Bonding through Code: Theoretical Models for Molecules and Materials

This timely and unique publication is designed for graduate students and researchers in inorganic and materials chemistry and covers bonding models and applications of symmetry concepts to chemical systems. The book discusses the quantum mechanical basis for molecular orbital concepts, the connections between molecular orbitals and localized views of bonding, group theory, bonding models for a variety of compounds, and the extension of these ideas to solid state materials in band theory. Unlike other books, the concepts are made tangible to the readers by guiding them through their implementation in MATLAB functions. No background in MATLAB or computer programming is needed; the book will provide the necessary skills. Key Features Visualization of the Postulates of Quantum Mechanics to build conceptual understanding MATLAB functions for rendering molecular geometries and orbitals Do-it-yourself approach to building a molecular orbital and band theory program Introduction to Group Theory harnessing the 3D graphing capabilities of MATLAB Online access to a growing collection of applications of the core material and other appendices Bonding through Code is ideal for first-year graduate students and advanced undergraduates in chemistry, materials science, and physics. Researchers wishing to gain new tools for theoretical analysis or deepen their understanding of bonding phenomena can also benefit from this text. About the Author 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.

Table of Contents:
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