Symmetry-Itemized Enumeration of Quadruplets of RS-Stereoisomers Derived from a Trigonal Pyramidal Skeleton

Abstract

Abstract Stereoisograms of five types are devised as graphical expressions of the action of an RS-stereoisomeric group on a trigonal pyramidal skeleton. A quadruplet of promolecules (i.e., a reference promolecule, its enantiomer, its RS-diastereomer, and its holantimer) contained in a stereoisogram is counted once as an equivalence class under the action of an RS-stereoisomeric group. An RS-stereoisomeric group <img align="middle" src="./Graphics/abst-20150015-1.gif"/> is constructed by starting from the point group C3v of a trigonal pyramidal skeleton, where a point group C3v, an RS-permutation group <img align="middle" src="./Graphics/abst-20150015-2.gif"/>, a ligand-reflection group C3Î are integrated. Fujita’s unit-subduced-cycle-index (USCI) approach for enumeration under point groups is extended to cover enumeration under RS-stereoisomeric groups. The fixed-point-matrix (FPM) method and the partial-cycle-index (PCI) method of the USCI approach are applied to symmetry-itemized and type-itemized enumeration of quadruplets of promolecules under the action of the RS-stereoisomeric group <img align="middle" src="./Graphics/abst-20150015-1.gif"/>. Theoretical foundations of stereochemical nomenclature are discussed on the basis of Fujita’s stereoisogram approach, where three aspects of absolute configuration are emphasized. Prochirality and pro-RS-stereogenicity are clarified to be conceptually distinct, just as chirality and RS-stereogenicity are conceptually distinct. Young’s tableaux for (pro)chirality and those for (pro-)RS-stereogenicity are compared to demonstrate such conceptual distinctions. It follows that a pair of R/S-stereodescriptors is assigned on the basis of RS-stereogenicity, not of chirality; and a pair of pro-R/pro-S-descriptors is assigned on the basis of pro-RS-stereogenicity, not of prochirality.