Unravelling the Origin of the Vibronic Spectral Signatures in an Excitonically Coupled Indocarbocyanine Cy3 Dimer

Abstract

The indocarbocyanine Cy3 dye is widely used to probe the dynamics of proteins and DNA. Excitonically coupled Cy3 dimers exhibit very unique spectral signatures that depend on the interchromophoric geometrical orientation induced by the environment, making them a powerful tool to infer the dynamics of their surroundings. Understanding the origin of the dimeric spectral signatures is a necessity for an accurate interpretation of the experimental results. In this work we simulate the vibronic spectrum of an experimentally well-studied Cy3 dimer, and we explain the origin of the experimental signatures present in its linear absorption spectrum. The Franck-Condon harmonic approximations among other tests are used to probe the factors contributing to the spectrum. It is found that the first peak in the absorption spectrum originates from the lower energy excitonic state while the next two peaks are vibrational progressions of the higher energy excitonic state. The polar solvent plays a crucial role in the appearance of the spectrum, being responsible for the localized S₁ minimum which results in increased intensity of the first peak.

Graphical Abstract

1. Introduction

The indocarbocyanine dye monomer, 1,1’-dimethyl-3,3,3’,3’-tetramethylindocarbocyanine (Cy3), and its dimers (see Figure 1 for structures) are commonly used as probes in optical studies on proteins and DNA.^1–6 The monomeric optical spectra have been shown to be mainly independent of the environment,⁷ while the dimeric spectra are very sensitive to the surroundings. Figure 2 shows the spectrum of the monomer compared to spectra of dimers in different environments. The monomeric spectrum has unique spectral bands around 18193 cm⁻¹, 19438 cm⁻¹, and 20694 cm⁻¹. This picture changes when Cy3 is excitonically coupled to another Cy3 chromophore, and the spectrum is highly dependent on the surroundings. As shown in Figure 2a, incorporating the dimers in a complex and relatively flexible environment like DNA (Dimers 2 and 3) changes the picture significantly compared to a rigidly linked dimer in methanol (Dimer 1). Dimers 2 and 3 incorporate Cy3 in two different DNA sequences, and the spectra in Figure 2a make it clear that the spectra are sensitive to the DNA sequence as well. Dimer 1, on the other hand, includes two rigidly linked Cy3 monomers and shows the most dramatic differences in the spectrum compared to the monomer, as shown in a closer look in Figure 2b. Compared to the monomeric spectrum, the spectrum of Dimer 1 exhibits three prominent and blue-shifted peaks with a clear change in the relative intensity order. This spectral sensitivity to the environment is a very useful tool to extract structural and dynamical information about these systems.^5,8–10

Figure 1:

(a) Indocarbocyanine dye Cy3 monomer optimized in the ground state showing the polymethine chain atoms numbered from 1 to 5 and the neighboring nitrogens. (b) Cy3 dimer (Dimer 1) optimized in the ground state showing the numbering of the polymethine chains and the neighboring nitrogens. d is the separation distance between the center of mass of the individual monomers. (c) Side view of the dimer showing the twist angle $\phi$ between the planes of the monomers. Both systems are optimized using $\omega$B97XD/Def2SVP in implicit methanol via PCM.

Figure 2:

(a) Experimental spectra of the monomeric⁹ and the dimeric Cy3 in three different experimental environments (1) methanol⁶ (Dimer 1) (2) DNA (5′-CAG TCA TAA TAT GCGA/Cy3/G CGA TTA TAT ATG CTT TTA CCA CTT TCA CTC ACG TGC TTA C-3′)⁹ (Dimer 2), (3) DNA (5′-ATT CAG ATT TTT TTT TTT TTT TTT TTT T/Cy3/TT TTT AGT TGA A-3′)¹⁵ (Dimer 3). Only the 5′−3′ sequence is shown. (b) Experimental spectra of the monomer⁹ and Dimer 1.⁶ Both experiments were done in methanol. The spectrum of Dimer 1 is reproduced from Ref.⁶ with permission from Nature Chemistry, copyright 2014, Springer Nature Limited. The spectra of the monomer and Dimer 2 are reproduced from Ref.⁹ with permission from the Journal of Royal Society of Chemistry, copyright 2019 RSC. The spectrum of Dimer 3 is adapted from Ref.¹⁵ with permission from the Journal of American Chemical Society, copyright 2016 ACS.

The unique spectra of the dimers in different environments can be explained by the excitonic coupling that depends on the relative orientation of the aggregate monomers.^5,8–10 The coupling between the electronic and vibrational degrees of freedom of two dipole-coupled cyanines leads to the formation of a new set of excitonic states whose properties depend on the relative orientation of the monomers.^8–14 Understanding the dynamics of the excited states and the excitonic and vibronic couplings between the interacting chromophores is crucial for the proper interpretation of the spectral signatures and for extracting information about the local conformations of the dyes and their surroundings.

There are several approaches that are used to explain the dimeric spectral shapes. The Frenkel exciton model¹² allows for a description of the spectral properties of excitonically-coupled systems using the electronic properties of the individual chromophores and the coupling between their individual vertical excitations.^7,12,16 Approximations to this model were developed and they are widely used. The simplest and most commonly used is the point-dipole approximation.^5,6,17–19 The extended-dipole,²⁰ transition charges,^21–24 and transition density^25,26 are better approximations of the electronic charge distributions improving the point dipole.⁷ Ab initio electronic excitonic coupling calculations on the whole aggregate systems have been possible due to the advancement in computational power, yet it is still very expensive to use for large systems at an adequate level of electronic structure theory.⁷ All these methods suffer from inadequate inclusion or complete ignorance of the vibrational and vibronic contributions. On the other hand, the Holstein-Frenkel Hamiltonian model^27–29 is a more appropriate model and it has been used by many research groups;^{5,6,8,9,30–33} it allows for the inclusion of the vibronic effects explicitly as it accounts for the internal electronic-vibrational motions within each chromophore, and the resonant electronic interaction that couples the two chromophores together. The drawback of this model is that the model parameters are obtained from the properties of the monomers, which might be different from the supermolecule, especially if the monomers are coupled to the surrounding environment. Another problem with this approach, and most of the previously discussed approaches, is that the coupling and geometrical parameters are obtained from the best fit of the Hamiltonian to the experimental spectra, which are subject to false positive results. For example, possible changes in the transition dipole moment magnitude and orientation are not taken into account. Since this model usually accounts for one effective vibrational mode, contributions from other modes, especially low-frequency modes, are not taken into account.³⁴

The Franck-Condon (FC) method^35–40 is widely used to simulate the spectra of molecules especially those with strong vibrational contributions, as it uses the vibrational wavefunctions of the ground and excited states to calculate FC overlaps.^40–43 The FC approximation accounts for the dipole-allowed transitions while it is necessary to use the Franck-Condon/Herzberg-Teller (FCHT) approach^44,45 to account for the contributions from weak transitions. In principle, the FC approach is superior to the previously discussed approaches because it takes into account the vibronic effects, the dynamics of the excited states, the multiple vibrational excitations, the combined vibrational excitations along with equal treatment to the contributions from all the normal modes.³⁴ Although it has been primarily applied to monomers, there have been some attempts to use the Franck-Condon approach to simulate the spectra of excitonically coupled systems. In the work of Nicoli et al.,¹⁵ they used the FC approach and B3LYP to simulate the spectrum of a Cy3 dimeric system in DNA. In their work, a weighted linear combination of the FC spectrum of the monomer and the dimer gave them the best fit to the dimeric experimental spectrum. To the best of our knowledge, there is no published work that benchmarked the proper way to use the FC approach to simulate the spectra of excitonically-coupled systems.

The Cy3 homodimer studied by Halpin et al.⁶ and Duan et al.⁵ (Dimer 1) consists of two Cy3 monomers covalently linked to each other (see Figure 1), and it was studied experimentally in methanol. The fact that the monomers are rigidly linked to each other and solvated in a homogeneous environment makes it an ideal controlled system to study. The linkers, in particular, constrain the conformation of the system making it easier to identify the correct structure. Understanding the spectral signatures of this system is of great importance to understanding the dynamics of Cy3 aggregates in more complicated and flexible environments, like DNA. The 2D spectroscopy experiments that have been done on Dimer 1 by Halpin et al.⁶ showed clear spectral signatures of the two peaks at 18,320 cm⁻¹ and 19,684 cm⁻¹, while the third peak around 21,031 cm⁻¹ could be explained on the basis of theoretical modeling.^5,6 According to Halpin et al., that third peak at 21,031 cm⁻¹ is due to a strongly radiant state that has a negligible contribution to the spectral dynamics they studied. They also predicted the presence of a state around 17,400 cm⁻¹ in the far red side of the spectrum.^5,6 Halpin et al.⁶ and Duan et al.⁵ used models based on the Holstein-Frenkel Hamiltonian to simulate the spectrum of this dimer (labeled Dimer 1 here), and they were able to obtain a spectrum that fits well to the experimental picture. Despite the fact that both groups could obtain good fits, Duan et al. emphasized that there is a significant difference between the stick-spectrum obtained by their group and those obtained by Halpin et al.⁶

In this contribution, we apply the FC method to Dimer 1, in an effort to obtain a spectrum based on ab initio calculations including all the vibrational modes. We will show that this approach leads to an alternative explanation of the origin of the vibrational bands observed in the spectrum.^5,6 The FC adiabatic and vertical approaches are used to simulate and assign the spectra, and the normal modes with the most contributions are identified. The temperature and solvent dependence are also tested.

2. Methods

The Cy3 dimer studied experimentally by Halpin et al.⁶ and Duan et al.⁵ was built using Chimera⁴⁶ version 1.14. The FC approach is used to simulate the spectrum. Based on a previous study from our group,³⁴ the FC approach could reproduce the monomeric spectrum with reasonable accuracy. It is well known that the FC approach is sensitive to the choice of the ab initio methods, and we showed that ωB97XD/Def2SVP⁴⁷ outperforms B3LYP’s performance for Cy3, where the later shouldn’t be used for the monomeric Cy3. Building on that, we continue to use ωB97XD/Def2SVP to simulate the FC spectrum of the Cy3 homodimer. The functional M11 and the ab initio method Configuration Interaction with Single excitation (CIS) with the same basis set (Def2SVP) were also used when necessary for comparisons. Solvation effects were accounted for using implicit methanol solvent via polarizable continuum model (PCM), and the equilibrium solvation formalism^48,49

In order to get the spectra first the structure was optimized on the ground state and first and second excited states. Frequency calculations using the same methods were run at the optimized geometries to ensure that true minima were obtained. The Franck-Condon calculations were run between the ground and first/second excited states at 0K and 298K. To obtain converged FC spectra at 298K, the FC calculations were run using the time-dependent (TD)^40,45,50 approach. To get information about the origin/contributions of the vibrational excitations, the time-independent (TI)⁴⁵ approach was used at 0K. For the harmonic approximation, the performance of the Adiabatic Hessian (AH), Adiabatic Shift (AS), Vertical Hessian (VH), and Vertical Gradient (VG) approaches were compared. The spectral visualization and extraction were done using Gaussview version 6.0.⁵¹ The ab initio excitonic coupling calculations were done on the ground state equilibrium structure using ωB97XD/Def2SVP. Throughout the study, the spectra are convoluted using a Gaussian function with width at half maximum = 150 cm⁻¹, except for the vertical excitation at equilibrium where a width of 500 cm⁻¹ gave a good fit to the experimental spectrum. The spectra are shifted so that the simulated maximum is matching the experimental maximum. The shift value and direction are included in parentheses in the legends of figures. The summation of the spectra was done using the “simple curve math” module of OriginLab Learning Edition 2023.⁵² The natural transition orbitals were calculated using the respective excited state method. All the calculations were done using the Gaussian16 package.⁵¹

3. Results and Discussion

3.1. Simulating Spectra

The spectrum of the homodimer was simulated using the vertical excitation at equilibrium (VEE) and the Franck-Condon approach using the adiabatic and vertical representations. Due to its good performance with the monomer,³⁴ we do the excited state calculations using ωB97XD/Def2SVP.

3.1.1. Vertical Excitation at Equilibrium

Vertical excitation at equilibrium (VEE) is the simplest method to obtain the electronic spectra of molecules. The excitation energies at the ground state equilibrium structure are plotted against the intensity using Gaussian or Lorentzian broadening functions.^42,53 VEE is not suitable for simulating the absorption spectrum of the monomeric Cy3 since it doesn’t account for the vibronic contributions,³⁴ but it is useful in the case of excitonically coupled systems since it can give the pure electronic contributions to the spectra and the magnitude of the electronic excitonic coupling. Table 1 shows the transition energies and oscillator strengths of the lowest two and four excited states at vertical excitation of the monomer and Dimer 1, respectively. The calculated monomeric S₁ state is bright and its VEE overestimates the experimental maximum by 4909 cm⁻¹. The monomeric S₂ state is dark and it lies outside of the experimental spectrum range in Figure 2b. For Dimer 1, we can see that only the S₁ and S₂ states have excitation energies that lie within the experimental range with S₂ being the brightest state. The energetics and properties (i.e oscillator strengths) of the lowest two dimeric excitonic states can be explained on the basis of the excitonic coupling theory of Kasha,^10,54 as we will elaborate on that later.

Table 1:

Vertical excitation energies (E/eV/cm⁻¹) and oscillator strengths (f) of the first two and four excited states of the Cy3 monomer and Dimer 1, respectively, at the minimum of their ground states optimized using wB97XD/Def2SVP. The excited state calculations are done using wB97XD/Def2SVP.

State	E/eV	E/cm−1	E/cm−1*	f	exp/cm−1
Monomer
S1	2.8643	23102	18193	1.5793	18193
S2	4.5341	36570	31661	0.0229
Dimer 1
S1	2.7584	22248	18388	0.0770	18320
S2	2.9190	23544	19684	2.9464	19684
S3	4.3545	35122	31262	0.0002
S4	4.3580	35150	31290	0.0005

^*The energy in cm⁻¹ is shifted by −4,909 for the monomer and −3860 cm⁻¹ for Dimer 1, so that the bright states match the experimental maxima in Figures 3 and 2b.

While the excitation energies of the first two excited states are somewhat higher than the experimental peaks, the electronic splitting of the S₁ and S₂ excitonic states predicted theoretically (1296 cm⁻¹) is very similar to the experimental splitting of the first two main peaks (1364 cm⁻¹). This suggests that the peaks may be originating from the electronic transitions. In that case, guided by the significantly high oscillator strength of S₂, we can assign this transition to the highest experimental peak at 19684 cm⁻¹, and the lower energy S₁ state to the 18193 cm⁻¹ peak, see the VEE spectra in Figure 3. The peak at 21031 cm⁻¹ however cannot originate from electronic transitions since the S₃ electronic state is much higher in energy, and it has to originate from the vibronic contributions that are ignored in the VEE formalism. A problem with this assignment is that the intensity of the first peak is predicted to be very low, contrary to what is seen experimentally. The higher intensity of the second peak compared to the first one can be explained based on the excitonic coupling theory of Kasha.^10,54 The dimer units are oriented in the form of an H-aggregate, where the individual transition dipole moments of the monomer are parallel to each other. This results in excitonic splitting where the higher excitonic state is brighter. A problem with this interpretation however is that the S₁ state is predicted to be dark (not visible) in the spectrum, but the experimental first peak has considerable intensity. Alternatively, one could assume that the structure in the experimental spectrum is entirely due to the vibronic structure on S₂. Including vibrations is necessary in order to make a proper assignment of the peaks and distinguish between electronic and vibrational contributions. Our results here suggest that the dramatic change between the spectrum of Dimer 1 and the monomeric spectrum originates from the pure electronic excitonic coupling between the aggregate monomers and it is governed by their relative geometrical orientation.

Figure 3:

Spectra obtained from S₁ and S₂ vertical excitation energies at the ground state minimum (VEE) of Dimer 1 using ωB97XD/Def2SVP in implicit methanol via PCM.

3.1.2. Franck-Condon Approach

The Franck-Condon approach is a powerful tool to simulate the spectra of molecules with strong vibronic contributions. The accuracy of this approach depends on the harmonic/anharmonic approximation used. In the current implementations, the harmonic approximation is more common.^55–57 In this approximation, the potential energy surface (PES) of the corresponding state is expanded about its true minimum using a quadratic function of the nuclear coordinates. Based on how the PES of the final state is expanded, the harmonic approximation can be subdivided into adiabatic and vertical representations.^55–58 In the adiabatic representation, the PESs of the reference and target states are expanded about the true minima of their respective states. Further approximations to the adiabatic representation have been developed and the main distinction between them is how the Hessian data (i.e. normal modes and frequencies) of the final state are obtained. In the Adiabatic Hessian (AH) approach, the Hessians of the initial and final states are calculated for their optimized geometries, and this is the most accurate approach. Calculating the Hessian for big systems, however, can be computationally challenging, especially if one needs to do that numerically on the excited states. The Adiabatic Shift (AS) approach offers a less computationally demanding, however, less accurate solution. In the AS approach, the final state can be assumed to have the same Hessian as the ground state which is convenient for systems where the spectrum is dominated by vibrational structure originating from displacements in the equilibrium structures between the two states. In the vertical approaches, the PES of the final state is expanded about the minimum geometry of the reference state. In the Vertical Hessian (VH) approach, the energy, gradient, and Hessian of the final state are calculated about the reference state equilibrium structure. A simpler approach is the Vertical Gradient (VG) approach which only needs the computation of the final state’s gradient.^55–58 Since these methods probe the parameters needed to simulate spectra, comparing their performance can give an idea about the importance of specific contributions. For example, for the monomeric Cy3, it has been shown that the adiabatic harmonic approximation is satisfactory; the AH and AS approaches at 0K showed reasonable performance.³⁴ This means that displacements in the equilibrium structures between the ground and excited states are the main source of the vibronic structure of the monomeric spectrum. Building on that, we will use the different harmonic approximations to get an idea about the major factors that contribute to the absorption spectrum of Dimer 1. We do that initially at 0K to eliminate the temperature effect.

Figure 4 compares the dimeric experimental spectrum to the shifted FC spectra obtained from the adiabatic and vertical approaches at 0K. The performance of the vertical approaches, VH and VG, resembles the performance of the VEE approach, and they fail to predict the third main peak at the blue side of the spectrum, see Figures 4a and 4b. Despite the improvement in the predicted relative intensities, they underestimate the bands’ separation. The VH and VG approaches predict peak separations of 476 cm⁻¹ and 593 cm⁻¹, respectively, compared to 1296 cm⁻¹ for VEE. The main difference between the VEE and the FC vertical approaches is that the VEE uses the ground state equilibrium structure while the FC vertical approaches expand the PES of the final states relative to that of the ground state. Hence, the improvement of the relative intensities in the vertical FC approaches can be attributed to the inclusion of the vibronic contributions and/or the excited state properties of the S₁ surface that are used for the FC calculations, while the poor prediction of the peak separation can be attributed to the formalism used to expand and relate the PESs of the ground and excited states. Unlike the vertical approaches, the adiabatic approaches account for the excited states’ structures, frequencies, and PESs’ curvature more properly. As shown in Figure 4c, the AS approach is capturing the first two bands with an accuracy comparable to the vertical approaches. However, it offers an explanation to the third band where it suggests that it is mainly originating from the combined electronic-vibrational contributions of the second excited state. The AH approach, see Figure 4d, is showing much better performance; it is able to predict the three spectral bands, their separations, and relative intensities with reasonable accuracy and it gives us a complete idea about the spectral contributions. It predicts that the first peak originates solely from the first excited state while the second peak is dominated by contributions from the second excited state and to a lower extent vibrational contributions from the first excited state. Similar to the AS approach, the third peak is well-predicted and it is mainly originating from the combined electronic-vibrational contributions of the second excited state. Since the main difference between the adiabatic and vertical approaches is that the latter ignores the final state’s displacements, we can conclude that the geometrical displacement between the ground and second excited state is the origin of the third peak. These comparisons demonstrate the importance of the proper inclusion of PES and frequencies of the final states, which are accounted for in the AH approach, for proper calculation of the energy separation between the different states. In summary, all the models predict that the first peak originates from the first electronic state which has now acquired some intensity compared to VEE, while the second and third peaks originate from S₂ and the highest intensity peak is its 0–0 transition.

Figure 4:

Performance of the different harmonic approximations at 0K on simulating the spectrum of the excitonically coupled Cy3 Dimer 1. FC: Franck-Condon, AH: Adiabatic Hessian, AS: Adiabatic Shift, VH: Vertical Hessian, VG: Vertical Gradient.

Now that we have an idea about the origin of the three main spectral bands from the FC simulation at 0K, we will try to capture the correct spectral shapes. For the Cy3 monomer, the inclusion of the temperature produced the correct spectral shapes suggesting the population and subsequent contributions of higher vibrational levels on the ground state,³⁴ and here we show that it is the same case for the homodimer. This is shown closely in Figure 5, where we compare the spectra obtained using the AH approach at 0K and room temperature. Including the temperature effects with the AH approach leads to excellent agreement with the experimental spectral shape.

Figure 5:

Temperature contribution to the AH spectrum of Dimer 1.

3.2. Origin of the Vibronic Structure

For the dimeric system, the vertical approaches, VEE, FC/VH, and FC/VG, suggested that the first two spectral bands are mainly due to electronic contributions, while the adiabatic approaches show that the third peak is originating from vibrational contributions. While simple models like the Holstein-Frenkel Hamiltonian assume a single effective mode being responsible for the vibronic structure, in reality the molecules have many normal modes and several of them may contribute. The stick spectra in Figure 6 show the S₁ and S₂ vibronic spectra of the homodimer with the normal mode contributions. Similar to the Cy3 monomer, the S₁ and S₂ vibronic spectra of the dimer are dominated by multiple electronic-vibrational excitations and combination bands.³⁴ Table 2 shows the normal modes with the most prominent contributions to the monomeric and dimeric vibronic spectra.

Figure 6:

The vibronic spectrum of Dimer 1 obtained from structures optimized with ωB97XD/Def2SVP in implicit methanol via PCM at 0K using the AH approach. The red sticks correspond to the vibrational excitations. The labels on the sticks are selected transitions. (a) Vibronic spectrum due to transitions S₀ → S₁. (b) Vibronic spectrum due to transitions S₀ → S₂. Notice the very different y axis scales on the two panels.

Table 2:

The most prominent normal modes contributing to the monomeric and dimeric spectra, see Figure 6. Frequencies in cm⁻¹.

Monomer		Dimer
S1		S2		S2
Mode	Freq	Mode	Freq	Mode	Freq
3	34.9	1	7.9	1	6.8
30	354.0	7	35.2	7	36.2
81	1136.0	27	165.3	23	145.0
100	1378.0	28	168.6	30	174.9
106	1415.0	67	377.3	67	382.8
125	1525.0	82	561.4	85	572.2
		178	1139.3	95	651.6
		239	1425.2	121	830.6
		247	1453.2	192	1203.0
				199	1268.0
				213	1324.6
				241	1440.1
				283	1634.4

As shown in Figure 6 and Table 2, there are several modes that contribute to the spectrum, quite differently from the picture of one effective mode being used when modeling the spectrum. The high frequency modes contribute mainly to the second peak in each electronic spectrum. These high-frequency modes in states S₁ and S₂ with prominent contributions are mainly stretching and bending of the polymethine chain. In addition to these modes, however, the spectra are characterized by many combination bands involving low frequency modes. Mode number 7 in the S₁ and S₂ states has the strongest contribution to the spectra, as part of combination bands. Mode 7 has frequencies of 35.2 cm⁻¹ and 36.2 cm⁻¹ in the S₁ and S₂ state, respectively. In the homodimer system, mode 7 is characterized by a symmetric bending of the indole-like rings around the polymethine chain of the monomer units. Interestingly, for the S₁ transitions, we found that this vibrational normal mode is localized on only one monomeric unit, and it is delocalized on the two monomeric units of the S₂ state. We were able to see other prominent dimeric normal modes similar to those in the monomer, such as mode 67 with frequencies of 377.3 cm⁻¹ and 382.8 cm⁻¹ on the first and second excited states, respectively. Mode 67 is similar to the monomeric mode number 30 with a symmetric twisting of the dimethyl groups. Similar to the behavior seen in the normal mode number 7, for the remaining normal modes, the vibrational motions in the S₁ contributions are mostly localized on only one monomeric unit, while they are delocalized on both the monomers for the S₂ normal modes. We will explain this interesting behavior in the next section. Movies of the dimeric normal modes can be found in the supplementary material. Movies of the monomeric normal modes can be found in previous work from our group.³⁴

From our calculations, the third experimental band at 21031 cm⁻¹ is originating from electronic-vibrational contributions due to structural displacements. This finding is aligning with the experimental findings of Mustroph et al.⁵⁹ Mustroph et al. concluded that, for the cyanines, the change in the molecular geometry of the excited state explains the appearance of vibronic sub-bands in addition to the 0–0 band. Based on the assumption that the vibronic sub-bands are determined by a unique dominant symmetric vibrational mode within the polymethine chain, they concluded that there is a displacement in the equilibrium configuration between electronic ground and excited states of the cyanine dyes, and that displacement is within the polymethine chain where the equilibrium C-C bonds are longer in the excited state, especially in molecules with lower symmetry. From our calculations, the geometrical displacements of the C-C bonds on the excited states were not very conclusive though, where we see no significant increase in the polymethine C-C bonds at the minima of the excited states (see Table 3). There is a slight increase in the bond length of the central C-C bonds, mainly, in the equilibrium structures of the first excited states. The most significant increase in bond length was identified to be between the terminal polymethine carbons and the neighboring nitrogen bonds. This suggests that the geometrical displacements in these bonds are likely to have more contributions to the vibronic spectra.

Table 3:

Monomeric and dimeric equilibrium bond lengths in Å of the ground state (S₀), first excited state (S₁), and second excited state (S₂). The atoms numbering is shown in Figure 1.

	Monomer			Dimer
	S0	S1	S1 - S0	S0	S1	S2	S1-S0	S2-S0
R(N1,1)	1.3435	1.3712	−0.0277	1.3447	1.3722	1.358	0.0275	0.0133
R(1,2)	1.3966	1.396	−0.0006	1.3966	1.3977	1.3974	0.0011	0.0008
R(2,3)	1.3938	1.4071	0.0133	1.3935	1.4069	1.4001	0.0134	0.0066
R(3,4)	1.3938	1.407	0.0132	1.3935	1.4069	1.4001	0.0134	0.0066
R(4,5)	1.3966	1.396	−0.0006	1.3966	1.3977	1.3974	0.0011	0.0008
R(5,N2)	1.3435	1.3712	−0.0277	1.3447	1.3722	1.358	0.0275	0.0133
R(N3,6)				1.3447	1.3451	1.358	0.0004	0.0133
R(6,7)				1.3966	1.3963	1.3974	−0.0003	0.0008
R(7,8)				1.3935	1.3936	1.4001	0.0001	0.0066
R(8,9)				1.3935	1.3936	1.4001	0.0001	0.0066
R(9,10)				1.3966	1.3963	1.3974	−0.0003	0.0008
r(10,N4)				1.3447	1.3451	1.358	0.0004	0.0133

3.3. S₁ Symmetry Breaking

The symmetry of the dimer in its ground state is D₂ (without taking into account small differences because of the methyl groups). The two excited states S₁ and S₂ belong to the B₁ and B₂ irreducible representations, respectively. The natural transition orbitals are shown in SI. A closer look at Table 3 shows that while the distortions at the S₂ minimum are delocalized on both monomers retaining the overall D₂ symmetry of the dimer, the distortions at the S₁ minimum are mostly localized on one monomer leading to breaking the symmetry. This suggests that the electronic excitation in S₁ is localized on only one monomeric unit. This is confirmed using the natural transition orbitals (NTOs) of the transitions contributing to electronic excitations. Figure 7 shows the NTOs of the most contributing transitions to the electronic excitations using the first/second excited states equilibrium structures. Corresponding NTOs at vertical excitation are given in SI. For the electronic excitations using the ground and second state equilibrium structures, S₀eq and S₂eq, the electronic transition is delocalized over the two monomeric units, while the first excited states equilibrium structure S₁eq, as shown in Figure 7a, shows a localization of the electronic excitation over one monomer. This explains the increase of central C-C/C-N bonds length of only one monomeric unit on the S₁ surface and the localization of vibrational motion on the same unit. Hence, we also can conclude that the PES of the first excited state is a double-well surface since either monomer can be distorted.

Figure 7:

Natural transition orbitals of the transitions with most contributions to the electronic excitations from the ground state to (a) the first excited state using the equilibrium structure of the S₁ S₁eq (b) the second excited state using the equilibrium structure of the S₂ S₂eq.

3.3.1. Solvent-Induced Breaking of Symmetry

Solvent-induced breaking of symmetry on the excited states has been proposed for cyanines.⁶⁰ Lutsyk et al.⁶⁰ suggested that, for Cy3, symmetry breaking is highly likely to occur in polar solvents, where the excited molecule relaxes to states featuring unsymmetrical charge distribution along the polymethine chain. So far we have performed our calculations in implicit solvent of moderate polarity (methanol) via PCM to account for the experimental environment. To verify our findings, we run the calculations in the gas-phase to see if there is any effect.

Figure 8 shows the FC spectrum in the gas phase. While the spectrum captures the correct electronic and vibronic contributions of the second excited state, the contribution from S₁ is entirely absent. Also, the correct separation between the two main peaks is poorly predicted. Upon investigating the optimized structures on the ground and excited states, we found that, unlike the case with methanol, the S₁ minimum didn’t undergo symmetry breaking post excitation. The picture for the S₂ minimum did not change compared to the minimum in methanol where the excitation is delocalized over the dimeric units with no breaking of symmetry upon relaxation. This suggests that the breaking of symmetry on the S₁ surface is driven by the solvent.

Figure 8:

The FC/AH spectrum of Dimer 1 obtained from structures optimized in gas-phase using ωB97XD/Def2SVP.

For extra verification, we found that ωB97XD and M11 predict the same behavior in gas-phase as in methanol. Also, we ran the excited state optimizations in methanol and gas-phase using CIS. CIS gave similar results to those obtained from the TD-DFT methods where it predicted the symmetry breaking on the S₁ surface in methanol and no symmetry breaking in the gas-phase. It also showed no localization for the S₂ state in the two environments.

3.3.2. Vibronic Coupling Model Leads to Symmetry Breaking

The localization observed here can be explained using the vibronic coupling model.^61–63 Diehl et al.⁶³ observed similar symmetry breaking on perylenediimide (PDI) dimers that could be controlled using the distance between the two monomers, which in turn controlled the excitonic coupling. Dreuw and coworkers explained the localization using the pseudo Jahn-Teller effect, while Diehl et al. used the vibronic coupling adapted to energy or electron transfer theory. In both cases, the appearance of the localized minimum and double well depends on the relative magnitudes of excitonic coupling $J$ in comparison to relaxation energy from the ${\mathrm{S}}_1$ energy at vertical excitation to its energy at the ${\mathrm{S}}_1$ minimum (or reorganization energy using energy transfer nomenclature). When $J$ is large compared to the relaxation energy there is no localization, but when the relaxation energy increases compared to $J$ then a double minimum and delocalization can occur.

Using vibronic coupling theory the energy of the two states can be described by the matrix

$$W=\left(\begin{array}{cc}-J& FQ\\ FQ& J\end{array}\right)$$

where $J$ is the excitonic coupling, and $F$ couples the two excitonic states ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ which have ${\mathrm{B}}_1$ and ${\mathrm{B}}_2$ symmetry, respectively.

$$F=⟨B_1|\frac{\partial H}{\partial Q}|B_2⟩$$ (1)

By symmetry, F is nonzero only for vibrations $Q$ of $B_3$ symmetry, which in this case is for displacements along the axis connecting the two monomers. Thus, the vibronic coupling will lead to distortions along that coordinate.

In our system, the excitonic coupling in gas-phase is 904.3 cm⁻¹ =0.11 eV compared to 647.6 cm⁻¹ =0.08 eV in solvent (see SI–section 2). The relaxation energy in the gas-phase is about 0.04 eV. In the solvent the relaxation increases to 0.2 eV. This increase is mainly due to the equilibration of the solvent to the electronic density of the excited state, consistent with the concept of reorganization energy. This explains why in the gas phase the relaxation energy is not sufficient to lead to localization while in the polar solvent, the increased relaxation energy leads to localization and the double well.

3.4. Electronic Structure Methods Effects

The FC spectral simulations are very sensitive to the choice of the electronic structure method, and it is important to compare the performance of more than one method.^10,64,65 For the optimizations and frequency calculations of a large system like a Cy3 dimer, especially on the excited states, TD-DFT is an optimum choice. So far, ωB97XD has shown very good performance with the monomeric and dimeric Cy3 systems. Our attempts to optimize the homodimer system using B3LYP and CAM-B3LYP among other functionals were not successful mainly because of poor convergence on the excited states. However, we were able to obtain well-converged true minima for the ground and excited states using M11. As shown in Figure 9, M11 gives spectra similar to that of ωB97XD. M11 even shows an improvement in the predicted energies with M11 over-predicting the energies by 1256 cm⁻¹ compared to 1630 cm⁻¹ for ωB97XD. M11, also, predicted the localization of excitation and symmetry breaking on the S₁ surface.

Figure 9:

Effect of different functionals on the Cy3 dimer FC/AH spectrum via comparing the performance of ωB97XD and M11.

In order to make a more detailed comparison, we superimpose the experimental Cy3 monomeric and dimeric spectra and compare them to the simulated FC/AH spectra obtained from ωB97XD and M11, shown in Figure 10. First, we note that M11 can predict the monomeric Cy3 absorption spectra with a good accuracy similar to ωB97XD. Figure 10a shows that despite the very good agreement between the simulated and experimental monomeric and dimeric spectral shapes using ωB97XD, the separation between the highest peaks of the dimer vs monomer is underestimated by about 200 cm⁻¹ compared to the experimental separation. The performance of M11 in describing the shifts of monomer vs dimer is significantly worse and underestimates the shift by about 600 cm⁻¹, as shown in Figure 10b. This shows that the overall performance of ωB97XD describing simultaneously the monomer and the dimer is better than M11.

Figure 10:

Comparing the experimental and theoretical FC/AH spectra of the Cy3 monomer and dimer optimized in implicit methanol via PCM using (a) ωB97XD (b) M11. The simulated monomer and dimer spectra were shifted by the same magnitude so that the simulated monomeric spectrum matches the experimental one.

3.5. Assignments and Comparison with Previous Work

In the previous sections, the Franck-Condon approach using the AH representation at 0K offered an ab initio explanation to the origin of the main bands of Dimer 1, while the FC/AH at room temperature gave an excellent agreement with the experimental absorption spectrum. Previous studies on the Cy3 dimer (Dimer 1) used the Holstein-Frenkel Hamiltonian and more sophisticated model Hamiltonians to explain the origin of the spectral features.^5,6 Although Halpin et al.⁶ and Duan et al.⁵ were able to reproduce the experimental absorption spectrum of Dimer 1 in good accuracy, the molecular Hamiltonians used by the two groups lead to significantly different stick spectra assignments. Halpin et al. interpreted the main peaks in the spectrum mainly as contributions from vibrational progressions on the S₂ state, while Duan et al. found that the main peak at 19684 cm⁻¹ has almost equal contributions from electronic and vibrational states and the main peak at 18193 cm⁻¹ is dominated by vibrational contributions. From the stick spectrum offered by Duan et al.,⁵ the third main peak at 21031 cm⁻¹ is dominated by vibrational contributions. Our work provides a more accurate alternative to model Hamiltonians since the spectrum is calculated from first principles including all vibrational modes. According to our results, the two main peaks at 18193 cm⁻¹ and 19684 cm⁻¹ originate from electronic transitions to the first and second electronic excitonically coupled states. Since this is an H-aggregate, the electronic transition to the second excitonic state is much more optically allowed compared to the first excitonic state. In our work, we found that the intensity of the first peak is increased because of the symmetry-broken excited state minimum. The third main peak at 21031 cm⁻¹ is due to a vibrational progression from the second excited state, in agreement with the Holstein-Frenkel models. Lastly, our work suggests that the spectral contributions around 17400 cm⁻¹, which were attributed to the lowest excitonic states in the previous work, originate from the first excited state due to thermal contributions at room temperature.

An important consequence of the nature of the peaks observed is the explanation of the nature of the observed oscillations in the two-dimensional spectra of Dimer 1.⁶ The motivation for exploring these oscillations is that they can offer insight into the origin of the long-lived coherences observed in light harvesting antenna complexes.⁶⁶ Halpin et al. predict that the Cy3 dimer could under specific circumstances lead to a long-lived vibronic coherence, but they only see experimentally the fast decay of electronic coherence. Duan et al.⁵ show that, by assuming reasonable numbers for the electronic dephasing time and the vibrational dephasing time in their model, there is a rapid dominant electronic dephasing component, but also a weak slowly dephasing component due to the vibrational contribution to the coherent states. Our results, on the other hand, suggest that the coherence between the states at 18300 and 19730 is electronic, and will decay rapidly on the time scale of electronic dephasing, since the two peaks come from different electronic states.

4. Conclusions

Accurate modeling of the spectral features of excitonically coupled systems is important for the inference of molecular structures and local dynamics of these systems. The cyanine dye Cy3 is one of the most widely used chromophores in experiments on proteins and DNA. Cy3 aggregates have unique spectral features that depend on the local conformation and the excitonic coupling between the interacting monomers. Previous studies on the Cy3 Dimer 1 have mostly used the Holstein-Frenkel Hamiltonian to explain the origin of the spectral features.^5,6 In this work, we used the FC approach to offer a more accurate description of the origin of the linear absorption spectral signatures of the Cy3 dimer studied experimentally.⁶

The spectrum generated using the FC/AH approach at room temperature is in excellent agreement with the experimental absorption spectrum, and provides assignments of the observed peaks. The comparisons with the other FC representations were indicative of the importance of including the frequencies and PES of the excited states for accurate predictions of the spectrum. The lower energy peak on the spectrum originates from the lower excitonic state while the other two peaks originate from vibrational progressions on the higher excitonic state.

A very interesting feature of our results is the solvent-induced symmetry breaking on the S₁ surface, which is responsible for the lower excitonic state not being completely dark, as one would expect from an H-aggregate. This broken symmetry is a result of an increased reorganization energy in polar solvent which competes with the excitonic coupling.

In this work, to the best of our knowledge, we introduce the first benchmark study on the proper usage of the FC approach for calculating the spectra of excitonically coupled systems. The excellent ability of the FC/AH approach to accurately capture the spectral features of this simple/clean system is an important step towards understanding and explaining the spectra and local dynamics of excitonically coupled systems in more complicated situations, like proteins and DNA.