Introduction of a Fully Relativistic Capable Basis Set in The
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INTRODUCTION OF A FULLY RELATIVISTIC CAPABLE BASIS SET IN THE AB INITIO ORTHOGONALIZED LINEAR COMBINATION OF ATOMIC ORBITALS METHOD A THESIS IN Physics Presented to the Faculty of the University of Missouri-Kansas City in partial fulfilment of the requirements for the degree MASTER OF SCIENCE by PATRICK RYAN THOMAS B. S., University of Missouri-Kansas City, 2013 Kansas City, Missouri 2014 2014 PATRICK RYAN THOMAS ALL RIGHTS RESERVED INTRODUCTION OF A FULLY RELATIVISTIC CAPABLE BASIS SET IN THE AB INITIO ORTHOGONALIZED LINEAR COMBINATION OF ATOMIC ORBITALS METHOD Patrick Ryan Thomas, Candidate for the Master of Science Degree University of Missouri-Kansas City, 2014 ABSTRACT Large simulation cell sizes, relativistic effects, and the need to correctly model excited state properties are major impediments to the accurate prediction of the optical properties of candidate materials for solid-state laser crystal and luminescent applications. To overcome these challenges, new methods must be created to improve the electron orbital wavefunction and interactions. In this work, a method has been developed to create new analytical four-component, fully-relativistic and single-component scalar relativistic descriptions of the atomic orbital wave functions from Grasp2K numerically represented atomic orbitals. In addition, adapted theory for the calculation of the relativistic kinetic energy contribution to Hamiltonian which bypasses directly solving the Dirac equation has been explicated. The orbital description improvements are tested against YAG, YBCO, SnO2 and BiF3. The improvements to the basis set reflect an improvement in both computational speed and accuracy. iii APPROVAL PAGE The Faculty listed below, appointed by the dean of the college of Arts and Sciences, have examined a thesis titled “Introduction of a Fully Relativistic Capable Basis Set in the ab initio Orthogonalized Linear Combination of Atomic Orbitals Method”, presented by Patrick Ryan Thomas, candidate for the Master of Science degree, and certify that in their opinion, it is worthy of acceptance. Supervisory Committee Paul Rulis, Ph.D., Committee Chair Department of Physics and Astronomy Wai-Yim Ching, Ph.D. Department of Physics and Astronomy Michael Kruger, Ph.D. Department of Physics and Astronomy iv TABLE OF CONTENTS ABSTRACT ....................................................................................................................... iii LIST OF ILLUSTRATIONS ............................................................................................. vi LIST OF TABLES ............................................................................................................ vii ACKNOWLEDGEMENTS ............................................................................................. viii Chapter 1. INTRODUCTION .......................................................................................................... 1 Context ........................................................................................................ 1 Motivation ................................................................................................... 3 Outline ........................................................................................................ 5 2. METHODS ..................................................................................................................... 7 The OLCAO Method .................................................................................. 8 The Discrete Variational Multi-Electron (DVME) Method ..................... 13 Grasp2K .................................................................................................... 14 3. SCALAR RELATIVISTIC BASIS SET IMPROVEMENTS ...................................... 20 Nature of Relativistic Orbitals and a Solution Method ............................. 20 Created Programs and Scripts ................................................................... 23 Tested Materials ........................................................................................ 28 Observations ............................................................................................. 34 4. ADAPTATION OF RELATIVISTIC KINETIC ENERGY THEORY ....................... 36 5. CONCLUSIONS AND FUTURE WORK ................................................................... 42 Conclusions ............................................................................................... 42 Future Work .............................................................................................. 43 REFERENCES ................................................................................................................. 46 VITA ................................................................................................................................. 49 v LIST OF ILLUSTRATIONS FIGURE PAGE 1. Effect of dopants on a band structure.. ........................................................................ 2 2. Example READRWF Output .................................................................................... 19 3. Comparison of single-component Schrödinger orbital to Dirac four-component orbital. ........................................................................................................................ 20 4. Divergence in fitting of Bismuth 4f(5/2) orbital. ...................................................... 22 5. Basis set generation flowchart.. ................................................................................. 24 6. Orbital Comparison between Grasp2K and DVME. ................................................. 28 7. Crystal Structure of YAG Primitive Cell .................................................................. 30 8. Band structures of YAG corresponding to three different atomic orbital basis sets.. 30 9. Crystal Structure of YBCO Primitive Cell ................................................................ 31 10. Band structures of YBCO corresponding to three different atomic orbital basis sets.. ................................................................................................................................... 31 11. Crystal Structure of SnO2 primitive cell. ................................................................... 32 12. Band structures of SnO2 corresponding to three different atomic orbital basis sets. 32 13. Crystal Structure of BiF3 primitive cell ..................................................................... 33 14. Band structures of BiF3 corresponding to three different atomic orbital basis sets.. 33 15. Difference in Area After Application of Derivatives to Gaussian Functions.. ......... 38 vi LIST OF TABLES TABLE PAGE 1. Main Program Attributes and Purposes ....................................................................... 8 2. YAG: Number of terms for each , SCF cycle iterations, and total energy .............. 30 3. YBCO: Number of terms for each , SCF cycle iterations, and total energy ........... 31 4. SnO2: Number of terms for each , SCF cycle iterations, and total energy .............. 32 5. BiF3: Number of terms for each , SCF cycle iterations, and total energy ............... 33 vii ACKNOWLEDGEMENTS I would like to thank all the people who have played important roles in reaching this milestone. I would first like to thank my advisor, Professor Paul Rulis. Professor Rulis’s guidance and direction are unparalleled in value in the completion of this work. I consider it a great honor to have studied and worked in his group. Next, I would like to thank Professor Michael Kruger. Professor Kruger gave me my first research position as an undergraduate in his experimental condensed matter group. He has been a continuous source of knowledge and advice. Lastly, I would like to thank Professor Fred Leibsle without whom I would not have finished an undergraduate degree in physics. The entirety of the physics faculty and students also are to thank for creating an atmosphere that makes studying and researching physics thoroughly enjoyable. I would like to thank my wife, Amanda, for her love, support, and patience while completing studies and research. I would like to thank my father without whom I never would have reached this goal. Your instruction in the universe, math, and logic from a small age paved the way to a fascination in studying physics. To all my family, without your faith and encouragement, I never would have become the person I am today. Thank you for everything you have done. viii CHAPTER 1 INTRODUCTION Context The demand for advanced technologies with ever increasing complexity and multi-functional capabilities has consequently increased the need for advanced materials with novel and/or highly tunable properties1–6. The process of optimizing a material for a specific set of properties can be very expensive in terms of both time and money because of the numerous physical fabrication trials that are required. However, systematic computational studies that accurately model the properties of interest can be used to bypass the need for physical fabrication to a great extent. Further, with the use of high performance computing (HPC) resources (as opposed to desktop workstations) not only is the time needed for a single calculation reduced, but an HPC based approach can allow for the use of entirely new predictive algorithms (e.g., genetic algorithms and data mining). With computationally driven predictions, it becomes possible to go from concept to market in a substantially shorter period