Bartók-Pártay The Gaussian Approximation Potential

An Interatomic Potential Derived from First Principles Quantum Mechanics
1. Auflage 2010
ISBN: 978-3-642-14067-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 90 Seiten

Reihe: Springer Theses

Simulation of materials at the atomistic level is an important tool in studying microscopic structures and processes. The atomic interactions necessary for the simulations are correctly described by Quantum Mechanics, but the size of systems and the length of processes that can be modelled are still limited. The framework of Gaussian Approximation Potentials that is developed in this thesis allows us to generate interatomic potentials automatically, based on quantum mechanical data. The resulting potentials offer several orders of magnitude faster computations, while maintaining quantum mechanical accuracy. The method has already been successfully applied for semiconductors and metals.

Autoren/Hrsg.

Bartók-Pártay, Albert

Inhaltsverzeichnis

1;Supervisor’s Foreword;6
2;Acknowledgements;9
3;Contents;10
4;1 Introduction;13
4.1;1.1 Outline of the Thesis;15
4.2;References;15
5;2 Representation of Atomic Environments;16
5.1;2.1 Introduction;16
5.2;2.2 Translational Invariants;17
5.2.1;2.2.1 Spectra of Signals;17
5.2.2;2.2.2 Bispectrum;18
5.2.3;2.2.3 Bispectrum of Crystals;19
5.3;2.3 Rotationally Invariant Features;20
5.3.1;2.3.1 Bond-order Parameters;21
5.3.2;2.3.2 Power Spectrum;23
5.3.3;2.3.3 Bispectrum;24
5.3.3.1;2.3.3.1 Radial Dependence;26
5.3.4;2.3.4 4-Dimensional Bispectrum;28
5.3.5;2.3.5 Results;30
5.4;References;32
6;3 Gaussian Process;34
6.1;3.1 Introduction;34
6.2;3.2 Function Inference;34
6.2.1;3.2.1 Covariance Functions;36
6.2.2;3.2.2 Hyperparameters;38
6.2.3;3.2.3 Predicting Derivatives and Using Derivative Observations;39
6.2.4;3.2.4 Linear Combination of Function Values;40
6.2.5;3.2.5 Sparsification;41
6.3;References;42
7;4 Interatomic Potentials;43
7.1;4.1 Introduction;43
7.2;4.2 Quantum Mechanics;44
7.2.1;4.2.1 Density Functional Theory;45
7.2.1.1;4.2.1.1 The Hohenberg--Kohn principles;45
7.2.1.2;4.2.1.2 The self-consistent Kohn--Sham equations;47
7.3;4.3 Empirical Potentials;49
7.3.1;4.3.1 Hard-sphere Potential;49
7.3.2;4.3.2 Lennard--Jones Potential;49
7.3.3;4.3.3 The Embedded-Atom Model;50
7.3.4;4.3.4 The Modified Embedded-Atom Model;51
7.3.5;4.3.5 Tersoff Potential;53
7.4;4.4 Long-range Interactions;54
7.5;4.5 Neural Network Potentials;54
7.6;4.6 Gaussian Approximation Potentials;55
7.6.1;4.6.1 Technical Details;57
7.6.2;4.6.2 Multispecies Potentials;58
7.7;References;59
8;5 Computational Methods;60
8.1;5.1 Lattice Dynamics;60
8.1.1;5.1.1 Phonon Dispersion;60
8.1.2;5.1.2 Molecular Dynamics;62
8.1.3;5.1.3 Thermodynamics;62
8.2;References;65
9;6 Results;66
9.1;6.1 Atomic Energies;66
9.1.1;6.1.1 Atomic Expectation Value of a General Operator;66
9.1.1.1;6.1.1.1 Mulliken Charges;67
9.1.2;6.1.2 Atomic Energies;68
9.1.3;6.1.3 Atomic Multipoles;69
9.1.4;6.1.4 Atomic Energies from ONETEP;69
9.1.4.1;6.1.4.1 Wannier Functions;70
9.1.4.2;6.1.4.2 Nonorthogonal Generalised Wannier Functions;70
9.1.5;6.1.5 Locality Investigations;71
9.2;6.2 Gaussian Approximation Potentials;73
9.2.1;6.2.1 Gaussian Approximation Potentials for Simple Semiconductors: Diamond, Silicon and Germanium;74
9.2.2;6.2.2 Parameters of GAP;75
9.2.3;6.2.3 Phonon Spectra;76
9.2.4;6.2.4 Anharmonic Effects;78
9.2.5;6.2.5 Thermal Expansion of Diamond;80
9.3;6.3 Towards a General Carbon Potential;82
9.4;6.4 Gaussian Approximation Potential for Iron;84
9.5;6.5 Gaussian Approximation Potential for Gallium Nitride;86
9.6;6.6 Atomic Energies from GAP;88
9.7;6.7 Performance of Gaussian Approximation Potentials;90
9.8;References;90
10;7 Conclusion and Further Work;91
10.1;7.1 Further Work;92
11;8 Appendices;93
11.1;8.1 A: Woodbury Matrix Identity;93
11.2;8.2 B: Spherical Harmonics;94
11.2.1;8.2.1 Four-dimensional Spherical Harmonics;94
11.2.2;8.2.2 Clebsch--Gordan coefficients;96