Exciton coupling effects on the two-photon absorption of squaraine homodimers with varying bridge units

Evripidis Michail , Maximilian H. Schreck , Marco Holzapfel and Christoph Lambert *
Institut für Organische Chemie and Center for Nanosystems Chemistry, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany. E-mail: christoph.lambert@uni-wuerzburg.de

Received 25th June 2020 , Accepted 5th August 2020

First published on 5th August 2020

We explored a series of squaraine homodimers with varying π-bridging centres to probe the relationship between the chemical structure and the two-photon absorption (2PA) characteristics. To this end, we designed and synthesised six linear homodimers based on two indolenine squaraine dyes with transoid configuration (SQA) which are connected by diverse bridges. In this regard, we investigated the effect of exciton coupling in these dimeric systems where the variation of the bridging units affects the magnitude of exciton coupling and leads to an alteration of their linear optical properties. Using two-photon absorption induced fluorescence measurements we determined the two-photon absorption cross section in this series of homodimers and found sizable values up to 5700 GM at ca. 11[thin space (1/6-em)]000 cm−1 and 12[thin space (1/6-em)]000 GM at 12[thin space (1/6-em)]500 cm−1. The 2PA strength roughly follows the exciton coupling interaction between the squaraine chromophores which therefore may be used as design criteria to achieve high 2PA cross sections. The results were substantiated by polarization dependent linear and nonlinear optical measurements and by density functional theory calculations based on time dependent and quadratic response theory.


In recent years, many studies have been devoted to a better understanding of the structure–property relationship of two-photon absorption (2PA) in organic chromophores. Hereby, a key aspect is to develop molecular design strategies in order to optimise novel organic multiphoton active materials and to explore their technical applications.1 Among the various functional dyes which are being used in a wide range of applications such as imaging, optical data storage, optical communication, and ion sensing, are squaraine dyes.2–5 They follow the well-established donor–acceptor–donor design rule for enhancing 2PA.6–8 The optical properties of these squaraine dyes have been in focus of research because they can be tuned by structural modifications to fit a specific application.9 This general approach is mainly based on the strong correlation between intra-molecular charge transfer processes and two-photon absorptivity as well as on the extent of conjugation in a molecular system with enhanced 2PA.10 It is known that increasing the π-conjugation length, thereby enhancing electronic delocalisation and charge-transfer phenomena, leads to an increase of the 2PA cross-section.11,12 On the other hand, there are several studies dealing with the existence of a limit for the two-photon absorptivity.13,14 Hence, diverse molecular design strategies were proposed for coupling multiple chromophores in oligomers and polymers.15,16 Therefore, a fundamental understanding of exciton coupling theory in the special case of non-linear optical interaction is without any doubt of utmost importance.17

Recently, we investigated the cooperative enhancement versus additivity of the 2PA cross section in linear and branched squaraine superchromophores.18 There, we could show that excitonic interactions lead to an enhancement of 2PA. In the present work we focus on engineering the exciton interaction by varying the spacer unit in a series of dimeric squaraine systems and probe the influence on the linear and non-linear photophysical properties. To this end, we have studied a series of six compounds built up from the well-investigated trans-squaraine parent chromophore SQA using a systematic length variation of the bridging unit, i.e. from a single bond in dSQA-1, over an acetylene bridge in dSQA-2, and a butadiyne-bridge in dSQA-3, to a tolan bridge in dSQA-5. But we also considered different conjugation pathways in e.g.dSQA-4vs.dSQA-3, and in dSQA-6vs.dSQA-1, as shown in Fig. 1.

image file: d0cp03410j-f1.tif
Fig. 1 Chemical structure of parent squaraine SQA and squaraine dimers. The blue and green arrows indicate the transition dipole moments of the lowest energy transitions of the individual chromophores and of the total homodimer, respectively.

Because of the flexible bridging units, in solution different conformers (rotamers) of the dimers exist, with exception of dSQA-6. The π-conjugated bridges allow us to consider essentially only two different stretched orientations, as indicated in Fig. 1. The arrangement where the long axes of the squaraine chromophores are parallel is called Type I and the one where they possess an angle of ca. 120° is called Type II. While for both conformers of all squaraine dimers besides dSQA-1, the two squaraine chromophores lie in the same plane, the biaryl dihedral angle in dSQA-1 leads to an out-of-plane arrangement of the two conformers. In fact, the two conformers of dSQA-1 possess C2 symmetry as the highest point group for Type I while that of all other dimers is higher (disregarding the symmetry lowering racemic alkyl chains R): D2h for dSQA-2, dSQA-3, and dSQA-6, Ci for dSQA-4 and dSQA-5. However, electronically all homodimers exhibit “pseudo centrosymmetric” symmetry, thus in the following discussion we treat all dimers as they belonged to the D2h point group and, for simplicity, we will use the symmetry assignments of states accordingly. Thus, considering the selection rules of 1PA and 2PA we assume that the one-photon states belong to the irreducible representation Bu and the two-photon allowed states belong to the Ag irreducible representation in all cases. Furthermore, Spano et al.19 recently investigated the optical properties of dSQA-1 by theoretical methods in detail and showed that the optical properties of the two conformers are rather similar. Thus, for most aspects, we do not further discriminate between these two conformers in this paper.

The linear optical absorption properties of this series of compounds were investigated by UV/vis absorption and emission spectroscopy and the 2PA spectra were determined by the two-photon induced fluorescence technique. One-photon fluorescence excitation anisotropy (1P-FEA) and two-photon fluorescence excitation anisotropy (2P-FEA) measurements were performed in viscous media (poly-THF) in order to assign the spectral positions of transitions and to determine the mutual orientation of transition dipoles, respectively. Experimental verification of the excited state symmetry has been carried out using two-photon polarization dependent spectroscopy with linear and circular polarised light. To gain a deeper insight into the evolution of the 2PA cross section as a function of the transition dipole moment as the exciton coupling decreases, transition dipole moment calculations were performed applying a three-state model.

We also performed theoretical calculations using time-dependent and quadratic response density functional theory (TD-DFT and QR-DFT) using the BHandHLYP hybrid functional for the optical properties and the B3LYP functional for optimisation of the geometries and the 6-31G* basis set. The calculations were carried out using the Gaussian and the DALTON program packages.20–22

A detailed description of the syntheses of all compounds and the experimental methodology of this photophysical study is presented in the ESI.

Results and discussion

Linear optical spectroscopy

In order to provide the basis to understand the nonlinear optical properties of the squaraine dimers we first present the linear optical properties. From these properties we will estimate the amount of exciton coupling between the two squaraine monomers within the dimers as this quantity turns out to be the design criterion to achieve high two-photon absorption cross sections.

The steady state linear absorption and normalised fluorescence spectra of the monomer and the homodimers in toluene are shown in Fig. 2, a logarithmic plot can be found in the ESI (Fig. S4). All the compounds investigated exhibit typical spectroscopic characteristics of squaraine dyes. The linear absorption spectrum of the monomer shows an intense sharp absorption peak at ca. 15[thin space (1/6-em)]500 cm−1 together with a weak vibronic shoulder on the high-energy side. The fluorescence spectrum behaves like a mirror image of the monomer absorption accompanied with a small Stokes shift.

image file: d0cp03410j-f2.tif
Fig. 2 Linear absorption (top) and normalised fluorescence (bottom) spectra of monomer SQA, and homodimers dSQA-1dSQA-6 in toluene.

The absorption spectra of the homodimers are more complex, they are generally red shifted compared to the monomer and show higher molar extinction coefficients compared to the parent monomer. Moreover, in the case of the dSQA-1 to dSQA-4 homodimers, additional absorption bands on the high energy side of the main absorption peak are observed. The most intense absorption peak (14[thin space (1/6-em)]500–15[thin space (1/6-em)]200 cm−1) originates from the lowest exciton state S1 which is bathochromically shifted with respect to the SQA monomer. In addition, two shoulders located at higher energies are attributed to the subsidiary vibronic progression and to a second electronic transition. The latter electronic transition is assigned to the upper exciton state S2. We interpret the presence of this S2 state within the series of squaraine dimers as being caused by exciton coupling of monomer localised transitions. This coupling produces a so-called Davydov splitting of the first excited state of the two coupled parent monomer states.23,24 According to exciton coupling theory the S1 excited state of the SQA monomer splits in two excited states (S1, S2) in the homodimers and the splitting energy refers to twice the electronic exciton coupling energy, J.25

The squaraine homodimer dSQA-6 with a rigid bridge was also investigated. Taking the advantage of its rigidity, the dimer cannot form rotamers and exhibits the features of a “J-type” dimer (i.e. head-to-tail arrangement of transition moments) exclusively. In this case, the upper exciton state vanishes, and the full transition strength is concentrated in the lower exciton state. The linear absorption spectra of the dSQA-6 in toluene shows an 1.5 times higher extinction coefficient (858[thin space (1/6-em)]000 M−1 cm−1 at 15[thin space (1/6-em)]100 cm−1) than the other homodimers.

The squared transition dipole moments (μ2 = dipole strength) between the ground state S0 and the two exciton states were determined by integration of these bands26 (see eqn (S7) in the ESI) and are summarised in Table 1. According to the Thomas–Reiche–Kuhn sum rule27 the square of the transition dipole moments shows nearly additive behaviour with respect to the SQA monomer, which gives similar values for all the homodimers (μ2 = 245–269 D2) proving that no other states besides the two exciton states are located in this energy range.

Table 1 The squaraine centre-to-centre distance, absorption maxima of S1 and S2 state, exciton coupling energy, dipole strength, emission maximum and fluorescence quantum yield of the parent monomer SQA and homodimers dSQA-1dSQA-6 in toluene
Squaraine centre-to-centre distance Absorption maximum of S1 Absorption maximum of S2 from 1PA spectra Exciton coupling energy as

image file: d0cp03410j-t3.tif

Absorption maximum of S2 from 1P-FEA spectra Exciton coupling energy as

image file: d0cp03410j-t4.tif

Dipole strengths of the total exciton manifold Emission maximum Fluorescence quantum yield
L [small nu, Greek, tilde] S1/cm−1 [ε/M−1 cm−1] [small nu, Greek, tilde] S2/cm−1 J/cm−1

image file: d0cp03410j-t5.tif

J r /cm−1 μ 2/D2 [small nu, Greek, tilde] em/cm−1 Φ fl
a See also ref. 34.
SQA 15[thin space (1/6-em)]530 [3.64 × 105] 127 15[thin space (1/6-em)]360 0.62
dSQA-1 17.1 14[thin space (1/6-em)]480 [4.66 × 105] 16[thin space (1/6-em)]140 830 16[thin space (1/6-em)]310 920 248 14[thin space (1/6-em)]240 0.80
dSQA-2 19.6 14[thin space (1/6-em)]550 [4.65 × 105] 16[thin space (1/6-em)]060 760 16[thin space (1/6-em)]290 870 254 14[thin space (1/6-em)]360 0.78
dSQA-3 22.0 14[thin space (1/6-em)]720 [6.04 × 105] 15[thin space (1/6-em)]400 340 15[thin space (1/6-em)]600 440 269 14[thin space (1/6-em)]530 0.74
dSQA-4 21.2 14[thin space (1/6-em)]910 [5.94 × 105] 15[thin space (1/6-em)]420 260 15[thin space (1/6-em)]690 390 260 14[thin space (1/6-em)]730 0.77
dSQA-5 27.7 15[thin space (1/6-em)]110 [5.94 × 105] 15[thin space (1/6-em)]280 90 252 14[thin space (1/6-em)]900 0.71
dSQA-6 18.7 15[thin space (1/6-em)]120 [8.58 × 105] 15[thin space (1/6-em)]900 390 245 15[thin space (1/6-em)]020 0.71

All homodimers display strong fluorescence with a small Stokes shift. The fluorescence spectra resemble that of SQA monomer but are red-shifted according to the respective absorption spectra. This proves that fluorescence is emitted from the lowest exciton state only. The quantum yield varies with the exciton coupling strength between 0.80 and 0.71 but is in all cases higher than that of the monomer which is the consequence of the coupled transition moments (superradiance).28 Likewise, the squared transition moment for the fluorescence as determined from the Strickler–Berg equation (see eqn (S8) in the ESI) varies between 147–177 D2 for the homodimers compared to 114 D2 for SQA (see Table S1 in the ESI). The strong fluorescence allowed the investigation of the 2PA cross section by the two-photon induced fluorescence method, see below.

The estimation of the exciton coupling strength is an important aspect in this study, as we will correlate the nonlinear optical properties with this quantity. Below, we will take symmetry arguments into account in order to assess the energy of the upper excitonic state, S2. Looking at the Type I conformer and assuming D2h symmetry, the first electric dipole allowed transition is S1 ← S0 (1Bu ← 1Ag), with 1Bu representing the lowest one-photon excitonic level. The next electronic transition is S2 ← S0 (2Ag ← 1Ag) which is forbidden by symmetry as a 1PA process. However this state is weakly allowed for two reasons, firstly, the symmetry of the corresponding S2 state in the Type II conformer, which is present in presumably equimolar amounts, is lower which breaks the centrosymmetry and, secondly, vibronic coupling of this 2Ag state to an asymmetric vibration also makes this transition weakly allowed.28,29

Going from dSQA-1 to dSQA-6 we observe a gradual decrease of the red-shift of the main absorption peak with respect to SQA which leads to an enhanced mixing of electronic and vibronic states.30 The conclusion from the latter result is that the exciton coupling strength of the homodimers decreases by changing the bridging units from dSQA-1 to dSQA-6. This interpretation is confirmed by estimation of the effective exciton coupling strength J between the two monomers (see Table 1) taking half of the S2–S1 energy as J for dSQA-1 to dSQA-4. For dSQA-5 and dSQA-6 the upper exciton state is hardly visible as a shoulder to the high energy side of the main peak and does not allow to extract J within reasonable accuracy. However, considering the absolute red-shift of the lowest energy absorption of the squaraine dimer vs. the one of the monomer allows to estimate J at least qualitatively. The fluorescence spectra show the same trend. Thereby, J decreases on going from dSQA-1 to dSQA-5 which has the same lowest energy absorption as dSQA-6. Attention is drawn to the fact that the spacer length between the two chromophores (see Table 1) does not completely correlate with the exciton coupling strength which varies from 830 cm−1 for dSQA-1 to 260 cm−1 for dSQA-4. This shows that the electronic coupling is not only due to dipole–dipole interactions but also depends to some extent on the chemical structure of the π-bridge and conjugative effects play a role.31

In order to estimate the energy of the S2 state and of J more accurately, especially for dSQA-5 and dSQA-6, and to assign the symmetry of higher-excited states we performed one-photon fluorescence excitation anisotropy measurements in viscous poly-THF (the 1P-FEA, spectra can be found in the ESI and the maximal anisotropy in Table 3). The 1PA anisotropy is related to the angle θ01 between the absorption and emission transition moment through eqn (1).32 Ideally, the lower energy exciton state of all dimers should show an anisotropy of r1PA = 0.4 (i.e. parallel transition moments for absorption and emission of the 1Bu state).33 In case of Type I conformer, the higher exciton state (2Ag symmetry) should possess a vanishing transition moment while that of Type II conformer should have perpendicular transition moments for absorption and emission, that is, r1PA = −0.2. In case of strong exciton coupling, i.e. delocalised excited state over both monomers, the limiting anisotropy of the 1Bu state should always be 0.4 even when the solution consists of different conformers of homodimers. Indeed, the 1P-FEA value r1PA for the dSQA-1 homodimer in poly-THF at the 1Bu state is 0.36, confirming strong exciton coupling. Even that of dSQA-2 is still pretty high (r1PA = 0.33). However, for weak exciton coupling the excitation becomes increasingly localised within the excited state lifetime because of vibronic coupling.34 This leads to a deviation of the angle of the transition moment for delocalised absorption (green transition moments, in Fig. 1) and localised emission (blue transition moments, in Fig. 1) for the Type II conformer. For the extreme case of total localisation of excitation we can estimate the limiting anisotropy as follows: while in Type I conformer the transition moment for absorption and emission are parallel even after localisation, in case of Type II conformer, the transition moments after localisation form an angle of ca. 60° which will lead to an anisotropy of −0.05. Averaging over the two conformers then leads to r1PA = 0.175. Indeed, for dSQA-5 the anisotropy is at the red side of the 1Bu band 0.18 which is a distinct hint towards complete exciton localisation after excitation. At the band maximum and the blue side the anisotropy is considerably lower because of band overlap with the 2Ag band. The anisotropies of the other dimers are in-between those of dSQA-1 and dSQA-5.

image file: d0cp03410j-t1.tif(1)

For the upper exciton state in case of strong coupling, we find a vanishing transition moment for Type I conformer (2Ag state). For the Type II conformer, an anisotropy of −0.2 results (this state is polarised along the C2 axis and thus is perpendicular to the emission transition moment). In case of weak exciton coupling, localisation leads to r1PA = 0.4 for Type I but to −0.05 for Type II. Again, averaging yields r1PA = 0.175. For the 2Ag state of dSQA-1 the experimental anisotropy is −0.02. The deviation from −0.2 is most likely caused by band overlap with the allowed 1Bu state which, if it were exclusively excited, would yield r = 0.4. Nevertheless, this low anisotropy helps to assign the 2Ag state to be at 16[thin space (1/6-em)]100 cm−1. The lowest anisotropy for the dSQA-2 homodimer is 0.016 and is at 16[thin space (1/6-em)]100 cm−1 which is therefore assigned to the S2 state. We therefore used the lowest anisotropy in the suspected S2 energy region to assign the energy of the S2 state more precisely. These values and the exciton couplings derived from these energies are also given in Table 1. The latter are in good agreement with those estimated from the absorption spectra. dSQA-6 is a special case as it is weakly coupled but cannot form a Type II conformer. Accordingly, the anisotropy is between 0.32–0.38 over the whole exciton state range. The latter finding also supports the interpretation of the anisotropy of the more flexible dimers to be caused by an equimolar mixture of two rotamers. Using the thereby obtained S2 energies image file: d0cp03410j-t2.tif we can now estimate the exciton coupling energy more accurately which ranges between Jr = 920 to 90 cm−1 for dSQA-1dSQA-5 but again increases for dSQA-6 with 390 cm−1 (see Table 1).

As will be shown below, the above mentioned exciton coupling energies for dSQA-1 to dSQA-4 are supported by quantum chemical time-dependent density functional calculations at BHandHLYP/6-31G* level (see Table 4). While the computed state energies for S1 and S2 are generally much too high by ca. 4000 cm−1, their energy difference divided by two gives an estimate for the exciton coupling energy. Here, for dSQA-1 to dSQA-4 we obtain values between 786 cm−1 and 430 cm−1 which are in reasonable agreement with experimental estimates. This gives confidence that the values for dSQA-5 and dSQA-6 are not far off the computed values of 238 and 482 cm−1. Thus, in this work we assume a qualitative order of exciton coupling decreasing from dSQA-1 to dSQA-5.

Non-linear optical spectroscopy

2PA of SQA monomer. Before we present the results of the non-linear absorption measurements of all homodimers, we provide a brief overview of the 2PA bands of the monomeric chromophore SQA as addressed in our previous studies by Ceymann et al.18 For the centrosymmetric monomeric compound SQA the selection rules between one- and two-photon transitions are complementary. Indeed, the lowest excited state with Bu symmetry (at 15[thin space (1/6-em)]500 cm−1) can only be reached by a one-photon process and the energetically higher-excited state Ag (at 24[thin space (1/6-em)]200 cm−1) is allowed by a two-photon process. Moreover, a weak 2PA band can be observed at 16[thin space (1/6-em)]700 cm−1 for the vibronic shoulder of the 1PA allowed absorption band. This electronically forbidden transition is allowed due to vibronic coupling to a bu asymmetric vibration (Bu × bu → Ag, Herzberg–Teller effect35,36). The coupling to an asymmetric mode also reduces the fluorescence excitation anisotropy from 0.4 for parallel absorption and emission transition moment to ca. 0.3, see ESI. This situation is sketched in Fig. 3.
image file: d0cp03410j-f3.tif
Fig. 3 Schematic energy state diagram for SQA monomer and the homodimers both with D2h symmetry. The blue arrows indicate 1PA allowed transitions, the red arrows 2PA allowed transitions. Uppercase symmetry assignments refer to electronic states, lowercase assignments to vibrations. Energies are not to scale.
2PA of squaraine dimers. Our recent work on dSQA-1 indicates that, besides the two exciton states that can be constructed using simple exciton coupling theory (1Bu and 2Ag), there are two further excited states that can be derived by an orbital interaction diagram and configuration interaction (3Ag and 2Bu, see ESI in ref. 18). One of these states is 2PA-allowed, the other 1PA-allowed. In fact, we observed a strong and structured 2PA band between 18[thin space (1/6-em)]000–24[thin space (1/6-em)]000 cm−1 (see Fig. 4) but only a very weak 1PA in this energy range (see logarithmic plots Fig. S6 in the ESI). Thus, the relative energy of these two states is not quite clear. In our earlier work we assigned the 2Bu state at lower energy than the 3Ag state. However, DFT calculations (see below) indicate the reversed order which is also represented in Fig. 3 but the energy of these two states is predicted to be quite similar and in fact, the relative order depends on the functional used for the DFT computations.
image file: d0cp03410j-f4.tif
Fig. 4 Plot of two-photon absorption cross-section spectra versus twice the laser excitation energy of monomer SQA and dSQA-1dSQA-6 in toluene.

All the homodimers in toluene solution show significant 2PA in the investigated spectral region between 15[thin space (1/6-em)]000–25[thin space (1/6-em)]000 cm−1 (see Fig. 4). We can roughly divide the 2PA spectra into three sections: the first section covers 2PA bands up to 18[thin space (1/6-em)]000 cm−1 which comprises the 2PA into the formally 2PA-forbidden lower-energy 1Bu exciton state and the 2PA-allowed higher-energy 2Ag exciton state. The 2PA cross section reaches maximum values of ca. 400 GM. The second section covers 2PA from roughly 18[thin space (1/6-em)]000 to 24[thin space (1/6-em)]000 cm−1. Here, several peaks are visible which show a distinct spectral evolution on going from dSQA-1 to dSQA-6. The 2PA cross section decreases from ca. 6000 GM for dSQA-1 to ca. 400 GM for dSQA-6, that is, their intensities roughly decrease upon decreasing exciton coupling. DFT computations (see below) yield only one 2PA-allowed transition in this spectral region, thus, the diverse peaks appear to be a vibronic progression of a single electronic transition. The third 2PA section is above 24[thin space (1/6-em)]000 cm−1 and shows an extremely high 2PA cross section for all homodimers on the order of 104 GM. A peak maximum cannot be discerned clearly in that case. For the sake of comparison, the 2PA together with linear absorption spectra and 1P-FEA are depicted in Fig. S6 in the ESI. The optical data are summarised in Table 2.

Table 2 2PA cross sections σ2PA at selected energiesc and 2PA strengths 〈δ2PA〉 for selected energy intervals of the parent monomer SQA and homodimers dSQA-1dSQA-6 in toluene
[small nu, Greek, tilde] S1/cm−1 σ 2PA/GM [small nu, Greek, tilde] S2/cm−1 σ 2PA/GM δ2PAa/a.u. [small nu, Greek, tilde] S3/cm−1 σ 2PA/GM δ2PAb/a.u.
a The 2PA strength cover that of the S1 and the S2 state. Integration limits 14[thin space (1/6-em)]600–17[thin space (1/6-em)]800 cm−1 (19[thin space (1/6-em)]300 cm−1 for dSQA-6). b Integration limits 17[thin space (1/6-em)]800–24[thin space (1/6-em)]000 cm−1 (23[thin space (1/6-em)]200 cm−1 for dSQA-6). c The wavenumbers for the assigned states are those with a maximum of 2PA cross section and do not necessarily agree with the wavenumbers in Table 1.
SQA 15[thin space (1/6-em)]750 24
dSQA-1 14[thin space (1/6-em)]390 47 16[thin space (1/6-em)]130 397 3.42 × 104 21[thin space (1/6-em)]980 5740 4.77 × 105
dSQA-2 14[thin space (1/6-em)]490 44 16[thin space (1/6-em)]000 360 2.53 × 104 21[thin space (1/6-em)]980 4660 3.53 × 105
dSQA-3 14[thin space (1/6-em)]270 121 16[thin space (1/6-em)]260 405 2.68 × 104 21[thin space (1/6-em)]980 4330 3.23 × 105
dSQA-4 15[thin space (1/6-em)]270 99 16[thin space (1/6-em)]130 397 2.07 × 104 21[thin space (1/6-em)]980 2670 2.22 × 105
dSQA-5 15[thin space (1/6-em)]040 36 16[thin space (1/6-em)]390 239 1.52 × 104 22[thin space (1/6-em)]470 1530 9.69 × 104
dSQA-6 15[thin space (1/6-em)]040 13 16[thin space (1/6-em)]260 269 1.58 × 104 22[thin space (1/6-em)]222 375 1.69 × 104

The 15[thin space (1/6-em)]000–18[thin space (1/6-em)]000 cm−1 2PA region. We now turn our attention to the three spectral sections in detail. As mentioned above, in the first section up to ca 18[thin space (1/6-em)]000 cm−1 two 2PA bands strongly overlap. The first band is caused by vibronic coupling of the otherwise 2PA-forbidden 1Bu (S1) state with bu intermolecular vibrations. This is clearly visible in the 2PA spectra of SQA monomer7,18,36 where a cross section of ca. 200 GM is reported. In the homodimers, this band is overlaid by the 2PA-allowed 2Ag state (S2) which has a similar 2PA cross section. In Table 2, the maximum 2PA cross section as well as the integrated 2PA strength of the spectral region are given. The latter was obtained by integrating the respective spectral region according to eqn (2)37–39 where the orientationally averaged transition strength 〈δ2PA〉 in a.u. is given by the integrated cross section divided by the squared photon energy ω in a.u. (1 eV = 0.03675 a.u.) and the prefactor image file: d0cp03410j-t6.tif = 39.92 (where c = 3 × 1010 cm s−1 is the speed of light, α = 7.297 × 10−3 is the fine structure constant, a0 = 5.292 × 10−9 cm is the Bohr radius) which converts the cross section in GM to a.u. The 2PA strength corresponds to the dipole strength (squared transition moment) and allows to compare the intensity of transitions irrespective of their band shape.
image file: d0cp03410j-t7.tif(2)

For the first energy region, a slight decrease of the 2PA strength from about 3.4 × 104 a.u. to 1.5 × 104 a.u. is observed on going from the strongly coupled dSQA-1 to the weakly coupled dSQA-5. In case of centrosymmetric molecules, the 2PA strength can be estimated by a three level model (eqn (3) with all quantities in a.u.)40 where μ012 and μ122 are the transition moments between the ground and the first 1PA allowed state (S1) and between first and second excited state (S2). Here, the transition moments are assumed to be parallel (for the general case of arbitrary angle between the transition moments, see eqn (4)). ω01 and ω02 are the energy differences between these excited states and the ground state. If we deal with excitonically coupled monomers in a dimer, the transition moment between the first and the second exciton state μ12 equals the dipole moment difference of ground and excited state of the monomer Δμ. Thus, for a centrosymmetric monomer without CT contribution in the first excited state such as SQA, μ12 = Δμ = 0 and for a dimer built from such a monomer the 2PA strength of the S2 state should vanish. In practice, because of symmetry breaking small CT contributions will lead to a small 2PA strength as observed in the series of dimers investigated in this work.

image file: d0cp03410j-t8.tif(3)

Polarization dependent 2PA measurements in the 15[thin space (1/6-em)]000–18[thin space (1/6-em)]000 cm−1 region. To verify the origin of the bands in the 15[thin space (1/6-em)]000–18[thin space (1/6-em)]000 cm−1 range we performed two-photon polarization-dependent 2PA measurements for SQA monomer, dSQA-1, dSQA-2, and dSQA-5 homodimers in fluid toluene solution with the results presented in Fig. S7 (ESI). The ratio Ω2PA of 2PA with linear and circular polarised light allows to distinguish whether the symmetry of states changes or not upon exciting the ground state to an excited state by 2PA even in fluid solution. This is based on the fact that the Ω2PA is described by the two-photon transition tensor patterns, which is a property of each excited state involved in the transition. For changing the state symmetry, Ω2PA = 3/2 and for retaining the symmetry 0 < Ω2PA < 3/2. Indeed, we observe two regions where the value of 2PA polarization ratio tends to a constant value, indicating two different type of transitions in this spectral region. In the first, lower-energy spectral region where the S0 → S1 transition is located; all three homodimers exhibit Ω2PA ∼ 3/2. Assuming a totally symmetric ground state, this result shows a transition to a nontotally symmetric excited state, that is, a change of symmetry. The origin of this transition is the same as described in case of the SQA monomer i.e. two-photon transition through vibrational coupling Bu × bu → Ag into an electronic state that is formally 2PA-forbidden. In the second, higher-energy spectral region where the S2 ← S0 transition is located, the value of Ω2PA varies from ∼1.3 (for dSQA-1) to 1 (for dSQA-5). This region where Ω2PA may adopt any value in the range of 0 up to 3/2 is assigned to a two-photon transition to a totally symmetric excited state.41
The 18[thin space (1/6-em)]000–24[thin space (1/6-em)]000 cm−1 2PA region. In the second spectral region between 18[thin space (1/6-em)]000–24[thin space (1/6-em)]000 cm−1 the experimental 2PA cross section decreases from 5740 GM for dSQA-1 to 375 GM for dSQA-6. Thus, upon modifying the molecular structure of the π-spacer and thereby decreasing the electronic exciton coupling going from dSQA-1 to dSQA-5, a decrease of the 2PA cross-section from dSQA-1 to dSQA-6 is observed and the 2PA spectra approach that of the monomer SQA. Here we stress that the experimental 2PA strength is associated with a major error as the integration limit especially at the high energy side is rather arbitrary. As DFT calculations (see below) using quadratic response theory suggest only one 2PA allowed state in this energy region, we assume that the structured 2PA absorption between ca. 18[thin space (1/6-em)]000-24[thin space (1/6-em)]000 cm−1 is caused by a single electronic transition. The unexpected large width of the band might be caused by vibronic progression as it is known that in 2PA spectra some vibronic transitions become enhanced, whereas the 1PA spectrum is dominated by pure electronic transitions.42,43 In this spectral region, according to the TD-DFT calculations (see below), an 1PA-allowed state (S4) should also be present but this state should possess negligible transition probability. We do not expect that vibronic coupling of this state to asymmetric vibrations leads to significant effective 2PA contributions to the 2PA-allowed S3 state. The integrated 2PA strength in that spectral region yields values between 3 × 105 a.u. for dSQA-1 to 2 × 104 a.u. for dSQA-6.
Three-state model to estimate μ13. In order to gain insight into the dependence of the 2PA strength on the exciton coupling strength of the S3 state we employed a three-state model for the symmetric homodimers as the excitation scheme is dominated by two major transitions. The first one occurs between the ground state S0 and the one-photon-allowed S1 state and the second is between the S1 state and the two-photon-allowed S3 state via an intermediate “virtual” state.44 Employing the three-level model the relation of the rotationally average 2PA strength 〈δ2PA〉 (in a.u.) is connected with the transition dipole moments for arbitrary angles θ13 between both transition moments as presented in eqn (4):41
image file: d0cp03410j-t9.tif(4)
where μ132 is the squared transition dipole moment between the one-photon-allowed excited state S1 and the two-photon-allowed state S3. The angle θ13 can be determined through 2P-FEA spectroscopy (see Fig. S9, ESI) in viscous poly-THF. The terms ω01 and ω03 are the transition energies from the ground state S0 to the one-photon-allowed excited state S1 and the two-photon-allowed state S3 respectively. The 2PA strength is related to the integrated 2PA cross section viaeqn (2). Thus replacing the 2PA strength 〈δ2PA〉 in eqn (4) by eqn (2) yields eqn (5) for the squared transition moment between the S1 and S3 state (all quantities in a.u., see also comments to eqn (2)):
image file: d0cp03410j-t10.tif(5)

As the input for eqn (5), the transition moment between S0 and S1 is known by integration of the S1 band (see Table 1). The energy ω01 is the S1 state energy (see Table 1) and ω03 is the S3 state energy (see Table 2). The integration of the 2PA allowed S3 band yields 〈δ2PA〉 in Table 2. The hitherto unknown angle θ13 is related to the 2PA anisotropy r2PAviaeqn (6):45

image file: d0cp03410j-t11.tif(6)

Here, the average angle for Type I and Type II isomer θ01 was obtained by 1P anisotropy measurements (see above and Fig. S6, ESI). The 2PA anisotropy r2PA was determined by measuring the 2PA cross section in viscous poly-THF (see above, Table 3 and Fig. S9, ESI) which yields the average angle θ03 through eqn (6). This angle was inserted in eqn (5) to evaluate μ132 which is listed in Table 3. One can immediately see that μ132 decreases from ca. 64 D2 to 3 D2 in the series from dSQA-1 to dSQA-6 which is, more or less, with decreasing exciton coupling strength. Here we have to take into account that these numbers are average values referring to an equimolar mixture of Type I and Type II conformers. Therefore, we also give μ132 for the extreme angles θ13 = 0° and 90°. Although the thereby evaluated μ132 is smaller/larger than in case of varying angles, they follow the same trend which clearly indicates that the differences in 2PA strength are caused by the difference of exciton coupling in this series of squaraine dimers.

Table 3 Input values for eqn (5) and (6): 1PA and 2PA anisotropy r, angles of transition moments θ and state energies ω, 2PA strength 〈δ2PA〉 and squared transition moments μ
r 1Pmax θ 01/deg r 2PA [at [small nu, Greek, tilde]/cm−1] θ 13/deg ω 01/cm−1 ω 03/cm−1 δ2PA〉/a.u. μ 13 2/D2 μ 13 2/D2θ13 = 0° μ 13 2/D2θ13 = 90°
a This angle refers to the minimum anisotropy of 0.14.
dSQA-1 0.35 16.8 0.41 [21 980] 71 14[thin space (1/6-em)]500 21[thin space (1/6-em)]980 477[thin space (1/6-em)]085 63.6 25.7 77.0
dSQA-2 0.33 20.0 0.28 [21 980] 81 14[thin space (1/6-em)]500 21[thin space (1/6-em)]980 353[thin space (1/6-em)]413 53.1 18.6 55.7
dSQA-3 0.23 32.2 0.17 [21 980] 89 14[thin space (1/6-em)]700 21[thin space (1/6-em)]980 322[thin space (1/6-em)]594 53.6 17.9 53.7
dSQA-4 0.29 25.4 0.38 [21 980] 77 14[thin space (1/6-em)]900 21[thin space (1/6-em)]980 222[thin space (1/6-em)]035 38.5 14.1 42.4
dSQA-5 0.18 37.3 0.13 [22 470] 90a 15[thin space (1/6-em)]100 22[thin space (1/6-em)]470 96[thin space (1/6-em)]886 18.7 6.22 18.7
dSQA-6 0.36 15.0 0.50 [22 220] 62 15[thin space (1/6-em)]120 22[thin space (1/6-em)]220 16[thin space (1/6-em)]886 2.7 1.27 3.82

The 24[thin space (1/6-em)]000–25[thin space (1/6-em)]000 cm−1 2PA region. The band with the highest 2PA cross-section in the accessible energy range is located at ca. 25[thin space (1/6-em)]000 cm−1 which is at even higher energies than the S3/S4 state. For this S6 state, all homodimers share the common feature of extremely large intrinsic 2PA cross section between 9000–13[thin space (1/6-em)]000 GM, which is caused by a double-resonance enhancement.46 In the same energy region the linear absorption spectra of the homodimers (as well as the monomer) also display another, less intense S5 ← S1 transition at energies around 26[thin space (1/6-em)]000 cm−1 (see Fig. S4, ESI). According to the three-level model (eqn (4)) when the term image file: d0cp03410j-t12.tif tends to zero, the rotationally average 2PA strength 〈δ2PA〉 and consequently the macroscopic 2PA cross-section σ(2PA) increases. This term represents the detuning energy of the transition (see Fig. 3), which is the energetic difference of the one-photon-allowed excited state and the virtual intermediate state of the two-photon transition.47 Thus, for states approaching twice the energy of the S1 state, the 2PA strength increases strongly. However, we were unable to cover this 2PA-allowed S6 state and determine its maximum cross section experimentally.
2PA from quadratic response DFT calculations. In order to further substantiate our experimental results, we performed DFT calculations. The 1PA states were calculated by time-dependent theory and the 2PA states by quadratic response theory.48–50 As mentioned above, the computed state energies of S1 are 4000 cm−1 too high compared with experiment. Thus, we shifted all state energies by 4000 cm−1 to make relative comparison easier. Such blue-shifted energies have also been found for DFT calculations of other squaraine dyes51–53 and are due to inherent DFT problems.54 These computed energies as well as the oscillator strengths of the 1PA-allowed states and the 2PA strengths are given in Table 4. For the S2 state we found significant 2PA strengths which decrease from ca. 1500 a.u. to 100 a.u. on going from dSQA-1 to dSQA-6. While the experimental values show the same trend, they are about one order of magnitude higher, even when considering that the S1 state contributes to the 2PA strength considerably in this energy region. While simple exciton coupling theory predicts only two exciton states when coupling two monomers, more elaborate orbital coupling and CI mixing schemes indicate the formation of four excited states, two of which correspond to the two exciton states (Bu and Ag). The other two states (S3 and S4) have also Ag and Bu symmetry. The DFT calculated S3 and S4 state energies are rather close in energy and, after shifting by 4000 cm−1, in reasonable agreement with experiment at ca. 23[thin space (1/6-em)]000 cm−1 for all homodimers. While the 1PA oscillator strength of S4 is quite small for most dimers, the 2PA cross sections of S3 are large for all dimers. Here, the experimental values are about one order of magnitude lower than the computed 2PA strengths. In the high energy region, the DFT calculations predict two further states, one with 3Bu symmetry (S5) and one with 4Ag symmetry (S6). In agreement with experiment, the DFT calculations, after renormalisation in energy, show that the 3Bu state has a moderate oscillator strength for all dimers (see Table 4). According to the DFT computations, the S6 state is 2PA-allowed and possesses large 2PA strength, on the same order as the S3 state (dSQA-1 to dSQA-3) or even larger (dSQA-4 to dSQA-6). Comparison with experiment is difficult as the whole energy region could not be measured but the 2PA cross section is extremely high at 25[thin space (1/6-em)]000 cm−1 for all dimers.
Table 4 TD-DFT (BHandHLYP/6-31G*) and QR-DFT calculated excitation energies [small nu, Greek, tilde] (shifted by 4000 cm−1), oscillator strengths f and TPA strengths 〈δa
S1 (1Bu) J = (ΔES1S2)/2/cm−1 S2 (2Ag) S3 (3Ag) S4 (2Bu) S5 (3Bu) S6 (4Ag)
[small nu, Greek, tilde]/cm−1 f [small nu, Greek, tilde]/cm−1 δ〉/a.u. [small nu, Greek, tilde]/cm−1 δ〉/a.u. [small nu, Greek, tilde]/cm−1 f [small nu, Greek, tilde]/cm−1 f [small nu, Greek, tilde]/cm−1 δ〉/a.u.
a The symmetry assignments refer to D2h symmetry although the actual symmetry might be different. n–π* excitations with zero oscillator strength and 2PA cross section are omitted. b For dSQA-1 with C2 symmetry there are very small oscillator strengths f also for the “pseudo” g state and, vice versa, TPA cross sections for the u states, but these are negligible. c The energetic order of states is reversed concerning symmetry.
SQA 15[thin space (1/6-em)]892 1.55 29[thin space (1/6-em)]147 1.12 × 105
dSQA-1 14[thin space (1/6-em)]666 3.55 786 16[thin space (1/6-em)]237 1540 22[thin space (1/6-em)]337 1.63 × 106 22[thin space (1/6-em)]613 0.01 27[thin space (1/6-em)]534 0.09 28[thin space (1/6-em)]583 1.38 × 106
dSQA-2 14[thin space (1/6-em)]442 3.98 759 15[thin space (1/6-em)]959 3880 22[thin space (1/6-em)]342 3.06 × 106 22[thin space (1/6-em)]577 0.05 26[thin space (1/6-em)]163 0.19 27[thin space (1/6-em)]454 1.64 × 106
dSQA-3 14[thin space (1/6-em)]460 4.21 657 15[thin space (1/6-em)]773 1810 22[thin space (1/6-em)]656c 3.39 × 106 22[thin space (1/6-em)]569c 0.14 25[thin space (1/6-em)]195 0.34 27[thin space (1/6-em)]131 2.09 × 106
dSQA-4 15[thin space (1/6-em)]037 3.83 430 15[thin space (1/6-em)]897 500 23[thin space (1/6-em)]281 7.40 × 105 23[thin space (1/6-em)]311 0.02 27[thin space (1/6-em)]050 0.23 27[thin space (1/6-em)]776 1.61 × 106
dSQA-5 15[thin space (1/6-em)]156 4.11 238 15[thin space (1/6-em)]632 198 24[thin space (1/6-em)]011c 5.30 × 105 23[thin space (1/6-em)]924c 0.14 25[thin space (1/6-em)]679 1.1 27[thin space (1/6-em)]050 1.75 × 106
dSQA-6 15[thin space (1/6-em)]222 3.72 482 16[thin space (1/6-em)]185 105 23[thin space (1/6-em)]238 2.02 × 104 23[thin space (1/6-em)]242 0 28[thin space (1/6-em)]744 0.04 28[thin space (1/6-em)]825 1.29 × 106


In conclusion, we have presented the linear and nonlinear optical properties of a series of squaraine dimers that are connected by diverse bridges allowing for tuning the extent of exciton coupling between the two squaraine chromophores. Thereby, the exciton coupling decreases on going from directly coupled squaraines in dSQA-1 to a dimer where the squaraine chromophores are connected via a tolan bridge in dSQA-5. In this series, the exciton coupling varies approximately along the centre-to-centre distance of the two chromophores (see Fig. 5). However in dSQA-6, where the two squaraines are connected by a short saturated bridge, the exciton coupling is more similar to dSQA-4 with a conjugative phenylene bridge which, at the same time, also places the squaraines at farther distance. This shows that conjugative effects may also increase the effective exciton coupling.
image file: d0cp03410j-f5.tif
Fig. 5 Plot of the two-photon absorption strength (black) for S3 and the dipole strength (red) between S1 and S3vs. exciton coupling energy for dSQA-1dSQA-5 in toluene. The linear correlation lines are only a guide to the eye.

Exciton coupling of the two squaraine chromophores leads to four excited states in the investigated energy region. While the first exciton state (1Bu) is strongly 1PA-allowed, the fourth (2Bu) is only weakly allowed. In contrast, the second exciton state (2Ag) is only weakly 2PA-allowed but the third (3Ag) is strongly 2PA-allowed. This could be rationalised applying a three-level model. Here, the squared transition moment between the first exciton state (1Bu) and the 2PA-allowed state plays the dominant role. In case of the weakly 2PA-allowed second exciton state, this transition moment is generally small in the series of squaraine dimers as we deal with chromophores lacking significant local charge-transfer contributions. This leads to overall small 2PA cross sections for the 2Ag states which do not vary greatly within the series of homodimers. However, in case of the third exciton state (3Ag) this squared transition moment is large and its magnitude follows qualitatively the exciton coupling strength (see Fig. 5). One exception is dSQA-6 which possesses the smallest 2PA cross section although its exciton coupling is quite significant and equals that of dSQA-4. As the homodimer dSQA-6 is the only one where the two squaraines within the investigated series of homodimers are connected by a saturated bridge, we conclude that a lack of conjugation leads to small 2PA cross sections of the 3Ag state. Thus, short and conjugated bridging units between the two squaraine chromophores increase the exciton coupling and thereby enhance the 2PA strength in such homodimers. In this way, substantial 2PA cross sections of up to 5700 GM at ca. 11[thin space (1/6-em)]000 cm−1 and 12[thin space (1/6-em)]000 GM at 12[thin space (1/6-em)]500 cm−1 could be obtained.

Conflicts of interest

There are no conflicts to declare.


We thank the Deutsche Forschungsgemeinschaft as well as the Bavarian State Ministry of Science, Research and the Arts – Collaborative Research Network “Solar Technologies Go Hybrid” for ongoing support of our work on squaraine dyes. We thank Dr M. Büchner for help with the Dalton program.


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