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SF6分子最高占据轨道对称性的判断

武瑞琪 郭迎春 王兵兵

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SF6分子最高占据轨道对称性的判断

武瑞琪, 郭迎春, 王兵兵

Determination of the symmetry of the highest occupied molecular orbitals of SF6

Wu Rui-Qi, Guo Ying-Chun, Wang Bing-Bing
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  • 量化计算是理论研究分子的重要手段, 对于具有高对称群的分子, 采用子群计算是常用的方法. 分子的电子态或分子轨道等的对称性在子群的表示中会出现重迭, 从而不能从子群的结果直接给出电子态或分子轨道对称性的归属. 本文以如何判断$ \rm {SF}_6 $基态$ ^1\rm A_{1\rm g} $的电子组态中最高占据轨道的对称性为例来解决这个问题. 针对某些文献中的$ \rm {SF}_6 $基态$ ^1\rm A_{1\rm g} $的电子组态中, 最高占据轨道对称性是$ T_{1g} $却写成$ T_{2g} $的问题, 采用Molpro量化计算软件, 对$ \rm {SF}_6 $基态的平衡结构, 进行了HF/6-311G*计算, 得到了能量三重简并的最高占据轨道的函数表达式, 进而运用$ O_h $群的对称操作作用在三个轨道函数上, 得到各操作的矩阵表示, 于是得到特征标, 最后确定了最高占据轨道为$ T_{1g} $对称性.
    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.
      通信作者: 郭迎春, ycguo@phy.ecnu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 61275128, 11774411, 11474348)资助的课题.
      Corresponding author: Guo Ying-Chun, ycguo@phy.ecnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61275128, 11774411, 11474348).
    [1]

    Tang B, Zhang L F, Han F Y, Luo Z C, Liang Q Q, Liu C Y, Zhu L P, Zhang J M 2018 AIP Adv. 8 015016Google Scholar

    [2]

    Zhang X, Gockenbach E, Liu Z L, Chen H B, Yang L H 2013 Electr. Power Syst. Res. 103 105Google Scholar

    [3]

    Okubo H, Beroual A 2011 IEEE Electr. Insul. M. 27 34

    [4]

    Yoshino K, Hayashi S, Kohno Y, Kaneto K, Okube J, Moriya T 1984 Jpn. J. Appl. Phys. 23 L198Google Scholar

    [5]

    Tachikawa H, Yamano T 2001 Chem. Phys. 264 81Google Scholar

    [6]

    Ravishankara A R, Solomon S, Turnipseed A A, Warren R F 1993 Science 259 194Google Scholar

    [7]

    Niessen W V, Kraemer W P, Diercksen G H F 1979 Chem. Phys. Lett. 63 65Google Scholar

    [8]

    Christophorou L G, Olthoff J K 2000 J. Phys. Chem. Ref. Data 29 267Google Scholar

    [9]

    Decleva P, Fronzoni G, Kivimaki A, AlvarezRuiz J, Svensson S 2009 J. Phys. B: At. Mol. Opt. Phys. 42 055102Google Scholar

    [10]

    Jose J, Lucchese R R, Rescigno T N 2014 J. Chem. Phys. 140 481

    [11]

    Hay P J 1977 J. Am. Chem. Soc. 8 1003

    [12]

    Weigold E, Zheng Y 1991 Chem. Phys. 150 405Google Scholar

    [13]

    Li Y, Agrena H, Carravettab V, Vahtrasa O, Karlssonc L, Wannbergc B, Hollandd D M P, MacDonald M A 1998 J. Electron. Spectrosc. 94 163Google Scholar

    [14]

    Zhao M F, Shan X, Yang J, Wang E L, Niu S S , Chen X J 2015 Chin. J. Chem. Phys. 28 539Google Scholar

    [15]

    Watanabe N, Yamazaki M, Takahashi M 2016 J. Electron. Spectrosc. 209 78Google Scholar

    [16]

    Hay P J 1982 J. Chem. Phys. 76 502Google Scholar

    [17]

    Tachikawa H 2002 J. Phys. B: At. Mol. Opt. Phys. 35 55Google Scholar

    [18]

    徐亦庄 1988 分子光谱理论 (北京: 清华大学出版社) 第75页

    Xu Y Z 1988 Theory of Molecular Spectroscopy (Beijing: Tsinghua University Press) p75 (in Chinese)

    [19]

    Werner H J, Knowles P J, Lindh R, Manby F R, Schutz M, Celani P, Korona T, Rauhut G, Amos R D, Bernhardsson A, Berning A, Cooper D L, Deegan M J O, Dobbyn A J, Eckert F, Hampel C, Hetzer G, Lloyd A W, McNicholas S J, Meyer W, Mura M E, Nicklass A, Palmieri P, et al. Molpro, A Package of ab initio Programs (Version 2006.1) http://www.molpro.net [2018-12-12 ]

    [20]

    Delhommelle J, Boutio A, Tavitian B, Mackie A D, Fuchs A H 1999 Mol. Phys. 96 719Google Scholar

    [21]

    Krishnan R, Binkley J S, Seeger R, Pople J A 1979 J. Chem. Phys. 72 650

  • 图 1  $\varPsi_{B_{1g}}$, $\varPsi_{B_{2g}}$$\varPsi_{B_{3g}}$波函数截面图 (a) $\varPsi_{B_{1g}}$x = 0.8 a.u.处的yz截面图; (b) $\varPsi_{B_{1g}}$y = 0.8 a.u.处的xz截面图; (c) $\varPsi_{B_{1g}}$z = 0 a.u.处的xy截面图;(d) $\varPsi_{B_{2g}}$y = 0 a.u.处的xz截面图; (e) $\varPsi_{B_{3g}}$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

    Fig. 1.  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 2  SF6对称操作$C_2'$, $C_3$, $C_4$$C_2$的对称轴

    Fig. 2.  Symmetric axes of symmetric operators $C_2'$, $C_3$,$C_4$ and $C_2$ on SF6

    表 1  SF6的分子结构

    Table 1.  Molecular structure of SF6

    kxk/a.u.yk/a.u.zk/a.u.
    S1000
    F22.92300
    F3–2.92300
    F402.9230
    F50–2.9230
    F6002.923
    F700–2.923
    下载: 导出CSV

    表 2  6-311G*基组中高斯函数的参数表

    Table 2.  Parameters of Gaussian functions of 6-311G* basis

    i123456
    ci0.0354610.2374510.8204581.01.01.0
    αi/a.u.55.444112.63233.717561.165450.3218921.75
    下载: 导出CSV

    表 3  $O_h$群的部分特征标表

    Table 3.  Part of character table of $O_h$ group

    $O_h$E$8C_3$$6C_2'$$6C_4$$3C_2$P$6S_4$$8S_6$$3\sigma_h$$6\sigma_d$
    $T_{1g}$30–11–1310–1–1
    $T_{2g}$301–1–13–10–11
    下载: 导出CSV
  • [1]

    Tang B, Zhang L F, Han F Y, Luo Z C, Liang Q Q, Liu C Y, Zhu L P, Zhang J M 2018 AIP Adv. 8 015016Google Scholar

    [2]

    Zhang X, Gockenbach E, Liu Z L, Chen H B, Yang L H 2013 Electr. Power Syst. Res. 103 105Google Scholar

    [3]

    Okubo H, Beroual A 2011 IEEE Electr. Insul. M. 27 34

    [4]

    Yoshino K, Hayashi S, Kohno Y, Kaneto K, Okube J, Moriya T 1984 Jpn. J. Appl. Phys. 23 L198Google Scholar

    [5]

    Tachikawa H, Yamano T 2001 Chem. Phys. 264 81Google Scholar

    [6]

    Ravishankara A R, Solomon S, Turnipseed A A, Warren R F 1993 Science 259 194Google Scholar

    [7]

    Niessen W V, Kraemer W P, Diercksen G H F 1979 Chem. Phys. Lett. 63 65Google Scholar

    [8]

    Christophorou L G, Olthoff J K 2000 J. Phys. Chem. Ref. Data 29 267Google Scholar

    [9]

    Decleva P, Fronzoni G, Kivimaki A, AlvarezRuiz J, Svensson S 2009 J. Phys. B: At. Mol. Opt. Phys. 42 055102Google Scholar

    [10]

    Jose J, Lucchese R R, Rescigno T N 2014 J. Chem. Phys. 140 481

    [11]

    Hay P J 1977 J. Am. Chem. Soc. 8 1003

    [12]

    Weigold E, Zheng Y 1991 Chem. Phys. 150 405Google Scholar

    [13]

    Li Y, Agrena H, Carravettab V, Vahtrasa O, Karlssonc L, Wannbergc B, Hollandd D M P, MacDonald M A 1998 J. Electron. Spectrosc. 94 163Google Scholar

    [14]

    Zhao M F, Shan X, Yang J, Wang E L, Niu S S , Chen X J 2015 Chin. J. Chem. Phys. 28 539Google Scholar

    [15]

    Watanabe N, Yamazaki M, Takahashi M 2016 J. Electron. Spectrosc. 209 78Google Scholar

    [16]

    Hay P J 1982 J. Chem. Phys. 76 502Google Scholar

    [17]

    Tachikawa H 2002 J. Phys. B: At. Mol. Opt. Phys. 35 55Google Scholar

    [18]

    徐亦庄 1988 分子光谱理论 (北京: 清华大学出版社) 第75页

    Xu Y Z 1988 Theory of Molecular Spectroscopy (Beijing: Tsinghua University Press) p75 (in Chinese)

    [19]

    Werner H J, Knowles P J, Lindh R, Manby F R, Schutz M, Celani P, Korona T, Rauhut G, Amos R D, Bernhardsson A, Berning A, Cooper D L, Deegan M J O, Dobbyn A J, Eckert F, Hampel C, Hetzer G, Lloyd A W, McNicholas S J, Meyer W, Mura M E, Nicklass A, Palmieri P, et al. Molpro, A Package of ab initio Programs (Version 2006.1) http://www.molpro.net [2018-12-12 ]

    [20]

    Delhommelle J, Boutio A, Tavitian B, Mackie A D, Fuchs A H 1999 Mol. Phys. 96 719Google Scholar

    [21]

    Krishnan R, Binkley J S, Seeger R, Pople J A 1979 J. Chem. Phys. 72 650

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  • 收稿日期:  2018-12-19
  • 修回日期:  2019-02-20
  • 刊出日期:  2019-04-20

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