Exploring the frequency-dependent nonlinear optical response of push-pull configured thianthrene-based chromophores: DFT/TD-DFT approach

2.1 Computational procedure

The quantum chemical calculations of the designed compound and its derivatives were conducted using Gaussian 16 program [25]. The molecular structures were optimized without any geometrical restrictions via GaussView 5.0 [26] software program. All computational analysis employed the M06 [27] functional combined with the 6–311G (d, p) basis set within the density functional theory (DFT) and time-dependent DFT (TD-DFT) frameworks. Frontier molecular orbitals (FMOs), transition density matrices (TDMs), density of states (DOS), global reactivity parameters (GRPs), absorption spectra, and hole-electron analysis were performed using the TD-DFT approach while NLO and natural bond orbital (NBO) analyses were carried out via DFT methodology at the aforementioned functional. The optimized geometries were further executed to compute the UV-Vis spectra, FMOs, DOS, NBO, and NLO properties at the same functional and basis set. Various software tools were employed to process the input files, such as Avogadro software [28] for FMO diagrams, NBO software package 3.1 [29] for NBO analyses, GaussSum [30] and Origin [31] for UV-Vis spectral analysis, and PyMOLyze [32] for DOS analysis.

The following Equations (1,2) were used to compute the linear polarizability 〈α〉 and the total first polarizability (β_tot):

Where:

The second hyperpolarizability (γ_tot) was computed using Eq. 3.

To provide valuable insights for experimentalists, we have computed the frequency-dependent NLO responses. These include the second harmonic generation (SHG) β(−2ω,ω,ω), the electro-optical Pockels effect (EOPE) β(−ω,ω,0), the electro-optic Kerr effect (EOKO) γ(−ω;ω,0,0) and the SHG γ(−2ω;ω,ω,0). The computations were conducted at the widely used laser wavelengths of 532 nm and 1064 nm [5].

The double-frequency first hyperpolarizability β(ω) is expressed as Eq 4:

Where ⟨α⟩, β_(tot), β(ω) are reported in esu.

3.1. Electronic properties

FMOs analysis is an important technique to understand the chemical stability and optoelectronic properties [33]. The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), are characterized by their electron donating and electron-withdrawing capabilities, respectively [34]. The FMO energy gap (ΔE) reflects the excitation energy required for electronic transitions and directly influences the NLO performance [35]. Table 1 outlines the HOMOs and LUMOs energy levels, and their energy gaps for the designed molecules PTMR1–PTMD7.

For the reference compound PTMR1, the HOMO and LUMO energy levels are -5.904 eV and -2.508 eV, respectively, with an energy gap of 3.396 eV (Table 1). Similarly, the E_HOMO of the designed molecules PTMD2 to PTMD7 are examined to be -6.221, -6.232, -5.641, -6.318, -6.331, and -6.318 eV, while the E_LUMO are found to be -2.590, -3.343, -2.606, -3.547, -3.702, and -3.526 eV, respectively. The corresponding energy gap values of the studied chromophores are calculated to be 3.631, 2.889, 3.035, 2.771, 2.629, and 2.729 eV, respectively.

The introduction of efficient electron-withdrawing moieties systematically lowers ΔE, enhancing ICT and thereby improving hyperpolarizability. The HOMO-LUMO energy gap (ΔE) reflects the excitation energy required for electronic transitions and directly influences the NLO performance. Similar DFT studies on donor–π–acceptor systems have reported ΔE values in the region of ∼2.5 to ∼3.5 eV for effective ICT chromophores [36-38]. Among all the designed derivatives, PTMD6, with nitro (–NO₂) and cyano (–CN) groups, exhibits the smallest bandgap (2.629 eV). These substituents exert strong mesomeric and inductive effects, stabilizing the LUMO while slightly destabilizing the HOMO. This intensifies the push-pull interaction and promotes efficient CT, consequently demonstrating a significantly higher NLO response. In contrast, PTMD2, containing a relatively weaker acceptor, shows the largest ΔE (3.631 eV), leading to reduced ICT and weaker NLO response.

Overall, the bandgap reduces in the following order: PTMD2 > PTMD4 > PTMD3 > PTMD5 > PTMD7 > PTMD6. This sequence confirms that the incorporation of electron-withdrawing groups is effective in reducing the energy gap, thus improving the NLO response. There is an inverse relationship between ΔE and ICT, which means that chromophores with strong charge transference capability among orbitals demonstrate a significant NLO response with a lower ΔE value. Similarly, higher ΔE values correlate with weaker ICT and lower NLO performance. This relationship is further validated by the trends observed for HOMO−1, HOMO−2, LUMO+1, and LUMO+2, as shown in Table S9 and Figure S3.

The distribution of electronic cloud in the studied compounds is illustrated via contour surface illustrations presented in Figure 3. It reveals that in the reference compound PTMR1 electronic cloud is concentrated on the HOMOs. In contrast, the designed compounds show a shift of the electronic cloud from the central core to the terminal units. This shift indicates that the presence of a π-bridge, which connects the electron donor and acceptor groups, facilitates the charge migration from HOMO to LUMO orbitals. Thus, this enhanced charge density migration, along with polarization, makes these compounds efficient NLO materials with significant potential for applications in nonlinear optics.

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Figure 3.

FMO surface diagrams of PTMR1 and PTMD2–PTMD7.

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3.2. Global reactivity parameters

Global reactivity indices can be determined using the E_HOMO, E_LUMO, and their energy gap. Koopman’s theorem [39] is employed to derive these parameters such as electronegativity (X) [40], electron affinity (EA) [41], global electrophilicity (ω) [42], global hardness (η), global softness (σ), chemical potential (μ), ionization potential (IP) and maximum charge transfer index (ΔN_max) [43,44]. The ionization potential (IP) reflects the electron-donating capacity of a molecule and represents the energy needed to extract an electron from the HOMO. While, the electron affinity (EA) describes electron accepting ability and equals to the energy released upon the addition of an electron. However, electronegativity (X) indicates the characteristic of a molecule to attract the electronic cloud towards itself. GRPs can be calculated using the standard Eqs. 5-11.

The findings reveal a substantial relationship between GRPs and HOMO–LUMO energy gaps. This energy gap is related inversely to chemical reactivity and directly to the stability, chemical potential, and hardness of a chromophore. The compounds having large energy band gaps are more stable, less reactive, and harder. However, less stability, more reactivity, and softness are associated with small energy gaps. Thus, molecules with lower energy band gaps were found to be less stable, more reactive, and polarized, which makes them suitable for enhanced NLO responses. The calculated GRP values for PTMR1 and PTMD2–PTMD7 have been shown in Table 2.

The chromophore PTMD6 shows the smallest band gap (2.629 eV) with the highest softness (0.380 eV) and the lowest hardness values (1.315 eV). These values show that PTMD6 is the most chemically reactive and polarizable system, which directly contributes to its significant NLO response. The strong electron-withdrawing substituents (–NO₂ and –CN) effectively stabilize the LUMO and enhance charge delocalization, lowering the energy barrier for charge transfer. The global softness of other derivatives PTMD2–PTMD5 and PTMD7 are 0.275, 0.346, 0.329, 0.361, and 0.358 eV^{- 1}, respectively. Their global hardness values are 1.816, 1.445, 1.518, 1.386, and 1.396 eV, respectively. The reactivity trend follows the order PTMD2 > PTMD4 > PTMD3 > PTMD5 > PTMD7 > PTMD6, which is consistent with their electronic structures and donor–acceptor strengths. Additionally, the electron affinity values (EA) of designed molecules range from (2.508–3.526 eV), while IP values range from (6.221–6.318 eV). The relatively high EA and low IP of PTMD6 further support its ability to facilitate ICT. Collectively, these GRPs confirm that PTMD6, owing to its reduced band gap, high softness, and strong polarization, is structurally optimized for advanced NLO applications.

3.3. Density of states analysis

The DOS investigation is conducted to validate findings obtained from FMO analysis and investigate the charge distribution patterns in the designed chromophores PTMR1 and PTMD2–PTMD7 [45]. To perform this analysis, the designed compounds are divided into three parts: acceptor, π-linker, and donor, which are correspondingly represented by red, blue, and green lines in the graphs (Figure 4). In DOS plots, these fragments display charge density, whereas the left side corresponds to HOMO and the right side to the LUMO. The DOS contributions at the HOMO and LUMO levels illustrate how different donor units influence the electronic charge distribution across molecular orbitals.

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Figure 4.

DOS pictographs of PTMR1 and PTMD2- PTMD7.

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For PTMR1 and PTMD2–PTMD7, the charge density percentages of acceptor at HOMO are 11.0 and 78.0, 4.5, 6.5, 7.1, 8.9, 7.5%, respectively. The charge density percentages of acceptor at LUMO are 86.2, 2.9, 88.0, 88.3, 87.4, 90.5, and 87.4%. The donor displayed 74.3, 9.1, 88.2, 84.2, 82.6, 84.9, and 81.6% electronic charge distribution pattern towards HOMOs, while 3.1,86.9, 2.2, 2.5, 2.7, 2.4, and 2.8% to LUMOs in all the designed chromophores, see Table S10. This donor-to-acceptor redistribution across the HOMO–LUMO boundary indicates the strong internal charge transmission that is essential for enhancing hyperpolarizability. The DOS investigation significantly supported the charge distribution patterns observed in the FMO contour surfaces (Figure 3). Peaks on the left (HOMO region) are primarily associated with donor orbitals, while those on the right (LUMO region) originate from acceptor orbitals. The clear spatial and energetic separation of donor and acceptor states, bridged by the π-spacer, reduces the bandgap and facilitates efficient charge migration. Notably, PTMD6 exhibits the most pronounced donor-to-acceptor charge relocation, consistent with its lowest bandgap and highest predicted NLO response. In a nutshell, both FMO and DOS analyses confirm that strong electron-withdrawing substituents reduce ΔE, enhance ICT, and thereby improve the β values of the designed chromophores.

3.4. UV–Visible analysis

UV-Visible analysis provides valuable insights related to the configurations of the nature of electronic transitions, contributing transitions, and the ability for charge transfer in the chromophores. This absorption analysis investigates the key parameters such as oscillation frequency (f_os), maximum absorption wavelength (λ_max), electron excitation energy (E), and the type of transitions in the studied systems. The UV–Vis spectral analysis is conducted using the TD-DFT approach at M06/6–311G (d, p) functional to calculate the aforementioned parameters in dichloromethane as well as in the gaseous phase. The absorption spectra, as shown in Figure 5, are used to explore the impact of different acceptor moieties on the absorption properties of the studied compounds PTMR1 and PTMD2-PTMD7. The computed values for the transition energy (E) in eV, maximum absorption wavelength λ_max (nm), oscillator strength (f_os), and contributions from molecular orbitals (MO) have been presented in Table 3. Other significant contributions from molecular orbitals, along with further details, have been listed in Tables S11 and S12.

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Figure 5.

The simulated absorption spectra of the PTMR1 and PTMD2-PTMD7.

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The polarity of the solvent strongly influences the absorption spectra by inducing a bathochromic shift [45]. The absorption spectra in dichloromethane (DCM) were computed using the conductor-like polarizable continuum model (CPCM) to account for solvent effects [46]. Due to its higher polarity, DCM stabilizes the excited state compared to the ground state (S₀), thereby influencing the spectral behavior [47]. In the solvent phase, the absorption values of PTMD2–PTMD7 range from 433.329 to 445.842 nm, with transition energies ranging from 2.861 to 2.781 eV and oscillation strength of 0.693 to 0.774. Notably, all derivatives exhibit higher λ_max values compared to the reference PTMR1 (430.995 nm). The λ_max values clearly show that the presence of electron-withdrawing groups in the acceptor part leads to absorbance shift towards longer wavelengths.

Among all the compounds, PMTD3 and PMTD6 exhibit the highest absorption maximum at 445.842 and 445.378 nm. The cyano (–CN), chloro (–Cl), and nitro (–NO₂) groups in PMTD3 introduce greater electronegativity (electron–pull effect), which enhances the red-shift effect in electronic transitions. The overall λ_max order in DCM is as follows: PMTR1 <PMTD2 < PMTD7 < PMTD4 < PMTD5 < PMTD6 < PMTD3.

Additionally, the analysis conducted in the gas phase shows that the λ_max values range from 425.785 to 456.361 nm. The increasing order of λ_max for the compounds studied in the gas phase is as follows: PTMD6 < PTMD3 < PTMD4 < PTMD7 < PTMD5 < PTMD2 < PTMR1. Among them, PTMD6 exhibits the most prominent wavelength (λ_max = 456.361 nm) in the gas phase, with a transition energy of 2.717 eV and an f_os of 0.362. It is concluded that PTMD6 and PTMD3 show reduced transition energies, narrow band gaps, and notable bathochromic shifts, highlighting them as a potent material for NLO applications.

3.5. Natural bond orbital study

NBO investigation provides an orbital representation that closely aligns with the classical Lewis structure of a molecule. This approach examines the delocalization of electron density as well as the intermolecular and intramolecular interactions between the occupied (donor) and unoccupied (acceptor) orbitals within a conjugated system [48]. Additionally, it elucidates the non-covalent interactions responsible for charge density transfer, orbital-orbital interactions within bonds and hyperconjugation between donor and acceptor moieties [49]. Second-order perturbation theory is applied to interpret the off-diagonal elements between donor and acceptor NBOs, which represent stabilization energies resulting from hyperconjugation [50].

Based on this approach, the stabilization energy E⁽²⁾ related to i (donor) → j (acceptor) delocalization is calculated as follows in Eq. 12.

Where i and j represent D and A moieties, respectively. The q_i, F_i, j and ε_j – ε_i corresponds to donor orbital occupancy, off-diagonal and diagonal NBO Fock matrix components.

For the investigated compounds PTMR1 and PTMD2-PTMD7, all the calculated values of E ⁽²⁾ have been recorded in Tables S13–S19, while some significant transitions have been discussed in Table 4. The occupied bonding orbitals and unoccupied non-bonding orbitals interact and result in some prominent transitions such as; π → π*, σ → σ*, LP → π* and LP → σ*. Among these, the least dominant is σ → σ* transitions and the highly dominant is π → π*transition depending upon the stabilization energy factor.

For PTMR, the highest stabilization energy (E²) values are 28.76, 2.05, 9.78 and 1.97 kcal mol^–1 for the transitions such as π(C40-C41) → π*(C42-C44), σ(C42-H43) → σ*(C40-S64), LP2 (S70) → π*(C29-C31) and LP (2) (O58) → σ*(C39-S64). The lowest stabilization energy values are recorded as 2.05, 1.97 kcal mol^–1 for transitions π(C42-C44) → π*(C42-C44) and σ(C3-C4) → σ*(C7-C8), respectively.

For PTMD2, the electronic transitions π(C41-C42) → π*(C43-C45), σ(C50-C52) → σ*(C52-N54), LP1 (S30) → π*(C14-C16) and LP1 (S30) → σ*(C14-H17) have the highest stabilizing energies as 29.59, 9.72, 0.77and 19.52 kcal mol^–1, respectively. However, the lowest stabilization energy values are recorded as 0.56, 1.96 kcal mol^–1 for transitions π(C62-C63) → π*(C59-C60) and σ(C47-C55) → σ*(C47-C48), respectively.

The transitions in derivative PTMD3 are π(C1-C22) → π*(C2-C3), σ(C1-C2) → σ*(C2-C6), LP2 (S65) → π* (C40-C66) and LP1 (N53) → σ* (C50-C51) have the highest stabilizing energy values as 29.03, 9.82, 28.43, and 12.61 kcal mol^–1, correspondingly. On the other hand, the lowest stabilization energy values are 0.74 and 2.78 kcal mol^–1 for the transitions π(C59-C60) → π*(C56-C58) and σ(C15-H19) → σ*(C4-C15), respectively.

In PTMD4, the transitions such as π(C41-C42) → π*(C43-C45), σ(C43-H44) → σ*(C45-C48), LP2 (S65) → π* (C40-C66) and LP1 (S31) → σ* (C29-C32) have the highest stabilizing energies as 30.21, 7.29, 28.76 and 3.92 kcal mol^–1, respectively. As for transitions π(C48-O49) → π*(C43-C45) and σ(C46-C55) → σ*(C45-C46) the lowest stabilization energy values are 3.65 and 1.98 kcal mol^–1, respectively. The chromophore PTMD4 exhibits the second highest value for π→ π* transitions and is considered a prominent chromophore.

In PTMD5, the highest stabilization energies are 29.7, 9.84, 28.63 and 12.63 kcal mol^–1for transitions π(C41-C42) → π*(C43-C45), σ(C43-H44) → σ*(C41-S65), LP2 (S65) → π* (C40-C66) and LP1 (N54) → σ* (C50-C52), respectively. As for transitions π(C46-C50) → π*(C46-C50) and σ(C3-C4) → σ*(C7-C8) the lowest stabilization energy values are 1.64 and 1.97 kcal mol^–1, respectively.

In PTMD6, the highest stabilization energies are 30.44, 9.9, 28.81and 12.64 kcal mol^–1for transitions π(C41-C42) → π*(C43-C45), σ(C43-H44) → σ*(C41-S65), LP2 (S65) → π* (C40-C66) and LP1 (N54) → σ* (C50-C52), respectively. As for transitions π(N72-O74) → π*(C60-C63) and σ(C60-N72) → σ*(C60-C63) the lowest stabilization energy values are 2.81 and 0.99 kcal mol^–1, respectively. The designed compound PTMD6 is recognized as a prominent chromophore as it has the highest stabilization energy value for the for π-conjugated transitions.

In PTMD7, the transitions are such as π(C41-C42) → π*(C43-C45), σ(C43-H44) → σ*(C41-S65), LP (2) (O69) → π* (O70-C73) and LP1 (N54) → σ* (C50-C52) have the highest stabilization energies as 29.58, 9.84, 48.98 and 12.63 kcal mol^–1, respectively. As for transitions π(C46-C50) → π*(C46-C50) and σ(C43-C45) → σ*(C47-C48) the lowest stabilization energy values are 1.66 and 0.86 kcal mol^–1, respectively.

It is evident that among all the fabricated compounds, the PTMD6 exhibit the greatest stability. The stability order in decreasing energy is represented as PTMD6 > PTMD4 > PTMD5 > PTMD1 > PTMD7 > PTMD2. PTMD6 shows the strongest π-conjugated stabilization, leading to pronounced electronic delocalization. This accounts for its reduced band gap, efficient ICT, and superior NLO response, confirming that –NO₂ and –CN substituents markedly enhance hyperpolarizabilites.

3.6. Hole-electron interaction analysis

Hole-electron interaction refers to the transition of an electron from the hole region to the electron region during the excitation process. This analysis is a highly efficient method to localize the charge density within a compound [51]. To investigate the charge flow in the designed compounds, electron excitation analysis was conducted using Multiwfn 3.8. In the studied compounds, the highest electron density is localized at the acceptor region, specifically at C (30) in PTMR, C (29) in PTMD2, C (34) in PTMD3 and C (32) in PTMD4–PTMD7. Figure S4 highlights that the electron withdrawing acceptor unit exhibits significant electron intensity at the carbon atoms, particularly those connected to the cyano (–CN) group. This observation is attributed to the strong electron withdrawing capability of the cyano (–CN) group which is further enhanced by its high electronegativity and resonance-stabilizing effects [52]. However, weaker hole density is observed at the donor region, located at C (20) in PTMR1, C (27) in PTMD2, C (20) in PTMD3 and C (17) in PTMD4–PTMD6 and C (20) in PTMD7.

The localization of the HOMO on the donor region across all the studied molecules confirms charge transfer from the donor to the acceptor region via hole transport. Similarly, the presence of the LUMO on the acceptor part of all the molecules provides strong evidence supporting this charge transfer mechanism. The reference compound PTMR1 and its derivatives PTMD2–PTMD7 are identified as electron-type materials due to the higher electron intensity in the electronic band compared to the hole intensity in the hole band. This study demonstrated that modifying the end-capped acceptor units is an effective strategy for developing high-performance chromophores with enhanced NLO properties.

3.7. Transition density matrix analysis

The TDM analysis is employed to investigate the phenomenon of electronic transitions in the studied chromophores. Moreover, this approach elucidates the interactions between donor and acceptor groups, as well as the characteristics of transitions from the ground state to the excited state (S₀–S₁) [53]. To perform this analysis, the compounds are categorized into the central core (π–bridge) and the terminal groups (donor and acceptor moieties). The influence of hydrogen atoms is excluded as they have a negligible contribution to electronic transitions. For TDM analysis, the investigated compounds are categorized into three segments: donor, acceptor, and π–spacer, which are denoted by green, red, and black lines, respectively. The TDM heat maps, shown in Figure 6, visually represent the pattern of electron flow as colored spots.

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Figure 6.

TDM heat maps of PTMR1, PTMD2-PTMD7 illustrated charge transferred between different parts of chromophores.

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The TDM heat map, charge density effectively flows diagonally from the donor portion to the acceptor region, thereby facilitating efficient charge transfer without restrictions in all the designed compounds. Overall, most of the charge density is concentrated in the π–bridge region and significantly transferred to the acceptor region.

3.8. Nonlinear optical analysis

The development of materials exhibiting remarkable NLO behavior is important for advanced photonic and optoelectronic technologies. These include high-order harmonic generation [54], high-speed electro-optic modulation [55], nonlinear photonic integrated circuits [56], nonlinear frequency transformation [57], and quantum optical computing [58]. The intensity of optical response by the material is intrinsically linked to its electronic characteristics, which are predominantly determined by parameters such as linear polarizability (<α>), first and second-order hyperpolarizabilities (β, γ), and the total dipole moment (μ_total). These electronic attributes are notably influenced by the molecular electronegativity. Table 5 presents the calculated μ_total, <α>, β_total, and γ_total values for the studied compounds PTMR1–PTMD7, whereas the details have been presented in Tables S20–S25.

Table S19 outlines the dipole polarizability parameters for the designed compounds (PTMR1-PTMD7), including the dipole moment components (μ_x, μ_y, μ_z) and the resultant total dipole moment (μ_tot) [59]. Among these, PTMR1 demonstrates the highest μ_total value (8.803 D), which is predominantly influenced by its significant μ_y component (6.558 D) and additional contributions from μ_x and μ_z (–5.868 D and 0.207 D, respectively), indicating a pronounced asymmetry in charge distribution. The compounds PTMD4 and PTMD6 follow closely while exhibiting total dipole moments of 8.390 D and 8.347 D, respectively, which reflect their strong polarization characteristics. A detailed examination of the dipole vector components reveals that the μ_x axis plays a major role in polarization of PTMD4 (μ_total = 8.390 D) and PTMD6 (μ_total = 8.347 D). This preferential charge delocalization along the x-axis suggests anisotropic electron density redistribution. In contrast, compounds PTMD3 and PTMD5 display comparatively lower total dipole moments, which reflect weaker polarization effects. The total dipole moments (μ_total) of the designed compounds follow a descending order as PTMR1 > PTMD4 > PTMD6 > PTMD7 > PTMD2 > PTMD5 > PTMD3.

Linear polarizability reflects the degree of electronic delocalization and molecular responsiveness to external electric fields [60]. Among the studied derivatives, PTMD7 shows the highest total polarizability (<α> = 1.57×10⁻²² esu), followed closely by PTMD4 (1.54×10⁻²² esu) and PTMD6 (1.53×10⁻²² esu), confirming their superior electronic delocalization. The αₓₓ component dominates the overall polarizability, highlighting the primary direction of charge displacement in these conjugated systems. Among the investigated derivatives, PTMD7 exhibits the highest total polarizability (<α> = 1.57×10⁻²² esu), followed by PTMD4 (1.54×10⁻²² esu) and PTMD6 (1.53×10⁻²² esu), indicating their superior electronic responsiveness. The αₓₓ component dominates the overall polarizability. The overall linear polarizability trend follows PTMD7 > PTMD4 > PTMD6 > PTMD5 > PTMD3 = PTMR1 > PTMD2, aligning with the electron-withdrawing strength of substituents (–COOCH₃ > –CN > –NO₂). These correlations confirm that strategic acceptor substitution effectively enhances molecular polarization and NLO response. The first hyperpolarizability (β_total) reflects the nonlinear distortion of a molecule’s electron cloud under an applied electric field, serving as a key indicator of second-order NLO efficiency [61]. Among the investigated derivatives, PTMD6 exhibits the highest β_total (5.36×10⁻²⁸ esu), confirming its superior second-order nonlinear response. This is followed by PTMD4 (4.90×10⁻²⁸ esu) and PTMD5 (4.31×10⁻²⁸ esu), highlighting the strong influence of the –CN and –CF₃ substituents, respectively. The β_xxx component dominates the tensor, indicating charge transfer predominantly along the x-axis, with PTMD6 showing the maximum contribution along β_xxx (5.42×10⁻²⁸ esu). The β_total trend follows PTMD6 > PTMD4 > PTMD5 > PTMD7 > PTMD2 > PTMD3 > PTMR1, underscoring the pivotal role of donor–acceptor substitution in tuning ICT and enhancing NLO performance. The second-order hyperpolarizability (γ_total) reflects the third-order NLO response, which is crucial for advanced photonic and optoelectronic applications [62]. Among the studied derivatives, PTMD6 exhibits the highest γ_total (3.91×10⁻³³ esu), attributed to the strong electron-withdrawing nature of the –NO₂ substituent. PTMD4 (3.60×10⁻³³ esu) and PTMD7 (3.10×10⁻³³ esu) also display significant third-order NLO activity. The γₓ component dominates the tensor, with PTMD6 showing the maximum γₓ (3.37×10⁻³³ esu), followed by PTMD4 (3.11×10⁻³³ esu) and PTMD7 (2.59×10⁻³³ esu), confirming preferential charge delocalization along the molecular x-axis. In contrast, γ_y and γ_z components are markedly weaker, indicating anisotropic polarization behavior. Notably, PTMD7 exhibits a relatively enhanced γ_z (1.02×10⁻³⁴ esu), suggesting minor out-of-plane polarization. The lowest γ_total values are observed for PTMD3 (2.57×10⁻³³ esu) and PTMR1 (2.59×10⁻³³ esu), consistent with their weaker nonlinear responses. The overall trend PTMD6 > PTMD4 > PTMD7 > PTMD5 > PTMD2 > PTMR1 > PTMD3.

To gain a deeper understanding of the NLO behavior of the designed compounds under realistic experimental conditions, frequency-dependent first (β) and second (γ) hyperpolarizability values were evaluated at two different incident wavelengths, 532 nm (0.042823 esu) and 1064 nm (0.085645 esu), as presented in Tables 6, 7, while details have been provided in Table S22 and S25. These frequencies correspond to commonly used laser sources in NLO characterization and allow insight into the dynamic response of the electron cloud under alternating electric fields [5].

The first (β) and second (γ) hyperpolarizabilites of the PTMR1–PTMD7 series were evaluated at two incident light frequencies, 532 nm and 1064 nm, to explore their NLO responses under different optical environments. The first hyperpolarizability governs the electro-optic Pockels effect β(–ω; ω, 0) and second-harmonic generation (SHG) [63] (Tables 6 and 7). At 532 nm, the β(–ω; ω, 0) values range from 4.38 × 10⁻²⁸ esu (PTMD3) to 8.45 × 10⁻²⁸ esu (PTMD6), indicating efficient second-order NLO activity at this visible wavelength. Among all compounds, PTMD6 exhibits the largest β(–ω; ω, 0) value, suggesting a stronger ICT interaction promoted by its donor–acceptor configuration. In contrast, PTMD3 and PTMR1 show relatively lower β values, implying weaker polarization under the same frequency. When the frequency is shifted to 1064 nm, a remarkable enhancement in β values is observed for most compounds, particularly PTMR1 (1.12 × 10⁻²⁴ esu), PTMD5 (7.71 × 10⁻²⁶ esu), and PTMD7 (7.99 × 10⁻²⁶ esu). This significant increase can be attributed to resonance effects, where the longer wavelength (lower photon energy) allows for greater molecular polarizability and electronic delocalization. The β(–2ω; ω, ω) components follow a similar trend, with PTMD6 and PTMD7 displaying enhanced values at 1064 nm compared to 532 nm, again emphasizing the role of dynamic frequency response in boosting second-order NLO efficiency. The second hyperpolarizability <γ> governs third-order effects like the dc-Kerr effect and electric field-induced second harmonic generation (ESHG) [64] At 532 nm, γ(–ω, ω, 0, 0) lies in the range of 10⁻³³ esu, indicating moderate third-order polarization. However, at 1064 nm, γ values increase significantly (up to the order of 10⁻²⁸ esu for PTMD4), suggesting stronger third-order nonlinearity at near-infrared excitation. Interestingly, some compounds (e.g., PTMR1, PTMD2, and PTMD5) show negative γ values at 1064 nm. Overall, the results highlight that frequency dispersion plays a crucial role in tuning the NLO response of the designed compounds. The enhanced β and γ values at 1064 nm indicate that these compounds possess promising dynamic NLO behavior.

To validate the NLO efficiency of the designed chromophores, their hyperpolarizabilities were compared with para-nitroaniline (p–NA), a standard NLO reference [65]. The β_total of p–NA (6.46 × 10⁻³⁰ esu) is greatly surpassed by all compounds, with PTMD6 showing the highest enhancement about 83 times greater than p–NA followed by PTMD4 (75 times), PTMD5 (66 times), PTMD7 (63 times), PTMD2 (54 times), PTMR1 (51 times), and PTMD3 (49 times). A similar pattern is seen for the γ_total values, where p–NA (7.29 × 10⁻³⁶ esu) is far exceeded by all chromophores. PTMD6 again leads with a 537-fold increase, followed by PTMD4, PTMD7, and PTMD5. These substantial enhancements confirm that strategic donor–acceptor substitution effectively strengthens charge transfer, making the designed compounds highly promising for advanced photonic and optoelectronic applications.