Written by a highly experienced author, this book fills the need for an accessible approach to grasping and working with the concepts of quantum theory. This is made possible by the use of MATLAB codes that allow the reader to experiment with the numerical techniques described. The book is also unique in its complete coverage of such hot topics as clusters, nanocrystals, transitions and organic molecules, while illustrating the DFT, the variational and various other methods, as well as approximate wave functions.

James Chelikowsky, PhD, holds the W.A. "Tex" Moncrief Chair at the University of Texas at Austin. He is a professor in the departments of physics, chemistry and chemical engineering. He also serves as the Director for the Center of Computational Materials in the Institute for Computational Engineering and Sciences.

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 CancellationTheorem 47

6.2 PseudopotentialsWithin Density FunctionalTheory 50

References 57

7 Methods for Atoms 59

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 FunctionalTheory 67

References 69

8 Methods for Molecules, Clusters, and Nanocrystals 71

8.1 The H2 Molecule: Heitler–LondonTheory 71

8.2 General Basis 76

8.2.1 PlaneWave Basis 79

8.2.2 PlaneWaves 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 DiagonalizationMethods: 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 FunctionalTheory 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

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 MassTheory 184

13.1.1.3 Vibrational Properties 187

13.1.1.4 Example of VibrationalModes 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

A Units 199

B A Working Electronic Structure Code 203

References 206

Index 207

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