1.School of Physics and Materials Science, East China Normal University, Shanghai 200241, China 2.Laboratory of Optical Physics, Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 61275128, 11774411, 11474348).
Received Date:19 December 2018
Accepted Date:20 February 2019
Published Online:20 April 2019
Abstract:Quantum chemical calculation is an important method to investigate the molecular structures for multi-atom molecules. The determination of electronic configurations and the accurate description of the symmetry of molecular orbitals are critical for understanding molecular structures. For the molecules belonging to high symmetry group, in the quantum chemical calculation the sub-group is always adopted. Thus the symmetries of some electric states or some molecular orbitals, which belong to different types of representations of high symmetry group, may coincide in the sub-group presentations. Therefore, they cannot be distinguished directly from the sub-group results. In this paper, we provide a method to identify the symmetry of molecular orbitals from the theoretical sub-group results and use this method to determine the symmetry of the highest occupied molecular orbitals (HOMO) of the sulfur hexafluoride SF6 molecule as an example. Especially, as a good insulating material, an important greenhouse gas and a hyper-valent molecule with the high octahedral $ O_h $ symmetry, SF6 has received wide attention for both the fundamental scientific interest and practical industrial applications. Theoretical work shows that the electronic configuration of ground electronic state $ ^1{\rm A_{1g}} $ of SF6 is ${({\rm {core}})^{22}}{(4{\rm a_{1\rm g}})^2}{(3{{\rm t}_{1\rm u}})^6}{(2{{\rm e}_{\rm g}})^4}{(5{{\rm a}_{1\rm g}})^2}{(4{{\rm t}_{1\rm u}})^6}{(1{{\rm t}_{2\rm g}})^6}{(3{{\rm e}_{\rm g}})^4}{(1{{\rm t}_{2\rm u}})^6}{(5{{\rm t}_{1\rm u}})^6}{(1{{\rm t}_{1\rm g}})^6} $ and the symmetry of the HOMOs is $ T_{1g} $. However, in some literature, the symmetry of HOMOs of SF6 has been written as $ T_{2g} $ instead of $ T_{1g} $. The reason for this mistake lies in the fact that in the ab initial quantum chemical calculation used is the Abelian group $ D_{2h} $, which is the sub-group of $ O_h $, to describe the symmetries of molecular orbitals of SF6. However, there does not exist the one-to-one matching relationship between the representations of $ D_{2h} $ group and those of $ O_h $ group. For example, both irreducible representations $ T_{1g} $ and $ T_{2g} $ of $ O_h $ group are reduced to the sum of $ B_{1g} $, $ B_{2g} $ and $ B_{3g} $ of $ D_{2h} $. So the symmetry of the orbitals needs to be investigated further to identify whether it is $ T_{1g} $ or $ T_{2g} $. In this work, we calculate the orbital functions in the equilibrium structure of ground state of SF6 by using HF/6-311G* method, which is implemented by using the Molpro software. The expressions of the HOMO functions which are triplet degenerate in energy are obtained. Then by exerting the symmetric operations of $ O_h $ group on three HOMO functions, we obtain their matrix representations and thus their characters. Finally, the symmetry of the HOMOs is verified to be $ T_{1g} $. By using this process, we may determine the molecular orbital symmetry of any other molecules with high symmetry group. Keywords:SF6/ the highest occupied molecular orbitals/ molecule with high symmetry group/ orbital's symmetry
这里$ w $代表电子坐标$ {r} $的任一分量$ x $, $ y $或$ z $; k = 1—7对应1个S原子及6个F原子; $ x_k $, $ y_k $, $ z_k $为1个S原子以及6 个F原子的核坐标, 取值如表1所列; $ r_k $是电子到第k个原子核的距离; $ c_i $表示组成原子轨道的高斯函数的系数, $ \alpha_i $是高斯函数参数, $ c_i $与$ \alpha_i $取值如表2所列.
k
xk/a.u.
yk/a.u.
zk/a.u.
S
1
0
0
0
F
2
2.923
0
0
F
3
–2.923
0
0
F
4
0
2.923
0
F
5
0
–2.923
0
F
6
0
0
2.923
F
7
0
0
–2.923
表1SF6的分子结构 Table1.Molecular structure of SF6
i
1
2
3
4
5
6
ci
0.035461
0.237451
0.820458
1.0
1.0
1.0
αi/a.u.
55.4441
12.6323
3.71756
1.16545
0.321892
1.75
表26-311G*基组中高斯函数的参数表 Table2.Parameters of Gaussian functions of 6-311G* basis
为更好地理解上面的三个轨道函数以及方便后面的分析, 图1给出了函数$ \varPsi_{B_{1g}} $, $ \varPsi_{B_{2g}} $和$ \varPsi_{B_{3g}} $的空间分布, 颜色代表函数的取值. $ \varPsi_{B_{1g}} $在$ xy $截面内为中心对称的八个花瓣形状的分布(图1(c)), 有四条节线, 另两个截面没有节线(图1(a), (b)). $ \varPsi_{B_{2g}} $和$ \varPsi_{B_{3g}} $相似, 分别对应$ xz $截面(图1(d))和$ yz $截面(图1(e))内的八个花瓣形状分布, 在另外的两个截面内没有节线. 图 1$\varPsi_{B_{1g}}$, $\varPsi_{B_{2g}}$和$\varPsi_{B_{3g}}$波函数截面图 (a) $\varPsi_{B_{1g}}$在x = 0.8 a.u.处的yz截面图; (b) $\varPsi_{B_{1g}}$在y = 0.8 a.u.处的xz截面图; (c) $\varPsi_{B_{1g}}$在z = 0 a.u.处的xy截面图；(d) $\varPsi_{B_{2g}}$在y = 0 a.u.处的xz截面图; (e) $\varPsi_{B_{3g}}$在x = 0 a.u.处的yz截面图; (f) $C_2'^1$作用在$\varPsi_{B_{1g}}$后取y = 0 a.u.处的xz截面图 Figure1. Functions of $\varPsi_{B_{1g}}$, $\varPsi_{B_{2g}}$ and $\varPsi_{B_{3g}}$: (a) $\varPsi_{B_{1g}}$ in the yz plane for x = 0.8 a.u.; (b) $\varPsi_{B_{1g}}$ in the xz plane for y = 0.8 a.u.; (c) $\varPsi_{B_{1g}}$ in the xy plane for z = 0 a.u.; (d) $\varPsi_{B_{2g}}$ in the xz plane for y = 0 a.u.; (e) $\varPsi_{B_{3g}}$ in the yz plane for x = 0 a.u.; (f) the function obtained by acting $C_2'^1$ on $\varPsi_{B_{1g}}$ in the xz plane for y = 0 a.u..
图 1 
图 2 SF6对称操作
