Davydov Splitting in Squaraine Dimers | The Journal of Physical

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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Davydov Splitting in Squaraine Dimers Chuwei Zhong,† David Bialas,† Christopher J. Collison,‡,§,∥ and Frank C. Spano*,† †

Department of Chemistry, Temple University, 130 Beury Hall, 1901 N. 13th Street, Philadelphia, Pennsylvania 19122, United States ‡ School of Chemistry and Materials Science, §Nanopower Research Laboratory, and ∥Microsystems Engineering, Rochester Institute of Technology, Rochester, New York 14623, United States

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S Supporting Information *

ABSTRACT: The essential states model (ESM) for donor− acceptor−donor (D−A−D) chromophores is used to explore absorption and photoluminescence (PL) in molecular dimers composed of centrosymmetric and non-centrosymmetric squaraine molecules. The spectral line shapes and shifts relative to the monomer spectrum are due to an interesting interplay between three-center charge distributions responsible for ground- and excited-state (permanent) dipole and quadrupole moments and two-center charge distributions responsible for transition dipole moments. The Davydov splitting is sensitive only to the interactions between the (extended) transition dipoles , whereas the permanent dipole-dipole and quadruple-quadrupole interactions impact the midpoint frequency of the two Davydov components, leading to a generally asymmetric splitting relative to the peak monomer transition frequency. The theory accurately reproduces the steady-state absorption and PL line shapes recently obtained for covalently bound squaraine dimers. The ESM also predicts an extreme type of non-Kasha behavior, where both Davydov components are blue-shifted above the monomer transition frequency.

1. INTRODUCTION Squaraine assemblies in the form of covalently bound dimer and trimer complexes,1−4 polymers,5,6 and crystals7−9 have attracted considerable attention as fundamental frameworks for studying aggregate photophysics and exciton transport.10 In squaraine polymers, for example, the exciton diffusion rate (10−100 ps−1) was found to be significantly larger than in conjugated polymers such as MEH-PPV.5 Such rapid transport has been attributed to strong Frenkel exciton coupling derived from transition dipole−dipole interactions. However, the pioneering work of Painelli and co-workers11−15 has shown that in aggregates of highly polarizable donor−acceptor−donor (D−A−D) chromophores, which include the squaraines, the Frenkel exciton model based on two-state (S 0 , S 1 ) chromophores may be overly simplistic as it does not account for the significant changes in the ground (S0) and excited-state (S1) electronic charge distributions induced by intermolecular interactions or chromophore−solvent interactions. For example, in the Frenkel exciton model, the permanent dipole or quadrupole moments corresponding to S0 and S1 remain fixed and independent of the intermolecular interactions. This is a good approximation for less polarizable chromophores such as oligoacenes and perylene diimides, where the S0 → S1 transition does not involve a significant electron transfer. Painelli and co-workers11−15 addressed the problem by introducing diabatic “essential” electronic states for each chromophore, which can mix together to form the molecular (adiabatic) states in response to intermolecular and © XXXX American Chemical Society

chromophore−solvent interactions. Hence, in the essential states model (ESM), properties such as the ground- and excited-state permanent dipole moments can change in response to interactions with other molecules or with the solvent. The ESM has also proven effective in evaluating the nonlinear optical response.14,16−18 Such sensitivity of the adiabatic molecular states to the environment will likely impact excitation transfer, which is important for a variety of applications. For example, squaraine molecules are representative of a larger class of push−pull D− A−D systems, pivotal for use in organic photovoltaics (OPV), whereby the specific natures of the donor and acceptor groups allow for energy level tuning,19 important for minimizing voltage losses, thereby optimizing the OPV power conversion efficiency.20,21 Nevertheless, modifying the donor−acceptor nature of push−pull materials, along with any associated changes in the molecular geometry or packing, will likely affect the rate of energy transfer, which must be fully understood in order to rationally design fully optimized optoelectronic devices.22 Therefore, an accurate prediction of the influence of molecular structure and associated crystalline packing on the exciton diffusion and energy transfer is critical for optoelectronic applications. Received: June 3, 2019 Revised: June 26, 2019

A

DOI: 10.1021/acs.jpcc.9b05297 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C In order to better understand the fundamental nature of electronic coupling in D−A−D aggregates, we investigate herein the Davydov splitting (DS) in squaraine dimers. DS arises from intermolecular interactions involving chromophores with nonparallel transition dipole moments (TDMs), resulting in two peaks in the absorption spectrum, referred to as the lower and upper Davydov components (LDC and UDC). In the Kasha model, both components are split symmetrically about the peak monomer absorption frequency, with the splittingthe DSequal to twice the transition dipole−dipole coupling. DS has been observed in the absorption spectra of squaraine dimers and higher order assemblies including thin films for OPV.1,7−9,23,24 The aim of the present work is to analyze the absorption spectral line shape in squaraine dimers using the ESM of Painelli and coworkers11−15 as a first step in a broader investigation to fundamentally understand the nature of energy transport in D−A−D complexes. Photophysical behavior diverging from the predictions of the conventional Kasha model (which is based on the Frenkel exciton theory) has been reported for aggregates of a watersoluble cationic squaraine dye by Belfield and co-workers.25 Such aggregates show a red-shifted absorption spectrum relative to the monomer (i.e., J-like) but are not fluorescent (i.e., H-like). Subsequently, Sanyal et al.14 and Zheng et al.26 showed that such “red-shifted H-aggregates” were natural outcomes of the ESM for aggregates of slipped D−A−D chromophores. In centrosymmetric chromophores, non-Kasha behavior derives from the intermolecular quadrupole−quadrupole interactions which stabilize the one-photon allowed excited states relative to the aggregate ground state.26 However, red-shifted H-aggregates have also been demonstrated in slipped dimers of bent D−A−D chromophores lacking inversion symmetry,27 thereby questioning the role played by permanent dipole−dipole interactions. The importance of permanent dipole−solvent interactions in asymmetric D−A−D′ squaraines was demonstrated by Shafeekh et al.,13 who showed that the inversion of the permanent dipole moment between the ground and bright state leads to inverse solvatochromism in the absorption band. In what follows, we use the ESM to analyze one-photon absorption in both linear (centrosymmetric) and bent (noncentrosymmetric) squaraine molecules and dimer complexes thereof. Specific applications are made to the covalently bound dimers of the SQA and SQB chromophores shown in Figure 1, recently investigated by Röhr et al.1 One of the advantages of working with the bent chromophores such as SQB is that optical transitions are now both one- and twophoton allowed, enabling one to fully parameterize the ESM using only the information obtained from linear spectroscopy in the visible and ultraviolet (UV) spectral regions.3,28−30 We will show that the ESM accounts quantitatively for the DS and general absorption line shape in both (SQA)2 and (SQB)2. The UDC and LDC in the dimer spectrum (which have orthogonal polarizations) are resolved theoretically, showing that the lower component is J-like, whereas the upper component is H-like with respect to vibronic signatures derived from the vibronic exciton theory.26,31−33 We further reveal a novel type of non-Kasha behavior arising from an interesting interplay between permanent dipole−dipole and transition dipole−dipole interactions in dimers of noncentrosymmetric chromophores.

Figure 1. Molecular structures of SQA and SQB.

2. ESM WITH VIBRONIC COUPLING: D−A−D MONOMERS In order to account for the electronic properties of a bent monomer, we appeal to the ESM,11−15 which identifies three essential states in a D−A−D chromophore: the “neutral” state with no charge-separation, |N⟩ and two degenerate zwitterionic states designated as |Z1⟩ (D+−A−−D) and |Z2⟩ (D−A−−D+). The charge distributions of the three states are indicated in Figure 2a. The arm length l is the distance between the negative charge center located at or near the squarillium center and either of the positive charge centers located at or near the nitrogen atoms of the indolenine moiety. The bend angle is α. The states |N⟩, |Z1⟩, and |Z2⟩ form a diabatic basis set with the energy-level diagram shown in Figure 2b; the neutral state

Figure 2. (a) Essential states in a bent D−A−D chromophore. The circles locate the positions of the nitrogen atom donors in the indolenine moieties. (b) Energy-level diagram for diabatic states (top) and adiabatic states (bottom) showing visible (g → c) and UV (g → e) transitions. B


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The Journal of Physical Chemistry C |N⟩ with the energy taken to be zero can generally couple to the zwitterionic states, (each having energy ηz) with the coupling strength, t z , as reflected in the monomeric Hamiltonian el Ĥ mon = ηz

∑

∑

|Za⟩⟨Za| − tz

a = 1,2

{|N⟩⟨Za| + h. c. }

a = 1,2

(1)

Diagonalizing the Hamiltonian yields three “adiabatic” states |g⟩ =

1 − ρ |N⟩ +

ρ /2 (|Z1⟩ + |Z 2⟩)

| c⟩ =

1/2 (|Z1⟩ − |Z 2⟩)

(2b)

| e⟩ =

ρ |N⟩ −

(2c)

(1 − ρ)/2 (|Z1⟩ + |Z 2⟩)

(2a)

where |g⟩ is the ground state and |c⟩ and |e⟩ are the first and second excited states, respectively, as shown in Figure 2b. The quantity, ρ, is defined as ρ = 0.5(1 − ηZ / ηz 2 + 8tz 2 )

Figure 3. Charge distributions defining the permanent dipole moments (left) and TDMs (right).

(3)

which represents the admixture of the zwitterionic component in the molecular ground state |g⟩. When there is a center of inversion (α = 180° in Figure 2a), the states, |c> and |e>, are respectively, one- and two-photon allowed. However, for the C2v “bent” geometry (α ≠ 180°), both states are one-photon allowed and therefore appear in a conventional absorption spectrum. This is fortunate as it allows us to uniquely determine ηz and tz from the peak positions of the visible and UV excitations (see below). Finally, the energies of the adiabatic states in eqs 2a−2c are given by Eg =

1 1 η − η 2 + 8tz 2 2 z 2 z

Ec = ηz Ee =

the lowest excited state |c⟩ is larger than that in the ground state. The dipole moment operator also gives rise to TDMs, μgj ≡ ⟨g|μ̂ |j⟩, j = c,e. For the |g⟩ → |c⟩ transition, the corresponding dipole moment is μgc =

μge =

(4a)

(|Z1⟩ + |Z 2⟩) and

1 2

(4c)

(|Z1⟩ − |Z 2⟩), positive charges (+e/

2) reside at the nitrogen atom centers of the indolenine moiety, whereas a negative charge (−e) resides at the squarylium center. The charge distribution results in permanent dipole moments, μjj ≡ ⟨j|μ̂ |j⟩, j = g,c,e, directed along the z-axis (C2 axis) for all three states, as depicted in Figure 3 with μgg = −ρel cos(α /2)k

(5a)

μcc = −el cos(α /2)k

(5b)

μee = −(1 − ρ)el cos(α /2)k

el Ĥ mon = Ĥ mon + ℏωvib

(7b)

∑

b†a ba

a = 1,2

(5c)

+ ℏωvibλdia

where k is the unit vector along the z-axis and the dipole moment operator is11,26 μ̂ = e l1|Z1⟩⟨Z1| + e l 2|Z 2⟩⟨Z 2|

ρ(1 − ρ) el cos(α /2)k

Hence, when α ≠ 180°, both transitions are one-photon allowed but with orthogonal relative polarizations, as depicted in Figure 3. Importantly, the TDM μgc for the lowest optical transition is normal to the permanent dipole moments in eqs 5a−5c. In what follows we focus entirely on the DS derived from the lowest optical transition. In D−A−D chromophores, there is also significant vibronic coupling involving the symmetric progression-forming vinyl stretching mode, as discussed in detail in ref 26. Two degenerate vibrational modes are included, one involving a D−A stretch and the other an A−D stretch, for the generally bent chromophore in Figure 2. The vibrational energy of each mode is ℏωvib. The complete monomer Hamiltonian including both electronic and vibrational degrees of freedom resembles a Holstein Hamiltonian

Importantly, in both symmetry-adapted states, 1 2

(7a)

where i is the unit vector along the x-axis (connecting the N atoms), while for the |g⟩ → |e⟩ transition

(4b)

1 1 η + η 2 + 8tz 2 2 z 2 z

ρ el sin(α /2)i

∑ a = 1,2

(b†a + ba + λdia)|Za⟩⟨Za| (8)

b†a (ba)

where creates (annihilates) a vibrational quantum of the D−A stretching model (a = 1) and the A−D stretching mode (a = 2). The vibronic coupling is quantified via the diabatic HR factor, λdia2, which is of the order unity for the vinyl stretching mode in most π-conjugated organic chromophores. Before introducing the dimer Hamiltonian (in Section 4), we pause to consider the monomer absorption spectrum based on the vibronic Hamiltonian in eq 8.

(6)

Here, l1 (l2) is the arm vector of magnitude l directed from the negative charge center to the “left” (“right”) positive charge center. For a linear molecule (α = 180°), the permanent dipole moments vanish, and the charge distribution in each state is dominated by the quadrupole moment Qxx.26 Note from eqs 5a−5c that μgg = ρμcc, so that the permanent dipole moment in C


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The Journal of Physical Chemistry C

3. MONOMER ABSORPTION SPECTRUM The absorption spectra of the centrosymmetric and noncentrosymmetric squaraine dyes, SQA and SQB dissolved in toluene, were measured by Lambert and co-workers1 and are reproduced in Figure 4 (>20 000 cm−1, unpublished). Both

reported in the Supporting Information of ref 3. As expected, our TD-DFT calculations (as well as those in ref 3) also showed much weaker UV activity for SQA. The presence of the UV peak in SQB conveniently allows us to assign the frequencies of both the red transition (|g⟩ → |c⟩) as well as the UV transition (|g⟩ → |e⟩) in the ESM (see Figure 2b). Using eqs 4a−4c and assuming no vibronic coupling, the two frequencies are related to ηz and tz in the following way g→c EM =

g→e EM =

1 1 η + η 2 + 8tz 2 2 z 2 z

(9a)

ηz 2 + 8tz 2

(9b)

By setting the left-hand-sides to the experimentally measured values, ηz and tz can be uniquely determined and appear in the caption of Figure 4. The corresponding value of ρ ≈ 0.46 for both SQA and SQB puts them in the so-called cyanine limit. The electronic parameters can be subsequently fine-tuned by adding vibronic coupling using the Hamiltonian in eq 8 and altering mainly the diabatic HR factor, λdia2, to reproduce the weak side-band (0−1), which is roughly 5 times less intense than the origin (0−0) band for both SQA and SQB, consistent with the cyanine limit. The 0−0/0−1 intensity ratio reflects the adiabatic HR factor, λ2, which was shown in ref 26 to be related to the diabatic HR factor by

Figure 4. Experimental absorption spectra of SQB and SQA in toluene from ref 1. Simulated spectra are also shown: for SQA, ηz = 2450 cm−1, tz = 9550 cm−1, ωvib = 1300 cm−1, and λdiab2 = 1; for SQB, ηz = 1900 cm−1, tz = 9200 cm−1, ωvib = 1200 cm−1, and λdiab2 = 1.4. In addition, the bend angle, α = 115°, was determined by equating the spectral areas of the calculated and measured UV peaks (not including the low-energy “pedestal” portion which arises from a B2-symmetry electronic transitionsee the Supporting Information).

λ2 =

(1 − ρ)2 2 λdia 2

(10)

Using the diabatic HR factor of 1.4 for SQB and the value of ρ = 0.46 from eq 3 gives λ2 = 0.20, from which the 0−0/0−1 ratio is 1/λ2 ≈ 5, in good agreement with the calculated spectrum in Figure 4. The final best fit spectrum for SQB is shown as the black dashed curve in Figure 4. Since the UV transition strength is sensitive to the bend angle α, the latter can also be approximately determined by fitting the integrated spectral area of the UV peak arising from just the state with the A1 symmetry, which gives α ≈ 115°. The simulated spectrum for SQA is also shown in Figure 4 (blue-dashed curve). Since there is no UV peak in the experimental spectrum, we obtained the best-fit spectrum by increasing ηz away from the SQB value in order to obtain the slightly higher-energy peak in the measured absorption spectrum (at ≈15 750 cm−1) and adjusted tz and λdia2 accordingly to obtain the measured 0− 0/0−1 ratio.

spectra are dominated by a peak in the red region of the spectrum, which is shifted slightly higher in energy for SQA versus SQB. As noted previously, the main difference in the spectra of the two chromophores lies in the UV region, where SQB displays a broadened transition peaking at approximately 26 000 cm−1 (very similar to the spectrum obtained using dichloromethane as the solvent29), while SQA shows virtually no UV activity. The difference is due to symmetry: In centrosymmetric chromophores such as SQA with the C2h point group symmetry, the state |e⟩ has gerade symmetry and can only be excited with two photons from the symmetric ground state. Conversely, SQB, with the C2v symmetry, lacks an inversion center, and the state |e⟩ becomes accessible via one-photon absorption. The ESM analysis in the previous section predicts |e⟩ to have an A1 symmetry, making the |g⟩ → |e⟩ UV transition polarized along the C2 axis (the z-axis in Figure 2) and therefore normal to the red transition, |g⟩ → |c⟩, which is polarized along the long axis containing the two nitrogen donors (x-axis in Figure 2). We also analyzed SQB and SQA using time-dependent density functional theory (TDDFT) calculations (CAM-B3LYP34 functional and the 6311+G(2d,p) basis set35), as described in greater detail in the Supporting Information. TD-DFT confirmed the presence of an A1 symmetry state in the UV region, approximately 1.4 × 104 cm−1 above the absorption origin in SQB, slightly greater than the experimental value. TD-DFT further showed the existence of a second UV state but with the B2 symmetry at a slightly lower energy than the A1 state, which is very likely responsible for the “pedestal-like” absorption feature in the measured spectrum in Figure 4. Similar polarization-dependent band assignments for SQB evaluated using TD-DFT were

4. ESM FOR D−A−D DIMERS In this section, we extend the ESM to D−A−D dimers with a geometry depicted in Figure 5a,b. The intermolecular orientations are obtained by rotating one chromophore by an angle θd about the axis indicated in Figure 5a with a vertical dashed line. The dimer geometry is chosen because of the similarity with the squaraine dimer complexes investigated in ref 1 and is discussed further in the following section. The two chromophores interact with each other Coulombically, as in the Kasha exciton model.36−38 As introduced by Painelli and co-workers,12,14,15 the dimer Hamiltonian is 2

Ĥdimer =

mon

∑ Ĥ n

+ V̂ (11a)

n=1

with the intermolecular Coulomb coupling given by D


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Figure 5. (a) Dimer geometry obtained through rotation by θd about the z-axis. In the unfolded (“linear”) geometry shown, θd = 180°, whereas in the folded geometry, θd = 0°. Note that the C2 axis, which rotates 1 into 1′ and 2 into 2′, is positioned at the red dot and directed along y (into the page). (b) View with z-axis out of the page for θd approximately 90°.

V̂ =

q (a , i)q2(b , j) 1 |ZaZb⟩⟨ZaZb| ∑∑ 1 4πεε0 a , b i , j |r1(a , i) − r2(b , j)|

Figure 6. (a) Energy-level diagram for D−A−D dimers showing only the lowest three energy levels necessary to describe optical absorption in the visible region. ΔDS and the midpoint transition energy Emp are indicated. Schematic representation of the charge distributions contributing to (b) the ground-state energy Egg, (c) the excitedstate energy, Egc, and (d) the transition dipole-transition dipole coupling.

(11b)

Here, the basis function |ZaZb⟩ in eq 11b indicates the electronic states of the two chromophores (Za for chromophore “1” and Zb for chromophore “2”). The Coulombic coupling term in eq 11b is calculated in the diabatic basis with the charge densities collapsed to the centers of the donor or acceptor sites, with the Coulombic interactions restricted to charges located on different molecules. The charge distribution which corresponds to the Za zwitterion (a = 1,2) on chromophore n is denoted as qn(a,i), where i = 1, 2, or 3 indicates the “left” donor, central acceptor, and “right” donor charge centers, respectively, as indicated in Figure 2. Hence, for the zwitterion Z1, we have, qn(1,i = 1) = +e, qn(1,i = 2) = −e, and qn(1,i = 3) = 0 for the nth chromophore. Finally, rn(a,i) is the position vector for the corresponding point charge, which is defined by the dimer geometry. ε and ε0 are the (relative) dielectric constant and vacuum permittivity, respectively. Here, ε accounts for screening effects derived from electronic excited states beyond the essential states. A detailed description of the electronic/vibrational basis set used to represent the Hamiltonian in eqs 11a and 11b can be found in the Supporting Information. 4.1. Weak Intermolecular Coupling. Following ref 26, we first consider the case with no vibronic coupling, treating the Coulomb coupling perturbatively. The latter is justified as long as intermolecular coupling remains small compared to the g → c and c → e transition energies (see Figure 2b). The zeroorder wave functions of importance for the lowest-energy optical transitions in a dimer are indicated in Figure 6a. The ground state is a direct product of the two adiabatic monomer ground states in eq 2a, that is, |gg⟩ ≡ |g⟩1 ⊗ |g⟩2. All dimer states are defined with respect to their symmetries under a C2 rotation, with the C2 axis defined in Figure 5b. Accordingly, |gg⟩ is symmetric while the (zeroth-order) excited states based on a single |c⟩ excitation can be symmetric or antisymmetric

|gc⟩S =

1 {|gc⟩ + |cg⟩} 2

(12a)

1 {|gc⟩ − |cg⟩} (12b) 2 ̂ where |cg⟩ ≡ C2|gc⟩. In a similar fashion, the highest-energy symmetric and antisymmetric states, |ge⟩S and |ge⟩AS, with energies in the UV can also be defined. The energies of the three lowest-energy states correct to first order are given by |gc⟩AS =

(1) Egg ≈ 2Eg + Egg

(13a)

ES ≈ Eg + Ec + ES(1)

(13b)

(1) EAS ≈ Eg + Ec + EAS

(13c)

where Eg and Ec are defined in eqs 4a−4c. First-order corrections derive from the diagonal and off-diagonal matrix elements of the Coulombic interaction term, V̂ , in eq 11b.26 Since the ground state is nondegenerate, its first-order correction is the diagonal matrix element ρ2 (1) Egg = ⟨gg|V̂ |gg⟩ = (V11 + V22 + V12 + V21) (14) 4 where Vij ≡ ⟨ZiZj|V̂ |ZiZj⟩. The right-hand-side is equivalent to the Coulombic interaction between the molecular charge distributions shown in Figure 6b, where each ground-state chromophore is represented by three charge centers. For centrosymmetric molecules (α = 180°), the dominant term in a multiple expansion is the quadrupole−quadrupole interaction, as discussed at length in ref 26, whereas a bend angle α < 180° leads to a dominant permanent dipole−dipole interaction. When the two chromophores are sufficiently far E


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The Journal of Physical Chemistry C apart that the point dipole approximation becomes accurate, one obtains (1) Egg ≈

2μgg

and (1) EAS = (⟨gc|V̂ |gc⟩ − ⟨gc|V̂ |cg⟩) =

2

4πεε0d3

d≫l

with μgg defined in eq 5a. Note the first-order correction is positive since the two permanent dipole moments point toward each other in the geometry of Figure 5 (see also Figure 3) and, for large separations, lie approximately along the line connecting the two mass centers. Note that in cases where μgg = 0, the leading term in the multipole expansion becomes the quadrupole−quadrupole term, which scales as the inverse fifth power of the intermolecular distance. By contrast, the zero-order symmetric and antisymmetric excited states in eqs 12a and 12b are degenerate, so that the first-order energy correction depends on both the diagonal26 ρ ⟨gc|V̂ |gc⟩ = ⟨cg|V̂ |cg⟩ = (V11 + V22 + V12 + V21) 4

5. DIMER ABSORPTION SPECTRUM AND DS The transitions, |gg⟩ → |gc⟩S and |gg⟩ → |gc⟩AS, indicated in Figure 6a have oscillator strengths proportional to |(μ(1) gc + 2 (1) (2) 2 (n) μ(2) )| and |(μ − μ )| , respectively, where μ is the g →c gc gc gc gc TDM for chromophore n (see Figure 5b). Hence, as long as the two dipole moments are not exactly parallel, two peaks will appear in the dimer absorption spectrum separated by the DS, as indicated in Figure 6a. To first order (and in the absence of vibronic coupling), the DS is given by ρ ΔDS = |ES − EAS| = |V11 + V22 − V12 − V21| (20) 2

(16a)

and off-diagonal

where we have used eqs 19a and 19b. In agreement with the Kasha model, the DS from eq 20 is twice the transition dipole−transition dipole coupling in Figure 6d, which reduces to 2μgc2|cos θd|/4πεε0d3 in the limit of well-separated chromophores. One can also evaluate the “midpoint” transition energy located half-way between the two Davydov components,

(16b)

matrix elements. The diagonal term in eq 16a represents the intermolecular Coulombic interaction between the groundstate and excited-state charge distributions shown in Figure 6c. The charge distribution for the unexcited molecule is identical to that in Figure 6b; the distribution for the excited molecule is similar but lacks the factor of ρ since the state |c⟩ is a pure zwitterionic state; see eq 2b. For well-separated chromophores, we get ⟨gc|V̂ |gc⟩ ≈

2μgg μcc 4πεε0d3

Emp ≡

d≫l (17)

Δmp =

4πεε0d3

(21a)

ρ − ρ2 (V11 + V22 + V12 + V21) 4

(21b)

Interestingly, the midpoint frequency shift, Δmp , is determined entirely by the coupling between the three-center charge distributions in Figure 6b,c, and unlike the DS, it is sensitive to the permanent dipole−dipole interactions for molecules which lack an inversion center. In this case, eq 21b reduces to Δmp =

2

⟨gc|V̂ |cg⟩ ≈

ES + EAS − Egg 2

as indicated in Figure 6a. The shift of Emp from the monomer transition energy is denoted as Δmp ≡ Emp − Eg→c M and is also a measure of intermolecular coupling. In the Kasha scheme, which includes only interactions between TDMs, Δmp = 0 and the two Davydov components split symmetrically about the monomer absorption frequency. Inserting the first-order corrected energies from eqs 13a−13c into eq 21a with subsequent use of eqs 14, 19a and 19b gives the midpoint frequency in the absence of vibronic coupling

which, because μgg = ρμcc, is larger than the first order energy shift experienced by the ground state in eq 15 by a factor of 1/ ρ. Hence, the combined influence of the two permanent dipole interactions in eqs 15 and 17 is to increase the transition frequencies of the upper and lower Davydov components in the dimer relative to the monomer (see Section 5, eq 22). The off-diagonal matrix element in eq 16b has a different structure all together because it is equivalent to the interaction between two chromophores, each represented by a two-center transition charge distribution, as indicated in Figure 6d, with opposite charges (±ρe/2) placed on the donor groups and with no charge on the central squarillium. The charge distribution is responsible for the TDM, equal to ρel sin(α/ 2) from eq 7a. Indeed the matrix element in eq 16b reduces to the point dipole−dipole interaction when d is very large compared to the arm length l μgc cos θd

(19b)

With the zeroth-order wave functions and (first-order) energies of the ground- and first-excited states in hand, we can now explore the nature of optical transitions in dimer complexes of D−A−D chromophores.

(15)

ρ ⟨gc|V̂ |cg⟩ = (V11 + V22 − V12 − V21) 4

ρ (V12 + V21) 2

2(μgg μcc − μgg 2 ) 4πεε0d3

d≫l (22)

for well-separated chromophores. Note that Δmp in eq 22 is independent of θd, as the two vectors μgg and μcc remain antiparallel for all θd (see Figure 6b,c). Because μgg = ρμcc from eqs 5a and 5b and ρ ≈ 0.46 for SQB, the midpoint shift, Δmp, is positive. Interestingly, if Δmp exceeds ΔDS/2, then both Davydov components are blue-shifted relative to the peak monomer absorption frequency, a highly non-Kasha behavior discussed in greater length in Section 7. Finally, when α = 180°, as in SQA, the midpoint frequency derives from

d≫l (18)

Taken together, the matrix elements in eqs 16a and 16b form a 2 by 2 submatrix, which, upon diagonalization, yields the proper first-order energy corrections for the symmetric and antisymmetric states26 ρ ES(1) = (⟨gc|V̂ |gc⟩ + ⟨gc|V̂ |cg⟩) = (V11 + V22) (19a) 2 F


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The Journal of Physical Chemistry C quadrupole−quadrupole interactions, which can induce blue or red shifts in Δmp (see below). Figure 7 shows a series of dimer absorption spectra for various torsion angles θd, as parameterized for SQB (see

In the linear configuration (θd = 180°) corresponding to Figure 7c, the TDMs are arranged head-to-tail (along the xaxis), as in a J-aggregate, and the spectrum responds accordingly with a dominant red-shifted peak arising from the antisymmetric exciton polarized along the x-axis. Although harder to appreciate, the midpoint frequency remains slightly blue-shifted relative to the monomer peak frequency. As opposed to the H-aggregate case (θd = 0°), including vibronic coupling now results in an increased A1/A2 ratio relative to the monomer value, also in agreement with the vibronic spectral signatures derived for J-aggregates from the Frenkel exciton theory.33 For intermediate angles (0 < θd < 180°), the spectrum displays both the LDC polarized along the x-axis and the UDC polarized along the y-axis. This arises because the nonparallel TDMs result in the oscillator strength divided over both the symmetric and antisymmetric excited-state transitions. The ratio of the oscillator strengths for the two components is set by the relative directions of the two monomeric TDMs, μ(1) gc and μ(2) gc , indicated in Figure 5b fUDC fLDC

≈

|(μgc(1) + μgc(2) )|2 |(μgc(1) − μgc(2) )|2

(23)

where f LDC (f UDC) is the spectrally integrated oscillator strength for the LDC (UDC) band and μ(i) gc is the TDM of the i-th chromophore (i = 1,2). The TDMs have equal magnitude (2) μ(1) gc = μgc = ρel sin(α/2)see eq 7bbut form an angle θd, as indicated in Figure 5b. Hence, when θd = 90°, as in Figure 7b, the TDMs are orthogonal, and the ratio of oscillator strengths from eq 23 is approximately unity, consistent with the roughly equal area peaks polarized along x and y in the limit of no vibronic coupling. However, even in the presence of vibronic coupling, where each Davydov component responds differently to the presence of vibrations, the integrated oscillator strengths remain approximately equal, as can be appreciated in the inset in Figure 7b. As shown in Figure 7b, the nonpolarized spectrum for θd = 90° consists of three main peaks when vibronic coupling is included. The vibronic structure can be traced back to the superposition of a J-like LDC and an H-like UDC: as the figure inset shows, the main peak which defines the absorption origin is due to the vibronic origin A1 of the LDC while the two remaining higher-energy peaks arise from the origin and first vibrational side-band peaks A1 and A2, respectively, of the UDC. Note that the A1/A2 ratio in the UDC remains attenuated compared to the monomer (i.e., “H-like”), while the opposite holds for the J-like LDC. Such polarized H/J behavior for the UDC and LDC has also been established using the Frenkel exciton theory.32 Because the UDC has two vibronic peaks (A1 and A2) of similar intensity, the assignment of the DS can be ambiguous. To remain consistent with Röhr et al.,1 we define the DS to be the spectral separation between the peak in the LDC spectrum (A1) and the main peak in the UDC spectrum (A2), as indicated in the Figure 7b inset. Figure 8 shows how the DS and the midpoint frequency shift, Δmp, vary as a function of torsion angle θd for the SQB dimer from Figure 7 (blue curves) and for a SQA-like dimer with a bend angle α = 180° (red curves). In both cases, vibronic coupling was neglected. Solid curves are first-order values evaluated from eq 21b, while the dots are exact values evaluated numerically. The DS in both dimers is maximum for the folded forms defined by small θd, diminishing by almost a

Figure 7. Calculated SQB dimer absorption spectra (blue, magenta, and red) in the visible region as a function of increasing dihedral angle θd in (a−c). Solid (dashed) spectra include (exclude) vibronic coupling. Black spectra in (a) correspond to the SQB monomer. The polarization directions of the peaks are also indicated with the x and y axes defined in Figure 5. The inset of (b) resolves the two polarization components. In all spectra, ηz = 1900 cm−1, tz = 9200 cm−, and α = 115°. The vibronic coupling is described using ωvib = 1200 cm−1 and λdia2 = 1.4. All dimer spectra assume the simplified geometry in Figure 5 with d = 0.7 nm. Dashed arrows in the top and bottom panels indicate the position of the exciton transition responsible for the other Davydov component (in the absence of vibronic coupling).

caption). Solid (dashed) spectra include (exclude) vibronic coupling. Vibronic coupling is included using the Hamiltonian in eq 11a represented in the vibronic basis set described in the Supporting Information. The expression for the absorption spectrum is also included in the Supporting Information. In the maximally folded configuration (θd = 0°) in Figure 7a, the spectrum is dominated by an intense, blue-shifted peak polarized along the y-axis (defined in Figure 5b) because of the symmetric exciton. This is the expected response from an Haggregate since the TDMs of the two chromophores are arranged in a side-by-side manner along the y-axis. Note that the transition to the antisymmetric exciton (indicated by the red arrow in the figure) remains dark since the TDMs cancel. Hence, there is no identifiable DS. The two excitons are split asymmetrically about the peak monomer frequency, with the midpoint frequency slightly blue-shifted above the monomer peak because of the three-center interactions from Figure 6b,c. Vibronic coupling (solid spectra) enhances the blue shift of both the monomer and dimer and leads to a very strong attenuation of the vibronic peak ratio A1/A2 compared to the monomer value, in agreement with the H-aggregate behavior derived from the exciton theory.33 G


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Figure 8. DS (ΔDS) and midpoint frequency shift (Δmp) as a function of the dihedral angle for dimers of centrosymmetric (red) and noncentrosymmetric (blue) squaraines in the geometry of Figure 5 (vibronic coupling is not included). Solid curves are first-order results, while dots are the exact values evaluated numerically. In both dimers, ηz = 1900 cm−1 and tz = 9200 cm−1. Note that the first-order DS for both dimers is identical since the arm lengths are taken such that the TDMs are equal for both chromophores (see eq 7a).

factor of 4 in the linear form (θd = 180°). Note that to first order the two dimer curves for ΔDS are perfectly overlapping, because the arm length in the SQA-like chromophore was slightly reduced compared to the bent form, in order that the transition dipole “length”, l sin(α/2), is the same for both chromophores. Figure 8 also shows that Δmp for SQB2 remains substantially smaller than ΔDS and is uniformly positive for all θd. For the dimer with the centrosymmetric chromophore, Δmp is dominated by quadrupole−quadrupole interactions and is considerably larger at small θd, eventually becoming negative as θd approaches 180°.

Figure 9. Measured absorption and PL spectra of (a) (SQA)2 and (b) (SQB)2 in toluene from ref 1. Also shown are the calculated spectra using the single-molecule electronic and vibrational parameters from Figure 4 and the dimer geometries from Figure 10 in the Hamiltonian of eq 11a. The final spectra represent an equal-weight average over the two minimal energy geometries in Figure 10. In order to bring the main absorption peaks in the measured and calculated spectra into agreement, the calculated spectra were red-shifted by 505 and 518 cm−1 for SQA and SQB, respectively. In addition, the arm length l was reduced by 5% in both cases for optimal agreement. Finally, the calculated PL spectra include a 240 cm−1 Stokes shift.

6. RESULTS: COMPARISON TO THE EXPERIMENT Lambert and co-workers have published several works dealing with covalently linked SQA and SQB dimers1−4 and polymers,5,6 demonstrating a pronounced DS for both molecules. The measured absorption and photoluminescence (PL) spectra for (SQA)2 and (SQB)2 are reproduced in Figure 9a,b, respectively. In both cases, the absorption spectrum has the three-peak form in agreement with the theory describing Figure 7b, with a dominant absorption origin. The DS, as evaluated from the spectral separation of the two main peaks, is approximately 1500 cm−1 for both molecules. Moreover, the PL is dominated by the 0−0 peak with a small Stokes shift of only a couple of hundred wave numbers. The PL spectrum also displays a smaller but significant 0−1 side-band, lower in energy by 1 vibrational quantum. In order to obtain the most accurate simulations, we utilized more refined molecular geometries than those in Figure 5, obtained by minimizing the ground-state energy of the dimer complexes using DFT calculations with the CAM-B3LYP functional and the def2-SVP basis set. Interestingly, there are essentially two minimal energy structures for both (SQB)2 and (SQA)2: an approximately linear structure with θd = 142° in agreement with ref 1 and a folded structure with θ d approximately 38°, as depicted in Figure 10. For both SQA

Figure 10. Minimal energy dimer structures of SQA (left) and SQB (right) evaluated using DFT calculations (CAM-B3LYP/def2-SVP).

and SQB dimers, the energy difference between the bent folded and linear geometries is less than kT at room temperature (see Supporting Information) so that both structures should be approximately equally represented. A mixture of both species in solution has also been suggested by Ceymann et al.3 H


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dimers. The additional red shift is justified by the enhanced charge delocalization associated with the covalent bond connecting the two phenylene units and completely masks the much smaller midpoint frequency blue shift, Δmp. The presence of the bond adds asymmetry in the donor charge distribution not accounted for in the present model. More sophisticated models would allow for interchromophore charge transfer39 but would come at the expense of additional parameters and a much larger basis set. Finally, the calculated PL spectra for both chromophore dimers (see ref 26 for details) also agrees well with the experiment. The spectrum is dominated by the 0−0 peak as expected for J-aggregates or, more generally, in aggregates with an allowed LDC.40 To better appreciate the vibronic structure of the LDC and UDC, we show in Figure 12a−d the polarization-resolved spectra for both folded and linear forms of (SQA)2 and (SQB)2 that were used to construct the unpolarized spectra in Figure 9. As in Figure 5, the y-axis coincides with the C2 rotation axis. Based on the analysis of Section 5, the symmetric exciton polarized along the y-axis is responsible for the UDC, whereas the antisymmetric exciton polarized normal to the yaxis is responsible for the LDC. As shown in Figure 12a−d, the LDC is dominated by a red-shifted peak A1 with a much weaker vibronic side-band, A2. The A1/A2 ratio of spectral intensities is significantly larger than what exists in the monomer (see Figure 4). By contrast, the blue-shifted UDC is H-like, with a significantly reduced A1/A2 ratio. Hence, the LDC (UDC) retains the same J-like (H-like) vibronic signatures as derived using the vibronic exciton theory,32 consistent with the conclusions of ref 26. The figure shows that the “three-peaked” spectrum measured for both dimer species by Röhr et al.1 derives from the A1 peak of the J-like LDC followed by the A1 and A2 peaks of the H-like UDC.

The most significant difference between the geometries in Figure 10 and the simplified one in Figure 5 is that the rotation axis (red line in Figure 10) is no longer normal to the line connecting the two donor centers of the same squaraine chromophore (red dots). As a result, when θd = 0°, the two (2) molecular TDMs (μ(1) gc and μgc , see Figure 5) are not parallel, that is, they are no longer “side-by-side” as was the case for the simpler H-aggregate geometry in Figure 5. From eq 23, this means that both Davydov components will contribute to the absorption spectrum when θd = 0°. To more clearly appreciate the absorption line shapes for the dimers in Figure 10, we show simulated spectra for both (SQA)2 and (SQB)2 in Figure 11a,b, respectively, as a function

7. CONCLUSIONS Using the ESM with vibronic coupling, we analyzed in detail the polarized absorption spectrum, and, in particular, DS, in dimers of D−A−D chromophores, making direct comparisons to the measured spectra of the covalently bound (SQA)2 and (SQB) 2 dimers of Rö hr et al. 1 The SQA and SQB chromophores in Figure 1 have C2h and C2v symmetry, respectively, resulting in permanent dipole moments (quadrupole moments) in the ground and excited states of SQB (SQA). In the ESM, the DS involving the lowest-energy optical transition (to state |c⟩) is determined by the two-center charge distributions shown in Figure 6d, which account for the interaction between the molecular (extended) TDMs. In both SQA and SQB dimers, the calculated DS is approximately 1500 cm−1, in very good agreement with the experiment.1 Dominant permanent dipole−dipole interactions in SQB and quadrupole−quadrupole interactions in SQA arise from interactions between the molecular three-center charge distributions in Figure 6b,c, resulting in the addition of a uniform spectral blue shift, Δmp, to both the UDC and LDC, which is not accounted for in the Kasha model. However, in both the (SQA)2 and (SQB)2 dimers, Δmp is masked by a larger red shift because of charge delocalization through the bonded phenyl groups. Using the ESM, the “three-peak” structure of the (unpolarized) absorption spectra of (SQA)2 and (SQB)2 from ref 1 is shown to consist of a superposition of a J-like LDC and an H-like UDC, as described in Section 6, with similar J-like and H-like vibronic signatures as those predicted from the Frenkel exciton theory.26,32,33 However, unlike the

Figure 11. Simulated dimer spectra of (SQA)2 (a) and (SQB)2 (b) as a function of θd based on the geometries in Figure 10 and the Hamiltonian in eq 11a with the monomer vibronic parameters from Figure 4. M indicates the peak monomer absorption frequency from Figure 4.

of the dihedral angle θd. We utilize the same vibronic parameters as were derived in the monomer analysis in Section 3 (see Figure 4). We have confirmed that in the folded structures (θd ≈ 0°), the two main peaks are associated with the upper (H-like) and lower (J-like) Davydov components. By contrast, the linear structures (θd ≈ 180°) are dominated by the J-like LDC, with a dominant red-shifted peak followed by a weak vibronic side-band. Interestingly, the figure shows that the DS is not that strongly dependent on the dihedral angle. Moreover, the midpoint frequency, Δmp, is only slightly blueshiftedby less than 200 cm−1relative to the peak monomer frequency (labelled “M”) in both dimers. In order to compare with the experimental spectra for (SQA)2 and (SQB)2, we show in Figure 9a,b the calculated unpolarized spectrum obtained by adding together the spectra corresponding to the folded and linear minimal energy configurations from Figure 10 and utilizing the full vibronic Hamiltonian from eqs 11a and 11b. In order for the position of the main absorption peak in the calculated and measured spectra to align, it was necessary to insert an overall spectral red shift of approximately 500 cm−1 for both SQA and SQB I


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Figure 12. Calculated polarization-resolved absorption spectra for folded (a) and linear (b) conformations of (SQA)2 and for folded (c) and linear (d) conformations of (SQB)2 from Figure 10. The y-axis is along the C2 rotation axis (see Figure 5). The blue-shifted (H-like) UDC is symmetric with respect to the C2 rotation, while the red-shifted (J-like) LDC is antisymmetric. All parameters defining the Hamiltonian in eq 11a are identical to those used in Figure 9 (including the 500 cm−1 red shift). The sum of all four spectra in (a,b) constitute the calculated spectrum in Figure 9a, while the sum of all four spectra in (c,d) constitute the calculated spectrum in Figure 9b.

2(1 − ρ) > tan 2(α /2)|cos θd|

exciton model, the ESM provides a self-consistent quantitative account of how the optically-induced charge redistribution characteristic of D−A−D chromophores impacts intermolecular interactions (and vice versa). As such, the ESM allows one to further analyze the interplay between permanent dipole− dipole (or quadrupole−quadrupole) interactions and transition dipole−dipole interactions in creating spectral shifts and splittings. In some cases, non-Kasha behavior results, such as in the slipped dimers of D−A−D chromophores studied previously,14,26,27 which behave like oxymoronic “red-shifted H-aggregates” over a range of slip angles between the conventional H and J-aggregate domains.26,27 For the twisted dimer geometries studied in the present paper, a very peculiar non-Kasha behavior manifests when the blue-shift, Δmp, of the Davydov components is so large, that both components lie higher in energy than the monomer transition frequency, somewhat akin to a blue-shifted J-aggregate. This happens when Δmp is greater than half the DS. For two well-separated SQB chromophores abiding by the simplified geometry in Figure 5, use of eqs 18 and 22 shows that the condition Δmp > ΔDS/2 is equivalent to 2(μgg μcc − μgg 2 ) 4πεε0d3

>

μgc 2 |cos θd| 4πεε0d3

(25)

Equation 25 shows that the non-Kasha behavior is easier to realize for smaller bend angles α which favor large permanent dipole moments. Assuming α = 115° and ρ = 0.46 as for SQB, condition (25) becomes 116° > θd > 64°. Interestingly, for bend angles α < 92°, condition (25) is satisfied for all θd, showing that the LDC and UDC remain higher in energy than the monomer frequency, independent of twist angle θd, a highly unusual result that cannot be obtained with the Kasha theory (which focusses entirely on transition dipole−dipole interactions). In our (SQB)2 simulations based on the geometries of Figures 5 and 10 (with d = 0.7 nm), we have not observed such highly divergent non-Kasha behavior; in all cases, the two Davydov components straddle the monomer transition energy, although not exactly symmetrically. This is mostly due to the large value of bend-angle α (115°) and the close proximity of chromophores so that the point dipole approximation assumed in eqs 24 and 25 is no longer justified. A successful observation of blue-shifted Davydov components will require dimers that are not covalently bound in order to prevent delocalization-induced red shifts. The ESM allows further exploration of nonlinear optical properties through the adiabatic state |e⟩, which in SQA can only be reached through two-photon absorption.12,14,16,17 In SQB, the lack of an inversion center allows |e⟩ to be also reached via one-photon (UV) absorption. Interestingly, unlike the lowest excited state |c⟩, the state |e⟩ in SQB shows no DS since the g → e transition is polarized normal to the long molecular axis (see Figure 5a). Overall, however, the main advantage over the Frenkel model in treating aggregates of D− A and D−A−D chromophores lies in its ability to account for changes in the ground- and excited-state molecular propertiesfor example, the permanent dipole or quadrupole momentsas a function of intermolecular interactions or solvent polarity. Indeed, Terenziani and Painelli41 have shown that the zwitterionic character of the ground molecular state

d≫l (24)

representing an interesting interplay between permanent dipole moments and TDMs. Note that the permanent dipole moments show no θd dependence, since for the geometry of Figure 5, they remain antiparallel for all dihedral angles. Moreover, the left side of condition (24) is overall a positive quantity since μgg < μcc (see eqs 5a and 5b), expressing the fact that the permanent dipole moment interactions blue shift the excited state more than the ground state, leading to an overall increase in the transition energy. Further inserting the expressions for the dipole moments from eqs 5a and 5b, 7a into eq 24 gives J


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can change dramatically with increasing intermolecular interaction strength in various one-dimensional lattices. We plan in the future work to compare the ESM and Frenkel exciton models with regard to energy transfer in larger aggregates in order to account for the rapid transfer observed in SQA and SQB polymers (N ≫ 2).5 With the promise of such an improved understanding of D−A materials, one can more accurately predict and more effectively design new candidates for a variety of optoelectronic applications such as OPV where, for example, increased energy transfer and greater exciton diffusion lengths (in addition to D−A based energy level control) will lead to a relaxation in the need for longrange morphology control and a greater potential for device reproducibility and commercial manufacture.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b05297.

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Basis set used to represent the vibronic Hamiltonian, calculation of the absorption and emission spectra, and quantum chemical calculations (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Frank C. Spano: 0000-0003-3044-6727 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was carried out with the financial support for C.Z. and F.C.S. from the National Science Foundation (SusChEM1603461), for C.J.C. from the National Science Foundation (CBET-1603372), and for D.B. from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 398287490. We would like to thank Christoph Lambert for generously providing us with the experimental data. Finally, we thank Chenyu Zheng for valuable discussions.

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REFERENCES

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Davydov Splitting in Squaraine Dimers | The Journal of Physical

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