FIGURE 6 Molecular orbitals corresponding to the dominant electron transitions of the OM3@GYs complexes. H and L represent the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), respectively
It is well-known that for the Li@calix[4]pyrrole electride with large β 0value,58 the electron involved in the critical excited state is in the HOMO, which is derived from the diffuse excess electron cloud of the alkali atom. This means that the excess electron makes a significant contribution to the large hyperpolarizability of electride. It can be found from Figure 6 that the electron clouds of HOMO and crucial excited states of OM3@(GDY/GTY) are mainly distributed over the (GDY/GTY) moiety. Therefore, we speculate that (GDY/GTY) unit is the main source of the large first hyperpolarizability value of the system. Obviously, electronic transition characteristics of the topic systems are different from those of electride molecules with excess electrons.
To show the contribution of (GDY/GTY) to the first hyperpolarizability of OM3+@(GDY/GTY), we take OLi3+@(GDY/GTY) as an example and calculate the first hyperpolarizabilities of (e @GDY) and (e @GTY). The structures of (e @GDY) and (e @GTY) were obtained by removing OLi3+ from the optimized structures of OLi3+@GDY and OLi3+@GTY, respectively. Subsequently, the β 0 values of the (e @GDY) and (e @GTY) anions were calculated to be 483232 and 711447 au at the CAM-B3LYP/6-31+G(d) level. Clearly, the large first hyperpolarizabilities of OLi3+@GDY (265248 au) and OLi3+@GTY(653420 au) are almost from the contributions of the graphyne anion units.
The spatial contribution of the electrons to the NLO response of OLi3@(GDY/GTY) is explored in terms of first hyperpolarizability density.61 Electron density ρ (r , F ) can be written as Taylor expansion with respect to the externally applied electric field F :
\(\rho\left(r,F\right)=\rho^{(0)}\left(r\right)+\sum_{j}\rho_{j}^{(1)}\left(r\right)F_{j}+\frac{1}{2!}\sum_{j}{\sum_{k}\rho_{\text{jk}}^{(2)}}\left(r\right)F_{j}F_{k}+\cdots\)(5)
From above equation and expansion formula of dipole moment in power of the field, a component of first hyperpolarizability can be expressed by:
\(\beta_{\text{ijk}}=-\frac{1}{2!}\int{r_{i}\rho_{\text{jk}}^{(2)}(r)dr^{3}}\)(6)
where
\({\rho_{\text{jk}}^{\left(2\right)}\left(r\right)=\left.\ \frac{\partial^{2}\rho(r)}{\partial F_{j}\partial F_{k}}\right|}_{F=0}\)(7)
The electron density was calculated at the CAM-B3LYP/6-31+G(d) level using F = 0.003 au. We focus on the βxxxand βyyy components which are the important components of β 0 value of OLi3@(GDY/GTY). The local contribution function\({-\text{xρ}}_{\text{xx}}^{(2)}\left(r\right)\)and\({-\text{yρ}}_{\text{yy}}^{(2)}\left(r\right)\)of the OLi3@(GDY/GTY) are displayed in Figure 7. Red and white regions denote positive and negative contributions, respectively, to theβxxx and βyyy values. It can be seen that the main contribution to the first hyperpolarizability comes from GDY and GTY moieties. The graphs of\(-x\rho_{\text{xx}}^{(2)}\left(r\right)\) and\(-y\rho_{\text{yy}}^{(2)}\left(r\right)\) for OLi3@GDY show the fact that the positive local contribution region is larger than the negative part. This is the reason why the OLi3@GDY has considerable positiveβxxx (266941 au) and βyyy (280948 au) values. For OLi3@GTY, the\(-x\rho_{\text{xx}}^{(2)}\left(r\right)\ \) function diagram shows that the positive contribution area is much larger than the negative part, while the opposite result is observed for the\(-y\rho_{\text{yy}}^{(2)}\left(r\right)\) function. Compared with the\(-y\rho_{\text{yy}}^{(2)}\left(r\right)\), the\(-x\rho_{\text{xx}}^{(2)}\left(r\right)\) shows a relatively larger regional scope for both positive and negative contributions. Thus, the OLi3@GTY possesses large positiveβxxx (521577 au) and relatively small negativeβyyy (–195843 au) values.