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Finite-Temperature Green's Function Methods for Ab-Initio Quantum

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Finite-Temperature Green's Function Methods for Ab-Initio Quantum Finite-Temperature Green’s Function Methods for ab-initio Quantum Chemistry: Thermodynamics, Spectra, and Quantum Embedding by Alicia Rae Welden A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Chemistry) in the University of Michigan 2018 Doctoral Committee: Assistant Professor Dominika Zgid, Chair Professor Eitan Geva Professor Kevin Kubarych Associate Professor Vanessa Sih Alicia Rae Welden [email protected] ORCID iD: 0000-0002-2238-9825 © Alicia Rae Welden 2018 All Rights Reserved DEDICATION To B. ii ACKNOWLEDGMENTS I am thankful to my advisor Dr. Dominika Zgid, for guiding me through my Ph.D. and believing in me (even if I didn’t believe in me). She continued to be supportive and nurturing throughout my graduate school years, and truely shaped me as a person and a scientist. I would like to thank Dr. Eitan Geva for serving on my committee and for being the advisor of the work we did together on Compute-to-Learn. That project was an excellent opportunity for me to develop as a teacher, and I am grateful for the time he spent advising it, and the opportunities it gave me. I would like to thank Dr. Kevin Kubarych and Dr. Vanessa Sih for serving on my committee and their helpful guidance during my Ph.D. work. I am thankful to Blair Winograd for being my colleague and friend. I would not have been able to navigate graduate school without you. I am thankful to Dr. Alexander Rusakov for being my deskmate, coauthor, and colleague. I would like to acknowledge my group mates Dr. Alexei Kananenka, Dr. Tran Nguyen Lan, Dr. Avijit Shee, and Yanbing Zhou for the opportunity to work and discuss science together. I would like to thank my former group member, Dr. Jordan Phillips, for his patience and for advising me during my first project in graduate school. I would not be where I am today if it were not for the love and support of my parents, Shirley Reynolds-Pincus and Rick Pincus, who have always encouraged me to do what makes me happy. I have made many new friends in Michigan and would like to thank them all for the warm and welcoming atmosphere they have created. I would like to thank my non-human supporters, Sterling, Mallory, and Romeo for injecting much cuteness into many of my days. iii Last but not at all least, I am thankful for my husband, Brian, and his love, support, and encouragement. Brian, I love you and would not have been able to do this without you. This thesis was advised by Dr. Dominika Zgid and completed in collaboration with mem- bers of my group and the Emanuel Gull’s group in the Department of Physics at the University of Michigan. Chapter 3 investigates the ability to obtain temperature dependent electronic ther- modynamics from a Green’s function and co-authored with Dr. Alexander Rusakov. Chapter 4 describes different ways to obtain the spectra from a temperature-dependent Green’s function and was co-authored by Dr. Jordan J. Phillips and Dr. Tran Nyugen Lan. Chapter 5 demonstrates the use of a temperature-dependent restricted active space configuration interaction impurity solver for quantum embedding, and was co-authored with Dr. Sergei Iskakov. iv PREFACE “But the computers have bad features. The main one is that students get so fascinated with them, so immersed with what they can do, that they never get away from the computer far enough to think about what they are doing. This in a way is probably what has kept the chemists from thinking beyond the LCAO method. Ever since they started with computers, they have been using this method, enlarging it more and more, using bigger and bigger computers, without stopping to ask whether there might not be some quite different approach to the problem.” -John C. Slater, 1975 “A thesis is not completed, but merely abandoned.” -Unknown v TABLE OF CONTENTS Dedication........................................ ii Acknowledgments.................................... ii Preface..........................................v List of Figures...................................... viii List of Tables.......................................x Abstract.......................................... xi Chapter 1 Introduction ....................................1 2 Background ....................................6 2.1 Overview of Quantum Chemistry.....................6 2.1.1 Correlation Energy........................6 2.1.2 Temperature in electronic structure................9 2.2 Green’s Function from Perturbation Theory................ 11 2.2.1 Temperature-Dependent Matsubara Green’s Function...... 12 3 Electronic Thermodynamics ........................... 15 3.1 Temperature-Dependent Thermodynamics from the Matsubara Green’s function................................... 15 3.1.1 Introduction............................ 15 3.1.2 Connection with thermodynamics................ 17 3.2 Properties of the Luttinger-Ward Functional................ 20 3.2.0.1 Self-energy as a functional derivative........... 20 3.2.0.2 Connection with the grand potential Ω .......... 20 3.2.0.3 Universality........................ 21 3.2.1 Evaluation of Luttinger-Ward Functional Within GF2...... 21 3.2.1.1 Description of the GF2 algorithm............. 21 3.2.1.2 Evaluation of the grand potential in the k-space..... 24 3.2.2 Results............................... 26 3.2.2.1 HF molecule........................ 26 3.2.2.2 Periodic calculation of 1D hydrogen........... 28 3.2.2.3 Short bond length/high pressure............. 30 3.2.2.4 Intermediate bond length/intermediate pressure..... 31 3.2.2.5 Long bond length/low pressure.............. 33 3.2.2.6 GF2 phase diagram for 1D hydrogen solid........ 35 3.2.2.7 Periodic calculation of 1D boron nitride......... 36 3.2.3 Conclusion............................ 40 vi 3.2.4 Appendix: High-frequency expansion of the Green’s function for evaluating the Luttinger-Ward functional........... 42 4 Spectra ....................................... 44 4.1 Introduction................................. 44 4.2 GF2: An imaginary axis formulation................... 47 4.3 Fundamental Gap Evaluation for GF2................... 48 4.3.1 Extended Koopmans’ Theorem.................. 49 (2) (2) 4.3.2 HF+Σii and HF+iter Σii ..................... 52 4.3.3 F(GF2)+Σ(2) ............................ 53 4.4 Results................................... 54 4.4.1 Ionization Potentials....................... 54 4.4.2 Electron Affinities......................... 58 4.4.3 Fundamental Gap......................... 58 4.4.4 DNA and RNA nucleobases................... 60 4.5 Singlet-Triplet Gap Evaluation....................... 62 4.6 Conclusions................................. 64 4.6.1 Supplemental Information.................... 65 5 Temperature-Dependent Configuration Interaction Impurity Solver for Quan- tum Embedding .................................. 68 5.1 Restricted Configuration Interaction (RASCI) Method.......... 69 5.2 Impurity Models.............................. 69 5.3 Finite temperature Lanczos......................... 70 5.3.1 Matsubara Green’s Function from Lanczos procedure...... 70 5.4 Results................................... 73 5.4.1 Hubbard Impurity Models.................... 73 5.4.2 Results for 1D Hubbard Impurity Model............. 73 5.4.3 Results for 2D Hubbard Impurity Model............. 74 6 Conclusions ..................................... 78 Bibliography....................................... 80 vii LIST OF FIGURES 3.1 The differences between GF2, finite temperature Hartree-Fock and FCI for the hydro- gen fluoride molecule at various temperatures...................... 28 3.2 Spectral functions and projections at different inverse temperatures for the metallic solution of 1D periodic hydrogen solid at R=0.75 Å................... 30 3.3 Spectral functions and projections at different inverse temperatures for the band insu- lator (“solution 1”) and the metallic solution (“solution 2”) of 1D periodic hydrogen solid at R=1.75 Å. Note the difference in scale between “solution 1” and “solution 2”. 32 3.4 Spectral functions and projections at different inverse temperatures for the band insu- lator (“solution 1”) and the metallic solution (“solution 2”) of 1D periodic hydrogen solid at R=2.5 Å...................................... 34 3.5 Spectral functions and projections at different inverse temperatures for the Mott insu- lator present in 1D periodic hydrogen solid at R=4.0 Å.................. 35 3.6 A phase diagram containing all distances and temperatures calculated for a 1D hydro- gen solid. Where multiple phases exist, the most stable phase is outlined in black. β is inverse temperature in units 1/a.u. and R is the separation between hydrogens in Å.. 37 3.7 Spectral projection in the vicinity of the Fermi energy at β=100 of periodic 1D boron nitride solid. Only the “correlated bands” corresponding to the conventional HOCO’s and LUCO’s (see Sec. V. B) are displayed. Note that the energy scale is in eV..... 38 3.8 Periodic 1D boron nitride solid spectrum for the “correlated HOCO and LUCO” (see Sec. V. B) for several temperatures at k=−π. The chemical potential was adjusted for ! = 0 to fall in the middle of the band gap........................ 39 4.1 Ionization potentials for atoms calculated with basis aug-cc-pVTZ. Shown here are differences from experimental values. Units in eV.................... 55 4.2 Electron affinities for atoms calculated with basis aug-cc-pVTZ. Shown here are dif- ferences from UCCSD(T) calculations in the same basis. Units in eV.......... 56 4.3 Fundamental gaps for atoms calculated with basis aug-cc-pVTZ. Shown here are dif-
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