Unsymmetrical squaraine dyes with extended conjugation for second-order nonlinear optics and TiO₂ sensitization to far-red light: a computational quantum chemical study†

Eman Nabil* and Mohamed Zakaria
Department of Chemistry, Faculty of Science, Alexandria University, Alexandria, Egypt. E-mail: eman.nabil@alexu.edu.eg

First published on 7th August 2025

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Design, System, Application Squaraine dyes, with their intense far-red absorption and tunable electronic properties, are propitious candidates for dye-sensitized solar cells (DSCs) and nonlinear optical (NLO) applications. However, their limited conjugation often restricts their charge transfer efficiency and light-harvesting range. In this work, we designed π-extended unsymmetrical benzoindole-based squaraine dyes by incorporating four π-conjugated spacers: P-acetyl dithienophosphole oxide (SQ-P), ethyl-dithienopyrrole (SQ-N), dimethyl-silolo-dithiophene (SQ-Si) and thienothiophene (SQ-Th). As computationally assessed for the free dyes and those adsorbed on a TiO2 (38-atom) nanocluster model, this molecular engineering enhanced the intramolecular charge transfer (ICT) and redshifted absorption toward the far-red/NIR domain, improving TiO2 sensitization. Additionally, their strong ICT character, EW-driven electron displacement and tunable dipole moments make them ideal for NLO applications, with computed first hyperpolarizabilities surpassing conventional squaraine chromophores.

1. Introduction

The relentless pursuit of developing materials for efficient solar energy conversion and nonlinear optical (NLO) applications has led to a surge of interest in organic chromophores with tailored optoelectronic properties. Only a few classes of dyes, including cyanines, squaraines, phthalocyanines, porphyrins, and BODIPY analogues, are easily synthesized or readily available for near-infrared (NIR) absorption (700–1500 nm).¹ Among these, squaraine dyes have emerged as a prominent class owing to their intense absorption in the visible-to-NIR regions, high molar absorptivity, remarkable fluorescence quantum yield, substantial polarizability, photochemical stability, and structural versatility.^2–4 Unsymmetrical squaraine dyes specialize at facile spectral tunability, which can be functionalized to achieve some of the deepest NIR and shortwave infrared [SWIR] >1000 nm photon-to-current conversion.⁵ These attributes position squaraine dyes as compelling candidates for various applications including dye-sensitized solar cells (DSCs) and nonlinear optics. The fundamental photophysical properties of squaraines are mainly dictated by the electronic communication between the electron-donating terminal groups and the electron-accepting squaric acid core.⁶ The integration of π-conjugated spacers within the squaraine molecular backbone has been identified as a strategic avenue to enhance this communication and hence modulate their optoelectronic properties.⁷ These π-spacers, typically aromatic or heteroaromatic molecular units, act as conduits for π-electron delocalization, leading to an augmented intramolecular charge transfer (ICT), in addition to a reduction in the HOMO–LUMO energy gap and a corresponding red-shift in the absorption spectrum, which are crucial for both NLO and DSC performance.

Dye-sensitized solar cells (DSCs), a third-generation photovoltaic technology, are distinguished by their cost-effectiveness and ambient sunlight harvesting, besides aesthetic appeal, making them applicable to portable electronics and indoor building integration.⁸ The three main components of a DSC device are: a dye-sensitized mesoporous semiconducting photoanode, typically TiO₂; a Pt counter electrode; and an electrolyte packed in between the two electrodes. The photosensitizer anchored to the TiO₂ nanoparticles absorbs photons and injects the photoexcited electrons into the conduction band (CB) of TiO₂. A critical barrier for maximizing photon harvesting efficacy is the inherent limitation of single-dye-sensitized TiO₂ in attaining panchromatic solar absorption. This limitation can be strategically mitigated by extending squaraine conjugation with π-spacers, which synergistically broaden spectral response into NIR and enhance electron injection dynamics. For instance, π-extension with each –CC– unit has been reported to exhibit ∼100 nm of redshift in the charge transfer transition for a series of unsymmetrical squaraine dyes, while maintaining high molar extinction coefficients.⁹ Similarly, thiophene-based spacers have been shown to contribute to the panchromatic response of unsymmetrical squaraine dye, with an onset of 850 nm.¹⁰ The role of π-spacers extends beyond mere conjugation lengthening; boosted molecular conjugation improves the electronic coupling between the dye's LUMO and CB of TiO₂, thus accelerating electron injection kinetics.¹¹ The π-spacers, incorporated with alkyl chains, can also suppress the charge recombination rate, thereby enabling smoother electron flow through the semiconductor material.⁶ Moreover, a π-spacer can introduce steric effects that prevent random dye aggregation on TiO₂ surfaces—a common issue that reduces photocurrent generation due to exciton quenching, further enhancing the DSC device's long-term stability.¹²

In the context of NLO, extended molecular conjugation increases the π-electron density participating in ICT, thereby elevating the nonlinear susceptibility.^13–15 NLO phenomena encompass a range of physical processes, including harmonic generation, electro-optic modulation, parametric frequency conversion (sum/difference frequency generation), and optical rectification. One fundamental second-order NLO phenomenon is the second-harmonic generation (SHG), a frequency-doubling effect (and consequently a wavelength-halved), which is utilized in laser technology to tune the colour of their outcoming light beam. Molecular-scale control of SHG has been previously established by several favourable structural axioms, including intramolecular polarizability, planar molecular geometry, a ‘push–pull’ system, and an effective ICT facilitated by an extended π-conjugated bridge.¹⁶ The NLO activity of squaraine chromophores with push–pull architecture has garnered sustained scholarly interest. Their high polarizability, amplified under high-intensity laser irradiation, renders them promising candidates for numerous advanced photonic technologies. In 1994, Chen et al. reported that extended π-conjugation of an unsymmetrical squaraine molecule resulted in an absorption maximum (λ_max) at 732 nm, a red-shift of over 160 nm, and a five-fold increase in the static first hyperpolarizability (β₀), a critical metric for NLO performance.¹⁷ Beverina et al. have synthesized π-extended symmetric squaraine derivatives as two-photon absorption (2PA) active photosensitizers, exhibiting a strong 2PA cross-section at 806 nm, attributed to π-rich heterocycles.¹⁸ Bondar et al. have presented new symmetrical squaraine molecules showing 2PA, with a maximum 2PA cross-section (δ2PA max) of approximately 400 GM at around 800 nm, in addition to efficient NIR-superluminescence and high photostability.¹⁹ Such multifunctional NLO performance is important for emerging technologies in telecommunications and biomedical imaging, where NIR compatibility and high nonlinearity are paramount.

This study presents a comprehensive quantum chemical investigation to elucidate the impact of π-extended conjugation on the performance of four modelled squaraine dyes as NLO-active chromophores and photosensitizers for TiO₂-based DSCs. A parent squaraine dye, coded as SQ-140,²⁰ featuring a benzoindole-squaraine-indole scaffold with a cyanoacrylic acid acceptor/anchoring group, and devoid of π-spacers, served as the reference system. The four modelled unsymmetrical squaraine derivatives (SQ-P, SQ-N, SQ-Si and SQ-Th) were computationally designed through structural tailoring of four π-conjugated spacers. The optoelectronic and NLO properties of the modelled dyes were rigorously evaluated at the molecular level via (time-dependent) density functional theory computations. Our dyes exhibited synergistic dipole alignment and 3D charge delocalization, i.e., a mixed dipolar–octupolar symmetry. Accordingly, this work establishes structure–property correlations demonstrating that squaraine dyes with mixed dipolar–octupolar symmetry enable simultaneous enhancement of NLO efficiency and TiO₂ charge injection. The study recommends an ethyl-dithienopyrrole π-linker as optimal for TiO₂-photosensitization and a P-acetyl dithienophosphole oxide π-linker for squaraine-based NLO chromophores, providing a rational molecular modeling blueprint for dual-functional organic materials in energy conversion and photonic technologies.

2. Computational methodology

Computational details for simulating ground state and linear optical properties

The geometrical/electronic structures and linear excitation characteristics of the four modelled squaraine (SQ) dyes were respectively obtained by density functional theory (DFT) and time-dependent DFT (TD-DFT) calculations. All quantum chemistry computations were performed using the Gaussian 09 (G09) software package.²¹ Full optimization of ground state geometries for isolated modelled SQ dyes along with the reference SQ-140 dye was carried out using the B3LYP hybrid density functional and the 6-311g(d,p) basis set. The solvent effects were taken into account by the conductor-like polarizable continuum model (C-PCM) with a dielectric constant of 24.85 to simulate the ethanol (EtOH) solvent, the solvent that was reported for the measured UV-vis spectrum. The nature of located energy minima on the potential energy surface was subsequently confirmed by calculating harmonic vibrational frequencies at the same level of theory. Ground-state optimizations employing B3LYP functional followed an established methodology for sulphur-rich squaraine dyes.^10,12,22 It is important to note, however, that omission of explicit dispersion corrections may influence planarity in sulphur-rich π-systems, particularly for flexible thioalkyl chains. The UV-vis electronic absorption spectra for all free molecules were simulated at the TD-B3LYP/6-311+G(d,p)/C-PCM(EtOH) computational level. The choice of B3LYP functional was justified by benchmarking 12 different exchange–correlation functionals (XCFs) against the experimental UV-vis spectrum of the SQ-140 reference dye, which was reported by Roy et al.²⁰ Using the G09 output and formatted check point files, Multiwfn 3.8 code^23,24 was subsequently utilized for assigning the excitation and intramolecular charge transfer characteristics, including natural transition orbitals (NTOs)²⁵ and charge-transfer spectra (CTS),²⁶ besides interfragment charge transfer (IFCT) and hole–electron analyses.²⁷

Computational details for simulating TiO₂ sensitization

The electronic properties and charge transfer features at the dye–TiO₂ interface were simulated for modeled SQ sensitizers in an adsorbed configuration onto the surface of the (TiO₂)₃₈ nanocluster model. While periodic models offer an extended surface representation, the cluster approach is well-suited for probing local adsorption, orbital interactions, and interfacial charge transfer at a manageable computational expense. Hence, a two-layer (TiO₂)₃₈ nanocluster model, representing the (101)-oriented anatase surface, was used, with each layer containing 19 Ti atoms. The model's geometry was constructed following the cleaving procedure outlined by Persson et al.²⁸ via systematic stripping off stoichiometric units (TiO₂)_n from an anatase supercell. The constructed (TiO₂)₃₈ cluster is stoichiometric, lacking permanent dipole moment, and with sufficiently high coordination to support the formal oxidation state of the nanocrystal. The studied SQ sensitizers have been anchored on the (TiO₂)₃₈ cluster surface thru the carboxylic group in a dissociative – homogeneous – bridged bidentate (BB) adsorption mode. In the considered BB mode, both oxygens of the carboxylate anchor are bonded to penta-coordinated (5c) Ti atoms on two successive rows, while the proton is transferred to the nearest 2c-oxygen atom.²⁹ The ground state geometries of the bare (TiO₂)₃₈ cluster and the adsorbed SQ@(TiO₂)₃₈ complexes were fully optimized via the Amsterdam density functional program (ADF) program,³⁰ as integrated into the AMS2020.103 package.³¹ The generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE)³² was employed along with the Slater-type orbital basis set TZP/DZP for Ti/H, C, N, O, S, P and Si atoms. The scalar relativistic approach of zero-order regular approximation (ZORA)³³ was included in optimization. Based on the optimized SQ@(TiO₂)₃₈ cluster complexes, a systematic investigation was conducted to analyze adsorption stability, orbital analysis, bonding energy decomposition, projected density-of-states (PDOS) and interfacial charge transfer (CT) characteristics.

Computational details for simulating the nonlinear optical properties

The second-order nonlinear optical (NLO) coefficient, termed first-order hyperpolarizability (β), quantifies the second-order NLO response of a molecular dipole moment to an external electric field. Therefore, the molecular (hyper)polarizabilities were calculated by the analytic derivatives of the system energy, specifically the coupled perturbed Kohn–Sham (CPKS) method, using the G09 code at the B3LYP/6-311+G(d,p) level of theory. The (hyper)polarizabilities were calculated in the gas phase to exclude solvent-induced amplification, allowing direct comparison of intrinsic molecular design effects. The static first-order hyperpolarizability β₀(∞) in the zero-frequency limit (λ = ∞ nm) was calculated, in addition to the dynamic first-order hyperpolarizability β(−2ω; ω, ω) in the frequency-dependent field, which is relevant for the second harmonic generation (SHG) effect. The dynamic DC-Pockels β(−ω; ω, 0) was calculated as well, which is significant in designing materials for electro-optic modulation. The simulated external electric field was assigned with frequencies ℏω = 1.18 eV (λ = 1064 nm) and ℏω = 0.84 eV (λ = 1460 nm), as the frequently employed incident light frequencies in experiments. Static and dynamic polarizabilities portraying the response of the molecular electron cloud and dipole moment to a static and oscillating external electric field were also calculated. To elucidate the molecular origin of NLO response, the depolarization ratio (DR) associated with hyper-Rayleigh scattering (HRS) β_HRS was analyzed.³⁴ The harmonic light intensity as a function of the polarization angle (Ψ) of incident light was investigated using polar plots. A 3D surface representation scanning the calculated beta hyperpolarizability, β(ω₁, ω₂), as a function of the input frequencies ω₁ and ω₂, was also plotted. Lastly, the unit sphere representations of molecular (hyper)polarizabilities were rendered by the visual molecular dynamics (VMD)³⁵ code based on the related files exported by G09 and Multiwfn codes. In which, NLO response is readily 3D visualized by calculating effective (hyper)polarizability at every vertex of a sphere surface enclosing a molecule.

XC-functional benchmark

The applicability of the TD-DFT domain depends on the approximate recovery of the exact −1/r asymptote of exchange–correlation (XC) potential V_XC, rather than a strongly decaying exponential function. The V_XC is the functional derivative of the XC-energy functional E_XC[ρ] with respect to the density [ρ(r)]. Accordingly, twelve XC-functionals, belonging to five categories, were benchmarked. The considered categories were: (i) four functionals based on hybrid generalized-gradient approximation (H-GGA): B3LYP^36,37 functional with 20% HFX (Hartree–Fock exact exchange), PBE0 (ref. 38) with 25% HFX, APF-D³⁹ functional with 23% HFX along with a dispersion correction term, and MPW1K⁴⁰ functional with 25% HFX and 1-parameter model for kinetics; (ii) two hybrid meta-GGA functionals: M06-2X⁴¹ (54% HFX) and TPSSh⁴² (10% HFX); (iii) four range-separated hybrid (RSH) long-range corrected GGA functionals, CAM-B3LYP⁴³ (short range 19% HFX and long range 65% HFX), LC-ωPBE⁴⁴ (short range 0% HFX and long range 100% HFX), the meta-GGA functional M11 (ref. 45) (short range 42.8% HFX and long range 100% HFX), and ωB97X-D⁴⁶ with dispersion effects (short range 22% HFX and long range 100% HFX); (iv) one among the RSH middle-range corrected functionals, HISSb^47,48 (HFX% = 0–60–0 at short-middle-long interelectronic separations, in that order); and (v) one among the RSH screened-exchange functionals, MN12-SX⁴⁹ (meta-NGA with HFX% = 25–0 at short-long range). For each XC functional considered, 20 excited states were computed using the 6-311+G(d,p) basis set. The accuracy of calculated wavelength of maximum absorption (λ_max) was assessed by computing the mean signed deviation, through , with respect to the reference experimental λ_max value of 668 nm. As depicted in Fig. 1, the results vividly show that the B3LYP conventional hybrid functional provided the most accurate representation of the absorption wavelength, with an MSD of ca. −12.15 nm (0.03 eV). The screened-exchange functional MN12-SX was the second-best performing functional with an MSD of ca. −23 nm (0.07 eV), followed by the hybrid functionals APF-D (−26 nm, 0.08 eV) and PBE0 (−34 nm, 0.10 eV). Alternatively, all four long-range corrected RSH functionals delivered the least accurate transition energies. Their corresponding MSD values were found within a range of ca. 0.24 to 0.27 eV (−78 to −84 nm) for CAM-B3LYP and LC-ωPBE functionals, respectively.

B3LYP is a hybrid functional that combines a limited amount of exact exchange (20% HFX) with a gradient-corrected correlation functional. The functional is known for its problematic over-polarization when a significant charge separation appears, which prompts errors in CT transition energies.⁵⁰ Herein, the superior performance showed by B3LYP could be attributed to the nature of transition in SQ-140 reference dye, being reminiscent of typical cyanine topology with a partial CT character. Indeed, the simulated charge-transfer spectrum of SQ-140 reference dye, as obtained by the Multiwfn code, revealed that the electronic excitation at 668 nm is a hybridized transition with π → π* local excitation (64%) and charge transfer (36%) characters. Accordingly, the balanced treatment of B3LYP between exact and dynamic exchange or local and non-local interactions could deliver a reasonable accuracy. In the same way, the RSH functionals are designed to partition the exchange–correlation energy into short-range and long-range components, with increased exact HFX% in long ranges to correct the dynamic behaviour of the exchange potential. This is crucial for pure long-range CT transition with significant extent of charge separation. However, for a hybrid excitation, as in the current system, the range-separation amplifies the CT component disproportionately, and neglects the significant local excitation contribution, leading to the observed overestimated excitation energies and large blue shifts in absorption wavelengths.

Considering that the four investigated dyes in the present study are the π-spacer modified analogues to SQ-140 reference dye, particular attention was given to the performance of B3LYP and CAM-B3LYP in predicting the resulting changes in excitation energies. Since the CT character is anticipated to become more prominent upon π-spacer insertion, CAM-B3LYP was selected for comparison due to its relative efficacy among the examined RSH functionals (Fig. 1), given that RSHs are generally considered suitable for describing CT transitions. A bathochromic shift in absorption spectra of modelled dyes is also anticipated owing to introducing a π-spacer reduces the energy gap by boosting π-conjugation. However, CAM-B3LYP predicted a counterintuitive blueshift, for all four modelled dyes (SQ-P, SQ-N, SQ-Si and SQ-Th) with calculated λ_max values of 595, 592, 594 and 589 nm, respectively. These values are physically improbable considering the extended π-systems compared to the parent SQ-140 dye (668 nm). Conversely, B3LYP maintained reasonable accuracy, with the corresponding wavelengths of 850, 695, 746 and 728 nm. This suggests that B3LYP benefits from its lower HF exchange and empirical parameterization in capturing the electronic interactions in the examined squaraine systems, thus effectively describing their red-shifted hybridized transitions, even when the CT character dominates. Indeed, a good agreement between the theoretical and experimental UV-vis spectra was reported for TD-B3LYP simulated structurally congeneric squaraine dyes.²² Lastly, the impact of augmenting the 6-311G(d,p) basis set with diffuse functions was evaluated for the best-performing B3LYP functional. As revealed in Fig. 1, upon utilizing the TD-B3LYP/6-311+G(d,p) level of theory, the absolute deviation attained is limited to ca. 0.016 eV (5.6 nm). It is worth mentioning that this result (0.016 eV) has attained the chemical accuracy, defined as an absolute error of less than 0.043 eV.⁵¹ Accordingly, this computational level can be deemed reliable for simulating both linear and nonlinear optical phenomena of the modelled squaraine dyes.

Photovoltaic performance parameters

The first step in operating DSC is the absorption of a photon of energy hν, which elevates the dye molecule to an excited state (D*). For efficient solar-to-electric energy conversion, the required fate of the photoexcited electron is to be injected into the conduction band (CB) of the semiconductor, leaving the dye in its oxidized state, D⁺. The overall power conversion efficiency (PCE) of the DSC-device therefore depends on photocurrent density, measured by the short circuit (J_SC). This J_SC is dependent on the absorption coefficient of the dye and electronic coupling of the dye–semiconductor surface.⁵² It is defined by the following expression:

JSC = ∫LHE(λ)Φinj(λ)ηcollectdλ	(1)

LHE(λ) is the light-harvesting efficiency at a wavelength λ, and Φ_inj(λ) is the quantum yield for electron injection. The η_collect is the efficiency of charge collection reflecting electron losses due to back-electron transfer and related processes occurring after electron injection in the semiconductor, which only depends on the architecture of the DSC-device. In contrast, LHE(λ) and Φ_inj(λ) are, in principle, the factors that depend on the excitation wavelength thus, they are strongly influenced by the structural and quantum phenomenon of the dye. LHE can be related to the oscillator strength f of the dye at the absorption wavelength according to the following equation:⁵³

LHE(λ) = 1 – 10–f	(2)

The electron injection quantum yield Φ_inj(λ) can be estimated as a function of the free energy change for the photogenerated electron injection process; Φ_inject ∝ f(−ΔG_inj). This electron injection driving force (ΔG_inj) is calculated by the following equation:^52,54

	(3)

where is the reduction potential of the CB of TiO₂, while and E_ox are the oxidational potential of the dye in the excited state and ground state, respectively. E_0–0 can be taken as the electronic vertical transition energy at the maximum absorption wavelength λ_max. Once injected into the CB of the semiconductor surface, the electron is transported through the semiconductor film by diffusion. A competing unwanted step can happen in parallel along with this diffusion step, namely recombination of the injected electron with the oxidized dye or electrolyte species. This loss pathway decreases the device efficiency by reducing the number of charge carriers transported in the electrical circuit. The free energy (ΔG_rec) driving force of such a recombination reaction can be theoretically estimated by the following equation:^55,56

	(4)

After completion of the injection process, the dye regeneration process takes place. In which, the electrolyte serves to close the electrical chain by capturing the photogenerated hole in the oxidized dye (D⁺), meanwhile, electrons are transferred from the counter electrode to the oxidized electrolyte species. Hence, the oxidized dye (D⁺) is restored to its neutral charge, allowing for a new photoexcitation cycle. The free energy (ΔG_reg) driving force for this dye regeneration step can be also theoretically evaluated by comparing the oxidation potential of dye in its ground state to the oxidation potential of the electrolyte using the following expression:^55,57

ΔGreg = E(I−/I3−) − Eox	(5)

Altogether, eqn (2)–(5) are to be employed in the present study for estimating the DSC-device photovoltaic (PV) performance based on TiO₂-sensitization by modelled squaraines.

3. Results and discussion

Molecular structures of modelled squaraine dyes

In the present study, four unsymmetrical dyes (SQ-P, SQ-N, SQ-Th and SQ-Si) featuring the D1-A1-D2-π-A2 architecture were modelled with four different rigidified π-conjugated spacer moieties. All the molecules were found asymmetric, belonging to the C1 point group symmetry i.e., lacking any symmetry operations other than the trivial identity (E). Thiophene and its derivatives are known to contribute to enhanced light-harvesting efficiency and electronic communication when utilized as π-spacers.^58,59 Accordingly, our incorporated π-conjugated bridges are 4-ethynylphospholo[3,2-b:4,5-b′]dithiophene 4-oxide or P-acetyl dithienophosphole oxide (DTPO), 4-ethyl-4H-dithieno[3,2-b:2′,3′-d]pyrrole (DTP), thieno[3,2-b]thiophene (TT), 4,4-dimethyl-4H-silolo[3,2-b:4,5-b′]dithiophene or dithienosilole (DTSi), respectively. The 2D chemical structures along with the significant structural attributes are portrayed in Fig. 2. The four-member sp²-carbon π-conjugated cycle (A1) that constitutes the core of squaraines is substituted by two oxygen atoms and two π-conjugated donors, benzoindole (D1) and indole (D2) rings. In the displayed mesomeric form, the SQ core is positively charged with two oxygen atoms bearing negative charge, while the positive charge is delocalized over the odd number of carbon atoms creating the π-conjugated pathway. Calculated data characterizing the conformation changes of ground-state/photo-excited geometries are listed in Table 1. The optimized 3D molecular geometries are displayed in Fig. S1.†

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DFT calculations showed that conjugation is exhibited via four of the π-conjugated spacers under study with vanishing bond length alteration (BLA) and quasi-planar conjugated pathway. This should allow electrons to move freely, creating a continuous conjugation pathway for π-electron delocalization. In particular, the calculated BLA for successive C–C bonds of the entire π-bridge and the vicinal C–C bond have shown an increasing trend: SQ-N (0.017 Å) < SQ-Si (0.030 Å) < SQ-Th (0.033 Å) < SQ-P (0.034 Å), which implies an improved degree of π-electron delocalization (smallest BLA) for the dithienopyrrole π-linker. A closely related pattern was found for the BLA of successive C–C bonds calculated over the entire length of the conjugated alkyl chain of molecules, namely SQ-N (0.009 Å) < SQ-Th (0.012 Å) < SQ-Si (0.013 Å) < SQ-P (0.014 Å). Further analysis of cyclic delocalization of mobile electrons over fused rings of the π-bridges was conducted by means of: ring-size normalized multi-centre index (MCI),⁶⁰ harmonic oscillator measure of aromaticity (HOMA)⁶¹ and Shannon aromaticity (SA)⁶² index at the position of bond critical points (3, −1). All indices were computed using the Multiwfn code based on G09-optimized geometries. Unlike MCI and HOMA indices, the smaller the SA index, the more aromatic the ring is. According to the geometry-based HOMA indicator, electron delocalization decreases in the following sequence: SQ-N (0.69) > SQ-Th (0.65) > SQ-P (0.42) > SQ-Si (0.38). Similarly, SQ-N (0.017) > SQ-Th (0.021) > SQ-P (0.027) > SQ-Si (0.051) was the order that the SA index indicated. Meanwhile, the MCI indicator has predicted the trend: SQ-Th (0.48) > SQ-N (0.46) > SQ-Si (0.41) > SQ-P (0.31). It is credible to draw the conclusion that a comparable electron delocalization potential is exhibited by dithienopyrrole and thienothiophene π-bridges, which is superior to those of dithienophosphole and dithienosilole. A promoted π-conjugation is anticipated for SQ-N and SQ-Th, accordingly. Regarding the conjugation pathway and molecular planarity, the D1-A1-D2 segment of all the molecules has attained planarity in both ground-state and excited-state geometries. A maximum deviation of ca. 1.7° was noted in SQ-Si ground-state for both β₁ and β₂ dihedral twist angles. These features indicate that donor moieties' orientation and thus, π-conjugation with the squaric ring is not affected by the insertion of π-bridges at the terminals. Planarity along π-linker rings was preserved with a maximum deviation of ca. 0.82°, 0.79° and 0.56° and 0.62° for the inner thiophene ring in SQ-P, SQ-N, SQ-Si and SQ-Th, respectively. Upon photoexcitation, this planarity was boosted with the corresponding values being 0.56°, 0.001°, 0.02° and 0.43°, implying enhanced conjugation for SQ-N due to the presence of the dithienopyrrole π-linker. It also can be detected that planarity and thus conjugation was conserved between the introduced π-spacer and terminal cyanoacrylic acid acceptor CAA (A2) in all the molecules, with maximum deviation recorded (φ₁ ≃ 3°) for excited geometries. On the other hand, the dihedral twist angle (φ₂) in ground-state geometries recorded the largest values (φ₂ = 23–25°). For excited-state geometries, results have shown a geometrical relaxation (φ₂ = 0.3–12°) concerning the photo-excited rotation about the C–C single bond, connecting the indole ring and π-spacer, with a tendency to align with the molecular plane. One exception is SQ-P dye (φ₂ = 24°), attributable to the relative bulkiness of the P-acetyl dithienophosphole oxide linker, and hence the space around the indole ring could not accommodate a planar alignment. This twist angle (φ₂) can be considered the most sensitive (φ₂ = 0.3–24°) to the nature of π-linkers during such molecular structure perturbations of relaxed optically excited geometries. For SQ-N dye, the greatest rotation recorded (25° → 0.3°) for the dithienopyrrole linker implies that the resonance effect dominates over the steric hindrance competing effect upon optical excitation. This is consistent with the dye displaying the lowest BLA (0.005 Å), corresponding to a higher degree of delocalization. Since enlarging the photo-induced geometrical relaxation ensures a large amount of torsional work and boosts the Stokes shift,⁶³ it is therefore reasonable to predict that SQ-N dye would display the largest Stokes shift and, hence, minimum self-quenching. Considering the negligible twisting about the spacer–acceptor dihedral angle (φ₁ = 0.3–3°), which reflects their maintained planarity and effective p-orbital overlap, one could argue that ease of charge flow and electron injection into the CB of TiO₂ are estimated for all modelled dyes.

Electronic structures of modelled squaraine dyes

The extrinsic properties of the photosensitizers are dependent on their inherent electronic structures. In particular, the energetics and spatial distribution of frontier molecular orbitals (FMOs) have underlying influences on the electron transport and quantum yield of electron injection, as well as the dye regeneration process. For a DSC's efficient functioning, two fundamental prerequisite energetics of photosensitizing dyes are: the lowest unoccupied molecular orbital (LUMO) of the dye being at least 0.2 eV (ref. 64) higher than the CB edge of the semiconductor for facile electron injection; and the highest occupied molecular orbital (HOMO) of the dye being beneath the oxidation potential of the redox shuttle (usually I/I₃⁻) for efficient dye regeneration. Fig. 3 depicts the energetics and spatial distributions of FMOs of the designed dyes, calculated at the B3LYP/6-311+G(d,p)/CPCM(EtOH) computational level. Also, the absolute energy alignment of the HOMO–LUMO energy levels is depicted with respect to the energy of the CB edge of TiO₂ at −4.00 eV (ref. 65) and the oxidation potential of the I/I₃⁻ redox mediator (−4.7 eV),⁶⁴ all referenced to the vacuum level. Obviously, all the modelled chromophores fulfil the basic prerequisite energetics of an operating DSC. The calculated energy of LUMO levels ranges from −3.40 to −3.04 eV, which are about 0.60, 0.96, 0.85 and 0.85 higher than the CB minima of TiO₂ respectively for SQ-P, SQ-N, SQ-Th and SQ-Si dyes. Accordingly, the thermodynamic feasibility of the electron injection process is expected upon photoexcitation. Also, the calculated HOMO levels, varying from −5.09 to −5.04 eV, are sufficiently lower than the redox potential of the electrolyte denoting the potential of oxidized dyes to gain electrons from the electrolyte. Thus, regeneration of the ground state of designed sensitizers is also predictable. Since stronger electron-accepting and electron-donating groups result in lower E_LUMO and higher E_HOMO levels, correspondingly, the comparable values of computed HOMO energy levels imply the little impact of the introduced π-bridges exerted on the donating ability of indole and benzoindole rings. Meanwhile the calculated energies of LUMO levels were slightly different, with a significant downward shift (E_LUMO = −3.4 eV) for SQ-P dye. This can be attributed to the introduction of the P-acetyl dithienophosphole oxide (DTPO) π-linker with relatively stronger electron-accepting strength compared with other π-linkers. DTPO has two electron-withdrawing groups: the acetyl group and the phosphine oxide group thus, attracting electrons away from the dithienophosphole ring, making DTPO a stronger auxiliary acceptor that stabilizes LUMO. On the contrary, the destabilized LUMO of SQ-N, with a relatively upward shift (E_LUMO = −3.04 eV), compared to SQ-140 reference dye, implies a relatively potent donating ability of the DTP π-bridge. This could be attributed to the improved electron-donating effect of the pyrrole ring due to the positive alkyl inductive effect (+I) of the ethyl group attached to the N atom. Regarding the HOMO–LUMO energy gap (E_g), it is obvious (Fig. 3) that insertion of π-spacers into the parent squaraine dye (SQ-Ref) consistently reduced the energy gap in all four modelled derivatives, attributable to stabilized LUMOs and improved π-delocalization. The E_g values decrease in the sequence of SQ-N > SQ-Th > SQ-Si> SQ-P, relating the capability of SQ-P dye to harvest light with longer wavelengths. In principle, E_g in conjugated systems is governed predominantly by four structural factors.⁶⁶ Namely, E_BLA due to BLA contribution; E_res concerning the resonance energy in ring-type conjugation; E_sub including the electron-releasing/-withdrawing power of the substituent; and E_θ for rotational disorder around inter annular single bonds. The SQ-N and SQ-Th molecules had lower values of BLA and delocalization indices (vide supra), suggesting better π-conjugation which should yield lower E_g. Nevertheless, the calculated lower E_g of SQ-P and SQ-Si could be ascribed to the strengthening of push–pull interactions in dyes, SQ-P and SQ-Si, thereby decreasing E_sub and concurrently localizing FMOs at distinct molecular parts. This resulted in increased intramolecular charge transfer (ICT) character and afforded lower E_res as well. Considering that molecular hardness is related to high energy gaps (E_g), all the molecules in the present study, with low E_g = 1.69 to 2.01 eV, can be characterized as soft molecules, thus, permitting perturbations in their electronic configurations, resulting in high chemical reactivity. The distribution pattern of FMOs for all the designed sensitizers satisfies the basic requirements of the molecular orbitals for photoinduced interfacial charge separation. As shown in Fig. 3, all the dyes have delocalized HOMOs across the benzoindole-squaraine core-indole molecular framework (D1-A1-D2). Alternatively, the electronic density distribution of LUMOs is mainly localized over the π-spacer and terminal cyanoacrylic acid acceptor (π-A2) groups of all the dyes, with a slightly large contribution on the squaraine core of SQ-N dye. Despite through-conjugation across the systems, this delocalization pattern of FMO topologies, namely HOMO (LUMO) being more localized on the donor (acceptor) unit, corroborates the existence of effective conjugation that is distinct from formal conjugation, measured via the BLA.⁶⁷

Photoabsorption spectra of modelled squaraine dyes

The incorporation of π-spacers into squaraine molecular architectures represents a promising approach for achieving panchromatic light absorption and extending the sensitization of TiO₂ towards red-NIR radiation. Herein, the UV-vis absorption spectra of the modelled SQ sensitizers are thoroughly investigated. The lowest 20 excited states were simulated at the TD-B3LYP/6-311G+(d,p)/CPCM(EtOH) computational level, as depicted in Fig. 4a. A full width at half maximum (FWHM) of 3000 cm⁻¹ was assigned for the simulated spectra. The Multiwfn code was utilized for estimating the coloration of the modelled dyes, based on the simulated UV-vis spectra. Fig. 4b displays the predicted absorbed and perceived colours for the dyes if dissolved in ethanol, with the transmitted colours representing the complementary hue expected. The excitation characteristics of the lowest two singlet excited states (S1 and S2) are tabulated in Table 2. Apparently, a strong broad absorption band in the visible-NIR spectrum domain at 500–800 nm can be observed for all the modelled SQ dyes, wherein SQ-N exhibits a strong hyperchromic shift of its low-energy transition (S1) peaking at 697 nm with ε⁶⁹⁷ ≥ 1.6 × 10⁵ L M⁻¹ cm⁻¹. A second most-intense absorption band (S2) can be observed at 586 nm with ε⁵⁸⁶ ≥ 9.4 × 10⁴ L M⁻¹ cm⁻¹. For other dyes, the molar extinction coefficients (L M⁻¹ cm⁻¹) for their maximum absorption S2 followed a decreasing order of SQ-P (ε⁵⁹⁹ ≥ 1.4 × 10⁵) > SQ-Si (ε⁵⁹⁶ ≥ 1.1 × 10⁵) > SQ-Th (ε⁵⁸¹ ≥ 1.0 × 10⁵). Despite having their maxima (S2) centered around 599, 596 and 581 nm, the spectral range of SQ-P, SQ-Si and SQ-Th extended even to the NIR region of 800 nm, with bathochromic shifted S1 of ε⁸⁴⁵ ≥ 6.1 × 10⁴, ε⁷⁴⁵ ≥ 9.1 × 10⁴ and ε⁷²⁸ ≥ 9.3 × 10⁴ L M⁻¹ cm⁻¹, respectively, in accordance with their narrower HOMO–LUMO gaps (1.69, 189 and 1.94 eV, Fig. 3). Moreover, less intense absorption peaks were detected for all the modelled SQ dyes in the 360–460 nm spectral range. Specifically, SQ-N, SQ-P, SQ-Si and SQ-Th dyes featured transitions with ε⁴⁰⁴ ≥ 1.0 × 10⁴, ε³⁷³ ≥ 1.9 × 10⁴, ε⁴²⁰ ≥ 1.0 × 10⁴ and ε³⁸⁴ ≥ 1.4 × 10⁴ L M⁻¹ cm⁻¹. These absorptions could potentially contribute to compensating for the competitive light absorption of I/I₃⁻ electrolyte in the 350–400 nm domain. Overall, the modeling results predict that incorporating dithienopyrrole π-spacers will induce both bathochromic and hyperchromic shifts, as evidenced by the comparison with the parent dye with λ_max = 662 nm and ε⁶⁶² = 1.3 × 10⁴ L M⁻¹ cm⁻¹. It should be noted that these spectral data of SQ-140 were calculated at the same computational level as used for the modelled dyes, ensuring a consistent methodological framework for assessing the impact of structural modifications. SQ-140 was reported to exhibit an intense absorption with λ of 668 nm and ε of 2.73 × 10⁵ dm³ mol⁻¹ cm⁻¹.²⁰ Still, extending the π-conjugation length of squaraines incorporating thiophene rings has been reported to cause a significant bathochromic shift of ca. 187 nm for squaraine dye maxima, namely 608 and 795 nm for correspondingly S1PEG (one thiophene ring) and S3PEG (two thienothiophene rings).⁶⁸

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The dominant character of main electronic transitions (S1 and S2) was assigned according to relocations of natural transition orbitals (Fig. 4c) and their calculated charge-transfer spectra (Fig. S2†). Natural transition orbitals (NTO) were simulated as a compact orbital representation of the electronic transition density matrix, since several MO pairs contribute non-negligibly to each electronic transition at the same time (Fig. S3†). Also, the charge-transfer spectrum (CTS) can intuitively reveal characters of various peaks of the UV-vis spectrum. The three types of excited states (ES) were estimated as percentages according to the quantitative inter fragment charge transfer (IFCT) analysis of CTS (Table 2), namely, intramolecular charge transfer (ICT), π → π* local excitation (LE) and hybridized local excitation-charge transfer (HLCT) states, wherein the percentage of π → π* LE accounts for the magnitude of intrafragment electron redistribution within the dye fragments. Meanwhile, the interfragment CT from D1-A1-D2 to the π-A2 fragment is denoted by the percentage of ICT (Table 2). It can be observed that all four of the modelled SQ dyes exhibit analogous patterns during the electronic transitions to the first and the second excited states. The first electronic transition S1 ← S0 is primarily (99.3–99.6%) contributed by HOMO → LUMO photoexcitation. In the same manner, two MO pairs were identified to be accountable for S2 ← S0 transition in all the SQ dyes, specifically HOMO → LUMO+1 (89.3–95.9%) and HOMO−1 → LUMO (3.3–9.1%). The molecular moieties D1-A1-D2 and π-A2 are correspondingly where HOMO and LUMO orbitals primarily spatially located in all the SQ dyes (vide supra). For the HOMO−1 orbital (Fig. S3†), the electron density in SQ-P and SQ-Th is distributed throughout the entire molecules, but in SQ-N and SQ-Si, it is mostly located over indole and π-A2 moieties. In turn, the LUMO+1 orbital is majorly confined to D1-A1-D2 in SQ-P and SQ-Si molecules, while being dispersed on entire SQ-N and SQ-Th molecular systems. Thus, all the examined S1 and S2 states can be characterized as hybridized states (HLCT), with contrary hybridization statuses. In other words, the dominance of ICT nature over π → π* LE for S1 and S2 states is in opposition to one another. Specifically, the S1 electronic state of SQ-P, SQ-Si, SQ-Th and SQ-N dyes is a CT-dominated hybridized state exhibiting 86, 77, 73 and 64% ICT character, alongside minor π → π* LE fractions of 10, 16, 23 and 28%, respectively. In contrast, their corresponding S2 states are LE-dominated hybridized, featuring predominantly (78, 71, 76 and 64%) π → π* LE, alongside minor ICT percentages of 14, 17, 19 and 26%. The approximate quantified amount of charge transferred to the π-A2 fragment upon S1 ← S0 CT-dominated photoexcitation is 0.62, 0.85, 0.76 and 0.71 |e⁻| for the sensitizers SQ-N, SQ-P, SQ-Si and SQ-Th, respectively. Alternatively, the corresponding quantity of transported electrons has diminished to 0.17, 0.06, 0.11 and 0.14 |e⁻| during S2 ← S0 LE-dominated transition. Remarkably, back charge transfer from the terminal cyanoacrylic acid acceptor (A2) to the π-spacer was found to be negligible in both transitions, with a maximum value of ca. 0.01 |e⁻| recorded for SQ-N dye's S1 state. These HLCT states have been regarded as unique excited states since they could harvest both high photoluminescence quantum yield sourced from the LE constituent, in addition to exciton utilization contributed by the ICT component.^69,70 The comprehensive CT metrics, obtained from both intramolecular (isolated dyes) and interfacial (dye–semiconductor) calculations, are compiled in Table 5, to enable a multiscale investigation of photoinduced electron transport.

Visualizing both hybridization statuses through depicted NTOs (Fig. 4c) validates the previous quantitative IFCT analysis of charge-transfer spectra. Upon red-NIR light photoexcitation to S1 of all the SQ dyes, the occupied NTOs are delocalized throughout the D1-A1-D2 horizontal backbone, exhibiting a slight shift towards thiophene rings of the π-spacers adjacent to the indole donor. Alternatively, the virtual NTOs are mostly localized over the π-spacer and CAA terminal acceptor (A2), with minor contributions originating from the benzoindole-squaraine core-indole (D1-A1-D2) unit. Furthermore, the shown minor contributions of the D1-A1-D2 unit to the virtual NTOs are much pronounced in SQ-N compared to SQ-P, in agreement with their corresponding calculated largest (28%) and smallest (10%) components of π → π* LE. It is therefore highly anticipated that the lowest energy HOMO → LUMO vertical excitation is accompanied by a considerable CT from all SQ sensitizers to the TiO₂ substrate upon red-NIR light photoexcitation, with an efficient promotion of exciton dissociation. Considering the second electronic transitions (S2), it is clear that the D1-A1-D2 molecular backbone is the foremost region for both occupied and virtual NTOs. This coincides with the approximated LE-dominant (64–78%) hybridization for S0 → S2 transitions in all four SQ dyes. Additionally, the displayed minor existence of the virtual NTOs over π-A2 fragments, following the order SQ-N > SQ-Th > SQ-Si > SQ-P, is in line with their computed ICT% being 26, 19, 17 and 13%. This implies the capability of photoinduced ICT to be exhibited from the D1-A1-D2 molecular unit to the π-spacer and CAA terminal acceptor upon light-harvesting in the yellow-orange range, especially for SQ-N dye. Moreover, a considerable blue-green light-induced ICT is also predictable for the third electronic transition S0 → S3 of all the SQ dyes, except SQ-Th. Specifically, the simulated UV-vis spectra of SQ-N, SQ-P, and SQ-Si dyes revealed their third absorptions peaking at 491 nm (ε⁴⁹¹ = 3.8 × 10⁴ M⁻¹ cm⁻¹), 523 nm (ε⁵²³ = 3.4 × 10⁴ M⁻¹ cm⁻¹) and 510 nm (ε⁵¹⁰ = 5.7 × 10⁴ M⁻¹ cm⁻¹), respectively. The corresponding ICT features of this S3 state were nearly 5% (0.05 |e⁻|), 17% (0.15 |e⁻|) and 13% (0.11 |e⁻|) from D1-A1-D2 to the CAA terminal acceptor (A2), in addition to 10% (0.11 |e⁻|), 11% (0.12 |e⁻|) and 10% (|e⁻|) from the π-spacers to the CAA acceptor (A2). Again, superior intramolecular CT accompanying photoexcitation to the S3 state is predictable for SQ-P and SQ-Si sensitizers featuring the dithienophosphole oxide and silolodithiophene π-spacers.

Regarding the optimal lifetime for the excited state, it should be sufficiently long to allow for electron injection (∼fs–ps) into the CB of the semiconductor, while being short enough to suppress recombination (∼μs) of the injected electrons with the cationic dye. Here, the lifetime of the excited state (τ_e) was estimated via: τ_e = 1.499/(f × E²), in which f is the electronic state's oscillator strength and E is the wavenumber in cm⁻¹ correspondingly.⁵² For SQ-N, SQ-P, SQ-Si, and SQ-Th sensitizers, the computed excitation energies to the S1 state were ca. 14345.4, 11837.8, 13414.6 and 13742.9 cm⁻¹, respectively. Calculated excitation energies of ca. 17108.6, 16681.2, 16768.3, and 17216.7 cm⁻¹ corresponded to the S2 state. Generally, the nanosecond scale of calculated τ_e for the four designed SQ dyes is in good agreement with the fs–ps time scale of charge injection, in addition to being faster than the μs time scale of the unwanted injected charge recombination. Table 2 shows τ_e decreasing in the order of SQ-P (12.68 ns) > SQ-Si (6.88 ns) > SQ-Th (6.35 ns) > SQ-N (4.07 ns) for S1 states, whereas SQ-N (4.74 ns) > SQ-Th (3.69 ns) ≥ SQ-Si (3.68 ns) > SQ-P (2.68 ns) for S2 states. The longer lifetime for the S1 state advocates a higher probability of radiative decay and reduced non-radiative decay pathways, making SQ-P a good candidate for fluorescent applications. On the other hand, the prolonged S2 of dye SQ-N could imply greater potential for effective electron transfer or charge separation processes before relaxing to S₁, which are crucial for DSCs and photocatalysis. Comparison of τ_e of the modelled dyes with the parent dye, with a calculated τ_e of ca. 3.44 ns, suggests an increase in excited-state lifetime (4.07–12.86 ns) upon incorporating π-spacers. This could be attributed to increased π-conjugation and enhanced delocalization of excited-state π-electrons. For dye SQ-P (12.86 ns), the highly stabilized LUMO (−3.4 vs. −3.1 eV, Fig. 3) provided an additional advantage, contributing to its greater lifetime.

Predicted photovoltaic performance of the modelled squaraine-based sensitizers

Assessing the DSC-device photovoltaic (PV) performance of the modelled SQ dyes was conducted according to eqn (2)–(5). The corresponding results are listed in Table 3. The calculated light harvesting efficiency LHE due to S0 → S1 transition has displayed an increasing trend of SQ-P (0.85) < SQ-Si (0.94) = SQ-Th (0.94) < SQ-N (0.98) < SQ-Ref (0.99). Conversely, a reverse pattern was noted for S0 → S2 transition: SQ-Ref (0.51) < SQ-N (0.92) < SQ-Th (0.96) = SQ-Si (0.96) < SQ-P (0.99). This implies that insertion of π-bridges enhances the overall LHE by strengthening the higher-energy transition S2, which broadens the absorption spectrum and compensates for any reduction in S1 oscillator strength. The heightened f of S2, so LHE, across all the dyes could be attributed to improved delocalization and extended orbital overlap of the second electronic state, in accordance with the relative localization of NTOs across D1-A1-D2 (Fig. 4c) and the increased π → π* LE contribution (64–78%, Table 2). Regarding the critical energy parameters, the negative values of ΔG_inj and ΔG_reg for all the studied dyes point to thermodynamical spontaneity (exoergic) of the photoinduced electron injection and the oxidized dye regeneration process, respectively. A greater stability was anticipated for electron injection dynamics. Meanwhile, the positive values of ΔG_rec, spanning from 1.04 to 1.09 eV (Table 2), indicate that the competitive electron loss mechanism for the electrolyte–oxide recombination reaction is thermodynamically nonspontaneous, which is advantageous for the operation of DSC devices. The calculated values of ΔG_reg were in the order of SQ-Si (−0.34 eV) < SQ-N (−0.35 eV) < SQ-Th = SQ-P (−0.39 eV) < SQ-Ref (−0.52 eV), implying that incorporating π-spacers did not improve ΔG_reg and the parent dye maintains its superior potential for regeneration. Among the modelled dyes, the oxidized SQ-P (SQ-Si) molecules are expected to have the greatest (least) affinity to be regenerated by accepting electrons from the I/I₃⁻ redox mediator. The thermodynamic feasibility of the injection process (ΔG_inj) was found to increase in the order of SQ-P (−0.38 eV) < SQ-Th (−0.61 eV) ≤ SQ-Si (−0.62 eV) < SQ-Ref (−0.65 eV) < SQ-N (−0.73 eV). This suggests that only the dithienopyrrole π-linker in dye SQ-N is correlated with enhanced ΔG_inj. Also, SQ-N (SQ-P) is predicted to exhibit the highest (lowest) gradient to the CB edge of TiO₂ and hence the greatest (weakest) energetic driving force for the photogenerated electron injection. Generally, all the modelled dyes are foreseeable to exhibit electron injection quantum yield (Φ_inject) approaching 1, since Φ_inject tends to approach 1 when |ΔG_inj| is greater than 0.20 eV.⁷¹

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Geometrical and electronic properties of modelled squaraine adsorbates on the (TiO₂)₃₈ nanocluster

The electronic coupling at the interface can alter the surface state energetic activity and trigger photoinduced electron injection. Therefore, the electronic properties of optimized SQ@(TiO₂)₃₈ complexes are to be explored for a deeper investigation of the adsorption impact of SQ dyes upon anchoring onto TiO₂ nanoparticles. Ground state optimized SQ@(TiO₂)₃₈ geometries are shown in Fig. S4,† as obtained using the ADF code via full relaxation of all atomic positions at the PBE/TZP(DZP) level of theory. The simulated FMOs topologies are displayed in Fig. 5, along with the calculated adsorption energies and critical interatomic distances for the equilibrium interface geometries.

In order to pursue precise energetics and density-of-states, more sophisticated single point calculations at the B3LYP/6-311G(d,p) level were achieved with G09 software for the SQ@(TiO₂)₃₈ optimized geometries. The adsorption energy (E_ads, kcal mol⁻¹) was calculated as E_ads = E_(dye@TiO2) − (E_dye + E_TiO2). In which, E_dye, E_TiO2 and E_(dye@TiO2) are the energies of isolated SQ dye, bare (TiO₂)₃₈ anatase nanocluster and SQ@(TiO₂)₃₈ systems, respectively. For all the studied adsorbed squaraines, the large negative values of E_ads authenticate spontaneity of dye coverage. Greater negative E_ads (−109.9 kcal mol⁻¹) for the SQ-N@(TiO₂)₃₈ complex implies relatively higher energy release as the SQ-N dye relaxes on the titania surface, thus stronger electronic coupling. It should be noted that, while the omission of explicit basis-set-superposition error (BSSE) correction could overestimate the values of E_ads, the consistent computational protocol applied across all SQ⋯TiO₂ complexes guarantees the validity of the comparative trend. The Ti–O bond interaction lengths spanning from 2.0 to 2.2 Å advocate the formation of stable complexes of tightly-adsorbed squaraines onto the TiO₂ surface. The bond lengths of O–H bonds due to a carboxylate anchor dissociation are ca. 1.02 Å, and the bond angles of the C–O–C bonds are ca. 126° for all the studied squaraines. Considering the expected deterioration of adsorbed molecular planarity due to induced steric hindrance, all dye segments have attained planarity upon adsorption except for the torsional angle φ₂ connecting π-linkers and the indole donor. In particular, φ₂ was found to increase in the order of SQ-N (3°) < SQ-Th (8°) < SQ-P (13°) < SQ-Si (17°), which implies that the dithienopyrrole linker provides a rigidified backbone that is less prone to bending, thus upholding the optoelectronic properties of the isolated SQ-N dye. For all the dyes, the geometric features (Fig. 5) reveal their vertical orientation above the TiO₂ cluster surface. The maximum inclination, and hence minimum distance between the donor and cluster surface, is exhibited by SQ-Th dye, making it more susceptible for the undesired charge recombination back to the oxidized dye. Therefore, SQ-P and SQ-N dyes, with the beneficial elongated distance between the cation hole and cluster surface, are projected to more strongly suppress this undesirable charge recombination. Regarding the simulated FMO topologies, for all the studied systems, the spatial distribution of electrons in the HOMO is predominantly delocalized over the benzoindole-squaraine core-indole (D1-A1-D2) molecular fragment. Alternatively, the LUMO's electron dispersion is confined to TiO₂ nanoparticles. Such an orbital relocation should construct the advantageous photogenerated exciton dissociation at the composite dye/TiO₂ interface.

Further elucidation of the electronic coupling and band alignment of modelled SQ⋯TiO₂ interfaces can be revealed via projected density-of-states (PDOS). As depicted in Fig. 6, sensitization of TiO₂ with squaraines has introduced HOMO and some π-occupied orbitals (magenta discrete lines) within the cluster band-gap. In particular, upon adsorption of SQ-N and SQ-Si dyes, there exist more plainly visible localized states within the band-gap between HOMO (−4.92 eV and −4.90 eV) to HOMO−12 (−7.86 eV) and HOMO−11 (−7.67 eV), respectively. Simulated PDOS for s, p and d orbitals (Fig. S5†) clearly reveals that HOMO corresponds to a purely dye p-orbital and is well distinguished from other TiO₂ valence band (VB) states. Predictably, it is also shown that Ti 3p and O 2p orbitals dominate the VB, while Ti 3d orbitals dominate the conduction band (CB). The inserted dye p-orbitals within the band-gap are discrete due to their “molecular” features, and their energies signify the new upper-domain of the VB, thus, minimizing the band-gap energy and broadening the absorption profile. Band gaps for SQ-P, SQ-Si, SQ-N and SQ-Th are 1.19, 1.21, 1.23 and 1.25, respectively. In comparison to that of isolated dye sensitizers (Fig. 3), there is a reduction in energy of the band-gap subsequent to dye adsorption, following the sequence of SQ-N (0.77 eV) > SQ-Th (0.69 eV) ≥ SQ-Si (0.68 eV) > SQ-P (0.49 eV). The new trend of band-gap energies for SQ@(TiO₂)₃₂ systems was SQ-P (1.19 eV) < SQ-Si (1.21 eV) < SQ-N (1.24 eV) ≤ SQ-Th (1.25 eV), implying a longer bathochromic shift in the absorption spectrum of SQ-P@(TiO₂)₃₂. The most pronounced reduction observed in the band-gap of SQ-N@(TiO₂)₃₂ (0.77 eV) can be attributed to stabilization of LUMO (−3.68 vs. −3.04 eV of free dye) upon adsorption, which suggests a greater redistribution of the negative charge over the titania nanocluster. Furthermore, in all the systems, virtual orbitals of the SQ adsorbates were coupled and overlapped within the manifold of titania CB states. The hybridized orbitals have predominant projections onto the p-orbitals of π-spacers as well as the 3d-orbitals of Ti⁴⁺ (Fig. S5†). Energetically, the LUMO of free SQ-P, SQ-Th and SQ-Si molecules corresponded to the energy of LUMO+13, LUMO+22 and LUMO+25 levels for the adsorbed dye-semiconductor systems. Meanwhile for SQ-N, the LUMO energy (−3.04 eV) of the free molecule goes much deeper in the CB of the TiO₂ supercell and transformed into LUMO+31 of the SQ-N@(TiO₂)₃₈ adsorbed complex. This validates the aforesaid stronger SQ-N⋯TiO₂ electronic coupling.

An insightful comprehension of the intermolecular chemical bonding at the surface was subsequently attained via bond energy decomposition (BED) analysis. Results are tabulated in Table 4, as obtained by the ADF code, in which the bonding energy is expressed as: ΔE_b = ΔE_oi + ΔE⁰. Herein, ΔE_oi is the attractive orbital interaction energy, while ΔE⁰ corresponds to the total steric repulsion. The latter consists of a repulsive term originating from the Pauli antisymmetrization energy (ΔE_Pauli) for fermion wavefunctions, in addition to a classical electrostatic attraction term (ΔV_elstat). In particular, ΔE⁰ = ΔV_elstat + ΔE_Pauli = ΔV_elstat + ΔV_Pauli + ΔT⁰, where the potential energy (ΔV_Pauli) and the kinetic energy (ΔT⁰) parts are summated into ΔE_Pauli. Of note, the repulsive character of ΔE_Pauli is usually attributed to the strong positive kinetic energy ΔT⁰, and hence the steric repulsion can be considered a kinetic repulsion.⁷² The computed contributions of steric ΔE⁰, presented in Table 4, clearly reveal that its positive repulsive character (114.01–121.30 Ha) is owing to the dominance of repulsive destabilizing ΔE_Pauli (160.47–169.71 Ha) over electrostatic attraction ΔV_elstat (−46.46 to −48.40 Ha). Nonetheless, the strong orbital attraction ΔE_oi (−169.14 to −178.72 Ha) prevailed Pauli repulsion, resulting in a considerable stabilization of the attractive binding energy ΔE_b, with a high negative value (−55.13 to −57.46 Ha). Importantly, ΔE_oi accounts for the combined effects of charge transfer (CT) and polarization interactions between two interacting closed-shell systems. Charge transfer involves electron donation from occupied orbitals of the donor to virtual orbitals of the acceptor, while polarization is attributed to the occupied-virtual orbital interactions within each fragment due to the influence of the neighbouring fragment. For the present adsorbed systems, the calculated ΔE_oi values (Ha) followed the order of −178.7 (SQ-N) > −177.8 (SQ-Si) > −177.5 (SQ-P) > −169.1 (SQ-Th). Consequently, it can be claimed that the SQ-N@(TiO₂)₃₈ adsorbed complex will exhibit the most effective polarization and interfacial CT characteristics.

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Sensitization and interfacial charge transfer features for squaraines at the (TiO₂)₃₈ nanocluster

In addition to the electronic coupling at the SQ⋯TiO₂ interface discussed above, the extent of outward-directed interfacial CT significantly triggers the photoinduced electron injection processes from sensitizers to the semiconducting oxide. Consequently, hole–electron analysis and IFCT modules were employed for propping the photoinduced interfacial CT characteristics for modelled SQ adsorbates. For the sake of comparability, Table 5 lists the calculated CT descriptors, corresponding to the most dominant electronic transitions, for SQ free molecules and molecular adsorbates. The charge density difference (CDD) upon photoexcitation of SQ@(TiO₂)₃₈ composites is depicted Fig. 7. The CT distance (D_CT) quantifies the CT length as a photoinduced spatial distance between centroids of holes and electrons in corresponding directions via the formula:

where X, Y and Z define the centroids' three-dimensional coordinates. For instance, Y_e and Y_h denote the electron density and hole density at the y-orientation as: Y_e = ∫yρ^ele(r)dr and Y_h = ∫yρ^hole(r)dr, respectively. The separation degree of holes and electrons was also estimated by the diagnostic tool: t-index = D_CT − H_CT, with H_CT being the average degree of spatial extension in the CT direction. In contrast, hole–electron overlap, S_r index, characterizing the extent to which hole and electron centroids converge, is calculated as: . From a theoretical standpoint, the upper limit of the S_r index is 1 a.u. implying a perfectly aligned hole and electron centroids with the absence of a detectable charge separation. Moreover, the exciton binding energy (E_C), defined as the Coulomb attractive energy between hole and electron densities; , was computed to assess the feasibility extent of exciton separation. The net interfacial charge transfer (Δq) was quantified as the effective total number of transported electrons from each SQ dye to the (TiO₂)₃₈ nanocluster, SQ → (TiO₂)₃₈, upon photoexcitation. It is generally noticed (Table 5) that the CT features of all the modelled squaraines improved upon adsorption onto the TiO₂ cluster. For isolated dye molecules, only the first state (S1) exhibited dominant CT characteristics, namely a significantly longer CT distance D_CT (9.79–12.96 Å), positive larger values of the t-index (3.49–8.29 Å), and a low hole–electron overlap S_r index (0.53–0.33 a.u.) besides lower exciton binding energies E_C (2.17–1.52 eV). This is in accordance with prior results of NTO, CTS, IFCT and MO analyses (vide supra), demonstrating that electronic states S1 are CT-dominated hybrids with major (64–86%) ICT nature and minor π → π* LE (28–10%) fractions. However, it is worth pointing out that only SQ-N showed S1 as the dominant transition (λ_max); LE-dominated S0 → S2 is λ_max for other dyes. Upon adsorption onto TiO₂, all inspected electronic states show promising CT features (Table 5), including long D_CT, positive t-index as well as low S_r and E_C, in addition to distinct separation of hole (cyan) and electron (purple) density isosurfaces (Fig. 7). As a result, the net transferred charge Δq to the CB of TiO₂ remains greater than ¾ of an electron, irrespective of the dye. In comparing the major states in adsorbed dyes to the CT state of free ones, the spatial distance D_CT has expanded by 39%, 41%, 42%, and 55%, respectively, for adsorbed SQ-P, SQ-Th, SQ-Si, and SQ-N sensitizers. This pattern is lined up with the 60%, 61%, 57% and 71% extent of decrease in exciton binding energy E_C.

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Explicitly, for the SQ-P@(TiO₂)₃₈ complex, the dominant electronic transitions (S10, S12 and S15 states) corresponded to photoabsorption at 903, 885 and 840 nm with an oscillator strength (f) of 0.08, 0.18 and 0.06, respectively. All excited states were characterized by separated electron–hole populations with a positive t-index (16.7–18.5 Å), low S_r index (0.09–0.16 a.u.), and E_C (0.59–0.64 eV), in addition to an average D_CT of ca. 21.99 Å, being about 14 times longer than the C–C bond length. Concerning SQ-N@(TiO₂)₃₈, three main absorption peaks at 777 nm (f = 0.057), 738 nm (f = 0.151) and 710 nm (f = 0.263) arise from the S0 to S19, S22 and S25 electronic transitions, respectively. Akin to SQ-P@(TiO₂)₃₈, CT descriptors demonstrated an evident CT nature of these states having a positive t-index (16.4–18.2 Å), a low S_r index (0.07–0.13 a.u.), and an average D_CT of ca. 21.93 Å, which is in close proximity to the physical distance separating the squaraine core from the surface Ti atoms. The drastic 71% extent of decrease in E_C for its major state S25 (0.64 eV) in comparison to the free molecule, being ca. 2.6 eV, implies that the strong orbital interaction with titania considerably improves the dye's CT characteristics. The S19, S22 and S24 dominant CT states in the SQ-Si@(TiO₂)₃₈ composite are due to photoexcitation at 794 nm (f = 0.096), 762 nm (f = 0.203) and 736 nm (f = 0.062). The corresponding values of D_CT were 21.6, 19.4 and 23.2 Å, each of which is longer than the average separation degree H_CT (5.4–5.9 Å), which was translated into a positive t-index (14.6–19.0 Å) and low overlapping S_r index (0.08–0.18 a.u.). For SQ-Th(TiO₂)₃₈, the excitation characteristics of S0 → S22 (λ = 749 nm, f = 0.166), S0 → S25 (λ = 719 nm, f = 0.115) and S0 → S30 (λ = 682 nm, f = 0.081) featured D_CT falling within the range of 18.2–20.5 Å, S_r of ca. 0.13–0.18 a.u. and E_C of 0.68–0.79 eV, a clear indication of outward-directed CT at the dye⋯TiO₂ interface as well.

A detailed analysis was conducted for the net charge Δq collected at the TiO₂ nanocluster due to the primary dominant state, namely S12 (f = 0.18) for SQ-P@(TiO₂)₃₈, S25 (f = 0.26) for SQ-N@(TiO₂)₃₈, S22 (f = 0.20) for SQ-N@(TiO₂)₃₈, and S22 (f = 0.17) for SQ-Th@(TiO₂)₃₈. Results revealed that the actual charge injected into the substrate, SQ → (TiO₂)₃₈, decreased in the following order: SQ-N (0.90 |e⁻|) > SQ-Th (0.83 |e⁻|) > SQ-P (0.76 |e⁻|) > SQ-Si (0.74 |e⁻|). Thus, the most pronounced interfacial CT can be predicted for SQ-N@(TiO₂)₃₈, exhibiting the maximum f and Δq. Additionally, SQ-N had the largest dipole moment, both as free (17.58 D) and adsorbed (13.95 D). This maximally allowed photoexcitation of SQ-N to be initiated from both HOMO and HOMO−1, where the electron density is distributed over D1-A1-D2 and π-spacers, respectively. This is in line with the anticipated electron-donating capability of the dithienopyrrole π-spacer, where HOMO−1 does not participate in the transitions that make up the main band for the remaining three dyes. For further details, Fig. S6† depicts all the simulated orbital contributions of the studied SQ@(TiO₂)₃₈ complexes upon photoexcitation. On a related aspect, the unpredicted outperformance of SQ-Th dye over SQ-P and SQ-Si, despite exhibiting promising intramolecular CT features, could be attributed to the stronger electron-withdrawing capability of P-acetyl dithienophosphole oxide (SQ-P) as well as dithienosilole (SQ-Si), which partially snatches the excited states and deteriorates the interfacial CT to TiO₂. This is consistent with the CDD maps in Fig. 7, with π-A2 fragments exhibiting greater accumulation of ca. 24% and 26% of the photoexcited electrons (purple isosurface), correspondingly, to S12 of SQ-P@(TiO₂)₃₈ and S22 of SQ-Si@(TiO₂)₃₈, compared to 17% in the SQ-Th@(TiO₂)₃₈ composite. Fig. 7 also evidences that photoexcitation of SQ-N@(TiO₂)₃₈, in the deep red-to-NIR range, triggers the most substantial depletion of electron density in SQ-N dye, with calculated hole densities (cyan isosurface) of ca. 90–97%. This could be attributable to a complex interplay of multiple factors: the strongest coupling of SQ-N with the TiO₂ surface, including the strongest spontaneous adsorption E_ads (−109.9 kcal mol⁻¹), orbital ΔE_oi (−178.72 Ha) and electrostatic ΔV_elstat (−48.40 Ha) attractive interactions; both HOMO and HOMO−1 contributing to its most intense transition; better alignment of energy levels with the CB of TiO₂, with deeper positioning of the dye's LUMO within the CB; besides the most favourable adsorption geometry and spatial orientation, with minimal adsorption-induced deterioration of SQ-N molecular planarity.

Electron injection mechanism upon photoexcitation

At the semiconductor–molecular heterointerface, the photoinduced electron transfer (ET) proceeds via either direct (one-step) or indirect (two-step) sequential mechanisms, depending on whether the electron is injected during or following the absorption of the photon. In the direct mechanism, photon absorption directly promotes ET from the molecule adsorbate to the nanostructured semiconductor. One way to conceptualize this pathway is as a photo-assisted quantum electron pumping process from the chromophore's HOMO directly to the CB states. Alternatively, the indirect mechanism proceeds through two steps: first, photon absorption excites the molecular sensitizer; subsequently, an electron is injected. The overall effect can be characterized by the creation of a localized photoexcited state on the dye, followed by ET into the CB states. The notion of an intermediate regime that incorporates various contributions from both direct and sequential mechanisms was also reported for some squaraine sensitizers.⁷³ Two fundamental differences between direct and indirect ET mechanisms are: firstly, the electronic coupling between the CB and the dye molecular orbitals, where strong electronic coupling enforces the direct injection mechanism. Secondly, the delocalization pattern of photoexcited electrons; whether the excited state is localized at the sensitizer or extends into the semiconductor phase. In the present study, BED (Table 4) and PDOS (Fig. 6) analyses discussed thus far have demonstrated considerable electronic coupling for all the studied SQ dyes, with apparent hybridized electronic states that are partially delocalized between the dye adsorbate and TiO₂ (Fig. S5†). The strength and nature of the coupling at the adsorption site can be further confirmed by the topological analysis of electron localization function (ELF) and localized orbital locator (LOL). Indeed, substantial coupling should be associated with moderate ELF and LOL values, which, respectively, indicate partial electron delocalization and significant orbital overlap. The four SQ sensitizers are all anchored with cyanoacrylic acid (CAA), and therefore, the chemistry of coupling with the adjacent TiO₂ units is comparable. In fact, the chemical bonding in a bidentate adsorbed CAA featured SQ dye onto a TiO₂ surface involves the donation of electron pairs from the oxygen atoms in the carboxylate group (–COO⁻) to empty d-orbitals of two Ti atoms, creating the bidentate coordination bonding interaction. According to ELF (Fig. 8) and LOL (Fig. S7†), the molecular anchor along with the two adjoining pentacoordinated Ti⁴⁺ ions exhibits analogous distribution of electron localization. Specifically, intermediate ELF values (≃0.4) indicate delocalized electron densities at O–Ti coordinate bonds and partial electron sharing at the adsorption site. This partial localization is essential for effective electronic coupling. The highest ELF values (close to 1) signify regions of localized electron pairs at C–C covalent bonds. Correspondingly, intermediate LOL values (≃0.33, Fig. S7†) demonstrate the areas where the dye and TiO₂ orbitals overlap, suggesting an effective coupling pathways for ET. The aforementioned factors provide clues about strong electronic coupling, which argue for the feasibility of a direct optical ET mechanism. Regarding the redistribution pattern of photoexcited electrons, the nature of all excited states has already been categorized as charge-transfer states, being predominantly (74–97%) localized on TiO₂, with minor contribution from chromophore adsorbate. Also, the IFCT results (Table 5) have detected the approximate Δq range of 0.74 to 0.97 |e⁻| as the net intermolecular ET received by TiO₂ during red-NIR photoexcitation of SQ adsorbates. In light of this, electron injection from the SQ modelled adsorbates is advocated to preferentially follow the direct ET mechanism from the HOMO of SQ adsorbates into the CB of the TiO₂ nanocluster. That is, electron injection is anticipated to occur via quantum mechanical tunnelling, in a single step, through the potential barrier at the heterointerface SQ-sensitized nanocrystalline titania under deep red-NIR irradiation.

Nonlinear optical (NLO) properties of modelled squaraine dyes

The molecular (hyper)polarizabilities were calculated for the modelled SQ dyes, with the analytical coupled perturbed Kohn–Sham (CPKS) method, using the G09 code at the B3LYP/6-311+G(d,p) level of theory under vacuum. Table 6 lists the computed static (λ = ∞ nm, ℏω = 0 eV) first hyperpolarizability β₀, in addition to the frequency-dependent (dynamic) hyperpolarizabilities; electro-optic Pockels (EOP) or dc-Pockels β(−ω; ω, 0) and second harmonic generation or SHG β(−2ω; ω, ω), with simulated electric fields of λ = 1064 (ℏω = 0.84) and λ = 1460 nm (ℏω = 1.18 eV). For brevity, β(−ω; ω, 0) and β(−2ω; ω, ω) will be denoted as β(ω) and β(2ω), respectively, throughout the remainder of this text. The molecular polarizabilities in zero-frequency limit and the corresponding frequency-dependent are tabulated in Table 7. Basically, the molecular nonlinearity exhibits a strong wavelength dependence, thereby the off-resonant β₀ can provide a measure of intrinsic second-order NLO response of different structured chromophores in the absence of an external oscillating field. Herein, total β₀ was also calculated for the parent SQ-140 dye, yielding a value of 411.1 × 10⁻³⁰ esu. It is apparent that all the modelled dyes exhibited significantly larger β₀ values (Table 6), with the enhancement being more than twofold (SQ-N) up to sixfold (SQ-P). This underscores the crucial role of extended π-conjugated spacer in modulating the NLO response of squaraine chromophores. Specifically, the total β₀ (×10⁻³⁰ esu) was detected to decrease in the sequence of SQ-P (2589.1) > SQ-Si (1392.6) > SQ-Th (1169.6) > SQ-N (1044.5), implying that β₀ has an uptrend with increasing EW strength of the π-bridge and decreasing E_g values (Fig. 3). The electron-rich pyrrole-derived spacer of SQ-N lacks strong EW groups, which leads to the largest E_g and weakest β₀, despite longer conjugation. It follows that the polarizing power of EW groups (P = O, Si) and low bandgaps dominate over conjugation length in determining β₀ for these dyes. Although the absolute β₀ values can exhibit basis-set dependence, the consistent level of theory applied to all computations ensures the reliability of the observed trends relative to the parent dye.

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The static isotropic average polarizability α_iso(∞) followed a similar pattern in the order of SQ-N and SQ-Th inverted. Namely, SQ-P (1204.8 a.u.) > SQ-Si (1178.4 a.u.) > SQ-N (1167.1 a.u.) > SQ-Th (1053.3 a.u.). Again, the conjugation length (SQ-N and SQ-Th) contributed less to α than the EW-driven electron displacement (in SQ-P and SQ-Si). Molecular volume offers another perspective on this trend, as polarizability in homologous molecules is predicted to increase with volume. An examination of the relatively higher eigenvalues of polarizability tensors α_x = α_xx + α_xy + α_xz (3.24 to 3.88 × 10⁻²² esu, Table 7) suggests that the molecule's principal polarizability axis is along the x-direction in all the dyes. In addition, the static and dynamic isotropic polarizability (α_iso ≈ 1.56 to 2.68 × 10⁻²² esu), exhibiting comparatively low values relative to polarizability anisotropy (α_aniso ≈ 2.50 to 5.49 × 10⁻²² esu), reflects the molecular anisotropy and electronic asymmetry with enhanced CT directionality. That is, the electron density is more easily delocalized along one molecular axis. A dominant hyperpolarizability (Table 6) along the x-axis was also anticipated for all the modelled dyes, with β_x = β_xxx + β_xyy + β_xzz being vastly larger (−1038.0 to 2584.6 × 10⁻³⁰ esu) than β_y (112.9 to −276.4 × 10⁻³⁰ esu) and β_z (3.4 to 57.4 × 10⁻³⁰ esu). This highly anisotropic NLO response can be attributed to the molecular quasi-planarity in the xy-plane, thus, the strongest π-delocalization and CT occur along x, making β_x the primary axial tensorial component. Notably, the permanent dipole moment, being mostly aligned along the x-axis (μ_x ≈ 88–96% of the total ||, Table S1†), also coincides with the primary polarizability and NLO-active direction (x-axis), which advocates maximization of β, efficient unidirectional CT, and robust macroscopic NLO responses.

Upon comparing the dominant diagonal components β_xxx (1035.2 to 2584.8 × 10⁻³⁰ esu) and the maximum off-diagonal β tensors in all the dyes; β_xxy (111.6 to −247.9 × 10⁻³⁰ esu, Table S2†), it is evident that minority of NLO response is associated with CT transitions that are polarized orthogonal to the molecular dipolar axis. Accordingly, β_prj (Table 6), which estimates the projection of the β tensor onto the dipole direction, is respectively 93%, 98%, 92% and 84% of β_tot for SQ-P, SQ-Si, SQ-Th and SQ-N, indicating strong alignment. Since β_‖ is scaled for experimental purposes (β_‖ = 3/5β_prj) with statistical averaging over random molecular orientations in isotropic media, β_‖ is then reduced, ending up near 50–59% of β_tot, implying high efficiency, with the solution-phase β retaining more than half of the intrinsic molecular β. Also, the synergy between the x-axis dipole moment and (hyper)polarizability makes the modelled SQ dyes highly responsive to electric fields applied along this direction. It suggests that orienting the electric field along the x-axis would harness the full potential of the molecule's intrinsic CT efficiency, and hence ensures minimal orientational losses and optimal NLO performance. Analysis of the dynamic β_tot/static β_tot ratios (Table 6) reveals a highly frequency-dependent polarization and dispersion-driven responses. The values of β(ω) and β(2ω) are greater than the static values for all the dyes, with β(ω) increasing with frequency and β(2ω) decreasing with frequency, reflecting opposing resonance alignment.

For the dc-Pockels form β(ω) at λ = 1460 nm (ℏω = 0.84 eV), the dynamic β enhancement ratios were modest (β(ω)/β₀ ≈ 1.8 to 2.5), signifying no direct one-photon resonance with S1/S2 states (all >1.47 eV, Table 2). Meanwhile at λ = 1064 nm (ℏω = 1.18 eV), the ratios were much higher (∼3.8 to 11.5), with dye SQ-P exhibiting the largest augmentation (11.5 × β₀), attributable to a stronger near-resonant electro-optic response at 1064 nm. For the SHG form, β(2ω) is massively amplified at lower frequency (ℏω = 0.84 eV, λ = 1460 nm), reaching 31 and 56 × β₀ for SQ-Si and SQ-Th, respectively, which validates their optimal applicability for frequency doubling telecom wavelengths (E-band) to red light (730 nm). The exact resonance between the second-harmonic frequency (2ℏω = 1.68 eV) and S1 (1.70 eV) of SQ-Th explains its SHG overestimation (β(2ω)/β₀ ≈ 56.1). At higher frequency (ℏω = 1.18 eV, λ = 1064 nm), SHG enhancements are generally lower (∼4 to 9) but still significant, suggesting the suitability of SQ-P (β(2ω) ≈ 10693.1 × 10⁻³⁰ esu) and SQ-N (β(2ω) ≈ 9531.5 × 10⁻³⁰ esu) for converting near-infrared to green light (532 nm). The dominance of the axial tensorial β_x over β_y and β_z along with the consistent ratio of β_prj/β_tot(84–98%), for both the Pockels effect (β(ω)) and SHG (β(2ω)) across both frequencies (0.84 and 1.18 eV), confirms the anisotropic electronic structure and alignment of μ with the primary CT axis and (hyper)polarizability along the x-direction The polarization-angle-resolved hyper-Rayleigh scattering (HRS) intensity profiles at λ = 1064 nm are depicted in Fig. 9. In which the radial distance of the pink curve corresponds to the intensity of scattered harmonic light (I^2ω_ΨV) with respect to the incident polarization angle (Ψ) scanned from −180° to +180°. It can be found that all the dyes exhibit HRS intensity peaks at Ψ = ±90° and minima at Ψ = 0°/180°.

The symmetrical decay of intensity between these angles reflects the alignment-dependent scattering, advocating a maximized scattering efficiency when the external field is optimally aligned with the molecule's dominant NLO axis. The HRS maximum at ±90° is attributable to the dominant influence of 〈β²_ZZZ〉 over 〈β²_XZZ〉 rotational averages in Bersohn's expression:

I2ωΨV ∝ 〈β2XZZ〉cos4Ψ + 〈β2ZZZ〉sin4Ψ + sin2Ψcos2Ψ〈(βZXZ + βZZX)2 − 2βZZZβXZZ〉

The corresponding orientational average β_HRS can be written as the sum of the two contributions:
The calculated values of β_HRS exhibited a decreasing order of SQ-P (2.2 × 10¹¹ a.u.) > SQ-N (1.8 × 10¹¹ a.u.) > SQ-Th (1.6 × 10¹¹ a.u.) > SQ-Si (1.1 × 10¹¹ a.u.), reflecting HRS intensity maxima at Ψ = ±90° (Fig. 9), which hold the same trend of the SHG β(−2ω; ω, ω) at λ = 1064 nm in Table 6. The ratio between rotational averages defines the depolarization ratio associated with β_HRS, a key diagnostic parameter to imitate the intrinsic molecular symmetry of the nonlinear response, where DR = 1.5 for molecules with the T_d point group. For all the dyes, the SHG resonance detuning can be evidenced by DR values exceeding 1.5 (Fig. 9), thereby validating the high values of off-resonant β_HRS as an indicative of inherent anisotropy of the molecular charge-transfer pathways. For dyes SQ-P, SQ-N and SQ-Th, the DR values (3.6–4.0), together with the relatively balanced contributions of octupolar Φ(β_J=3) at 52–55% and dipolar Φ(β_J=1) at 45–48%, provide evidence for a simultaneous presence of a notable dipolar and octupolar character. Meanwhile a lower DR value of 2.9 for SQ-Si indicates a more pronounced octupolar response. Also, the angular distribution (nearly four-lobed pattern) of the HRS intensity profile, in conjunction with the higher quantified fractions of octupolar (Φ(β_J=3) = 61%) and the lower dipolar (Φ(β_J=1) = 39%), underscores that the octupolar symmetry dominates the total second-order NLO response of SQ-Si.

Scanning ω₁ and ω₂ of β(−(ω₁ + ω₂); ω₁, ω₂), depicted in Fig. 10, can serve as a spectral signature of how the molecules respond to any pair of input frequencies. The discrepancy between the previous analytical calculations (Table 6), which predicted the existence of strong SHG peaks satisfying ω₁ = ω₂, and their absence in this plot is attributed to the insufficient number of excited states (20) for the sum-over-states (SOS) calculation owing to computational constraints, thereby failing to capture the full first hyperpolarizability response. However, the scanning still provides valuable insights into the frequency-dependent electro-optic Pockels (EOP) and other NLO properties. In particular, the abundant peaks appearing for SQ-N and SQ-Th at ω₁ + ω₂ and ω₁ − ω₂ imply the two photon processes: sum frequency generation (SFG) and difference frequency generation (DFG), respectively. For instance, SQ-Th exhibits two strong (β ≈ 5.32 × 10⁸ a.u.) peaks at (0.014, 0.064 a.u.) and (0.064, −0.078 a.u.), signifying SFG and DFG output, respectively. That is, SFG involves combining the two photons to create a visible green light with a higher-frequency of ω ≈ 0.078 a.u. (∼580 nm), while DFG generates a lower-frequency photon with ω ≈ 0.014 a.u. (∼3230 nm) corresponding to mid-IR. For SQ-N, two SFG and DFG peaks of β ≈ 1.95 × 10⁸ a.u., with input frequencies of (0.092, 0.016 a.u.) and (0.092, −0.108 a.u.), advocate violet-blue light (ω ≈ 0.108 a.u., λ ≈ 418 nm) and mid-IR (ω ≈ 0.016 a.u., λ ≈ 2830 nm) generation. The six most dominant peaks far exceeding others in height of dyes SQ-P, SQ-Si, SQ-N and SQ-Th exhibited a β of approximately 3.08 × 10¹², 3.43 × 10⁹, 8.12 × 10⁸ and 3.47 × 10⁸ a.u. In each dye, four of the six highest peaks correspond to ω₁ or ω₂ values of 0, implying a significant electro-optics Pockels effect. Meanwhile, the remaining two correspond to ω₁ = −ω₂, which indicates a remarkable optical rectification (OR) effect i.e., rectifying the optical field into a DC signal. As demonstrated in Fig. 10, for dye SQ-P, the four EOP peaks are observable at the frequencies (a.u.) of (0, ±0.08) and (±0.08, 0), while the two OR peaks appear at (−0.08, 0.08) and (0.08, −0.08). Similarly, dye SQ-Si exhibits EOP at (±0.14, 0) and (0, ±0.14), with OR occurring at (−0.14, 0.14) and (0.14, −0.14). The EOP response of SQ-N is detectable at (±0.11, 0) and (0, ±0.11), while the OR is located at (−0.11, 0.11) and (0.11, −0.11). Dye SQ-Th displays EOP at (±0.06, 0) and (0, ±0.06) and OR output at (−0.06, 0.06) and (0.06, −0.06). These findings advocate the suitability of SQ-P, SQ-Si, SQ-N and SQ-Th dyes for correspondingly processing green light (ω ≈ 0.08 a.u., λ ≈ 566 nm), UV light (ω ≈ 0.14 a.u., λ ≈ 323 nm), violet light (ω ≈ 0.11 a.u., λ ≈ 414 nm) and NIR (ω ≈ 0.06 a.u., λ ≈ 754 nm) in both electro-optic modulation and optical rectification.

Lastly, the unit sphere representation of static α and β, as developed by Tuer et al.,⁷⁴ is displayed in Fig. 11 so as to rationalize the directionality of the tensors of (hyper)polarizabilities in a static field. It is perceptible that the elongated α ellipsoids and the β spheres clearly show the anisotropy of polarizability and hyperpolarizability. That is, the (hyper)polarizability tensorial components are primarily dispersed in the molecular plane (xy-plane), while the z-axis components, orthogonal to the molecular plane, are negligible (blue radial arrows), in agreement with the domination of α_x and β_x tensors, as presented in Tables 6 and 7. For all the dyes, the β spheres highlight, with thick arrows, the induced SHG regions if subjected to an external electric field (EEF). Upon applying two EEFs along the cyan arrow, their combination effect will result in the occurrence of a substantial SHG dipole in the same direction, as exhibited by the red radial arrows. Meanwhile, an induced SHG dipole will appear directed oppositely if two EEFs are exerted simultaneously along the green arrow. If the two EEFs are imposed from top to bottom, as illustrated by the pink arrow, their combination effect will lead to a generated SHG dipole pointing to the left. Overall, the directional maxima of all unit sphere representations reveal the non-centrosymmetry of second-order NLO response and the pronounced push–pull character, which are critical for high NLO efficiency.

4. Conclusion

This paper reports a thorough quantum chemical investigation into the various functionalities of four modelled squaraine dyes. The four benzoindole-based squaraine dyes were designed, each with a unique π-conjugated spacer group: P-acetyl dithienophosphole oxide (SQ-P), ethyl-dithienopyrrole (SQ-N), dimethyl-silolo-dithiophene (SQ-Si) and thienothiophene (SQ-Th). The study aimed to investigate the effect of these π-spacers on the dyes' performance as TiO₂-photosensitizers in DSCs and NLO-chromophores. In the context of DSCs, all the modelled sensitizers demonstrated applicability, as evidenced by their calculated electronic structures, energetic prerequisites, orbital distribution, optical properties, and electrochemical performance. Nevertheless, the SQ-N photosensitizer has consistently demonstrated superior performance compared to other sensitizers, and it can be judiciously judged as the most promising TiO₂-photosensitizer accordingly. In particular, SQ-N exhibited the most advantageous structural attributes in both the ground and excited states as well as when adsorbed onto the TiO₂ nanocluster. That is, the resonance effect dominated over the steric hindrance competing effect upon optical excitation, and adsorption induced minimal deterioration of molecular planarity. Electronically, SQ-N showed a propitious spatial distribution of FMO electron densities, either as a free dye or an adsorbate, and had the longest conjugation length with vanishing BLA (0.009 Å) over the entire molecular alkyl chain. Additionally, SQ-N had the largest dipole moment, both as a free (17.58 D) and adsorbed (13.95 D) molecule. The dithienopyrrole π-spacer of SQ-N exhibited the most efficient cyclic delocalization of mobile electrons, with optimal aromatic indices of HOMA (0.69) and SA (0.017). Energetically, SQ-N achieved the highest thermodynamical spontaneity of electron injection (ΔG_inj of −0.73 eV), with an accepted affinity (ΔG_reg of −0.35 eV) for regenerating its cationic form via an iodine redox mediator. Optically, SQ-N revealed a hyperchromic shift (ε⁶⁹⁷ ≥ 1.6 × 10⁵ M⁻¹ cm⁻¹) of its S0 → S1 main peak (λ_max) at 697 nm, attributable to a 99% HOMO → LUMO and a CT-dominated (64%) hybridized state. Thus, SQ-N had the highest light harvesting capability (LHE = 0.98) and highest electric transition dipole moment (1.78 a.u.). SQ-N displayed an excited state lifetime of ∼4 ns, being sufficiently long to allow electron injection (∼fs–ps) but short enough to suppress recombination (∼μs). Lastly, the interfacial optical and electronic features of the SQ-N@(TiO₂)₃₈ composite were undoubtably superior to the other ones. SQ-N displayed the most favourable spatial orientation, with an elongated distance between the cation hole and cluster surface. The strongest electronic coupling was estimated for the SQ-N⋯TiO₂ composite, with the greatest spontaneous adsorption E_ads (−109.9 kcal mol⁻¹), orbital ΔE_oi (−178.7 Ha) and electrostatic ΔV_elstat (−48.4 Ha) attractive interactions. Moreover, SQ-N achieved the most effective sensitization via introducing additional occupied molecular p-orbitals within the nanocluster band-gap, hybridization between virtual molecular π* orbitals and 3d-orbitals of Ti atoms, and deepest positioning (LUMO+31) of the dye's photoexcited state in the CB. Most importantly, SQ-N featured superior interfacial CT properties, including the longest space-through CT distance D_CT (21.77 Å), lowest overlap S_r index (0.13 a.u.), and lowest exciton binding energy E_C (0.64 eV), which implied a minimum thermodynamic barrier for light-activated interfacial exciton dissociation, and reciprocated in the maximum hole population (97%) over the molecular excited state, along with the maximum injected electrons Δq (0.97 |e⁻|) via the direct ET pathway from the HOMO of the SQ-N adsorbate into the CB of the TiO₂ nanocluster.

The computational investigation of the modelled NLO-chromophores was achieved through a dual approach: analytical (DFT-based) and numerical sum-over-states (TD-DFT-based) methodologies. The π-spacer-functionalized modelled SQ dyes exhibited a twofold up to sixfold enhancement of the off-resonant β₀, when compared to dyes' counterpart lacking the π-spacer. The intrinsic second-order NLO response, as evaluated by total β₀ (×10⁻³⁰ esu), revealed a decrease in the sequence of SQ-P (2589.1) > SQ-Si (1392.6) > SQ-Th (1169.6) > SQ-N (1044.5). The permanent dipole, charge-transfer, polarizability and hyperpolarizability were all predominantly aligned along the molecular plane (x-axis), creating highly anisotropic molecules with strong electronic and optical responses along this direction, as evidenced through both calculated numerical parameters and the visualized unit sphere representations. The larger values of EOP β (−ω; ω, 0) and SHG β(−2ω; ω, ω) relative to the static β₀ values, with EOP β(ω) increasing and SHG β(2ω) decreasing with frequency, reflected a dispersion-driven response with opposing resonance alignment. Dye SQ-P exhibited the greatest EOP response at both 1064 nm (β(ω) ≈ 29781.7 × 10⁻³⁰ esu) and 1460 nm (β(ω) ≈ 6592.1 × 10⁻³⁰ esu), and the strongest SHG signal (β(2ω) ≈ 10693.1 × 10⁻³⁰ esu at 1064 nm). The SHG activity of the modelled dyes, as estimated by analytical-DFT, showed a simultaneous potential for NIR-to-green and telecom E-band (1460 nm) to red light conversion. Polarization-angle-resolved HRS intensity profiles predicted the maximum intensity of harmonic light peaking at Ψ = ±90° for all the dyes, with SQ-P exerting maximum off-resonant β_HRS of ca. 2.2 × 10¹¹ a.u. For all the dyes, the molecular symmetry of the nonlinear response displayed both dipolar and octupolar characters, with SQ-Si having more pronounced octupolar response (Φ(β_J=3) = 61%). Scanning ω₁ and ω₂ of β(−(ω₁ + ω₂); ω₁, ω₂) predicted the suitability of all the dyes in both electro-optic modulation (EOP) and optical rectification (ω₁ = −ω₂) effects, with the highest signal intensity (β ≈ 3.08 × 10¹² a.u.) exhibited by SQ-P, if the nonzero external wavelength corresponds to the green light (ω ≈ 0.08 a.u., λ ≈ 566 nm). Additional abundant peaks appeared for SQ-N and SQ-Th at ω₁ + ω₂ and ω₁ − ω₂ corresponding to sum frequency generation (SFG) and difference frequency generation (DFG), respectively. The divergent trends in β₀, α, and μ addressed the need for multifactorial optimization in dye design, with SQ-P exceling in β₀ and α owing to strong EW-driven polarization, while SQ-N leveraging asymmetry for high μ and significantly advantageous interfacial CT features. Hence, incorporation of the P-acetyl dithienophosphole oxide spacer and ethyl-dithienopyrrole spacer into squaraines predicts a dependable performance in nonlinear optics and dye-sensitized solar cells, respectively. In conclusion, the current study highlights the role of extended π-conjugation as a critical design strategy in optimizing linear and nonlinear optical activity of squaraine chromophores. While limited to four π-spacers, our quantum mechanical framework established here is readily extensible to broader spacer classes. Future work could explore strong donor–acceptor systems (e.g., carbazole derivatives) and planar π-extended architectures (e.g., indacenodithiophene) to further generalize the derived structure–property relationships and design principles for squaraine-based optoelectronics.

Data availability

The supplementary data related to this article contains the optimized geometries of modeled dyes, simulated charge-transfer spectra, major molecular orbitals contributing to main electronic transitions, optimized geometry of modeled dyes adsorbed onto the (TiO₂)₃₈ cluster along with the structure of the anatase 101 (TiO₂)₃₈ cluster before and after optimization, simulated PDOS for s, p and d orbitals of dyes adsorbed onto the (TiO₂)₃₈ cluster, major molecular orbital contributions of dyes adsorbed onto (TiO₂)₃₈ upon photoexcitation, topological analysis of the localized orbital locator (LOL) at the dye–TiO₂ interface, calculated total dipole moment (μ_total) and its components (μ_x, μ_y, μ_z), calculated diagonal components and off-diagonal β tensors, in addition to the atomic coordinates of optimized modeled dyes.

Any data not included in the ESI† is available upon request to E. N. at E-mail: eman.nabil@alexu.edu.eg.

Conflicts of interest

There are no conflicts to declare.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5me00071h

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