Graph Theoretical Approaches to Chemical Reactivity

290 Pages·1994·10.22 MB·English

by  Haruo Hosoya (auth.), Danail Bonchev, Ovanes Mekenyan (eds.)

GRAPH THEORETICAL APPROACHES TO CHEMICAL REACTIVITY
Understanding Chemical Reactivity
Volume 9
Series Editor
Paul G. Mezey, University of Saskatchewan, Saskatoon, Canada
Editorial Advisory Board
R. Stephen Berry, University of Chicago, IL, USA
John I. Brauman, Stanford University, CA, USA
A. Welford Castleman, Jr., Pennsylvania State University, PA, USA
Enrico Clementi, IBM Corporation, Kingston, NY, USA
Stephen R. Langhoff, NASA Ames Research Center, Moffet Field, CA, USA
K. Morokuma, Institute for Molecular Science, Okazaki, Japan
Peter J. Rossky, University of Texas at Austin, TX, USA
Zdenek Slanina, Czechoslovak Academy of Sciences, Prague, Czechoslovakia
Donald G. Truhlar, University of Minnesota, Minneapolis, MN, USA
Ivar Ugi, Technische Universitat, Munchen, Germany
The titles published in this series are listed at the end of this volume.
Graph Theoretical Approaches
to
Chemical Reactivity
edited by
Danail Bonchev
and
Ovanes Mekenyan
Higher Institute of Chemical Technology,
Burgas, Bulgaria
SPRINGER SCIENCE+BUSINESS MEDIA, BV.
Library of Congress Cataloging-in-Publication Data
Graph thearetical appraaches ta chemical react,v,ty edited by Danai I
Banchev and Ovanes Mekenyan.
p. cm. -- (Understand,ng chemical reactiv,ty ; v. 91
Includes bibl iagraphical references and ,ndex.
ISBN 978-94-010-4526-1 ISBN 978-94-011-1202-4 (eBook)
DOI 10.1007/978-94-011-1202-4
1. Reactivity (Chemistryl 2. Graph theary. 1. Banchev. Danai 1.
II. Mekenyan. Ovanes. III. Series.
OD505.5.G73 1994
541.3' 94--dc20 94-14280
ISBN 978-94-010-4526-1
Printed on acid-free paper
AII Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint of the hardcover 1s t edition 1994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONfENTS
1. INTRODUCTION TO GRAPH THEORY
Haruo Hosoya ..... 1
1. Chemical Graph Theory ..... 1
2. Representation and Characterization of a Graph ..... 3
3. Realization of a Graph .... 10
4. Operations on Graphs .... 23
5. References .... 32
2. THE INTERPLAY BETWEEN GRAPH THEORY AND MOLECULAR
ORBITAL THEORY
Nenad Trinajstic, Zlatko Mihalic, and Ante Graovac .... 37
1. Introduction .... 37
2. Fundamentals of Graph Theory .... 38
3. Isomorphism of Graph Spectral Theory and Huckel Molecular
Orbital Theory ... .42
4. Huckel Spectrum ... .45
5. Topological Effect on Molecular Orbitals .... 46
6. The HOMO-LUMO Separation .... 50
7. Topological Charge Stabilization .... 56
8. Localization Energy .... 63
9. Concluding Remarks .... 67
10. References .... 68
3. TOPOLOGICAL CONTROL OF MOLECULAR ORBITAL THEORY:
A COMPARISON OF I'l-SCALED HUCKEL THEORY AND RESTRICTED
HARTREE-FOCK THEORY FOR BORANES AND CARBORANES
Roger Rousseau and Stephen Lee .... 73
1. Introduction .... 73
2. Calculational Method .... 74
3. The Method of Moments .... 77
4. Elemental Boron .... 78
5. BaHt Clusters .... 86
6. The a-Parameter of BlOHlO2 89
o ....
7. Reaction Pathways .... 97
8. Conclusion ... 105
9. References ... 106
4. POLYHEDRAL DYNAMICS
Robert B. King ... 109
1. Introduction ... 109
2. The Topology of Polyhedra ... 111
3. Polyhedral Isomerizations ... 116
4. Microscopic Models: Diamond-Square-Diamond Processes and Gale
Diagrams 00.116
5. Macroscopic Models: Topological Representations ... 126
6. Literature References ... 134
5. REACTION GRAPHS
Alexandru T. Balaban ... 13 7
vi TABLE OF CONTENTS
1. Introduction 138
2. Reaction Graphs of Rearrangements Via Carbocations ... 138
3. Automerization of Bulvalene, Other Valence Isomers of Annulenes,
and Azabullvalene ... 155
4. Rotation in Molecular Propellers ... 158
5. Reaction Graphs for Rearrangements of Metallic Complexes ... 159
6. Xenon Hexafluoride ... 175
7. Heptaphosphide Trianion ... 175
8. Kinetic Graphs, Synthon Graphs, and Graph Transforms ... 176
9. Conclusions ... 177
10. References ... 177
6. DISCRETE REPRESENTATIONS OF THREE-DIMENSIONAL
MOLECULAR BODIES AND TIIEIR SHAPE CHANGES IN
CHEMICAL REACTIONS
Paul G. Mezey ... 181
1. Introduction and Review of Basic Topological Concepts of Molecular
Shape Representation ... 181
2. Molecular Shape Representation by Nuclear Potential Contours
(NUPCO's) ... 193
3. Topological Patterns of NUPCO Sequences ... 195
4. Shape Changes of NUPCO Sequences Along Reaction Paths and in
Conformational Domains ... 199
5. Shape Changes of NUPCO's in Conformational Changes and in
Molecular Deformations; NUPCO Shape Invariance Domains of the
Configuration Space ... 201
6. Local Shape Invariance of NUPCO's and the Transfer of Functional
Groups in Chemical Reactions ... 203
7. Summary ... 208
8. References ... 208
7. THE INVA RIANCE OF MOLECULAR TOPOLOGY IN CHEMICAL
REACTIONS
Eugeny V. Babaev ... 209
1. Introduction ... 209
2. From a Lewis Diagram to the Pseudo-Graph and Graphoid ... 210
3. From Graph (Graphoid) to Surface ... 212
4. What Is the Topological Homeomorphism from the Chemical Point
of View? ... 212
5. The Invariance of the Euler Characteristic in Chemical Reactions ... 215
6. The Main Theorem ... 215
7. Conclusion ... 219
8. References and Notes ... 219
8. TOPOLOGICAL INDICES AND CHEMICAL REACTIVITY
Ovanes Mekenyan and Subhash C. Basak ... 221
1. Introduction ... 221
2. Basic Principles Underlying the Topological Nature of Chemical
Reactivity ... 221
3. Molecular Topology and Topological Invariants ... 223
4. Applications of Topological Indices to Chemical Reactivity ... 232
5. Conclusions ... 236
6. References ... 237
TABLE OF CONTENTS vii
9. GRAPH-THEORETICAL MODELS OF COMPLEX REACTION
MECHANISMS AND THEIR ELEMENTARY STEPS
Oleg N. Temkin, Andrey V. Zeigamik, and Danai! Bonchev ... 241
1. Introduction ... 241
2. Graph-Theoretical Approach to Studies in the Elementary Steps
of Complex Reactions ... 242
3. Classification and Coding of Linear Reaction Mechanisms By Using
Kinetic Graphs ... 252
4. Application of Bipartite Graphs and Stoichiometric Matrices to the
Description of Linear and Nonlinear Reaction Mechanisms ... 261
5. Topological Aspects of Complex Reaction Mechanisms ... 269
6. References ... 273
INDEX ... 277
PREFACE
The progress in computer technology during the last 10-15 years has enabled the
performance of ever more precise quantum mechanical calculations related to structure
and interactions of chemical compounds. However, the qualitative models relating
electronic structure to molecular geometry have not progressed at the same pace. There
is a continuing need in chemistry for simple concepts and qualitatively clear pictures that
are also quantitatively comparable to ab initio quantum chemical calculations.
Topological methods and, more specifically, graph theory as a fixed-point topology,
provide in principle a chance to fill this gap.
With its more than 100 years of applications to chemistry, graph theory has proven to
be of vital importance as the most natural language of chemistry. The explosive
development of chemical graph theory during the last 20 years has increasingly
overlapped with quantum chemistry. Besides contributing to the solution of various
problems in theoretical chemistry, this development indicates that topology is an
underlying principle that explains the success of quantum mechanics and goes beyond it,
thus promising to bear more fruit in the future.
As a part of the series "Understanding Chemical Reactivity", this volume is designed
to introduce the reader to the graph-theoretical and, more generally, topological
elucidation of chemical reactivity. The nine chapters of the volume are written by 15
authors from seven countries who have contributed largely to the development of this
area of science. This emphasizes the importance and complexity of chemical reactivity
studies whose elucidation requires the broad cooperation of scientists from allover the
world, as well as from various branches of chemistry. The chapters are well illustrated
and provide an extensive reference to the problems discussed, in line with the scope of
not teaching but intriguing and guiding the reader.
The introductory chapter on graph theory by H. Hosoya is not just a collection of
terms, definitions, and formulae. The basic notions and concepts of graph theory are
specifically conveyed in a way that benefits from the numerous personal contributions of
the author in this area. Emphasis is put on the matrix and polynomial representations,
symmetry, isomorphism, and operations on graphs. Indeed, a single chapter could not
cover all aspects of the very rich graph-theoretical formalism, and the reader may find
additional information on the subject in the introductory sections of the other chapters.
It is traditional to connect chemical reactivity to graph theory via Hiickel molecular
orbital theory (HMO), which provided the first reactivity indices (atomic charges, bond
orders, free valences, localization energy, superdelocalizability indices, frontier orbital
indices). In Chapter 2, Trinajstic, Mihalic, and Graovac go beyond reviewing the
isomorphism between graph-spectral theory and the HMO theory, and beyond the
discussion on the structure of the Huckel eigenvalue spectrum. They present the modern
view on the interplay between graph theory and molecular orbital theory by reviewing
the achievements of recent years. Several major topics are included. The TEMO principle
(developed by the late Oskar Polansky, one of the pioneers of chemical graph theory)
allow, among other things, reactivity predictions for a special class of topological isomers
(topomers). The rule of topological charge stabilization of Gimarc is a reliable guide in
predicting relative stabilities of various heteroatomic isomers. The graph-theoretical
assessments of the HOMO-LUMO separation and absolute hardness of altemant
hydrocarbons pave the way for future achievements in this area.
In Chapter 3, Rousseau and Lee present a form of Huckel theory, termed second
moment scaling, which had been developed earlier by Burdett and Lee and has proven
ix
x PREFACE
successful in both rationalizing and optimizing the structure of molecules and solids.
Roussseau and Lee go beyond the reviewed previous work in this area and apply the
method to obtain the actual shape of the electronic energy surface as a function of
geometry for two important classes of compounds (boranes and carboranes). The
minimum energy geometries, electron density contour maps, and reaction paths thus
calculated are shown to be in reasonable accord with the ab initio method. The
topological method used may be regarded as a third-generation Huckel method,
applicable to covalent and metallic (but not ionic) compounds that are formed of main
group atoms and transition metals.
Chapter 4 by R. B. King summarizes topological and graph-theoretical aspects of
isomerization reactions of polyhedral molecules (both coordination and cluster
polyhedra). The microscopic approach is discussed, in which the details of polyhedral
topology are used to help elucidate which types of single isomerization steps are possible.
The reader may gain experience in using specific techniques and processes, such as the
Schlegel diagrams, the Gale diagrams, and the dsd-processes, as well as learn about
exciting developments in this area in which the author is one of the major contributors.
The earlier macroscopic approach, which uses the so-called topological representations
(reaction graphs) to show the relationships between different permutational isomers, is
also reviewed.
The macroscopic approach just mentioned is further detailed by Balaban in Chapter 5,
a chapter devoted to reaction graphs in both organic and inorganic chemistry. The author
was the first chemist to apply graph theory to isomerization processes (interconversions
of carbenium ions), doing so as early as 1966. The reader will find in this chapter the
complete and intriguing story of the rearrangements via carbocations with a particular
emphasis on those leading to diamond hydrocarbons and their derivatives. Reaction
graphs dealing with inorganic compounds describe different classes of rearrangements of
complexes (mainly metallic ones) with various geometries. The chapter is rich in
illustrative examples demonstrating that a chemist applying this graph-theoretical
technique may gain a closer insight into rearrangement mechanisms and be enabled to
indicate likely intermediates. .
Chapter 6 by Mezey initially offers a summary of the previously developed topological
methodology for treating three-dimensional molecular shapes and their changes. The
mathematical formalism developed is formulated in terms of contour surfaces of
electronic charge densities or, alternatively, of molecular electrostatic potential contours.
Chemical reactivity is thus regarded as the strongest change in molecular shape within
a series of similarly treated but less pronounced changes like conformational and
vibrational-rotational ones, as well as electronic excitations. The second part of the
chapter presents a newly developed method for representing molecular shapes by the
topological patterns of contour surfaces of three-dimensional nuclear potentials.
Computationally simple, the new technique is extended for the modeling of shape
changes (reaction paths) in chemical processes.
A quite different topological approach to chemical reactions is advanced in Chapter 7
by Babaev. Proceeding from the classical picture of molecules with localized bonds,
described by multigraphs with loops and then by graphoids, Babaev introduces the
concept of two-dimensional manifolds of surfaces. This novel concept characterizes
chemical species in a highly generalized manner by several topological invariants. The
well-known empirical types of chemical similarity (e.g., isoelectronic, isostructural,
homological) are thus shown to result from topological homeomorphisms; they all
conserve the Euler characteristic of the respective surfaces. Moreover, the author proves
the invariance of this topological characteristic in any chemical reaction involving
PREFACE xi
molecules with localized bonds, a result that might be termed a principle of conservation
of molecular topology in chemical reactions. The reader may thus gain an exciting and
unusually general view on reactivity in chemistry as a whole.
Chapter 8 by Mekenyan and Basak begins by reviewing some of the basic principles
underlying the topological nature of chemical reactivity. Topological indices, one of the
powerful tools of graph theory, ~e then introduced on this basis. The most common
indices are classified and formulated in several large groups which also distinguish
between global molecular, fragment, and atomic indices. Some of the first electronic
indices of reactivity, derived within the HMO theory, are also mentioned, owing to the
topological origin of the Hiickel matrix. Examples are presented of successful modeling
of various reactivity effects in different branches of chemistry including environmental
chemistry, toxicology, and drug-receptor interaction, along with some topology-based
reactivity rules and relationships.
Chapter 9 by Temkin, Zeigarnik, and Bonchev centers on progress in the mechanistic
and kinetic studies of complex reactions as a source of information on the reactivity of
intermediates and their elementary steps. The graph-theoretical concept of the topological
identifier, which produces two general principles (simple bond-change topology and
bond-change compensation topology), is first developed for identifying, classifying, and
enumerating elementary reactions. Then, the formalism of kinetic graphs and bipartite
graphs is applied to the classification, coding, and enumeration of linear and nonlinear
reaction mechanisms. The authors also introduce and discuss the concept of mechanistic
topological structure (the reactant interrelations, the number and kind of reaction routes,
and their mutual connectedness), an aspect of complex chemical reactions that was
largely neglected in the past. A topological characterization of the four major classes of
complex reactions (noncatalytic, noncatalytic conjugated, chain, and catalytic reactions)
is presented on this basis.
In conclusion, we would like to thank our authors and express the hope that the
material presented will find resonance with our readers and prompt their own
contributions to the field. If this proves to be the case, then the aim of this volume will
be fulfilled. .
Danai! Bonchev and
Ovanes Mekenyan