ENABLING ELECTRONIC STRUCTURE CALCULATIONS OF HIGH Z ELEMENT CONTAINING MATERIALS USING DIRAC RELATIVISTIC DFT METHODS A DISSERTATION IN Physics and Mathematics Presented to the Faculty of the University of Missouri–Kansas City in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY by PATRICK RYAN THOMAS B. S., University of Missouri - Kansas City, USA, 2012 M. S., Univeristy of Missouri - Kansas City, USA, 2014 Kansas City, Missouri 2020 © 2020 PATRICK RYAN THOMAS ALL RIGHTS RESERVED ENABLING ELECTRONIC STRUCTURE CALCULATIONS OF HIGH Z ELEMENT CONTAINING MATERIALS USING DIRAC RELATIVISTIC DFT METHODS Patrick Ryan Thomas, Candidate for the Doctor of Philosophy Degree University of Missouri–Kansas City, 2020 ABSTRACT Novel properties may be induced in a host material by doping it with high Z elements to alter its underlying electronic states. Presently, no method exists that can accurately capture both large system sizes and the intricate fundamental physics of the induced multiplet states in the electronic structure. As part of a collaborative effort to merge a configuration interaction (CI) and density functional theory (DFT) method into one method that is capable of calculating the properties of high Z doped materials, an entirely new scheme for relativistic consideration of a localized orbital basis set and en- ergy calculation was devised. This work presents a novel scheme for the creation of single-component scalar relativistic and four-component fully relativistic atomic orbital basis sets of Gaussian-type functions. A minimal norm least squares method was used to fit numerically represented, four-component Dirac spinors into a set of Gaussian basis iii functions possessing exponential coefficients expressed over a geometric series. A si- multaneous parameter sweep of both the maximum range of exponential coefficients and number of terms in the expansion for each quantum number was used to optimize the basis set efficiency. An algorithm that circumvents the error prone direct calculation of relativistic kinetic energy terms has been adapted from the CI method of our collabora- tor into one compatible with the Orthogonalized Linear Combination of Atomic Orbitals DFT package. Lastly, an application of the current state-of-the-art in high-throughput relativistic DFT to a well-known material science problem will be discussed. iv APPROVAL PAGE The faculty listed below, appointed by the Dean of the School of Graduate Studies, have examined a dissertation titled “Enabling Electronic Structure Calculations of High Z Ele- ment Containing Materials Using Dirac Relativistic DFT Methods,” presented by Patrick Ryan Thomas, candidate for the Doctor of Philosophy degree, and hereby certify that in their opinion it is worthy of acceptance. Supervisory Committee Paul Rulis, Ph.D., Committee Chair Department of Physics & Astronomy Michael Kruger, Ph.D. Department of Physics & Astronomy Elizabeth Stoddard, Ph.D. Department of Physics & Astronomy Noah Rhee, Ph.D. Department of Mathematics & Statistics Xianping Li, Ph.D. Department of Mathematics & Statistics v CONTENTS ABSTRACT . iii List of Figures . viii List of Tables . x ACKNOWLEDGEMENTS . xi Chapter 1 INTRODUCTION . 1 1.1 Context . 1 1.2 Motivation . 3 1.3 Outline . 5 2 METHODS . 6 2.1 Overview . 6 2.2 The OLCAO Method . 6 2.3 The Discrete Variational Multi-Electron (DVME) Method . 11 2.4 GRASP2K . 11 3 FULLY RELATIVISTIC CAPABLE BASIS SET CREATION . 20 3.1 Context . 21 3.2 Fitting Method . 23 3.3 Python Scripts . 29 4 INCLUSION OF RELATIVISTIC KINETIC ENERGY THEORY . 36 vi 4.1 Nature of the Problem . 36 4.2 Kinetic Energy Algorithm . 40 5 HYDROGEN TRAPPING IN FECRNI ALLOYS . 47 5.1 Background . 47 5.2 Methods and Techniques . 51 5.3 Hydrogen Trapping . 57 5.4 Conclusion . 70 6 CONCLUSIONS AND FUTURE WORK . 76 6.1 Conclusions . 76 6.2 Future Work . 77 Appendices . 80 A Python Codes . 80 A.1 org radwavefn.py . 80 A.2 make veusz graph.py . 84 A.3 interFit.py . 84 B Code Operation Examples . 90 B.1 Standard Operation . 91 B.2 Modular Operation . 92 REFERENCE LIST . 96 VITA . 104 vii List of Figures Figure Page 1 Ideal Four Step Lasing Process. 2 2 Non-relativistic vs. Relativistic Uranium 2p Orbital Comparison . 3 3 Uranium Isodata File . 13 4 Uranium CSL Sample Program Output . 14 5 Radial Wavefunction Output File . 19 6 Section of Isotope Data File for Uranium . 30 7 Area Differences for Single Gaussian with One Percent Change . 39 8 Augmented Self-Consistent Field Cycle . 45 9 ICME System Design Chart Explanation . 50 10 Hydrogen Trapping PSP Linkage . 51 11 fcc-bcc Crystallographic Orientation . 56 12 Headley-Brooks Crystallographic Orientation . 57 13 MedeA UNCLE Calculation of Fe0:7Cr0:2Ni0:1 . 58 14 Single Fe-Cr-Ni Structure . 59 15 UNCLE Parsed Structure Chemistry Distribution . 60 16 Sample Lattice Parameter Distribution . 61 17 Sample Binding Energy Analysis . 62 18 Binding Energy Nearest Neighbor Ternary by Chemistry . 63 viii 19 Trapping Energy Scheme . 64 20 fcc Lattice Vacancy . 65 21 Trapping Energy Flowchart . 72 22 Sample MedeA Workflow . 73 23 Total Energy Ternary . 73 24 Trapping Energy by Interstitial-Vacancy Distance . 74 25 Trapping Energy by Vacancy Element . 74 26 Trapping Energy Dependence on Interstitial and Vacancy Chemistry . 75 ix List of Tables Tables Page 1 Main Program Attributes and Purpose . 7 2 Atomic Orbital Basis Function Types . 21 3 Orbital Spectroscopic Labels . 24 4 Parameter Sweeping Values . 35 5 Observed Austenite (fcc) and Ferrite (bcc) Interfaces . 55 6 Statistics on the trapping energy by vacancy . 69 x ACKNOWLEDGEMENTS I would like to thank all the people that have played a vital role in finishing this dissertation. In truth, I could fill volumes with the names and contributions of each person in reaching this milestone. First, I would like to thank, my advisor, Professor Paul Rulis. Any praise of the fantastic job Professor Rulis has done would fall well short of describing its actual value. Professor Rulis is as equally knowledgeable as he is kind in his consistent advising to his students. I cannot thank him enough for all the opportunities he has given me to do excellent research and prepare for the future. Next, I would like to thank Professor Michael Kruger. Professor Kruger is nothing short of family to my wife and I. Professor Kruger gave us both our first research jobs and championed my first academic post as adjunct faculty at UMKC. Every time I see Professor Kruger I learn or am reminded about how fascinating the field of physics is. His council is always welcomed and valued. I would also like to thank Professor Elizabeth Stoddard who has been my boss, supervisor, and friend. Professor Stoddard promoted me to the position of co-head teach- ing assistant which I had the pleasure of working as for four years. Professor Stoddard encouraged me to pursue research in relativistic quantum theory. I am also immeasurably thankful for Professor Anthony Caruso who continuously supplied the most interesting and complex problems while providing unparalleled support and guidance. I would like to thank both Professors Rhee and Li for agreeing to be part of my committee and for providing pivotal conversations around some of the mathematics prob- lem that were necessary to solve for this work. Last but most absolutely not least, I would like to thank Professor Fred Leibsle very much for all that he has done for me and my family. Among the many blessings in this life, Professor Leibsle ranks amongst the highest. Without his selfless compassion and concern, I would never had made it through undergraduate studies let alone a doctorate. I would like to thank my wife, Amanda, for her constant love and support while finishing this work. I would like to thank my parents, Pat and Debbie, for the most encouraging up- bringing in both faith and scientific inquiry. Dad gave me my first physics and chemistry book at an early age that sparked a lifetime love of math and physics. I would like to thank two of my closest friends, James Curie and Jake Pursley, who provided important conversations and incredible friendship. I would like to thank my professional mentor, friend, and colleague, Dr. Ben Sikora, for all of your help and advice on wrapping up this work. Lastly, a countless number of students are to thank for fostering an environment at UMKC that encourages scientific minds and research. xii CHAPTER 1 INTRODUCTION 1.1 Context The development and discovery of complex materials with tunable properties is key to maintaining the rapid rate of advancement of technology [1–6]. The physical search for suitable candidate materials with precise properties incurs a large cost in both time and money as numerous materials must be fabricated over a range of each possi- ble parameter. The theoretical and computation studies of condensed matter systems can mitigate a sizable portion of the cost by eliminating the need for physically creating and testing numerous candidate materials. With access to high performance computing envi- ronments becoming more easily obtainable, advanced programming techniques applied to extremely large mathematical problems have made it possible to calculate the properties of systems previously thought impossible. The so-called ”materials-by-design” approach can guide a material from inception to production by cost effectively eliminating unfavor- able materials prior to fabrication and providing insights not limited to just the material but also the fabrication process. Laser host crystals and lighting phosphors are a unique family of materials whose energy output directly depends on a complex arrangement of the electron energy states in the crystal. Solid state lasers have a broad range of ap- plications including but not limited to scientific research, national defense, and medical procedures [7–11]. There are multiple lasing methods [12–18], but at the fundamental 1 Figure 1: Ideal Four Step Lasing Process.