Investigating functional performance and substituent effect in modelling molecular structure, UV-visible spectra, and optical properties of D-π-A conjugated organic dye molecules: a DFT and TD-DFT study

The molecular structure, UV-visible spectra, and optical properties of D-π-A conjugated organic dye molecules (Disperse Red 1 (DR1) and Disperse Red 73 (DR73)) were analyzed using density functional theory (DFT) and time-dependent density functional theory (TD-DFT) and compared with azobenzene molecule to study the effect of donor and acceptor substituents on the molecular properties. The performance of DFT functionals is investigated using B3LYP hybrid functional and three long-range corrected functionals (CAM-B3LYP, LC-ω PBE, and ω B97XD) in conjunction with 6-31G(d,p) basis set. Using TD-DFT, we calculate the vertical excitation energies and transition dipole moment values for 100 excited states. These values were further utilized to calculate frequency dependent polarizability under sum-over-states (SOS) formalism and refractive index of these molecular systems. We observe that for azobenzene and DR1 molecules, ω B97XD predicted wavelengths corresponding to peak absorbance closest to the experimental results, while for DR73 molecule, B3LYP gave better prediction. Large polarizability response is also observed for these molecules (DR1 and DR73) in comparison to parent azobenzene structure due to charge transfer between donor and acceptor groups. For DR1 and DR73 molecules, αxx component of polarizability dominates in contrast to azobenzene where αyy dominates. The HOMO → LUMO transition during excitation contributes to the peak molecular response in simulated UV-visible spectra. The high polarizability response of selected D-π-A conjugated molecules in comparison to parent molecule suggests that these molecules are promising candidates for tailor-made photonic and optoelectronic device development. Graphical Abstract Functional and substituent effect on the optical response of D-π-A conjugated molecules modelled using DFT and TDDFT. Functional and substituent effect on the optical response of D-π-A conjugated molecules modelled using DFT and TDDFT.


Introduction
Novel organic optical materials have garnered significant scientific interest due to their potential applications in optical data storage [1], molecular photovoltaics [2,3], laser technology [4], organic photonics [5], and nonlinear optics [6]. It has been observed that organic molecules exhibit large intrinsic optical responses and are therefore promising candidates for optoelectronic and photonic applications [7,8].
due to molecular-level changes in the pendant DR1 units in the film [13]. Polymer composite poly(vinyl carbazole):trinitrofluorenone/DR1 has been shown to possess dual grating formation due to simultaneous photorefractivity and photoisomerization [14]. The substantial third-order non-linear effect is experimentally observed for DR1 dye covalently doped organic-inorganic hybrid films [15]. Thus, polymers functionalized with disperse red molecules can be used to control the electrical and optical properties at the molecular levels and thereby help develop new technologies for high data storage applications [16].
Quantum chemical methods act as an excellent computational tool to predict the optical properties of these organic molecules and identify the structure-property relationship within these molecules [17]. Some computational research studies have been previously performed for DR1 molecule. The molecular structure and absorption spectrum of the DR1 dye molecule has been computationally studied by Ojanen et al. [18]. Poprawa-Smoluch [19] investigated the photoisomerization property of DR1 with the help of transient absorption spectroscopy and quantum chemical calculations.
In the present study, azobenzene (C 12 H 10 N 2 ) is the molecular mainframe upon which substituents are attached resulting in Disperse Red 1 (C 16

H 18 N 4 O 3 ) and Disperse
Red 73 (C 18 H 16 N 6 O 2 ) molecules. Azobenzene consists of an azo group (N=N) connecting two benzene molecules and therefore has a delocalized charge density. DR1 and DR73 molecules consist of a donor and an acceptor group attached to the azobenzene molecule. Both molecules have the same acceptor group -NO 2 attached to the azobenzene parent structure. On the other hand, they differ in the donor group attached to each molecule. DR1 has -N(C 2 H 5 )CH 2 CH 2 OH donor group whereas DR73 has -N(C 2 H 5 )CH 2 CH 2 CN group instead (Ref. Fig. 1). We study the molecular response of these organic molecules using time-dependent density functional theory (TD-DFT) and compare it with parent azobenzene structure to highlight the effect of substituents. We focus on the selected organic molecules due to their unique optical properties which makes them suitable candidates for nanophotonics, terahertz photonics, and optical image processing [20][21][22][23].
For accurate prediction of response properties of these D-π -A molecular systems, electron correlation within the molecular system must be taken into account. Under timedependent density functional theory (TD-DFT) formalism, functionals are used to approximate electron correlation within the system; and hence, the selection of appropriate functional is critical for accurate prediction of optical properties. Conventional functionals overestimate the optical properties for large π-conjugated systems. However, long-range corrected functionals incorporate long-range effects and therefore describe the diffuse regions of charge distribution adequately [24]. Consequently, we have considered long-range corrected functionals for estimating the structure and response of selected molecules. The predictive performance of functionals is theoretically estimated and compared with experimental data available.
Our objective is to perform a systematic computational analysis on Disperse Dyes (DR1 and DR73) and azobenzene molecules using both long-range corrected functionals (CAM-B3LYP, LC-ωPBE, and ωB97XD) and, for comparison purposes, the popular hybrid B3LYP functional in conjunction with 6-31G(d,p) basis set [25]. The molecular structure, vertical excitation energy, transition dipole moments, optical properties (static and dynamic polarizability, refractive index), and UV-visible spectrum have been reported. We have compared the calculated results with experimental data wherever available. We aim to assess the quality of these functionals in predicting the optical properties. In addition, we also study the substituent effect on the optical properties of target molecules.

Computational details
The ground-state molecular structures of azobenzene, DR1, and DR73 molecules were optimized using density functional theory (DFT) along with B3LYP, CAM-B3LYP, LC-ωPBE, and ωB97XD functionals and 6-31G(d,p) basis set. These structural calculations were performed in vacuum. Excited-state calculations for all molecules were performed using the TD-DFT method for 100 excited states. The calculated vertical excitation energies and oscillator strengths were used to calculate the static and dynamic polarizability of target molecules using all functionals under sum-over-states formalism. The calculated values of dynamic polarizability were further employed to assess the refractive index of target molecules. To simulate the UV-visible spectrum of molecules, DFT calculations were performed on selected molecules in solvent medium. The solvents used in our study are chloroform, n-hexane, and water for azobenzene, DR1, and DR73 molecules, respectively. All calculations were performed using Gaussian 16 Software package [26].

DFT functionals and their effect in simulating molecular optical properties
To model the optical properties of molecules, DFT has emerged as a popular method due to its lower computational need and improved accuracy than traditional methods like Hartree-Fock Theory. Within the DFT formalism, functionals are used to estimate the exchange-correlation energy of electrons within the molecule. A variety of functionals with different levels of approximations from the simplest to highly sophisticated have been developed and arranged in the order of Jacob's ladder of density functional approximations [27]. The computational cost increases with the level of sophistication of approximation. Local density approximation (LDA) is the most basic approximation that takes into account only local densities for calculating the exchange-correlation energy. The next hierarchical approximation is the generalized gradient approximations (GGA), which considers both the local densities and its gradients to estimate the exchange-correlation energies. Champagne et al. [28] studied the accuracy of LDA and GGA in predicting the linear polarizability of p-nitroaniline (PNA) and 4amino-4'-nitrostilbene(ANS) conjugated systems and found that the calculated values did not match the experimental results. They suggested that to estimate the polarizability of such systems, the contribution of exchange energy is much more significant than the correlation term within the exchange-correlation energy. Moreover, the exchange contribution of the LDA and GGA functionals has been reported to overestimate the molecule's optical properties. The overestimation of these properties is due to the poor asymptotic behavior of exchange-correlation functionals.
Hybrid functionals were introduced to alleviate these shortcomings by combining general gradient approximations (GGA) with Hartree-Fock (HF) exchange integral at a constant rate. B3LYP [29] is a hybrid functional and has been popular in the field of organic chemistry for calculating molecular geometry and thermochemical properties. It is composed of Becke 1988 (B88) exchange, Lee-Yang-Parr (LYP) correlation functional, Slater (S) exchange, Vosko-Wilk-Nusiar (VWN) correlation LDA functional, and HF exchange integral with three parameters. The values of parameters a 0 , a x , and a c are 0.2, 0.72, and 0.81 respectively. However, it has been observed that it fails to predict the response of molecules to external electric fields and charge-transfer excitations within the molecule due to its inaccurate asymptotic behavior, which decays as −0.2 r instead of exact asymptotic decay of −1 r . This necessitates improvement of asymptotic correction to exchange functional.
For this, range-separated hybrid (RSH) functionals were first introduced by Savin et al. [30] where the interelectron Coulomb operator 1 r 12 of LDA exchange functional was split into short-range (SR) and a long-range (LR) contribution.
where ω is a constant. However, this scheme was inapplicable to conventional GGA functionals. To overcome this deficiency, a "long-range correction scheme" was developed for GGA functionals [31]. This technique was utilized by Yanai et al. [32] to develop CAM-B3LYP functional which uses "Coulomb-attenuating method (CAM)" approach to split the Coulomb operator. Mathematically, it can be represented by: where α + β = 0.65. For B3LYP functional, the Hartree-Fock (HF) exchange term remains fixed (20%) in response to variations in the interelectronic distance (r 12 ). However, for range-separated functionals (ref. Eq. 2), short-range (r 12 →0) and longrange(r 12 → ∞) operators of interelectronic Coulomb operator 1 r 12 contributes differently in response to variations in r 12 . In the case of CAM-B3LYP functional, short-range and long-range operators model 19% and 65% HF exchange interaction, respectively. CAM-B3LYP provides a good estimation of excitation energies as compared to GGA and hybrid functionals [33]. Other functionals included in this study, LC-ωPBE [34] and ωB97XD [35], have been developed as long-range-corrected approximations to PBE and B97X functionals with ω = 0.4 Bohr −1 and 0.2 Bohr −1 values respectively (ref. Eq. 2). Both these functionals provide 100% long-range HF exchange (coefficient of LR term in Eq. 2 is 1) in comparison to CAM-B3LYP, which includes 65% long-range HF exchange (coefficient of LR term in Eq. 3 is α + β=0.65). However, they simulate short-range exchange interactions differently. LC-ωPBE cannot model short-range HF exchange interactions. On the other hand, ωB97XD models 22% of short-range HF exchange and includes a dispersion correction term to account for van der Waals interactions, which is absent in LC-ωPBE functional.
In our calculations, we have considered four functionals (B3LYP, CAM-B3LYP, LC-ωPBE, and ωB97XD) to approximate the exchange-correlation energy within a molecular system. The contribution of long-range HF exchange within each functional required to provide improved estimates of our target molecules' linear polarizability is further analyzed. To achieve this objective, we performed the sum-over-states calculation of polarizability for 100 excited states. Therefore, we have focused on the performance of functionals in predicting optical properties of targeted organic π-conjugated molecules (azobenzene, Disperse Red 1, and Disperse Red 73). We have also explored the effect of various substituent groups in modifying the π electron delocalization of parent molecule by studying DR1 and DR73 molecular systems. Our study aims to provide novel insights in innovating new materials with tailor-made optical properties.

Molecular structure
All molecules considered in this study are π-conjugated systems. Azobenzene (C 12 H 10 N 2 ) molecule has electron distribution confined within the xy plane. In contrast, Disperse Red 1 (C 16 H 18 N 4 O 3 ) and Disperse Red 73 (C 18 H 16 N 6 O 2 ) molecules have substituents attached to their parent azobenzene backbone which introduces electron density distribution along the z-axis. The structures of these molecules are illustrated in Fig. 1. The presence of substituent groups in DR1 and DR73 modifies the structure of the parent azobenzene molecule. The calculated optimized structures for azobenzene and DR73 molecules, using all four functionals, are compared with the experimental bond parameters of the molecules [36,37] (Ref. Table 1).
For all molecules, the bond lengths and bond angles are broadly classified into four groups: C-C bonds, C-N bonds, N=N azo bond, and central C-C-N bond angles connecting the two benzene rings of azobenzene mainframe structure. Table 1(a) and (b) list the optimized bond lengths and bond angles, respectively, of these molecules calculated using B3LYP, CAM-B3LYP, LC-ωPBE, and ωB97XD functionals and 6-31G(d,p) basis set. The order of calculated C-C and N=N bond lengths follows the trend: B3LYP > ωB97XD > CAM − B3LYP > LC−ωPBE for all these molecules considered in this study containing azobenzene mainframe. There are two C-N bonds (N1-C4 and N2-C7) present in the azobenzene molecule and four C-N bonds (N1-C4, N2-C7, N3-C1, and N4-C10) present in DR1 and DR73 molecules. The N1-C4 and N2-C7 bond lengths for all molecules are in the order LC−ωPBE > ωB97XD > CAM − B3LYP > B3LYP. These bonds are equal in the case of the azobenzene molecule, thus making the structure symmetric. However, this bond symmetry is broken due to attachment of substituents in DR1 and DR73 with N2-C7 bond lengths longer than N1-C4 bond lengths. In Table 1(a), the reported calculated values are compared with available experimental results. The calculated values are close to experimental values; however, we would like to mention that our calculations are done in vacuum but the experimental results are in crystalline form, so perfect match could not be obtained. For azobenzene molecule, LC-ωPBE gave the closest result to experimental C-N bond length. Furthermore, we observe that all functionals predicted azobenzene molecule's planar structure, exemplified by the C4-N1=N2-C7 dihedral angle reported to be 180 • . The calculated bond angle C3-C4-N1 matched with C12-C7-N2 and C5-C4-N1 angle is equal to C8-C7-N2, indicating symmetry in bond angles of azobenzene molecule. The donor and acceptor groups in DR1 and DR73 introduce an asymmetry in electronic distribution; therefore, these angles are no longer equivalent in magnitude (Ref.

Excitation energies and optical properties
For any unperturbed molecular system, the ground and excited electronic states can be approximated using TD-DFT formalism. In this work, we have approximated the electronic structure using four exchange-correlation functionals: B3LYP, CAM-B3LYP, ωB97XD, and LC-ωPBE for three molecular systems (azobenzene, DR1, and DR 73) to evaluate the ground and excited states for each molecule. In the presence of an external field, the molecule is in a perturbed ground state, which can be expressed as a combination of unperturbed ground and excited states of the molecule and exhibits different electron density distribution than the unperturbed ground state. When the molecule interacts with the external field, instantaneous transitions occur between molecular states. This response of the molecule to the external electric field can be modelled using perturbation theory, which ultimately describes this behavior quantitatively in terms of polarizability tensor.
Molecular polarizability is one of the most fundamental optical properties of conjugated systems and can be calculated using sum-over-states (SOS) formalism. Within this method, for all single excited electronic states, their corresponding vertical excitation energy and transition dipole moment from the ground state are used to calculate the static and dynamic polarizability of the molecule as follows: where α ij (0) and α ij (ω) are static and dynamic polarizability respectively, μ gn i is the transition dipole moment from ground state to n th excited state in the i th direction, μ ng j is the transition dipole moment from n th excited state to ground state in the j th direction, and ω gn = E n − E g is the transition energy and ω is the photon energy.
The components of molecular polarizability tensor thus calculated can be utilized further to estimate the isotropic polarizability (α) and anisotropic polarizability ( α) using the following equations: The sum-over-states calculation requires the number of states (n) to be specified to calculate the polarizability tensor components of a molecule. For this purpose, we calculated the isotropic (α(ω)) and anisotropic ( α(ω)) polarizability of azobenzene molecule using B3LYP functional and 6-31G(d,p) basis set for n=10, 20, 30, 50, 100, 150, 200, 250 excited states. The results are plotted in Fig. 2. From the figure, we infer that the calculated values of α(ω) and α(ω) remain almost constant beyond n=100. This indicates that all the important transitions responsible for the linear polarizability response of the molecule lie within 0-100 states. The transitions occurring to states above n=100 do not contribute appreciably to our calculations. Therefore, to maintain a balance between the computational cost of our calculations and the accuracy of our calculations, we have calculated polarizability response of our target molecules by sum-over-states calculations for n=100 excited states.
The polarizability tensor components, as defined in Eqs. 4-7, depends on two factors: transition dipole moment between the ground and excited states of the molecule and the excitation energy of corresponding transitions (i.e., the energy gap between the ground and excited states). During the electronic transition within the molecule, transition dipole moment provides a measure of the extent of the oscillating movement of electrons within the molecule resulting from interaction with the oscillating field. If the extent of interaction is large, then from Eq. 5, we observe that the substantial value of TDM will result in significant polarizability tensor component. The polarizability tensor components calculated using the sum-over-state method can be further utilized to estimate the isotropic (α(ω)) and anisotropic ( α(ω)) polarizability of the molecules using (8) and (9). Thus, the behavior of α(ω) and α(ω) response functions is determined by the polarizability components α xx (ω), α yy (ω), and α zz (ω).
In Figs. 3a, b, and c, we have reported the TDM values of all 100 excited states for azobenzene, DR1, and DR73 molecules, respectively. It has been observed from these graphs that the maximum TDM corresponds to | 0 →| 2 transition.  Figure 4 shows the polarizability curves of tensor components α xx (ω), α yy (ω), and α zz (ω) as well as the net isotropic (α(ω)) and anisotropic ( α(ω)) polarizability variation of    azobenzene molecule as a function of frequency of external field (ω). These curves demonstrate the effect of functional in modelling the response of azobenzene to external electric field. In the case of azobenzene molecule, the contribution from α zz (ω) component is negligible when compared with the contribution from α xx (ω) and α yy (ω). Among all the polarizability components, α yy (ω) has the largest polarizability value and therefore plays a major role in determining the response of α(ω) and α(ω) with respect to the external field. For α yy (ω) response curve (Fig. 4), all functionals predict peak polarizability at ω=0.152 a.u.. Corresponding to | 0 →| 2 maximum transition (Ref. Fig. 3), the TDM values are reported in Table 2 where | μ y |=2.890 for B3LYP, 2.852 for CAM-B3LYP, 2.851 for ωB97XD, and 2.761 for LC-ωPBE functionals. The excitation frequency (calculated from excitation energy of Table 2) corresponding to | 0 →| 2 transition is calculated to be 0.139 a.u. for B3LYP, 0.150 a.u. for CAM-B3LYP, 0.150 a.u. for ωB97XD, and 0.161 a.u. for LC-ωPBE functionals. The excitation frequency (ω 02 ) of | 0 →| 2 transition matches closely with frequency corresponding to peak polarizability frequency (ω). Thus, | 0 →| 2 transition plays an important role in determining sum-over-state polarizability of α yy (ω) component for azobenzene molecule. We observe that as the percentage of long-range HF exchange increases from 20% in B3LYP to 65% in CAM-B3LYP and subsequently to 100% in ωB97XD and LC-wPBE functionals, the predicted excitation frequency increases and TDM values decrease for | 0 →| 2 transition. In the case of ωB97XD and LC-ωPBE functionals, although both have the same % of long-range HF exchange, the predicted values of excitation energy and TDM differ. This difference in values might be due to the variation of short-range HF exchange in ωB97XD and LC-ωPBE functionals. From Fig. 4, we also observe that some functionals give positive values of TDM while others predict negative values. The positive values indicate that the change in dipole moment in response to oscillating electric field is along the direction of the  applied electric field. However, the negative values of signify change in dipole moment opposite to the direction of the applied electric field. Figure 5 describes the response of molecular polarizability tensors (α xx (ω), α yy (ω) and α zz (ω)) and isotropic (α(ω)) and anisotropic( α(ω)) polarizability tensor of Disperse Red 1 molecule as a function of frequency of external electric field. From the graph, it is evident that the response of DR1 predicted by CAM-B3LYP functional for α xx (ω) is larger as compared to the response predicted by B3LYP, ωB97XD, and LC-ωPBE functionals. For α xx (ω) vs. ω curve, two positive peaks are observed at external  4 Plot of polarizability tensor components (α xx (ω), α yy (ω) and α zz (ω)) and isotropic (α(ω)) and anisotropic ( α(ω)) polarizability of azobenzene molecule as a function of frequency (ω) of external electric field calculated using four different functionals B3LYP, CAM-B3LYP, ωB97XD, and LC-ωPBE field frequencies ω=0.09 a.u. and 0.15 a.u. for B3LYP functional. In the case of this functional, the | 0 →| 2 and | 0 →| 9 transitions occur at excitation frequency 0.097 a.u. and 0.152 a.u.. Thus, the one positive peak in the curve of α xx (ω) vs. ω at frequency 0.09 a.u. is due | 0 →| 2 transition with corresponding TDM | μ x |=3.8 (Ref. Table 2). The other positive peak in the curve at frequency 0.15 a.u. is due to | 0 →| 9 transition occurring within the molecular system with TDM | μ x |=1.04. The calculated plots of α xx (ω) using CAM-B3LYP, ωB97XD, and LC-ωPBE functionals show peak polarizability at 0.114 a.u.. The excitation frequencies for | 0 →| 2 transition (calculated from excitation energy reported in Table 2) are 0.115 a.u., 0.126 a.u., and 0.117 a.u. for CAM-B3LYP, ωB97XD, and LC-ωPBE functionals respectively. Thus, in contrast with B3LYP func-tional which predicts two positive peaks corresponding to transitions | 0 →| 2 and | 0 →| 9 , other functionals exhibit peak polarizability only due to | 0 →| 2 transition for α xx (ω). From the curves of polarizability tensor components in Fig. 5, it is evident that significant contribution to α(ω) and α(ω) is due to α xx (ω), while α zz (ω) contributes the least. The peaks for α xx (ω) are confined within the frequency 0.09-0.2 a.u., while for α yy (ω) component, response extends within the range of 0.1-0.2 a.u. This variation of each polarizability tensor components (α xx (ω), α yy (ω), and α zz (ω)) ultimately affects the isotropic (α(ω)) and anisotropic ( α(ω)) response of Disperse Red 1 molecule. The polarizability response of α(ω) closely mimics the response of α xx (ω) due to the dominant contribution of Fig. 5 Plot of polarizability tensor components (α xx (ω), α yy (ω) and α zz (ω)) and isotropic (α(ω)) and anisotropic ( α(ω)) polarizability of Disperse Red 1 molecule as a function of frequency (ω) of external electric field calculated using four different functionals B3LYP, CAM-B3LYP, ωB97XD, and LC-ωPBE this tensor component. The maximum peak polarizability is computed by CAM-B3LYP functional, and the least peak polarizability for α(ω) and α(ω) is calculated for LC-ωPBE functional. Figure 6 represents variation in polarizability of Disperse Red 73 molecule in response to an external electric field of frequency ω. The contribution of polarizability tensor components in the estimation of α(ω) and α(ω) increases as α xx (ω) > α yy (ω) > α zz (ω). α xx (ω) polarizibility tensor component has the maximum positive peak responses at external field frequencies ω= 0.09 a.u. and 0.12 a.u. for B3LYP and ω=0.12 a.u. for CAM-B3LYP, wB97XD, and LC-wPBE. Since α xx (ω) is the major contributing factor to α(ω), we observe the maximum positive peak responses of α(ω) at the same values as α xx (ω) for each functional (Ref. Fig. 6).
In the case of B3LYP functional, the excitation frequencies for DR73 are calculated to be ω 02 = 0.1 a.u. (calculated from | 0 →| 2 excitation energy in Table 2) and ω 09 = 0.15 a.u.. These values of excitation frequencies are close to the frequencies of positive peak sum-over-state α(ω) polarizability (Ref equation 4) observed for B3LYP functional. Thus, in the case of B3LYP functional, instantaneous transitions | 0 →| 2 and | 0 →| 9 give rise to positive peak polarizability (α(ω)) of DR73 molecule. For functionals, with increased % of long-range HF exchange (CAM-B3LYP, wB97XD, and LC-wPBE), the excitation frequency of | 0 →| 2 transition is estimated to be 0.11 a.u. for CAM-B3LYP, 0.12 a.u. for ωB97XD, and 0.13 a.u. for LC-wPBE (Ref. Table 2). In the presence of external field, the variation in TDM for azobenzene molecule is observed along the xy plane for all functionals, i.e., J Mol Model (2021) 27: 229 Fig. 6 Plot of polarizability tensor components (α xx (ω), α yy (ω) and α zz (ω)) and isotropic (α(ω)) and anisotropic ( α(ω)) polarizability of Disperse Red 73 molecule as a function of frequency (ω) of external electric field calculated using four different functionals B3LYP, CAM-B3LYP, ωB97XD and LC-ωPBE μ z =0 (Ref Table 2). However, substituted azobenzene molecules (DR1 and DR73) have donor and acceptor groups attached to the parent azobenzene backbone. These substituent groups introduce electron density distribution in the z plane, which results in μ z = 0 (Ref. Table 2). Therefore, substituent group attachment to the azobenzene molecule introduces a slight variation of dipole moment along the zaxis, thereby increasing the net TDM value of substituted molecules compared to the parent molecule. Moreover, for DR1 and DR73, the substituent groups decrease the excitation energy of the molecule. Changes in excitation energy and transition dipole moment directly affect the static and dynamic polarizability of these molecules. Table 3 reports the static polarizability of target molecules. Analysis with different functionals reveals that B3LYP consistently predicts a large value of static polarizability, whereas LC-ωPBE predicts the least values. The static polarizability predicted using CAM-B3LYP, and ωB97XD functionals are numerically larger than LC-ωPBE but are in close approximation to each other. This behavior is observed for all studied molecules. It indicates that an increase in long-range HF exchange from 65 to 100% resulted in slight decreased value of linear static response of the molecule. We also observe that both ωB97XD and LC-ωPBE functionals have 100% long-range HF exchange functional. However, LC-ωPBE values are lower than ωB97XD values. This behavior may be observed in molecules due to the different variations of % short-range HF exchange in ωB97XD and LC-ωPBE functionals.
Another important property that we have studied is refractive index. When the electromagnetic field passes through a molecular gas, it induces motion in the electrons within the molecule. The moving electrons generate a field which modifies the source field within the molecular gas. Thus, the total field experienced by an electron is due to the source field and the field of all other moving electrons within the molecule. Thus, the field within the molecular gas medium appears to be moving with different velocity than the source field. This variation in field velocity is quantifiable in terms of the refractive index of the molecular gas medium. This property is of practical importance for optical fibers, optical switching devices, and photorefractive materials.
The refractive index of a molecular medium can be estimated using the calculated value of isotropic polarizability (α(ω)) of an isolated molecule using the Lorentz-Lorenz equation [38] as: where N is the average number of molecules per unit volume (Avogadro number=N A ), α(ω) is the isotropic polarizability of an isolated molecule and n(ω) is the refractive index of medium for a field of frequency ω passing through the molecular medium. The frequency-dependent isotropic polarizability directly determines the molecule's refractive index and, therefore, can be altered by substituent effect and by using different wavelengths of the field to design materials whose refractive index can be externally controlled. Figure 7 shows the variation of the refractive index of molecules (calculated using Eq. 10) as a function of frequency ω of the external field. For azobenzene, DR1, and DR73, ωB97XD predicts very large refractive index values at ω=0.12 a.u.. in comparison to other functionals and thus might not be suitable to predict refractive index response within the theoretical framework considered in Fig. 7 Refractive index (n(ω)) of molecules a azobenzene, b Disperse Red 1, and c Disperse Red 73 as a function of frequency (ω) of external electric field calculated using four different functionals B3LYP, CAM-B3LYP, ωB97XD, and LC-ωPBE our study . The refractive index is dependent on the isotropic polarizability α(ω) of the molecule whose response is governed by α yy (ω) in case of azobenzene molecule and α xx (ω) in case of DR1 and DR73. These polarizability responses are due to molecular electronic transitions within the molecule among which | 0 →| 2 transition contributes significantly. Each functional simulates this molecule transition uniquely and thus predicts different responses of polarizability and refractive index of the molecule. Therefore, choosing an adequate functional is necessary to study the response of π-conjugated molecular systems of practical importance.

UV-visible spectra
Using properties calculated using TD-DFT, we can also simulate the UV-Vis spectra of molecules according to the relation [39]: (λ) = (1.3062974 × 10 8 ) f i 10 7 /3099. 6 .exp where (λ) is the absorbance corresponding to external field of wavelength λ, f i is the calculated oscillator strength, and λ 1 is the excitation wavelength of the molecule. Figures 8, 9, and 10 show the calculated UV-visible spectra of azobenzene, Disperse Red 1, and Disperse Red 73 molecules in chloroform, n-hexane, and water solvents, respectively. These solvents have been considered as these do not absorb radiation in the region under investigation. Moreover, for these solvents, the experimental results are available in the literature. In Table 4, we report the calculated and experimental wavelengths corresponding to peak absorbance for azobenzene, Disperse Red 1, and Disperse Red 73 molecules in chloroform, n-hexane, and water solvents, respectively, along with the relative orbital contributions for each calculated peak absorbance wavelengths. For azobenzene and substituted azobenzene molecules, two major transitions occur, i.e., π → π * and n→ π * transitions. In the trans configuration of the azobenzene molecule (the one we have considered in our study), π → π * is the dominant transition occurring at 325 nm and n→ π * is a much weaker transition occurring at 450 nm [40]. For the UV-visible spectra of azobenzene molecule simulated in chloroform solvent, wavelength correponding to peak absorbance due to | 0 →| 2 transition occurs at 344 nm. CAM-B3LYP, ωB97XD, and LC-ωPBE predict maximum absorbance excitation wavelength at 307 nm, 324 nm, and 302 nm, respectively. In the case of substituted DR1 and DR73 molecules, similar π → π * dominant transitions take place at wavelengths 445 nm [41] and 533 nm [42] in n-hexane and water solvents, respectively. In the case of DR1 molecule in n-hexane solvent, B3LYP predicts wavelength corresponding to peak absorbance at wavelength 490 nm. CAM-B3LYP, ωB97XD, and LC-ωPBE predict maximum absorbance excitation wavelengths at 396 nm, 413 nm, and 346 nm. For DR73 molecule, B3LYP predicts wavelength corresponding to peak absorbance at wavelength 517 nm. CAM-B3LYP, ωB97XD, and LC-ωPBE predict maximum absorbance excitation wavelengths at 422 nm, 413 nm, and 367 nm. Thus, for azobenzene and DR1 molecules, ωB97XD predicted wavelengths corresponding to peak absorbance closest to the experimental results, while for DR73 molecule, B3LYP gave better prediction. We also observed a general trend that B3LYP predicts maximum peak absorbance wavelength for all molecules while LC-wPBE predicts the least. For all functionals, calculated wavelengths corresponding to peak absorbance were red shifted due to substituent effect (for DR1 and DR73 molecules) in comparison to peak absorbance wavelength calculated for azobenzene molecule. Although both ωB97XD and LC-ωPBE have the same percentage of long-range HF exchange, we observe a shift in maximum excitation wavelength of the molecules due to the variation in short-range HF exchange-correlation in both functionals. Our analysis demonstrates that HF exchange interaction plays a crucial role in modeling the response of molecules.
Additionally, in case of azobenzene and Disperse Red 1 molecules, the calculations were performed using nonpolar solvents chloroform and n-hexane respectively since experimental data for comparison of our results were available with these solvents only. In the case of nonpolar solvents, the dispersion interactions between solvent molecules are modelled by ωB97XD functional. This functional takes into account the dispersion interactions and calculates the total energy of molecule as [35]: In our study, ωB97XD gives better result in comparison to other functionals in the case of azobenzene and Disperse Red 1 molecules as it was able to model the dispersion effect (E dispersion ) which are not included in other functionals, thus predicting UV-visible maximum absorption wavelength closest to the experimental result. In the case of Disperse Red 73 molecule, the calculation is performed using water as a solvent which is a polar solvent as experimental data for comparison is available with this solvent. For this polar solvent, B3LYP models the intermolecular interactions of solvent better than ωB97XD due to which B3LYP predicts the UV-visible maximum absorption wavelength closest to the experimental results. However, in the case of Disperse Red 73, ωB97XD predicts a difference of 120 nm in the UV-visible wavelength. Thus, solvent effect also plays an important role in choosing the appropriate functional for modeling molecular behavior.
From results reported in "Excitation energies and optical properties" section regarding sum-over-states polarizability curves for azobenzene, DR1, and DR73 molecules, we infer that it is | 0 →| 2 transition, which contributes to maximum polarizability of molecules. We compare this experimentally observed dominant transition π → π * in azobenzene, DR1, and DR73 with | 0 →| 2 transition observed in our calculated results. This comparison is reported in Table 4. These excitations are observed due to orbital transitions occurring within the molecule. For the excitation wavelength corresponding to maximum absorbance ( ), the percentage contribution of the transition H(HOMO) → L(LUMO) is the dominant orbital contribution to the electron excitation. We also observe that the percentage contribution of the transition H→ L decreases with increase in the % of long-range HF exchange.
To verify the validity of our theoretical results, we compared the calculated excitation wavelength with available experimental wavelengths reported in Table 4. Figure 11 shows the correlation plot comparing the calculated and experimental wavelengths, where we report the correlation factor (R 2 ) and root mean square error (RMSE) parameters of the best fit for all functionals. It is clearly evident from the graph that long-range functionals report better R 2 values in comparison to B3LYP functionals. Moreover, among the long-range functional, CAM-B3LYP results exhibit the least RMS error and therefore performs better suited for modelling molecular excitations of D-π -A molecules like azobenzene, DR1, and DR73. In the case of D-π -A molecules, intramolecular charge transfer occurs from the donor to the acceptor group, which is mediated by the central azobenzene backbone. The donor group feeds the electron contribution to the azobenzene mainframe, which is further withdrawn by the acceptor group. Thus, D-π -A system has charge distribution over an extended outer region in comparison to the azobenzene parent molecule. The electrons residing in the outer regions (donor and acceptor electrons) interact with the exchange hole, which is situated Fig. 11 Comparison between theoretical (λ theo ) and experimental wavelengths (λ expt ) of UV-visible spectra for all dye molecules (where R 2 is the correlation coefficient and RMSE is the root mean square error) for four DFT functionals B3LYP, CAM-B3LYP, ωB97XD, and LC-ωPBE at a large distance from the electron. This electrostatic interaction of electrons with its holes in the donor and acceptor groups separated by substantial distance can be modelled only by the inclusion of non-local HF interaction. This is due to the fact that accompanying maximum contributing orbital transition H→L (or π → π * transition) results in charge transfer upon excitation over long-range and can therefore be modelled only by the inclusion of appropriate balance of short-range and long-range HF exchange in the functionals. Therefore, long-range functionals yield better performance in modelling the molecular response of D-π-A molecules.

Conclusion
In this study, we used DFT to study molecular structure and TD-DFT to analyze the excitation energies, transition dipole moments, molecular polarizability (static and dynamic), UV-visible spectra, and refractive index of azobenzene and its substituted molecules (DR1 and DR73). The analysis was made using hybrid B3LYP functional as well as rangeseparated DFT functionals (CAM-B3LYP, ωB97XD, and LC-ωPBE). The range-separated functionals had a different percentage of long-range HF exchange character associated with each functional. In the case of the azobenzene molecule, the maximum change in polarizability is due to α yy (ω) component of polarizability tensor, whereas for substituted molecules DR1 and DR73, α xx (ω) contributes to maximum polarizability response. It can be concluded that for azobenzene, DR1, and DR73, | 0 →| 2 transition is the dominant transition contributing to dynamic polarizability response of these π-conjugated molecule to an external field. For azobenzene and DR1 molecules, ωB97XD predicted wavelengths corresponding to peak absorbance closest to the experimental results, while for DR73 molecule B3LYP gave better prediction. The performance of functionals indicates its ability to model the charge transfer processes in substituted molecules upon excitation. This inclusion of long-range HF exchange energy is also crucial for accurate prediction of polarizability response where | 0 →| 2 transition contributed to the dominant response. We also observed that introducing substitution within the parent molecule red shifted the peak absorbance wavlengths and can be used to control the molecular polarizability and tune it for practical applications.
The π-electron conjugation of D-π-A conjugated organic dye molecules contributes to the remarkable performance of these molecules for optoelectronic applications. Our study demonstrates that selecting the right theoretical model is essential for predicting optical properties of organic conjugated molecules. Moreover, this study also sheds light on tailoring the optical properties by attaching donor and acceptor groups to azobenzene-based molecules. These studies are essential for studying the interaction of organic conjugated molecules with light. These light-molecule interactions are at the heart of many novel technologies like optoelectronic logic circuits, piezoelectric materials, organic molecular circuits, optical storage devices, and noninvasive imaging techniques.