Stereoisograms for three-membered heterocycles: I. Symmetry-itemized enumeration of oxiranes under an RS-stereoisomeric group

Abstract

For the purpose of characterizing the stereochemistry and stereoisomerism of oxirane derivatives, the RS-stereoisomeric group $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ of order 8 has been defined by starting point group $$\mathbf{C}_{2v}$$ of order 4, which specifies the geometric features of an oxirane skeleton. The isomorphism between $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ and the point group $$\mathbf{D}_{2h}$$ of order 8 is discussed algebraically and diagrammatically. The data necessary to combinatorial enumeration under $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ , e.g., the non-redundant set of subgroups, the subduction of coset representations, and the inverse of the mark table, are prepared by referring to the data of $$\mathbf{D}_{2h}$$ . The fixed-point-matrix method and the partial-cycle-index method, which have been originally developed to accomplish combinatorial enumeration under point groups in the unit-subduced-cycle-index approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), are extended and applied to the combinatorial enumeration of oxirane derivatives under the RS-stereoisomeric group $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ . Thereby, the numbers of inequivalent quadruplets are calculated in an itemized fashion with respect to the subgroups of $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ , where each quadruplet contained in a stereoisogram is counted once. Such quadruplets are further categorized into five types (type I–V). The enumeration of oxiranes under the point group $$\mathbf{C}_{2v}$$ as well as under the RS-permutation group $$\mathbf{C}_{2\widetilde{\sigma }}$$ is also conducted and the results are compared with those of the RS-stereoisomeric group $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ .