Systematic Classification of Molecular Symmetries

Point groups are frequently incomplete to afford full symmetry information about molecules; various molecules belong to the same point group. For example, a list of C_2v molecules contains water, hydrogen sulfide, formaldehyde, phosgene, chlorobenzene, fluorobenzene, 1,2- and 1,3-difluorobezenes, 1,2-benzoquinone, 1,4-dichloronaphthalene, 1,4-naphthoquinone, pyridine, 4-chloropyridine, 2,6-dichloropyridine, pyridine N-oxide, furan, thiophene, cyclopropanone, cyclobutanone, cyclopentanone, tetrahydrofuran, tetrahydrothiophene, dichloromethane, difluoromethane, oxirane, phenanthrene, bicyclo[2.2.1]heptane, basketane, adamantanone, noradamantane and so on. Such a list should be classified into several categories in the light of a rational criterion. For this purpose, Pople[1] has proposed the concept of “framework group”. By this method, difluoromethane is designated as C_2v [C₂(C), σ _v (F₂), σ ’_2v (H₂)]. Flurry[2] pointed out that the framework group is related to local (site) symmetries and proposed his notation based on the local symmetries. Thus, the difluoromethane is designated as C_2v [C_2v ,(C), C _s (F₂), C’ _s (H₂)]. Although Flurry’s method has a potential applicability, it is not so easy to determine such local symmetries especially in the cases of complex molecules. We here propose the SCR (set-of-coset-representation) notation for classifying molecular symmetry.