Density Functional Theory: A Practical Introduction

Demonstrates how anyone in math, science, and engineering can master DFT calculations Density functional theory (DFT) is one of the most frequently used computational tools for studying and predicting the properties of isolated molecules, bulk solids, and material interfaces, including surfaces. Although the theoretical underpinnings of DFT are quite complicated, this book demonstrates that the basic concepts underlying the calculations are simple enough to be understood by anyone with a background in chemistry, physics, engineering, or mathematics. The authors show how the widespread availability of powerful DFT codes makes it possible for students and researchers to apply this important computational technique to a broad range of fundamental and applied problems. Density Functional Theory: A Practical Introduction offers a concise, easy-to-follow introduction to the key concepts and practical applications of DFT, focusing on plane-wave DFT. The authors have many years of experience introducing DFT to students from a variety of backgrounds. The book therefore offers several features that have proven to be helpful in enabling students to master the subject, including: Problem sets in each chapter that give readers the opportunity to test their knowledge by performing their own calculations Worked examples that demonstrate how DFT calculations are used to solve real-world problems Further readings listed in each chapter enabling readers to investigate specific topics in greater depth This text is written at a level suitable for individuals from a variety of scientific, mathematical, and engineering backgrounds. No previous experience working with DFT calculations is needed.

Table of Contents:
Preface xi 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 4 1.2.3 Materials Properties for Modeling Planetary Formation 6 1.3 The Schrodinger Equation 7 1.4 Density Functional Theory—From Wave Functions to Electron Density 10 1.5 Exchange–Correlation Functional 14 1.6 The Quantum Chemistry Tourist 16 1.6.1 Localized and Spatially Extended Functions 16 1.6.2 Wave-Function-Based Methods 18 1.6.3 Hartree–Fock Method 19 1.6.4 Beyond Hartree–Fock 23 1.7 What Can DFT Not Do? 28 1.8 Density Functional Theory in Other Fields 30 1.9 How to Approach This Book (Revisited) 30 References 31 Further Reading 32 2 DFT Calculations for Simple Solids 35 2.1 Periodic Structures Supercells and Lattice Parameters 35 2.2 Face-Centered Cubic Materials 39 2.3 Hexagonal Close-Packed Materials 41 2.4 Crystal Structure Prediction 43 2.5 Phase Transformations 44 Exercises 46 Further Reading 47 Appendix Calculation Details 47 3 Nuts and Bolts of DFT Calculations 49 3.1 Reciprocal Space and k Points 50 3.1.1 Plane Waves and the Brillouin Zone 50 3.1.2 Integrals in k Space 53 3.1.3 Choosing k Points in the Brillouin Zone 55 3.1.4 Metals—Special Cases in k Space 59 3.1.5 Summary of k Space 60 3.2 Energy Cutoffs 61 3.2.1 Pseudopotentials 63 3.3 Numerical Optimization 65 3.3.1 Optimization in One Dimension 65 3.3.2 Optimization in More than One Dimension 69 3.3.3 What Do I Really Need to Know about Optimization? 73 3.4 DFT Total Energies—An Iterative Optimization Problem 73 3.5 Geometry Optimization 75 3.5.1 Internal Degrees of Freedom 75 3.5.2 Geometry Optimization with Constrained Atoms 78 3.5.3 Optimizing Supercell Volume and Shape 78 Exercises 79 References 80 Further Reading 80 Appendix Calculation Details 81 4 DFT Calculations for Surfaces of Solids 83 4.1 Importance of Surfaces 83 4.2 Periodic Boundary Conditions and Slab Models 84 4.3 Choosing k Points for Surface Calculations 87 4.4 Classification of Surfaces by Miller Indices 88 4.5 Surface Relaxation 94 4.6 Calculation of Surface Energies 96 4.7 Symmetric and Asymmetric Slab Models 98 4.8 Surface Reconstruction 100 4.9 Adsorbates on Surfaces 103 4.9.1 Accuracy of Adsorption Energies 106 4.10 Effects of Surface Coverage 107 Exercises 110 References 111 Further Reading 111 Appendix Calculation Details 112 5 DFT Calculations of Vibrational Frequencies 113 5.1 Isolated Molecules 114 5.2 Vibrations of a Collection of Atoms 117 5.3 Molecules on Surfaces 120 5.4 Zero-Point Energies 122 5.5 Phonons and Delocalized Modes 127 Exercises 128 Reference 128 Further Reading 128 Appendix Calculation Details 129 6 Calculating Rates of Chemical Processes Using Transition State Theory 131 6.1 One-Dimensional Example 132 6.2 Multidimensional Transition State Theory 139 6.3 Finding Transition States 142 6.3.1 Elastic Band Method 144 6.3.2 Nudged Elastic Band Method 145 6.3.3 Initializing NEB Calculations 147 6.4 Finding the Right Transition States 150 6.5 Connecting Individual Rates to Overall Dynamics 153 6.6 Quantum Effects and Other Complications 156 6.6.1 High Temperatures/Low Barriers 156 6.6.2 Quantum Tunneling 157 6.6.3 Zero-Point Energies 157 Exercises 158 Reference 159 Further Reading 159 Appendix Calculation Details 160 7 Equilibrium Phase Diagrams from Ab Initio Thermodynamics 163 7.1 Stability of Bulk Metal Oxides 164 7.1.1 Examples Including Disorder—Configurational Entropy 169 7.2 Stability of Metal and Metal Oxide Surfaces 172 7.3 Multiple Chemical Potentials and Coupled Chemical Reactions 174 Exercises 175 References 176 Further Reading 176 Appendix Calculation Details 177 8 Electronic Structure and Magnetic Properties 179 8.1 Electronic Density of States 179 8.2 Local Density of States and Atomic Charges 186 8.3 Magnetism 188 Exercises 190 Further Reading 191 Appendix Calculation Details 192 9 Ab Initio Molecular Dynamics 193 9.1 Classical Molecular Dynamics 193 9.1.1 Molecular Dynamics with Constant Energy 193 9.1.2 Molecular Dynamics in the Canonical Ensemble 196 9.1.3 Practical Aspects of Classical Molecular Dynamics 197 9.2 Ab Initio Molecular Dynamics 198 9.3 Applications of Ab Initio Molecular Dynamics 201 9.3.1 Exploring Structurally Complex Materials: Liquids and Amorphous Phases 201 9.3.2 Exploring Complex Energy Surfaces 204 Exercises 207 Reference 207 Further Reading 207 Appendix Calculation Details 208 10 Accuracy and Methods beyond “Standard” Calculations 209 10.1 How Accurate Are DFT Calculations? 209 10.2 Choosing a Functional 215 10.3 Examples of Physical Accuracy 220 10.3.1 Benchmark Calculations for Molecular Systems—Energy and Geometry 220 10.3.2 Benchmark Calculations for Molecular Systems—Vibrational Frequencies 221 10.3.3 Crystal Structures and Cohesive Energies 222 10.3.4 Adsorption Energies and Bond Strengths 223 10.4 DFTX Methods for Improved Treatment of Electron Correlation 224 10.4.1 Dispersion Interactions and DFT-D 225 10.4.2 Self-Interaction Error Strongly Correlated Electron Systems and DFTU 227 10.5 Larger System Sizes with Linear Scaling Methods and Classical Force Fields 229 10.6 Conclusion 230 References 231 Further Reading 232 Index 235

About the Author :
David S. Sholl is a Professor of Chemical & Biomolecular Engineering at the Georgia Institute of Technology, where he holds the Michael Tennenbaum Family Chair and is a GRA Eminent Scholar in Energy Sustainability. Janice A. Steckel is a Physical Scientist at the U.S. Department of Energy, National Energy Technology Laboratory in Pittsburgh, Pennsylvania.