UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Improving Wavefunction Efficiency by Tessellating Correlation Factors and Coupled State- Specific Optimization Permalink https://escholarship.org/uc/item/4j57f3q0 Author Van Der Goetz, Beatrice Weston Publication Date 2021 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Improving Wavefunction Efficiency by Tessellating Correlation Factors and Coupled State-Specific Optimization by Beatrice Van Der Goetz 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 Eric Neuscamman, Chair Professor Martin Head-Gordon Professor James Demmel Spring 2021 Abstract Improving Wavefunction Efficiency by Tessellating Correlation Factors and Coupled State-Specific Optimization by Beatrice Van Der Goetz Doctor of Philosophy in Chemistry University of California, Berkeley Professor Eric Neuscamman, Chair Rearranging chemical bonds is chemistry. Simulating chemical reactions is an expen- sive and complex process, and is necessary to understand the photochemical reactions that drive processes like chemical light-harvesting. The electronic many-body physics describing the bonds that participate in these processes becomes complicated and expensive even on modestly sized molecules, and computationally affordable approximations can qualitatively fail. One approach to this problem relies on devising compact and expressive wavefunction forms that are simple enough to be efficiently computed yet complex enough to capture the subtleties of many-electron physics. Due to their relaxed integrability conditions, position-space Quantum Monte Carlo meth- ods permit the use of flexible wavefunction components like Jastrow correlation factors that can be used to exactly express wavefunction cusps, which are otherwise difficult to describe. As many existing factors used in these calculations focus primarily on short-range or gen- eral correlation effects, we aim to augment the library of real-space correlation factors by developing one designed to handle the strong electronic correlation of bond-breaking. These factors do this by accounting for correlations between populations of electrons in different pockets of space using a set of indicator-like functions fashioned into a tessellation of Voronoi cells. These Voronoi cells can be automatically constructed around atomic coordinates or further tailored to the chemical system, as the math describing them is flexible enough to allow more subtle intra-atomic subdivisions and curved interfaces. In simple test systems, this factor is able to correctly cleave chemical bonds when applied to a single-determinant reference, and the resulting wavefunction is competitive with highly-accurate but expensive methods. Just as wavefunctions can be limited by their functional form, so too they can be limited by the definition of their reference point. Many excited state methods are based on a linear response formalism, in which excited states are generated in the language of perturbations applied to a fixed ground state. When the wavefunction qualitatively changes upon elec- tronic excitation – as they do in charge-transfer states and core excitations – these methods can fail to predict excitation energies with errors on the scale of several electron-volts. Op- timizing molecular orbitals for individual excited states is one particularly efficient way to make the necessary zeroth-order changes to capture these states, but these state-specific 1 methods can suffer from instabilities in their optimization. One particular benefit of this state-specific approach is that these tailored orbitals naturally compress the wavefunction, but in certain cases, the ground state can interfere when minimal representations of the excited state are sought, ultimately causing variational collapse. We show that this collapse behavior occurs in two different state-specific approaches, and show how this arises from an inadequately-modeled avoided crossing, and argue that orbital-CI coupling plays a key role in its prevention. Newer excited state methods have also seen the use of target functions, where functions of the energy such as the square-gradient magnitude are minimized in place of the energy in order to stabilize state-specific optimization by transforming saddle points on the energy surface into minima on the target function surface. While pilot implementations of square- gradient-based optimizers for the Excited-State Mean-Field (ESMF) wavefunction are able to obtain state-specific orbitals at low cost, these are still new and have yet to benefit from numerical accelerations, limiting their use. For instance, while stable, the quasi-Newton optimizer in the ESMF-GVP implementation uses no Hessian preconditioner, and while stable to variational collapse due to its inclusion of orbital-CI coupling, is much slower relative to the occasionally-unstable ESMF-SCF implementation. Using the exact Hessian and full Newton-Raphson optimization as a benchmark, we explore a variety of Hessian approximations, and find that an approximate diagonal Hessian can accelerate the ESMF- GVP square-gradient minimization to match the speed of the gradient-only ESMF-SCF at mean-field cost-scaling while resisting variational collapse. 2 for mom Contents Acknowledgements iv List of Figures v List of Tables xi 1 Introduction 1 1.1 Quantum Mechanics . .2 1.2 Electronic Wavefunctions . .3 1.3 Second Quantization . .6 1.4 Mean-Field Theory . .8 1.5 H2 Dissociation . .9 1.6 Density Matrices . 12 1.7 Twisted Ethene . 13 1.8 Strong Correlation Overview . 16 1.9 Weak Correlation and Wavefunction Cusps . 18 1.10 Quantum Monte Carlo . 21 1.11 Variational Monte Carlo . 25 1.12 Diffusion Monte Carlo . 27 2 Number-Counting Jastrow Factors 32 2.1 Abstract . 32 2.2 Introduction . 32 2.3 Mimicking Hilbert Space Jastrows . 33 2.4 Existing 4-body Jastrows . 35 2.5 Mathematical formulation . 36 2.6 Results . 38 2.7 Hydrogen Molecule . 39 2.8 Ethene . 40 2.9 Curvature Hiding . 43 2.10 Basis Construction Schemes . 45 2.11 Conclusions . 47 3 Automated Spatial Tessellations for Correlation Factors 49 3.1 Abstract . 49 3.2 Introduction . 49 3.3 Review . 51 3.4 Normalized Counting Functions . 59 3.5 Classical Voronoi Partitioning . 61 3.6 Spherical Voronoi Partitioning . 62 3.7 Region Composition . 64 3.8 Results . 66 3.9 Ethene . 67 ii 3.10 Random Planar H4 ................................ 70 3.11 Calcium Oxide . 74 3.12 Conclusions . 81 4 Stable and Fast Excited State Orbital Optimization 83 4.1 Introduction . 83 4.2 Variational Collapse . 86 4.3 Root Flipping . 88 4.4 Coupled Collapse . 91 4.5 Methanimine . 93 4.6 Excited State Mean Field . 100 4.7 ESMF-SCF Collapse . 102 4.8 Quasi-Newton Preconditioning . 104 4.9 Hessian Preconditioning . 105 4.10 Conclusions . 109 5 Concluding Remarks 111 Bibliography 114 A Linear Dependencies 140 B Spherical Voronoi Partitioning 142 C Region Composition 143 D ESMF Derivatives 146 D.1 Wavefunction and Notation . 146 D.2 Energy . 147 D.3 First Derivatives . 147 D.4 Second Derivatives . 148 iii Acknowledgements It has taken nearly a decade and finding myself here feels all but impossible. Many deserve acknowledgement, for their mentorship, support, and friendship. Edward Perry, whose class in organic chemistry opened my eyes to its beauty; James Vyvyan and Paul Deck, for introducing me to the world of research; Daniel Crawford, for an incomparable introduction to quantum chemistry; Jim Tanko and Hervè Marand, for wonderful lectures and the longest semester of my undergraduate career; Bill Floyd, for precise and lucid lectures on Real Analysis; Susan Bell, Maria Cristina-Mendoza, and Linda Carlson for all their help. Eric Neuscamman, for personal support, dedication to the craft of research, insightful mentorship, and our many, many enlightening conversations; Luning, Jacki, Sergio, Becky, Connie, Tarini, Scott, and Rachel, for being a wonderful research group; Brad, Alex, Will, Jessalyn, Harvey, and Hazel, who I will never forget; Alexia, for everything, and who I wish nothing but the best; Kali and Taeko, for being there for me; Dad and Julie, for their constant encouragement; Erik, Brittany, and Lauren, for being the best siblings I could ever have, Mom, for all your love and support, without which I would not be here. iv List of Figures 1.1 Dissociation of the dihydrogen molecule as a function of the distance between two hydrogen atoms. The upper curve, calculated by spin-restricted Hartree- Fock (RHF), has a wavefunction determined entirely by spin-symmetry, and is given by equation (1.43) at both points A and B. The lower curve corresponds to a near-exact result obtained by exhaustively diagonalizing the Hamiltonian in the full determinant space with a large single-particle basis (FCI in cc- pV5Z). This near-exact wavefunction at equilibrium (point C) is qualitatively captured by the single-determinant wavefunction (point A) while the near- exact wavefunction at stretched geometries (point D), given approximately by equation (1.48) is not. 10 1.2 Plot of electronic energy as a function of HCCH dihedral angle in ethene. Single determinant wavefunctions with doubly occupied π and π∗ orbitals (red and orange curves respectively, wavefunctions given in equation (1.58)) are good approximations to the equilibrium ground state wavefunction at points A and C respectively. These single-determinant