Models for Bonding in Chemistry

Size: px

Start display at page:

Download "Models for Bonding in Chemistry"

Patricia Nichols
6 years ago
Views:

1 Models for onding in hemistry Valerio Magnasco University of Genoa, Italy

3 Models for onding in hemistry

5 Models for onding in hemistry Valerio Magnasco University of Genoa, Italy

6 This edition first published Ó John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, hichester, West Sussex, PO9 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the opyright, Designs and Patents Act 988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK opyright, Designs and Patents Act 988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom. Library of ongress ataloging-in-publication Data Magnasco, Valerio. Models for bonding in chemistry / Valerio Magnasco. p. cm. Includes bibliographical references and index. ISN (cloth) ISN (pbk.). hemical bonds. I. Title. QD46.M dc 39 A catalogue record for this book is available from the ritish Library. ISN (cloth) (paper) Set in.5/3pt Sabon-Roman by Thomson Digital, Noida, India Printed and bound in United Kingdom by TJ International., Padstow, ornwall

7 To Deryk

9 ontents Preface xi Mathematical Foundations. Matrices and Systems of Linear Equations. Properties of Eigenvalues and Eigenvectors 6.3 Variational Approximations.4 Atomic Units 5.5 The Electron Distribution in Molecules 7.6 Exchange-overlap Densities and the hemical ond 9 Part : Short-range Interactions 7 The hemical ond 9. An Elementary Molecular Orbital Model 3. ond Energies and Pauli Repulsions in Homonuclear Diatomics 34.. The Hydrogen Molecular Ion H þ (N¼) 35.. The Hydrogen Molecule H (N¼) The Helium Molecular Ion He þ (N¼3) The Helium Molecule He (N¼4) 36.3 Multiple onds s p Description of the Double ond ent (or anana) Description of the Double ond Hybridization Effects Triple onds 46.4 The Three-centre Double ond in Diborane 47.5 The Heteropolar ond 49.6 Stereochemistry of Polyatomic Molecules 55

10 viii ONTENTS.6. The Molecular Orbital Model of Directed Valency Analysis of the MO ond Energy 58.7 sp-hybridization Effects in First-row Hydrides 6.7. The Methane Molecule 6.7. The Hydrogen Fluoride Molecule The Water Molecule The Ammonia Molecule 87.8 Delocalized onds The Ethylene Molecule The Allyl Radical The utadiene Molecule.8.4 The yclobutadiene Molecule.8.5 The enzene Molecule 4.9 Appendices 8.9. The Second Derivative of the Hückel Energy 8.9. The Set of Three oulson Orthogonal Hybrids alculation of oefficients of Real MOs for enzene 3 An Introduction to onding in Solids 9 3. The Linear Polyene hain 3.. utadiene N ¼ 4 3. The losed Polyene hain enzene N ¼ A Model for the One-dimensional rystal Electronic ands in rystals Insulators, onductors, Semiconductors and Superconductors Appendix: The Trigonometric Identity 43 Part : Long-Range Interactions 45 4 The van der Waals ond Introduction Elements of Rayleigh Schrödinger (RS) Perturbation Theory Molecular Interactions Non-expanded Energy orrections up to Second Order Expanded Energy orrections up to Second Order The Two-state Model of Long-range Interactions 57

11 ONTENTS ix 4.5 The van der Waals Interactions Atom Atom Dispersion Atom Linear Molecule Dispersion Atom Linear Dipolar Molecule Induction The 6 Dispersion oefficient for the H H Interaction The van der Waals ond The Keesom Interaction 69 5 The Hydrogen ond A Molecular Orbital Model of the Hydrogen ond Electrostatic Interactions and the Hydrogen ond The Hydrogen Fluoride Dimer (HF) The Water Dimer (H O) The Electrostatic Model of the Hydrogen ond The Rg HF Heterodimers 97 References Author Index 9 Subject Index 3

13 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 oulson 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 99s to the present day. The book should serve as a complement to a -hour university lecture course in Physical and Quantum hemistry. The book consists of two parts, where essentially two models have been proposed, mostly requiring the solution of quadratic equations with real roots. Part explains forces acting at short range, typical of localized or delocalized chemical bonds in molecules or solids; Part 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 derived for H-bonded and VdW dimers, which explains in a simple 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, hapter presents the essential elements of the variation approach to either total or second-order molecular energies,

14 xii PREFAE 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 is based on such premises. Using mostly H uckel secular equations, hapter 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 (H 4, HF, H O and NH 3 ) are examined in some detail in Section.7. In hapter 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 determining the different properties of insulators, conductors, semiconductors and superconductors. Turning to Part, after a short introduction to stationary Rayleigh Schr odinger (RS) perturbation theory and its use for the classification of long-range intermolecular forces, hapter 4 deals with a simple twostate 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, hapter 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. esides 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 7 February 8. 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

15 PREFAE xiii some calculations using the SYSMO programme at the University of Modena and Reggio, and my son Mario who prepared the drawings on the computer. Finally, I acknowledge the support of the Italian Ministry for Education University and Research (MIUR) and the University of Genoa. Valerio Magnasco Genoa, December 9

17 Mathematical Foundations. Matrices and Systems of Linear Equations. Properties of Eigenvalues and Eigenvectors.3 Variational Approximations.4 Atomic Units.5 The Electron Distribution in Molecules.6 Exchange-overlap Densities and the hemical ond In physics and chemistry it is not possible to develop any useful model of matter without a basic knowledge of some elementary mathematics. This involves use of some elements of linear algebra, such as the solution of algebraic equations (at least quadratic), the solution of systems of linear equations, and a few elements on matrices and determinants.. MATRIES AND SYSTEMS OF LINEAR EQUATIONS We start from matrices, limiting ourselves to the case of a square matrix of order two, namely a matrix involving two rows and two columns. Let us denote this matrix by the boldface capital letter A: A ¼ A! A ð:þ A A Models for onding in hemistry Ó John Wiley & Sons, Ltd Valerio Magnasco

18 MATHEMATIAL FOUNDATIONS where A ij is a number called the ijth element of matrix A. The elements A ii (j ¼ i) are called diagonal elements. We are interested mostly in symmetric matrices, for which A ¼ A.IfA ¼ A ¼, the matrix is diagonal. Properties of a square matrix A are its traceðtr A ¼ A þ A Þ; the sum of its diagonal elements, and its determinant, denoted by jaj ¼ det A; a number that can be evaluated from its elements by the rule: jaj ¼A A A A ð:þ Two matrices can be multiplied rows by columns by the rule: A ¼ ð:3þ!! A A ¼! ð:4þ A A the elements of the product matrix being: ( ¼ A þ A ; ¼ A þ A ; ¼ A þ A ; ¼ A þ A : ð:5þ So, we are led to the matrix multiplication rule: ij ¼ X k¼ A ik kj ð:6þ If matrix is a simple number a, Equation (.6) shows that all elements of matrix A must be multiplied by this number. Instead, for a A, we have from Equation (.): ajaj ¼aðA A A A Þ¼ aa aa A A ¼ aa A aa A ; ð:7þ so that, multiplying a determinant by a number is equivalent to multiplying just one row (or one column) by that number. We can have also rectangular matrices, where the number of rows is different from the number of columns. Particularly important is the column vector c: or the row vector ~c: c ¼ c! ¼ c! c c ð:8þ

19 MATRIES AND SYSTEMS OF LINEAR EQUATIONS 3 ~c ¼ ðc c Þ ¼ ðc c Þ ð:9þ where the tilde means interchanging columns by rows or vice versa (the transposed matrix). The linear inhomogeneous system: ( A c þ A c ¼ b ð:þ A c þ A c ¼ b can be easily rewritten in matrix form using matrix multiplication rule (.3) as: Ac ¼ b ð:þ where c and b are column vectors. Equation (.) is a system of two algebraic equations linear in the unknowns c and c, the elements of matrix A being the coefficients of the linear combination. Particular importance has the case where b is proportional to c through a number l: Ac ¼ lc ð:þ which is known as the eigenvalue equation for matrix A. l is called an eigenvalue and c an eigenvector of the square matrix A. Equation (.) is equally well written as the homogeneous system: ða lþc ¼ ð:3þ where is the diagonal matrix having along the diagonal, called the identity matrix, and is the zero vector matrix, a column of zeros. Written explicitly, the homogeneous system (Equation.3) is: ( ða lþc þ A c ¼ ð:4þ A c þða lþc ¼ Elementary algebra then says that the system of equations (.4) has acceptable solutions if and only if the determinant of the coefficients vanishes, namely if: ja lj ¼ A l A A l ¼ A ð:5þ Equation (.5) is known as the secular equation for matrix A. If we expand the determinant according to the rule of Equation (.), we obtain

20 4 MATHEMATIAL FOUNDATIONS for a symmetric matrix A: ða lþða lþ A ¼ ð:6þ giving the quadratic equation in l: l ða þ A Þl þ A A A ¼ ð:7þ which has the two real solutions (the eigenvalues, the roots of the equation): 8 l ¼ A þ A þ D >< l ¼ A þ A D ð:8þ >: where D is the positive quantity: h i = D ¼ ða A Þ þ 4A > ð:9þ Inserting each root in turn in the homogeneous system (Equation.4), we obtain the corresponding solutions (the eigenvectors, our unknowns): 8! c ¼ D þða =! A Þ ; c ¼ D ða = A Þ >< D D! c ¼ D ða =! ð:þ A Þ ; c ¼ D þða = A Þ >: D D where the second index (a column index, shown in bold type in Equations.) specifies the eigenvalue to which the eigenvector refers. All such results can be collected in the square matrices: L ¼ l l! ; ¼ ðc c Þ ¼ c c c c! ð:þ the first being the diagonal matrix of the eigenvalues (the roots of our secular equation.7), the second the row matrix of the eigenvectors (the unknowns of the homogeneous system.4). Matrix multiplication rule shows that: ~A ¼ L; ~ ¼ ~ ¼ ð:þ This is a mathematical property of real symmetric matrices.

21 MATRIES AND SYSTEMS OF LINEAR EQUATIONS 5 We usually say that the first of Equations (.) expresses the diagonalization of the symmetric matrix A through a transformation with the complete matrix of its eigenvectors, while the second equations express the normalization of the coefficients (i.e., the resulting vectors are chosen to have modulus ). Equations (8 ) simplify noticeably in the case A ¼ A ¼ a. Then, putting A ¼ A ¼ b, we obtain: 8 l ¼ a þ b; l ¼ a b >< pffiffiffi = >: c ¼ p!; = ffiffiffi c ¼ pffiffiffi! = p = ffiffiffi ð:3þ Occasionally, we shall need to solve the so called pseudosecular equation for the symmetric matrix A arising from the pseudoeigenvalue equation: Ac ¼ lsc Y ja lsj ¼ A l A ls A ls A l ¼ ð:4þ where S is the overlap matrix: S ¼ S S S S! ¼ S! S Solution of Equation (.4) then gives: 8 l ¼ A þ A A S D ð S >< Þ ð S Þ >: l ¼ A þ A A S ð S Þ þ D ð S Þ ð:5þ ð:6þ h i = D ¼ ða A Þ þ 4ðA A SÞðA A SÞ > ð:7þ The eigenvectors corresponding to the roots (Equations.6) are rather complicated (Magnasco, 7), so we shall content ourselves here by giving only the results for A ¼ A ¼ a and A ¼ A ¼ b: The length of the vectors. A matrix satisfying the second of Equations (.) is said to be an orthogonal matrix.

22 6 MATHEMATIAL FOUNDATIONS 8 l ¼ a þ b þ S ; c ¼ðþSÞ = ; c ¼ðþSÞ >< = >: l ¼ a b S ; c ¼ ð SÞ = ; c ¼ð SÞ = ð:8þ under these assumptions, these are the elements of the square matrices L and (Equations.). Matrix multiplication shows that these matrices satisfy the generalization of Equations (.): ~A ¼ L; S ~ ¼ S ~ ¼ ð:9þ so that matrices A and S are simultaneously diagonalized under the transformation with the orthogonal matrix. All previous results can be extended to square symmetric matrices of order N, in which case the solution of the corresponding secular equations must be found by numerical methods, unless use can be made of symmetry arguments.. PROPERTIES OF EIGENVALUES AND EIGENVETORS It is of interest to stress some properties hidden in the eigenvalues ðl l Þ and eigenvectors, (Equations.3), of the symmetric c c matrix A of order with A ¼ A ¼ a and A ¼ A ¼ b: In fact, Equation (.7) can be written: ðl lþðl lþ ¼l l ðl þ l Þl þ l ¼ ð:3þ so that: l l ¼ A A A ¼ a b ¼ det A l þ l ¼ A þ A ¼ a ¼ tr A ð:3þ ð:3þ In Equation (.7), therefore, the coefficient of l, the determinant of matrix A, is expressible as the product of the two eigenvalues; the coefficient of l, the trace of matrix A, is expressible as the sum of the two eigenvalues. From the eigenvectors of Equations (.3) we can construct the two square symmetric matrices of order :

23 P ¼ c ~c ¼ ffiffiffi p ffiffiffi p A ffiffiffi p ffiffiffi p! ¼ A ð:33þ P ¼ c ~c ¼ ffiffiffi p ffiffiffi p A ffiffiffi p ffiffiffi p! ¼ A ð:34þ The two matrices P and P do not admit inverse (the determinants of both are zero) and have the properties: P ¼ A A ¼ A ¼ P ð:35þ P ¼ A A ¼ A ¼ P ð:36þ P P ¼ A A ¼! ¼ ð:37þ P P ¼ A A ¼! ¼ ð:38þ PROPERTIES OF EIGENVALUES AND EIGENVETORS 7

24 8 MATHEMATIAL FOUNDATIONS P þ P ¼ A þ A ¼! ¼ ð:39þ In mathematics, matrices having these properties (idempotency, mutual exclusivity, completeness 3 ) are called projectors. In fact, acting on matrix of Equation (.) since: P c ¼ P ¼ P c þ P c ¼ c p ffiffiffi p ffiffiffi þ p ffiffiffi p ffiffiffi ¼ A p ffiffiffi A p ffiffiffi þ ¼ ¼ c p ffiffiffi A p ffiffiffi A ð:4þ ð:4þ P c ¼ pffiffiffi p ffiffiffi þ p ffiffiffi ¼ A p ffiffiffi A p ffiffiffi þ ¼! ¼ p ffiffiffi A ð:4þ so that, acting on the complete matrix of the eigenvectors, P selects its eigenvector c, at the same time annihilating c. In the same way: P ¼ P c þ P c ¼ c ð:43þ This makes evident the projector properties of matrices P and P. Furthermore, matrices P and P allow one to write matrix A in the socalled canonical form: A ¼ l P þ l P ð:44þ 3 Often referred to as resolution of the identity.

PRACTICAL RAMAN SPECTROSCOPY AN INTRODUCTION

PRACTICAL RAMAN SPECTROSCOPY AN INTRODUCTION PRACTICAL RAMAN SPECTROSCOPY AN INTRODUCTION Peter Vandenabeele Ghent University, Belgium This edition first published 2013 2013 John Wiley & Sons, Ltd Registered