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Development of Full Configuration Interaction Quantum Monte Carlo Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems Von der Fakult¨atf¨ur Chemie der Universit¨at Stuttgart zur Erlangung der W¨urdeeines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung Vorgelegt von Werner Dobrautz aus Rottenmann, Osterreich¨ Hauptberichter: Prof. Dr. Ali Alavi Mitberichter: Prof. Dr. Hans-Joachim Werner Mitberichter: Prof. Dr. Andreas K¨ohn Pr¨ufungsvorsitzender: Prof. Dr. Wolfgang Kaim Tag der m¨undlichen Pr¨ufung: 26. M¨arz, 2019 Institut f¨urTheoretische Chemie der Universit¨atStuttgart 2019 An meinen Vater und Bruder... Contents Preface V Acronyms VII Abstract IX Abstract (German) / Kurzfassung XI 1 Introduction 1 1.1 Scope and Overview of the Thesis . .3 1.2 Electronic Structure Theory . .5 1.3 Symmetry in Electronic Structure Calculations . .8 1.4 The Hubbard Model . 10 1.5 Hartree-Fock Theory . 12 1.6 Correlation Energy . 13 1.7 Post-Hartree-Fock Methods . 15 1.7.1 Multi-Configurational Self-Consistent Field Method . 15 1.7.2 Configuration Interaction . 16 1.7.3 Perturbation Theory . 18 1.7.4 The Coupled Cluster Approach . 19 1.7.5 Density Matrix Renormalization Group . 20 1.7.6 Full Configuration Interaction . 22 1.7.7 Explicitly Correlated Methods . 23 2 The (Quantum) Monte Carlo Method 25 2.1 The Monte Carlo Method . 25 2.1.1 Importance Sampling . 27 2.1.2 Markov Chains and the Metropolis-Hastings Algorithm . 27 2.2 Bringing Quantum into Monte Carlo . 28 2.2.1 The Variational Monte Carlo Method . 29 2.2.2 The Projector Monte Carlo Method . 30 2.2.3 The Diffusion Monte Carlo Method . 31 I 2.2.4 The Green's Function Monte Carlo method . 32 2.2.5 Auxiliary Field Quantum Monte Carlo . 33 2.2.6 The Sign Problem in Quantum Monte Carlo . 35 2.3 The Full Configuration Interaction Quantum Monte Carlo Method (FCIQMC) 37 2.3.1 Derivation of FCIQMC . 37 2.3.2 The Three Ingredients to FCIQMC . 39 2.3.3 The Initiator Approximation . 41 2.3.4 Energy Estimators . 43 2.3.5 Stochastic Sampling of Density Matrices . 46 2.3.6 Excited States within FCIQMC . 46 2.3.7 Automated Timestep Optimization . 47 2.3.8 Optimized Excitation Generation . 48 3 Spin as a Symmetry Property 51 3.1 A Brief History of Spin . 52 3.2 The Use of Spin Symmetry in Electronic Structure Calculations . 53 3.3 The Unitary Group Approach (UGA) . 58 3.3.1 The Generators of U(n)........................ 60 3.3.2 The Gel'fand-Tsetlin Basis . 63 3.3.3 The Pure Spin Space - U s(2) . 66 3.3.4 Matrix Representation of U(2) and SU(2) . 68 3.3.5 Connection of U(2) and SU(2) . 71 3.3.6 The Pure Orbital Space - U o(n).................... 73 3.3.7 The Paldus tableau . 73 3.4 The Graphical Unitary Group Approach (GUGA) . 76 3.4.1 Evaluation of Nonvanishing Hamiltonian Matrix Elements . 78 3.4.2 Two-Body Matrix Elements . 81 4 Spin-Adapted Full Configuration Interaction Quantum Monte Carlo 87 4.1 Excitation Generation: Singles . 88 4.1.1 The Branching Tree . 89 4.1.2 Remaining Switches . 93 4.1.3 On-The-Fly Matrix Element Calculation . 93 4.1.4 Branch Weights . 94 4.2 Excitation Generation: Doubles . 95 4.3 Histogram based Timestep Optimization . 101 4.4 Results from the Spin-Adapted FCIQMC Approach . 103 4.4.1 Nitrogen Atom . 103 4.4.2 Nitrogen Dimer Binding Curve . 106 4.4.3 One-Dimensional Hydrogen Chain . 110 4.4.4 The Hubbard Model . 112 II 4.4.5 Computational Effort and Scaling of GUGA-FCIQMC . 115 4.4.6 3d Transition Metal Atoms . 117 4.5 Conclusion and Outlook . 122 5 Explicitly Correlated Ansatz in FCIQMC 127 5.1 Nonunitary Similarity Transformations . 128 5.2 Introduction . 128 5.3 The Similarity Transformed Hamiltonian . 130 5.3.1 Recap of the Derivation of H¯ ..................... 132 5.3.2 Optimisation of J and Analytic Results for the Hubbard Model . 135 5.3.3 Exact Diagonalization Study . 138 5.4 The Similarity Transformed FCIQMC Method . 139 5.4.1 Excited States of non-Hermitian Operators with ST-FCIQMC . 142 5.5 Results of the ST-FCIQMC Method . 146 5.5.1 18-site Hubbard Model . 147 5.5.2 Results for the 36- and 50-site Hubbard Model . 152 5.6 Conclusion and Outlook . 155 5.6.1 The real-space Hubbard Formulation . 157 5.6.2 The Heisenberg Model . 158 5.6.3 The t-J Model . 158 6 Discussion and Conclusion 161 A The Graphical Unitary Group Approach 165 A.1 CSF Excitation Identification . 165 A.2 Detailed Matrix Element Evaluation in the GUGA . 169 A.3 Weighted Orbital Choice with GUGA Restrictions . 180 B The Similarity Transformed Hamiltonian 189 B.1 Analytic Optimization of J in the Thermodynamic Limit at Half-Filling . 189 Bibliography 192 Acknowledgements 221 III Preface This dissertation contains the results of my research carried out at the Max Planck Insti- tute for Solid State Research between November 2014 and November 2018 in the group of and under supervision of Professor Ali Alavi. Parts of Chapter 5 are included in work that is published as: Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations. Werner Dobrautz, Hongjun Luo and Ali Alavi in Phys. Rev. B 99, 075119 (2019) This dissertation is a result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This disser- tation has not been submitted in whole or in part for any other degree or diploma at this or any other university. Stuttgart, Date Werner Dobrautz V Erkl¨arung¨uber die Eigenst¨andigkeit der Dissertation Ich versichere, dass ich die vorliegende Arbeit mit dem Titel Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems selbst¨andigverfasst und keine anderen als die angegebenen Quellen und Hilfsmittel be- nutzt habe; aus fremden Quellen entnommene Passagen und Gedanken sind als solche kenntlich gemacht. Declaration of Authorship I hereby certify that the dissertation entitled Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems is entirely my own work except where otherwise indicated. Passages and ideas from other sources have been clearly indicated. Name/Name: Unterschrift/Signed: Datum/Date: Acronyms AFQMC Auxiliary-field Quantum Monte Carlo AO Atomic Orbital aug-cc-pVnZ Augmented Correlation-Consistent Polarized Valence-only n-Zeta basis set aug-cc-pwCVnZ-DK Relativistic-contracted Augmented Correlation-Consistent Polarized Weighted Core-Valence n-Zeta Douglas-Kroll basis set CAS Complete Active Space CASSCF Complete Active Space Self-Consistent Field CBS Complete Basis Set CC Coupled Cluster CCSD Coupled Cluster Singles and Doubles CCSD(T) Coupled Cluster Singles and Doubles with perturbative Triples cc-pVnZ Correlation-Consistent Polarized Valence-only n-Zeta basis se CISD Configuration Interaction Singles and Doubles CNDO Complete Neglect of Differential Overlap CP Constrained Path CPMC Constrained Path Quantum Monte Carlo CSF Configuration State Function DMC Diffusion Monte Carlo DMRG Density Matrix Renormalization Group DRT Distinct Row Table ED Exact Diagonalization ESC Electronic Structure Calculation FCI Full Configuration Interaction FCIQMC Full Configuration Interaction Quantum Monte Carlo GFMC Green's Function Monte Carlo GTO Gaussian-Type Orbital GUGA Graphical Unitary Group Approach HF Hartree-Fock VII HS Hubbard-Stratonovich i-FCIQMC Initiator Full Configuration Interaction Quantum Monte Carlo irrep Irreducible Representation KY Kotani-Yamanouchi LCAO Linear Combination of Atomic Orbitals MB Minimal Basis MC Monte Carlo MCMC Markov Chain Monte Carlo MCSF Multi-Configurational Self-Consistent Field MO Molecular Orbital MPn nth order Møller-Plesset MPPT Møller-Plesset Perturbation Theory MPS Matrix Product State MRCI Multi-Reference Configuration Interaction OBC Open Boundary Conditions PBC Periodic Boundary Conditions PDF Probability Distribution Function PMC Projector Monte Carlo PPP Pariser-Parr-Poble PT Perturbation Theory QMC Quantum Monte Carlo RDM Reduced Density Matrix RHF Restricted Hartree-Fock ROHF Restricted Open-shell Hartree-Fock SCF Self-Consistent Field SD Slater Determinant SGA Symmetric Group Approach STO Slater-Type Orbital SU(n) Special Unitary Group of order n U(n) Unitary Group of order n UGA Unitary Group Approach UHF Unrestricted Hartree-Fock VMC Variational Monte Carlo ZDO Zero Differential Overlap VIII Abstract Full Configuration Interaction Quantum Monte Carlo (FCIQMC) is a prominent method to calculate the exact solution of the Schr¨odingerequation in a finite antisymmetric basis and gives access to physical observables through an efficient stochastic sampling of the wavefunction that describes a quantum mechanical system. Although system-agnostic (black-box-like) and numerically exact, its effectiveness depends crucially on the compact- ness of the wavefunction: a property that gradually decreases as correlation effects become stronger. In this work, we present two|conceptually distinct|approaches to extend the applicability of FCIQMC towards larger and more strongly correlated systems. In the first part, we investigate a spin-adapted formulation of the FCIQMC algorithm, based on the Unitary Group Approach. Exploiting the inherent symmetries of the non- relativistic molecular Hamiltonian results in a dramatic reduction of the effective Hilbert space size of the problem. The use of a spin-pure basis explicitly resolves the different spin-sectors, even when degenerate, and the absence of spin-contamination ensures the sampled wavefunction is an eigenfunction of the total spin operator S^2. Moreover, tar- geting specific many-body states with conserved total spin allows an accurate description of chemical processes governed by the intricate interplay of them. We apply the above methodology to obtain results, not otherwise attainable with conventional approaches, for the spin-gap of the high-spin cobalt atom ground- and low-spin excited state and the electron affinity of scandium within chemical accuracy to experiment.
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