Towards Large-Scale Accurate Kohn-Sham DFT for the Cost of Tight-Binding
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Towards large-scale accurate Kohn-Sham DFT for the cost of tight-binding Tiago Jose Remisio Marinheiro Newcastle University Newcastle upon Tyne, UK. A thesis submitted to the School of Engineering for the degree of Doctor of Philosophy June 2019 Dedicado aos meus pais sem eles nada disto seria possivel. And to Carmen and Alelo who have always been there and to whom I am eternally grateful, Sete anos de pastor Jacob servia Lab˜ao,pai de Raquel, serrana bela; mas n˜aoservia ao pai, servia a ela, e a ela s´opor pr´emiopretendia. Os dias na esperan¸cade um s´odia passava, contentando-se com vˆe-la; por´emo pai, usando de cautela, em lugar de Raquel, lhe dava Lia. Vendo o triste pastor que por enganos lhe fora assi negada sua pastora, como se a n˜aotivera merecida, Come¸caa servir outros sete anos, dizendo: Mais servira, se n˜aofora para t˜aolongo amor t˜aocurta vida. Luis Vaz de Cam˜oes Acknowledgements: First I would like to thank my supervisor Mark Rayson for giving me this opportunity. I have benefited immensely not only from his knowledge but from his ability to explain things at a fundamental level, this is what has always drawn me to Physics and I feel lucky to have had a supervisor who had the patience to do so. Thanks must also go to Patrick Briddon who helped me when I needed and from whom I have also learned much, I stand in awe at the breath of his knowledge and his ability to impart it. I must also extend thanks to John Goss who always had an open door and a willingness to answer any questions. Further, I must thank all the people who made these 4 years less stressful, particularly Faiz. A big thanks also to James, Johan and Luke. I would like to sincerely thank my parents who made all this possibly, their patience and help knows no bounds. They both went beyond what anybody could reasonably ask and nothing could ever repay what they did. I only hope that one day Alelo will look at me and feel the same level of love and admiration as I do for them. Finally, my greatest thanks must go to my wife, Carmen, who has endured by my side “though thick and thin” and listened to my lamentations. I could never thank you enough and I will spend the rest of my life trying. Serendipity brought us together and love pulled us through. We now go on to better things. And of course I must thank Alelo for his patience even if he didn’t realize it. Grow beautiful and true. It is to them who I dedicate this thesis. Abstract Density functional theory (DFT) is a widely used ab initio quantum mechanical method to study the properties of materials. Over the past 20 years a huge amount of work has been done developing codes that are able to tackle calculations containing large numbers of atoms. AIMPRO, a DFT code which uses Gaussian type orbitals (GTO) as a basis set, uses a filtration methodology which makes calculations with a few thousand atoms routinely possible on desktop machines. Previous implementations of filtration have focused on the time saving aspect of the methodology and performed calculations on structures containing only atoms from a small subset of the periodic table. In this thesis a novel basis set generation routine is presented and the filtration method- ology is modified and expanded to include most of the atoms in the periodic table. The focus of this work lies in demonstrating the potential gains in accuracy, in addition to effi- ciency, available through use of the filtration algorithm and shows that results comparable to codes using systematic basis set can be achieved for each of the elements considered across the periodic table. Two huge advantages present themselves using this scheme; firstly, the time to solution is essentially decoupled from the basis size; secondly, basis sets that would be unstable in a conventional calculation can be used allowing for more accurate calculations. The work presented here is assessed using a recently developed benchmark, the ∆-test. This, together with the increases in speed previously demonstrated, shows that a filtered basis calculation can now achieve the accuracy of a plane wave calculation at the asymp- totic cost, with respect to system size, of a tight-binding calculation, enabling Kohn-Sham calculations of unprecedented size to be performed at the basis set limit. Contents 1 Introduction 11 1.1 Summary of the thesis . 12 1.2 Abbreviations . 13 2 Quantum Mechanics 15 2.1 Introduction . 15 2.1.1 The Single particle Hamiltonian . 15 2.2 Operators . 17 2.2.1 Finite space . 17 2.2.2 The variational principle . 19 2.2.3 The Rayleigh-Ritz method . 20 2.3 Angular momentum . 20 2.4 Fermions and Bosons . 22 2.5 Spin . 23 2.6 The Many-Body Hamiltonian. 24 2.6.1 The Born-Oppenheimer approximation . 26 2.7 Independent electron approximations . 27 2.7.1 Hartree approximation . 27 2.8 Hartree-Fock theory . 28 2.8.1 Exchange interaction . 28 2.8.2 Hartree-Fock approximation . 29 2.9 Summary . 30 3 Density Functional Theory 31 3.1 Density Functional Theory . 31 3.2 Kohn-Sham equations . 35 3.2.1 The auxiliary system . 36 3.2.2 Solving the Kohn-Sham Equations . 38 1 3.2.3 Local Density Approximation . 39 3.3 Spin Polarization . 40 3.4 Fractional Occupation . 41 3.5 Boundary Conditions . 42 3.5.1 Crystal structure . 42 3.5.2 Periodic boundary conditions . 43 3.6 Summary . 44 4 Kohn-Sham DFT implementation 45 4.1 Basis sets . 45 4.1.1 Slater-Type orbitals . 46 4.1.2 Plane Waves . 47 4.1.3 Discrete variable representation . 50 4.1.4 Gaussian Basis sets . 50 4.2 Pseudopotentials . 51 4.2.1 Operator approach (Non-locality) . 52 4.2.2 Norm-conserving pseudopotentials (NCPPS) . 53 4.2.3 Goedecker-Tetter-Hutter (GTH) potentials . 54 4.3 The Hartree potential . 55 4.3.1 The Generalized Eigenvalue Problem . 55 4.4 AIMPRO code . 57 4.4.1 The kinetic energy . 59 4.4.2 External potential . 59 4.4.3 Exchange-Correlation . 60 4.4.4 Hartree terms . 60 4.4.5 Brillouin sampling . 61 4.5 Summary . 62 5 New method for generating Gaussian Basis Sets for filtration. 63 5.1 Introduction . 63 5.2 Preamble . 65 5.2.1 Atomic energy minimization with respect to primitives . 65 5.2.2 Contracted GTOs . 66 5.2.3 Even-tempered Gaussians. 67 5.3 History of Gaussian basis sets . 69 5.3.1 Pople and Dunning basis sets . 70 5.3.2 Summary . 71 5.4 Theoretical preamble . 72 5.4.1 The intermediate value theorem . 73 2 5.4.2 The bisection method . 73 5.4.3 The downhill simplex method . 75 5.5 Basis generation procedure . 76 5.5.1 Assignment of angular momentum l.................. 78 5.5.2 Assignment of n, α and β ....................... 79 5.5.3 Basis set generation; Main loop . 81 5.6 Results . 85 5.6.1 Output of the hydrogen run. 86 5.6.2 Convergence of the Gaussian atomic energy . 87 5.6.3 Behaviour of even-tempered parameters . 88 5.6.4 Behaviour of β ............................. 89 5.6.5 Regarding the choice of high first exponent (α)........... 91 5.7 Summary . 94 6 An updated filtration methodology tailored for large primitive basis sets 97 6.1 Introduction . 97 6.2 Linear-scaling methods and locality . 99 6.3 Filtration . 100 6.3.1 KSDFT summary . 101 6.3.2 Contracted Basis sets . 102 6.3.3 Filtration . 103 6.4 Localization constraints . 104 6.4.1 Trial functions . 105 6.4.2 Computational method . 107 6.5 Primitive to filtered space transformation. 108 6.6 Filtered to primitive subspace. 109 6.7 Summary . 109 7 The ∆ benchmark as an assessment of the filtration methodology 111 7.1 Introduction . 111 7.1.1 The ∆-benchmark . 112 7.1.2 The ∆-test . 113 7.2 Methodology . 116 7.3 Standard ∆-values . 117 7.4 Results . 118 7.4.1 Convergence of the energy with respect to basis-ecut ......... 118 7.4.2 AIMPRO’s standard ∆-value . 119 7.4.3 ∆-value with respect to systematic basis sets . 120 3 7.5 Precision of the filtration methodology . ..