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Hierarchical enumeration of octahedral complexes by using combined-permutation representations

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Abstract

Hierarchical enumerations of octahedral derivatives are conducted in accord with the hierarchy of groups for characterizing an octahedral skeleton, i.e., point groups (\({\varvec{O}}\) and \({\varvec{O}}_{h}\); orders 24 and 48) \(\subset \) RS-stereoisomeric group (\({\varvec{O}}_{h\widetilde{\sigma }\widehat{I}}\); order 96) \(\subset \) stereoisomeric group (\(\widetilde{{\varvec{O}}}_{h\widetilde{\sigma }\widehat{I}} \,(= {\varvec{S}}^{[6]}_{\sigma \widehat{I}})\); order 1440) \(=\) isoskeletal group (\(\widetilde{\widetilde{{\varvec{O}}}}_{h\widetilde{\sigma }\widehat{I}} \,(= {\varvec{S}}^{[6]}_{\sigma \widehat{I}})\); order 1440). The corresponding cycle indices with chirality fittingness (CI-CFs) are calculated by using combined-permutation representations (Fujita in MATCH Commun Math Comput Chem 76:379–400, 2016). Then, a set of three ligand-inventory functions for 3D enumeration is introduced into the CI-CFs for the enumerations under the point groups and the RS-stereoisomeric group, while a single ligand-inventory function for 2D (graph) enumeration is introduced into the CI-CFs for the enumerations under the stereoisomeric group and the isoskeletal group. The expansion of the resulting equations gives generating functions, in which the coefficients of respective terms show the numbers of octahedral derivatives. They are discussed by drawing isomer-classification diagrams after they are categorized into octahedral derivatives with achiral proligands, those with achiral and chiral proligands, and those with chiral proligands. Type-I and type-V stereoisograms are drawn to demonstrate the conceptual distinction between RS-stereogenicity and stereogenicity, where the combination of configuration indices and C/A-descriptors is discussed.

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Notes

  1. The symbols \([\theta ]_{15}^{*}\), \([\theta ]_{19}^{*}\), \([\theta ]_{27}^{*}\), \([\theta ]_{30}^{*}\), \([\theta ]_{40}^{*}\), and \([\theta ]_{45}^{*}\) in Table 2 should be corrected to be \([\theta ]_{15}\), \([\theta ]_{19}\), \([\theta ]_{27}\), \([\theta ]_{30}\), \([\theta ]_{40}\), and \([\theta ]_{45}\).

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  1. Shonan Institute of Chemoinformatics and Mathematical Chemistry, Kaneko 479-7 Ooimachi, Ashigara-Kami-Gun, Kanagawa-Ken, 258-0019, Japan

    Shinsaku Fujita

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  1. Shinsaku Fujita
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Fujita, S. Hierarchical enumeration of octahedral complexes by using combined-permutation representations. J Math Chem 56, 2845–2875 (2018). https://doi.org/10.1007/s10910-018-0921-6

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