1 Introduction

In recent years, there has been an increasing interest in developing organic materials (OM) with nonlinear optical (NLO) responses, such as second harmonic generation (SHG), aiming for photonics applications, e.g., optoelectronics and optical communications [1,2,3]. Through the engineering of molecular structures, OM exhibiting linear and NLO properties have been optimized for the development of photonics crystals [4,5,6,7,8,9,10,11,12].

Indeed, the search for high performance NLO materials, organic or inorganic, has grown significantly as the need for faster, reliable and efficient optical devices increases [3, 13]. Aiming photonics applications, these materials should exhibit ideally high NLO responses, good mechanical, chemical and thermal stabilities, fast response time, good optical transparency, and high damage threshold [14]. Low dimension materials [15], metal-organic frameworks [16], inorganic crystals [17], and transparent conducting oxides [18] are examples of classes of materials that have been recently reported with outstanding NLO responses. Nevertheless, owing to their structural versatility and large NLO responses, OM are still amongst the most promising systems aiming the development specific applications based on NLO [19,20,21]. Moreover, OM can be also optimized by incorporating electrons donors (D) and acceptors (A) in a push-pull configuration, which improves the charge transfer and enhances their NLO responses [22].

There are two primary criteria for an OM to become a potential photonics crystal with a robust SHG response: (a) it should exhibit large first-order molecular hyperpolarizability, and (b) it should possess a noncentrosymmetric crystalline structure and phase-match to optimize the efficiency of light conversion. The former is inherently linked to the compound’s molecular structure and electrical polarization properties, whereas the latter pertains mainly to the molecular packing within the solid material. Experimental and theoretical techniques are available and well consolidated to evaluate the first criterion. For example, hyper-Rayleigh scattering (HRS) [23] and electric field-induced second harmonic (ESFISH) [24] are the two most widely used experimental techniques to evaluate the first-order molecular hyperpolarizability in OM. On the other hand, quantum-chemical calculations (QCC) have been widely used to predict the first-order molecular hyperpolarizability of single molecules [25,26,27,28,29]. The second criterion is the most costly process, as the mass of the final product of the organic synthesis must be at least tens of grams for efficient microcrystal growth and its respective NLO characterization [17, 30,31,32,33]. Even with relatively low-cost production and versatile synthesis techniques, such as Schiff bases and solution growth [34,35,36], the amount of mass needed to produce a final cut phase-match crystal with a couple of centimeters could be inviable costly in terms of synthesis final product just for testing the SHG efficiency. Consequently, extensive research has been dedicated to discovering novel molecular structures that could be used to build efficient SHG bulk materials, mainly by predicting the single molecules’s first-order molecular hyperpolarizability magnitude via QCC.

Empirical studies, especially those that measure phase-matching and SHG efficiency in bulk crystals, are the gold standard for validating the performance of photonic crystals. However, QCC are invaluable during the initial stages of developing new OMs with NLO responses, complementing the experimental approaches in different ways. For instance, these theoretical approaches offer a cost-effective and versatile means to explore a wide array of molecular structures and allow for rapid hypothesis testing and optimization before committing to the more expensive and time-consuming empirical validation processes. In this sense, theoretical results can be used as a guide to the experimentalists, identifying the most promising systems to be synthesized and investigated. Moreover, by leveraging various computational frameworks, such as ab initio methods, including Hartree-Fock (HF) [37, 38], semiempirical (SE) Parametric Methods (PM) [39,40,41,42,43,44,45,46,47,48,49], and the computationally more demanding but robust density functional theory (DFT) [25, 50,51,52], researchers can draw correlations between molecular structure, dipole moment, and hyperpolarizabilities. These are crucial information that allow a deep understating and accurate interpretation of the experimental results. In particular, the computation of the first-order molecular hyperpolarizability is particularly telling; this value is the microscopic analog of SHG [53] and, therefore, can predict, to some degree, the SHG potential in its crystalline form. Numerous studies demonstrate the significant impact of the -NO2 group in enhancing first-order hyperpolarizability (NLO) properties, such as [54,55,56,57,58,59]. These calculated parameters provide predictive values for novel materials. Over the years, such theoretical methodology has enhanced our understanding of OM’s NLO behavior [40, 42, 43, 60].

Amongst the myriad classes of OM identified for SHG applications, benzophenones (BP) and benzophenone hydrazones (BPH) derivatives are particularly noteworthy. These compounds exhibit exceptional optical transparency above 400 nm and a natural tendency towards noncentrosymmetric crystallization, attributes crucial for efficient SHG, as highlighted in the literature [61,62,63,64,65,66,67,68,69]. The fundamental structures of unsubstituted BP and BPH share a common architecture, comprising two aromatic rings, each attached to distinct functional groups; BP features a carbonyl group (C = O), whereas BPH is characterized by a hydrazone group (C = N-NH2). Em nosso trabalho, buscamos identificar alguns destes materiais promissores, otimizando as respostas ópticas não lineares de sistemas já conhecidos por este potencial.

For instance, Arivanandhan et al. [70] and Sankaranarayanan et al. [71] reported the SHG efficiency of the grown BP-0 crystal, experimentally measured by the Kurtz and Perry powder method [72]. The result showed that this material has SHG efficiency nearly three times larger than the potassium dihydrogen phosphate (KDP) and optical transparency with short cut-off wavelengths UV region (< 400 nm). It is also known that second [35, 64, 65] and third-order [73, 74] NLO responses of BP-0 can be optimized by adding electron donors, such as -Cl, -Br, -F, -NH2, -N(CH3)2, -CH3, -OCH3, and -OH in the phenyl ring. Nevertheless, due to the extensive array of molecular modification options, there is space for the NLO response enhancement. On the other hand, Babu et al. [36] verified experimentally that the crystalline BPH-0 presents SHG efficiency ten times larger than KDP and excellent optical transparency with short cut-off wavelengths in the UV region. The SHG response reported is associated with a one-dimensional charge transfer process between NH2 and the phenyl group. In addition, first-order molecular hyperpolarizability values were calculated using ab-initio theoretical framework [75].

The theoretical study carried out by Hong et al. [76] showed the similarities between the molecular structures of BP and BPH, providing new information about the vibrational spectrum and molecular parameters. However, the authors did not describe or investigate any NLO properties nor explore new BPH derivatives.

Owing to their abovementioned remarkable chemical and physical properties, BP and BPH are two classes of close related chemical compounds with a still underexploited potential for the development of photonic applications. In this sense, pursuing novel BP and BPH derivatives is not just an academic endeavor but a steppingstone toward next-generation NLO materials. Therefore, this work reports an estimated first-order molecular hyperpolarizability in gas-phase (\(\beta\)) of 185 compounds (60 BP and 125 BPH derivatives). The dynamic first-order molecular hyperpolarizability \({( {\beta }_{2\omega })}\) is deemed essential once it retrieves the \(\beta\) value when an external electromagnetic (EM) field is applied. The absence of an EM will retrieve the static first-order molecular hyperpolarizability (\({\beta }_{0}\)). The frequency of the EM simulated in this work corresponds to the wavelength of 1064 nm, commonly used in experimental techniques, such as the Kurtz and Perry powder method [72]. Both \({\beta }_{0}\) and \({\beta }_{2\omega }\) values were acquired by employing two theoretical frameworks, the SE method using the PM7 [77] and DFT using the CAM-B3LYP [78] levels of theory. By establishing this theoretical groundwork, we aim to enhance molecular engineering efforts to optimize the experimental procedures, such as synthesis, crystal growth, and NLO characterization, which have high experimental costs in exploring extensive molecular configurations [22, 53, 79,80,81].

2 Compounds

Figure 1 delineates the strategic modifications applied to the BP and BPH molecular structures to augment their NLO responses. Specifically, these strategies include the introduction of electron-donating and -accepting groups at para positions on both mono- and disubstituted aromatic rings [82,83,84]. Various electron-donating groups have been explored, such as methyl (-CH3), methoxy (-OCH₃), and different amines, including dimethylamine (-N(CH3)2), methylamine (-NHCH₃), ethylamine (-NHC₂H₅), and amine (-NH₂). Additionally, halide groups like chlorine (-Cl), bromine (-Br), and fluorine (-F), along with the electron-accepting nitro group (-NO2), were considered. We also assessed the influence of substituting the carbonyl functional group (C = O) with a hydrazone core (C = N-NH2). Through these structural modifications, we aimed to identify the higher \({\beta }_{2\omega }\) values, revealing the most suitable compounds to become severe candidates for photonics crystal-enhanced SHG response.

Fig. 1
figure 1

Schematic molecular structure of (a) BP and (b) BPH

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Within our investigation’s scope, the unsubstituted BP molecular structure is denoted as BP-0, with both the R1 and R2 positions occupied by hydrogen atoms explicitly indicated as BP-0 (R1 = H, R2 = H). Similarly, the structure of unsubstituted BPH is labeled as BPH-0 (R1 = H, R2 = H). The nomenclature for the derivatives of these molecules follows an organized scheme, with BP-i (where i ranges from 1 to 65) and BPH-j (where j spans from 1 to 120). The specific nature of the substituents at the R1 and R2 positions for each derivative, i and j, can be found detailed in Tables A-1 and A-3, respectively, in the Supporting Material (SM). This systematic approach to naming ensures clarity and precision in identifying and discussing each compound’s NLO properties throughout our study.

3 Computational methodologies

Several QCC methods are available for predicting \(\varvec{\beta }\)values, each offering varying degrees of reliability. As mentioned before, this study employs two theoretical frameworks, SE and DFT, using two distinct software packages, MOPAC 2016 [85] and Gaussian 16 (rev. B.01) [86], respectively. Therefore, henceforth, all the values calculated with MOPAC, using SE approach with PM7 level of theory, will have in superscript the letters PM7, e.g., the dynamic first-order molecular hyperpolarizability will be denoted as \({\varvec{\beta }}_{2\varvec{\omega }}^{\varvec{P}\varvec{M}7}\). On the other hand, all the parameters calculated with Gaussian software, which uses DFT with CAM-B3LYP/6-311 + + G(2d, dp) will have the CAM letter in the superscript, e.g., \({\varvec{\beta }}_{2\varvec{\omega }}^{\varvec{C}\varvec{A}\varvec{M}}\).

On the other hand, Gaussian 16 offers a comprehensive suite of advanced quantum chemistry functions, including DFT, which provides a more rigorous treatment of electron correlation effects compared to semi-empirical methods. Although at a higher computational cost, gaussian 16 can deliver highly accurate molecular-level calculations, including geometric optimizations, electronic transitions, and NLO properties. In this work, we have employed time-dependent DFT with the CAM-B3LYP functional and extensive basis sets, such as 6-311 + + G(2d,2p); this software package allows for a detailed and accurate assessment of the NLO properties, closely matching experimental data, as reported in several works [25, 29, 87,88,89,90]. Thus, while MOPAC provides a faster, more cost-effective approach, Gaussian 16 is used to obtain high-precision predictions of the NLO responses of OM.

The computational methodology employed to analyze the NLO properties of the investigated OM can be summarized in four main steps: (1) Molecular structure drawing and pre-optimization: Initially, molecular structures were sketched using chemical structure editors and pre-optimized using the Avogadro software [91], ensuring the molecules have a reasonable conformation before detailed QCC; (2) Geometry optimization: Following pre-optimization, a comprehensive structural optimization was performed using two quantum chemistry packages: MOPAC 2016 for SE and Gaussian 16 for DFT calculations; (3) Calculating permanent dipole moments and first-order molecular hyperpolarizabilities: The \(\varvec{\mu }\), \({\varvec{\beta }}_{0}\), and \({\varvec{\beta }}_{2\varvec{\omega }}\) were calculated in the gas phase using both PM7 and CAM-B3LYP/6-311 + + G(2d,2p) levels of theory; (4) Evaluation of the third-order rank tensor (χ²): The core of our NLO property analysis is the assessment of \({\varvec{\chi }}^{2}(2\varvec{\omega };\varvec{\omega }, \varvec{\omega })\), a third-order rank tensor, in which all components must be considered to predict \({\varvec{\beta }}_{0}\)and \({\varvec{\beta }}_{2\varvec{\omega }}\). In step 3, by calculating \({\varvec{\beta }}_{0}\), and \({\varvec{\beta }}_{2\varvec{\omega }}\), both MOPAC and Gaussian return the third-order rank components, which will be analyzed. Such analysis is done by Hyper-QCC v.1.3.2 [92], a free-distributed post-processing software (photonicsresearchgroup.org/hyper-qcc), which interprets the output log files of Gaussian and MOPAC. This methodology allows the extraction of the tensor components rapidly and reliably to consolidate the final \({\varvec{\beta }}_{0}\) and \({\varvec{\beta }}_{2\varvec{\omega }}\) values.

Further details on the computational methodology employed for calculating first-order molecular hyperpolarizabilities using MOPAC are available in [40, 48]. In contrast, a comprehensive description using Gaussian 16 and Hyper-QCC for this purpose can be found in [25, 53].

4 Results and discussion

We have compared 185 values of \({\beta }_{2\omega }^{PM7}\) with \({\beta }_{2\omega }^{CAM}\), which corresponds to the 65 different configurations of the BP’s molecular base structure plus 120 BPH’s molecular BS configurations. In ascending order, such configurations were based on the donation strength nature of the selected substituents, shown in Table 1. The BP and BPH base structures feature a central carbonyl group (C = O) and hydrazone core (C = N-NH2), bifurcating the conjugated system and delineating two relatively independent molecule segments. The BP and BPH base structures feature a central carbonyl group (C = O) and hydrazone core (C = N-NH2), bifurcating the conjugated system and delineating two relatively independent molecule segments. Consequently, substituents added in either the R1 or R2 position of the phenyl rings, as depicted in Fig. 1, allow us to understand the \(\mu\) and \(\beta\) values changes. BPH allows for a wider array of spatial arrangements, each potentially uniquely influencing the \(\mu\) or the \(\beta\) values. As a result, the hydrazone moiety has 120 distinct configurations. These BPH configurations reflect varied orientations and substitutions that can modulate the NLO properties more substantially than in the BP base structure.

Table 1 Hierarchical donor strength of substituents in BP and BPH molecular frameworks. This table categorizes substituents by their electron-donating capabilities, ranging from the weakest to the strongest donor, as employed in the molecular designs of BP and BPH compounds investigated in this work
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Figure 2 illustrates the \({\beta }_{2\omega }\) values for 11 different configurations of the BP system as calculated using both PM7 and the CAM-B3LYP/6-311 + + G(2d,2p) methodologies. The trends observed are consistently aligned between the two theoretical approaches. To the best of our knowledge, there is a lack of corresponding experimental results, that could serve as a benchmark for comparing the efficacy/reliability of these computational methods. Nevertheless, the CAM-B3LYP functional has garnered support in the literature for being well-suited to estimating \({\beta }_{2\omega }\) values for similar OM, as reflected in various published studies [26, 27, 29, 53, 92, 93].

Fig. 2
figure 2

Variation of dynamic first-order hyperpolarizability (\({\beta }_{2\omega }\)) with substituent electron-donating ability in BP. Each circle represents the computed result for a different substituent, plotted against the substituent’s electron-donating strength. The red circles are the \({\beta }_{2\omega }^{PM7}\)values obtained via SE method (PM7 via MOPAC 2016), while the blue circles are the\({\beta }_{2\omega }^{CAM}\) values via DFT (CAM-B3LYP/6-311 + + G(2d,2p) via Gaussian 16). The dashed-curves serves as a visual aid, illustrating the overall trend observed across the series

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Using CAM-B3LYP/6-311 + + G(2d,2p) as a provisional standard, PM7’s performance is noteworthy—its \({\beta }_{2\omega }^{PM7}\) values are in the same order of magnitude as those calculated by CAM-B3LYP. The largest \({\beta }_{2\omega }^{PM7}\) discrepancy observed for BP derivatives calculations is about 38% (BP-22, R1 = Br, R2 = N(CH3)2) compared to the CAM-B3LYP. For the other calculated compounds, it is observed a \({\beta }_{2\omega }^{PM7}\) variation ranging up to 30%. Accounting for computational cost (CC), the PM7 methodology is a notably more economical option than CAM-B3LYP. This positions PM7 as a viable and efficient preliminary tool for estimating the \({\beta }_{2\omega }\) of such OM in the gas phase. Upon completion of the geometry optimization, the CC per core for the β calculations of BP-25 (R1 = NO2, R2 = N(CH3)2) was approximately 3 s with the PM7 method, whereas the CAM-B3LYP/6-311 + + G(2d,2p) method required roughly 780 min. Remarkably, geometry optimization represented the most resource-intensive step when employing CAM-B3LYP/6-311 + + G(2d,2p), with the optimization of BP-25 consuming about 5760 min. In stark contrast, the same process took merely 10 s with PM7. DFT calculations were distributed across 64 cores (2.35 GHz) for our investigations, whereas SE calculations were efficiently carried out on a single core (2.40 GHz). Although this order of magnitude in CC was consistent for this study’s investigated family of compounds, it may vary with different classes of materials or other computational challenges. Therefore, its broader applicability should be evaluated on a case-by-case basis.

This trend aligns with the established push-pull strategy in molecular design, where the strategic positioning of electron donors and acceptors can significantly amplify a molecule’s NLO behavior. Moreover, it was observed that \({\beta }_{2\omega }\) values could also be achieved by the concurrent addition of electron-donating groups at both the R1 and R2 positions, further underscoring the nuanced interplay between molecular structure and NLO properties, mainly \({\beta }_{2\omega }\).

Compounds featuring substituents with weaker electron-donating capabilities at the R1 position, like -H, -F, -CH3, -Cl, and -Br, may still manifest considerable \({\beta }_{2\omega }\) values if paired with a significantly strong electron-donor at the R2 position. It is observed that the lowest\(\beta\) values tend to occur when the substituents at R1 and R2 are either identical or have comparable electron-donating abilities. Conversely, when stronger electron-donating groups, such as -OCH3, are attached at R1, the first hyperpolarizability follows a similar trend, albeit with some fluctuations in value (see Fig. 2b). Moreover, the highest \({\beta }_{2\omega }\) magnitudes are associated with pairs of substituents exhibiting markedly different donating characteristics, whereas a dip in the values is noted for symmetrical disubstitution with identical substituents at both para positions in BP or when substituents possess closely related but distinct electron-donating properties. In SM (Table A-1 and A-2), one can find all calculated values for dipole moment, static, and dynamic first-order hyperpolarizability for both levels of theory.

The analysis extended to encompass 120 BPH configurations. The BPH results show the same behavior as the one reported for BP compounds. Figure 3 underscores the \({\beta }_{2\omega }\) values for select BPH combinations explored, reflecting a consistent trend across both used computational methods. In SM (Table A-3 and A-4), one can find all calculated values for dipole moment, static, and dynamic first-order hyperpolarizability for both levels of theory.

Fig. 3
figure 3

Variation of \({\beta }_{2\omega }\) with substituent electron-donating ability in BP. Each circle represents the computed result for a different substituent, plotted against the substituent’s electron-donating strength. The red circles are the \({\beta }_{2\omega }^{PM7}\)values obtained via semi-empirical method (PM7 via MOPAC 2016), while the blue circles are the\({\beta }_{2\omega }^{CAM}\) values via density functional theory (CAM-B3LYP/6-311 + + G(2d,2p) via Gaussian 16). The dashed curves serves as a visual aid, illustrating the overall trend observed across the series

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This figure systematically explores variations by fixing the substituent at the R1 position while altering the group at R2. For BPH compounds, a pattern emerges where \({\beta }_{2\omega }\) tends to be lower when either symmetrical di-substitution occurs with identical donor groups at both para positions or when substituents share close donor characteristics, underscoring the nuanced interplay between substituent identity and molecular NLO responses. Like the BP compounds, BPH derivatives substituted with groups of markedly different electron-donating abilities exhibited the highest \({\beta }_{2\omega }\).

We reviewed findings from Sankaranarayanan et al. [71]. and Babu et al. [36] concerning the SHG efficiency of the crystalline form of BP-0 (R1 = H, R2 = H) and BPH-0 (R1 = H, R2 = H). Sankaranarayanan et al. observed that the SHG efficiency of BP-0 was approximately three times that of KDP, a standard reference for SHG materials. Conversely, Babu et al. reported an even more striking result, indicating that BPH-0’s SHG efficiency could be as much as ten times greater than KDP. These discrepancies highlight the need for further research and verification but collectively suggest BPH-0 as a potentially superior material for NLO applications due to its high SHG efficiency.

The asymmetric molecular structure of BPH compounds significantly influences their \({\beta }_{2\omega }\) values, as detailed in Table A4 of the SM—this dependency is pronounced by the nature and positioning of substituents at R1 and R2. An illustrative comparison between BPH and BP derivatives, particularly those substituted with the electron-accepting group -NO2, elucidates this effect. Figure 4 compiles these compounds’ calculated \({\beta }_{2\omega }\) using the PM7 and CAM-B3LYP methodologies. The data reveal that BPH derivatives with the -NO2 group positioned at R2 exhibit substantially higher \({\beta }_{2\omega }\) values than configurations with reversed substituent positions or analogous BP derivatives equipped with identical substituents. This finding underscores the critical role of substituent positioning in enhancing the NLO properties of BPH compounds, offering strategic guidance for the molecular engineering of NLO materials.

Fig. 4
figure 4

Comparison values of \({\beta }_{2\omega }\) with PM7 and CAM-B3LYP/6-311 + + G (2d,2p) levels of theory as a function of the order of donation of the substituents for a series of BP and BPH in different configurations, such as, BP-25(R1 = NO2, R2 = N(CH3)2)), BPH-39(R1 = N(CH3)2, R2 = NO2), and BPH-40(R1 = Cl, R2 = NH2)

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The compounds BP-25 (R1 = NO2, R2 = N(CH3)2)) and BPH-39 (R1 = N(CH3)2, R2 = NO2), they were those with the highest values. In the case of BPH-40 the calculated \({\mu }^{PM7}\) is around 5.3 D and the \({\beta }_{2\omega }^{PM7}\) of about\({\text{ 15}\times{10}}^{\text{-30}}\text{ }{\text{cm}}^{\text{4}}\text{ }{\text{statvolt}}^{\text{-1}}.\)Altering the positions of these substituents for compound BPH-39 results in an increase in both the \({\mu }_{2\omega }^{PM7}\) and \({\beta }_{2\omega }^{PM7}\) to ~ 9 D and \({\text{~}\text{23}\times{10}}^{\text{-30}}\text{ }{\text{cm}}^{\text{4}}{\text{statvolt}}^{\text{-1}}\), respectively. This variation underscores the impact of substituent positioning on the molecular properties. Specifically, BPH derivatives with -NO2 at R2 show a minimum \(\beta\)value when -NO2 is also at R1, whereas derivatives with -NO2 at R1 do not exhibit a well-defined minimum \({\beta }_{2\omega }\) across substituents with weaker donor characteristics like -H, -F, -CH3, -Cl, and -Br. Across all derivatives, the largest \({\beta }_{0}\) and \({\beta }_{2\omega }\) values are observed when substituents of significantly differing donor characters are used (see Table A-1 a A-4 in SM).

While hydrogen bonding within BP and BPH molecules and some substituent groups may reduce the microscopic SHG response, they can contribute positively to the macroscopic NLO response, as previously reported [35, 63,64,65]. organic compounds featuring hydrogen bonds often crystallize in noncentrosymmetric point groups, enabling potential strong second-order NLO responses. Despite the observed influences on molecular structure and electrical properties, both computational methods indicate consistent trends.

As widely reported in the literature, D-π-A molecular systems are recognized for inducing effective intramolecular charge transfer (ICT) from the donor region (D) to the acceptor (A) region, enhancing the permanent dipole moment [94]. Due to this dipole enhancement, OM compounds can exhibit a strong ICT, promoting, e.g., a significant increase in first-order molecular hyperpolarizability [95]. Therefore, a \(\mu\) comparative analyses of the investigated compounds have been made.

Table 3 reveals that the dipole moments of the studied BP and BPH compounds, compared with known SHG benchmark compounds like para-nitroaniline (PNA), KDP, and urea, demonstrate notable differences. Specifically, BP-0 and BPH-0 exhibit lower dipole moments relative to these standards. However, when the -NO2 group is positioned at R2 and various donor groups such as -NH2, -NHCH3, -NHC2H5, and -N(CH3)2 are introduced at R1, e.g., BP-25 and BPH-39, the resulting \(\mu\)PM7 is around 9 D, surpass all the references compounds.

Focusing on dynamic first-order hyperpolarizability, the PM7 SE reveals that BP-0’s calculated \({ \beta }_{2\omega }^{PM7}\)value of \({\text{ 1.4}\times{10}}^{\text{-30}}\text{ }{\text{cm}}^{\text{4}}\text{ }{\text{statvolt}}^{\text{-1}}\)is significantly lower than BPH-0’s\({\text{ 6.0}\times{10}}^{\text{-30}}\text{ }{\text{cm}}^{\text{4}}\text{ }{\text{statvolt}}^{\text{-1}}\). Conversely, CAM-B3LYP/6-311 + + G(2d,2p) calculations yield identical \({\beta }_{2\omega }^{CAM}\) values for BP-0 and BPH-0, each at \({\text{ 1.9}\times{10}}^{\text{-30}}\text{ }{\text{cm}}^{\text{4}}\text{ }{\text{statvolt}}^{\text{-1}}\). These values and additional comparative hyperpolarizability analyses for the studied and reference compounds are thoroughly detailed in Tables A-1 to A-4 in SM.

Among the numerous structures analyzed, compounds BP-25 (R1 = N(CH3)2, R2 = NO2) and BPH-39 (R1 = N(CH3)2, R2 = NO2), stand out as the highest \({\beta }_{2\omega }\) responses, surpassing standard benchmarks such as PNA, KDP and Urea. Moreover, these values are comparable or higher than other promising NLO materials reported in literature [96, 97]. This enhanced performance indicates the push-pull electronic effect in their molecular design, underscoring the strategy’s efficacy in augmenting NLO properties beyond conventional reference materials See Table 2.

Table 2 Comparative dipole moments (\(\mu\)) and dynamic first-order hyperpolarizabilities (\({\beta }_{2\omega }\)) of BP-0 and its highest \({\beta }_{2\omega }\) derivative BP-25 (R1 = N(CH3)2, R2 = NO2), as well as the BPH-0 and its highest \({\beta }_{2\omega }\) derivative BPH-39 (R1 = N(CH3)2, R2 = NO2)). This table includes reference compounds paranitroaniline (PNA) and potassium dihydrogen phosphate (KDP) for benchmarking purposes. The dipole moments are presented in Debye (D), and the hyperpolarizabilities are listed in units* of \({\text{10}}^{\text{-30}}{\text{cm}}^{\text{4}}\text{ }{\text{sta}\text{t}\text{volt}}^{\text{-1}}\)
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Additionally, to explain the origin of \(\beta\), Oudar and Chemla [99] established a direct connection between β and the intramolecular charge transfer, indicating that hyperpolarizability values increase as the \({E}_{gap}\) decreases. In fact, the relationship between \(\beta\)and the energy difference \({( \varDelta \epsilon )}\) between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) has been a topic of significant interest in the research of organic compounds with NLO properties. Particularly, in push-pull systems where intramolecular charge transfer is prominent, the \(\beta\) of molecules plays a crucial role in determining their optical properties. Several theoretical studies have shown an inverse relationship between \(\varDelta \epsilon\) and \(\beta ,\) indicating that molecules with a small HOMO-LUMO gap tend to exhibit higher \(\beta\) values [56, 100, 101]. In this study, we specifically focused on analyzing the β concerning molecular structure, mainly due to the addition of substituents. However, we have not conducted additional calculations to confirm this relationship between \(\varDelta \epsilon\) and \(\beta\). Future studies could expand this analysis, providing a more comprehensive understanding of the NLO properties of the investigated compounds.

5 Conclusions

In this extensive computational investigation, we have harnessed the capabilities of two QCC methods—SE and DFT—to scrutinize the second-order NLO response of 185 organic compounds. The NLO investigated in this work is \({\beta }_{2\omega }\), the molecular counterpart of the SHG, and the organic compounds investigated are BP and BPH derivatives. All QCC were performed in a gas-phase environment. The two most significant \({\beta }_{2\omega }\) responses are attributed to BP-25 (R1 = N(CH3)2, R2 = NO2) and BPH-39 (R1 = N(CH3)2, R2 = NO2), with values two, seven and one hundred and fifteen times superior to standard references PNA, KDP, and urea, respectively. We have also demonstrated that using the SE (PM7) framework can achieve almost the same accuracy as DFT (CAM-B3LYP) for this specific family of compounds. Such a result is highly relevant once the PM7 employed in MOPAC, a freely distributed software, has the lowest CC compared with CAM-B3LYP employed in Gaussian 16. Thus, using the SE method for material screening provides a cost-effective approach to predicting \({\beta }_{2\omega }\), response in OM structures, mainly this family class, avoiding significant computational and experimental expenses. However, it may vary with different classes of materials or other computational challenges. Therefore, its broader applicability should be evaluated on a case-by-case basis.

Finally, our results indicate a favorable space for future empirical investigations—particularly through techniques such as HRS or the Kurtz and Perry method—and they also underline the crucial role of computational methods in predicting NLO properties, which may significantly economize resources in the development of new photonic materials.