Bonding through Code

Theoretical Models for Molecules and Materials

Daniel C. Fredrickson

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9781498762212
• eBook ISBN: 9780429531873
• Topic: Physical Sciences
• Subtopic: Medical Theory, Practice & Reference

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.

CHAPTER 1

The Postulates of Quantum Mechanics

Chemical bonding is a quantum mechanical phenomenon. As such, we will need familiarity with the properties and behavior of quantum mechanical systems as we work our way from the electronic structure of atoms to increasingly complex molecules and solids. In this first chapter, we will review the fundamental assumptions underlying quantum mechanics (as are covered in more detail in textbooks such as I. N. Levine, Quantum Chemistry, Pearson), while simultaneously seeing how they can be realized in the language of MATLAB®, which we will be using to express and explore bonding concepts throughout this book. To illustrate these postulates, we will make use of a common model system: The particle in a box.

Postulate 1: The state of a system is given by its wavefunction. All that can be known about the system is obtainable from this function.

There are several ways in which this wavefunction can be expressed. When writing equations down on paper, we might express it as a function of the coordinates of the particles in a system: ψ(x,y,z) for a single electron, ψ(x,y,z,σ) for an electron with a specific spin, or ψ(x ₁ ,y ₁ ,z ₁ ,σ ₁ ,x ₂ ,y ₂ ,z ₂ ,σ ₂ , …, x _N ,y _N ,z _N ,σ _N ) for an N-electron system. Alternatively, the wavefunction ψ can be expressed in terms of a series of basis functions, ϕ _n: ψ = ∑ n c n ϕ n, where the state of the system can be completely expressed with the set of coefficients {c _n}. When using such basis sets, ψ can then be written as a vector of coefficients,

ψ = ( c 1 c 2 c 3 ⋮ ),

and manipulated using matrix math. Finally, a more general notation can be used, which does not refer to any specific mathematical form for the state of the system: The bra-ket notation of Dirac. Here, the state of the system is specified as | ψ 〉 = ∑ n c n | ϕ n 〉 .

Both the functional and vector formats for the wavefunction can be implemented in MATLAB. To see this, let’s consider the stationary states for an electron placed in a one-dimensional box of length a, whose potential energy is 0 within the box and ∞ outside of it. As we will see in more detail below, the stationary states of this system are given by the functions ψ n = ( 2 a ) 1 2 sin ( π n a x ), where n is a positive integer going over the different possible states of the system. Such a function can be created in MATLAB by opening the editor, and typing in the following:

function psi = particle_in_box_psi(x,a,n)

psi = (2/a)^0.5*sin(pi*n*x/a);

Here, the first line serves to define the properties of the function, with “psi” being the output variable, “particle _ in _ box _ psi” being the name of the function, and “x,a,n” being the names of parameters that are given when the function is called. The second line does the actual work of the function, computing the value of psi from the values that are given for x,a,n. Once this function is typed in, it can be saved as the file “particle _ in _ box _ psi.m”.

Whenever this file is in your active directory (or a path is given to its location), y...

Table of contents

• Cover
• Half-Title
• Title
• Copyright
• Dedication
• Contents
• Acknowledgments
• About the Author
• CHAPTER 1 The Postulates of Quantum Mechanics
• CHAPTER 2 Atoms and Atomic Orbitals
• CHAPTER 3 Overlap between Atomic Orbitals
• CHAPTER 4 Introduction to Molecular Orbital Theory
• CHAPTER 5 The Molecular Orbitals of N2
• CHAPTER 6 Heteronuclear Diatomic Molecules
• CHAPTER 7 Symmetry Operations
• CHAPTER 8 Symmetry Groups
• CHAPTER 9 Group Theory and Basis Sets
• CHAPTER 10 The MOs of H2O
• CHAPTER 11 MOs of the Trigonal Planar Geometry
• CHAPTER 12 Walsh Diagrams and Molecular Shapes
• CHAPTER 13 Getting Started with Transition Metals
• CHAPTER 14 Translational Symmetry and Band Structures
• INDEX