Determination of the symmetry of the highest occupied molecular orbitals of SF<sub>6</sub>
Authors: Wu Rui-Qi , Guo Ying-Chun , Wang Bing-Bing ,   School of Physics and Materials Science, East China Normal University, Shanghai 200241, China,   Laboratory of Optical Physics, Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Published: December 2020

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 SF<sub>6</sub> molecule as an example. Especially, as a good insulating material, an important greenhouse gas and a hyper-valent molecule with the high octahedral <inline-formula><tex-math id="M11">\begin{document}$ O_h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M11.png"/></alternatives></inline-formula> symmetry, SF<sub>6</sub> has received wide attention for both the fundamental scientific interest and practical industrial applications. Theoretical work shows that the electronic configuration of ground electronic state <inline-formula><tex-math id="M13">\begin{document}$ ^1{\rm A_{1g}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M13.png"/></alternatives></inline-formula> of SF<sub>6</sub> is <inline-formula><tex-math id="M15">\begin{document}${({\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} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M15.png"/></alternatives></inline-formula> and the symmetry of the HOMOs is <inline-formula><tex-math id="M16">\begin{document}$ T_{1g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M16.png"/></alternatives></inline-formula>. However, in some literature, the symmetry of HOMOs of SF<sub>6</sub> has been written as <inline-formula><tex-math id="M18">\begin{document}$ T_{2g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M18.png"/></alternatives></inline-formula> instead of <inline-formula><tex-math id="M19">\begin{document}$ T_{1g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M19.png"/></alternatives></inline-formula>. The reason for this mistake lies in the fact that in the ab initial quantum chemical calculation used is the Abelian group <inline-formula><tex-math id="M20">\begin{document}$ D_{2h} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M20.png"/></alternatives></inline-formula>, which is the sub-group of <inline-formula><tex-math id="M21">\begin{document}$ O_h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M21.png"/></alternatives></inline-formula>, to describe the symmetries of molecular orbitals of SF<sub>6</sub>. However, there does not exist the one-to-one matching relationship between the representations of <inline-formula><tex-math id="M23">\begin{document}$ D_{2h} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M23.png"/></alternatives></inline-formula> group and those of <inline-formula><tex-math id="M24">\begin{document}$ O_h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M24.png"/></alternatives></inline-formula> group. For example, both irreducible representations <inline-formula><tex-math id="M25">\begin{document}$ T_{1g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M25.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M26">\begin{document}$ T_{2g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M26.png"/></alternatives></inline-formula> of <inline-formula><tex-math id="M27">\begin{document}$ O_h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M27.png"/></alternatives></inline-formula> group are reduced to the sum of <inline-formula><tex-math id="M28">\begin{document}$ B_{1g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M28.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M28.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M29">\begin{document}$ B_{2g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M29.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M29.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M30">\begin{document}$ B_{3g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M30.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M30.png"/></alternatives></inline-formula> of <inline-formula><tex-math id="M31">\begin{document}$ D_{2h} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M31.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M31.png"/></alternatives></inline-formula>. So the symmetry of the orbitals needs to be investigated further to identify whether it is <inline-formula><tex-math id="M32">\begin{document}$ T_{1g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M32.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M32.png"/></alternatives></inline-formula> or <inline-formula><tex-math id="M33">\begin{document}$ T_{2g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M33.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M33.png"/></alternatives></inline-formula>. In this work, we calculate the orbital functions in the equilibrium structure of ground state of SF<sub>6</sub> by using HF/6-311G* method, which is implemented by using the Molpro software. The expressions of the HOMO functions which are triplet degenerate in energy are obtained. Then by exerting the symmetric operations of <inline-formula><tex-math id="M35">\begin{document}$ O_h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M35.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M35.png"/></alternatives></inline-formula> group on three HOMO functions, we obtain their matrix representations and thus their characters. Finally, the symmetry of the HOMOs is verified to be <inline-formula><tex-math id="M36">\begin{document}$ T_{1g} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M36.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182231_M36.png"/></alternatives></inline-formula>. By using this process, we may determine the molecular orbital symmetry of any other molecules with high symmetry group.