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Advances in Physical Chemistry
Volume 2013 (2013), Article ID 394697, 8 pages
http://dx.doi.org/10.1155/2013/394697
Research Article

Second Harmonic Generation, Electrooptical Pockels Effect, and Static First-Order Hyperpolarizabilities of 2,2′-Bithiophene Conformers: An HF, MP2, and DFT Theoretical Investigation

Department of Chemistry, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Received 29 September 2013; Accepted 30 October 2013

Academic Editor: Miquel Solà

Copyright © 2013 Andrea Alparone. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The static and dynamic electronic (hyper)polarizabilities of the equilibrium conformations of 2,2′-bithiophene (anti-gauche and syn-gauche) were computed in the gas phase. The calculations were carried out using Hartree-Fock (HF), Møller-Plesset second-order perturbation theory (MP2), and density functional theory methods. The properties were evaluated for the second harmonic generation (SHG), and electrooptical Pockels effect (EOPE) nonlinear optical processes at the typical nm of the Nd:YAG laser. The anti-gauche form characterized by the dihedral angle of 137° (MP2/6-311G**) is the global minimum on the potential energy surface, whereas the syn-gauche rotamer ( = 48°, MP2/6-311G**) lies ca. 0.5 kcal/mol above the anti-gauche form. The structural properties of the gauche structures are rather similar to each other. The MP2 electron correlation effects are dramatic for the first-order hyperpolarizabilities of the 2,2′-bithiophenes, decreasing the HF values by ca. a factor of three. When passing from the anti-gauche to the syn-gauche conformer, the static and frequency-dependent first-order hyperpolarizabilities increase by ca. a factor of two. Differently, the electronic polarizabilities and second-order hyperpolarizabilities of these rotamers are rather close to each other. The syn-gauche structure could be discriminated from the anti-gauche one through its much more intense SHG and EOPE signals.

1. Introduction

Thiophene-based oligomers and polymers are an interesting class of π-conjugated materials for the development and construction of conductive and nonlinear optical (NLO) devices [15]. Electronic properties of -conjugated polymeric systems are significantly affected by the twisting degree of the backbone and extension of the electron delocalization although molecular structure and physicochemical properties of extended oligomers and polymers are usually modelled through smaller oligomeric chains [6, 7].

The Cα–Cα bonded bithiophene oligomer, 2,2′-bithiophene (Figure 1), is the principal building block of polythiophene chains, extensively characterized by experimental and theoretical studies. In the solid state, 2,2′-bithiophene predominantly exists as a planar anti-structure together with a nonnegligible fraction of planar syn-conformation (ca. 15%) [8]. A slightly different picture occurs in the gas phase: two nonplanar minimum-energy structures are observed to coexist (Figure 1), which are characterized by S–C2–C2’–S dihedral angles of 148° ± 3° (anti-gauche) and 36° ± 5° (syn-gauche) [9]. On the basis of electron diffraction data [9] and experimental fluorescence spectra [10], the anti-gauche is the minimum-energy form on the potential energy surface of the 2,2′-bithiophene, being more stable than the syn-gauche rotamer by  kcal/mol (enthalpy difference extrapolated from the dependence of the relative abundance in the 58–130°C range of temperatures) [10] and 0.18 kcal/mol at 100°C (energy difference) [9]. On the theoretical side, there are many contributions in the literature about relative stabilities and torsional potentials of 2,2′-bithiophene conformations [1115]. In agreement with experiment, correlated ab initio and density functional theory (DFT) levels concordantly predict both the gauche structures as stationary points on the PESs of 2,2′-bithiophene, the anti-gauche being predicted to be the global minimum [1115]. The torsional potentials for the rotation around the C2–C2’ bond are characterized by flat double-well potentials, allowing a high degree of conformational flexibility to oligothiophenes [1115]. In addition, the influence of the torsional potential on the electronic polarizabilities of 2,2′-bithiophene rotamers was analyzed by Lukeš et al. [14] using HF/aug-cc-pVDZ and MP2/aug-cc-pVDZ computations. However, the average electronic polarizabilities of 2,2′-bithiophene are little affected by the conformation, varying within 5-6% [14]. Therefore, this response electric property is little informative for identification of conformations. On the other hand, electronic first-order hyperpolarizabilities (), the related NLO electrooptical Pockels effect (EOPE), and second harmonic generation (SHG) properties are usually much more influenced by structural characteristics than electronic polarizabilities, being of potential utility for discrimination of conformers [1619].

394697.fig.001
Figure 1: Structures of thiophene and 2,2′-bithiophene rotamers. Colours: white (hydrogen), grey (carbon), and orange (sulphur).

In the present investigation, we have calculated the static and dynamic electronic (hyper)polarizabilities of the most stable conformations of 2,2′-bithiophene in the gas phase, aiming to identify physicochemical properties useful to discriminate the different rotamers. The frequency-dependent first-order hyperpolarizabilities were obtained for the SHG and EOPE NLO processes at the typical experimental wavelength of the Nd:YAG laser (1064 nm). Neither experimental nor theoretical electronic values of 2,2′-bithiophenes are known so far, whereas some computational studies were previously carried out on the first-order hyperpolarizabilities of the monomer of thiophene [2025].

2. Computational Details

The geometries of the anti-gauche and syn-gauche 2,2′-bithiophene rotamers (Figure 1) were optimized using ab initio Hartree-Fock (HF) and Møller-Plesset second-order perturbation theories (MP2) with the 6-311G** basis set. The vibrational analysis obtained under the harmonic approximation confirmed that all the investigated structures are stationary points (no imaginary frequencies). The calculations were also carried out on the thiophene molecule for comparison.

For thiophene and 2,2′-bithiophenes, we computed the dipole moments (), static and dynamic electronic polarizabilities (), first-order hyperpolarizabilities (), and second-order hyperpolarizabilities (). To this purpose, the HF and MP2 levels as well as a series of hybrid DFT methods such as B3LYP [26, 27], PBE0 [28], BH&HLYP [26], and B97-1 [29] were employed. Additionally, we investigated the performances of the long-range corrected ωB97X-D functional [30], recently employed with success for response electric property calculations [3134]. The present computations were thoroughly carried out using the polarised and diffuse Sadlej’s POL basis set [35]. There are many indications in the literature showing that this basis set is adequate for predicting response electric properties of organic compounds [31, 3639]. However, for thiophene as a test case, we also performed (hyper)polarizability calculations using the larger correlation-consistent Dunning’s triple-zeta basis set (aug-cc-pVTZ) [40]. At the HF level, the static and values were determined analytically by means of the coupled-perturbed HF (CP-HF) theory [41, 42], whereas the MP2 and DFT and data were obtained through a finite-field (FF) numerical scheme illustrated in detail by Kurtz and coworkers [43]. For the FF computations, we used a field strength amplitude of 0.005 a.u.. The accuracy of the numerical procedure was verified at the HF level by comparing the FF-HF and CP-HF (hyper)polarizability values. The static values were determined at the HF/POL level by means of the FF procedure. Frequency-dependent polarizabilities [] and first-order hyperpolarizabilities were computed for the 2,2′-bithiophenes by using the CP-HF procedure at the characteristic Nd:YAG laser wavelength () of 1064 nm ( a.u.). Specifically, the SHG [] and EOPE [] NLO processes were investigated. At this λ value, resonance enhancement effects for the SHG phenomenon are expected to be rather negligible, since the 2ω value of 0.08564 a.u. (2.33 eV) is rather far from the lowest-energy absorption of planar 2,2′-bithiophene which is observed in gas at 3.86 eV [44, 45].

As commonly used in the literature, the calculated physicochemical properties are here expressed as dipole moment (), average polarizability (), polarizability anisotropy (), first-order hyperpolarizability aligned along the direction (), and average second-order hyperpolarizability (), which are orientationally invariant quantities: where is given by For the (hyper)polarizabilities, atomic units are used throughout the work. Conversion factors to the SI are 1 a.u. of α (e2) = 1.648778 × 10−41 C2m2J−2; 1 a.u. of β (e3) = 3.206361 × 10−53 C3m3J−2; 1 a.u. of γ (e4) = 6.235377 × 10−65 C4m4J−3. All calculations were performed using the GAUSSIAN 09 program [46].

3. Results and Discussion

3.1. Geometries and Energetics of Thiophene and 2,2′-Bithiophene Rotamers

The structural parameters of the equilibrium conformations of 2,2′-bithiophene (anti-gauche and syn-gauche) as well as those of the monomer obtained in the gas phase at the MP2/6-311G** level are displayed in Figure 2. On the basis of the MP2/6-311G** calculations, the anti-gauche form is the global minimum, while the syn-gauche form is less stable by 0.51 kcal·mol−1, in excellent agreement with previous correlated ab initio results [1115] and in particular with the highest CCSD(T)/6-31G** and CCSD(T)/cc-pVDZ levels (0.49 kcal·mol−1) [13]. Thus following the MP2/6-311G** results, the anti-gauche rotamer is the prevailing form of 2,2′-bithiophene in vacuum (ca. 70%), the syn-gauche being, however, an important conformation (ca. 30%).

394697.fig.002
Figure 2: Gas phase MP2/6-311G** geometrical parameters of thiophene and 2,2′-bithiophene anti-gauche and syn-gauche rotamers. The experimental data reported in parentheses are taken from [47] (thiophene) and [9] (2,2′-bithiophene). The MP2/6-311G** (experimental) S–C2–C2’–S dihedral angles are 137° (148°, see [9]) and 48° for the anti-gauche and syn-gauche conformation, respectively.

The MP2/6-311G** geometry of thiophene is in good agreement with the experimental gas phase microwave structure [47]. In particular, the experimental C–S, C–C, and C=C bond lengths are well reproduced by the present computations. As can be appreciated from the data reported in Figure 2, the agreement between the observed [9] and calculated geometries for the 2,2′-bithiophene is slightly less satisfactorily. In particular, the present computations underestimate the experimental typical S–C2–C2’–S dihedral angle (148°) by 11° and the C–C interring bond length (1.46 Å) by ca. 0.01 Å. However, the MP2/6-311G** S–C2–C2’–S dihedral angles of 137° (anti-gauche rotamer) and 48° (syn-gauche rotamer) are in reasonable agreement with the CCSD(T)/6-31G* estimates (142° and 44°, resp.) [13]. Some additional observations can be made: (i) the equilibrium geometry of the thiophene ring shows only little differences when passing from the monomer to the dimer; (ii) the geometries of the gauche conformations are very similar to each other. Indeed, the bond lengths and angles of the anti-gauche rotamer differ, respectively, by no more than 0.01 Å and 1° from the data of the syn-gauche form.

3.2. Effects of the Basis Set and Theoretical Level on the Electronic (Hyper)polarizabilities of Thiophene

As widely documented in the literature, accurate estimates of electronic (hyper)polarizabilities need polarized and diffuse basis sets as well as introduction of electron correlation contributions [4852]. In particular for the series of thiophene (C4H4X, X = O, S, Se, Te) [23] and pyrrole (C4H4YH, Y = N, P, As, Sb, Bi) [53] homologues, the correlated MP2 method reproduces reasonably well the response electric properties obtained using high-level CCSD(T) and MP4-SDTQ calculations. In the current study, we explored the effects of the basis set and computational method on the , , and values of thiophene as a case test, for which some experimental and high-level theoretical data are known. Specifically, we compared the POL and the aug-cc-pVTZ basis sets. The POL basis set consists of [3s2p] functions for hydrogen atom, [5s3p2d] for carbon and [7s5p2d] for sulphur, giving a total of 164 basis functions on thiophene, which are ca. half of the functions for the aug-cc-pVTZ basis set ([4s3p2d] for H, [5s4p3d2f] for C, and [6s5p3d2f] for S). The calculations were performed at the HF level and the data are collected in Table 1. The results show that, when passing from the HF/POL to HF/aug-cc-pVTZ level, only marginal variations are observed on the calculated properties. Indeed, the μ, , Δα, and βμ values vary by 1.43, 0.22, 0.04, and 10.41%, respectively. It is worth mentioning that the above results are in line with previous (hyper)polarizability studies using the two basis sets [22, 31, 54]. However, it is important to notice that for thiophene the HF/aug-cc-pVTZ  hyperpolarizability computations require a CPU demand by one order of magnitude greater than that for the HF/POL calculations. Thus, the use of the POL basis set can be considered a valid compromise between accuracy and computational cost and will be entirely adopted for the subsequent MP2 and DFT calculations on thiophene and on the 2,2′-bithiophenes.

tab1
Table 1: Dipole moments (D), static electronic polarizabilities (a.u.), and first-order hyperpolarizabilities (a.u.) of thiophenea.

The effects of electron correlation as evaluated at the MP2/POL level are rather significant for the dipole moment but negligible for the polarizabilities, decreasing the HF/POL datum by ca. 0.3 D (−40%) and increasing the value by  ca. 2.5% and the value by ca. 2.9%. All the DFT methods give similar and values to each other, being in reasonable agreement with the MP2/POL data. The observed gas phase dipole moment of 0.55 D [55] is underestimated by the MP2 and DFT calculations (by 0.03–0.13 D), the smallest deviation being obtained by the ωB97X-D functional. The experimental () data, comprised between 61 and 66 a.u. (23 and 32 a.u.) [5659], are reasonably well reproduced by all the present theoretical methods, including the HF level. More importantly, the introduction of electron correlation contributions is crucial for the first-order hyperpolarizability. In fact, when passing from the HF/POL to MP2/POL level, the βμ value reduces by ca. a factor of three. The effect is especially conspicuous for the component, which decreases by one order of magnitude. Note that our HF versus MP2 comparison for thiophene agrees with those previously obtained with other basis sets [22, 23]. Unfortunately, the experimental first-order hyperpolarizability of thiophene is not available so far. However, it is of interest mentioning that the present MP2/POL βμ(thiophene) value is in good agreement with the datum previously predicted by the high-level MP4-SDTQ calculations (βμ = 13.5 a.u.) [23], showing a difference of 1.3 a.u. (−9.6%). On the other hand, similar to the HF/POL computations, all the DFT methods overestimate the MP2/POL βμ value by a factor between 3.0 and 3.4, the BH&HLYP functional giving the closest value. Note that the failure of the traditional DFT methods in the prediction of the electronic (hyper)polarizabilities especially of π-conjugated compounds is well known and has been exhaustively illustrated by Champagne and co-workers [60]. Quite surprisingly, it is worth noting that the use of the long-range corrected ωB97X-D functional does not improve significantly the performances obtained using the conventional functionals.

3.3. Static and Frequency-Dependent (Hyper)polarizabilities of 2,2′-Bithiophene Rotamers

Table 2 lists the dipole moments and static (hyper)polarizabilities of the anti-gauche and syn-gauche 2,2′-bithiophene forms calculated at the HF/POL, MP2/POL, and BH&HLYP/POL levels. By analogy to the monomer, both the 2,2′-bithiophene forms exhibit a somewhat low polarity. Nevertheless, when passing from the syn-gauche to the anti-gauche conformer, the μ value decreases by ca. a factor of two, due to the mutual disposition of the monomeric thiophene rings. Note that, although the syn-gauche conformation reveals a rotated structure, its μ value is slightly less than 2 × μ(thiophene). On the other hand, in the case of the polarizabilities, the variations between the studied rotamers are significantly minor. In fact, the static MP2/POL value for the syn-gauche form is only 0.51 a.u. smaller than the corresponding value for the anti-gauche conformation (−0.4%). For both the dimers, is the largest component, recovering ca. 50% of the total polarizability ( + + ). The MP2/POL data overestimate the experimental datum obtained in tetrahydrofuran solution by ca. 10 a.u. (+7.7%) [59]; however, they are in good agreement with those previously computed at the MP2/aug-cc-pVDZ level for conformations in the 0–180° S–C2–C2’–S dihedral angle range [14]. Note that the present values are slightly more influenced by the structure, increasing by ca. 3 a.u. (+3.6%) when passing from the syn-gauche to the anti-gauche rotamer. By analogy to the results found for the monomer, in comparison to the MP2/POL data, the HF/POL and BH&HLYP/POL methods furnish good and estimates (within 2.5–4.4% and 0.4–4.6%, resp.). Similarly to the electronic polarizabilities, the values are little dependent on the conformation, decreasing by ca. 5% when going from the anti-gauche to the syn-gauche form. In addition, as for the calculated polarizabilities, the dominant component lies along the x-axis for both the gauche structures, amounting to ca. 50% of the total second-order hyperpolarizability ( + + + + + ).

tab2
Table 2: Dipole moments (D), static electronic polarizabilities (a.u.), first-order hyperpolarizabilities (a.u.), and second-order hyperpolarizabilities (a.u.) of 2,2′-bithiophene rotamersa.

The dispersion effects (Table 3) evaluated at and 0.08564 a.u. increase the static () values of both the conformations, respectively, by ca. 2 and 7% (4 and 18%). It is of interest noting that the and data of the 2,2′-bithiophenes are greater than twice the corresponding values for the monomer, suggesting a some degree of interring -conjugation in the dimers.

tab3
Table 3: Static and frequency-dependent ( a.u.) electronic polarizabilities (a.u.) and first-order hyperpolarizabilities (a.u.) of 2,2′-bithiophene rotamersa.

Differently from the calculated polarizabilities and second-order hyperpolarizabilities, both the magnitude (Table 1) and direction (Figure 1) of the vector are noticeably affected by the structural features, almost following the behaviour found for the dipole moments. In fact, for the anti-gauche form, the property which is aligned along the -axis is predicted to be smaller than that for the syn-gauche rotamer (with aligned along the -axis) by ca. a factor of two. This result is almost independent from the theoretical level and is observed for the static first-order hyperpolarizabilities as well as for the NLO SHG and EOPE processes. As for the monomer, the introduction of the MP2 electron correlation contributions is negative for , reducing the HF/POL data by 50–60%, the largest effect being found for the syn-gauche rotamer. Similar to the HF behavior, the BH&HLYP functional overestimates the MP2/POL values by 95–165%. It is of interest to note that, at the HF/POL level when passing from the monomer to the syn-gauche 2,2′-bithiophene, the static value increases by a factor of 2.6, although the dimer exhibits a nonplanar arrangement. Note that the corresponding monomer→dimer increases obtained at the BH&HLYP/POL and MP2/POL levels are greater, being calculated to be 2.9 and 3.6, respectively.

Not surprisingly, for both the rotamers, (SHG) > (EOPE) > (static) (Table 3). The dispersion effects estimated at  a.u. increase the static values by 5-6% for the EOPE and 12–16% for the SHG NLO phenomenon, the largest percentages being predicted for the syn-gauche conformation.

4. Conclusions

The dipole moments and static and frequency-dependent electronic (hyper)polarizabilities of the anti-gauche and syn-gauche minimum-energy conformations of 2,2′-bithiophene were studied in the gas phase using ab initio HF, MP2, and DFT methods. The NLO properties for the SHG and EOPE phenomena were explored at  nm. The effects of electron correlation at MP2 level are remarkable especially for the first-order hyperpolarizabilities, reducing the HF data by 50–60%. The DFT methods, although furnishing good performance for the dipole moments and polarizabilities, significantly overestimate the MP2 first-order hyperpolarizabilities. The polarizabilities and second-order hyperpolarizabilities are little influenced by the conformation. By contrast, both the magnitude and direction of the dipole moment and first-order hyperpolarizabilities are strongly affected by the structural characteristics, the magnitudes increasing when going from the anti-gauche to the syn-gauche form by ca. a factor of two. On the basis of the present findings, the 2,2′-bithiophene rotamers might be identified through experimental SHG and EOPE NLO measurements.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

  1. J. Roncali, “Conjugated poly(thiophenes): synthesis, functionalization, and applications,” Chemical Reviews, vol. 92, no. 4, pp. 711–738, 1992. View at Scopus
  2. J. Zyss, Molecular Nonlinear Optics: Materials, Physics and Devices, Academic Press, London, UK, 1994.
  3. R. D. McCullough, “The chemistry of conducting polythiophenes,” Advanced Materials, vol. 10, no. 2, pp. 93–116, 1998. View at Scopus
  4. D. Fichou, Ed., Handbook of Oligo- and Polythiophenes, Wiley-VCH, Weinheim, Germany, 1999.
  5. A. Mishra, C.-Q. Ma, and P. Bäuerle, “Functional oligothiophenes: molecular design for multidimensional nanoarchitectures and their applications,” Chemical Reviews, vol. 109, no. 3, pp. 1141–1176, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. V. Hernandez, C. Castiglioni, M. Del Zoppo, and G. Zerbi, “Confinement potential and π-electron delocalization in polyconjugated organic materials,” Physical Review B, vol. 50, no. 14, pp. 9815–9823, 1994. View at Publisher · View at Google Scholar · View at Scopus
  7. J. L. Brédas, G. B. Street, B. Thémans, and J. M. André, “Organic polymers based on aromatic rings (polyparaphenylene, polypyrrole, polythiophene): evolution of the electronic properties as a function of the torsion angle between adjacent rings,” The Journal of Chemical Physics, vol. 83, no. 3, pp. 1323–1329, 1985. View at Scopus
  8. P. A. Chaloner, S. R. Gunatunga, and P. B. Hitchcock, “Redetermination of 2, 2′-bithiophene,” Acta Crystallographica C, vol. 50, pp. 1941–1942, 1994.
  9. S. Samdal, E. J. Samuelsen, and H. V. Volden, “Molecular conformation of 2,2′-bithiophene determined by gas phase electron diffraction and ab initio calculations,” Synthetic Metals, vol. 59, no. 2, pp. 259–265, 1993. View at Scopus
  10. J. E. Chadwick and B. E. Kohler, “Optical spectra of isolated s-cis- and s-trans-bithiophene: torsional potential in the ground and excited states,” Journal of Physical Chemistry, vol. 98, no. 14, pp. 3631–3637, 1994. View at Scopus
  11. A. Karpfen, C. H. Choi, and M. Kertesz, “Single-bond torsional potentials in conjugated systems: a comparison of ab initio and density functional results,” Journal of Physical Chemistry A, vol. 101, no. 40, pp. 7426–7433, 1997. View at Scopus
  12. S. Millefiori, A. Alparone, and A. Millefiori, “Conformational properties of thiophene oligomers,” Journal of Heterocyclic Chemistry, vol. 37, no. 4, pp. 847–853, 2000. View at Scopus
  13. G. Raos, A. Famulari, and V. Marcon, “Computational reinvestigation of the bithiophene torsion potential,” Chemical Physics Letters, vol. 379, no. 3-4, pp. 364–372, 2003. View at Publisher · View at Google Scholar · View at Scopus
  14. V. Lukeš, M. Breza, and S. Biskupič, “Structure and electronic properties of bithiophenes. I. Torsional dependence,” Journal of Molecular Structure, vol. 618, no. 1-2, pp. 93–100, 2002. View at Publisher · View at Google Scholar · View at Scopus
  15. M. A. V. Ribeiro Da Silva, A. F. L. O. M. Santos, J. R. B. Gomes et al., “Thermochemistry of bithiophenes and thienyl radicals. A calorimetric and computational study,” Journal of Physical Chemistry A, vol. 113, no. 41, pp. 11042–11050, 2009. View at Publisher · View at Google Scholar · View at Scopus
  16. J. S. Salafsky, “Second-harmonic generation as a probe of conformational change in molecules,” Chemical Physics Letters, vol. 381, no. 5-6, pp. 705–709, 2003. View at Publisher · View at Google Scholar · View at Scopus
  17. G. J. M. Velders, J.-M. Gillet, P. J. Becker, and D. Feil, “Electron density analysis of nonlinear optical materials. An ab initio study of different conformations of benzene derivatives,” Journal of Physical Chemistry, vol. 95, no. 22, pp. 8601–8608, 1991. View at Scopus
  18. E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Brédas, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” Journal of the American Chemical Society, vol. 117, no. 12, pp. 3547–3555, 1995. View at Scopus
  19. J. Lipiński and W. Bartkowiak, “Conformation and solvent dependence of the first and second molecular hyperpolarizabilities of charge-transfer chromophores. Quantum-chemical calculations,” Chemical Physics, vol. 245, no. 1–3, pp. 263–276, 1999. View at Scopus
  20. V. Keshari, W. M. K. P. Wijekoon, P. N. Prasad, and S. P. Karna, “Hyperpolarizabilities of organic molecules: Ab initio time-dependent coupled perturbed Hartree—fock—roothaan studies of basic heterocyclic structures,” Journal of Physical Chemistry, vol. 99, no. 22, pp. 9045–9050, 1995. View at Scopus
  21. S. Millefiori and A. Alparone, “(Hyper)polarizability of chalcogenophenes C4H4X (X = O, S, Se, Te) conventional ab initio and density functional theory study,” Journal of Molecular Structure: THEOCHEM, vol. 431, no. 1-2, pp. 59–78, 1998. View at Scopus
  22. S. Millefiori and A. Alparone, “Theoretical determination of the vibrational and electronic (hyper)polarizabilities of C4H4X (X = O, S, Se, Te) heterocycles,” Physical Chemistry Chemical Physics, vol. 2, no. 11, pp. 2495–2501, 2000. View at Publisher · View at Google Scholar · View at Scopus
  23. K. Kamada, M. Ueda, H. Nagao et al., “Molecular design for organic nonlinear optics: polarizability and hyperpolarizabilities of furan homologues investigated by ab initio molecular orbital method,” Journal of Physical Chemistry A, vol. 104, no. 20, pp. 4723–4734, 2000. View at Publisher · View at Google Scholar · View at Scopus
  24. K. Jug, S. Chiodo, P. Calaminici, A. Avramopoulos, and M. G. Papadopoulos, “Electronic and vibrational polarizabilities and hyperpolarizabilities of azoles: a comparative study of the structure-polarization relationship,” Journal of Physical Chemistry A, vol. 107, no. 20, pp. 4172–4183, 2003. View at Publisher · View at Google Scholar · View at Scopus
  25. B. Jansik, B. Schimmelpfennig, P. Norman, P. Macak, H. Ågren, and K. Ohta, “Relativistic effects on linear and non-linear polarizabilities of the furan homologues,” Journal of Molecular Structure, vol. 633, no. 2-3, pp. 237–246, 2003. View at Publisher · View at Google Scholar · View at Scopus
  26. A. D. Becke, “A new mixing of Hartree-Fock and local density-functional theories,” The Journal of Chemical Physics, vol. 98, no. 2, pp. 1372–1377, 1993. View at Scopus
  27. C. Lee, W. Yang, and R. G. Parr, “Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density,” Physical Review B, vol. 37, no. 2, pp. 785–789, 1988. View at Publisher · View at Google Scholar · View at Scopus
  28. J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Physical Review Letters, vol. 77, no. 18, pp. 3865–3868, 1996. View at Scopus
  29. F. A. Hamprecht, A. J. Cohen, D. J. Tozer, and N. C. Handy, “Development and assessment of new exchange-correlation functionals,” Journal of Chemical Physics, vol. 109, no. 15, pp. 6264–6271, 1998. View at Publisher · View at Google Scholar · View at Scopus
  30. J.-D. Chai and M. Head-Gordon, “Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections,” Physical Chemistry Chemical Physics, vol. 10, no. 44, pp. 6615–6620, 2008. View at Publisher · View at Google Scholar · View at Scopus
  31. A. Alparone, “Comparative study of CCSD(T) and DFT methods: electronic (hyper)polarizabilities of glycine,” Chemical Physics Letters, vol. 514, no. 1–3, pp. 21–25, 2011. View at Publisher · View at Google Scholar · View at Scopus
  32. A. Alparone, “Evolution of electric dipole (hyper)polarizabilities of β-strand polyglycine single chains: an ab initio and DFT theoretical study,” Journal of Physical Chemistry A, vol. 117, pp. 5184–5194, 2013.
  33. M. Alipour and A. Mohajeri, “Assessing the performance of density functional theory for the dynamic polarizabilities of amino acids: treatment of correlation and role of exact exchange,” International Journal of Quantum Chemistry, vol. 113, pp. 1803–1811, 2013.
  34. A. Alparone, “Theoretical study on the static and dynamic first-order hyperpolarizabilities of adenine tautomers,” Molecular Physics, 2013. View at Publisher · View at Google Scholar
  35. A. J. Sadlej, “Medium-size polarized basis sets for high-level correlated calculations of molecular electric properties,” Collection of Czechoslovak Chemical Communications, vol. 53, pp. 1995–2016, 1988.
  36. U. Eckart, M. P. Fülscher, L. Serrano-Andrés, and A. J. Sadlej, “Static electric properties of conjugated cyclic ketones and thioketones,” Journal of Chemical Physics, vol. 113, no. 15, pp. 6235–6244, 2000. View at Publisher · View at Google Scholar · View at Scopus
  37. T. Pluta and A. J. Sadlej, “Electric properties of urea and thiourea,” Journal of Chemical Physics, vol. 114, no. 1, pp. 136–146, 2001. View at Publisher · View at Google Scholar · View at Scopus
  38. U. Eckart, V. E. Ingamells, M. G. Papadopoulos, and A. J. Sadlej, “Vibrational effects on electric properties of cyclopropenone and cyclopropenethione,” Journal of Chemical Physics, vol. 114, no. 2, pp. 735–745, 2001. View at Publisher · View at Google Scholar · View at Scopus
  39. A. Alparone and S. Millefiori, “Gas and solution phase electronic and vibrational (hyper)polarizabilities in the series formaldehyde, formamide and urea: CCSD(T) and DFT theoretical study,” Chemical Physics Letters, vol. 416, no. 4–6, pp. 282–288, 2005. View at Publisher · View at Google Scholar · View at Scopus
  40. D. E. Woon and T. H. Dunning Jr., “Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties,” The Journal of Chemical Physics, vol. 100, no. 4, pp. 2975–2988, 1994. View at Scopus
  41. H. Sekino and R. J. Bartlett, “Frequency dependent nonlinear optical properties of molecules,” The Journal of Chemical Physics, vol. 85, no. 2, pp. 976–989, 1986. View at Scopus
  42. S. P. Karna and M. Dupuis, “Frequency dependent nonlinear optical properties of molecules: formulation and implementation in the HONDO program,” Journal of Computational Chemistry, vol. 12, pp. 487–504, 1991.
  43. H. A. Kurtz, J. J. P. Stewart, and K. M. Dieter, “Calculation of the nonlinear optical properties of molecules,” Journal of Computational Chemistry, vol. 11, pp. 82–87, 1990.
  44. D. Birnbaum and B. E. Kohler, “Lowest energy excited singlet state of 2,2′:5′,2′- terthiophene, an oligomer of polythiophene,” The Journal of Chemical Physics, vol. 90, no. 7, pp. 3506–3510, 1989. View at Scopus
  45. S. Siegert, F. Vogeler, C. M. Marian, and R. Weinkauf, “Throwing light on dark states of α-oligothiophenes of chain lengths 2 to 6: radical anion photoelectron spectroscopy and excited-state theory,” Physical Chemistry Chemical Physics, vol. 13, no. 21, pp. 10350–10363, 2011. View at Publisher · View at Google Scholar · View at Scopus
  46. M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 09, Revision A. 02, Gaussian, Wallingford, Conn, USA, 2009.
  47. B. Bak, D. Christensen, L. Hansen-Nygaard, and J. Rastrup-Andersen, “The structure of thiophene,” Journal of Molecular Spectroscopy, vol. 7, no. 1–6, pp. 58–63, 1961. View at Scopus
  48. J. E. Rice, R. D. Amos, S. M. Colwell, N. C. Handy, and J. Sanz, “Frequency dependent hyperpolarizabilities with application to formaldehyde and methyl fluoride,” The Journal of Chemical Physics, vol. 93, no. 12, pp. 8828–8839, 1990. View at Scopus
  49. H. Sekino and R. J. Bartlett, “Molecular hyperpolarizabilities,” Journal of Chemical Physics, vol. 98, no. 4, pp. 3022–3037, 1993. View at Scopus
  50. F. Sim, S. Chin, M. Dupuis, and J. E. Rice, “Electron correlation effects in hyperpolarizabilities of p-nitroaniline,” Journal of Physical Chemistry, vol. 97, no. 6, pp. 1158–1163, 1993. View at Scopus
  51. G. Maroulis, “Hyperpolarizability of H2O revisited: accurate estimate of the basis set limit and the size of electron correlation effects,” Chemical Physics Letters, vol. 289, no. 3-4, pp. 403–411, 1998. View at Scopus
  52. D. Xenides and G. Maroulis, “Basis set and electron correlation effects on the first and second static hyperpolarizability of SO2,” Chemical Physics Letters, vol. 319, no. 5-6, pp. 618–624, 2000. View at Scopus
  53. A. Alparone, H. Reis, and M. G. Papadopoulos, “Theoretical investigation of the (hyper)polarizabilities of pyrrole Homologues C4H4XH (X = N, P, As, Sb, Bi). A coupled-cluster and density functional theory study,” Journal of Physical Chemistry A, vol. 110, no. 17, pp. 5909–5918, 2006. View at Publisher · View at Google Scholar · View at Scopus
  54. A. Alparone, “Dipole (hyper)polarizabilities of fluorinated benzenes: an ab initio investigation,” Journal of Fluorine Chemistry, vol. 144, pp. 94–101, 2012.
  55. T. Ogata and K. Kozima, “Microwave spectrum, barrier height to internal rotation of methyl group of 3-methylthiophene, and dipole moments of 3-methylthiophene and thiophene,” Journal of Molecular Spectroscopy, vol. 42, no. 1, pp. 38–46, 1972. View at Scopus
  56. C. G. Le Fèvre, R. J. W. Le Fèvre, B. Purnachandra Rao, and M. R. Smith, “Molecular polarisability. Ellipsoids of polarisability for certain fundamental heterocycles,” Journal of the Chemical Society, pp. 1188–1192, 1959. View at Scopus
  57. M. H. Coonan, I. E. Craven, M. R. Hesling, G. L. D. Ritchie, and M. A. Spackman, “Anisotropic molecular polarizabilities, dipole moments, and quadrupole moments of (CH2)2X, (CH3)2X, and C4H4X (X = O, S, Se). Comparison of experimental results and ab initio calculations,” Journal of Physical Chemistry, vol. 96, no. 18, pp. 7301–7307, 1992. View at Scopus
  58. G. R. Dennis, I. R. Gentle, G. L. D. Ritchie, and C. G. Andrieu, “Field-gradient-induced birefringence in dilute solutions of furan, thiophen and selenophen in cyclohexane,” Journal of the Chemical Society, Faraday Transactions 2, vol. 79, no. 4, pp. 539–545, 1983. View at Publisher · View at Google Scholar · View at Scopus
  59. M.-T. Zhao, B. P. Singh, and P. N. Prasad, “A systematic study of polarizability and microscopic third-order optical nonlinearity in thiophene oligomers,” The Journal of Chemical Physics, vol. 89, no. 9, pp. 5535–5541, 1988. View at Scopus
  60. B. Champagne, E. A. Perpète, S. J. A. Van Gisbergen et al., “Assessment of conventional density functional schemes for computing the polarizabilities and hyperpolarizabilities of conjugated oligomers: an ab initio investigation of polyacetylene chains,” Journal of Chemical Physics, vol. 109, no. 23, pp. 10489–10498, 1998. View at Publisher · View at Google Scholar · View at Scopus