Bonding through Code

Daniel Fredrickson is a Professor in the Department of Chemistry at the University of Wisconsin–Madison, where his research group focuses on understanding and harnessing the structural chemistry of intermetallic phases using a combination of theory and experiment. His interests in crystals, structure and bonding can be traced to his undergraduate research at the University of Washington (B.S. in Biochemistry, 2000) with Prof. Bart Kahr, his Ph.D. studies at Cornell University (2000–2005) with Profs. Stephen Lee and Roald Hoffmann, and his post-doctoral work with Prof. Sven Lidin at Stockholm University (2005–2008). As part of his teaching at UW–Madison since 2009, he has worked to enhance his department’s graduate course Physical Inorganic Chemistry I: Symmetry and Bonding, through the incorporation of new material and the development of computer-based exercises.

Contents

Acknowledgments, xi

About the Author, xiii

Chapter 1 ◾ The Postulates of Quantum Mechanics 1

Chapter 2 ◾ Atoms and Atomic Orbitals 23

INTRODUCTION 23

THE RADIAL WAVEFUNCTION 24

VISUALIZING ATOMIC ORBITALS WITH MATLAB: THE

ANGULAR WAVEFUNCTIONS 28

COMBINING THE RADIAL AND ANGULAR FUNCTIONS 35

FOCUSING ON THE VALENCE ELECTRONS: SLATER-TYPE

ORBITALS 38

Chapter 3 ◾ Overlap between Atomic Orbitals 41

INTRODUCTION 41

PARAMETERS FOR SLATER-TYPE ORBITALS 41

COMBINING THE RADIAL AND ANGULAR FUNCTIONS 42

VISUALIZING ISOSURFACES OF SLATER-TYPE ORBITALS 44

PROGRAMMING OVERLAP INTEGRALS IN MATLAB 47

EXERCISES FOR EXPLORING OVERLAP INTEGRALS 49

REFERENCES 53

Chapter 4 ◾ Introduction to Molecular Orbital Theory 55

INTRODUCTION 55

CONSTRUCTION OF THE HAMILTONIAN MATRIX 58

SOLVING FOR THE MOLECULAR ORBITALS 61

VISUALIZING ISOSURFACES OF MOS IN MATLAB 63

EXTENDED HÜCKEL VS. SIMPLE HÜCKEL 69

A SIMPLIFIED REPRESENTATION OF MOs IN MATLAB 72

REFERENCES 76

Chapter 5 ◾ The Molecular Orbitals of N2 77

INTRODUCTION 77

SOLVING THE GENERAL PROBLEM OF BUILDING THE

HAMILTONIAN 77

THE BRUTE FORCE SOLUTION OF THE MOs OF N2 84

SYMMETRIZED BASIS FUNCTIONS 85

Chapter 6 ◾ Heteronuclear Diatomic Molecules 93

INTRODUCTION 93

DRAWING MOLECULAR STRUCTURES 93

HeH: ELECTRONEGATIVITY PERTURBATION 97

HeH: INTERATOMIC INTERACTIONS AS A PERTURBATION 103

THE MOs OF CO AND CN− 106

Chapter 7 ◾ Symmetry Operations 109

INTRODUCTION 109

APPLYING SYMMETRY OPERATIONS IN MATLAB 109

THE IDENTITY OPERATION 112

INVERSION THROUGH A CENTRAL POINT 113

REFLECTIONS THROUGH A PLANE 114

ROTATIONS ABOUT AN AXIS 115

IMPROPER ROTATIONS 117

CREATING MORE COMPLICATED OPERATIONS 118

Chapter 8 ◾ Symmetry Groups 123

INTRODUCTION 123

PROPERTIES OF MATHEMATICAL GROUPS 123

DEMONSTRATION OF MATHEMATICAL GROUPS WITH

MATLAB 124

GENERATING OPERATIONS 128

APPLYING GROUP OPERATIONS 132

BUILDING THE MOLECULAR SYMMETRY GROUPS 135

Chapter 9 ◾ Group Theory and Basis Sets 139

INTRODUCTION 139

sp3 HYBRID ORBITALS OF H2O AS A BASIS FOR

REPRESENTING POINT GROUP SYMMETRY 139

BASIS SETS AS REPRESENTATIONS OF POINT GROUP

SYMMETRY 143

CHARACTERS OF A MATRIX REPRESENTATION 146

REDUCIBLE AND IRREDUCIBLE REPRESENTATIONS 147

REDUCTION OF REDUCIBLE REPRESENTATIONS 148

TRANSFORMATION OF BASIS SET TO IRREDUCIBLE

REPRESENTATIONS 151

Chapter 10 ◾ The MOs of H2O 153

INTRODUCTION 153

THE MOs OF H2O BY BRUTE FORCE 155

THE MOs OF H2O FROM SP3 HYBRID SYMMETRY

ADAPTED LINEAR COMBINATIONS (SALCs) 157

PERCEIVING LOCALIZED BONDING IN H2O 165

BONUS CODE: BETTER BALL-AND-STICK MODELS 166

Chapter 11 ◾ MOs of the Trigonal Planar Geometry 171

INTRODUCTION 171

CONSTRUCTION OF NH3 GEOMETRIES 171

MOs AT SPECIFIC GEOMETRIES 173

SALCs FOR THE TRIGONAL PLANAR GEOMETRY 175

BUILDING THE MO DIAGRAM FROM THE SALCs 182

Chapter 12 ◾ Walsh Diagrams and Molecular Shapes 185

INTRODUCTION 185

GEOMETRIES OF THE AL3 MOLECULE 185

CONSTRUCTING WALSH DIAGRAMS 186

Chapter 13 ◾ Getting Started with Transition Metals 191

INTRODUCTION 191

NORMALIZATION OF DOUBLE-ZETA FUNCTIONS 192

INCLUSION OF D ORBITALS INTO MATLAB FUNCTIONS 193

THE MOs OF AN OCTAHEDRAL COMPLEX WITH

σ-LIGANDS; THE 18-ELECTRON RULE 200

Chapter 14 ◾ Translational Symmetry and Band Structures 205

INTRODUCTION 205

TRANSLATIONAL SYMMETRY AND BLOCH’S THEOREM 205

CONSTRUCTING SALCs 208

HAMILTONIAN MATRICES 209

A SIMPLE EXAMPLE: THE CHAIN OF H ATOMS 210

UNIQUE VALUES OF K: THE 1ST BRILLOUIN ZONE 212

BUILDING THE HAMILTONIAN MATRICES FOR PERIODIC

STRUCTURES 213

EXAMPLE: THE BAND STRUCTURE OF GRAPHENE 220

DETERMINING THE FERMI ENERGY FOR GRAPHENE 223

INDEX, 227