AN INTRODUCTION TO MOLECULAR ORBITALS

by YVES JEAN and FRANCOIS VOLATRON

translated and edited by Jeremy Burdett

New York Oxford OXFORD UNIVERSITY PRESS

1993

Contents

Introduction, xiii

I INTRODUCTION TO ATOMIC AND MOLECULAR STRUCTURE

1. From the periodic table to molecules, 3

1.1. The periodic Classification, 3 1.1.1. Short description of the first three periods, 4 1.1.2. Core and valence electrons, 4 1.1.3. Systems with eight valence electrons, 5 1.1.4. The electronegativity concept, 5

1.2. Lewis' theory and Lewis structures, 6 1.2.1. Bond pairs and lone pairs, 6 1.2.2. The octet rule, 8 1.2.3. Formal charges, 10 1.2.4. Classification of reactants, 11 1.2.5. Dipole moments of diatomic molecules, 13

1.3. Resonance or mesomerism, 14 1.3.1. Exampies of mesomeric structures: carbonate ion (C0 3

2~) and benzene (C6H6), 15

1.3.2. Selection of mesomeric or resonance structures, 17 1.3.3. Application to the structure of aniline, 17

1.4. Molecular geometry, 18 1.4.1. Spatial representation of molecules: Cram's model, 18 1.4.2. VSEPR theory, 19 1.4.3. Extensions of VSEPR theory, 23 1.4.4. Dipole moments of polyatomic molecules, 24

1.5. Conclusion, 25 Exercises, 27

2. Properties of atoms, 29

2.1. Elements of quantum mechanics, 29 2.1.1. Schrödinger's equation, 29 2.1.2. Some important properties of the eigenfunctions, 31

2.2. The hydrogen atom, 32 2.2.1. Solutions of the Schrödinger equation, 32 2.2.2. Description of the eigenfunctions, 37

Vlll CONTENTS

2.2.3. Electron spin, 44 2.2.4. Hydrogen-like atoms, 44

2.3. Many-electron atoms, 45 2.3.1. The orbital approximation, 45 2.3.2. Mathematical description and nomenclature of atomic orbitals, 46 2.3.3. Atomic orbital energies, 46 2.3.4. The electronic configuration of atoms, 47 2.3.5. Hund's rule, 48 2.3.6. Core and valence electrons, 50

2.4. The periodic Classification of the elements, 51 2.4.1. Organization by rows, 51 2.4.2. Organization by column: chemical families, 53

2.5. Electronic parameters of many-electron atoms, 54 2.5.1. Screening, 54 2.5.2. The effective charge: Slater's rules, 55 2.5.3. Orbital radii and atomic size, 56

2.6. Evolution of atomic properties, 57 2.6.1. Atomic orbital parameters, 57 2.6.2. Relationship with measurable properties, 60 2.6.3. Electronegativity scales, 66 2.6.4. Electronegativity, orbital energy and orbital radius, 68

Exercises, 69

II BUILDING UP MOLECULAR ORBITALS AND ELECTRONIC STRUCTURE

3. Interaction of two atomic orbitals on different centers, 73

3.1. Basic approximations, 73 3.1.1. The Born-Oppenheimer approximation, 73 3.1.2. The orbital approximation, 74 3.1.3. The form of the MOs: the LCAO approximation, 74

3.2. Construction of MOs, 75 3.2.1. Interaction of two identical AOs, 76 3.2.2. Interaction of two different AOs, 81 3.2.3. Orbitals with zero overlap, 83

3.3. Application to some simple diatomic molecules, 83 3.3.1. Level Alling rules, 83 3.3.2. Systems with two or four electrons, 83 3.3.3. Total energy of the molecule: the Morse curve for H2, 85 3.3.4. Systems with one or three electrons, 85

3.4. Overlap and symmetry, 87 3.4.1. ls/ls overlap, 87 3.4.2. Overlap between 'parallel' 2p orbitals (rc-type overlap), 88 3.4.3. ls/2p overlap, 88 3.4.4. Symmetry ideas, 90

CONTENTS IX

3.5. Application of symmetry ideas to some polyatomic molecules, 93 3.5.1. a/n Separation, 93 3.5.2. The n MOs of ethylene, 94 3.5.3. n System of formaldehyde, 95 3.5.4. A comparison between ethylene and formaldehyde, 96 3.5.5. The n orbitals of acetylene, 96

3.6. Conclusions, 97 Exercises, 99

4. The fragment orbital method; application to some model Systems, 101

4.1. Molecular orbitals of some model Systems, H„, 102 4.1.1. Square planar H4, 102 4.1.2. Rectangular H4, 107 4.1.3. Linear H3, 109 4.1.4. Linear H4, 110 4.1.5. Triangulär H3, 112 4.1.6. Tetrahedral H4, 115 4.1.7. HexagonalH6, 118

4.2. Influence of electronegativity on the form and energy of the molecular orbitals, 120

Exercises, 123 Appendix: Degenerate orbitals, 126

5. Interactions between two fragment orbitals: linear AH2, trigonal AH3

and tetrahedral AH4, 128

5.1. Linear AH2 molecules, 129 5.1.1. Symmetry properties of the fragment orbitals, 129 5.1.2. MOs for linear AH2 molecules, 131 5.1.3. Application to BeH2, 133

5.2. Trigonal planar molecules, 133 5.2.1. Symmetry properties of the fragment orbitals, 133 5.2.2. Molecular orbitals of trigonal planar AH3, 137 5.2.3. Application to the electronic structure of BH3, 138

5.3. Tetrahedral AH4 molecules, 139 5.3.1. Symmetry properties of the fragment orbitals, 139 5.3.2. MOs of tetrahedral AH4 molecules, 142 5.3.3. Application to the electronic structure of CH4, 143

Exercises, 145 Appendix: Analogous orbitals, 148

6. Interactions between three fragment orbitals: AH, bent AH2 and pyramidal AH3, 150

6.1. Rules for the interaction of three orbitals, 150 6.1.1. Outline of the problem, 150 6.1.2. Rules for the construction of the MOs, 150

X CONTENTS

6.2. Electronic structure of AH molecules, 152 6.2.1. Outline of the problem, 152 6.2.2. Form of the MOs, 153 6.2.3. Electronic structure of LiH, 155 6.2.4. Electronic structure of BH, 156 6.2.5. Electronic structure of FH, 158 6.2.6. Conclusions for AH molecules, 161

6.3. Bent AH2 molecules, 162 6.3.1. Symmetry of the fragment orbitals, 163 6.3.2. Interaction diagram and form of the MOs: H 2 0 as an example, 164 6.3.3. Electronic structure of H 2 0 , 166

6.4. Pyramidal AH3 molecules, 167 6.4.1. Symmetry of the fragment orbitals, 168 6.4.2. Interaction diagram and form ofthe MOs: the example ofNH3, 169 6.4.3. Electronic structure of NH3 , 171

Exercises, 173

7. Interactions between four fragment orbitals: the diatomic molecules A2 and AB, 176

7.1. Homonuclear diatomics, A2, 177 7.1.1. Outline of the problem, 177 7.1.2. Construction of the rc-type MOs, 178 7.1.3. Construction of the cx-type MOs, 179 7.1.4. MO diagrams for A2 molecules (A = L i , . . . , Ne), 183 7.1.5. Electronic structure of the A2 molecules (A = L i , . . . , Ne), 185 7.1.6. Bond lengths and bond energies, 189

7.2. Heteronuclear diatomic molecules, AB, 190 7.2.1. Construction of the n MOs, 190 7.2.2. Construction of the <x MOs, 191 7.2.3. MOs and electronic structure of CO, 192

Exercises, 195 Appendix: The number of bonds (bond order) in diatomic molecules, 196

8. Large molecules, 197

8.1. MOs of acetylene, ethylene and ethane, 197 8.1.1. MOs and electronic structure of acetylene, 198 8.1.2. MOs and electronic structure of ethylene, 200 8.1.3. MOs and electronic structure of ethane, 204

8.2. Conjugated polyenes, 207 8.2.1. MOs and electronic structure of allyl, 208 8.2.2. MOs and electronic structure of butadiene, 209 8.2.3. MOs and electronic structure of cyclopropenyl, 211 8.2.4. MOs and electronic structure of cyclobutadiene, 211 8.2.5. MOs and electronic structure of benzene, 212

CONTENTS XI

8.2.6. Aromatic and antiaromatic Compounds and Hückel's rule, 213 Exercises, 215 Appendix: Bond localization, 218

III INTRODUCTION TO THE STUDY OF THE GEOMETRY AND REACTIVITY OF MOLECULES

9. Orbital correlation diagrams: the model Systems H 3+ and H3~, 223

9.1. Rules for drawing orbital correlation diagrams, 224 9.1.1. Stabilization or destabilization of the MOs, 224 9.1.2. Conservation of orbital symmetry, 225 9.1.3. The non-crossing rule for MOs of the same symmetry, 225

9.2. Orbital correlation diagram for bending H3, 226 9.2.1. Geometrical model, 226 9.2.2. Symmetry of the MOs, 227 9.2.3. Energetic evolution of the MOs, 227

9.3. Geometry o f H 3+ , 228

9.4. Geometry of H3~ and the rule of the highest occupied MO, 230 9.4.1. Rule of the highest occupied MO (HOMO), 230 9.4.2. Geometry ofH3~, 230 9.4.3. The Jahn-Teller effect, 232

9.5. Conclusion, 232 Exercises, 234

10. Geometry of AH2 and AH3 molecules, 236

10.1. AH2 molecules, 236 10.1.1. MOs of linear AH2, 237 10.1.2. Orbital correlation diagram, linear to bent AH2, 237 10.1.3. Geometry of AH2 molecules, 242

10.2. AH 3 molecules, 243 10.2.1. MOs of trigonal planar AH3, 243 10.2.2. Orbital correlation diagram for trigonal planar to pyramidal

AH3, 244 10.2.3. Planar or pyramidal geometries?, 248

10.3. Extension to more complex molecules, 249 10.3.1. The geometries of AX2 and AX3 molecules, 249 Exercises, 251

11. Molecular geometry using fragment molecular Orbitals, 253

11.1. Two- and four-electron interactions, 253 11.1.1. Energetic consequences, 254 11.1.2. Electron transfer, 255

11.2. Model examples of H 3+ and H3~, 255

11.2.1. Geometry o fH 3+ , 256

Xll CONTENTS

11.2.2. Geometry ofH 3~, 257 11.3. Hyperconjugation, 258

11.3.1. Conformation of the C 2 H 42 + dication, 258

11.3.2. The ethyl cation CH 3CH 2+ , 261

11.4. The s-cis and s-trans conformations of butadiene, 265 Exercises, 268

12. An introduction to the study of chemical reactivity, 271

12.1. Description of a chemical reaction, 271 12.1.1. The reaction scheme and elementary processes, 271 12.1.2. Reaction mechanism, 272 12.1.3. Reaction coordinate, 272 12.1.4. Energy profiles, transition states and reaction intermediates, 273

12.2. The frontier orbital approximation, 274 12.2.1. The method, 275 12.2.2. Electrophilic and nucleophilic reactants, 276 12.2.3. The validity of the approximation, 277

12.3. Cycloaddition reactions, 279 12.3.1. The [4s + 2s] thermal cycloaddition: the Diels-Alder

reaction, 279 12.3.2. Thermal [2s + 2s] cycloaddition: the dimerization of ethylene, 281 12.3.3. Generalization to [ms + ns] cycloadditions, 282

12.4. Further aspects of the [2 + 2] cycloaddition, 282 12.4.1. Concerted mechanisms, 282 12.4.2. Non-concerted mechanisms, 284

12.5. Examples of ionic reactions, 286 12.5.1. The SN2 mechanism, 286 12.5.2. Markovnikov's rule, 288

Exercises, 292

IV PROBLEMS

Problem 1. Stabilization of a planar tetravalent carbon atom via n effects, 297 Problem 2. Nucleophilic attack on a carbonyl group, 300 Problem 3. Structure and reactivity of substituted cyclopropanes, 304 Problem 4. Conformational consequences of hyperconjugation, 307

Bibliography, 311

Answers to Exercises, 313

Index, 333