Models for Bonding in Chemistry

Contents

Preface

1 Mathematical Foundations

1.1 Matrices and Systems of Linear Equations

1.2 Properties of Eigenvalues and Eigenvectors

1.3 Variational Approximations

1.4 Atomic Units

1.5 The Electron Distribution in Molecules

1.6 Exchange-overlap Densities and the Chemical Bond

Part 1: Short-range Interactions

2 The Chemical Bond

2.1 An Elementary Molecular Orbital Model

2.2 Bond Energies and Pauli Repulsions in Homonuclear Diatomics

2.2.1 The Hydrogen Molecular Ion H₂⁺ (N=1)

2.2.2 The Hydrogen Molecule H₂(N=2)

2.2.3 The Helium Molecular Ion He₂⁺ (N=3)

2.2.4 The Helium Molecule He₂ (N=4)

2.3 Multiple Bonds

2.3.1 σ²π² Description of the Double Bond

2.3.2 B₁²B₂² Bent (or Banana) Description of the Double Bond

2.3.3 Hybridization Effects

2.3.4 Triple Bonds

2.4 The Three-centre Double Bond in Diborane

2.5 The Heteropolar Bond

2.6 Stereochemistry of Polyatomic Molecules

2.6.1 The Molecular Orbital Model of Directed Valency

2.6.2 Analysis of the MO Bond Energy

2.7 sp-Hybridization Effects in First-row Hydrides

2.7.1 The Methane Molecule

2.7.2 The Hydrogen Fluoride Molecule

2.7.3 The Water Molecule

2.7.4 The Ammonia Molecule

2.8 Delocalized Bonds

2.8.1 The Ethylene Molecule

2.8.2 The Allyl Radical

2.8.3 The Butadiene Molecule

2.8.4 The Cyclobutadiene Molecule

2.8.5 The Benzene Molecule

2.9 Appendices

2.9.1 The Second Derivative of the Hückel Energy

2.9.2 The Set of Three Coulson Orthogonal Hybrids

2.9.3 Calculation of Coefficients of Real MOs for Benzene

3 An Introduction to Bonding in Solids

3.1 The Linear Polyene Chain

3.1.1 Butadiene N = 4

3.2 The Closed Polyene Chain

3.2.1 Benzene N = 6

3.3 A Model for the One-dimensional Crystal

3.4 Electronic Bands in Crystals

3.5 Insulators, Conductors, Semiconductors and Superconductors

3.6 Appendix: The Trigonometric Identity

Part 2: Long-Range Interactions

4 The van der Waals Bond

4.1 Introduction

4.2 Elements of Rayleigh–Schrüdinger (RS) Perturbation Theory

4.3 Molecular Interactions

4.3.1 Non-expanded Energy Corrections up to Second Order

4.3.2 Expanded Energy Corrections up to Second Order

4.4 The Two-state Model of Long-range Interactions

4.5 The van der Waals Interactions

4.5.1 Atom-Atom Dispersion

4.5.2 Atom-Linear Molecule Dispersion

4.5.3 Atom-Linear Dipolar Molecule¹⁰ Induction

4.6 The C₆ Dispersion Coefficient for the H—H Interaction

4.7 The van der Waals Bond

4.8 The Keesom Interaction

5 The Hydrogen Bond

5.1 A Molecular Orbital Model of the Hydrogen Bond

5.2 Electrostatic Interactions and the Hydrogen Bond

5.2.1 The Hydrogen Fluoride Dimer (HF)₂

5.2.2 The Water Dimer (H₂O)₂

5.3 The Electrostatic Model of the Hydrogen Bond

5.4 The Rg–HF Heterodimers

References

Author Index

Subject Index

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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
Title. QD461.M237 2010 541’.224-dc22

2010013109

Preface

Experimental evidence shows that molecules are not like ‘liquid droplets’ of electrons, but have a structure made of bonds and lone pairs directed in space. Even at its most elementary level, any successful theory of bonding in chemistry should explain why atoms are or are not bonded in molecules, the structure and shape of molecules in space and how molecules interact at long range. Even if modern molecular quantum mechanics offers the natural basis for very elaborate numerical calculations, models of bonding avoiding the more mathematical aspects of the subject in the spirit of Coulson’s Valence are still of conceptual interest for providing an elementary description of valence and its implications for the electronic structure of molecules. This is the aim of this concise book, which grew from a series of lectures delivered by the author at the University of Genoa, based on original research work by the author and his group from the early 1990s to the present day. The book should serve as a complement to a 20-hour university lecture course in Physical and Quantum Chemistry.

The book consists of two parts, where essentially two models have been proposed, mostly requiring the solution of quadratic equations with real roots. Part 1 explains forces acting at short range, typical of localized or delocalized chemical bonds in molecules or solids; Part 2 explains forces acting at long range, between closed-shell atoms or molecules, resulting in the so-called van der Waals (VdW) molecules. An electrostatic model is further derivedfor H-bonded and VdW dimers, which explainsinasimple way the angular shape of the dimers in terms of the first two permanent electric 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 of the variation approach to either total or second-order molecular energies, the system of atomic units (au) necessary to simplify all mathematical expressions, and an introductory description of the electron distribution in molecules, with particular emphasis on the nature of the quantum mechanical exchange-overlap densities and their importance in determining the nature of chemical bonds and Pauli repulsions.

The contents of Part 1 is based on such premises. Using mostly 2x2 Hückel secular equations, Chapter 2 introduces a model of bonding in homonuclear and heteronuclear diatomics, multiple and delocalized bonds in hydrocarbons, and the stereochemistry of chemical bonds in polyatomic molecules; in a word, a model of the strong first-order interactions originating in the chemical bond. Hybridization effects and their importance in determining shape and charge distribution in first-row hydrides (CH₄, HF, H₂O and NH₃) are examined in some detail in Section 2.7.

In Chapter 3, the Huckel 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 determining the different properties of insulators, conductors, semiconductors and superconductors.

Turning to Part 2, after a short introduction to stationary Rayleigh-Schrödinger (RS) perturbation theory and its use for the classification of long-range intermolecular forces, Chapter 4 deals with a simple two-state model of weak interactions, introducing the reader to an easy way of understanding second-order electric properties of molecules and VdW bonding between closed shells. The chapter ends with a short outline of the temperature-dependent Keesom interactions in polar gases.

Finally, Chapter 5 studies the structure of H-bonded dimers and the nature of the hydrogen bond, which has a strength intermediate between a VdW bond and a weak chemical bond. Besides a qualitative MO approach based on HOMO-LUMO charge transfer from an electron donor to an electron acceptor molecule, a quantitative electrostatic approach is presented, suggesting an electrostatic model which works even at its simplest pictorial 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 my colleagues Gian Franco Musso and Giuseppe Figari for useful discussions on different topics of this subject, Paolo Lazzeretti and Stefano Pelloni for some calculations using the SYSMO programme at the University of ModenaandReggio,andmysonMariowhopreparedthedrawingsonthe computer. Finally, I acknowledge the support of the Italian Ministry for Education University and Research (MIUR) and the University of Genoa.

Valerio Magnasco
Genoa, 20 December 2009