Chemical Physics of Free Molecules Chemical Physics of Free Molecules

CHEMICAL PHYSICS OF FREE MOLECULES CHEMICAL PHYSICS OF FREE MOLECULES

Norman H. March Oxford Univenity Oxford, England and Joseph F. Mucci Vassar College Poughkeepsie, New York

Springer Science+Business Media, LLC Library of Congress Cataloging-in-Publication Data

March, Norman H. (Nor~an Henry), 1927- Chemical physics of free molecules I Norman H. March ana ~oseph F. Mucc i. p. cm. Includes bibliographical references and index.

1. Chemical bonds. 2. Intermolecular forces. 3. Molecules. I. Mucci, Joseph F. II. Title. 00461. M26 1994 541.2'24--dc20 93-24193 CIP

ISBN 978-1-4757-9648-3 ISBN 978-1-4757-9646-9 (eBook) DOI 10.1007-978-1-4757-9646-9

C 1993 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1993. Softcover reprint of the hardcover 1st edition 1993

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

The aim of this book is to provide a basic treatment of both electronic and nuclear motions in molecules, which is suitable for chemical physicists and for chemists. The book may also be of use, as an introduction to molecules, to materials scientists who wish to employ the methods of quantum chemistry to study con densed molecular matter, though the present work is almost exclusively about the gas phase. * The level is for advanced undergraduate students and postgraduates. From the outset, attention is focused on the molecule, with its attendant multinuclear character. Thus, the simplest problem treated in the body of the text is the hydrogen molecule ion. However, some background is presented in the Appendixes, regarding the electronic structure of atoms and their binding energies. The main tool employed in treating electronic behavior is the variation method. While this is widely used in current undergraduate texts for dealing with wave functions, a unique feature of the present account is that it is also used, within the framework of the "inhomogeneous electron gas,"t to treat the ground state electron density. Both variational approaches, wave function and electron density, are brought to full fruition when used to establish a self-consistent field. This concept, going back to Hartree, is a recurring theme. We therefore elaborate a little on its meaning immediately below. Only in the one problem of the H~ ion referred to above do we have a situ.ation of chemical interest where electron repulsion does not occur. It is our view in this work that the existence of localized chemical bonding is a rather direct manifestation of the importance of the correlated motion of electrons, both because the electrons have spin, and parallel spin electrons avoid one another as a consequence of the Pauli Exclusion Principle; and also because all electrons, whether with parallel or antiparallel spins, avoid one another by virtue of Cou lomb repulsion. Nevertheless, certain properties of molecules, most importantly the ground state electron density (the pioneers having been Thomas, Fermi, and Dirac), can be calculated by one-electron (orbital) methods, provided the electron correlation, both Pauli principle and Coulombic repulsion, is built into a suitable self-consistent field in which the electrons move independently. This recognition means that the "conflict" which sometimes appeared historically to exist between orbital methods on the one hand and chemical, valence bond, methods on the

*A few examples taken from solids and liquids have been used to illustrate points concerned with chemical bonding in free molecules. tFor the reader who becomes deeply interested in the electron density description, advanced accounts can be found in The Theory of the Inhomogeneous Electron Gas, S. Lundqvist and N. H. March, eds., Plenum Press, New York (1983); Density Functional Theory of Atoms and Molecules by R. G. Parr and W. Yang, Oxford University Press (1989); and Electron Density Theory of Atoms and Molecules by N. H. March, Academic Press, New York (1992).

v VI other is only apparent. For some purposes, orbital methods are very convenient Preface and can be formulated, at least in principle, exactly. For specified (i.e., fixed) nuclear positions, knowledge of the ground state electron density suffices (via the so-called Hellmann-Feynman theorem described in the Appendix) to calculate the forces on all nuclei, which are evidently needed to discuss vibrational degrees of freedom. Though this approach is not explicitly utilized in the present work (see Chapter 3 for an alternative approach to molecu lar force fields), it serves to highlight further the long-term importance for chemical physics of the ground state electron density. * While we are both practicing theorists, we have also sought in our presenta tion here to emphasize the crucial interplay between experiment and theory in this subject. This, we believe, is made especially clear in Chapters 1, 5, and 7, the last constituting a brief introduction to the now vast field of chemical reactions. Finally we note (i) that much of the material presented here has been used in undergraduate courses at Vassar College and at Oxford University, and (ii) that if our book should find favor with university teachers of such courses, we shall be most grateful for constructive criticism, to which we shall do our best to respond if the opportunity arises in the future. We gratefully acknowledge the encouragement and help of members of the Editorial Staff at Plenum; namely, Amelia McNamara, Laura Troup, and Barbara Sonnenschein. We especially want to thank our copyeditor, Evelyn Grossberg, for her understanding and expertise. In addition, we thank Marilyn Bontempo for the production of the art work.

Norman H. March Joseph F. Mucci

*In principle a directly observable property, unlike the many-electron wave function, via X-ray or electron diffraction experiments. ACKNOWLEDGMENTS

We gratefully acknowledge permission to reproduce figures and tables from other sources as follows:

Chapter 1 Figure 1.2. Chemical Bonding Clarified Through Quantum Mechanics by G. C. Pimentel and R. D. Spratley, Holden-Day Inc., San Francisco (1969). Reprinted by permission of the authors, the copyright owners. Figure 1.4. High Resolution Spectroscopy by J. M. Hollas, Butterworth!fPublish ers (1982). Reprinted by permission of Butterworth-Heinemann Publishers. Figure 1.5. ECSA Applied to Free Molecules by K. Siegbahn, C. Nordling, G. Johansson, P. F. Heden, K. Hamerin, U. Gelius, T. Bergmark, L. O. Worme, R. Manne, and Y. Baer, copyright (c) 1969 by North-Holland Publishing Co., Amsterdam. Reprinted by permission of Elsevier Science Publishers. Figures 1.7 and 1.8. Quanta by P. W. Atkins, 2nd Ed., Oxford University Press, Oxford (1991). Reprinted by permission of Oxford University Press. Figures 1.9 and 1.10. H. L. Chen and C. B. Moore, J. Chem. Phys. 54, 4072 (1971). Reprinted by permission of the American Institute of Physics and the authors. Figure 1.11. Molecular Structure and Dynamics by W. H. F1ygare, Prentice-Hall Inc., Englewood Cliffs, N.J. (1978). Reprinted by permission of Ruth M. Flygare, the current copyright owner. Figure 1.12. Modern Spectroscopy by J. M. Hollas, copyright (c) 1987 by John Wiley and Sons Ltd., Chichester. By permission of copyright owner. Tables 1.1 and 1.4. Reprinted from The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chem istry 3rd. Ed. by Linus Pauling. 3rd edition copyright (c) 1960 by Cornell Univer sityPress. Used by permission of the publisher, Cornell University Press. Tables 1.2 and 1.3. Chemical Structure and Bonding by R. L. DeKock and H. B. Gray, Benjamin/Cummings Publishing Co., Menlo Park, CA. (1980). Reprinted by permission of the current copyright holder, Roger DeKock. Table 1.5. Chemical Constitution, 2nd. Ed. by J. A. A. Ketelaar, Elsevier Science Publishers, Amsterdam, 1958. Reprinted by permission of Elsevier Science Publi shers and the author. Table 1.6. J. L. Hollenberg, J. Chem. Educ. 47, 2 (1970). Reprinted by permission of the Division of Chemical Education of the American Chemical Society.

vii V111 Chapter 2 Acknowledgments Figure 2.4. Physical Chemistry by R. S. Berry, S. A. Rice, and J. Ross. Copyright (c) 1980 by John Wiley and Sons, Inc., New York. Reprinted by permission of John Wiley and Sons, Inc. Figures 2.5, 2.7, 2.9, 2.10, 2.11, and 2.13. Coulson's Valence, 3rd Ed. by R. McWeeny, Oxford University Press, Oxford, 1979. Reprinted by permission of Oxford University Press. Figure 2.14. R. F. W. Bader, 1. Keaveney, and P. E. Cade, J. Chem. Phys. 47, 3381 (1967). Reprinted by permission of the American Institute of Physics and the authors. Table 2.2. Wave Mechanics for Chemists by C. W. N. Cumper, Academic Press Inc., 1966. Reprinted by permission of the current copyright owner Butterworth Heinemann Ltd. and the author. Table 2.5. The Structure of Small Molecules by W. J. Orville-Thomas, Elsevier Science Publishers, 1966. Reprinted by permission of the author and the current copyright owner.

Chapter 3 Figures 3.1,3.9,3.10,3.15,3.17, and Table 3.4. The Forces Between Molecules by M. Rigby, E. B. Smith, W. A. Wakeham, and G. C. Maitland, Oxford Univer sity Press, Oxford, 1986. Reprinted by permission of Oxford University Press. Figure 3.2. Introduction to Physical Chemistry by Arthur M. Lesk, (c) 1982. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Figure 3.8. M. Ross, H. K. Mao, P. M. Bell, and J. A. Xu, J. Chem. Phys. 85, 1028 (1986). Reprinted by permission of the American Institute of Physics and the authors. Figure 3.11. Bonding Theory by D. J. Royer, McGraw-Hill, New York, 1986. Copyright (c) 1968 by McGraw-Hill Inc. Reprinted by permission by McGraw Hill, Inc. Figure 3.12. Advanced Physical Chemistry by J. C. Davis, Ronald Press Co., N.Y., 1965. Reprinted by permission of John Wiley and Sons, Inc., the current copyright owners. Figure 3.13. Kinetic Theory of Gases by W. Kauzmann, W. A. Benjamin Inc., New York, 1966. Reprinted by permission of Benjamin/Cummings Publishing Co. Figures 3.14 and 3.16. Physical Chemistry, 2nd Ed. by P. W. Atkins. Copyright (c) 1982 by P. W. Atkins. Reprinted by permission of W. H. Freeman and Company, New York. Figure 3.18. P. A. Egelstaffand F. Barocchi, Phys. Lett. 3, 313 (1985). Reprinted by permission of Elsevier Science Publishers and the authors. Table 3.5. The Chemical Bond, 2nd Ed. by J. N. Murrell, S. F. A. Kettle, and J. M. Tedder. Copyright (c) 1985 by John Wiley and Sons Ltd. Reprinted by permission of John Wiley and Sons Ltd. Chapter 4 IX Figure 4.2. N. L. Allan, D. L. Cooper, C. G. West, P. J. Grout, and N. H. Acknowledgments March, J. Chem. Phys. 83, 239 (1983). Reprinted by permission of the American Institute of Physics and the authors. Figures 4.3 and 4.4. J. F. Mucci and N. H. March, J. Chem. Phys. 71, 5270 (1979). Reprinted by permission of the American Institute of Physics.

Chapter 5 Figure 5.1. Fundamentals of Molecular Spectroscopy by C. M. Banwell, McGraw-Hill Book Co. Reprinted by permission of McGraw-Hill, Inc. Figure 5.5. Modern Spectroscopy by J. M. Hollas, 1987. Copyright (c) by John Wiley and Sons Ltd., Chichester. By permission of the copyright owner. Figures 5.6, 5.7, and 5.10. High-Resolution Spectroscopy by J. M. Hollas, Butter worths Publishers (1982). Reprinted by permission of Butterworth-Heinemann Publishers.

Figure 5.11. E. A. Ballik and D. A. Ramsay, Astrophys. J. 137, 84 (1963). Reprinted by permission of the publisher and the authors. Figure 5.13. L. Karlsson, K. Mattsson, R. Jadrny, T. Bergmark, and K. Siegbahn, Physica Scripta 14, 230 (1976). Reprinted by permission of the Royal Swedish Academy of Sciences. Figure 5.14. Molecular Structure and Dynamics by W. H. Flygare, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1978. Reprinted by permission of Ruth M. Flygare the current copyright owner. Figure 5.17. Spectroscopy and Structure by R. N. Dixon, John Wiley and Sons Inc., 1965. Reprinted by permission of Chapman and Hall, London the current copyright owner. Figure 5.18. A. A. Bothner-By and C. Naar-Co1in, Ann. N. Y. Acad. Sci. 70, 833 (1958). Reprinted by permission of the New York Academy of Science and the authors. Figure 5.22. Quantum Chemistry by J. P. Lowe, copyright (c) Academic Press Inc., New York, 1978. Reprinted by permission of Academic Press and the author.

Chapter 6 Figure 6.3. Molecular Photoelectron Spectroscopy by D. W. Turner, C. Baker, A. D. Baker, and C. R. Brund1e, Copyright (c) 1970 by John Wiley and Sons Ltd. Reprinted by permission of John Wiley and Sons Ltd. Figures 6.4, 6.15, and 6.30. High Resolution Spectroscopy by J. M. Hollas, Butterworths Publishers, 1982. Reprinted by permission of Butterworth Heinemann Publishers. Figure 6.10. Valence, 2nd Ed, by C. A. Coulson, Oxford University Press, Oxford, 1961. Reprinted by permission of Oxford University Press. X Figure 6.27. L. Karlsson, L. Mattson, R. Jadrny, T. Bergmark, and K. Siegbahn, Acknowledgments Physica Scripta, 230 (1976). Reprinted by permission of the Royal Swedish Academy of Sciences. Figure 6.29. M. A Coplan, J. H. Moore, and J. A Tossell, J. Chem. Phys. 68, 329 (1978). Reprinted by permission of the American Institute of Physics and the authors.

Chapter 7 Figures 7.5, 7.6, 7.7, 7.8, 7.9, and Tables 7.2, 7.3, and 7.4. Physical Organic Chemistry by K. B. Wiberg. John Wiley and Sons Inc., New York 1964. Reprinted by permission of the author the current copyright owner. Figures 7.10, 7.11, 7.12, 7.13, 7.14, 7.15, 7.16, and 7.17, 7.19, 7.20. Quantum Chemistry by J. P. Lowe, Academic Press 1978. Reprinted by permission of the publisher and author. Figure 7.18. Molecular Orbital Theory by A Streitwieser Jr., John Wiley and Sons Inc., N.Y., 1961. Reprinted by permission of the author the cum,nt copyright owner. Figure 7.21. R. N. Zare and R. B. Bernstein, Physics Today, Vol. 33, No. 11 (1980). Reprinted by permission of the American Institute of Physics and the authors. Figure 7.22 and Table 7.6. Chemical Kinetics 3rd Ed. by Keith J. Laidler. Copy right (c) 1987 by Harper and Row Publishers, Inc., N.Y. Reprinted by permission of Harper Collins Publishers. Table 7.5. Quanta, 2nd Ed. by P. W. Atkins, Oxford University Press, Oxford, 1991. Reprinted by permission of Oxford University Press. . Appendix Figure A.5. C. A Coulson and I. Fischer, Philosophical Magazine (London) 40, 386 (1949). Reproduced with permission of the publisher, Taylor and Francis, Inc. Figure A9. L. S. Bartell and L. O. Brockway, Phys. Rev. 90, 833 (1953). Reprinted by permission of the American Institute of Physics and the authors. Figures A.ll, A.12, and A. 13. J. A. Alonso and N. H. March, Chem. Phys. 76, 121 (1983). Reprinted by permission of Elsevier Science Publishers BV and the authors. Figures A.14. K. E. Banyard and N. H. March, Acta. Cryst. 9, 385 (1956). Reprinted by permission of International Union of Crystallography. Figures A.20, A2l, and A22 and Table A. 1. Quantum Chemistry by J. P. Lowe. Copyright (c) Academic Press, Inc.,·N.Y. 1978. Reprinted by permission of the publishers and the author. Figures A.23 and A.24 and Tables A.3 through A.8. M. Raimondi, M. Simonetta, and J. Gerratt, Chem. Phys. Letts. 77, 12 (1981). Reprinted by permis sion of Elsevier Science Publishers BV and the authors. While the above makes abundantly clear our indebtedness to many authors, we Xl want to acknowledge the extensive use we have made of the fine books High Acknowledgments Resolution Spectroscopy (1982) by J. M. Hollas, Quantum Chemistry (1978) by J. P. Lowe, and Physical Organic Chemistry (1964) by K. B. Wiberg. Finally, one of us (NHM) chaired sets of splendid pedagogical lectures by Professor N. C. Handy at consecutive Coulson Summer Schools in Theoretical Chemistry in Oxford. Three of the Advanced General Appendixes have their origins directly in these lectures; however, it must be emphasized that the sole responsibility for the final presentation of this and, of course, all other material rests with the present authors. CONTENTS

Chapter One CHEMICAL CONCEPTS AND EXPERIMENTAL TECHNIQUES

1.1. Chemical Bonds: Types and Strengths ...... 1 1.2. Single, Double, and Triple Bond Formation via Octet Rule...... 1 1.2.1. Homonuclear Examples ...... '... 2 1.2.2. Heteronuclear Examples ...... 3 1.3. Bond Dissociation ...... 4 104. Ionization Potential and Electron Affinity ...... 5 1.5. The Concept of Electronegativity ...... 9 1.6. Dipole Moments and Electronegativity ...... 13 1.7. Bond Type and Length: Associated Force Constants ...... 13 1.8. Some Important Experimental Methods ...... 15 1.8.1. Background ...... 15 1.8.2. Mainly Thermodynamics: The Born-Haber Cycle...... 15 1.8.3. Techniques for Probing Electronic Structure ...... 18 1.804. Molecular Spectroscopy and Molecular Properties ...... 22 1.8.5. Coherent Radiation: Laser Studies ...... 22 Problems...... 34 References ...... 35

Chapter Two THE NATURE OF BONDING IN DIATOMS

2.1. Valence Bond and Molecular Orbital Methods ...... 37 2.2. Nature of Wave Functions for H~ and H2 ...... 37 2.3. The LCAO Approximation to Ground States of H~ and H2 ...... 39 2.3.1. LCAO Treatment of the H2 Molecule ...... 41 2.3.2. The Molecular Energy of the H2 Molecule in Its Ground State...... 42

xiii XlV 2.3.3. The United Atom: He ...... 44 Contents 2.3.4. Coulomb and Multicenter Integrals in H2 ...... 45 2.4. Valence Bond Theory of the H2 Molecule ...... 46 2.5. Comparison of Valence Bond and Molecular Orbital Theories for H2 ...... 49 2.6. Chemical Bonding, Molecular Orbital Energy Levels, and Correlation Diagrams for Homonuc1ear Diatoms ...... 52 2.6.1. Symmetry Classification ...... 52 2.6.2. Correlation Diagrams ...... 54 2.6.3. Electron Configurations ...... 57 2.7. Heteronuc1ear Diatomic Molecules ...... 60 Problems ...... 66 References ...... 67

Chapter Three MOLECULAR INTERACTIONS

3.1. Introduction ...... 69 3.2. Long-Range Forces...... 69 3.2.1. The London Dispersion Force ...... 70 3.2.2. The Drude Model...... 71 3.2.3. Introduction of Molecular Polarizability into the Dispersion Interaction ...... 73 3.2.4. Van der Waals Equation of State Related to the London Dispersion Force ...... 74 3.3. Short-Range Forces ...... 76 3.3.1. The Hard-Sphere Model...... 76 3.3.2. Point Centers of Inverse Power Law Repulsion ...... 77 3.3.3. The Square-Well Model ...... 77 3.3.4. The Lennard-Jones Potential ...... 78 3.3.5. The Exponential -6 Potential ...... 79 3.4. Short-Range Energies and Quantum-Chemical Studies...... 80 3.5. The Hydrogen Bond ...... 82 3.6. The Relation to Experimental Studies ...... 85 3.6.1. The Virial Equation: Imperfect Gases ...... 85 3.6.2. Transport Properties of Gases ...... 88 3.6.3. Pair Potentials and Transport Coefficients ...... 90 3.6.4. Molecular Beam Studies ...... 93 3.6.5. Spectra of Molecular Pairs ...... 95 3.6.6. The Inversion of Measured Fluid Structure to Extract Pair Potentials ...... 96 Problems...... 98 References ...... 98 Chapter Four XV ELECTRON DENSITY DESCRIPTION OF MOLECULES Contents

4.1. Approximate Electron Density~Potential Relation...... 101 4.2. Energy Relations in Molecules at Equilibrium ...... 103 4.3. Can the Total Energy of a Molecule Be Related to the Sum of Orbital Energies? ...... 106 4.4. Foundations of Walsh's Rules for Molecular Shape ...... 108 4.5. Self-Consistent Field Treatment for Diatomic Molecules and the Roothaan~Hall Formulation ...... 110 4.6. Roothaan's Approach Compared with the Electron Density Description...... 112 4.6.1. Kinetic Energy ...... 112 4.6.2. Test of Energy Scaling Relations ...... 114 4.7. Chemical Potential in Relation to Electronegativity ...... 115 Problems ...... 119 References ...... 121

Chapter Five MOLECULAR PARAMETERS DETERMINED BY SPECTROSCOPIC METHODS

5.1. Rotational Spectroscopy...... 123 5.1.l. Principal Moments ofInertia of Molecules ...... 125 5.1.2. The Symmetric Rotor or Symmetric Top...... 127 5.1.3. Spherical Rotors ...... 127 5.1.4. Rotational Spectra of Diatomic and Linear Polyatornic Molecules ...... 128 5.l.5. Rotational Selection Rules...... 128 5.2. Centrifugal Distortion...... 132 5.3. Vibrational Spectroscopy of Diatomic Molecules: Harmonic Approximation ...... 133 5.4. Anharmonicity ...... 134 5.5. Vibrational Selection Rules...... 135 5.6. Vibration~Rotation Spectrum...... 136 5.7. Electronic Spectroscopy ...... 138 5.7.l. Rydberg Orbitals ...... 138 5.7.2. Classification of Electronic States ...... 138 5.7.3. Electric Dipole Selection Rules for Transitions with I1S = 0 .. 140 5.7.4. Transitions with I1S # 0 ...... 141 5.8. Excited-State Potential Energy Curves ...... 142 5.8.l. Progressions and Sequences ...... 144 5.8.2. Vibronic Transitions ...... 144 XVI 5.9. Determination of Ionization Potentials of Polyatomic Molecules. . .. 145 Contents 5.10. Nuclear Magnetic Resonance Spectra ...... 147 5.10.1. Chemical Shifts ...... 149 5.10.2. Chemical Applications ...... 150 5.10.3. Interaction Constants: Spin-Spin Coupling ...... 151 5.10.4. Comments on Analyses of NMR Spectra ...... 151 5.10.5. Rules for Interpreting First-Order Spectra ...... 152 5.10.6. The Relation between Chemical Shift and Electronegativity 153 5.11. Electron Spin Resonance and 1r-Electron Densities: Hyperfine Structure ...... 154 Problems ...... 158 References ...... 159

Chapter Six MOLECULAR ORBITAL METHODS AND POLYATOMIC MOLECULES

6.1. Directed Bonds: Conformation of H20 and NH3 ...... 161 6.2. Bent Molecules: Walsh Diagrams and Rules ...... 162 6.2.1. HeI UPS Spectrum of the H20 Molecule...... 163 6.2.2. HeI UPS Spectrum of H2S ...... 165 6.3. UPS of AH3 Molecules ...... 166 6.4. The Need for Hybrid Orbitals ...... 168 6.5. Some Carbon Compounds with spn Hybridization (n = 1, 2, 3) .... 173 6.6. Molecular Orbital Theory of 1r-Electron Systems ...... 175 6.6.1. The Hiickel Molecular Orbital Method ...... 177 6.6.2. An Example of 1r-Level Spectra: Ethylene ...... 180 6.6.3. The HeI UPS Spectrum of Ethylene...... 181 6.6.4. Other Examples of 1r-Level Spectra ...... 183 6.6.5. The UPS Spectrum of Benzene ...... 192 6.7. Determination of Molecular Orbitals and LCAO Coefficients ...... 195 6.8. Molecular Indexes ...... 197 6.8.1. Electron Density on Atoms ...... 198 6.8.2. Charge Density on Atoms ...... 199 6.8.3. Bond Order ...... 201 6.9. Some Quantum-Mechanical and Semiempirical Methods Applied to Polyatomic Molecules ...... 202 6.9.1. The Pariser-Parr-Pople Method ...... 205 6.9.2. The Ab Initio Method...... 207 6.10. Molecular Orbitals in Acetylene and the (e,2e) Experiment...... 207 6.10.1. The Technique and Theory of (e, 2e) Studies ...... " . .. 207 6.10.2. Results of Measurements on Acetylene ...... 209 6.11. Slater-Kohn-Sham One-Electron Equations ...... '...... 211 6.12. Renner-Teller Effect ...... 211 Problems...... 213 References ...... 216 Chapter Seven XVll CHEMICAL REACTIONS, DYNAMICS, Contents AND LASER SPECTROSCOPY

7.1. Introduction and Background ...... 219 7.1.1. Rates of Chemical Reactions: Reaction Rates and Rate Laws 219 7.1.2. Largely Qualitative Considerations ...... 224 7.1.3. The Transition State: Reaction Coordinate and Potential Energy Diagrams...... 224 7.2. Rates of Chemical Reactions: Absolute Rate Theory ...... 227 7.2.1. The Methane-Chlorine-Atom Reaction ...... 228 7.2.2. A Reaction Coordinate Representation ...... 232 7.2.3. Determination of the Equilibrium Constant ...... 233 7.3. The Woodward-Hoffmann Rules ...... 236 7.3.1. Introduction...... 236 7.3.2. Cycloaddition Reactions...... 241 7.3.3. Other Types of Chemical Reactions...... 242 7.4. Localization Energy and the Rate of Reaction for Aromatic Hydrocarbons ...... 246 7.5. Oxidation-Reduction and Orbital Energies of Hydrocarbons ...... 247 7.6. Laser Spectroscopy and Chemical Dynamics ...... 248 7.6.1. Rate Coefficient Analysis ...... 248 7.6.2. State-to-State Chemical Reaction Dynamics: Hydrogen Exchange Reaction ...... 251 7.7. Electron Density Theory and Chemical Reactivity...... 252 7.7.1. Hardness and Softness ...... 252 7.7.2. Acids and Bases ...... 253 7.8. Other Studies in Chemical Kinetics ...... :.. 254 7.8.1. Electron Impact Studies...... 254 7.8.2. Photons and Chemical Reactions...... 255 7.8.3. Rate Studies for Reactions of the Hydrated Electron 256 7.8.4. Topics for Further Study Involving Chemical Reactivity. . .. 257 Problems ...... 257 References ...... 259

APPENDIX

AI. 1. Wave Functions for the Hydrogen Atom ...... 261 A1.2. The Periodic Table and Atomic Ground States...... 271 A1.3. Total Energies of Heavy Atomic Ions-Coulomb Field Model .. 273 Al.4. Orthogonality of Solutions of the Schrodinger Equation...... 275 A2.1. Variation Principle ...... 276 A2.2. Integrals Involved in LCAO Treatment of the H; Ground State 277 XVlll A2.3. Born-Oppenheimer Approximation...... 278 Contents A2.4. Slater- and Gaussian-Type Orbitals...... 280 A2.5. The Coulson-Fischer Wave Function for the Ground State of the H2 Molecule...... 281 A2.6. Virial and Hellmann-Feynman Theorems ...... 282 A3.I. Topics Relevant to the Treatment of Intermolecular Forces .... 289 A3.2. Bibliography for Further Study of Intermolecular Forces ...... 290 A4.1. The Correspondence between Cells in Phase Space and Quantum-Mechanical Energy Levels ...... 290 A4.2. The Kinetic Energy Density of an Inhomogeneous Electron Gas 294 A4.3. The Chemical Potential, Teller's Theorem, and Scaling of Energies of Homonuclear Diatoms ...... 295 A4.4. The Self-Consistent Field in the Helium Atom...... 296 A4.5. The Self-Consistent Field Treatment of Binding Energies of Heavy Positive Atomic Ions ...... 297 A4.6. The Hartree-Fock Self-Consistent Field Method...... 299 A4.7. The Dirac-Slater Exchange Energy and Existence of a One-Body Potential Including Both Exchange and Correlation.. 308 A4.8. Proof that the Ground-State Energy of a Molecule Is Uniquely Determined by the Electron Density ...... 312 A4.9. Modeling of the Chemical Potential in Hydrogen Halides and Mixed Halides ...... 314 A4.10. X-Ray Scattering by Neon-Like Molecules ...... 318 A4.1I. Two-Center Calculations from the Thomas-Fermi Theory...... 322 A5.I. Rotational Energy Levels of Some Simple Classes of Molecules: The Symmetric Rotor ...... 326 A5.2. The Rotational Partition Function in Relation to Spectroscopic Intensities ...... 328 A5.3. Normal Modes of Vibration of Molecules ...... 330 A5.4. The Franck-Condon Principle ...... 335 A5.5. Time-Dependent Perturbation Theory and Selection Rules for Electric Dipole Transitions in Atoms ...... 338 A5.6. Spin-Orbit Coupling ...... 342 A5.7. Dirac's Relativistic Wave Equation for One Electron ...... 345 A5.8. Relativistic Electron Density Theory for Molecules Composed of Heavy Atoms ...... 349 A6.1. The Jahn-Teller Effect ...... 350 A6.2. Koopmans' Theorem and Its Use in Interpreting Photoelectron Spectra ...... 351 A7.I. Symmetry Arguments for Electrocyclic Reactions ...... 353 A7.2. Chemical Reactions: Arrhenius' Empirical Work, the Collision Theory, and the Absolute Rate or Transition State Theory ...... 356 Advanced Problems ...... 360 AI. Some Advanced Aspects of Quantum-Mechanical Perturbation Theory ...... 362 All. The Formation of Acetylene from Two CH Fragments ...... 367 AIII. MacDonald's Theorem ...... 374 AIVo Spectroscopic Nomenclature 0 0 0 00000000000000000000000000000 375 XIX AVo The Wentze1-Kramers-Brillouin Semiclassical Method for Contents

Calculating Eigenvalues for Central Fields 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 377 AVI. Electron Correlation in the Helium Atom and the Slow

Convergence of Configuration Interaction 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 379

Further Problems 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 380

References 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 386

INDEX 389