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+@GTY–series, 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)+@GTY–have the electron cloud distributions close to those of HOMOs.
Therefore, the charge transfer of
(ONa3/OK3)+@GTY–is 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 .