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