FIGURE 5 Effects of the atomic number of alkali atom in superalkali OM3 unit (a) and the pore size of the graphyne (b) on the first hyperpolarizability (β 0), respectively
The rough two-level model59 can help to understand the reason why the introduction of superalkali dopant can effectively increase the β 0 value of GYs. This model can be expressed as:
\(\text{\ β}_{0}\propto\frac{{\mu\bullet f}_{0}}{{E}^{3}}\) (4)
where ΔE , f 0, and Δµ are the transition energy, the oscillator strength, and the difference of the dipole moment between the ground state and the crucial excited state, respectively. Since the static β 0 inversely varies with the third power of ΔE , low ΔE is usually a decisive factor for large β 0 values of molecules. Meanwhile, the f 0 and Δµ are also important influencing factors on β 0 for some compounds.60 The TD-DFT calculations at the CAM-B3LYP/6-31+G(d) level were performed to obtain the crucial excited states of the studied systems, and the above three parameters are listed in Table 2. The pristine GY/GDY/GTY molecules with very small β 0 values (0.07–0.47 au) have high ΔE values (4.504, 3.763, and 3.526 eV, respectively). Conversely, the introduction of OM3 can bring much lower ΔEvalues (1.307–1.361 for OM3+@GY, 2.061–2.217 for OM3+@GDY, and 0.604–0.692 eV for OM3+@GTY) and large β 0 values. These results may help us understand the significant superalkali doping effect onβ 0 values of graphynes. Nevertheless, the ΔE value is not the only factor that determines theβ 0 of a system. It is necessary to further consider the influence of f 0 and Δµ on the first hyperpolarizability. From Table 2, the f 0values of each superalki-doped series are close to each other, so they are not the crucial factor that results in the difference ofβ 0 among the studied compounds. It should be noted, however, that the systems with large Δµ also present relatively large NLO responses. For instance, the computed Δμvalue of OLi3+@GTY(8.687 Debye) is much larger than those of ONa3+@GTY (1.557 Debye) and OK3+@GTY (0.673 Debye), indicating that the Δμ value may be the main reason for the largest β 0 value of OLi3+@GTY among these salts.
The dominant electron transitions (the transition with relatively low transition energy and high oscillator strength) of OM3+@(GY/GDY/GTY)also help to further explore the reason why the complexes have largeβ 0 values (see Figure 6). Now, we focus on the nature of crucial transition of the OM3+@GTY series. For the OM3+@GTYseries, the crucial transition is all from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO). Their electron clouds of HOMOs are almost of the same shape, which are mainly centralized on the large GTY surface. The LUMOs of (ONa3/OK3)+@GTYhave the electron cloud distributions close to those of HOMOs. Therefore, the charge transfer of (ONa3/OK3)+@GTYis small and the corresponding Δμ values are small in the electron transitions. However, the electron cloud distribution of LUMO of OLi3+@GTY lacks a part in orange box in Figure 6. Therefore, OLi3+@GTY has a large charge transfer from the ground state to the crucial transition state, resulting in the largest Δμ .