Theoretical Diagnostics of Second and Third-order Hyperpolarizabilities of Several Acid Derivatives

Abstract The density functional theory (DFT) at B3LYP/6-31G(d) level has been utilized to achieve the electric dipole moment (μ), $\left( \mu \right),$static dipole polarizability (α) $\left( \alpha \right)$and first hyperpolarizability (β) $\left( \beta \right)$values for ferulic acid (1) and chenodeoxycholic acid (2). The time-dependent Hartree-Fock (TDHF) technique as a powerful quantum chemical method has been implemented to reveal the dynamic α, β $\alpha ,\,\beta $and third-order hyperpolarizabilities (γ) $\left( \gamma \right)$of the examined compounds. Our computational conclusions have been compared with the results of similar materials in the literature. The first and second frontier molecular orbitals (MOs) and their band gaps have also been investigated by means of DFT. Graphical Abstract


Introduction
To determine the magnitudes of first hyperpolarizabilities is quite important for the devices provided the second-harmonic generation (SHG) and quadratic electrooptic responses. So, the push-pull type π -electron arrangements associated with aromatic chains and unsaturated bonds and also unsymmetrically substituted donors and acceptors have been especially designed to obtain the SHG processes [1]. It has been also shown that the molecules with octupolar symmetries are among the efficient molecular materials [2]. The third-order macroscopic NLO susceptibilities are directly related to their corresponding microscopic cubic responses. The images of biological structures are provided by the thirdharmonic generation (THG) technique indicating the variations on the third-order NLO susceptibilities [3].
In this work, one of our aims is to focus on theoretically evaluating the second and third-order NLO behaviour of the title molecules in Figure 1. The m , dispersion-free a and b values have been produced using DFT calculations.
The dynamic dipole polarizabilities, quadratic and cubic hyperpolarizabilities have been also computed by means of ab-initio quantum mechanical approach (TDHF). Besides, the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied molecular orbitals (LUMOs) have been defined by DFT/ B3LYP.

Theoretical Calculations
We have firstly performed the optimization studies on the examined structures. After the geometry optimizations, we have calculated the m , static a and b for 1-2 utilizing the finite field (FF) procedure [4]. The GAUSSIAN03W [5] package program using DFT method at B3LYP/ 6-31G(d) level has carried out the computations of optimization, m , static a and b . To build the molecular models of examined compounds shown in Figure 2, GaussView [6] program has been utilized as the interface program for GAUSSIAN03W [5]. The total electric dipole moments in terms of electric dipole moment components ( first static hyperpolarizability) are evaluated as follows, respectively [7,8]:

Theoretical Calculations
We have firstly performed the optimization studies on the examined structures. After the geometry optimizations, we have calculated the  , static  and  for 1-2 utilizing the finite field (FF) procedure [4]. The GAUSSIAN03W [5] package program using DFT method at B3LYP/ 6-31G(d) level has carried out the computations of optimization,  , static  and  . To build the molecular models of examined compounds shown in Figure 2, GaussView [6] program has been utilized as the interface program for GAUSSIAN03W [5]. The total electric dipole moments in terms of electric dipole moment components ( z y x , ,    ), the orientationally averaged (isotropic) dipole polarizabilities  and the magnitudes of tot  (total first static hyperpolarizability) are evaluated as follows, respectively [7,8]: The TDHF procedure of the GAMESS [9] package program has obtained the ) The TDHF procedure of the GAMESS [9] package program has obtained the Using the following equations, we have computed the which is the vector part of the secondorder hyperpolarizability and the averaged (isotropic) second hyperpolarizability is given by: yyzz xxzz xxyy zzzz yyyy xxxx g g g g g g g The GAUSSIAN03W [5] package program at DFT/ B3LYP level with 6-31G(d) basis set has also derived the HOMOs, LUMOs and HOMO-LUMO energy band gaps. The HOMO-LUMO energy gaps ) E ( g are achieved by the following expression:  [10]. Our computed data on m of 1 (Table 1) is almost 1.6 times lower than the estimated value in Ref. [10]. The m value of 1 was calculated utilizing DFT technique by Kumar et al. as m =3.22 D [11]. The reported m of 3.22 D by Ref. [11] is in good agreement with our calculated value of 2.807 D (Table 1). Calaminici et al. calculated the dipole moment ( m =3 D) with DFT method of a phosphonic acid stilbene derivative containing the conjugated stilbene backbone, which is p-substituted by a methoxy electron donor group and p′ -substituted by a phosphonic acid electron acceptor moiety [12]. Our computed data on m for 1-2 (Table 1) are quite close to the evaluated result by Ref. [12]. It has been found that the DFT results on m values of 1-2 have given rather consistent results with the computed DFT data for similar structures in Refs. [11,12], while the other technique (ab-initio method) used in Ref.

Computational Results And Discussion
[10] yield a numerical diversity. Tables 2-4, respectively, show a few important computed components for the static dipole polarizabilities, first and second hyperpolarizabilities of 1-2. The dispersion-free a values in Table 2 have an apparent reduction in sort order 2 > 1. The static a value of a phosphonic acid stilbene derivative with a conjugated stilbene backbone was computed by DFT with a triple zeta valence basis set (TZVP) to be 41.854 × 10 -24 esu [12]. The calculated static a value reported by Calaminici et al. [12] are about 2 and 1.1 times, respectively, higher than that of 1 and 2 in Table 2. The static first hyperpolarizability of a phosphonic acid stilbene derivative containing a conjugated stilbene backbone was reported at DFT/ TZVP level to be 44.075 × 10 -30 esu by Ref. [12]. Our result on tot b for 1 (Table 3) is almost a factor of 2 lower than the presented data by Calaminici et al. [12]. To change the basis sets (TZVP) in the same method (DFT) for a similar acid reported by Ref. [12] has found out numerically discrepancies with the static a and b results of 1-2 (DFT method and 6-31G(d) basis set). It is seen from Table   4 that the dispersion-free g values show a reduction in sort order 1 > 2. Tables 5-7, respectively, give a few important calculated components for dynamic dipole polarizabilities, second and third-order hyperpolarizabilities of 1-2. The dynamic a and g values display the same reduction in sort order 2 > 1 (for a values) and 1 > 2 (for g values) as their corresponding static ones (see Tables 2,4 Table 6 are in accordance with the experimental results of similar acids in the literature reported by Refs. [13,14]. Our computed results on dynamic first hyperpolarizabilities could be compared with the result of urea which is well-known an efficiency standard in order to find out second-order NLO characterization. The        with non-zero values predict that the title compounds might acquire microscopic second and third-order NLO responses. As was to be expected from the comparisons with the urea standard, compound 1 might also possess macroscopic second-order NLO responses with non-zero values in NLO measurements. So, compound 1 could be a promising material having quadratic electro-optic responses in second-order NLO applications, such as SHG experiments. Hence, compared with compound 2, compound 1 may fulfill many of quadratic optical nonlinearity requirements and could have potential applications in NLO and electro-optic devices. Table 8 presents the computed first and second frontier MO energies and also band gaps for 1-2. Figure 3 shows the first and second frontier MOs. Since the charge transfer stimulations containing HOMO and LUMO affect the second and third-order optical nonlinearities, the lower HOMO-LUMO energy band gaps should generate the higher first and second hyperpolarizability values. In this work, the HOMO-LUMO and (HOMO-1)-(LUMO+1) energy band gaps have a reduction in sort order 2 > 1, while the calculated values of dynamic V − b and g show an inverse reduction in sort order 1 > 2 (see Tables 6-8). It is obvious that the HOMO-LUMO energy band gaps and hyperpolarizabilities introduce an opposite correlation [17]. The LUMO for 1 is localized on almost the whole molecule, while the HOMO is mainly localized on methoxy and hydroxyl groups, consequently the HOMO → LUMO transition implies an electron density transfer to aromatic part and propenoic acid of π -conjugated system from methoxy and hydroxyl group (Figure 3). The HOMO, HOMO-1, LUMO+1 for 2 are located over the benzene rings. By contrast, the LUMO of 2 is mainly located over the side chain ( Figure 3).

Conclusions
We have determined the dispersion-free and frequencydependent dipole polarizabilities, quadratic and cubic hyperpolarizabilities utilizing DFT and TDHF approaches, respectively. The microscopic second and third-order

CONCLUSIONS
We have determined the dispersion-free and frequency-dependent dipole polarizabilities quadratic and cubic hyperpolarizabilities utilizing DFT and TDHF approaches, respectively. The