Chirality and RS-Stereogenicity as Two Kinds of Handedness. Their Aufheben by Fujita’s Stereoisogram Approach for Giving New Insights into Classification of Isomers

Abstract

Chirality under point-group symmetry and RS-stereogenicity under RS-permutation-group symmetry are discussed from a viewpoint of two kinds of handedness, which are proposed on the basis of whether or not ligand reflections are taken into consideration. After the additional formulation of sclerality under ligand-reflection-group symmetry, the three groups are integrated to generate RS-stereoisomeric groups, which are represented by stereoisograms as diagrammatic expressions. The vertical direction of a stereoisogram is concerned with (self-)enantiomeric relationships or chirality/achirality for supporting Le Bel’s way, while the horizontal direction is concerned with (self-)RS-diastereomeric relationships or RS-stereogenicity/RS-astereogenicity for supporting van’t Hoff’s way. By taking account of the diagonal direction for characterizing (self-)holantimeric relationships or sclerality/asclerality, the two kinds of handedness (chirality and RS-stereogenicity) are integrated to develop the concept of RS-stereoisomerism, which is an intermediate concept for mediating between enantiomerism and stereoisomerism. The creation of RS-stereoisomerism as an intermediate concept means the Aufheben of van’t Hoff’s way and Le Bel’s way. Thereby, Fujita’s stereoisogram approach has brought about a paradigm shift, so that modern stereochemistry has been restructured substantially on the basis of mathematical formulations, where true remedies for the misleading terminology of modern stereochemistry have been developed in a rational fashion. In particular, the hierarchy of isomers and stereoisomers has been thoroughly revised by adding RS-stereoisomerism, so as to develop a new flowchart based on equivalence relationships and equivalence classes, as exemplified by using various skeletons as probes.

Application of GAP (groups, algorithms and programming) system to stereoisograms for characterizing RS-stereoisomers of cubane derivatives

“…gap> gen_1 := [(1,2,3,4) (5,6,7,8), (2,4,5) (3,8,6)];; #generators for O gap> O_cube := Group(gen_1); Group([ (1,2,3,4) (5,6,7,8), (2,4,5) (3,8,6) ]) gap> Display(Size(O_cube));; 24 gap> tom_O_cube := TableOfMarks(O_cube);; gap> gap> list1 := [1,2,3,4,5,6,7,8,9,10,11];; gap> list2 := [1,2,3,4,6,5,7,8,9,10,11];; gap> gap> perm := PermListList(list1, list2); (5,6) gap> Display 1 1 1 1 1 1 1 1 1 1 1 Such standardizations of mark tables and USCI-CF (unit subduced cycle indices with chirality fittingness) tables have been discussed by using different O h -skeletons (e.g., octahedron (OC-6), cube (CU-8), cuboctahedron, truncated octahedron, and truncated hexahedron) as probes [30] after various GAP functions were newly developed t...…”

“…The effect of the RS -permutation σ h(1) onto the cubane skeleton 1 is depicted in Figure 3. If we obey the conventions of the GAP system, any 1-cycle can be omitted, so that we can use ( 1 or the deletion of two 1-cycles (1)(2) from (1 5)(2 6)(3 7)(4 8)(9) (10), which is generated from a 2-cycle (9 10). The resulting numbered skeleton 2 is not accompanied with ligand reflections, so that the eight positions are numbered without using an overbar.…”

“…Fujita's RS -stereogenicity is differentiated from Mislow-Siegel's stereogenicity by considering five types of stereoisograms. In a previous account article [10], the author (Fujita) has pointed out two aspects of symmetry, i.e., chirality and RS -stereogenicity, as two kinds of handedness. After he has proposed sclerality as an additional aspect, he has accomplished the Aufheben of the three aspects so as to propose the concept of RS -stereoisomerism.…”

“…8),( 9,10),(1,2,3,4)(5,6,7,8) ] #29 D4^I=====9 gen[30] := [ (1,8)(2,7)(3,6)(4,5), (1,3)(2,4)(5,7)(6,8), (2,5,4)(3,6,8), (1,2,3,4)(5,6,7,8) ] #30 O---11 gen[31] := [ (1,8)(2,7)(3,6)(4,5), (1,3)(2,4)(5,7)(6,8), (2,5,4)(3,6,8), ( 9,10) ] #31 T^I====10 gen[32] := [ (1,8)(2,7)(3,6)(4,5), (1,3)(2,4)(5,7)(6,8), (2,5,4)(3,6,8), ( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10) ] #32 ^O gen[33] := [ (1,2,3,4)(5,6,7,8), (2,4,5)(3,8,6), ( 9,10) ] #33 O^I====11 gap>…”

Mathematical Stereochemistry as the Theoretical Foundations of Organic and Inorganic Chemistry

“…Fujita's stereoisogram approach [1] indicates the presence of two kinds of handedness, i.e., chirality and RS-stereogenicity [27]. As a result, a stereoisogram specifies a quadruplet of RSstereoisomers, which exhibits a net interaction between chirality and stereogenicity.…”