Structural, linear, and nonlinear optical properties of ortho-carboranyl luminophores: insights from DFT and TD-DFT studies

Abstract

Three non-centrosymmetric molecular series, namely, iM, iH, and iC, are systematically investigated, each incorporating a donor fragment substituted with various functional groups (R = –CF₃, –F, –H, –CH₃, –tBu, –OMe, –OH, –NH₂, and –NMe₂) and differing in the nature of the electron-accepting core, with a trimethylsilyl-acetylene unit in iM, an o-carborane cage in iH, and a trimethylsilyl-functionalized o-carborane cage in iC. The geometries of the ground and first excited states, the absorption and emission electronic transitions, and intrafragment charge transfer are fully characterized using DFT and TD-DFT methods. Furthermore, first- and second-order NLO responses are examined under both static and dynamic regimes. The results show that the iC derivatives exhibit slightly higher variations in dipole moment (Δµ), oscillator strength (f), Coulomb attractive energy (E_CA), net electron transfer between the substituent (R) and the o-carborane cage, Stokes shift, and NLO responses compared with the corresponding iH derivatives. In contrast, the iM molecules display consistently lower values for these parameters. For all series, the magnitude of these properties increases with the electron-donor strength of the R group, with the iC series showing an ∼854% rise in β₀ from 1C to 9C. A strong correlation is observed between the first hyperpolarizability and both the net electron transfer between fragments (1 → 3) (R² > 0.97) and the Coulomb attractive energy of iH and iC (R² > 0.95). For the iC compounds (i = 2–6), an excellent linear relationship is also found between the photoluminescence quantum yield (Φ_em) and the static first hyperpolarizability (R² = 0.92). Notably, the o-carborane derivatives bearing an –NMe₂ substituent demonstrate the potential to serve as highly efficient second-order NLO materials.

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Introduction

Organic and inorganic π-conjugated materials with tailored donor–acceptor (D–A) architectures have attracted significant research attention in recent years owing to their potential in organic photovoltaic (OPV),¹ optoelectronic² and photonic applications, including photodetectors,³ organic light-emitting diodes (OLEDs), dyes,^4–6 fluorescence sensors⁷ and nonlinear optical (NLO) devices.^8–10 Furthermore, D–A systems are known to exhibit a strong intramolecular charge transfer (ICT) process from the donor to the acceptor, which makes them promising candidates for near-infrared (NIR) NLO applications.^1,11–13

Three-dimensional electron acceptors constitute an important class of molecular architectures for nonlinear optical (NLO) applications, among which fullerenes have been extensively studied owing to their exceptional electron affinity, highly delocalized π-conjugated spherical surface, and remarkable capability to stabilize long-lived charge-separated states.^14,15 Since the pioneering characterization of C₆₀, numerous fullerene–chromophore dyads have been developed by covalently integrating C₆₀ with a wide range of electron-donating units,¹⁶ including porphyrins,^17,18 tetrathiafulvalenes (TTF),¹⁹ ferrocenes²⁰ and carbazoles.²¹ These donor–acceptor conjugates have demonstrated substantial NLO activity, highlighting their efficiency in promoting intramolecular charge-transfer (ICT) processes.^22,23

Muhammad et al.²⁴ systematically investigated a series of C₆₀ fulleropyrrolidine and fulleropyrrolidine–tetrathiafulvalene derivatives and reported strong second-order NLO responses, with their β values ranging from 1.73 × 10⁻³⁰ to 15.69 × 10⁻³⁰ esu. The highest value was obtained for compound {2c}, which is attributed to its enhanced intramolecular charge transfer and extended π-conjugation. In parallel, Fouejio et al.²⁵ demonstrated that the functionalization of C₆₀ with dihydroartemisinin dramatically boosts its optical response, increasing its first hyperpolarizability from 1.21 × 10⁻³⁰ to 3.47 × 10⁻³⁰ esu. These studies confirm that the chemical modification of fullerenes significantly enhances their CT character and NLO activity, positioning them as benchmark 3D acceptors in computational NLO design.

Despite these achievements, fullerenes still face intrinsic limitations that restrict their broader technological impact. These include their high production cost, combined with the centrosymmetric nature of pristine C₆₀ (which necessitates asymmetric functionalization to induce a non-zero first hyperpolarizability)²⁶ and processability challenges in devices.²⁷ Consequently, they have motivated the development of alternative 3D acceptors possessing improved physicochemical and electronic characteristics.

In this context, the ortho-closo-carborane cluster (C₂B₁₀H₁₀) has emerged as one of the most promising substitutes for fullerene-based electronic acceptors. This cluster, defined by its icosahedral σ-aromatic boron cage, is frequently described as a three-dimensional analogue of benzene, but it exhibits a distinct and highly polarized electronic structure.^28–30 Unlike fullerene derivatives, carboranes combine a substantial permanent dipole moment (4–5 Debye for ortho-carborane),³¹ exceptional thermal stability (>400 °C) and superior synthetic accessibility,^32,33 thereby positioning them as attractive scaffolds for advanced photonic materials.³⁴ The high polarizability of the boron cage induces a pronounced electron-withdrawing effect on carbon-bound substituents, enabling efficient charge separation within donor–acceptor frameworks.^35–39 Recent investigations have further highlighted the versatility of carborane–chromophore conjugates, demonstrating their promising performance across a broad spectrum of functional materials, including NLO systems,^40–42 electro-optic materials,^40–42 light-emitting diodes^43,44 and organic solar cells.⁴⁵ Beyond photonic applications, they have found use in medicine,^33,46 supramolecular and materials chemistry, and coordination and organometallic chemistry.^47,48

In 2023, You et al.⁴⁹ synthesized and characterized a series of six o-carboranyl luminophores functionalized with biphenyl moieties bearing systematically varied para-substituents (R = –CF₃, –F, –H, –CH₃, –tBu, and –OCH₃) to elucidate the influence of electronic effects on ICT behavior and radiative decay efficiency. Optical studies revealed progressive redshifts in both absorption and emission maxima with increasing electron-donating ability, accompanied by a systematic increase in photoluminescence quantum yield. The radiative decay rate constants remained essentially invariant, whereas nonradiative decay was markedly suppressed for electron-rich derivatives. Collectively, these findings demonstrate that precise modulation of the electronic environment in o-carborane-biphenyl systems enable control over the ICT efficiency and emission output, providing valuable design principles for next-generation o-carborane-based emissive materials.

The discovery of new nonlinear optical materials remains a central challenge in advanced materials science for both theoretical and experimental researchers. Quantum chemical calculations have proven to be a powerful approach, providing not only a means to screen and identify promising molecular candidates but also to reveal previously unexplored parameters that can be exploited to predict and control NLO responses with greater accuracy. In this study, DFT and TD-DFT calculations were performed on three non-centrosymmetric molecular series, iM, iH, and iC, of compounds (i = 1–9) from six o-carboranyl luminophores (1C to 6C) originally synthesized by You et al.,⁴⁹ as depicted in Fig. 1. The primary objective of this study, which is carried out on 27 compounds, is to examine the effect of various donor substituents [R = –CF₃, –F, –H, –Me, –CMe₃, –OMe, –OH, –NH₂ and –NMe₂] on the nonlinear optical properties of the target compounds. In addition, this work aims to evaluate how substituting the o-carborane acceptor with a –C≡C–SiMe₃ fragment influences the NLO response, and to determine the impact of incorporating a bulky trimethylsilyl (TMS) group into the o-carborane cage on both the linear and nonlinear optical properties of the iH and iC series. The principal goal of this study is to establish new parameters that can serve as reliable predictors of NLO behavior, thereby enabling the precise control and rational design of advanced materials with an enhanced NLO performance.

Fig. 1 Chemical structures of the title compounds iM, iH and iC (i = 1 to 9).

Computational methods

All calculations were carried out using density functional theory (DFT) and time-dependent DFT (TD-DFT) methodologies, as implemented in the Gaussian 16 suite.⁵⁰ The theoretical framework of this study utilized molecular structures derived from X-ray crystallography of the o-carboranyl compounds synthesized by You and collaborators.⁴⁹ Calculations were executed using the CAM-B3LYP functional,⁵¹ a long-range corrected functional developed to handle the inaccuracies of the non-Coulomb part of the exchange functional at long distances. All calculations were conducted in a tetrahydrofuran solvent environment utilizing a polarizable continuum model (PCM), specifically through the integral equation formalism variant (IEFPCM),⁵² to obtain the geometrical structure inputs used for simulation of the static and dynamic components of the polarizability and the first- hyperpolarizability tensor within a solvent medium.

The average polarizability (α) and its anisotropy (Δα) are defined through the following relations:⁵³

The first hyperpolarizability is outlined as follows:

Also, β_vec may be defined as:

where the components β_x, β_y and β_z are defined as:

On the other hand, the hyper-Rayleigh scattering (HRS) method is employed for the detection of incoherent second harmonic generation (SHG) at frequency 2ω, induced by the incidence of a laser operating at frequency ω. This technique facilitates the determination of the first-order hyperpolarizability (β_HRS) and its associated depolarization ratios (DR). These parameters can be quantified using the following analytical expressions:^54–57

〈β_ZZZ²〉 and 〈β_XZZ²〉 can be decomposed into two symmetry-allowed contributions, namely, the dipolar (J = 1) and octupolar (J = 3) components of the first hyperpolarizability, respectively, which can be calculated as follows:⁵⁷

The two components can be evaluated as follows:

The relative dipolar and octupolar contributions to the first hyperpolarizability, denoted as φ_J=1 and φ_J=3, respectively, are defined as follows:

where ρ = |β_j=3|/|β_j=1|, clearly φ_J=1 + φ_J=3 = 1.

Additionally, the hole–electron distribution was analyzed to investigate the characteristics of electronic excitation within the compounds. The spatial distributions of holes and electrons are defined as follows:^58–60

where ϕ represents the orbital wave function, W denotes the excitation coefficient, i and j refer to the indices of occupied orbitals, while a and b correspond to the indices of virtual orbitals.

The overlap distribution between the hole and electron can be mathematically represented as follows:^59,60

To quantify the degree of overlap between the hole and electron distributions, the S_r index is defined as follows:

The overall charge transfer (CT) length is quantified by the D_index, which is determined using the following expression:

D_index = [(D_x)² + (D_y)² + (D_z)²]^1/2 (18)

The charge transfer direction (H_CT) can be calculated as follows:

H_CT = |H·u_CT| (19)

where u_CT is the unit vector in CT direction and H_index represents the spatial extent of the average distribution of holes and electrons.

The variations in the dipole moment of the excited state with respect to the ground state in the X, Y and Z directions can be simply calculated as follows:

where, Δµ_x = (X_ele − X_hole), Δµ_y = −(Y_ele − Y_hole), and Δµ_Z = −(Z_ele − Z_hole).

Results and discussion

The effect of basis sets on the NLO responses

In contemporary computational chemistry research, DFT and TD-DFT have proven to be instrumental in the detailed description of the optical characteristics of compounds. Nevertheless, selecting an appropriate level of computational precision is crucial for accurately determining both linear and nonlinear optical properties. To this end, seven Pople basis sets, specifically 6-31++G(d,p), 6-31+G(d,p), 6-31++G, 6-31+G, 6-31G(d,p), 6-31G(d) and 6-311G(d), were chosen to discuss the influence of the basis sets on the electronic absorption wavelength and NLO responses by utilizing the CAM-B3LYP functional.

The calculated principal absorption bands of 1C, as evaluated across different basis set levels, are illustrated in Fig. 2. Intriguingly, the variation in absorption wavelength (λ₁) between the 6-31++G(d,p) and 6-31G(d,p) basis sets is approximately 8 nm. This finding indicates that the effect of the basis set size on the estimated absorption wavelength is negligible (Fig. 2).

Fig. 2 Simulated absorption spectrum of 1C.

The effect of the basis set on the NLO responses was tested for compound 1C. As can be seen in Table S2, the polarizability and anisotropy polarizability values change slightly with an increase in the number of basis functions. Overall, α(0,0) and Δα(0,0) increase in the following order:

α(0,0): 6-31G(d) ≈ 6-31G(d,p) < 6-311G(d) < 6-31+G ≈ 6-31++G < 6-31+G(d,p) ≈ 6-31++G(d,p),

Δα(0,0): 6-31++G ≈ 6-31+G < 6-31+G(d,p) = 6-31++G(d,p) < 6-311G(d) ≈ 6-31G(d) < 6-31G(d,p).

From this observation, it is apparent that incorporating diffuse functions leads to an increase in the α(0,0) value and a decrease in the Δα(0,0) value, each by approximately 40 a.u. A secondary insight derived from these results is that the static first hyperpolarizabilities of 1C are significantly overestimated when the basis set excludes diffuse functions. This overestimation decreases in the following sequence: 6-31G(d,p), 6-31++G(d,p), 6-31+G(d,p) and (6-31++G) ≈(6-31+G), a trend that also holds for the anisotropy of polarizability. Going from a (+) diffuse to a (++) diffuse basis set results in a lower enhancement in both polarizability and first hyperpolarizability (Table S1). These studies show that including one or two diffuse functions results in almost unchanged values for polarizability and first hyperpolarizability. This occurs when the 6-31+G(d,p) or 6-31+G basis set is compared with 6-31++G(d,p) or 6-31++G, respectively. For instance, β_HRS^∞ (6-31++G) ≈ β_HRS^∞ (6-31+G) > β_HRS^∞ (6-31++G(d,p)) ≈ β_HRS^∞ (6-31+G(d,p)).

On the other hand, the calculated static first hyperpolarizability values obtained with polarization-type basis sets exhibit the following decreasing trend: β_HRS^∞ (6-311G(d)) > β_HRS^∞ (6-31G(d,p)) ≈ β_HRS^∞ (6-31G(d)). This observed trend indicates that the triple-split valence basis set 6-311G(d) yields the largest first hyperpolarizability, while the double-split sets 6-31G(d) and 6-31G(d,p) give smaller and nearly equivalent values.

For 1C, the 6-31G(d,p) and 6-31G(d) basis sets yield moderate first hyperpolarizability values, positioned between the higher values obtained with triple-split valence sets and the lower values from diffuse basis sets (Table S1). Previous works,^61–65 including studies by You et al.,⁴⁹ have demonstrated the effectiveness of the 6-31G(d,p) basis set in linear and nonlinear optical calculations, confirming its suitability for providing an optimal balance between accuracy and computational efficiency. Consequently, the 6-31G(d,p) basis set was chosen for this investigation, as it has proven reliable in determining both linear and nonlinear optical properties for similar compounds.

To verify that our computational approach can reproduce the experimental UV/vis behavior of the studied molecules, we compared the TD-CAM-B3LYP/6-31G(d,p) absorption (λ_abs) and emission (λ_em) wavelengths of compounds 1C–6C with their measured values. As shown in Table S2, the calculated λ_abs values follow the experimental trend very closely, with deviations of 6.8–9.3%, which is fully consistent with the expected accuracy of this functional. For the emission spectra, the calculated λ_em wavelengths show slightly larger deviations (11.57–12.96%), but still reproduce the ordering and relative evolution of the fluorescence bands across the series. Despite the experimental red-shift, the theoretical results capture the correct magnitude and progression of the optical transitions. Overall, these comparisons confirm that the CAM-B3LYP/6-31G(d,p) methodology provides a reliable description of the electronic excitations and excited-state properties of the targeted compounds.

Geometrical study

The optimized geometries of the ground state (S₀) and first excited (S₁) states of iM, iH and iC (i = 1 to 9) in THF were obtained at the CAM-B3LYP/6-31G(d,p) level of theory. Based on the examination of the geometric parameters of the ground state (S₀) summarized in Table 1, it is evident that there is outstanding concordance between the simulated and experimental data. In all the optimized molecular geometries (S₀), the biphenyl entities, designated as Frag-2, manifested a uniform angular distortion of 37° (φ₁), as listed in Table 1. On the other hand, the φ₂ angle, which quantifies the exo–π-interaction magnitude between the cage framework, termed Frag-1, and the conjugated aromatic moiety, was observed to be perpendicular (∼90°) in the series of 1C–9C compounds. This angle approximated perpendicularity (∼82°) for the o-carborane compounds, with the notable exceptions of compounds 6H and 7H, wherein φ₂ was recorded to be 30°. The analysis further reveals that the C1′–C2′ bond length in the trimethylsilyl-o-carborane compounds is elongated by approximately 0.046 Å compared to its counterparts in the H-o-carborane compounds (1H–9H). This elongation is attributed to the incorporation of the SiMe₃ group within the o-carborane cage. Additionally, it is noteworthy that the computed C2–C3 bond length in the studied compounds is 1.418 Å, indicating that Frag-1 and Frag-3 do not influence the C2–C3 bond length within Frag-2.

Table 1 Calculated values from the ground- (S₀) and first-excited-singlet- (S₁) state optimized structures: dihedral angles between the rings Ph1 and Ph2 (φ₁ = C1–C2–C3–C4) and between the C1′–C2′ bond axis in the o-carborane cage and Ph1 (φ₂ = C1′–C2′–C6–C5), including the bond lengths for C2–C3, C6–C2′ and C1′–C2′ of the compounds 1–9. The experimental results reported for the iC compounds (values provided in brackets) are from ref. 49

	S0	S1	S0	S1	S0	S1	S0	S1	S0	S1
	C2–C3	C2–C3	C6–C2′	C6–C2′	C1′–C2′	C1′–C2′	φ1	φ1	φ2	φ2
1M	1.483	1.413	1.431	1.388	—	—	38	2.82	—	—
1H	1.483	1.401	1.509	1.481	1.649	1.707	38	0.094	81	91.74
1C	1.483	1.440	1.509 (1.506)	1.435	1.689 (1.709)	2.329	38 (42.6)	17.46	91 (77.2)	91.35
2M	1.483	1.414	1.431	1.387	—	—	37	3.173	—	—
2H	1.483	1.435	1.509	1.437	1.650	2.319	38	15.059	83	91.16
2C	1.483	1.436	1.509 (1.505)	1.439	1.690/1.710	2.336	38 (24.6)	16.21	92 (86)	89.28
3M	1.483	1.414	1.431	1.387	—	—	38	3.111	—	—
3H	1.483	1.437	1.509	1.437	1.649	2.319	38	15.70	81	91.328
3C	1.483	1.438	1.509 (1.506)	1.438	1.690 (1.708)	2.336	38 (19.0)	16.36	93 (87.9)	90.168
4M	1.483	1.412	1.431	1.389	—	—	36	2.60	—	—
4H	1.482	1.433	1.509	1.438	1.650	2.318	37	13.14	84	88.92
4C	1.482	1.434	1.509 (1.511)	1.439	1.690 (1.715)	2.335	37 (30.5)	14.54	92 (79.2)	89.80
5M	1.482	1.412	1.431	1.389	—	—	37	2.67	—	—
5H	1.482	1.433	1.509	1.437	1.650	2.317	37	12.43	83	88.92
5C	1.482	1.434	1.509 (1.504)	1.439	1.690 (1.697)	2.334	37 (36.9)	14.53	93 (83)	89.29
6M	1.481	1.412	1.430	1.391	—	—	36	2.47	—	—
6H	1.481	1.406	1.507	1.480	1.631	1.644	36	1.55	30	31.55
6C	1.481	1.431	1.509 (1.503)	1.441	1.691 (1.718)	2.336	36 (34.5)	13.28	92 (82.6)	90.424
7M	1.482	1.390	1.430	1.413	—	—	37	2.96	—	—
7H	1.481	1.406	1.507	1.479	1.632	1.643	37	2.29	30	30.86
7C	1.481	1.432	1.508	1.441	1.691	2.337	36	−13.69	92	90.07
8M	1.480	1.415	1.430	1.395	—	—	35	−3.071	—	—
8H	1.480	1.434	1.508	1.442	1.652	2.326	35	13.74	84	89.18
8C	1.480	1.433	1.508	1.445	1.692	2.341	35	−14.09	92	90.06
9M	1.479	1.414	1.430	1.397	—	—	34.17	2.89	—	—
9H	1.478	1.435	1.507	1.441	1.652	2.325	34.06	−13.20	84	91.12
9C	1.478	1.434	1.508	1.444	1.692	2.341	33.72	−14.81	91.78	89.90

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The analysis of the optimized S₁ excited-state structures of the title compounds reveals a slight contraction in the C2–C3 bond length compared to their corresponding ground states (S₀). For the iH and iC compounds, this contraction is approximately 0.04 Å. Interestingly, in the 1H, 6H, and 7H derivatives, this contraction is nearly doubled, reaching approximately 0.07 Å. The iM compound exhibits an even more pronounced reduction in the C2–C3 bond length in the S₁ state, measuring approximately 0.07 Å shorter than that in the ground state (Table 1). Moreover, a comparable trend is observed for the C6–C2′ bond. In the S₁ excited state, this bond shortens by approximately 0.062–0.074 Å for the iH and iC compounds. However, the iM compound demonstrates a less pronounced contraction, with a reduction ranging from 0.033 to 0.044 Å. Distinct exceptions are noted in the 1H, 6H, 6M, and 7H compounds, where the C6–C2′ bond contracts by 0.028 Å, 0.027 Å, 0.017 Å, and 0.028 Å, respectively.

Additionally, the optimized structures of the iH and iC compounds demonstrate substantial differences in the C1′–C2′ bond length in the cage for the S₀ and S₁ optimized structures. Computational analysis reveals that in the S₁ state, the C1′–C2′ bond length extends by approximately 0.6 Å compared to its value in the S0 state. However, distinct deviations are noted for the 1H, 6H, and 7H compounds, where the bond length increases by only 0.058 Å, 0.013 Å, and 0.011 Å, respectively.

On the other hand, the S₁-optimized structure exhibits a noticeably distorted geometry in the biphenyl rings of the o-carboranyl compounds, with an observed dihedral angle (φ₁) of approximately 13°–17°, except for 1H, 6H, and 7H, where φ₁ = 0°, 1° and 2°, respectively. This is accompanied by an orthogonal arrangement between the C–C bond within the o-carborane cage and the biphenyl plane (φ₂ ≈ 90°), contrasting sharply with the ground-state geometry, where φ₁ measures 36°. This is with the exception of compounds 6H and 7H, which do not exhibit perpendicular structures, φ₂ = 30°. According to this analysis, it can be concluded that the results obtained for the S1-optimized structures of the o-carboranyl compounds indicate that the C1′–C2′ bond distance and the angle φ₁ are primary contributors to the emission properties.

UV-vis and IFCT analyses (electron–hole analysis)

In this phase of our research, we focused on analyzing the UV-vis spectra of the title compounds. To accomplish this objective, we employed the TD-DFT method, a robust computational approach widely acknowledged for its effectiveness in scrutinizing the electronic properties of molecules. Furthermore, we conducted theoretical calculations for the compounds under investigation to determine their inter-fragment charge transfer (IFCT)⁶⁶ properties. Tables S3–S7 in the SI provide comprehensive information regarding the spectroscopic attributes of the electronic transitions, encompassing absorption wavelengths, excitation energies, oscillator strengths, and significant contributions from each transition.

The TD-DFT analysis indicates that the first-excited-state transition S₀ → S₁ in the iM molecules shifted to a longer wavelength compared to the corresponding transition in the iC and iH o-carboranyl compounds for i = 1 to 9. Furthermore, the absorption spectrum of the iC compounds closely resembles that of the iH compounds, as illustrated in Fig. 3 and S1 in the SI. All the series of compounds, iM, iH, and iC, exhibited two prominent absorption bands. The first band is centered within the wavelength range of 240–300 nm, while the second band is observed at approximately 185 nm (Table S4). Additionally, the lowest-energy electronic transitions (S₀ → S₁) of the iM compounds (i = 1–9) exhibit a systematic redshift of approximately 21 nm in their absorption maximum (λ_max) relative to those of the corresponding iC and iH analogues (see Table S4). On the other hand, the title compounds show no absorption within the UV-vis range (300 nm to 700 nm) and the near-infrared (NIR) range (700 nm to 2500 nm), indicating their transparency throughout the UV-vis-NIR spectrum. This transparency, extending beyond 300 nm, positions these compounds as promising candidates for applications in UV-vis-NIR NLO materials.

Fig. 3 Simulated UV-vis absorption spectra in THF for the compounds 1 and 2 at the TD-CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory.

By analyzing the electron–hole wave-function overlap integral (S_r), the distance between the centroids of holes and electrons (D_index), the variation in dipole moment relative to the ground state (Δµ), the hole delocalization index (HDI), and the electron delocalization index (EDI) as descriptors of electron excitation types, it is possible to infer the nature of these transitions.^67–70

The TD-DFT calculations for the title compounds indicate that the low-energy transitions (S₀ → S₁) are predominantly characterized by an electronic excitation from the HOMO to the LUMO, with a 90% contribution (Table S4). As detailed in Table 2 (see also Table S5 in the SI), it is evident that the electron–hole wave-function overlap integral (S_r) for all the first excited states (S₁) is close to 1 (S_r > 0.7). The centroid distance between electrons and holes ranges from 0.2 to 1.2 Å for the iH, iM, and iC compounds (i = 1 to 5), while for compounds i = 6 to 8, D varies from 1.7 to 2.3 Å. The delocalization index (H) is approximately 3.4 Å for the iM molecules and 3.1 Å for the iH and iC compounds. The hole delocalization index (HDI) is slightly larger than the electron delocalization index (EDI) across all the compounds. The t index is negative and significantly less than zero, indicating the absence of charge separation. This suggests that the first excited state (S₁) is a typical locally excited (LE) state. This conclusion is further supported by the analysis of the charge density difference (CDD) between the ground and excited states, as shown in Fig. 4 and S2 in the SI.⁵⁹

Table 2 Calculated overlap (S_r), D_index (Å), H_index (Å), variation in dipole moment with respect to the ground state (Δµ, a.u.), hole delocalization index (HDI), electron delocalization index (EDI), oscillator strength (f), excitation energy (ΔE, eV), and Coulomb attractive energy (E_CA, eV) for the iM, iH and iC compounds in the first excited state (S₀ → S₁) at the CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory

	Sr	D	H	Δµ	HDI	EDI	ΔE	f	ECA
1M	0.845	0.577	3.417	1.090	6.84	6.28	4.613	1.557	5.274
1H	0.851	0.250	3.072	0.472	6.91	6.57	5.052	1.352	5.572
1C	0.850	0.240	3.114	0.453	6.86	6.50	5.02	1.15	5.450
2M	0.847	0.193	3.380	0.364	6.63	6.64	4.65	1.507	5.347
2H	0.830	1.091	3.021	2.061	6.96	6.95	5.043	1.225	5.540
2C	0.828	1.113	3.073	2.103	6.90	6.86	5	1.111	5.392
3M	0.849	0.048	3.352	0.091	6.68	6.57	4.655	1.233	5.368
3H	0.839	0.877	2.997	1.656	6.90	6.82	5.065	0.994	5.587
3C	0.837	0.899	3.045	1.698	6.85	6.74	5.0238	1.103	5.441
4M	0.8455	0.290	3.438	0.548	6.44	6.58	4.6006	1.608	5.274
4H	0.826	1.160	3.073	2.191	6.88	6.81	4.9501	1.299	5.458
4C	0.824	1.182	3.113	2.233	6.81	6.73	4.9093	1.1787	5.320
5M	0.846	0.291	3.465	0.549	6.42	6.56	4.5915	1.674	5.237
5H	0.827	1.173	3.095	2.217	6.83	6.80	4.9381	1.339	5.417
5C	0.8253	1.201	3.142	2.269	6.78	6.69	4.9074	1.216	5.290
6M	0.827	0.935	3.514	1.767	6.33	6.67	4.501	1.651	5.152
6H	0.794	1.750	3.147	3.306	7.30	6.87	4.791	1.431	5.235
6C	0.788	1.811	3.137	3.421	7.26	6.85	4.731	1.223	5.156
7M	0.828	0.858	3.475	1.621	6.35	6.71	4.537	1.589	5.212
7H	0.795	1.714	3.090	3.238	7.20	7.00	4.836	1.367	5.322
7C	0.790	1.754	3.096	3.314	7.21	6.92	4.774	1.178	5.224
8M	0.798	1.591	3.536	3.005	6.47	6.70	4.355	1.694	5.055
8H	0.751	2.249	3.11	4.251	7.67	6.98	4.5337	1.356	5.158
8C	0.750	2.281	3.162	4.309	7.57	6.84	4.4982	1.235	4.996
9M	0.769	2.165	3.602	4.090	7.20	6.56	4.089	1.798	4.808
9H	0.731	2.551	3.231	4.821	8.33	6.73	4.243	1.4319	4.873
9C	0.723	2.648	3.222	5.003	8.36	6.70	4.187	1.296	4.764

---

Fig. 4 CDD analysis for the first excited states (with the largest oscillator strength) of the compounds 1 and 8. CDD was calculated as the difference between the corresponding excited state and the ground state of the considered system at the CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory. Blue regions indicate negative electron density while orange regions correspond to positive electron density. The isosurface level is set to be 0.001807.

It is important to emphasize that the iM series exhibits the highest S_r values for weak donors (3M: 0.849) but converges toward values comparable to the iH and iC series for strong donors (9M: 0.769, 9H: 0.731, and 9C: 0.723). TMS functionalization (iC) slightly reduces S_r compared to iH, with a maximum difference of −1.1% for compounds 9. This decrease in S_r with increasing donor strength confirms the reduced hole–electron overlap, which is favorable for charge separation. The D_index increases drastically with donor strength, establishing the hierarchy iC > iH > iM, except for the –CF₃ substituent, where the order is the opposite (1M: 0.577 Å > 1H: 0.250 Å > 1C: 0.240 Å). This atypical behavior arises from the strongly electron-withdrawing character of –CF₃, which reverses the electron flow and alters the charge distribution. In the iM series, the D_index varies from 0.048 Å (3M, –H) to 2.165 Å (9M, –NMe₂), corresponding to a ∼45-fold increase. The iH and iC series show higher values for donor substituents, reaching 2.551 Å for 9H and 2.648 Å for 9C (+3.8% vs. 9H and +22% vs. 9M). Therefore, TMS functionalization enhances spatial separation, with the effect being maximized for strong donors. The dipole moment variation (Δµ) follows the same trend as D_index, confirming the proportional relationship between spatial separation and dipole moment variation. In the iC series, Δµ increases from 1.698 a.u. (3C) to 5.003 a.u. (9C), corresponding to a ∼3-fold enhancement. The hierarchy iC > iH > iM is maintained, as follows: 9C (5.003 a.u.) > 9H (4.821 a.u.) > 9M (4.090 a.u.). TMS functionalization further amplifies Δµ by +3.8% (9C vs. 9H) and +22% (9C vs. 9M), demonstrating the synergistic effect between the o-carborane cage and TMS substitution.

Generally, the H_index follows the reverse trend (iM > iC > iH) with the exception of compounds 6 and 9, where iH > iC. The values remain within a moderate range (from 2.997 (3H) to 3.602 (9M)) corresponding to an increase of roughly 20%. This trend suggests that the iH and iC series confine the electronic distributions more strongly while maximizing vectoral charge separation.

The coulombic energy (E_CA), which quantifies the electrostatic attraction between the hole and the electron, systematically decreases with donor strength and follows the hierarchy iC < iH < iM. For weak donors, the iM series presents moderate values (3M: 5.368 eV), while the iH and iC series display higher values (3H: 5.587 eV and 3C: 5.441 eV). However, with strong donors, a remarkable inversion occurs, as follows: 9C (4.764 eV) < 9M (4.808 eV) < 9H (4.873 eV). TMS functionalization consistently reduces the E_CA (9C vs. 9H: −2.2%), confirming its role in optimizing charge separation.

Our results show that the integration of the o-carborane cage (iH and iC) significantly enhances the D_index and Δµ, while concurrently reducing the S_r overlap and the E_CA relative to the iM series (i = 2–9). These effects are further amplified when strong electron-donating groups, such as –NH₂ and –NMe₂, are incorporated. This observation indicates that the combination of o-carborane with TMS substitution constitutes a synergistic strategy for enhancing spatial charge separation and promoting dipolar asymmetry. The atypical behavior of compounds 1M, 1H, and 1C can be attributed to the presence of the strongly electron-withdrawing –CF₃ group,⁷¹ which reverses the charge-transfer direction typically observed with donor substituents. This highlights the critical role of substituent electronic character in modulating charge-transfer properties.

The hole–electron descriptors (Sr, HDI, and EDI) for the second absorption band at around 180 nm, as presented in Table S5, show no significant difference from those of the S₁ transition, with the D_index values following the same trend. The complementary CDD analysis (see Fig. S2) confirms that all the excitations at around 180 nm correspond to local charge transfer within the biphenyl fragment.

In this study, the IFCT approach was employed to compute the net electron transfer between various segments during electronic excitation.^58,59,72 This is achieved by analyzing the difference between electron-donating and electron-accepting contributions in each segment. Additionally, calculations were performed to determine the charge transfer percentage (CT%) and its complement, the local excitation percentage (LE%). The results of these calculations are presented in Fig. 5 and Table S6.

Fig. 5 Transferred electrons between fragments, calculated for the first excited states of the iM, iH and iC compounds at the CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory in THF.

The analysis of the S₀ → S₁ excitation reveals that the percentages of local excitation (LE) significantly exceed those of charge transfer (CT) across the iM, iH, and iC compounds, indicating that LE predominantly characterizes the transitions. The contribution from CT follows the trend iM > iH ≈ iC, with the iM compounds exhibiting relatively consistent values (∼40% CT and ∼60% LE) independent of the R group. In contrast, the R substituent has a more pronounced effect on the CT character in the iH and iC compounds. Among the substituents, –NMe₂ and –NH₂ lead to the highest CT percentages, followed by –OMe, –OH, –CMe₃, –Me, and –H. Taking the iH compounds as an example, CT (%) increases in the following order: 9H (37%) > 8H (29%) > 6H (26%) > 7H (24%) ≈ 2H (24%) > 5H (20%) > 4H (19%) > 3H (17%) > 1H (9%), highlighting the significant influence of the R substituent on the CT characteristics.

The interfragment analysis (Table S6) reveals that intrafragment electron redistribution on the biphenyl is moderate for the iM series (0.52–0.59) and more pronounced for the iH and iC series (0.60–0.83), with the trend of iH > iC > iM. Notably, these values decrease with an increasing donor strength of the substituents. The variation in population numbers supports these trends, as follows: in the iM series, biphenyl gains electrons, while acetylene shows variable behavior. In contrast, the iH and iC series consistently exhibit electron gain by the o-carborane, with strong donors such as –NMe₂ efficiently donating electrons, as seen in 9H (−0.198). The fragment contributions further support these findings, as follows: Frag-2 (biphenyl) contributes 68–75% to the hole distribution in iM versus 71–91% in iH and iC, and 75–78% to the electron distribution in iM compared to ∼85% in iH and iC. These results highlight the optimization of charge transfer (CT) in the o-carborane-based architectures by minimizing local electronic reorganization, emphasizing the role of both the o-carborane cage and the R group in controlling electronic distribution.

In the 1H[1C], 2H[2C], 3H[3C], 4H[4C] and 5H[5C] compounds, Frag-2 donates 0.039[0.042], 0.041[0.069], 0.059[0.068], 0.062[0.070] and 0.061[0.071] electrons to Frag-1, respectively. For the iM, iH, and iC non-centrosymmetric molecular series (i = 1 to 5), the S₀ → S₁ excitation results in minimal net electron transfer between fragments 1 and 3, and between fragments 2 and 3 in the iH series. The net electron transfer in the iC compounds is slightly larger than that in the iH compounds, likely due to the influence of the SiMe₃ substituent on the o-carborane cage. In the iM compounds (i = 6–9), electron transfer from Frag-2 to Frag-1 varies, as follows: 6M (0.007), 7M (0.005), 8M (0.035), and 9M (0.042). The Frag-3 → Frag-2 transfer also varies with the R group, as follows: –OH (0.033), –OMe (0.026), –NH₂ (0.064), and –NMe₂ (0.129), with –NMe₂ showing the largest transfer due to its strong electron-donating nature (Table S7). For the iH[iC] compounds (i = 6–9), electron transfer occurs via dual pathways, as follows: Frag-2 → Frag-1 and Frag-3 → Frag-2. The Frag-2 → Frag-1 transfers are 0.063[0.069], 0.068[0.075], 0.061[0.073] and 0.053[0.062] for 6H[6C], 7H[7C], 8H[8C], and 9H[9C], respectively. The Frag-3 → Frag-2 donations increase with donor strength, as follows: 0.068[0.066] for –OMe, 0.050[0.051] for –OH, 0.109[0.103] for –NH₂, and 0.173[0.175] for –NMe₂. This dual electron flow toward Frag-1 highlights a cooperative mechanism where strong donors enhance charge transfer through the biphenyl spacer to the o-carborane acceptor.

Stokes shift

The analysis of the Stokes shift variations across the iM, iH, and iC compounds, as depicted in Fig. 6, reveals distinct structure–property relationships governed by both the geometry and the electronic nature of the substituents. The iM compounds consistently display the smallest Stokes shifts (typically <1.09 eV), indicating limited excited-state relaxation and a weak ICT character. In contrast, the iH and iC series exhibit significantly larger Stokes shifts (ranging from ∼1.1 eV to 2.1 eV), reflecting enhanced ICT and substantial excited-state reorganization. However, an exception is observed for 6H (R = –OMe) and 7H (R = –OMe), which exhibit unusually low Stokes shifts, which are even comparable to or lower than those of the iM series. This deviation in 6H and 7H can be ascribed to the dihedral angle φ₂ and the ground-state π-conjugation specific to the iH compounds bearing oxygen donors. In these molecules, the oxygen lone pair (n) engages in conjugation with the π system of the biphenyl rings (φ₁ ≈ 30°), which in turn reduces the dihedral angle between the o-carborane cage and the biphenyl to φ₂ ≈ 30°. This partial coplanarity enhances S₀ conjugation, thereby minimizing the structural and electronic reorganization required upon excitation, resulting in a smaller geometry relaxation and a markedly reduced Stokes shift. This deviation is absent in the iC analogues because the bulky TMS group on the cage enforces orthogonality (φ₂ ≈ 90°) through steric hindrance, suppressing n(O)-induced planarization, maintaining donor–acceptor decoupling in S₀, and enabling a larger ICT and geometry relaxation, thus restoring the large Stokes shifts for –OMe and –OH. In the iM compounds, the absence of a strong cage-mediated ICT pathway leads to uniformly small and relatively constant shifts across the substituents. On the other hand, nitrogen donors (–NH₂ and –NMe₂), despite possessing lone pairs, do not induce the same φ₂ collapse in iH due to differences in their lone-pair energy, orbital orientation, and steric effects; they preserve greater donor–acceptor separation in S₀ and thus undergo more pronounced ICT and relaxation upon excitation, yielding large Stokes shifts.

Fig. 6 Variation in Stokes shift (eV) as a function of substituent nature for the iM, iH, and iC compounds.

Linear and nonlinear optical parameters

The calculated static isotropic polarizability (α), polarizability anisotropy (Δα), first hyperpolarizability (β_vec and β₀), hyper-Rayleigh scattering (β_HRS), depolarization ratio (DR), dipolar contribution to the beta (φ_J=1) values and octupolar contribution to the beta (φ_J=3) values of the title compounds (Fig. 1) are presented in Table 3 and Fig. 7.

Table 3 Static isotropic polarizability (α), polarizability anisotropy (Δα), first hyperpolarizability (β_vec and β₀), hyper-Rayleigh scattering (β_HRS), depolarization ratio (DR), dipolar contribution to beta (φ_J=1) and octupolar contribution to beta (φ_J=3) calculated at the CAM-B3LYP/6-31G(d,p) level of theory in THF solvent (values in a.u.)

	α	Δα	β0	βvec	βHRS	DR	ϕJ=1	ϕJ=3
1M	273	272	415	409	166	5.685	0.595	0.405
1H	327	217	850	637	329	6.546	0.652	0.348
1C	394	203	828	545	322	6.341	0.639	0.361
2M	258	260	1255	−1254	590	3.555	0.446	0.554
2H	313	208	2005	1740	830	4.985	0.550	0.450
2C	381	194	2041	1561	842	5.069	0.555	0.445
3M	258	260	673	672	341	3.090	0.404	0.596
3H	313	208	1482	1355	603	5.299	0.570	0.430
3C	381	194	1493	1252	605	5.388	0.576	0.424
4M	276	280	1444	1444	680	3.556	0.446	0.554
4H	331	227	2188	2033	915	4.832	0.539	0.461
4C	398	213	2236	1918	930	4.919	0.545	0.455
5M	318	295	1378	1377	617	3.991	0.480	0.520
5H	372	240	2172	2021	884	5.318	0.571	0.429
5C	440	227	2200	1887	892	5.367	0.575	0.425
6M	283	294	3085	2205	1364	4.116	0.489	0.511
6H	337	238	3332	3299	1394	4.824	0.539	0.461
6C	406	228	3744	3090	1564	4.855	0.541	0.459
7M	267	280	2836	1522	1254	4.121	0.490	0.510
7H	321	224	3080	3056	1283	4.899	0.544	0.456
7C	390	214	3471	2758	1443	4.928	0.546	0.454
8M	278	306	5094	4675	2202	4.380	0.508	0.492
8H	333	253	5440	4958	2278	4.816	0.538	0.462
8C	401	240	5627	5260	2349	4.869	0.542	0.458
9M	314	339	7364	7335	3179	4.395	0.509	0.491
9H	369	286	7666	7336	3235	4.700	0.530	0.470
9C	437	274	7902	7316	3326	4.741	0.533	0.467

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Fig. 7 Variation in molecular dipole moment as a function of substituents (i = 1–9) for the iM, iH, and iC compounds.

Dipole moments and polarizability

Fig. 7 presents the ground-state dipole moments for the three molecular families, iM, iH, and iC, as a function of substituent (i = 1–9). For each substituent, the dipole moment increases systematically in the order of iM < iH < iC, indicating that the iC compounds produce the strongest permanent polarization, the iH family shows intermediate values, and the iM series displays the weakest dipolar character. Across the substituent series, the dipole moment follows a clear monotonic trend consistent with donor strength. Electron-withdrawing groups (–CF₃ and –F) yield the lowest dipole moments, followed by –H, –Me, –CMe₃, –OMe, and –OH, while the strongest electron-donating groups (–NH₂ and –NMe₂) produce the largest dipole moments, reaching values up to ∼12 debye for 9C. This ordering reflects the progressive enhancement in intramolecular charge separation as the donor ability increases. Furthermore, analysis of the dipole-moment components shows that µ_y ≈ µ_z ≈ 0 for all the molecules, whereas µ_x is distinctly non-zero, demonstrating that the permanent dipole is strictly aligned along the longitudinal molecular axis. This confirms that the substituent-induced charge redistribution is strongly oriented along the x-direction of these donor–acceptor dyads.

For the iM series of compounds (i = 1 to 9), which incorporate a trimethylsilylacetylene acceptor, the calculated average polarizability (α) values range from 258 to 318 a.u. Upon substitution with the o-carborane unit in the iH and iC series, a consistent enhancement in polarizability is observed. In particular, the iC compounds exhibit significantly higher (α) values compared to both their iH and iM analogs, reflecting the strong electron-accepting nature and high electronic delocalization capacity of the o-carborane cage in combination with the extended π-conjugation. The highest polarizability is found in compound 5C, reaching 440 a.u., followed closely by 9C (437 a.u.), 7C (406 a.u.), and 8C (401 a.u.). The corresponding iH derivatives show intermediate values (369 a.u. for 9H, 333 a.u. for 8H, and 321 a.u. for 7H), whereas their iM counterparts yield the lowest polarizabilities of 314 a.u. (9M), 278 a.u. (8M), and 267 a.u. (7M), respectively. This trend highlights the synergistic role of both π-extension and electron-rich donor groups in enhancing the electronic cloud responsiveness. Notably, the increase in isotropic polarizability (α) closely parallels the trend observed in the computed dipole moments, underscoring the role of molecular asymmetry and charge-transfer character in governing the overall electronic response. In contrast, the polarizability anisotropy (Δα) exhibits inverse behavior, where the iM derivatives display the largest anisotropy, while the iC derivatives show the smallest. This opposite trend indicates that, although the iC family possesses both higher polarizability and larger dipole moments, its electronic density is redistributed more uniformly, resulting in a comparatively isotropic response to an external electric field.

The static first hyperpolarizability

Regarding the static first hyperpolarizability, the title compounds exhibit a consistent and parallel trend for both β(0;0,0) and β⁰_HRS, with a near-perfect correlation between these parameters (R² = 0.999; see Fig. S3). Notably, the iC compounds display the highest values for both β(0;0,0) and β⁰_HRS compared to their iH and iM counterparts (see Fig. 8). For example, in the 1C–5C series, the static first hyperpolarizabilities are elevated by approximately 100%, 63%, 122%, 55%, and 60% relative to their corresponding iM counterparts (1M–5M), respectively. A more moderate increase is observed for compounds 6C and 7C, with β(0;0,0) values approximately 21% higher than those of 6M and 7M, respectively. In the case of 8C and 9C, the enhancement is more modest at around 9% over 8M and 9M.

Fig. 8 Static first hyperpolarizability (triangles: β(0;0,0) and stars: β⁰_HRS) calculated for the compounds iM, iH and iC (i = 0 to 9) at the CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory in THF.

These results clearly indicate that substitution of the –C≡C–SiMe₃ fragment by the o-carborane cage significantly enhances the second-order NLO response of the investigated molecules. Furthermore, a comparison between the iC and iH series reveals that β(0;0,0) for the iC compounds is generally ∼2% higher than that of their iH analogues, suggesting that the additional trimethylsilyl (TMS) group on the o-carborane cage exerts only a marginal influence on the first hyperpolarizability. However, notable exceptions are observed for compounds 6C and 7C, where their β(0;0,0) values increase by approximately 12% relative to that of 6H and 7H, respectively. This deviation can be attributed to geometric factors, particularly the φ₂ angle, which adopts a value of ∼30° for R = −OMe and –OH substituents. This analysis indicates that the incorporation of a bulky trimethylsilyl (TMS) group into the cage structure has a minimal effect on the molecular hyperpolarizability (β(0;0,0) and β⁰_HRS) of the iC compounds (6C and 7C are exceptions).

An in-depth analysis of the static first hyperpolarizability parameters (β(0;0,0) and β⁰_HRS) across the iM, iH, and iC non-centrosymmetric molecular series reveals a pronounced dependence on the electronic nature of the R substituent at the donor site. Among the studied molecules, compound 1M, featuring a strong electron-withdrawing –CF₃ group, exhibits the lowest first hyperpolarizability values (β(0;0,0) = 828 a.u. and β⁰_HRS = 322 a.u.). This trend persists consistently in the iH and iC analogues, suggesting that the –CF₃ substituent severely limits the efficiency of intramolecular charge transfer (ICT), thereby diminishing the overall NLO response.

Progressive substitution of –CF₃ with less electron-withdrawing or electron-donating groups results in substantial enhancements in first hyperpolarizability, following the trend of –H (3) < –F (2) < –Me (4) < –CMe₃ (5) < –OH (7) < –OMe (6) < –NH₂ (8) < –NMe₂ (9). Across the iM, iH, and iC families, the compounds with R = –OMe, –OH, –NH₂, and –NMe₂ exhibit markedly higher β values, reflecting the superior electron-donating capability of these substituents, which facilitates stronger ICT from the donor to acceptor moieties through the π-conjugated bridge.

Quantitatively, the β(0;0,0) values for compounds i = 2–7 are typically 2–5-times larger than those of compound 1 (R = –CF₃), while 8M and 9M show enhancements by factors of ∼12 and ∼18, respectively, relative to those of 1M. Similar enhancements are observed in compounds 8-9C and 8-9H, which exhibit first hyperpolarizabilities approximately 7- and 10-times greater than those of their corresponding 1C and 1H analogs, respectively. These results clearly demonstrate that electron-donating substituents significantly enhance the NLO response by increasing the electron density at the donor end, thereby promoting polarization across the D–A framework.

The diminished β values in the –CF₃-substituted compounds can be ascribed to the strong electron-withdrawing character of the –CF₃ group, which depletes electron density from the donor fragment and suppresses ICT in the A–A configuration. Conversely, electron-donating groups such as –NH₂ and –NMe₂ enable efficient push–pull electronic interactions in the D–A configuration, resulting in enhanced charge delocalization and elevated first hyperpolarizability. This substitution-driven modulation of electronic distribution highlights a powerful design strategy for tuning the NLO properties of conjugated molecular systems.

On the other hand, the β_vec analysis reveals a hierarchy of iH > iC > iM, with maximum values of 7316–7336 a.u. for the NMe₂ series. The o-carborane architectures significantly enhance the vectoral first hyperpolarizability compared to the linear acetylene fragment, while the naked o-carborane (iH) slightly outperforms the TMS analogue (iC), suggesting that the TMS group partially attenuates the directional charge transfer response. This parameter represents the projection of the first hyperpolarizability tensor onto the dipole moment µ, where positive values indicate alignment with the permanent dipole (angle <90°) and negative values reflect the antiparallel orientation typical of inverted donor–acceptor polarity.

Furthermore, the analysis of the depolarization ratio combined with the dipolar–octupolar decomposition of first hyperpolarizability (Table 3) shows that only the first members of each series (1M, 1H, and 1C) display a pronounced dipolar nonlinear response, characterized by high DR values (5.685–6.546) and dominant dipolar contributions (φ_J=1 > 0.59). In contrast, derivatives 2M–5M exhibit lower DR values (3.090–3.991) together with φ_J=3 > φ_J=1, indicating an octupolar-dominated response, with 3M being the most octupolar compound in the dataset (DR = 3.090). For the iH and iC families with i ≥ 2, the NLO behavior shifts toward a dipolar configuration, as reflected by their DR values close to 5 and moderately dominant dipolar contributions (φ_J=1 ≈ 0.53–0.58). It is also noteworthy that 2H (4.985) and 2C (5.069) exhibit DR values essentially coincident with the dipolar reference (DR ≈ 5), a feature typically associated with ideal dipolar systems.

Urea is widely regarded as a prototypical reference compound in the study of second-order nonlinear optical responses and is frequently utilized as a benchmark for comparative analysis.⁷³ In the present work, the first hyperpolarizabilities (β) of the iM, iH, and iC series were compared with that of urea (β_HRS^∞ = 38 a.u.).⁷⁴ The comparison reveals that all the investigated derivatives exhibit β_HRS^∞ values that are significantly larger than that of urea. For instance, the first hyperpolarizabilities of 5M, 5H, and 5C are approximately 16-, 23-, and 24-times greater, respectively, than that of urea. This highlights the markedly enhanced NLO responses of the iM, iH, and iC derivatives relative to the standard urea molecule, emphasizing the superior nonlinear optical characteristics of the title compounds. On the other hand, a quantitative comparison with the fullerene-based benchmarks reported by Muhammad et al.²⁴ and Fouejio et al.²⁵ reveals the pronounced superiority of the o-carborane derivatives investigated in this study. For instance, the first hyperpolarizability of title compound 9C surpasses those of compounds {2 and 2c}²⁴ and {fC₆₀ }²⁵ by multiplicative factors of 4, 37, and 20, respectively. These substantial enhancements demonstrate that the o-carborane cage furnishes a far more efficient three-dimensional electronic environment for intramolecular charge redistribution than the fullerene frameworks typically employed in nonlinear-optical design.

Frequency dependence

The absorption spectra of the investigated compounds, calculated in THF at the TD-CAM-B3LYP/6-31G(d,p) level of theory, are presented in Fig. 9 and S1 (in the SI). These spectra confirm the absence of an electronic absorption at 532 nm, indicating that two-photon absorption (2PA) is negligible when the molecules are excited at 1064 nm. Additionally, all the compounds exhibit a broad optical transparency window beginning at around 350 nm, ensuring that no significant linear absorption interferes with the high-resolution hyper-Rayleigh scattering measurements in the visible and near-infrared regions. Our calculations show that the maximum absorption wavelength (λ_max) corresponds to the S₀ → S₁ transition, predominantly of ICT character (Table 2). Based on these results, the frequency-dependent NLO properties, including dynamic polarizabilities and first hyperpolarizabilities, were computed at both non-resonant wavelengths (1064, 1340 and 1906 nm) and near-resonant conditions (556 nm, approximately twice the λ_Max).

Fig. 9 Dynamic first hyperpolarizabilities (β_SHG^λ and β_HRS^λ) at 1064, 1340 and 1906 nm for the iM, iH, and iC (i = 1–9) series at the CAM-B3LYP/6-31G(d,p) level of theory.

The calculated dynamic polarizability (α) and polarizability anisotropy (Δα) values for the iM, iH, and iC compounds exhibit a consistent trend of iC > iH > iM across all wavelengths (1906–556 nm), reflecting the increasing electronic polarizability induced by the o-carborane substitution (Table S8). For all series, both (α) and Δα show a modest increase with decreasing wavelength, which is attributed to mild dispersion effects under non-resonant conditions. The relatively small variation in values confirms that their optical response remains stable across the infrared-to-visible range and is primarily governed by their ground-state electronic structure rather than frequency-dependent effects.

On the other hand, frequency-dependent NLO calculations were carried out to examine three important first hyperpolarizability coefficients: the electro-optical Pockels effect (β_EOPEλ) β(−ω;ω,0), second harmonic generation (β_SHG^λ) β(−2ω;ω,ω), and hyper-Rayleigh scattering (β_HRS^λ). Our findings demonstrate a strong correlation between β_HRS^λ and β_SHG^λ; additionally, we observe a robust linear relationship between the static (λ = ∞) and dynamic first hyperpolarizability (β_SHG^λ ↔ β_SHG^∞ and β_HRS^λ ↔ β_HRS^∞, where R² = 0.995) at λ = 1064, 1340, 1906 nm, as shown in Fig. S4 in the SI.

The dynamic first hyperpolarizability (β_SHG^λ and β_HRS^λ) exhibits a pronounced increase as the incident wavelength is shortened, with a gradual increase observed from 1906 to 556 nm (Fig. 9). Furthermore, in the static regime, the first hyperpolarizability displays comparatively lower values than those recorded in the dynamic regime. Notably, a significant enhancement in the β_SHG^λ and β_HRS^λ values is observed at the incident wavelength of 556 nm, where the sharp increase is attributed to the resonant two-photon absorption process. This resonance occurs at the characteristic wavelength (λ_max) leading to substantial amplification of the nonlinear optical response (Table 4).

Table 4 Calculated dynamic first hyperpolarizabilities (β_EOPE^λ, β_SHG^λ and β_HRS^λ) at 1064 and 556 nm for the iM, iH, and iC (i = 1–9) non-centrosymmetric molecular series at the CAM-B3LYP/6-31G(d,p) level of theory

	λ (nm)	βEOPE^λ	βSHG^λ	βHRS^λ
1M	1064	462	744	293
	556	959	26 937	11 144
1H	1064	773	827	322
	556	956	2803	1129
1C	1064	716	750	293
	556	892	2823	1138
2M	1064	1181	1347	616
	556	1575	12 940	5475
2H	1064	1895	2183	904
	556	2558	10 723	4458
2C	1064	1898	2185	904
	556	2587	11 717	4870
3M	1064	617	690	337
	556	811	5488	2348
3H	1064	1384	1602	655
	556	1898	8076	3331
3C	1064	1358	1568	641
	556	1886	8680	3581
4M	1064	1387	1670	756
	556	1991	28 245	11 808
4H	1064	2098	2495	1040
	556	2949	15 575	6463
4C	1064	2110	2512	1046
	556	3000	17 350	7197
5M	1064	1368	1698	742
	556	2012	32 418	13 514
5H	1064	2139	2604	1064
	556	3059	17 349	7174
5C	1064	2132	2598	1062
	556	3082	18 842	7792
6M	1064	3023	3796	1642
	556	4590	209 096	86 890
6H	1064	3251	3982	1660
	556	4739	36 712	15 252
6C	1064	3614	4446	1854
	556	5323	48 832	20 280
7M	1064	2739	3374	1461
	556	4074	116 244	48 356
7H	1064	2964	3565	1482
	556	4241	28 399	11 798
7C	1064	3310	3988	1658
	556	3046	37 084	15 403
8M	1064	4978	6436	2739
	556	7963	180 941	74 927
8H	1064	5296	6647	2771
	556	8066	252 257	104 624
8C	1064	5457	6884	2866
	556	8399	503 536	208 779
9M	1064	7460	10 271	4360
	556	12 894	114 154	47 146
9H	1064	7209	9613	4027
	556	11 818	201 648	83 520
9C	1064	7972	10 721	4486
	556	13 245	154 910	64 118

---

On the other hand, DFT calculations indicate that the magnitude of β(−ω; ω, 0) slightly increases as the incident light wavelength decreases, following the trend of β_EOPE⁵⁵⁶ > β_EOPE¹⁰⁶⁴ > β_EOPE¹³⁴⁰ β > β_EOPE¹⁹⁰⁷ (Fig. 10). Additionally, the β(−ω; ω, 0) values show a sequential increase from the iM to iH to iC compounds, except for compound 9, where β(−ω; ω, 0) [9C] > β(−ω; ω, 0) [9M] > β(−ω; ω, 0) [9H]. This trend mirrors the order observed in the dynamic first hyperpolarizability. Overall, stronger NLO responses are seen at shorter wavelengths, with the iC compounds outperforming both the iH and iM compounds in NLO responses.

Fig. 10 Dynamic first hyperpolarizabilities (β_EOPE^λ) at 556, 1064, 1340 and 1906 nm for the iM, iH, and iC (i = 1–9) series at the CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory in THF.

Frequency dispersion factor

The frequency dispersion factor for hyper-Rayleigh scattering (β_HRS) in tetrahydrofuran was systematically calculated and presented in Fig. 11. At wavelengths of 1064, 1340, and 1906 nm, the frequency dispersion factor (FDF) is approximately unity. This value indicates that the intensity of the incident light has no significant impact on the NLO response of the title compounds. This phenomenon arises from the mismatch between the excitation frequencies and the intrinsic electronic transitions of the molecular system.

Fig. 11 Variation in the frequency dispersion factor of iM, iH and iC (i = 1 to 9) calculated at the CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory in THF.

In contrast, as illustrated in Fig. 11, compounds 8H, 1M, 6M, 7M, and 8M display elevated FDF values at 556 nm (2.229 eV), corresponding to 102, 67, 64, 39, and 34, respectively. These values signify strong proximity to resonance conditions. The first resonance energy, defined as half of the first excitation energy, was determined for each compound, as follows: 8H (4.534 eV), 1M (4.614 eV), 6M (4.510 eV), 7M (4.537 eV), and 8M (4.355 eV). On the other hand, compounds 1H, 1C, 2H, 2C, 3H, and 3C exhibit minimal FDF⁵⁵⁶ values, ranging between 3 and 6, which indicates their positioning within the off-resonance regime, where dispersion effects are negligible. Additionally, compounds iC and iH (where i = 1 to 7) display moderate FDF values, generally below 12. These results reflect the modest influence of incident wavelength on the NLO response, suggesting that these compounds are approaching resonance conditions at 556 nm. Overall, it can be observed that the iH and iC compounds exhibit FDF⁵⁵⁶ values that are similar in magnitude and significantly smaller compared to those of the iM molecules.

Two-level model analysis of first hyperpolarizability

The sum-over-states (SOS) method represents one of the prevailing approaches for theoretically estimating first hyperpolarizability.^75–80 The comprehensive expression for the SOS formula pertaining to the component denoted as ABC of first hyperpolarizability is precisely defined as follows:where

The symbol ω represents the energy associated with external fields. Δ_i signifies the excitation energy of state i in relation to the ground state (0). P̂ represents the permutation operator, which is responsible for the manipulation of the xyz indices within the β-components. Furthermore, µ_ij^x denotes the x-component of the transition dipole moment, characterizing the transition between states i and j.

The molecular NLO responses of the compounds exhibit a profound connection to their electronic absorption characteristics. This intricate relationship was elucidated through the application of the well-established two-level model developed by Oudar and Chemla. Within this theoretical framework, one can articulate the static hyperpolarizabilities with precision and relevance.^81,82

In the context of the specified variables, where ΔE represents the excitation energy (S₀ → S_n), f stands for the oscillator strength, and Δµ signifies the difference in dipole moment between the ground state (S₀) and the excited state (S_n), a critical relationship emerges. According to this relationship, a clear and noteworthy trend is observed that the static first hyperpolarizability (β₀) displays inverse proportionality to ΔE³. This observation underscores the pivotal role played by lower excitation energies (ΔE_0→nn), larger dipole moments (Δµ_0→n), and higher oscillator strengths (f_0→n) in generating the highest values of β₀ for materials.

It is well established that the molecular NLO response, as evaluated via the SOS method, is significantly influenced by the number of electronically excited states incorporated into the calculation. In this context, we conducted a systematic investigation into the dependence of the first hyperpolarizability (β_SOS) on the number of excited states considered, with particular emphasis on the inclusion of up to 120 excited states, as illustrated in Fig. S5. The resulting data clearly demonstrate that the β_SOS values exhibit pronounced convergence behavior upon the incorporation of 120 excited states, thereby validating the adequacy of this level of state inclusion for accurate and reliable first hyperpolarizability estimations.

As illustrated in Fig. 12, the β_SOS values exhibit qualitative agreement with the overall trends observed for the β₀ values obtained at the CAM-B3LYP level of theory. However, notable discrepancies are observed, which are primarily attributed to the inherent approximations and limitations of the SOS methodology. Specifically, the SOS approach consistently underestimates the first hyperpolarizability magnitudes relative to those calculated via the CAM-B3LYP functional.^56,83 Additionally, a two-level model analysis reveals that the first excited state exerts (S₀ → S₁) a predominant influence on the first hyperpolarizability values of the studied compounds iM, iH, and iC (i = 1 to 9). This predominant excited state is distinguished by a pronounced dipole moment variation relative to the ground state (Δµ) and a markedly elevated oscillator strength (f), parameters that significantly enhance its contribution to the overall NLO response.

Fig. 12 Computed static first hyperpolarizabilities of the title compounds using various computational methods.

The analysis of the parameters associated with the first excited state reveals insightful trends related to the electronic behavior of the studied molecules. Notably, a strong inverse linear relationship is observed between the first hyperpolarizability and the net number of electrons transferred from fragment 1 to fragment 3 (net 1 → 3) across the iM, iH, and iC series (Fig. 13). The high regression coefficients (R² = 0.985, 0.977 and 0.989 for iM, iH and iC, respectively) confirm the robustness of this correlation within each molecular class. This trend suggests that as the extent of electron transfer toward Frag-3 increases (more negative net 1 → 3), the first hyperpolarizability value significantly increases, implying enhanced ICT. This enhancement in ICT contributes to a greater asymmetry in electron distribution, a key factor in boosting the second-order NLO response. Notably, the iC compounds display the highest net electron transfer values (net 1 → 3), followed by iH and iM. This consistent trend across all the series highlights the key role of fragment-based charge redistribution in tuning NLO properties. Thus, the direction and magnitude of electron flow emerge as effective predictors for designing efficient D–A-type o-carborane-based NLO chromophores.

Fig. 13 Correlation between first hyperpolarizability and net electron transfer between fragments (1 → 3) for the iM, iH and iC compounds (i = 1 to 9) of the first excited state (S₀ → S₁).

On the other hand, in all the investigated compounds, the Coulomb attractive energy (E_CA) associated with the first excited state consistently exceeds the corresponding vertical excitation energy (see Tables 2 and S4), emphasizing the critical role of electron–hole Coulomb interactions in stabilizing the excited-state manifold.^84,85 The magnitude of E_CA provides a direct measure of electron–hole binding strength and the degree of charge localization within the excited state. As depicted in Fig. 14, an unambiguous inverse correlation is observed between E_CA and the computed first hyperpolarizability, indicating that enhanced coulombic attraction tends to suppress charge delocalization and intramolecular polarization, thereby diminishing the NLO response. These observations underscore the fundamental interplay between electronic structure and NLO properties and highlight the importance of modulating excitonic interactions to optimize molecular first hyperpolarizability in the design of advanced optoelectronic materials. Furthermore, a clear linear relationship was observed between the first hyperpolarizability and the dipole moment variation relative to the ground state upon first excitation (Fig. S6). These results are in good agreement with previous reports, which also identified Δµ as a key electronic descriptor for predicting first hyperpolarizability in compounds.^56,86

Fig. 14 Correlation between the Coulomb attractive energy (E_CA) of first excited state and the first hyperpolarizability for the iM, iH, and iC non-centrosymmetric molecular series.

Our results confirm the expected trend that the first hyperpolarizability of the iM, iH, and iC compounds reaches its highest values when Δµ_0–1 and f_0–1 are simultaneously maximized, while the transition energy (ΔE_0–1) and the Coulomb attractive energy (E_CA) of the first excited state are minimized. This combined electronic profile promotes efficient intramolecular charge transfer and enhances the overall nonlinear optical response of the systems under study.

To elucidate the electronic factors influencing NLO activity, compounds 1C and 9C were selected as representative case studies. Compound 9C exhibits a significantly higher static first hyperpolarizability (β₀ = 7902 a.u. [β_HRS = 3326 a.u.]) compared to compound 1C ((β₀ = 828 a.u.) [(β_HRS = 322 a.u.)]), corresponding to a nearly nine-fold enhancement in β₀. This pronounced increase in NLO response for 9C is attributed to its larger dipole moment (Δµ_0–1 = 5.003 a.u.), lower vertical excitation energy (ΔE_0–1 = 4.187 eV), and reduced Coulomb attractive energy (E_CA = 4.764 eV). In contrast, compound 1C is characterized by a markedly smaller dipole moment change (Δµ_0–1 = 0.453 a.u.) and higher values of ΔE_0–1 (5.02 eV) and E_CA (5.450 eV). The results indicate that excellent nonlinear optical materials are characterized by lower excitation energies, significant dipole moment variations in the first excited state, and reduced Coulomb attractive energy values (Fig. 14). These observations highlight E_CA as a novel and promising parameter for predicting and optimizing the NLO properties of materials.

Photoluminescence and second-order NLO response

To predict experimentally accessible parameters that correlate with first hyperpolarizability, we investigated the relationship between the photoluminescence quantum yield (Φ_em)⁴⁹ and the calculated first hyperpolarizability for iC non-centrosymmetric compounds (i = 2 to 6). As shown in Fig. 15, a strong linear correlation (R² = 0.926) is observed, revealing that Φ_em can serve as a predictive indicator for first hyperpolarizability in D–A-type o-carboranyl luminophores. The resulting regression equation, β_HRS = 603 + 2238Φ_em, quantitatively describes this correlation and can be employed to estimate second-order NLO responses based on experimental emission efficiencies.

Fig. 15 Linear correlation between the photoluminescence quantum yield Φ_em (Φ_em value from ref. 49) and static first hyperpolarizability (β_HRS) for the iC non-centrosymmetric compounds (i = 2 to 6).

The unsubstituted compound 3H (R = –H) exhibits the lowest photoluminescence quantum yield (Φ_em)⁴⁹ and first hyperpolarizability (β) values, indicating its inefficient intramolecular charge transfer (ICT) behavior and minimal charge redistribution upon excitation. As shown in Fig. 5 and Table S6, the net transferred electron density between molecular fragments in this derivative is nearly negligible, consistent with its weak NLO response.

Substitution with a fluorine atom (R = –F) induces a slight increase in both (Φ_em) and β_HRS. This limited enhancement can be attributed to the strong inductive electron-withdrawing effect of fluorine, which mildly perturbs the electronic distribution and stabilizes the excited state (see Fig. 5 and Table 2). However, the absence of a mesomeric effect and the inherently low polarizability of –F restrict its ability to promote effective ICT.

On the other hand, the introduction of electron-donating alkyl groups (–Me and –CMe₃) leads to moderate improvements in both (Φ_em) and β_HRS (Fig. 15). These substituents increase the electronic density within the π-system through hyperconjugative and inductive effects, facilitating partial ICT. The corresponding increase in ICT, particularly between fragments 1 → 3 and 2 → 3 (as evidenced in Fig. 6 and Table S6-7), supports a greater degree of polarization in the excited state.

Notably, the methoxy-substituted derivative (R = –OMe) displays the highest (Φ_em) and β_HRS (0.45 and 1564 a.u., respectively) among the studied compounds (Fig. 15). This pronounced enhancement is attributed to the strong +M mesomeric effect of the –OMe group, which significantly increases the extent of ICT. Quantitative analysis reveals the highest net electron transfer between the donor and acceptor fragments for this compound, substantially exceeding those observed for –H, –F, –Me, and –CM₃. The resulting increase in Δµ (variation in dipole moment with respect to the ground state) reinforces both the photoluminescence quantum yield (Φ_em) and first hyperpolarizability.

Collectively, these findings confirm that increasing the electron-donating strength of the R substituent enhances the ICT efficiency, which in turn amplifies both the photoluminescence and 2nd-order NLO responses of non-centrosymmetric compounds. The strong linear correlation (Fig. 15) established between Φ_em and β_HRS underscores the feasibility of using photoluminescence parameters as predictive descriptors of the NLO performance of π-conjugated non-centrosymmetric molecules.

Conclusions

A comprehensive computational investigation was carried out in three non-centrosymmetric molecular series, iM, iH, and iC, at the CAM-B3LYP/6-31G(d,p) level of theory using the IEFPCM solvation model in THF. These series differ in the electron-accepting core, with trimethylsilyl-acetylene in iM, an o-carborane cage in iH, and a trimethylsilyl-functionalized o-carborane cage in iC, and incorporate various donor substituents (R = –CF₃, –F, –H, –Me, –tBu, –OMe, –OH, –NH₂, and –NMe₂). Their ground- and excited-state geometries, absorption and emission characteristics, intrafragment charge transfer, and first- and second-order NLO responses were systematically analyzed. The iC series consistently showed an enhanced dipole moment variation, oscillator strength (f), Coulomb attractive energy (E_CA), net electron transfer between the substituent and the o-carborane cage, Stokes shift, and NLO activity compared to the iH analogues, while the iM series exhibited the lowest values. Across all series, these properties increased with the electron-donor strength of the substituent, with the exception of 6H (–OMe) and 7H (–OH) in the iH series, which showed a marked decrease in Stokes shift due to their ground-state conjugation induced by the oxygen lone pair and reduced dihedral angle, φ₂. Strong correlations were found between the first hyperpolarizability and both the net electron transfer between fragments (1 → 3) and E_CA, as well as between the photoluminescence quantum yield (Φ_em) and β for the iC derivatives (i = 2–6). The o-carborane derivatives bearing –NMe₂ substituents exhibit the largest first hyperpolarizabilities in this study and thus appear suitable for future development as second-order NLO materials. These results validate that enhancing the electron-donating ability of the R substituent improves the efficiency of intramolecular charge transfer, thus boosting both the photoluminescence and second-order nonlinear optical responses in o-carborane-based non-centrosymmetric molecules.

Author contributions

Conceptualization and methodology: D. H. and H. C.; investigation: D. S., M. Z., D. H., O. A. and G. H.; writing – original draft preparation: D. H. and H. C.; writing – review and editing: D. S., M. Z., D. H., G. H. and H. C.; and data curation: D. S., M. Z., D. H., A. O., G. H. and H. C. All authors have read and agreed to the published version of the manuscript.

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information (SI): calculated static polarizability and its anisotropy polarizability and first hyperpolarizability of 1C at CAM-B3LYP/basis sets/IEFPCM level in THF; TD-CAM-B3LYP/6-31G(d,p)) absorption (abs), emission (em) wavelengths and transition dipole moment (µ_0→1) for compounds iM, iH and iC; experimental values are given in parentheses from the work of Dong Kyun You et al.; total energy (E, a.u.), HOMO and LUMO energies (eV) of the ground state (S₀), first excited state (S₁) wavelength (nm), and oscillator strength of the emission transition S₁ → S₀ for compounds iM, iH, and iC, calculated at the CAM-B3LYP/6-31G(d,p)/IEFPCM level in THF; TD-DFT calculated excited-state (S_n) parameters of the title compounds: transition energies (eV), wavelengths (nm), oscillator strengths, and electronic configurations, calculated at the CAM-B3LYP/6-31G(d,p)/IEFPCM level in THF; calculated overlap (S_r), D index (Å), H index (Å), variation of dipole moment with respect to ground state (Δµ, a.u.), hole delocalization index (HDI), electron delocalization index (EDI), ghost-hunter index 1st and 2nd, Coulomb attractive energy (ECA) for iM, iH and iC compounds in the first excited state at the CAM-B3LYP/6-31G(d,p)/IEFPCM level; contribution of each fragment to hole and electron (I), variation of population number of fragment (II) and intrafragment electron redistribution of fragment (III) calculated in tetrahydrofuranat the CAM-B3LYP/6-31G(d,p)/IEFPCM level; intrinsic charge transfer percentage, CT (%), intrinsic local excitation percentage, LE (%) and transferred electrons between fragments, calculated for the electronic transition S_0→i of the iM, iH and iC compounds at the CAM-B3LYP/6-31G(d,p)/IEFPCM level in tetrahydrofuran; calculated dynamic polarizability 〈α〉 and polarizability anisotropy (Δα); simulated absorption spectrum of the title compounds; CDD for different excited states of the title compounds. CDD was calculated as a difference between the corresponding excited state and the ground state of the considered system using the CAM-B3LYP/6-31G(d,p)/IEFPCM level of theory; correlation between static first hyperpolarizability β_HRS and (0;0,0) for iM, iH and iC compounds (i = 1 to 9), calculated at the CAM-B3LYP/6-31G(d,p)/IEFPCM level in THF; correlation between dynamic and static first hyperpolarizability β_HRS for iM, iH and iC compounds (i = 1 to 9), calculated at the CAM-B3LYP/6-31G(d,p)/IEFPCM level in THF; plots of static first hyperpolarizability values as computed in the SOS formalism as a function of the number of excited states for for iM, iH and iC compounds (i = 1 to 9), calculated at the CAM-B3LYP/6-31G(d,p)/IEFPCM level in THF; linear correlation between first hyperpolarizability and dipole Moment Variation (Δµ) in the first excited state for iM, iH and iC (i = 1 to 9) compounds. See DOI: https://doi.org/10.1039/d6ra00681g.

Acknowledgements

The authors gratefully acknowledge the GENCI/IDRIS for the HPC resources/computer time through the grant AD010814932 on the supercomputer Jean Zay and the PSMN of the ENS-Lyon for the computing resources.

Notes and references

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