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C Copyright 2017 Joshua James Goings c Copyright 2017 Joshua James Goings Breaking symmetry in time-dependent electronic structure theory to describe spectroscopic properties of non-collinear and chiral molecules. Joshua James Goings A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2017 Reading Committee: Xiaosong Li, Chair David Masiello Stefan Stoll Program Authorized to Offer Degree: Chemistry University of Washington Abstract Breaking symmetry in time-dependent electronic structure theory to describe spectroscopic properties of non-collinear and chiral molecules. Joshua James Goings Chair of the Supervisory Committee: Professor Xiaosong Li Department of Chemistry Time-dependent electronic structure theory has the power to predict and probe the ways electron dynamics leads to useful phenomena and spectroscopic data. Here we report several advances and extensions of broken-symmetry time-dependent electronic structure theory in order to capture the flexibility required to describe non-equilibrium spin dynamics, as well as electron dynamics for chiroptical properties and vibrational effects. In the first half, we begin by discussing the generalization of self-consistent field methods to the so-called two-component structure in order to capture non-collinear spin states. This means that individual electrons are allowed to take a superposition of spin-1=2 projection states, instead of being constrained to either spin-up or spin-down. The system is no longer a spin eigenfunction, and is known a a spin-symmetry broken wave function. This flexibility to break spin symmetry may lead to variational instabilities in the approximate wave function, and we discuss how these may be overcome. With a stable non-collinear wave function in hand, we then discuss how to obtain electronic excited states from the non-collinear reference, along with associated challenges in their physical interpretation. Finally, we extend the two-component methods to relativistic Hamiltonians, which is the proper setting for describing spin-orbit driven phenomena. We describe the first implementation of the explicit time propagation of relativistic two-component methods and how this may be used to capture spin-forbidden states in electronic absorption spectra. In the second half, we describe the extension of explicitly time-propagated wave functions to the simulation of chiroptical properties, namely circular dichroism (CD) spectra of chiral molecules. Natural circular dichroism, that is, CD in the absence of magnetic fields, originates in the broken parity symmetry of chiral molecules. This proves to be an efficient method for computing circular dichroism spectra for high density-of-states chiral molecules. Next, we explore the impact of allowing nuclear motion on electronic absorption spectra within the context of mixed quantum-classical dynamics. We show that nuclear motion modulates the electronic response, and this gives rise to infrared absorption as well as Raman scattering phenomena in the computed dynamic polarizability. Finally, we explore the accuracy of several perturbative approximations to the equation-of-motion coupled-cluster methods for the efficient and accurate prediction of electronic absorption spectra. TABLE OF CONTENTS Page List of Figures . iii List of Tables . vii Glossary . ix Chapter 1: Perspectives on Non-Collinear and Two-Component Electronic Structure Theory . .1 Chapter 2: Stability of the Complex Generalized Hartree-Fock Equations . .9 Chapter 3: Approximate Singly Excited States from a Two-Component Hartree-Fock Reference . 24 Chapter 4: Real-Time Propagation of the Exact Two-Component Time-Dependent Density Functional Theory . 37 Chapter 5: An Atomic Orbital Based Real-Time Time-dependent Density Functional Theory for Computing Electronic Circular Dichroism Band Spectra . 51 Chapter 6: Can Quantized Vibrational Effects be Obtained from Ehrenfest Mixed Quantum-Classical Dynamics? . 64 Chapter 7: Assessment of Low-Scaling Approximations to the Equation-of-Motion Coupled-Cluster Singles and Doubles Equations . 75 Bibliography . 95 Appendix A: Brief Outline of (Generalized) Hartree-Fock Theory . 114 Appendix B: Broken Symmetries in the Hartree-Fock Model . 120 i Appendix C: Proof of Thouless' Theorem . 130 Appendix D: Derivation of the EOM-MBPT2 Equations . 135 ii LIST OF FIGURES Figure Number Page 1.1 Atomic magnetic moments of H2 during dissociation from equilibrium with non-collinear Ehrenfest method (B3LYP/6-31G). The molecule had an initial vibrational kinetic energy of 7.25 eV. As the hydrogen atoms separate over time, the local spins take on non-zero values, so that by full dissociation around 25 fs each has an atomic magnetic moment of 1=2 µB. The oscillatory phenomenon between 8 fs and 20 fs corresponding to an energy of 2.7 eV, corresponding to coherence between the singlet ionic state and the triplet ground state in the avoided crossing region. Dissociation follows along a mean Ehrenfest potential between different spin states. Adapted from Reference 1. .6 2.1 Local GHF energetic minima of hydrogen rings sizes 3 to 15. Odd-membered rings break the axial symmetry of the axis normal to the molecular plane; this is unique to GHF solutions and minimizes Pauli repulsion. The spin density is the 2-norm of the magnetization vector at each point in space, with blue being the greatest in magnitude. Charge density is represented as the electrostatic potential mapped to the total density, with red color showing a greater electronic population. 16 2.2 Local UHF energetic minima of hydrogen rings sizes 3 to 15. Unlike GHF, the ^ UHF cannot break Sz symmetry (hence collinear magnetizations), resulting in asymmetric spin distribution through the odd-numbered rings. This raises the energy compared to their GHF counterparts. The spin density is the 2-norm of the magnetization vector at each point in space, with blue being the greatest in magnitude. Charge density is represented as the electrostatic potential mapped to the total density, with red color showing a greater electronic population. 17 iii ^ ^2 2.3 Expected values of Sz and S operators on GHF (left) and UHF (right) solutions according to ring size. For n = 6 and n = 10, the solutions converge to a solution on the RHF manifold. These solutions are aromatic, and have a diamagnetic ring current. The anti-aromatic solutions at n = 4j (where j is an integer) show the greatest expected value S^2, consistent with a paramagnetic ring current. For odd-membered rings, GHF, unlike UHF, has a zero expected ^ value of Sz, the result of net cancellation of individual spin magnetic moments which is only possible with non-collinear methods. 20 2.4 GHF magnetizations of several degenerate microstates for H3 (a{d) and H4 (e,f). The uppermost magnetization vector of each microstate is arbitrarily aligned to the \laboratory z-axis". For H3, several microstates were found that ^ break the collinearity characteristic to Sz symmetry invariance. For H3, (a,b,d) lie in the plane of the page, however, (c) has magnetization vectors that go out of plane. For H4, GHF and UHF give the same solution and the microstates ^ are invariant to Sz symmetry. For this reason only two collinear microstates can be found. 21 2.5 Connection between higher order GHF solutions in the neutral chromium trimer (LANL2DZ, D3h, 2.89A).˚ There exists a high spin GHF solution (left), which also lies in the UHF manifold, that preserves collinearity. However, GHF stability tests indicate a lower solution exists, and the update scheme breaks the collinearity, slightly lowering the energy in the process (center). Subsequent iterations in the SCF bring this non-collinear rotated guess to the true GHF non-collinear minima (right). 23 4.1 Computed optical absorption spectra of (a) Zn, (b) Cd, and (c) Hg using RT-X2C-TDDFT within the SVWN5/Sapporo-DKH3-2012 level of theory with diffuse-sp functions. 47 4.2 Computed optical absorption spectra of AuH using RT-X2C-TDDFT within the SVWN5/Sapporo-DKH3-2012 level of theory with diffuse-sp functions. The AuH bond length corresponds to an experimental equilibrium length of 1.52385 A......................................˚ 49 4.3 Computed optical absorption spectra of TlH using RT-X2C-TDDFT within the SVWN5/Sapporo-DKH3-2012 level of theory with diffuse-sp functions. The TlH bond length corresponds to an experimental equilibrium length of 1.8702 A.˚ ..................................... 49 5.1 The chiral molecules considered in this study, (a) 2,3-(S,S)-dimethyloxirane (DMO) and, (b) α-1,3-(R,R)-pinene. They contain 166 and 332 basis functions, respectively. 62 iv 5.2 Computed ECD spectrum of (a) 2,3-(S,S)-dimethyloxirane (DMO) and, (b) α-1,3-(R,R)-pinene at PBEPBE/6-311+G**. The vertical lines correspond to analogous LR-TDDFT results for the first 100 states. An artificial Lorentzian damping was applied to give the RT-TDDFT peaks a line width of 0.1 eV. Lorentzian functions were added to the LR-TDDFT stick spectra, and the RT- TDDFT and LR-TDDFT lineshapes exactly coincide. Thus in the weak-field limit, LR and RT-TDDFT methods give essentially the same results, though RT-TDDFT computes the whole ECD band spectrum. 63 6.1 Computed optical absorption spectra of H2 and D2 using the Ehrenfest dy- namics approach at the RT-TDHF/STO-3G level of theory. Each molecule was displaced from its equilibrium bond length of R = 0:7122 A˚ to a slightly shorter bond length of R = 0:7100 A˚ (maximum displacement of 0:0022 A).˚ p Theoretically, D2 sidebands should be 1=2 eV less than those of H2, which H2 D2 is observed here. ~!N = 0:69 eV and ~!N = 0:48. Because displacement is small, the intensity of the central peak and sidebands is dominated by the first order terms. Thus, it is possible to see !e ± 2!N sidebands, though they are vanishingly small. 72 6.2 Computed optical absorption spectra of several stretched H2 using the Ehrenfest dynamics approach at the RT-TDHF/STO-3G level of theory.
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