Density functional theory and nuclear quantum effects Lin Lin A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Program in Applied and Computational Mathematics Advisers: Roberto Car and Weinan E September, 2011 c Copyright by Lin Lin, 2011. All Rights Reserved Abstract This dissertation consists of two independent parts: density functional theory (Part I), and nuclear quantum effects (Part II). Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory in condensed matter systems. The computational time of KSDFT increases rapidly with respect to the number of electrons in the system, which hinders its practical application to systems of large size. The central quanti- ties in KSDFT are the electron density and the electron energy, which can be fully characterized by the diagonal elements and the nearest off-diagonal elements of the single particle density matrix. However, methods that are currently available require the calculation of the full density matrix. This procedure becomes highly inefficient for systems of large size. Part I of this dissertation develops a new method for solving KSDFT, which directly targets at the calculation of the diagonal and the nearest off- diagonal elements of the single particle density matrix. The new method is developed under the framework of Fermi operator expansion. The new method achieves the optimal expansion cost in the operator level. The electron density and the electron energy is then evaluated from a serires of Green’s functions by a new fast algorithm developed in this dissertation. This dissertation also develops a novel method for discretizing the Hamiltonian of the system that achieves high accuracy with a very small number of basis functions. Combining all these components together, we ob- tain a unified, accurate, efficient method to solve KSDFT for insulating and metallic systems. Nuclear quantum effects play an important role in a large variety of hydrogen bonded systems such as water and ice due to the small mass of protons (the nuclei of hydrogen atoms). The equilibrium proton dynamics is reflected in the quantum momentum distribution and is the focus of intense research. The standard open path integral formalism for computing the quantum momentum distribution requires the iii calculation of quantum momentum distribution for one particle at a time, which is an inefficient process especially when the potential energy surface is generated from Kohn-Sham density functional theory. The information of the proton dynamics is reflected in the momentum distribution in a highly averaged way and the interpre- tation of the momentum distribution can involve significant amount of ambiguity. Part II of this dissertation develops the displaced path integral formalism which al- lows the computation of quantum momentum distribution for all particles at the same time and therefore greatly enhances the computational efficiency. Part II of this dissertation unambiguously interprets the quantum momentum distribution in two representative systems: ice Ih and high pressure ice. For ice Ih in which the potential is quasi-harmonic, this disseration clarifies the previously unclear relation between anisotropic and anharmonic effects in shaping the momentum distribution by analyzing the 3D proton momentum distribution and the associated vibrational dynamics. For high pressure ice in which the potential is strongly anharmonic and proton tunneling occurs, this dissertation assesses the important role of proton corre- lation effects by means of spectral decomposition of the single particle density matrix. The concerted proton tunneling process directly observed and quantified in this study reduces significantly the number of ionized configurations, and avoids the ionization catastrophe predicted by the mean field theory, which was used in previous studies to interpret the path integral simulation results. iv Acknowledgements I would like to thank my thesis advisers Professor Roberto Car and Professor Weinan E. Without their guidance and encouragement, I cannot imagine how I would over- come the problems and difficulties in my research. They have taught me how to ask questions, how to solve problems, and how to think as an applied mathematician and as a computational scientist. Their thoughts have deeply influenced me in the past four years. I would like to thank my collaborators: Professor Carla Andreani, Professor Weiguo Gao, Dr. Juan Meza, Dr. Joseph Morrone, Professor Michele Parrinello, Dr. Antonino Pietropaolo, Dr. Amit Samanta, Dr. Roberto Senesi, and Dr. Chao Yang. Special thanks are given to Dr. Jianfeng Lu and Professor Lexing Ying for their numerous support and the fruitful work and discussions we had together. Without their help, my achievements would not be possible. I would also like to thank the tremendous support and encouragement I re- ceived from other professors at Princeton University, especially from Professor Robert Calderbank and Professor Ingrid Daubechies. Last but not least, my wife Dongxu Lu holds all my gratitude for her patience and for the love that she gives to me every day in my life. This thesis is dedicated to her, together with my beloved parents, my mother Xiaolan Liu and my father Chao Lin. v Contents Abstract..................................... iii Acknowledgements ............................... v I Density functional theory 21 1 Introduction 22 1.1 Overview.................................. 22 1.2 Quantum many body problem and electronic structure . 26 1.3 Kohn-ShamDensity functional theory (KSDFT) . 30 1.4 KSDFT:pseudopotentialframework. 36 1.5 MathematicalpropertiesofKSDFT . 38 1.6 Existing methods and software packages for solving KSDFT . 42 1.6.1 Cubic scaling methods . 42 1.6.2 Linear scaling methods . 44 1.6.3 All-electron methods . 46 1.7 Unified, accurate and efficient method for solving KSDFT . 47 2 Discretization of the Hamiltonian matrix: adaptive local basis func- tions 52 2.1 Introduction................................ 52 vi 2.2 Discontinuous Galerkin framework for Kohn-Sham density functional theory ................................... 55 2.3 Basis functions adapted to the local environment . 58 2.4 Implementationdetails .......................... 63 2.4.1 Grids and interpolation . 63 2.4.2 Implementation of the discontinuous Galerkin method . 65 2.4.3 Parallelization . 66 2.5 Numericalexamples............................ 68 2.5.1 Setup ............................... 69 2.5.2 Periodic Quasi-1D system . 71 2.5.3 Quasi-1Dsystemwithrandomperturbation . 73 2.5.4 Quasi-2Dand3DBulksystem. 74 2.5.5 Thepenaltyparameter . .. .. 76 2.5.6 Computationalefficiency . 78 2.6 Conclusion................................. 80 3 Representation of the Fermi operator: Pole expansion 82 3.1 Introduction................................ 82 3.2 Multipole expansion . 84 3.2.1 Formulation............................ 84 3.2.2 Numerical calculation and error analysis . 88 3.2.3 Numericalexamples ....................... 91 3.3 Poleexpansion .............................. 96 3.3.1 Pole expansion: basic idea . 96 3.3.2 Gappedcase: insulatingsystem . 98 3.3.3 Gapless case: metallic system . 100 3.3.4 Numericalexamples ....................... 104 3.4 Discussion................................. 108 vii 3.5 Conclusion................................. 112 4 Evaluation of the Fermi operator: Selected inversion 113 4.1 Introduction................................ 113 4.2 Selected inversion: Basic idea . 116 4.2.1 Densematrix ........................... 116 4.2.2 Sparsematrix ........................... 119 4.3 SelInv – An algorithm for selected inversion of a sparse symmetric matrix122 4.3.1 BlockAlgorithmsandSupernodes. 122 4.3.2 Implementation details . 125 4.3.3 Performance............................ 130 4.3.4 Application to electronic structure calculation of aluminum . 135 4.4 Parallel selected inversion algorithm . 137 4.4.1 Algorithmic and implementation . 137 4.4.2 Performance of the parallel selected inversion algorithm . 148 4.4.3 Application to electronic structure calculation of 2D rectangu- larquantumdots ......................... 159 4.5 Conclusion................................. 161 5 Fast construction of matrix 164 H 5.1 Introduction................................ 164 5.1.1 Motivation and applications . 165 5.1.2 Randomized singular value decomposition algorithm . 166 5.1.3 Top-down construction of -matrix............... 167 H 5.1.4 Relatedworks........................... 172 5.2 Algorithm ................................. 175 5.2.1 Geometricsetupandnotations . 175 5.2.2 Hierarchical matrix . 178 viii 5.2.3 Peeling algorithm: outline and preparation . 181 5.2.4 Peeling algorithm: details . 185 5.2.5 Peeling algorithm: variants . 192 5.3 Numericalresults ............................. 193 5.4 Conclusion................................. 199 6 Conclusion of Part I 202 II Nuclear quantum effects 205 7 Introduction 206 8 Displaced path integral formalism 212 8.1 Introduction................................ 212 8.2 Displaced path integral formalism . 216 8.3 Application of the displaced path integral formalism to water . 228 8.4 A new way of interpreting the momentum distribution . 231 8.5 Semiclassical limit of displaced path integral formalism . 235 8.6 Anewkineticestimator ......................... 244 8.7 Displacedpathformalismforbosons . 252 8.8 Conclusion................................. 254 9 Momentum distribution, vibrational dynamics and the potential