Methods for the Electronic Structure of Large Chemical Systems Hong-Zhou Ye
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Methods for the Electronic Structure of Large Chemical Systems by Hong-Zhou Ye B.S., Peking University (2015) Submitted to the Department of Chemistry in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 ○c Massachusetts Institute of Technology 2020. All rights reserved. Author............................................................................ Department of Chemistry May 7, 2020 Certified by. Troy Van Voorhis Haslam and Dewey Professor of Chemistry Thesis Supervisor Accepted by....................................................................... Robert W. Field Haslam and Dewey Professor of Chemistry Chairman, Department Committee on Graduate Theses 2 This doctoral thesis has been examined by a Committee of the Department of Chemistry as follows: Professor Adam P. Willard. Chairman, Thesis Committee Associate Professor of Chemistry Professor Troy Van Voorhis . Thesis Supervisor Haslam and Dewey Professor of Chemistry Professor Heather J. Kulik . Member, Thesis Committee Associate Professor of Chemical Engineering 4 Methods for the Electronic Structure of Large Chemical Systems by Hong-Zhou Ye Submitted to the Department of Chemistry on May 7, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry Abstract An accurate description of the electronic structure of chemical systems is crucial to under- standing the atomistic mechanisms of many functional materials and designing new ones. In this thesis, we present methods that can potentially be used to computationally study the electronic structure of large chemical systems. The thesis can be roughly divided into two parts. In the first part, we focus on using quantum embedding (QE) theory to reduce the scaling of traditional electron correlation methods while maintaining their accuracy. Specifically, we extend bootstrap embedding (BE), a QE scheme that has displayed high performance for model Hamiltonians, to general chemical systems. Two challenges arise in such an extension. First, unlike the model systems, the basis functions of a real chemical system do not possess a uniquely defined connectivity and often lack high symmetries, which pose challenges on partitioning a chemical system into fragments that are necessary for a BE calculation (or other QE schemes). Second, the key to the success of BE on model Hamiltonians is the density matching between fragments that overlap with each other, which requires one to identify fragment centers and edges. Unlike model Hamiltonians where the fragments are regularly shaped – rendering the identification trivial – a real chemical system can have fragments that display complex patterns that make the distinction not obvious. To that end, we propose two fragment choices for partitioning a chemical system, one based on individual orbitals and the other on atoms, and systematically benchmark their effects on the performance of BE. Our finding is that the atom-based fragments are a better choice for BE, leading to fast convergence with the fragment size to the full-system calculations. We then develop an efficient implementation of atom-based BE using coupled cluster with singles and doubles (CCSD) as the local solver, and benchmark it on a series of conjugated molecules containing up to ∼ 2900 basis functions. Numerical tests confirm both the accuracy and computational efficiency of BE, rendering it a potential alternative to the more established local correlation methods. At the end of the first part, we also present another QE scheme, incremental embedding (IE), that does not rely on the connectivity between the units of a system (orbitals or atoms). In the second part, we switch our gear to develop self-consistent methods for locating excited states, which are complementary to the linear response-based methods for excited state calculations. Here, the difficulty is to avoid the variational collapse: unlike the ground state, which is the global minimum of the energy, excited states are often saddle points, making regular algorithms aimed at minimizing the energy collapse down to the ground state. To that end, we propose minimizing the energy variance, which is a minimum for all states and hence promises a numerically robust algorithm for locating them. To target a 5 specific state, we couple the variance minimization to a direct energy-targeting functional. The resulting method, which we dub 휎-SCF, can in principle locate any excited state by specifying a guess of the energy of the state. Numerical tests confirm that 휎-SCF solutions behave like the energy-stationary solutions. More importantly, for single excitations, 휎- SCF displays the pertinent spin symmetry breaking, which motivates us to improve it using spin projection. This effort leads to the half-projected (HP) 휎-SCF, which maintains the ability of 휎-SCF to effectively locate excited states, and significantly improves the results for singlet and triplet single excitations. In the conclusion, we comment on how self-consistent excited state methods could be combined with QE, hence enabling large-scale, correlated calculations for excited states. Thesis Supervisor: Troy Van Voorhis Title: Haslam and Dewey Professor of Chemistry 6 Acknowledgments First and foremost, I would like to thank my research advisor, Prof. Troy Van Voorhis, for his consistent support and patient encouragement on virtually every aspect of my research at MIT. It is under his guidance that I have gradually acquired interest in my current research area and prepared myself for exploring much more. I would like to thank my thesis committee, Prof. Adam P. Willard and Prof. Heather J. Kulik, for their time and advice, on not only my research, but also my academic career in the long run. I am also grateful to my undergraduate research advisor, Prof. Hong Jiang, for showing me the door of chemical theory and computation and supporting me for further pursuing that road in graduate school. The Van Voorhis group has always been a supportive place for me, and I would like to thank everyone in the group for making that happen. Particularly, I am grateful to Dr. Matthew Welborn and Dr. Nathan Ricke for their great mentorship in my rookie seasons, to Dr. Tianyu Zhu, Diptarka Hait, Prof. Zhou Lin and Prof. James Shepard for many useful discussions on research, to Henry Tran for his curiosity and hard work, to Alexandra McIsaac, Changhae Andrew Kim, Dr. Yael Cytter, Dr. Tamar Goldzak, Natasha Seelam and all other zoo residents for making me feel home in the zoo. Special thanks to Prof. Takashi Tsuchimochi: although you had left the group before I came to MIT, a big portion of my research has been inspired by your excellent works! Also, special thanks to Lexie, Andrew, and Henry for proofreading this thesis. I am also grateful to all my friends at MIT, especially Tian Xie, Jiayue Wang, Xinhao Li, Zhiwei Ding, Hejin Huang, Hongzi Mao, Manxi Wu, Ge Liu, and Danhao Ma, for their support and encouragement during my hard time at MIT. I would also like to thank my experimental collaborators, including Dr. Michelle MacLeod, Dr. Nathan Oldenhuis, Dr. Nolan Gallagher, and Dr. Yayuan Liu. It is the collaboration and discussion with them that has opened up a new world of chemistry in action to me. Special thanks to Prof. Jeremiah Johnson for many inspiring and enlightening discussions. Last but by no means the least, I am deeply grateful to my parents for their love and consistent support on all the decisions I have made. I am also greatly indebted to my girlfriend, Weiwei, for her accompanying through all the tough time of graduate school and being supportive to my career. Special thanks also go to Caomei, for helping me find bugs 7 in my code. This thesis is dedicated to all of you! 8 Contents 1 Theoretical background 25 1.1 Overview of electronic structure theory . 25 1.1.1 The Born-Oppenheimer approximation . 27 1.1.2 Representation of a state . 27 1.1.3 Second quantization . 29 1.1.4 Atomic orbitals and integral transform . 30 1.1.5 Density matrices . 31 1.2 Methods for electronic ground states . 32 1.2.1 Variational principle . 32 1.2.2 The Hartree-Fock theory . 33 1.2.3 Full configuration interaction, electron correlation, and size-consistency 34 1.2.4 Methods for weak electron correlation . 36 1.2.4.1 Truncated configuration interaction . 36 1.2.4.2 Møller-Plesset perturbation theory . 37 1.2.4.3 Coupled cluster theory . 38 1.2.5 Methods for strong electron correlation . 40 1.3 Methods for electronic excited states . 42 1.3.1 Linear response-based methods . 42 1.3.1.1 Linear response theory . 42 1.3.1.2 Time-dependent Hartree-Fock and configuration interaction with singles . 43 1.3.2 Self-consistent methods . 45 1.3.2.1 Single-reference methods . 45 1.3.2.2 Multi-reference methods . 46 9 1.4 Other theoretical tools . 47 1.4.1 Orbital localization . 47 1.4.2 Spin symmetry breaking and restoration . 47 1.5 Structure of this thesis . 48 2 Bootstrap embedding: general theory 53 2.1 Introduction . 53 2.2 Schmidt decomposition and embedding Hamiltonians . 56 2.3 Matching conditions and the BE equations . 57 2.4 The BE iteration algorithm . 59 2.5 Expectation values in BE . 61 2.6 Relation to other methods . 62 2.6.1 Complete active space methods . 62 2.6.2 Density matrix embedding theory . 63 2.6.3 Local correlation methods . 64 3 Bootstrap embedding using orbital-based fragments 65 3.1 Introduction . 65 3.2 Theory . 66 3.2.1 Orbital-centered fragments . 66 3.2.2 Normalized Coulomb metric . 68 3.2.3 Computational scaling . 68 3.3 Computational details . 69 3.4 Results and discussion . 71 3.4.1 Correlation energy at equilibrium geometry . 71 3.4.2 Homolytic cleavage of covalent bonds . 72 3.4.3 Polyacene chains . 74 3.5 Conclusion .