys. 88 (1988) 4957.

[39] Z. Chen, V. Stepanenko, V. Dehm, P. Prins, L. Siebbeles, J. Seibt, P. Marquetand, V. Engel, F. Wu¨rthner, Chem. Eur. J., submitted. [40] A. Hager, V. de Halleux, A. Ko¨hler, A. El-Garoughy, E. Meijer, J. Barbera´, J. Tant, J. Levin, M. Lehmann, J. Gierschner, J. Cornil, Y.H. Geerts, J. Phys. Chem. B 110 (2006) 7653. [41] E.G. McRea, M. Kasha, in: L. Augenstein, B. Rosenberg, S.F. Mason (Eds.), Physical Processes in Radiation Biology, Academic Press, New York, 1963, p. 23. [42] R. Schinke, Photodissociation Dynamics, Cambridge University Press, Cambridge, 1993.