SF_6分子最高占据轨道对称性的判断
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摘要
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SF6基态1 A_(1g)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SF6基态1 A1g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SF6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 multiatom 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 Oh 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 A_(1 g) of SF6 is(core)~(22)(4 a_(1 g))2(3 t_(1 u))6(2 eg)~4(5 a_(1 g))2(4 t_(1 u))6(lt_(2 g))6(3 e_g)~4(lt2 u)6(5 t_(1 u))6(lt_(1 g))6 and the symmetry of the HOMOs is T_(1 g). However, in some literature, the symmetry of HOMOs of SF6 has been written as T_(2 g) instead of T_(1 g). The reason for this mistake lies in the fact that in the ab initial quantum chemical calculation used is the Abelian group D_(2 h), 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_(2 h) group and those of Oh group. For example, both irreducible representations T_(1 g) and T_(2 g) of Oh group are reduced to the sum of B_(1 g), B_(2 g) and B_(3 g) of D_(2 h). So the symmetry of the orbitals needs to be investigated further to identify whether it is T_(1 g) or T_(2 g). In this work, we calculate the orbital functions in the equilibrium structure of ground state of SF6 by using HF/6-311 G* 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 Oh 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_(1 g). By using this process, we may determine the molecular orbital symmetry of any other molecules with high symmetry group.
Quantum chemical calculation is an important method to investigate the molecular structures for multiatom 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 Oh 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 A_(1 g) of SF6 is(core)~(22)(4 a_(1 g))2(3 t_(1 u))6(2 eg)~4(5 a_(1 g))2(4 t_(1 u))6(lt_(2 g))6(3 e_g)~4(lt2 u)6(5 t_(1 u))6(lt_(1 g))6 and the symmetry of the HOMOs is T_(1 g). However, in some literature, the symmetry of HOMOs of SF6 has been written as T_(2 g) instead of T_(1 g). The reason for this mistake lies in the fact that in the ab initial quantum chemical calculation used is the Abelian group D_(2 h), 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_(2 h) group and those of Oh group. For example, both irreducible representations T_(1 g) and T_(2 g) of Oh group are reduced to the sum of B_(1 g), B_(2 g) and B_(3 g) of D_(2 h). So the symmetry of the orbitals needs to be investigated further to identify whether it is T_(1 g) or T_(2 g). In this work, we calculate the orbital functions in the equilibrium structure of ground state of SF6 by using HF/6-311 G* 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 Oh 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_(1 g). By using this process, we may determine the molecular orbital symmetry of any other molecules with high symmetry group.
引文
[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 015016
[2] Zhang X, Gockenbach E, Liu Z L, Chen H B, Yang L H 2013Electr. Power Syst. Res. 103 105
[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 L198
[5] Tachikawa H, Yamano T 2001 Chem. Phys. 264 81
[6] Ravishankara A R, Solomon S, Turnipseed A A, Warren R F1993 Science 259 194
[7] Niessen W V, Kraemer W P, Diercksen G H F 1979 Chem.Phys. Lett. 63 65
[8] Christophorou L G,Olthoff J K 2000 J. Phys. Chem. Ref.Data 29 267
[9] Decleva P, Fronzoni G, Kivimaki A, AlvarezRuiz J, Svensson S 2009 J. Phys. B:At. Mol. Opt. Phys. 42 055102
[10] Jose J, Lucchese R R, Rescigno T N 2014 J. Chem. Phys. 140481
[11] Hay P J 1977 J. Am. Chem. Soc. 8 1003
[12] Weigold E, Zheng Y 1991 Chem. Phys. 150 405
[13] Li Y, Agrena H, Carravettab V, Vahtrasa O, Karlssonc L,Wannbergc B, Hollandd D M P, MacDonald M A 1998 J.Electron. Spectrosc. 94 163
[14] Zhao M F, Shan X, Yang J, Wang E L, Niu S S, Chen X J2015 Chin. J. Chem. Phys. 28 539
[15] Watanabe N, Yamazaki M, Takahashi M 2016 J. Electron.Spectrosc. 209 78
[16] Hay P J 1982 J. Chem. Phys. 76 502
[17] Tachikawa H 2002 J. Phys. B:At. Mol. Opt. Phys. 35 55
[18] Xu Y Z 1988 Theory of Molecular Spectroscopy(Beijing:Tsinghua University Press)p75(in Chinese)[徐亦庄1988分子光谱理论(北京:清华大学出版社)第75页]
[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. 96 719
[21] Krishnan R, Binkley J S, Seeger R, Pople J A 1979 J. Chem.Phys. 72 650
[2] Zhang X, Gockenbach E, Liu Z L, Chen H B, Yang L H 2013Electr. Power Syst. Res. 103 105
[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 L198
[5] Tachikawa H, Yamano T 2001 Chem. Phys. 264 81
[6] Ravishankara A R, Solomon S, Turnipseed A A, Warren R F1993 Science 259 194
[7] Niessen W V, Kraemer W P, Diercksen G H F 1979 Chem.Phys. Lett. 63 65
[8] Christophorou L G,Olthoff J K 2000 J. Phys. Chem. Ref.Data 29 267
[9] Decleva P, Fronzoni G, Kivimaki A, AlvarezRuiz J, Svensson S 2009 J. Phys. B:At. Mol. Opt. Phys. 42 055102
[10] Jose J, Lucchese R R, Rescigno T N 2014 J. Chem. Phys. 140481
[11] Hay P J 1977 J. Am. Chem. Soc. 8 1003
[12] Weigold E, Zheng Y 1991 Chem. Phys. 150 405
[13] Li Y, Agrena H, Carravettab V, Vahtrasa O, Karlssonc L,Wannbergc B, Hollandd D M P, MacDonald M A 1998 J.Electron. Spectrosc. 94 163
[14] Zhao M F, Shan X, Yang J, Wang E L, Niu S S, Chen X J2015 Chin. J. Chem. Phys. 28 539
[15] Watanabe N, Yamazaki M, Takahashi M 2016 J. Electron.Spectrosc. 209 78
[16] Hay P J 1982 J. Chem. Phys. 76 502
[17] Tachikawa H 2002 J. Phys. B:At. Mol. Opt. Phys. 35 55
[18] Xu Y Z 1988 Theory of Molecular Spectroscopy(Beijing:Tsinghua University Press)p75(in Chinese)[徐亦庄1988分子光谱理论(北京:清华大学出版社)第75页]
[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. 96 719
[21] Krishnan R, Binkley J S, Seeger R, Pople J A 1979 J. Chem.Phys. 72 650