Mathematical concepts in organic chemistry


Mathematical concepts in organic chemistry

Ivan Gutman, Oskar E. Polansky

Springer-Verlag, c1986

  • : us
  • : gw

大学図書館所蔵 件 / 14



Bibliography: p. [203]-207

Includes index



The present book is an attempt to outline some, certainly not all, mathematical aspects of modern organic chemistry. We have focused our attention on topological, graph-theoretical and group-theoretical features of organic chemistry, Parts A, B and C. The book is directed to all those chemists who use, or who intend to use mathe- matics in their work, and especially to graduate students. The level of our exposition is adjusted to the mathematical background of graduate students of chemistry and only some knowledge of elementary algebra and calculus is required from the readers of the book. Some less well-known. but still elementary mathematical facts are collected in Appendices 1-4. This, however, does not mean that the mathematical rigor and numerous tedious, but necessary technical details have been avoided. The authors' intention was to show the reader not only how the results of mathematical chemistry look, but also how they can be obtained. In accordance with this, Part 0 of the book contains a few selected advanced topics which should give the reader the flavour of the contemporary research in mathe- matical organic chemistry. One of the authors (I.G.) was an Alexander von Humboldt fellow in 1985 when the main part of the book was written. He gratefully acknowledges the financial support of the Alexander von Humboldt Foundation which enabled his stay at the Max-Planck-Institut fUr Strahlenchemie in M iilheim and the writing of this book.


A Chemistry and Topology.- 1 Topological Aspects in Chemistry.- 1.1 Topology in Chemistry.- 1.2 Abstraction in Science and How Far One Can Go.- 2 Molecular Topology.- 2.1 What is Molecular Topology?.- 2.2 Geometry, Symmetry, Topology.- 2.3 Definition of Molecular Topology.- B Chemistry and Graph Theory.- 3 Chemical Graphs.- 4 Fundamentals of Graph Theory.- 4.1 The Definition of a Graph.- 4.1.1 Relations.- 4.1.2 The First Definition of a Graph.- 4.1.3 The Second Definition of a Graph.- 4.1.4 Vertices and Edges.- 4.1.5 Isomorphic Graphs and Graph Automorphisms.- 4.1.6 Special Graphs.- 4.2 Subgraphs.- 4.2.1 Sachs Graphs.- 4.2.2 Matchings.- 4.3 Graph Spectral Theory.- 4.3.1 The Adjacency Matrix.- 4.3.2 The Spectrum of a Graph.- 4.3.3 The Sachs Theorem.- 4.3.4 The ?-Polynomial.- 4.4 Graph Operations.- 5 Graph Theory and Molecular Orbitals.- 6 Special Molecular Graphs.- 6.1 Acyclic Molecules.- 6.1.1 Trees.- 6.1.2 The Path and the Star.- 6.1.3 The Characteristic Polynomial of Trees.- 6.1.4 Trees with Greatest Number of Matchings.- 6.1.5 The Spectrum of the Path.- 6.2 The Cycle.- 6.3 Alternant Molecules.- 6.3.1 Bipartite Graphs.- 6.3.2 The Pairing Theorem.- 6.3.3 Some Consequences of the Pairing Theorem.- 6.4 Benzenoid Molecules.- 6.4.1 Benzenoid Graphs.- 6.4.2 The Characteristic Polynomial of Benzenoid Graphs.- 6.5 Hydrocarbons and Molecules with Heteroatoms.- 6.5.1 On the Question of the Molecular Graph.- 6.5.2 The Characteristic Polynomial of Weighted Graphs.- 6.5.3 Some Regularities in the Electronic Structure of Heteroconjugated Molecules.- C Chemistry and Group Theory.- 7 Fundamentals of Group Theory.- 7.1 The Symmetry Group of an Equilateral Triangle.- 7.2 Order, Classes and Representations of a Group.- 7.3 Reducible and Irreducible Representations.- 7.4 Characters and Reduction of a Reducible Representation.- 7.5 Subgroups and Sidegroups - Products of Groups.- 7.6 Abelian Groups.- 7.7 Abstract Groups and Group Isomorphism.- 8 Symmetry Groups.- 8.1 Notation of Symmetry Elements and Representations.- 8.2 Some Symmetry Groups.- 8.2.1 Rotation Groups.- 8.2.2 Groups with More than One n-Fold Axis, n > 2.- 8.2.3 Groups of Collinear Molecules.- 8.3 Transformation Properties and Direct Products of Irreducible Representations.- 8.3.1 Transformation Properties.- 8.3.2 Rules Concerning the Direct Product of Irreducible Representations.- 8.4 Some Applications of Symmetry Groups.- 8.4.1 Electric Dipole Moment.- 8.4.2 Polarizability.- 8.4.3 Motions of Atomic Nuclei: Translations, Rotations and Vibrations.- 8.4.4 Transition Probabilities for the Absorption of Light.- 8.4.5 Transition Probabilities in Raman Spectra.- 8.4.6 Group Theory and Quantum Chemistry.- 8.4.7 Orbital and State Correlations.- 9 Automorphism Groups.- 9.1 Automorphism of a Graph.- 9.2 The Automorphism Group A(G1).- 9.3 Cycle Structure of Permutations.- 9.4 Isomorphism of Graphs and of Automorphism Groups 112..- 9.5 Notation of some Permutation Groups.- 9.6 Direct Product and Wreath Product.- 9.7 The Representation of Automorphism Groups as Group Products.- 10 Some Interrelations between Symmetry and Automorphism Groups.- 10.1 The Idea of Rigid Molecules.- 10.2 Local Symmetries.- 10.3 Non-Rigid Molecules.- 10.4 What Determines the Respective Orders of the Symmetry and the Automorphism Group of a Given Molecule?.- D Special Topics.- 11 Topological Indices.- 11.1 Indices Based on the Distance Matrix.- 11.1.1 The Wiener Number and Related Quantities.- 11.1.2 Applications of the Wiener Number.- 11.2 Hosoya's Topological Index.- 11.2.1 Definition and Chemical Applications of Hosoya's Index.- 11.2.2 Mathematical Properties of Hosoya's Index.- 11.2.3 Example: Hosoya's Index of the Path and the Cycle.- 11.2.4 Some Inequalities for Hosoya's Index.- 12 Thermodynamic Stability of Conjugated Molecules.- 12.1 Total ?-Electron Energy and Thermodynamic Stability of Conjugated Molecules.- 12.2 Total ?-Electron Energy and Molecular Topology.- 12.3 The Energy of a Graph.- 12.4 The Coulson Integral Formula.- 12.5 Some Further Applications of the Coulson Integral Formula.- 12.6 Bounds for Total ?-Electron Energy.- 12.7 More on the McClelland Formula.- 12.8 Conclusion: Factors Determining the Total ?-Electron Energy.- 12.9 Use of Total ?-Electron Energy in Chemistry.- 13 Topological Effect on Molecular Orbitals.- 13.1 Topologically Related Isomers.- 13.2 Interlacing Rule.- 13.3 PE Spectra of Topomers.- 13.4 TEMO and a-Electron Systems.- 13.5 TEMO and Symmetry.- Appendices.- Appendix 1 Matrices.- Appendix 2 Determinants.- Appendix 3 Eigenvalues and Eigenvectors.- Appendix 4 Polynomials.- Appendix 5 Characters of Irreducible Representations of Symmetry Groups.- Appendix 6 The Symbols Used.- Literature.- References.

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