Introductory Quantum Mechanics with MATLAB® Introductory Quantum Mechanics with MATLAB® For Atoms, Molecules, Clusters, and Nanocrystals James R. Chelikowsky Author All books published by Wiley-VCH Prof. James R. Chelikowsky are carefully produced. Nevertheless, University of Texas at Austin authors, editors, and publisher do not Institute for Computational Engineering warrant the information contained in and Sciences these books, including this book, to Center for Computational Materials be free of errors. Readers are advised 201 East 24th Street to keep in mind that statements, data, Austin, TX 78712 illustrations, procedural details or other United States items may inadvertently be inaccurate. Library of Congress Card No.: applied for Cover: © agsandrew/iStockphoto British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. 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Print ISBN: 978-3-527-40926-6 ePDF ISBN: 978-3-527-65501-4 ePub ISBN: 978-3-527-65500-7 oBook ISBN: 978-3-527-65498-7 Cover Design Tata Consulting Services Typesetting SPi Global, Chennai, India Printing and Binding Printed on acid-free paper 10987654321 To Ellen vii Contents Preface xi 1 Introduction 1 1.1 Different Is Usually Controversial 1 1.2 The Plan: Addressing Dirac’s Challenge 2 Reference 4 2 The Hydrogen Atom 5 2.1 The Bohr Model 5 2.2 The Schrödinger Equation 8 2.3 The Electronic Structure of Atoms and the Periodic Table 15 References 18 3 Many-electron Atoms 19 3.1 The Variational Principle 19 3.1.1 Estimating the Energy of a Helium Atom 21 3.2 The Hartree Approximation 22 3.3 The Hartree–Fock Approximation 25 References 27 4 The Free Electron Gas 29 4.1 Free Electrons 29 4.2 Hartree–Fock Exchange in a Free Electron Gas 35 References 36 5 Density Functional Theory 37 5.1 Thomas–Fermi Theory 37 5.2 The Kohn–Sham Equation 40 References 43 6 Pseudopotential Theory 45 6.1 The Pseudopotential Approximation 45 6.1.1 Phillips–Kleinman Cancellation Theorem 47 viii Contents 6.2 Pseudopotentials Within Density Functional Theory 50 References 57 7MethodsforAtoms59 7.1 The Variational Approach 59 7.1.1 Estimating the Energy of the Helium Atom. 59 7.2 Direct Integration 63 7.2.1 Many-electron Atoms Using Density Functional Theory 67 References 69 8 Methods for Molecules, Clusters, and Nanocrystals 71 8.1 The2 H Molecule: Heitler–London Theory 71 8.2 General Basis 76 8.2.1 Plane Wave Basis 79 8.2.2 Plane Waves Applied to Localized Systems 87 8.3 Solving the Eigenvalue Problem 89 8.3.1 An Example Using the Power Method 92 References 95 9 Engineering Quantum Mechanics 97 9.1 Computational Considerations 97 9.2 Finite Difference Methods 99 9.2.1 Special Diagonalization Methods: Subspace Filtering 101 References 104 10 Atoms 107 10.1 Energy levels 107 10.2 Ionization Energies 108 10.3 Hund’s Rules 110 10.4 Excited State Energies and Optical Absorption 113 10.5 Polarizability 122 References 124 11 Molecules 125 11.1 Interacting Atoms 125 11.2 Molecular Orbitals: Simplified 125 11.3 Molecular Orbitals: Not Simplified 130 11.4 Total Energy of a Molecule from the Kohn–Sham Equations 132 11.5 Optical Excitations 137 11.5.1 Time-dependent Density Functional Theory 138 11.6 Polarizability 140 11.7 The Vibrational Stark Effect in Molecules 140 References 150 12 Atomic Clusters 153 12.1 Defining a Cluster 153 Contents ix 12.2 The Structure of a Cluster 154 12.2.1 Using Simulated Annealing for Structural Properties 155 12.2.2 Genetic Algorithms 159 12.2.3 Other Methods for Determining Structural Properties 162 12.3 Electronic Properties of a Cluster 164 12.3.1 The Electronic Polarizability of Clusters 164 12.3.2 The Optical Properties of Clusters 166 12.4 The Role of Temperature on Excited-state Properties 167 12.4.1 Magnetic Clusters of Iron 169 References 174 13 Nanocrystals 177 13.1 Semiconductor Nanocrystals: Silicon 179 13.1.1 Intrinsic Properties 179 13.1.1.1 Electronic Properties 179 13.1.1.2 Effective Mass Theory 184 13.1.1.3 Vibrational Properties 187 13.1.1.4 Example of Vibrational Modes for Si Nanocrystals 188 13.1.2 Extrinsic Properties of Silicon Nanocrystals 190 13.1.2.1 Example of Phosphorus-Doped Silicon Nanocrystals 191 References 197 AUnits199 B A Working Electronic Structure Code 203 References 206 Index 207 xi Preface It is safe to say that nobody understands quantum mechanics. Richard Feynman. Those who are not shocked when they first come across quantum theory cannot possibly have understood it. Niels Bohr. The more success the quantum theory has, the sillier it looks. Albert Einstein. When some of the scientific giants of the twentieth century express doubt about a theory, e.g. referring to it as a “silly” or “shocking” theory, one might ask whether it makes sense to use such a theory in practical applications. Well, it does – because it works. Bohr, Einstein, and Feynman were reflecting on the exceptional nature of quantum theory and how it differs so strongly from previous theories. They did not question the success of the theory in describing the microscopic nature of matter. Indeed, quantum theory may be the most successful and revolutionary theory devised to date. The theory of quantum mechanics provides a framework to accurately predict the spatial and energetic distributions of electronic states and the concurrent properties in atoms, molecules, clusters, nanostructures, and bulk liquids and solids. As a specific example, consider the electrical conductivity of elemental crystals. The ratio of the conductivities for a metal crystal such as silver to an insulating crystal such as sulfur can exceed 24 orders of magnitude. A comparable ratio is the size of our galaxy divided by the size of the head of a pin, i.e. an astronomical difference! Classical physics provides no explanation for such widely different conductivities; yet, quantum mechanics, specifically energy band theory, does. That is the good news. The bad news is that quantum theory can be extraordinarily difficult to apply to real materials. There are a variety of reasons for this. One notable characteristic of quantum theory is the wave –particle duality of an electron, i.e. sometimes an electron behaves like a point particle, other times it behaves like a wave. When asked if an electron is a particle or a wave, an expert in quantum theory might respond with the seemingly incongruous answer: yes. xii Preface This is one reason why quantum theory seems so paradoxical and strange within the framework of traditional physics. Using quantum mechanics, we cannot ascertain the spatial characteristics of an electron by enumerat- ing three spatial coordinates. Rather, quantum mechanics can only give us information on where the electron is likely to be, not where it actually is. The probabilistic nature of quantum theory adds many degrees of freedom to the problem. We need to specify a probability for finding an electron at every possible point in space. As a first pass, this issue alone would appear to doom any quantitative approach to solving for the quantum behavior of an electron. Yet, the intellectual framework and the computational machinery have developed in spurts over the last several decades, which makes practical approaches to quantum mechanics feasible. Numerous textbooks often discuss the machinery of how to apply quantum mechanics, but fail to give the reader practical tools for accomplishing this goal. This is an odd situation whose origin likely resides in intellectual “latency.” In the not too distant past, it would require a huge and complex computer code, run on a large mainframe, to do quantum mechanics for a relatively small molecule. This is no longer the case. A modest amount of computing power, such as that available with a laptop computer or in principle even a “smartphone,” is sufficient to allow one to implement quantum theory for many systems of interest such as atoms, molecules, clusters, and other nanosacle structures. The goal of this book is to illustrate how this framework and machinery works. We will endeavor to give the reader a practical guide for applying quantum mechanics to interesting problems. Many people are owed thanks to this effort. Yousef Saad helped me frame many of the electronic structure codes and gave guidance about algorithms for solv- ing complex numerical problems. I also express a deep appreciation to a number of mentors and friends, including Marvin Cohen, Jim Phillips, and Steve Louie. Of course, I also thank my students and postdocs who did much of the heavy lifting. Austin, Texas James R. Chelikowsky 1 1 Introduction The great end of life is not knowledge but action. –T.H.Huxley 1.1 Different Is Usually Controversial Perhaps all breakthroughs in science are initially clouded with controversies. Consider the discovery of gravity. Isaac Newton invoked the concept of “action at a distance” when he developed his theory of gravity. Action at distance couples the motion of objects; yet, the objects possess no clear physical connection.