c Copyright 2016 Huihuo Zheng.

FIRST PRINCIPLES QUANTUM MONTE CARLO STUDY OF CORRELATED ELECTRONIC SYSTEMS

BY

HUIHUO ZHENG

DISSERTATION

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2016

Urbana, Illinois

Doctoral Committee:

Associate Professor Shinsei Ryu, Chair Research Assistant Professor Lucas K. Wagner, Director of Research Professor David M. Ceperley Professor S. Lance Cooper Professor So Hirata Abstract

The many-body correlation between electrons is the origin of many fascinating phenomena in condensed

matter systems, such as high temperature superconductivity, superfluidity, fractional quantum Hall effect,

and Mott insulator. Strongly correlated systems have been an important subject of condensed matter physics

for several decades, especially after the discovery of high temperature cuprate superconductors. In this thesis,

we apply first principles quantum Monte Carlo (QMC) method to several representative systems to study the

electron correlations in transition metal oxides (vanadium dioxide) and low dimensional electronic systems

(graphene and graphene-like two dimensional systems).

Vanadium dioxide (VO2) is a paradigmatic example of a strongly correlated system that undergoes a metal-insulator transition at a structural phase transition. To date, this transition has necessitated significant

post-hoc adjustments to theory in order to be described properly. We apply first principles quantum Monte

Carlo (QMC) to study the structural dependence of the properties of VO2. Using this technique, we simulate the interactions between electrons explicitly, which allows for the metal-insulator transition to naturally

emerge, importantly without ad-hoc adjustments. The QMC calculations show that the structural transition directly causes the metal-insulator transition and a change in the coupling of vanadium spins. This change in the spin coupling results in a prediction of a momentum-independent magnetic excitation in the insulating state. While two-body correlations are important to set the stage for this transition, they do not change significantly when VO2 becomes an insulator. These results show that it is now possible to account for electron correlations in a quantitatively accurate way that is also specific to materials.

Electron correlation in graphene is unique because of the interplay of the Dirac cone dispersion of π

electrons with long range Coulomb interaction. The random phase approximation predicts no metallic

screening at long distances and low energies because of the zero density of states at Fermi level. It is

thus interesting to see how screening takes place in graphene at different length scales. We addressed this

problem by computing the structure factor S(q) and S(q, ω) of freestanding graphene using ab initio fixed-

node diffusion Monte Carlo and the random phase approximation. The X-ray measured structure factor

is reproduced very accurately using both techniques, provided that σ-bonding electrons are included in

ii the simulations. Strong dielectric screening from σ electrons are observed, which redshifts the π plasmons resonance frequency at long distance and reduces the effective interactions between π electrons at short distance. The short distance screening makes suspended graphene a weakly correlated semimetal which otherwise would be an insulator.

The third piece of works is dedicated to studying the low energy excitation of many-body systems using extended Koopmans’ theorem (EKT). The EKT provides a straight forward way to compute charge excitation spectra, such as ionization potentials, electron affinities from any level of theory. We implemented the EKT within the QMC framework, and performed systematic benchmark studies of ionization potentials of the second- and third-row atoms, and closed- and open-shell molecules. We also applied it to compute the quasiparticle band structure of solids (graphene).

For complex correlated systems, identifying relevant low energy physics degrees of freedom is extremely important to understanding the system’s collective behavior at different length scales. In this sense, bridging the realistic systems to lower energy effective lattice models that involve fewer but important degrees of freedom is significant to understanding correlated systems. We have formulated three ab initio density matrix based downfolding (AIDMD) methods to downfold the ab initio systems into effective lattice models.

We have demonstrated the successfulness of these methods by applying them to molecules (H2) and periodic systems (hydrogen chain and graphene).

iii To my parents, and Shasha Miao

iv Acknowledgements

This thesis cannot be completed without the support from many people. First, I would like to thank my advisor Lucas K. Wagner for his support and encouragement throughout these years. Special thanks for many hand-by-hand teachings in the early days of my PhD study. Lucas’s enthusiasm for physics, his down- to-earth style of approaching a physics problem, have deeply influenced me. His helpful critiques always motivate me to push on in many of my studies and make me never be satisfied with superficial understanding.

He is an excellent advisor especially in helping his students like me to develop things that are necessary for us to survive in academia.

I also thank other group members for helpful discussions inside and outside the group meetings. Prof.

David M. Ceperley’s physics intuition always impresses me. His deep and broad knowledge on quantum

Monte Carlo and theory of electronic structure as general are always helpful to me. I also would like to thank Hitesh Changlani for teaching me the exact diagonization of lattice models, and Awadhesh Narayan for collaboration on transition metal dichalcogenides. Thanks to other group members for many enlightening discussions as well.

Thanks to many of my classmates, including En-Chuang Huang, Wenchao Xu, Xueda Wen, Xiao Chen,

Can Zhang, and Hong-Yan Shih, and Taylor Byrum, for running the PhD race together with me. Thanks for the mutual encouragement throughout these years.

Thanks to all the brothers and sisters in the Church in Champaign for the bountiful supply. Thanks to their prayers and petition, and all the fellowship and leading in the Body. Their firm standing on their faith is a great encouragement to me. I live if they stand firm in the Lord! The Body of Christ has been my great supply during my PhD studies.

Thanks to my parents for the long term supply. Thanks to them for allowing me to leave far away from them and pursue something that they are not able to understand. Thanks for their encouragement. Thanks to my beloved fianc´ee,Shasha Miao, for being with me in the final stage of my PhD. Thanks for her prayer and encouragement in love in Christ.

Finally, thanks to the Lord for the shepherding! His grace is always sufficient to me. He is the treasure

v in my earthen vessel. Thanks to the Lord for His presence and being with me throughout the days. It is

Christ that makes the whole PhD process meaningful to transform me and to conform me to His image for the eternal glory ready to be revealed at His second coming. Surely goodness and lovingkindness will follow me all days of my life. And I will dwell in the temple of Jehovah forever and ever. I would like to end the acknowledgements by praising Him for His old creation and new creation through the following two verses:

Col. 1:19 For in Him all the fullness was pleased to dwell.

Col. 2:3 In Whom all the treasures of wisdom and knowledge are hidden.

vi Table of Contents

List of Tables ...... x

List of Figures ...... xi

List of abbreviations ...... xv

Chapter 1 Correlated electronic systems and ab initio approaches ...... 1 1.1 A brief historical overview of condensed matter physics ...... 1 1.2 Understanding correlated systems from ab initio approaches ...... 4 1.2.1 Hartree-Fock approximation and post-Hartree-Fock methods ...... 5 1.2.2 Density functional theory ...... 9 1.2.3 GW approximation ...... 12 1.2.4 LDA + U ...... 14 1.2.5 DFT + DMFT ...... 16 1.2.6 General remarks of the electronic structure methods ...... 19 1.3 Contents of this thesis ...... 20

Chapter 2 Quantum Monte Carlo method and its application to strongly correlated electronic systems ...... 21 2.1 Monte Carlo integration and the Metropolis algorithm ...... 21 2.2 Variational Monte Carlo (VMC) ...... 22 2.3 Diffusion Monte Carlo (DMC) ...... 24 2.3.1 The fermion sign problem and the fixed-node approximation ...... 26 2.3.2 Mixed and extrapolated estimators ...... 27 2.3.3 Pseudopotential approximation ...... 28 2.3.4 Comparison of QMC with other methods ...... 29 2.4 Application to strongly correlated systems ...... 29 2.4.1 NiO ...... 29 2.4.2 MnO ...... 30 2.4.3 FeO ...... 30 2 2.4.4 Cuprate: Ca2CuO3, La2CuO4, CaCuO2, CuO2−, Ca2CuCl2O2 ...... 31 2.4.5 Defect formation energy of ZnO ...... 31 2.4.6 Magnetic order in FeSe ...... 31 2.4.7 Metal-insulator transition in VO2 ...... 32 2.4.8 Rare earth: cerium ...... 32 2.5 Summary ...... 32

Chapter 3 The metal-insulator transition in vanadium dioxide ...... 34 3.1 Summary of previous studies ...... 36 3.1.1 Important experimental facts ...... 36 3.1.2 Theoretical and numerical studies ...... 38 3.2 Method ...... 41

vii 3.3 DFT calculations with different functionals ...... 42 3.4 QMC energetic results ...... 44 3.4.1 Ground state energy ...... 44 3.4.2 Magnetic properties ...... 44 3.4.3 Energy gap ...... 47 3.4.4 Mechanism of metal-insulator transition ...... 48 3.5 Conclusion ...... 50

Chapter 4 The importance of σ bonding electrons for the accurate description of electron correlation in graphene ...... 52 4.1 Introduction ...... 52 4.2 Structure factor, dielectric constant, and random phase approximation ...... 54 4.3 Computation setup ...... 55 4.4 Results and discussion ...... 56 4.4.1 Structure factor of graphene compared with other systems ...... 56 4.4.2 Long distance screening effects ...... 57 4.4.3 Short distance screening and emergent low energy effective field theory for π electrons 60 4.5 Conclusion ...... 61

Chapter 5 The extended Koopmans’ theorem for charge excitations in atoms, molecules and solids ...... 62 5.1 Introduction ...... 62 5.2 Extended Koopmans’ theorem and its implementation in QMC ...... 64 5.2.1 Valence band energy and ionization potential ...... 64 5.2.2 Conduction band energy and electron affinity ...... 67 5.2.3 Regularization of the Coulomb potential in a periodic system ...... 69 5.3 Factors that affect the accuracy of EKT ...... 72 5.3.1 Stochastic error and orbital cutoff ...... 72 5.3.2 Basis set error ...... 73 5.3.3 Relation to quality of the correlated wave function in capturing static and dynamic correlation ...... 74 5.4 Benchmarks of EKT ...... 74 5.4.1 Ionization potentials (IPs) of atoms ...... 75 5.4.2 Electron affinity ...... 78 5.4.3 EKT valence band structure for solids ...... 78 5.5 Conclusion ...... 79

Chapter 6 Effective model construction from first principles simulation ...... 81 6.1 Introduction ...... 81 6.2 Downfolding procedures ...... 84 6.2.1 Eigenstate AIDMD method (E-AIDMD) ...... 84 6.2.2 Non-eigenstate AIDMD method (NE-AIDMD) ...... 84 6.2.3 EKT ab intio density matrix downfolding methods (EKT-AIDMD) ...... 86 6.2.4 Generating states for AIDMD methods ...... 87 6.3 H2 molecule as a toy model ...... 88 6.3.1 E-AIDMD method ...... 89 6.3.2 NE-AIDMD method ...... 94 6.3.3 EKT-AIDMD method ...... 95 6.3.4 Comparisons of the three methods ...... 96 6.4 Application of downfolding methods to solid systems ...... 98 6.4.1 One dimensional hydrogen chain (E-AIDMD) ...... 98 6.4.2 Graphene and hydrogen lattice (NE-AIDMD) ...... 100 6.5 Conclusion ...... 101

viii Chapter 7 Summary ...... 103

Chapter 8 References ...... 105

Appendix A Supplementary information for VO2 ...... 121 A.1 Inter-chain coupling in Ising model simulation ...... 121 A.2 Inter-site electron covariance ...... 121 A.3 Pseudoptential, finite size, and time step errors ...... 122 A.4 Pseudopotential and basis set ...... 124

Appendix B Supplementary information for graphene ...... 125 B.1 Electron energy loss spectroscopy ...... 125 B.1.1 Extraction of the response function for graphene from graphite ...... 125 B.2 Tight-binding model for graphene and the polarization function computed by RPA ...... 126 B.3 Screening effect from σ electrons ...... 128 B.4 Joint density of states ...... 130 B.5 Two dimensional Fourier transformation of the Coulomb potential ...... 130 B.6 Downfolding of graphene and hydrogen lattice to Hubbard models ...... 130 B.7 Setup of first principles calculations ...... 132 B.8 Pseudopotential and basis set ...... 132

Appendix C Extended Koopmans’ theorem ...... 134 C.1 Finite size analysis for graphene ...... 134

Appendix D Derivation details related to downfolding ...... 135 D.1 Two-site extended Hubbard model: effective model for H2 molecule ...... 135 D.2 Derivation of EKT matrix, and forming linear least problem ...... 137 D.2.1 Valence band matrix element ...... 137 D.2.2 Conduction band matrix elements ...... 140

ix List of Tables

3.1 Comparison among different phases of VO2...... 38

4.1 Systems/models investigated ...... 55 4.2 Effective parameters of an on-site Hubbard model for graphene with and without σ electrons. 60

5.1 Ionization potential evaluated from different wave functions...... 74 5.2 Quasiparticle dispersion of graphene near Dirac point. GW, LDA, and experimental results are from [1]...... 79

6.1 Two-body reduced density matrix of a two-site generalized Hubbard model ...... 90 6.2 Comparison of different downfolding methods ...... 96 6.3 Trace of the one-body reduced density matrix of H2 at different states ...... 97 6.4 Comparison of different downfolding methods using scaled RDMs ...... 97 6.5 Parameters of effective Hamiltonian [mHa], and error of RDMs and energies [mHa]...... 99 6.6 Downfolding parameters for graphene and hydrogen...... 101

B.1 Downfolding parameters for graphene with and without σ electrons, and hydrogen...... 132

x List of Figures

1.1 Binding curve of H2 molecule: HF –restricted Hartree-Fock, CI2 – CI with 2 CSF’s, UHF – unrestricted Hartree-Fock CISD – Single and double CI, DMC – Diffusion Monte Carlo calculation...... 7 1.2 Molecular orbitals for H2 with interatomic distance d = 2.5A.˚ ...... 8 1.3 H2 binding curve: HF – Hartree-Fock, RDFT(PBE) – restricted DFT with PBE functional, RDFT(PBE0) – restricted DFT with PBE0 functional, UDFT(PBE) – unrestricted DFT with PBE functional, CI2 – CI with two determinants, CISD – CI with single and double excitations. 11

2.1 H2 binding curve: HF: Hartree-Fock, PBE – DFT with PBE functional; PBE0 – DFT with PBE0 functional; CISD – CI with single and double excitation; CI2 – CI with two determi- nants; SJ – VMC with single Slater-Jastrow trial wave function; CI2-J – VMC with CI2 Slater determinant multiplied with a Jastrow factor; DMC – DMC with single Slater-Jastrow trial wave function...... 24 2.2 Illustration of the walker evolution in a one dimensional potential well in the diffusion Monte Carlo (DMC) method. Figure is taken from [2]...... 26

3.1 Crystal structures of VO2 in different phases. Blue balls are vanadium atoms, and red balls are oxygen atoms. (a) high temperature rutile structure, with vanadium atoms sitting at the center of octahedra forms by surrounding oxygens; (b) low temperature monoclinic structure: vanadium atoms dimerized and shifted away from the centers of the octahedra and form zig-zag chain...... 35 3.2 Schematic band structure from molecular orbital picture: (a) splitting of vanadium d orbitals due to the presence of crystal fields (hybridization with oxygen p orbitals) (b)changing of the band structure due to structural change from rutile to monoclinic...... 39 3.3 FN-DMC energies of VO2 with trial wave functions from DFT of different hybrid functionals. We computed the energies for different magnetic states: antiferromagnetic (AFM), ferromag- netic (FM), spin-unpoloarized (UNP). All the energies are relative to monoclinic AFM state: 2830.836(2)eV...... 41 3.4 DFT− band structures of different exchange-correlation functionals. For FM states, both of the two spin channels (spin majority and spin minority) are plotted on the figure. Three different mixing of Hartree-Fock are used in the computations, 0%, 10% and 25%. The k- point coordinates are: B-[1/2, 0, 0], G-[0, 0, 0], Y-[0, 1/2, 0], C-[0, -1/2, 1/2], Z-[0, 0, 1/2], D-[0, -1/2, 1/2], E-[1/2,-1/2, 1/2], A-[1/2, 1/2, 0]...... 43 3.5 DFT energies for various magnetic states of the two structures. (a)DFT with PBE functional; (b)DFT with PBE0 functional. The energy is relative to monoclinic AFM state...... 44 3.6 (Color online) FN-DMC energetic results: (a) per VO2 unit for different magnetically ordered states, spin-unpolarized, ferromagnetic (FM), and antiferromagnetic [AFM and AFM (intra)]. Energies are referenced to AFM monoclinic VO2.(b) Optical gaps for various states. (c) Spin density for various states...... 45 3.7 Antiferromagnetic coupling between two vanadium atoms through Kramers-Anderson su- perexchange mechanism...... 46

xi 3.8 Temperature dependence of magnetic susceptibility obtained by Ising model simulation on VO2 lattice, with magnetic coupling from FN-DMC compared to results from Zylbersztejn[3] and Kosuge[4]. There has been an overall scale applied on the experimental data to match the maximum χ of Ising model. The transition temperature is indicated by the vertical dashed line...... 47 3.9 Band structures of VO2 in different phases. These band structures correspond to a unit cell containing 4-VO2 formula units. The DFT functional used is PBE0...... 47 3.10 (Color online) Change of V-O hybridizations manifested through: (a) Inter-site charge (ni), and unlike-spin (ni or ni ) covariance quantified as OiOj Oi Oj , where Oi is the onsite value of specific physical↑ ↓ quantities. The inter-siteh covariancei−h betweenih i a chosen vanadium center and the surrounding atoms is plotted as a function of the inter-atomic distance. (b)

Onsite spin-resolved probability distribution on vanadium atoms – ρ(ni↑, ni↓). (c) Spin and charge density of the rutile and monoclinic VO2. Figures on the left panel are 3D isosurface plots of spin density, and on the right panel are contour plots of spin and charge density on the [110] plane...... 48

4.1 Band structure of graphene, hydrogen and tight-binding model (with hopping constant t = 2.7eV)...... 55 4.2 S(q) of different systems obtained through different methods. G: graphene; Gπ: π electrons only graphene; H: hydrogen; TB: π-band tight-binding model with 1/r interactions; Hubbard: the Hubbard model with U/t = 1.6...... 56 4.3 Effect of σ electrons at long range. (a) imaginary part of the response function χ(q, ω) of graphene (IXS and RPA), tight-binding model with 1/r interaction, and hydrogen lattice at different momentum transfer. -Imχ has included local field corrections, and has been scaled in order to compare with experiment. The different curves have been scaled for comparison. (b) S(q) from integration of S(q, ω) with different energy cutoff ωc. (c) dispersion of π π∗ plasmon resonance peaks. (d) effective screening function from σ electrons...... → . . . . 57 4.4 The isosurface plot of Wannier orbitals constructed by a unitary rotation of Kohn-Sham orbitals: (a) σ orbital; (b) π orbital. Different colors represent different signs. The root- p mean-square radius ( r2 ) is 2.69A˚ for σ orbitals and 3.37A˚ for π orbitals...... 58 4.5 Joint density of states (JDOS)h i of graphene (G), hydrogen (H), and tight-binding (TB) model. The JDOS of graphene and hydrogen are computed from PBE band structure...... 59

5.1 Ionization potentials for oxygen atom and H2 molecule with different cutoff orbitals. V5Z basis set from [5] is used for both oxygen and hydrogen. For oxygen atom, we use a multiple Slater-determinant wave function from CISD. For H2 molecule, we use a single Slater-Jastrow wave function. The dash line is the experimental ionization potential from NIST database [6]. 72 5.2 Ionization potentials calculated using different basis sets. From VS1 to V5Z, the basis set is more and more complete. The black line is the exact result (16.45eV)...... 73 5.3 Ionization potentials of atoms on the 1st, 2nd and 3rd rows of the periodic table (from He to Ar) using different methods: (a) by direct subtraction of energy between atoms and corresponding ions (A and A+). We denote as ∆ methods; (b) Box-and-whisker plot of errors of ∆ methods, the purple dots are the root-mean-square errors of each methods; (c) EKT results based on different wave functions; (d) Box-and-whisker plot of errors of EKT based on different wave functions, the purple dots are the root-mean-square errors of each methods. (e) errors of ionization potentials for each atoms. For the CISD (MSL) results, we set the number of active orbitals to be 10 and include those CSF’s whose coefficients are larger than 0.01...... 75 5.4 Correlation energy of atoms and ions, computed by the difference of DMC energy and HF energy...... 76 5.5 Ionization spectra of atoms (C, O, and Ne) using different methods). ∆ DMC: directly subtract energies between two states; EKT(SJ, DMC), EKT from DMC using a Slater-Jastrow trial wave function; EKT(ML): EKT using multiple Slater-Jastrow wave function. The black solid lines are experimental results from NIST database [6]...... 77

xii 5.6 Electron affinity of ions. The experimental results are negative of the ionization potential of corresponding neutral atom...... 78 5.7 Graphene energy momentum dispersion near Dirac point using different first principle meth- ods. Experimental results, LDA and GW results are from [1]. GGA and EKT results are from this work...... 79

6.1 Schematic for downfolding. The full Hamiltonian H is defined in the space of active (partially occupied), core (mostly occupied) and virtual (mostly unoccupied) orbitals. These orbitals have been arranged according to their energy (E) in the figure. The objective is to map the physics of the original system to that of the effective one, defined only in the active space, with Hamiltonian H˜ ...... 82 6.2 Schematic of ab initio density matrix downfolding (AIDMD) methods employed for deter- mining the effective Hamiltonian parameters. (a) In the eigenstate (E)-AIDMD, the reduced density matrices and energies of eigenstates of the model are matched to the ab initio coun- terparts. (b) The non-eigenstate (N)-AIDMD method uses RDMs and energies of arbitrary low-energy states to construct a matrix of relevant density matrices and performs a least square fit to determine the optimal parameters...... 85 6.3 Iso-surface plot of Hartree-Fock orbitals and Wannier orbitals. Different colors represent different signs (yellow, positive; blue, negative). Wannier orbitals are constructed through a unitary transformation of Hartree-Fock orbitals...... 88 6.4 ab initio information that can be used for determining the model parameters: (a) ground state parameter λ; (b) energies of various eigenstates (from CISD)...... 91 6.5 Model parameters of H2 obtained through E-AIDMD method...... 92 6.6 Asymptotic behavior of model parameters...... 93 6.7 Error of 1-RDM for H2 with difference distance...... 93 6.8 Downfolding of H2 using NE-AIDMD method: (a) fitted model parameters; (b) r.m.s error of energies of all the states...... 94 6.9 H2 molecule downfolding results for different distance...... 95 6.10 H2 molecule downfolding results for different distance. The model includes hoping term Tˆ, onsite Coulomb and nearest neighbor Coulomb interaction (Uˆ, Vˆ and Jˆ)...... 96 6.11 Kohn-Sham orbitals (upper panel) from DFT calculations with PBE exchange-exchange cor- relation functional, and Wannier orbitals (lower panel) constructed through a unitary trans- formation of Kohn-Sham orbitals...... 98 6.12 Errors of RDMs and energy for hydrogen chains with different number of sites: (a) relative error of two-body reduced density matrix; (b) error of eigen energy for S = 0 and S = 1 states per atom. the parameters used in model calculations are from 4-site chain...... 100 6.13 Wannier orbitals constructed from Kohn-Sham orbitals: (a) graphene π orbital; (c) hydrogen S orbital...... 100 6.14 Comparison of ab initio (x-axis) and fitted energies (y-axis) of the 3 3 periodic unit cell of graphene and hydrogen lattice: (a) graphene; (b) hydrogen lattice. .× ...... 101

A.1 Temperature dependence of magnetic susceptibility obtained by Ising model simulation on VO2 lattice, with magnetic coupling from FN-DMC compared to results from Zylbersztejn[3] and Kosuge[4]. There has been an overall scale applied on the data, and the transition temperature is indicated by the vertical dashed line...... 121 A.2 Pseudoptential, finite size, and timestep error analysis: (a) Energetic results of 4-VO2 cells for different pseudopo- tentials: BFD and Yoon. PBE functional was employed in DFT to optain the trial wave function for FN-DMC. (b) Energetic results for different cell size: 4-VO2 cell and 16-VO2 cell; (c) Energetic results of of 4-VO2 cells for different time step: 0.01 and 0.02. BFD pseudopotential and PBE0 functional were used in (b),(c)...... 122

xiii B.1 EELS (-Imχ) of various systems. (a) checking of the conversion formula. G3: 3D graphite; G2: monolayer graphene; Converted: result converted from G3 obtained through conversion formula Eq. (B.1). (b) EELS spectrum for different systems. G(IXS): inelastic X-ray scat- tering result; H: hydrogen system; TB: tight-binding model using RPA; TBσ: tight-binding model with screening from σ electrons. We have scale the EELS spectrum such that the π plasmon resonance peak of different systems have the same height...... 126 B.2 Comparison of ab initio (x-axis) and fitted energies (y-axis) of the 3?3 periodic unit cell of graphene and hydrogen lattice: (a) graphene with σ electrons included; (b) graphene without σ electrons; (c) hydrogen lattice...... 132

C.1 Graphene energy momentum dispersion near Dirac point using different first principle meth- ods. Experimental results, LDA and GW results are from [1]. GGA and EKT results are from this work. EKT(1 1), EKT(2 2), EKT(3 3), EKT(4 4) are results corresponding to different sizes of supercell.× . . . .× ...... × ...... × ...... 134

xiv List of abbreviations

HF Hartree-Fock

DFT Density Functional Theory

CI Configuration Interactions

QMC Quantum Monte Carlo

VMC Variational Monte Carlo

DMC Diffusion Monte Carlo

AFQMC Auxiliary field quantum Monte Carlo

FCIQMC Full configuration quantum Monte Carlo

VO2 Vanadium dioxide RPA Random Phase Approximation

DMFT Dynamic mean field theory

KT Koopmans’ theorem

EKT Extended Koopmans’ theorem

IP Ionization potential

EA electron affinity

xv Chapter 1

Correlated electronic systems and ab initio approaches

This thesis will be focused on building a theoretical framework on understanding correlated condensed matter systems from a first principles perspective. On one hand, we apply first principles techniques, including the

Density Functional Theory and quantum Monte Carlo method, to understand the microscopic electronic structure of correlated systems from ab initio level. On the other hand, we will formulate systematic approaches to construct low energy effective lattice models from ab initio simulations, so as to identify low energy degrees of freedom relevant to physical properties of interest.

In the first chapter of the thesis, I will first give an overview on current understanding of condensed matter physics. I will then narrow down to introduce several electronic structure techniques that are developed in the past decades. By doing this, I am hoping that the readers of the thesis will have a broad view regarding the big framework that my thesis belongs to, thus to have an objective sense regarding the significance of all the works that are presented in the following chapters of this thesis.

1.1 A brief historical overview of condensed matter physics

Let me trace back the history to 1900, when Drude proposed his classical electron gas model, known as Drude model [7] (extended later on by Lorentz [8]). This model treats electrons in a solid as classical particles moving around in a background of immobile ions. With concepts such as mean free path, relaxation time, and the assumption of Boltzmann distribution of electron kinetic energy, Drude model provides a classical interpretation of many transport phenomena in metals, for example, the Ohm’s law (electric current is proportional to the voltage), the Wiedemann-Franz law [9] (the ratio of the thermal to electrical conductivity of the metals is proportional to the temperature). However, despite these successes, the theory has many problems and limitations. For example, the relaxation time τ is unknown from the theory itself; it cannot explain why the mean field path in some systems is several orders of magnitude larger than the interatomic distance; it does not take into account the electron-electron correlation; it is unable to explain the low temperature behavior of the specific heat of solids; it cannot explain the negative Hall coefficient (indicating

1 the charge of carriers is positive). Some of these were solved after the introduction of quantum mechanics into the theory of solids, through the works of following people:

Einstein, Debye [10], Born and van K´arm´an[11, 12]: specific heat of solid; • Fermi and Dirac [13]: Fermi-Dirac statistics; • Pauli [14]: applying Fermi-Dirac statistic to electron gas to explain the paramagnetism; • Sommerfeld [15]: a complete application of Fermi-Dirac statistics to electron gas. A semiclassical • theory of metal was developed.

The second stage of the development starts from Bloch’s theorem [16], which laid the foundation of the band theory. A complete theoretical framework is developed after several successive works from Peierls [17],

Bethe [18], Brillouin [19], and Wilson [20]. Band theory provides a clear intuitive understanding of metals, insulators and semiconductors from quantum mechanics. According to the band theory, the energy levels in a periodic solid form bands. If all the bands are either completely filled or empty, the system will be an insulator; on the other hand, if there are partially filled bands, the system will be a metal. A semiconductor is essentially an insulator with a small energy gap. It is insulating at zero temperature but has finite conductivity at room temperature because of the particle-hole excitations due to phonons. Band theory answers many mysterious questions which cannot be understood in the classical Drude model framework.

For example, for the negative Hall coefficient, the concept of “hole” naturally emerges in the band theory framework.

However, there was a long standing question during that time: in many metals and semiconductors, the electrons with Coulomb interaction behave like noninteracting particles that are well described by the band theory. This was not understood until Landau’s Fermi-liquid theory had been proposed [21]. The Landau

Liquid theory identifies the elementary excitation of the interacting system as one-to-one correspondence with that of the noninteracting electron gas, through adiabatically turning on the pair interactions [22].

The fundamental excitations in a Fermi-liquid system are called quasiparticles. The basic hypothesis of

Landau’s theory is that as the electron-electron interaction is adiabatically switched on, the system does not undergo a phase transition. This phenomenological concept later on was rigorously proved by Abrikosov, and Kalatnikov using diagrammatic perturbation theory to all orders [23]. Landau’s Fermi-liquid theory establishes the conceptual basis characterizing the spectrum of low-lying excitations.

Accompanying the conceptual development, numerical methods for computing electronic structure of realistic systems are also developed. Wigner and Seitz performed the first realistic calculation on the cohesive energy of solid [24]. Besides, many approximate methods were developed in the early twenty century,

2 including the Thomas-Fermi theory [25], Hartree-Fock approximations [26], which laid the foundation of ab initio approaches in quantitive description of solids. A landmark achievement in the electronic structure

field is the development of the Density Functional Theory (DFT) in the mid 1960s by Hohenberg, Kohn, and

Sham [27, 28]. DFT with a simple exchange-correlation functional generalized from electron gas problem

(local density approximation) is able to describe metals, insulators, and semiconductors, whose physical properties are rather distinct from the electron gas. It has achieved great success in the electronic structure calculations of atoms, molecules [29], metals [30], structure and phase properties of semiconductor [31], transition metal and transition metal chemistry [32].

Up to this point, we have a conceptual understanding of simple metals and semiconductors within the band theory framework, and the low energy excitations of a many-body system according to Landau’s

Fermi liquid theory. We also have quantitative ab initio techniques for electronic structure calculations of realistic materials. It seems that the theoretical framework is complete. However, there are some systems that cannot be understood in the above framework, for example, many transition metal oxides such as

FeO, NiO, CoO, MnO, and rare earth heavy fermion systems [33]. These systems have partially filled conduction bands. According to band theory, they should be conductors instead of insulators. DFT with local density approximations also predicts many of these systems to be metals even though all of them are large gap insulators. Meanwhile, many conductors behave differently from a Fermi-Liquid system. To understand these, we need to go beyond the band theory framework and carefully take into account the effect of electron-electron correlations. These systems are later on termed as Mott insulator, charge transfer insulator, and non-Fermi-liquid metals. They are generally called as strongly correlated systems. The physics of strongly correlated systems gradually gain physicists’ attention in the last three decades, especially after the discovery of fractional quantum Hall effect in 1982 [34], high temperature superconductivity in 1986 [35], and giant magnetoresistance [36, 37] in 1988 and many others. They demonstrate a great variety of physics phenomena that emerge because of strong electron-electron correlations.

Most transition metal oxides and rare-earth compounds are strongly correlated. They are interesting because the localized nature and multiple degeneracy of 3d, 4f, and 5f orbitals [33]. In these systems, it is generally believed that the electron-electron interactions are comparable to the bandwidth, therefore, the general band structure framework does not work. Meanwhile, this type of systems exhibit complex behaviors because of the intertwining of different degrees of freedom, such as spin, charge, orbitals, and lattice. It is the competition/cooperation of many degrees of freedom that gives rise to many interesting phenomena.

Understanding the electron correlations in transition metal oxides will be the focus of this thesis. However, the whole zoo of transition metal oxides is very diverse. A complete survey of different categories of systems

3 is beyond my limited PhD time. What I do here is to study a representative system (VO2) to get a glimpse of the whole zoo. In this representative study, I am developing a theoretical framework for understanding

correlated systems from an ab initio perspective, and hoping that many fruitful studies will come out through

the application of the theoretical framework.

1.2 Understanding correlated systems from ab initio approaches

The phrase “ab initio” means starting from the very beginning. However, “beginning” is always relative, and depends on the energy/length scales that we are interested. In condensed matter physics, the “beginning” is the many-body Schr¨odingerequation of electrons and ions,

2 N 2 2 2 2 ˆ ¯h X 2 X ZI e X e ¯h X 2 X ZI ZJ e H = i + + I + , (1.1) −2me ∇ ri RI ri rj − 2M ∇ RI RJ i=1 i,I | − | i,j | − | I=1 I,J | − |

where i, and I (and J) are the indices of electrons and nucleus respectively, and ri and RI are positions of electrons and ions respectively. This contains the kinetic energy of electrons and ions, the potential energy of

electron-electron, electron-ion, and ion-ion interaction. All macroscopic properties of the systems arise from

this many-body Hamiltonian (1.1). As long as one is able to solve this many-body Schr¨odingerequation,

one can understand essentially all phenomena in condensed matter systems at the energy scales that we are

interested. This in principle, is just a matter of complexity.

Unfortunately, exactly solving Eq. (1.1) for condensed matter systems is almost impossible. Various

ways of approximation are introduced to simplify the problem. The first one is the Born-Oppenheimer

approximation [38]. In most of the cases, the motion of nuclei and electrons are of different time scales, one

can separate these two motions. We first fix the position of ions and solve the dynamics of electrons. The

effect of lattice might be included in a perturbative way later on. The Hamiltonian in Eq. (1.1) is simplified

as follows,

2 N 2 2 ˆ ¯h X 2 X ZI e X e H = i + + . (1.2) −2me ∇ ri RI ri rj i=1 i,I | − | i

which includes the kinetic energy of electrons, external potential from ions, and electron-electron interactions.

However, even Eq. (1.2) is still impossible to solve exactly. The major complexity comes from the electron-

electron Coulomb interaction. Different ab initio methods have different ways of treating the electron-electron

interactions so as to further simplify Eq. (1.2) and solve it in different approximate ways.

Before moving to the introduction of various electronic structure methods, let me make some general

4 comments on the significance of ab initio simulations. Physicists usually have a tradition of using intuitive

models to capture relevant physics of a system. In the past, many phenomenological models have been in-

troduced in condensed matter physics, including the Ising model and Heisenberg model for magnetism [39],

Jellium model for electron gas [40], Mott-Hubbard model [41] and t-J model [42] for strongly correlated sys-

tems. Besides these, there are many phenomenological field theories, for example, Ginzburg-Landau theory

of superconductivity [43], Chern-Simon’s theory for topological phase of matter [44]. All of these capture

certain ingredients that are crucial to macroscopic phenomena while ignoring other irrelevant microscopic

details. Many numerical methods have been developed to solve these models. These models do provide us

a basic understanding of many physics phenomena. Then why do we need ab initio simulation in physics?

There are many reasons that we still need ab initio simulation. A very obvious one is that in applied science, one requires highly accurate quantification of physics properties rather than a mere qualitative understanding. Ab initio simulations are needed for accurate prediction of structural, optical, magnetic properties of materials. Second, in complex systems, there are many different degrees of freedom competing with each other, giving rise to macroscopic phenomena. It is sometimes difficult to identify which is the dominant one. For example, in the case of high Tc superconductivity, we do not know what is the dominant contribution to the formation of Cooper pairs. Ab initio simulation which includes all details of the systems, is potentially capable to simultaneously take into account many relevant physics and identify their relative importance. In this sense, ab initio simulation also provides a way to verify whether many existing models describe the true mechanism of superconductivity. Therefore, ab initio simulation is not only important to chemistry and materials science for engineering application, but also extremely significant in physics for understanding the fundamental mechanism behind various phenomena in complex systems.

In the following, I will give a brief survey of several widely used electronic structure methods, ranging from mean-field based methods, such as Hartree-Fock approximation, density functional theory (DFT), to perturbative many-body methods, such as GW approximations, dynamic mean field theory (DMFT). I will leave the quantum Monte Carlo method to Chapter. 2.

1.2.1 Hartree-Fock approximation and post-Hartree-Fock methods

The Hartree-Fock (HF) approximation is a starting point for more accurate quantum chemistry methods such as configuration interactions (CI) and coupled cluster. Here, we merely give a brief introduction of the formulation and comment on its applications in both molecular and solid systems. For a much more comprehensive introduction, one can refer to [45].

In the Hartree-Fock approximation, one approximates the ground state of a many-body system using a

5 Slater determinant wave function,

φ1(r1) φ1(r2) φ1(rN ) ···

1 φ2(r1) φ2(r2) φ2(rN ) ··· Ψ(RN ) = , (1.3) √N!

··· φ (r ) φ (r ) φ (r ) N 1 N 2 ··· N N where φ ’s are single-body orbitals. φ ’s are variationally optimized with an orthonormal constraint φ φ = i i h i| ji δij, so as to lower the energy expectation value:

X δS[φi] S[φi] = Ψ Hˆ Ψ  φλ φλ , = 0 . (1.4) h | | i − h | i δφλ λ

This leads to the following set of self-consistent equations,

Z 2 Z h 1 2 i X φi(r0) 1 + V (r) φ (r) + dr0 | | φ (r) dr0φ∗(r0)φ (r0) φ (r) =  φ (r) , − 2∇ ext λ r r λ − i λ r r i λ λ i=λ 0 0 6 | − | | − | λ = 1, N. (1.5) ···

Each equation in (1.5) is an effective single-body Schr¨odingerequation with a mean-field averaged potential

from other electrons. The whole operator in Eq. (1.5) is called Fock operator. The first term is the kinetic

energy plus the potential energy from ions; the second term is the Hartree potential, a mean field averaged

Coulomb interaction from other electrons; and the third term is the exchange potential originated from the

fermion statics of electrons. Eq. (1.5) can be solved iteratively, resulting in a set of optimized single particle

orbitals φi with the corresponding eigenvalue i. Here, we have not taken into account the spin degree of freedom. If it is included, there will be two determinants, one for spin-up electrons and the other for spin-

down electrons. If there are equal number of spin-up and spin-down electrons, and they occupy the same set

of orbitals, the calculation is called restricted close shell Hartree-Fock (RHF). There is also restricted open

shell Hartree-Fock (ROHF) calculation when the numbers of spin-up electrons and spin-down electrons are

not equal. If they occupy different set of orbitals, the calculation is called unrestricted Hartree-Fock (UHF).

HF has been widely applied in chemistry in calculating the electronic structure of atoms and molecules

[45]. It has also been applied to study various solid systems, to compute the cohesive energy, lattice constant,

band structure, and Compton profiles of semiconductors [46, 47].

There is an inherent limitation of the Hartree-Fock approximation: it completely neglects the correlation

between electrons of opposite spins. Thus the correlation energy is completely absent in the total Hartree-

6 Fock energy. The correlation energy is defined as = E E . This part of energy, though is only Ecorr exact − HF a small fraction of the total energy, is crucial to many properties of the system, such as cohesive energy, binding energy, ionization potential and electron affinity. Because it fails to capture correlation energy, HF poorly describes the chemical bonding in molecules (the error of binding energy is generally about 50%

[2, 48]).

Conceptually, in quantum chemistry, one differentiates into two parts [45]: Ecorr

dynamic correlation: describing the effect of instantaneous electron repulsion (electron avoiding each • other to lower the Coulomb interaction energy). This is mainly between opposite-spin electrons. For

electrons of the same spin, Pauli exclusion already introduces similar mutual avoiding effect. HF,

especially RHF, poorly describes the dynamic correlation.

static (or non-dynamic) correlation: describing the inadequacy of a single Slater determinant in de- • scribing the system due to nearly degenerate states or rearrangement of electrons within partially filled

shells.

There is however, no clear boundary between these two types of correlations. In many cases, including more determinants, reduces not only the static correlation error, but also the dynamic correlation error.

0.6 − HF CISD 0.7 − CI2 DMC UHF 0.8 −

0.9 −

Energy [Ha] 1.0 −

1.1 −

1.2 − 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 distance [A]

Figure 1.1: Binding curve of H2 molecule: HF –restricted Hartree-Fock, CI2 – CI with 2 CSF’s, UHF – unrestricted Hartree-Fock CISD – Single and double CI, DMC – Diffusion Monte Carlo calculation.

To illustrate the limitation of the Hartree-Fock approximation and to get a semi-quantitive understanding of correlation energy, we computed the binding energy of H2 molecule at different interatomic distances. As one can see in Fig. 1.1, RHF calculation poorly describes the molecular binding especially at long distance.

The binding energy given by RHF is around 0.32 Ha; however, the exact result is 0.175 Ha [from diffusion

7 Monte Carlo and CISD (with be discussed later)]. At long distance, the system is essentially two isolated hydrogen atoms, therefore the total energy should be exactly 1.0 Ha. However, in RHF, the spin-up and − spin-down electrons are forced to occupy the same set of orbitals. In H2, both the two electrons occupy the bonding orbital shown in Fig. 1.2(a). There is an unphysical large probability for electrons to stay in between the two hydrogens. This constraint increases the total Coulomb energy significantly at long distance.

If we add an extra Slater determinant [both the two electrons occupy the anti-bonding orbital shown in

Fig. 1.2(b)], the total energy is significantly lowered [see the CI2 curve in Fig. 1.1]. This extra determinant mainly reduces the static correlation error, since at large distance the bonding and anti-bonding orbitals are nearly degenerate. At the same time, the avoiding effect between the two electrons is also captured to some extent in CI2. Thus, it also improves the description of dynamic correlation. To fully capture the correlation energy, especially the dynamic correlation energy, One needs much more determinants. This can be done within configuration interactions framework. The other way is to perform UHF calculations, allowing spin-up and spin-down electrons to occupy different orbitals [see Fig. 1.2(c) and (d)]. This however will introduce local magnetic moments to hydrogen atoms and break the time reversal symmetry explicitly.

Nevertheless, it gives correct energetic results at large distance [see the green curve in Fig. 1.1].

(a) RHF bonding orbital (b) RHF antibonding orbital

(c) UHF spin up orbital (d) UHF spin down orbital

Figure 1.2: Molecular orbitals for H2 with interatomic distance d = 2.5A.˚

The configuration interactions (CI) method is a direct extension of the HF approximation [45]. In this method, one represents the Schr¨odingerequation Hˆ Ψ = E Ψ in many-body basis. These many-body basis | i | i can be written as excitations of the Hartree-Fock ground state Ψ , i.e., | 0i

X X X Ψ = c Φ + cr Φr + crs Φrs + Φrst + ..., (1.6) | i 0| 0i a| ai ab| abi | abci ra a

8 where Φr denotes the Slater determinant formed by replacing orbital a in Φ with orbital r. Every Slater | ai | 0i determinant in Eq. (1.6) corresponds to a certain occupation of the Hartree-Fock orbitals. A set of orbital occupation is referred to as a “configuration” or a “configuration state function” (CSF). The full CI method provides numerically exact solutions (within the infinitely flexible complete basis set). It is essentially the exact diagonization method in electronic structure theory.

In principle, the dimension of the problem is infinity. In practice, however, one only employs an incomplete set of one-particle basis functions. In this case, the full CI corresponds to solving Schr¨odingerequation exactly within the space spanned by the specified one-electron basis. However, even in such a finite basis set, the number of CSF needed to provide an accurate description of the system is very large. For example, for

H2O, using the DZP basis set, the number of CSF is 256,473 [49]. It grows exponentially with the number of electrons in the system, and will reach the limit very quickly. One can introduce a cutoff in the number of excitation in the configurations in Eq. (1.6). For example, the widely-employed CI singles and doubles

(CISD) method includes only the single and double excitations relative to the Hartree-Fock ground state configuration. This will typically recover 95% correlation energy for small molecules at their equilibrium geometries [49]. One can also include higher order “excitation” such as CISDT, and CISDTQ (T – triple, Q

– quadruple).

Instead of directly evaluating the matrix element of the Hamiltonian in the configuration space and solving a linear algebra equation, a stochastic way of determining the coefficients of CSF’s has also been developed. This is called full configuration interactions quantum Monte Carlo method (FCIQMC) [50], which might have the potential of polynomial scaling. FCIQMC has been applied to various solid state systems and shown great success [51], including rare gas (Ne, Ar), ionic and covalent solids (LiH, LiF, BN,

AlP), and the charge transfer insulator (NiO). The accuracy for cohesive energy surpasses 1 kcal/mol [51].

In conclusion of this subsection, HF provides a zeroth order solution that many post-Hartree-Fock meth- ods can base on. The CI methods can provide very high accurate results. But their application to large systems is limited by the exponential scaling with respect to the number of electrons. There are also a few successful attempts in applying these quantum chemistry methods including FCIQMC to solid states systems [50]. However, it is still under further investigation. Hopefully, with the technical, algorithmic and methodological advancement in the near future, these methods can be more widely applied to solids.

1.2.2 Density functional theory

As we already see from the CI method in Subsection. 1.2.1, the dimension of the Fock space grows expo- nentially with the size of the system. We will face a so called exponential wall (N 10) [52], beyond which '

9 the wave functional approach fails. A general question is whether it is possible to describe the system with quantities other than wave function. In fact, shortly after the introduction of Schr¨odingerequation, Thomas and Fermi took the first step and formulated the many-body problem using electron density [25, 53], which later on is called Thomas-Fermi theory. However, the Thomas-Fermi theory is unable to describe the shell structures of atoms and binding of molecules [54]. Nevertheless, this can be regarded as the most rudimen- tary form of a“density functional” theory, and in fact it directly inspired Kohn to formulate the standard

Density Functional Theory (DFT) [52]. Hohenberg and Kohn rigorously proved that it is indeed sufficient to describe the system merely using the charge density of the system [27, 52].

The Hohenberg-Kohn theorem states that: the ground-state density n(r) of a bound system of interacting electrons in some external potential Vext(r) determines this potential uniquely [27]. In other words, the ground state density determines the Hamiltonian uniquely, and one can in principle know all the information of the system once the charge density of the system is known. In particular, the ground state energy of the interacting electronic system is a functional of the charge density: EGS = E[n(r)]. Based on this theorem, one can transform the 3-N dimensional Schr¨odingerequation to a functional minimization problem.

Unfortunately, this theorem only proves the existence of a universal energy functional but does not provide any information to find it. Kohn and Sham made a great progress in finding an approximate form of the functional [28]. They introduced an ansatz: the charge density of the interacting system can be represented using the density of a noninteracting system n(r) = P φ (r) 2. By further introducing the so called i | i | exchange-correlation functional Exc[n(r)], they arrived at the following self-consistent Kohn-Sham equation,

" Z # 1 2 n(r0) + V (r) + dr0 + V [n(r)] φ (r) =  φ (r), λ = 1, N, (1.7) − 2∇ ext r r xc λ λ λ ··· | − 0|

This looks very similar to the HF approximation in Eq. (1.5), except that the non-local exchange interaction in Eq. (1.5) is replaced by the exchange-correlation potential Vxc = δExc/δn(r). The exact form of the exchange-correlation functional is unknown. In practical calculations, different kinds of approximate forms have been introduced in the literature [54]: the local density approximation

(LDA) [28] and local spin density approximation (LSDA), various forms of generate gradient approximation

(GGA) [55, 56, 57], meta generate gradient approximation (meta-GGA) [58], and many flavors of hybrid functionals such as PBE0 [59] and HSE [60]. A comparison of different exchange-correlation functionals can be found in [61]. Unfortunately, there is no known systematic way to achieve an arbitrarily high level of accuracy. Different functionals might be suitable to different systems.

DFT has achieved great success in the electronic structure calculations of atoms, molecules [29], metals

10 [30], structure and phase properties of semiconductor [31], transition metal and transition metal chemistry

[32]. It has become the Standard model in electronic structure calculations. However, despite its great success, there are two major problems [62]:

(a) delocalization error: for bonding problem, there is an underestimation of the chemical reaction barrier,

band gap (LDA and GGA) for semiconductors and large band gap insulators, the cohesive energy, and

ionization energy.

(b) static correlation error: error due to inadequacy of single Slater determinant wave function, for nearly

degenerate systems, strongly correlated systems, such as Mott insulators.

Let us again consider the H2 molecule. At this time, we computed the binding energy using DFT. The results are shown in Fig. 1.3. We see that at large distance, a pure spin unpolarized DFT calculation (similar to restricted Hartree-Fock calculation) still has large static correlation error. The total correlation error does decrease compared to HF. Compared with RHF, restricted DFT does capture electron correlation energy to some extent. However, because the spin-up and spin-down electrons still occupy the same orbital, they cannot effectively avoid each other to reduce the total Coulomb energy. Unrestricted DFT calculation which allows symmetry breaking between spin-up and spin-down orbitals, does give very good results.

0.6 − HF UDFT(PBE) CISD RDFT(PBE) CI2 DMC 0.7 − RDFT(PBE0)

0.8 −

0.9 −

Energy [Ha] 1.0 −

1.1 −

1.2 − 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 distance [A]

Figure 1.3: H2 binding curve: HF – Hartree-Fock, RDFT(PBE) – restricted DFT with PBE functional, RDFT(PBE0) – restricted DFT with PBE0 functional, UDFT(PBE) – unrestricted DFT with PBE func- tional, CI2 – CI with two determinants, CISD – CI with single and double excitations.

Because of the delocalization error, DFT with LDA or GGA underestimates the fundamental gap (or conductivity gap) by about 40% for semiconductors and insulators [63, 64]. One might hope that with exact exchange-correlation functional, DFT will predict insulating gap correctly. In fact, it is proved that

11 even with the exact exchange-correlation functional, the Kohn-Sham gap is not the fundamental gap. The difference is caused by the derivative discontinuity of the exchange-correlation functional,

" #

KS δExc[n] δExc[n] gap = EN+1 + EN 1 2EN = gap + lim . (1.8) − − δ 0 δn(r) − δn(r) → N+δ N δ −

The correct quasiparticle gap in principle can be computed in GW. In GW, an energy-dependent self inter- action energy is introduced to replace the exchange-correlation function when calculating the quasiparticle band gap. This will be discussed in the Subsection. 1.2.3.

One the other hand, because of the static correlation error, DFT fails to describe strongly correlated electronic systems where static correlation is significant and a single Slater deteriminant wave function is no longer a good description [65, 66]. Such systems usually contain transition metal or rare-earth metal ions

(Kondo systems for example) with partially filled d (or f) shells. When applying DFT with LDA or GGA to transition metal compounds, one has as a result a partially filled d band with metallic type electronic structure and itinerant d electrons [65, 66]. For example, the late transition-metal monoxides, such as MnO,

FeO, CoO, and NiO, which are regarded as Mott insulators, are poorly described by DFT with LDA or

GGA. Many methods, such as orbital dependent functional approach, LDA + U, and LDA + DMFT, are introduced to specifically target the electron-electron correlations in d (or f) orbitals. We will discuss the

LDA + U and LDA + DMFT in following subsections.

1.2.3 GW approximation

GW approximation determines the excited states based on the quasiparticle concept and the Green function methods [67]. In Coulomb system, the repulsion between electrons leads to a depletion of negative charge around a given electron. The ensemble of this electron and its surrounding positive screening charge forms a quasiparticle. The mathematical description of quasiparticles is based on the single-particle Green function

G, whose exact determination requires complete knowledge of the quasiparticle self-energy Σ. The self-energy

Σ is a non-Hermitian, energy-dependent, and non-local operator that describes exchange and correlation effects beyond the Hartree-Fock approximation or DFT [66].

GW approximation is based on the quasiparticle concept and the Green function methods [67]. In

Coulomb system, the Coulomb interaction is screened by forming particle-hole pairs surrounding the elec- trons. An electron with its surrounding particle-hole pairs forms a quasiparticle. The mathematical descrip-

12 tion of quasiparticles is based on the Green function G,

G(r, t; r0, t) = N, 0 T [Ψ(r, t)Ψ†(r0, t0)] N, 0 , (1.9) h | | i where N, 0 denotes the ground state of an N-electrons system, and Ψ is the Heisenberg operator in the | i second quantization framework. Green function represents the probability amplitude of a particle at r0 at t0 to propagate to r at t. “T []” is the time ordering operator which preserves causality. In many-body theory, we can use the Green function to develop the quasiparticle Hamiltonian,

Z 1 2 Hψˆ (r) = ψ(r) + v (r)ψ(r) + v (r)ψ(r) + Σ(r, r0, )ψ(r0) = ψ(r) . (1.10) −2∇ ext H

where vH is the Hartree potential, Σ is the self-energy which is energy-dependent, and non-local operator that describes exchange and correlation effects beyond the Hartree-Fock approximation or DFT [66]. ψ is the quasiparticle wave function.  in general will be complex, with its real part be the quasiparticle energy and its imaginary part determines the quasiparticle lifetime.

G and W are determined by the following five self-consistent many-body equations together [67],

Dyson’s equation : G(12) = G0(12) + G0(13)Σ(34)G(42), (1.11a)

Screening equation : W (12) = v(12) + v(13)P (34)W (42), (1.11b)

Self energy : Σ = iΓ(4; 1, 3)G(3, 2)W (4, 2), (1.11c)

Polarizability : P (1, 2) = 2iΓ(1; 4, 5)G(2, 4)G(5, 2), (1.11d) − δΣ(23) Vertex Correction : Γ(1; 2, 3) = δ(12)δ(34) + Γ(1; 4, 7)G(6, 4)G(78) . (1.11e) δG(68)

Here, W is the screened Coulomb interaction. For simplicity, we have used the standard shorthand notation for the coordinates, f(12) = f(r1, t1; r2, t2). One can solve these self-consistent equations iteratively. In practical calculations, various different types of approximations are introduced:

(1) Plasmon-pole form for the dielectric function [68]. This assumes that all the weight in ImW resides in

the plasmon excitation. The inverse dielectric function is written as

1 Im− (q, ω) = AGG (q)δ(ω ωGG (q)) , (1.12) GG0 0 − 0

1 where G and G’ are reciprocal lattice vectors. A and ωq are determined from the static limit of − and the f-sum rule. The plasmon-hole form is exact in the long wavelength limit q 0. For finite q, → 13 the spectrum also contains particle-hope excitations at low energies. This approximation gives good

description in semiconductors [68]. One problem of this approximation is that the imaginary part of

the self-energy is zero except at the plasmon poles. Thus it is unable to compute the quasiparticle life

time.

(2) Non self-consistent GW [69]: the G0W0 method tends to underestimate the gap for semiconductors. It

is known that, a single iteration (G0W0) is sufficient only if the quasiparticle energy shifts are relatively small. One can use a suitable density functional such as a hybrid functional to obtain KS eigen energies

that are already close to the GW quasiparticle energies [70].

(3) assuming the locality of GW self-energy. It is shown that in main-group compounds, the nondiagonal

components of the GW self-energy has a rather limited effect [71].

With these approximations, GW obtains band gaps and band structures for main-group semiconductors and insulators that are in good agreement with experiment [71, 72]. In combining with the Bethe-Salpeter equation, the GW-BSE approach also describes the exciton effects in transition metal dichalcogenides mono- layers [73, 74, 75].

GW approximation in many strongly correlated transition metal oxides is still problematic [71]. For example, G0W0 using LDA as a reference underestimates the gap of ZnO by 1eV and Fe2O3 by 0.7 eV [70] but overestimates the gap of TiO2 by 0.3 eV and SrTiO3 by 0.6 eV [71]. The GW quasi-particle gap strongly depends on the reference DFT calculations. For example, in Fe2O3, G0W0 with PBE0 and HSE as references overestimates the band gap by 2.0 eV, while it underestimates the band gap by 0.7 eV when it uses LDA as a reference. G0W0 using LDA + U(4.3eV) as a reference does give reasonable gap in agreement with experiment [70]. On the other hand, the self-consistency in the wave functions is essential in some systems

(Cu2O [76], Fe2O3 [70]) while in other systems, the self-consistency does not improve the calculation [77]. It is also argued that in transition metal oxides, one needs to employ LDA + U (to be discussed in later section) or DFT with hybrid functionals to provide a reasonable good zeroth order reference that GW can start with [71].

1.2.4 LDA + U

As we mentioned above, DFT (with LDA or GGA) and GW often fails for strongly correlated materials such as transition metal oxides or Kondo systems, in which the electronic ground state requires a superposition of multiple Slater determinants. There were several attempts to improve on the LDA in order to take into account the strong electron-electron correlations. One of the approaches is the so called LDA + U

14 method [78, 65, 66], in which one includes a Hubbard U interaction term for localized d orbitals,

1 X Hˆ = Hˆ UN(N 1)/2 + U nˆ nˆ + , (1.13) LDA − − 2 i j ··· i=j 6 where ni is the number of d electrons on site-i. The inclusion of the Hubbard U interaction results in an orbital dependent potential,

1  V = V (r) + U n + . (1.14) i LDA 2 − i ···

The LDA + U orbital-dependent potential does produce upper and lower Hubbard bands with a gap related to U, thus qualitatively describe the physics of Mott insulators. Eq. (1.13) in principle can include other

Coulomb interaction terms, such as exchange interaction Jσ σ , to capture more specific properties of the i · j correlated systems.

It has been demonstrated that when applied to the transition metal oxides, such as CaCuO2, CuO, NiO, CoO, FeO, MnO, the LDA + U method gives improvement compared with LDA not only for excited- state properties such as energy gaps but also for ground-state properties such as magnetic moments and interatomic exchange parameters [78, 66, 79]. For example, it successfully describes the charge ordering in Fe3O4 by including an intersite Coulomb interaction [80]; it gives a correct orbital polarization and a corresponding Jahn-Teller distortion in KCuF3 [66]; it captures the polaron formation in La2 xSrxCuO4 − and La2 xSrxNiO4 [66]. LDA+U has also been applied to produce the optimal trial wave function for − quantum Monte Carlo simulations for cuprate (Ca2CuO3) [81]. However, apart from the success of LDA + U in qualitative description of strongly correlated transition metal compounds, there are some inherent limitations of this approach. A serious one is the double counting of Coulomb interactions [82]. In LDA, all electron-electron interactions have already been taken into account in a mean field way. A including of an extra Hubbard Hamiltonian leads to a double counting of the portion of Coulomb energy related to the electrons in d orbitals. In principle, one could subtract the mean-field energy of the Hubbard Hamiltonian, leaving only a correction to the LDA-type mean-field solution, as is done in Eq.(1.13). However, since the LDA Hamiltonian is written in terms of the total density and the

Hubbard Hamiltonian in the orbital representation, there is no well defined approach to build a direct link between the two. Several recipes do exist; however it has been shown that different recipes give different results [82].

The second problem is related to the choice of parameters: the Hubbard U and the exchange interaction

J whose value is not known a-priori. What one normally does is to determine the value by seeking a

15 good agreement of the calculated properties with the experimental results in a semiempirical way. Many of

the studies have chosen the optimal parameter to match the Kohn-Sham band gap with experiment [83].

However, as we have discussed earlier, it is unknown whether the Kohn-Sham gap equal the fundamental

gap within the LDA + U framework, therefore tuning U to match the Kohn-Sham gap with experiment in a

na¨ıve way is not acceptable. Meanwhile, this approach greatly limits the method in application to materials

prediction and design. Several approaches have been developed in the literature, including the constraint

density functional theory approach [84] (changing the occupancy of localized orbitals and calculating the

energy cost), the constraint RPA approach [85], and the linear response approach [86].

Third, LDA + U is not a good theory for characterizing correlated metals [87]. Particularly in describing

the metal-insulator transition, LDA + U can characterize the insulating phase, however it fails to describe

the correlated metallic phase. For example, in vanadium dioxide, LDA with a reasonable U will produce

insulating states for both the monoclinic and rutile structures (the latter is a paramagnetic metal) [83]. The

reason is that the Hubbard U Hamiltonian is still treated in a mean-field way. Therefore, the energy gain

due to the formation of Hubbard bands is so large in LDA + U that for realistic values of U, it always

artificially splits bands and makes the system an insulator, even if it is a conductor.

1.2.5 DFT + DMFT

Within the LDA + U framework, the Coulomb interaction in Eq. (1.13) is treated within the Hartree-

Fock approximation. Hence, LDA + U does not contain the many-body effect in a dynamical way. As

we know, LDA + U is good in describing insulating phase of strongly correlated systems (for example the

antiferromagnetic Mott insulator); however it fails to describe correlated metallic states [87]. It generally

produce insulating state for correlated metals. It is claimed that LDA with dynamic mean field theory

(DMFT) can potentially describe systems for all Coulomb interactions [87], especially in the intermediate

regime where the Coulomb interaction is comparable with the bandwidth which LDA and LDA+U fails to

describe.

DMFT is first developed to study model Hamiltonians. To understand the general ideas of DMFT, let

us start from the Weiss mean-field theory in Ising model 1

X X H = J S S + h h . (1.15) − ij i j i ij i h i

The Weiss mean-field theory treats each site So as a spin feeling the presence of an averaged magnetic field

1We follow [88] in describing the DMFT formalism.

16 from nearby spins,

X H = h S , h = h + J m = h + zJm . (1.16) mean − eff o eff oi i i

With this, one can determin the mean field equation for magnetization

m = tanh(βh + zβJm) . (1.17)

This mean field equation is an approximation of the true solution. It is shown that if z , the mean field → ∞ solution is exact.

DMFT is a direct extension of the Weiss mean-field theory to quantum many-body systems. Let us use the Hubbard model as an example to illustrate the main ideas. The Hubbard model is,

X X H = tij(ciσ† cjσ + h.c.) + U ni ni . (1.18) − ↑ ↓ ij ,σ i h i

In the path integral formalism, the partition function is,

Z Z β   S h X ∂ X X i = dci†dciσe− ,S = dτ ciσ† (τ) µ ciσ(τ) tciσ† (τ)cjσ(τ) + Uni (τ)ni (τ) . Z ∂τ − − ↑ ↓ 0 iσ ijσ i (1.19)

To obtain a mean field description of this Hamiltonian, we identify a specific site o, and integrate out degrees of freedom of all the other sites c , i = o, to obtain an effective action for c , iσ 6 oσ

Z β Z β Z β X 1 Seff = dτ dτ 0 coσ† (τ) 0− (τ τ 0)coσ(τ 0) + U dτno (τ)no (τ) , (1.20a) − G − ↑ ↓ 0 0 σ 0   1 ∂ X − (τ τ 0) = µ δ t t G (τ τ 0) (1.20b) G0 − − ∂τ − τ1,τ2 − io oj ij − ij

Here, other higher order terms (higher than quadratic order) are neglected. The higher order terms vanish in the infinite dimension limit D . (τ τ 0) plays the role of the Weiss effective field. Its physical → ∞ G0 − content is the effective amplitude for a fermion created on the isolated site at time τ (coming from the

“eternal bath”) and destroyed at time τ 0 (going back to the bath). Gij is the Green function on the whole crystal with site-o being excluded,

1 G (τ τ 0) = T c (τ)c† (τ 0) ,G(k, iω ) = . (1.21) ij − −h iσ jσ i n iω + µ  Σ(k, iω ) n − k − n 17 Given G (τ τ 0), we know (τ τ 0), thus we can solve the single-site interacting model (let us called ij − G0 − “impurity” model), and obtain the onsite interacting Green function Go

Z β iωnτ (2n + 1)π Go(τ τ 0) = T co(τ)co†(τ 0) Seff ,Go(iωn) = dτGo(τ)e , ωn . (1.22) − −h i 0 ≡ β

The self-consistency requires

X 1 Go = Gii = . (1.23) iωn + µ k Σ(k, iωn) k − −

Here, one neglects the spatial fluctuation of the self-interaction and equals the self interaction of the back-

ground lattice Σ(k, iωn) with the self interaction from the impurity model Σo(iωn),

Σ(k, iω ) Σ (iω ) . (1.24) n ' o n

Σ (iω ) is related to and G as, o n G0 o

1 1 Σ (iω ) = − (iω ) G− (iω ) . (1.25) o n G0 n − o n

In practical calculations, one usually solve the problem iteratively:

(1) choose an initial guess of the self interaction for Gij:Σin(iωn) (can be set to zero);

(2) solve the impurity model and obtain Go and Σo(iωn) (by quantum Monte Carlo, exact diagonalization, etc);

(3) use Σo(iωn) as a new self interaction for Gij and repeat the process until Σo = Σin

From DMFT formulation, we see that there are two approximations in DMFT: (1) the neglect of higher

order interacting terms from the “bath”; (2) the neglect of the k-dependence of the self interaction. Both

approximations will be exact as D [89]. → ∞ In the LDA + DMFT approach, one considers the following Hamiltonian [89]

0 X X H = HLDA + U nim nim + (V δσσ0 J)nimσnim0σ0 , (1.26) ↑ ↓ − m,i i,m=m ,σ,σ 6 0 0

0 where nim is the number operator for m-th d orbital on site i. HLDA is the one-particle part of the

18 Hamiltonian,

X X H0 = H ∆ n . (1.27) LDA LDA − d imσ i mσ

The energy term containing ∆d is a shift of the one-particle potential of the interacting orbitals. This cancels the Coulomb contribution to the LDA results, which is the double-counting correction. ∆d can be calculated by constrained RPA. Within this framework, Eq. (1.21) changes to

1 G(k, iω ) = . (1.28) n iω + µ H0 (k) Σ(iω ) n − LDA − n

This approach has been applied to many correlated systems and reproduces the photoemission data qualitatively agree with experiments in many materials, such as SrVO3 and CaVO3 [90], V2O3 [91], and

VO2 in monoclinic phase [92]. Despite its success in capturing the spectral properties of correlated systems. There are several issues in the LDA+DMFT approach: (1) similar to LDA+U, the results depend on the parameters being used. For example, in the case of La0.94 Sr0.06TiO3, different values of the Hubbard U give rise to different quantitative results [87]. One generally needs to tune the parameter U to match the experimental data. (2) The results depend on different double counting correction schemes in a lot of systems such as MnO, NiO, and CoO

[93]. (3) In some systems, the spatial fluctuation of the self energy might be relevant to the system’s physical properties. However, this is inherently neglected in DMFT. One can in principle enlarge the size of

“impurity” to account for the spatial fluctuation to some extent. This is called the cluster DMFT method

(cDMFT). For example, in monoclinic VO2, it is shown that the single-site DMFT is unable to produce an insulating solution. One needss to use cluster DMFT to correctly captured the spatial fluctuation of the self energy which is related to the dimerization [94]. (4) The physical quantity that can be computed is still limited to spectral property, since it arise naturally in the DMFT self-consistency scheme [87].

1.2.6 General remarks of the electronic structure methods

Up to now, we have discussed several electronic structure methods. Let us give some general remarks.

Hartree-Fock: a variational method which provides starting points for post-Hartree-Fock methods; It • poorly describes electron correlations. The ground state wave function is pure non-interacting.

CI: exact in the limit of infinite basis set; exponential scaling with system size. Many cutoff schemes • can be applied, such as CISD, CISTD, CISTDQ, etc.

19 DFT: successfully describes the electronic structure for main group compounds; underestimates the • fundamental gap; fails to describe transition metal compounds due to its inadequacy in capturing

static correlations.

GW: solves the fundamental gap problems in DFT; fails in many transition metal oxides. •

LDA + U: qualitatively describes the effect of Coulomb interaction of d/f orbitals in Mott-Hubbard • and charge transfer insulators; fails to describe correlated paramagnetic metals; has double counting

problem; involves tuning parameters (U, J, etc).

LDA + DMFT: qualitatively describes both Mott-Hubbard/charge transfer insulators and paramag- • netic metals; has double counting problem; involves tuning parameters (U, J, etc).

1.3 Contents of this thesis

In Chapter. 2, we will focus on the formalism of the quantum Monte Carlo method and its application to correlated electronic systems. In Chapters. 3 and 4, we apply QMC to two representative correlated systems,

VO2, and graphene. VO2 is a transition metal compound with a metal-insulator transition at T=340K. We will demonstrate how QMC correctly describes the electronic structure of VO2, and unveils the mechanism of the phase transition which had been under debate for decades. On the other hand, graphene is special because the interplay of its linear quasiparticle dispersion with the long ranged Coulomb interaction. Using high accurate QMC calculations as well as random phase approximations (RPA), we reveal the importance of the electron screening from bonding σ electrons to the low energy physics of graphene. Chapters. 5 and 6 will be dedicated to two methodology developments: (1) the implementation of the extended Koopmans’ theorem in QMC for characterizing quasiparticle excitation spectrum; (2) constructing low energy effective models from ab intitio simulations to identify relevant low energy physics for correlated systems. Finally,

Chapter. 7 will be a summary.

20 Chapter 2

Quantum Monte Carlo method and its application to strongly correlated electronic systems

In this chapter, we will present the main idea of the quantum Monte Carlo method and its application

to strongly correlated electronic systems. There are many different ab initio quantum Monte Carlo tech-

niques, including the path integral quantum Monte Carlo (PIMC) [95], auxiliary field quantum Monte Carlo

(AFQMC) [96], full configuration interactions quantum Monte Carlo (FCIQMC) [51], variational quantum

Monte Carlo (VMC), and fixed-node diffusion Monte Carlo (DMC) [2]. In this chapter, we will mainly focus

on the last two methods: VMC and DMC. Both of them are ground state Monte Carlo methods.

2.1 Monte Carlo integration and the Metropolis algorithm

The complexity of any deterministic integration techniques using quadrature rules, grows exponentially as

the number of dimensions increases. This exponential scaling makes all the deterministic integration methods

fail to be applied to multidimensional integrals. The Monte Carlo method, on the other hand, is useful for

multi-dimensional integration. In the Monte Carlo method, one first generates a sample of random numbers

Ri (multdimensional) with a certain probability distribution ρ(R), and calculates the integral as follows,

Z R dR[f(R)/ρ(R)]ρ(R) 1 X f(Ri) I = dRf(R) = = , (2.1) R dRρ(R) M ρ(R ) i i

where M is the number of sampling points. One striking feature of the Monte Carlo integration is that the

complexity is independent of the number of dimension, but depends on the variance,

σ 2 2 2 δI = , σ = Var(g f/ρ) = g ρ g . (2.2) √N ≡ h i − h iρ where we define g = R g(R)ρ(R)dR, assuming R dRρ(R) = 1. If we choose a good sampling function h ig ρ(R), we can reduce the variance. This method of choosing a proper probability distribution function ρ

is called importance sampling. In general, to reduce the variance of f/ρ, we should choose a function sufficiently close to f. It is easy to see that if we choose ρ(R) = f(R) , the variance will be very small. In | |

21 fact if f > 0 (or f < 0) always, choosing ρ = f , the variance will be zero. | | The Metropolis algorithm [97] is a way to generate samples of a certain unnormalized probability distribu-

tion using random walk approach. It essentially generates a set of numbers Xi, with the relative probability

for Xi and Xj given by ρ(Xi)/ρ(Xj). In this algorithm, one chooses an initial position of the walker X1 and

then perform a random move to X2. This move is either accepted or rejected based on certain acceptance probability P (X X ). Imposing the detailed balance condition: 1 → 2

ρ(X )P (X X )T (X X ) = ρ(X )P (X X )T (X X ) . (2.3) 1 1 → 2 1 → 2 2 2 → 1 2 → 1

where T is the probability of the trial move: X X 1. P satisfies 1 → 2

 ρ(X )T (X X ) P (X X ) = min 1, 2 2 → 1 . (2.4) 1 → 2 ρ(X )T (X X ) 1 1 → 2

If we want to calculate the expectation value of certain observable Oˆ, we can use the Metropolis algorithm

to generate the sampling with probability Ψ(R) 2, and compute the expectation value of O as follows | |

Z 1 X O = dR Ψ(R) 2O(R) = O(R ) . (2.5) h i | | M i i

This is extremely efficient compared with deterministic integration approaches as the number of particles

increases.

2.2 Variational Monte Carlo (VMC)

The variational Monte Carlo is based on the variational principle, that for any trial wave function ΨT (p)(p is the adjustable parameter in ΨT ), the energy expectation value is an upper bound of ground state energy: Ψ H Ψ E . Therefore, one can adjust the parameter p to get a lowest upper bound for ground state h T | | T i ≥ 0 energy. Monte Carlo integration is employed in evaluation the energy expectation, which is a 3N dimension integral (N is the number of electrons in the system),

R R 2 HΨT dRN Ψ∗ HΨT dRN ΨT 1 X E = T = | | ΨT = E (R ), (2.6) R dR Ψ 2 R dR Ψ 2 M L i N | T | N | T | i 1This in principle can be any positive function. For example, if one simulate the trial move as a diffusion process, one can choose T to be a Gaussian function. One can adjust the diffusion rate to maximize the acceptance rate.

22 where we define the “local energy” as,

Hˆ ΨT (R) EL(R) = . (2.7) ΨT (R)

Instead of minimizing the energy expectation value, one can also minimize the variance of the local energy,

σ2 = E2 E 2 , (2.8) h Li − h Li

which goes to zero as ΨT approach to a certain eigenstate. One can also minimize the mixed quantity

c[Ψ ] = γE2 + (1 γ)σ2 , 0 < γ < 1. (2.9) T VMC − VMC

In practice, one would choose a trial wave function sufficiently close to the exact ground state. For

electronic systems, one normally chooses a Slater-Jastrow wave function,

" # X u ψ(R) = ciDi e− . (2.10) i

Here, Di’s are Slater determinants obtained from previous calculations such as HF and DFT. u is the Jastrow factor which is to capture the electron-electron correlations. The following is a general form of u (up to three-body) [98]

X X (1b) X X (2b) X X 3b u(ri, rj, p) = pk ak(riα) + pk bk(rij) + pklm[ak(riα)al(rjα) α,i k(α) i

+al(riα)ak(rjα))bm(rij] . (2.11)

where α is nucleus index and i, j are electron indices; riα is the electron-nucleus distance; rij is the electron- electron distance; and a and b are basis functions. All the p with different indices are the parameters to be optimized. The Jastrow factor takes care of the electron-electron and electron-ion cusp conditions:

  1 , like spin electrons  4 du − = 1 , unlike spin electrons . (2.12) dr − 2   Z, electron ions

The singe Slater Jastrow wave function can recover about 80% correlation energy [2]. It is particularly

good for capturing the dynamic correlation energy. The cusp condition force electrons to avoid each other,

23 0.6 − RHF CI2 PBE 0.7 − CISD CI2-J PBE0 RHF-SJ DMC 0.8 −

0.9 −

Energy [Ha] 1.0 −

1.1 −

1.2 − 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 distance [A]

Figure 2.1: H2 binding curve: HF: Hartree-Fock, PBE – DFT with PBE functional; PBE0 – DFT with PBE0 functional; CISD – CI with single and double excitation; CI2 – CI with two determinants; SJ – VMC with single Slater-Jastrow trial wave function; CI2-J – VMC with CI2 Slater determinant multiplied with a Jastrow factor; DMC – DMC with single Slater-Jastrow trial wave function. thus reduces the total Coulomb interaction. To get a sense what a Jastrow factor can do, let us go back to the stretched H2 example. The QMC results for H2 binding are shown in Fig. 2.1. First, we find that the single Slater-Jastrow wave function (RHF Slater determinant multiplied with a Jastrow factor) captures the correlation energy to some extent. And interestingly, it is close to restricted DFT results. This implies that in this particular system, a Jastrow factor capture the correlation energy in a similar way to DFT. However, the static correlation error is still very large in single Slater-Jastrow wave function (restricted orbitals).

Adding an extra determinant in the trial wave function captures the static correlation (CI2). We also see an improvement of CI2-J compared with CI2 because the dynamic correlation has been properly counted with the Jastrow factor.

2.3 Diffusion Monte Carlo (DMC)

After VMC, one already obtains a relative good trial wave function. The next step is to apply the diffusion

τH Monte Carlo (DMC). The basic idea is to act the imaginary time evolution operator e− on the trial wave function,

τ(H ET ) Ψ0 = lim e− − Ψ T , (2.13) τ | i →∞ | i

24 where ET is an adjusting parameter to keep the wave function normalized. As the imaginary time goes to infinity, we will arrive at the ground state Ψ , and E E . This evolution process corresponds to an 0 T → 0 imaginary time Schr¨odingerequation, which is formally the same as a diffusion equation. This diffusion equation can then be solved using Monte Carlo algorithm. One identifies the wave function with the density distribution of random walkers. Given an initial distribution ΨT , the random walkers undergo a diffusion process according to certain rules. The equilibrium distribution of the walkers give us the final ground state

Ψ0. To see the rules of the diffusion process, let us define the Green function,

τ(H E ) G(R0 R, τ) = R0 e− − T R . (2.14) ← h | | i

For small time step τ, by using the Trotter-Suzuki formula, one can get

 2  3N/2 (R R0) 3 G(R0 R, τ) = (2πτ)− exp − exp[ τ[V (R) + V (R0) 2E ]/2] + O(τ ) . (2.15) ← − 2τ − − T

The Gaussian part can be interpreted as a transition probability for the evolution of the random walkers,

and the factor

P = exp[ τ(V (R) + V (R0) 2E )/2] (2.16) − − T

acts as a time-dependent reweighting of the random particles. One can implement this in two different ways:

(a) assign each walker a weight and accumulate the product of weights during the propagation.

(b) each walker has the same weight, but will undergo a branching process ( birth/death ). P determines

the number of walkers that survive to the next step: (1) if P < 1, the walker continues its evolution

with probability P [i.e., dies with probability (1-P)]; (2) if P 1 the original walker continues, and at ≥ the same time, a new walker is created with probability P 1. −

The energy offset ET controls the total number of walkers (or total weight of walkers), which is occasionally adjusted during the simulation. Method (a) is easier to implement in parallel computation architecture,

since the number of walkers is the same throughout the whole simulation. One can distribute equal number

of walkers in each processor.

The whole process can be schematically depicted in Fig. 2.2. Here we have one dimensional potential.

The walkers are initialized according to a uniform distribution. As the imaginary-time propagation proceeds,

25 the distribution converges towards the ground state. The branching process is applied, as one can see the death and birth of walkers in the process.

Figure 2.2: Illustration of the walker evolution in a one dimensional potential well in the diffusion Monte Carlo (DMC) method. Figure is taken from [2].

2.3.1 The fermion sign problem and the fixed-node approximation

In diffusion Monte Carlo, one identifies the wave function with the density/weight distribution of the walkers.

However, the many-body fermionic wave function (more than one electron) of a system is not always positive.

In principle, one can allow negative weight of walkers, however, this will lead to the fermion sign problem. The signal-to-noise ratio decay exponentially as the number of particles increases. The fermion quantum Monte

Carlo problem is a nondeterministic polynomial (NP) hard problem [99]. Therefore, a na¨ıve implementation with negative weights applies only for small systems.

In the current state of the art, the fixed-node approximation is employed. In the fixed-node approx- imation, a trial many-electron wave function is used to define a trial many-electron nodal surface. One introduces a new function f(R, t) = φ(R, t)ΨT (R), which satisfies the following differential equation,

1 ∂ f(R, t) = 2f(R, t) + [v (R)f(R, t)] + [E (R) E ]f(R, t), − t −2∇ ∇ · D L − T 1 1 v (R) = Ψ (R)− Ψ (R),E (R) = Ψ (R)− HΨ (R) . (2.17) D T ∇ T L T T

26 Eq. (2.17) is a diffusion equation with drift velocity vD. The corresponding Green function is

 2  3N/2 (R0 R τvD(R)) G˜(R0 R, τ) = (2πτ)− exp − − exp[ τ[E (R) + E (R0) 2E ]/2] . (2.18) ← − 2τ − L L − T

This importance sampling using trial wave function improves the efficiency of the simulation in the following ways:

(1) The fixed-node DMC algorithm produces the lowest-energy many-electron state with the given nodal

surface. Fixed-node DMC may therefore be regarded as a variational method that gives exact results

if the trial nodal surface is exact.

(2) The reweighting function P now contains the local energy EL instead of V (R). A good trial wave

function will have roughly constant EL, therefore it significantly reduces the population fluctuation of walkers.

In current state of the art, Slater-Jastrow type wave function has been widely employed [2]. In most of the cases, even single Slater-Jastrow wave function produce very good results. For example, in the case of

H2, although the single Slater-Jastrow wave function still has large static correlation error at large distance, but the DMC energy is almost exact, implying that the nodal structure of the single Slater-Jastrow wave function is close to the exact ground state in the case of H2. There are some known cases where a single

Slater determinant wave function does not give correct nodal structure, for example Be2. In these cases, One needs to use multiple Slater-Jastrow wave functions to make the nodal structure correct. More complicated wave functions have also been applied for specific purpose, including coupled-cluster wave function [100], backflow wave function [101], Pfaffian pair wave function [102], resonating valence bond wave function [103].

2.3.2 Mixed and extrapolated estimators

The energy expectation can be evaluated as

τH/2 τH/2 e− Ψ H e− Ψ E = lim h T | | T i τ τH/2 τH/2 →∞ e− ΨT e− ΨT h τH | i e− Ψ H Ψ = lim h T | | T i τ τH →∞ e− ΨT ΨT h | iR Ψ0 H ΨT f(R, τ)EL(R)dR = h | | i = lim R Ψ Ψ τ f(R, τ)dR h 0| T i →∞ 1 X = E (R ) . (2.19) M L m m

27 Here we have used the fact that limτ f(R, τ) = ΨT (R)Ψ0(R), which is the equilibrium distribution of →∞ the walkers in the drift diffusion problem. For observables other than energy, one can use the extrapolated estimator,

  Ψ O Ψ = 2 Ψ O Ψ Ψ O Ψ + Ψ Ψ 2] =: 2 O O . (2.20) h 0| | 0i h 0| | T i − h T | | T i O | 0 − T | h iDMC − h iVMC

Notice that if O is a diagonal operator, we have

Z Ψ O Ψ = f(R)O(R)dR . (2.21) h 0| | T i

2.3.3 Pseudopotential approximation

The complexity of QMC calculation increases rapidly with the atomic number Z (roughly Z5.5 [104] Z6.5 ∼ [105]). On the other hand, the presence of core electrons also cause two problems [2]: (1) Shorter length scales associated with variations in the wave function near a high Z nucleus requires small time step; (2) the

fluctuation in the local energy is large near the nucleus, which induces large variance of the total energy.

Introducing the pseudopotential to replace the core electrons reduces the effective value of Z thus improves the computational efficiency. The underlying reason why we can use pseudopotential is that many properties of interest, including the interatomic bonding and the low-energy excitations, are determined by the behavior of valence electrons.

A generic form of the pseudopotential is

X V = f (r) + g (r) L L . (2.22) loc L | ih | L

Here, the first term is the local part which is the same for all angular moments. The second term is the nonlocal part, which exerts different potentials on electrons of different angular moments. When applying to a single body wave function, it acts as,

X Z V ψ(r) = floc(r)ψ(r) + gL(r)YLM (Ω) YLM (Ω0)ψ(r0)dΩ0 . (2.23) L 4π

The nonlocal pseudopotential will make the branching term R exp( τV ) R0 negative in DMC. One h | − | i needs to apply approximations such as localization approximations, T-moves [2]. The accuracy of these approximations depends on the quality of the trial wave function.

28 2.3.4 Comparison of QMC with other methods

Here I would like make some remarks about several favorable characteristics of QMC:

(a) QMC explicitly samples many-body electronic configurations using Coulomb’s law for interactions,

which allows for the description of correlation effects without effective parameters.

(b) Both VMC and DMC are variational. The accuracy depends on the quality of the trial wave function.

One can in principle unbiasedly approach the exact results through improving the quality of trial wave

function.

(c) The computation cost scales as N3 which is significantly cheaper than other correlated methods such

as CI, CCSD.

(d) Monte Carlo calculations are inherently parallel in nature. This will make QMC eventually surpass

other correlated methods as the advancing of the computation power.

2.4 Application to strongly correlated systems

QMC has been applied to a variety of continuum systems, including homogeneous electron gas [106, 107,

108, 109, 110, 111], equation of state and properties of liquid, band structure of solids [112], the cohesive energy of solids [113], transition metal oxide molecules [114], defects in solids [115, 116]. Here in this section, we would like to focus on strongly correlated systems transition metal oxides and rare earth compounds.

We are hoping that through this relatively detailed yet still very brief review, the reader can have a sense of what QMC has achieved and what techniques have been employed.

2.4.1 NiO

NiO is known to be a charge-transfer insulator with an antiferromagnetic ordering with alternating spin

(111) planes in cubic rocksalt structure (AF II) [117]. DFT with LDA or GGA predicts a gap of 0.4 eV which is much smaller than the experimental value of 4.3 eV. LDA and GGA also give the wrong order of single orbital eigenvalue. Hartree-Fock on the other hand, gives a gap that is too large. However, Hartree-

Fock does give correct energetic order of the single particle orbitals. It indicates the importance of exchange interaction in NiO. Needs and Tower performed the first DMC calculations using a Slater-Jastrow trial wave function with Hartree-Fock orbitals, and predicts a cohesive energy of 9.442(2)eV in agreement with the experimental value (9.5 eV) [117]. Later on, Mitra et al [118] employed a Slater-Jastrow trial wave function with orbitals from LDA+U, also gives consistent result for cohesive energy [9.44(7) eV]. Their calculations

29 also obtained high accuracy in computing the lattice constant (∆a = 0.01A), bulk modulus, quasiparticle band gap [4.72(15) eV].

2.4.2 MnO

MnO is also a charge transfer insulator similar to NiO. DFT again predicts a band gap that is too small.

Lee et al performed DMC calculations using Slater-Jastrow wave functions with orbitals from unrestricted

Hartree-Fock and DFT with B3LYP exchange-correlation functional [119]. It is shown that both the two trial wave functions give comparable results for total energy. But B3LYP predicts a gap [4.8(2)eV] that is closer to experiment (4.1eV). The predicted cohesive energy is within 0.1 eV difference from the experiment.

The predicted magnetic moment is also consistent with experiment. Their calculation supports the fact that

MnO is a charge transfer insulator rather than a Mott-Hubbard insulator.

Schiller et al studied the relative phase stability of the rocksalt structure with the zincblende structure

[120]. The DMC calculations find that Rocksalt MnO is 0.35(1) eV lower than Zincblende MnO. They studied the DMC energy of different trial wave functions. The Slater determinants are from DFT calculations with hybrid functionals of different mixing of Hartree-Fock exchange. It is found that PBE0 functional with 25% mixing produces lowest DMC energy.

2.4.3 FeO

Kolorenˇcand Mitas [121] studied the equation of state of FeO under pressure. FeO is an antiferromagnetic insulator with rocksalt structure (B1) at ambient conditions. Below N´eeltemperature (198K), a small rhombohedral distortion occurs. The distorted structure is called rB1. The shockwave experiment found that at 70 GPa, FeO turns into NiAs type structure, denoted as iB8 structure. DFT predicts the transition pressure to be 500 GPa, far beyond the experimental measured value. Using a Slater-Jastrow trial wave function with orbitals from DFT calculations with PBE020 (20% GGA exchange is replaced with the exact exchange), they computed the total energy of B1 and iB8 under different pressures, and found a transition pressure [65(5) GPa] consistent with experiment (70 GPa). The P(V) curves for B1 and iB8 phases are given.

At the same time, DMC also accurately predicts lattice constants (error within 0.01A) and bulk module

(error within 10 GPa), cohesive energy [9.66(4) eV/FeO, experimental value is 9.7eV], and quasiparticle gap

[2.8(3)eV, experimental value 2.4 eV].

30 2− 2.4.4 Cuprate: Ca2CuO3, La2CuO4, CaCuO2, CuO2 , Ca2CuCl2O2

Various high Tc cuprates have also been studied using QMC. Foyevtsova et al [81] performed DMC calcu- lations for Ca2CuO3 at ferromagnetic and antiferromagnetic states, and obtained the spin super-exchange coupling constants in good agreement with experiment.

2 Wagner and Abbamonte [122] performed complete studies for La2CuO3, CaCuO2, and CuO2− on su- perexchange coupling constant, magnetic moment, optical gaps, magneto-phonon (spin-lattice) coupling effect, that are in reasonably good agreement with available experimental data.

Wagner [123] studied 25% hole doped cuprate (CaCuO2 and Ca2CuO2Cl2). By investigating the DMC energies of different spin ordered states, Wagner found that the doped ground state is a spin polaron state, in which charge is localized through a strong interaction with the spin. Similar study has also been performed for Ca2NiO2Cl2, but the spin polaron is not found, indicating that the spin polaron state is unique in cuprate.

2.4.5 Defect formation energy of ZnO

DMC has also been applied to study the structural stability of ZnO of three different structures: rock salt, zinc blende, and wurtzite [124]. At low pressure, rock salt is the most stable structure. DMC predicts a structural transition from rock salt to wurtzite at P = 9.17(8)GPa consistent with experiment (8.7 10 ∼ GPa). The errors of the predicted lattice constants for the three structures are within 0.005 A.˚ The cohesive energy for rock salt structure [7.42(2)eV] agrees with experiment within 0.1 eV. The quasiparticle gap is predicted to be 3.22(16) 3.43(9) eV consistent with the experimental value (3.44eV). The energetics of ∼ different types of defects in ZnO are also investigated, including the formation energy of oxygen vacancy, donor like defects.

2.4.6 Magnetic order in FeSe

FeSe is an ion-based superconductor with the superconducting transition temperature (Tc) equals to 8K at ambient pressure. The transition temperature is strongly dependent on pressure and reaches 37 K at

7 GPa. It is argued that the spins and magnetism might be important to the superconductivity [125].

Busemeyer et al [126] applied DMC to studied energetic properties of FeSe of different magnetic orders: collinear, ferromagnetic, checkerboard, and staggered dimer configurations. It is found that the collinear type magnetic configurations (including collinear, bycollinear, collinear flip 1) are the lowest energetic orderings up to 10GPa. Among the three, the collinear is the lowest energy state, but it is degenerate with bicollinear and collinear flip 1 at around 7GPa, coincident with the optimal pressure at which the Tc is maximum. This implies that the magnetic fluctuation might be closely related to superconductivity.

31 On the other hand, DMC predicts lattice constants in very good agreement with experiment (within

0.05A).˚ In particular, the interlayer distance is overestimated by about 0.5A˚ because of its inadequacy in describing the van der Waals interaction for DFT with GGA functional. DMC instead describes the van der

Waals interactions very well.

2.4.7 Metal-insulator transition in VO2

Vanadium dioxide (VO2) is a paradigmatic example of a strongly correlated system that undergoes a metal- insulator transition at a structural phase transition (from high temperature rutile to low temperature mon- oclinic structure). Many DFT based methods demonstrate that the correlation effects sensitively affects the properties of the two phases. Unfortunately, due to inadequate description of the electron-electron correla- tions of this system, the transition mechanism remained unclear for decades. We applied DMC method to perform energetic calculation for VO2 with different magnetic orders [127]. The DMC calculations correctly describe the conductivity (quasiparticle gap) and magnetic properties of the two phases. By explicitly inves- tigating the charge-charge, spin-spin spatial correlations, we revealed the microscopic transition mechanism.

The structural distortion induces a significant change of the superexchange coupling, which eventually drives the system to a spin-singlet insulating state.

2.4.8 Rare earth: cerium

Cerium is a prototypical f-electron system. There is a α γ phase transition. Across the transition, the − volume collapses along, which has been a puzzle since its discovery in 1927 [128]. Devaux et al investigated the electronic structure modifications across the volume collapse [128] using DMC. Through a close examination of the occupancy of the atomic orbitals in the two structure, it is found that the transition results from a subtle competition between local Coulomb repulsion and bandwidth. On the other hand, the overall onsite charge-charge correlation and spin-spin correlation do not have significant change across the transition, which disproved the Mott transition picture.

2.5 Summary

Throughout the above brief review of the application of QMC to correlated systems, let me comment on

QMC:

In most of the cases, a Slater-Jastrow wave function is accurate enough to capture the correlation • energy, 80% in VMC and 95% in DMC;

32 Trial-wave functions can be generated from HF, DFT (LDA, GGA, PBE0, HSE, ...), LDA + U calcu- • lations;

In transition metal oxides, DFT with PBE0 (GGA with 25% mixing of Hartree-Fock exchange) usually • provides optimal orbitals for trial wave function;

Physical observables that can be computed: total energy, cohesive energy, quasiparticle band gap, • lattice constant, charge and spin density, spatial correlation of physical quantity, many-body reduced

density matrix, magnetic moment, superexchange constant;

33 Chapter 3

The metal-insulator transition in vanadium dioxide

The contents of this chapter are based on our publication [127].

Systems of strongly correlated electrons at the border between a metal-insulator transition can result in a variety of unique and technologically useful behavior, such as high-temperature superconductivity [129] and colossal magnetoresistance [130]. In vanadium dioxide (VO2), the metal-insulator transition (MIT) occurs at T = 340K [131], at which the conductivity decreases by more than 4 orders of magnitude. This MIT is accompanied by a structural change from rutile (P42/mnm) to monoclinic (P21/c) [132] (see Fig. 3.1), as well as a transition in the magnetic susceptibility from a paramagnet-like Curie-Weiss law to a temperature- independent form. In the rutile phase, the vanadium atoms are located at the centers of octahedra formed by the oxygen atoms; chains of equidistant vanadium atoms lie along the c axis ([001]). In the low-temperature monoclinic phase, vanadium atoms shift from the centers of the oxygen octahedra and form a zig-zag pattern consisting of V dimers. It is a long-standing question whether the MIT is primarily caused by the structural change that doubles the unit cell (Peierls distortion), or by correlation effects that drive the system to become insulating [133, 3, 134], or potentially some mixture of the two.

The metal-insulator transition in VO2 is unusually challenging to describe. Standard density functional theory (DFT) [135, 83] obtains metallic states for both structures, while corrections based on an effective

Hubbard U[83, 136] or hybrid functionals[137] often obtain insulating states for both structures. Similarly,

DFT+DMFT[138, 139, 134] and DFT+GW[140, 141, 142] calculations indicate that how the correlation is treated changes the calculated properties dramatically. Since there is no a-priori guideline for these theories, while they are descriptive and valuable techniques, their reliability is uncertain in a predictive capacity. This issue is a severe constraint for the design and study of correlated electron systems.

In this study, we use the explicitly correlated fixed-node diffusion quantum Monte Carlo (FN-DMC) [2] to investigate the electronic structures of the rutile and monoclinic VO2 from first principles. FN-DMC has been shown to be a highly accurate method on other transition metal oxides[121, 122]. In this method, one explicitly samples many-body electronic configurations using Coulomb’s law for interactions, which allows for the description of correlation effects without effective parameters. We show that FN-DMC correctly

34 (a) (b)

Figure 3.1: Crystal structures of VO2 in different phases. Blue balls are vanadium atoms, and red balls are oxygen atoms. (a) high temperature rutile structure, with vanadium atoms sitting at the center of octahedra forms by surrounding oxygens; (b) low temperature monoclinic structure: vanadium atoms dimerized and shifted away from the centers of the octahedra and form zig-zag chain.

characterizes the electronic structure and magnetic response of VO2 in the two phases. Our calculations provide quantitative microscopic details of the structure-dependent spin couplings between vanadium atoms.

It clearly reveals how the structural distortion changes the interatomic hybridization between vanadium and oxygen, which results in a significant change of the superexchange magnetic coupling between vanadium atoms. Monoclinic VO2 is in a non-magnetic singlet state consisting of spin dimers due to strong intradimer coupling. The calculations contain a singlet-triplet spin excitation of 123(6) meV in monoclinic VO2, which can be verified in experiment to test the predictive power of this method.

The content of this chapter is as follows: in Section 3.1, I will give an overview of previous experimental and theoretical studies; then in Section 3.2, I will discuss our computation setup; In Sections 3.3 and 3.4, I will present the main results and conclusions of this work.

35 3.1 Summary of previous studies

3.1.1 Important experimental facts

Carrier concentrations and effective mass

The conductivity changes by more than four orders of magnitude across the transition [131]. The infrared

optical properties and the Hall experiment [143] shows that the mobility changes from 16cm2/V sec in rutile

2 VO2 (T=353K) to 0.5cm /V sec in monoclinic VO2 (T = 295K), and the carrier concentration changes from 30 1020/cm3 in rutile VO to 0.09 1020/cm3 in monoclinic VO . × 2 × 2 The effective mass is m∗ 0.75m at high temperature (T=550K), and increases as temperature lowers ' e [144]. It is shown that the effective mass diverges when approaching the transition temperature [145].

Magnetic properties

4 4 The magnetic susceptibility drops from 8 10− e.m.u./g (rutile) to 1 10− e.m.u./g (monoclinic) [146, 147]. × × In the high temperature metallic phase, the susceptibility decreases as temperature increases. This behavior, intermediate between Pauli and Curie-Weiss, is probably related to rather strong electron correlations.

Mossbauer effect studies of doped VO2 [4] shows that effective magneton number is ranging from 1.58 to

2.30µB in the rutile phase. In the low temperature monoclinic phase, susceptibility is much weaker and nearly temperature independent from the transition temperature (340K) down to 77K. NMR experiments prove that this is not an antiferromagnetic phase [148, 133].

Latent heat: lattice or electrons?

There is a nonzero latent heat of 1, 020 cal/mole [44.2meV/VO2] across the transition, corresponding to an entropy change ∆S of about 3 cal/mole C [149]. Therefore, the transition is first order. It is possibly made up of the following two contributions: (1) change of phonon spectrum due to crystalline distortion,

(2) discontinuity in the carrier density. Berglund and Guggenheim argued that a simple free-electron model based on reasonable carrier density can only account for approximately 1/6 of the entropy change [149]. The discrepancy might come from either the electron correlations effect or the lattice degree of freedom.

Raman spectroscopy measurement [150] shows that the phonon modes exhibit softening and damping across the transition from the monoclinic phase to the rutile phase. In the monoclinic phase only sharp phonon structure is observed, where as the rutile phase exhibits a pronounced softening and damping in the observed phonon spectrum [150]. The temperature-dependent phonon density of states (PDOS) was measured using neutron spectrometer by Budai et al [151]. The rutile PDOS at temperature above Tc are

36 considerably softer (lower in energy) than M1 PDOS. It contributes to a free energy change (T ∆S) of 28.9(3) meV/VO2, account for two thirds of latent heat. Therefore, there is still considerable fraction of latent heat coming from electron-electron correlations.

Strain dependence of the phase transition

Structure dependence of the metal-insulator transition is studied by using samples of different geometries, including bulk, nano-particle [152], nanowire [153, 154], thin film [155], etc. It was found that the mechanical effects can change the transition temperature, induce hysteresis, and even change the magnetic properties, indicating that the lattice degree of freedom couples strongly to the electronic degree of freedom.

Phase diagram under doping: existence of strongly correlated M2 phase

The addition of Ti, Cr, Al, or Fe leads to phases at room temperature which have different structures from the normal monoclinic phase of pure VO2. Doping the system with Cr will lead to another phase M2.

VO2 under pressure has similar phase diagram [156]. In the M2 phase, half of the V atoms form pairs and the other half form zig-zag chains. While the V atoms in the pairs are nonmagnetic, those in the zig-zag chains have local moments with antiferromagnetic order and are regarded as one-dimensional Heisenberg antiferromagnetic chains according to NMR studies [157]. Hence, M2 VO2 is a typical Mott insulator. This suggests strong electron correlations in VO2.

Rutile VO2: correlated metal

The resistivity of VO2 at high temperature violates the Ioffe-Regel-Mott limit [144]. The Ioffe-Regel-Mott limit is the threshold of the resistivity of materials when the mean free path is comparable to the interatomic distance. In rutile VO2, the resistivity increases linearly with temperature up to 600K. It goes beyond the Ioffe-Regel-Mott limit (400µΩcm) at 350K. At high temperature (above 350K), the mean free path is shorter than the interatomic distance; hence the resistivity is mainly due to electron-electron scattering instead of electron-phonon scattering. This behavior is common in cuprate [158]. This suggests that rutile VO2 is a correlated non Fermi-liquid metal.

Summary of important experimental facts

The conductivity and magnetic properties of different phases are summarized in Table. 3.1. The fact • that both the rutile and M2 phases are strongly correlated, indicates that electron-electron correlations

might play important role in the metal-insulator transition.

37 Table 3.1: Comparison among different phases of VO2. Structure rutile monoclinic M2

Transport correlated metal insulator (Eg = 0.7eV) Mott insulator (Eg = 0.5 eV) Magnetic paramagnetic nonmagnetic antiferromagnetic

Applying strain to the system strongly affects the metal-insulator transition temperature, as well as • the physical properties of the two phases, indicating that the lattice degree of freedom is strongly

coupled to the charge, spin degree of freedoms.

Changes of Raman spectra and phonon density of states across the transition indicates contribution • from phonons in driving the phase transition.

3.1.2 Theoretical and numerical studies

In the past decades, there have been tremendous theoretical efforts on understanding the properties of the two phases and the transition mechanism, ranging from phenomenological modeling to first principles simulations.

In the following, we will selectively introduce some representative studies. For the phenomenological models, we will introduce the main ideas of Goodenough’s molecular band theory [133, 159] which attributes the transition completely to structural change, in other words, electron-lattice coupling or Perierls instability.

The Mott-Hubbard picture attributes the opening of the insulating gap to the electron-electron correlations in the d orbitals. We will then comment on numerical studies, including DFT, DFT+U, DFT+GW, and

DFT+DMFT (or cluster DMFT).

Phenomenological models

Goodenough proposed a schematic energy band diagram based on the molecular orbitals picture [133, 159].

In this picture, vanadium 3d orbitals are subject to octahedral crystal field from the surrounding six oxygen atoms, and split into lower t2g and higher eg levels [see Fig. 3.2(a)]. Furthermore, due to the tetragonal crystal fields from the surrounding vanadium atoms, as well as the imperfectness of the octahedrons, the three t2g states located near the Fermi level, split into a d state, which is directed along the rutile cr k axis, and the remaining two π∗ states [see Fig. 3.2(a)]. In the monoclinic phase, V-V pairing within the vanadium chains parallel to the rutile c axis causes a splitting of the d band into filled bonding and empty k antibonding states. In contrast, the π bands move to higher energies due to the antiferroelectric zigzag-type displacement of the vanadium atoms. As a result, a band gap opens between the bonding d band and the π∗ k bands. Goodenough suggested that the metal-insulator transition results predominantly from the increased p-d overlap coming with the antiferroelectric distortion of the VO6 octahedra.

38 (a)

(b)

Figure 3.2: Schematic band structure from molecular orbital picture: (a) splitting of vanadium d orbitals due to the presence of crystal fields (hybridization with oxygen p orbitals) (b)changing of the band structure due to structural change from rutile to monoclinic.

A mechanism different from the molecular orbital picture of Goodenough was proposed by Zylbersztejn and Mott [160]. They attributed the coupled metal-insulator and structural transition to the strong electron- electron correlations in the vanadium d bands instead of electron-lattice interaction. These correlations are efficiently screened by the π∗ bands which overlaps with d band in rutile VO2. However, in monoclinic VO2, k the screening of the d electrons is suppressed since the π∗ bands shift upward due to the antiferroelectric displacement of the V atoms. Consequently, the narrow d bands at the Fermi energy are susceptible to strong Coulomb correlations; thus an insulating gap is opened and the system undergoes a Mott transition.

Zylbersztejn and Mott regarded rutile VO2 as a normal metal in which the electronic correlations has been screened.

In resolving the relative contribution of the lattice and electronic degree of freedom, Paquet and Leroux-

Hugon proposed a multi-band tight-binding model for VO2 (includes d and π∗ bands), aiming at incorpo- rating both the electron-electron interactions and the electron-lattice coupling simultaneously [161]. Using experimentally derived parameters, Paquet and Leroux-Hugon concluded that the metal-insulator transi- tion is primarily driven by electron-electron correlations. However, these correlations would be strongly renormalized by the lattice distortion and the electrostatic interaction between the d and the π∗ electrons.

39 First principles calculations

Wentzcovitch et al performed DFT(LDA) calculations of VO2 [162]. The optimal lattice constants found by their ab initio molecular dynamic studies agree very well with experiments. However, the band structure of monoclinic VO2 contradicts the experimental finding of an 0.6eV insulating gap. It predicts both the two structures to be metals.

LDA+U approach produces the correct optical band gap if one tunes the on-site coulomb repulsion U term appropriately (U 4eV, and J 0.12U) [163, 164, 83]. On the contrary, the rutile structure remains ∼ ∼ metallic if U < 7eV [163]. However, LDA+U predicts an antiferromagnetic state in the monoclinic phase which contradicts the fact that monoclinic VO2 is nonmagnetic. Similar problem also occurs for DFT using hybrid exchange functional (HSE) [83, 137]. HSE predicts an antiferromagnetic monoclinic VO2 ground state with a gap of 2.23 eV, and a ferromagnetic rutile VO2 ground state with a gap of 1.43 eV [137]. The energy difference between the low-temperature and high-temperature phases is much larger than latent heat at the transition [137].

A model GW [164] scheme with a static dielectric constant as an input (taken from experiment) does produce a good description of the quasiparticle density of states [164]. It is found that the vanadium d bands shift rigidly in GW band structure compared to LDA band structure. Full ab initio GW calculations are also performed [141, 165]. It is showed that a one-shot GW calculation (G0W0) using LDA as a reference fails to open up an insulating gap for monoclinic VO2 [141]. A full self-consistent (self-consistency obtained on the COHSEX level [76]) GW does give a correct quasiparticle band gap [141, 165]. GW calculation seems to support that the quasiparticle concept is more or less correct in VO2. It is shown that a single site LDA+DMFT calculation is not able to open up an insulating gap for monoclinic VO2 [166]. DFT+cluster DMFT with U = 4eV and J = 0.68eV correctly reproduces the energy gap in the insulator phase and at the same time matches the spectral weight with the photoemission experiment [166]. However, several critiques can be made for calculation of [166]: (a) the U and J are tuned to match the spectral function. It is unknown whether it can describe other properties as well; (b)

DMFT based on an approximation that the self energy is momentum independent, which is not true in VO2 according to the photoemission study [167].

In spite of these studies, the mechanism of the transition remain unclear. Since there is no a-priori guideline for these theories, while they are descriptive and valuable techniques, their reliability is uncertain in a predictive capacity. However, they do indicate the importance of electron-electron correlations to the electronic structure of VO2. How the correlation is treated changes the calculated properties. Therefore, employing a method that can accurately capture the electron-electron correlations in the system is crucial

40 500 Monoclinic(AFM) Monoclinic(FM) ]

V Monoclinic(UNP) e 400 Rutile(AFM) m

[ Rutile(FM)

2

O 300 V

r e

p 200

y g r 100 e n E 0

0 10 20 30 40 50 60 70 Fock mixing %

Figure 3.3: FN-DMC energies of VO2 with trial wave functions from DFT of different hybrid functionals. We computed the energies for different magnetic states: antiferromagnetic (AFM), ferromagnetic (FM), spin-unpoloarized (UNP). All the energies are relative to monoclinic AFM state: 2830.836(2)eV. −

for us to understand the underlying mechanism of the phase transition as well as the electronic structure of

the two phases.

3.2 Method

The calculations were performed as follows. The crystal structure of the rutile and monoclinic phases were

taken from experiment [168]. DFT calculations were performed using the CRYSTAL package In order to

find the true ground states with correct magnetic properties, we initialize spin configurations to be aligned,

anti-aligned, or unpolarized on vanadium atoms and compare their energy difference to find the lowest

energy states for the two structures. The different magnetic orderings are shown in Fig. 3.6 (c). Different

exchange-correlation functionals with varying levels of Hartree-Fock exchange were used:

E = (1 p)EPBE + pEHF + EPBE , (3.1) xc − x x c

PBE PBE where Ex and Ec are the Perdew-Burke-Ernzerhof (PBE) exchange and correlation functionals re- spectively. The simulations were performed on a supercell including 16 VO2 formula units with 400 valence electrons. A 4 4 8 Monkhorst-Pack k-grid was chosen for sampling the first Brillouin zone of the sim- × × ulation cell. A Burkatzki-Filippi-Dolg (BFD) pseudopotential [5, 169] was used to represent the He core in oxygen and Ne core in vanadium. The band structure obtained using the BFD pseudopotential shows good agreement with the all-electron DFT calculation [170] (using PBE functional).

The result of the DFT calculations is a set of Slater determinants made of Kohn-Sham orbitals that

41 have varying transition metal-oxygen hybridization and spin orders. A Jastrow correlation factor was then added to these Slater determinants as the trial wave functions for quantum Monte Carlo calculations [98].

Total energies were averaged over twisted boundary conditions and finite size errors were checked to ensure that they are negligible (see Supplementary Information). The fixed-node error in FN-DMC was checked by comparing the energetic results from different trial wave functions from DFT calculations with different p in

Eqn 3.1. The trial wave functions corresponding to the 25% Hartree-Fock mixing (PBE0 functional [171]) produce the minimum FN-DMC energy (Fig 3.3). The behavior in VO2 is commonly seen in other transition metal oxides [172]. Thus, all our FN-DMC results in the main manuscript were produced using 25% mixing.

The gap was determined by promoting an electron from the highest occupied band to the lowest unoccupied orbital in the Slater determinant, then using that determinant as a trial function for FN-DMC.

3.3 DFT calculations with different functionals

The DFT band structures for the two structures with different magnetic orders are shown in Fig. 3.4. First, we know that using PBE functional, it is unable to open up an energy gap for monoclinic VO2. If we increase the amount of Fock mixing in the hybrid functionals in Eq. (3.1), both the two structures start to open up an energy gap for FM and AFM states. However, using the well employed 25% percentage mixing (PBE0 functional) [171], the Kohn-Sham gap is too large for both the two structures.

The relative DFT energy of various phase is showed in Fig. 3.5. The total energies are sensitive to the exchange-correlation functional used in DFT calculation. In particular, DFT with PBE functional predicts the FM state to be the lowest energy state in both the two structures. To roughly estimate the magnetic property, we can assume an Ising model to the vanadium magnetic moments, considering only the nearest neighbor coupling. In rutile VO2, the ferromagnetic coupling strength is about 114.25 meV, given by the energy difference between FM and AFM states. This corresponds to a temperature of 1,326 K. According to this, rutile VO2 should be ferromagnetic below 1,326K. This however contradicts the experiments. In monoclinic VO2, the ferromagnetic coupling strength is 23.3 meV (corresponding to 270 K). Therefore, according to this, monoclinic VO2 should also be ferromagnetic, which again contradicts the experiments.

Meanwhile, DFT with PBE also predicts a the total energy of rutile VO2 which is 150 meV lower than that of monoclinic VO2. This contradicts to the fact that monoclinic VO2 is the low temperature state. DFT with PBE0 functional however predicts the AFM state to be the lower energy magnetic ordering in both the two structures. In rutile VO2, the antiferromagnetic coupling strength is 15.45 meV (179.40K).

According to this, rutile VO2 which is above this temperature is ferromagnetic. This consistent to the

42 e ht vrl,terltv nryfo F ihPE svr iia oQCrsl.Hwvr swe as are. However, result. VO QMC monoclinic to for similar gap very VO is rutile PBE0 predicts with PBE0 DFT see, from already energy relative AFM expect experiment, the range the overall, long VO with that, rutile agree no see of to is energy seems There total This the coupling. that other. interdimer fact each weak the a to such coupled with weakly system the are in singlets order these weak and and singlets coupling forms intradimer dimers strong the of VO Because monoclinic dimers. coupling, nearby interdimer two between 0]. (35.44K) 1/2, meV A-[1/2, 15.45 1/2], and E-[1/2,-1/2, VO 0], 1/2], 0, of -1/2, monoclinic B-[1/2, D-[0, mixing In 1/2], different are: 0, experiment. Three coordinates Z-[0, k-point 1/2], figure. The -1/2, the C-[0, 25%. the 0], on and of 1/2, plotted 10% Y-[0, both 0%, are 0], states, computations, minority) 0, FM the spin G-[0, For in and used functionals. majority are exchange-correlation Hartree-Fock (spin different channels of structures spin band two DFT 3.4: Figure FM AFM Unpolarized -1 -2 -1 -2 -1 -2 0 3 1 4 2 0 3 1 4 0 3 2 1 4 2 a Rutile (a) 2 sas o ag 3V.Teeoe ti nnw o eibeteeegtcresults energetic the reliable how unknown is it Therefore, (3eV). large too also is 2 efidta h F opigi 0.3mV(,3K ihnadimer a within (1,235K) meV 106.43 is coupling AFM the that find we , 2 2 ilb nasnltsae nwihtemgei oet ihnthe within moments magnetic the which in state, singlet a in be will ob nisltrwt ag a 3V,ada h aetm the time same the at and (3eV), gap large a with insulator an be to 2 ssil3.5e oe hnta fmncii VO monoclinic of that than lower 30.25meV still is 43 [eV] -1 -2 -1 -2 -1 FM AFM -2 Unpolarized 0 3 1 4 2 0 3 1 4 0 3 2 1 4 2 %Fc PE 25%Fock (PBE0) 0% Fock (PBE) b Monoclinic (b) 2 sw will we As . 1000 50 Rutile Monoclinic Rutile Monoclinic 800

[meV] 0 [meV] 2 2 600 50 400 − 100 200 − 0 150 Energy per VO − Energy per VO 200 UNP FM AFM AFM(intra) − UNP FM AFM AFM(intra) (a) DFT(PBE) (b) DFT(PBE0)

Figure 3.5: DFT energies for various magnetic states of the two structures. (a)DFT with PBE functional; (b)DFT with PBE0 functional. The energy is relative to monoclinic AFM state.

3.4 QMC energetic results

3.4.1 Ground state energy

The energetic results of the quantum Monte Carlo calculations are summarized in Fig 3.6. Both the rutile

and monoclinic structures have lowest energy with antiferromagnetic ordering of the spins. The unpolarized

trial function has sufficiently high energy to remove it from consideration of the low energy physics. The

energy difference between the ferromagnetic and antiferromagnetic orderings changes from 24(6) meV to

123(6) meV from the rutile to monoclinic structure. In the monoclinic structure, the vanadium atoms are

dimerized, which allows for a type of magnetic ordering in which the intra-dimer vanadium dimers are

aligned. This ordering increases the energy by 13(6) meV.

The energy difference between the lowest energy spin orderings for rutile and monoclinic is 10(6) meV,

which is within statistical uncertainty of zero. Which is not expected since monoclinic is the low temperature

phase. The possible reason is that our FN-DMC might suffer from different fixed-node errors in the two

structures, or there might be strains in the two structures. The latent heat 44.2(3) meV across the transition

[149]. It is found experimentally that the phonon entropy contributes 28.9(3)meV. The change of the density

of states when crossing the transition will also cause a change of electronic entropy, which is about 1/6 of the latent heat. There might be contributions from electron-electron correlation effects.

3.4.2 Magnetic properties

The lowest energy wave functions all have local magnetic moments on the vanadium atoms close to 1 Bohr magneton. The spin density is of d orbital profile, aligned on the c axis. In fact, it is d , one of the k t2g orbitals proposed by Goodenough [159]. The spin moments are coupled through ligand oxygens along

44 800 Ru t ile Monoclinic

[meV] 600 (a) 2

400

200

0 Energy per VO 1.0 0.8 (b) 0.6 0.4 0.2

Energy gap [ eV]0.0

(c) Spin density

UNP FM AFM AFM(intra)

Figure 3.6: (Color online) FN-DMC energetic results: (a) per VO2 unit for different magnetically ordered states, spin-unpolarized, ferromagnetic (FM), and antiferromagnetic [AFM and AFM (intra)]. Energies are referenced to AFM monoclinic VO2.(b) Optical gaps for various states. (c) Spin density for various states.

the c axis through a Kramers-Anderson superexchange mechanism [173] (see Fig. 3.7). The superexchange

coupling is due to the hybridization between the vanadium d orbitals and oxygen pz orbitals. k There is a significant change of the magnetic coupling strength across the transition. In the rutile structure, the spins are coupled with a small superexchange energy along the c axis [10(6)meV]. In the monoclinic structure, the spins are coupled strongly within the vanadium dimers and weakly between them.

Therefore, the magnetic moments tightly bind together with each other within the dimers, forming spin singlets. Because of the week coupling between the dimers, there will not be a long range AFM order.

The should give rise to a spin excitation with little dispersion at approximately 123(6) meV, which could potentially be observed with neutron spectroscopy. This excitation has been proposed in the past by Mott

[174]; our results here provide a precise number for this excitation.

To study whether the magnetic behavior from the energies in Fig 3.6 are consistent with experiment, we

45 Figure 3.7: Antiferromagnetic coupling between two vanadium atoms through Kramers-Anderson superex- change mechanism.

make a simple Ising model. In our model, the spins are on vanadium sites, and we only consider couplings

between adjacent sites: (1)J1 – intra-dimer coupling; (2) J2 – inter-dimer coupling; (3) Jint – nearest inter- chain coupling (see Fig. 3.8). We assume that the energy for a magnetic state takes the following form,

X X X E = J1 σiσj + J2 σiσj + Jint σiσj + E0 . (3.2) intradimer interdimer interchain

We fit J1 and J2 to the energetic results of the FM, AFM, and AFM(intra) orderings. For the monoclinic structure, J1 = 123 meV, J2 = 13 meV, while for the rutile structure J1 = J2 = 12 meV. Since we did not compute the inter-chain coupling strength Jint, we set it to be 2.5 meV which is reasonably small compared to the intra-chain coupling. The result is not sensitive to the inter-chain coupling as long as it is small. We then perform a Monte Carlo simulation is on a 10 10 10 super cell, in which the finite size effect has × × been checked to be small. The results are presented in Fig 3.8. The Curie-Weiss behavior of the magnetic susceptibility above the transition temperature is reproduced very well, while the flat susceptibility below the transition is also reproduced. In experiment, there is still a nonzero magnetic susceptibility around

4 1 10− emu/mol due to the contribution from orbitals or magnetic impurities, which cannot be captured × by our Ising model simulations.

The change of magnetic property will contribute to the change of free energy across the transition. In monoclinic VO2, the magnetic moments tightly bind together to form singlets. The entropy of such a state is essentially zero. Whereas above the transition temperature, rutile VO2 is in a magnetic disordered state. This will contribute roughly T ∆S = k T ln(2) 20 meV per VO to the latent heat. If we add the phonon B ' 2 entropy 28.9 meV with the magnetic entropy 20meV, it is close to the latent heat 44.2(3) meV.

46 5 Monoclinic Zylbersztejn, 1975 4 Rutile Kosuge, 1967

3 TR→M em u/m ol] 2 − 4

[10 1 m χ 0 0 200 400 600 800 1000 1200 Temperature [K]

Figure 3.8: Temperature dependence of magnetic susceptibility obtained by Ising model simulation on VO2 lattice, with magnetic coupling from FN-DMC compared to results from Zylbersztejn[3] and Kosuge[4]. There has been an overall scale applied on the experimental data to match the maximum χ of Ising model. The transition temperature is indicated by the vertical dashed line.

3.4.3 Energy gap

[eV] UNP AFM FM

4 (a) (b) (c) 3 2 1 ALPHA BETA 0 Monoclinic -1

4 (d) (e) (f) 3 2 1 Rutile ALPHA 0 BETA -1

DFT DMC

Figure 3.9: Band structures of VO2 in different phases. These band structures correspond to a unit cell containing 4-VO2 formula units. The DFT functional used is PBE0.

Moving to the gap properties (Fig 3.6b). We computed the DMC energy gap by replacing the highest occupied Kohn-Sham orbital with unoccupied ones and compute the energy change from the original ground state. The detailed band structures for the two structures at different magnetic states are shown in Fig. 3.9.

The gap of the low-energy AFM ordering in the rutile structure is zero, while the gap in the monoclinic is

0.8(1) eV. This compares favorably to experiments, which have gaps of zero and 0.6–0.7 eV[175]. Meanwhile, in the higher energy FM ordering, the monoclinic structure has a minimal gap of 0.42(7) eV and the rutile

47 structure has a gap of 0.0(1) eV, both in the spin-majority channel. If there are not unpaired electrons on

the vanadium atoms, then the gap is zero.

The spin unpolarized states (restricted closed shell) are not the ground states. Recalling our previous

discussion on stretched H2 in Chapter.1, at long distance, when electron-electron interaction dominates over the kinetic energy, a closed shell Hartree-Fock or DFT calculation has large static/dynamic correlation error.

One need an unrestricted calculation with unpaired spins because of the localized nature of d orbitals.

3.4.4 Mechanism of metal-insulator transition

0.0 Spin density Spin density Charge density − 0.1 3D [110] [110] Charge [ e2] − 0.2 V − 0.3 Ru t ile V Monoclinic V O Ru t ile O Monoclinic O − 0.4

0.06 am

Covariance c 2 ar m Unlike spin [ e ] b

0.04 Rutile r ar

dm 0.02 (a) 0.00 c 1.6 2.0 2.4 2.8 3.2 b Distance [Å] a Rutile Monoclinic

8 0.02 0.01 0.02 0.02 0.01 Pr o b a b i l i t y 0.16 0.01 0.07 0.07 0.03 0.01 0.07 0.07 0.03 0.01 7 dm 0.12 c 6 0.01 0.14 0.16 0.07 0.02 0.01 0.13 0.16 0.08 0.02 0.08 m 0.04 Monoclinic a electrons 5 0.01 0.10 0.14 0.08 0.02 0.01 0.09 0.14 0.08 0.02 m 0.00 bm 4 (b) Number of spin-up 3 4 5 6 7 3 4 5 6 7 (b) (c) Number of spin-down electrons 0.015 0 0.125

Figure 3.10: (Color online) Change of V-O hybridizations manifested through: (a) Inter-site charge (ni), and unlike-spin (ni or ni ) covariance quantified as OiOj Oi Oj , where Oi is the onsite value of specific physical quantities.↑ The↓ inter-site covariance betweenh ai chosen− h ih vanadiumi center and the surrounding atoms is plotted as a function of the inter-atomic distance. (b) Onsite spin-resolved probability distribution on vanadium atoms – ρ(ni↑, ni↓). (c) Spin and charge density of the rutile and monoclinic VO2. Figures on the left panel are 3D isosurface plots of spin density, and on the right panel are contour plots of spin and charge density on the [110] plane.

From the above energy considerations, a few things become clear about the results of the FN-DMC calculation. The gap formation is not due to a particular spin orientation, so the transition is not of Slater type. However, it is dependent on the formation of unpaired electrons on the vanadium atoms. These behaviors might lead one to suspect that the transition is of Mott type, since the gap in the monoclinic structure is a d d transition. In the classic Mott-Hubbard model, the metal-insulator transition is a →

48 function of U/t, where U is the on-site repulsion and t is the site-to-site hopping.

To make the physics more clear, we connect the detailed quantum results to an approximate low-energy

Hubbard-like model. In our calculation, we evaluate the number of up spins ni↑ and the number of down spins ni↓ on a given site i. We define the V sites and O sites by the Voronoi polyhedra surrounding the nuclei. For

σi σj a given sample in the FN-DMC histogram the joint probability to obtain a set of functions ρ(ni , nj ), where i, j are site indices and σi, σj are spin indices. The connection to t and U are made through covariances of these number operators. t is connected to the covariance in the total number of electrons on a site i, n = n↑ + n↓ with site j: (n n )(n n ) . If this charge covariance is large, then the two sites share i i i h i − h ii j − h ji i electrons and thus t between those two sites is large. U/t¯, where t¯ is the average hopping, is connected to the covariances in the number of up electrons and down electrons on a given site: (n↑ n↑ )(n↓ n↓ ) . h i − h i i i − h i i i This quantity, which we term the onsite spin covariance, is zero for Slater determinants and is a measure of the correlation on a given site.

The onsite unlike spin covariance does not change within stochastic uncertainties. In Fig 3.10b, the joint probability function of n and n averaged over vanadium atoms with a net up spin are shown for ↑ ↓ both rutile and monoclinic phases. The correlations between up and down electrons are identical within statistical uncertainties in the two structures. The onsite unlike spin correlations are 0.127(6) and 0.137(6) for rutile and monoclinic VO2 respectively. Therefore, U/t¯ does not change very much between the two structures. A one dimensitional Hubbard model would have U/t¯ 2.0 to obtain the unlike spin covariance ' that we observe, which puts VO2 in the moderately correlated regime.

The fact that n↑ n↓ n↑ n↓ does not change significantly when crossing the phase transition h V V i − h V ih V i disproved Zylbersztejn and Mott’s picture [160]. That picture states that in the rutile structure, because of the overlap of d bands with the π∗, the electron-electron correlation of d orbitals are screened; while in k monoclinic VO2, π∗ is lifted up therefore no long overlaps with d bands, thus electron-electron correlation k will be large. Here we shows that the overall electron-electron correlations does not change significantly.

The microscopic details of the transition is revealed in Fig. 3.10. Fig 3.10a shows the covariances evaluated for the AFM ordering of the two structures of VO2. The most striking feature is that the charge covariance changes dramatically between the two structures. This feature can also be seen in the charge density in

Fig 3.10c. The dimerization and the antiferrielectric distortion causes a large change in the hybridization between vanadium and oxygen atoms. The dimerization enhances the hybridization between vanadium atoms and oxygen atoms within the dimers, while reducing the hybridization in between the dimers. There are thus large rearrangements in the effective value of t, although the average value t¯ is approximately constant. The zig-zag distortion (ferroelectric distortion) shortened the distance between vanadium atoms

49 and oxygen atoms located at the interdimer regions in the the adjacent chain, denoted am in Fig 3.10b. The largest hybridization is now not within the dimers, but between the vanadium atoms and the oxygen atoms in the adjacent chain. This effect detaches the interdimer oxygen atoms from the two nearby dimers, making them incapable to mediate superexchange magnetic coupling between the two dimers. Overall, the two aspects of the structural distortion enhance the intra-dimer superexchange coupling while suppressing the inter-dimer superexchange coupling.

3.5 Conclusion

By performing detailed calculations of electron correlations within VO2, we have shown that it is possible to describe the metal-insulator transition by simply changing the structure. To obtain the essential physics, it appears that the change in structure is enough to cause the metal-insulator transition. As has been noted before[141], the calculated properties of VO2 are exceptionally sensitive to the way in which correlation is treated. It is thus a detailed test of a method to describe this transition. Fixed node diffusion quantum

Monte Carlo passed this test with rather simple nodal surfaces, which is encouraging for future studies on correlated systems. This method, historically relegated to studies of model systems and very simple ab- initio models, can now be applied to ab-initio models of correlated electron systems such as VO2 and other Mott-like systems [121, 122, 176].

From the quantitatively accurate simulations of electron correlations, a simple qualitative picture arises.

In both phases, there are net spins on the vanadium atoms with moderate electron-electron interactions compared to the hopping. In the rutile phase, the vanadium oxide chains are intact with large hopping and small superexchange energy and thus the material is a correlated paramagnetic metal. In the mono- clinic phase, the dimerization reduces the interdimer hopping, primarily by interchain V-O coupling. The intradimer magnetic coupling increases because of an increase of intradimer V-O coupling. The spins then condense into dimers and a gap forms. This can be viewed as a spin Peierls-like transition.

The results contained in this work, alongside other recent results show that the dream of simulating the many-body quantum problem for real materials to high accuracy is becoming achievable. This accom- plishment is a lynchpin for the success of computational design of correlated electron systems, since these calculations can achieve very high accuracy using only the positions of the atoms as input. We have demon- strated that clear predictions for experiment can be made using ab-initio quantum Monte Carlo techniques, in particular the value of the singlet-triplet excitation in the spin-dimers of VO2. If this prediction is veri- fied, then it will be clear that these techniques can provide an important component to correlated electron

50 systems design.

51 Chapter 4

The importance of σ bonding electrons for the accurate description of electron correlation in graphene

This chapter is based on our paper on arXiv: [177].

Electron correlation in graphene is unique because of the interplay of the Dirac cone dispersion of π electrons with long range Coulomb interaction. The random phase approximation predicts no metallic screening at long distance and low energies because of the zero density of states at Fermi level, so one might expect that graphene should be a poorly screened system. However, empirically graphene is a weakly interacting semimetal, which leads to the question of how electron correlations take place in graphene at different length scales.

We address this question by computing the equal time and dynamic structure factor S(q) and S(q, ω) of freestanding graphene using ab-initio fixed-node diffusion Monte Carlo and the random phase approximation.

We find that the σ electrons contribute strongly to S(q, ω) for relevant experimental values of ω even distances up to around 80 A˚ and modify the effective onsite interactions strongly enough to prevent the system from becoming a Mott insulator. These findings illustrate how the emergent physics from underlying Coulomb interactions result in the observed weakly correlated semimetal.

4.1 Introduction

Graphene has drawn much attention in the last decade because of its unusual electronic and structural properties and its potential applications in electronics [178, 179, 180, 181, 182, 183, 184, 185]. Although many electronic properties of graphene can be successfully described in a noninteracting electron picture [185], electron-electron interactions do play a central role in a wide range of electronic phenomena [186]. The interplay of the effective Dirac cone with Coulomb interactions make the correlation effects in graphene unique.

There are a number of phenomena that arise from correlations in graphene. For example, the reshaping of the Dirac cone was first predicted [187, 188, 189] and later observed experimentally [190]. The fractional quantum hall effect has been observed under high magnetic field [191]. Collective plasmon and plasmaron

52 excitation have also been observed [192, 193, 194, 195]. On the theoretical side, the random phase approx-

imation (RPA) predicts no screening in graphene at long distance and static limit because the density of

states is zero at the Fermi level. This is in contrast to normal metals, where charge carriers and impurities

are highly screened by the Fermi sea through a formation of virtual electron-hole pairs according to RPA [40].

Therefore, it is an interesting question how electronic response takes place in graphene and how to describe

it accurately.

In recent years, it has become possible to obtain very high resolution inelastic X-ray (IXS) experiments

on graphite [196, 197], which were then modified to obtain information about the graphene planes. These

experiments directly probe the dynamical structure factor S(q, ω), which allows for a detailed look at the electron correlations. The main purpose of these experiments was to investigate the role of screening at long wave lengths, and particularly whether the random phase approximation (RPA) obtains an accurate representation of the physics at long range. While these studies obtained unprecedented detail for the low- energy charge excitations, their interpretation is challenging because of limited experimental resolution and uncertainties about the reference for RPA; both whether the σ bonding electrons are included or not, and the underlying theory, which has large effects on the result [198]. For small enough wave vector q and small enough energy ω, the effect of the σ electrons should be small, but it is unclear whether the experiments have reached that regime.

In this manuscript, we address both the question of the suitability of RPA perturbation theory and the effect of σ electrons by applying highly accurate first principles diffusion quantum Monte Carlo (DMC) to a series of planar systems including graphene. DMC is a non-perturbative method with minimal approxima- tions [2, 172, 110] and explicit representation of the electron-electron interactions. It has been shown to be a highly accurate method on both molecular systems and solids [2, 98, 114, 172, 110, 111, 199, 127, 200]. We compute the structure factor S(q) which gives information about the long-range density-density correlations in the material and compare it directly to the X-ray data, obtaining agreement within the experimental error bars. We find that the bonding σ electrons are surprisingly important even at the longest range accessible to experiment, and if the RPA is performed including the σ electrons from a DFT reference, it is in good agreement with the experimental data, although the peak locations do depend on the reference as noted previously [198]. We further examine the short-range effective interactions using DMC and find that, if the σ electrons were removed, graphene would be an antiferromagnetic Mott insulator due to strong on-site interaction.

53 4.2 Structure factor, dielectric constant, and random phase

approximation

The structure factor S(q) is a measure of the equal-time charge-charge correlations of the system, defined

as

1 S(q) := ρ qρq , (4.1) N h − i

iq rˆ where N is the number of electrons in the system, and ρq = e · is the density operator in reciprocal space. S(q) is directly related to the Coulomb interactions of the system [201],

1 Z V = dq[S(q) 1]v(q), (4.2) (2π)D −

where V is the Coulomb energy per particle, D is the dimension of the system, and v(q) is the Coulomb

2πe2 interaction in reciprocal space. In two dimensions, v(q) = q . S(q) is the integral of the dynamic structure factor over frequency space S(q, ω):

2 R Z ∞ dω ¯hq ∞ dωS(q, ω) S(q) = S(q, ω) = 0 . (4.3) R ∞ 0 2π 2m 0 dωωS(q, ω)

Here we have applied the f-sum rule for S(q, ω). The dynamic structure factor describes the dielectric

response of the system [40]. It can be directly measured through inelastic X-ray scattering experiment [196],

and can also be computed using RPA [40, 202]. It is related to the retarded response function χ(q, ω) as

follows,

1 1 S(q, ω) = Imχ(q, ω) . (4.4) −π 1 e βhω¯ − −

The dielectric function (q, ω) can be derived from χ(q, ω),

1 (q, ω) = , (4.5) 1 + U(q)χ(q, ω)

where U(q) is the Coulomb interaction in Fourier space.

54 4.3 Computation setup

The first-principles calculations were performed as follows. DFT calculations were first performed using

the CRYSTAL package [203] with Perdew-Burke-Ernzerhof (PBE) exchange and correlation functional [57].

The simulations were performed on a 16 16 supercell including 512 atoms. Burkatzki-Filippi-Dolg (BFD) × pseudopotentials [5, 169] were used to remove the core electrons. The result of the DFT calculations is

a set of Slater determinants made of Kohn-Sham orbitals. A Jastrow correlation factor was then added

to these Slater determinants as the trial wave functions for DMC calculations. DMC calculations were

performed using the QWalk package [98] to obtain S(q). RPA calculations were performed using the GPAW

package [204, 205] to obtain S(q, ω). The Hubbard model was solved by auxiliary-field quantum Monte

Carlo method (AFQMC) using the QUEST package [206].

15

10

5

0 EF

5 − Energy [eV] 10 − Graphene 15 Hydrogen − TB(t=2.7 eV) 20 − Γ M K Γ

Figure 4.1: Band structure of graphene, hydrogen and tight-binding model (with hopping constant t = 2.7eV).

Table 4.1: Systems/models investigated system/model electrons method 1 1 Graphene (G): a=2.46 A˚− σ & π DMC, S-J , RPA 2 π-only graphene (Gπ) π S-J 1 Hydrogen (H): a=2.46 A˚− s DMC, RPA Tight-binding (TB): t = 2.7 eV3 π RPA Hubbard: U/t = 1.6 π AFQMC

In order to disentangle different contributions to S(ω) from π and σ electrons in graphene, we compared

S(q) among the five systems listed in Table 4.1. All systems have similar low energy band structure but differ

in the presence or absence of σ electrons, and in the interaction between electrons (Fig. 4.1). The s orbital

55 of the hydrogen lattice has almost the same dispersion as the π orbitals in graphene [see Fig. 4.1], which

provides a way to understand the behavior of π electrons in graphene in the absence of σ electrons while

still retaining a 1/r interaction. Graphene and hydrogen system are studied using DMC. The tight-binding

model (t=2.7eV) is studied using RPA with 1/r interactions. S(q) is obtained by integration of S(q, ω) according to Eq. (4.3).

1.2 1 e/site sys. 1.0 H(DMC) Gπ (S-J) TB(RPA) 0.8 Hubbard 4 e/site sys.

) G(Slater) G(S-J) q ( 0.6 G(DMC) G(RPA) S G(IXS) 0.4

0.2

0.0 0 1 2 3 4 5 6 7 8 1 q [A˚ − ]

Figure 4.2: S(q) of different systems obtained through different methods. G: graphene; Gπ: π electrons only graphene; H: hydrogen; TB: π-band tight-binding model with 1/r interactions; Hubbard: the Hubbard model with U/t = 1.6.

4.4 Results and discussion

4.4.1 Structure factor of graphene compared with other systems

Let us first consider the S(q) results for ab-initio graphene, denoted by G in Fig 4.2. For comparison, we have plotted S(q) of a non-interacting Slater determinant of Kohn-Sham orbitals [G(Slater)] in that plot, and that of a Slater-Jastrow wavefunction [G(S-J)]. Both RPA and DMC results are very close to the experimental IXS results, but there is a significant difference between the correlated calculations and the

Slater determinant, as expected. A Slater-Jastrow wavefunction is indistinguishable from DMC results. It thus appears that the experimental S(q) is well reproduced by any of these three correlation techniques

(RPA, S-J and DMC) for ab-initio wave functions. Quantitatively, this change of S(q) from the Slater determinant reflects a reduction of the Coulomb energy by 1.31(5)eV per electron.

56 4 G(IXS) 0.10 G(IXS12) ωπ ωσ+π G(RPA) TBσ(RPA) 3 (a) TB(RPA) 0.08 (b) ) q

[a.u.] H(RPA)

( 0.06

χ 2 S 0.04 -Im 1 1 0.02 G(IXS) q = 0.21A˚ − G(DMC) 0 0.00 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 1 ω [eV] q [A˚ − ] 14 2.5 TB(RPA) H(RPA) TB (RPA) 12 (c) σ 2.0 (d) )

10 q [eV]

( 1.5

κ qd π m κ1+1 (κ1 1)e− ω 8 κ(q) = − − qd κ1 κ1+1+(κ1 1)e m G(IXS) 1.0 − − 6 G(RPA) 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1 q [A˚ − ]

Figure 4.3: Effect of σ electrons at long range. (a) imaginary part of the response function χ(q, ω) of graphene (IXS and RPA), tight-binding model with 1/r interaction, and hydrogen lattice at different momentum transfer. -Imχ has included local field corrections, and has been scaled in order to compare with experiment. The different curves have been scaled for comparison. (b) S(q) from integration of S(q, ω) with different energy cutoff ωc. (c) dispersion of π π∗ plasmon resonance peaks. (d) effective screening function from σ electrons. →

Now consider the models H, TB, Hubbard, and Gπ in Fig. 4.2, which only contain one electron per site, in contrast to the four electrons per site of ab-initio graphene. Each of these models has a computed S(q) quite close to the others. Therefore, regardless of the computational method and interaction, if one electron per site is considered, S(q) is about the same, while for four electrons per site, S(q) is about the same if a correlated method is used. We can thus assess the importance of the lower energy electrons, the σ electrons, at different wavelengths by comparing the one electron per site curve to the four electron per site curve.

To within the limits of both the computational and experimental techniques, the 1e/site S(q) differs from the 4e/site S(q), while at long range/small q, all 1e/site and 4e/site models are equivalent. If it were the case that the σ electrons did not contribute to the long-range density density fluctuations, we would expect those S(q) values to coincide for q small enough. We thus conclude that the σ electrons contribute to the

1 density density fluctuations even for q 0.1 0.2A˚− . ∼ −

4.4.2 Long distance screening effects

Now let us move to the dynamic response of graphene, S(q, ω). Fig. 4.3(a) shows the computed imaginary part of the response function χ(q, ω) of graphene in comparison with the X-ray experiment. There are mainly two resonance peaks in the spectrum [193, 207, 195]: (1) π π∗ interband transition (π plasmons); →

57 (a) (b)

Figure 4.4: The isosurface plot of Wannier orbitals constructed by a unitary rotation of Kohn-Sham orbitals: p (a) σ orbital; (b) π orbital. Different colors represent different signs. The root-mean-square radius ( r2 ) is 2.69A˚ for σ orbitals and 3.37A˚ for π orbitals. h i

(2) σ π∗ and π σ∗ interband transition (σ + π plasmons), denoted as ω and ω in Fig. 4.3(a). → → π σ+π RPA accurately reproduces the experimental IXS data, although the level of agreement for RPA may be fortuitous [198]. This is also why the S(q) of G(RPA) matches G(DMC) and G(IXS) very well as is shown in Fig. 4.2, from which we know that there are no large peaks that are missing in RPA. It thus appears that the long-range response, at least to the limits of experimental resolution and potentially with small errors in the peak positions, is well-described by RPA calculations that include the σ electrons.

The π + σ plasmons are inherently missing in TB model and hydrogen lattice. This is why they overesti- mate S(q) for graphene as shown in Fig. 4.2. To see this, let us recall how we compute S(q) from the X-ray measured S(q, ω) using Eq. (4.1). The frequency cutoff in our experimental data is 2, 000 eV which is high enough to include relevant excitations of valence electrons. If we set ωc = 12 eV, it includes π plasmons but excludes σ + π plasmons. The S(q) of ωc = 12 eV matches the S(q) of TB model [see Fig. 4.3(d)]. The π plasmon resonance peak ω of the TB model and hydrogen is 1 3 eV larger than that of π ∼ graphene [Fig. 4.3(a) and (d)]. This 1 3 eV discrepancy indicates strong screening effects from σ electrons ∼ in graphene which are not present in TB model nor in hydrogen system. To take into account the screening from σ electrons in TB model, we include a dielectric function κσ(q) to the calculation of the response function χ(q, ω) [208, 197]

Π (q, ω) χ(q, ω) = TB , κσ(q) V (q)ΠTB(q, ω) − qL κ + 1 (κ 1)e− κ (q) = 1 − 1 − κ . (4.6) σ κ + 1 + (κ 1)e qL 1 1 1 − −

ΠTB(q, ω) is the polarization function for TB model computed using the Lindhard function [208]. κσ [plotted in Fig. 4.3(d)] is the effective dielectric function from σ electrons. κ 2.4 is the dielectric constant 1 ' of graphite, and L = 2.8A˚ is the effective thickness for a single layer graphene. This correction reduces the discrepancy by about 1 2 eV [see the TB (RPA) curve in Fig. 4.3(b)], which is similar to the reduction ∼ σ

58 0.14 0.12 G 0.10 H TB 0.08

JDOS 0.06 0.04 0.02 0.00 0 5 10 15 20 ω [eV]

Figure 4.5: Joint density of states (JDOS) of graphene (G), hydrogen (H), and tight-binding (TB) model. The JDOS of graphene and hydrogen are computed from PBE band structure.

from H(RPA) to G(RPA). The remaining small discrepancy between TBσ(RPA) and G(IXS) [or G(RPA)]

is partially due to the deviation of tight-binding band dispersion from ab-initio graphene. The π π∗ → transition energy at M point is 5.4 eV for tight-binding model, 4.2 eV for ab-initio graphene, and 4.0 eV

for hydrogen (see Fig. 4.1). This deviation of the band dispersion will cause an artificial overestimation of

ωπ by about 1 eV. The discrepancy induced by a pure band effect can be, visualized from the π-band joint density of states defined as,

1 X jdos(ω) = δ( (k)  (k) ω) , (4.7) N π∗ − π − k

where π(k) and π∗ (k) are the eigenvalues of π band and π∗ band respectively, and N is the number of k points sampled in the first Brillouin zone. The result for different systems are shown in Fig. 4.5. Because

of the the van Hove singularity at M point, there is a peak located near  (M)  (M). The peak shifts π∗ − π towards high energy from graphene to the tight-binding model (about 1eV).

In the limit as q 0, the response from the σ electrons also goes to zero. This is reflected in κ : κ 1 → σ σ → if q 0, and κ κ for large q [see Fig. 4.3(d)]. However, even at the lowest momentum that the X-ray → σ → 1 1 experiment has access to (q = 0.21A˚− ) [197], the screening effect is not small (κ (q) 1.5). It shifts ω σ ' π by 0.8 eV, which is comparable to estimations of excitonic effects [196, 198, 197]. If it is desired to isolate

1 1 the π electrons from the σ electron screening, q 0.21A˚− must be accessed. For q = 0.07 A˚− , the shift | | 

59 reduces to 0.25 eV [see Fig. 4.3(c)].

4.4.3 Short distance screening and emergent low energy effective field theory for π electrons

We now turn to the short-range effects of σ electrons in graphene. Here we will show that the σ electrons

reduce the effective interactions between π electrons and makes graphene a semimetal which otherwise would

be an antiferromagnetic Mott insulator without σ electrons. We demonstrate this by explicitly downfolding

the ab initio system into a Hubbard model of π orbitals,

ˆ X X H = C + t ci,σ† cj,σ + h.c. + U ∗ ni, ni, . (4.8) ↑ ↓ i,j i h i

The effect of the long range part can be considered as a renormalization to the onsite Coulomb interaction

U at low energy [209, 210].

To assess the value of U, we used the non-eigenstate ab initio density matrix downfolding (NAIDMD) [210] method, which allows the use of QMC results to fit an effective model. We first constructed π Wannier orbitals from Kohn-Sham orbitals (shown in Fig. 4.4). We then determined the effective parameters C, t and U ∗ using energies and reduced density matrices of various low lying states through a least squares fit [210]

(please see also the Supplementary Materials for details). The results of the fit are shown in Table 4.2. The root mean square of the deviation between ab-initio energies and model predicted energies is small, which confirmed that the Hubbard model is indeed a relatively good description of graphene both with and without

σ electrons, although worse for the without σ model. If σ electrons are included, U ∗/t 1.9 [209, 210]. This ∼ is smaller than the metal-insulator transition critical value 3.869 0.013 for the honeycomb lattice [211], ± which is consistent to the fact that suspended graphene is a weakly correlated semimetal. If σ electrons are not included, U ∗ is doubled, and U ∗/t = 4.9(1) which is beyond the critical value. In this case, π electrons will form an antiferromagnetic Mott insulating state. The enlargement of U ∗ by a factor of 2 from former to latter is consistent with the effective dielectric constant κσ(q) in Eq. (4.6).

Table 4.2: Effective parameters of an on-site Hubbard model for graphene with and without σ electrons. with σ [eV] without σ [eV] t 3.61(1) 4 2.99(1) U ∗ 7.16(3) 14.8(2) ∆Erms 0.12 0.44

60 4.5 Conclusion

In conclusion, using the first principles quantum Monte Carlo approach and the random phase approximation with DFT as the reference, we are able to describe the electron correlation in graphene accurately and reproduce the X-ray data very well for all q available, provided that the σ electrons are included in the calculations. The level of agreement for RPA may be fortuitous [198], but it is clear that the σ electrons are important even at ranges up to around 30A.˚ Quantum Monte Carlo as a check on RPA and experiment confirms this.

With the accurate first principles descriptions, we have demonstrated the effects of σ electrons to the low energy physics of graphene in both short range and long range. At short range, they reduce the effective onsite interaction for π electrons to allow a semimetallic state to form, without which the system would be an antiferromagnetic insulator. On the other hand, at long range, the σ electrons response through π + σ plasmon. The screening from σ electrons reduces the π plasmons resonance frequency for about 1 2 eV ∼ comparable to other effects that will cause similar shift such as excitonic effect. It has an observable effect

1 even at q = 0.21A˚− , the lowest momentum that current X-ray experiments have accessed to. It is thus

1 necessary to access momentum q 0.2A˚− to isolate the π electrons from the screening of σ electrons. 

61 Chapter 5

The extended Koopmans’ theorem for charge excitations in atoms, molecules and solids

5.1 Introduction

Koopmans’ theorem [212] is widely used to estimate ionization potentials (IP) of atoms and molecules within

the framework of Hartree-Fock approximation. It equates the first ionization potential of a system to the

negative of the eigen energy of the highest occupied one-electron orbital, and the electron affinity to the

energy of the lowest unoccupied one-electron orbital. The ionization potentials calculated using Koopmans’

theorem are in qualitative agreement with experiments [45]. The errors are often less than 2eV [45, 6]. It

has been used to interpret or predict photoelectron spectra [213, 214, 215].

There are two major sources of errors in Koopmans’ theorem. (1) orbital relaxation: the theorem assumes

that the orbitals of an ion are the same with the corresponding neutral atom (frozen orbital approximation).

However, after one electron is removed from or added to the system, the mean field potential changes; thus

the rest of the orbitals will change accordingly. This effect usually is significant in small systems and becomes

smaller as the size of the system increases. Because of ignoring the orbital relaxation, Koopmans’ theorem

will introduce a positive error to IPs. (2) electron correlation: the added or removed electron is correlated

with the rest of the system. The Hartree-Fock approximation poorly describes the electron correlations. The

correlation energy although is only a small fraction of the total energy, it has much more contribution to the

ionization potential. The error due to ignoring the correlation effect will generally be negative. It is seen

that Koopmans’ theorem usually breaks down in systems where electron correlation effect is important [216].

Examples of the breakdown include N2 [217], F2 [218], and several organic molecules [219, 220, 221], where Hartree-Fock predicts wrong energetic order of the single-body orbitals in ionization. Nevertheless, in most cases, the errors caused by these two effects partially cancel each other, resulting in approximate ionization potentials that are within 10% error compared with experiment [45, 6].

Considerable efforts have been made improvement on the Koopmans’ theorem, including various ways of perturbative correction [222, 223, 224, 225, 226], and the extended Koopmans’ theorem (EKT) [227, 228,

229, 230, 231]. Among these theories, the EKT approach offers systematic ways of computing ionization

62 potentials and electron affinities from a correlated wave function. It has been demonstrated both analytically

[232, 233] (perturbation expansion up to 12-th order) and computationally [234, 235, 232, 236, 237] that

EKT is exact for the lowest IP (or the first IP) as long as the correlated wave function is exact.

EKT has been implemented in various quantum chemistry methods such as CI [234], MCSCF [232], MP2,

MP3 [238], orbital-optimized methods [239, 240], self-consistent Green function theory [241]. However, most

of the existing implementations of EKT in quantum chemistry are limited to atoms and molecules. They

are difficult to be generalized to extended systems because of the exponential growth of the computation

cost. The goal of this study is to implement EKT for extended systems. In such, one needs to go beyond the

quantum chemistry methods and find a correlated electronic structure method that is applicable to solids.

The accuracy of EKT lies in the correlated wave function that the theorem is based on. Therefore, a

good theory of correlated systems should be able to produce highly accurate many-body wave functions in

a relative cheap wave (compared with quantum chemistry methods). At the same time, all the operations

to the wave function in EKT should be easy to compute within this framework. One ideal option is the

quantum Monte Carlo (QMC) method. QMC is an explicitly correlated wave function based method [2]. It

captures the electron-electron correlation very accurately through direct sampling the electron configurations

with Coulomb interaction. On the other hand, QMC with fixed-node approximation scales with the system

size very favorably [O(N3)] [2]. All the wave function operations in EKT are easy to compute stochastically in QMC. In this sense, QMC is a good candidate to implement EKT for extended systems. In fact, EKT has been implemented in QMC previously [112], and was applied to compute the quasiparticle band structure of silicon. However, its validity to other more complex solids requires more systematic studies.

The goal of this chapter is to formulate EKT in QMC [98] and investigate its validity in various systems.

We will first discuss the formulation of EKT in QMC, and then present our benchmark studies for atoms and molecules, and solids. In QMC, single Slater-Jastrow type wave functions are used. This type of wave function is generally very good in capturing electron correlations [2, 122, 127, 200]. However, we find that in molecule systems, a single Slater-Jastrow wave function is not good enough to capture the relaxation effect.

One generally needs to use multiple Slater determinants in order to get accurate results. In solids, on the contrary, a single Slater-Jastrow wave function can already produce very accurate quasiparticle dispersion.

63 5.2 Extended Koopmans’ theorem and its implementation in

QMC

5.2.1 Valence band energy and ionization potential

Starting from the ground state wave function of an N-particle system ΨN , one can get the wave function for the ionized (N 1)-electron system by eliminating an orbital from ΨN , −

N Z N 1 N 1 X RN 1 Ψ − = RN 1 av Ψ = dr0φv∗(r0)ψ(r1 rn, r0, rn+1 rN 1) , h − | i h − | | i √ ··· ··· − N n=1

RN 1 := r1, r1, , rN 1 . (5.1) − ··· −

where a is an annihilation operator, and φ (r) is the corresponding wave function, φ (r) = r a† 0 ( 0 v v v h | v| i | i is the vacuum). Notice that this is not the only possible way to construct an (N-1)-electron wave function.

Generally, one should consider states like

N 1 X N X N Ψ − = c a Ψ + c a a†a Ψ + .... (5.2) | i i i| i i,j,k i j k| i i i,j,k

Eq. (5.1) holds only when the higher order terms can be neglected. In this sense, Eq. (5.1) is more to be considered as an ansatz rather than a theorem. This ansatz has been proved to be exact for the first ionization potential of a Coulomb system [232, 233]. In other words, one will be able to find an orbital av

N 1 to produce Ψ − an exact ground state of the N-particle system. | i Now, assuming that we have a complete basis set of single-particle orbitals, we can construct the elimi-

P i nated orbital av using a linear superposition of these single-particle orbitals: av† = i avci†. ci’s can be DFT Kohn-Sham orbitals or Hartree-Fock orbitals. The ionization potential is the energy difference between the

N-particle system and the (N-1)-particle system,

N 1 ˆ N 1 N ˆ N N N Ψ − H Ψ − Ψ av† [H, av] Ψ v = Ψ Hˆ Ψ h | | i = h | | i . (5.3) N 1 N 1 N N h | | i − Ψ − Ψ − − Ψ av† a Ψ h | i h | v| i

Notice that in deriving Eq. (5.3), we have used the fact that

N N Ψ a† a Hˆ Ψ ΨN Hˆ ΨN = h | v v | i . (5.4) N N h | | i Ψ av† a Ψ h | v| i

This is exact as long as ΨN is an exact eigenstate of Hˆ . | i

64 Eq. (5.3) can be reorganized into the following form,

X i j a ∗[  ρ ]a = 0 , (5.5) v Vij − v ij v i,j

where ρij is the one-body reduced density matrix in the orbital basis,

N Z   N N X ψ∗(RN,n 0) ρij = Ψ ci†cj Ψ = dr0 φi(r0)φj∗(rn) → . (5.6) h | | i ψ∗(RN ) ψ 2 n=1 | |

2 Here, O Ψ 2 the expectation value of certain quantity O under probability distribution ρ(R) = Ψ(R) , h i| | | | Z 2 O Ψ 2 =: dRN O(R) Ψ(R) . (5.7) h i| | | |

can be evaluated as follows in QMC, Vij

N N N N = Ψ c†[H,ˆ c ] Ψ = Ψ c†c Hˆ c†Hcˆ Ψ Vij −h | i j | i h | i j − i j| i N Z " # X ˆ ˆ = dRN+1 ψ∗(RN,n 0)φi(r0)φj∗(rn)Hψ(RN ) ψ∗(RN,n 0)φi(r0)Hφj∗(rn)ψ(RN ) → − → n=1 N X Z = dRN+1φi(r0)φj∗(rn)ψ∗(RN,n 0)Hnψ(RN ) → n=1 N Z   X ψ∗(RN,n 0) Hnψ(RN ) = dRN+1 φi(r0)φj∗(rn) → . (5.8) ψ∗(RN ) ψ(RN ) ψ 2 n=1 | | where we have defined RN , RN,n 0, and RN+1 as, →

R =: r , , r , (5.9a) N 1 ··· N

RN,n 0 =: r1, , rn 1, r0, rn+1, , rN , (5.9b) → ··· − ··· R = r , r , , r . (5.9c) N+1 0 1 ··· N and dR =: dr dr . H is the effective Hamiltonian related to the n-th electron, N 1 ··· N n

N 1 2 X 1 Hn = n + Vext(rn) + . (5.10) −2∇ rn rj j=n 6 | − |

In the derivation, we have employed the antisymmetry property of ΨN .

Once we have all the matrices evaluated in QMC, the problem becomes the following generalized eigen-

65 value problem,

( ρ)α = 0 . (5.11) V −

Let us make some remarks about the matrix : V

(1) Hermiticity of matrix: the definition of in Eq. (5.8) is generally not hermitian, as we can see V Vij

N N N N ∗ = Ψ c†c Hˆ c†Hcˆ Ψ ∗ = Ψ Hcˆ †c c†Hcˆ Ψ = . (5.12) Vji h | j i − j i| i h | i j − i j| i 6 Vij

N ∗ and equal to each other if and only if the wave function Ψ is exact. We will always assume Vji Vij N N N N N that Ψ is sufficiently close to the exact ground state, such that Ψ a† a Hˆ Ψ Ψ Haˆ † a Ψ . h | v v | i ' h | v v| i With such, we can use the following new definition for , V

v 1 h i = Hcˆ †c c†Hcˆ + c†c Hˆ c†Hcˆ Vij 2 h i j − i ji h i j − i ji N Z "   1 X ψ∗(RN,n 0) Hnφ(Rn) = dr0 φi(r0)φj∗(rn) → 2 ψ∗(RN ) φ(RN ) ψ 2 n=1 | |   # ψ(RN,n 0) Hnψ∗(Rn) + φj∗(r0)φi(rn) → , (5.13) ψ(RN ) ψ∗(RN ) ψ 2 | |

which is guaranteed to be hermitian.

(2) Nonlocal pseudopotential: the pseudopotentials employed in DFT and QMC calculations generally

have the following form,

Z X V (r) = + f(r) + g (r) l l . (5.14) ext − r l | ih | l=0

In our implementation (Qwalk package [98]), we take the following local approximation,

Z Vext(rn, rn )ψT (RN,n n ) 0 → 0 Vnonloc(rn) = drn0 . (5.15) ψT (RN )

where ψT (RN ) is the trial wave function in QMC.

66 (3) Different spin channels:

N ↑ Z N 1 N 1 X Ψ − = av, Ψ = dr0φv∗(r0)ψ(r1, rn, r0, rn+1, rN ); (5.16a) | i↑ ↑| i pN ··· ↑ n=1 N Z N 1 N 1 X Ψ − = av, Ψ = dr0φv∗(r0)ψ(r1 rn, r0, rn+1, rN ) . (5.16b) | i↓ ↓| i p ··· N n=N +1 ↓ ↑

The and ρ matrices for the two spin channels are, V

N  N  ↑ Z   v X X ψ∗(RN,n 0) Hnψ(Rn) Vij, (or ) = or  dr0 φi(r0)φj∗(rn) → , (5.17a) ↑ ↓ ψ (RN ) ψ(RN ) 2 n=1 n=N +1 ∗ ψ ↑ | | N  N  ↑ Z   X X ψ∗(RN,n 0) ρij, (or ) = or  dr0 φi(r0)φj∗(rn) → . (5.17b) ↑ ↓ ψ (RN ) 2 n=1 n=N +1 ∗ ψ ↑ | |

Identifying ionization potential

Solving Eq. (5.11) will produce a set of eigenvalues and eigenvectors. The number of eigenvalues and eigenvectors depends on the dimension of the matrix, which equals to the number of basis function φi(r). However, only N of them have physical meaning (N is the number of occupied orbitals). In most of the

weakly correlated electronic systems, if we use Kohn-Sham or Hartre-Fock orbitals , the eigenvectors will be

very close to [δ , , 1 + δ , , δ ] where δ are small numbers. The corresponding eigenvalue  might 1 ··· n ··· N i n also be close to the Kohn-Sham energy or Hartree-Fock single-particle eigen energy. Suppose in DFT or HF, the highest occupied orbital is φm, then m will be the first ionization potential. However, there might be

some particular cases, in which m0 (m0 < m) is even smaller in magnitude in m. In this case, m0 should be the first ionization potential.

5.2.2 Conduction band energy and electron affinity

Similar with the valence band case, we assume the following ansatz,

" N # N+1 N 1 X ψ (RN+1) =: RN+1 ac† ψ = φc(r0)ψ(RN ) φc(rm)ψ(RN,m 0) , h | | i √ − → N + 1 m=1 R =: r , r , , r . (5.18) N+1 0 1 ··· N

The electron affinity is

N ˆ N ψ ac[H, ac†] ψ c = h | | i . (5.19) N N ψ a ac† ψ h | c | i 67 P Again, if we expand φc in a set of single-body orbitals, φc = i ac,iφi, we arrive at the following generalized eigenvalue problem,

( S)α = 0, S = 1 ρ , (5.20) C − c −

where

N N N N N N = ψ c [H,ˆ c†] ψ = ψ c Hcˆ † ψ ψ c c†Hˆ ψ . (5.21) Cij h | i j | i h | i j| i − h | i j | i

Using the antisymmetry property of the wave function with respect to the interchange of ri we have

Z " # X ˆ ciHcj† = dRN+1 φi∗(r0)ψ∗(RN ) φi∗(rm)φ∗(RN,m 0) Hφj(r0)ψ(RN ); (5.22a) h i − → m Z " # ˆ X ˆ cicj†H = dRN+1 φi∗(r0)ψ∗(RN ) φi∗(rm)ψ∗(RN,m 0) φj(r0)Hψ(RN ) , (5.22b) h i − → m

The Hamiltonian in Eq. (5.22a) is an (N+1)-electron Hamiltonian,

N X 1 Hˆ = Hˆ + v(r , r ) + hˆ , hˆ = 2 + V (r ) . (5.23) N 0 i 0 0 −2∇ ext 0 i=1

Hence,

" N # X Hφˆ j(r0)ψ(RN ) = φj(r0)HˆN ψ(RN ) + v(r0, rm) + hˆ0 φj(r0)ψN (RN ); (5.24) m=1

whereas the Hamiltonian in Eq. (5.22b) is simply HN . Therefore, one have,

Z " #" N # X ˆ ij = dr0dRN φi∗(r0)ψ∗(RN ) φi∗(m)ψ∗(RN,m 0) v(r0, rm) + h0 φj(r0)ψ(RN ) C − → m=1 Z *" N #" N # + X ψ∗(RN,m 0) ˆ X = dr0 φi∗(r0) φi∗(rm) → h0 + v(r0, rn) φj(r0) . (5.25) − ψ∗(RN ) m=1 n=1 ψ 2 | |

There are two remarks I want to make here,

(1) If the wave function is antisymmetrized only within the same spin channel, one should change the first PN PN PN summation in Eq. (5.25) ( m=1) to m↑=1 and m=N +1 for spin-up channel and spin-down channel ↓ respectively.

68 (2) Regarding the hermiticity of , one can also consider an alternative definition of , C Cij

N N N N = ψ c Hcˆ † ψ ψ Hcˆ c† ψ , (5.26) Cij h | i j| i − h | i j| i

where

Z N N h ψ Hcˆ c† ψ = dr dR ψ∗(R )Hφˆ ∗(r )φ (r )ψ(R ) h | i j| i 0 N N i 0 j 0 N i X ˆ ψ∗(RN,m 0)Hφi∗(rm)φj(r0)ψ(RN ) . (5.27) − → m

Substracting Eq. (5.27) from Eq. (5.22a), we get

Z * N + X ψ∗(RN,m 0) Hmψ(RN ) ij = dr0 φi∗(r0)H0φj(r0) φi∗(rm)φj(r0) → . (5.28) C − ψ∗(RN ) ψ(RN ) m=1 ψ 2 | |

where Hn is the part of Hamiltonian that is related to electron n,

N X Hn = hˆn + v(rn, rm) . (5.29) m=0,m=n 6

If we combine the two definitions, we get a hermitian , C

Z * N " # + 1 X ψ∗(RN,m 0) H0φj(r0) Hmψ(RN ) ij = dr0 φi∗(r0)H0φj(r0) φ∗(rm) → + φj(r0) . C − 2 ψ∗(RN ) φj(r0) ψ(RN ) m=1 ψ 2 | | (5.30)

In our calculations, we mainly use Eq. (5.30).

Identifying electron affinity

In Eq. (5.20), the number of eigenvalues and eigenvectors equals to the dimension of the basis set M. However, the first N eigenvalues and eigenvectors are meaningless (N is the occupied orbitals). After solving Eq. (5.20), one should pick the eigenvalues  corresponding to eigenvectors [δ , , 1 + δ , δ ](n > N, and δ’s are n 1 ··· n ··· M small numbers).

5.2.3 Regularization of the Coulomb potential in a periodic system

For both adding one electron to or removing one electron from the system, the new system a ψ or a† ψ are c| i c| i charged. However, even in this situation, and is still finite for solid in principle. However, practically, V C

69 since we are simulating finite unit cell with periodic boundary condition, we are not adding one electron to

(or removing from) the whole system. Instead we are adding one electron to the finite unit cell. Therefore,

the matrix elements and will diverge in a periodic system because of the Coulomb interaction between Vij Cij images in different unit cells.

To resolve this, we add a constant charge background to the electron to be removed or added to cancel

the divergence. This is valid because we are only interested in the relative energy difference. Adding a

constant charge background just shifts the zero energy. It shifts the same amount of energy for both the

conduction band and valence band (if the system is sufficiently large that the finite size effect is small). The

removed electron corresponds to the following charge density,

q ρ (r) = q δ(r r ) i , (5.31) i i − i − V

where V is the volume of the system. With this, all the Coulomb interaction terms, V (ri, rj) is no longer 1/ r r . This kind of treatment is very much related to the Ewald summation. | i − j| Notice that, in the Ewald summation, we separate the charge density into short range and long range parts,

 3  S α α2 r r 2 ρ (r) = q δ(r r ) e− | − i| ; (5.32a) i i − i − π3/2  3  L α α2 r r 2 1 ρ (r) = q e− | − i| . (5.32b) i i π3/2 − V

We will evaluate the short range term in real space, and the long range term in the reciprocal space.

For the long range part, we have

Z h k2 i X L ik r X ik rj ρ(k) = ρ (r)e− · dr = q e− · e− 4α2 δk . (5.32c) i i − ,0 i i

The total Coulomb energy is

N N 0 1 X X X qiqj V = erfc(α r r + R ) tot 2 r r + R | i − j | R i=1 j=1 | i − j | k2 N 2 2π X e− 4α δk,0 α X + − S(k) 2 q2 . (5.33) V k2 | | − √π i k i=1

70 where, in the first term, j = i if and only if R = 0. The k = 0 term in the reciprocal sum is 6

k2 2 2π e− 4α δk,0 2 π X 2 lim − S(k) = qi . (5.34) k 0 V k2 | | −2V α2 | | →

This will be zero for neutral system. Therefore, eventually, one has

2 X qiqj 4π X qiqj ik (r r ) k πqiqj r r r r R i j − 4α2 v( i, j) = erfc(α i j + ) + 2 e · − e 2 , (5.35) ri rj + R | − | V k − V α R k=0 | − | 6

where erfc is the complementary error function. The self interacting term is

2 2 2   1 X qi 2π X qi k π α 2 v(r ) = erfc(αR) + e− 4α2 + q . (5.36) i 2 R V k2 −2V α2 − √π i R=0 k=0 6 6

We have qi = 1 for electrons, and qi = Zi for ions. The interaction between two electrons is

2 X 1 4π X 1 ik (r r ) k π r r r r R i j − 4α2 v( i, j) = erfc(α i j + ) + 2 e · − e 2 . (5.37) ri rj + R | − | V k − V α R k=0 | − | 6

The external potential of an electron is

2 X X ZI 4π X X ZI ik (r r ) k r r r R i I − 4α2 vext( i) = erfc(α i I + ) 2 e · − e − ri rI + R | − | − V k R I | − | k=0 I 6   π X 1 X 1 2π X 1 k2 π α + Z + erfc(αR) + e− 4α2 + . (5.38) V α2 I 2 R V k2 −2V α2 − √π I R=0 k=0 6 6

Note that we have put all the self interacting term in the single-body potential.

We would like to mention that,

2π X 1 k2 α e− 4α2 = . (5.39) V k2 √π k=0 6

This is easy to prove by changing the lattice sum into an integral P V R dk3. Using this relation, k → (2π)3 one can further simplify the formula listed above if one likes.

Remarks about relative position of valence bands and conduction bands

In general, if we treat the regularization carefully, we would be able to compute the quasiparticle band gap,

Eg = EN+1 + EN 1 2EN , (5.40) − −

71 where EN , EN+1, and EN 1 are the ground state energies of the N-, (N+1)- and (N-1)-particle systems. − However, based on our limited testing examples (graphene, silicon and hydrogen chain), the calculated quasiparticle gap is not correct for solid. More studies are needed to understand whether this is finite size error or the problem of this decomposition of Coulomb interaction. However, the relative position of the bands within the group of valence bands or conduction bands is reasonable.

5.3 Factors that affect the accuracy of EKT

Now let us start to discuss the application of EKT to real systems. There are several factors that affect the accuracy of the calculations: stochastic errors, basis set error, and correlation error of the ground state wave function.

5.3.1 Stochastic error and orbital cutoff

16.0 19 15.5 18 15.0 17 14.5 14.0 16

IPs [eV] 13.5 IPs [eV] 15 13.0 14 12.5 12.0 13 2 4 6 8 10 12 14 1 2 3 4 5 6 7 8 Cutoff orbital Cutoff orbital

(a) Oxygen atom (b) H2 molecule

Figure 5.1: Ionization potentials for oxygen atom and H2 molecule with different cutoff orbitals. V5Z basis set from [5] is used for both oxygen and hydrogen. For oxygen atom, we use a multiple Slater-determinant wave function from CISD. For H2 molecule, we use a single Slater-Jastrow wave function. The dash line is the experimental ionization potential from NIST database [6].

As we have already seen in last section, EKT finally give rises to generalized eigenvalue problems of

Eqs. (5.5) and (5.20). In QMC, all the matrix elements are evaluated stochastically. Therefore, the final eigenvalues depend on the stochastic errors of each matrix elements in a nontrivial way1. The stochastic

1To reduce the influence of the stochastic fluctuation of unimportant matrix elements, if the stochastic error is larger than 3 times of the absolute value of a matrix element, We set both the specific matrix element and its corresponding stochastic error to be zero.

72 error of the eigenvalues are calculated from a large samples of matrices which are generated randomly with each elements sampled from a Gaussian distribution with mean and standard deviation obtained from QMC.

Meanwhile, the dimension of the matrix is generally very large. In principle, it is infinite, but is finite in practical calculation because a finite basis set is used. However, even when using a finite basis set, the number of single orbitals is still relatively large. We might have to include only a smaller number of orbitals in the calculation. In this way, there will be an orbital cutoff so that the dimension of , , and ρ will be V C small. This might introduce systematic errors. Ideally, as we increase the number of orbitals, we will get increasingly accurate results. However, because of the stochastic error of the matrix elements, increasing the number of orbitals will also accumulate the overall stochastic error of ionization potential. For oxygen and H2 molecule shown in Fig. 5.1 (see blue curves), as we increase the dimension of the matrices, the error of ionization potential increases.

So here we encounter a dilemma: increasing the orbital cutoff will generally include more information and make ionization potential more accurate; however, on the other hand, it also accumulates more and more stochastic errors. We should always seek a balance between the two. In producing the results in this chapter, we restrict the stochastic error to be within 0.01Ha. We normally keep increasing the number of orbitals until the stochastic error reaches 0.01 Ha.

5.3.2 Basis set error

The ionization potential depends on the completeness of the basis set, especially the diffused ones which governs the asymptotic behavior of the wave function of the removed electron. In the example of H2, as we add more diffused gaussian basis functions, we get increasingly better results [see Fig.5.2]. In our following calculations we always use V5Z basis set. All the basis sets are taken from [5]. Ionization potential

Figure 5.2: Ionization potentials calculated using different basis sets. From VS1 to V5Z, the basis set is more and more complete. The black line is the exact result (16.45eV).

73 5.3.3 Relation to quality of the correlated wave function in capturing static and dynamic correlation

The accuracy of the results also depends on the quality of the ground wave function. As we know that

one failure of the original Koopmans’ theorem is that it is based on a Hartree-Fock non-interacting wave

function which can not capture the electron-electron correlations, neither the static correlation nor the

dynamic correlation. Therefore, one can generally improve the wave function in two ways:

adding more determinants; •

including a Jastrow factor. •

Let us consider H2 molecule with d= 2.0A,˚ in which case the static correlation energy is relatively large. Both adding one more determinant and including a Jastrow factor will decrease the overall energy. However, we see that although single Slater-Jastrow wave function has lower energy, the two-Slater wave function gives us more accurate ionization potential than the single Slater-Jastrow wave function. Multiple Slater wave function generally gives us better results for ionization potential than single Slater-Jastrow wave function.

We will see more in our benchmark results for atoms.

The possible reason for this might be that, the multiple Slater determinants help to describe the relaxation effect after ionization. It also improve the description of electron-electron correlation. Adding a Jastrow factor, although improve the description of electron-electron correlation [as is seen by the amount of energy lowered from a Slater determinant], it does not improve the description of relaxation effect. Therefore, adding more determinants performs better than adding a Jastrow factor.

Table 5.1: Ionization potential evaluated from different wave functions. E(VMC) E(DMC) IP(VMC) IP(DMC) Single Slater determinant -0.921(1) – 0.38004(5) – Single Slater-Jastrow -1.01238(1) -1.02217(2) 0.417(6) 0.447(2) Two Slater determinants -1.0051(3) – 0.456(4) – Two Slater determinants with Jastrow -1.01702(7) -1.02202(3) 0.463(2) 0.476(2) Exact ionization energy by E(A) E(A+) = 0.46972(9). In the two-Slater-determinant wave function, there are two Slater determinants with− both electrons occupy the bonding orbital and antibonding orbital respectively.

5.4 Benchmarks of EKT

Let us apply the implemented EKT to various systems. We will compute the ionization potentials and electron affinities of atoms, ions, and the band structure of solids.

74 5.4.1 Ionization potentials (IPs) of atoms

30 1.5 (a) (b) 25 1.0

0.5 20 0.0 15 0.5 − 10 Error of IPs [eV] 1.0 Theoretical IPs [eV] ∆ HF − ∆ DFT 5 1.5 ∆ VMC − ∆ DMC 0 2.0 − 0 5 10 15 20 25 30 HF ∆ ∆DFT VMC DMC Experimental IPs [eV] ∆ ∆

30 3.5 (c) 3.0 (d) 25 2.5 20 2.0 1.5 15 EKT(HF-SL) 1.0 EKT(DFT-SL) 10 EKT(SJ, VMC) 0.5 Error of IPs [eV]

Theoretical IPs [eV] EKT(SJ, DMC) 0.0 5 EKT(MSL) EKT(MSJ, VMC) 0.5 EKT(MSJ, DMC) − 0 1.0 0 5 10 15 20 25 30 − HF DFT Experimental IPs [eV] VMC(SJ)DMC(SJ)VMC(MSL)VMC(MSJ)DMC(MSJ)

3.5 3.0 HF-SL VMC(MSL) 2.5 DFT-SL VMC(MSJ) 2.0 (e) VMC(SJ) DMC(MSJ) DMC(SJ) 1.5 1.0 0.5

Error of IPs [eV] 0.0 0.5 − 1.0 − He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar

Figure 5.3: Ionization potentials of atoms on the 1st, 2nd and 3rd rows of the periodic table (from He to Ar) using different methods: (a) by direct subtraction of energy between atoms and corresponding ions (A and A+). We denote as ∆ methods; (b) Box-and-whisker plot of errors of ∆ methods, the purple dots are the root-mean-square errors of each methods; (c) EKT results based on different wave functions; (d) Box-and- whisker plot of errors of EKT based on different wave functions, the purple dots are the root-mean-square errors of each methods. (e) errors of ionization potentials for each atoms. For the CISD (MSL) results, we set the number of active orbitals to be 10 and include those CSF’s whose coefficients are larger than 0.01.

Fig. 5.3 shows the calculated ionization potentials of atoms on the 1st, 2nd, and 3rd rows of the periodic table. Here, we compare EKT with other different methods, such as ∆ methods (direct subtraction of the energy of atoms and ions), EKT using different correlated wave functions as a reference (single Slater determinant, single Slater-Jastrow, and multiple Slater determinant, and multiple Slater-Jastrow), EKT with VMC and DMC. As we can see from Fig. 5.3(a) and (c), all the methods agree with the experiments

75 within 1eV.

Among the ∆ methods, ∆DMC is highly accurate with root-mean-square error smaller than 0.2eV.

∆DFT is comparable to ∆VMC regarding the total root-mean-square error ( 0.3eV). ∆HF is not very ∼ good compared with others (error is around 1eV). This is mainly because that HF can not describe the electron-electron correlation.

Now let us see the EKT with different correlated wave functions as a reference. EKT on HF is the original

Koopmans’ theorem. The ionization potentials using EKT (or Koopmans’ theorem) is comparable with ∆HF regarding the root-mean-square error. However, if we look at the box-whisker plot, we find that they behave differently. ∆HF turns to underestimate the ionization potentials. This is because of the correlation error in HF. Atoms and ions have different correlation energy [see Fig. 5.4]. In general, the correlation energy of a specific atom is larger in magnitude than its corresponding ions. This induces a negative correlation error of IPs. Compared with EKT(HF), ∆HF corrects the relaxation effect, but not the correlation error. This is why the errors of ∆HF are negative.

0

2 −

4 −

6 −

8 atom Correlation energy [eV] − ion 10 − He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar

Figure 5.4: Correlation energy of atoms and ions, computed by the difference of DMC energy and HF energy.

Compared with ∆HF, ∆DFT describe the electron-electron correlation much better. Therefore ∆DFT has smaller errors. The errors are also centered around zero. EKT using DFT Slater determinant as a reference is comparable with EKT using HF Slater determinant. Although ∆DFT is better than ∆HF, the corresponding non-interacting wave function of DFT does not perform better than that of HF.

Single Slater-Jastrow (SJ) wave functions [VMC(SJ) and DMC(SJ)] are slightly better than Slater deter- minant wave function. The errors of IPs from VMC(SJ) and DMC(SJ) are mostly positive. This behavior is also seen in VMC(MSJ) and DMC(MSJ) [multiple Slater-Jastrow wave function] compared with VMC(MSL).

In both cases, adding Jastrow factor to the Slater-determinant makes the errors become positive. The reason is that, a single Slater-Jastrow wave function can describe the electron correlation well. However, EKT on

76 a single Slater-Jastrow wave function still has large relaxation error. The relaxation error is positive. This is why EKT(SJ) has positive errors [see Fig. 5.3(e)].

Compared with adding a Jastrow factor, increasing the number of Slater determinants can improve the results in a more significant way. This is very significant in some strongly correlated atoms such as C, N, F, and Ne, in which Single-Slater-determinant wave functions produce very large errors. This again confirms what we have found in H2 molecule. Increasing the number of Slater determinant can reduce the correlation error and relaxation error simultaneously.

Approximate ionization potential for inner shell electrons

The generalized eigenvalue problem in Eq. (5.11) not only gives us a first ionization potential, but in fact a whole ionization spectrum. However, whether the other eigenvalues of the spectrum can be associated with ionization of inner shell electron is well studied. We compare the ionization spectra computed from EKT with experimental photoemission spectra (from NIST database [6]) for several atoms. The results are showed in Fig. 5.5. We see that EKT does produce energy levels that qualitatively agree with the photoemission spectra. The errors of the energy levels are large (1 2 eV). The errors are also system dependent. For ∼ carbon and neon, the results are in good agreement with experiments (errors are smaller than 1eV). But the agreement is not as good in oxygen. We see there is an extra energy level ( 28 eV) that is not present in ∼ − experimental spectra.

10 10 20 − 2 − − 2S 2P 2S22P3 12 15 25 − − − 2S22P5 30 14 20 − − − 2S22P3 35 16 2S2P2 25 − − − 40 − 18 30 − − 45 2S12P6 − 2S2P2 Ionization spectrum [eV] 20 Ionization spectrum [eV] 35 2S12P4 Ionization spectrum [eV] 50 − − − 22 40 55 − ∆ DMC EKT(SJ, DMC) EKT(ML) − ∆ DMC EKT(SJ, DMC) EKT(ML) − ∆ DMC EKT(SJ, DMC) EKT(ML)

(a) C (b) O (c) Ne

Figure 5.5: Ionization spectra of atoms (C, O, and Ne) using different methods). ∆ DMC: directly subtract energies between two states; EKT(SJ, DMC), EKT from DMC using a Slater-Jastrow trial wave function; EKT(ML): EKT using multiple Slater-Jastrow wave function. The black solid lines are experimental results from NIST database [6].

77 5.4.2 Electron affinity

EKT is exact for first ionization potential, but up to our knowledge, there is no rigorous proof of the ansatz

for electron affinity. We verify this by computing the electron affinity of ions (A+), which should be the

negative of the first ionization potential of neutral ions.2 The results are showed in Fig. 5.6. As one can

see, EKT overestimates the electron affinity by 1 5 eV. EKT is not as good for electron affinity as for ∼ ionization potential.

5 −

10 −

15 −

20 Experiment EKT(SJ, DMC)

Electron affinity [eV] − EKT(SL) EKT(MSL) EKT(SJ, VMC) 25 − He+ Be+ B+ C+ N+ O+ F+ Ne+

Figure 5.6: Electron affinity of ions. The experimental results are negative of the ionization potential of corresponding neutral atom.

5.4.3 EKT valence band structure for solids

Fermi velocity in suspended graphene

It is interesting to see whether EKT band dispersion results agree with the photoemission results, since EKT

essentially describes the photoemission process for valence bands. We apply it to calculate the graphene

band structure in which the experimental data is available. We focus on the dispersion near Dirac point. It

is noted from [1] that, LDA underestimates the Fermi velocity by about 20%, whereas GW overestimates ∼ it by 50% 70%. The Fermi velocity can be easily calculated with EKT since it is related to the dispersion ∼ of highest occupied orbitals. The theorem is exact for first ionization potential. As long as we have accurate

ground state wave function, EKT will predict accurate fermi velocity. The dispersion curve is showed in

2The electron affinity of neutral atoms are generally small. For benchmark purpose, we choose to compute the electron affinity of ions.

78 0.0 EXP GW LDA EKT 0.2 − GGA

0.4 −

0.6 −

Energy [eV] 0.8 −

1.0 −

1.2 − 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 − − − − − − − 1 k K [A˚ − ] | − |

Figure 5.7: Graphene energy momentum dispersion near Dirac point using different first principle methods. Experimental results, LDA and GW results are from [1]. GGA and EKT results are from this work.

Fig. 5.7(b). Our EKT with a single Slater-Jastrow wave function obtains a Fermi velocity that is in very good agreement with experiment [see Table. 5.2].

Table 5.2: Quasiparticle dispersion of graphene near Dirac point. GW, LDA, and experimental results are from [1]. 6 method fermi velocity vf (10 m/s) experiment 1.10 - 1.13 GGA/LDA 0.907 GW 1.71 EKT3 1.19(5)

5.5 Conclusion

In summary, we have implemented the extended Koopmans’ theorem within quantum Monte Carlo frame- work. We have demonstrated its success in study the ionization process of atoms, molecules, and solids.

Several conclusions come out of our studies:

(1) the ionization potential is sensitive to how good the wave function is in describing the static electron-

electron correlation. Multiple Slater determinant wave function is generally better than single Slater-

Jastrow wave functions for computing ionization potential.

(2) For solids, even a single Slater-Jastrow wave function is able to give accurate quasiparticle dispersion.

79 (3) EKT is exact for the ionization potential of the outermost shell electron. For the inner shell ionization,

it is only an approximation.

(4) EKT normally overestimates the electron affinity.

Given the accuracy of EKT in describing quasiparticle dispersion in the simple systems that have been studied (silicon in [112], hydrogen chain, and graphene from this work), it will be interesting to see how it performs for more complex systems, for example, transition metal oxides, high Tc cuprates. Therefore, in future, we will apply it to transition metal oxides, such as MnO, NiO, CoO.

Having a theory that can correctly capture the low energy physics of adding and removing electrons is essential for us to understand the transport properties of physical systems. In the next chapter, we will see that EKT can also be used for constructing low energy effective models. The constructed models are expected to be able to describe low lying charge excitations.

80 Chapter 6

Effective model construction from first principles simulation

The introduction of the downfolding methods are partially based on our publication [210].

6.1 Introduction

Physicists usually like to have an intuitive understanding of physical phenomena using simplified models.

These models are expected to capture the most relevant physical degrees of freedom related to the observed

phenomena. For example, at high temperature and low density when quantum effect is not significant, the

23 ideal gas model can successfully capture the statistical properties of 10 H2 molecules in a box, without any detailed knowledge of the fundamental constituent of H2. This approach is valid when we are interested in phenomena at certain energy scale (or length scale), while the degrees of freedom at other length scales

which are not dynamically excited simply renormalize the dynamics of the low energy degrees of freedom.

This concept/principle has been widely employed in condensed matter physics. For example, in the

past decades, there are a lot of studies on describing complex systems (high Tc Cuprate and many other

transition metal oxides) using models such as Hubbard model [242], t-J model [243], etc. However, very

few studies have addressed the effectiveness of those models in describing the real complex systems. In

complex systems such as high Tc cuprates, it is unclear to what extent can these models describe the reality

[244]. Generally, in strongly correlated systems, the macroscopic phenomena are strongly dependent on

material-specific properties, motivating the need to determine the effective Hamiltonians that can capture

all necessary details.

On the other hand, the reliable simulation of systems for which the large-scale physics is not well-

approximated by a non-interacting model, is a major challenge in physics, chemistry, and materials science.

These systems appear to require a multi-scale approach in which the effective interactions between electrons

at a small distance scale are determined, which then leads to a coarse-grained description of emergent

correlated physics. This reduction of the Hilbert space is often known as “downfolding”.

A schematic of “downfolding” is depicted in Fig. 6.1. The full Hamiltonian H is defined in the space of

81 active (partially occupied), core (mostly occupied) and virtual (mostly unoccupied) orbitals. These orbitals

have been arranged according to their energy (E) in the figure. The objective is to map the physics of the

original system to that of the effective one, defined only in the active space, with Hamiltonian H˜ . Once an

effective model Hamiltonian in the reduced Hilbert space is obtained, it can be used to perform a calculation

on a larger system using techniques designed for lattice models [245, 246, 247, 248, 249, 250, 251, 252, 253,

254, 255, 256, 257, 258]. This multi-step modeling procedure is needed since the ab initio calculations for a

given system size are, in general, computationally more expensive than the equivalent lattice calculations.

Large sizes are crucial to study finite size effects, and in turn theoretically establish the presence of a phase.

In addition, excited states and dynamical correlation functions have traditionally been difficult in ab initio

approaches, but have seen progress for lattice model methods [259, 260, 261].

virtual

E active downfold ~H core

H

Figure 6.1: Schematic for downfolding. The full Hamiltonian H is defined in the space of active (partially occupied), core (mostly occupied) and virtual (mostly unoccupied) orbitals. These orbitals have been ar- ranged according to their energy (E) in the figure. The objective is to map the physics of the original system to that of the effective one, defined only in the active space, with Hamiltonian H˜ .

One can loosely categorize downfolding techniques into two strategies. The first strategy is based on

performing ab initio calculations and then matching them state by state to the effective model. Alternately,

some approaches employ a model for the screening of Coulomb interactions, for which the ab initio single particle wavefunctions provide the relevant inputs. Techniques that fall into this class include the constrained density functional theory [262, 263], the constrained random phase approximation (cRPA) [69], fitting spin models to energies [264, 265], and fitting reduced density matrices of quantum Monte Carlo (QMC) calcula- tions to that of model Hamiltonian [122]. The second class is based on L¨owdindownfolding [266, 267, 268] and canonical transformation theory [269, 270, 271, 272, 273], which involves a sequence of unitary transfor- mations on the bare Hamiltonian, chosen in a way that minimize the size of the matrix elements connecting the relevant low energy (valence) space to the high energy one.

82 Downfolding by fitting has the advantage that it is conceptually straightforward to perform, although

it demands an a priori parameterization of the effective Hamiltonian. The methods have been applied to

complex bulk systems [274, 275, 276, 277, 278, 279], but it is only recently that their accuracy is being

rigorously checked [280]. On the other hand, canonical transformations do not need such parameterizations

and can discover the relevant terms in an automated way. However, their application to complex materials

remains to be carried out and tested.

In this chapter, we demonstrate three downfolding methods that use data from ab initio QMC techniques to derive an effective coarse-grained Hamiltonian.

(1) Eigenstate ab initio density matrix downfolding (E-AIDMD): we optimize model parameters to match

the density matrix and energy spectrum of the model Hamiltonian to that of the ab initio system. In

this method, we build a one to one correspondence between the states of the model Hamiltonian and

that of the ab initio Hamiltonian. To accomplish this, we need to solve both the model Hamiltonian

and the ab initio Hamiltonian. Generally, we need to find a cheaper way to solve model Hamiltonian

such that we can efficiently solve it multiple times for different parameters in order to find the optimized

set of parameters that can match the ab initio system.

(2) Non-eigenstate ab initio density matrix downfolding (NE-AIDMD): this is based on the fact that, once

a model Hamiltonian is assumed, the energy expectation of a certain state is directly related to the

reduced density matrix on that state. Thus, each state imposes a constraint for the model parameters.

We can choose several low lying states (not necessary to be the eigenstates of the system), and form a

linear least fitting problem for the model parameters. Solving the linear least problem will determine

the value of the parameters. Meanwhile, analyzing the residue of the fitting also give us a quantitative

estimate of how good the model is.

(3) Non-eigenstate ab initio EKT (extended Koopmans’ theorem) matrix downfolding (EKT-AIDMD):

this is similar to the NE-AIDMD, except that now we use the EKT matrices and [see Eqs. (5.8) V C and (5.25)] instead of energy. Compared with NE-AIDMD, we have much more information (all the

matrix elements of and ) in EKT-AIDMD to impose constraints on the parameters, instead of just V C the energy expectation for a particular state. Through the quality of the linear least fit, one can also

assess the quality of the low energy model in describing the low energy physics of the system.

In the following section, we will first provide detailed formulation of the three downfolding methods. We will then use the three methods to study the most simple “many-body” system, H2 molecule, to demonstrate the idea/procedure of the downfolding process. We will then apply these methods to solids, in particular,

83 the one-dimensional hydrogen chain and graphene.

6.2 Downfolding procedures

6.2.1 Eigenstate AIDMD method (E-AIDMD)

In the first method, schematically depicted in Fig. 6.2(a), eigenstates from an ab initio calculation are used to

match density matrices and energies of the corresponding model. The parameters of the model Hamiltonian

are obtained by minimizing a cost function that is a linear combination of the energy and density matrix

errors,

X a m 2 X X a m 2 (E E ) + f ( c†c†c c c†c†c c ) . (6.1) R ≡ s − s h i j l kis − h i j l kis s s i,j,k,l

where the subscript s is an eigenstate index, i, j, k, l are orbital indices and the superscripts a and m refer to

ab initio and model calculations respectively. Here, we only include the two-body reduced density matrix,

but not the one-body reduce density matrix, since the latter is completely determined by the former. There

is no definite prescription for choosing the weight f; a good heuristic is to choose a value that gives roughly the same size of errors for the two terms that enter the cost function. The cost minimization is performed with the Nelder Mead simplex algorithm.

This method is limited by the number of available eigenstates and the accuracy of true estimators. This method requires accurate model solvers. Currently, it is only applicable to models that are exactly solvable.

6.2.2 Non-eigenstate AIDMD method (NE-AIDMD)

Consider a set of ab initio energy averages E˜s, i.e. expectation values of the Hamiltonian, and corresponding

1- and 2-RDMs c†c , c†c†c c for arbitrary low-energy states characterized by index s. Assume a h i jis h i j l kis model 2-body Hamiltonian with effective parameters tij (1-body part) and Vi,j,k,l (2-body part) along with

a constant term C; the total number of parameters being Np. Then for each state s, we have the equation,

X X E˜ H = C + t c†c + V c†c†c c , (6.2) s ≡ h is ijh i ji ijklh i j l ki ij ijkl

where we have made the assumption that the chosen set of operators corresponding to single particle wave-

functions or orbitals, explain all energy differences seen in the ab initio data. The constant C is from

energetic contributions of all other orbitals which are not part of the chosen set.

We then perform calculations for M low-energy states which are not necessarily eigenstates. These states

84 are not arbitrary in the sense that they have similar descriptions of the core and virtual spaces. Each state

satisfies the criteria (1) its energy average does not lie outside the energy window of interest and (2) the

trace of its 1-RDM matches the electron number expected in the effective Hamiltonian.

The objective of choosing a sufficiently big set of states is to explore parts of the low-energy Hilbert space

that show variations in the RDM elements. Since the same parameters describe all M states, they must

satisfy the linear set of equations,

  E˜1     1 c†cj 1 .. c†c†clck 1 ..  ˜  h i i h i j i  E2             1 ci†cj 2 .. ci†cj†clck 2 ..  C  E˜   h i h i     3         1 c†cj 3 .. c†c†clck 3 ..   tij     h i i h i j i     ...        =  1 c†c .. c†c†c c ..   ..  (6.3)    i j 4 i j l k 4     ...   h i h i             1 ......   Vijkl   ...             1 ......  ..      ...      1 c†c .. c†c†c c .. h i jiM h i j l kiM E˜M which is compactly written as,

E = Ax , (6.4) where E (E˜ , E˜ , ...E˜ )T is the M dimensional vector of energies, A is the M N matrix composed of ≡ 1 2 M × p density matrix elements and x (C, t ....V ...)T is a N dimensional vector of parameters. This problem ≡ ij ijkl p is over-determined for M > Np, which is the regime we expect to work in.

get eigenstates E match Ax = E eigenstate RDMs (non eigenstates)

H ~H H ~H (a) (b)

Figure 6.2: Schematic of ab initio density matrix downfolding (AIDMD) methods employed for determining the effective Hamiltonian parameters. (a) In the eigenstate (E)-AIDMD, the reduced density matrices and energies of eigenstates of the model are matched to the ab initio counterparts. (b) The non-eigenstate (N)- AIDMD method uses RDMs and energies of arbitrary low-energy states to construct a matrix of relevant density matrices and performs a least square fit to determine the optimal parameters.

In the case of any imperfection in the model, which is the most common case, the equality will not hold

85 exactly and one must then instead minimize the norm of the error, : R

Ax E 2 . (6.5) R ≡ || − ||

This cost function can be minimized in a single step by using the method of least squares, employing the

singular value decomposition of matrix A. This matrix also encodes exact (or near-exact) linear dependences.

Thus, the quality of the fit is directly judged by assessing (1) the singular values of the A matrix and (2) the value of the cost function itself i.e. the deviations of the input and fitted energies. We will refer to this as the non eigenstate NE-AIDMD method throughout the rest of the paper. This idea is schematically depicted in Fig. 6.2(b).

The matrix A gives a very natural basis for understanding renormalization effects. For example, consider a set of wavefunctions that show that the correlator n n does not change significantly. This would lead h i ji to the corresponding column of matrix A being identical (up to a scale factor) to the first column of 1’s.

Physically, this would correspond to the coupling constant Vijji being irrelevant for the low-energy physics; it can take any value including zero and can be absorbed into the constant shift term. This could also alternatively mean that the input data is correlated and does not provide enough information about Vijji, so care must be taken in constructing the set of wavefunctions.

In summary, the N-AIDMD method performs the following operation. The 1- and 2-RDMs and en- ergy expectation values of many non-eigenstate correlated states are calculated. Then, given an effective

Hamiltonian parameterization, linear equations (6.4) are constructed and solved. Standard model fitting principles apply, and we can evaluate the goodness of fit to determine whether a given effective Hamiltonian can sufficiently describe the data.

6.2.3 EKT ab intio density matrix downfolding methods (EKT-AIDMD)

The extended Koopmans’ theorem (EKT) provides a straight forward way to compute charge excitation spectra, such as ionization potentials, electron affinities from any level of theory. EKT is based on the following two types of matrix elements,

σ σ = ψ c† [H,ˆ c ] ψ , = ψ c [H,ˆ c† ] ψ . (6.6) Vmn −h | m,σ n,σ | i Cmn h | m,σ n,σ | i

These two matrices can be represented by reduced density matrices.

86 Suppose the Hamiltonian includes one-body and two-body terms as follows,

X X σ;σ0 Hˆ = t c† c + V c† c† c c . (6.7) ij iσ jσ ij;lk iσ jσ0 kσ0 lσ i,j,σ i,j,l,k,σ,σ0

One can derive that

  σ X σ X α;σ α;σ X σ;β σ;β = ρ tnj +  V M + V M  , (6.8a) Vmn mj in;lk im;lk nj;lk mj;lk j i,l,k,α j,l,k,β X h X X X i σ = ρ¯σ t + 2V σ;β ρβ 2 V σ;σ ρσ 2 V σ;β M σ;β . (6.8b) Cmn mi in mj;nk jk − im;nk ik − ij;nk ij;mk j j,k,β i,k i,j,k,β

σ σ,τ where ρmn and Mij,lk are the one-body and two-body reduced density matrices on a certain state defined as

σ σ σ;τ ρ = ψ c† c ψ , ρ¯ = ψ c c† ψ ,M = ψ c† c† c c ψ . (6.9) mn h | m,σ n,σ| i mn h | m,σ n,σ| i ij;lk h | i,σ j,τ k,τ l,σ| i

One can evaluate ρ, M, and for each state using ab initio methods. Each state will impose 2N2 V C mo constraints on t and V according to Eq. (6.8), where Nmo is the number of single particle orbitals. If we have N state available, we will have 2N N constraints for t and V. We then organize these constraints s s× mo into a linear least fitting problem.

6.2.4 Generating states for AIDMD methods

We now address the central issue of generating states to be used as inputs for the AIDMD methods.

For the E-AIDMD, the near-eigenstates were obtained by performing CISD calculations with multiple roots and optimizing a multi-determinant Jastrow wavefunction with each CISD guess as a starting point.

This is known to be approximate, especially for higher excited states. It is the inherent uncertainty about the accuracy of this procedure, along with the fact that only a small number of eigenstates are accessible, that limits the utility of E-AIDMD.

From the point of view of the NE-AIDMD and EKT-AIDMD methods too, automating the construction of the database of wavefunctions may not be completely straightforward and here we offer some heuristics for doing this within QMC methods. We re-emphasize that any state described by the effective Hamiltonian must be one that does not involve large contributions from the core and virtual orbitals i.e. single particle degrees of freedom outside the chosen active space. This check can be imposed at the ab initio level by monitoring the RDMs, for example, the trace of the 1-RDM taken over the active orbitals must equal the number of electrons in the effective Hamiltonian description.

87 One way to generate new states is to perturb near-eigenstates. For example, after optimizing the multi-

determinantal-Jastrow trial wavefunction, we artificially change the determinantal coefficients by small

amounts. This procedure changes the nodal surface and gives energies close to, but different from, the

optimized ground state. A second source of data is spin excitations of the DFT reference Slater determi-

nant, generated within the space of orbitals that play an important role in the active space; for benzene and

graphene these involve the Kohn-Sham orbitals with π symmetry. Finally, in the case of extended systems, we chose a linear combination of determinants which, in spite of being not size-extensive, reveal additional properties of the effective Hamiltonian.

6.3 H2 molecule as a toy model

Let us take the simplest “many-body” (two-body) system, H2 molecule, as an example to demonstrate the downfolding procedures and test their reliability. We would assume that the distance between the two

hydrogen atoms is not too small, such that the Hilbert space of the system can be well expanded by the

bonding (B) and anti-bonding (AB) orbitals [see Fig. 6.3(a) and (b)]. In our model Hamiltonian, we use two

localized orbitals - L and - R [see Fig. 6.3(c) and (d)] which are constructed from the bonding ( B ) and | i | i | i antibonding ( AB ) orbitals in the following way, | i r r 1 1 L = ( B + AB ) , R = ( B AB ) . (6.10) | i 2 | i | i | i 2 | i − | i

Using localized basis is generally more convenient when studying solid, in which we can make use of the

translation symmetry to reduce the number of parameters in the effective model.

(a) Bonding (b) Anti-bonding (c) Wannier left (d) Wannier right

Figure 6.3: Iso-surface plot of Hartree-Fock orbitals and Wannier orbitals. Different colors represent different signs (yellow, positive; blue, negative). Wannier orbitals are constructed through a unitary transformation of Hartree-Fock orbitals.

Let us further propose the following effective low energy Hamiltonian,

Hˆ = tc† cLσ + V nRnL + U(nR nR + nL nL ) + JσL σR + C. (6.11) − Rσ ↑ ↓ ↑ ↓ ·

88 which includes the hopping term, nearest neighbor Coulomb interaction, the onsite Coulomb interaction,

exchange interaction, and a constant term. The exchange interaction can be written in a second quantized

form,

X m m Jˆ = JσL σR = J σ σ c† cLβc† cRβ · αβ α0β0 Lα Rα0 0 m=x,y,z

= 2J(cL† cR† cL cR + cR† cL† cR cL ) + J(nL nL )(nR nR ) , (6.12) − ↑ ↓ ↓ ↑ ↑ ↓ ↓ ↑ ↑ − ↓ ↑ − ↓ where σx, σy, and σz are Pauli matrices.

In the following, we will demonstrate how to use the three downfolding methods to obtain the effective parameters for H2 molecule with different interatomic distance.

6.3.1 E-AIDMD method

The E-AIDMD method requires us to solve the model Hamiltonian exactly, and tune the parameters t, V,

U, J, and C, so that the density matrices and energy spectrum match that of H2 molecule.

Solving the model Hamiltonian

Let us focus in the case of Sz = 0, in which case we have one spin up electron and one spin down electron in the system. The dimension of the Hilbert space is 4. The many-body eigenstates are 1,

1 1 Ψ = ( S + λ D ), Ψ+ = (λ S + D ), TS , TD , (6.13) − √1 + λ2 | i | i √1 + λ2 | i | i | i | i

where,

2t λ = , U E + C r − − r 1 1 S = ( , , ), D = ( , 0 + 0, ), | i 2 | ↑ ↓i − | ↓ ↑i | i 2 | ↑↓ i | ↑↓i r r 1 1 TS = ( , + , ), TD = ( , 0 0, ) . (6.14) | i 2 | ↑ ↓i | ↓ ↑i | i 2 | ↑↓ i − | ↑↓i

We use the following notation for the two-body states,

, := cL† cR† 0, 0 = cR† cL† 0, 0 , (6.15a) | ↑ ↓i ↑ ↓| i − ↓ ↑| i

0, := cR† cR† 0, 0 = cR† cR† 0, 0 = 0, . (6.15b) | ↑↓i ↑ ↓| i − ↓ ↑| i −| ↓↑i 1Please refer to Appendix. D for detailed derivation.

89 0, 0 is the vacuum state. The corresponding eigenvalues of the four eigenstates in Eq. (6.13) are 2 | i r U + V 3J (U V + 3J)2 E = − 4t2 + − + C, (6.16a) ± 2 ± 4

ETS = V + J + C, (6.16b)

ETD = U + C. (6.16c)

The one-body reduced density matrix is,

Ψ cR† cR Ψ = Ψ cL† cL Ψ = Ψ cR† cR Ψ = Ψ cL† cL Ψ = 0.5, h −| ↑ ↑| −i h −| ↑ ↑| −i h −| ↓ ↓| −i h −| ↓ ↓| −i λ † † † † Ψ cR cL Ψ = Ψ cL cR Ψ = Ψ cR cL Ψ = Ψ cL cR Ψ = 2 . (6.17) h −| ↑ ↑| −i h −| ↑ ↑| −i h −| ↓ ↓| −i h −| ↓ ↓| −i 1 + λ

The two-body reduced density matrix is,

σ;σ0 † † M = Ψ ci,σc cl,σ0 ck,σ Ψ . (6.18) ijkl h −| j,σ0 | −i

σ;σ0 1) if σ = σ0 = or , M = 0; ↑ ↓ ijkl

2) if σ = , σ0 = , ↑ ↓

λ 1 cL cL Ψ = 0, 0 , cL cR Ψ = 0, 0 , ↓ ↑| −i √1 + λ2 | i ↓ ↑| −i √1 + λ2 | i 1 λ cR cL Ψ = 0, 0 , cR cR Ψ = 0, 0 . (6.19) ↓ ↑| −i √1 + λ2 | i ↓ ↑| −i √1 + λ2 | i

All the two-body reduced density matrix elements are listed in Table. 6.1. We see that all the RDMs for

Table 6.1: Two-body reduced density matrix of a two-site generalized Hubbard model ; Mijkl↑ ↓ LL LR RL RR λ2 λ λ λ2 LL 1+λ2 1+λ2 1+λ2 1+λ2 λ 1 1 λ LR 1+λ2 1+λ2 1+λ2 1+λ2 λ 1 1 λ RL 1+λ2 1+λ2 1+λ2 1+λ2 λ2 λ λ λ2 RR 1+λ2 1+λ2 1+λ2 1+λ2 the ground state are functions of λ. The model Hamiltonian is so simple that that a single parameter λ determines all the RDMs. 2see Appendix. D.

90 ab initio eigenstates and eigen energies

Let us see what ab initio data can be used to determinant the parameters:

the ground state of S = 0, Sz = 0 sector: Ψ . This is a Slater-Jastrow wave function with Slater • − determinant from CISD calculation. Using CISD wave function instead of a single Slater determinant

is necessary because of the static correlation at long distance.

Here, we have mentioned that, all the reduced density matrices are function of λ. Therefore, we can

tune λ such that the two-body reduced density matrices of the model system match that of ab initio

3 simulation of H2. The results are shown in Fig. 6.4(a) .

Eabinit – from CISD, or DMC; • −

Eabinit – from CISD; • +

Eabinit – from CISD, or DMC of S = 1, S = 1 calculations. • TS z

The eigen energies are shown in Fig. 6.4(b).

1.0 1.0 E − 0.8 0.5 E+

ETS 0.6 0.0 abinit

λ 0.4 0.5

Energy [Ha] −

0.2 1.0 −

0.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 − 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Distance [A]˚ Distance [A]˚ (a) (b)

Figure 6.4: ab initio information that can be used for determining the model parameters: (a) ground state parameter λ; (b) energies of various eigenstates (from CISD).

If we fix C = 1.0 Ha, which corresponds to the total energy of H with d , we have only t, U, V, − 2 → ∞ J, four parameters to be determined. Essentially, we are minimizing the following cose function,

abinit 2 abinit 2 abinit 2 2 = (E E ) + (E+ E+ ) + (ETS ETS ) + f(λ λabinit) . (6.20) R − − − − − − 3 We have used symmetry property of the system. For example, we consider ρLL = ρRR, ρLR = ρRL. Therefore, we consider the equivalent matrix elements to be one parameter and take the averaged value to compare with result from model Hamiltonian.

91 with the density matrices being “encoded” in λ. Tuning the four parameters can make = 0 if the following R set of nonlinear equations has a least one solution.

E (t, U, V, J) = Eabinit, (6.21a) − − abinit E+(t, U, V, J) = E+ , (6.21b)

abinit ETS(t, U, V, J) = ETS , (6.21c)

λ(t, U, V, J) = λabinit . (6.21d)

The final solution for the values of t, U, V, and J is plotted in Fig. 6.5.

400 350 300 t 250 U 200 V 150 J 100

Model parameters [mHa] 50 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Distance [A]˚

Figure 6.5: Model parameters of H2 obtained through E-AIDMD method.

Let us investigate the asymptotic behavior of t, U, V, and J to check whether they are physical. First, we find that t, V , and J decay as the distance increases. For t and J, they are directly related to the overlap integral of the two Wannier orbitals on the two sites. Therefore, they should decay exponentially with distance. This is essentially what we see from the asymptotic behavior of t and J [see Fig. 6.6(a)].

J decays faster than t (approximately twice) because it involves two integrals of Wannier orbitals. V should decay as 1/r6 at long distance; since at long distance, the Coulomb interaction manifests as van der

Waals interaction (induced dipole dipole interaction) [see Fig. 6.6(b)]. U, however, will eventually approach

+ U 1/r + E(H−) + E(H ) 2E(H), which is the energy when two electrons occupy the same hydrogen → − − atom leaving the other hydrogen atom unoccupied, minus the energy of two isolated hydrogen atoms [see

Fig. 6.6(c)].

92 400

2 350 10 (a) (b) 102 (c) 300

1 250 10 1 200 10 150 100 100 J U 100 V Model paramters [mHa] Model parameters [mHa] t 50 1/r + 474.6 Model parameters [mHa] C/r6 1 − 10− 0 1 0 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 10− 10 10 Distance [A]˚ Distance [A]˚ Distance [A]˚

Figure 6.6: Asymptotic behavior of model parameters.

How close the model is to the low energy physics of H2

In general, we can look at the value of the cost function to see whether the downfolded model is a good

description of H . However, in this particular case, we have made = 0. The eigen energies are completely 2 R matched between model Hamiltonian and ab initio system. The quality of the model lies in how well the density matrices are matched. We can check the difference of the one-body reduced density matrix ρ between

the model Hamiltonian and H2 molecule,

s 1 X Err(ρ) = ρabinit ρmodel 2 , (6.22) N | ij − ij | i,j

where N is the total number of matrix elements. The error of ρ with distance is shown in Fig. 6.7. The

error decays with distance. The model is a good description of H2 at long distance, in particular at distance d > 2A.˚

0.14

0.12

0.10 )

ρ 0.08 ( 0.06 Err

0.04

0.02

0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Distance [A]˚

Figure 6.7: Error of 1-RDM for H2 with difference distance.

93 6.3.2 NE-AIDMD method

In this method, we fit the energy expectation values,

; ; E = t(ρ↑ + ρ↑ + ρ↓ + ρ↓ ) + U(M ↑ ↓ + M ↑ ↓ ) − LR RL LR RL LL;LL RR;RR ; ; ; +(V + J)(M ↑ ↑ + M ↓↓ ) + (V J)(M ↑ ↓ + M ↑ ↓ ) LR;LR LR;LR − LR;LR RL;RL ; ; 2J(M ↑ ↓ + M ↑ ↓ ) + C. (6.23) − LR;RL RL;LR

We use six different states:

U U U U Single Slater-Jastrow : Ψ1 = φ1 φ1 e , Ψ2 = φ1 φ2 e , Ψ3 = φ1 φ2 e , Ψ4 = φ2 φ2 e , (6.24) ↑ ↓ ↑ ↓ ↑ ↑ ↑ ↓ 1 1 Double occupied state : TD = ( , 0 0, )eU , D = ( , 0 + 0, )eU . (6.25) | i √2 | ↑↓ i − | ↑↓i | i √2 | ↑↓ i | ↑↓i

U where φ1 and φ2 are bonding and antibonding orbitals respectively, and e is the optimized Jastrow factor. The fitting results are shown in Fig. 6.8. Again the fit is not good at small distance (d < 1.5A)˚ [see

6.8(b)]. The r.m.s. error is large. At large distance (d > 1.5A),˚ the small r.m.s errors are near chemical

accuracy, suggesting that the model is a very good description of H2 molecule. In Fig. 6.9, we plot the fitted

400 100 300 (a) (b) 80 200

100 60 0 100 40 −

200 r.m.s error [mHa] − t V 20

Model parameters [mHa] 300 − U J 400 0 − 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Distance [A]˚ Distance [A]˚

Figure 6.8: Downfolding of H2 using NE-AIDMD method: (a) fitted model parameters; (b) r.m.s error of energies of all the states.

energies with respect to DMC energies for some distances. We can see that for large distance, the fitted

energies match better with ab initio energies.

94 2000 700 (a)d = 0.2A˚ − (b)d = 1.5A˚

1500 800 −

1000 900 − t=333.00 [mHa] t=99.08 [mHa] 500 U=1869.47 [mHa] U=308.48 [mHa] V =1851.82 [mHa] 1000 V =37.16 [mHa] Predicted energy [mHa] J=-83.55 [mHa] Predicted energy [mHa] − J=0.55 [mHa] δErms=755.26 [mHa] δErms=4.56 [mHa] 0 1100 0 500 1000 1500 2000 − 1100 1000 900 800 700 − − − − − Ab initio DMC energy [mHa] Ab initio DMC energy [mHa]

700 − (c)d = 2.0A˚ 700 (d)d = 3.0A˚ 750 − − 750 800 − − 800 850 − − 850 900 − t=47.26 [mHa] t=8.59 [mHa] − 900 U=304.06 [mHa] − U=341.17 [mHa] 950 − V =8.48 [mHa] V =1.05 [mHa]

Predicted energy [mHa] Predicted energy [mHa] 950 J=1.00 [mHa] − J=0.41 [mHa] 1000 δErms=3.25 [mHa] δErms=1.84 [mHa] − 1000 − 1050 − 1050 1000 950 900 850 800 750 700 1000 950 900 850 800 750 700 − − − − − − − − − − − − − − − Ab initio DMC energy [mHa] Ab initio DMC energy [mHa]

Figure 6.9: H2 molecule downfolding results for different distance.

6.3.3 EKT-AIDMD method

In the EKT-AIDMD method, since it describes the charge removing and adding process, we need to replace P the constant C with a chemical potential term µ ni.

Hˆ = tc† cLσ + V nRnL + U(nR nR + nL nL ) + JσL σR + µ(nR + nL) . (6.26) − Rσ ↑ ↓ ↑ ↓ ·

In fact for a general Hamiltonian,

ˆ X X X X H = tijciσ† cjσ + Vijninj + Uni ni + Jijσi σj + µ ni . (6.27) ↑ ↓ · i,j,σ i ij i

We have

X ; ; X ; ; ↑ = t ρ↑ + U(M ↑ ↓ + M ↑ ↓ ) + (V + V )(M ↑ ↑ + M ↑ ↓ ) Vmn nj mj mn;nn nm;nn nj jn mi;ni mi;ni j j X h ; ; i σ + (J + J ) 2M ↑↓ + M ↑ ↑ M ↑ ↓ + µρ . (6.28) nj jn − mj;jn mj;nj − mj;nj mn j

95 Using the six states available at each distance, we can fit the EKT matrices σ , and get the parameters Vmn for the models. The results are shown in Fig. 6.10. As we can see, at small distance (d = 0.2A),˚ the model is not very good for describing charge excitations (δE 70mHa). But at large distance (d = 2.0A˚ and rms ∼ 3.0A),˚ it becomes better (δE 5mHa). rms ∼

200 (a)d = 0.2A˚ 200 (b)d = 1.5A˚ 0 100 0 200 − 100 − 400 200 − t=463.37 [mHa] − t=84.30 [mHa] U 300 U =559.24 [mHa] − =277.50 [mHa] 600 V =190.01 [mHa] V =26.31 [mHa]

Predicted energy [mHa] − Predicted energy [mHa] 400 J=-37.39 [mHa] − J=1.52 [mHa] mu=-1110.56 [mHa] mu=-561.16 [mHa] 500 800 δErms=73.20 [mHa] δErms=11.45 [mHa] − − 800 600 400 200 0 200 500 400 300 200 100 0 100 200 − − − − − − − − − Ab initio DMC energy [mHa] Ab initio DMC energy [mHa]

200 200 (c)d = 2.0A˚ (d)d = 3.0A˚ 100 100

0 0

100 100 − − 200 200 − t=33.44 [mHa] − t=2.49 [mHa] 300 U=277.42 [mHa] 300 U=286.35 [mHa] − − V =16.18 [mHa] V =1.68 [mHa]

Predicted energy [mHa] 400 Predicted energy [mHa] 400 − J=1.31 [mHa] − J=0.56 [mHa] mu=-535.30 [mHa] mu=-504.86 [mHa] 500 500 − δErms=5.98 [mHa] − δErms=4.32 [mHa] 500 400 300 200 100 0 100 200 500 400 300 200 100 0 100 200 − − − − − − − − − − Ab initio DMC energy [mHa] Ab initio DMC energy [mHa]

Figure 6.10: H2 molecule downfolding results for different distance. The model includes hoping term Tˆ, onsite Coulomb and nearest neighbor Coulomb interaction (Uˆ, Vˆ and Jˆ).

6.3.4 Comparisons of the three methods

Now let us compare the results from the three different methods. Let us focus on d = 3.0A˚ in which case all the three methods have very good fit to H2. The results are shown in Table. 6.2. They are in reasonable agreement with each other. However, there are still some discrepancies among the three methods.

Table 6.2: Comparison of different downfolding methods [mHa] E-AIDMD NE-AIDMD EKT-AIDMD t 7.62 8.59 2.49 U 311.76 341.17 286.35 V 0.74 1.05 1.68 J 0.24 0.41 0.56 µ -500 -500 -504.86

The third method is different from the other two, because they describe different low energy excitation

96 physics. E-AIDMD and NE-AIDMD mainly describe particle-hole or spin excitation due to the specific

choice of states for fitting parameters, whereas EKT-AIDMD mainly describes charge excitations. Because

of this, there is a chemical potential term in EKT-AIDMD method which is not present in E-AIDMD and

NE-AIDMD. Therefore, in general the effective parameters for these two types of excitations are different.

However, why there is so much difference between E-AIDMD and NE-AIDMD? Both of them describe

the same low energy physics. At d = 3.0, the lattice model fits ab initio data very well in both of the two methods according to their own criterions (the cost functions). But, there is a significant difference of U

(much larger than the r.m.s error).

The reason is that in E-AIDMD, we are trying to match the eigenstates of the model Hamiltonian with that of the ab initio Hamiltonian. However, we know in the model Hamiltonian, all the electrons are in the

Hilbert space expanded by the two Wannier orbitals. In other words, all the electrons occupy the bonding and antibonding orbitals. However, in real H2, there are always a small fraction of electrons occupying

P σ σ high energy orbitals (such as Pz, 2S orbitals). We can check σ ρLL + ρRR for different states for H2 with d = 3.0A˚ [see Table. 6.3]. However, in E-AIDMD, all the states from model Hamiltonian have Trρ = 2.

Table 6.3: Trace of the one-body reduced density matrix of H2 at different states State Ψ Ψ1 Ψ2 Ψ3 Ψ4 D TD − | i | i Trρabinit 1.956(2) 1.987(2) 1.984(2) 1.956(2) 1.987(2) 1.889(2) 1.885(2)

NE-AIDMD starts directly from Trρabintio. This is why there are discrepancies between the two set of downfolded parameters. We can scale the one-body reduced density matrix from ab initio Hamiltonian first, so that it will satisfied Trρ = 2 (the two-body reduced density matrix need to be scaled accordingly). We then use the scaled RDMs to do the downfolding. In this case, E-AIDMD and NE-AIDMD should match up. Indeed, we find that E-AIDMD(scale) and NE-AIDMD(scale) in Table. 6.4 are close to each other; while

EKT-AIDMD(scale) remains different from the other two as we expect.

Table 6.4: Comparison of different downfolding methods using scaled RDMs [mHa] E-AIDMD(scale) NE-AIDMD(scale) EKT-AIDMD(scale) t 6.26 8.31 2.31 U 312.00 303.78 254.21 V 0.74 0.68 1.45 J 0.30 0.32 0.49 µ -500 -500 -504.02

We would like to mention that for NE-AIDMD, using non-scaled RDMs is more physical. The scaled

results are just for sanity check. In the following, if we are using NE-AIDMD, we will use the original RDMs.

97 6.4 Application of downfolding methods to solid systems

In this section, we apply our downfolding methods to solid systems. Because of our limited time, we only use one of the three methods for each system.

6.4.1 One dimensional hydrogen chain (E-AIDMD)

Let us still consider the following Hamiltonian,

X n o H = t[ciσ† ci+1σ + h.c.] + V nini+1 + Uni ni + Jσi σi+1 + C. (6.29) − ↑ ↓ · i

Here, ci’s are Wannier orbitals generated from Kohn-Sham orbitals. We will use E-AIDMD method in this section. The model Hamiltonian is solved by exact diagonization, whereas the ab initio system is solved using diffusion Monte Carlo method with a single Slater-Jastrow trial wave function.

We start from the 4-site hydrogen chain with bond length equals to 2A.˚ Fig. 6.11 shows the Kohn-Sham orbitals and Wannier orbitals, In our calculations, we used energies and RDMs of the following three states:

(a) KS 1 (b) KS 2 (c) KS 3 (d) KS 4

(e) Wannier 1 (f) Wannier 2 (g) Wannier 3 (h) Wannier 4

Figure 6.11: Kohn-Sham orbitals (upper panel) from DFT calculations with PBE exchange-exchange cor- relation functional, and Wannier orbitals (lower panel) constructed through a unitary transformation of Kohn-Sham orbitals.

S = 0 : E = 2047.8(6)mHa; (6.30a) − S = 1 : E = 2038.9(5)mHa; (6.30b) − S = 2 : E = 1957.8(3)mHa. (6.30c) −

98 In order to understand the relative importance of various two-body terms [(a) the onsite Hubbard interaction

– Uˆ; (b) nearest neighbor Coulomb interaction – Vˆ ; (c) nearest neighbor exchange – Jˆ], we compare the

parameters obtained when downfolding the system to the following three different models with different

two-body interactions,

(a) UVJ model: onsite Hubbard interaction(U), nearest neighbor Coulomb (V) and exchange (J);

(b) UV model: onsite Hubbard interaction(U), nearest neighbor Coulomb (V);

(c) U model: onsite Hubbard interaction(U).

The quality of downfolding is measured by the relative error of the two-body reduced density matrices,

and the error of the eigen energies of the three states.

v uP ab initio model 2 s u i,j,k,l(Mijkl Mijkl ) X R = t − , ∆E = Eab initio Emodel 2 . (6.31) err P (M ab initio)2 | i − i | i,j,k,l ijkl i

The resulting effective parameters are showed in Table. 6.5. We see that all the three models can match the ab initio data accurately. U model has relatively larger error in energy, but it is still comparable to the stochastic error from QMC (0.3 0.5 mHa). ∼

Table 6.5: Parameters of effective Hamiltonian [mHa], and error of RDMs and energies [mHa]. Model t U V J err(RDM) err(energy) 13 UVJ 23.68 34.58 0.03 -3.11 0.25% 10− 13 UV 32.76 130.63 65.31 / 0.75% 10− U 37.45 114.62 / / 0.26% 1.8

Transferability of the model parameters to larger systems

In order to verify the transferability of the parameters for larger systems. We study longer chains (6-site,

8-site and 10-site) with the same interatomic distance (2A),˚ and examine whether our parameters obtained from the 4-sites hydrogen chain is able to match the low energy physics of longer chains. We therefore, solve the model Hamiltonian with the parameters from Table. 6.5, and examine the errors of the RDMs of

S=0 and S=1 states. The results are shown in Fig. 6.12. As we can see, the error of the RDMs is around

10%, indicating that the downfolding parameters from a smaller system is transferable to larger systems.

Therefore, at d = 2A,˚ the hydrogen chain can be described by the extended Hubbard model (6.29) very well.

99 (a) (b)

Figure 6.12: Errors of RDMs and energy for hydrogen chains with different number of sites: (a) relative error of two-body reduced density matrix; (b) error of eigen energy for S = 0 and S = 1 states per atom. the parameters used in model calculations are from 4-site chain.

6.4.2 Graphene and hydrogen lattice (NE-AIDMD)

In this subsection, we will downfold graphene and hydrgoen (with the same lattice constant a = 2.46A)˚ into the following effective Hubbard model,

ˆ X X H = C + t ci,σ† cj,σ + h.c. + U ni, ni, . (6.32) ↑ ↓ i,j i h i where we only include π orbitals in the effective model of graphene, and s orbitals in the effective model of hydrogen. ci0 s are Wannier orbitals constructed from Kohn-Sham orbitals, shown in Fig. 6.13.

(a) (b)

Figure 6.13: Wannier orbitals constructed from Kohn-Sham orbitals: (a) graphene π orbital; (c) hydrogen S orbital.

We choose a set of Slater-Jastrow wave functions corresponding to the electron-hole excitations within the π channel. The energy expectation expressed in terms of the density matrices is,

X σ σ X , E = C + t( ρij + ρji) + U Mii↑;↓ii , (6.33) i,j ,σ i h i

100 If we have n wave functions, we can form a linear least problem of n 3 from the n constraints of Eq. (6.33). ×     h P σ σ i h P , i E1 1 i,j ρij + ρji i Mii↑;↓ii    1 1     h h i i h i  C    P σ σ P ,    E2  1 i,j ρij + ρji i Mii↑;↓ii    2 2      =  h i   t  . (6.34)             ··· ·········  U  h P σ σ i h P , i  En 1 i,j ρij + ρji i Mii↑;↓ii h i n n

Table 6.6 shows the final results using 25 wave functions. The errorbar is calculated using Jackniff method.

Fig. 6.14 shows fitted energies versus the ab initio VMC energies. We find that the onsite Hubbard model

Table 6.6: Downfolding parameters for graphene and hydrogen. parameters [eV] graphene hydrogen t 3.61(1) 3.73(1) U 7.16(3) 9.47(5) describes graphene and hydrogen very well. The ratio of t/U is small than the semimetal-insulator transition critical value (3.8) in both the two systems, consistent to the fact that both the two systems are semimetals.

2.768 103 0 − × 255 − 1 256 − T=-3.61 0.00 [eV] − T=-3.73 0.01 [eV] ± 257 2 U=7.16 0.03 [eV] − U=9.47 ±0.05 [eV] − ± 258 ± 3 − − 259 4 − − 260 5 − − 261 E (Fitted) [eV] E (Fitted) [eV] − 6 262 − − 7 ∆Erms=0.12 [eV] 263 ∆Erms=0.12 [eV] − − 8 264 − 8 7 6 5 4 3 2 1 0 − 264 263 262 261 260 259 258 257 256 255 − − − − − − − − − − − − − − − − − − 2.768 103 E (VMC) [eV] − × E (VMC) [eV] (a) (b)

Figure 6.14: Comparison of ab initio (x-axis) and fitted energies (y-axis) of the 3 3 periodic unit cell of graphene and hydrogen lattice: (a) graphene; (b) hydrogen lattice. ×

6.5 Conclusion

We have demonstrated three AIDMD methods where ab initio QMC data are used to fit simple effective

Hamiltonians. We have elaborated on the fitting procedures. The limitations of the model are judged by

101 assessing the quality of the fitted energies and two-body density matrices or EKT matrix elements. This feature is useful for constructing refined models needed for the accurate simulation of real materials.

This approach can be used to identify important physics degrees of freedom in complex systems for understanding macroscopic phenomena. We are at the position of applying it to more complex systems, such as transition metal oxides, high Tc cuprates.

102 Chapter 7

Summary

In summary, this thesis have combined four major pieces of works:

(a) the first two pieces of works are mainly on the application of quantum Monte Carlo method to realistic

quantum simulations. We have demonstrated that the ground state quantum Monte Carlo method

is highly accurate in describing correlated systems. In VO2, quantum Monte Carlo provides highly accurate energetic results for different states, which helps to determine the magnetic coupling precisely.

Through computing microscopic electronic configuration (the charge and spin density) and spatial

correlation, we clearly revealed the mechanism for metal-insulator transition. Our study demonstrates

that the quantum Monte Carlo method can be applied to strongly correlated transition metal oxides

in the vicinity of phase transition.

(b) For graphene, quantum Monte Carlo with its high accuracy in capturing the electron correlation in all

length scales, produces the structure factor that is in excellent agreement with experiment. Combining

with other ab intio techniques, such as the random phase approximation, we are able to characterize

the electron screening effect from σ electrons at different length scales. We clearly showed that at

long distance, σ screening reduces the π plasma resonance frequency; whereas at short distance, it

reduces the onsite Coulomb interaction between π electrons, thus makes the system a weakly correlated

semimetal. This study demonstrates that, quantum Monte Carlo, combined with other first principles

techniques, is able to describe the microscopic electron-electron correlation physics that gives rise the

emergent physics.

(c) The implementation of the extended Koopmans’ theorem into quantum Monte Carlo expands its

functionality in computing the spectral properties of correlated materials. Quantum Monte Carlo is

known to be a accurate ground state method, but with the extended Koopmans’ theorem, it can

explore the low lying charge excitation and describe the quasiparticle dispersion. It also releases the

extended Koopmans’ theorem from the “dungeon” of quantum chemistry and make it applicable to

condensed matter physics. We are looking for more realistic application to correlated solid to manifest

103 its potential in understanding low energy excitations of correlated materials.

(d) Finally, we have the downfolding techniques. These connect the realistic ab initio simulation to low

energy effective lattice models, paving the way for multiscale simulations. It also helps to identify

important physics degrees of freedom that are relevant to the macroscopic phenomena in complex

systems, for intuitive understanding of complex correlated systems.

Overviewing the whole thesis, my PhD work is not merely a glimpse of the electronic structure of correlated systems, but much more on the building up a systematic framework to understand correlated physics.

104 Chapter 8

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120 Appendix A

Supplementary information for VO2

A.1 Inter-chain coupling in Ising model simulation

In our Ising model simulation, we take the following form,

X X X E = J1 σiσj + J2 σiσj + Jint σiσj + E0 . (A.1) intradimer interdimer interchain

The only undetermined term is Jint, which is not too sensitive to the qualitative behavior of the magnetic susceptibility. The results for different Jint are shown in Fig.SM. A.1.

Figure A.1: Temperature dependence of magnetic susceptibility obtained by Ising model simulation on VO2 lattice, with magnetic coupling from FN-DMC compared to results from Zylbersztejn[3] and Kosuge[4]. There has been an overall scale applied on the data, and the transition temperature is indicated by the vertical dashed line.

A.2 Inter-site electron covariance

We define the V sites and O sites by the Voronoi polyhedra surrounding the nuclei. For a given sample in the FN-DMC calculation, we evaluate the number of spin-up electrons n ,i and the number of spin- ↑ down electrons n ,i on a given site i. We then histogram the joint probability to obtain a set of functions ↓

121 σi σj ρi,j,σi,σj (ni , nj ). Using this joint probability we are able to compute the inter-site electron covariance. First, the expectation value of the number of electrons with specific spin is

σi X σi σj ni = ρi,j,σi,σj (ni , nj ) . (A.2) h i σ σi j ni ,nj

σi σj The covariance between ni and nj is

σi σi σj σj X σi σj σi σj (ni ni )(nj nj ) = ρi,j,σi,σj (ni , nj ) ni nj . (A.3) h − h i − h i i σ − h ih i σi j ni ,nj

The total charge covariance is computed as

X σ σ (n n )(n n ) = (nσi nσi ) (n j n j ) . (A.4) h i − h ii j − h ji h i − h i i ih j − h j i i σi,σj

A.3 Pseudoptential, finite size, and time step errors

0.03 0.015 Structure 0.020 0.01 Monoclinic 0.02 Rutile 0.010 BFD 4 VO2 cell 0.015 0.010 0.005 0.01 0.005

0.0 0.00 0.0 0.03 0.015

Energy 0.020 0.02 0.02 Yoon 0.015 0.010 16 VO2 cell 0.010 0.005 0.01 0.005

0.0 0.00 0.0 AFM FM UNP AFM AFM(I) FM UNP AFM FM UNP (a) (b) (c)

Figure A.2: Pseudoptential, finite size, and timestep error analysis: (a) Energetic results of 4-VO2 cells for different pseudopo- tentials: BFD and Yoon. PBE functional was employed in DFT to optain the trial wave function for FN-DMC. (b) Energetic results for different cell size: 4-VO2 cell and 16-VO2 cell; (c) Energetic results of of 4-VO2 cells for different time step: 0.01 and 0.02. BFD pseudopotential and PBE0 functional were used in (b),(c).

We computed fixed-node diffusion quantum Monte Carlo (FN-DMC) energy for two different super-

1 1 cells (4-VO2 cell and 16-VO2 cell), with different timestep (0.02 Ha− and 0.01 Ha− .), and with different pseudopotentials (BFD and Yoon). Noted from Fig.SM. A.2(a), the two different pseudopotentials show qualitatively same behavior. We should mention that in DFT level, both the two pseudopotentials give the same band structure that is in agreement with all electron calculations. For finite size error, We find that

122 for PBE0 functional, 4-VO2 cell and 16-VO2 cell give us the same results (the difference is within stochastic

1 errorbars)[see Fig.SM. A.2(b)]. We also find that if a time step of 0.02 Ha− already give us converged results [see Fig.SM A.2(c)].

123 A.4 Pseudopotential and basis set

The following are the pseudopotential and basis set used in our calculations in CRYSTAL input format.

Vanadium Oxygen 223 11 208 6 INPUT INPUT 13.0 3 1 1 0 0 0 6 . 0 3 1 0 0 0 0 2.163618 13.000000 1 9.297939 6.000000 1 4.079018 28.127028− 1 8.864922 55.787634− 1 3.214364 48.276563 0 8.629257 38.819785 0 8.443260− 96.232266 0 8.719245− 38.419141 0 6.531361 41.580435 0 0 0 7 2 . 0 1 . 0 0 0 6 2 . 0 1 . 0 0.573098 0.453752 18.360298 0.023212 1.225429 0.295926 11.577461 0.200288− 2.620277 0.019567 6.390046 0.55432 5.602818 0.128627 1.528787 0.565283− 11.980245− 0.012024 0.72883 0.543474 25.616801 0.000407 0.349741 0.134989 54.775216 7.6e 05 0 0 6 0 . 0 1 . 0 0 2 7 6 . 0 1 . 0− − 18.360298 0.002873 0.333673 0.255999 11.577461 0.043911 0.666627 0.281879 6.390046 0.144148− 1.331816 0.242835 1.528787 0.165823 2.660761 0.161134 0.72883 −0.313756 5.315785 0.082308 0.349741 − 0.055699 10.620108 0.039899 0 2 6 6 . 0 1− . 0 21.217318 0.004679 13.883264 0.003648 0 3 1 0 . 0 1 . 0 6.174 0.175046 0.66934 1 . 0 4.772897− 0.155473 0 1 1 0 . 1 . 2.182411 0.346594 0 . 2 1 . 1 . 1.049492 0.44341 0 1 1 0 . 1 . 0.49268 0.263803 0 . 6 1 . 1 . 0 3 4 1 . 0 1 . 0 0 1 1 0 . 1 . 7.384196 0.067593 1 . 8 1 . 1 . 3.350355 0.246802 1.398721 0.336404 0.571472 0.349348 0 4 1 0 . 0 1 . 0 0.850084 1.0 0 1 1 0 . 1 . 0 . 2 1 . 1 . 0 1 1 0 . 1 . 0 . 6 1 . 1 . 0 1 1 0 . 1 . 1 . 8 1 . 1 . 0 3 1 0 . 1 . 0 . 2 1 . 0 3 1 0 . 1 . 0 . 6 1 . 0 3 1 0 . 1 . 1 . 8 1 .

124 Appendix B

Supplementary information for graphene

B.1 Electron energy loss spectroscopy

B.1.1 Extraction of the response function for graphene from graphite

The experiment is performed on 3D graphite [196, 197], and we used the following formula to extract out

the response function for 2D,

χ (q, ω)d χ (q, ω) = 3D , (B.1) 2D 1 V (q)[1 F (q)]χ (q, ω)d − 2D − 3D

2 where V2D = 2πe /q is the Fourier transformation of Coulomb interaction in 2D, and where d = 3.35 A˚ is the distance between the layers. F (q) is the graphite form factor

sinh(q d) F (q) = k , (B.2) cosh(q d) cos(qzd) k −

In the IXS experiment, we measured imaginary part of the response function,

1 1 S(q, ω) = Imχ(q, ω) . (B.3) −π 1 e βhω¯ − −

we computed the real part of χ(q, ω) using Kramers-Kronig relation,

Z 2 ∞ ω0Imχ(q, ω0) Reχ(q, ω) = dω0 , (B.4) π P ω 2 ω2 0 0 −

where is the principal part. P

1 EELS = Im(q, ω)− = V (q)Imχ (q, ω), n = 2, 3 . (B.5) − − nD nD

To check whether the conversion formula works, we performed RPA calculations for graphene and graphite,

125 and directly compare the converted results with directly computed ones. The result is showed in Fig. B.1

(a). We found that the converted results from 3D to 2D agrees well with the direct computed results.

G(IXS) H(RPA) TB(RPA)

G3(RPA) G2(RPA) Converted G(RPA) TBσ(RPA)

1 1 q=0.070 A− q=0.070A−

1 1 q=0.140 A− q=0.140A−

1 1 q=0.210 A− q=0.210A−

1 1 q=0.280 A− q=0.280A−

1 1 q=0.351 A− q=0.351A− EELS EELS

1 1 q=0.421 A− q=0.421A−

1 1 q=0.491 A− q=0.491A−

1 1 q=0.561 A− q=0.561A−

0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 ω [eV] ω [eV]

(a) (b)

Figure B.1: EELS (-Imχ) of various systems. (a) checking of the conversion formula. G3: 3D graphite; G2: monolayer graphene; Converted: result converted from G3 obtained through conversion formula Eq. (B.1). (b) EELS spectrum for different systems. G(IXS): inelastic X-ray scattering result; H: hydrogen system; TB: tight-binding model using RPA; TBσ: tight-binding model with screening from σ electrons. We have scale the EELS spectrum such that the π plasmon resonance peak of different systems have the same height.

B.2 Tight-binding model for graphene and the polarization

function computed by RPA

In 2D honeycomb lattice, suppose d is the nearest neighbor distance, and a = √3d is the length of the primary lattice vector. The two primary vectors can be defined as

√3 1 √3 1 α = a( , ), α = a( , ) . (B.6) 1 2 −2 2 2 2

126 In this way, the two atoms are positioned at (0, 0), (1/3, 1/3). The reciprocal lattice is

2π 1 2π 1 β1 = ( , 1), β2 = ( , 1) . (B.7) a √3 a √3 −

1 We have two sub lattice: A-site (a) and B-site: (b) Let us define d = 3 (α1 + α2). The Hamiltonian is,

X = ar† br+d + ar† br α1+d + ar† br α2+d + h.c. (B.8) − − H − r

Define ak and bk to be the Fourier transformation of ar and br ,

1 X ik r 1 X ik r a := a e− · , b := b e− · . (B.9) k √ r k √ r N r N r

Therefore, we have

X ik d ik (d α1) ik (d α2) = t a† bk[e− · + e− · − + e− · − ] + h.c. H − k k X = t ψ† [ (k)σ + (k)σ ]ψk . (B.10) − k A 1 B 2 k

Here, we have define

(k) = cos(k d) + cos[k (d α )] + cos[k (d α )] ; (B.11a) A · · − 1 · − 2 (k) = sin(k d) + sin[k (d α )] + sin[k (d α )] ; (B.11b) B · · − 1 · − 2 1 (k) φ(k) = tan− B . (B.11c) (k) A

The eigen spectrum is

r h i E(k) = (k), (k) = 3 + 2 cos(k α ) + cos(k α ) + cos[k (α α )] . (B.12) ±E E · 1 · 2 · 1 − 2

The eigenvectors corresponding to E+/ are, −     e iφ(k)/2 e iφ(k)/2  −   −  ψk, =   , ψk,+ =   . (B.13) − eiφ(k)/2 eiφ(k)/2 −

127 Now ρ(q) can be written as

X iq r N X iq r X iq r ρ(q) = ρ(r)e− · = ar† are− · + br† bre · V r r N X = [a† a + b† b ]δ . (B.14) V k k0 k k0 k+q+G,k0 k,k0,G where N is the number of electrons in the unit cell (N=2) and V is the area of the unit cell. Now in RPA, the polarization function can be calculated as

2 0 gs V V X ψk0,+ ρ(q) ψk, Π (q, ω) = lim |h | | −i| 2 η 0+ 2 2 k k −(2π) → (2π) (2π) E+( 0) E ( ) ω + iη k,k0 − − − 2 h φ(k0) φ(k) i sin − 16 X 2 = 5 δk0,k+q+G , (B.15) −(2π) (k) + (k0) ω + iη k,k0,G E E −

where

hφ(k0) φ(k)i 1 (k) (k0) + (k) (k0) sin2 − = A A B B . (B.16) 2 2 − 2 (k) (k ) E E 0

B.3 Screening effect from σ electrons

In the tight-binding model, we only include π electrons in the calculations. However, the σ electrons and

π electrons are dynamically correlated with each other. Therefore, it is important to study the effective

screening from σ electrons as a background to the low energy π orbitals.

In general, we can write the Hamiltonian for such a system as

H = H0 + HI + Hext, (B.17)

where H0 is the noninteracting kinetic term for σ and π electrons. HI is the Coulomb interaction term,

1 X X HI = V (q)ˆρπ( q)ˆρπ(q) + V (q)ˆρπ( q)ˆρσ(q) 2 q − q − 1 X + V (q)ˆρσ( q)ˆρσ(q), (B.18) 2 q −

whereρ ˆπ andρ ˆσ are the density operators of the π and σ electrons, V (q) is the Coulomb interaction in

128 Fourier space. Hext is the external potential,

X Hext = U(q) [ˆρσ(q) +ρ ˆπ(q)] (B.19) q

At the RPA level, the equations of motion for the density of π and σ electrons

d ρˆ (q) i h π i = [H, ρˆ (q)] (B.20) dt h π i

d ρˆ (q) i h σ i = [H, ρˆ (q)] (B.21) dt h σ i can be calculated in linear response theory [40] and written in matrix form as

    ρˆσ(q, ω) U(q) Πσ(q, ω)   = M   , (B.22)   (q, ω)   ρˆπ(q, ω) Ππ(q, ω) where   1 V (q)Π (q, ω) V (q)Π (q, ω)  − π σ  M =   (B.23) V (q)Π (q, ω) 1 V (q)Π (q, ω) π − σ and the dielectric function is

(q, ω) = det M = 1 V (q)Π (q, ω) V (q)Π (q, ω) , − π − σ

where Ππ and Πs are the RPA polarization functions of the π- and σ-electrons, respectively. The dielectric function can be equivalently written as

(q, ω) = κ (q) V (q)Π (q, ω), σ − π where we have defined

κ (q) = 1 V (q)Π (q, ω) . (B.24) σ − σ

We have ignored the ω dependence of κσ(q).

129 B.4 Joint density of states

We define the joint density of states to be

1 X jdos(ω) = δ( (k)  (k) ω) , (B.25) N π∗ − π − k

where N is the number of k-points sampled in the first Brillouin zone. In our calculation, we use a smeared

δ-function, a Gaussian function with finite width (we set it to be 0.5eV),

1 x2/2/σ2 δ(x) e− . (B.26) ∼ √2πσ

The summation is evaluated on a 32 32 uniform k-grid. × Now the problem is to identity π bands for graphene and s bands for hydrogen. Because they overlap with

σ orbitals (or Pz orbitals for hydrogen). What we do is essentially to check the wave functions of each bands.

In fact, it is much easier. We use crystal2qmc to generate input. This will output the expansion coefficients of each orbital using atomic orbitals. For π We simply select those orbitals with large Pz coefficients. For hydrogen, the band structure is simple, the s orbitals lies in 1, 2, and 3. The non-s orbitals will have nonzero

Pz atomic orbital coefficients. Removing those orbitals, we can get s and s∗.

B.5 Two dimensional Fourier transformation of the Coulomb

potential

Z Z Z Z 2 1 iq r iqr cos θ ∞ 2π ∞ 2π V (q) = dr e− · = e rdrdθ = 2π J0(qr) = J0(x)dx = . (B.27) r 0 q 0 q

B.6 Downfolding of graphene and hydrogen lattice to Hubbard

models

In this section, we explain the details of constructing low energy effective model of graphene and hydrogen

lattice, using density-matrix based downfolding method. Part of these contents have already been presented

in Chapter. 6.

The detailed formulation is described in Ref. [210]. Here, we downfold the two ab inito systems into the

130 following effective Hubbard model,

ˆ X X H = C + t ci,σ† cj,σ + h.c. + U ni, ni, , (B.28) ↑ ↓ i,j i h i

where we only include π orbitals in the effective Hamiltonian. ci0 s were constructed from Kohn-Sham orbitals. The resulting orbitals are showed in Fig. 6.13,

We know that due to a lack of screening, the Coulomb interaction might be long range, instead of Hubbard

onsite interaction. Nevertheless, one consider the effect of the long range part as a renormalization to the

onsite Coulomb interaction U [209]. We hope that to some extend, a Hubbard model with renormalized U

is still a good description of the system [209, 210].

The general method is that, one can choose a set of wave functions, which corresponds to low energy

excitations of the system (not necessary to be the eigenstate of the system). Here we mainly focus on the

electron-hole type excitations within the π orbitals. Then we can compute the energy expectation value,

and density matrices from first principles. Assuming the underlying model is Hubbard model described in

Eq. (6.32), we have

X σ σ X , E = C + t( ρij + ρji) + U Mii↑;↓ii , (B.29) i,j i h i

where ρ and M are the one-body and two-body reduced density matrices respectively,

σ , ρij = ψ ci,σ† cj,σ ψ ,Mii↑;↓ii = ψ ni, ni, ψ . (B.30) h | | i h | ↑ ↓| i

If we have n wavefunctions, we can form a linear least problem of n 3 from the n constraints of Eq. (B.29). ×     h P σ σ i h P , i E1 1 i,j ρij + ρji i Mii↑;↓ii    1 1     h h i i h i  C    P σ σ P ,    E2  1 i,j ρij + ρji i Mii↑;↓ii    2 2      =  h i   t  . (B.31)             ··· ·········  U  h P σ σ i h P , i  En 1 i,j ρij + ρji i Mii↑;↓ii h i n n

To understand the effect of the σ electrons to the effective model of π electrons, we do two calculations. In the first calculation, all electrons are included; where as in the second calculation, we take out the σ electrons and replace it with a constant charge background. By comparing the two results one can understand the effect of the sigma electrons to the effective model of graphene. Table 4.2 shows the final results. As one

131 Table B.1: Downfolding parameters for graphene with and without σ electrons, and hydrogen. parameters [eV] graphene with σ1 graphene without σ hydrogen t 3.61(1) 2.99(1) 3.73(2) U 7.16(3) 14.8(2) 9.47(5) can clearly tell that, including σ electrons reduced the effective onsite interaction from 14.8(4) eV to 7 eV.

Fig. B.2 shows fitted energy versus the ab initio energy.

2.768 103 0 − × 3800 255 − 1 256 − T=-3.61 0.00 [eV] 3798 − T=-3.73 0.01 [eV] ± T=-2.99 0.01 [eV] 257 2 U=7.16 0.03 [eV] ± − U=9.47 ±0.05 [eV] − ± 3796 U=14.78 0.18 [eV] 258 ± 3 ± − − 259 4 3794 − − 260 5 − − 3792 261 E (Fitted) [eV] E (Fitted) [eV] E (Fitted) [eV] − 6 262 − 3790 − 7 ∆Erms=0.12 [eV] ∆Erms=0.44 [eV] 263 ∆Erms=0.12 [eV] − − 8 3788 264 − 8 7 6 5 4 3 2 1 0 3788 3790 3792 3794 3796 3798 3800 − 264 263 262 261 260 259 258 257 256 255 − − − − − − − − − − − − − − − − − − 2.768 103 E (VMC) [eV] − × E (VMC) [eV] E (VMC) [eV] (a) (b) (c)

Figure B.2: Comparison of ab initio (x-axis) and fitted energies (y-axis) of the 3?3 periodic unit cell of graphene and hydrogen lattice: (a) graphene with σ electrons included; (b) graphene without σ electrons; (c) hydrogen lattice.

B.7 Setup of first principles calculations

DFT (CRYSTAL[203]) QMC (Qwalk[98]) • •

1 – Functional: PBE – Timestep: 0.01 Ha− (DMC)

– k-grid: 16 16 – Trial wavefunction: Single Slater-Jastrow × – System size: 16 16 supercell – System size: 16 16 supercell × ×

– Pseudopotential: BFD TDDFT-RPA (GPAW[204, 205, 202]) • AFQMC (QUEST[206]) – k-grid: 54 54 • × – Timestep: 0.1 – Energy cutoff: 100 eV

– System size: 8 8 supercell – Number of bands included: 60 × B.8 Pseudopotential and basis set

In order for other people to reproduce our results, we herein provide the pseudopotential and basis set used in our calculations in CRYSTAL input format.

132 Hydrogen Oxygen 201 7 206 8 INPUT INPUT 1.00000000 3 1 0 0 0 0 4 . 0 3 1 0 0 0 0 4.47692410 1.00000000 1 8.359738 4.000000 1 2.97636451 4.47692410− 1 4.483619 33.438953− 1 3.01841596 4.32112340 0 3.938313 19.175373 0 1 0 0 − 5.029916 22.551642− 0 0 0 6 1 1 . 0 0 0 7 2 . 0 1 . 0 6.359201 0.004943 0.2051 0.397529 3.546637 0.049579− 0.409924 0.380369 1.493442 0.037176 0.819297 0.180113 0.551948 0.287908 1.637494 0.033512 0.207203 0.009543 3.272791 −0.121499 0.079234 0.770084 6.541187 0.015176− 0 0 6 0 1 . 0 13.073594 0.000705 6.359201 0.016672 0 0 1 0 . 1 . − 3.546637 −0.005774 0.25 1 . 1.493442 −0.227982 0 0 1 0 . 0 1 . 0 0.551948 −0.285652 0.75 1 . 0 0.207203 −1.071579 0 2 6 2 . 0 1 . 0 0.079234 1.423767− 0.234064 0.302667 0 0 6 0 1 . 0 0.468003 0.289868 6.359201 0.018886 0.935757 0.210979 3.546637 −0.058854 1.871016 0.112024 1.493442 −0.556988 3.741035 0.054425 0.551948 −1.084022 7.480076 0.021931 0.207203 2.721525− 0 2 1 0 . 0 1 . 0 0.079234 1.458091 0.75 1 . 0 0 1 0 1 . 0 − 0 2 1 0 . 0 1 . 0 0.102700 1.000000 0.25 1 . 0 0 2 1 0 1 . 0 0 3 1 0 . 0 1 . 0 1.407000 1.000000 0.329486 1.0 0 2 1 0 1 . 0 0 3 1 0 . 0 1 . 0 0.388000 1.000000 1.141611 1.0 0 3 1 0 1 . 0 . 1.057000 1.000000 .

133 Appendix C

Extended Koopmans’ theorem

C.1 Finite size analysis for graphene

To see the finite size effect, we calculate the energy momentum dispersion using EKT for different size of supercell. As we can see from Fig. C.1, up to 4x4, the result converges within stochastic error.

0.0 EXP EKT(1x1) LDA EKT(2x2) 0.2 − GGA EKT(3x3) GW EKT(4x4) 0.4 −

0.6 −

Energy [eV] 0.8 −

1.0 −

1.2 − 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 − − − − − − − 1 k K [A˚ − ] | − |

Figure C.1: Graphene energy momentum dispersion near Dirac point using different first principle methods. Experimental results, LDA and GW results are from [1]. GGA and EKT results are from this work. EKT(1 1), EKT(2 2), EKT(3 3), EKT(4 4) are results corresponding to different sizes of supercell. × × × ×

134 Appendix D

Derivation details related to downfolding

D.1 Two-site extended Hubbard model: effective model for H2 molecule

Let us consider the following a two site model,

ˆ H = t[c1†σc2σ + h.c.] + U(n1 n1 + n2 n2 ) + V n1n2 + Jσ1 σ2 . (D.1) − ↑ ↓ ↑ ↓ · where we have included the hoping term, the onsite and nearest neighbor Coulomb interaction term, and the exchange interaction term. Let us first write the exchange interaction term in ci, using the fact that

x,y,z σˆx,y,z = σαβ c1†αc1β . (D.2)

Therefore,

σ1 σ2 = (c1† c1 + c1† c1 )(c2† c2 + c2† c2 ) (c1† c1 c1† c1 )(c2† c2 c2† c2 ) + (n1 n1 )(n2 n2 ) · ↑ ↓ ↓ ↑ ↑ ↓ ↓ ↑ − ↑ ↓ − ↓ ↑ ↑ ↓ − ↓ ↑ ↑ − ↓ ↑ − ↓

= 2(c1† c2† c1 c2 + c1† c2† c1 c2 ) + n1 n2 + n1 n2 n1 n2 n1 n2 . (D.3) − ↑ ↓ ↓ ↑ ↓ ↑ ↑ ↓ ↑ ↑ ↓ ↓ − ↑ ↓ − ↓ ↑

Let us set up the basis of the Hilbert space. Let me define the vacuum state to be 0, 0 , with all the two | i sites be empty. We regard the following state as

c1† c2† 0, 0 = , , c1† c1† 0, 0 = , 0 = , 0 . (D.4) ↑ ↓| i | ↑ ↓i ↑ ↓| i | ↑↓ i −| ↓↑ i

Let us express the Hamiltonian in terms of these two-body states.

Tˆ , = t( , 0 + 0, ), Tˆ , = t( , 0 + 0, ) , | ↑ ↓i − | ↑↓ i | ↑↓i | ↓ ↑i | ↑↓ i | ↑↓i Tˆ , 0 = t( , , ), Tˆ 0, = t( , , ) . (D.5) | ↑↓ i − | ↑ ↓i − | ↓ ↑i | ↑↓i − | ↑ ↓i − | ↓ ↑i

135 The application of Uˆ on the four states are

Uˆ , = 0, Uˆ , = 0, Uˆ , 0 = U , 0 , Uˆ 0, = U 0, . (D.6) | ↑ ↓i | ↓ ↑i | ↑↓ i | ↑↓ i | ↓↑i | ↓↑i

The application of Jˆ on the four states are

Jˆ , = J , + 2J , , Jˆ , = 2J , J , , | ↑ ↓i − | ↑ ↓i | ↓ ↑i | ↓ ↑i | ↑ ↓i − | ↓ ↑i Jˆ , 0 = 0, Jˆ 0, = 0 . (D.7) | ↑↓ i | ↓↑i

The application of Vˆ on the four states are

Vˆ , = V , , Vˆ , = V , , Vˆ , 0 = 0, Vˆ 0, = 0 . (D.8) | ↑ ↓i | ↑ ↓i | ↓ ↑i | ↓ ↑i | ↑↓ i | ↓↑i

Therefore, one can further define the following states

r r 1 1 S = ( , , ), TS = ( , + , ) , (D.9) | i 2 | ↑ ↓i − | ↓ ↑i | i 2 | ↑ ↓i | ↓ ↑i r r 1 1 D = ( , 0 + 0, ), TD = ( , 0 0, ) . (D.10) | i 2 | ↑↓ i | ↑↓i | i 2 | ↑↓ i − | ↑↓i

Therefore, we have

Jˆ S = 3J S , Jˆ TS = J TS , Jˆ D = Jˆ TD = 0 . (D.11) | i − | i | i | i | i | i therefore

H S = 2t D + (V 3J) S ,H D = 2t S + U D . (D.12) | i − | i − | i | i − | i | i

Therefore, in the sub hilbert space, we have

  V 3J 2t  − −  H :=   . (D.13) 2t U −

Therefore, we get the ground state is

r V + U 3J U V + 3J 2 E = − 4t2 + | − | . (D.14) ± 2 ± 4

136 The eigenvectors corresponding to E is ±

S + λ D 2t Ψ = | i | i, λ = . (D.15) − √1 + λ2 U E − −

The other two eigenstates are

H TS = V + J, H TD = U TD . (D.16) | i | i | i

D.2 Derivation of EKT matrix, and forming linear least problem

Let us consider Hamiltonian with the form of

H = Tˆ + Uˆ + Vˆ + J.ˆ (D.17) where

ˆ X ˆ X ˆ 1 X ˆ 1 X T = tijciσ† cjσ, U = U ni ni , V = Vijninj, J = Jijσi σj . (D.18) ↑ ↓ 2 2 · ij i ij ij

Let us calculate the matrix one by one. In the following, for convenience, we will use Greek letter α, β, ..., σ to denote the spin indices, and i, j, k, l to denotes the site/orbital indices.

D.2.1 Valence band matrix element

The valance band matrix elements are defined as

σ = c [H, c† ] . (D.19) Vmn −h mσ nσ i

One-body term

ˆ P For the one-body term T = ij,σ tijciσ† cjσ, we have

X X X c† [Tˆ , c ] = c† [c† c , c ] = c† t c† , c c = c† c t . (D.20) − mσ nσ − mσ iα jα nσ mσ ij{ iα nσ} jα mσ jσ nj ij,α ij,α j

137 Two-body term

ˆ P α;β For the two-body term V2 = ijkl,αβ Vij;lkciα† cjβ† ckβclα . Let us see some important properties of the two- α;β body tensor Vij;lk. Because of the symmetry, we have

α;β β;α Vij;lk = Vji;kl . (D.21)

For some specific Coulomb terms we have,

1 Uˆ : V α;β = Uδ δ δ (1 δ ) (D.22a) ij;lk 2 ij lk il − αβ 1 Vˆ : V α;β = V δ δ (D.22b) ij;lk 2 ij il jk

V σ α;β = c† [V, c ] = c† [V c† c† c c , c ] Vmn − mσ nσ − mσ ij;lk iα jβ kβ lα nσ α;β = c† V [c† c† , c ]c c − mσ ij;lk iα jβ nσ kβ lα α;β = c† V [c† δ δ δ δ c† ]c c − mσ ij;lk iα jn βσ − in ασ jβ kβ lα α;σ σ;β = Vin;lkciα† cmσ† ckσclα + Vnj;lkcmσ† cjβ† ckβclσ . (D.23)

U terms

For U terms, we have

1 V α;β = Uδ δ δ (1 δ ) . (D.24) ij;lk 2 ij kl il − αβ

Therefore, we have

U U c† [V, c ] = c† c† c c + c† c c c . (D.25) − mσ nσ 2 nσ¯ mσ nσ nσ¯ 2 mσ nσ¯ nσ¯ nσ

Here we have defineσ ¯ as the opposite direction of σ. Finally, we have

Uσ = U c† c† c c . (D.26) Vmn h mσ nσ¯ nσ¯ nσi

We can directly calculate this using the definition of U,

U X X mn↑ = U cm† [ni† ni , cn ] = U cm† [ni† , cn ]ni = U cm† nn cn . (D.27) V − h ↑ ↑ ↓ ↑ i − h ↑ ↑ ↑ ↓i h ↑ ↓ ↑i i,σ i

138 This is consistent with Eq. (D.26).

V terms

For V terms, we have

1 V α;β = V δ δ . (D.28) ij;lk 2 ij il jk

The corresponding term is V

V σ α;σ σ;β = V c† c† c c + V c† c† c c Vmn in;lk iα mσ kσ lα nj;lk mσ jβ kβ lσ 1 1 = V c† c† c c + V c† c† c c 2 in iα mσ nσ iα 2 ni mσ iβ iβ nσ 1 = (V + V )c† c† c c . (D.29) 2 in ni mσ iα iα nσ

Let us directly verify Eq. (D.29),

V 1 X 1 X † † † mn↑ = Vij cm [niσnjσ0 , cn ] = Vij cm cn δinδσ, njσ0 + cm cn δjnδσ0, niσ V −2 h ↑ ↑ i 2 h ↑ ↑ ↑ ↑ ↑ ↑ i ,σ,σ0 1 X = (Vin + Vni) cm† niσcn . (D.30) 2 h ↑ ↑i σ,i=n 6

J terms

The exchange term is defined as

h i ˆ 1 1 J =: Jijσi σj = Jij 2(ci† cj† ci cj + ci† cj† ci cj ) + ni nj + ni nj ni nj ni nj . (D.31) 2 · 2 − ↑ ↓ ↓ ↑ ↓ ↑ ↑ ↓ ↑ ↑ ↓ ↓ − ↑ ↓ − ↓ ↑

Let us first directly compute the corresponding term from the definition of Eq. (D.31). For the first term, V

cm† [ci† cj† ci cj , cn ] = cm† cj† ci cj δin . (D.32) ↑ ↓ ↓ ↑ ↑ − ↑ ↓ ↓ ↑

There second term can be obtain simply by exchange of i and j. The other two terms can be written as

X X Jijcm† [ni nj + ni nj , cn ] = Jijcm† ([ni , cn ]nj + [nj , cn ]ni ) − ↑ ↑ ↓ ↓ ↑ ↑ − ↑ ↑ ↑ ↓ ↑ ↑ ↓ ij ij X = (Jin + Jni)cm† ni cn . (D.33) ↑ ↓ ↑ i

139 Therefore, eventually, the J term will contribute

J X 1 mn↑ = (Jin + Jni) cm† ci† cn ci + (Jin + Jni) cm† (ni + ni )cn . (D.34) V − h ↑ ↓ ↓ ↑i 2 h ↑ ↓ ↑ ↑i i

For spin down channel, one can easy obtained by interchange and . ↑ ↓ Now let us start from Eq. (D.23). We have

n 1 1 o Jˆ = J c† c† c c + n n n n ij − iα jα¯ iα¯ jα 2 iα jα − 2 iα jα¯ n 1 1 o = J δ δ (1 δ ) + δ δ δ (1 δ )δ δ c† c† c c . (D.35) ij − ik jl − αβ 2 αβ il jk − 2 − αβ il jk iα jβ kβ lα

Therefore, we have

n 1 1 o V α;β = J δ δ (1 δ ) + δ δ δ (1 δ )δ δ . (D.36) ij;lk ij − ik jl − αβ 2 αβ il jk − 2 − αβ il jk

Jσ α;σ σ;β = V c† c† c c + V c† c† c c Vmn in;lk iα mσ kσ lα nj;lk mσ jβ kβ lσ n 1 1 o = J c† c† c c + c† c† c c c† c c c in − iσ¯ mσ iσ nσ¯ 2 iσ mσ nσ iσ − 2 iσ¯ mσ nσ iσ¯ n 1 1 o +J c† c† c c + c† c† c c c† c c c nj − mσ jσ¯ nσ¯ jσ 2 mσ jσ jσ nσ − 2 mσ jσ¯ jσ¯ nσ 1 1 = (J + J ) c† c† c c + c† c† c c c† c† c c (D.37) in ni {− mσ iσ¯ nσ¯ iσ 2 mσ iσ iσ nσ − 2 mσ iσ¯ iσ¯ nσ}

D.2.2 Conduction band matrix elements

The conduction band matrix is defined as

σ = c [H,ˆ c† ] . (D.38) Cmn h mσ nσ i

One-body terms

For the one-body term: T = tijciτ† cjτ ,

c [Tˆ , c† ] = t c [c† c , c† ] = t c c† c , c† = t c c† δ δ = t c c† . (D.39) mσ nσ ij mσ iτ jτ nσ ij mσ iτ { jτ nσ} ij mσ iτ jn τσ in mσ iσ

Therefore, for the one-body term we have

c [Tˆ , c† ] = c c† t . (D.40) h mσ nσ i h mσ iσi in

140 Two-body terms

τ;λ For the two-body terms: V = Vij;klciσ† cjλ† clλckσ:

ˆ α;β cmσ[V , cnσ† ] = cmσ[Vij;lkciσ† cjβ† ckβclα, cnσ† ]

α;β = Vij;lkcmσciα† cjβ† [ckβclα, cnσ† ]

α;β = V c c† c† [c δ δ c δ δ ] ij;lk mσ iα jβ kβ ln ασ − lα kn βσ σ;β α;σ = V c c† c† c V c c† c† c . (D.41) ij;nk mσ iσ jβ kβ − ij;ln mσ iα jσ lα

Let us change the two terms into normal orders.

c c† c† c = (δ c† c )c† c mσ iσ jβ kβ mi − iσ mσ jβ kβ

= c† c δ c† (δ δ c† c )c jβ kβ mi − iσ mj σβ − jβ mσ kβ

= c† c δ c† c δ δ c† c† c c ; (D.42a) jβ kβ mi − iσ kβ mj σβ − iσ jβ kβ mσ

c c† c† c = (δ δ c† c )c† c mσ iα jσ lα mi ασ − iα mσ jσ lα

= δ δ c† c c† (δ c† c )c mi ασ jσ lα − iα mj − jσ mσ lα

= δ δ c† c c† c δ + c† c† c c . (D.42b) mi ασ jσ lσ − iα lα mj iα jσ mσ lα

Therefore, we have

σ;β σ;σ σ;β c [Vˆ , c† ] = V c† c V c† c V c† c† c c mσ nσ mj;nk jβ kβ − im;nk iσ kσ − ij;nk iσ jβ kβ mσ σ;σ α;σ α;σ V c† c + V c† c V c† c† c c − mj;ln jσ lσ im;ln iα lα − ij;ln iα jσ mσ lα σ;β σ;σ α;β = 2V c† c 2V c† c 2V c† c† c c . (D.43) mj;nk jβ kβ − im;nk iσ kσ − ij;nk iσ jβ kβ nσ

141