Postmodern Electronic Structure Theory by Joonho Lee A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Martin P. Head-Gordon, Chair Professor Eran Rabani Professor Lin Lin Summer 2019 Postmodern Electronic Structure Theory Copyright 2019 by Joonho Lee 1 Abstract Postmodern Electronic Structure Theory by Joonho Lee Doctor of Philosophy in Chemistry University of California, Berkeley Professor Martin P. Head-Gordon, Chair This dissertation is concerned with the development and applications of approaches to the electron correlation problem. We start with an introduction that summarizes modern approaches to the electron correlation problem. In our view, there are two remaining challenges that modern density functional theory cannot satisfactorily solve. The first challenge is due to self-interaction error and the second is due to strong correlation. We discuss two methods developed by the author that attempt to make progress to address the second challenge. The first approach is useful in distinguishing strong and weak correlation in a computationally economical way. It is based on orbital optimization in the presence of regularized second-order Møller-Plesset perturbation theory (κ-OOMP2), which is an approximate method to obtain Br¨uckner orbitals. κ-OOMP2 includes weak correlation while attenuating strong correlation. As such, it distinguishes artificial and essential symmetry breaking which occur at the Hartree-Fock (HF) level. Ar- tificial symmetry breaking appears due to the lack of weak correlation, not due to the lack of strong correlation. Therefore, the common wisdom in quantum chem- istry, which equates symmetry breaking at the HF level and strong correlation, can result in a wrong understanding of the system. Essential symmetry breaking, on the other hand, signals strong correlation that is beyond the scope of simple perturba- tion theory. κ-OOMP2 has been shown to reliably distinguish these two: symmetry breaking in the κ-OOMP2 orbitals is essential. This has been applied to a recent con- troversy about whether C60 is strongly correlated. Starting from a broken-symmetry HF solution, κ-OOMP2 restores every symmetry. As such, C60 is not strongly cor- related. Moreover, κ-OOMP2 successfully predicts strong correlation for a known biradicaloid, C36, by showing essential symmetry breaking in its orbitals. We also exploited essential symmetry breaking in singlet biradicaloids using κ-OOMP2 and 2 showed quantitative accuracy in obtaining singlet-triplet gaps of various molecules. This new approach should be helpful for redefining the common wisdom in quantum chemistry. The second method is an exact, spin-pure, polynomial-scaling way to describe strong spin-correlation (SSC). SSC is present when there are many spatially separate open-shell electrons with small spin-flip energy cost. Describing SSC exactly requires the inclusion of all essential spin-couplings. The number of such spin-couplings scales exponentially with the number of electrons. Because of this, SSC was thought to require an exponential number of wavefunction parameters in general. However, new development suggests that there is an efficient way to obtain all these spin- couplings with only a quadratic number of wavefunction parameters, which is called the coupled-cluster valence-bond (CCVB) method. We discuss different challenges in CCVB: (1) its non-black-box nature and (2) its inability to describe SSC in spin- frustrated systems. We present two improved CCVB approaches that address these two challenges. These approaches were applied to describe emergent strong corre- lation in oligoacenes and SSC in spin-frustrated systems such as single molecular magnets and metalloenzymes. The remaining challenges in CCVB are the inclu- sion of ionic excitations which are not relevant for SSC, but crucial for obtaining quantitative accuracy. i To my wife Soojin ii Acknowledgments The first to thank is my advisor, Prof. Martin Head-Gordon. The best academic decision I made so far was to move from Caltech to Berkeley to work with Martin. The transfer was very difficult and the intellectual stimuli provided by Martin were the only things that I could hold on to in the beginning. I am proud of having learned and contributed to his scientific philosophy and legacy. Among many things, I would like to thank Martin for having taught me how to critically assess not only other people's work but also my own work. Most importantly, I thank Martin for giving me intellectual freedom for the last 4 years. Without this freedom, I would not have become the scientist that I am now. I also thank Martin for proofreading every chapter of this dissertation carefully. Any remaining errors are my own. I would like to thank everyone with whom I interacted during my time in Berkeley. I must thank Prof. Eran Rabani and Prof. Lin Lin for giving me career advice and being approachable whenever I needed their help. I must thank Dr. Dave Small for teaching me everything I know about coupled-cluster theory. The second best thing that happened to me during my time in Berkeley was to work with Dave early on. I must thank Dr. Evgeny Epifanovsky for encouraging me to reimplement anything that interests me in Q-Chem. Without his encouragement, I would not have learned as much about scientific programming as I know now. I must thank Luke Bertels, Srimukh Prasad, and Matthias Loipersberger for being best friends, always standing by me, and working with me on projects that have taught me a lot. I spent more time with you guys than with my wife Soojin during Ph.D. and I am grateful for the quality time we had together. I must thank Evgeny once more, Dr. Jerome Gonthier, and Dr. Eloy Ramos for sharing the courage I had about periodic boundary conditions and following me down the path of endless implementations of missing features when no publication was guaranteed. Lastly, I thank other people with whom I interacted through Martin's group including Eric Sundstrom, Paul Horn, Narbe Mardirossian, Jonathan Thirman, Yuezhi Mao, Jon Witte, Alec White, Jon Weisberg, Brad Ganoe, Leo Cunha, Dr. Shane Yost, Dr. Udi Tsivion, Dr. Roberto Pevrati, Dr. Xintian Feng, Dr. Cameron Mackie, Dr. Tim Stauch, and Dr. Christopher Stein for being good co-workers. I must thank Eric once again for convincing me to transfer to Berkeley when I was unhappy about my scientific development. I thank Jessie Woodcock for making sure that I get paid on time. Outside the Head-Gordon group, I have many individuals to thank. First, I thank Prof. Eric Neuscamman, Luning Zhao, Brett Van Der Goetz, Sergio Flores, Dr. Peter Walters, and Leon Otis for welcoming me to the Neuscamman group's meetings and teaching me some aspects of variational Monte Carlo. I thank Luning once more for good friendship and always giving me good scientific arguments. I thank Lin again, iii Jiefu Zhang, Leo Zepeda, Michael Lindsey, and others in the Lin group for warm hospitality when I joined their meetings at Evans and teaching me useful applied math tricks from time to time. I thank Prof. Birgit Whaley and Donghyun Lee for welcoming me back to Berkeley and helping me finish my undergraduate project with them. I thank Donghyun again for sharing his dissertation template which saved me weeks when writing this dissertation. I must thank Birgit again and Bill Huggins for teaching me many things (mostly things that are not yet possible!) about quantum computing. I thank Dr. Miguel Morales and Fionn Malone for teaching me almost everything I know about quantum Monte Carlo (QMC). I thank Fionn once more for being patient with me when I ask millions of questions about QMC. I would also like to thank people from my Caltech time who helped me prepare for my Ph.D. journey at Berkeley. I thank Prof. Fred Manby for teaching me tricks in density functional theory and joys of working on quantum chemistry early on. I thank Mike Webb, Michiel Niesen, Dr. Pengfei Huo, Mark Fornace, and Kevin Shen for being good colleagues with me. I would like to thank Luke, Srimukh, and Fionn for proofreading the abstract, introduction, and conclusions of this dissertation. Finally, I am truly grateful to my family for all their support and patience. Thank you Soojin for your love, patience, understanding, encouragement, and everything else. I look forward to building our new home together (briefly in Berkeley and) in New York City and our future afterwards. Thank you my dad Hyungkeun, my mom Jungsook, and my sister Eunkyung for consistent support throughout my academic journey. iv Contents Contents iv List of Figures viii List of Tables xiii Note on Previously Published Work xix 1 Introduction 1 1.1 Modern Quantum Chemistry . 2 1.2 Author's Views on Pople's Criteria . 5 1.3 Challenges Addressed in This Dissertation . 8 1.4 Works Not Mentioned in This Dissertation . 13 1.5 Outline . 14 2 Regularized Orbital-Optimized Second-Order Møller-Plesset Per- turbation Theory 17 2.1 Introduction . 17 2.2 Theory . 20 2.2.1 OOMP2 Lagrangian in Spin-Orbital Basis . 20 2.2.2 Regularized OOMP2 . 24 2.3 Computational Details . 28 2.4 Results and Discussion . 29 2.4.1 ROOMP2 to UOOMP2 Stability Analysis . 29 2.4.2 Training the Regularization Parameter and Its Validation . 31 2.4.3 Application to Organic Singlet Biradicaloids . 37 2.5 Conclusions . 42 3 Distinguishing Artificial and Essential Symmetry Breaking in a Single Determinant 49 v 3.1 Introduction . 49 3.2 Theory . 52 3.2.1 Classification of HF Solutions . 52 3.2.2 Generalized MP2 . 53 3.2.3 Regularized Orbital-Optimized MP2 . 54 3.2.4 Classification of OOMP2 Solutions .