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Excited-state absorption of conjugated polymers in the near-infrared and visible
Ling, Sanliang; Schumacher, Stefan; Galbraith, Ian; Paterson, Martin J.
Published in:
Journal of Physical Chemistry C
DOI:
10.1021/jp401359a
Publication date:
2013
Document Version
Accepted author manuscript
Link to publication in Heriot-Watt University Research Gateway
Citation for published version (APA):
Ling, S., Schumacher, S., Galbraith, I., & Paterson, M. J. (2013). Excited-state absorption of conjugated
polymers in the near-infrared and visible: a computational study of oligofluorenes. Journal of Physical Chemistry
C, 117(13), 6889-6895. DOI: 10.1021/jp401359a
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Excited-State Absorption of Conjugated Polymers in the NearInfrared and Visible: A Computational Study of Oligofluorenes
Sanliang Ling1,2, Stefan Schumacher3, Ian Galbraith,4 Martin J. Paterson1
1
Institute of Chemical Sciences, School of Engineering and Physical Sciences, Heriot-Watt
University, Edinburgh EH14 4AS, UK.
2
Department of Physics and Astronomy, University College London, Gower Street, London
WC1E 6BT, UK.
3
4
Department Physik and Center for Optoelectronics and Photonics Paderborn (CeOPP),
Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany.
Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, UK.
Abstract
Excited-state properties of conjugated polymers play a central role in applications ranging from
organics-based photovoltaics to nonlinear photonics. From a theoretical and computational point
of view however, an accurate first-principles description poses a formidable task. Typical
molecule sizes go well beyond the size limits for which highly-reliable wavefunction based
electronic-structure methods can be applied. In the present work we demonstrate, that the
excited-state absorption (ESA) in large conjugated molecules, computed with non-linearresponse density-functional theory can be used to accurately model this process in an important
class of conjugated materials. We compute transitions between up to 100 excited states for
fluorene oligomers containing up to about 100 conjugated atoms. Furthermore we demonstrate
that this approach can explain the nature of near-infrared and visible excited absorption bands,
which play leading roles of such materials in device physics. These systems are large enough that
we approach the polymer limit in terms of electronic properties of excited states. The results
obtained are in good agreement with available experimental data.
1
Table of Contents Figure
Keywords: conjugated polymers, excited-state absorption, oligofluorenes, nonlinear optical
properties, time-dependent density functional theory
2
1. Introduction
Organic semiconductors show great potential for optoelectronic applications. For example, it
has been demonstrated that such conjugated polymers can be used in organic solar cells,1 lasers,2
light-emitting diodes,3 optical amplifiers,4 and chemical sensors.5 Excited states play a central
role in the operation of these devices, either directly, or indirectly through various loss channels.
Moreover, optoelectronic devices are driven devices and so operate far from equilibrium where
transitions amongst excited states are integral to the device operation. A number of detrimental
processes involve higher lying excited states and their interactions, such as exciton-exciton
annihilation,6 exciton-polaron quenching,7 electric-field induced ionization of excitons,8 and reabsorption of emitted light by polarons.9 A better understanding of the excited states and their
interactions with charge carriers and light fields will therefore directly inform mitigation
strategies enabling systematic design of new conjugated polymer devices and materials with
improved performance.
To improve performance of conjugated polymers in these applications, one of the most
important tasks is to understand the underlying electronic structures at both molecular and
polymeric levels. The latter is often achieved through an extrapolated procedure based on studies
of simpler molecular oligomers.10 The oligomer approach has been quite successful in
elucidating the electronic structures of conjugated polymers, and many experimental and
theoretical studies have been reported.10-14 However, with increasing number of repeat units, the
size of the molecular oligomers becomes extremely large, and a comprehensive theoretical
description of the excited states becomes increasingly challenging. Therefore, many existing
studies stop at a point where only the linear optical properties with respect to the electronic
ground state are included and even semi-empirical methods are still being used to reduce
3
numerical complexity.15-18 Even fewer studies have reported on nonlinear optical properties of
conjugated polymers,19 in which quite often sum-over states approaches were employed.20-24
Polyfluorenes are one of the prototypical semiconducting conjugated polymers which show
promising properties for applications in optoelectronic devices. For example, it has been shown
that polyfluorenes are efficient blue emitters with high photoluminescence quantum
efficiencies.25 Moreover, one can tune the emission wavelength over the entire visible range
through chemical modifications of main or side chains.26 Oligofluorenes have been a highly
investigated model system in ab initio studies of the photophysics of conjugated polymers. In a
previous study,27 we have investigated optical transitions between the ground and lowest energy
singlet excited state in fluorene oligomers using both experiment and theoretical simulations. In
the current contribution, we extend our calculations to dipole-allowed optical transitions which
occur between the lowest and higher-lying singlet excited states of fluorene oligomers. As
mentioned above, such transitions are important to understand some of the dominant loss
channels for optoelectronic devices constructed from such materials.
While the transition energies and transition dipoles for one-photon absorption (OPA) from the
ground state are readily obtained as the poles and residues of linear response functions for a
given quantum chemical model, calculations of the transition dipole moments for OPA from
excited states (ESA) require a higher computational effort, and are obtained as the double residue
of the quadratic response function with dipole operators. With the recent developments in both
computational software and hardware, first-principles ESA calculations for medium sized
molecules (~100 conjugated atoms) have become feasible. These theoretical calculations can
provide additional details, e.g. the nature of the excited states involved in the excited state
absorption, which cannot be directly gleaned from experiment. In the current study, we take
4
fluorene oligomers as an example and show that nonlinear response time dependent density
functional theory (TD-DFT) can successfully be applied to even relatively large systems,
approaching the polymer limit in terms of electronic and optical properties.
In experiments, excited state absorption can often not be easily separated from other photoinduced signatures which contribute in the same spectral range, e.g., polaron and triplet
absorption.9, 28, 29 In our calculations, the excited-state absorption alone is obtained over a wide
spectral range. We believe that such calculations will be a very valuable tool to more reliably
disentangle photoinduced spectral signatures of different origin in future studies.
Figure 1: (a) molecular structure of fluorene heptamer and schematic illustration of (b) onephoton transitions and (c) excited-state absorption. Subscripts in (b) and (c) refer to state
numbers for fluorene heptamer. (d) Molecular-orbital characteristics of selected higher-lying
excited states that are predominantly involved in the excited-state absorption from S1 in fluorene
heptamer. We find that these excited states S6, S8, S81 and S82 have strongly mixed excitation
character with different orbital contributions as indicated.
2. Methodology
5
Theoretical calculations at the density functional theory level have been performed for a set of
fluorene oligomers, including the fluorene dimer (F2), trimer (F3), tetramer (F4), pentamer (F5),
hexamer (F6) and heptamer (for F7, see Figure 1), for molecular geometries in both the α-phase
(twisted backbone)30 and β-phase (planar backbone)31, 32 conformations. While an alternating
structure was considered for all oligomers in the α-phase, we also considered a helical structure
for several oligomers, including F3, F4 and F5, to investigate how the calculated excited state
absorption spectra are affected by oligomer conformation.
The ground state geometries were optimized with the Coulomb-attenuated B3LYP (CAMB3LYP) exchange-correlation functional, and all other calculations involving excited states were
based on time-dependent DFT, via the response function formulation.33 We calculate excitation
energies and transition dipoles with CAM-B3LYP, which was designed to better describe the
long-range contributions to the electronic exchange interaction and give a more accurate
description for charge-transfer contributions to electronic excitations. While B3LYP shows good
agreement with experiments in predicting linear optical properties,34 its performance in
prediction of non-linear optical properties has been criticized.35 CAM-B3LYP has been used
successfully in the predication of two-photon absorption, which is obtained as the single residue
of the quadratic response function. Indeed it was found that CAM-B3LYP accuracy could
approach that of coupled cluster theory for non-linear optical properties in small systems,35 and
has been applied to a diverse range of large conjugated systems.36-38 As we show below, the use
of the CAM-B3LYP functional is crucial in finding ESA spectra in agreement with experiments.
To obtain further insight on how geometry relaxation affects excited states, we computed ESA
spectra of both the ground (S0) and optically active lowest excited (S1) state geometries, obtained
using analytical CAM-B3LYP gradient optimization on the excited surface. Based on our
6
previous studies which were performed for fluorene oligomers27 and star-shaped fluorene
molecules,34 we used the 6-31G basis set throughout the present work. We also tested 3-21G and
6-31G* basis sets for F2 and F3, and we found positions of the first peak in ESA spectra differ
by less than 0.02 eV. Such relatively minor sensitivity to one-electron basis set has been
observed previously in the TD-DFT calculation of non-linear optical properties.37
The calculated excitation energies and transition dipoles were used to calculate the oscillator
strengths for electronic transitions from both the ground state (OPA) and optically active lowest
excited state (ESA) to higher lying excited states (cf. Figure 1). A homogeneous broadening is
included in all the spectra plotted, and is chosen to be 0.2 eV. The main tool we use to visualize
the character of electronic transitions is the introduction of natural transition orbitals (NTOs).39
The NTOs cast each electronic transition into a minimum number of pairs of effective singleparticle orbitals. Ideally, a reduction to only one relevant pair of NTOs can be achieved for each
transition. If correlations play a significant role, however, a larger number of NTOs remains to
be analyzed. A number of the more commonly used canonical Kohn-Sham molecular orbitals are
additionally visualised in Supporting Information. All geometry optimizations and linear
response calculations were performed with the Gaussian 09 package.40 All ESA calculations
were performed with a parallel version of the Dalton 2011 package.41
3. Results and Discussions
3.1 Excited state absorption spectra
In one of our previous studies, we have calculated the OPA spectra of a series of fluorene
oligomers at B3LYP/6-31G level, and excellent agreement with experiment was obtained for the
vertical transition energies. In the present study, we focus on excited state absorption.
7
Figure 2: Calculated excited-state absorption spectra for fluorene oligomers with (a) S0 and (b)
S1 (right panel) geometries. Both broadened (black curves, left vertical axis)) and stick (red
sticks, right vertical axis) spectra are shown. The same scales are used for absorptivity (arbitrary
units) of broadened spectra and oscillator strength (atomic units) of stick spectra.
We have calculated the ESA spectra for the series of fluorene dimer to fluorene heptamer. To
begin with, in Figure 2, we concentrate on the low-energy range in the calculated ESA spectra as
this can be directly compared with available experimental data. Results are shown for both S1
and S0 geometries. As experimentally the excited-state absorption is likely to be measured
dominantly in the S1 geometry (as sufficiently fast time resolution in the detection is typically
not obtained), the computed ESA spectra in the S1 geometries ought to be comparable with the
experimental data. The ESA spectra of F2 and F3 were measured by Hayes and Silva in a
transient absorption spectroscopy experiment.42 They observed photoinduced low-energy
absorption peaks attributed to excited-state absorption from the S1 electronic state into higherlying states at 1.87 eV and 1.66 eV, respectively, for F2 and F3. These absorption bands were
assigned to the mAg←1Bu transitions. We find that our calculated excitation energies for the
lowest optically active transition from S1 for the F2 and F3 oligomers are in very good agreement
with these experimental data. The calculated positions of the first peaks in the ESA in Figure 2
are 1.65 eV and 1.45 eV for F2 and F3 oligomers, respectively. Compared with experiment our
calculations underestimate the transition energies by only about 0.2 eV. The redshift between F2
8
and F3 observed experimentally at about 0.2 eV is also well reproduced in our calculations.
Although the availability of further significant experimental data is limited, from this comparison
we conclude that a slight underestimation of the transition energy of the lowest absorption
feature in the calculated ESA is systematic in the calculations. In this context it is important to
note that we found that the use of the very popular B3LYP functional completely fails in
predicting the ESA spectra correctly. It underestimates the positions of the first peaks in the ESA
by more than 1 eV (deviating by about 200% from the experimental values and CAM-B3LYP
results). We note here that the experimental ESA spectra, referred to above, were measured at a
finite temperature, where multiple conformations might coexist for a single oligomer. The ESA
spectra in this energy range are not sensitive to small conformational changes as obtained from
our theoretical calculations, which we will discuss in more detail below. Thus the neglect of
finite temperature effects can be justified.
In Figure 2, we show that the effect of geometry relaxation on ESA is significant for F2; the
energy of the first peak in the ESA spectrum for S0 geometry is redshifted by about 0.2 eV from
the corresponding spectral feature in S1 geometry. For longer oligomers (F3-F7) we find that this
effect is much less pronounced such that the lowest optically active ESA transitions for S0 and S1
geometries almost coincide. For all molecules we only find a very small change in oscillator
strength upon relaxation from S0 to S1 geometry.
In the following we further analyze the nature of the low-energy spectral features in the ESA.
For the F4 oligomer as an example, the first peak in the ESA spectrum in S1 geometry has two
contributions (Fig 2(b) ) belonging to the S1→S5 and S1→S7 transitions. In the discussion below
we focus on the character of S1→S5 as it clearly dominates in terms of oscillator strength. The
nature of the lowest excited state (S1 state) of the oligomer can be easily characterized as it is
9
dominated (>80%) by a single electronic transition from the highest occupied molecular orbital
(HOMO) to the lowest unoccupied molecular orbital (LUMO). In contrast, the electronic nature
of the higher-lying S5 excited state is less simple as multiple electronic contributions are
involved. We find the dominant contributions to be HOMO-1→LUMO and HOMO→LUMO+1
for the S1 to S5 excitation.
Figure 3: Natural transition orbitals for relevant excited states involved in the ESA peak of F4:
(a) HONTO→LUNTO (88.2%) of S1, (b) HONTO→LUNTO (48.3%) of S5, and (c) HONTO1→LUNTO+1 (41.6%) of S5. The percentages indicate relative weight of each set of NTOs.
We find that the dominant S1 to S5 transition cannot be represented by a pure single-particle
electron-hole excitation picture. If we analyze the relevant natural transition orbitals, we need to
take into account both the HONTO→LUNTO and the HONTO-1→LUNTO+1 transitions. These
are depicted in Figure 3 for F4 as an example. In this case for the S1→S5 transition
HONTO→LUNTO and HONTO-1→LUNTO+1 have almost the same weight, namely 48.3%
and 41.6%, respectively. We find that the excitation of S1 has no obvious charge transfer
character (cf. Figure 3a). For S5, however, two complementary charge transfers occur, one
(HONTO→LUNTO, see Figure 3b) from the periphery of the oligomer to the center, and another
(HONTO-1→LUNTO+1, see Figure 3c) from the center of the oligomer to the periphery.
Although the overall transfer of charges in the S1→S5 transition is small as the two contributions
10
partly cancel each other, the charge transfer character clearly visible in the orbital picture of the
high-lying S5 excited state (cf. Figures 3b and 3c) appears to be very relevant. Here we
emphasize again that it is only with the long-range corrected DFT functional CAM-B3LYP that
these excitations are correctly represented well in our calculations and compare well with
experiment. This also explains why the conventional B3LYP fails to give an accurate prediction
on the peak energy in the ESA spectra; B3LYP cannot capture the nature of excited states with
pronounced charge transfer character. Such failure of more conventional DFT functionals in TDDFT calculations of excited state properties has previously been discussed in a number of other
polyaromatic hydrocarbons.43-45 In these systems, two simultaneously
low-lying π→π*
transitions into excited states can exist, one with S1-like HOMO→LUMO excitation character,
commonly denoted as
1
La, and another with S5-like mixed HOMO-1→LUMO and
HOMO→LUMO+1 excitation characters, commonly denoted as 1Lb. It has been suggested that
the 1La state is of “ionic” nature and the 1Lb state is of “covalent” nature.45 Usually the 1La
excitation energy is substantially underestimated and the 1Lb excitation energy is very-well
predicted by TD-DFT using the conventional B3LYP hybrid functional.44 In addition, Wong et al.
argued that CAM-B3LYP is not sufficient to give an accurate description of both states either.45
However, our good agreement with experiment seems to indicate that CAM-B3LYP is indeed
capable of describing both states correctly for the oligofluorenes studied here. A possible
explanation could likely be that in other polyaromatic hydrocarbons such as Wong et al.
discussed previously,45 both states are low-lying excited states and they are not well separated in
energy but only by about 0.2 eV. In our fluorene oligomers, the 1Lb state is a higher-lying excited
state and it is well separated from the 1La state, with an excitation energy of >1 eV.
11
For the other oligomers studied here, upon closer inspection of the dominant transitions that
constitute the low-energy ESA, we find that the nature of the higher-lying excited states
contributing to the ESA is very similar to what we have discussed above in detail for F4, and
most of the relevant excitations are dominated by HOMO-1→LUMO and HOMO→LUMO+1
transitions.
With respect to the oligomer length dependence, we find that there is a small red shift of the
energies of the first peak in the ESA spectra of the S1 geometries with increasing oligomer length.
For the S0 geometries, this shift is less pronounced. Based on the convergence visible for the
peak positions in Figure 2 when moving to longer oligomers, it is reasonable to conclude that for
the ESA we are already approaching the polymer limit with the F7 oligomer. This claim can also
be verified in comparison with experimental ESA data for polyfluorene. The experimental value
of the vertical transition energy of the first peak in the ESA spectra of polyfluorene is about 1.5
eV.46 The computed value for F7 differs by less than 0.2 eV from this value. Considering that we
have found CAM-B3LYP to systematically underestimate the transition energy of the first peak
in the ESA as discussed above by about 0.2 eV, our theoretically calculated transition energy for
F7 is thus in excellent agreement with the experimental value for polyfluorene.46
3.2 Conformational dependence
As mentioned above, in addition to the alternating structure, we also considered a helical
structure for several of the oligomers studied here. For F3 we also considered a helical structure
with C1 symmetry to study the effect of the lowered symmetry on the computed ESA spectra.
We find that all these different molecular conformations give very similar ESA. Small
conformational changes do not seem to significantly influence the ESA spectra in the low-energy
range discussed so far.
12
We further considered the differences between S0 and S1 geometries, in which the dihedral
angles between adjacent fluorene units are significantly different as the oligomers planarize in
the center upon geometry relaxation. While the dihedral angles between adjacent fluorene units
in S0 geometry of all oligomers considered are around 38°, relaxation from S0 to S1 geometry
leads to significant planarization of the oligomers in the center. Such planarization has also been
found in previous experiments on photoexcited oligofluorenes47 and polyfluorene47, 48 as well as
other similar conjugated oligomers.49,
50
We did not find this geometry transformation to
influence the appearance of the ESA in the energy range around 1.5 eV much. In particularly for
the longer oligomers, we find very similar overall ESA in S0 and S1 geometries as shown in
Figure 2. However, for S0 we consistently find that this feature is only related to a single
electronic transition, whereas in the S1 geometry the total oscillator strength gets split up
between one to three electronic transitions depending on the oligomer length. This can be seen in
the stick spectra included in Figure 2. We recall that adjacent fluorene units have very similar
dihedral angles in the S0 geometry, while dihedral angles between central fluorene units are close
to zero in the S1 geometry. Therefore it is reasonable to assume that the S0 geometry only a
single higher-lying excited state is delocalized over the entire oligomer, whereas in the S1
geometry, multiple higher-lying excited states with near degenerate excitation energies
contribute to the overall absorption feature.
13
Figure 4: Theoretical excited state absorption spectra for fluorene oligomers with planar β phase
conformation. The same style as Figure 2 is used.
To conclude the study on conformational changes, we studied the ESA in a series of fluorene
oligomers in the important planar β phase conformation.31, 32 Results are shown in Figure 4. As
in the non-planar fluorene oligomers, for the β-phase we also find a dominant ESA feature at
about 1.5 eV. This is further evidence that conformational effects in fluorenes are not of
particular importance for ESA spectral signatures in the low-energy range around 1.5eV.
3.3 Excited state absorption in the high energy range
We have shown excited state absorption in fluorene oligomers in the low energy range (~1.5
eV). While this energy range can be easily accessed by experiment, excited state absorption in
the higher energy range, e.g. up to 4eV, is also of considerable interest. At high excitation
densities in extended fluorene systems, the spectral density in the ESA at higher energies
contributes to re-absorption of emitted photons and more importantly is a crucial ingredient to
understand efficiency of Auger-type exciton-exciton annihilation processes, a major loss
mechanism in, e.g., optical amplifiers. To access the high-energy range a large number of excited
14
states has to be included in the calculations which greatly increase the numerical cost in
particular for the longer oligomers studied.
In Figure 5 we present the ESA spectra calculated for a greatly extended spectral range for F4,
F5, and F7. For the heptamer 60 states with symmetry element A' needed to be included to cover
the spectral range shown, due to the fact that the density of excited states is much higher in F7
compared with those in small fluorene oligomers. Clearly visible is a pronounced ESA feature at
energies of about 3.3eV.
Figure 5: Calculated excited-state absorption spectra in the energy range of 0.5~4.0 eV. Results
are shown for F4, F5, and F7 in the S1 geometry. The same style as Figure 2 is used.
This high-energy peak in the ESA spectra in Figure 5 involves one or two high-lying electronic
transitions that share the total oscillator strength. Taking the F7 oligomer as an example, these
two transitions correspond to transitions from S1 into the high-lying excited states, S81 and S82.
As for the low-energy ESA feature these high-lying excited electronic states cannot be well
represented by a simple single-particle electron-hole excitation picture. Multiple orbitals
contribute significantly to these transitions. In contrast to the S5 state of the F4 oligomer, which
is dominated by HOMO-1→LUMO and HOMO→LUMO+1 transitions, the S81 state of the F7
15
oligomer is dominated by several electronic transitions, with major contributions from HOMO9→LUMO and HOMO-10→LUMO+3 transitions. The situation for the S82 state of F7 is even
more complicated. We find ten different orbital contributions to this electronic transition, all of
similar weight. We show the most relevant natural transition orbitals of the S81 state of F7 in
Figure 6, from which we can find that similar to the S5 state of F4 (see Figure 3), this high-lying
excited state also shows significant long-range charge transfer character, e.g. HONTO2→LUNTO+2 (Figure 6c) involves charge transfer from the ends to the middle of the oligomer,
and HONTO-3→LUNTO+3 (Figure 6d) involves charge transfer from the middle to the ends of
the oligomer.
Figure 6: Natural transition orbital for S81 state of F7 oligomer which is involved in excited state
absorption in the high energy range: a. HONTO→LUNTO (λ=21.2%), b. HONTO1→LUNTO+1 (λ=19.3%), c. HONTO-2→LUNTO+2 (λ=16.8%) and d. HONTO3→LUNTO+3 (λ=13.5%). The value of λ indicates relative weight of each set of NTOs.
These results indicate that electronic excitations from deeper-lying occupied electronic orbitals
to higher-lying unoccupied electronic orbitals might play an important role in excited state
absorption in the high energy range. The convergence and nature as well as the underlying
principle of excited state absorption in fluorene oligomers in the high-energy range is still being
16
investigated and will be discussed in our future work. We note that despite the obvious trend in
the high-energy ESA feature visible in Figure 5 for the series of F4, F5, and F7, we did not find a
similar high-energy ESA features for F3 and F6. So far we were not able to trace the absence of
this peak in some of the oligomers back to symmetry or conformational issues. We hope to be
able to shed some more light on this aspect in a future study. We would also like to note that
computational cost for the calculation of the ESA spectra greatly increases with oligomer length.
Additionally, the longer the oligomers the more excited states need to be included in the
calculations to cover the entire spectral range of interest up to about 4eV in the ESA in the
present study. For the longest oligomer studied here, we are at about the limit of what is
computationally feasible with current HPC clusters.
4. Conclusions
We performed a systematic computational study on the excited state absorption spectra of a
series of fluorene oligomers. Our calculated ESA spectra of F2 and F3 oligomers are in very
good agreement with previously reported experiments. Our results demonstrate that excited state
absorption occurs in fluorene polymers at a low excitation energy of around 1.5 eV. The nature
of excited state absorption in this energy range is characterized as an electronic transition from
the lowest excited state with a single HOMO→LUMO excitation character, to high-lying excited
states with mixed HOMO-1→LUMO and HOMO→LUMO+1 excitation characters which
involve simultaneous complementary intra-molecular charge transfer processes. Moreover, our
system represents one of the largest systems for which nonlinear optical properties have been
investigated by theory so far, even taking into consideration those studies which were based on
semiempirical methods.18 Taking fluorene oligomers as an example, the longest oligomer which
we have studied, F7, has a total of 149 atoms and 935 basis functions, when the 6-31G basis set
17
is employed. Our results show that nonlinear response TD-DFT with CAM-B3LYP functional
can be applied to large conjugated systems, for which high-level wave-function based quantum
chemical methods cannot be employed due to the high computational cost.
In addition, we demonstrate excited state absorption can also be expected for some of the
fluorene oligomers in the high energy range, around 3.5 eV. Although measurements in this
energy range are challenging with current pump-probe techniques, it certainly has technological
relevance due to the proximity of this energy to the one photon absorption and emission
resonances. It should also be pointed out here that high-lying excited states are extremely
difficult to be describe accurately via the (temporally) adiabatic TD-DFT in its present
implementations, employed throughout the current work, and our current results on excited state
absorption in the high energy range should be further verified by more advanced theoretical
methods that are currently under development.51-53 Nevertheless, we anticipate such excited state
absorption processes will play a very important role in performance degradation in
optoelectronic devices, and thus they should be taken into account in future design of new
materials for such applications.
Acknowledgements
IG acknowledges financial support from the Engineering and Physical Sciences Research
Council (U.K.). MJP acknowledges the European Research Council (ERC) for funding under the
European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant No. 258990.
Supporting Information
Cartesian coordinates (including F2-F7), calculated excitation energies and oscillator strengths of
both the low-energy (including F2-F7) and the high-energy (including F4, F5 and F7) peaks in
18
ESA spectra of S0 and S1 minima of fluorene oligomers; a comparison of the calculated
excitation energies and oscillator strengths of some of the low energy ESA transitions with 631G and 6-31G* basis sets at CAM-B3LYP level, (F2 at S0 geometry and F3 at S1 geometry);
relevant canonical molecular orbitals involved in the low energy peak in the ESA spectrum of F4
at S1 geometry. This material is available free of charge via the Internet at http://pubs.acs.org.
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