2. COMPUTATIONAL DETAILS
The Becke’s three-parameter hybrid exchange functional with the Lee-Yang-Parr correlation functional (B3LYP) was reported to perform well for the calculation of alkali metal-adsorbed graphdiyne structures.32,33In this work, the optimization and frequency analysis for all the geometrical structures were performed at the B3LYP/6-31+G(d) level. Merz-Kollman (MK) charges,45 vertical ionization potentials (VIP), and interaction energies (E int) of the superalkali-doped graphynes were evaluated at the same level based on the optimized geometries. The VIP is defined as the total energy difference between the cationic and neutral compound with the same geometry as the neutral compound. The counterpoise (CP) procedure46 was applied in interaction energy calculation to eliminate the basis set superposition error (BSSE). That is, the E int is computed according to:
\(E_{\text{int}}{=E}_{\text{AB}}\left(\chi_{\text{AB}}\right)-E_{A}\left(\chi_{\text{AB}}\right)-E_{B}\left(\chi_{\text{AB}}\right)\)(1)
where the same basis set, χAB , is used for both the monomer energy calculations (EA andEB ) and the complex energy (EAB ) calculation.
Previous studies have shown that a new density functional Coulomb-attenuated hybrid exchange-correlation functional (CAM-B3LYP)47,48can provide more reasonable and accurate predictions for the evaluation of the NLO properties.31,35,49Furthermore, the values of first hyperpolarizability (β 0) are quite sensitive to the basis set used. Therefore, taking OLi3@GDY as an example, the test calculations for basis set effect are shown in Figure 2. Results showed that the dispersion basis set greatly influenced the first hyperpolarizability. The result obtained with the medium-sized 6-31+G(d) basis set was very close to those with the other dispersion basis sets and much larger with respect to that with 6-31G(d). Therefore, the 6-31+G(d) basis set was chosen to calculate theβ 0 values of all the complexes in this study.
The total energy of a molecular system in the weak and homogeneous electric field can be expressed as:
\(E=E_{0}-\mu_{i}F_{i}-\frac{1}{2!}\alpha_{\text{ij}}F_{i}F_{j}-\frac{1}{3!}\beta_{\text{ijk}}F_{i}F_{j}F_{k}+\)(2)
where E 0 is the system energy in the absence of an electric field; Fi is the Cartesian component of the applied electric field along the i direction;μi , αij , andβijk are the permanent dipole moment, polarizability, and first hyperpolarizability tensors, respectively; andi , j , and k designate the different components along the x , y , and z directions, respectively. The static first hyperpolarizability (β 0) is expressed as:
\(\beta_{0}={(\beta_{x}^{2}+\beta_{y}^{2}+\beta_{z}^{2})}^{\frac{1}{2}}\)(3)
where\(\beta_{i}={\left(\frac{1}{3}\right)\sum}_{j}\left(\beta_{\text{ijj}}+\beta_{\text{jji}}+\beta_{\text{jij}}\right),\ \ i,j=\left\{x,y,z\right\}\).
It is known that the time-dependent density functional theory (TD-DFT) method is one of the most widely used methods to calculate the excitation energies owing to its efficiency and accuracy. Thus, crucial excited states and ultraviolet-visible (UV-vis) absorption spectrums of the related systems were obtained by means of the TD-CAM-B3LYP method in conjunction with the 6-31+G(d) basis set.
All the above calculations were carried out using the Gaussian 09 program package.50 The dimensional plots of molecular geometries and molecular orbitals were generated with the GaussView program.51 The hyperpolarizability density diagrams were obtained by employing the Multiwfn 3.7 (dev) code52 and Visual Molecular Dynamics (VMD) software.53