Models for Bonding

in Chemistry

Models for Bonding in Chemistry Valerio Magnasco© 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-66702-6

Models for Bonding

in Chemistry

Valerio MagnascoUniversity of Genoa, Italy

This edition first published 2010

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Library of Congress Cataloging-in-Publication Data

Magnasco, Valerio.

Models for bonding in chemistry / Valerio Magnasco.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-66702-6 (cloth) – ISBN 978-0-470-66703-3 (pbk.) 1. Chemical bonds. I. Title.

QD461.M237 2010

541’.224–dc22

2010013109

A catalogue record for this book is available from the British Library.

ISBN 978-0-470-66702-6 (cloth) 978-0-470-66703-3 (paper)

Set in 10.5/13pt Sabon-Roman by Thomson Digital, Noida, India

Printed and bound in United Kingdom by TJ International., Padstow, Cornwall

To Deryk

Contents

Preface xi

1 Mathematical Foundations 1

1.1 Matrices and Systems of Linear Equations 11.2 Properties of Eigenvalues and Eigenvectors 61.3 Variational Approximations 101.4 Atomic Units 151.5 The Electron Distribution in Molecules 171.6 Exchange-overlap Densities and the Chemical Bond 19

Part 1: Short-range Interactions 27

2 The Chemical Bond 29

2.1 An Elementary Molecular Orbital Model 302.2 Bond Energies and Pauli Repulsions

in Homonuclear Diatomics 342.2.1 The Hydrogen Molecular Ion H2

þ (N¼1) 352.2.2 The Hydrogen Molecule H2(N¼2) 352.2.3 The Helium Molecular Ion He2

þ (N¼3) 352.2.4 The Helium Molecule He2 (N¼4) 36

2.3 Multiple Bonds 372.3.1 s2p2 Description of the Double Bond 382.3.2 B1

2B22 Bent (or Banana) Description

of the Double Bond 402.3.3 Hybridization Effects 422.3.4 Triple Bonds 46

2.4 The Three-centre Double Bond in Diborane 472.5 The Heteropolar Bond 492.6 Stereochemistry of Polyatomic Molecules 55

2.6.1 The Molecular Orbital Model of Directed Valency 552.6.2 Analysis of the MO Bond Energy 58

2.7 sp-Hybridization Effects in First-row Hydrides 602.7.1 The Methane Molecule 612.7.2 The Hydrogen Fluoride Molecule 642.7.3 The Water Molecule 752.7.4 The Ammonia Molecule 87

2.8 Delocalized Bonds 962.8.1 The Ethylene Molecule 982.8.2 The Allyl Radical 982.8.3 The Butadiene Molecule 1002.8.4 The Cyclobutadiene Molecule 1022.8.5 The Benzene Molecule 104

2.9 Appendices 1082.9.1 The Second Derivative of the Huckel Energy 1082.9.2 The Set of Three Coulson Orthogonal Hybrids 1092.9.3 Calculation of Coefficients

of Real MOs for Benzene 110

3 An Introduction to Bonding in Solids 119

3.1 The Linear Polyene Chain 1203.1.1 Butadiene N¼ 4 122

3.2 The Closed Polyene Chain 1233.2.1 Benzene N¼ 6 126

3.3 A Model for the One-dimensional Crystal 1313.4 Electronic Bands in Crystals 1333.5 Insulators, Conductors, Semiconductors

and Superconductors 1383.6 Appendix: The Trigonometric Identity 143

Part 2: Long-Range Interactions 145

4 The van der Waals Bond 147

4.1 Introduction 1474.2 Elements of Rayleigh–Schrodinger (RS)

Perturbation Theory 1494.3 Molecular Interactions 151

4.3.1 Non-expanded Energy Correctionsup to Second Order 152

4.3.2 Expanded Energy Correctionsup to Second Order 153

4.4 The Two-state Model of Long-range Interactions 157

viii CONTENTS

4.5 The van der Waals Interactions 1594.5.1 Atom–Atom Dispersion 1614.5.2 Atom–Linear Molecule Dispersion 1624.5.3 Atom–Linear Dipolar Molecule10 Induction 163

4.6 The C6 Dispersion Coefficient for the H�H Interaction 1654.7 The van der Waals Bond 1674.8 The Keesom Interaction 169

5 The Hydrogen Bond 177

5.1 A Molecular Orbital Model of the Hydrogen Bond 1785.2 Electrostatic Interactions and the Hydrogen Bond 179

5.2.1 The Hydrogen Fluoride Dimer (HF)2 1825.2.2 The Water Dimer (H2O)2 185

5.3 The Electrostatic Model of the Hydrogen Bond 1865.4 The Rg–HF Heterodimers 197

References 201

Author Index 209

Subject Index 213

CONTENTS ix

Preface

Experimental evidence shows thatmolecules are not like ‘liquid droplets’of electrons, but have a structure made of bonds and lone pairs directedin space. Even at its most elementary level, any successful theory ofbonding in chemistry should explainwhy atoms are or are not bonded inmolecules, the structure and shape of molecules in space and howmolecules interact at long range. Even if modern molecular quantummechanics offers the natural basis for very elaborate numerical calcula-tions, models of bonding avoiding the more mathematical aspects of thesubject in the spirit of Coulson’s Valence are still of conceptual interestfor providing an elementary description of valence and its implicationsfor the electronic structure of molecules. This is the aim of this concisebook, which grew from a series of lectures delivered by the author at theUniversity of Genoa, based on original research work by the author andhis group from the early 1990s to the present day. The book should serveas a complement to a 20-hour university lecture course in Physical andQuantum Chemistry.The book consists of two parts, where essentially twomodels have been

proposed, mostly requiring the solution of quadratic equations with realroots. Part 1 explains forces acting at short range, typical of localized ordelocalized chemical bonds in molecules or solids; Part 2 explains forcesacting at long range, between closed-shell atoms ormolecules, resulting inthe so-called van der Waals (VdW) molecules. An electrostatic model isfurther derived forH-bonded andVdWdimers,which explains in a simpleway the angular shape of the dimers in terms of the first two permanentelectric moments of the monomers.The contents of the book is as follows. After a short self-contained

mathematical introduction, Chapter 1 presents the essential elements ofthe variation approach to either total or second-order molecular energies,

the system of atomic units (au) necessary to simplify all mathematicalexpressions, and an introductory description of the electron distributionin molecules, with particular emphasis on the nature of the quantummechanical exchange-overlap densities and their importance in determin-ing the nature of chemical bonds and Pauli repulsions.The contents of Part 1 is based on such premises. Using mostly 2�2

H€uckel secular equations, Chapter 2 introduces a model of bonding inhomonuclear and heteronuclear diatomics, multiple and delocalizedbonds in hydrocarbons, and the stereochemistry of chemical bonds inpolyatomic molecules; in a word, a model of the strong first-orderinteractions originating in the chemical bond. Hybridization effectsand their importance in determining shape and charge distribution infirst-row hydrides (CH4, HF, H2O andNH3) are examined in some detailin Section 2.7.In Chapter 3, the H€uckel model of linear and closed polyene chains is

used to explain the origin of band structure in the one-dimensional crystal,outlining the importance of the nature of the electronic bands in deter-mining the different properties of insulators, conductors, semiconductorsand superconductors.Turning to Part 2, after a short introduction to stationary Rayleigh–

Schr€odinger (RS) perturbation theory and its use for the classification oflong-range intermolecular forces, Chapter 4 deals with a simple two-state model of weak interactions, introducing the reader to an easy wayof understanding second-order electric properties of molecules andVdW bonding between closed shells. The chapter ends with a shortoutline of the temperature-dependent Keesom interactions in polargases.Finally, Chapter 5 studies the structure of H-bonded dimers and the

nature of the hydrogen bond,which has a strength intermediate between aVdWbondandaweak chemical bond. Besides a qualitativeMOapproachbased on HOMO-LUMO charge transfer from an electron donor to anelectron acceptor molecule, a quantitative electrostatic approach is pre-sented, suggesting an electrostatic model which works even at its simplestpictorial level.A list of alphabetically ordered references, and author and subject

indices complete the book.The book is dedicated to the memory of my old friend and colleague

Deryk Wynn Davies, who died on 27 February 2008. I wish to thank mycolleagues Gian FrancoMusso and Giuseppe Figari for useful discussionson different topics of this subject, Paolo Lazzeretti and Stefano Pelloni for

xii PREFACE

some calculations using the SYSMO programme at the University ofModena andReggio, andmy sonMariowhoprepared the drawings on thecomputer. Finally, I acknowledge the support of the Italian Ministry forEducation University and Research (MIUR) and the University of Genoa.

Valerio MagnascoGenoa, 20 December 2009

PREFACE xiii