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Approaching Exact Quantum Chemistry by Stochastic Wave Function Sampling and Deterministic Coupled-Cluster Computations

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Approaching Exact Quantum Chemistry by Stochastic Wave Function Sampling and Deterministic Coupled-Cluster Computations APPROACHING EXACT QUANTUM CHEMISTRY BY STOCHASTIC WAVE FUNCTION SAMPLING AND DETERMINISTIC COUPLED-CLUSTER COMPUTATIONS By Jorge Emiliano Deustua Stahr A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Chemistry – Doctor of Philosophy 2020 ABSTRACT APPROACHING EXACT QUANTUM CHEMISTRY BY STOCHASTIC WAVE FUNCTION SAMPLING AND DETERMINISTIC COUPLED-CLUSTER COMPUTATIONS By Jorge Emiliano Deustua Stahr One of the main goals of quantum chemistry is the accurate description of ground- and excited-state energetics of increasingly complex polyatomic systems, especially when non-equilibrium structures and systems with stronger electron correlations are exam- ined. Size extensive methods based on coupled-cluster (CC) theory and their extensions to excited electronic states via the equation-of-motion (EOM) framework have become the de facto standards for addressing this goal. In the vast majority of chemistry prob- lems, the traditional single-reference CC hierarchy, including CCSD, CCSDT, CCSDTQ, etc., and its EOM counterpart provide the fastest convergence toward the exact, full configuration interaction (FCI), limit, allowing one to capture the leading many-electron correlation effects, in a systematic manner, by employing particle-hole excitations from a single reference determinant. Unfortunately, computational costs associated with the incorporation of higher-than-two-body components of the cluster and excitation opera- tors of CC and EOMCC, which are required to achieve a fully quantitative description of molecular systems, are usually prohibitive. This has exacerbated the need for new ideas in quantum chemistry leading to computationally affordable methodologies that do not suffer from failures of lower-order, less expensive approximations. In this dissertation, I propose two novel and computationally cost-effective strategies for acquiring accurate electronic energetics, equivalent to those obtained with high-level CC/EOMCC meth- ods, such as CCSDT, EOMCCSDT, and CCSDTQ, and the exact, FCI, theory, even when wave function quasi-degeneracies and other higher-order correlation effects become sig- nificant. The first strategy consists in merging the deterministic CC framework, abbreviated as, CC(P;Q), with the stochastic CI Quantum Monte Carlo (QMC) approaches, such as CISDT-MC, CISDTQ-MC, FCIQMC, and their CCSDT-MC counterpart with the intention of using wave functions resulting from QMC propagations to identify the leading higher- than-doubly excited components entering deterministic CC and EOMCC calculations. The second proposed methodology provides a direct way of recovering FCI energetics following the ideas of the externally corrected CC theories. In the resulting approach, that we denominate CAD-FCIQMC, we utilize the cluster analysis of FCIQMC wave functions to obtain the connected three- and four-body cluster components, which are the only clusters that directly couple to the one- and two-body components of the cluster operator. In general, the externally corrected CC methods are guaranteed to attain the exact, FCI, ground-state energy, if the triply and quadruply excited cluster components extracted from a non-CC source entering the suitably corrected CCSD-like equations ap- proach their exact values. Since the long-time FCIQMC propagations converge the exact FCI wave functions, our new CAD-FCIQMC method is assured to produce numerically exact energies. All of the above approaches, including the semi-stochastic CC(P;Q) and CAD- FCIQMC methodologies, are discussed in this dissertation. This includes their math- ematical foundations, computer implementation, and numerical tests using bond break- + ing in the F2,H2O, and CH molecules and the automerization of cyclobutadiene. In memory of My Aunt, Monica Klien. iv ACKNOWLEDGEMENTS First, I would like to thank my doctoral advisor, Professor Piotr Piecuch. With his guid- ance and determination, he has taken me from an almost non-existent knowledge of quantum mechanics to the writing of this dissertation. His enriching attitude toward “doing things right” has been fundamental during my years in this graduate program and will remain in me for the years to come. Professor Piecuch has always motivated me to improve myself as a person, as a professional, and as a scientist, and for this I am truly thankful. I will always be grateful for all the kind opportunities he has given me in his research group and the huge amount of knowledge he has shared with me. In addition, I would like to extend my gratitude to the MolSSI and National Science Foundation for awarding me the Phase I and II Software Fellowships that supported my research and software development during my Ph.D. studies. Also, I want to acknowl- edge the Department of Energy for funding most projects I have worked on during my Ph.D. studies. I would also like to thank the remaining members of my committee, Professor Ben- jamin G. Levine, Professor James K. McCusker, and Professor Kenneth M. Merz for their support, time, and advice. In particular, I am grateful for the opportunity of having Pro- fessor Levine and Professor McCusker as instructors. I learned a great deal from their courses. Also, I would like to extend my thanks to Professor Ali Alavi and his group, for their help in interfacing their QMC software, NECI, with our group codes, as well as for introducing the FCIQMC methodology, which inspired me to investigate into the topics of this thesis. I would also like to acknowledge Dr. Alexander J.W. Thom and his team for the HANDE Quantum Monte Carlo software package, which I have extensively used throughout my work, and which has been fundamental for my research. v I am deeply grateful for the opportunity I have had in Professor Piecuch group in meeting and befriending my fellow group members: Dr. Adeayo O. Ajala, Dr. Nicholas P. Bauman, Mr. Arnab Chakraborty, Mr. Karthik Gururangan, Dr. Jared A. Hansen, Mr. Ilias Magoulas, Dr. Jun Shen, and Mr. Stephen H. Yuwono. My experience in the Ph.D. program would not have been the same without their support, help, encouragement, and amazing discussions. Furthermore, I would like to thank Dr. Jonathan Moussa, who has been my MolSSI mentor for the past two years, and has helped me to a great extend in my software development efforts. Likewise, I would like to express my gratitude toward Professor Katharine C. Hunt for her encouraging attitude toward teaching and scholarship. Being able to interact with her as a scientist, Ph.D. student, and teaching assistant, has been a truly enriching and pleasurable experience. Finally, I would like to thank my family and friends for the continuous support throughout my career, their kindness, and their love. Having such a wonderful group of people supporting me is invaluable and deeply moving. Thank you all! vi TABLE OF CONTENTS LIST OF TABLES ....................................... ix LIST OF FIGURES ....................................... xvii CHAPTER 1 INTRODUCTION .............................. 1 CHAPTER 2 THEORY ................................... 17 2.1 Coupled-Cluster Theory and its Equation-of-Motion Extension to Ex- cited States . 22 2.2 The Active-Space Coupled-Cluster Approaches . 25 2.3 The CC(P;Q) Formalism . 29 2.4 The FCIQMC and its Initiator Approximation (i-FCIQMC) . 36 2.5 Merging the FCIQMC Ideas with the CC(P;Q) Formalism: The Semi- Stochastic CC(P;Q) Approach . 43 2.6 The CAD-FCIQMC Method . 48 CHAPTER 3 APPLICATION OF THE SEMI-STOCHASTIC CC(P;Q) AND CAD-FCIQMC APPROACHES TO MOLECULAR MODEL SYSTEMS 52 3.1 Evaluating the Semi-Stochastic CC(P;Q) Approach by Recovering High- Level Ground-State CC Energetics Characterizing the F2 and H2O Molecules, and the Automerization Reaction of Cyclobutadiene . 52 3.1.1 Computational Details . 55 3.1.2 Numerical Results . 57 3.2 Applying the Extension of the Semi-Stochastic CC(P) Approach to Sin- glet Excited Electronic States of the CH+ Ion . 75 3.2.1 Computational Details . 76 3.2.2 Numerical Results . 77 3.3 Approaching the Exact Quantum Chemistry via Cluster Analysis of the FCIQMC Wave Function . 81 3.3.1 Computational Details . 82 3.3.2 Numerical Results . 82 CHAPTER 4 COMPUTATIONAL IMPLEMENTATION AND ALGORITHMIC DETAILS OF THE SEMI-STOCHASTIC CC(P;Q) AND CAD- FCIQMC APPROACHES ......................... 87 4.1 Taking Advantage of the Sparse Stochastically Determined P Spaces to Improve the Performance of the Semi-Stochastic CC(P;Q) Method . 92 4.2 Optimizing the CAD-FCIQMC Approach by Exploiting On-The-Fly Contractions of the T4 Operator . 103 CHAPTER 5 CONCLUSIONS AND FUTURE OUTLOOK .............. 121 vii APPENDIX ........................................... 124 REFERENCES ......................................... 158 viii LIST OF TABLES Table 3.1: Convergence of the CC(P), CC(P;Q)MP, and CC(P;Q)EN energies to- ward CCSDT, where the P spaces consisted of all singles and doubles and subsets of triples identified during the i-FCIQMC, i-CISDTQ-MC, or i-CISDT-MC propagations with Dt = 0.0001 a.u. and where the corresponding Q spaces consisted of the triples not captured by the corresponding QMC simulations, for the F2/cc-pVDZ molecule in which the F–F distance R was set at Re, 1.5Re, 2Re, and 5Re, with Re = 2.66816 bohr representing the equilibrium geometry. The i- FCIQMC, i-CISDTQ-MC, and i-CISDT-MC calculations preceding the CC(P) and CC(P;Q) steps were initiated by placing 100 walkers on the RHF determinant and the na parameter of the initiator algorithm was set at 3. In all post-RHF calculations, the lowest two core orbitals were kept frozen and the Cartesian components of d orbitals were employed throughout. 62 Table 3.2: Convergence of the CC(P), CC(P;Q)MP, and CC(P;Q)EN energies to- ward CCSDT, where the P spaces consisted of all singles and doubles and subsets of triples identified during the i-FCIQMC, i-CISDTQ-MC, or i-CISDT-MC propagations with Dt = 0.0001 a.u. and where the corresponding Q spaces consisted of the triples not captured by the corresponding QMC simulations, for the F2 molecule in which the F–F distance R was set at twice the equilibrium bond length, using the cc-pVTZ and aug-cc-pVTZ basis sets, abbreviated as VTZ and AVTZ, respectively.
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