Grounds for Grounding

Elya B. Joffe is President of Elya Joffe – Electromagnetic Solutions, Ltd. He holds a B.ScEE from Ben Gurion University, Israel, and is a Registered Professional Engineer. Elya is an IEEE Life Senior Member, and Past President of the IEEE EMC and Product Safety Engineering Societies. Elya has received many awards from the IEEE and EMC Society, particularly the prestigious IEEE EMC Society 2002 Laurence G. Cumming Award for Outstanding Service and the 2006 IEEE RAB Larry K. Wilson Transnational Award. He is an iNARTE-certified EMC and ESD Control Engineer, and is a Member of IEEE-HKN and dB Society. Kai-Sang Lock, PhD, is a Professor of Engineering at the Singapore Institute of Technology. He has been a practicing Professional Engineer for over 20 years. He is a Fellow of the Academy of Engineering Singapore, a Fellow of the Institution of Engineering and Technology, UK, an Honorary Fellow of the Institution of Engineers, Singapore, as well as a Life Senior Member of IEEE. He was a President of the Institution of Engineers, Singapore, a past Board Member of the Professional Engineers Board, and a past Chairman of the Singapore Standards Council.

Preface to the Second Edition ix

Preface to First Edition xi

About the Companion Website xiii

1 What is Density Functional Theory? 1

1.1 How to Approach This Book 1

1.2 Examples of DFT in Action 2

1.2.1 Ammonia Synthesis by Heterogeneous Catalysis 2

1.2.2 Embrittlement of Metals by Trace Impurities 3

1.2.3 Materials Properties for Modeling Planetary Formation 4

1.2.4 Screening Large Collections of Materials to Develop Photoanodes 5

1.3 The Schrödinger Equation 7

1.4 Density Functional Theory – From Wavefunctions to Electron Density 9

1.5 The Exchange-Correlation Functional 12

1.6 The Quantum Chemistry Tourist 13

1.6.1 Localized and Spatially Extended Functions 13

1.6.2 Wavefunction-Based Methods 15

1.6.3 The Hartree–Fock Method 15

1.6.4 Beyond Hartree–Fock 18

1.7 What Can DFT Not Do? 22

1.8 Density Functional Theory in Other Fields 23

1.9 How to Approach This Book (Revisited) 24

1.10 Which Code Should I Use? 25

Further Reading 26

References 27

2 DFT Calculations for Simple Solids 29

2.1 Periodic Structures, Supercells, and Lattice Parameters 29

2.2 Face-Centered Cubic Materials 31

2.3 Hexagonal Close-Packed Materials 32

2.4 Crystal Structure Prediction 35

2.5 Phase Transformations 35

Exercises 37

Further Reading 37

Appendix – Calculation Details 38

Reference 38

3 Nuts and Bolts of DFT Calculations 39

3.1 Reciprocal Space and k-Points 40

3.1.1 Plane Waves and the Brillouin Zone 40

3.1.2 Integrals in k-Space 42

3.1.3 Choosing k-Points in the Brillouin Zone 43

3.1.4 Metals – Special Cases in k-Space 47

3.1.5 Summary of k-Space 48

3.2 Energy Cutoffs 49

3.2.1 Pseudopotentials 50

3.3 Numerical Optimization 51

3.3.1 Optimization in One Dimension 52

3.3.2 Optimization in More Than One Dimension 54

3.3.3 What Do I Really Need to Know About Optimization? 57

3.4 DFT Total Energies – An Iterative Optimization Problem 58

3.5 Geometry Optimization 59

3.5.1 Internal Degrees of Freedom 59

3.5.2 Geometry Optimization with Constrained Atoms 61

3.5.3 Optimizing Supercell Volume and Shape 61

Exercises 62

Further Reading 63

Appendix – Calculation Details 64

References 64

4 Accuracy of DFT Calculations 65

4.1 How Accurate are DFT Calculations? 65

4.2 Choosing a Functional 69

4.3 Examples of Physical Accuracy 73

4.3.1 Benchmark Calculations for Molecular Systems – Energy and Geometry 74

4.3.2 Benchmark Calculations for Molecular Systems – Vibrational Frequencies 75

4.3.3 Crystal Structures and Cohesive Energies 75

4.3.4 Adsorption Energies and Bond Strengths 76

4.4 When Might DFT Fail? 77

Exercises 78

Further Reading 79

References 79

5 DFT Calculations for Surfaces of Solids 81

5.1 Why Surfaces are Important 81

5.2 Periodic Boundary Conditions and Slab Models 82

5.3 Choosing k-Points for Surface Calculations 85

5.4 Classification of Surfaces by Miller Indices 85

5.5 Surface Relaxation 88

5.6 Calculation of Surface Energies 91

5.7 Symmetric and Asymmetric Slab Models 92

5.8 Surface Reconstruction 93

5.9 Adsorbates on Surfaces 95

5.9.1 Accuracy of Adsorption Energies 98

5.10 Effects of Surface Coverage 99

5.11 DFT Calculations for Grain Boundaries 101

Exercises 102

Further Reading 103

Appendix – Calculation Details 104

References 105

6 DFT Calculations of Vibrational Frequencies 107

6.1 Isolated Molecules 107

6.2 Vibrations of a Collection of Atoms 110

6.3 Molecules on Surfaces 112

6.4 Zero-Point Energies 114

6.5 Reaction Energies at Finite Temperatures 118

6.6 Phonons and Delocalized Modes 119

Exercises 120

Further Reading 120

Appendix – Calculation Details 121

Reference 122

7 Calculating Rates of Chemical Processes Using Transition State Theory 123

7.1 One-Dimensional Example 124

7.2 Multidimensional Transition State Theory 128

7.3 Finding Transition States 131

7.3.1 Elastic Band Method 132

7.3.2 Nudged Elastic Band Method 134

7.3.3 Initializing NEB Calculations 135

7.4 Finding the Right Transition States 137

7.5 Connecting Individual Rates to Overall Dynamics 139

7.6 Quantum Effects and Other Complications 141

7.6.1 High Temperatures/Low Barriers 142

7.6.2 Quantum Tunneling 142

7.6.3 Zero-Point Energies 142

Exercises 143

Further Reading 144

Appendix – Calculation Details 145

Reference 146

8 Predicting Equilibrium Phase Diagrams and Electrochemistry Using Open Ensemble Methods 147

8.1 Stability of Bulk Metal Oxides 148

8.1.1 Examples Including Disorder – Configurational Entropy 152

8.2 Stability of Metal and Metal Oxide Surfaces 154

8.3 DFT for Electrochemistry: The Computational Hydrogen Electrode 156

8.4 Using DFT to Predict Dissolution of Solids in Electrochemical Environments 159

Exercises 161

Further Reading 162

Appendix – Calculation Details 163

References 163

9 Electronic Structure and Magnetic Properties 165

9.1 Electronic Density of States 165

9.2 Local DOS and Atomic Charges 170

9.3 Magnetism 172

Exercises 174

Further Reading 174

Appendix – Calculation Details 175

10 Ab Initio Molecular Dynamics 177

10.1 Classical Molecular Dynamics 177

10.1.1 Molecular Dynamics with Constant Energy 177

10.1.2 Molecular Dynamics in the Canonical Ensemble 179

10.1.3 Practical Aspects of Classical Molecular Dynamics 180

10.2 Ab Initio Molecular Dynamics 180

10.3 Applications of Ab Initio MD 182

10.3.1 Exploring Structurally Complex Materials: Liquids and Amorphous Phases 182

10.3.2 Exploring Complex Energy Surfaces 183

Exercises 186

Further Reading 186

Appendix – Calculation Details 188

References 188

11 Methods Beyond “Standard” Calculations 189

11.1 Estimating Uncertainties in DFT 189

11.2 DFT+X Methods for Improved Treatment of Electron Correlation 191

11.2.1 Dispersion Interactions and DFT-D 191

11.2.2 Self-Interaction Error, Strongly Correlated Electron Systems and DFT+U 192

11.3 Random Phase Approximation 194

11.4 TD-DFT 196

11.5 Larger System Sizes with Linear Scaling Methods and Classical Forcefields 197

11.6 Conclusion 197

Further Reading 198

References 199

Index 201