b. School of Physics and Electronics, Shandong Normal University, Jinan 250014, China
Two-photon absorption (TPA) is a nonlinear optical process describing the simultaneous absorption of two photons by a particular molecule. As described in Ref., TPA of organic materials has been under the spotlight of diversified research areas, such as chemistry, photonics, and biology, with a wide variety of applications of two-photon technology emerging from such studies. Upconverted lasing , optical power limiting , photodynamic therapy , and three-dimensional (3D) microfabrication [5, 6] are some examples given in Ref.. A strong effort on performing experimental synthesis and theoretical calculations has been made in order to find novel non-linear optical materials with strong TPA. A recent study  enumerates some advantages of organic nonlinear optical materials superior to inorganic materials: large NLO coefficients, greater ease of synthetic design and lower cost [8-13]. Such benefits may trigger new developments in optical communications, information processing, frequency doubling, and integrated optics . However, there remains some uncertainty in the details of design criteria for molecules with large TPA cross sections at desired wavelengths, so the need to understand the structure-property relationships is of significant importance in a research area . In recent work  we have stated that the TPA activities are related to a number of factors: conjugated length, π center property, symmetrical/asymmetrical arrangement of the electron-donor (D) and electron-acceptor (A) attached to the π center and its strength to push and pull electrons, molecular dimensionality [16-19] and planarity [20-22]. These listed factors are nowadays considered for the design of molecular materials possessing strong TPA.
In diverse TPA materials, various π-bridging chromophores have been considered for increasing the TPA cross section. Among them, alkene π-bridging (C=C) is the most widely studied, with the double bond between carbons being an excellent conjugation bridge for the intramolecular charge transfer from donor to acceptor. In the recent years, conjugated organic polymers containing acetylene units have been thoroughly both experimentally and theoretically investigated [23-25]. In order to reach a more detailed understanding of the effect of π-bridging and conjugation length on the TPA cross section of alkyne and alkene chromophores, we investigated alkyne π-bridging molecules (A1-A4) synthesized by Liu et al.  and also designed four similar molecules with alkene π-bridging (noted as B1-B4), as shown in Fig. 1.
While this specific molecular design has been proven to be the most effective way to modify the TPA cross sections of molecules , the choice of solvent has also been suggested to play a similar role [28-33]. Solvent effects on the molecular geometrical structures and geometric distortion can be measured by the bond-length-alternation (BLA) parameter, which can be defined as the difference between average single and double bond distance in the conjugated pathway . In order to investigate the solvent effect on alkyne π-bridging (C≡C), we tried to expand this concept and define the BLA as the difference between average single and triple bond distance, including the alternation of C≡C for alkyne π-bridging compounds.II. Computational methods
where ωf is the excited state energy at the final state and μα is the dipole moment operator. The summation runs over the molecular axes: α∈(x, y, z).
TPA cross sections of randomly oriented systems can be directly related to the imaginary part of the third susceptibility. Alternatively, the TPA cross section can be obtained by computing the individual TP transition matrix elements Sαβ between the initial state $\mid$$0\rangle$ and final state $\mid$$f\rangle$ [2, 36],
Where α, β∈(x, y, z), ωi and ωf are the excitation energy to the intermediate state |i＞ and the final state |f＞ , respectively. The summation here includes all intermediate, initial, and final states. The TPA probability of excited molecules by a linearly polarized monochromatic beam can be calculated by
The TPA cross section that can be directly compared with the experiment is defined as:
where a0 is the Bohr radius, c0 is the speed of light in vacuum, α is the fine structure constant, ω is the photon energy of the incident light, g(ω) denotes the spectral line profile that is assumed to be a δ function here, and the level broadening of final state is assumed to have the commonly used value  Γf=0.1 eV, corresponding to a lifetime of a few femtoseconds.III. RESULTS AND DISCUSSION A. Computational details
In previous studies, Masunov et al. have shown that Hartree-Fock (HF) optimizations yield better results than optimizations performed at the DFT level with the B3LYP functional, while studying the ground-state geometries of similar conjugated systems and reproducing the BLA [37, 38]. In previous researches, Zhao et al.  and Li et al.  have also reached three conclusions: (i) HF optimizations yield nonplanar structures; (ii) the BLA value obtained with the HF method is larger than the one obtained with B3LYP; (iii) excitation energies based on HF optimized geometries agree very well with experimental results, with the opposite happening between B3LYP calculations and experimental results. Based on this information, we decided to optimize molecular geometries at the HF level using the Gaussian program . For the calculation of the one-photon absorption (OPA) and TPA properties we relied on response theory  as implemented in the DALTON program suite . This was done initially at the DFT/B3LYP level. Solvent effects on geometric structures and on OPA and TPA properties were studied via PCM. HF and B3LYP calculations were performed with the 6-31G basis set. However, for the purpose of assuring some degree of convergence in the numerical results, we performed additional calculations at the HF/6-31G* and B3PW91/6-31G levels of theory.B. Molecular structures
The studied molecular structures (B1-B4) are shown in Fig. 1, which differ from those (A1-A4) in Ref., by replacing the hexyls at the nitrogen atoms by methyl (CH3) groups. This alteration has the merit of achieving a significant reduction of computational time while maintaining the essential characteristics of the single chromophores. Our optimizations show that the choice of π-bridging affects the molecular geometry. For the alkyne π-bridging compounds, the backbones of molecules A1-A4 lie virtually on the same plane, while the alkene counterparts yield a nonplanar structure, with "both side phenyl groups rotated in opposite directions by a torsion angle with respect to the middle benzene ring" . Furthermore, B4 has the largest torsion angle of about 90°. We present the optimized geometries of A4 and B4 in Fig. 2 for comparison.
As stated by Masunov and coworkers , there are two geometric parameters that play a significant role in characterizing the electronic properties of conjugated molecules: planarity, which relates to the dihedral angle along the backbone, and BLA. We have found that the same solvent effects exist on both of the alkyne and alkene π-bridging molecules. To make a comparison, we list some selective bond lengths and dihedral angles of A1 (A2) and B1 (B2) in the gas phase and in varying solutions, in Table I and II. The triple bond length of the optimized A1 structure (1.200 Å) can be seen to be similar to the one obtained by other groups (1.209 Å) , (1.216 Å) , (1.200 Å)  and to a bond length resulting from a measurement from a crystallographic structure (1.206 Å)  for dipheny-lacetylene. The type of substituent is seen to have a small influence on the triple bong length, which slightly shortens with an increase in the alkyne π-bridging length , as A2 (1.201 Å).
As previously mentioned above in the introduction, the extent of changes in molecular geometry induced by solvents can be characterized by BLA, which is defined as the average between single and double (or triple) bond distances in the conjugated pathway . It can be seen from Table I that the bond length of single and double bonds for the alkene π-bridging compounds show opposite behavior via increasing solvent polarization. Same effects are also found in the alkyne counterparts with decreased single bond length and increased triple bond length. For example, the single bond length decreases from 1.432 Å to 1.420 Å for A2, and the triple bond length increases from 1.201 Å to 1.228 Å. The value of BLA decreases from 0.203 Å to 0.162 Å for gas and DMF solvent, respectively. As similar in B2, single C-C bond length decreases from 1.468 Å to 1.467 Å, double bond length increases from 1.336 Å to 1.336 Å, and BLA decreases from 0.126 Å to 0.125 Å. We noted that it shows a larger solvent effect on the alkyne π-bridging compounds geometries.
In Table II, the dihedral angle defined as along the chains C4-C5-C1-C2 and C4-C5-C11-C12, are both 179.9° for alkyne π-bridging molecule A1, and 147.5°, 163.6° for alkene π-bridging counterpart B1 in the gas phase, respectively. Larger solvent effects on the dihedral angles were found for alkene π-bridging molecules: the dihedral angle for B1 is 179.6° in toluene, 157.5° in CHCl3 and 167.5° in DMF. When the number of double bonds in the π-bridging center increases from one to two, it significantly decreases the dihedral angle, and improves the planarity, such as B2.C. One-photon absorption
In Table III we have listed the OPA properties of the five lowest excited states of eight compounds as calculated with the TDDFT method. For all compounds, the first excited states have the largest OPA strength, which is noted as a charge transfer (CT) state. For alkene and alkyne π-bridging derivatives, the substituents in the π-bridging center cause a red shift in the absorption spectrum, compared with the parent compound. The largest observed shift is attributed to the A4 compound (λ=493.1 nm), compared to A1 (λ=337.2 nm). One notes that the expected enhancement in the oscillator strength on going from A1 to A3 and from B1 to B3 is caused by the increase in the molecular conjugation length. However, the oscillator strength gets lower in A4 and B4: A4 has two CT states occurring at the first (δop=1.17) and fifth (δop=0.76) states, which increases the extent of intramolecular charge transfer in two dimensions; B4 also has two CT states but with very small oscillator strength, which is mainly due to the fact that B4 has the largest torsion angle of about 90° with respect to the middle benzene ring, and the nonplanar geometry significantly reduces intramolecular charge transfer effects.
The OPA peak for alkyne π-bridging compounds are blue-shifted relative to their alkene counterparts, which is consistent with the results obtained by Bhaskar et al. . The oscillator strengths for alkyne π-bridging compounds are slightly larger than those of alkene analogues except B2. This is mainly due to the fact that all of the alkyne π-bridging compounds have perfect planarity, while the alkene counterparts yield nonplanar structures, except B2, which shows a good planarity. That is, nonplanar structures are unfavorable to π electron delocalization, decreasing oscillator strengths of the alkene π-bridging compounds.
With the objective of testing the convergence of our results with respect to the choice of hybrid functionals and basis sets, we have performed calculations for A4 based on the hybrid functional B3PW91 and a different basis set (6-31G*). The results, which can be seen in Table IV, show a good degree of similarity between the three sets of calculations, thus suggesting convergence.
The calculated excited energies and oscillator strengths of the CT states in various solutions are presented in Table V. The polarity of the solvent has a small effect on the position and strength of the absorption spectrum. Compared with the experimental results, it can be seen that the numerical calculations correctly described the solvent effects for the absorption spectrum of the measured counterparts. Naturally, the agreement between theoretical predictions and experiments is not always perfect. Such disagreements can be attenuated by considering the hexyls substituents at the nitrogen atoms, more advanced theoretical methods, and also the vibronic effect, which is known to contribute not only to OPA magnitudes, but also to the profile of the OPA spectra .D. Two-photon absorption
Table VI presents the TPA cross section in the gas phase referring to excited states of all compounds. The four alkyne containing systems (A1-A4) differ only by an increase in the conjugation length of the π-bridging. When increasing π-bridging length with two triple bonds, a one-fold increase in TPA for A2 in comparison with A1 is detected. It is clearly seen that the TPA of A3 is approximately 1.5 times higher than A2 via inserting a phenyl into the π-bridging, and yields a 2-fold increase in comparison with A1. Comparing compounds A3 and A4, which differ only by an increase in the number of phenyl as the center, it is noted that the maximal TPA of A4 is about 1000 GM (1 GM=10-50 cm4·S/photon) lower than that of A3. This is mainly due to the fact that A4 existes in two stronger TPA states which lie at the second and fifth state, and the TPA cross section are 2911.0 and 1205.1 GM, respectively. The same trends can be noted to the alkene π-bridging counterparts. However, in the case of the alkene π-bridging molecule B4, the TPA cross sections are rather small when compared to the other cases. This is another example of how important the planarity of the molecules can be for enhancing TPA ability. Another factor contributing to the increase in the TPA cross sections is the increase in conjugation, which has two well-known effects [48-50]: reduction of the detuning term and increase of the transition dipole moment. Comparing the TPA for the two series of molecules, we found that the alkyne π-bridging chromophores have larger TPA properties over the alkene π-bridging counterparts. This may be attributed to the better planarity of the alkyne π-bridging molecules over the alkene π-bridging counterparts.
Results listed in Table VII show that TPA cross sections are slightly enhanced upon solvation. For instance, the TPA cross section of A3 is 4021.5 GM in toluene, 3989.8 GM in CHCl3, and 4010.8 GM in DMF, which are larger than the TPA cross section in the gas phase. Furthermore, the effect on the TPA cross section exhibits a nonmonotonic behavior with respect to the polarity of the solvent, a behavior consistent with experimental results. However, the calculated results are larger than the experimental values, by about two orders of magnitude. These discrepancies may be due to the fact that the experimental TPA cross section values are given at 720 nm for molecule A3. We found that our calculated results at about 740 nm agree well with the measured values.E. Charge-transfer process
CT states are known to have an important contribution in the determination of optical properties of molecules. In order to get a better understanding of the processes involved in charge-transfer, we have calculated, for all compounds and much in the spirit of Ref., the charge variation of the donor and the acceptor between the first CT and the ground states in the gas phase, as shown in Fig. 3. The peak and valley parts represent the enhance and decrease of the electron densities, respectively. We also calculate the charge variation of the donor and the π-bridging in all compounds between the first CT and the ground states (see Table VIII). It is easily seen that electrons mainly move from the terminal electron donors to the central electron acceptor group . When the molecular conjugation length is increased, charge transfer is increased in a greater degree, which thereby shows higher OPA and TPA properties. An additional significant feature is that the amount of intramolecular charge transfer for the alkene π-bridging compounds is larger than that for the alkyne π-bridging systems, which is in agreement with the early observation for alkene and alkyne branched systems . However, the TPA cross section in the alkene π-bridging molecules is smaller than the one observed for alkyne π-bridging compounds, and we believe that this is mainly related to the decreased charge transfer region for the nonplanar molecular geometry of alkene π-bridging systems.IV. Conclusion
In this work we have performed calculations with the purpose of obtaining OPA and TPA properties of the two series of (alkyne and alkene) π-bridging molecules. Such calculations were mainly based on the response theory employing DFT with the B3LYP functional. Our calculations show that, in the optical region, the maximal OPA intensity for the studied compounds is in the first excited state. Two series of alkyne and alkene π-bridging molecules present relatively strong TPA activities. Furthermore, alkyne π-bridging chromophores show larger TPA cross sections over the alkene π-bridging counterparts, which is mainly due to the fact that alkyne π-bridging molecules yield better planar structures. Again, it is shown  that the planarity of the molecules is helpful for enhancing the OPA and TPA intensities. Two effective ways are suggested to improve the molecular geometric planarity: substituting alkene with triple bonds (C≡C) as π-bridging, or increasing the number of central double bonds.
The polarized continuum model (PCM) was employed with the intention of studying solvent effects on the molecular geometrical structures and TPA properties for those molecules. The obtained results show larger solvent effects on the alkyne π-bridging compounds geometries than alkene π-bridging ones, with larger bond length alternation BLA. TPA cross sections are slightly enhanced upon solvation, exhibiting a nonmonotonic behavior with respect to the polarity of the solvent , which is consistent with experimental results.V. Acknowledgments
This work was supported by the National Natural Science Foundation of China (No.11604179 and No.11304185), Shandong Natural Science Foundation (No.ZR2016AQ18). The National Supercomputer Center (NSC) in Linköping, Sweden, for grants in terms of CPU time is appreciated.
|||C. K. Wang, P. Macak, Y. Luo, and Ågren H., J. Chem. Phys. 114 , 9813 (2001). DOI:10.1063/1.1371258|
|||A. Mukherjee, Appl. Phys. Lett. 622 , 3423 (1993).|
|||W. Tutt and T. F. Boggess L., Prog. Quant. Electron. 17 , 299 (1993). DOI:10.1016/0079-6727(93)90004-S|
|||E. A. Wachter, W. P. Partridge, W. G. Fisher, H. C. Dees, and M. G. Petersen, Proc. SPIE Int. Soc. Opt. Eng. 3269 , 68 (1998).|
|||S. Maruo, O. Nakamura, and S. Kawata, Opt. Lett. 22 , 132 (1997). DOI:10.1364/OL.22.000132|
|||H. B. Sun, S. Matsuo, and H. Misawa, Appl. Phys. Lett. 74 , 786 (1999). DOI:10.1063/1.123367|
|||S. P. Chou and C. Y. Yu S., Synth. Met. 142 , 259 (2003).|
|||T. Kogel, D. Beljonne, F. Meyers, J. Perry, S. Marder, and J. Bredas, Chem. Phys. Lett. 298 , 1 (1998). DOI:10.1016/S0009-2614(98)01196-8|
|||K. D. Belfield, A. R. Morales, J. Hales, D. Hagan, W. Van Stryland E., V. Chapela, and J. Percino, Chem. Mater. 16 , 2267 (2004). DOI:10.1021/cm035253g|
|||A. R. Morales, K. D. Belfield, J. Hales, W. Van Stryland E., and D. D. Hagan, Chem. Mater. 18 , 4972 (2006). DOI:10.1021/cm061406z|
|||S. Marder, Chem. Commun. 2 , 131 (2006).|
|||A. R. Morales, A. Frazer, A. W. Woodward, Y. AhnWhite H., A. Fonari, P. Tongwa, T. Timofeeva, and K. D. Belfield, J. Org. Chem. 78 , 1014 (2013). DOI:10.1021/jo302423p|
|||G. W. Githaiga, A. W. Woodward, A. R. Morales, M. V. Bondar, and K. D. Belfield, J. Phys. Chem C119 , 21053 (2015).|
|||W. V. Moreshead, O. V. Przhonska, M. V. Bondar, A. D. Kachkovski, I. H. Nayyar, A. E. Masunov, A. W. Woodward, and K. D. Belfield, J. Phys. Chem C117 , 23133 (2013).|
|||J. Li, Y. P. Sun, Z. L. Li, X. N. Song, and C. K. Wang, Chem. Phys. Lett. 464 , 9 (2008). DOI:10.1016/j.cplett.2008.08.041|
|||P. Macak, Y. Luo, P. Norman, and Ågren H., J. Chem. Phys. 113 , 7055 (2000). DOI:10.1063/1.1313559|
|||J. Yoo, S. K. Yang, M. Y. Jeong, H. C. Ahn, S. J. Jeon, and B. R. Cho, Org. Lett. 5 , 645 (2003). DOI:10.1021/ol027343h|
|||M. Albota, D. Beljonne, L. Brédas J., J. E. Ehrlich, J. Y. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, McCord-Maughon D., J. W. Perry, Röckel H., M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, Science 281 , 1653 (1998). DOI:10.1126/science.281.5383.1653|
|||S. K. Pati, T. J. Marks, and M. A. Ratner, J. Am. Chem. Soc. 123 , 7287 (2001). DOI:10.1021/ja0033281|
|||C. K. Wang, P. Macak, Y. Luo, and Ågren H., J. Chem. Phys. 114 , 9813 (2001). DOI:10.1063/1.1371258|
|||C. K. Wang, K. Zhao, Y. Su, R. Yan, X. Zhao, and Y. Luo, J. Chem. Phys. 119 , 1208 (2003). DOI:10.1063/1.1579680|
|||Y. H. Sun, K. Zhao, C. K. Wang, Y. Luo, Y. X. Yan, X. T. Tao, and M. H. Jiang, Chem. Phys. Lett. 394 , 176 (2004). DOI:10.1016/j.cplett.2004.07.003|
|||T. McQuade D., A. E. Pullen, and T. M. Swager, Chem. Rev. 100 , 2537 (2000). DOI:10.1021/cr9801014|
|||Günes S., H. Neugebaueer, and N. S. Sariciftci, Chem. Rev. 107 , 1324 (2007). DOI:10.1021/cr050149z|
|||Kivala and F. Diederich M., Acc. Chem. Res. 42 , 235 (2009). DOI:10.1021/ar8001238|
|||B. Liu, J. Liu, H. Q. Wang, Y. D. Zhao, and Z. L. Huang, J. Mole. Stru. 833 , 82 (2007). DOI:10.1016/j.molstruc.2006.09.007|
|||K. Zhao, L. Ferrighi, C. K. Wang, and Y. Luo, J. Chem. Phys. 126 , 204509 (2007). DOI:10.1063/1.2740641|
|||H. Y. Woo, B. Liu, B. Kohler, D. Korystov, A. Mikhailovsky, and G. C. Bazan, J. Am. Chem. Soc. 127 , 14721 (2005). DOI:10.1021/ja052906g|
|||Johnsen and P. R. Ogilby M., J. Phys. Chem A112 , 7831 (2008).|
|||Y. Zhao, A. M. Ren, J. K. Feng, X. Zhou, X. C. Ai, and W. J. Su, Phys. Chem. Chem. Phys. 11 , 11538 (2009). DOI:10.1039/b908415k|
|||H. Wang, Z. Li, P. Shao, J. Qin, and Z. L. Huang, J. Phys. Chem B114 , 22 (2010).|
|||J. J. Shao, Z. P. Guan, Y. L. Yan, C. J. Jiao, Q. H. Xu, and C. Y. Chi, J. Org. Chem. 76 , 780 (2011). DOI:10.1021/jo1017926|
|||M. M. Alam, M. Chattopadhyaya, S. Chakrabarti, and K. Ruud, J. Phys. Chem. Lett. 3 , 961 (2012). DOI:10.1021/jz300198y|
|||R. Cammi, B. Mennucci, and J. Tomasi, J. Am. Chem. Soc. 120 , 8834 (1998). DOI:10.1021/ja980823c|
|||Y. H. Sun, K. Zhao, C. K. Wang, Y. Luo, Y. Ren, X. T. Tao, and M. H. Jiang, J. Mol. Struct:THEOCHEM 682 , 185 (2004). DOI:10.1016/j.theochem.2004.05.030|
|||S. R. Marder, C. B. Gorman, F. Meyers, J. W. Perry, G. Bourhill, K. Brédas J., and B. M. Pierce, Science 265 , 632 (1994). DOI:10.1126/science.265.5172.632|
|||A. Masunov and S. Tretiak A., J. Phys. Chem B108 , 899 (2004).|
|||A. Masunov, S. Tretiak, J. W. Hong, B. Liu, and G. C. Bazan, J. Chem. Phys. 122 , 224505 (2005). DOI:10.1063/1.1878732|
|||Gaussian, References in http://www.gaussian.com|
|||Olsen and P. Jφrgensen J., J. Chem. Phys. 82 , 3235 (1985). DOI:10.1063/1.448223|
|||DALTON, References in http://www.kjemi.uio.no/software/dalton/.|
|||M. Wierzbicka, Bylińska L., C. Czaplewski, and W. Wiczk, RSC Adv. 5 , 29294 (2015). DOI:10.1039/C5RA01077B|
|||Nagano Y Y., T. Ikoma, K. Akiyama, and Tero Kubota S., J. Am. Chem. Soc. 125 , 14103 (2003). DOI:10.1021/ja035173d|
|||C. Ferrante, U. Kensy, and B. Dick, J. Phys. Chem. 97 , 13457 (1993). DOI:10.1021/j100153a008|
|||R. Thomas, S. Lakshmi, S. K. Pati, and G. U. Kulkarni, J. Phys. Chem B110 , 24674 (2006).|
|||T. Vreven, B. Mennucci, C. O. Silva, K. Morokuma, and J. Tomasi, J. Chem. Phys. 115 , 62 (2001). DOI:10.1063/1.1376127|
|||A. Bhaskar, G. Ramakrishna, Z. K. Lu, R. Twieg, J. M. Hales, D. J. Hagan, E. V. Stryland, and T. Goodson, J. Am. Chem. Soc. 128 , 11840 (2006). DOI:10.1021/ja060630m|
|||J. Strickler and R. A. Berg S., J. Chem. Phys. 37 , 814 (1962). DOI:10.1063/1.1733166|
|||D. Beljonne, W. Wenseleers, E. Zojer, Z. G. Shuai, H. Vogel, J. K. Pond S., J. W. Perry, S. R. Marder, and L. Brédas J., Adv. Funct. Mater. 12 , 631 (2002). DOI:10.1002/1616-3028(20020916)12:9<631::AID-ADFM631>3.0.CO;2-W|
|||Rubio-Pons O., Y. Luo, and Ågren H., J. Chem. Phys. 124 , 094310 (2006). DOI:10.1063/1.2178790|
|||Y. Z. Song, D. M. Li, X. N. Song, X. M. Huang, and C. K. Wang, J. Mol. Struct:THEOCHEM 772 , 75 (2006). DOI:10.1016/j.theochem.2006.06.024|
b. 山东师范大学物理与电子科学学院, 济南 250014