Static and Dynamic Electronic (Hyper)polarizabilities of Dimethylnaphthalene Isomers: Characterization of Spatial Contributions by Density Analysis

Static and frequency-dependent electronic (hyper)polarizabilities of the dimethylnaphthalene (DMN) isomers were computed in vacuum using the Coulomb-attenuating Density Functional Theory method. The nonlinear optical Second Harmonic Generation (SHG) and Electro-Optical Pockels Effect (EOPE) were investigated at the characteristic Nd:YAG laser wavelength of 1064 nm. The response electric properties especially the longitudinal polarizability, polarizability anisotropy, and first-order hyperpolarizability are significantly affected by the position of the methyl groups. The SHG and EOPE techniques can be potentially useful to discriminate the α,α-DMN isomers (2,6-DMN < 2,7-DMN < 2,3-DMN) as well as the β,β-DMN isomers (1,5-DMN < 1,4-DMN < 1,8-DMN). The (hyper)polarizability differences among the investigated DMNs were elucidated through density analysis calculations. The predicted polarizabilities exhibit good linear relationships with the experimental first-order biomass-normalized rate coefficient, a physicochemical property connected to the rates of biodegradation processes of polycyclic aromatic hydrocarbons.

The main focus of this study is to determine the static and frequency-dependent electronic and values of the series of the DMN isomers, aiming to explore the effects of the position of the CH 3 groups on these electric properties, potentially helpful for the isomeric discrimination. The electronic (hyper)polarizabilities, are commonly predicted by means of ab initio and/or Density Functional Theory (DFT) computations. However, as well-known in the literature for an accurate determination of the electronic (hyper)polarizabilities, the choice of the functional is critical, especially in the case of -conjugated compounds [33]. In fact, on the whole, the conventional DFT methods tend to systematically overestimate the electronic (hyper)polarizabilities obtained by highlevel correlated ab initio levels. On the other hand, the longrange corrected DFT methods incorporating nonlocal effects [34,35], describe adequately the diffuse regions of the charge distributions, giving much more satisfactory performances for the prediction of the response electric properties. In the present study we used the Coulomb-attenuating hybrid exchange-correlation functional (CAM-B3LYP) [36], which has been recently employed with success for computing electronic (hyper)polarizabilities of organic compounds [37][38][39][40][41][42][43][44][45][46][47][48].

Basis Set and Level of Calculation Effects: Response Electric
Properties of Toluene. For an accurate prediction of the electronic (hyper)polarizabilities, the choice of the basis set and level of computation is of crucial importance [57][58][59][60][61][62][63][64][65][66][67]. In the present study we explored the effects of the basis set and theoretical level on toluene as a test case. Table 1 reports the ⟨ ⟩, Δ , and vec values of toluene calculated using the CAM-B3LYP and MP2 levels with the 6-31+G * and Sadlej's POL basis set. The latter basis set was specifically constructed for polarizability computations and has been recently employed with success to predict the electronic polarizabilities of naphthalene (N) [68]. However, it is well-demonstrated that for -conjugated compounds the smaller 6-31+G * basis set furnishes an adequate alternative to the POL as well as more extended basis sets for predicting response electric properties, but at significantly minor computational costs [39,[45][46][47][48][52][53][54].
The present results show that when passing from the 6-31+G * to the POL basis set, only marginal effects are observed. In fact, the ⟨ ⟩ and Δ values increase by 0.75Å 3 (+6.5%) and 0.25Å 3 (+3.9%), respectively, whereas the vec decreases by 37.3 × 10 −53 C 3 m 3 J −2 (−13.3%). Note that the (hyper)polarizability calculations carried out using the 6-31+G * basis set require noticeably minor CPU resources than those with the POL basis set (by about a factor of twenty!).
Thus, considering the above results, the CAM-B3LYP/6-31+G * level can be judged as an acceptable compromise between accuracy and computational cost and has been entirely employed for the subsequent calculations on the static and frequency-dependent (hyper)polarizabilities of the DMN isomers. and 0.40-0.55Å 3 (+3%), respectively. Table 2 also reports the data of the unsubstituted compound N for which some experimental and high-level correlated ab initio values are available in the literature [68,70]. The static CAM-B3LYP/6-31+G * , ⟨ ⟩ and Δ values of N agree satisfactorily with both the observed (within −0.8, −2.8, and +2.3%, resp.) [70] and CCSD/POL data (within −2.0, −4.0, and +2.4%, resp.) [68].

Static and Dynamic Polarizabilities of the DMN Isomers.
Not surprisingly, the static CAM-B3LYP/6-31+G * ⟨ ⟩ values of the DMN isomers underestimate the previously calculated B3LYP/6-31+G * figures [14] by 0.42-0.51Å 3 (2.0-2.4%), principally owing to the introduction of nonlocal effects. The order of the static and dynamic CAM-B3LYP/6-31+G * values is the following: For the ⟨ ⟩ values, the above order is slightly modified, with the 1,8-DMN and 2,6-DMN isomers being, respectively, the less and more polarizable along the series: The order of the Δ values is rather similar to that found for the data, except for the inversions between 1,6-DMN and 1,7-DMN and between 2,6-DMN and 2,7-DMN: All the predicted polarizabilities concordantly increase on passing from the , -DMNs to the , -DMNs and then to the , -DMNs, in agreement with the previous ⟨ ⟩ data obtained at the B3LYP/6-31+G * level in gaseous and aqueous phases [14]. Specifically, when we compare the 1,4-DMN and 2,6-DMN isomers, the static CAM-B3LYP/6-31+G * , ⟨ ⟩ and Δ values enhance by 4.51Å 3 (+17.0%), 0.53Å 3 (+2.6%), and 3.32Å 3 (+24.1%), respectively. However, whereas the static ⟨ ⟩ data slightly change along the series of isomers (within 0.65Å 3 , 3.2%), the Δ and values are distributed over larger ranges being within 3.32Å 3 (24.1%) and 4.54Å 3 (17.1%), respectively. On the whole, these results suggest that, in comparison to the average polarizabilities, the and Δ properties are much more affected by the position of the methyl substituent, being potentially useful to identify the DMN isomers. Additionally, in agreement with a previous study on the average polarizabilities [14], the present and Δ data of DMNs are found to be linearly related to the biodegradation experimental biomass-normalized firstorder rate coefficients [12], confirming the crucial role of the polarizabilities in the biodegradation process of this group of organic pollutants. The / , ⟨ ⟩/ , and Δ / linear relationships are displayed in Figure 2 showing good statistics since the value is predicted between 0.97 and 1.00 (following the discussion reported in [14], the 2,7-DMN isomer was excluded from the relationships).
In order to explain the polarizability differences among the DMN isomers, we analyzed the spatial contributions of electrons to the polarizabilities by using the concept of density of polarizability (1) ( ) [71,72]. The  All calculations were carried out at the CAM-B3LYP/6-31+G * level on the B3LYP/6-31G * geometry taken from [14]. b Experimental first-order biomass-normalized rate coefficient ((mg of protein/L) −1 (h) −1 ) in aqueous systems, [12]. c [70].
to the applied fields ( is the position vector). The ( , ) is commonly expanded in powers of as For a certain positive-negative (1) ( ) pair, the sign is positive when the direction of the positive to negative densities coincides with the positive direction of the chosen coordinate system, whereas the magnitude is proportional to the distance between the two densities. In the present study, we determined the (1)  from the plots in Figure 3, the 2,3-DMN isomer exhibits additional relevant (1) ( ) densities localized on both the CH 3 groups. These methylic (1) ( ) amplitudes are much more conspicuous than those found for the 1,4-DMN isomer, contributing to the increase of the component by ca. 17%.

Static and Dynamic First-Order Hyperpolarizabilities of the DMN Isomers.
The calculated static and vec values of the DMN isomers obtained in vacuum at the CAM-B3LYP/6-31+G * level are collected in Table 3. As for the computed dipole moments [14], due to the mutual disposition of the methyl substituents, the 1,5-DMN and 2,6-DMN isomers are nonpolar compounds. In the present study, besides to the static first-order hyperpolarizabilities we also determined the frequency-dependent properties for the SHG and EOPE NLO phenomena since observed data are nearly always obtained at incident optical fields. In order to minimize resonance enhancements, the dynamic hyperpolarizabilities were evaluated at the value of 1064 nm (ℏ = 0.04282 a.u.), which is sufficiently apart from the observed lowest-energy absorption of DMNs (the experimental max values are in the range 274-289 nm) [73]. The dynamic vec (− ; 1 , 2 ) data are included in Table 3. As should be expected, vec (−2 ; , ) > vec (− ; , 0) for all the isomers. The dispersion effects enhance the static vec (0;0,0) values by 6-18% and 17-57%, for the EOPE and SHG processes, respectively, the largest increases being found for 1,4-DMN, in agreement with its highest max value among the DMN isomers.
The static and dynamic vec values concordantly predict the following order: Note that the above order is rather different from those obtained for the polarizabilities; a vec versus relationship cannot be established. In particular, the static and dynamic CAM-B3LYP/6-31+G * vec (2,3-DMN) values are calculated to be 5-6 times higher than the corresponding data obtained for the 1,4-DMN isomer. More importantly, we notice that the first-order hyperpolarizabilities of the , -DMNs are somewhat different from each other for the 2,6-DMN isomer the vec is zero since it belongs to the 2ℎ symmetry point group, whereas the vec (2,3-DMN)/ vec (2,7-DMN) ratios are predicted to be 2.5-2.7 by the present static and dynamic computations. Similarly, the vec values of the , -DMN As for the polarizabilities, we explored the hyperpolarizability differences between the 1,4-DMN and 2,3-DMN isomers through hyperpolarizability density analyses. The density, (2) ( ), is expressed as follows [71,74]: By analyzing the main contributing components, for the 1,4-DMN isomer, the largest hyperpolarizability component lies along the -axis. At the CAM-B3LYP/6-31+G * level, the static (1,4-DMN) value is calculated to be 62.5 × 10 −53 C 3 m 3 J −2 , recovering ca. 35% of the vec (0; 0, 0) value. Similarly, the component dominates the first-order hyperpolarizability of 2,3-DMN, the (0; 0, 0)/ vec (0; 0, 0) ratio being calculated to be 0.76. Thus, we computed the (2) ( ) distributions for the component at the CAM-B3LYP/6-31+G * level using a numerical procedure described in detail in [74]. Figure 4 displays the (2) ( ) densities for the 1,4-DMN and 2,3-DMN isomers. As can be seen from the graphical representations, for both the isomers, the most significant (2) ( ) amplitude is furnished by the N skeleton, although it is much more expanded in the case of the 2,3-DMN isomer. In addition, different from the 1,4-DMN isomer, for 2,3-DMN, a conspicuous positive (2) ( ) contribution is also provided by both the CH 3 groups. On the whole, the above (2) ( ) results are in qualitative agreement with the calculated static hyperpolarizabilities, the (2,3-DMN)/ (1,4-DMN) and vec (2,3-DMN)/ vec (1,4-DMN) ratios being predicted to be 14 and 6, respectively.

Conclusions
In this study, we investigated the electronic polarizabilities and first-order hyperpolarizabilities of the series of DMN isomers. The computations were performed in vacuum using the CAM-B3LYP functional and the 6-31+G * basis set. The response electric properties were obtained in the static and dynamic regimes for the SHG and EOPE NLO phenomena at the value of 1064 nm. The average polarizability varies a little along the series of isomers, whereas both the longitudinal polarizability and anisotropy of polarizability are much more affected by the position of the methyl substituents. In agreement with recent theoretical studies on the polarizabilities [14] and Raman spectra [15], linear relationships are established between the calculated polarizabilities and the experimental biodegradation rates of DMNs, confirming the important role of dispersive and/or inductive interactions for the biodegradative mechanisms of this group of substituted PAH isomers.
The static and frequency-dependent first-order hyperpolarizabilities are strongly dependent on the relative position of the CH 3 groups. This is especially evident for the , -DMN and , -DMN isomers: 2,6-DMN < 2,7-DMN < 2,3-DMN and 1,5-DMN < 1,4-DMN < 1,8-DMN. The present results suggest that some DMN isomers might be distinguished through SHG and EOPE NLO measurements. Density analysis computations are useful for qualitative interpretations of the (hyper)polarizabilities of the DMN isomers.