The Gaussian Approximation Potential (eBook)

Supervisor’s Foreword 6
Acknowledgements 9
Contents 10
1 Introduction 13
1.1 Outline of the Thesis 15
References 15
2 Representation of Atomic Environments 16
2.1 Introduction 16
2.2 Translational Invariants 17
2.2.1 Spectra of Signals 17
2.2.2 Bispectrum 18
2.2.3 Bispectrum of Crystals 19
2.3 Rotationally Invariant Features 20
2.3.1 Bond-order Parameters 21
2.3.2 Power Spectrum 23
2.3.3 Bispectrum 24
2.3.3.1 Radial Dependence 26
2.3.4 4-Dimensional Bispectrum 28
2.3.5 Results 30
References 32
3 Gaussian Process 34
3.1 Introduction 34
3.2 Function Inference 34
3.2.1 Covariance Functions 36
3.2.2 Hyperparameters 38
3.2.3 Predicting Derivatives and Using Derivative Observations 39
3.2.4 Linear Combination of Function Values 40
3.2.5 Sparsification 41
References 42
4 Interatomic Potentials 43
4.1 Introduction 43
4.2 Quantum Mechanics 44
4.2.1 Density Functional Theory 45
4.2.1.1 The Hohenberg--Kohn principles 45
4.2.1.2 The self-consistent Kohn--Sham equations 47
4.3 Empirical Potentials 49
4.3.1 Hard-sphere Potential 49
4.3.2 Lennard--Jones Potential 49
4.3.3 The Embedded-Atom Model 50
4.3.4 The Modified Embedded-Atom Model 51
4.3.5 Tersoff Potential 53
4.4 Long-range Interactions 54
4.5 Neural Network Potentials 54
4.6 Gaussian Approximation Potentials 55
4.6.1 Technical Details 57
4.6.2 Multispecies Potentials 58
References 59
5 Computational Methods 60
5.1 Lattice Dynamics 60
5.1.1 Phonon Dispersion 60
5.1.2 Molecular Dynamics 62
5.1.3 Thermodynamics 62
References 65
6 Results 66
6.1 Atomic Energies 66
6.1.1 Atomic Expectation Value of a General Operator 66
6.1.1.1 Mulliken Charges 67
6.1.2 Atomic Energies 68
6.1.3 Atomic Multipoles 69
6.1.4 Atomic Energies from ONETEP 69
6.1.4.1 Wannier Functions 70
6.1.4.2 Nonorthogonal Generalised Wannier Functions 70
6.1.5 Locality Investigations 71
6.2 Gaussian Approximation Potentials 73
6.2.1 Gaussian Approximation Potentials for Simple Semiconductors: Diamond, Silicon and Germanium 74
6.2.2 Parameters of GAP 75
6.2.3 Phonon Spectra 76
6.2.4 Anharmonic Effects 78
6.2.5 Thermal Expansion of Diamond 80
6.3 Towards a General Carbon Potential 82
6.4 Gaussian Approximation Potential for Iron 84
6.5 Gaussian Approximation Potential for Gallium Nitride 86
6.6 Atomic Energies from GAP 88
6.7 Performance of Gaussian Approximation Potentials 90
References 90
7 Conclusion and Further Work 91
7.1 Further Work 92
8 Appendices 93
8.1 A: Woodbury Matrix Identity 93
8.2 B: Spherical Harmonics 94
8.2.1 Four-dimensional Spherical Harmonics 94
8.2.2 Clebsch--Gordan coefficients 96