Organic nonlinear optical materials have attracted considerable attention in recent years because of their potential applications in photonic devices and optical information processing. Recent studies have shown that annulene derivatives exhibit good second-order nonlinear optical properties, but their third-order nonlinear optical properties are studied little. In this paper, the values of molecular static linear polarizability α and second hyperpolarizability γ of substituted annulenes have been investigated with different levels of HF, B3LYP, BHandHLYP and CAM-B3LYP at different basis sets, respectively. Their ultraviolet spectra have also been calculated by using the TD-B3LYP method. It is found that the quality of the basis set is important for the hyperpolarizability calculations, and diffuse functions are important to obtain accurate results for the second hyperpolarizability. We also study the structure-optical property relationship for annulene. It is found that annulene molecular structure has a significant influence on third-order nonlinear optical response. Increasing the conjugation length and introducing push-pull electronic groups can enhance the second hyperpolarizability. But the introduction of push-pull electronic groups can enhance the hyperpolarizability more remarkably than increasing the conjugation length dose, which may be due to the fact that the introduction of push-pull electronic groups can provide a large number of polarizable electrons whereas increasing the conjugation length can only enhance the electron delocalization. Meanwhile the push-pull electronic group substituted annulenes can also exhibit high transparency in visible region. Thus, this work has a good reference for designing nonlinear optical material with high, nonlinear optical coefficient and good transparency. In addition, for the same push-pull electronic groups, the higher conjugation degree and the longer πup -conjugated bridge result in the decrease of HOMO-LUMO energy gap and transition energy which benefits the enhancement of nonlinear optical response. Our results demonstrate that annulene derivative shows both high transparency and large second hyperpolarizability, and thus becomes a promising candidate for third-order nonlinear optical material. In addition, the dynamic (hyper) polarizabilities of considered annulene molecules are calculated by using CAM-B3LYP method. It is found that in near-infrared region, with the increase of frequency of incident light, α (ω; ω), γ (-ω; ω, 0, 0) and γ (-2ω; ω, ω, 0) are all increased, and the near-resonance enhancement effect occurs. Under the condition of far resonance, α (ω; ω), γ (-ω; ω, 0, 0) and γ (-2ω; ω, ω, 0) change little. This dispersion effect may be helpful for the experimental study and applications as well.
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