All-Electron Ground-State and Time-Dependent Density Functional Theory: Fast Algorithms and Better Approximations by Bikash Kanungo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering and Scientific Computing) in The University of Michigan 2019 Doctoral Committee: Associate Professor Vikram Gavini (Chair) Professor Krishna Garikipati Associate Professor Emanuel Gull Associate Professor Shravan Veerapaneni Associate Professor Paul M. Zimmerman Bikash Kanungo [email protected] ORCID iD: 0000-0002-3911-9354 c Bikash Kanungo 2019 Dedicated to Bou and Baba (mom and dad) ii ACKNOWLEDGEMENTS This work rests on the implicit and explicit support of many different people to whom I owe my deepest gratitude. First, to Vikram (Prof. Vikram Gavini) for being an immensely enterprising and exceedingly caring advisor, and for steadily and very patiently trying to make a mature researcher out of me. I can certainly vouch for the fact that he has lost as much sleep as me for some of the particularly challenging aspects of in this thesis. I thank all my committee members for the few yet valuable discussions. I thank Prof. Paul Zimmerman for some of the crucial data used in this thesis as well as for his contribution to one of the manuscripts. I thank Prof. Krishna Garikipati, Prof. Emanuel Gull and Prof. Shravan Veerapaneni for not only being generous committee members but also for being wonderful teachers in some of my graduate courses, which provided me with a sound understanding of the key subjects involved in this work. I thank all my current and former labmates|Ian, Nelson, Nathan, Paavai, Krish- nendu, Mrinal, Janakiraman, Bala|for the pleasure of an encouraging and joyous lab experience. I specially thank Sambit and Phani for the innumerable technical discussions, and, more importantly, for being two extraordinary human debuggers for my code. It has been my good fortune to have many supportive and caring friends. With a terrified sense of unwarranted omissions, I would like to extend my gratefulness to some of them. To Ram, Srayan, Isha, and Ananda for keeping me delectably fed as well as for tolerating my idiosyncrasies. To Vikas, Anurag, Aahana, and Suren, whose iii erudition has immensely shaped my philosophical perspective. To the friends at the Association for India's Development (AID) for all the parallel sociopolitical wisdom that has sharpened my overall critical outlook. Lastly, to my parents and my sister for their unconditional love and affection. iv TABLE OF CONTENTS DEDICATION :::::::::::::::::::::::::::::::::: ii ACKNOWLEDGEMENTS :::::::::::::::::::::::::: iii LIST OF FIGURES ::::::::::::::::::::::::::::::: vii LIST OF TABLES :::::::::::::::::::::::::::::::: x ABSTRACT ::::::::::::::::::::::::::::::::::: xii CHAPTER I. Introduction ..............................1 II. Density Functional Theory ..................... 11 2.1 Time-independent many-electron Schr¨odingerequation . 11 2.2 Ground-state density functional theory . 14 2.3 Time-dependent many-electron Schr¨odingerequation . 21 2.4 Time-dependent density functional theory . 22 III. Large-scale All-electron Density Functional Theory Calcula- tions Using an Enriched Finite Element Basis ......... 27 3.1 All-electron and Pseudopotenial DFT . 28 3.2 Enriched finite element basis|Motivation . 30 3.3 Real space formulation of Kohn-Sham DFT . 36 3.4 Classical finite element method . 39 3.5 Enriched finite element method . 43 3.6 Self-consistent field iteration and Chebyshev filtering . 57 3.7 Results and Discussion . 61 3.8 Summary . 71 v IV. Real Time Time-Dependent Density Functional Theory Us- ing Higher-order Finite Element Methods ............ 74 4.1 Introduction . 75 4.2 Real space time-dependent Kohn-Sham equations . 80 4.3 Semi- and full-discrete solutions . 84 4.4 Discretization Errors . 89 4.5 Efficient spatio-temporal discretization . 113 4.6 Numerical Implementation . 117 4.7 Results and discussion . 120 4.8 Summary . 141 V. Exact Exchange-correlation Potentials from Ground-state Elec- tron Densities ............................. 144 5.1 Introduction . 145 5.2 PDE-constrained optimization for inverse DFT . 147 5.3 Numerical Implementation . 149 5.4 Verification with LDA-based densities . 152 5.5 Removing Gaussian basis-set artifacts . 156 5.6 Exact vxc from CI densities for molecules . 158 5.7 Comparison with existing methods . 162 5.8 Summary . 164 VI. Conclusions ............................... 170 6.1 Summary . 170 6.2 Future directions . 175 APPENDIX :::::::::::::::::::::::::::::::::::: 179 BIBLIOGRAPHY :::::::::::::::::::::::::::::::: 186 vi LIST OF FIGURES Figure psp 3.1 Schematic of a pseudopotential. The vext is constructed such that the psp matches its corresponding ae, beyond a cutoff radius (rcut). 29 3.2 Schematic of truncated atomic orbital (radial part). 51 3.3 Convergence of energy with respect to element size for methane molecule 63 3.4 Convergence of energy with respect element size for carbon monoxide molecule . 63 3.5 Parallel scalability of the enriched finite element implementation. 71 4.1 Adaptive finite element mesh for all-electron Al2 (slice shown on the plane of the molecule) ............................. 116 4.2 Rates of convergence with respect to spatial discretization for LiH (all- electron) ................................. 124 4.3 Rate of convergence with respect to temporal discretization for LiH (all- electron) ................................. 124 4.4 Rates of convergence with respect to spatial discretization for CH4 (pseu- dopotential) ............................... 125 4.5 Rate of convergence with respect to temporal discretization for CH4 (pseu- dopotential) ............................... 125 4.6 Computational efficiency of various orders finite elements for LiH (all- electron) ................................. 127 4.7 Computational efficiency of various orders finite elements for CH4 (pseu- dopotential) ............................... 128 vii 4.8 Absorption spectra for Al2 ........................ 132 4.9 Absorption spectra for Al13. ....................... 133 4.10 Absorption spectra of Buckminsterfullerene ............... 135 4.11 Dipole power spectrum of Mg2 ...................... 137 4.12 Absorption spectra of methane ...................... 139 4.13 Absorption spectra of benzene ...................... 140 4.14 Parallel scalability of the higher-order finite element implementation. .. 141 5.1 Schematic of the inverse DFT problem. The exact ground-state many- body wavefunction (Ψ(r1; r2;:::; rNe )), and hence, the ground-state elec- tron density (ρ(r)) is obtained from configuration interaction calculation. The inverse DFT calculation evaluates the exact exchange-correlation po- tential (vxc(r)) that yields the given ρ(r). The ability to accurately solve the inverse DFT problem, presented in this work, presents a powerful tool to construct accurate density functionals (vxc[ρ(r)]), either through con- ventional approaches or via machine-learning. The schematic shows the ground-state density and the exact exchange-correlation potential for H2O obtained in this work. .......................... 146 5.2 Illustration of two adjacent 1D cubic finite elements|e1 and e2. The vertical dashed lines denote the boundary between adjacent elements. The black circles highlight the C0 continuity (cusp) of the basis at the element boundary. ................................. 150 5.3 The exchange-correlation potential (vxc) for various atomic systems, each corresponding to the local density approximation (LDA) based density (ρdata). The dashed line corresponds to the exchange-correlation poten- tial obtained from the inverse DFT calculation, and the solid line corre- sponds to the LDA exchange-correlation potential. The atomic systems considered are: (a) He; (b) Be; (c) Ne. ................. 154 5.4 (a): The vxc(r) (in a.u.) for 1,3-dimethylbenzene (C8H10) obtained from the inverse DFT calculation with the local density approximation (LDA) based ρdata, displayed on the plane of the benzene ring. (b): Relative error in the vxc for 1,3-dimethylbenzene (C8H10) obtained from the inverse DFT calculation with the local density approximation (LDA) based ρdata (refer Table A.3 in Appendices for coordinates). ................ 155 viii 5.5 Artifact of Gaussian basis-set based density. The exchange-correlation potential (vxc) is evaluated from inverse DFT, using ρdata obtained from a Gaussian basis-set based configuration interaction (CI) calculation for the equilibrium hydrogen molecule (H2(eq)). The lack of cusp in ρdata at the nuclei induces wild oscillations in the vxc obtained through inversion. The two atoms are located at r = 0:7 a.u. ............... 158 ± 5.6 Exchange-correlation potentials (vxc) for equilibrium hydrogen molecule (H2(eq)). A comparison is provided between the exact and the LDA based vxc potential. The exact exchange-correlation potential is evaluated using the cusp-corrected configuration interaction (CI) density. The effect of the choice of the functional used in evaluating the cusp correction is demon- strated using two different functionals|LDA (exact-∆ρLDA) and GGA (exact-∆ρGGA). .............................. 159 5.7 Exact vxc for stretched H2 molecules. The exact vxc is provided for two stretched hydrogen molecules: one at twice the equilibrium bond-length (H2(2eq)) and the other at dissociation (H2(d)). The H atoms for H2(2eq) and H (d) are located at r = 1:415 a.u. and r = 3:78 a.u., respectively. 160 2 ± ± 5.8 Nature and extent of electronic correlations in H2 molecules.