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

摘要

The RS-stereoisomeric group T_(dσ?) is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group O_h of order 48. The non-redundant set of subgroups (SSG) of T_(dσ?) is obtained by referring to the non-redundant SSG of O_h. The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be T_(dσ?) (/C_(3σ?)), which is closely related to the O_h(/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 T_(dσ?)(/C_(3σ?)) 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 T_(dσ?). After the subgroups of T_(dσ?) are categorized into types I-V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.