Mathematical Stereochemistry 2nd Edition by Shinsaku Fujita 3110728338 9783110728330

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

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

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 Based on Stereoskeletons

3.2 Point Groups for Characterizing Geometric Symmetry of Molecular Entity

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 Various 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

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

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)

6 Symmetry-Itemized Enumeration Under Point Groups

6.1 Fujita’s USCI Approach

6.1.1 USCI-CFs for Itemized Enumeration

6.1.2 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

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 Sphericities of Cycles

7.3.2 Products of Sphericity Indices

7.3.3 Practices of Fujita’s Proligand Method

7.3.4 Enumeration of Achiral and Chiral Promolecules

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

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

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 C3vσ̃Î

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

11 Stereoisograms for Tetrahedral Derivatives

11.1 RS-Stereoisomeric Group Tdσ˜I^ 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 Tdσ˜l^

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 Tdσ̃Î

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

12 Stereoisograms for Allene Derivatives

12.1 RS-Stereoisomeric Group D2dσ^I^ 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 D2dσ^

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 D2dσ̃Î

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

13 Combined-Permutation Representations (CPRs)

13.1 Chemical Compounds as Graphs vs. as 3D Structures

13.2 Combined-Permutation Representations (CPRs) as Computer-Oriented Representations of Point Groups

13.2.1 Coset Representations

13.2.2 Mirror-Permutation Representations

13.2.3 Combined Representations and The Corresponding Groups

13.2.4 Partition into Conjugacy Classes

13.3 Correspondence Between Oh-Skeletons

13.3.1 Combined-Permutation Groups for Oh-Skeletons

13.3.2 Isomorphism between Combined-Permutation Groups for Oh-Skeletons

13.4 Calculation of CI-CFs

13.4.1 Products of Sphericity Indices

13.4.2 Definition of CI-CFs

13.4.3 Functions for Calculating CI-CFs

13.4.4 Practices of Calculation of CI-CFs

13.5 Combinatorial Enumeration

13.5.1 Generating Functions Derived From CI-CFs

13.5.2 Selective Calculation of Coefficients of Generating Functions

13.5.3 Merits of the Present Procedure in Enumeration by Fujita’s Proligand Method

13.6 Conclusive Remarks on Applicability of Combined-Permutation Representations

13.7 Appendix A. Source Code of CICFgenCC.gapfunc for Calculating CI-CFs

13.8 Appendix B. Source Code of Oh-Enum-cube.gap for Enumerating Cubane Derivatives

14 Stereochemical Nomenclature

14.1 Absolute Configuration

14.1.1 Single Pair of Attributes ‘Chirality/Achirality’ in Modern Stereochemistry

14.1.2 Three Pairs of Attributes in Fujita’s Stereoisogram Approach

14.1.3 Three Aspects of Absolute Configuration

14.2 Quadruplets of RS-Stereoisomers as Equivalence Classes

14.2.1 Three Types of Pairwise Relationships in a Quadruplet of RS-Stereoisomers

14.2.2 Formulation of Stereoisograms as Quadruplets of RS-Stereoisomers

14.3 Inner Structures of Promolecules

14.3.1 Inner Structures of RS-Stereogenic Promolecules

14.3.2 Inner Structures of RS-Astereogenic Promolecules

14.4 Assignment of Stereochemical Nomenclature

14.4.1 Single Criterion for Giving RS-Stereodescriptors

14.4.2 RS-Diastereomers: the CIP Priority System

14.4.3 R/S-Stereodescriptors and Stereoisograms

14.4.4 Chirality Faithfulness

14.4.5 Stereochemical Notations for Other Skeletons

15 Pro-RS-Stereogenicity Based on Orbits

15.1 Prochirality vs. Pro-RS-Stereogenicity

15.1.1 Prochirality as a Geometric Concept

15.1.2 Pro-RS-Stereogenicity as a Stereoisomeric Concept

15.1.3 Prochirality and Pro-RS-Stereogenicity for Tetrahedral Derivatives

15.2 Orbits under RS-Permutation Groups

15.2.1 RS-Tropicity

15.2.2 Pro-RS-Stereogenicity as a Stereoisomeric Concept

15.3 pro-R/pro-S-Descriptors

15.3.1 RS-Diastereotopic Relationships

15.3.2 Single Criterion for Giving pro-R/pro-S-descriptors

15.3.3 Probe Stereoisograms for Assining pro-R/pro-S-Descriptors

15.3.4 Misleading Interpretation of ‘Prochirality’ in Modern Stereochemistry

15.4 Pro-RS-Stereogenicity Distinct From Prochirality

15.4.1 Simultaneity of Prochirality and Pro-RS-Stereogenicity in a Type-IV Promolecule

15.4.2 Coincidence of Prochirality and Pro-RS-stereogenicity

15.4.3 Prochiral (but Already RS-Stereogenic) Promolecules

15.5 Pro-RS-Stereogenicity for pro-R/pro-S-Descriptors

Supplementary Remark: Mathematical Stereochemistry and STEM

16 Perspectives

16.1 Enumeration of Highly Symmetric Molecules

16.2 Interaction of Orbits of Different Kinds

16.3 Correlation Diagrams of Stereoisograms

16.4 Group Hierarchy

16.5 Combined-Permutation Representations (CPRs) and the GAP System

16.6 Non-Rigid Molecules and Conformations

16.7 Interdisciplinary Nature of Mathematical Stereochemistry

16.7.1 Mathematical and Stereochemical Barriers In Practical Levels

16.7.2 Mathematical and Stereochemical Barriers In Conceptual Levels

16.8 Stereochemistry and Stereoisomerism. Theoretical Foundations

16.9 Hints for Further Investigations

Supplementary Remark: for Chemical Equations and for Methematical Equations

Index

Symbols