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 к Points .......................... 50
3.1.1 Plane Waves and the Brillouin Zone .............. 50
3.1.2 Integrals in к Space ............................ 53
3.1.3 Choosing A: Points in the Brillouin Zone ........ 55
3.1.4 Metals—Special Cases in k Space ................. 59
3.1.5 Summary of к 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 к 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
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