Mathematical Stereochemistry

Description

Mathematical Stereochemistry uses both chemistry and mathematics to present a challenge towards the current theoretical foundations of modern stereochemistry, that up to now suffered from the lack of mathematical formulations and minimal compability with chemoinformatics. The author develops novel interdisciplinary approaches to group theory (Fujita’s unit-subduced-cycle-index, USCI) and his proligand method before focussing on stereoisograms as a main theme. The concept of RS-stereoisomers functions as a rational theoretical foundation for remedying conceptual faults and misleading terminology caused by conventional application of the theories of van’t Hoff and Le Bel. This book indicates that classic descriptions on organic and stereochemistry in textbooks should be thoroughly revised in conceptionally deeper levels. The proposed intermediate concept causes a paradigm shift leading to the reconstruction of modern stereochemistry on the basis of mathematical formulations. • Provides a new theoretical framework for the reorganization of mathematical stereochemistry. • Covers point-groups and permutation symmetry and exemplifies the concepts using organic molecules and inorganic complexes. • Theoretical foundations of modern stereochemistry for chemistry students and researchers, as well as mathematicians interested in chemical application of mathematics. • Provides a new theoretical framework for the reorganization of mathematical stereochemistry. • Covers point-groups and permutation symmetry and exemplifies the concepts using organic molecules and inorganic complexes. Cover Half Title Also of interest Mathematical Stereochemistry Copyright Preface Contents 1. Introduction 1.1 Two-Dimensional versus Three-Dimensional Structures 1.1.1 Two-Dimensional Structures in Early History of Organic Chemistry 1.1.2 Three-Dimensional Structures After Beginning of Stereochemistry 1.1.3 Arbitrary Switching Between 2D-Based and 3D-Based Concepts 1.2 Problematic Methodology for Categorizing Isomers and Stereoisomers 1.2.1 Same or Different 1.2.2 Dual Definition of Isomers 1.2.3 Positional Isomers as a Kind of Constitutional Isomers 1.3 Problematic Methodology for Categorizing Enantiomers and Diastereomers 1.3.1 Enantiomers 1.3.2 Diastereomers 1.3.3 Chirality and Stereogenicity 1.4 Total Misleading Features of the Traditional Terminology on Isomers 1.4.1 Total Misleading Flowcharts 1.4.2 Another Flowchart With Partial Solutions 1.4.3 More Promising Way 1.5 Isomer Numbers 1.5.1 Combinatorial Enumeration as 2D Structures 1.5.2 Importance of the Proligand-Promolecule Model 1.5.3 Combinatorial Enumeration as 3D Structures 1.6 Stereoisograms 1.6.1 Stereoisograms as Diagrammatic Expressions of RS-Stereoisomeric Groups 1.6.2 Theoretical Foundations and Group Hierarchy 1.6.3 Avoidance of Misleading Standpoints of R/S-Stereodescriptors 1.6.4 Avoidance of Misleading Standpoints of pro-R/pro-S-Descriptors 1.6.5 Global Symmetries and Local Symmetries 1.6.6 Enumeration under RS-Stereoisomeric Groups 1.7 Aims of Mathematical Stereochemistry References 2. Classification of Isomers 2.1 Equivalence Relationships of Various Levels of Isomerism 2.1.1 Equivalence Relationships and Equivalence Classes 2.1.2 Enantiomers, Stereoisomers, and Isomers 2.1.3 Inequivalence Relationships 2.1.4 Isoskeletomers as a Missing Link for Consistent Terminology 2.1.5 Constitutionally-Anisomeric Relationships vs. Constitutionally-Isomeric Relationships 2.2 Revised Flowchart for Categorizing Isomers 2.2.1 Design of a Revised Flowchart for Categorizing Isomers 2.2.2 Illustrative Examples 2.2.3 Restriction of the Domain of Isomerism 2.2.4 Harmonization of 3D-Based Concepts with 2D-Based Concepts References 3. Point-Group Symmetry 3.1 Stereoskeletons and the Proligand-Promolecule Model 3.1.1 Configuration and Conformation 3.1.2 The Proligand-Promolecule Model 3.2 Point Groups 3.2.1 Symmetry Axes and Symmetry Operations 3.2.2 Construction of Point Groups 3.2.3 Subgroups of a Point Group 3.2.4 Maximum Chiral Subgroup of a Point Group 3.2.5 Global and Local Point-Group Symmetries 3.3 Point-Group Symmetries of Stereoskeletons 3.3.1 Stereoskeletons of Ligancy 4 3.3.2 Stereoskeletons of Ligancy 6 3.3.3 Stereoskeletons of Ligancy 8 3.3.4 Stereoskeletons Having Two or More Orbits 3.4 Point-Group Symmetries of (Pro)molecules 3.4.1 Derivation of Molecules from a Stereoskeleton via Promolecules 3.4.2 Orbits in Molecules and Promolecules Derived from Stereoskeletons 3.4.3 The SCR Notation 3.4.4 Site Symmetries vs. Coset Representations for Symmetry Notations References 4. Sphericities of Orbits and Prochirality 4.1 Sphericities of Orbits 4.1.1 Orbits of Equivalent Proligands 4.1.2 Three Kinds of Sphericities 4.1.3 Chirality Fittingness for Three Modes of Accommodation 4.2 Prochirality 4.2.1 Confusion on the Term ‘Prochirality’ 4.2.2 Prochirality as a Geometric Concept 4.2.3 Enantiospheric Orbits vs. Enantiotopic Relationships 4.2.4 Chirogenic Sites in an Enantiospheric Orbit 4.2.5 Prochirality Concerning Chiral Proligands in Isolation 4.2.6 Global Prochirality and Local Prochirality References 5. Foundations of Enumeration Under Point Groups 5.1 Orbits Governed by Coset Representations 5.1.1 Coset Representations 5.1.2 Mark Tables 5.1.3 Multiplicities of Orbits 5.2 Subduction of Coset Representations 5.2.1 Subduced Representations 5.2.2 Unit Subduced Cylce Indices (USCIs) References 6. Symmetry-Itemized Enumeration Under Point Groups 6.1 Fujita’s USCI Approach 6.1.1 Historical Comments 6.1.2 USCI-CFs for Itemized Enumeration 6.1.3 Subduced Cycle Indices for Itemized Enumeration 6.2 The FPM Method of Fujita’s USCI Approach 6.2.1 Fixed-Point Vectors (FPVs) and Multiplicity Vectors (MVs) 6.2.2 Fixed-Point Matrices (FPMs) and Isomer-Counting Matrices (ICMs) 6.2.3 Practices of the FPM Method 6.3 The PCI Method of Fujita’s USCI Approach 6.3.1 Partial Cycle Indices With Chirality Fittingness (PCI-CFs) 6.3.2 Partial Cycle Indices Without Chirality Fittingness (PCIs) 6.3.3 Practices of the PCI Method 6.4 Other Methods of Fujita’s USCI Approach 6.4.1 The Elementary-Superposition Method 6.4.2 The Partial-Superposition Method 6.5 Applications of Fujita’s USCI Approach 6.5.1 Enumeration of Flexible Molecules 6.5.2 Enumeration of Molecules Interesting Stereochemically 6.5.3 Enumeration of Inorganic Complexes 6.5.4 Enumeration of Organic Reactions References 7. Gross Enumeration Under Point Groups 7.1 Counting Orbits 7.2 Pólya’s Theorem of Counting 7.3 Fujita’s Proligand Method of Counting 7.3.1 Historical Comments 7.3.2 Sphericities of Cycles 7.3.3 Products of Sphericity Indices 7.3.4 Practices of Fujita’s Proligand Method 7.3.5 Enumeration of Achiral and Chiral Promolecules References 8. Enumeration of Alkanes as 3D Structures 8.1 Surveys With Historical Comments 8.2 Enumeration of Alkyl Ligands as 3D Planted Trees 8.2.1 Enumeration of Methyl Proligands as Planted Promolecules 8.2.2 Recursive Enumeration of Alkyl ligands as Planted Promolecules 8.2.3 Functional Equations for Recursive Enumeration of Alkyl ligands 8.2.4 Achiral Alkyl Ligands and Pairs of Enantiomeric Alkyl Ligands 8.3 Enumeration of Alkyl Ligands as Planted Trees 8.3.1 Alkyl Ligands or Monosubstituted Alkanes as Graphs 8.3.2 3D Structures vs. Graphs for Characterizing Alkyl Ligands or Monosubstituted Alkanes 8.4 Enumeration of Alkanes (3D-Trees) as 3D-Structural Isomers 8.4.1 Alkanes as Centroidal and Bicentroidal 3D-Trees 8.4.2 Enumeration of Centroidal Alkanes (3D-Trees) as 3D-Structural Isomers 8.4.3 Enumeration of Bicentroidal Alkanes (3D-Trees) as 3D-Structural Isomers 8.4.4 Total Enumeration of Alkanes as 3D-Trees 8.5 Enumeration of Alkanes (3D-Trees) as Steric Isomers 8.5.1 Centroidal Alkanes (3D-Trees) as Steric Isomers 8.5.2 Bicentroidal Alkanes (3D-Trees) as Steric Isomers 8.5.3 Total Enumeration of Alkanes (3D-Trees) as Steric Isomers 8.6 Enumeration of Alkanes (Trees) as Graphs or Constitutional Isomers 8.6.1 Alkanes as Centroidal and Bicentroidal Trees 8.6.2 Enumeration of Centroidal Alkanes (Trees) as Constitutional Isomers 8.6.3 Enumeration of Bicentroidal Alkanes (Trees) as Constitutional Isomers 8.6.4 Total Enumeration of Alkanes (Trees) as Graphs or Constitutional Isomers References 9. Permutation-Group Symmetry 9.1 Historical Comments 9.2 Permutation Groups 9.2.1 Permutation Groups as Subgroups of Symmetric Groups 9.2.2 Permutations vs. Reflections 9.3 RS-Permutation Groups 9.3.1 RS-Permutations and RS-Diastereomeric Relationships 9.3.2 RS-Permutation Groups vs. Point Groups 9.3.3 Formulation of RS-Permutation Groups 9.3.4 Action of RS-Permutation Groups 9.3.5 Misleading Features of the Conventional Terminology 9.4 RS-Permutation Groups for Skeletons of Ligancy 4 9.4.1 RS-Permutation Group for a Tetrahedral Skeleton 9.4.2 RS-Permutation Group for an Allene Skeleton 9.4.3 RS-Permutation Group for an Ethylene Skeleton References 10. Stereoisograms and RS-Stereoisomers 10.1 Stereoisograms as Integrated Diagrammatic Expressions 10.1.1 Elementary Stereoisograms of Skeletons with Position Numbering 10.1.2 Stereoisograms Based on Elementary Stereoisograms 10.2 Enumeration Under RS-Stereoisomeric Groups 10.2.1 Subgroups of the RS-Stereoisomeric Group C3vsI 10.2.2 Coset Representations 10.2.3 Mark Table and its Inverse 10.2.4 Subduction for RS-Stereoisomeric Groups 10.2.5 USCI-CFs for RS-Stereoisomeric Groups 10.2.6 SCI-CFs for RS-Stereoisomeric Groups 10.2.7 The PCI Method for RS-Stereoisomeric Groups 10.2.8 Type-Itemized Enumeration by the PCI Method 10.2.9 Gross Enumeration Under RS-Stereoisomeric Groups 10.3 Comparison with Enumeration Under Subgroups 10.3.1 Comparison with Enumeration Under Point Groups 10.3.2 Comparison with Enumeration Under RS-Permutation Groups 10.3.3 Comparison with Enumeration Under Maximum-Chiral Point Subgroups 10.4 RS-Stereoisomers as Intermediate Concepts References 11. Stereoisograms for Tetrahedral Derivatives 11.1 RS-Stereoisomeric Group TdsI and Elementary Stereoisogram 11.2 Stereoisograms of Five Types for Tetrahedral Derivatives 11.2.1 Type-I Stereoisograms of Tetrahedral Derivatives 11.2.2 Type-II Stereoisograms of Tetrahedral Derivatives 11.2.3 Type-III Stereoisograms of Tetrahedral Derivatives 11.2.4 Type-IV Stereoisograms of Tetrahedral Derivatives 11.2.5 Type-V Stereoisograms of Tetrahedral Derivatives 11.3 Enumeration Under the RS-Stereoisomeric Group TdsI 11.3.1 Non-Redundant Set of Subgroups and Five Types of Subgroups 11.3.2 Subduction of Coset Representations 11.3.3 The PCI Method for the RS-Stereoisomeric Group TdsI 11.3.4 Type-Itemized Enumeration by the PCI Method 11.4 Comparison with Enumeration Under Subsymmetries 11.4.1 Enumeration of Tetrahedral Promolecules Under the Point-Group Symmetry 11.4.2 Enumeration of Tetrahedral Promolecules Under the RS-Permutation-Group Symmetry 11.4.3 Comparison with Enumeration Under Maximum-Chiral Point Subgroups 11.4.4 Confusion Between the Point-Group Symmetry and the RS-Permutation-Group Symmetry References 12. Stereoisograms for Allene Derivatives 12.1 RS-Stereoisomeric Group D2dsI and Elementary Stereoisogram 12.2 Stereoisograms of Five Types for Allene Derivatives 12.2.1 Type-I Stereoisograms of Allene Derivatives 12.2.2 Type-II Stereoisograms of Allene Derivatives 12.2.3 Type-III Stereoisograms of Allene Derivatives 12.2.4 Type-IV Stereoisograms of Allene Derivatives 12.2.5 Type-V Stereoisograms of Allene Derivatives 12.3 Enumeration Under the RS-Stereoisomeric Group D2dsI 12.3.1 Non-Redundant Set of Subgroups and Five Types of Subgroups 12.3.2 Subduction of Coset Representations 12.3.3 The PCI Method for the RS-Stereoisomeric Group D2d s I 12.3.4 Type-Itemized Enumeration by the PCI Method 12.4 Comparison with Enumeration Under Subsymmetries 12.4.1 Enumeration of Allene Promolecules Under the Point-Group Symmetry 12.4.2 Enumeration of Allene Promolecules Under the RS-Permutation-Group Symmetry References 13. Stereochemical Nomenclature 13.1 Absolute Configuration 13.1.1 Single Pair of Attributes ‘Chirality/Achirality’ in Modern Stereochemistry 13.1.2 Three Pairs of Attributes in Fujita’s Stereoisogram Approach 13.1.3 Three Aspects of Absolute Configuration 13.2 Quadruplets of RS-Stereoisomers as Equivalence Classes 13.2.1 Three Types of Pairwise Relationships in a Quadruplet of RS-Stereoisomers 13.2.2 Formulation of Stereoisograms as Quadruplets of RS-Stereoisomers 13.3 Inner Structures of Promolecules 13.3.1 Inner Structures of RS-Stereogenic Promolecules 13.3.2 Inner Structures of RS-Astereogenic Promolecules 13.4 Assignment of Stereochemical Nomenclature 13.4.1 Single Criterion for Giving RS-Stereodescriptors 13.4.2 RS-Diastereomers: the CIP Priority System 13.4.3 R/S-Stereodescriptors and Stereoisograms 13.4.4 Chirality Faithfulness 13.4.5 Stereochemical Notations for Other Skeletons References 14. Pro-RS-Stereogenicity Based on Orbits 14.1 Prochirality vs. Pro-RS-Stereogenicity 14.1.1 Prochirality as a Geometric Concept 14.1.2 Pro-RS-Stereogenicity as a Stereoisomeric Concept 14.1.3 Prochirality and Pro-RS-Stereogenicity for Tetrahedral Derivatives 14.2 Orbits under RS-Permutation Groups 14.2.1 RS-Tropicity 14.2.2 Pro-RS-Stereogenicity as a Stereoisomeric Concept 14.3 pro-R/pro-S-Descriptors 14.3.1 RS-Diastereotopic Relationships 14.3.2 Single Criterion for Giving pro-R/pro-S-descriptors 14.3.3 Probe Stereoisograms for Assining pro-R/pro-S-Descriptors 14.3.4 Misleading Interpretation of ‘Prochirality’ in Modern Stereochemistry 14.4 Pro-RS-Stereogenicity Distinct From Prochirality 14.4.1 Simultaneity of Prochirality and Pro-RS-Stereogenicity in a Type-IV Promolecule 14.4.2 Coincidence of Prochirality and Pro-RS-stereogenicity 14.4.3 Prochiral (but Already RS-Stereogenic) Promolecules 14.5 Pro-RS-Stereogenicity for pro-R/pro-S-Descriptors References 15. Perspectives 15.1 Enumeration of Highly Symmetric Molecules 15.2 Interaction of Orbits of Different Kinds 15.3 Correlation Diagrams of Stereoisograms 15.4 Group Hierarchy 15.5 Non-Rigid Molecules and Conformations 15.6 Interdisciplinary Nature of Mathematical Stereochemistry 15.6.1 Mathematical and Stereochemical Barriers In Practical Levels 15.6.2 Mathematical and Stereochemical Barriers In Conceptual Levels 15.7 Reorganizing the Theoretical Foundations of Stereochemistry and Stereoisomerism References Index