Sphericity Governs Both Stereochemistry in a Molecule and Stereoisomerism Among Molecules

Abstract

The concept of sphericity and relevant fundamental concepts that we have proposed have produced a systematized format for comprehending stereochemical phenomena. Permutability of ligands in conventional approaches is discussed from a stereochemical point of view. After the introduction of orbits governed by coset representations, the concepts of subduction and sphericity are proposed to characterize desymmetrization processes, with a tetrahedral skeleton as an example. The stereochemistry and stereoisomerism of the resulting promolecules (molecules formulated abstractly) are discussed in terms of the concept of sphericity as a common mathematical and logical framework. Thus, these promolecules are characterized by point group and permutation group symmetry. Prochirality, stereogenicity, prostereogenicity, and relevant topics are described in terms of the concept of sphericity.

Mathematical Stereochemistry as the Theoretical Foundations of Organic and Inorganic Chemistry

“…under the name Fujita's USCI approach, the four methods of symmetry-itemized enumerations were developed and their versatility was testified by using various skeletons as probes [10]. In addition to these quantitative applications, Fujita's USCI approach also enables us to develop qualitative discussions on prochirality and symmetric properties of various molecules [11,12].…”

Systematic Enumeration and Symmetries of Cubane Derivatives

“…On the other hand, the terms defined newly by the author (Fujita) [24] such as enantiospheric, homospheric, and hemispheric are concerned with equivalence classes, which stem from mathematical formulations such as coset representations and their subductions. These terms should be used for qualitative applications [25,26] as well as for quantitative applications in stereochemistry. [27,28] In particular, the subductions of coset representations and the concepts of sphericities result in the formation of unit subduced cycle indices without and with chirality fittingness (USCIs and USCI-CFs), so that this course of examining symmetric properties is referred to under the name Fujita's USCI approach.…”

“…[44,45] From a qualitative point of view, Fujita's USCI approach enables us to rationalize various stereochemical phenomena in a systematic fashion, [24,29] e.g., derivation from methane and adamantane skeletons of T d symmetry; [46,47] general treatments of local chirality and prochirality; [25] chirogenic sites in an enantiospheric orbit; [48] design of molecules of high symmetry; [49] the subduction-of-coset-representation (SCR) notation for systematic classification of molecular symmetries; [50] systematic characterization of prochirality, prostereogenicity, and stereogenicity by means of the sphericity concept; [51] an approach to topic relationships; [52] classification of meso compounds; [53] systematic design of chiral molecules of high symmetry; [54] desymmetrization of achiral skeletons by monosubstitution; [55] stereochemistry and stereoisomerism characterized by the sphericity concept; [56] chirality and stereogenic-ity for square-planar complexes; [57] approaches for restructuring stereochemistry by novel terminology; [58,59] and sphericity beyond topicity. [21,26] Fujita's Proligand Method and Related Enumeration Tools Although Fujita's USCI approach enables us to accomplish symmetry-itemized enumeration, gross enumeration without such symmetry itemization is desirable if a brief perspective is necessary. Because inverse mark tables can be effectively restricted to cyclic subgroups, USCIs and USCI-CFs are reduced into cycle indices with and without chirality fittingness (CIs and CI-CFs).…”

Numbers of Alkanes and Monosubstituted Alkanes. A Long-Standing Interdisciplinary Problem over 130 Years

“…5961 In addition to such quantitative applications as stereoisomer enumerations, Fujita's USCI approach enhanced by the concept of sphericities is also useful for qualitative discussions in stereochemistry, as summarized in reviews. 62,63 Several examples of its qualitative applications are as follows: systematic classification of molecular symmetry, 64 systematic investigation on local chirality and prochirality, 57,65 and characterization of anisochrony and symmetry non-equivalence. 66 However, algebraic features of Fujita's USCI approach were criticized with respect to their popularity among organic chemists.…”