Quantum Chemistry: Fundamentals to Applications

Quantum Chemistry: Fundamentals to Applications

Tamas Veszpremi Department of Inorganie Chemistry,

Teehnieal University of Budapest 1521 Budapest, Gellertter 4, Hungary

and

Mik16s Feher Nanodesign, [ne. Researeh Park Centre,

150 Researeh Lane, Guelph, Ontario N1G 4T2, Canada

Springer Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data

Veszpremi, Tamas Quantum chemistry : fundamentals to applications / Tamas

Veszpremi and Mikl6s Feher p. cm.

Includes bibliographical references and index. ISBN 978-1-4613-6879-3 ISBN 978-1-4615-4189-9 (eBook) DOI 10.1007/978-1-4615-4189-9 1. Quantum chemistry. 1. Feher, Mikl6s. II. Title

QD412.V48 1999 541.2'8--dc21 99-42895

CIP

ISBN 978-1-4613-6879-3

© 1999Springer Science+Business Media New York Origina1ly published by Kluwer Academic/Plenum Publishers, New York in 1999 Softcover reprint of the hardcover 1 st edition 1999

http:/ jwww.wkap.com

1098765432 1

AH rights reserved.

No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means, electronic, mechanica1, photocopying, microfilming, recording, or otherwise, without written pennission from the Publisher.

To John A. Pople and Walter Kohn, whose giant contribution helped chemistry to be no longer

a purely experimental science.

Foreword

Quantum chemistry is one of the most dynamic fields of contemporary chemistry, providing a solid foundation for all of chemistry, and serving as the basis for practical, computational methodologies with applications in virtually all branches of chemistry. The scope of this book by Veszpremi and Feher is extremely ambitious; it ranges from a thorough treatment of the fundamentals of quantum chemistry to a wide variety of important applications. The authors have succeeded very well in covering all the basics and providing enough intriguing detail to compel readers to appreciate the power and utility of modem quantum chemistry.

All the mathematical tools necessary are discussed separately in the first chapter as well as in Appendix I. The prominence of group theory justifies its special treatment in the first chapter, and the introduction to linear algebra is the main subject of Appendix I. Following the postulates of quantum mechanics and some of the standard simple problems involving analytical solutions of the SchrOdinger equation, the discussion quickly turns to the chemically important topics of Hartree-Fock methods, semiempirical methods, and density functional theory.

The second part of the book, which is devoted to applications, reflects the emphasis of the authors on those fields that they find the most important, including molecular structure, spectroscopy, thermodynamics, chemical reactions, solvent effects, and excited-state chem­istry. The coverage and depth of discussion emphasize those aspects which are useful in applied fields, such as the study of reaction mechanisms and biochemical structure.

The topics discussed demonstrate the major contributions quantum chemistry has made in recent decades to both original chemical research and routine investigations of molecular properties. Chemistry has become an understandable science, and quantum chemistry is one of the main components of the contemporary theory of chemistry. Whereas experiments are both the source and the ultimate test of any theoretical framework, individual experiments provide only isolated facts and data; the interpretation of such facts and data involves fitting them within a theoretical model. We usually claim understanding when the experimental data are found to match a theoretical model. The increased sophistication, accuracy, and scope of the theory of chemistry are due to a large extent to the spectacular development of quantum chemistry, and in this book the authors have made a remarkable effort to provide a modem account of the field.

Paul G. Mezey

Department of Chemistry and Department of Mathematics and Statistics

University of Saskatchewan, 110 Science Place, Saskatoon,

SK, Canada S7N 5C9

Vll

Preface

We are witnesses of a breathtaking revolution in our way of thinking. Although the first computers were built only half a century ago, they have completely changed our way of life. Similarly to other areas of science, they also had a profound impact on chemistry. Computers did not simply automate what had previously been performed manually, like model building and visualization, but made it possible to create an entirely new approach to chemistry: computational chemistry. This book is an introduction to one of the major branches of computational chemistry: quantum chemistry.

Quantum chemistry is the science that attempts to apply the laws of quantum mechanics to chemical systems. We can trace its origins to an article by Heitler and London on the hydrogen molecule that was published in 1927. * The major phase of development in the theory and methods in this area took place between the 1930s and 1950s of this century. By the 1960s, nearly all of today's theories had been developed, but there has been a continual development in the armory of methods ever since. An important characteristic of quantum chemistry is that it is fundamentally dependent on computers. With the limited computing power of those early times, it is a wonder it could still be used and could provide the theoretical basis for a number of chemical principles and phenomena.

In the 1980s supercomputers appeared on the scene, making it possible to carry out highly sophisticated calculations. The results demonstrated that quantum chemistry is not only capable of providing qualitative answers but can also allow the accurate determination of physical and physicochemical properties. However, because supercomputers were prohibi­tively expensive, such calculations could be performed by only a few research groups. The area was revolutionized in the 1990s with the introduction of high-performance workstations. Although these are less powerful than mainframe supercomputers, they cost only a tiny fraction of the latter. Parallel to this, suites of user-friendly programs for these workstations have also been developed. Quantum chemistry is now accessible to a wide range of chemists.

The importance of the area is amply demonstrated by the fact that the 1998 Nobel Prize for chemistry was awarded for the development of this field of knowledge. John Pople received the award for showing that computational quantum chemistry is a practical, viable proposi­tion, and Walter Kohn received it for the development of density functional theory.

Dirac's famous words from 1929 were quoted in the recommendation of the Nobel Committee: "The fundamental laws necessary for the mathematical treatment of large parts of physics and the whole chemistry are thus fully known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved." The continual development of ideas since that time has made this pessimistic statement obsolete. In the words of the Nobel Prize committee: "Quantum chemistry is today a mature science, which penetrates all of chemistry. Who could believe 10 years ago that quantum chemistry would play an important role in biochemistry, for example, in the understanding of mechanisms for enzymatic reactions, or explain the relation between structure and spectro­scopic properties in transition metal containing proteins. Theory works today hand in hand with experimental chemistry in the exploration of the properties of matter."

This book attempts to demonstrate the use of quantum chemistry through practical examples. We will see how it can be used to determine the structure of molecules, to

*W Heitler and F. London, Z. Physik 44,455 (1927). ix

x PREFACE

calculate spectroscopic properties, to understand the mechanism of chemical reactions, or to characterize the products. It can assist in devising the synthesis of novel molecules or in assessing the effect of a newly introduced substituent. In other words, quantum chemistry has indeed grown to be an important tool of chemistry, on a par with experimental methods.

Many organic and inorganic chemists are baffled by these developments and decide to stick to the familiar ball-and-stick and space-filling models and curly arrows in their explanations. The authors of this volume feel that with the wide availability of computers, it is time to change this philosophy. When writing this book, we had those chemists in mind who would be willing to employ results from the wealth of quantum chemistry literature but who are unsure about how to approach the subject and what information they can possibly gather from it.

The book is constructed in two parts. The first part gives an overview of the principles of modern quantum chemistry and tries to explain to those with little background in this area what many of the buzzwords mean that are often encountered in publications. It was our aim to provide an introduction to quantum chemistry while simultaneously clarifying the physical origin of some everyday chemical concepts. We wished to present the theoretical basis of the most important models, their applicability, and the limitations of their use. We primarily dealt with standard methods. An entire chapter is devoted to group theory to ease the understanding of some of the more difficult concepts. Also, for those less familiar with mathematics, a short summary of the basics of linear algebra is presented in Appendix I.

The second part of the book seeks to demonstrate the use of the principles discussed and contains examples of the calculation of quantities and properties of practical interest. The examples have been carefully selected from the current quantum chemistry literature to introduce real-life chemical problems for which computational chemistry provides the key to solution. To achieve this, the discussion often remains more qualitative than in the original article and no in-depth description of the methods and algorithms is given. The selection of over two dozen papers from the thousands of relevant articles was a daunting task and any choice was necessarily arbitrary. We sought recent articles that best demonstrated the concept discussed. These articles are not necessarily the best papers on the subject, nor are they always the original sources for the given problem. However, their reasoning is usually sufficiently simple for a beginner to see both the merits and pitfalls of the methods applied.

It must be admitted that the choice of suggested readings to illustrate the important points in the theoretical discussions in Part I was equally arbitrary. There we tried to quote the two to three papers that made an impact on the field, as well as ones that explained the ideas clearly and at the right level. Hence we often chose well-written articles over others with perhaps a greater depth but that were more difficult to follow. Finally, we recommend a few books that the reader could use to acquire a deeper understanding of a certain area. To compl~ment this selection, in Appendix N we give a more complete selection of quantum chemistry books on the market. This list contains no popular science books nor any specialized high-level monographs. We sincerely hope that the carefully selected references will enhance the understanding of the principles in this current book.

It is often said that each mathematical formula may reduce the potential readership of a book. We believe that this might also apply in an area as heavily dependent on math as quantum chemistry. Therefore it was our aim to reduce the number of formulas and simplify the explanations. On the other hand, the logic of the discussion would have often required more in-depth explanations and proofs. We thought that these conflicting requirements could be best fulfilled if only the main points of the explanations remained in the text and the boring details were collected at the end of each chapter in a Notes section. The casual reader can simply ignore these notes without any major loss of understanding.

PREFACE xi

Finally, what can one expect from a book of this modest size, dealing with an area as immense as quantum chemistry? It is intended to demonstrate that a compromise is necessary between quality of results and computational costs. It is meant to introduce the major approximations in the theory. A further objective is to show that the calculated results should always be judged in view of the approximations involved. It is hoped that the lack of rigorous derivations will be more than compensated for by the pleasure of appreciating the abundance of valuable information available from quantum chemistry. This book has achieved its aim if it encourages readers to use the quantum chemistry literature and to do calculations as a source for ideas and inspiration, as well to support their everyday work.

Acknowledg,ments

We express our thanks to those distinguished colleagues and friends who helped us with their criticism and suggestions: Dr. E.P.F. Lee, Dr. G. Pongor, and Dr. G. Csonka. We are grateful to those authors and copyright owners who gave us permission to use their illustrations in our book. We gratefully acknowledge the kind assistance of Prof. R. Herges, Dr. A. Kovacs, Prof. 1. M. Lluch, Prof. A. Gonzales-Lafont, Dr. A. Perczel, Prof. W Thiel, and Dr. L. Turi. We are very much indebted to Mr. B. Hajgato for drawing the lion's share of the figures.

September 1998 Tamas Veszpremi Miklos Feher

xiii

Contents

Part 1. The Fundamentals. . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Fundamentals of Group Theory. . . . • . . . . . . . • . . . . . • . . • . • . . . . . . . . 3

1.1. Symmetry Elements and Symmetry Operations . . . . . . . . . . . . . . . . . . . 3 1.2. Concept of Symmetry Point Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3. Procedure for Symmetry Classification. . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4. Multiplication of Symmetry Operations. . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5. Classes in a Point Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6. Matrix Representation of Symmetry Operations . . . . . . . . . . . . . . . . . . . 11 1.7. How to Transform Objects? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.8. Direct Product Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2. Postulates of Quantum Mechanics • . . . . . . . . . . . . . • • . . • . . . • • . . . . . . 29

Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3. Playing with the SchrOdinger Equation. . . . . . . . . . . . . . . • . . . • . • . . . . . 43

3.1. The Free Particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2. The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3. Motion in a Coulomb Field: The Hydrogen Atom. . . . . . . . . . . . . . . . . 47 3.4. Magnetic Moment and Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4. Quantum Chemistry: A Hierarchy of Approximations. . . . • . . . . . . . . . . . 65

4.1. The First Approximation: Neglect of Relativistic Effects. . . . . . . . . . . . . 66 4.2. The Sixth Postulate of Quantum Mechanics: The Pauli Principle. . . . . . . . 68 4.3. The Bom-Oppenheimer approximation. . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4. One-Electron Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5. Methods of Solution . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . • 83

5.1. Variation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2. The Elements of Time-Independent Perturbation Theory. . . . . . . . . . . . . 87 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

xv

xvi CONTENTS

6. The Bartree-Fock Method and Its Consequences . • • • • • • • • • • • • • • • • • • 93

6.1. The Hartree-Fock method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2. Electronic Structure of Many-Electron Atoms . . . . . . . . . . . . . . . . . . . . 97 6.3. The Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 104 6.4. The Hartree-Fock-Roothaan Method. . . . . . . . . . . . . . . . . . . . . . . . .. 105 6.5. Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 106 6.6. The Molecular Orbital Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113 6.7. Ab initio Hartree-Fock-Roothan Calculations. . . . . . . . . . . . . . . . . . .. 123 6.8. Localized Molecular Orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135 6.9. Symmetry of the Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137 6.10. How to Carry on? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 140 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 141 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 144

7. Beyond the Hartree-Fock Method. • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 147

7.1. Configuration Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 148 7.2. Methods for Calculating the Electron Correlation. . . . . . . . . . . . . . . . .. 152 7.3. Correlation Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 159 7.4. Theoretical Model Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165 7.5 .. Inclusion of Relativistic Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170

8. The Theory of Electron Density. • • • • • • • • . • . • . • • • • • • • • • • • • • • • • •• 173

8.1. Mulliken Population Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175 8.2. Natural Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 178 8.3. Molecular Electrostatic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179 8.4. The Shape of Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181 8.5. Topological Analysis of Electron Density. . . . . . . . . . . . . . . . . . . . . .. 184 8.6. Principles of Density Functional Theory. . . . . . . . . . . . . . . . . . . . . . .. 191 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 196 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 198

9. Semiempirical Methods. • . • • • • • • • • • • • • . • . • • • • • • • • • • • • • • • • • • •• 201

Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204

Part II. Applications. • • . • . • • . • • • • • • • • • • • • • . • . • • • • • • • • • • • • • • • • •• 205

10. Molecular Structure. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • .• 207

10.1. Structure of Perftuorocyclopropene. . . . . . . . . . . . . . . . . . . . . . . . . .. 209 10.2. Structure of [lO]annulenes: The Failure of Second-order Perturbation

Theory and Density Functional Schemes. . . . . . . . . . . . . . . . . . . . . .. 211

CONTENTS xvii

10.3. Theoretical Studies of a 2: 1 Bromine-Olefin 7t-Complex. . . . . . . . . . .. 214 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 217 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 217

11. Vibrational Frequencies . . . . . . . . . . . • . . . . . • . . . . . . . . . . . . . . • . . .. 219

11.1 Harmonic Vibrational Frequencies of Perfluorocyclopropene . . . . . . . . .. 222 11.2. Identification ofIsomers from Calculated Vibrational Spectra. . . . . . . .. 224 11.3. Vibrational Analysis of Highly Symmetric Molecules . . . . . . . . . . . . .. 227 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 231 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 233

12. Thermodynamic Properties ...............•..............•... , 235

12.1. Relative Energies of Different CsHt Isomers. . . . . . . . . . . . . . . . . . .. 236 12.2. Hydrogen Bond Energy of the Water Dimer. . . . . . . . . . . . . . . . . . . .. 238 12.3. Calculations of Low-Barrier Hydrogen Bonds in the

Hydrogen Maleate Anion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., 240 12.4. Conformational Energies and Rotational Energy Barriers of n-Butane . .. 243 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 247 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 248

13. Properties Related to Electron Density •......................... , 249

13.1. Topological Analysis of Low-Barrier Hydrogen Bonds ............. , 251 13.2. Dipole Moments and Polarizabilities of Oxazoles . . . . . . . . . . . . . . . .. 253 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 257 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 257

14. Chemical Reactions. . . . . . . . . . . . • . . . • . . . . . . • . • . . . . . . . . . . . . . .. 259

14.1. Reactions of the Peroxy Radical Complexed to the Lithium Cation .... , 261 14.2. How Does Helium get into Buckminsterfullerene? . . . . . . . . . . . . . . .. 264 14.3. Ring Opening of Dewar Benzenes. . . . . . . . . . . . . . . . . . . . . . . . . .. 270 14.4. Ring Opening of Cyclopropylidene. . . . . . . . . . . . . . . . . . . . . . . . . .. 273 14.5. Base-Promoted Hydrolysis of N-Methylformamide . . . . . . . . . . . . . . .. 277 14.6. Mechanism and Rate Constants of the HCO + HN02 ~HCHO + N02

Reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 281 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 284 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 286

15. Calculations on Liquids and Solids .. . . • . . . . . . . . . . . . . • . . . . . . . . .. 289

15.1. Tautomeric Equilibria of Hydroxypyridines in Different Solvents ...... , 291 15.2. Dissociation of Hydrogen on Platinum. . . . . . . . . . . . . . . . . . . . . . .. 295 15.3. Structure of Crystalline Nitroanilines . . . . . . . . . . . . . . . . . . . . . . . .. 299 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 304

xviii CONTENTS

16. Molecular Dynamics Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 307

16.1. Ionization of Hydrogen Chloride in Water Clusters . . . . . . . . . . . . . . .. 309 16.2. Ab initio Molecular Dynamics Simulation of the Solvation and

Transport of Hydronium Ions in Water. . . . . . . . . . . . . . . . . . . . . . .. 312 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 315 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 317

17. Ions and Excited Electronic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 319

17.1. Excited Electronic States of Benzaldehyde. . . . . . . . . . . . . . . . . . . . .. 321 17.2. Lowest Excited States of the Imidazole Molecule . . . . . . . . . . . . . . . .. 323 17.3. Ionization Energies of HFCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 327 17.4. Thermochemistry ofIron Chlorides and their Positive and Negative

Ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 330 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 335 Suggested Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 337

Appendix I. The Basics of Linear Algebra .......................... 339

1. Vectors in Three-Dimensional Space. . . . . . . . . . . . . . . . . . . . . . . . . . .. 339 2. Vectors in n-Dimensional Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 340 3. Simultaneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 342 4. Matrices and Matrix Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 344 5. The Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 346 6. Scalar Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 347 7. Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 349 8. Eigenvalue Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 351 9. The Space of Square Integrable Functions. . . . . . . . . . . . . . . . . . . . . . .. 352

Appendix II. Selected Character Tables 355

Appendix III. Units .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 367

Appendix Iv. Recent Books in Quantum Chemistry ........... . .. . . . . .. 369

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 373

Index ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 375

Quantum Chemistry: Fundamentals to Applications