Introduction

The properties at the molecular or microscopic level of materials allow the description of their properties at the macroscopic level. Understanding the mechanisms that govern its properties and their expressions at the macroscopic level from molecular properties is an important requirement when developing devices with a desired property. With rapid developments of first‐principles methods in studying materials, now it is effective and efficient to predict without any experimental data at different stages. The compounds incorporating a phosphazene bond (–P = N–) are interchangeably called phosphinimines, phosphazenes, or iminophosphoranes. Under these different names are grouped different families of molecules incorporating the P = N bond, opening a wide field in the chemistry of phosphorus compounds. Many organophosphorus polymers have been studied [1,2,3], which are able to bind with metal surfaces [4,5,6], and for biological applications [7]. Their clinical utility against microbial activity gives them an increasing interest. Also, the broad range of properties they exhibit and the attached group to the phosphorus [8], make the phosphazene materials very interesting. This material type has low glass-transition temperatures, is resistant to degradation by hydrocarbons and can be used as arctic fuel lines [9]. Also, this material may have applications such as solid-state batteries [10]. Several theoretical and experimental papers [11,12,13,14] show the suitability of phosphazenes in NLO field. The phosphazene compounds like spirocyclophosphazenes studied by Maturana et al. [12] have charge transfer which mainly depends on the electron delocalization. We note that the cyclotriphosphazenes can act as host specie in the host–guest interaction system [15,16,17]. The cyclotriphosphazenes can use in polymer [16] and be used to obtain systems with transition metal [18]. The presence of the cyclotriphosphazene in the design of materials was expected to increase the flame resistance and the thermal stability in polyphosphazenes [19,20,21,22] such as the hexakis(4–aminophenoxy)cyclotriphosphazene-based polyimide matrices [22]. Cyclotriphosphazenes belong to a growing family of channel-type inclusion compounds [23, 24]. Indeed, the tris(o–phenylenedioxy)cyclotriphosphazene can act as channel in supramolecular-wire confinement of I2. Nevertheless, the most important works, which interest us, have concerned the chlorocyclotriphosphazene group participation to increase the NLO properties in materials [25,26,27]. Following Kang et al. [28], the large polarizability anisotropy \(\Delta \alpha \) and the first-order hyperpolarizability β of our organocyclo–phosphazenes can reflect the macroscopic birefringence and SHG like phosphazenes with skeletal architectures –P = N–P = N–P = N–, and like cyano-based materials. The recent works of phosphazene [12, 13, 29, 30] showed the suitability of phosphazenes and organophosphazenes for the NLO applications. In this study, we investigated the inorganic ring hexachlorocyclotriphosphazene (HCCP), organic comonomer tris(4–hydroxyphenyl) ethane (THE), and their derived core organo–cyclotriphosphazene structure series where the cyclotriphosphazene ring substituted with THE at different positions [31]. These compounds are noted (Fig. 1): HCCP (hexachlorocyclotriphosphazene), THE (tris(4–hydroxyphenyl) ethane), PCCP–THE (pentachclorocyclotriphosphazene–co–tris(4–hydroxyphenyl) ethane)), TCCP–gem2THE (tetrachclorocyclotriphosphazene–co–gem2(tris(4–hydroxyphenyl) ethane, TCCP–2THE (tetrachclorocyclotriphosphazene–co–2(tris(4–hydroxyphenyl) ethane), TCCP–3THE: trichlorocyclotriphosphazene–co–3(tris(4–hydroxyphenyl) ethane), DCCP–4THE (dichlorocyclotriphosphazene–co–4(tris(4–hydroxyphenyl) ethane)), 3PCCP–THE 3(pentachclorocyclotriphosphazene)–co–tris(4–hydroxyphenyl) ethane.

Figure 1
figure 1

Structures of the studied cyclophosphazene, tris 4–(hydroxyphenyl) ethane and their derived organocyclo–phosphazenes

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Methods

Computational chemistry methods are used to predict the linear and NLO properties of organocyclo–phosphazenes. The study reveals that cyclotriphosphazene compounds linked to tris(4–hydroxyphenyl) ethane groups are a remarkably large first-order NLO response. The results obtained from the density functional theory (DFT) will provide the electronic properties of this important class of phosphazenes. At first time, the geometry optimizations for all organocyclophosphazenes are performed within DFT method at the B3LYP functional using the 6–311 + G(d) basis set. No constraints on geometries were implemented. Real values for all vibrational frequencies confirmed that the optimized geometries of the studied phosphazenes correspond to minima on the potential energy surface. All linear (dipole moment µ, mean polarizability \(\langle \alpha \rangle \), and anisotropy polarizability \(\Delta \alpha \)) and NLO properties (gas phase electric field-induced second harmonic generation (EFISHG) \( \beta _{{//}} \left( { - 2\omega ;\omega ,\,\omega } \right) \), the hyper–Rayleigh scattering (HRS) first hyperpolarizability βHRS (− 2ω; ω, ω), and the depolarization ratios (DR)) are calculated in the gas phase at different DFT levels (B3LYP, ωB97X–D, CAM–B3LYP, and M05–2X functionals [32,33,34,35,36,37]) in association with the 6–311 + G(d) and aug–cc–pVDZ basis sets. The efficiency of range separated and DFT hybrid functionals in polarizability and first hyperpolarizability calculations is studied less extensively. Hyperpolarizabilities considerable improvements were reported when the fraction of Hartree–Fock exchange in the hybrid functional increased up to 56% [38,39,40]. In Fig. 2, we present the optimized geometries of the cyclophosphazene and their derived organocyclo–phosphazenes at the B3LYP/6–311 + G(d) level.

Figure 2
figure 2

Optimized geometries of the cyclophosphazene and their derived organocyclo–phosphazenes at the B3LYP/6–311 + G(d) level

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The total dipole moment was defined as follows:

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

Using the polarizability components obtained by the Gaussian out file, the average polarizability \(\langle \alpha \rangle \) was calculated using the polarizability components by the formula

$$\langle \alpha\rangle =\frac{1}{3}\sum_{i = x,y,z}{\alpha }_{ii}$$
(2)

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

$${\Delta \alpha }=\sqrt{\frac{1}{2}\left[\left({\left({\alpha}_{\text{xx}}-{\alpha}_{\text{yy}}\right)}^{2}+{\left({\alpha}_{\text{xx}}-{\alpha}_{\text{zz}}\right)}^{2}+{\left({\alpha}_{\text{yy}}-{\alpha}_{\text{zz}}\right)}^{2}\right)\right]}$$
(3)

The βijk components were calculated using the finite field approach as implemented in Gaussian 09 [41]. The two properties \({\beta }_{\text{HRS}}\)(–2ω;ω,ω) and \({\beta }_{//}\)(–2ω;ω,ω) can be deduced from the EFISHG measurements. The \({\beta }_{\text{HRS}}\)(–2ω;ω,ω) is here related to the HRS intensity for non-polarized incident light and observation of plane-polarized scattered light made perpendicularly to the propagation plane. The HRS method allows obtaining different independent tensor components of the first hyperpolarizability, which is not possible by EFISHG [42].

$${\beta }_{//}\left(-2\upomega ;\upomega ,\upomega \right)={\beta }_{//}=\frac{1}{5}\sum_{\text{i}}\frac{{\upmu }_{\text{i}}}{\left|\overrightarrow{\upmu }\right|}\sum_{\text{j}}\left({\beta }_{\text{ijj}}+{\beta }_{\text{jij}}+{\beta }_{\text{jji}}\right)$$
(4)

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

and so,

$$ \beta_{//} = \frac{3}{5}\sum\limits_{i = x,y,z} {\frac{{\mu_{i} \beta_{i} }}{{\left| {\left. {\vec{\mu }} \right|} \right.}}} $$
(5)

\(\left|\overrightarrow{\mu }\right|\) is the norm of the dipole moment.

\({\mu }_{\text{i}}\) and \({\beta }_{\text{i}}\) are the ith components of the µ and \(\beta \) vectors, respectively.

In HRS, the depolarization ratios DR were also calculated. For a non-polarized incident signal, and for the reason that both polarizations are detected with equal sensitivity, the orientational average over \(\beta \) is

$$ \left| {\left. {\beta_{{{\text{HRS}}}} } \right|} \right. = \sqrt {\left\langle {\left. {\beta_{{{\text{HRS}}}}^{2} } \right\rangle } \right.} = \sqrt {\left\langle {\left. {\beta_{{{\text{ZZZ}}}}^{2} } \right\rangle } \right. + \left\langle {\beta_{{{\text{ZXX}}}}^{2} } \right\rangle } $$
(6)

and

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

The \(\langle {\beta }_{\text{ZZZ}}^{2}\rangle \) and \(\langle {\beta }_{\text{ZXX}}^{2}\rangle \) are orientational averages of the \(\beta \) tensor components, describing the HRS intensities when the incident light is vertically or horizontally polarized, respectively. These two invariants \(\langle {\beta }_{\text{ZZZ}}^{2}\rangle \) and \(\langle {\beta }_{\text{ZXX}}^{2}\rangle \) are calculated without assuming Kleinman’s conditions. Their full expressions can be find in Ref. [43]

Results and discussions

Linear and NLO properties

Dipole moment

Figure 2 lists the relevant optimized geometries of the studied phosphazenes HCCP, THE, PCCP–THE, TCCP–3THE, TCCP–gem2THE, DCCP–4THE, TCCP–2THE, and 3PCCP–THE optimized at the B3LYP/6–311 + G(d,p) level. In Table 1, we summarize the dipole moment values of our phosphazenes performed at DFT level for the geometry optimized at the B3LYP/6–311 +  + G(d,p) level of theory. The accurate calculation of dipole moment, polarizability and first hyperpolarizability requires extended basis sets. Indeed, it is obvious that for the computational reasons, moderate size basis sets are preferable [45,46,47]. In our study, the choice of these two 6–311 + G(d,p) and aug–cc–pVDZ basis sets is to find a good compromise between precision and calculation needs. Nandi et al. showed in several studies [48,49,50] of the hyperpolarizability of medium size molecules that the 6–311 +  + G(d,p) basis set can give almost identical results of \(\beta \) as obtained with the Dunning’s correlation consistent basis set higher angular momentum diffuse functions. High µ values were obtained for the TCCP–3THE and PCCP–THE (Table 1, Fig. 4, and Fig. 5). On the other hand, for TCCP–gem2THE and 3PCCP–THE, low values have been obtained compared to the µ values for the TCCP–3THE and PCCP–THE. Because of their high symmetry confirmed by the D3h symmetry group, and the planar cyclic arrangement [51], the HCCP shows zero µ values. This µ value was confirmed in our recent study [52]. The THE and TCCP–3THE are octupolar by symmetry, and their symmetry group is C3. In the case of (PCCP–THE, TCCP–gem2THE, TCCP–2THE, DCCP–4THE, and 3PCCP–THE), and after the optimization of geometric structures, the reorientation of THE groups linked to the HCCP cycle creates asymmetric structures of these compounds, and then, these compounds belong to the C1 point group. The B3LYP and ωB97X–D DFT functionals give the highest µ values in comparison with the M05–2X and CAM–B3LYP functionals (Table 1 and Fig. 3). The B3LYP/aug–cc–pVDZ \(\mu \) values of the TCCP–3THE which gets the high \(\mu \) values differ by about 31%, 70%, 14%, 41%, 41%, and 63% for TCCP–2THE, TCCP–gem2THE, PCCP–THE, DCCP–4THE, THE, and 3PCCP–THE, respectively. The obtained \(\mu \) values at the B3LYP/aug–cc–pVDZ level evolve in the following order: (µ = 0) HCCP < µ TCCP–gem2THE < µ 3PCCP–THE < µ THE < µ DCCP–4THE < µ TCCP–2THE < µ PCCP–THE < µ TCCP–3THE.

Table 1 B3LYP, ωB97X–D, CAM–B3LYP, and M05–2X dipole moment µ, mean polarizability \(\langle {\alpha }\rangle \), polarizability anisotropy \(\Delta \alpha\), EFISHG first hyperpolarizability \({\beta }_{//}\), HRS first hyperpolarizability \({\beta }_{\text{HRS}}\), and the depolarization ratios DR (in parentheses) for the HCCP and their derived materials
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Figure 3
figure 3

Mean polarizablility \(\left\langle {\left. \alpha \right\rangle } \right.\) of HCCP, THE, PCCP–THE, TCCP–gem2THE, TCCP–3THE, TCCP–3THE, DCCP–4THE, and 3PCCP–THE compounds determined at different levels of approximation using the 6–311 + G(d) (a) and aug–cc–pVDZ basis set (b)

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Theoretical papers [52, 53] confirmed the null dipole moment for the HCCP. For similar chair-like phosphazenes ((PHNH)3, (PFNF)3, (PClNCl)3 and (PBrNBr)3), theirs µ values ranged from 1.23, 2.28, 2.73, and 3.09 D, respectively. The calculated µ value (3.69 D) for the THE at the CAM–B3LYP/aug–cc–pVDZ level is in close agreement with the theoretical data (3.67 D) showed by Zheng et al. [44] for the diazonaphthoquinone photoactive compound (DNQ PAC)-based THE compound calculated using the PW91 level. For all studied phosphazenes, the augmented correlation consistent basis set aug–cc–pVDZ gives highest µ values compared to the 6–311 + G(d) basis set. Polyphosphazene/polynitrile copolymers present µ comparable to polyphosphazenes, and to the one of standard push–pull systems [54].

Polarizability

In phosphazenes, the highly polarized PN backbone of the phosphazene ring [8], and the direct relation with the ligand electronegativity on the phosphorus atom [54], makes our phosphazenes carry high polarizability values. In the PN bond, the phosphorus d orbitals play a very important role in the formation of the PN bonds. The system \({d}_{\uppi }(\text{P})-{p}_{\uppi }(\text{N})\) has been generally accepted and confirmed by several studies [55,56,57]. From the results presented in Table 1 and Fig. 3, we note that the DFT functionals give higher mean polarizabilities \(\langle \alpha\rangle \) and \(\Delta \alpha \) values of our studied phosphazenes except the HCCP.

The mean polarizability \(\langle \alpha \rangle \) values of our phosphazenes showed that the DCCP–4THE gets the high \(\langle \alpha \rangle \) values (1099.41 Arb. Units) compared to other phosphazene compounds. This phosphazene is connected by four THE. Similar trend has been obtained for the 3PCCP–THE, TCCP–gem2THE, and TCCP–2THE. Their values at the B3LYP/aug–cc–PVDZ level are 638.86, 629.31, and 627.11 Arb. Units, respectively. The B3LYP/aug–cc–PVDZ level gives the high \(\langle \alpha \rangle \) values. Theoretical studies [14, 52] show the reliability of the B3LYP functional for the prediction of polarizability for this compounds type. The results show that the number of THE affects the \(\langle \alpha \rangle \) values. High \(\Delta \alpha\) values (239.54 Arb. Units) have been obtained for the DCCP–4THE using the CAM–B3LYP/aug–cc–PVDZ level. Using the ωB97X–D/aug–cc–PVDZ level, the lowest \(\Delta \alpha\) values (0.22 and 52.65 Arb. Units) have been obtained for HCCP and 3PCCP–THE, respectively. A little difference has been obtained for the HCCP values using the DFT levels, and their values ranged from 0.15 to 1.89 Arb. Units. Semi-empirical \(\langle \alpha \rangle \) calculation of similar cyclic phosphazenes [30] shows an acceptable agreement to our \(\langle \alpha \rangle \) value for the HCCP. The results show that the mean polarizability is driven by the size of the phosphazenes, but the polarizability anisotropy presents more variations as a function of the phosphazenes. Our recent papers [58,59,60,61] showed the same assessment. We note that the B3LYP/aug–cc–pVDZ level gives the high \(\langle \alpha \rangle \) values compared to other levels used in this study. The calculated \(\langle \alpha \rangle \) value of the DCCP–4THE at the B3LYP/aug–cc–pVDZ level which is the high \(\langle \alpha \rangle \) value differs by about 40%, 36%¸43%¸42%, 62%, 76%, and 83% for 3PCCP–THE, TCCP–3THE, TCCP–2THE, TCCP–gem2THE, PCCP–THE, THE, and HCCP, respectively. For \(\Delta \alpha \), the calculated value of the DCCP–4THE at the same B3LYP/aug–cc–pVDZ level which is the obtained high \(\Delta \alpha \) value differs by about 77%, 3%¸ 53%¸ 46%, 51%, 67%, and 99% for 3PCCP–THE, TCCP–3THE, TCCP–2THE, TCCP–gem2THE, PCCP–THE, THE, and HCCP, respectively. At the same level, the \(\alpha\) values evolve in the following order: \(\langle \alpha \rangle \) HCCP < \(\langle \alpha \rangle \) THE < \(\langle \alpha \rangle \) PCCP–THE < \(\langle \alpha \rangle \) TCCP–2THE < \(\langle \alpha \rangle \) TCCP–gem2THE < \(\langle \alpha \rangle \) 3PCCP–THE < \(\langle \alpha \rangle \) TCCP–3THE < \(\langle \alpha \rangle \) DCCP–4THE and \(\Delta \alpha\) HCCP < \(\Delta \alpha\) 3PCCP–THE < \(\Delta \alpha\) THE < \(\Delta \alpha\) TCCP–2THE < \(\Delta \alpha\) PCCP–THE < \(\Delta \alpha\) TCCP–2THE < \(\Delta \alpha\) TCCP–3THE < \(\Delta \alpha\) DCCP–4THE.

Hyperpolarizability

The results shown in Table 1 present gas phase EFISHG \({\beta }_{//}\), the HRS first hyperpolarizability \({\beta }_{\text{HRS}}\) and the depolarization ratios (DR) for all investigated compounds; the values of the hyperpolarizability contributions of Gaussian 09 output are reported in Arb. Units.

The passage from the traditional B3LYP functional to the long-range functional CAM–B3LYP increased the hyperpolarizability \({\beta }_{\text{HRS}}\) of these compounds by 0.5 to 3% (Fig. 4 and Fig. 5). Comparing the obtained \({\beta }_{\text{HRS}}\) values using the B3LYP and CAM–B3LYP levels at the same 6–311 + G(d) basis set, the \({\beta }_{\text{HRS}}\) are increased from (382.74 to 384.65 Arb. Units) and from (354.01 to 356.10 Arb. Units) for the TCCP–2THE and TCCP–3THE, respectively. It is now well recognized that the electron correlation may strongly influence the values of hyperpolarizability [62,63,64,65,66]. The \({\beta }_{//}\) and \({\beta }_{\text{HRS}}\) results show also that M05–2X values using aug–cc–pVDZ basis set are higher. Further, it is surprising that the choice of the DFT functional is very required to get the correct tendency of β. According to the phosphazenes surveyed in Table 1, Fig. 4, and Fig. 5, there is little difference among the \({\beta }_{\text{HRS}}\) values calculated using the B3LYP, CAM–B3LYP, and M05–2X functionals, while the ωB97X–D level shows a somewhat greater propensity to deliver results divergent from these DFT levels. For \({\beta }_{//}\), there is also little difference between the calculated values using the B3LYP, CAM–B3LYP, and M05–2X functionals, while the ωB97X–D shows a weaker propensity except for PCCP–THE where the difference is clearly marked. The calculated \({\beta }_{\text{HRS}}\) value of the DCCP–4THE at the ωB97X–D/aug–cc–pVDZ level which is the obtained high \({\beta }_{\text{HRS}}\) value differs by about 20%, 23%¸ 29%¸ 30%, 52%, 73%, and 82% for TCCP–2THE, TCCP–3THE, TCCP–gem2THE, PCCP–THE, THE, 3PCCP–THE, and HCCP, respectively. For \({\beta }_{//}\), and at the same ωB97X–D/aug–cc–pVDZ level, the calculated \({\beta }_{//}\) value of the TCCP–2THE differs by about 18%, 28%¸ 80%¸ 96%, 99%, 100%, and 167% for DCCP–4THE, THE, TCCP–3THE, 3PCCP–THE, TCCP–gem2THE, HCCP, and PCCP–THE, respectively. We can note that at the ωB97X–D/aug–cc–pVDZ level, the \({\beta }_{\text{HRS}}\) decreases in the order: \({\beta }_{\text{HRS}}\) DCCP–4THE > \({\beta }_{\text{HRS}}\) TCCP–2THE > \({\beta }_{\text{HRS}}\) TCCP–3THE > \({\beta }_{\text{HRS}}\) > \({\beta }_{\text{HRS}}\) PCCP–THE > \({\beta }_{\text{HRS}}\) THE > \({\beta }_{\text{HRS}}\) 3PCCP–THE > \({\beta }_{\text{HRS}}\) HCCP and the \({\beta }_{//}\) decreases in the order: \({\beta }_{//}\) TCCP–2THE > \({\beta }_{//}\) DCCP–4THE > \({\beta }_{//}\) THE > \({\beta }_{//}\) TCCP–3THE > \({\beta }_{//}\) 3PCCP–THE > \({\beta }_{//}\) TCCP–gem2THE > (\({\beta }_{//}\) = 0) HCCP > \({\beta }_{//}\) PCCP–THE.

Figure 4
figure 4

\({\beta }_{\text{HRS}}\) (a) and \({\beta }_{//}\) (b) of HCCP, THE, PCCP–THE, TCCP–gem2THE, TCCP–3THE, TCCP–3THE, DCCP–4THE, and 3PCCP–THE compounds determined at different levels of approximation using the 6–311 + G(d) basis set

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Figure 5
figure 5

\({\beta }_{\text{HRS}}\) (a) and \({\beta }_{//}\) (b) of HCCP, THE, PCCP–THE, TCCP–gem2THE, TCCP–3THE, TCCP–3THE, DCCP–4THE, and 3PCCP–THE compounds determined at different levels of approximation using the aug–cc–pVDZ basis set

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The results from our calculations for the gas phase EFISHG \({\beta }_{//}\) and the HRS first hyperpolarizability \({\beta }_{\text{HRS}}\) suggest these compounds as a potential compound for NLO applications. Dendritic NLO chromophores-based THE have been synthesized by Choi et al. [67]; the authors showed that these compound-based THE are thermally stable up to 275 °C in a nitrogen atmosphere, showed as large as 19 pm/V electro–optic coefficient for the poled film and had a glass transition temperature of 76 °C. The dependence between the dipole moment μ and the hyperpolarizability β (M05–2X with the two 6–311 + G(d) and aug–cc–pVDZ basis sets shown in Table 1) is well established. Indeed, a high value of the first-order hyperpolarizability β corresponds to a high value of the dipole moment μ. This high value of the first-order hyperpolarizability β gives rise to large second-order effects, which has applications in electro-optic devices. There is not a large difference between the two functionals CAM–B3LYP and M05–2X for the values of all calculated properties. Several theoretical [53] and experimental [68, 69] studies show the efficiency of phosphazenes for NLO applications. Jacquemin has shown the efficiency of polyphosphazene polynitrile alternating copolymers in their theoretical study [53], using the Møller–Plesset perturbation theory MP2 with the 6–31G(d) basis set. He reported the evolution with the chain length of the geometry, charges, dipole moments, polarizability, and first hyperpolarizability. Theirs results showed that the polyphosphazene copolymers present larger dipole moment, polarizability, and first hyperpolarizability. Li et al. [69] showed that polyphosphazenes containing indole-based dual chromophores have demonstrated the second order NLO property. Our theoretical study is likely to be of value in guiding subsequent efforts toward the reliable predictive calculation of linear and NLO responses for multifunctional phosphazenes-based cyclotriphosphazene of interest to experimentalists. It appears through the study of this series of phosphazenes that the symmetry plays a lot on the NLO response of compounds. Indeed, the depolarization ratios DR cover a broad range of dipolar and octupolar characters, which is directly depends on the symmetry. For our studied phosphazenes, the HCCP and TCCP–3THE are octupolar by symmetry, and their symmetry groups are D3h and C3, respectively. The DR is exactly equal to 1.5 for HCCP and close to 1.5 for TCCP–3THE. The THE is octupolar by symmetry with C3 symmetry group, and present intermediate values varied from 2.8 to 3.2. The dominant hyperpolarizability component for octupolar molecule is βxyz, which is in good agreement with our βxyz and DR values. The 3PCCP–THE gets the lowest DR values (DR = 0.45 to 0.5) at the four DFT levels using the 6–311 + G(d) and aug–cc–pVDZ basis sets. The dipolar character of the TCCP–gem2THE increase when adding a THE group (DCCP–4THE), their DR values are ranged from 4.05 to 4.05. For the PCCP–THE, and compared to HCCP, the octupolar character decreases when adding a THE group (DR = 3.18 to 3.64). The TCCP–2THE present intermediate values (DR = 2.45 to 2.82), it is dominantly octupolar.

Frontier Molecular Orbitals (FMOs) and their energy gaps

The largest and smallest values of the gaps are obtained for HCCP and TCCP–3THE, respectively (Table 2 and Table 3). The corresponding gaps and \({\beta }_{\text{HRS}}\) values are 5.72, 81.05 and 3.15 eV, 383.01 Arb. Units, respectively. Good agreement has been obtained between our energy gap value obtained at the M05–2X/aug–cc–pVDZ level for the HCCP (Table 2) and that obtained by Zoghaib et al. [70], the difference is 6%. The deviations with respect to experiment reported in our recent paper [71] amount to 15%. Direct correlation has been found between the energy gaps and \({\beta }_{\text{HRS}}\). The results show an inverse relationship between the Δε and \({\beta }_{\text{HRS}}\). Similar studies show the same finding [59, 72,73,74,75,76].

Table 2 The HRS first hyperpolarizability \({\beta }_{\text{HRS}}\) in Arb
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Table 3 The HRS first hyperpolarizability \({\beta }_{\text{HRS}}\) in Arb
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The shapes of the LUMO orbital have a π character, and their electron density is centered on the cyclotriphosphazene ring in all studied derivatives (Fig. 6). For the HOMO orbital, the delocalization of the molecular orbital appeared mostly on the tris(4–hydroxyphenyl) ethane group in all the studied compounds (Fig. 6). This suggests the important role of the cyclotriphosphazene ring in the optical properties of its partially substituted derivatives. The intramolecular charge transfer process guaranteed formed in large conjugated system like in the case of the DCCP–4THE. In addition, the results show that correlations are found between the increases of the \({\beta }_{\text{HRS}}\) values and the changes of the isosurfaces of the HOMO and the LUMO for these phosphazenes. The LUMO in the case of DCCP–4THE has a more diffuse on central cyclotriphosphazene and the adjacent carbon atoms, which correlates with the higher \({\beta }_{\text{HRS}}\) value compared to other studied phosphazenes. The isosurface of the HOMO of PCCP–THE is almost identical to that obtained in THE which is also consistent as small \({\beta }_{\text{HRS}}\) values. Our findings are contrast with earlier theoretical results [77] using the CAM–B3LYP/aug–cc–PVDZ, which showed that the HOMO of Li@B10H14 is identical to that of B10H14 and corresponds to small \({\beta }_{\text{HRS}}\).

Figure 6
figure 6

HOMO and LUMO orbitals of the HCCP, THE, PCCP–THE, TCCP–gem2THE, TCCP–2THE, TCCP–3THE, DCCP–4THE, and 3PCCP–THE and their Δε (in eV) estimated by the B3LYP/aug–cc–pVDZ level

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Conclusion

To discover the possibilities of phosphazenes in NLO application, we have studied electrical properties such as dipole moment µ, polarizability α, and first-order hyperpolarizability β for the THE, PCCP–THE, TCCP–3THE, TCCP–gem2THE, and DCCP–4THE selected phosphazenes. DFT studies were performed (B3LYP, CAM–B3LYP, ωB97X–D, and M05–2X functionals) extensively to study and comprehend the change in frontier molecular orbital HOMO and LUMO and their gaps of these compounds. Our results show different values from one to another DFT functionals. The calculations are made on isolated molecules and do not take into account the influence of the environment. However, the β values are strongly influenced by the surrounding solvent [78, 79]. According to the simulations that are made on these phosphazenes, we can predict if one phosphazene will be better than another, which is already very interesting for directing the synthesis toward more efficient molecules in nonlinear optic. The phosphazenes TCCP–3THE, PCCP–THE, DCCP–4THE, and TCCP–gem2THE get the high \({\beta }_{\text{HRS}}\) compared to other phosphazenes. The HCCP could not attain optical nonlinearities as high compared to other phosphazenes. The DR values of the HCCP and TCCP–3THE are exactly equal to 1.5, which are congruent with their chemical topology. An inverse relationship between the \({\beta }_{\text{tot}}\) and the \( {\text{gap}}\left| {\left. {{\text{HOMO}} - {\text{LUMO}}} \right|} \right. \) was obtained. The most effective of these phosphazenes are those which have a shorter band gap. To increase the second-order molecular NLO properties for these phosphazenes, a method consists in probing the push–pull disposition to design good NLO chromophores. This strategy was carried out in several studies [80, 81]. This study will provide new direction to the design of new materials for the NLO technologies and their applications.