Abstract
The RS-stereoisomeric group \(\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}\) is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group \(\mathbf{O}_{h}\) of order 48. The non-redundant set of subgroups (SSG) of \(\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}\) is obtained by referring to the non-redundant SSG of \(\mathbf{O}_{h}\). The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be \(\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})\), which is closely related to the \(\mathbf{O}_{h}(/\mathbf{D}_{3d})\). According to the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), the subdution of \(\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})\) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). The fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of \(\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}\). After the subgroups of \(\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}\) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.
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Fujita, S. Symmetry-itemized enumeration of quadruplets of RS-stereoisomers: I—the fixed-point matrix method of the USCI approach combined with the stereoisogram approach. J Math Chem 52, 508–542 (2014). https://doi.org/10.1007/s10910-013-0276-y
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DOI: https://doi.org/10.1007/s10910-013-0276-y