Introduction

Over the recent few decades, organic aromatic delocalized π-electron systems and other analogue fragments combined with strong electron donating and strong electron-withdrawing groups have become one of the largest classes of organic materials. It has been the subject of significant interest to several researchers worldwide in recent decades, where about 50% of all chemical compounds synthesized by the end of the second millennium are aromatic derivatives. The first synthesis of chalcone by Claisen and Claparede [1] was prepared by aldol condensation reactions between aromatic aldehydes and acetophenones using sodium methoxide as a condensing agent. In recent years, much attention has been given to organic compounds containing chalcone scaffolds due to their interesting structure, which is the main essential constituent of many natural or synthetic sources. Chemically, the chalcone derivatives as another’s electronically rich system have two aromatic rings that are joined by a three-carbon \(\alpha\), β-unsaturated C=O group of plant-derived polyphenolic compounds belonging to the flavonoids and isoflavonoids family [2, 3]. Moreover, the chalcone parent ring with raw formula C6H5C(O)CH=CHC6H5, therefore, exists in two stereoisomers (Z and E) depending on the arrangement of substituents around the central double bond with planar conformation structure. In chalcones (Chals), the π-electrons are delocalized and are easily polarizable. Chals and their other associated derivatives allow good work, so a range of new aromatic with good optical materials [4]. Chals class molecules play an essential role in chemical reactions [5]. Chalcone scaffold represents a common motif in many pharmaceutical active and remarkable compounds demonstrating a wide range of pharmacological activities; the essential activities are antioxidant [6], antifungal [7], antimicrobial [8], and anti-inflammatory [9], in addition, the chalcone fragment and their privileged analogues are a class of compounds with cross conjugated (nonlinear optical) NLO chromophores that are reported to exhibit with efficient blue and yellow light transmittance [10], good crystallizability [11], transparency [12], and large second harmonic generation SHG efficiency [11]. Chemical properties with π-conjugated chalcone and its derivatives are of interest due to their wide excellent applications in fields such as molecular switching, frequency doubling, and optical limiting [13]. The molecular structure of organic compounds greatly influences their optoelectronic properties by affecting conjugation length, the presence of functional groups, molecular packing, and resonance effects. The comprehension and regulation [14, 15] of these structural parameters are essential for optimizing the performance of organic optoelectronic devices [16]. The degree of π-conjugation along the molecular backbone significantly impacts the optoelectronic properties [17]. Longer conjugation lengths allow for greater delocalization of π-electrons, resulting in narrower band gaps and enhanced absorption of light across a wider range of wavelengths. Moreover, the presence of functional groups in those materials can alter their optoelectronic properties [18, 19] by modifying the energy levels of molecular orbital and introducing additional electronic transitions. Electron-donating or electron-withdrawing functional groups can shift the energy levels of the HOMO and LUMO, thereby affecting the band gap and absorption/emission spectra of the material [18, 19]. Structural engineering of materials serves as a cornerstone for advancing the field of NLO optics, driving innovation in optical materials design, device technology, and photonic applications [20, 21]. By harnessing the power of molecular and crystal engineering, researchers can unlock the full potential of NLO materials for next-generation optical technologies. Altering the molecular architecture through strategies such as incorporating electron-donating and electron-withdrawing groups [22] into the molecular structure creates a push–pull architecture. Also, the introduction of functional groups [18] that promote favourable intermolecular interactions, enhance solubility, or improve film-forming properties can optimize the performance of NLO materials, additionally, combining different types of NLO-active components, such as metals [23], complexes [20], organic molecules, or polymers, can lead to synergistic effects and tailored NLO responses. By applying these strategies, researchers can tailor the structural properties to achieve enhanced NLO optical responses. Various electron-withdrawing groups, including F, Cl, Br, I, CF3, and CN, have been included at the end and lateral terminal positions of terpyridine-substituted hexamolybdates to build different molecular systems [19]. The \(\beta\) values are considerably influenced by the addition of halogens at the terminus of the terpyridine ligand, as it establishes a robust D–bridge–A structure. Conversely, the presence of CF3 and CN substantially impacts the second-order molecular reaction. Similarly, Janjua et al. [24] have documented the occurrence of additional POM-clusters connected to terpyridine units by bridging groups. In addition, the research [25] employs machine-learning techniques to evaluate the suitability of small-molecule donors for organic solar cells. Moreover, an alternative methodology [26] has been employed to address the development of novel dye designs through the structural modification of π-spacers/conjugated systems and terminal acceptors. The computational studies of NLO properties of materials based on chalcone derivatives have been studied at various DFT levels and compared with the experimental results [4]. From the literature, it has been found that among the reported NLO chalcone derivatives, only a few Chals have been investigated both experimentally and theoretically due to their potential applications in photonics and optoelectronics [27, 28]. In the present work, we focused on the theoretical analysis of linear and NLO properties of seven chalcones (Chal1–Chal7) [29] (Scheme 1) using density functional theory (DFT) at five levels of theory. The methods based on DFT represent an important road since they allow the simulation of a wide range of molecular properties with high precision. Recently, DFT calculations have successfully been used to evaluate the NLO properties of chalcone derivatives [30, 31] to contribute to this series of investigations, in the present study, we investigate the impact of various substituted donor/acceptor strength on the dipole moment, polarizability, first hyperpolarizability, and energy gap \({E}_{\text{g}}\). This new study will focus on the relation between the higher hyper-Rayleigh scattering (HRS) hyperpolarizability \({\beta }_{\text{HRS}}\) and the electric field-induced second harmonic generation (EFISHG) \({\beta }_{//}\), and the relation between the first hyperpolarizability \(\beta\) and the energy gaps of these Chals. While these properties are related to the electronic structure of Chals, they are not directly proportional to each other. However, they can be influenced by similar factors such as molecular size, shape, symmetry, and electronic configuration. There might be correlations between these properties in certain cases, but they are not simple proportional relationships. We also investigated the frontier molecular orbitals (FMOs) HOMO and LUMO, their energies, the location, and the nature of the isosurfaces. The FMOs analysis has been used to elucidate information regarding the ionization potential (I), electron affinity (A), global hardness (η), global softness (S), chemical potential (V), electronegativity (χ), and global electrophilicity index, (ω), nucleophilicity index (N). The examined Chals demonstrate attractive NLO characteristics that can be utilised for practical purposes in many domains. The present study is anticipated to yield valuable insights for researchers in the field of optoelectronics, specifically in the areas of greenhouse-integrated solar cells [32] and hole transport materials [33, 34]. These materials have the potential to enhance the power conversion efficiency and stability of organic solar cells [35].

Scheme 1
scheme 1

Structure of seven chalcone-based molecules Chal1–Chal7

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Methodologies

All theoretical computations were performed with density functional theory at the 6-311++G** basis set implemented in Gaussian 09 [36]. The optimized structures and their coordinates are performed using the B3LYP/6-311++G** level and presented in Fig. 1 and Tables S1-S7 in supporting information SI, respectively. The B3LYP balanced treatment of exchange and correlation effects allows for accurate predictions of molecular geometries, vibrational frequencies, and electronic properties [37, 38]. Additionally, the B3LYP has been found to perform well for organic compared to many modern-day XC functionals [37, 38]. The visualization of the Chal1, Chal2, Chal3, Chal4, Chal5, Chal6, and Chal7 optimized geometries, HOMO, and LUMO plots has been carried out using GaussView 5.1 [39]. Linear (dipole moment \(\mu\), mean polarizability \(\langle \alpha \rangle\), polarizability anisotropy \(\Delta \alpha\)) and NLO properties (total first hyperpolarizability \({\beta }_{\text{tot}}\), electric-field-induced second harmonic generation (EFISHG) \({\beta }_{//}\), hyper-Rayleigh scattering (HRS) first hyperpolarizability \({\beta }_{\text{HRS}}\), and the depolarization ratio DR) were calculated at the B3LYP [40, 41], PBE0 [42], CAM-B3LYP [43], ωB97X–D [44], and M06–2X [45] methods using the 6-311+G** basis set. The scattering intensity \({I}_{\Psi V}^{2\omega }\) corresponding to the polarization angle Ψ was obtained by the Multiwfn 3.7 [46]. Studies show the suitability of the corrected XC functionals to calculate \(\alpha\) and \(\beta\) [47, 48].

Fig. 1
figure 1

Chal1–Chal7 optimized structure at the B3LYP∕6-311+G** level

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\(\mu\) is defined as:

$$\mu =\sqrt{\left({\mu }_{x}^{2}+{\mu }_{y}^{2}+{\mu }_{z}^{2}\right)}$$
(1)

\(\langle \alpha \rangle\) was calculated from as [49]:

$$\langle \alpha \rangle =\frac{1}{3}\sum_{i = x,y,z}{\alpha }_{ii}$$
(2)
$$\langle \alpha \rangle =\frac{1}{3}\left({\alpha }_{xx}+{\alpha }_{yy}+{\alpha }_{zz}\right)$$
(3)

and the \(\Delta \alpha\) as:

$$\Delta \alpha =\sqrt{\frac{1}{2}\left({\left({\alpha }_{xx}-{\alpha }_{yy}\right)}^{2}+{\left({\alpha }_{xx}-{\alpha }_{zz}\right)}^{2}+{\left({\alpha }_{yy}-{\alpha }_{zz}\right)}^{2}\right)}$$
(4)

For β, we are interested in \({\beta }_{//}\) and βHRS. The \({\beta }_{//}\) is the central quantity of \(\beta\) vector projected along the \(\mu\)-axis and calculated as:

$${\beta }_{//}\left(-2\omega ,\omega ,\omega \right)={\beta }_{//}=\frac{1}{5}\sum_{\text{i}}\frac{{\mu }_{\text{i}}}{\left|\overrightarrow{\mu }\right|}\sum_{\text{j}}\left({\beta }_{ijj}+{\beta }_{jij}+{\beta }_{jji}\right)$$
(5)

In the static limits (\({\beta }_{\text{ijj}}={\beta }_{\text{jij}}={\beta }_{\text{jji}})\), \({\beta }_{//}\) becomes

$${\beta }_{//}=\frac{3}{5}\sum_{i = x,y,z}\frac{{\mu }_{\text{i}}{\beta }_{\text{i}}}{\left|\overrightarrow{\mu }\right|}$$
(6)

\(\left|\overrightarrow{\mu }\right|\) is the norm of the dipole moment, \({\mu }_{\text{i}}\) and \({\beta }_{\text{i}}\) the ith components of \(\mu\) and \(\beta\) vectors, respectively. The \({\beta }_{\text{HRS}}\) is related to the \({I}_{\Psi V}^{2\omega }\) for nonpolarized incident light of frequency \(\omega\), and observation of plane-polarized scattered light made perpendicular to the propagation plane. The \({\beta }_{\text{HRS}}\) is calculated as

$${\beta }_{\text{HRS}}=\langle {\beta }_{\text{HRS}}^{2}\rangle =\langle {\beta }_{\text{ZZZ}}^{2}\rangle +\langle {\beta }_{\text{ZXX}}^{2}\rangle$$
(7)
$$DR=\frac{{I}_{\text{VV}}^{2\omega }}{{I}_{\text{HV}}^{2\omega }}=\frac{\langle {\beta }_{\text{ZZZ}}^{2}\rangle }{\langle {\beta }_{\text{ZXX}}^{2}\rangle }$$
(8)

\(\langle {\beta }_{\text{ZZZ}}^{2}\rangle\) and \(\langle {\beta }_{\text{ZXX}}^{2}\rangle\) given by the molecular (x, y, z) reference are averages of \(\beta\) components that describe the isotropic distribution of molecular orientations in dilute solutions. \(\langle {\beta }_{\text{ZZZ}}^{2}\rangle\) and \(\langle {\beta }_{\text{ZXX}}^{2}\rangle\) were calculated without assuming Kleinman’s conditions, and the temperature convention was used for defining the \(\beta\) quantities. The detailed expressions of \(\langle {\beta }_{\text{ZZZ}}^{2}\rangle\) and \(\langle {\beta }_{\text{XZZ}}^{2}\rangle\) [50] are shown in the SI.

$$\langle {\beta }_{ZZZ}^{2}\rangle =\frac{1}{7}{\sum }_{\zeta }^{x,y,z}{\beta }_{\zeta \zeta \zeta }^{2}+\frac{4}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \zeta \eta }^{2}+\frac{2}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \zeta \zeta }{\beta }_{\zeta \eta \eta }+\frac{4}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\eta \zeta \zeta }{\beta }_{\zeta \zeta \eta }+\frac{4}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \zeta \zeta }{\beta }_{\eta \eta \zeta }+\frac{1}{35}{\sum }_{\zeta \ne \eta }^{x, y,z}{\beta }_{\eta \zeta \zeta }^{2}+\frac{4}{105}{\sum }_{\zeta \ne \eta \ne \varepsilon }^{x,y,z}{\beta }_{\zeta \zeta \eta }{\beta }_{\eta \varepsilon \varepsilon }+\frac{1}{105}{\sum }_{\zeta \ne \eta \ne \varepsilon }^{x,y,z}{\beta }_{\eta \zeta \zeta }{\beta }_{\eta \varepsilon \varepsilon }+\frac{4}{105}{\sum }_{\upzeta \ne\upeta \ne\upvarepsilon }^{\text{x},\text{y},\text{z}}{\upbeta }_{{\zeta \zeta \eta }}{\upbeta }_{{\varepsilon \varepsilon \eta }}+\frac{2}{105}{\sum }_{\upzeta \ne\upeta \ne\upvarepsilon }^{\text{x},\text{y},\text{z}}{\upbeta }_{{\zeta \eta \varepsilon }}^{2}+\frac{4}{105}{\sum }_{\upzeta \ne\upeta \ne\upvarepsilon }^{\text{x},\text{y},\text{z}}{\upbeta }_{{\zeta \eta \varepsilon }}{\upbeta }_{{\eta \zeta \varepsilon }}$$
(9)
$$\langle {\beta }_{XZZ}^{2}\rangle =\frac{1}{35}{\sum }_{\zeta }^{x,y,z}{\beta }_{\zeta \zeta \zeta }^{2}+\frac{4}{105}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \zeta \zeta }{\beta }_{\zeta \eta \eta }-\frac{2}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \zeta \zeta }{\beta }_{\eta \eta \zeta }+\frac{8}{105}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \zeta \eta }^{2}+\frac{3}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \eta \eta }^{2}-\frac{2}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \zeta \eta }{\beta }_{\eta \zeta \zeta }+\frac{1}{35}{\sum }_{\zeta \ne \eta }^{x,y,z}{\beta }_{\zeta \eta \eta }{\beta }_{\zeta \varepsilon \varepsilon }-\frac{2}{105}{\sum }_{\zeta \ne \eta \ne \varepsilon }^{x,y,z}{\beta }_{\zeta \zeta \eta }{\beta }_{\eta \varepsilon \varepsilon }+\frac{2}{35}{\sum }_{\zeta \ne \eta \ne \varepsilon }^{x,y,z}{\beta }_{\zeta \eta \varepsilon }^{2}-\frac{2}{105}{\sum }_{\zeta \ne \eta \ne \varepsilon }^{x,y,z}{\beta }_{\zeta \eta \varepsilon }{\beta }_{\eta \zeta \varepsilon }$$
(10)

Results and discussion

Linear and NLO results

Dipole moment

Generally, the double bond between the \(\alpha\) and β carbon C atoms of the carbonyl group C =O influenced the \(\mu\) in Chal molecules. \(\mu\) arises due to the difference in electronegativity between the atoms involved in the bond. The C=O group consists of a carbon–oxygen double bond, where O is more electronegative than C. The O atom pulls the electron density towards itself, creating a \(\mu\) with a partial negative charge on the O and a partial positive charge on the C. The presence of substituents in the Chal can also affect the overall \(\mu\). In this exploration, we study seven Chals where the substituent has a significant electronic effect that modulates some overall molecular properties (\(\mu\), \(\alpha ,\) and \(\beta\)), which were corroborated by combining theoretical results. The most fundamental property of the electrical responses is \(\mu\). Therefore, determining these values has given rise to a solid foundation for discussing the calculation accuracy of the different theoretical methods. The different \(\mu\) values of seven Chals obtained using the B3LYP, PBE0, CAM-B3LYP, ωB97X–D, and M06–2X levels at the 6-311+G** basis set are displayed in Table 1. In Fig. 2, we showed the \(\mu\) vector of our studied Chals. An accurate analysis of these results shows that the B3LYP and PBE0 functionals give the highest \(\mu\) values, while the M06-2X functionals give the lowest \(\mu\) values. The CAM-B3LYP and ωB97X–D levels show systematically smaller \(\mu\) values. The calculated \(\mu\) values vary between 1.60 and 5.38 D. The variation of \(\mu\) as a function of DFT functionals attain a maximum of ~ 5% for Chal1, ~ 6% for Chal2, ~ 7% for Chal3, ~ 22% for Chal4, ~ 9% for Chal5, ~ 4% for Chal6, and ~ 5% for Chal7 (Table 1 and Fig. 3). The B3LYP and PBE0 functionals show similar \(\mu\) values of less than 1% for most Chals. The investigated compounds have a high \(\mu\) values compared with many similar chalcones (Table 1) computed at different levels of theory. The \(\mu\) values follow the order:

Table 1 \(\mu\), \(\langle \alpha \rangle\), \(\Delta \alpha\), \({\beta }_{//}\), \({\beta }_{\text{tot}}\), \({\beta }_{\text{HRS}}\), and DR for Chal1–Chal7 obtained using five DFT levels at the 6-311+G** basis set
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Fig. 2
figure 2

\(\mu\) vector of Chal1–Chal7

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Fig. 3
figure 3

\(\mu\) of Chal1–Chal7 calculated using five functionals at the 6-311+G** basis set

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Chal1 and Chal2

\(\mu\) M06–2X < \(\mu\) PBE0 < \(\mu\) ωB97X–D < \(\mu\) CAM-B3LYP < \(\mu\) B3LYP

Chal3 and Chal4

\(\mu\) M06–2X < \(\mu\) ωB97X–D < \(\mu\) CAM-B3LYP < \(\mu\) B3LYP < \(\mu\) PBE0

Chal5

\(\mu\) M06–2X < \(\mu\) CAM-B3LYP ≤ \(\mu\) ωB97X–D < \(\mu\) B3LYP < \(\mu\) PBE0

Chal6

\(\mu\) M06–2X < \(\mu\) PBE0 < \(\mu\) B3LYP < \(\mu\) ωB97X–D ≤ \(\mu\) CAM-B3LYP

Chal7

\(\mu\) M06–2X < \(\mu\) CAM-B3LYP < \(\mu\) ωB97X–D < \(\mu\) B3LYP ≤ \(\mu\) PBE0

\(\mu\) Chal4 < \(\mu\) Chal6 < \(\mu\) Chal1 < \(\mu\) Chal2 < \(\mu\) Chal7 < \(\mu\) Chal3 < \(\mu\) Chal5

The \(\mu\) values converge for each Chal individually, whereas Chal5 has the strongest \(\mu\), Chal4 has the lowest \(\mu\), where the substitutions were introduced on both aryl rings. Chal5 exhibits one methoxy group OCH3 and a chlorine Cl atom, while Chal4 has two OCH3. We notice that the two OCH3 as terminal groups in Chal4 reduced the quantity of partial positive and negative charges in this molecule and the separations between these charges. In the case of Chal5, the high \(\mu\) values can be explained by the presence of a Cl atom and one OCH3 on both aryl rings as a powerful electron-withdrawing group with strong Pauling electronegativity and vacant 3d orbitals [51] and one OCH3 as an electron-donating group. Zhao et al. [51] studied some of the inherent characteristics of the Cl atom and compared them with those of the fluorine (F) atom. Their results [51] showed that Cl atoms can form strong noncovalent interactions; therefore, intense \(\mu\). Salim et al. [52] have observed a general increase in \(\mu\) values of pyrazoline analogues with Cl and F atoms as an electron-attractor group. Zhang et al. [53] found that in Cl-modified non-fullerene acceptors, the \(\mu\) rises from 2.26 to 2.77 D when F atoms are substituted with Cl atoms. The results showed that, like our Chal5, electron donor–acceptor groups on opposite sides of the Chal form the basis for constructing strong NLO organic materials. Chal3 has a higher \(\mu\) than Chal1, Chal2, Chal4, Chal6, and Chal7. In the case of Chal2 and Chal3, the substitutions were introduced in the benzaldehyde ring (A position) (Scheme 1). Chal2 contains an electron-withdrawing F atom, whereas Chal3 possesses a high electron-donating NH2 at the para position (Table 1 and Fig. 3). Electron-donating groups can increase the electron density in the vicinity, potentially influencing the \(\mu\) [54]. In contrast, electron-withdrawing groups may decrease electron density. The lower \(\mu\) value of Chal6 compared to Chal7 could be related to electron-donating and electron-withdrawing power. Chal6 is a bromo-substituted chalcone, and Chal7 displays a furane in ring B and a chloroarene at the benzaldehyde ring. Chal1represents the non-substituted pattern. The decrease of \(\mu\) values by more than 50% in Chal4 and Chal6 proves the low intramolecular charge transfer, which occurs due to the push–pull of electrons between donor and attractor units. Introducing Cl with strong electronegativity can affect the electron density distribution along the conjugated backbone of all Chals. Hence, the type, number, and position of these units are what generate a clear variation in \(\mu\).

Polarizability

This section scrutinizes the B3LYP, PBE0, ωB97X-D, CAM-B3LYP, and M06-2X \(\alpha\) results. The calculated mean polarizability \(\langle \alpha \rangle\) and polarizability anisotropy \(\Delta \alpha\) of the selected Chal1–Chal7 are presented in Figs. 4 and 5. The PBE0 method shows the highest \(\alpha\) values. Their \(\langle \alpha \rangle\) and \(\Delta \alpha\) varied between (206.65 to 269.60) and (193.71 to 331.26 a.u.). Cha4 gets the highest \(\langle \alpha \rangle\) and \(\Delta \alpha\) varied between (236.60 and 269.60) and (253.77 and 331.26 a.u.) obtained at the ωB97X-D and PBE0 levels, respectively. Theoretical studies [58,59,60,61,62] showed that the PBE0 gives high \(\alpha\) compared to some functionals. Chal1 without substitution gets the lowest \(\langle \alpha \rangle\) and \(\Delta \alpha\). In this case \(\langle \alpha \rangle\) and \(\Delta \alpha\) are 187.60 and 162.35 a.u. at the M06-2X and ωB97X-D, respectively. A careful analysis of \(\alpha\) components of these Chals shows that the contributions along the x and y axis (\({\alpha }_{\text{xx}}\) and \({\alpha }_{\text{yy}}\)) are the majority. The calculated \(\alpha\) showed that the Chals size drove the average polarizability, and \(\Delta \alpha\) depends on the nature of each Chals. Several \(\alpha\) studies [63,64,65,66,67] show the same relationship. Andrade-Filho and co-workers [56] proposed new metabolites chalcone with \(\langle \alpha \rangle\) between 37 × 10−24 and 53.5 × 10−24 esu, which are quite similar than those of the selected Chal1 and Chal2. The authors showed that the structural effect of the substitution of HC=CH by H2C–CH2 acts decreasing \(\langle \alpha \rangle\). Proposed new chemicals which are the (2E)-3-(4-methylphenyl)-1-(3-nitrophenyl)prop-2-en-1-one and (2E)-3-[4-(dimethylamino)phenyl]-1-(3-nitrophenyl)prop-2-en-1-one [57] with \(\langle \alpha \rangle\) between 32.15 × 10−24 and 38.76 × 10−24 esu, respectively, similar to those of selected Chals in this new study.

Fig. 4
figure 4

\(\langle \alpha \rangle\) of Chal1–Chal7 calculated using five levels at the 6-311+G** basis set

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Fig. 5
figure 5

\(\Delta \alpha\) of Chal1–Chal7 calculated using five levels at the 6-311+G** basis set

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Hyperpolarizability

Hyperpolarizabilities \({\beta }_{//}\), \({\beta }_{\text{tot}}\), \({\beta }_{\text{HRS}}\), and DR of chalcone-based derivatives Chal1–Chal7 are computed and presented in Table 1, Figs. 7 and 8. The obtained hyperpolarizability contributions show that the β ones along the x-axis are the majority (Fig. 6). High \({\beta }_{\text{tot}}\) values are obtained for these Chals. The \({\beta }_{\text{tot}}\) values ranged from 1484.54 to 5766.25 a.u. using the PBE0 level. This approach gives the highest \({\beta }_{\text{tot}}\) values of all studied Chals compared to other DFT levels B3LYP, M06-2X, CAM-B3LYP, and ωB97X–D used in this investigation while the ωB97X–D functional with 22% of HF exchange level gives the lowest \({\beta }_{\text{tot}}\) values. The functionals B3LYP give close \({\beta }_{\text{tot}}\) values to that of PBE0. The difference does not exceed 13.90, 10.04, 33.92, 15.47, 20.38, 31.16, and 1.94% for Chal1, Chal2, Chal3, Chal4, Chal5, Chal6, and Chal7, respectively (Table 1). The variation of \({\beta }_{\text{tot}}\) as a function of the DFT functionals attains a maximum of ~ 34% for Chal1, ~ 30% for Chal2, ~ 60% for Chal3, ~ 41% for Chal5, ~ 57% for Chal6, and ~ 11% for Chal7. For the \({\beta }_{//}\), the highest and the lowest values are –3.19 and –3575.87 a.u., respectively, which were obtained using the ωB97X–D and PBE0 functionals, respectively. The same behavior was also found for [68,69,70]. These findings lead to the following orderings:

Fig. 6
figure 6

Chal1–Chal7 orientation during calculations

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Chal1-Chal7 \({\beta }_{\text{tot}}\) ωB97X–D < \({\beta }_{\text{tot}}\) CAM-B3LYP < \({\beta }_{\text{tot}}\) M06-2X < \({\beta }_{\text{tot}}\) B3LYP < \({\beta }_{\text{tot}}\) PBE0

\({\beta }_{\text{tot}}\) Chal2 < Chal1 < Chal7 < Chal4 < Chal3 < Chal6 < Chal5.

In this section, we also investigated the correlation between the Chal structures and the second-order NLO responses \({\beta }_{\text{HRS}}\) and \({\beta }_{//}\). The relationship between the \({\beta }_{\text{HRS}}\) and \({\beta }_{//}\) may depend on the specific characteristics of the Chals (electronic structure) under investigation. Researchers typically use a combination of experimental measurements and computational simulations, like our study, to understand better the NLO properties and the connection between different \(\beta\). A comparative study of selected Chal1–Chal7 indicates that the higher \({\beta }_{\text{HRS}}\) values were obtained for these Chals using hybrid DFT functionals, including a minor fraction of the Hartree–Fock (HF) exchange. Whereas the XC functionals containing larger amounts of HF exchange, such as M06-2X, ωB97X-D, and CAM-B3LYP, give the lowest \({\beta }_{\text{HRS}}\) values. The CAM-B3LYP, M06-2X, and ωB97X-D values are convergent and smaller than the B3LYP, which contains the smallest amount of HF exchange. The B3LYP and PBE0 functional with 20% and 40% HF exchange gives the highest \({\beta }_{\text{HRS}}\) values equal to 2150.65 and 2651.23 a.u, respectively, whereas the ωB97X-D DFT global hybrid functional with 22% HF exchange yields the smallest \({\beta }_{\text{HRS}}\) values equal 600.74, 602.21, 902.32, 1301.25, 1558.54, and 971.30 a.u. for Chal1, Chal2, Chal3, Chal4, Chal5, and Chal6, respectively. As an exception, the B3LYP gives the lowest \({\beta }_{\text{HRS}}\) values for Chal7 (896.65 a.u). In the meantime, the PBE0 functional yields the greatest \({\beta }_{\text{HRS}}\) values equal to 902.23, 863.58, 1930.48, 2049.65, 2651.23, 2148.36, and 1222.23 a.u. for Chal1, Chal2, Chal3, Chal4, Chal5, Chal6, and Chal7, respectively. A glance at Table 1, Figs. 7 and 8 is enough to determine that we state opposite variations between the \({\beta }_{\text{HRS}}\) and \({\beta }_{//}\) in addition to the presence of a direct correlation between the \({\beta }_{\text{HRS}}\) and \({\beta }_{\text{tot}}\). Many studies [71] have reached the same findings. Computations lead us to propose an increasing/decreasing classification relative to the \({\beta }_{\text{HRS}}\) and \({\beta }_{//}\), respectively. The established order is as follows:

Fig. 7
figure 7

\({\beta }_{\text{tot}}\) of Chal1–Chal7 calculated using five levels at the 6-311+G** basis set

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Fig. 8
figure 8

\({\beta }_{//}\) and \({\beta }_{\text{HRS}}\) of Chal1–Chal7 calculated using five levels at the 6-311+G** basis set

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Chal1, Chal2, Chal4 and Chal5

\({\beta }_{\text{HRS}}\) ωB97X–D < \({\beta }_{\text{HRS}}\) M06-2X < \({\beta }_{\text{HRS}}\) CAM-B3LYP < \({\beta }_{\text{HRS}}\) B3LYP < \({\beta }_{\text{HRS}}\) PBE0

Chal1, Chal3 and Chal6

\({\beta }_{\text{HRS}}\) ωB97X–D < \({\beta }_{\text{HRS}}\) CAM-B3LYP < \({\beta }_{\text{HRS}}\) M06-2X < \({\beta }_{\text{HRS}}\) B3LYP < \({\beta }_{\text{HRS}}\) PBE0

Chal7

\({\beta }_{\text{HRS}}\) B3LYP < \({\beta }_{\text{HRS}}\) ωB97X–D < \({\beta }_{\text{HRS}}\) CAM-B3LYP < \({\beta }_{\text{HRS}}\) M06-2X < \({\beta }_{\text{HRS}}\) PBE0

\({\beta }_{\text{HRS}}\)

Chal2 < Chal1 < Chal7 < Chal3 < Chal4 < Chal6 < Chal5

Chal1, Chal2, Chal3 and Chal4

\({\beta }_{//}\) ωB97X–D > \({\beta }_{//}\) M06-2X > \({\beta }_{//}\) CAM-B3LYP > \({\beta }_{//}\) B3LYP > \({\beta }_{//}\) PBE0

Chal5

\({\beta }_{//}\) ωB97X–D > \({\beta }_{//}\) M06-2X > \({\beta }_{//}\) CAM-B3LYP > \({\beta }_{//}\) B3LYP > \({\beta }_{//}\) PBE0

Chal6

\({\beta }_{//}\) M06-2X > \({\beta }_{//}\) ωB97X–D > \({\beta }_{//}\) CAM-B3LYP > \({\beta }_{//}\) B3LYP > \({\beta }_{//}\) PBE0

Chal7

\({\beta }_{//}\) PBE0 > \({\beta }_{//}\) ωB97X–D > \({\beta }_{//}\) CAM-B3LYP > \({\beta }_{//}\) M06-2X > \({\beta }_{//}\) B3LYP

\({\beta }_{//}\)

Chal6 > Chal1 > Chal7 > Chal3 > Chal4 > Chal5

Our computations show that electron donating and withdrawing groups can effectively increase and decrease the first hyperpolarizability \(\beta\), respectively. The results in Tables 1 and 2 establish the influence of different substituents and signify the importance of position and substitution number on the \(\beta\) Chals. The values are the largest for the disubstituted Chal5 than the mono-substituted ones. \({\beta }_{\text{HRS}}\)(OCH3/Cl)Chal5 > \({\beta }_{\text{HRS}}\)(Br)Chal6 > \({\beta }_{\text{HRS}}\)(NH2)Chal3 > \({\beta }_{\text{HRS}}\)(F)Chal2. \({\beta }_{\text{tot}}\)(OCH3/Cl)Chal5 > \({\beta }_{\text{tot}}\)(Br)Chal6 > \({\beta }_{\text{tot}}(\) NH2)Chal3 > \({\beta }_{\text{tot}}\)(F)Chal2. These high \({\beta }_{\text{HRS}}\) and \({\beta }_{\text{tot}}\) values are due to the electron transfer from the electron-donating OCH3 via the Chals to the electron-withdrawing Cl. These values decrease significantly in the absence of electron-donating or/and electron-withdrawing group (D-π-(A or D) and A-π-(A or D)) as illustrated by Chal4 with two OCH3. Chal1 represents the non-substituted pattern, and Chal7, with the furane in rings B and A, has a chloroarene at the benzaldehyde ring. The \(\beta\) values are influenced by the distribution of electrons within a molecule. Electron-donating groups, by increasing electron density, and electron-withdrawing groups, by decreasing electron density, can affect the NLO response without ignoring the role of the overall molecular architecture (the HC=CH bridge π-character). The effectiveness of the targeted compounds as the basis for constructing effective NLO organic materials can be confirmed by comparison to other studies (Table 1). \({\beta }_{\text{tot}}\) values of Chal5 are nearly five times higher than that of para-nitroaniline p-NA and one hundred and thirty-five times greater than that reported for urea [55] as well-known prototype NLO molecules add to that all targeted Chals in the present investigation possess larger \({\beta }_{\text{tot}}\) values compared to that of p-NA. Most importantly, among all Chal1–Chal7; Chal5 has a high \({\beta }_{\text{tot}}\) value compared with those of similar chalcone derivatives [57] designed with rotation of OCH3 at three different possible ortho, meta, and para positions on the phenyl ring. In addition, Chals have high \({\beta }_{\text{tot}}\) values compared with those of some Chals biotransformed using the endophytic fungus Aspergillus flavus (an agent naturally found in the Amazon biome), 2,3,4,4′-tetramethoxychalcone [56] obtained at the CAM-B3LYP/6-311++G** level, and (1E,4E)-1,5-di-p-tolylpenta-1,4-dien-3-one obtained at the B3LYP/6-31G level of theory [30].

Table 2 EHOMO, ELUMO, \({E}_{\text{g}}\) in eV and \({\beta }_{\text{tot}}\) in a.u. of Chal1–Chal7 derivatives obtained using five DFT levels at the 6-311+G** basis set
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The four DFT functionals used in this study give close DR values for the selected Chal1-Chal7. Their DR values were close to 3. The DR results, in combination with the chemical topology of the Chal structures, are congruent with the representative \({C}_{\text{s}}\) the molecular point group of these Chals. The \({C}_{\text{s}}\) point group symmetry is well marked by the presence of a single \(\sigma\) plane of symmetry (a planar structure) for this type of Chals. The DR property covers a broad range of dipolar-octupolar characters of our Chals and makes it possible to know the symmetry of the Chals when it has a few elements of the non-zero \(\beta\) tensor. The evolution of \({I}_{\Psi V}^{2\omega }\) with respect to the elliptical polarization of the incident light denoted by the polarization angle Ψ of the incident beam was also investigated in this study. To further understand the \({\beta }_{\text{HRS}}\) results, we show the calculated \({I}_{\Psi V}^{2\omega }\) at the PBE0 functional (Fig. 9). \({I}_{\Psi V}^{2\omega }\) intensity corresponding to the Ψ varied from –180 to 180° with a step-size of 1°, their shapes agree quantitatively with the more polar Chal3 and Chal5. The results show that the trends of \({I}_{\Psi V}^{2\omega }\) are completely consistent with the \({\beta }_{\text{HRS}}\) of Chlas. The same assessment has been obtained by Lu et al. [72] for the 1,3-thiazolium-5-thiolates mesoionic.

Fig. 9
figure 9

\({I}_{\Psi V}^{2\omega }\) in a.u. of Chal1–Chal7 as a function of Ψ at 1064 nm obtained with the B3LYP/6-311+G** level

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Molecular orbitals analysis and their energy gaps

The FMO analysis provides valuable insights into the electronic structure, reactivity, and properties of molecules, serving as a fundamental tool for advancing our understanding of chemical phenomena and guiding the rational design of new materials and processes [16, 73,74,75]. Frontier electron density predicts the most reactive position in π-electron systems and clarifies several types of reactions in conjugated systems [38, 76]. Compounds exhibiting a significant \({E}_{\text{g}}\) demonstrate stability, hence exhibiting greater chemical hardness than those with a smaller \({E}_{\text{g}}\) [77, 78]. The study of the occupied and the virtual orbitals helps provide more information about energy gaps \({E}_{\text{g}}\), orbital surfaces, and the possibility of the charge for Chals. Table 2 and Figs. 10 and 11 summarize the \({E}_{\text{HOMO}}\), \({E}_{\text{LUMO}}\) and their \({E}_{\text{g}}\) for Chals. The substituent effect on FMOs is straightforward; the NH2 and OCH3 donor groups cause increasing orbital energy, while the Cl and Br acceptor groups cause decreasing orbital energy. The DFT \({E}_{\text{g}}\) results show that the PBE0 gives the lowest \({E}_{\text{g}}\) values compared to the B3LYP, ωB97X–D, CAM-B3LYP, and M06-2X levels. Their values are 3.17, 3.08, 2.18, 2.17, 2.16, 2.15, and 3.15 eV for Chal1, Chal2, Chal3, Chal4, Chal5, Chal6, and Chal7, respectively. Also, the B3LYP gives low \({E}_{\text{g}}\) values compared to ωB97X–D, CAM-B3LYP, and M06-2X levels. Their values are 4.13, 4.13, 3.44, 3.89, 3.74, 4.07, and 3.98 eV for Chal1, Chal2, Chal3, Chal4, Chal5, Chal6, and Chal7, respectively. Recent studies [68,69,70,71, 79,80,81,82] show the suitability of the PBE0 for the \({E}_{\text{g}}\) calculation. For similar Chals [83], the PBE0 provide reliable \({E}_{\text{g}}\). Their calculated values range from 3 to 4 eV for the (E)-1-(5-bromo-2-thienyl)-3-(4-bromophenyl)prop-2-en-1-one,(E)-1-(5-bromo-2-thienyl)-3-(3-chlorophenyl)prop-2-en-1-one, and (E)-1-(5-bromo-2-thienyl)-3-(2-methoxyphenyl)prop-2-en-1-one [84], the B3LYP level gives similar \({E}_{\text{g}}\) values [85] ranged between 3.71 and 3.93 eV. The ωB97X–D method overestimates \({E}_{\text{g}}\) values compared to the B3LYP and PBE0. Their obtained values are 5.42, 5.39, 5.49, 5.48, 5.47, and 4.47 eV, respectively. These high \({E}_{\text{g}}\) values may be due to the dispersion effect at the ωB97X–D level compared to some DFT functionals [54, 86, 87]. The substitution with NH2 in the case of Chal3 decreases the \({E}_{\text{g}}\) with 1 eV compared to Chal1 at the PBE0 level.\({E}_{\text{g}}\) showed the same variation in the case of Chal4, where the OCH3 group was present. The presence of the push–pull system, which is represented by the OCH3 as donor and Cl as attractor (Chal5), decreases the \({E}_{\text{g}}\) value and increases the \({\beta }_{\text{tot}}\) by 68.48% compared to Chal1, which confirms the significant effect of the donor–acceptor group on enhancing the NLO properties. The presence the Br in the case of Chal6 decreases the \({E}_{\text{g}}\) value and increase considerably the \({\beta }_{\text{tot}}\) by 62.42% compared to Chal1. In the case of Chal7 or the presence of Cl alone, a slight decrease in the \({E}_{\text{g}}\) followed by a significant increase in \({\beta }_{\text{tot}}\), but not like the other Chals, clearly shows the crucial effect of the substitution on the \({E}_{\text{g}}\) and NLO properties. Study [49] highlighted the chlorination influence on the photophysical and photovoltaic properties and the decreasing effect of both HOMO and LUMO simultaneously in donor and acceptor molecules. The presence of donor and attractor electron groups alters the nature of delocalization and consequently affects the electronic properties of the Chals [83]. The analysis of FMO surfaces shows that the HOMOs of Chals are more localized on the cinnamoyl group (Fig. 11), whereas strong delocalization over the whole Chals was shown in the LUMO isosurfaces. The same distribution was demonstrated by Xue et al. [83] for similar Chals, and in the biphenyl based difluorochalcones ((2E)-1-[4-(2,4-difluorophenyl)phenyl]3-arylprop 2-en-1-ones) [88]. The FMO isosurfaces of the selected Chals demonstrate the π-type MO characteristics (Fig. 11). Similar Chals [69] demonstrate the same orbital natures. For this type of Chals, the relatively small \({E}_{\text{g}}\) obtained at the PBE0 approach and high dipole moments contribute to high NLO responses. Changes in the electronic structure, such as alterations in orbital energies or the presence of electronic states near the band edges, can influence both binding energy and \({E}_{\text{g}}\). The functional groups that donate or withdraw electrons can affect both its binding affinity and the energy levels of molecular orbitals [89,90,91,92,93,94].

Fig. 10
figure 10

\({\beta }_{\text{tot}}\) (upper panel) and \({E}_{\text{g}}\) (lower panel) of Chal1–Chal7 calculated using five levels at the 6-311+G** basis set

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Fig. 11
figure 11

HOMO, LUMO, and \({E}_{\text{g}}\) in eV of Chal1–Chal7 estimated by five DFT functionals

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To bolster our noticed trends in linear properties and NLO properties (EHOMO, ELUMO, \({E}_{\text{g}}\), \(\mu\) and \(\alpha\), and \(\beta\)) as well as in NLO properties (\(\beta\)). Here, we have correlated our observed results for linear and NLO properties with the global relativity descriptors of Chal5, such as chemical hardness (η) [95], chemical potential (μ), chemical softness (S), electronegativity (χ), electrophilic index (ω), and nucleophilicity index (N) [96, 97] (Table 3). The small \({E}_{\text{g}}\) of Chal5 (2.16 eV) indicates that charge transfer easily occurs in it, which influences the biological activity of the molecules [98]. While the EHOMO and ELUMO values are all negative, confirming this derivative's stability [99]. A molecule with a small \({E}_{\text{g}}\) is more polarized and characterized by good bioactivity, high chemical reactivity, and low kinetic stability, known as a soft molecule [100, 101]. The stability of Chal5 is validated by the chemical hardness, which is 2.76 eV. The computed significant η value and the small \({E}_{\text{g}}\) value for Chal5 exhibits promising charge transfer in this Chal. ω represents the energy reduction measure caused by the maximum charge transfer between the acceptor and donor [102]. ω is related to the stabilization of energy when the system gains an additional electronic charge from the surroundings and quantifies the global electrophilic power of the molecule [103]. These descriptors were evaluated from the following equations:

$$\begin{aligned} & A = \, -\;{E_{{\text{LUMO}}}};I = \, -{E_{{\text{HOMO}}}}\;;V = \, \left( {I + A} \right)/{2 };S = { 1}/{2}\eta \\ & \eta = \, \left( {-A + I} \right)/{2 };\chi = \, \left( {I + A} \right)/{2 };\omega = {V^{2}}/{2}\eta ;N = { 1}/\omega \\ \end{aligned}$$
Table 3 \({E}_{\text{HOMO}}\), \({E}_{\text{LUMO}}\), \({E}_{\text{g}}\), I, A, η, χ, ω in eV, S, V, and N in eV−1 of Chal5 computed at five DFT approaches using the 6-311+G** basis set
Full size table

According to the absolute scale of electrophilicity based on the ω index, Chal5 may be classified as strong electrophiles with ω > 1.5 eV, moderate electrophiles with 0.8 < ω < 1.5 eV, and marginal electrophiles with ω < 0.8 eV [104]. The higher values of ω described Chal5 as a good electrophile. The strong electrophilic nature of Chal5 supports their usability as bioactive substances [105]. For this purpose this part has been added, as this information will be valuable for examining the biological activity of Chal5 for its exciting NLO properties.

Conclusion

The current study describes for the first time seven chalcone-based derivatives where the substituent has a significant electronic effect that modulates some overall molecular properties (\(\mu\), \(\alpha\), \(\beta\), and \({E}_{\text{g}}\)), which was corroborated by combining theoretical results. Various factors, including molecular geometry, position, number of substitutions, and choice of computational level, influence the NLO parameters. The order of \(\mu\) and \(\alpha\) varies based on the different approximations and treatments of the electronic interactions and exchange–correlation effects implemented by each functional. Different methods have various levels of sophistication and accuracy in describing electronic interactions, giving different predictions for linear and NLO parameters. The study's functionals also differ in capturing the electronic structure and molecular interactions, leading to variations in the predicted NLO parameters. We carefully consider these factors and validate the results using multiple computational approaches (B3LYP, PBE0, M06-2X, CAM-B3LYP, and ωB97X-D) and compare them with experimental data to enhance the reliability of our findings. We found that these Chals get low \({E}_{\text{g}}\) and possess efficient \({\beta }_{\text{tot}}\) and \({\beta }_{\text{HRS}}\) responses. The PBE0 functional gives the highest \({\beta }_{\text{HRS}}\) and \({\beta }_{\text{tot}}\) values and is the most satisfying functional for predicting the NLO properties of these Chals. Chal5 has shown superior NLO properties as compared to other selected Chals. In addition to the direct link between the \({\beta }_{\text{HRS}}\) and \({\beta }_{\text{tot}}\), we also indicate the opposite variations between the \({\beta }_{\text{HRS}}\) and \({\beta }_{//}\). An inverse relationship has been obtained between the \({\beta }_{\text{tot}}\) and \({E}_{\text{g}}\). This quantum chemical exploration of Chals with high hyperpolarizabilities will direct numerous scientists to use computational data for rational design of novel chalcone-based materials with potential applications in optoelectronic devices, such as light-emitting diodes (LEDs) and photovoltaic cells.