CiNii Books - Theory of coronoid hydrocarbons

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Theory of coronoid hydrocarbons

S.J. Cyvin, J. Brunvoll, B.N. Cyvin

（Lecture notes in chemistry, 54, 62）

Springer-Verlag, c1991-c1994

[1] : Berlin
[1] : New York
2

Other Title: Coronoid hydrocarbons

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Note

Includes bibliographical references (p. [159]-170) and index

Description and Table of Contents

Volume: [1] : Berlin ISBN 9783540535775

Description

The present studies of coronoid systems is a natural continua- tion of the corresponding studies of benzenoid systems. Both topics are rooted in organic chemistry through certain polycyclic conjugated hydrocarbons, which are the chemical counterparts of the systems in question. HO'v'Tever, the scope of the present work and corresponding works on benzenoids goes far beyond a chemical motivation. These works are classified under mathematical chemistry, a relatively new designa- tion. The book is supposed to have an interest for organic chemists within certain specialities, but still more for theoretical and mathe- matical chemists. The last category has been characterized as enfants terribles in the foreword of the first issue of the Journal of Mathe- matical Chemistry (1987). Finally, this book may have a considerable interest for mathematicians within combinatorics and gra~h theory. It is supposed that the book will be most useful for researchers, including graduate students, in the pertinent fields. The text contains no advanced mathematics whatsoever and should as such not represent any barrier even for undergraduate students. Here we wish to make some comments on the terminology, which is not standardized and is partly controversial in this field of topolo- gical studies of polyhex (benzenoid and coronoid) systems.

Table of Contents

I - Introduction.- 1 - Coronoid Hydrocarbons.- 1.1 Introduction.- 1.2 Cycloarenes.- 1.2.1 The Story of Kzkulen.- 1.2.2 Other Cycloarene.- 1.3 Corannulenes.- 1.4 Other Macrocyclic Conjugated Hydrocarbons.- 1.4.1 Introductio.- 1.4.2 Annulene.- 1.4.3 Annulenoannulene.- 1.4.4 Annelated Annulene.- 2 - Coronoids and Coronoid-Like Systems.- 2.1 Coronoid System.- 2.1.1 Basic Definitions.- 2.1.2 Terminology.- 2.2 Interlude.- 2.2.1 Coronoid Systems and Coronoid Hydrocarbons.- 2.2.2 Kekule Structures.- 2.3 Non-Coronoid Systems.- 2.4 Degenerate Coronoid Systems.- 2.5 Quasi-Coronoid Systems.- 2.5.1 Coronene and Some of Its Derivatives.- 2.5.2 Helicenic Systems.- 2.6 Concluding Remarks.- II - Anatomy.- 3 - Anatomy: General Treatment.- 3.1 Introduction.- 3.2 Basic Concepts.- 3.2.1 Hexagon Modes and Some Structural Features.- 3.2.2 Vertices and Edges. Some Invariants.- 3.2.3 Extremal Values of Some Invariants.- 3.2.4 Coloring of Vertices. More Invariants.- 3.3 Classification.- 3.3.1 Introduction.- 3.3.2 Classification According to Internal Vertices.- 3.3.3 Kekulean and non-Kekulean Coronoids.- 3.3.4 The "neo" Classification.- 3.3.5 Introduction to Regular and Half Essentially Disconnected Coronoids.- 3.3.6 Regular Addition and Regular Tearing Down.- 3.3.7 The "rheo" and "rio" Classifications.- 3.3.8 Symmetry.- 3.4 Aromatic Sextets and All-Coronoids.- 3.4.1 Aromatic Sextet.- 3.4.2 Clar Structure.- 3.4.3 All-Coronoid.- 4 - Primitive Coronoids (I) and Annulenes: Anatomy.- 4.1 Introduction.- 4.2 Some Topological Properties of Primitive Coronoids.- 4.2.1 Definition.- 4.2.2 Corona Hole.- 4.2.3 Corners.- 4.2.4 Segments.- 4.2.5 Perimeters ".- 4.2.6 Symmetry.- 4.3 Notation.- 4.4 Associated Chain with a Primitive Coronoid.- 4.4.1 Definition.- 4.4.2 Single Unbranched Chains.- 4.5 Hollow Hexagon.- 4.6 Extremal Properties of the Corona Holes of Primitive Coronoids.- 4.7 Annulene.- III - Enumeration.- 5 - Enumeration: General Treatment and Coarse Classifications.- 5.1 Introduction.- 5.2 General Principles.- 5.3 Survey of Polyhexes.- 5.4 Some Coarse Classifications of Coronoids.- 5.4.1 Catacondensed and Pericondensed.- 5.4.2 The vio Classification.- 5.4.3 Classification According to Color Excess Benzenoids p. 67
Color Excess for Coronoids p. 68
Forms of Extremal Coronoids p. 70
Conjectures p. 72.- 5.4.4 Symmetry.- 5.5 Conclusion.- 6 - Primitive Coronoids (II) and Annulenes: Enumeration.- 6.1 Previous Work.- 6.2 Hollow Hexagons.- 6.2.1 Method of Generation (I).- 6.2.2 Symmetry Considerations.- 6.2.3 Enumeration Results.- 6.2.4 Method of Generation (II).- 6.2.5 Forms.- 6.3 Primitive Coronoids.- 6.3.1 Method of Generation (I).- 6.3.2 Method of Generation (II).- 6.3.3 Extended Computations.- 6.3.4 Forms.- 6.4 Primitive Coronoids with Constant Segment Lengths.- 6.4.1 Introductory Remarks.- 6.4.2 Primitive Coronoids with 2-Segments Only.- 6.4.3 Primitive Coronoids with Z-Segments Only, Especially for Z > 2.- 6.5 Annulenes.- 7 - Enumeration and Classification of Non-Primitive Coronoids.- 7.1 Non-Primitive Kekulean Coronoids.- 7.1.1 Regular Systems.- 7.1.2 Irregular Systems.- 7.2 Classification According to the Color Excess.- 7.3 Concealed Non-Kekulean Coronoids.- 8 - Enumeration and Classification of Coronoids with Definite Holes.- 8.1 Introduction.- 8.2 Supplementary Topological Properties.- 8.2.1 Outside and Inside Features.- 8.2.2 Basic Coronoid.- 8.3 Basic Coronoids.- 8.3.1 Generation and Enumeration.- 8.3.2 Forms.- 8.4 Definite Corona Holes.- 8.4.1 First Survey.- 8.4.2 Second Survey.- 8.4.3 Detailed Classification.- 9 - Enumeration: Numbers of Internal Vertices.- 9.1 Introduction.- 9.2 Coronoid Chemical Isomers.- 9.3 Computational Results.- 9.4 Corona Holes and their Invariants.- 9.4.1 Numbers of Corona Holes.- 9.4.2 Extremal Values Associated with Corona Holes.

Volume: 2 ISBN 9783540581383

Description

The present monograph is a continuation of Cyvin SJ, Brunvoll J and Cyvin (1991c), a reference to be found in Bibliography. Naturally, the previous volume is cited frequently here. For the sake of brevity, it is referred to as "Volume I". References to different chapters, sections or paragraphs are given like Vol. 1-1, 1-1.2 or 1-1.2.2, respectively. Also tables and equations in "Volume I" are cited; the very last equation therein, for instance, is Vol. I-{9.9). The present text spans from references to organic syntheses or attempted organic syntheses - - to stringent mathematical theorems proved by graph-theoretical methods. Enumerations of coronoid systems is a substantial part of the work. Algebraic methods involving combinatorics and generating functions are employed on one hand, and computer programming on the other. The whole book is supposed to demonstrate a piece of mathematical chemistry, which can be characterized as lying on the "interfaces between mathematics, chemistry and computer science", a formulation used for the MATH/CHEM/COMP Conferences; d. Cyvin SJ, Brunvoll and Cyvin (1989d) in Bibliography. Financial support to BNC from the Norwegian Council for Science and the Humanities is gratefully acknowledged.

Table of Contents

1 - Introduction and Chemical Relevance.- 1.1 Reiteration.- 1.2 Motivation.- 1.2.1 General Viewpoints.- 1.2.2 Alkane Isomers.- 1.3 Cycloarenes.- 1.3.1 The Story of Kekulene Revisited.- 1.3.2 Other Cycloarenes.- 1.3.3 The Story of Kekulene Continues.- 1.4 Annulenes, Annulenoannulenes, and Annulene Derivatives.- 1.5 Antikekulene.- 1.6 Corannulene.- 1.7 Cyclacenes.- 1.8 Buckminsterfullerene.- 1.9 Nomenclature and Coding.- 1.10 Conclusion.- 2 - Classification of Polygonal Systems, and Some Aspects of KekulE Structures.- 2.1 Introduction.- 2.2 Classification of Single Coronoids in Relation to Kekule Structures.- 2.3 Degenerate and Generalized Single Coronoids.- 2.4 Examples of Single Coronoids, and Their Kekule Structure Counts.- 2.4.1 Regular Single Coronoids.- 2.4.2 Essentially Disconnected Single Coronoids.- 2.4.3 Additional Instructive Examples.- 2.4.4 Irregular Single Coronoids With Isolated Internal Vertices.- 2.5 Survey of Kekule Structure Counts for Single Coronoids.- 2.5.1 Combinatorial Formulas.- 2.5.2 Algorithm and Annulenoid Kekule Structures for Primitive Coronoids.- 2.5.3 General Solution Convenient for Computer Programming.- 2.5.4 Supplementary References.- 2.6 Isospectral Single Coronoids.- 2.7 Some Main Classes of Polyhexes.- 2.7.1 Introduction.- 2.7.2 Hexagonal and Trigonal Lattices, and the Dualist.- 2.7.3 Helicenes and Corohelicenes.- 2.7.4 Planarity and Nonplanarity.- 2.8 Examples of Graph-Theoretically Nonplanar Polyhexes.- 2.8.1 Cyclohelicenes.- 2.8.2 Moebius-Polyhexes.- 2.9 Polygonal Systems.- 2.9.1 Introduction.- 2.9.2 Mono-q-Polyhexes.- 2.9.3 Holes and Polygons.- 2.9.4 Cluster Systems.- 3 - Benzenoids, Single Coronoids and Multiple Coronoids.- 3.1 General Considerations, Basic Definitions, and Terminology.- 3.2 Invariants and Relations Between Them.- 3.2.1 Specifications.- 3.2.2 Relations.- 3.2.3 Outer and Inner Perimeters.- 3.3 Additional Definitions, Terminology and Relations.- 3.3.1 Corona Holes.- 3.3.2 Associated Benzenoid and Perforated Benzenoid.- 3.3.3 Naphthalenic Coronoid.- 3.3.4 Extremal Coronoid.- 3.4 First Enumeration Results for Benzenoids and Coronoids.- 3.5 Smallest Multiple Coronoids.- 3.5.1 Introduction.- 3.5.2 Basic Assumptions.- 3.5.3 Algorithm for Construction of Smallest Multiple Coronoids.- 3.5.4 Discussion and Depiction of Forms.- 3.5.5 Pericondensed Smallest Multiple Coronoids.- 3.6 Perfect and Imperfect Extremal Coronoids.- 3.6.1 Introduction.- 3.6.2 Numbers of Hexagons and of Internal Vertices.- 3.6.3 Catacondensed Extremal Coronoids.- 3.6.4 Extension to Pericondensed Extremal Coronoids.- 3.7 Chemical Formulas.- 3.7.1 Introduction and Notation.- 3.7.2 Inequalities for the Formula Coefficients.- 3.7.3 Table of Formulas.- 3.8 Numbers of Isomers.- 3.8.1 Definition and Notation.- 3.8.2 Numerical Values.- 4 - Invariants of Single Coronoids.- 4.1 Introduction.- 4.2 Summary of Invariants and Relations Between Them.- 4.2.1 Summary of Relations.- 4.2.2 Connectivity and the Dias Parameter.- 4.3 Maximum Number of Internal Vertices, and Minimum Number of Hexagons.- 4.3.1 Maximum Number of Internal Vertices, and Extremal Single Coronoids.- 4.3.2 Minimum Number of Hexagons.- 4.3.3 Spiral Walk.- 4.3.4 Perforated Polycircumcoronenes.- 4.4 Possible Values of Invariants.- 4.5 Upper and Lower Bounds for Some Invariants.- 4.5.1 General.- 4.5.2 Functions of the Number of Hexagons and of the Number of Internal Vertices.- 4.5.3 Functions of Invariants Other Than the Number of Hexagons and the Number of Internal Vertices.- 4.6 Minimum Number of Vertices of Degree Two, and Maximum Number of Hexagons.- 4.6.1 Minimum Number of Vertices of Degree Two.- 4.6.2 Maximum Number of Hexagons, and Circular Single Coronoids.- 4.6.3 Spiral Walk.- 4.6.4 Detailed Analysis.- 5 - Chemical Formulas of Single Coronoids.- 5.1 Introduction.- 5.2 Terminology.- 5.3 Inequalities in Terms of the Formula Coefficients.- 5.3 1 When is a Given Formula Compatible With a Single Coronoid?.- 5.3.2 Supplementary Inequalities.- 5.4 Circumscribing and Excising.- 5.4.1 Introduction.- 5.4.2 Definitions of Circumscribing and Excising for Coronoids.- 5.4.3 Possibilities of Circumscribing and Excising.- 5.4.4 Generalization for k-Fold Circumscribing and k--Fold Excising.- 5.4.5 Core Coronoids.- 5.4.6 Algebraic Treatment.- 5.4.7 Catacondensed Single Coronoids.- 5.4.8 Naphthalenic Single Coronoids.- 5.5 Extremal Single Coronoids and Some of Their Subclasses.- 5.5.1 Extremal Single Coronoids.- 5.5.2 Circumextremal Single Coronoids.- 5.5.3 Circular Single Coronoids.- 5.6 Extreme Single Coronoids.- 5.6.1 Definition.- 5.6.2 Formula.- 5.6.3 Nonextremal Extreme Single Coronoids.- 5.7 Table of Formulas.- 5.8 Single Coronoid Isomers and Number of Edges.- 6 - Further Studies of The Chemical Formulas of Single Coronoids.- 6.1 Introduction.- 6.2 Ground Forms and Higher Members.- 6.2.1 Introduction.- 6.2.2 Definitions and Preliminary Treatment.- 6.2.3 Perfect and Imperfect Ground Forms.- 6.2.4 Formulas for Perfect Ground Forms and for Higher Members.- 6.2.5 Formulas for Imperfect Ground Forms.- 6.2.6 Depictions.- 6.2.7 Mapping of Formulas.- 6.3 Formula Index.- 6.4 A Property of Circumscribing.- 6.5 Building-Up.- 6.5.1 Introduction and Basic Concepts.- 6.5.2 Fundamental Building-Up Principle.- 6.5.3 Addition Units.- 6.5.4 Application of the Fundamental Building-Up Principle to Isomers.- 6.5.5 Simplified Building-Up Principles.- 6.5.6 Formations Available for Addition.- 6.5.7 Further Deductions.- Supplement.- Introductory Remarks.- Summary of Formulas and Definitions.- Conclusion.- 7 - Generation and Enumeration of Single Coronoid Isomers.- 7.1 Tabulation of Complete Data.- 7.1.1 Introduction and Survey.- 7.1.2 Classification According to neo and the Color Excess.- 7.1.3 Classification According to the Corona Holes.- 7.2 Forms.- 7.3 Methods of Generation.- 7.3.1 Introduction.- 7.3.2 Application of the Fundamental Building-Up Principle.- 7.3.3 Perforating Benzenoids.- 7.3.4 Examples and Discussion.- 7.4 Circular Single Coronoids.- 7.4.1 Introduction.- 7.4.2 Methods.- 7.4.3 Sample Analysis.- 7.4.4 Extension to All Circular Single Coronoids.- 7.5 Circular Benzenoids Perforated by Phenalene Hole.- 7.5.1 Introduction.- 7.5.2 The Systems.- 7.5.3 Results of Enumeration.- 7.6 Circular Benzenoids Perforated by Coronene Hole.- 7.6.1 Introduction.- 7.6.2 The Systems.- 7.6.3 Results of Enumeration.- 7.7 Supplementary Enumeration Data.- 8 - Theorems for Single Coronoids.- 8.1 Introduction.- 8.2 Some Basic Concepts.- 8.2.1 Additions.- 8.2.2 Alternating Cycles.- 8.2.3 Edge Cuts.- 8.3 A Property of Kekule Structures.- 8.4 Essentially Disconnected Single Coronoids.- 8.4.1 Properties and Lemmas.- 8.4.2 Main Theorem.- 8.4.3 "Essentially Disconnected Coronoids" are Essentially Disconnected.- 8.5 Normal Single Coronoids.- 8.6 Regular Single Coronoids.- 8.6.1 Definition.- 8.6.2 A Criterion for a Single Coronoid to be Regular.- 8.7 Half Essentially Disconnected Single Coronoids.- 8.7.1 Introductory Remark.- 8.7.2 Main Theorem.- 8.7.3 New Definition.- 8.7.4 Example.- 8.8 Kekulean and Non-Kekulean Single Coronoids.- 8.8.1 Introduction.- 8.8.2 Old Theorem.- 8.8.3 Additional Basic Concepts and Properties.- 8.8.4 Main Theorem.

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