Linear-scaling methods in ab initio quantum-mechanical calculations A dissertation submitted for the degree of Doctor of Philosophy at the University of Cambridge Peter David Haynes Christ’s College, Cambridge July 1998 Preface ThisdissertationdescribesworkdonebetweenOctober1995andJune1998intheTheoryof Condensed Matter group at the Cavendish Laboratory, Cambridge, under the supervision of Dr. M. C. Payne. Except where stated otherwise, this dissertation is the result of my own work and contains nothing which is the outcome of work done in collaboration. This dissertation has not been submitted in whole or in part for any degree or diploma at this or any other university. Peter Haynes Cambridge, July 1998 i O Lord, our Lord, how majestic is your name in all the earth! You have set your glory above the heavens. From the lips of children and infants you have ordained praise because of your enemies, to silence the foe and the avenger. When I consider your heavens, the work of your fingers, the moon and the stars, which you have set in place, what is man that you are mindful of him, the son of man that you care for him? You made him a little lower than the heavenly beings and crowned him with glory and honour. You made him ruler over the works of your hands; you put everything under his feet: all flocks and herds, and the beasts of the field, the birds of the air, and the fish of the sea, all that swim the paths of the seas. O Lord, our Lord, how majestic is your name in all the earth! Psalm 8 iii Acknowledgements The research described in this dissertation was supported financially by an EPSRC stu- dentship, and it is a great pleasure to be able to thank some of the people who have helped me in various ways, and whose support has been invaluable over the past three years. My supervisor, Mike Payne, has been a faithful source of wisdom and encouragement throughout, as well as a generous supplier of wine. I am also grateful to Richard Needs and Roger Haydock for stimulating discussions which have helped to broaden my knowledge and understanding of condensed matter physics. I am indebted to Nicola Marzari for his hospitality and also for pointing out several k-points, not least the concept of the roto-occupations (the f ). I also benefitted from ij discussions with Eric Sandr´e concerning computational minimisation schemes. Thanks are due to those who have had the dubious pleasure of sharing offices with me: Murray Jarvis, Duncan Kerr, Yong Mao and Matt Segall have all made life at the Cavendish much more interesting than it would have been on my own. Ian Bung Bung White has been a continual source of entertainment and diversion, whose initiative in proposing frequent experimental investigations into the aerodynamics of spinning disks and ellipsoids made the days pass much more quickly. There is insufficient space to thank all my friends at Christ’s College and St. Andrew the Great individually, but I value those times spent together which have contributed to making Cambridge a great place to live as well as to work. Finally, I thank my family for their love and support, without which I would never have got here in the first place. v Contents 1 Introduction 1 1.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Computer simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Many-body Quantum Mechanics 5 2.1 Principles of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Wave-functions and operators . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 Stationary states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . 9 2.3 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Spin and statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Variational principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Quantum Mechanics of the Electron Gas 19 3.1 Density-functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.1 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . 20 3.1.2 The constrained search formulation . . . . . . . . . . . . . . . . . . 22 3.1.3 Exchange and correlation . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.4 The Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.5 The local density approximation . . . . . . . . . . . . . . . . . . . . 26 3.2 Periodic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.2 Brillouin zone sampling . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 The pseudopotential approximation . . . . . . . . . . . . . . . . . . . . . . 32 3.3.1 Operator approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Scattering approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 vii Linear-scaling methods in ab initio quantum-mechanical calculations 3.3.3 Norm conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.4 Kleinman-Bylander representation . . . . . . . . . . . . . . . . . . . 37 4 Density-Matrix Formulation 39 4.1 The density-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Partial occupation of the Kohn-Sham orbitals . . . . . . . . . . . . . . . . 42 4.3 Density-matrix DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Constraints on the density-matrix . . . . . . . . . . . . . . . . . . . . . . . 45 4.4.1 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4.2 Idempotency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4.3 Penalty functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.4 Purifying transformation . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4.5 Idempotency-preserving variations . . . . . . . . . . . . . . . . . . . 48 4.5 Requirements for linear-scaling methods . . . . . . . . . . . . . . . . . . . 51 4.5.1 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5.2 Spatial localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6 Non-orthogonal orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5 Localised basis-set 57 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Origin of the basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.3 Fourier transform of the basis functions . . . . . . . . . . . . . . . . . . . . 60 5.4 Overlap matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.5 Kinetic energy matrix elements . . . . . . . . . . . . . . . . . . . . . . . . 64 5.6 Non-local pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.6.1 Green’s function method . . . . . . . . . . . . . . . . . . . . . . . . 66 5.6.2 Kleinman-Bylander form . . . . . . . . . . . . . . . . . . . . . . . . 71 5.7 Computational implementation . . . . . . . . . . . . . . . . . . . . . . . . 72 6 Penalty Functionals 75 6.1 Kohn’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.1.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.1.2 Implementation problems . . . . . . . . . . . . . . . . . . . . . . . 78 6.2 Corrected penalty functional method . . . . . . . . . . . . . . . . . . . . . 80 6.2.1 Derivation of the correction . . . . . . . . . . . . . . . . . . . . . . 80 6.2.2 Further examples of penalty functionals . . . . . . . . . . . . . . . . 86 6.2.3 Minimisation efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 87 viii