CARTESIAN AXIS AND PLANE CONVENTIONS IN C<sub>nv</sub> SYMMETRY GROUPS

Dias, Lucas A. L.; Faria, Roberto B.

doi:10.21577/0100-4042.20170790

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Educação • Quim. Nova 44 (10) • 2021 • https://doi.org/10.21577/0100-4042.20170790 copy

CARTESIAN AXIS AND PLANE CONVENTIONS IN C_nv SYMMETRY GROUPS

Authorship SCIMAGO INSTITUTIONS RANKINGS

In this work, we call the attention to the ambiguity found in the literature when labeling vibrations and molecular orbitals as B₁ and B₂ for molecules belonging to the C_2v point group as, for example, the water molecule. A survey of several books and some articles shows that this ambiguity comes from a long time ago and persists today, being a source of misunderstanding and a waste of time for students and teachers. It means that, in the case of the point groups C_nv, D_n, and D_nh (n = 2, 4, 6), it is very important to draw students’ attention to this ambiguity that exists in the literature. It is unfortunate that the recommendation made by Mulliken, more than sixty years ago, to always place the water molecule in the yz plane, has not been followed.

Keywords:
symmetry; vibrational spectroscopy; electronic spectroscopy

C_2v	E	C₂	σ_v^(xz)	σ'_v^(yz)
A₁	1	1	1	1	z	x²2 Kettle, S. F. A.; Symmetry and Structure - Readable Group Theory for Chemists, 3rd ed., John Wiley & Sons: New York, 2007., y², z²
A₂	1	1	-1	-1	R _z	xy
B₁	1	-1	1	-1	x, R_y	xz
B₂	1	-1	-1	1	y, R_x	yz

Author	σ(xz) Figure 1a $Γ = 2 A_{1} + B_{1}$ HOMO nonbonding (b₂)	σ(yz) Figure 1b $Γ * = 2 A_{1} + B_{2}$ HOMO nonbonding (b₁)	Reference	Comments
Albright and Burdett		x	¹⁰10 Albright, T. A.; Burdett, J. K.; Problems in Molecular Orbital Theory, Oxford University Press: Oxford, 1992.	pg. 119
Barrow		x	⁷7 Barrow, G. M.; Introduction to Molecular Spectroscopy, McGraw-Hill Book Company: Singapore, 1962.	pg. 163
Bernath		x	⁸8 Bernath, P. F.; Spectra of Atoms and Molecules, 3rd ed., Oxford University Press: Oxford, 2016.	pg. 174, 177, 238
Bishop		x	¹¹11 Bishop, D. M.; Group Theory and Chemistry, Dover: New York, 1993.	pg. 182
Carter		x	¹²12 Carter, R. L.; Molecular Symmetry and Group Theory, John Wiley & Sons: New York, 1998.	pg. 13
Cotton	x		¹1 Cotton, F. A.; Chemical Applications of Group Theory, 2nd ed., John Wiley & Sons: New York, 1971.	pg. 310
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Jaffé, H.H.; Orchin, M.		x	¹⁹19 Jaffé, M.; Orchin, H. H.; Symmetry in Chemistry, John Wiley & Sons: New York, 1966.	pg. 14
Kettle		x¹	²2 Kettle, S. F. A.; Symmetry and Structure - Readable Group Theory for Chemists, 3rd ed., John Wiley & Sons: New York, 2007.	pg. 21
Lever	x		²⁰20 Lever, A. B. P.; Inorganic Electronic Spectroscopy, Elsevier Publishing Company: Amsterdam, 1968.	pg. 36
Levine		x	²¹21 Levine, I. N.; Quantum Chemistry, 4th ed., Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1991.	pg. 456
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Miessler, Fischer and Tarr	x		²⁴24 Miessler, G. L.; Fischer, P. J.; Tarr, D. A.; Inorganic Chemistry, 5th ed., Pearson Education Limited: Harlow, United Kingdom, 2013.	pg. 97
Murrell, Kettle, and Tedder		x	²⁵25 Murrell, J. N.; Kettle, S. F. A.; Tedder, J. M.; Valence Theory, 2nd ed., John Wiley & Sons Ltd.: London, 1970.	pg. 190
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Pfennig		x	²⁶26 Pfennig, B. W.; Principles of Inorganic Chemistry, John Wiley & Sons, Inc.: Hoboken, NJ, 2015.	pg. 194
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Wilson, Decius, and Cross	x		³⁰30 Wilson, E. B.; Decius, J. C.; Cross, P. C.; Molecular Vibrations, Dover: New York, 1980.	pg. 79

Author	σ(xz) Figure 1a $Γ = 2 A_{1} + B_{1}$ HOMO nonbonding (b₂)	σ(yz) Figure 1b $Γ * = 2 A_{1} + B_{2}$ HOMO nonbonding (b₁)	Reference	Comments
Brundle and Turner		x	³¹31 Brundle, C. R.; Turner, D. W.; Proc. R. Soc. London, Ser. A 1968, 307, 27.	pg. 28
Orchin and Jaffé		x	³²32 Orchin, M.; Jaffé, H. H.; J. Chem. Educ. 1970, 47, 372.	pg. 374
Greinacher	x		³³33 Greinacher, Von E.; Lüttke, W.; Mecke, R.; Z. Elektrochem. 1955, 59, 23.	v₃ (antisymmetric stretching vibrational mode) = B₁
Jayachander Rao and Varandas		x	³⁴34 Jayachander Rao, B.; Varandas, A. J. C.; J. Phys. Chem. A 2015, 119, 12367.	pg. 12372
Luo et al.		x	³⁵35 Luo, Z.; Chang, Y.; Zhao, Y.; Yang, J.; Chen, Z.; Cheng, Y.; Che, L.; Wu, G.; Yang, X.; Yuan, K.; J. Phys. Chem. A 2021, 125, 3622.	HOMO = b₁
Mulliken		x	⁴4 Mulliken, R. S.; J. Chem. Phys. 1955, 23, 1977.	pg. 2002
Shimanouchi	x		⁵5 Shimanouchi, T.; J. Phys. Chem. Ref. Data 1977, 6, 993.	pg. 1001
Walrafen	x		³⁶36 Walrafen, G. E.; J. Chem. Phys. 1964, 40, 3249.	v₃ (antisymmetric stretching vibrational mode) = B₁
Walsh		x	³3 Walsh, A. D.; J. Chem. Soc. 1953, 2260.	pg. 2262

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CARTESIAN AXIS AND PLANE CONVENTIONS IN Cnv SYMMETRY GROUPS

CARTESIAN AXIS AND PLANE CONVENTIONS IN C_nv SYMMETRY GROUPS