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Original Article
:19;
4012025
doi:
10.25259/AJC_401_2025

Molecular modeling of indole-based materials with efficient electron withdrawing moieties to enhance optical nonlinearity: Quantum chemical investigation

, , , , ,
aInstitute of Chemistry, Khwaja Fareed University of Engineering & Information Technology, Rahim Yar Khan, Pakistan
bIndustry Solutions, Northern Alberta Institute of Technology, Edmonton, Alberta, Canada
cDepartment of Infectious Diseases, The Affiliated Hospital of Southwest Medical University, Luzhou, China
dDepartment of Chemistry, College of Science, King Saud University, Riyadh, Saudi Arabia

*Corresponding author: E-mail address: suvash_ojha@swmu.edu.cn (S.C. Ojha)

Received: , Accepted: ,
© 2026 Arabian Journal of Chemistry - Published by Scientific Scholar
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This is an open-access article distributed under the terms of the Creative Commons Attribution-Non Commercial-Share Alike 4.0 License, which allows others to remix, transform, and build upon the work non-commercially, as long as the author is credited and the new creations are licensed under the identical terms.

Abstract

Herein, a series of seven indole-based donor-π-acceptor (D-π-A) chromophores (3aiR and 3ai2-3ai8) was designed via strategic modification of the terminal acceptor moieties. Density functional theory (DFT) and time-dependent DFT (TD/DFT) calculations were executed at M06/6-311G(d,p) level. To investigate the optoelectronic properties of 3aiR and 3ai2-3ai8, various analyses were conducted, including UV-Visible, frontier molecular orbitals (FMO), density of states (DOS), transition density matrix (TDM), natural bond orbitals (NBOs), and nonlinear optical (NLO) properties. Among all the designed derivatives, 3ai6 exhibited the narrowest band gap (2.497 eV), the most significant redshift (478.907 nm), and the highest chemical reactivity, as indicated by its maximum softness value of 0.401 eV⁻1. Moreover, DOS and TDM analyses further supported the charge delocalization observed in FMO analysis. The NBO analysis revealed significant hyperconjugative interactions and enhanced intramolecular charge transfer (ICT), contributing to the stability of the designed compounds. While all derivatives demonstrated promising NLO properties, 3ai6 exhibited the highest static first-order (βtotal = 2.89×10-28 cm⁴ statvolt⁻1) and second-order (γtotal = 2.32×10-33 cm5 statvolt⁻1) hyperpolarizability values. The dynamic first-order hyperpolarizability β(−ω;ω,0) EOPE, shows enhanced values at 532 nm, with 3ai5 exhibiting the maximum βtotal of 599 × 10⁻2⁸ statvolt⁻1 cm⁴. Similarly, the second harmonic generation (SHG) (−2ω;ω,ω) is more pronounced at this shorter wavelength. In contrast, the second-order hyperpolarizability γ(−ω;ω,0,0) shows slightly higher values at 532 nm, peaking at -421000 × 10⁻33 statvolt⁻1 cm5 for 3ai8, indicating a frequency-dependent NLO response. These findings provide crucial insights into the structure-property relationships of indole-based materials, facilitating further advancements in the NLO domain.

Keywords

Density functional theory
Frontier molecular orbitals
Indole-based materials
Non-linear optical properties

1. Introduction

Nonlinear optical (NLO) materials have garnered considerable attention in recent years for their ability to modulate the light propagation properties, such as phase, frequency, amplitude, and polarization [1,2]. The first-order hyperpolarizability (βtot), plays a crucial role in facilitating NLO processes such as second harmonic generation (SHG) and electro-optic modulation (EOM). Meanwhile, the second-order hyperpolarizability (γ) contributes to higher-order phenomenon, which includes the optical Kerr effect (EOKE) and third-harmonic generation (THG), enabling intensity-dependent refractive index changes and efficient frequency conversion in photonics systems [3,4]. Currently, researchers emphasize both theoretical and experimental efforts toward developing novel materials with enhanced NLO responses through diverse design strategies [5-9]. Organic optoelectronic materials have emerged as promising alternatives to their inorganic counterparts due to their inherently higher polarizability, structural flexibility, faster NLO responses, and ease of fabrication [10-14]. These features collectively accelerate progress in advanced photonic and optoelectronic technologies. Moreover, organic chromophores exhibit strong intramolecular charge transfer (ICT) through delocalized π-electron systems, enabling an efficient push-pull mechanism that enhances NLO properties [15,16]. The π-conjugated spacers in D-π-A systems facilitate efficient ICT between donor and acceptor groups, thus enhancing first, second, and third-order NLO response [17,18]. Among various organic materials, fullerenes and their derivatives are recognized as highly effective NLO molecules [19,20]. However, a key limitation is the weak absorption of these systems in the visible region, which can hinder their performance in applications that rely on visible light excitation [14]. Unlike fullerenes, non-fullerene acceptors (NFAs) offer tunable band gaps, better stability, adjustable energy levels, and easier synthesis, making them ideal for optoelectronic applications [21]. A significant second-order NLO response can be achieved by strengthening the donor-acceptor (D-A) interaction [22]. Thiophene, used as the π-spacer, is indicated to facilitate charge transport and enhance the push-pull interaction between the donor and acceptor. Additionally, thiophene derivatives are widely recognized in optoelectronic applications for their favorable electronic properties and ability to support charge transport, making them valuable components in NLO-active materials [23,24].

In this study, 9-methyl-9H-dibenzo[e,g]benzo[5,6][1,4]dithiino[2,3-b]indole, an indole-based donor chromophore, was selected as the parent compound, based on a previous report by Kawahara and colleagues [25]. To enhance its NLO performance, a reference compound (3iaR) was designed by introducing a conjugated thiophene ring and an acceptor group, 2-(2-methylene-1-oxo-1H-benzo[b]cyclopenta[d]thiophen-3(2H)-ylidene)malononitrile to the parent donor molecule. Building upon 3iaR, a series of novel chromophores (3ai2-3ai8) were designed by systematically modifying the acceptor units, all featuring a donor–π–acceptor (D–π–A) configuration. Density functional theory (DFT)/time dependent density functional theory (TD-DFT) calculations were conducted to investigate the electronic structures, optical absorption characteristics, and first and second order hyperpolarizabilities of the designed compounds. The results provide valuable insights into structure-property relationships and key electronic factors influencing NLO behavior. These chromophores could be promising candidates for next-generation optoelectronic technologies.

2. Materials and Methods

All quantum chemical calculations in this study were carried out using the Gaussian 16 software package [26]. Gauss View 6.0 [27] was utilized to build molecular structures and generate Gaussian input files. Ground-state geometry optimizations of the studied chromophores were performed using DFT calculations at M06/6-311G(d,p) basis set and level of theory. This combination was selected based on its proven accuracy in describing noncovalent interactions, charge transfer, and NLO properties, particularly in π-conjugated organic systems [28]. All optimized structures were confirmed as true minima by the absence of imaginary frequencies. TD-DFT calculations were used for UV-Visible absorption spectra, frontier molecular orbitals (FMOs), transition density matrix (TDMs), and electron hole analyses. To investigate stabilization patterns and donor-acceptor interactions, natural bond orbital (NBO) analysis was also performed at the same level of theory. FMOs surface plots and energy gaps were obtained by using Avogadro, [29] facilitating analysis of charge distribution across the molecules. Global reactivity descriptors (GRDs) were calculated from the highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) energy differences to assess chemical reactivity trends among the designed chromophores. To complement the FMO analysis, density of states (DOS) was calculated using PyMOlyze [30] to quantify electron distribution across D–π–A frameworks. UV–Visible spectra were extracted from Gaussian.log files using GaussSum [30], and the resulting absorption data were plotted using Origin [31].

Key NLO-related molecular properties, including dipole moment ( μ ), [32] average polarizability α , [33] the first hyperpolarizability ( β total), and second hyperpolarizability (γtotal) [34] values were calculated using Eqs. (1-5), respectively.

(1)
μ = μ x 2 + μ x 2 + μ x 2 1 / 2

(2)
a = ( a x x + a y y + a z z ) 3

(3)
β t o t a l = ( β x     2 +   β y 2 + β z 2 ) 1 / 2

Whereas βx = βxxx+βxyy+βxzz, βy = βyxx +βyyy+βyzz, and βz = βzxx+βzyy+βzzz

(4)
β i = ∑   β ijj + β jji + β jij i ,   j = x ,   y ,   z

(5)
γ total = γ x     2 +   γ y 2 + γ z 2

Whereas γ i =   1 15   ∑ j ( γ i j j i + γ i j i j + γ i i j j )                             i , j   =   x ,   y ,   z

The main tensor components contributing to β and γ were extracted from Gaussian output and analyzed using Multiwfn 3.8. The βtotal and γtotal were computed by contracting selected tensor elements. The frequency-dependent NLO properties investigated in this study correspond to key photonic effects such as second-harmonic generation (SHG), THG, electro-optic Pockels effect (EOPE), and electro-optic Kerr effect (EOKE). Specifically, SHG is governed by the first-order hyperpolarizability β(−2ω; ω, ω), THG is related to the second-order hyperpolarizability γ(−3ω; ω, ω, ω), EOPE is associated with β(−ω; ω, 0), and EOKE corresponds to γ(−ω; ω, 0, 0) [35-37]. The frequency-dependent (dynamic) NLO response was calculated using the Coupled Perturbed Kohn-Sham (CPKS) [38] approach, and results were extracted using the Multiwfn software. This dynamic first hyperpolaraizability can be represented by Eq. (6).

(6)
β ω = β x 2 + β y 2 + β z 2 1 / 2

The SHG coefficients are calculated using Eq. (7).

(7)
β i = β i i i ( − 2 ω , ω , ω )   + β i j j ( − 2 ω , ω , ω )   + β i k k ( − 2 ω , ω , ω )

The electro-optical Pockels effect (EOPE) coefficients are calculated by Eq. (8).

(8)
β i = β i i i ( − 2 ω , ω , 0 )   + β i j j ( − 2 ω , ω , 0 )   + β i k k ( − 2 ω , ω , 0 )

The frequency-dependent second-order hyperpolarizability is calculated as follows (Eq. 9).

(9)
γ ( ω ) = γ x     2   ( ω ) + γ y 2 ( ω ) + γ z 2 ( ω )  

2.1. Frontier molecular orbitals (FMOs)

FMOs are essential parameters that describe a molecule’s optoelectronic properties, including its light absorption capability and chemical reactivity [39,40]. Quantum orbitals (HOMOs and LUMOs) play a crucial role in ICT, where their spatial redistribution significantly influences charge flow from higher to lower energy levels [41]. The chemical and kinetic stability of compounds is closely linked to the energy gap between their electron-donating (HOMO) and electron-accepting (LUMO) orbitals [42].

The compounds with narrow energy gaps are softer, more reactive, and highly polarizable, leading to enhanced NLO responses, whereas larger band gaps indicate chemical hardness and stability [43-45].

Figure 1 shows that LUMO has charge density aligned over the acceptor moiety, while in the HOMO, charge density is dispersed on the donor part in all the studied molecules. The HOMO LUMO energies and their energy gaps are presented in Table 1. The reference compound (3iaR), having the D–π–A configuration, has an energy gap of 2.719 eV. The incorporation of various electron-withdrawing substituents on the acceptor moieties effectively reduced the energy gap in all the designed compounds compared to the reference chromophore. The compounds 3ai2-3ai8 exhibit ΔE values of 2.662, 2.643, 2.650, 2.536, 2.497, 2.585, and 2.609 eV, respectively. 3ai6 displays the smallest HOMO/LUMO energy gap, due to the presence of a highly electron-withdrawing nitro group, lowering the LUMO energy level. 3ai2, 3ai3, and 3ai4 incorporate highly electronegative halogen atoms (F, Cl, and Br, respectively) within their acceptor segments, thereby reducing energy gaps. Similarly, the electronegative moieties, such as cyano (–CN) in 3ai5, trifluoromethyl (–CF₃) in 3ai7, and ester (–COOCH₃) in 3ai8, enhance ICT by lowering the LUMO energy levels. Moreover, the π linker also contributes to the electron transfer from the donor to the acceptor in all chromophores. Overall declining trend of bang gap of entitled chromophores is 3ai2 > 3ai4 > 3ai3 > 3ai8 > 3ai7 > 3ai5 > 3ai6. The observed trend corresponds to an increase in chemical reactivity, softness, and enhanced ICT across the series. Table shows calculated energies (E) and energy gap (∆E) of HOMO-1, LUMO+1, HOMO-2, and LUMO+2 of all designed compounds (Tables S1-S9).

HOMO and LUMO surface diagrams of the reference 3aiR and designed chromophores 3ai2-3ai8.
Figure 1.
HOMO and LUMO surface diagrams of the reference 3aiR and designed chromophores 3ai2-3ai8.
Table 1. Energies of FMOs.
Compounds EHOMO ELUMO ΔE
3aiR -5.877 -3.158 2.719
3ai2 -5.878 -3.216 2.662
3ai3 -5.886 -3.243 2.643
3ai4 -5.888 -3.238 2.650
3ai5 -5.896 -3.360 2.536
3ai6 -5.900 -3.403 2.497
3ai7 -5.882 -3.297 2.585
3ai8 -5.867 -3.258 2.609

Energy gap (ΔE) = ELUMO−EHOMO, units in eV.

Supplementary Tables S1-S9

2.2. Optical properties

The UV-Visible analysis of 3aiR and 3ai2-3ai8 was performed using TD-DFT calculations in both gas and chloroform phases to evaluate their optoelectronic characteristics. Tables 2 and 3 show the computed excitation energy (E), oscillator strength (fos), transition type, and maximum absorption wavelength (λmax) in the gaseous and solvent phases (chloroform). Additional information has been provided in supplementary data (Tables S10-S25). The polar solvent stabilizes the excited state more effectively than the ground state, resulting in a red shift (longer wavelengths) [46]. Figure 2 displays the spectra of the entitled compounds in the solvent (chloroform) and the gaseous phase.

Table 2. Wavelength (λmax), excitation energy (E), oscillator strength (fos), and nature of molecular orbital contributions of 3aiR and 3ai2-3ai8 in the gaseous phase.
Compounds DFT λ (nm) E(eV) fos MO contributions
3aiR 428.329 2.895 0.510 H-2→L+1 (64%)
3ai2 430.905 2.877 0.459 H-2→L+1 (58%)
3ai3 436.272 2.842 0.506 H-2→L+1 (61%)
3ai4 436.042 2.843 0.486 H-2→L+1 (58%)
3ai5 464.639 2.668 0.586 H-2→L+1 (57%)
3ai6 474.073 2.615 0.486 H-2→L+1 (49%)
3ai7 446.484 2.777 0.513 H-2→L+1 (55%)
3ai8 435.812 2.845 0.513 H-2→L+1 (62%)

MO: Molecular orbitals, H: Highest occupied moleucular orbital, L: Lowest unoccupied molecular orbital, fos: oscillator strength

Table 3. Wavelength (λmax), excitation energy (E), oscillator strength (fos), and nature of molecular orbital contributions of 3aiR and 3ai2-3ai8 in the solvent phase.
Compounds DFT λ (nm) E(eV) fos MO contributions
3aiR 435.002 2.850 0.463 H-2→L+1 (73%)
3ai2 437.952 2.831 0.509 H-2→L+1 (77%)
3ai3 443.434 2.796 0.529 H-2→L+1 (77%)
3ai4 444.467 2.790 0.552 H-2→L+1 (77%)
3ai5 467.125 2.654 0.545 H-2→L+1 (72%)
3ai6 478.907 2.589 0.434 H-2→L (42%)
3ai7 449.153 2.760 0.487 H-2→L+1 (74%)
3ai8 443.529 2.795 0.544 H-2→L+1 (78%)

MO: Molecular orbital, H: Highest occupied moleucular orbital, L: Lowest unoccupied molecular orbital, fos: oscillator strength

Supplementary Tables S10-S25
Absorption spectra of 3iaR and 3ia2-3ia8 in (a) gas and (b) chloroform solvent.
Figure 2.
Absorption spectra of 3iaR and 3ia2-3ia8 in (a) gas and (b) chloroform solvent.

The simulated absorption spectra of the studied compounds in the gas phase have been shown in Figure 2. The absorption range for derivatives 3ai2-3ai8 spans from 430.905 to 474.073 nm (3ai2-3ai8), which is red shifted compared to the reference compound 3aiR, (λmax = 428.329 nm). Similarly, in the solvent phase, the absorption spectra of compounds 3ai2-3ai8 range from 437.952-478.907 nm, showing a red shift compared to the reference compound (λmax = 435.002 nm). Among all the derivatives 3ai6 observed with the highest absorption maxima at 478.907 nm in the solvent phase, with corresponding excitation energy of 2.589 eV due to the strong electron-withdrawing -NO2 group. 3ai5 and 3ai7, containing -CN and -CF3 groups, respectively, show absorption maxima at 464.639 and 449.153 nm with excitation energies of 2.654 and 2.760 eV, respectively. Furthermore, 3ai2, 3ai3, and 3ai4 which feature highly electronegative halogen atoms in their acceptor moieties show maximum absorption at 437.952, 443.434, and 444.467 nm, respectively. 3ai8 bearing -COOCH3 group, shows a λmax value of 443.529 nm, which facilitates charge transfer due to its withdrawing group. The increasing order of λmax in the solvent phase is given as 3ai2 < 3ai4 < 3ai3 < 3ai8 < 3ai7 < 3ai5 < 3ai6, while for the gaseous phase, the order is 3ai2 < 3ai8 < 3ai4 < 3ai3 < 3ai7 < 3ai5 < 3ai6 with little variation at a lower range of wavelength. The transition energies in both phases follow the same trend as the λₘₐₓ values, but in the inverse order. As a result, 3ai6 exhibits the highest absorption wavelength at the lowest excitation energy, highlighting its potential as a promising candidate for NLO applications.

2.3. Global reactivity parameters (GRPs)

The chemical reactivity and kinetic stability of compounds 3iaR and 3ia2-3ia8 were evaluated through global reactivity parameters (GRPs), which were derived from the FMO energy gap [47,48]. Eqs. (10, 11) are used to compute ionization potential (IP) [49] and electron affinity (EA).

(10)
E A = − E LUMO

(11)
I P = − E HOMO

It has been observed that molecular stability is strongly influenced by hardness (η), whereas softness (σ) is associated with chemical reactivity. Additionally, the stability trend is consistent with the increasingly negative chemical potential (μ), indicating reduced tendency for electron transfer and enhanced electronic stability [45]. Eqs. (12-17) were applied to compute the chemical potential (μ), [50]softness (σ) [51], electronegativity (X) [52], global hardness (η), [51] and global electrophilicity index (ω) [53] electronic charge transfer (∆Nmax) [54] using Koopmans’s theorem [55].

(12)
η = I P − E A 2

(13)
σ = 1 2 η

(14)
X = I P + E A 2

(15)
μ = E HOMO + E LUMO 2

(16)
ω = μ 2 2 η

(17)
Δ N max = − μ / η  

Table 4 represents all the computed values of the entitled compounds. The IP values for the 3aiR and 3ai2-3ai8 are 5.887, 5.878, 5.886, 5.896, 5.900, 5.882, and 5.867 eV, while their EA values are 3.258, 3.216, 3.243, 3.238, 3.360, 3.403, 3.297, and 3.258 eV. The increasing order of ionization potential is 3aiR < 3ai8 < 3ai2 < 3ai3 < 3ai4 < 3ai7 < 3ai5 < 3ai6 and that of electron affinity is given as 3aiR < 3ai2 < 3ai4 < 3ai3 < 3ai8 < 3ai7 < 3ai5 < 3ai6 representing their ability to lose and gain an electron. The ascending order of global hardness is given as 3ai6 (1.249 eV) < 3ai5 (1.268 eV) < 3ai7 (1.293 eV) < 3ai8 (1.305 eV) < 3ai3 (1.322 eV) < 3ai4 (1.325 eV) < 3ai2 (1.331 eV) < 3aiR (1.360 eV), showing their increasing stability. Whereas, the order of global softness in eV-1 is 3aiR (0.368) < 3ai2 (0.376) < 3ai4 (0.377) < 3ai3 (0.378) < 3ai8 (0.383) < 3ai7 (0.387) < 3ai5 (0.394) < 3ai6 (0.401), showing 3ai6 as the most reactive compound. In terms of global electrophilicity, the ascending order of accepting additional charge is given as: 3aiR (7.506 eV) < 3ai2 (7. 767 eV) < 3ai4 (7.857 eV) < 3ai3 (7.883 eV) < 3ai8 (7.979 eV) < 3ai7 (8.149 eV) < 3ai5 (8.446 eV) < 3ai6 (8.665 eV). Similarly, the order of ΔNmax explains the charge transfer capabilities as 3aiR (3.323 eV) < 3ai2 (3.416 eV) < 3ai4 (3.444 eV) < 3ai3 (3.454 eV) < 3ai8 (3.498 eV) < 3ai7 (3.551 eV) < 3ai5 (3.650 eV) < 3ai6 (3.726 eV). The chemical potential (μ) values, arranged in decreasing order, are as follows: 3ai6 (-4.652 eV) > 3ai5 (-6.628 eV) > 3ai7 (-4.590 eV) > 3ai3 (-4.565 eV) > 3ai8 = 3ai4 (-4.563 eV) > 3ai2 (-4.547 eV) > 3aiR (-4.518 eV). Among all the investigated compounds, 3ai6 exhibits the most favorable global reactivity parameters (GRPs), indicating superior chemical potential, stability, and reactivity. This aligns with FMO and UV-Vis results, reinforcing its potential as the most promising chromophore in the series.

Table 4. Computed GRPs of 3aiR and 3ai2-3ai8.
Compounds IP EA X η μ ω σ ΔNmax
3aiR 5.877 3.158 4.518 1.360 -4.518 7.506 0.368 3.323
3ai2 5.878 3.216 4.547 1.331 -4.547 7.767 0.376 3.416
3ai3 5.886 3.243 4.565 1.322 -4.565 7.883 0.378 3.454
3ai4 5.888 3.238 4.563 1.325 -4.563 7.857 0.377 3.444
3ai5 5.896 3.360 4.628 1.268 -4.628 8.446 0.394 3.650
3ai6 5.900 3.403 4.652 1.249 -4.652 8.665 0.401 3.726
3ai7 5.882 3.297 4.590 1.293 -4.590 8.149 0.387 3.551
3ai8 5.867 3.258 4.563 1.305 -4.563 7.979 0.383 3.498

I: Ionization potential, A: Electron affinity, X: Electronegativity, global hardness (η), μ: chemical potential, ω: Global electrophilicity, σ: Global softness, and ΔNmax: Maximum charge transfer index. All values are in eV, Softness is in eV-1

2.4. Density of states (DOS)

The DOS analysis was conducted to explore the electronic properties of the compounds, confirming FMO results and providing deeper insights into the contributions of the HOMO and LUMO [56,57]. To gain a deeper understanding, all the chromophores were divided into donor moiety, π-linker core, and acceptor unit. Each part is denoted by a distinct color: blue, green, and red for donor, π-spacer, and acceptor, respectively. The DOS analysis reveals how the distribution of charge pattern is affected by variations in the acceptor subunits, as highlighted by the HOMO-LUMO contributions. The graphs in Figure 3 display the HOMO on the left side and the LUMO on the right side of the x-axis, with the energy gap represented by the distance between them. In Tables S26-S28, the charge density analysis for the HOMO of 3aiR reveals that most of the charge (97.9%) is localized on the donor, while for the LUMO, the charge is predominantly located on the acceptor units (85.2%), as shown in Table S28. The charge distribution for the donor across HOMO and LUMO for compounds 3ai2-3ai8 is as follows: HOMO contributions range from 98.1% to 98.5%, while LUMO contributions range from 2.0% to 4.0%. Similarly, the π-linker exhibits a small but noticeable charge contribution to the HOMO (ranging from 1.0% to 1.3%) and a more significant contribution to the LUMO (ranging from 8.1% to 15.5%). The acceptor’s charge distribution is minimal in the HOMO (0.5% to 0.7%), but its contribution to the LUMO is substantial, varying from 80.5% to 89.9% for 3ai2-3ai8, as presented in Table S26. This efficient charge transfer through the donor–π–acceptor architecture is a critical factor in enhancing the NLO response.

Graphical representation of DOS for reference and designed molecules 3aiR and 3ai2-3ai8.
Figure 3.
Graphical representation of DOS for reference and designed molecules 3aiR and 3ai2-3ai8.

Supplementary Tables S26-S28

2.5. TDM investigation

The TDM analysis is a powerful tool for visualizing the excited-state charge transfer and evaluating interactions among donor, acceptor, and π-spacer units in molecular systems [58]. Moreover, it allows the understanding of electron-hole coherence, mobility, and the extent of ICT, all of which are closely linked to NLO performance [59]. Hydrogen atoms, due to their minimal contribution to electronic transitions, were excluded. As illustrated in Figure 4, the TDM heat maps display the diagonal and near-diagonal transitions, indicating coherent electron-hole delocalization and efficient charge transport. Chromophores 3aiR and 3ai2-3ai8 show pronounced charge flow from the electron-rich donor region to the electron-deficient acceptor through the π-conjugated linker. This spatially extended ICT character is known to enhance polarizability and contribute significantly to first- and second-order hyperpolarizabilities (βtot and γ) [60], which are key to NLO phenomena such as SHG and the EOKE.

TDM plots of 3aiR and 3ai2-3ai8.
Figure 4.
TDM plots of 3aiR and 3ai2-3ai8.

2.6. Binding energy

The binding energy (Eb) is a key parameter for evaluating charge separation and electronic properties. It is inversely related to exciton dissociation and charge mobility, linking the energy gap to efficient exciton dissociation and enhanced charge transport. A lower binding energy corresponds to a weaker Columbic attraction between the electron and hole, facilitating more efficient exciton separation [16]. It can be determined using the Egap and first singlet excitation energy, as given in Eq. (18).

(18)
E b = E H − L − E opt

According to Table 5, the calculated Eb values for 3aiR and 3ai2-3ai8 are 0.376, 0.176, 0.215, 0.199, 0.193, 0.132, 0.118, 0.192, and 0.236 eV, respectively. This is the ascending order of binding energies (Eb): 3ai6 > 3ai5 > 3aiR> 3ai7 > 3ai4 > 3ai3 > 3ai2 > 3ai8. Out of all designed chromophores, 3ai6 exhibited the lowest dissociation energy (0.118 eV), suggesting more efficient exciton dissociation and enhanced charge carrier density.

Table 5. Exciton binding energies (Eb) of 3aiR and 3ai2-3ai8.
Compounds EH-L Eopt Eb
3aiR 2.719 2.895 0.176
3ai2 2.662 2.877 0.215
3ai3 2.643 2.842 0.199
3ai4 2.650 2.843 0.193
3ai5 2.536 2.668 0.132
3ai6 2.497 2.615 0.118
3ai7 2.585 2.777 0.192
3ai8 2.609 2.845 0.236

Units in eV

2.7. Hole-electron analysis

The hole-electron analysis offers a profound understanding of the nature of the excitation process within the compounds [61]. This study performed electron excitation analysis using Multiwfn 3.8 [62]. Figure 5 depicts the transmission of holes across several atoms in the substances under study at a specific intensity. The color spectrum on the right-hand side shows corresponding strength levels. The pictographs show that all the studied compounds have hole-electron interactions. The significant hole intensity originates from the donor moiety, indicating electron migration toward the acceptor regions. These interactions reveal that electron-accepting groups contribute to charge transfer and enhance NLO response.

Pictorial representation of hole-electron transport analysis for 3aiR and 3ai2-3ai8.
Figure 5.
Pictorial representation of hole-electron transport analysis for 3aiR and 3ai2-3ai8.

2.8. Natural bond orbital (NBO)

NBO investigation gives insight into charge transfer, intramolecular delocalization, and hyperconjugative interactions, providing comprehensive insights into bond interactions within π-conjugated systems [63,64]. The HOMO-LUMO energy gap (ΔE) between donor and acceptor orbitals influences the ease of electron density transfer, with a smaller gap indicating more efficient charge transfer and favorable orbital interaction [65]. Eq. (19) was used to compute the stabilization energy.

(19)
E 2 = Δ E i j = q i F i , j 2 E j − E i

Here, i and j represent the donor and acceptor moieties, respectively, while E(2) denotes the associated stabilization energy arising from their interaction. In addition, the donor orbital occupancy is denoted by qi, diagonal elements, while Ei and Ej represent the diagonal and off-diagonal elements corresponding to the acceptor and donor orbitals, respectively. The term Fi,j refers to the off-diagonal element of the NBO Fock matrix that describes the interaction between these orbitals [66]. Key electronic transitions are highlighted in Table 6, while detailed and comprehensive data for all investigated compounds are provided in Tables S27-S34.

Table 6. Selected values of NBO analysis for 3aiR and 3ai2–3ai8.
Compounds Donor (i) Type Acceptor (j) Type E(2) [kcal/mol] E(j)–E(i) [a.u] F(i,j) [a.u]
3aiR C1–C22 π C23–C24 π* 21.84 0.27 0.071
C54–C56 π C52–C53 π* 9.12 0.32 0.05
C48–H49 σ C50–C54 σ* 9.57 0.99 0.087
C51–O55 σ C51–C52 σ* 2.78 1.57 0.06
N36 LP(1) C23–C24 π* 45 0.28 0.101
O55 LP(1) C50–C51 σ* 22.42 0.72 0.115
3ai2 C30–C33 π C25–C32 π* 24.97 0.29 0.076
C41–C42 π C30–C33 π* 12.36 0.32 0.058
C58–N59 σ C56–C58 σ* 8.38 1.57 0.026
C1–N36 σ C1–C22 σ* 1.85 1.38 0.045
N36 LP(1) C23–C24 π* 45.04 0.28 0.101
O55 LP(2) C50–C51 σ* 22.38 0.72 0.115
3ai3 C30–C33 π C25–C32 π* 24.94 0.29 0.076
C48–C50 π C54–C56 π* 17.18 0.31 0.066
C1–C22 σ C23–S26 σ* 7.6 0.88 0.073
C1–C22 σ C1–N36 σ* 1.77 1.12 0.04
N36 LP(1) C23–C24 π* 45.07 0.28 0.101
O55 LP(2) C50–C51 σ* 22.35 0.72 0.115
3ai4 C30–C33 π C25–C32 π* 24.92 0.29 0.076
C58–N59 π C57–N60 π* 0.68 0.48 0.016
C48–H49 σ C50–C54 σ* 9.47 0.99 0.087
C18–H21 σ C15–C18 σ* 0.98 1.13 0.03
N36 LP(1) C23–C24 π* 45.05 0.28 0.101
O55 LP(2) C50–C51 σ* 22.33 0.72 0.115
3ai5 C30–C33 π C25–C32 π* 24.8 0.29 0.075
C58–N59 π C57–N60 π* 0.67 0.48 0.016
C48–H49 σ C50–C54 σ* 9.34 0.99 0.086
C58–N59 σ C54–C56 σ* 0.51 1.64 0.026
N36 LP(1) C23–C24 π* 45.1 0.28 0.101
O55 LP(2) C51–C52 σ* 22.55 0.73 0.116
3ai6 C62–C63 π C64–C67 π* 26.45 0.27 0.076
C41–C42 π C41–C42 π* 0.5 0.31 0.011
C48–H49 σ C50–C54 σ* 9.26 0.99 0.086
C8–H11 σ C7–C8 σ* 0.99 1.13 0.03
O73 LP(3) N74–O75 π* 190.76 0.17 0.161
O55 LP(2) C51–C52 σ* 22.68 0.73 0.116
3ai7 C30–C33 π C25–C32 π* 24.87 0.29 0.076
C57–N60 π C58–N59 π* 0.66 0.47 0.016
C48–H49 σ C50–C54 σ* 9.3 0.99 0.086
C8–H11 σ C7–C8 σ* 0.99 1.13 0.03
N36 LP(1) C23–C24 π* 45.14 0.28 0.101
O55 LP(2) C51–C52 σ* 22.36 0.73 0.116
3ai8 C30–C33 π C25–C32 π* 24.9 0.29 0.075
C61–C62 π C61–C62 π* 0.76 0.28 0.013
C48–H49 σ C50–C54 σ* 9.46 0.99 0.087
C8–H11 σ C7–C8 σ* 0.99 1.13 0.03
O70 LP(2) O71–C83 π* 50.35 0.37 0.123
S26 LP(1) C14–H17 σ* 0.98 1.06 0.029

Supplementary Tables S27-S34

The π conjugated system in entitled compounds having D–π–A framework significantly facilitates the π → π* electronic transition, thus boosting their NLO response. The feeble σ → σ* transitions are observed due to minimal interaction between electron-donating and electron-withdrawing units. Table 6 highlights the electronic transitions in compound 3aiR. The strongest π → π* transition occurs from π(C1–C22) to π*(C23–C24) at 21.84 kcal/mol, while the weakest is between π(C54–C56) and π*(C52–C53) at 9.12 kcal/mol. The most significant σ → σ* transition is from σ(C48–H49) to σ*(C50–C54) at 9.57 kcal/mol, with the weakest being from σ(C51–O55) to σ*(C51–C52) at 2.78 kcal/mol. The dominant LP → π* interaction is LP1(N36) to π*(C23–C24) at 45 kcal/mol, and the lowest LP → σ* transition is LP1(O55) to σ*(C50–C51) at 22.42 kcal/mol. In compound 3ai2, the highest π → π* transition is from π(C30–C33) to π*(C25–C32) at 24.97 kcal/mol, while the weakest is from π(C41–C42) to π*(C30–C33) at 12.36 kcal/mol. The strongest σ → σ* transition is from σ(C58–N59) to σ*(C56–C58) at 8.38 kcal/mol, and the weakest is from σ(C1–N36) to σ*(C1–C22) at 1.85 kcal/mol. The most significant LP → π* interaction is LP1(N36) to π*(C23–C24) at 45.04 kcal/mol, while the lowest LP → σ* transition is LP2(O55) to σ*(C50–C51) at 22.38 kcal/mol. In compound 3ai3, the strongest π → π* transition occurs from π(C30–C33) to π*(C25–C32) with an energy of 24.94 kcal/mol, while the weakest is from π(C48–C50) to π*(C54–C56) at 17.18 kcal/mol. The most significant σ → σ* transition is between σ(C1–C22) and σ*(C23–S26) at 7.6 kcal/mol, with the lowest occurring from σ(C1–C22) to σ*(C1–N36) at 1.77 kcal/mol. The highest LP → π* interaction is LP1(N36) to π*(C23–C24) at 45.07 kcal/mol, and the weakest LP → σ* transition is LP2(O55) to σ*(C50–C51) at 22.35 kcal/mol. In 3ai4, the most prominent π → π* transition is from π(C30–C33) to π*(C25–C32) at 24.92 kcal/mol, with the weakest occurring between π(C58–N59) and π*(C57–N60). The highest σ → σ* transition is from σ(C48–H49) to σ*(C50–C54) at 9.47 kcal/mol, while the lowest is from σ(C18–H21) to σ*(C15–C18) at 0.98 kcal/mol. LP1(N36) to π*(C23–C24) represents the dominant LP → π* interaction at 45.05 kcal/mol, while the lowest LP → σ* value is LP2(O55) to σ*(C50–C51) at 22.33 kcal/mol. For 3ai5, the largest π → π* transition is from π(C30–C33) to π*(C25–C32) at 24.8 kcal/mol, with the smallest being from π(C58–N59) to π*(C57–N60) at 0.67 kcal/mol. The most significant σ → σ* transition occurs between σ(C48–H49) and σ*(C50–C54) at 9.34 kcal/mol, and the weakest is from σ(C58–N59) to σ*(C54–C56) at 0.51 kcal/mol. LP1(N36) to π*(C23–C24) shows the highest LP → π* interaction at 45.1 kcal/mol, while the lowest LP → σ* value is LP2(O55) to σ*(C51–C52) at 22.55 kcal/mol. In 3ai6, the largest π → π* transition occurs from π(C62–C63) to π*(C64–C67) at 26.45 kcal/mol, while the weakest is from π(C41–C42) to π*(C41–C42) at 0.5 kcal/mol. The highest σ → σ* transition is from σ(C48–H49) to σ*(C50–C54) at 9.26 kcal/mol, and the weakest is from σ(C8–H11) to σ*(C7–C8) at 0.99 kcal/mol. The strongest LP → π* interaction is LP3(O73) to π*(N74–O75) at 190.76 kcal/mol, while the weakest LP → σ* transition is LP2(O55) to σ*(C51–C52) at 22.68 kcal/mol. For 3ai7, the highest π → π* transition is from π(C30–C33) to π*(C25–C32) at 24.87 kcal/mol, and the lowest is from π(C57–N60) to π*(C58–N59) at 0.66 kcal/mol. The greatest σ → σ* transition is from σ(C48–H49) to σ*(C50–C54) at 9.3 kcal/mol, while the weakest is from σ(C8–H11) to σ*(C7–C8) at 0.99 kcal/mol. LP1(N36) to π*(C23–C24) shows the strongest LP → π* transition at 45.14 kcal/mol, with the lowest LP → σ* value at LP2(O55) to σ*(C51–C52) at 22.36 kcal/mol. In the final compound, 3ai8, the largest π → π* transition is from π(C1–C7) to π*(C2–C3) at 24.9 kcal/mol, while the weakest is from π(C10–C20) to π*(C10–C20) at 0.76 kcal/mol. The highest σ → σ* transition occurs from σ(C73–C75) to σ*(C77–N79) at 9.46 kcal/mol, and the lowest is from σ(C9–N13) to σ*(C3–C9) at 0.99 kcal/mol. The strongest LP → π* interaction is LP2(S74) to π*(C72–C73) at 50.35 kcal/mol, while the lowest LP → σ* value is LP1(N79) to σ*(S74–C77) at 0.98 kcal/mol. NBO analysis reveals that the hyperconjugation and internal charge transfer significantly enhance stability and NLO properties, achieved through effective structural modifications using π-linkers in the studied molecules.

2.9. Non-linear optics (NLO)

NLO materials are crucial for advanced technologies such as optoelectronics, data storage, optical signal processing, and laser protection systems [67,68]. These organic molecules can effectively interact with electromagnetic (EM) forces due to their structural variation, which in turn causes the formation of new EM waves with varying frequencies, phases, and amplitudes [69,70]. These waves’ propagation characteristics are very different from the incident wave’s [71]. Advancements in EOM, SHG, and other means of communication have led to the creation of materials with stronger NLO properties [58,72]. NLO phenomena originate from the nonlinear polarization response of materials to an applied electric field, typically expressed as a power series expansion: P = αE + βE2 + γE3 +..., where α, β, and γ are the linear, second-, and third-order polarizabilities, respectively. These NLO parameters are tensor quantities: β is a third-rank tensor with 27 components, and γ is a fourth-rank tensor with 81 components in Cartesian space [73,74]. β governs second-order effects such as second-harmonic generation (SHG) and the Pockels effect, while γ contributes to third-order processes like THG, the EOKE, and two-photon absorption.

2.10. Static NLO properties

In Table 7, the calculated values for important parameters focus on the NLO properties of the examined materials. These include total dipole moment (µtotal), average linear polarizability ⟨α⟩, and first- and second-order hyperpolarizabilities (βtotal and γtotal) calculated at static frequencies. The values of linear and NLO responses at static frequencies, along with the contributing tensor, have been provided in Tables S35-S40. The first- and second-order hyperpolarizabilities are reported in the centimeter-gram-second (CGS) unit system as cm⁴·statvolt⁻1 and cm⁵·statvolt⁻1, respectively, for experimental comparison.

Table 7. Calculated linear and NLO responses of 3aiR and 3ai2-3ai8.
Compounds µtotal ⟨α⟩×10-22 βtotal×10-28 γtotal×10-33
3aiR 7.5436 1.35 1.60 1.65
3ai2 5.8657 1.34 1.72 1.66
3ai3 5.9697 1.40 1.71 1.78
3ai4 5.8750 1.43 0.21 1.46
3ai5 8.2955 1.43 2.65 2.19
3ai6 7.9180 1.41 2.89 2.32
3ai7 6.3132 1.36 2.01 1.67
3ai8 2.5176 1.45 1.85 1.76

µtotal in Debye (D), ⟨α⟩ in cm3, βtotal in statvolt⁻1 cm⁴, and γtotal in statvolt⁻1 cm5.

Supplementary Tables S35-S40

The degree of charge dispersion inside a molecule is reflected by the total dipole moment (µtotal) influenced by a change in electronegativity. The significant dipole moments shown by compounds 3ai5 (8.2955 D) and 3ai6 (7.9180 D) reflect their enhanced NLO responses. These compounds exhibit significant charge-transfer interactions, leading to enhanced polarization, particularly when influenced by an applied electric field. It is evident from Table 7 that linear polarizability <αtotal> values vary from 1.34×10−22 to 1.45×10−22 statvolt⁻1 cm⁴ with 3ai8 showing the highest value in this range. The first hyperpolarizability (βtot), a measure of second-order NLO processes, exhibits substantial variation over the series. 3ai5 and 3ai6 display with the highest values of βtotal as 2.65×10⁻2⁸ statvolt⁻1 cm⁴ and 2.89×10⁻2⁸ statvolt⁻1 cm⁴ correspondingly. The second hyperpolarizability (γtotal) with a similar pattern to βtotal shows that 3ai5 and 3ai6 have higher efficiency over other compounds, with respective γtotal values of 2.19×10−33 statvolt⁻1 cm⁴ and 2.32×10−33 statvolt⁻1 cm⁴. The compound 3ai4 shows the lowest value of βtotal 0.21×10⁻2⁸ statvolt⁻1 cm⁴, and γtotal value of 1.46×10⁻33 statvolt⁻1 cm⁴. Following the greatest βtotal and γtotal values, compounds 3ai5 (ΔE=2.536 eV) and 3ai6 (ΔE=2.497 eV) have the lowest ΔE. In contrast, 3aiR and 3ai8, with relatively larger ΔE (2.719 eV and 2.609 eV, respectively), show lower NLO responses.

2.11. Frequency-dependent NLO properties

The frequency-dependent first hyperpolarizability, β(ω), and second hyperpolarizability γ (ω), were evaluated at two dispersion frequencies 0.042823 Eh (1064 nm) and 0.085645 Eh (532 nm) using M06/6-31G(d,p) functional. These values correspond to two key processes: electro-optic Pockels effect (EOPE) (−ω; ω, 0) and second-harmonic generation (SHG) (−2ω; ω, ω). The EOPE-related β(−ω; ω, 0) values at 532 nm show a significant enhancement across the series, ranging from 2.33 to 4.47 × 10⁻2⁸ statvolt⁻1 cm⁴., with the highest response observed for 3ai6 due to the presence of a strong electron-withdrawing –NO₂ group. At the lower dispersion frequency (1064 nm), the β(−ω; ω, 0) values are notably smaller in comparison, confirming that the EOPE is more pronounced at higher photon energy (shorter wavelength). Similarly, the SHG component β(−2ω; ω, ω) displays frequency-dependent behavior. At 532 nm, SHG values span from 6.11 to 179 × 10⁻2⁸ statvolt⁻1 cm⁴, again showing enhanced hyperpolarizability relative to values computed at 1064 nm, where the response is generally reduced. Overall, the results confirm the strong frequency dispersion of dynamic first-order hyperpolarizability, with both EOPE and SHG effects being more pronounced at 532 nm, attributed to more effective donor–acceptor charge transfer at higher photon energies. The comparative variation of β(−ω; ω, 0) and β(−2ω; ω, ω) across both frequencies has been summarized in Table S38. The second-order hyperpolarizability (γtot) was dynamically computed through two key tensor components: γ(−ω; ω, 0, 0), associated with the EOKE, and γ(−2ω; ω, ω, 0), relevant to THG. At a dispersion frequency of 1064 nm, the γ(−ω; ω, 0, 0) values range from 2.44 to 4.26 × 10⁻33 statvolt⁻1 cm⁴, with 3ai6 exhibiting the highest value, likely due to its strong donor-acceptor interaction and extended π-conjugation. In contrast, at 532 nm, these values show substantial variation, including large negative magnitudes (e.g., −421000 statvolt⁻1 cm⁴, for 3ai8), indicating possible resonance effects and complex dispersive behavior at higher excitation energies. The second component, γ(−2ω; ω, ω, 0), also reflects notable frequency dependence. At 1064 nm, values span from –3830 to 10600 × 10⁻33 statvolt⁻1 cm⁴, while at 532 nm, the variation persists, with 3ai6 again showing the highest positive value of 771 × 10⁻33 statvolt⁻1 cm⁴. These findings, summarized in Table S36, highlight the strong influence of both frequency and molecular structure on the dynamic second-order hyperpolarizability across the compound series.

3. Results and Discussion

This study explores the electronic structure and NLO properties of non-fullerene compounds comprising an indole-based donor and benzothiophene-based acceptor unit linked by a thiophene π-bridge. To further enhance the NLO response, a series of seven derivatives (3ai2-3ai8) was rationally designed by systematically altering the terminal acceptor groups. The ChemDraw structures of the parent compound (3ai), the reference molecule (3aiR), and its derivatives (3ai2-3ai8) have been shown in Figure S1-S3, while the Cartesian coordinates of all optimized structures have been provided in Tables S1-S8. Figure 6 presents a schematic representation of acceptor modification in 3aiR, and Figure 7 displays the optimized geometries of 3aiR and its derivatives.

Supplementary Figures S1-S3
Schematic illustration of acceptor unit modifications in reference compound 3aiR.
Figure 6.
Schematic illustration of acceptor unit modifications in reference compound 3aiR.
Optimized structures of 3aiR and 3ai2-3ai8.
Figure 7.
Optimized structures of 3aiR and 3ai2-3ai8.

4. Conclusions

In this study, NLO responses of indole-based D-π-A compounds (3ai2-3ai8) were systematically investigated through strategic variations of the terminal acceptor groups. Quantum-chemical analysis at the M06 level provided detailed insights into the molecular geometry and electronic behavior of the designed chromophores. Among all the derivatives, compound 3ai6 exhibited the most favorable performance, characterized by the lowest band gap (2.497 eV), the longest wavelength absorption (478.907 nm), and notable reactivity with a softness value of 0.401 eV⁻1. TDM and DOS analyses validated the charge migration from the donor function to the acceptor group through the π-spacer, highlighting the enhanced electronic interactions. Furthermore, 3ai6 showed exceptional βtotal and γtotal values, showing its potential as an efficient NLO material. The findings provide crucial insights for designing indole-based compounds with optimized NLO properties for advanced applications.

Acknowledgment

The authors extend their sincere appreciation to Ongoing Research Funding Program (ORF-2025-253), King Saud University, Riyadh, Saudi Arabia. S.C.O. acknowledges the support from the doctoral research fund of the Affiliated Hospital of Southwest Medical University. MI expresses appreciation to the Deanship of Scientific Research at King Khalid University Saudi Arabia through the research groups program under Grant Number RGP-2/695/46.

CRediT authorship contribution statement

Sadia Jamal: Supervision; Investigation; Resources; software; project administration; Methodology. Muhammad Arshad: Data Curation; Writing - original draft: Formal analysis. Maham Mehfooz: Data Curation; Writing - original draft: Formal analysis. Wen Qin: Formal analysis; Investigation; Writing - review & editing, funding. Norah Alhokbany: Formal analysis; Validation; Visualization, funding. Suvash Chandra Ojha: Formal analysis; Investigation; Validation; Visualization, funding.

Declaration of competing interest

The authors have no relevant financial or non-financial interests to disclose.

Data availability

All data generated or analyzed during this study are included in this published article and its supplementary information files.

Declaration of generative AI and AI-assisted technologies in the writing process

The authors confirm that there was no use of artificial intelligence (AI)-assisted technology for assisting in the writing or editing of the manuscript and no images were manipulated using AI.

Supplementary data

Supplementary material to this article can be found online at https://dx.doi.org/10.25259/AJC_401_2025.

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