Symmetry-itemized enumeration of quadruplets of RS-stereoisomers: I—the fixed-point matrix method of the USCI approach combined with the stereoisogram approach

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.