Stereochemistry and Stereoisomerism Characterized by the Sphericity Concept

The sphericity concept proposed for specifying stereochemistry in a molecule (S. Fujita, J. Am. Chem. Soc., 112, 3390 (1990)) has been extended to investigate stereoisomerism among molecules. The new matter of the present approach is to characterize the global symmetries of molecules as the “local symmetries of stereoisomerism”. Thereby, stereochemistry and stereoisomerism have been discussed on a common basis. Promolecules, which have been generated as stereochemical models of molecules by placing proligands (structureless ligands with chirality/achirality) on the vertices of a tetrahedral skeleton, have been analyzed by a permutation-group approach as well as by a point-group one. The skeleton has been considered to belong to the symmetric group of degree 4 (S[4]) as well as to the isomorphic point group Td. The chirality fittingness derived from the sphericity concept has been applied to the characterization of local symmetries of promolecule, where two types of Young’s tableaus have been compared. Thus, Young’s tableaus of symmetry have been introduced to treat the ligand packing based on the chirality fittingness. These tableaus have been compared with Young’s tableaus of permutation, which have taken no account of such chirality fittingness. The two types of Young’s tableaus have been applied to the enumeration of tetrahedral isomers under the observance and the violation of chirality fittingness. This enumeration has enabled us to clarify the quantitative aspect of the sphericity concept in characterizing isomer equivalence. Thereby, equivalent isomers under a point-group symmetry have been shown to construct an orbit of stereoisomers that is ascribed to a coset representation. Homomeric, enantiomeric, and diastereomeric relationships between stereoisomers have been discussed by means of homospheric, enantiospheric and hemispheric orbits of stereoisomers. Skeleton-based and ligand-based categories for enantiomers and diastereomers have been discussed. The stereogenicity and the prostereogenicity of the Chan–Ingold–Prelog system have been related to Young’s tableaus of permutation.