Establishing Quantum Monte Carlo and Hybrid Density Functional Theory As Benchmarking Tools for Complex Solids

Home , Quantum Monte Carlo, Variational Monte Carlo

Establishing Quantum Monte Carlo and Hybrid Density Functional Theory as benchmarking tools for complex solids

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Kevin P. Driver, B.S., M.S. Graduate Program in Physics

The Ohio State University 2011

Dissertation Committee: John W. Wilkins, Advisor

Richard J. Furnstahl

Ciriyam Jayaprakash

Arthur J. Epstein c Copyright by

Kevin P. Driver

2011 Abstract

Quantum mechanics provides an exact description of microscopic matter, but predictions require a solution of the fundamental many-electron Schr¨odingerequation. Since an exact solution of Schr¨odinger’sequation is intractable, several numerical methods have been developed to obtain approximate solutions. Currently, the two most successful methods are density functional theory (DFT) and quantum Monte Carlo (QMC). DFT is an exact theory which, which states that ground-state properties of a material can be obtained based on functionals of charge density alone. QMC is stochastic method which explicitly solves the many-body equation.

In practice, the DFT method has drawbacks due to the fact that the exchange-correlation functional is not known. A large number of approximate exchange-correlation functionals have been produced to accommodate for this deﬁciency. Conceptual systematic improvements known as “Jacob’s Ladder” of functional approximations have been made to the standard local density approximation (LDA) and generalized gradient approximation (GGA).

The traditional functionals have many known failures, such as failing to predict band gaps, silicon defect energies, and silica phase transitions. The newer generation functionals including meta-GGAs and hybrid functionals, such as the screened hybrid, HSE, have been developed to try to improve the ﬂaws of lower-rung functionals. Overall, approximate functionals have generally had much success, but all functionals unpredictably vary in the quality and consistency of their predictions.

Often, a failure of one type of DFT functional can be ﬁxed by simply identifying another

DFT functional that best describes the system under study. Identifying the best functional for the job is a challenging task, particularly if there is no experimental measurement to

ii compare against. Higher accuracy methods, such as QMC, which are vastly more computationally expensive, can be used to benchmark DFT functionals and identify those which work best for a material when experiment is lacking. If no DFT functional can perform adequately, then it is important to show more rigorous methods are capable of handling the task.

QMC is high accuracy alternative to DFT, but QMC is too computationally expensive to replace DFT. Hybrid DFT functionals appear to be a good compromise between QMC and standard DFT. Not many large scale computations have been done to test the feasibility or benchmark capability of either QMC or hybrid DFT for complex materials. This thesis presents three applications expanding the scope of QMC and hybrid DFT to large, scale complex materials. QMC computes accurate formation energies for single-, di-, and tri- silicon-self-interstitials. QMC combined with phonon energies from DFT provide the most accurate equations of state, phase boundaries, and elastic properties available for silica. The

HSE DFT functional is shown to reproduce QMC results for both silicon defects and high pressure silica phases, establishing its benchmark accuracy compared to other functionals.

Standard DFT is still the most efficient and useful for general computation. However, this thesis shows that QMC and hybrid DFT calculations can aid and evaluate shortcomings associated the exchange-correlation potential in DFT by offering a route to benchmark and improve reliability of standard, more efficient DFT predictions.

iii To my family and friends for guidance, help, and love.

iv Acknowledgments

The research and underlying educational enlightenment represented by this thesis are most notably a product of the continuous support and encouragement of my advisor, John

Wilkins. John provided impeccable guidance and maintained a high standard of excellence in developing my scientiﬁc career.

Other than my advisor, a few others deserve specific mention for their critical guidance and support. Richard Hennig was an excellent mentor and source of scientific inspiration through out my entire graduate career. My office mate and group partner, William Parker, provided invaluable amounts of feedback and support throughout my entire graduate career as well. I am also indebted to the help and guidance of Ronald Cohen, whose training made significant portions of this research (silica) possible.

There are many other people have taught me or played some role in my scientiﬁc ed- ucation. I would like to thank Cyrus Umrigar, Burkhard Militzer, Hyoungki Park, Amita

Wadhera, Mike Fellinger, Jeremy Nicklas, Ken Esler, Neil Drummond, Yaojun Du, Jeong- nim Kim, Thomas Lenosky, Shi-Yu Wu, Chakram Jayanthi, David Brown, P. J. Ouseph, and my high school physics teacher – Robert Rollings for general support and advice.

Many departmental staﬀ members oﬀered important assistance me in some manner while carrying out this work. I’d like to thank Trisch Longbrake, Shelly Palmer, Carla Allen, Tim

Randles, Brian Dunlap, and John Heimaster.

I owe much gratitude to my family: Gerald and Patricia and Silvia Driver, Diane and

Gerald Link, Jeremy and Leah Driver, Betty and Tom Wells, Robert and Dorothy Driver, and Irene Muller. Thanks for all of the love and support, and the opportunities provided that made my academic career possible.

v I also want to acknowledge many important friends that have helped me personally and/or academically persevere in somewhat of a chronological order: Roseanne Cheng,

Chuck and Danna Pearsall, Jeﬀ Stevens, Sheldon Bailey, Julia Young, Grayson Williams,

Nick and Barbara Harmon, Jake and Nichole Knepper, Brandon and Ester Parks, Chad and

Nikki Morris, James Morris, Charlie Ruggiero, Greg Mack, Yuhfen Lin, Becky Daskalova,

Kevin Knobbe, Mark and Sara Murphey, Matt Fisher, Yi Yang, Louis Nemzer, Kaden Haz- zard, Shawn Walsh, Justin Link, Chen Zhao, Qiu Weihong, Jia Chen, David Daughton,

Kent Qian, Eric Jurgenson, Daniel Clark, Taeyoung Choi, Kerry Highbarger, Anthony

Link, Emily Sistrunk, Mike Boss, Mike Hinton, Fred Kuehn, Iulian Hetel, Jen White, Va- lerie Bossow, Jim Potashnik, Deniz Duman, Greg Sollenberger, Patrick Smith, Anastasios

Taliotis, Steven Avery, Aaron Sander, Eric Cochran, Hayes Lara, Adam Hauser, Luke

Corwin, Srividya Iyer Biswas, Colin Schisler, Borun Chowdhury, Reni Ayachitula, Neesha

Anderson, Rakesh Tivari, Nicole De Brabandere, Jim Davis, Rob Guidry, Lee Mosbacker,

Don Burdette, Mehul Dixit, Dave Gohlke, Alex Mooney, Greg Vieira, James and Veronica

Stapleton, John Draskovic, Mariko Mizuno, Yuval DaYu, Emily Harkins, Sabine Shaikh,

Angie Detrow, The two Ashers, The HCGs, Meghan Ruck, Chiaki Ishikawa, April Brown,

Kim Pabilona, Eumie Carter, Liesen Parkus, Cassandra Plummer, Nadia Ahmad, Natalie,

Emma Brownlee, Claudia Veltze, Savannah Laurel-Zerr, Tom Steele, Alex Gray, Michelle

Oglesbee, Kimberly Rousseau, Carlos Rubio, JC Polanco and all my friends from La Fogata,

Patrick Roach, Heather Doughty, and many more whose names I’ve forgotten.

I also have much appreciation for several ﬁnancial agencies that supported me and my work. I was supported for two years at The Ohio State physics department as a Fowler fellow and further supported mostly by the DOE. I’d also like to thank the NSF for supporting my stay at the Carnegie Institution of Washington at the Geophysical Laboratory during the summers of 2007 and 2008. This work was also made possible by generous computational resources from OSC, NERSC, NCSA, and CCNI.

vi Vita

April 14, 1980 ...... Born—New Albany, Indiana, USA

2003 ...... B.S., University of Louisville, Louisville, Kentucky 2003-2005 ...... Fowler Fellow, Department of Physics, Ohio State University, Columbus, Ohio 2006 ...... M.S., Ohio State University, Columbus, Ohio 2005-Present ...... Graduate Research Associate, Depart- ment of Physics, Ohio State University, Columbus, Ohio

Publications

K. P. Driver, R. E. Cohen, Zhigang Wu, B. Militzer, P. L´opez R´ıos,M. D. Towler, R. J. Needs, and J. W. Wilkins, Quantum Monte Carlo computations of phase stability, equations of state, and elasticity of high-pressure silica, Proc. Natl. Acad. Sci. USA, 107, 9519 (2010).

R. G. Hennig, A. Wadehra, K. P. Driver, W. D. Parker, C. J. Umrigar, and J. W. Wilkins, Phase transformation in Si from semiconducting diamond to metallic beta-Sn phase in QMC and DFT under hydrostatic and anisotropic stress, Phys. Rev. B, 82, 014101 (2010).

M. Floyd, Y. Zhang, K. P. Driver, Jeﬀ Drucker, P.A. Crozier, and D.J. Smith, Nanometer- scale composition variations in Ge/Si(100) islands, Appl. Phys. Lett. 82, 1473 (2003).

Y. Zhang, M. Floyd, K. P. Driver, Jeﬀ Drucker, P.A. Crozier, and D.J. Smith, Evolution of Ge/Si(100) island morphology at high temperature, Appl. Phys. Lett. 80, 3623 (2002).

P. J. Ouseph, K. P. Driver, J. Conklin, Polarization of Light By Reﬂection and the Brewster Angle, Am. J. Phys. 69, 1166 (2001).

vii Fields of Study

Major Field: Physics

Studies in quantum Monte Carlo claculations of solids: J. W. Wilkins

viii Table of Contents

Page Abstract ...... ii Dedication ...... iv Acknowledgments ...... v Vita ...... vii List of Figures ...... xii List of Tables ...... xiv

Chapters

1 Introduction 1 1.1 Quantum Mechanics Correctly Explains Matter ...... 1 1.2 Modeling Matter with Numerical, Quantum Simulations ...... 2 1.3 Organization and Summary of Thesis Accomplishments ...... 4

2 Methods of Solving the Schr¨odingerEquation 7 2.1 The Many Body Problem ...... 7 2.1.1 The Born Oppenheimer Approximation ...... 8 2.2 Mean-Field-based Ab Initio Methods ...... 9 2.2.1 The Hartree Approximation: No Exchange, Averaged Correlation . . 9 2.2.2 The Hartree-Fock Approximation: Explicit Exchange, Averaged Cor- relation ...... 10 2.3 Density Functional Theory ...... 12 2.3.1 Self-interaction Error in DFT ...... 14 2.4 Many Body Ab Initio Methods ...... 15 2.4.1 Conﬁguration Interaction ...... 15 2.4.2 Quantum Monte Carlo ...... 16

3 Coping with DFT Approximations: Benchmarking with Hybrid functionals and QMC 30 3.1 Introduction: Approximations and Weaknesses of Density Function Theory 30 3.1.1 Exchange-Correlation Approximations: Categorization of functionals in Jacob’s ladder ...... 31 3.1.2 Basis Set Approximations ...... 36 3.2 Benchmarking functionals With Hybrid DFT and Quantum Monte Carlo . 38

ix 4 Results for Silicon Self-Interstitials 40 4.1 Introduction ...... 40 4.2 Calculation Details ...... 47 4.2.1 Results ...... 48 4.2.2 Tests for errors in QMC ...... 58 4.3 Conclusions ...... 68

5 Results for Silica 71 5.1 Introduction ...... 71 5.2 Previous Work and Motivation ...... 74 5.3 Computational Methodology ...... 74 5.3.1 Pseudopotential Generation ...... 74 5.3.2 DFT Calculations ...... 75 5.4 QMC Calculations ...... 78 5.4.1 Wave-function Construction and Optimization ...... 78 5.4.2 DMC Calculations ...... 79 5.5 Results ...... 80 5.5.1 Free Energy ...... 80 5.5.2 Thermal Equations of State and Fit Parameters ...... 82 5.5.3 Phase Stability ...... 88 5.5.4 Thermodynamic Parameters ...... 92 5.5.5 Stishovite Shear Constant ...... 117 5.6 Geophysical Implications ...... 122 5.7 Conclusions ...... 122

6 Hybrid DFT Study of Silica 124 6.1 Introduction ...... 124 6.2 Previous Work ...... 125 6.2.1 Hybrid Calculations of Silica ...... 125 6.2.2 Hybrid B3LYP and PBE0 Calculations of Solids ...... 126 6.2.3 Screened Hybrid (HSE) Calculations of Solids ...... 126 6.3 Computational Methodology ...... 126 6.3.1 CRYSTAL Calculations ...... 127 6.3.2 ABINIT Calculations ...... 132 6.3.3 VASP Calculations ...... 132 6.4 Results ...... 133 6.4.1 Energy Versus Volume ...... 133 6.4.2 Pressure Versus Volume ...... 135 6.4.3 Equilibrium Quartz and Stishovite Volume from Vinet Fits . . . . . 138 6.4.4 Equilibrium Quartz and Stishovite Bulk Moduli from Vinet Fits . . 140 0 6.4.5 Equilibrium Quartz and Stishovite K0 from Vinet Fits ...... 142 6.4.6 Enthalpy Versus Pressure ...... 144 6.4.7 Quartz-Stishovite Transition Pressures ...... 148 6.5 Conclusions ...... 150

7 Conclusions 151 7.1 Summary ...... 151 x 7.2 Future Research ...... 156

Appendices

A Error Propagation in QMC Thermodynamic Parameters 169 A.1 Taylor Expansion Method ...... 169 A.2 Monte Carlo Method ...... 170

B Optimized cc-pVQZ Gaussian Basis Set used for Silica 172

C Details of the Ewald and MPC Interaction in Periodic Calculations 177 C.1 Ewald Interaction ...... 177 C.2 Model Periodic Coulomb (MPC) Interaction ...... 184

D Summary of CODES Used in This Work 186 D.1 ABINIT ...... 186 D.2 Quantum ESPRESSO ...... 186 D.3 VASP ...... 186 D.4 CASINO ...... 187 D.5 CHAMP ...... 187 D.6 OPIUM ...... 187 D.7 CRYSTAL ...... 187 D.8 WIEN2K ...... 187 D.9 ELK ...... 187

E Strong and Weak Scaling in the CASINO QMC Code 188

xi List of Figures

Figure Page

4.1 Images of single Si interstitial defects ...... 42 4.2 Images of Si di-self-interstitial defects ...... 42 4.3 Images of Si tri-self-interstitial defects ...... 43 4.4 QMC and DFT band gaps of Si ...... 44 4.5 QMC and DFT cohesive energy of Si ...... 45 4.6 QMC and DFT diamond to β-tin energy difference in Si ...... 46 4.7 I1 16-atom formation energy ...... 50 4.8 I1 64-atom formation energy ...... 51 4.9 I1 diffusion path ...... 53 4.10 I2 64-atom formation energy ...... 55 4.11 I3 64-atom formation energy ...... 57 4.12 DMC time step convergence for Si ...... 59 4.13 DMC finite size convergence for X defect ...... 61 4.14 DMC formation energy for LDA and GGA pseudopotentials ...... 63 4.15 Convergence of Jastrow polynomial order for Si ...... 65 4.16 QMC pseudopotential dependence ...... 67

5.1 Schematic silica phase diagram ...... 73 5.2 Silica energy vs. volume curves ...... 81 5.3 Silica equations of state ...... 83 5.4 Silica Vinet fit parameter: Zero pressure free energy vs. temperature. . . . 84 5.5 Silica Vinet fit parameter: Zero pressure volume vs. temperature ...... 85 5.6 Silica Vinet fit parameter: Zero pressure bulk modulus vs. temperature . . 86 5.7 Silica Vinet fit parameter: Pressure derivative of the bulk modulus vs. temperature at zero pressure...... 87 5.8 Silica enthalpy curves ...... 90 5.9 Silica phase boundaries ...... 91 5.10 Thermal pressure of silica...... 94 5.11 Changes in thermal pressure in silica...... 96 5.12 Bulk moduli of silica...... 98 5.13 Pressure derivative of the bulk modulus of silica...... 100 5.14 Thermal expansivity of silica...... 102

xii 5.15 Heat capacity of silica...... 104 5.16 Percentage volume differences of silica ...... 106 5.17 Grüneisenratio of silica...... 108 5.18 Volume dependence of the Grüneisenratio of silica...... 110 5.19 Anderson-Grüneisenparameter of silica...... 112 5.20 Sound Velocity and Density profile of Earth...... 114 5.21 Bulk Sound Velocity and Density of Silica...... 116 5.22 Energy vs. b/a strain in stishovite ...... 119 5.23 Stishovite shear constant softening ...... 121

6.1 Gaussian basis set convergence ...... 131 6.2 HSE energy versus volume of quartz and stishovite ...... 134 6.3 DFT pressure versus volume of quartz ...... 136 6.4 DFT pressure versus volume of stishovite ...... 137 6.5 DFT zero pressure volumes of silica...... 139 6.6 DFT Bulk Moduli of silica...... 141 6.7 DFT K0 of silica...... 143 6.8 HSE enthalpy versus pressure of quartz and stishovite ...... 147 6.9 DFT quartz-stishovite transition pressures ...... 149

E.1 CASINO VMC weak scaling ...... 189 E.2 CASINO DMC weak scaling ...... 190 E.3 CASINO DMC strong scaling ...... 191

xiii List of Tables

Table Page

4.1 Separation of bulk and defect e-n Jastrows ...... 68

5.1 Silica thermal equation of state parameters ...... 92

6.1 Quartz DFT equation of state parameters ...... 145 6.2 Stishovite DFT equation of state parameters ...... 146

xiv Chapter 1 Introduction

1.1 Quantum Mechanics Correctly Explains Matter

In the late 17th century, Isaac Newton’s Principia [1] solidified mathematical physics as the precise and formal language to describe the nature of the universe. Mathematical theories, in check with experimental observations, allow the logical connection of one fact to another, giving a comprehensive, disillusioned picture of the universe. Most often, progress in the scientific conception of the universe comes about from the reciprocal relationship of experiment and theory: mathematical predictions inspire new experiments and experiments inspire modifications to mathematical theory. In the case of Newton’s Principia, the mathematical foundation was laid for an entire field of physics known as classical mechanics.

Classical mechanics prevailed as the theory of motion of macroscopic objects for over two centuries when its inadequacies to describe the microscopic world became apparent in the atomic era.

A modern, updated theory, quantum mechanics [2, 3], emerged in the early 20th century

as science progressed into the atomic era with the help of many important scientiﬁc ﬁgures

(Moseley, Thompson, Rutherford, Bohr, Heisenberg, Einstein, Dirac, De Broglie, Millikan,

Stern, Gerlach, Pauli, and Schr¨odinger,to name a few). Quantum theory has prevailed

as a rigorously tested and robust mathematical description of the behavior of microscopic,

atomic matter. There is no doubt that it is a theory that allows scientists to accurately

predict and understand properties of materials.

Quantum mechanics takes into account the wave-particle duality of microscopic matter

1 and interactions of energy and matter. It accurately describes the structure of atoms, bonding of atoms in molecules and solids, behavior of electrons and, in fact, can describe all properties of matter. Electrons are the important particles for binding matter together and their mathematical treatment is at the heart of computations presented in this thesis.

The main issue is not whether quantum theory correctly describes matter, but whether the quantum mathematical equations can be adequately and feasibly solved to successfully predict properties of interest. The main aim of this thesis is to investigate high accuracy techniques of solving the equations of quantum mechanics to predict properties of complex matter.

1.2 Modeling Matter with Numerical, Quantum Simulations

The fundamental equation of matter in quantum mechanics, known as Schr¨odinger’swave equation [4], relates the wave properties (wave function) Ψ of a particle to its energy, E, through the action of a Hamiltonian operator, Hˆ . There are two forms of the equation: a

time-independent form,

Hˆ Ψ = EΨ, (1.1) and a time-dependent form, ∂Ψ Hˆ Ψ = i¯h . (1.2) ∂t In practice, these equations are very diﬃcult to solve, and, in fact, they only have analytic solutions for a single particle that is not interacting with any others. Interacting electrons [5, 6] have a correlation energy because of Coulomb interactions and electrons have a quantum mechanical exchange energy based on Pauli’s exclusion principle, whose role is to help minimize the Coulomb energy. The Coulomb interaction causes Schr¨odinger’s equation to be inseparable and, hence, the wave function cannot be written as an analytically solvable product of independent functions. This fact rules out any simple approach to a highly accurate solution.

The only tractable solution to the problem of solving Schr¨odinger’sequation for real materials is to use sophisticated numerical simulations, often requiring massively parallel

2 supercomputers. A number of numerical techniques have been developed oﬀering various levels of treatment of the troublesome exchange and correlation interactions of electrons.

These are sometimes referred to as electronic structure calculations [5, 6]. This most accurate electronic structure methods are classified as first principles or ab initio, which means they are numerical simulations of Schrödinger’sequation that have no experimental input or adjustable parameters.

However, exact, unapproximated ﬁrst-principles simulations are still too diﬃcult for all but the smallest systems. Although ab initio methods are technically able to exactly compute properties of materials, the computational time required for such a calculation often scales exponentially with system size, taking longer than the lifetime of the researcher.

The calculations are said to be computationally expensive. In general, a method trades oﬀ accuracy for the ability to study larger system sizes. Quickly advancing computer technology allows more expensive calculations, but it is human nature to always seek beyond what is easily done.

Consequently, a highly active area of computational physics research involves developing approximations that speed up ab initio methods, but only negligibly reduce the accuracy

and predictive power. It is the electron interactions in materials that are most computa-

tionally cumbersome and require approximations. One of the most popular and successful

ab initio methods is density functional theory (DFT) [7], which is an exact theory. How-

ever, in practice, the functional describing exchange and correlation must be approximated

and allows DFT computation time to scale with the cube of the number of particles sim-

ulated. DFT is an extremely successful and predictive method for thousands of published

calculations. However, DFT functionals have also been a source of skepticism for DFT,

as they sometimes are unreliable and unexpectedly fail for certain properties or materi-

als. The systematic improvement of functionals is sought in various ways. One type of

functionals, called hybrids, which include exact exchange properties of the electrons show

particular promise for computing properties of materials. Part of this thesis involves testing

and evaluating performance of several exchange-correlation functionals in DFT, including

hybrids.

3 This thesis also focuses on a highly accurate, stochastic ab initio method called quantum

Monte Carlo (QMC) [8]. QMC explicitly computes the exchange and correlation of electrons eﬃciently enough to produce benchmark accuracy for solids [9], but the computational expense of QMC makes it intractable to replace DFT. One of the important applications of QMC is for benchmarking the predictions of density functionals. A particular density functional may fail for a given system, but the failure can be overcome by identifying a density functional that is capable of describing the system of interest. However, if there is no experimental data, a highly accurate and reliable method, such as QMC, must be used to benchmark the functionals.

Part of the aim of this thesis is to investigate whether hybrid DFT functionals can reliably match the accuracy of QMC and also be used as a benchmarking tool for standard functionals. QMC and hybrid DFT are too computationally expensive to replace the computational demand standard DFT fulfills for materials science. QMC calculations are generally heroic computational efforts (millions of CPU hours), hundreds of times more expensive than standard DFT. Hybrid DFT is about thirty times more expensive than standard DFT, capable of computing energies of larger systems. Neither QMC or hybrid calculations are efficient enough to compute atom dynamics in solids, such as forces or phonons. QMC and hybrid DFT have only been used to publish perhaps a couple dozen simple solid calculations, compared to tens of thousands of DFT calculations. The main aim of this thesis is to significantly expand the scope of QMC and hybrid functionals for solids by applying them to large, complex structures and establish them as a benchmarking tools for more efficient DFT exchange-correlation functionals.

1.3 Organization and Summary of Thesis Accomplishments

The research presented in this thesis makes use of enormous amounts of knowledge and tools developed by other researchers in various scientiﬁc communities. Chapter 2-3 introduces the electronic structure methods and concepts that set the foundation for the research in this thesis. The ﬁrst sections of chapter 2 discusses the many-body problem and presents

4 mean-ﬁeld-based Hartree-Fock and density functional approaches to solving the many-body

Schr¨odingerequation. The ﬁnal section introduces fully many-body approaches to solving

Schr¨odinger’sequation, with focus on the quantum Monte Carlo method.

This thesis aims to be critical of density functionals and further strengthen their reliability and accuracy. Therefore, chapter 3 turns to focus on weaknesses of DFT exchange- correlation functionals and develop strategies to cope with their bias and unreliability. The conceptual systematic improvement of “Jacob’s ladder” of functionals is discussed. Particu- lar attention is given to a new type of screened hybrid functional, HSE, which appears to be the most accurate and reliable functional to date. A discussion of basis sets – particularly localized (Gaussian) basis sets – and their convergence is included for their association with hybrid functionals. Part of the original work in this thesis involved signiﬁcant time converging Gaussian basis sets to work with hybrid functional calculations of silica. The ﬁnal section discusses the concept of benchmarking lower-rung functionals on “Jacob’s Ladder” with hybrid functionals and QMC.

The majority of new and original research presented within this thesis is presented in chapters 4-6. These three chapters involve the application and evaluation of DFT and

QMC for complex materials. Chapter 4 examines the performance of selected “Jacob’s ladder” DFT functionals and QMC for computing formation energies of silicon single, di- and tri-self-interstitial defects. Several QMC tests for sources of error are performed to ensure reliable results. Chapter 5 presents QMC calculations of high pressure phases of silica. QMC is combined with DFT phonon computations to provide QMC-based thermal properties of silica. This work provides the best constrained equations of state, phase boundaries, and thermodynamic parameters for silica, and demonstrates the feasibility of computing elastic constants with QMC for the ﬁrst time. Chapter 6 investigates reliability of hybrid functionals for silica. In the spirit of “Jacob’s Ladder,” the performance of various exchange-correlation functionals, basis sets, pseudopotentials, and codes is benchmarked against QMC and experiments to determine which are most accurate.

It is important to note that the QMC and hybrid calculations are heroic in scope and eﬀort compared to standard DFT. These are calculations are only performed when a highly

5 accurate benchmark is needed. QMC calculations are needed for both silicon defects and silica because experiment and other methods are not capable of providing a reliable answer for the properties of interest. Additionally, QMC is used to determine that the hybrid HSE functional is one capable of producing benchmark accuracy results for silicon and silica. The hybrid functional allows a more eﬃcient approach to benchmark standard DFT functionals.

The QMC calculations required roughly 10 million CPU hours in total. Hybrid calculations used roughly 10 thousand CPU hours to compute only a few equations of state. Each project signiﬁcantly expands the scope of QMC and hybrid DFT methods and establishes their usefulness as benchmarking tools for complex solids.

The ﬁnal chapter of the thesis, chapter 7, concisely summarizes the results and conclusions of the thesis, and provides some thoughts on future work. Several appendices follow chapter 7 providing details on error propagation techniques in QMC, optimization of Gaus- sian basis sets, details of ﬁnite size error in QMC, scaling in QMC, and summaries of the codes used.

6 Chapter 2 Methods of Solving the Schrodinger¨ Equation

2.1 The Many Body Problem

The properties of most matter that is of interest to physicists and materials scientists arise from interactions of electrons and nuclei. Using only fundamental particle properties such as charge, Z, and mass, m, materials properties can be determined by solving the many- particle, time-independent Schr¨odingerequation [6, 10] (Equation 1.1),

 N N  ˆ h ˆ ˆ i X 1 2 X ZiZj HΨ(R) T + V Ψ(R) = − ∇r +  Ψ(R) = EΨ(R), (2.1) 2m i |r − r | i=1 i i>j i j which is a 3N-dimensional eigen-problem, where R is the collective coordinate for all N

particles ri,...,rN . Ψ(R) is a square integrable wave-function, which is anti-symmetric under exchange of two electrons, obeying the Pauli exclusion principle. The equation is written in

atomic units (e = me =h ¯ = 4π0 = 1), and Tˆ and Vˆ are the kinetic and Coulomb potential energy, respectively.

The Coulomb potential term forces Schr¨odinger’sequation to be inseparable for more

than one particle. The simplest possible solution technique of separation of variables is

ruled out, which means the form of the wave-function is not a simple product of one-

electron orbitals. This is why most introductory quantum mechanics texts never go beyond

one particle, “Hydrogen-like” problems.

Of course, most materials of interest contain a large number of interacting protons

7 and electrons, which means approximations must be made in order to reduce the complexity allow one to solve for the wave-function and energy. Once the wave-function and energy is known for a system, many properties may be calculated. However, the various approximations made in a particular method have signiﬁcant impact on the accuracy of the predictions.

The following sections discuss three principal approaches to approximating the solution of the Schr¨odingerequation for real materials (i.e. methods for more than just a few electrons): 1) orbital based methods that approximate Ψ(R) as a Slater determinant of single particle orbitals (Hartree-Fock (HF) Theory), 2) density functional theory (DFT), which is based fundamentally on the charge density rather than a many-body wave-function, and 3) the stochastic approaches such as quantum Monte Carlo (QMC).

2.1.1 The Born Oppenheimer Approximation

A common approximation that all methods discussed in this thesis take advantage of is the

Born-Oppenheimer Approximation [11, 6, 10]. In this approximation, for the purpose of constructing the Hamiltonian, the nuclei are held in ﬁxed position in order to separate out the electronic and nuclear degrees of freedom. This approximation is reasonable because the mass of the nuclei are several thousand times larger than the mass of electrons. There- fore, the nuclei have much lower velocities than the electrons and the nuclei are relatively stationary compared to the electrons. The Hamiltonian many-body Hamiltonian reduces to   ˆ X 1 2 X X Zα X 1 1 X ZαZβ H = − ∇ri + + +  , (2.2) 2mi |ri − dα| |ri − rj| 2 |dα − dβ| i i α i>j α>β where the terms for electrons of charge -1 at positions ri and ions of charge Zα at positions dα have been separated. This Hamiltonian is still diﬃcult to solve, with still no analytic solution for more then one electron, but excellent approximations can be made starting with the Born-Oppenheimer Hamiltonian.

8 2.2 Mean-Field-based Ab Initio Methods

2.2.1 The Hartree Approximation: No Exchange, Averaged Correlation

A common approach for an ab initio treatment of the many-body problem is to break down the many-electron Schr¨odingerequation into many simpler one-electron equation [6, 10, 12].

In order to accomplish this feat, the behavior of each electron is described in the net ﬁeld of all other electrons. That is, each electron experiences a mean-ﬁeld potential,

Z 1 U el(r) = −e dr0ρ(r0) , (2.3) |r − r0| where and each one-electron equation will yield a single-electron wave-function, ψi, called an orbital, and an orbital energy. The total electronic charge would be

X 2 ρ(r) = −e |ψi(r)| , (2.4) i where the sum is over all occupied levels. And, the ion potential is

X 1 U ion(r) = −Z , (2.5) r − R R where R is the nuclear position and U = U ion + U el.

Since the electrons are assumed to be independent (non-interacting), the N-electron wave-function can be written as a product of one-electron wave-functions:

Ψ(r1, r2, ..., rN) = ψ1(r1)ψ2(r2)...ψN(rN) (2.6)

Employing the variational principle and minimizing the expectation value of the Hamilto- nian with respect to variations in the wave functions produces a set of one-electron equations, called the Hartree equations:

  Z 1 2 ion 2 X 0 0 2 1 − ∇ ψi(r) + U (r)ψi(r) + e dr ψj(r ) ψi(r) = iψi(r) (2.7) 2  r − r0  j

9 Self-interaction Error Arises the Hartree Method

A subtle, important feature to notice about the Hartree equations is that the electron potential term (Equation 2.3) includes an unphysical repulsive interaction between the electron and itself. This is because each electron interacts with the average potential computed from

2 | ψi | , which includes the average eﬀect of itself. The error in the energy due to the spurious interaction is called the self interaction error. This point is mentioned now because it will be an important source of error later in the discussion of density functional theory.

2.2.2 The Hartree-Fock Approximation: Explicit Exchange, Averaged Correlation

The Hartree equations are a good ﬁrst attempt at solving the many-body Schr¨odinger equation, but inadequately describe a few very important properties of electrons: quantum mechanical indistinguishability of particles and exchange, and explicit, unaveraged Coulomb correlation. Quantum mechanics demands that the wave-function be noncommittal as to which electron is in which state because all electrons are identical. This gives rise to two types of quantum particles: bosons and fermions. Electrons are fermions whose wave- function must be antisymmetric under the interchange of two particles, obeying the Pauli exclusion principle. The Pauli exclusion principle leads to the exchange energy of electrons, which can be thought of as another means of minimizing the Coulomb energy. The term correlation energy refers to the explicit electron-electron Coulomb interaction, which mean-

ﬁeld approaches only compute as the average eﬀect of Coulomb repulsion.

The Hartree-Fock approximation [6, 10, 12] extends the Hartree approximation to incorporate the indistinguishability and exchange properties of electrons, but still keeps the mean-ﬁeld approach to electron correlation in order to use the one-electron equations. In fact, the term correlation energy is usually deﬁned based on the amount of correlation

Hartree-Fock overlooks:

Ecorrelation = Eexact,non−relativistic − EHartree−Fock (2.8)

10 The Pauli exclusion principle requires the wave-function to be antisymmetric under exchange, such that when two electrons are interchanged the wave-function changes sign:

Ψ(r1, r2, ..., ri, rj, ..., rN) = −Ψ(r1, r2, ..., rj, ri, ..., rN) (2.9)

In the Hartree-Fock method, the indistinguishability and exchange properties of electrons are included mathematically by representing the wave-function as a Slater determinant of one-electron orbitals instead of a simple product of orbitals as in the Hartree method. The determinant is a antisymmetric function of all permutations of one-electron wave-functions:

ψ1(r1) ψ2(r1) ··· ψN (r1)

ψ1(r2) ψ2(r2) ··· ψN (r2)

Ψ(r1, r2, ..., rN) = . . . . (2.10) ......

ψ1(rN) ψ2(rN) ··· ψN (rN) The quantum spin variables have been left out for clarity, but they are easily included with the position dependence.

Minimizing the expectation value of Hˆ with respect to variations in the one-electron wave-functions results in the one-electron, Hartree-Fock equations:

1 X Z 1 − ∇2ψ (r)+U ion(r)ψ (r)+U el(r)ψ (r)− δ dr0 ψ∗(r0)ψ∗(r0)ψ∗(r0) = ψ (r), 2 i i i sisj |r − r0| j i j i i j (2.11) where si represents the spin state. The additional term on the left side compared to the Hartree equations (Equation 2.7) is known as the exchange term. The exchange term is only non-zero when considering like spins and causes like-spin electrons to avoid each other.

The exchange term adds considerable complexity to the one-electron equations, making the

Hartree-Fock equations diﬃcult to solve except for special cases.

11 Self-interaction Error Exactly Cancels in Hartree-Fock

Just as in the Hartree Equations, the Hartree-Fock equations have a Hartree potential

(classical Coulomb) term that includes a spurious self interaction of an electron with itself.

However, in the Hartree-Fock equations, the self-interaction energy is exactly cancelled by the Exchange (Fock) term.

2.3 Density Functional Theory

DFT is currently one of the most successful and popular electronic methods available for computing properties of real solids. It allows for a great simpliﬁcation in solving the many- body problem based on functionals of the electron density. The theory, while based on a mean-ﬁeld approach, is formally exact and, as a result, some consider DFT as its method class. The framework consists of two major parts:

The ﬁrst part is a theorem developed by Hohenberg and Kohn [13] which says that the total energy, Etot, of a system is a unique functional of the electron density, n(r).

Furthermore, the functional Etot[n(r)] is minimized for the ground state density, nGS(r). In short, this aﬀords the possibility of calculating electronic properties based on the electron

density (3 spatial variables), instead of the 3N-variable many-body wave function.

The second part of the theory involves the construction and variational minimization

of the total energy functional, Etot[n(r)], with respect to variations in the electron density. This part of the theory, developed by Kohn and Sham [14] states that the many-body

Schrödingerequation can be mapped onto the problem of solving an effective single-particle wave equation with an effective potential, Veff . The first step is to write the total energy functional as

Z Etot[n(r)] = T [n(r)] + Ee−e[n(r)] + Exc[n(r)] + Vext(r)n(r)dr (2.12)

where T is the kinetic energy of a noninteracting system, Ee−e is the electron-electron interaction energy, Exc is the exchange-correlation energy, and Vext represents an external potential including the ions. By minimizing this total energy functional with respect to

12 variations in the electron density, subject to the constraint that the number of electrons are

ﬁxed, an eﬀective one-particle equation is obtained:

¯h2 − ∇2 + V [n(r)] ψ (r) = ψ (r), (2.13) 2m eff i i i where Veff is an eﬀective potential given by

Veff [n(r)] = Vext[n(r)] + Ve−e[n(r)] + Vxc[n(r)], (2.14)

where Ve−e is the Hartree potential,

Z n(r) V (r) = −e dr0, (2.15) e−e r − r0 and δE [n(r)] V [n(r)] = xc . (2.16) xc δ[n(r)]

Equations 2.13 and 2.14, are known as the Kohn-Sham equations. The quantities ψi and

i are auxiliary quantities used to calculate the electron density and total energy, not the wave function and energy of real electrons.

A formally exact expression for the exchange-correlation energy [15], Exc[n(r)], is given by 1 Z Z n¯(r, r0) E = dr dr0n(r) , (2.17) xc 2 |r0 − r| where, if we introduce a coupling constant λ, which varies from 0 (real-interacting system) to 1 (Kohn-Sham noninteracting system), then

Z 1 0 λ 0 0 0 n¯xc(r, r ) = dλnxc(r, r ) = nx(r, r ) +n ¯c(r, r ) (2.18) 0 is the average over the coupling constant λ of the density at r0 of the exchange-correlation hole about an electron at r :

hΨ | nˆ(r)ˆn(r) | Ψ i nλ (r, r0) = α λ − δ(r − r0), (2.19) xc n(r)

0 α=0 0 and nx(r, r ) = nxc (r, r ) is the exchange hole. Here, Ψα is the correlated ground state wave-function for a system with the same spin densities as the real system but with the

13 electron-electron interaction reduced by a factor α. Due to Pauli exclusion and Coulomb repulsion, an exchange-correlation hole forms satisfying the sum rule

Z 0 α 0 dr nxc(r, r ) = −1 (2.20)

The main ﬂaw of DFT is that, while the theory is exact, the form of the exchange- correlation potential is unknown for all but the simplest systems. In practice, approximations are made for the exchange-correlation potential. A large number of approximations have been made with varying, and sometimes inconsistent performance. The functional approximations are discussed more in Chapter 3.

The Kohn-Sham equations are then solved self-consistently. One starts by assuming a charge density n(r), calculates Vxc[n(r)], and then solves Eq. (2.13) for the wave functions,

ψi(r), using a standard band theory technique. From the wave functions obtained, one calculates a new charge density:

occ. X 2 n(r) = |ψi(r)| . (2.21) i This procedure is then repeated until the charge density is converged.

2.3.1 Self-interaction Error in DFT

Similar to Hartree-Fock, DFT contains a Hartree potential term (Equation 2.15) with a spurious self-interaction error. In the formal DFT theroy, an exact exchange-correlation functional potential cancels the self-interaction energy, just as the Fock term cancels the error in Hartree-Fock. However, in all practical calculations, DFT approximates the exchange and correlation with an approximate exchange-correlation functional. The approximate functional does not likely cancel the self-interaction error in the Hartree term, which may introduce a signiﬁcant error in some calculations.

14 2.4 Many Body Ab Initio Methods

Fully many-body methods abandon the mean field approach of reducing the full Schrödinger equation to a set of one-electron equations and compute exchange and correlation explicitly using a many-body wave-function. The following section mentions the Configuration Inter- action approach, which is too expensive for solid calculations, and Quantum Monte Carlo, which is the only many-body method capable of efficiently simulating solids.

2.4.1 Conﬁguration Interaction

The Conﬁguration Interaction (CI) Method [6] is a general technique of going beyond the

Hartree-Fock Approximation. The Hartree-Fock approximation uses a single Slater determinant to represent the many-electron ground state wave-function. In the CI method, the many-electron wave-function is written as a linear combination of many slater determinants representing energetically higher orbitals:

N XCI Φ = CI ΦI , (2.22) I=0 where CI are the CI expansion coefficients and ΦI are the different configurations of orbitals.If there are N electrons and M basis states, then there are

M! N = (2.23) CI N!(M − N)!

conﬁgurations that can be constructed form the orbitals. Setting all CI to 0 except C0 = 1 reduces the CI method to the Hartree-Fock method.

In order to simulate solids, a very large number of determinants is required (perhaps

millions or billions). The computational cost scales exponentially with the number of elec-

trons. Typically the CI method is only useful for 20 electrons or less. Quantum Monte

Carlo is a method which can achieve the same level of accuracy using stochastic methods,

and is eﬃcient enough to study large solids (up to 500 atoms).

15 2.4.2 Quantum Monte Carlo

Unlike Hartree-Fock or DFT, the quantum Monte Carlo (QMC) method is a stochastic method of solving the many-electron Schr¨odinger equation using an explicitly correlated wave-function. The two main method variational Monte Carlo (VMC) and diﬀusion Monte

Carlo (DMC), which are essentially diﬀerent approaches to evaluating quantum mechanical expectation values. This section of the thesis describes the background and use of VMC and

DMC for continuum systems (periodic solids). The main attraction of the QMC methods is that they are accurate, many-body methods with a computational time that scales favorably with the number of particles simulated, making it possible to deal with large, periodic systems (500 atoms). The discussion that follows is based on the works of Foulkes et al. [9] and Needs et al. [16]

Statistical Foundations: Monte Carlo Methods

Monte Carlo is most eﬃcient method for evaluating large dimensional integrals. The method randomly samples points according to some probability distribution of a function to numerically compute its average value. The main advantage of Monte Carlo integration over other forms of integration is that the error in the result is independent of the dimension of the problem

In order to evaluate the integral

Z I = dRg(R), (2.24)

an “importance function,” P (R) is introduced such that

Z I = dRf(R)P (R), (2.25)

where the importance function is a probability density such that P (R) > 0 and R dRP (R) =

1, and f(R) = g(R)/P (R). The mean value theorem from calculus asserts that the exact vaule of the integral can now be evaluated by sampling inﬁnitely many points from P(R)

16 and computing the sample average:

" M # 1 X I = lim f(Rm) . (2.26) M→∞ M m=1 However, Monte Carlo only estimates the integral by averaging over a ﬁnite sample of

points from P(R): M 1 X I ≈ f(R ), (2.27) M M m m=1 which provides a result with a certain statistical conﬁdence error. The variance of the

2 estimate of I is σf /M, which is estimated as

M " M #2 σf 1 X 1 X ≈ f(R ) − f(R ) , (2.28) M M(M − 1) m M n m=1 n=1 such that the statistical standard deviation decreases with the square root of the number of samples: σf I ≈ IM ± √ . (2.29) M The importance of this result is that Monte Carlo error is independent of the dimension

of the problem. Other methods of numerically evaluating integrals, such as quadrature

trapezoid or Simpson’s methods of weighted grids of points have errors which scale with

the dimension of the problem. Since Schr¨odinger’s equation is 3N dimensional, the number

of dimensions can grow quite large, making Monte Carlo integration indispensable.

In order to sample the points of the probability distribution eﬃciently when the number

of dimensions is large, a technique was developed called The Metropolis algorithm [17]. The

Metropolis algorithm uses an accept-reject algorithm to generate the set of sampling points.

Initially, a random sample is made from the probability distribution and then a trial move

is made to a new position. The ratio of the probability density function at the two points

is examined is examine. If the ratio of the new to old sample is greater than one, then the

algorithm accepts the sample into the set of sampling points. If the ratio of new to old

sample is less than one, then the ratio is compared to a random number between zero and

one. If the ratio is greater than the random number, then the algorithm accepts the sample

into the set of sampling points.

17 Variational Monte Carlo

The variational Monte Carlo (VMC) method is the less rigorous of the two QMC methods discussed in this thesis. VMC provides a less expensive estimate of the total energy and is used to optimize the trial wave-function, which diﬀusion Monte Carlo (DMC) uses to project out the true ground state. VMC is essentially the evaluation of the variational principle using Monte Carlo integration and a many-body wave-function, ΨT .

In order for VMC to work, ΨT must be a reasonably good approximation to the ground state. Details of generating a good trial wave-function will be discussed in a later section.

In general,ΨT and ∇ΨT must be continuous when the potential is ﬁnite and the integrals R ∗ R ∗ ˆ R ∗ ˆ ΨT ΨT and ΨT HΨT must exist. It is also convient that ΨT HΨT exist in order to keep the variance of the energy ﬁnite.

The variational theorem of quantum mechanics states that the expectation value of Hˆ

evaluated with any trial wave-function ΨT is an upper bound on the ground-state energy

E0 : R ∗ ˆ ΨT (R)HΨT (R)dR EV = R ∗ (2.30) ΨT (R)ΨT (R)dR In order to evaluate this integral with Monte Carlo methods via the Metropolis algorithm, it is written in terms of a probability density function, p(R), and a local energy, EL(R): Z EV = p(R)EL(R)d(R), (2.31) where 2 ΨT (R) p(R) = R 2 0 0 , (2.32) ΨT (R )d(R ) and −1 ˆ 2 EL(R) = ΨT (R)HΨT (R). (2.33)

The Metropolis algorithm is used generate a set of electron conﬁgurations, sometimes

called walkers, {(Rm : m = 1,M)} from the conﬁguration-space probability density. The

18 local energy is evaluated for each walker and the average energy is computed:

M 1 X E ≈ E (R ), (2.34) V M L m m=1 with a statistical error of

r 1 σ ≈ (hE (R )2i − hE (R )i2). (2.35) VMC M L m L m

Diﬀusion Monte Carlo

Diﬀusion Monte Carlo (DMC) is a stochastic, projector-based method that solves the time-

dependent, many-body Schr¨odingerequation by allowing the wave-function to decay to the

ground state from some initial state in imaginary time. In imaginary time (t → it), the

Schr¨odingerequation becomes

1 −∂ Φ(R, t) = (Hˆ − E )Φ(R, t) = (− ∇2 + V (R) − E )Φ(R, t), (2.36) t T 2 R T where t measures progress in imaginary time, R = (r1, r2, ..., rN ) is a 3N-dimensional vector

(also called a conﬁguration or walker) providing the coordinates of the N electrons, and ET is an energy oﬀset or interaction strength. In order to propagate the walkers in imaginary

time, Equation 2.36 is written in integral form using a Green’s function, G(R ← R0, t) and

time-step, τ : Z Φ(R, t + τ) = G(R ← R0, τ)Φ(R0, t)d(R0), (2.37)

where

0 0 G(R ← R , τ) = hR | exp(−τ(Hˆ − ET )) | R i. (2.38)

It follows, that the Green’s function obeys Equation 2.36,

0 0 −∂tG(R ← R , t) = (Hˆ (R) − ET )G(R ← R , t), (2.39)

satisfying the initial condition G(R ← R0, 0) = δ(R − R0). Using the spectral expansion,

X exp(−τHˆ ) = | Ψii exp(−τEi)hΨi |, (2.40) i

19 the Green’s function can be written in a manner which reveals important physics:

0 X ∗ 0 G(R ← R , τ) = Ψi(R) exp(−τ(Ei − ET ))Ψi (R ) (2.41) i This expression reveals that the critical feature of DMC is that the Green’s function operator exp(−τ(Hˆ − ET )) projects out the lowest energy eigenstate | Ψ0i in the limit of inﬁnite time steps (τ → ∞):

limτ→∞ Φ(R, t) lim hR | exp(−τ(Hˆ − ET )) | Φiniti (2.42) τ→∞ Z 0 0 0 = lim G(R ← R , τ)Φinit(R )dR (2.43) τ→∞ X = lim Ψi(R) exp(−τ(Ei − ET ))hΨi | Φiniti (2.44) τ→∞ i = lim Ψ0(R) exp(−τ(E0 − ET ))hΨ0 | Φiniti (2.45) τ→∞

The last step (the crux of DMC) follows from the fact that Ei > Ei−1 ··· > E2 > E1 > E0,

where Ei are excited states and E0 is the ground state energy. That is, the excited states are all exponentially damped compared to the ground state and will decay to zero in the

limit of inﬁnite time steps.

Unfortunately, the expression for the exact Green’s function solving Equation 2.36 is

not known except for a few special, simplistic cases. An approximate Green’s function

must be constructed. Some important insight can be gain if, for the moment, the potential

term is neglected in Equation 2.36. Neglecting the potential term reduces the imaginary-

time Schrödingerequation to a diffusion equation in the configuration space, and, in fact,

this is where DMC gets its name. On the other hand, if the kinetic term is neglected,

Equation 2.36 reduces to a rate equation. This information suggests that a imaginary-time

evolution can be simulated by subjecting a population of conﬁgurations {Ri} to a random hops to simulate the diﬀusion process and an ability to undergo a birth-death process to

simulate the rate process, which is sometimes called branching.

Let Hˆ = Vˆ +Vˆ , where Tˆ is the N-electron kinetic energy operator and Vˆ is the N-electron

potential energy operator. Then, the Trotter-Suzuki formula [18] for the exponential sum

of operators applied to Equation 2.38leads to an approximate Green’s function for small τ :

20 (R − R0)2 G(R ← R0, τ) ≈ (2πτ)3N/2 exp − × exp −τ V (R) + V (R0) − 2E /2, 2τ T (2.46)

0 0 0 G(R ← R , τ) ≈ G˜D(R ← R , τ)G˜B(R ← R , τ), (2.47)

where the ﬁrst factor,

(R − R0)2 G (R ← R0, τ) = (2πτ)3N/2 exp − , (2.48) D 2τ is the Green’s function for a diﬀusion equation, while the second factor,

0 0 GB(R ← R , τ) = exp −τ V (R) + V (R ) − 2ET /2 , (2.49) is a time-dependent renormalization (re-weighting) of the diﬀusion Green’s function known as the branching-factor, which determines the number of walkers that survive in the birth- death algorithm.

Fermion Sign Problem

There is one ﬂaw in the DMC method unmentioned up to this point. Thus far, the wave- function is assumed to be purely positive. However, the fermion antisymmetry demands the wave-function have negative and positive regions. However, DMC uses the wave-function

(Green’s function) as a probability distribution from which conﬁgurations are sampled. If the wave-function has both positive an negative regions, then it is not possible to interpret it as probability distribution. This is the so called, fermion sign problem.

One solution to the fermion sign problem is the so called, ﬁxed-node approximation.

In the fixed-node approximation, the sign problem is evaded by fixes the nodes (zeros) of the wave-function to that of the initial trial wave-function. This is equivalent to placing an infinite potential barrier on the nodal surface of the trail wave-function, such that any walkers that approach are removed. DMC projects out the ground state consistent with the nodes of the trial (VMC) wave-function. The size of the error depends on how close the

21 nodes of the trail function are to that of the exact ground state. It follows that the DMC energy is always less than or equal to the VMC energy using the same trial wave-function, and DMC energy is always greater than or equal to the exact ground state. Techniques to relax the node positions, called backﬂow, have been developed in an eﬀort to reduce the

ﬁxed-node error in systems [19].

Importance Sampling Transformation

An impotance sampling transformation vastly improves eﬃciency of the DMC method.

The importance sampling function is a mixed distribution of the trial wave-function ΨT and DMC wave-function Φ,

f(R, t) = ΨT (R)Φ(R, t), (2.50) is purely positive if the nodes of both wave-functions are equal. Substituting into Equa- tion 2.36 gives ∂f 1 − = − ∇2 f + ∇ · [v f] + [E − E ] f, (2.51) ∂t 2 R R D L T where

−1 vD(R) = ΨT (R)∇RΨT (R) (2.52)

is a 3-N dimensional drift velocity. The three terms on the right-hand side of Equation 2.51

represent diﬀusion, drift, and branching processes, respectively. The importance sampling

transformation has several important consequences. First, conﬁgurations multiply where

the probability is large. Second, the branching is now controlled more smoothly by the local

energy instead of the potential. Thirdly, the statistical error bar on the energy estimate is

reduced.

Just as the imaginary-time Schr¨odingerequation (Equation 2.36) was written in inte-

gral form, the importance-sampled imaginary-time Schr¨odingerequation may be written in

integral form: Z f(R, t + τ) = G˜(R ← R0, τ)f(R0, t)d(R0), (2.53)

where G˜ is the modiﬁed Green’s function for the importance sampled wave-function, which

22 amounts to a similar expression for the Green’s function as in Equation 2.46, but updated for the importance sampling:

0 0 0 G˜(R ← R , τ) ≈ G˜D(R ← R , τ)G˜B(R ← R , τ), (2.54)

where, again, the ﬁrst factor,

(R − R0 − τv (R0))2 G˜ (R ← R0, τ) = (2πτ)3N/2 exp − D , (2.55) D 2τ is the Green’s function for a diﬀusion equation which now has a new drift term, while the second factor,

0 0 G˜B(R ← R , τ) = exp −τ EL(R) + EL(R ) − 2ET /2 , (2.56) is the branching-factor, where the local energy replaced the potential.

0 0 The Green’s function, G˜D(R ← R , τ), makes each configuration drift a distance τv(R ) and then diffuse by a random distance drawn from Gaussian noise on τ. Each configuration

0 is then copied or deleted according to G˜B(R ← R , τ). The trail wave-function and initial configurations are typically taken from a VMC calculations. The configurations undergo an equilibration period within DMC and then the importance-sampled DMC algorithm generates configurations according to the importance- sampled mixed distribution, f(R) = ΨT (R)φ0(R), where φ0 is the ground state for the P wave-function expanded in eigenstates,φi of the Hamiltonian: Φ(R, t) = i ci(t)φi(t)(R).

The ﬁxed-node DMC energy is evaluated using HΨT = ELΨT , where EL is the local energy:

ˆ R M hφ0 | H | ΨT i f(R)EL(R)dR 1 X E = E = = ≈ E (R ). (2.57) DMC 0 hφ | Ψ i R f(R)dR M L i 0 T i

Trial Wave-functions and Optimization

In principle, the accuracy of VMC depends on the entire trial wave-function, the accuracy of

DMC only depends on the nodes of the trial wave-function. However, in practice, the quality of the trial wave-function is important in both methods: The trial wavefunction introduces

23 importance sampling, controls the statistical eﬃciency, and limits the ﬁnal accuracy of both

VMC and DMC.

VMC and DMC algorithms repeatedly evaluate the trial-wave-function, which demands a form that is compact and can be evaluated rapidly. While most quantum chemistry methods use linear combinations of determinants for the wave-function, they converge slowly due to their diﬃculty in describing cusps associated with two electrons coming in close contact. QMC instead uses a Slater-Jastrow form of the wave-function consisting of a pair of up and down spin determinants multiplied by a Jastrow correlation factor:

↑ ↓ ΨSJ (R) = exp[J(R)] det[ψn(ri )] det[ψn(rj )], (2.58)

↑ where exp[J] is the Jastrow factor and det[ψn(ri )] is the up-spin determinant. The determinant is made up of single particle electron-orbitals, which are most often from high quality, converged DFT calculations.

The Jastrow factor provides explicit correlation dependence in the wavefunction. The

Jastrow is a symmetric function such that the over all wave-function is anti-symmetric. Un- like quantum chemistry methods, the Jastrow ensures the cusp conditions as two electrons approach one another. The form of the Jastrow involves an exponential of polynomials as functions of particle separation. The basic form used for electrons and ions is a sum of an electron-electron term, u, an electron-nucleus term χ, and an electron-electron-nucleus

term, f:

N N N N N X Xions X Xions X J({ri}, {rI}) = u(rij) + χI (riI ) + fI (riI , rjI , rij), (2.59) i>j I=1 i=1 I=1 i>j

where N is the number of electrons, Nions is the number of ions, rij = ri − rj, riI = ri − rI ,

ri is the position of the electrons, and rI is the position of the nucleus I. The functions u, χ, and f are power expansions with a variety of optimizable coeﬃcients. Reasonable choices

for the form of the functions have been developed by Drummond et al. [20]

Each function comprising the Jastrow has a number of parameters which are optimized

through minimizing either the variance of the local energy, the energy itself, or some com-

24 bination. The first step of the optimization procedure [21] is to generate a set of electron configurations distributed according to the square of the trial wave-function. The Jastrow parameters are then passed to an algorithm which adjusts the parameters to minimize the desired quantity. Following the initial optimization, a set of configurations is generated using the optimized wave-function and the minimization is performed again. Such an it- erated procedure can be carried out many times until the VMC energy and error bars are converged. Typically, the biggest changes are in the first two to three optimization steps and these are sufficient to produce a well-optimized Jastrow.

Approximations in Quantum Monte Carlo

All ab inito solutions to Schrödinger’sequation require some form of approximation in order to make computation feasible using a finite amount of computing resource. Approximations are classified into two groups: 1) controlled and 2) uncontrolled. Controlled approximations are those whose errors are explicitly known and can be converged to arbitrarily small values.

Uncontrolled errors are those whose errors are unknown. In QMC, the controlled approximations include statistical error, form of the wave-function in VMC, using a numerical grid for DFT orbitals, DMC time-step size, ﬁnite-size errors, and the walker population size in

DMC. The uncontrolled approximations include the use of a pseudopotential, pseudopotential locality, and ﬁxed-node error in DMC. The following sections brieﬂy discuss these errors.

Controlled Approximations:

Statistical Error

The Monte Carlo algorithm in both VMC and DMC samples the wave-function as a probability distribution a for a ﬁnite amount of Monte Carlo steps. The size of the statistical error (standard deviation) in the ﬁnal result scales inversely with the square root of the number of samples used to do the Monte Carlo evaluation. To reduce the error by two from a given point, the calculation must be run for four times as many Monte Carlo steps.

25 The error can be made as small as one needs for the problem at hand. Typical errors for cohesive or formation energies are around 0.0001 Ha per formula unit. Elastic constants may require 0.00004 Ha per formula unit.

Wave-function Form

The Slater-Jastrow form of the VMCM wave-function described in the previous section is an approximation of the true many-body wave-function. Electron correlation is accounted for through the ﬁxed form of the Jastrow factor, which has several optimizable parameters.

The exchange is accounted for via a single Slater determinant with single-electron orbitals from DFT. In principle, there are better forms of the Jastrow, perhaps with more parameter

ﬂexibility, and more than one determinant can be used for the exchange part, as is popular in quantum chemistry. The ﬁxed wave-function form limits how close VMC can estimate the true ground state energy. In general, the form used in the codes for this thesis are extremely good, and often match the results obtained with DMC.

Numerical orbitals

The Slater determinant part of the wave-function uses single-electron orbitals that are typically produced from a DFT calculation. The orbitals are most often represented in a plane-wave form. However, the repeated evaluation of plane-wave orbitals is expensive in QMC because it requires a sum over all plane-waves in the cell. A significant increase in speed is achieved by re-representing the plane-wave orbitals with a localized polynomial basis that approximates a grid of plane-wave orbitals. The speed up (typically by a factor of the number of particles simulated) comes from the fact that while using a localized basis set, the orbital evaluation has a computational cost independent of the system size. The type of polynomials that work most efficiently are B-splines. The grid spacing is a approximation that must be checked for convergence, but typically the so called, natural grid spacing [22] is well-converged corresponding to about five grid points per angstrom.

26 DMC time-step

The progress of the time-dependent projection algorithm in DMC uses a small, ﬁnite, imaginary time step. The size of the time step aﬀects the accuracy of the total energy. Before a production calculation, the time step must be converged by computing the DMC energy for a range of increasing small time steps until suitable convergence is found. Typically convergence in the energy to chemical accuracy or better is the goal.

Finite-Size Errors

Calculations of solids with periodic boundary conditions give the effect of infinitely repeating the simulation cell in all directions. The finite size of the repeated simulation cell introduces

fictitious errors into the energy in various ways [23]. In fact, there are three types of finite size errors: 1) The error associated with QMC choosing only a single k-point to evaluate single particle orbitals of the wave function, 2) Spurious correlation effects due to replication of electron behavior in mirroring cells, and 3) if the calculation includes a defect, then the defects can interact in mirroring cells.

The single kpoint approximation can be dealt with a number of ways. The first is to use so called, twist-averaging, where the DMC energy is computed at several different k- points and averaged. The k-point error can also simply be estimated from the DFT energy difference between a converged k-point calculation and one with the equivalent number of k-points that correspond to the simulation cell with one k-point in each primitive cell.

Alternatively, increasing the size of the simulation cell eventually makes the k-point error negligible.

The correlation error due to mirroring cells is reduced by modifying the Hamiltonian.

The so called model periodic Coulomb (MPC) interaction [24] alters the Ewald interaction such that the pure Coulomb interaction is restored within the simulation cell. Alternatively, a structure factor method is used to provide a correction for both potential and kinetic energies [25]. A third method [26], which uses the difference in energy of a DFT calculation of a finite-sized and infinite-sized model of the exchange-correlation functional corrects the

27 ﬁnite size error of the total energy.

The defect interaction from periodic cells is estimated either by studying DFT calculations of a range of cells or performing DMC for a range of cell sizes. The error can be made arbitrarily small with a large enough simulation cell size. Often, the defect interaction error will cancel when diﬀerences in energies are taken.

Walker Population Size

DMC uses a statistical form of the many-body wave-function, which is represented by a

finite number of electron configurations. The number of configurations naturally fluctuate in the birth-death algorithm for DMC. The total number of walkers must be managed such that the population of walkers does not diverge or vanish. The control of the population introduces a bias in the energy, which is made small by using a large number (hundreds to thousands) of configurations.

Uncontrolled Approximations:

Pseudopotential

QMC is too computationally expensive to simulate all electrons in atoms fora solid. Since valence electrons are the most important for determining bonding properties, a so called pseudopotential is used to replace the core electrons with an eﬀective potential. Pseu- dopotentials are generally produced from solving the inverted Schr¨odingerequation for the potential of atoms. Various types of pseudopotentials are possible, including Hartree-Fock,

LDA, GGA, and others. Typical tests for systems studied in this thesis do not show variation in DMC energies outside of statistical error.

Pseudopotential Locality

The DMC algorithm never produces an analytic wave-function. Instead, a statistical representation of the wave-function is made up of a distribution of walkers. Therefore, an issue with the non-local channels in pseudopotentials arises in DMC when a wave-function

28 is needed to evaluate the integrals projecting out the nonlocal angular momentum components of the wave-function. The nonlocal propogator matirx elements can cause the branching Green’s function to change sign, creating yet another sign problem. A reasonable solution [27] is to use the trial wave-function to evaluate the nonlocal components. How- ever, there is an error in doing so, since the trial wave-function is not necessarily accurate.

Casula [28] developed a lattice regularized DMC technique to include nonlocal potentials whose lowest energy is an upper bound of the true ground state energy. The lattice method appears to work well for complex solid calculations.

Fixed-node Error

The Fixed-Node error was discussion in the previous section outlining the DMC method.

The nodes of the DMC wave-function are ﬁxed to those of the trial wave-function. The size of the error depends on how accurate the trial nodes are with respect to the true ground state wave-function. The size of the error is typically unknown, but backﬂow [19] techniques attempt to estimate the size of the error.

29 Chapter 3 Coping with DFT Approximations: Benchmarking with Hybrid functionals and QMC

3.1 Introduction: Approximations and Weaknesses of Den- sity Function Theory

Density functional theory (DFT) is currently the standard model for computing materials properties and has been successful in thousands of studies for a wide range of materials. As described in Chapter 2, DFT is an exact theory, which states that ground-state properties of a material can be obtained from functionals of the charge density alone. In practice, exchange-correlation functionals describing many-body electron interactions must be approximated. A early, common choice still in use is the local density approximation

(LDA). Alternate exchange-correlation functionals, such as generalized gradient approximations (GGAs) and hybrids are also useful, but there is no usable functional that can provide exact results. This fact had lead to a weakness in the predictive power of DFT.

Various DFT functionals produce a variety of results for reasons that often unclear or depend subtle physics. DFT functionals are often expected to only have problems with materials exhibiting exotic electronic structures, such as highly correlated materials. However,

DFT has notoriously failed to compute band gaps for any material, for example. Initially the quantum chemistry would not use DFT because the LDA function failed for atomization

30 energies. Material properties are often dependent on the functional, which has generated a source of skepticism in DFT, especially where there is not reliable experimental data to compare against.

An important example for this thesis is that of DFT functionals failing compute properties of silica. Silica has simple, closed shell, covalent/ionic bonding [29] and should be ideal for DFT. However, while LDA provides good results for properties of individual silica polymorphs, it incorrectly predicts stishovite as the stable ground-state structure rather than quartz. The GGA improves the energy diﬀerence between quartz and stishovite [30, 31], giving the correct ground-state structure, and this discovery was one of the results that popularized the GGA. However, almost all other properties of silica, such as structures, equations of state, and elastic constants are often worse within the GGA [30, 31] and other alternate approximations. Indeed, a drawback of DFT is that there exists no method to estimate the size of functional bias or to know which functional will succeed at describing a given property.

The following sections review exchange-correlation approximations and typical basis sets used with them. A conceptual hierarchy of approximations referred to as “Jacob’s

Ladder” of approximations adds various improvements to the functional approximations.

Functionals, at the highest rungs, called hybrids, should be most accurate, but also most computationally expensive. One means of improving the reliability of the more eﬃcient functionals is to benchmark their predictions with the hybrids, or even QMC. Hybrids or

QMC can help identify the best functional for a given system, reducing computational time and improving the quality of the predictions. One of the goals of this thesis is to test the ability of hybrids and QMC to benchmark lower-rung DFT functionals for complex solids.

3.1.1 Exchange-Correlation Approximations: Categorization of functionals in Jacob’s ladder

This section discusses a hierarchy of density functional approximations for the exchange- correlation energy, Exc [15]. The hierarchy is described by a ladder of approximations for

31 the exchange-correlation energy as a function of the electron density:

Z Exc[n↑, n↓] = drn(r)xc([n↑, n↓]; r), (3.1)

where the integrand nxc is an exchange-correlation energy density, and xc is an exchange- correlation energy per electron.

The ladder is referred to as Jacob’s Ladder for a biblical analogy aiming to reach the goal of chemical accuracy (1 kcal/mol = 0.434 eV). At the lowest rung of the ladder, the energy is simply determined by the local density, n↑(r), n↓(r). The second level incorporates density gradients ∇n↑(r) and ∇n↓(r). Higher rungs incorporate increasingly complex features constructed from the density or the Kohn-Sham orbitals around the volume element dr. Each new level makes it possible to satisfy additional exact or nearly exact formal properties of

Exc[n↑, n↓]. Higher level rungs are typically more accurate and computationally demanding the lower rungs. Although, the increase in accuracy is not always true in practice.

Local Density Approximation

The base approximation of Jacob’s ladder is the local density approximation (LDA).:

Z LDA unif Exc [n↑, n↓] = drn(r)xc (n↑(r), n↓(r)), (3.2)

uniform where xc (n↑, n↓) is the exchange-correlation energy per particle of an electron gas with unif uniform spin densities n↑ and n↓. xc [n] is the exchange and correlation energy per particle in the uniform electron gas known from QMC calculations [32]. The exchange contribution

is from the Xα functional energy with α = 2/3. LDA has has surprising accuracy for a large

range of materials, but also some notable failures.

Generalized Gradient Approximation

The generalized gradient approximation (GGA),

Z GGA GGA Exc [n↑, n↓] = drn(r)xc (n↑(r), n↓(r), ∇n↑(r), ∇n↓(r)), (3.3)

32 initially oﬀered great improvements over LDA for atomization energies and became a standard method in chemistry. The leading gradient correction for exchange and correlation is second order for slowly varying densities. Initially, the PW91 [33] GGA functional used analytic expansions to second order. However, PW91 was found to violate exact properties of the exchange-correlation holes, which spurred the development of a numerically-deﬁned

GGA parametrized to satisfy exact hole constraints, called PBE [34]. GGA is a semi-local functional of density since it requires the density in an inﬁnitesimal neighborhood around r.

Meta-Generalized Gradient Approximation

The meta-GGA (MGGA) functional adds an additional dependence on the Kohn-Sham kinetic energy densities, τσ(r), beyond GGA. The kinetic energy densities are implicit functionals of the spin densities n↑(r) and n↓(r). The MGGA exchange-correlation energy is given by Z MGGA MGGA Exc [n↑, n↓] = drnxc (n↑, n↓, ∇n↑, ∇n↓, τ↑, τ↓). (3.4)

The meta-GGA is the highest-rung which avoids full computational expense of non-locality.

Meta-GGA is fully nonlocal in density, but a semi-local functional of orbitals. The MGGA

exchange-correlation energy has a fourth order gradient expansion. MGGA generally im-

proves atomization energies, lattice constants, and surface energies over GGA. However,

bond lengths can be worsened, especially for hydrogen bonds.

Exact Exchange: Hybrid Functionals

Hybrid Functionals are those which incorporate exact exchange by combining Hartree-Fock

and the density functional treatments of exchange. Correlation eﬀects are still treated only

within the density-functional scheme. Hybrid functionals are the only functionals on the

ladder which are fully nonlocal in the density and orbitals. That is, exact exchange depends

on the density and orbitals at points r0 around r. The exact DFT expression for exchange

33 energy is Z Ex = drn(r)x(r), (3.5) where Z 0 2 1 X |ρσ(r, r )| n(r) (r) = − dr0 , (3.6) x 2 |r − r0| σ and

0 ∗ 0 ρsigma(r, r ) = σαθ(µ − ασ)ψασ(r)ψασ(r ) (3.7) is from the Kohn-Sham one particle density matrix, where α is the orbital and σ is the spin [15]. The exchange functional is an implicit density functional because it is written in terms of the Kohn-Sham orbitals. Hybrid functionals have historically provided some of the most accurate energies and structures relative to lower-rung functionals [7].

The construction of Hybrid density functionals was ﬁrst by Becke [35, 36]. In Becke’s approach, the Hamiltonian operator is thought of as one which can be tuned with a coupling constant, λ, from one that represents a fully interacting system to one that represents a non- interaction Kohn-Sham system constructed such that both systems have the same density

(See reference [6] for a clear discussion of DFT and hybrid methods):

Hˆλ = Tˆ0 + λVˆee + Vˆλ + VˆC + Vˆext, (3.8)

where Tˆ0 is the non-interacting kinetic energy operator, VC is the Hartree Coulomb potential

(i.e. includes average e-e correlations), Vext is the external (nuclei) potential. Vee is a true many-body operator containing both electron exchange-correlation and the missing kinetic energy of the interacting system, while Vxc is a local one-particle exchange-correlation potential of the ﬁctitious, non-interacting Kohn-Sham system.

Becke showed that the lower, non-interacting limit of the associated coupling-constant integral for the density (Equation 2.18), sometimes called the adiabatic connection, must mix some exact exchange into Exc. In terms of the potential, the adiabatic connection is written as Z 1 hΨ1 | Vˆee | Ψ1i = hψλ | Vˆee | Ψλidλ, (3.9) 0

34 where the left-hand side is the expectation value for the true interacting system and the right-hand side is an integral over a whole class of matrix elements for varying strength of e-e interactions.

A key idea of the hybrid method is to approximate the right-hand side of Equation 3.9.

This is done by first breaking up the potential: Vêe = Vˆx + Vˆc. Then, for correlation effects, hybrids use the standard DFT approach, but for exchange effects the integral is approx-

imated. The approximation is based on a crude average between Hartree-Fock exchange

and LDA exchange. More ﬂexibility can be included by using GGA exchange-correlation

approximations. A common combination motivated by Becke, known as B3LYP, is given

by:

B3LY P LDA HF LDA GGA LDA GGA LDA Exc = Exc + a0(Ex − Ex ) + ax(Ex − Ex ) + ac(Ec − Ec ), (3.10)

where a0 = 0.20, ax = 0.72, and ac = 0.81 are three empirical parameters determined in order to reproduce energies set of molecules as accurately as possible; the GGA exchange

is that of Becke [37] and the GGA correlation is that of Lee, Yang, and Parr (LYP) [38],

and the LDA correlation is the VWN approximation [39] A number of other possible hybrid

functionals exist. Another popular hybrid that performs well is the PBE0 functional:

PBE0 HF PBE PBE Exc = aEx + (1 − a)Ex + Ec , (3.11) where HF is Hartree-Fock PBE is the DFT functional [34].

Hybrid Screen Exchange functionals (HSE)

The advantage of hybrid functionals is that they tend to signiﬁcantly improve the quality of

DFT predictions. The disadvantage is they they are much more computationally expensive due to the fully non-local nature of the exchange. In order to speed up the exact exchange computation, Hyde, Scuseria and Ernzerhof [40, 41, 42, 43] have exploited the fact that the range of the exchange interaction decays exponentially in insulators, and algebraically in metals. A new, approximate hybrid functional, called HSE, applies a screened Coulomb potential to the exchange interaction in order to screen the long-range part of the HF

35 exchange:

HSE HF,SR P BE,SR P BE,LR PBE Exc = aEx + (1 − a)Ex + Ex + Ec , (3.12)

HF,SR P BE,SR P BE,LR where Ex is the short-range Hartree-Fock exchange, Ex and Ex are the short-range and long-range components of PBE exchange, and a = 0.25 is the Hartree-Fock mixing parameter. The short-range and long-range parts are determined by splitting the

Coulomb operator into short-range and long-range parts:

1 erfc(ωr) erf(ωr) = + , (3.13) r r r

where the left term is short-range and the right term is long range, and erf and erfc and

the error and complementary error functions, respectively, and ω = 0.11Bohr−1 [44] is the

screening parameter based on molecular basis tests.

3.1.2 Basis Set Approximations

There are essentially two types of basis sets: Extended and Localized. Most solid-state codes

use plane-waves (extended basis set). Quantum Chemistry codes tend to use localized basis

sets (better for molecules and clusters). Localized, usually Gaussian basis are a zoo sets

must be converged.

Extended Basis Sets

In order to make ab initio calculations possible for realistic systems, the single-particle

orbitals used in DFT and QMC are expanded in a set of pre-deﬁned basis functions [6].

The most convenient basis functions to use for periodic solids are plane waves:

X ψn(r) = cnk exp(ik · r), (3.14) k

where cnk are the coeﬃcients, and the wave vectors, k, go over the reciprocal lattice vectors

of the super lattice, with k less than the plane wave cutoﬀ, kmax. The plane waves are completely delocalized and not ascribed to individual atoms.

Plane waves have two advantages: 1) Matrix elements involving plane waves are com-

36 puted very eﬃciently using fast-Fourier-transform techniques and 2) the size of the basis set can easily be converged by simply increasing the maximum value of the wave vector.

Localized (Gaussian) Basis Sets

Many quantum chemistry codes, which focus on atomic and molecular calculations rather than periodic solids use a basis that consists of functions localized on speciﬁc atoms. Many codes available to perform hybrid DFT calculations of solids used localized basis sets and, thus, it is important to discuss localized basis sets here.

The localized basis sets [6] come from a picture based on constructing molecular orbitals from atomic orbitals. Naturally, a set of basis functions consisting of atomic orbitals is useful. One possibility is the radial and spherical harmonics from the one electron Schr¨odinger euqation:

χ(r) = Rnl(r)Ylm(θ, φ), (3.15)

where χ belongs to a given atom.

So called Slater-type orbitals choose the radial part as calculated for hydrogen-like

atoms, but these are complicated to evaluate. A more practical basis function is simply

to use Gaussian-type orbitals (GTOs). GTOs are deﬁned as

α(2n+1)/4 χ(r) = χ (r) = 2n+1 rn−1 exp(−αr2Y (θ, φ)) (3.16) R,α,n,l,m [(2n − 1)!!]1/2(2π)1/4 lm

An enormous amount of modified Gaussians basis set types can be constructed. A minimum basis set consists of one basis function for each inner-shell and valence shell. A double-zeta basis set replaces each function in a minimal basis set by two functions that differ in orbital exponents, called zeta. Split valence basis sets use two functions for valence shells but one function for inner shells. Polarization functions are 3d functions which are added to account for distortion of the atomic orbitals in molecule formation. Additionally, basis sets are contracted in order reduce the number of variational coefficients determined to optimize the basis set. Basis sets are optimized within the Hartree-Fock-Roothaan calculations. The

37 contracted function form looks like

X χR,k,n,l,m(r) = ukiχR,αi,n,l,m(r), (3.17) i

where the uki are fixed constants. Only the coefficients to the contracted functionsχ ˜ are optimized and k distinguishes different contracted functions. The corresponding notation is

something like [X,Y,Z], where X, Y, and Z are numbers that give the number of contracted

functions (eg. 421 implies 4 s-type orbitals contracted, 2 p-type orbitals contracted, and 1

d-type orbital.

Unlike a plane-wave basis, where the basis set is easily converged by adjusting the

maximum wave vector, Gaussian basis sets are notoriously diﬃcult to converge, especially

for solids. Diﬀuse exponents which are important for long range interaction in solids often

cause numerical instabilities. The exponents must be linearly optimized and sometimes

removed by hand in order to study a speciﬁc system with a particular basis set. For any

given system, the size of the basis set must be converged. Generally, it is a good idea to

converge to a value that can be checked against in a plane wave code. Unfortunately, one

may need to try several increasingly large basis sets, all of which will have exponents that

need to be optimized for the system being studied. The optimization is a time consuming

process, but perhaps worth it for eﬃcient hybrid calculations of solids.

3.2 Benchmarking functionals With Hybrid DFT and Quan- tum Monte Carlo

Progress in computational materials science requires accurate and reliable methods capa-

ble of studying large, complex materials. The lower-rung DFT functionals (LDA, GGA,

mGGA) provide the highest eﬃciency to accuracy ratio for studying large systems. How-

ever, the lower-rung functionals are not always reliable enough to fully trust their predic-

tions. More computationally expensive benchmark accuracy methods test the accuracy of

lower-rung functionals. Benchmark methods can help identify the mores accurate lower-

rung functional for a given material or property, such that large calculations can be done

38 eﬃciently.

QMC is the most obvious choice to benchmark DFT functionals. QMC is the only many body method that explicitly computes both exchange and correlation to a high level of accuracy and is able to simulate sizable, solid systems (up to hundreds of atoms). However,

QMC is very expensive and doing more than a few calculations or calculations for a very large system isn’t always possible with ﬁnite computer resources.

Hybrid functionals [42, 41, 43, 34, 45, 46, 47, 48, 49, 50, 51, 52, 53] offer a possible more efficient alternative benchmark method to QMC. In fact, one of the aims of this thesis is to test the ability of hybrid functionals to benchmark complex solids. Hybrid methods use exact (HF) exchange and DFT correlation, which tends to significantly improve their predictive power over lower-rung DFT functionals. In addition, recent developments of the screened hybrid functionals (HSE) have made hybrid functional calculations more efficient.

HSE calculations are roughly 30 times more expensive than LDA or GGA, while QMC is

100-1000 times more expensive than LDA and GGA. HSE has already shown great predictive power for band gaps and solid properties in small, simple systems [54, 55, 56, 44, 42, 41,

57, 40, 43, 58, 59, 60]. In the chapters that follow, the benchmark ability hybrids are tested for complex solids: silicon defects and high pressure silica phases.

39 Chapter 4 Results for Silicon Self-Interstitials

4.1 Introduction

Silicon is one of the most important materials in the semiconductor and microelectronics industry. Due to silicon’s electronic properties, abundance, and cheap cost, silicon wafers are the basis of all electronics. Fabrication of computer chips and integrated circuits requires doping silicon wafers to make certain regions n-type or p-type. For example, boron is commonly used to make p-type regions in silicon. During ion-implantation of dopants, a large number of silicon self-interstitial defects are formed as the silicon lattice is damaged by the dopant ions [61, 62]. The silicon interstitials may be made up of a single atom, two atoms, or three or more. The interstitials may be charged, but in the work shown here all defects are assumed neutral. During annealing, which is used to repair the damaged lattice after ion-implantation, the interstitials condense to form larger defects, such as the {311}

planar defect. Evidence suggests that the planar defects limit the performance of electronic

devices. For example, the planar defects facilitate boron transient enhanced diﬀusion, an

undesirable broadening of the boron doped concentration proﬁles. This eﬀect limits how

small devices can be made.

Figures 4.1, 4.2,and 4.3 show ball-stick images of the most stable defects as determined

by molecular dynamics simulations [63, 64]. The three lowest energy single-interstitial

defects are named Split-<110> (X), Hexagonal (H), and Tetrahedral (T), named after their

40 geometry. The di- and tri-interstitials are simply name alphabetically based on stability ordering found by Richie et al. [63].

Direct detection of small interstitials and measurement of their properties is not currently possible [61, 62], though their presence may be inferred indirectly. Numerical simulations oﬀer an aid to experiments to help determine their formation and diﬀusion properties.

Indeed, the technological importance of silicon defects and diﬀusion have motivated a number of theoretical studies [65, 66, 63, 67, 68, 69, 70, 71]. However, even among the theoretical work, including various treatments of exchange-correlation in DFT, and QMC, there is con- fusion. For single self-interstitials, GGA predicts formation of the defects by roughly 0.5 eV higher than LDA, and QMC predicts formation energies to be 1-1.5 eV larger than GGA.

The activation energy of self-diﬀusion depends on the formation energy plus the migration energy. For a simulation of cell size of N atoms, the formation energy for a single interstitial is computed as follows (di- and tri- are computed with a similar expression):

N + 1 E = E(defect) − E . (4.1) f N bulk The current QMC results indicate that DFT in the standard LDA and GGA approximations are not accurate for silicon interstitial defects. QMC is expected to be highly accurate and reliable for silicon calculations, based on band gap [72] (Figure 4.4), cohesive energy

(Figure 4.5), and high pressure phase transition studies [73] (Figure 4.6). The ﬁrst aim of this work is to do an independent check of previous QMC calculations [71] and examine possible sources of error. The second aim of this work is to look beyond single interstitials, and study di- and tri-self-interstitials in silicon with QMC and DFT. The following sections present calculation details and results of this work. This silicon work was produced 6 years ago. Further investigations for single interstitials have been done since this work was produced [74, 75].

41 (a) Bulk (b) Split-<110>(X) (c) Hexagonal (H) (d) Tetrahedral (T)

Figure 4.1: Single interstitial defects in silicon

(a) I2a (b) I2b

Figure 4.2: Double interstitial defects in silicon

42 (a) I3a (b) I3b (c) I3c

Figure 4.3: Triple interstitial defects in silicon

43 Figure 4.4: QMC and DFT band gaps of Si. Results follow the trend of Jacob’s ladder of functionals, where LDA is least accurate and HSE agrees best with QMC and experiment. The black and white diagonal box on the QMC result respresents plus and minus sigma about the mean.

44 Figure 4.5: QMC and DFT cohesive energy of Si. LDA overestimates the energy and GGA improves DFT agreement with QMC and experiment. The black and white diagonal box on the QMC result respresents plus and minus sigma about the mean.

45 Figure 4.6: QMC and DFT diamond to β-tin energy diﬀerence in Si. Results nearly follow the trend of Jacob’s ladder of functionals, where LDA is least accurate and HSE agrees best with QMC and experiment. GGA performs slightly better than mGGA. The black and white diagonal box on the QMC result respresents plus and minus sigma about the mean.

46 4.2 Calculation Details

DFT computations are performed within the CPW2000 DFT code [76], which was the only plane-wave pseudopotential DFT interfaced to work with the QMC code we used for this project, CHAMP [77]. Calculations use a Ceperely-Alder LDA, norm-conserving pseudopotential. All silicon calculations use a converged plane-wave energy cutoﬀ of 16 Ha and Monkhorst-Pack k-point meshes such that total energy converged to tenths of milli-

Ha/SiO2 accuracy. Silicon QMC calculations were done for cell sizes of 8-, 16-, 32-, and 64-atoms, requiring 5×5×5, 7×7×7, 7×7×7, and 3×3×3 k-point mesh sizes, respectively.

All meshes were shifted to the L-point from the origin by (0.0, 0.5, 0.5), corresponding to

the reduced coordinates in the coordinate system deﬁning the k-point lattice. The defect

structures studied are the most stable candidates as predicted by tight-binding MD [63].

They are further optimized in VASP [78] at the experimental lattice constant of silicon

(5.432 A˚) until all forces on the atoms are smaller than 10−4 Ha/Bohr.

QMC calculations are performed in the CHAMP [77] code using a Slater-Jastrow type

of wave-function, where the Slater determinant is made of single particle orbitals from a

converged DFT-LDA calculation. The same LDA pseudopotential as used in DFT is used

in QMC calculations. Calculations were repeated with GGA to check for functional bias

and results (shown below) indicated none. The Jastrow used in for this project describes

correlations by including only two-body terms: electron-electron and electron-nuclear with

a total of 12 parameters. The parameters in the Jastrow are optimized by minimizing

the VMC energy [79, 80] over thousands of electron conﬁgurations. Several iterations of

computing the VMC energy and re-optimizing the parameters are done until the VMC

energy changes are typically inside of two sigma statistical error.

DMC calculations use the optimized trial wave functions to compute highly accurate

energies. A typical silicon DMC calculation uses 1000 electron conﬁgurations, a time step

of 0.1 Ha−1, and several thousand Monte Carlo steps. In order for QMC calculations of

solids to be feasible, a number of approximations [9, 16] are usually implemented that

can be classiﬁed into controlled and uncontrolled approximations, which are discussed in

47 detail in Chapter 2. The controlled approximations for silica include statistical Monte Carlo error, numerical grid interpolation (∼5 points/A)˚ of the DFT orbitals [22], ﬁnite system size, the number of conﬁgurations used, and the DMC time step. All of the controlled approximations are converged to within at least 1 milli-Ha/SiO2. Finite size errors are reduced to less then chemical accuracy by simulating cell sizes up to 64 atoms. Additional

finite size errors are corrected due to insufficient k-point sampling based on a converged k-point DFT calculation. The uncontrolled approximations include the pseudopotential, nonlocal evaluation of the pseudopotential, and the fixed node approximation. These errors are difficult to estimate and generally assumed to be small [28, 81, 19].

4.2.1 Results

Figure 4.7 shows QMC and DFT formation energies for 16-atom bulk and 16(+1)-atom self-interstitial defects, X, H, and T, while Figure 4.8 shows the QMC and DFT formation energy for 64-atom bulk and 64(+1)-atom self-interstitial defects, X, H, and T. Results display the expectation of Perdew’s Jacob’s ladder, with LDA, GGA, mGGA, and HSE progressively agreeing better with QMC. In both simulation cell sizes, LDA and GGA calculations generally agree with QMC ordering, but lie 1-1.75 eV below QMC formation energies. HSE closely agrees with QMC compared to LDA, GGA, and meta-GGA (mGGA) functionals for 16 atom simulation cell calculations.

QMC results agree well with QMC results of Leung et al. [71]. Leung et al. studied

16-atom and 54-atom cells, while work presented here studied 16-atom and larger 64-atom

cells. Leung et al. predict X and H are degenerate within statistical error in both 16- and

54-atom cells and T lies 0.4-0.5 eV higher in energy, but is within two-sigma statistical

error. Work presented here predicts X and T are degenerate in the 16-atom cell and T lies

about 0.25 eV higher. Finite size errors are expected to be large for the 16-atom cell and,

thus, convergence is not expected. In the 64-atom cell, results agree with Leung etal.. Data

presented here goes beyond Leung et.al. by studying the larger 64-atom cell and obtaining

QMC statistical error that is roughly a factor of 10 smaller. Later in this chapter, several additional sources or error are also investigated to check results more carefully beyond the

48 work of Leung etal.. The results of Leung et.al. are signiﬁcantly improved by this work, but the conclusion remains the same: X and H are degenerate, T is most unstable, GGA predicts energies roughly 1 eV below QMC, and LDA predicts energies roughly 1.5 eV below

QMC.

49 Figure 4.7: Formation energy of the three lowest-energy single self-interstitials in silicon (X, H, and T) in a 16-atom cell. The black and white diagonal box on the QMC results respresents plus and minus sigma about the mean. QMC predicts the H defect formation energy lies 0.2 eV higher than the X and T defects, whose energies agree within two-sigma. HSE agrees well with QMC compared to other DFT functional types, lying only about 0.25 eV below QMC, but predicts T to lie highest. LDA and GGA results predict a similar energy ordering as QMC, but lie 1-1.75 eV lower. The mGGA predicts T lies highest and lies about 1 eV lower than QMC.

50 Figure 4.8: Formation energy of the three lowest-energy single self-interstitials in silicon (X, H, and T) in a 64-atom cell. The black and white diagonal box on the QMC results respresents plus and minus sigma about the mean. QMC predicts the formation energy of T is 0.6 eV higher than degenerate X and H defects. LDA and GGA predict similar energy ordering, but lie 1-1.75 eV lower than QMC.

51 Figure 4.9 shows the QMC and DFT energy barriers (migration energies) for the self- diﬀusion path from the X to H to T and back to X. The barriers are the energy required for a diﬀusive hope between X, H, and T. The calculations are for 64-atom simulation cells.

X-H and T-X labels correspond to saddle point structures determined from Nudged Elastic

Band (NEB) calculations of diﬀusion from X to H and T to X, respectively. The QMC

X-to-H diﬀusion barrier is 355(100) meV and the X-to-T barrier is about 720(100) meV.

There is essentially no barrier for H-to-T diﬀusion. The X-to-H barrier is similar in QMC and GGA, while the X-to-T barrier is about 400 meV larger in QMC. Previous LDA and

GGA estimates of the migration energy gave 100-300 meV [70], which is similar to GGA results in Figure 4.9.

52 Figure 4.9: Single interstitial diﬀusion path in 64 atom cell. QMC(DMC) benchmarks GGA-NEB calculations. The QMC error bar respresents plus and minus sigma about the mean value. The lowest barrier from X to H is similar in QMC and DFT. The T defect formation energy and barrier are larger in QMC.

53 The experimental estimates of the diﬀusion activation energy (formation + migration) for defects in silicon are near 4.7-4.9 eV [82, 83]. Using the estimates of migration energy computed above (about 280 meV for GGA and 355 meV for QMC for the lowest path), indicates that GGA (3.87 + 0.280 = 4.15 eV) underestimates the experimental activation energy range by about 0.6 eV and the lowest QMC estimate (4.9(1) + 0.3(1) = 5.2(1) eV) is about 0.3 eV above the experimental range. QMC provides a signiﬁcant improvement over GGA. Some of the QMC error could be due to using the DFT(GGA) geometry for the defects.

Figure 4.10 shows QMC and DFT formation energies for the two lowest energy di-self-

a b interstitial defects, I2 and I2 in a 64 atom cell. Results again display the expected functional a b Jacob’s ladder trend. QMC predicts the I2 defect lies 1.2 eV lower in energy than I2. LDA and GGA predict a similar energy ordering, but LDA lies up to 3 eV below QMC and GGA lies up to 2 eV below QMC.

54 Figure 4.10: Formation energy of the two lowest-energy di-self-interstitials in silicon (I2a and I2b) in a 64-atom cell. The black and white diagonal box on the QMC results respresents a plus and minus sigma about the mean. QMC predicts the I2 defect lies 1.2 eV lower in b energy than I2. LDA and GGA predict a similar energy ordering, but LDA lies up to 3 eV below QMC and GGA lies up to 2 eV below QMC.

55 Figure 4.11 shows QMC and DFT formation energies for the three most stable tri-self-

a b c interstitial defects, I3 , I3, and I3 in a 64-atom cell. Results again display the Jacob’s ladder a c b trend. QMC predicts the stability ordering is I3 < I3 < I3. GGA predicts the stability b a c a b c a c ordering I3 < I3 < I3, while LDA predicts I3 < I3 < I3. For the I3 and I3 defects, GGA a c closely agrees with QMC. The improved GGA results may be because the I3 and I3 defects b are less distorted (smaller coordination number) than the I3 defect. LDA energies lie up to 3.5 eV below QMC and GGA lies up to 2 eV below QMC.

56 a Figure 4.11: Formation energy of the three lowest-energy tri-self-interstitials in silicon (I3 , b c I3, and I3) in a 64-atom cell. The black and white diagonal box on the QMC results respresents plus and minus sigma about the mean.

57 Physical Explanation for Results

4.2.2 Tests for errors in QMC

Part of the aim of this work is to do an independent check of previous QMC calculations [71] and examine possible sources of error. Sources of error include DMC time step convergence,

finite size convergence, dependence on exchange-correlation functional, Jastrow polynomial order, pseudopotential choice, or allowing independent Jastrow correlations for the interstitial atom. Ultimately, results here do not find any significant unexpected error and, thus, improve the confidence in the Leung etal. results.

Figure 4.12 shows the convergence of the DMC time step τ. The time step is converged

within chemical accuracy by 0.1 Ha−1.

58 Figure 4.12: Convergence of the DMC time step for Si. The QMC error bars represent plus and minus sigma about the mean value. The energy diﬀerence on the vertical axis is with respect to a very small time step energy that has been set to zero. The time step, τ in units of Ha−1 is converge within one-sigma statistical accuracy by 0.1.

59 Figure 4.13 shows the ﬁnite size convergence of the simulation cell for LDA, GGA, VMC, and DMC calculations of the X interstitial. LDA and GGA calculations are converged by

16-atom simulation cell sizes. VMC and DMC converge by the 64-atom simulation cells size, when neighboring 54 and 64 atom cells agree within one-sigma statistical error. It is also interesting to note the VMC calculations are in agreement with much more computationally expensive DMC in this case.

60 Figure 4.13: Convergence of DMC ﬁnite size error Si X defect. The QMC error bars represent plus and minus sigma about the mean value.

61 Figure 4.14 shows a check for DMC energy dependence on choice of functional used to produce orbitals in DFT. Two identical QMC calculations are performed except for the exchange correlation functional used to produce the orbitals. In one calculation the orbitals are produced with LDA and GGA for the other calculation. An LDA pseudopotential is used in both sets of calculations. Since both sets of calculations agree within one-sigma, there is negligible dependence on functional choice.

62 Figure 4.14: DMC formation energy using LDA and GGA orbitals. The black and white diagonal box on the QMC results respresents plus and minus sigma about the mean. Re- sults agree within one-sigma statistical error, indicating negligible dependence on functional choice.

63 Figure4.15 shows convergence of the Jastrow electron-nuclear and electron-electron polynomial expansion order. Formation energy of the X defect in a 16-atom cell is converged by 5th order polynomials.

64 Figure 4.15: VMC convergence of X-defect formation energy versus Jastrow polynomial order in 16-atom Si. The QMC error bars represent plus and minus sigma about the mean value. The notation MN0 indicates M order electron-nuclear polynomial and N order electron-electron polynomial, and no electron-electron-nuclear polynomial. Formation energy is converge for 5th order polynomials.

65 Figure 4.16 shows tests of the eﬀect of using a LDA versus Hartree-Fock pseudopotential in VMC and DMC as a function of simulation cell size. By a cell size of 32 atoms, the X defect VMC and DMC formation energy using both LDA and Hartree-Fock pseudopotentials agrees within one sigma. Results indicate choice of pseudopotential does not change the results.

66 Figure 4.16: DMC and VMC ﬁnite size convergence of X-defect formation energy with LDA and Hartree-Fock pseudopotentials. The QMC error bars represent plus and minus sigma about the mean value. By 32 atoms, when ﬁnite-size error is small, it is clear that both types of pseudopotentials produce the same result.

67 Table 4.1: VMC and DMC calculations of Si single self-interstitial formation energies. One set of calculations uses the same e-n Jastrow for bulk and the defect atom. A second set set of calculations uses independent Jastrow for bulk and defect e-n Jastrows. Results agree within one-sigma error, indicating that a single Jastrow is suﬃcient. All atoms have same e-n Jastrow VMC X H T VMC 5.44(16) 6.22(15) 4.90(15) DMC 4.60(7) 5.10(8) 4.15(8) Defect and bulk atoms have independent e-n Jastrows VMC 5.55(14) 5.72(14) 4.56(15) DMC 4.60(7) 4.99(8) 4.19(6)

Table 4.1 shows results of calculations comparing the eﬀects of using independent electron-nuclear Jastrows for the defect and bulk atoms. Typically, one electron-nuclear

Jastrow with a single set of parameters is used to model correlations for both bulk and defect atoms. However, one may ask if the correlations are significantly different for the defect atom, then independent electron-nuclear Jastrows are needed with independently optimized parameters. Results show that using independent Jastrows does not make a detectable difference in the VMC or DMC formation energy.

All of the above test for sources of error increase the confidence of our Si interstitial results. However, it is important to note that other possible sources of error not been studied here, such as pseudopotential locality approximation and fixed node error, may also affect results. Further work studying the effect of such sources of error are included in the work of Parker [75].

4.3 Conclusions

This chapter presents the most accurate results available for QMC and DFT computations of silicon self-interstitial defect formation energies. For single interstitials, formation energies and self-diﬀusion barriers of the three most stable defects were computed. QMC (DMC) provides a benchmark for various DFT functionals, which follow the expected trend of

Jacob’s ladder: LDA in least agreement with QMC, improved by GGA, mGGA, and HSE

68 in best agreement with QMC. LDA underestimates single interstitial formation energy by roughly 2 eV, which GGA underestimates the formation energy by about 1.5 eV. The best

QMC results predict the X and H defects are degenerate and more stable than T by about

0.6 eV. Additionally, the lowest path migration energies for GGA and QMC are estimated to be 280 meV and 355 meV, respectively, for single interstitials. The single interstitial activation energies (formation + migration) are predicted to be 4.15 eV and 5.2(1) eV in

GGA and QMC, respectively. QMC agrees best with the experimental activation energy range of 4.7-4.9 eV The di- and tri-interstitials also display the Jacob’s ladder trend, but

a a LDA and GGA energies lie 2-3.5 eV below QMC. QMC predicts I2 and I3 are the most stable of the di- and tri-interstitials.

The QMC calculations indicate that DFT is not satisfactory for studying self-interstitial diﬀusion in silicon. It could be that more accurate experiments are needed. The experiments are challenging and cannot easily diﬀerentiate between interstitials and vacancies, for example. However, there is also reason to suspect DFT functionals to be inadequate for silicon defects. First, the self-interaction error could potentially be large in these systems.

In addition, there are a wide range of coordination numbers from 3 (X) to 4 (T) to 6 (H), compared to the bulk coordination number of 4. The charge in the bulk structure is likely much more uniform than in the defect structures. LDA is based on the uniform electron gas, and GGA is based on slowly varying gradients in the density. Therefore, GGA is expected to estimate the energy difference between different structures with very different interatomic bonding better than LDA. However, apparently the bulk-defect density difference is still too stark for GGA to predict the correct energy difference. QMC explicitly computes exchange and correlation, providing the best estimate of energy differences.

Various tests check of previous QMC calculations of single Interstitials [71] and examine possible sources of error. Sources of error checked include DMC time step convergence, ﬁnite size convergence, dependence on exchange-correlation functional, Jastrow polynomial order, pseudopotential choice, or allowing independent Jastrow correlations for the interstitial atom. Of all possible sources of QMC error checked, none aﬀected results presented outside of a one-sigma error bar. Further tests of Jastrow optimization, pseudopotential locality

69 error, ﬁxed-node error are needed [75].

70 Chapter 5 Results for Silica

Silica (SiO2) is an abundant component of the Earth whose crystalline polymorphs play key roles in its structure and dynamics. Experiments are often too difficult to probe extreme conditions that calculations can easily probe. However, DFT calculations may unexpected fail for silica due to bias of the exchange correlation functional choice. This chapter describes calculations of highly accurate ground state QMC plus phonons within the quasiharmonic approximation of density functional perturbation theory to obtain benchmark thermal pressure and equations of state of silica phases up to Earth’s core-mantle boundary [84]. The chapter starts with an introduction to the significance of silica and goes over previous theoretical work and challenges. In the final sections, computational details and results are discussed. The QMC Results provide the best constrained equations of state and phase boundaries available for silica. QMC indicates a transition to the most dense α-PbO2 structure above the core-insulating D00 layer, but the absence of a seismic signature suggests

the transition does not contribute signiﬁcantly to global seismic discontinuities in the lower

mantle. However, the transition could still provide seismic signals from deeply subducted

oceanic crust. Computations also identify the feasibility of QMC to ﬁnd an accurate shear

elastic constants for stishovite and its geophysically important softening with pressure.

5.1 Introduction

Silica is one of the most widely studied materials across the ﬁelds of materials science,

physics, and geology. It plays important roles in many applications, including ceramics,

71 electronics, and glass production. As the simplest of the silicates, silica is also one of the most ubiquitous geophysically important minerals. It can exist as a free phase in some portions of the Earth’s mantle. In order to better understand geophysical roles silica plays in Earth, much focus is placed on improving knowledge of fundamental silica properties.

Studying structural and chemical properties [29] oﬀers insight into the bonding and electronic structure of silica and provides a realistic testbed for theoretical method development.

Furthermore, studies of free silica under compression [85, 86, 87, 88, 89, 90] reveal a rich variety of structures and properties, which are prototypical for the behavior of Earth minerals from the surface through the crust and mantle. However, the abundance of free silica phases and their role in the structure and dynamics of deep Earth is still unknown.

Free silica phases may form in the Earth as part of subducted slabs [91] or due to chemical reactions with molten iron [92]. Determination of the phase stability ﬁelds and thermodynamic equations of state are crucial to understand the role of silica in Earth. The ambient phase, quartz, is a fourfold coordinated, hexagonal structure with nine atoms in the primitive cell [85]. Compression experiments reveal a number of denser phases. The mineral coesite, also fourfold coordinated, is stable from 2–7.5 GPa, but is not studied here due to its large, complex structure, which is a 24 atom monoclinic cell [86]. Further compression transforms coesite to a much denser, sixfold coordinated phase called stishovite, stable up to pressures near 50 GPa. Stishovite has a tetragonal primitive cell with six atoms [87]. In addition to the coesite-stishovite transition, quartz metastably transforms to stishovite at a slightly lower pressure of about 6 GPa. Near 50 GPa, stishovite undergoes a ferroelastic transition to a CaCl2-structured polymorph via instability in an elastic shear constant [88, 93, 94, 95, 96, 97, 98, 99]. This transformation is second order and displacive, where motion of oxygen atoms under stress reduces the symmetry from tetrahedral to orthorhombic. Experiments [89, 90, 100] and computations [101, 102, 103] have reported a further transition of the CaCl2-structure to an α-PbO2-structured polymorph at pressures near the base of the mantle. Figure 5.1 shows a schematic version of the silica pressure- temperature phase diagram. Note the pressure scale is no linear, and the dashed phase boundaries indicate they are not well known.

72 Figure 5.1: Schematic version of the silica phase diagram.

73 5.2 Previous Work and Motivation

The importance of silica as a prototype and potentially key member among lower mantle minerals has prompted a number of theoretical studies [93, 96, 98, 99, 101, 102, 103, 31,

30, 104] to investigate high pressure behavior of silica. Density functional theory (DFT) successfully predicts many qualitative features of the phase stability [101, 102, 103, 31,

30], structural [31], and elastic [93, 96, 98, 99, 104] properties of silica, but it fails in fundamental ways, such as in predicting the correct structure at ambient conditions and/or accurate elastic stiﬀness [31, 30]. Work presented here instead uses the quantum Monte

Carlo (QMC) method [9, 16] to compute silica equations of state, phase stability, and elasticity, documenting improved accuracy and reliability over DFT. This work signiﬁcantly expands the scope of QMC by studying the complex phase transitions in minerals away from the cubic oxides [105, 106]. Furthermore, the QMC results have geophysical implications for the role of silica in the lower mantle. Though QMC ﬁnds the CaCl2-α-PbO2 transition is not associated with any global seismic discontinuity, such as D00, the transition should be

detectable in deeply subducted oceanic crust.

5.3 Computational Methodology

This section discusses the general computational methods and choices made for all silica

calculations. The ﬁrst section describes pseudopotential generation choices for Si and O

used for all silica calculations. The second section discusses types of silica calculations using

DFT: geometry optimization, wave-function generation, and phonon calculations. The last

section addresses QMC calculations including wave-function optimization and VMC/DMC

choices.

5.3.1 Pseudopotential Generation

In order to improve computational eﬃciency, pseudopotentials replace core electrons of the

atoms with an eﬀective potential. The Opium code [107] produces optimized nonlocal,

norm-conserving pseudopotentials for Si and O. Both pseudopotentials are generated using

74 the appropriate exchange-correlation functional (LDA, PBE, WC) for DFT calculations.

All QMC calculations use pseudopotentials generated with the WC functional. In all cases, the silicon potential has a Ne core with equivalent 3s, 3p, and 3d cutoﬀs of 1.80 a.u. The oxygen potential has a He core with 2s, 2p, and 3d cutoﬀs of 1.45, 1.55, and 1.40 a.u., respectively.

5.3.2 DFT Calculations

All DFT computations are performed within the ABINIT package [108]. Silica calculations use a converged plane-wave energy cutoﬀ of 100 Ha and Monkhorst-Pack k-point meshes such that total energy converged to tenths of milli-Ha/SiO2 accuracy. Converged silica calculations require k-point mesh sizes of 4×4×4, 4×4×6, and 4×4×4 for quartz,

stishovite/CaCl2, and α-PbO2, respectively, and all meshes were shifted from the origin by (0.5, 0.5, 0.5), corresponding to the reduced coordinates in the coordinate system deﬁning

the k-point lattice.

Geometry Optimization

The ABINIT code allows various types of structural optimizations that are useful for phase

stability and elasticity calculations. This work utilizes several diﬀerent types of optimiza-

tions: 1) optimization of forces on the ions only, 2) simultaneous optimization of ions and

cell shape at a ﬁxed volume, and 3) full optimization of ions, cell shape, and volume. Ion-

only optimization is used in shear constant calculations of stishovite after an initial full

optimization of the cell. For equations of state of all silica phases, total energies are com-

puted for six or seven cell volumes ranging from roughly 10% expansion to 30% compression

about the fully-optimized equilibrium volume. Constant volume optimization of compressed

and expanded cell geometries relaxes forces on the atoms to less than 10−4 Ha/Bohr.

Wave-function Generation

All QMC calculations start with a trial wave-function that is partially made up of the

Slater determinant of single electron orbitals from corresponding DFT calculations. The

75 DFT wave-function is produced from a ﬁxed geometry calculation after all variables are optimized to produce a converged charge density. The converged charge density is then used in a non-self-consistent calculation to output the orbitals.

Phonon Free Energy Calculations

There are two commonly used free energies in thermodynamic calculations: 1) Helmholtz and 2) Gibbs free energies. The Helmholtz free energy is used for the derivation of most thermodynamic quantities, where volume, V, and temperature, T, are the independent variables. Gibbs free energy is important for equilibrium studies for determining phase boundaries, where the convenient independent variables are pressure, P, and T.

The Helmholtz free energy [109] is deﬁned as,

F (V,T ) = Ustatic(V,T ) − TSvib(V,T ), (5.1)

where Ustatic is the static internal energy of the crystal lattice and TSvib is the vibrational

(phonon) contribution of the thermal atomic motion to the free energy and Svib is the vibrational entropy. The Gibbs free, G = F + PV energy is constructed from Helmholtz

energy. Therefore, this section focuses on Helmholtz free energy, while Gibbs free energy

will be discussed in more detail in the phase stability results section below.

Before any thermodynamic properties may be computed from the Helmholtz free en-

ergy, one must decide how to treat the temperature and volume dependence of the phonon

frequencies, ωi of the lattice. Statistical mechanics allows a system’s vibrational quantum mechanical energy levels to completely determine the vibrational Helmholtz free energy

(F = Ustatic + Fvib) via a partition function, Z:

Fvib = −kT lnZ, (5.2)

where Z is a sum over all quantum energy levels given by,

X −i Z = exp ( ), (5.3) kT i

1 3 where k is the Boltzmann constant, and the i are the microstate energies: 2 ¯hωi, 2 ¯hωi, 76 5 2 ¯hωi, ect. Therefore, for each mode, there are many energy levels

all modes 1 ¯hωi X s¯hωi Z = exp ( ) exp (− ), (5.4) i 2 kT kT s=0 exp (− 1 ¯hωi ) = 2 kT , (5.5) −¯hωi 1 − exp ( kT ) which when combined with Equation 5.2 gives

1 −¯hω F = ¯hω + kT ln(1 − exp ( i )). (5.6) vib,i 2 i kT

The total expression for the Helmholtz free energy (F = Ustatic + Fvib) in the harmonic state approximation is then

all modes all modes X 1 X −¯hωi F = U + ¯hω + kT ln(1 − exp ( )), (5.7) static 2 i kT i=1 i=1 where the second term is the zero-temperature quantum vibrational energy and the third term is the thermal vibrational energy.

In the pure harmonic approximation, one assumes all the ωi’s are constant. This choice causes F to be independent of V, so that all volume derivatives are automatically zero. This is clearly disastrous for any thermodynamic properties computed with a volume derivative of F, such as thermal expansivity. In addition, all thermodynamic quantities will not have a volume dependence, as they should with temperature.

A successful alternative is the so-called quasiharmonic approximation (QHA) [109]. In the QHA, the phonons frequencies are assumed to depend on volume, but not temperature

(ω = ω(V )). This allows all thermodynamic properties to depend on both T and V using the harmonic expression in Equation 5.7, eﬀectively giving rise to low order anharmonicity terms. The phonon frequencies don’t directly depend on T, but the harmonic sum does.

In general, QHA is valid for low temperatures and becomes less valid towards the melting temperature of materials, where thermal atomic motion is least harmonic-like.

The main novelty of the work presented in this chapter is that QMC (not DFT) is used to compute the internal energy of the static lattice, Ustatic, while DFT linear response

77 calculations in the QHA provide the much smaller vibrational energy contribution, TSvib. DFT is a ground-state (zero temperature) method used to compute phonons for each volume point in the equation of state, corresponding to phonon frequencies that depend on volume, but not temperature. As a side note, there is also generally an electronic contribution to the entropy due to thermal excitations in materials with small band gaps or metals. Since silica is a large band gap insulator, electronic entropy may be ignored for temperature ranges considered.

The ABINIT code produces phonon free energies by modeled lattice dynamics using the linear response method [110] within the QHA. Phonon free energies for silica were computed over a large range of temperatures for each cell volume and added them to ground-state energies in order to obtain equations of state at various temperatures. Phonon energies were computed up to the melting temperature in steps of 5 K in order to compute thermodynamic properties. Converged silica calculations require a plane-wave cutoﬀ energy of 40 Hartree with matching 4×4×4 q-point and k-point meshes.

5.4 QMC Calculations

The CASINO code [21, 16] facilitates computation of various types of QMC calculations.

The QMC calculations for silica are composed of three major steps: (i) DFT calculation producing a relaxed crystal geometry and single particle orbitals, (ii) construction of a trial wave function and optimization within VMC, and (iii) a DMC calculation to determine the ground-state wave-function accurately. Production of a DFT Slater determinant of orbitals was discussed above.

5.4.1 Wave-function Construction and Optimization

Construction of the trial QMC wave function is done by multiplying the determinant of single particle DFT orbitals with a Jastrow correlation factor [9, 16]. As a check for dependence on DFT functional choice, QMC total energies for stishovite were compared using various functionals for the orbitals and found the energies were equivalent within one-sigma

78 statistical error (tenths of milli-Ha). The Jastrow describes various correlations by including two-body (electron-electron electron-nuclear), three body (electron-electron-nuclear), and plane-wave expansion terms, with a total of 44 parameters. Parameters in the Jastrow are optimized by minimizing the variance of the VMC total energy over several hundred thousand electron conﬁgurations. Several iterations of computing the VMC energy and re-optimizing the parameters are done until the VMC energy changes are typically inside of two sigma.

5.4.2 DMC Calculations

DMC calculations use the optimized trial wave functions to compute highly accurate energies. A typical silica DMC calculation uses 4000 electron conﬁgurations, a time step of

0.003 Ha−1, and several thousand Monte Carlo steps. In order for QMC calculations of solids to be feasible, a number of approximations [9, 16] are usually implemented that can be classified into controlled and uncontrolled approximations, which are discussed in detail in Chapter 2. The controlled approximations for silica include statistical Monte Carlo error, numerical grid interpolation (5 points/A)˚ of the DFT orbitals [22], finite system size, the number of configurations used, and the DMC time step. All of the controlled approximations are converged to within at least 1 milli-Ha/SiO2. Finite size errors are reduced by using a model periodic Coulomb Hamiltonian [24] while simulating cell sizes up to 72 atoms

(2×2×2) for quartz, 162 atoms (3×3×3) for stishovite/CaCl2, and 96 atoms (2×2×2) for

α-PbO2. Additional ﬁnite size errors are corrected due to insuﬃcient k-point sampling based on a converged k-point DFT calculation. The uncontrolled approximations include

the pseudopotential, nonlocal evaluation of the pseudopotential, and the ﬁxed node approx-

imation. These errors are diﬃcult to estimate, but the scheme of Casula [28] minimizes the

nonlocal pseudopotential error and some evidence suggests the ﬁxed node error may be

small [81, 19].

79 5.5 Results

This section presents and discusses all of the results produced from the silica free energy

QMC and DFT calculations: Helmholtz Free energy, Equation of State and Vinet Fit Pa- rameters, Phase Stability, Thermodynamic parameters, and stishovite shear constant softening. Thermodynamic parameters computed include the bulk modulus, pressure derivative of the bulk modulus, thermal expansivity, heat capacity, percent change in volume, Gr¨uneisen parameters, and the Anderson-Gr¨uneisenparameter. Note that all QMC calculations for silica use orbitals from DFT within the Wu-Cohen (WC) GGA functional approximation.

5.5.1 Free Energy

Figure 5.2 shows the zero temperature, Helmholtz free energy versus volume curves computed using QMC. For each phase, the QHA DFT phonon energies are added to the ground state, static QMC energy curves, producing a set of free energy isotherms for each silica phase. In this work, an isotherm was ﬁt in temperature increments of 5 K, ranging from 0

K to the melting point of silica (∼2000-4000 K). Such small increments allow for the construction of a fine T-V grid for computing thermodynamic functions with finite differences.

80 (a) Quartz (b) Stishovite

Figure 5.2: Computed Ground state (static) QMC free energy as a function of volume for (a) quartz, (b) stishovite and (c) α-PbO2.

81 5.5.2 Thermal Equations of State and Fit Parameters

Figure 5.3 shows the computed equations of state compared with experimental data for quartz [85, 111], stishovite/CaCl2 [112, 113], and α-PbO2 [89, 90]. Thermal equations of state are computed from the Helmholtz free energy [109]. Pressure is determined from the

expression P = − (∂F/∂V )T. The analytic Vinet [114] equation of state ﬁts isotherms of the Helmholtz free energies and is deﬁned as

9K (T )V (T ) E(V,T ) = E (T ) + 0 0 [1 + [ξ(1 − x) − 1] exp [(1 − x)]] , (5.8) 0 ξ2 where E0 and V0 are the zero pressure equilibrium energy and volume, respectively, x =

1/3 3 0 0 (V/V0) and ξ = 2 (K0 − 1)), K0(T) is the bulk modulus, and K0(T) is the pressure derivative of the bulk modulus. The subscript 0 indicates zero pressure. E0,V0,K0, and

0 K0 are the four ﬁtting parameters. Pressure is obtained analytically as

3K (T )(1 − x) P (V,T ) = 0 exp [ξ(1 − x)] . (5.9) x2 Figures 5.4, 5.5, 5.6,and 5.7 show the four Vinet fit parameters as a function of temperature. Vinet fits are made for free energy isotherms in increments of 5 K and the zero pressure fit parameters (free energy, F0, volume, V0, bulk modulus K0, and pressure deriva-

0 tive of the bulk modulus, K0) from each ﬁt form the curves in the plots. The gray shading of the QMC curves indicates one standard deviation of statistical error from the Monte

Carlo data. The QMC results generally agree well with diamond anvil-cell measurements at room temperature, as do the corresponding DFT calculations using the GGA functional of Wu and Cohen (WC) [115].

82 Figure 5.3: Thermal equations of state of (A) quartz, (B) stishovite and CaCl2, and (C) α-PbO2. The lower sets of curves in each plot are at room temperature and the upper sets are near the melting temperature. Gray shaded curves are QMC results, with the shading indicating one-sigma statistical errors. The dashed lines are DFT results using the WC functional. Symbols represent diamond-anvil-cell measurements (Exp.) [85, 89, 90, 111, 112, 113].

83 (a) Quartz (b) Stishovite

Figure 5.4: QMC and WC free energy as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-PbO2. The QMC energies are constructed by adding QMC energy for the static lattice to the WC phonon energy. Vinet ﬁts are made for free energy isotherms in increments of 5 K and the zero pressure free energy parameter from each ﬁt forms the curve in the plot. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

84 (a) Quartz (b) Stishovite

Figure 5.5: QMC and WC volume as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-PbO2. The QMC energies are constructed by adding QMC energy for the static lattice to the WC phonon energy. Vinet ﬁts are made for free energy isotherms in increments of 5 K and the zero pressure volume parameter from each ﬁt forms the curve in the plot. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

85 (a) Quartz (b) Stishovite

Figure 5.6: QMC and WC bulk modulus, K0 as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-PbO2. The QMC energies are constructed by adding QMC energy for the static lattice to the WC phonon energy. Vinet ﬁts are made for free energy isotherms in increments of 5 K and the zero pressure bulk modulus parameter from each ﬁt forms the curve in the plot. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

86 (a) Quartz (b) Stishovite

0 Figure 5.7: Pressure derivative of the bulk modulus, K0 as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-PbO2. The QMC energies are constructed by adding QMC energy for the static lattice to the WC phonon energy. Vinet ﬁts are made for free energy isotherms in increments of 5 K and the zero pressure K0 parameter from each ﬁt forms the curve in the plot. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

87 5.5.3 Phase Stability

A phase transition occurs at the pressure where the Gibbs free energy (or enthalpy at T=0) of two phases is equal, or, equivalently where the diﬀerence in Gibbs free energy changes sign. The Gibbs free energy is given by

G(V,T ) = Ustatic(V,T ) − TSvib(V,T ) + P (V,T )V (P,T ). (5.10)

At phase equilibrium (i.e. at the transition pressure), PT),

Gphase1(PT,V1(PT)) = Gphase2(PT,V2(PT))

F1(PT) + PTV1(PT) = F2(PT) + PTV2(PT),

which gives the expression for the transition pressure as

− [F2(PT) − F1(PT)] PT = , [V2(PT) − V1(PT)] which is equivalent to the so-called common-tangent [116] slope of the two energy versus volume curves.

Figure 5.8 shows zero temperature phase transitions via changes in the enthalpy. Sta- tistical uncertainty in the energy differences determines how well phase boundaries are constrained. The one-sigma statistical error on the QMC enthalpy difference is 0.5 GPa for the quartz-stishovite transition and 8 GPa for the CaCl2-α-PbO2 transition. The error on the latter is larger because the scale of the enthalpy difference between the quartz and stishovite phases is about a factor of 10 larger than for CaCl2 and α-PbO2. In both transitions, variation in the DFT result with functional approximation is large. For the metastable quartz-stishovite transition, LDA incorrectly predicts stishovite to be the stable ground state, WC underestimates the quartz-stishovite transition pressure by 4 GPa,

88 and the GGA of Perdew, Burke, and Ernzerhof (PBE) [117] matches the QMC result. For the CaCl2-α-PbO2 transition, the same three DFT approximations lie within the statistical uncertainty of QMC. The variability of the present calculations is less than diﬀerent experimental determinations of this transition (Figure 5.9). The experimental variability may be due to the diﬃculty in demonstrating rigorous phase transition reversals as well as pressure and temperature gradients and uncertainties in state-of-the-art experiments.

Figure 5.9a compares QMC and DFT predictions with measurements for the quartz- stishovite phase boundary. The QMC boundary agrees well with thermodynamic modeling of shock data [118, 119] and thermocalorimetry measurements [120, 121], while the

WC boundary is about 4 GPa too low in pressure. The melting curve shown is from a classical model [122], which agrees well with available experiments collected in the reference. The triple point seen in the melting curve is for the coesite stishovite transition, and not the metastable quartz-stishovite transition that computed boundaries represent. The geotherm [123] is shown for reference.

Figure 5.9b shows similar QMC and DFT calculations compared with experiments for the CaCl2-α-PbO2 phase boundary. The WC boundary lies within the QMC statistical error. Previous DFT work also shows that the LDA boundary lies near the upper range of the QMC boundary and PBE produces a boundary 10 GPa higher than the LDA [102].

The two diamond-anvil-cell experiments [89, 90] constrain the transition to lie between 65 and 120 GPa near the mantle geotherm (2500 K), while QMC constrains the transition to

105(8) GPa. The boundary inferred from shock data [100] agrees well with QMC and WC.

The boundary slope measured by Dubrovinsky et al. [89] is negative, which is in contrast to the positive slope inferred by Akins et al. [100]. QMC and WC as well as previous DFT studies [102, 103] predict a positive slope.

89 Figure 5.8: Enthalpy diﬀerence of the (A) quartz-stishovite, and (B) CaCl2-α-PbO2 transitions. The DMC transition pressures are 6.4(2) GPa and 88(8) GPa for quartz-stishovite and CaCl2-α-PbO2, respectively. Gray shaded curves are QMC results, with the shading indicating one-sigma statistical errors. The dashed, dot-dashed, and dotted lines are DFT results using the WC, PBE, and LDA functionals, respectively.

90 Figure 5.9: (a) Computed phase boundary of the quartz-stishovite transition. The gray shaded curve is the QMC result, with the shading indicating one-sigma statistical errors. The dashed line is the boundary predicted using WC. The dash-dot and solid lines represent shock [118, 119] analysis, while dotted and dash-dash-dot lines represent thermochemical data (Thermo.) [120, 121]. (b) Computed phase boundary of the CaCl2-α-PbO2 transition. Gray shaded curves are QMC results, with the shading indicating one-sigma statistical errors. The dashed, dotted, and dash-dot lines are DFT boundaries using WC, LDA [102], and PBE [102] functionals, respectively. The dark green shaded region and the solid blue line are diamond-anvil-cell measurements (Exp.) [89, 90]. The dash-dot-dot line is the boundary inferred from shock data [100]. The vertical light blue bar represents pressures in the D region. Circles drawn on the geotherm [123] indicate a two-sigma statistical error in the QMC boundary.

91 Table 5.1: Computed QMC thermal equation of state parameters at ambient conditions 3 (300 K, 0 GPa). The units are as follows: F0 (Ha/SiO2), ∆F (Ha/SiO2), V0 (Bohr /SiO2), −5 −1 K0 (GPa), α10 (K ). All other quantities are unitless. QMC one-sigma statistical error on F0 is 0.0002 Ha/SiO2. Phase quartz stishovite α-PbO2 Method QMC Exp. QMC Exp. QMC Exp. F0 -35.7946 -35.7764 -35.7689 ∆F 0.0182(4) i0.020(1) 0.0257(3) a d,e h V0 254(2) 254.32 159.0(4) 157.1(2) 154.8(1) 157.79 a d,e h K0 32(6) 34 305(20) 291–310 329(4) 313(5) 0 a d,e h K0 7(1) 5.7(9) 3.7(6) 4.29–4.59 4.1(1) 3.43(11) α 3.6(1) b3.5 1.2(1) f 1.26(11) 1.2(1) b f Cp/R 1.82(1) 1.80 1.71(1) 1.57(38) 1.69(1) γ 0.57(1) c0.57 1.22(1) f,g1.35-1.33 1.27(1) q 0.40(1) c0.47 2.22(1) f,g2.6(2)-6.1 2.05(1) b f,g δT 6.27(1) 3.3-8.9 5.98(1) 6.6-8.0(5) 6.40(1) aRef. [111] bRef. [124] cRef. [125] dRef. [112] eRef. [113] f Ref. [126] gRef. [127] hRef. [89] iRef. [120]

5.5.4 Thermodynamic Parameters

Thermal equations of state facilitate the computation of all desired thermodynamic parameters. Table 1 summarizes ambient computed and available experimentally measured values [89, 111, 112, 113, 124, 125, 126, 127, 120] of the Helmholtz free energy, F0 (Ha/SiO2), the Helmholtz free energy diﬀerence relative to quartz, ∆F (Ha/SiO2), volume, V0 (Bohr/SiO2),

0 −1 bulk modulus, K0, pressure derivative of the bulk modulus, K0, thermal expansivity,α (K ), heat capacity, Cp/R, Grüneisenratio, γ, volume dependence of the Grüneisenratio, q, and the Anderson-Grüneisenparameter, δT , for the quartz, stishovite, and α-PbO2 phases of silica. QMC generally offers excellent agreement with experiment for each of these parameters.

92 Thermal Pressure

Thermal pressure is the thermal energy eﬀect on the pressure [109, 128, 129]. It is computed by taking the volume derivative (P = − (∂F/∂V )T ) of the thermal vibrational energy term in Equation 5.7. However, through thermodynamic relations thermal pressure can be written as Z T Pth = P (V,T ) − P (V,T0) = αKTdT, (5.11) T0 where αKT is nearly constant at high temperatures for most materials, making the equation linear in T.

In essence, the expression for thermal pressure provides another form of an equation of state that can be checked against experimental measurements. From calculations, the thermal pressure equation of state is simply obtained by computing diﬀerences in pressures between a given isotherm and the ground state isotherm.

Figure 5.10 shows the computed QMC and DFT(WC) variation of thermal pressure with volume and temperature for quartz, stishovite, and α-PbO2. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines. All phases show a nearly constant dependence on the volume, but linear dependence on the temperature, especially at higher temperatures as expected. Quartz and

Stishovite experiments [112, 126] agree well with the predicted thermal pressure equations of state.

93 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.10: Computed QMC and WC thermal pressure of (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2as functions of volume and temperature. Experiments [112, 126] compare favorably with quartz and stishovite calculations. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

94 Changes in Thermal Pressure (αKT)

Figure 5.11 shows changes in thermal pressure [109, 128, 129], given by αKT = (∂P/∂T )T. −3 Changes in αKT are quite small (note the scale is 10 ) as expected for most materials.

95 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.11: Computed QMC and WC variation of αKT with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

96 Bulk Modulus

Figure 5.12 shows the calculated temperature and pressure dependencies of the bulk moduli [109] of quartz, stishovite, and α-PbO2. The bulk modulus is denoted as

∂2F ∂P KT = V 2 = −V . (5.12) ∂V TT ∂V T

The bulk moduli decrease linearly with temperature and increase linearly with pressure

for all pressure-temperature ranges. Although it is well known that DFT bulk moduli can

vary signiﬁcantly with choice of functional, results with the WC functional tend to lie only

slightly below QMC. Both DFT and QMC agree with the experimental data for quartz [85]

and stishovite [126] at zero pressure. No measurements are yet available for the elastic

moduli of the α-PbO2 phase.

97 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.12: Computed QMC and WC variation of the bulk modulus with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well with available experimental data [85, 126]. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

98 Pressure Derivative of the Bulk Modulus

Figure 5.13 shows the rate of change in the bulk modulus with respect to pressure [109],

0 denoted as K = (∂KT/∂P )T . It is an important quantity in many thermodynamic expressions and also occurs as a parameter in most universal equations of state. K0 is a unitless quantity and shows only slight variation with both temperature and pressure.

99 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.13: Computed QMC and WC variation of the pressure derivative of bulk modulus,K0, with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α- PbO2. Results agree well with available experimental data [85, 126]. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

100 Thermal Expansivity

Figure 5.14 shows the computed QMC and WC thermal expansivity [109] for quartz, stishovite, and α-PbO2. Thermal expansivity is denoted as

1 ∂2F ∂2F 1 ∂V α = − / 2 = (5.13) V ∂T ∂V ∂V T V ∂T P

QMC and WC both agree well with experimental measurements [130, 131, 132, 126, 133,

134, 135, 136] at zero pressure. Experimentally, the quartz structure transforms to the

β-phase around 846 K, when the volume thermally expands to about 900 Bohr3 and the thermal expansivity becomes negative. However, the computations presented here consider only the α-quartz phase. QMC and DFT also show good agreement with the measured stishovite expansivity. There have been no expansivity measurements for α-PbO2.

101 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.14: Computed QMC and WC variation of thermal expansivity with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well with available experimental data [130, 131, 132, 126, 133, 134, 135, 136]. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

102 Heat Capacity

Figure 5.15 shows the computed QMC and WC heat capacities for quartz, stishovite, and

α-PbO2. Heat capacity [109] may be computed at constant volume or pressure:

∂U Cv = (5.14) ∂T V or ∂H Cp = . (5.15) ∂T P For solids, the two expressions are nearly identical. For all silica phases, the WC results almost exactly match QMC. The quartz and stishovite results agree well with experiment [126, 130]. There have been no heat capacity measurements for the α-PbO2 phase.

103 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.15: Computed QMC and WC variation of heat capacity with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well with available experimental data [130, 126]. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

104 Change in Volume

Figure 5.16 shows the computed QMC and WC volume diﬀerences for quartz-stishovite and

CaCl2-α-PbO2 at 0 K. The quartz-stishovite volume change is large due to the four to six fold coordination change. The volume change in the CaCl2-α-PbO2 transition is roughly a factor of ten smaller.

105 Figure 5.16: Computed QMC and WC percentage volume diﬀerence of (A) quartz-stishovite and (B) CaCl2-α-PbO2 transitions. Gray shaded curves are QMC results, with the shading indicating one-sigma statistical errors. The dashed lines are DFT results using the WC functional.

106 Gr¨uneisenParameters

The Gr¨uneisenratio [109], γ, quantiﬁes the relationship between thermal and elastic properties of a solid. It is a very important parameter to Earth scientists because it sets limitations for thermoelastic properties of the mantle and core of Earth, the adiabatic temperature gradient, and the geophysical interpretation of shock (Hugoniot) data. γ is approximately constant, dimensionless parameter that varies slowly with pressure and temperature [137].

Experimental measurement of γ is diﬃcult, and accurate calculations are useful for con- straining possible values.

γ has both microscopic and macroscopic deﬁnitions. The microscopic deﬁnition is based on the volume dependence of the ith phonon mode of the crystal lattice, and is given by:

∂lnω γ = i . (5.16) i ∂lnV Evaluation of the microscopic deﬁnition is diﬃcult because knowledge of all phonon modes

requires a dynamical lattice model or inelastic neutron scattering.

Summing all of γi over the ﬁrst Brillouin zone leads to a macroscopic (thermodynamic) deﬁnition, written as

αK V γ = T , (5.17) CV where α is the thermal expansivity, KT is bulk modulus, V is volume, and CV is the heat capacity at constant volume. Both the microscopic and macroscopic deﬁnitions are diﬃcult to analyze experimentally because the former requires knowledge of the phonon dispersion spectrum and the latter requires measurements of thermodynamic properties at high temperatures and pressure.

Figure 5.17 shows the computed QMC and DFT(WC) Gr¨uneisenratios, providing accurate values to help constrain experiments. The computations show reasonable agreement with quartz data [125] available at low pressures. In general, γ initially decreases with pressure and increases at high pressures. For quartz, results show γ decreases with temperature, but for stishovite and α-PbO2, γ increases with temperature. 107 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.17: Computed QMC and WC variation of the Gr¨uneisen ratio with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well with available experimental data [125]. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

108 An additional parameter, q, is used to describe the volume dependence of γ, and is deﬁned as

∂lnγ q = . (5.18) ∂lnV The q parameter is often assumed to be constant. However, ﬁgure 5.18 show that q is

both temperature and pressure dependent. Temperature dependence of all phases tends to

ﬂuctuate at low temperatures and become constant at high temperatures. All phases show

a strong decrease in q with increasing pressure. DFT(WC) and QMC predict very similar

results for all phases.

109 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.18: Variation of q with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well with available experimental data [125]. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines.

110 Anderson-Gr¨uneisenParameter

The Anderson-Gr¨uneisenparameter [109], δT, which characterizes the relationship between thermal expansivity and pressure is deﬁned as

∂lnα −1 ∂KT δT = = . (5.19) ∂lnV T αKT ∂T P

Figure 5.19 shows δT may initially increase or decrease at low temperatures, and eventually become constant at high temperatures. At all temperatures, δT shows a strong decrease with pressure. DFT(WC) and QMC predict very similar results in all cases.

111 (a) Quartz (b) Quartz

(e) α-PbO2 (f) α-PbO2

Figure 5.19: Computed QMC and WC variation of the Anderson-Gr¨uneisenparameter with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. QMC results are represented by black curves and WC results are represented by red dashed lines.

112 Bulk Sound Velocity and Density

The differences in bulk sound velocity and density are particularly important properties for phases of minerals because they indicate the strength of corresponding seismic signals. If the change in bulk sound velocity and density between two phases is large, then there will be strong discontinuity in seismic data. For example, Figure 5.20 shows the sound velocity and density profiles for Earth [138]. There are major discontinuities in the profiles as the phases change form the solid mantle, to the liquid outer core, to the solid Fe inner core.

113 (a) Quartz

Figure 5.20: Proﬁle of the p-wave, α, s-wave, β, and density, ρ, in Earth. The discontinuities correspond to major compositional transitions inside Earth [138].

114 The bulk sound velocity is deﬁned as

1 K 2 V = S , (5.20) BS ρ where ρ is the density, and KS is the adiabatic bulk modulus. The adiabatic bulk modulus is deﬁned as KS = −V (∂P/∂V )S, where S indicates adiabatic conditions. However, for solids, the adiabatic bulk modulus generally agrees with the isothermal bulk modulus,

KT = −V (∂P/∂V )T within about one percent at room temperature. The two types of bulk moduli are related by KS = KT (1 + αγT ), where α is the thermal expansivity and γ is the Gr¨uneisenratio.

Regarding silica, the important question is whether the CaCl2 to α-PbO2 transition is seismically visible. Since phase stability work described above indicates the transition is

00 not associated with the D layer, localized quantities of α-PbO2 may be visible seismically if the diﬀerence in bulk sound velocity and density is large.

Figure 5.21 shows the QMC predicted bulk sound velocity and density as a function of pressure for room temperature and for a typical lower mantle temperature near the base. The pairs of lines in the bulk sound velocity plot indicate the one-sigma error bars on the QMC calculations. With respect to postperovskite, MgSiO3, (the dominate D00

00 material) measurements [139, 140] at 120 GPa in D , QMC predicts α-PbO2 has 12% lower density and 67% larger bulk sound velocity, which may provide enough contrast to be seen seismically if present in appreciable amounts.

115 (a) Quartz

(b) Quartz

Figure 5.21: QMC calculations of the variation in (a) bulk sound velocity and (b) density with pressure for CaCl2 and α-PbO2. Proﬁles are plotted at both room temperature and the mantle-base temperature and compared with the experimental values of perovskite/post- perovskite at the base of the mantle. The pairs of lines in the bulk sound velocity plot indicate one-sigma statistical error.

116 5.5.5 Stishovite Shear Constant

Most information about the deep Earth comes from the study of seismic waves, and elastic constants determine sound velocities of those waves in the Earth. Much work has been done using DFT to compute and predict elastic constants for minerals in the Earth [104], but there is much uncertainty in the predicted elastic constants because different density functionals predict significantly different values. Here, computations test the feasibility of using QMC to predict softening of the shear elastic constant, c11-c12, in stishovite, which drives the ferroelastic phase transition to CaCl2 [88, 93, 99]. While there are many methods of computing elastic constants, the strain-energy density relation outlined by Barron and Klein [141, 93] is particularly convenient when working with volume conserving strains. The full expression for strain-energy density is given by

∆E 1 1 = −p + c − p (2δ δ − δ δ ) , (5.21) V ii 2 ijkl 2 ij kl il jk ij kl

where E is the free energy, V is the volume, ij is the Eulerian strain, cijkl is the elastic

constant tensor, and δij is the Kronecker delta. The pressure terms vanish for volume conserving strains leaving

∆E 1 = c . (5.22) V 2 ijkl ij kl It’s clear from this expression that the elastic constants are second energy derivatives of the energy with respect to strain, given by

1 ∂2E cijkl = . (5.23) V ∂ij∂kl

It is the c11 − c12 shear constant related to the B1g Raman mode that becomes unstable at the phase transition to CaCl2. The elastic constant is found by computing the energy versus b/a strain in the tetragonal stishovite lattice for a constant volume and c lattice parameter. The volume conserving strain matrix applied to the lattice vectors to produce

117 c11 − c12 is given by   δ 0 0     =  0 −δ 0  (5.24)     0 0 0 The strain matrix is applied to the matrix of lattice vectors, R, to produce a new set of

lattice vectors R0 corresponding to a structure with the same volume as follows:

   δ 0 0    0    R = I +  0 −δ 0  R (5.25)       0 0 0

Values of δ are chosen to produce structures with 0%, 1%, 2%, and 3% strain in the lattice

for a given volume. DFT and QMC compute energies of the strained structures at various

ﬁxed volumes, with the ions relaxed for each structure.

Figure 5.22 shows the energy versus strain curves for WC, VMC, and DMC calculations.

Points are shown at 0%, 1%, 2%, and 3% strain for a few slected volumes. WC calculations

optimized the ion positions of each structure. Those optimized structures were subsequently

used in the QMC calculations.

118 (a) Quartz (b) Quartz

Figure 5.22: Computed QMC energy versus b/a strain with ion positions optimized by WC at each point. Energies are shifted to be equal at b/a=1.

119 Plugging the strain matrix elements into Equation 5.22 gives the expression needed to evaluate c11 − c12:

1 ∂2E c − c = , (5.26) 11 12 2V ∂δ2

∂2E where ∂δ2 is the curvature of a polynomial ﬁt to the DFT or QMC energy versus strain data.

Figure 5.23 shows the shear softening of c11 − c12 as a function of pressure at zero temperature. At low pressures c11 −c12 is almost constant, but softens as pressure increases and becomes unstable near 50 GPa. For a well-optimized trial wave function, VMC often comes close to matching the results of DMC. Due to the large computational cost, VMC computes c11 − c12 at several pressures and the more accurate DMC checks only the endpoints. The ﬁgure also shows the result of WC and previous LDA computations [93]. Both QMC and

DFT results correctly describe the softening of c11-c12, indicating the zero temperature transition to CaCl2 near 50 GPa. Radial X-ray diﬀraction data [94] lies lower than calculated results. However, discrepancies can arise in the experimental analysis depending on

the strain model used. Recent Brillouin scattering data [97] agrees well with DMC. Accu-

rate computation of the QMC total energies on a small strain scale is very computationally

expensive, requiring roughly 100-1,000 times more CPU time than a corresponding DFT

calculation. The QMC calculations for this feasibility test require over 3 million CPU hours,

which NERSC made available during alpha and beta testing of their Cray-XT4 (Franklin).

120 Figure 5.23: Softening of the c11-c12 shear constant for stishovite with pressure. Down triangles and circles are the DMC and VMC results, respectively. Diamonds and up triangles represent DFT results within the WC and LDA [93], respectively. Squares represent radial X-ray diﬀraction data [94] and stars represent Brillouin scattering data [97]. The shear constant in all methods softens rapidly with increasing pressure and becomes unstable near 50 GPa, signaling a transition to the CaCl2 phase.

121 5.6 Geophysical Implications

The QMC CaCl2-α-PbO2 boundary indicates that the transition to α-PbO2, within a two- sigma conﬁdence interval, occurs in the depth range of 2,000-2,650 km (86122 GPa) and in the temperature range of 2,300-2,600 K in the lower mantle. This places the transition

50650 km above the D00 layer, a thin boundary surrounding Earth’s core ranging from a depth of ∼2,700 to 2,900 km [142]. The DFT boundaries all lie within the QMC two-sigma

conﬁdence interval, with PBE placing the transition most near the D00 layer. Free silica in

D00, such as in deeply subducted oceanic crust or mantlecore reaction zones, would have the

α-PbO2 structure. However, based on QMC results, the absence of a global seismic anomaly above D00 suggests that there is little or no free silica in the bulk of the lower mantle. The

α-PbO2 phase is expected to remain the stable silica phase up to the coremantle boundary. Bulk sound velocity and density proﬁles indicate that the transition should be seismically

visible for large enough concentrations.

5.7 Conclusions

This chapter has presented QMC (using WC orbitals) computations of silica equations of

state, phase stability, and elasticity. This work provides highly accurate values for thermal

properties for silica and expands the scope of QMC by studying the complex phase transi-

tions in minerals away from the cubic oxides. The DMC zero temperature quartz-stishovite

transition pressure is 6.4(2) GPa and the QMC zero temperature CaCl2-α-PbO2 transition

pressure is 88(8) GPa. Results show the CaCl2-α-PbO2 transition is not associated with the global D00 discontinuity, indicating there is not signiﬁcant free silica in the bulk lower

mantle. However, the transition should be detectable in deeply subducted oceanic crust.

A number of thermodynamic properties are computed by combining QMC with DFT pho-

non energies. The QMC thermodynamic parameters and their dependence on pressure and

temperature agree well with experimental data. QMC also provides an accurate description

of shear constant softening in stishovite.

Additionally, results document the improved accuracy and reliability of QMC relative

122 to DFT. As expected, LDA is the worst for predicting properties based on energy differences for structures that have large differences in interatomic bonding (similar for silicon interstitials). For example, LDA fails to predict the quartz-stishovite transition, while PBE and QMC agree with experiment. Other GGA functionals do not predict the transition well though. DFT currently remains the method of choice for computing material properties because of its computational efficiency, but results show that QMC is feasible for computing thermodynamic and elastic properties of complex minerals. DFT is generally successful but does display failures independent of the complexity of the electronic structure and sometimes shows strong dependence on functional choice. With the current levels of computational demand and resources, one can use QMC to spot-check important DFT results to add confidence at extreme conditions or provide insight into improving the quality of density functionals. In any case, QMC is bound to become increasingly important and common as next generation computers appear and have a great impact on computational materials science.

123 Chapter 6 Hybrid DFT Study of Silica

One of the main themes of this thesis is improving reliability of DFT through coping methods for weaknesses caused by approximating the exchange-correlation functional. In Chapter 3, general methods of coping were discussed: benchmarking with hybrid functionals or QMC.

Chapter 4 discussed an application of both hybrid functionals and QMC to benchmark DFT calculations of silicon interstitial defects. Notably, hybrid silicon results generally matched the QMC results. Chapter 5 discussed benchmarking of DFT silica calculations with QMC.

This chapter focuses on the hybrid DFT calculations of silica.

6.1 Introduction

Standard local (LDA) and semi-local (GGA) DFT has been extremely successful for tens of thousands of published calculations of various materials. However, occasionally, DFT lacks the accuracy and reliability to predict experimental results. Sometimes results are heavily biased depending on which exchange-correlation functional is employed, and the best functional for the given problem is only identified after comparison with experiment. Reliable predictive power is needed when desired properties are challenging for experiments to measure. QMC provides very high computational accuracy, but at an enormous computational cost that is not always feasible or convenient. QMC can also not afford to optimize geometries or compute phonon properties. A computational method that has the computational ease of standard DFT with and the accuracy of QMC is needed. Hybrid DFT functionals have offered a significant increase in accuracy over standard LDA and GGA functionals for

124 the Quantum Chemistry community. However, they are also tend to be computationally expensive for solids. New generations of hybrid functionals which are screened may be the best compromise between accuracy and speed, but they remain largely untested.

Silica is one well-known material for which DFT makes unreliable predictions. LDA generally predicts structural and elastic properties well, while GGA predicts structural energetics and phase stability better. In fact, LDA fails to predict the lowest pressure, quartz to stishovite transition, while the GGA functional known as PBE predicts the transition perfectly. Other GGA functionals do not predict the transition well, however. Chapter 5 showed that QMC can generally be used to predict all properties of silica accurately when accuracy is paramount. This chapter investigates whether hybrid functionals or perhaps some newly developed semi-local functionals can aﬀord the same accuracy as QMC for silica and oﬀer a reliable and computationally inexpensive route to future silica predictions.

Namely, this work studies silica with the following functionals: 1) the local LDA [32, 143] functional; 2) the semi-local PBE [117], PW91 [33], PBEsol [144], and WC [115, 145] functionals; and 3) the hybrid B3LYP [146, 147], PBE0 [34, 45, 148, 46], and HSE [40, 41,

42, 43] functionals. The screened hybrid, HSE, is generally found to match the accuracy of QMC for all properties of silica and the best compromise between standard DFT and

QMC.

6.2 Previous Work

6.2.1 Hybrid Calculations of Silica

There have been small number of studies of certain phases or properties of silica using the hybrid B3LYP functional [149, 150, 151], all using the CRYSTAL code. Civalleri et al. [149] studied all-silica zeolite framework stability. Zicovich-Wilson et al. [150] studied vibrational frequencies of quartz. Ottonello et al. [151] studied vibrational properties of stishovite. There have been no studies using other hybrid functionals, such as PBE0 or

HSE. B3LYP results typically show B3LYP is superior in performance to LDA and GGA, and the expectation of this result based on molecular studies is often why B3LYP is chosen.

125 While these studies indicate B3LYP may be a reliable functional for computing properties of silica, more extensive testing and comparisons need to be done. A thorough comparison of local, nonlocal, and hybrid functional performance is needed for a wide range of properties to determine which functional works best.

6.2.2 Hybrid B3LYP and PBE0 Calculations of Solids

The Quantum Chemistry community has a long and proliﬁc history of publishing B3LYP and PBE0 DFT calculations for molecules and non periodic systems. Indeed, the accuracy of the B3LYP functional is what convinced most quantum chemists to switch to using DFT instead of CI. A couple of codes (CRYSTAL and GAUSSIAN) were also developed to study solids with Gaussian basis sets, but still relatively few solid systems have been studied with hybrid B3LYP and PBE0 functionals due the large computational expense. Even so, a small number of periodic solids have been studied with the B3LYP functional [47, 48, 49, 50, 51,

52, 53] and PBE0 [34, 45, 46]. Until recently, all calculations employed Gaussian basis sets.

6.2.3 Screened Hybrid (HSE) Calculations of Solids

With the development of more computationally eﬃcient screened hybrids, plane-wave pseudopotential codes (VASP and PWSCF) have begun to incorporate hybrid functionals into calculations of periodic solids. There have been roughly a couple dozen published calculations for solids since these developments occurred using the HSE functional [54, 55, 56, 44,

42, 41, 57, 40, 43, 58, 59, 60].

6.3 Computational Methodology

The overall aim of this project is to systematically study and compare the major inﬂuences of ab initio calculations on properties of quartz and stishovite phases of silica and the quartz-stishovite phase transition. Namely, the aim is to compare performance of various exchange-correlation functionals, basis sets, pseudopotentials, and codes, and benchmark against QMC and experiments to determine which are most accurate. Calculations compare

126 local (LDA), semi-local GGA (PBE, PW91, PBEsol, WC), and hybrid (B3LYP, PBE0, and HSE) functionals. All electron calculations are compared against projector augmented wave (PAW) pseudopotentials and nonlocal, norm-conserving pseudopotentials. In addition, results from the CRYSTAL, ABINIT, and VASP codes are compared.

The general protocol for every calculation is to ﬁrst fully relax the cell shape and volume until forces are smaller than 10−4Ha/Bohr. The optimized cell determines the zero pressure lattice constants. The equation of state is then determined by computing total energy as a function of volume for on the order of ten volumes ranging form 10% expansion to 30% compression about the fully optimized equilibrium volume. Constant volume optimization of the cell is done for each point to relax forces on the atoms to less than 10−4Ha/Bohr.

The analytic Vinet [114] equation of state fits the energy versus volume data. Zero pressure volume and bulk modulus are determined from the energy versus volume data. A pressure versus volume curve is generated by taking the derivative of the Vinet energy expression with respect to volume: P = − (∂F/∂V )T. Analysis of phase stability requires constructing the enthalpy, H = U + PV, versus pressure for quartz and stishovite. The quartz-stishovite transition is determined by the crossing enthalpy curves. This protocol is carried out for various functionals, pseudopotentials, and basis sets in different codes. The following subsections address details of the calculations specific to each code.

6.3.1 CRYSTAL Calculations

The CRYSTAL [152] code allows for all-electron calculations using Gaussian Basis sets, and a large variety of exchange-correlation functionals, including LDA, PBE, PW91, B3LYP, and PBE0 used in this study. CRYSTAL does not allow for plane-wave basis set calculations.

K-point sampling

A k-space sampling converged to at least chemical accuracy for quartz uses Monkhust-Pack grids with a shrinking factor in reciprocal space and a Gilat shrinking factor of 7. Converged stishovite calculations use shrinking factors of 9 and 9, respectively.

127 Coulomb and Exchange Series Truncation and Additional Convergence Param- eters

In order to allow for eﬃcient computation of periodic systems with Gaussian basis sets,

CRYSTAL adopts a bipolar expansion to compute the Coulomb integrals when two distributions do not overlap. However, tests indicated that truncation of the integrals at any level resulted in inaccurate energies and non-smooth energy versus volume curves. There- fore, all energy versus volume calculations presented here activated the nobipola ﬂag, forcing CRYSTAL to exactly compute all bi-electronic integrals, avoiding noisy numerical error. In addition, convergence for the density matrix, toldep, was set high to 18

(10−18) and convergence on the total energy, toldee was increased from default to 7 (10−7 Ha). Energy convergence was also made more stable by using the eigenvalue level shifting

technique, activated with the ﬂag choices as follows: levshift 6 1. Mixing of Fock and

Kohn-Sham matrices was also used to accelerate convergence with the fmixing 30 ﬂag, indicating a choice of 30% mixing.

Convergence of Gaussian Basis Sets

The most crucial and signiﬁcant challenge in performing a periodic solid calculation that uses a Gaussian orbital basis set is determining the type, size, and parameters of the basis set that converge the energy to chemical accuracy. Generally, one must try a range of basis set types and sizes, optimizing the parameters for each one such that the calculation will remain stable and converge the total energy.

Diffuse Gaussian exponents for periodic solid systems are very important for contributions of long-range interactions. However, Gaussian basis sets, generally designed for molecules and atoms, have exponents which are too diffuse for numerical stability in periodic solids. The overlap among orbitals is overestimated and numerical linear dependencies become problematic. Adjusting the calculation constraints to achieve higher accuracy (higher integral, density, and energy tolerances, ect) allow one to overcome numerical instability to some extent. However, if the exponents are too diffuse to overcome with accuracy tolerances

128 or included in an uncontrolled way [153], then some of the diffuse exponents must be modified or removed. In fact, in the CRYSTAL code, the most diffuse exponent allowed is 0.12.

In the basis sets convergence calculations, the diﬀuse exponents of each basis set were either modiﬁed as needed and re-optimized using line optimization techniques, or simply removed if numerical stability could not be achieved otherwise. This aspect makes converging the basis particularly challenging.

In order to make the basis set optimization and convergence tests eﬃcient, the convergence calculations did not use the production ﬂag option NOBIPOLA, and, in addition, default options were used for TOLDEP and TOLDEE. However, levshift 6 1 and fmixing

30 were still used, as they increase stability and eﬃciency of the calculations. Figure 6.1 shows convergence of the quartz-stishovite energy diﬀerence as a function of

increasingly large Gaussian basis sets. The converged LDA, PW91, and B3LYP plane-wave

energies of the quartz-stishovite energy diﬀerence provide a benchmark to compare Gaussian

basis set calculations against. The CRYSTAL (LDA, PW91, and B3LYP) quartz-stishovite

energy diﬀerence was converged to the plane-wave value by using a series of modiﬁed basis

sets that systematically add polarization functions to account for higher angular momentum

orbitals. The series of basis sets tested range from ones including only s and p orbitals:

(3-21GSP, 3-21G, 6-311G), to ones including a single d orbital (cc-pVDZ, 66-21G∗(Si)/6-

31G∗(O), 65-111G∗(Si)/8-411G(O)), to two d orbitals (6-311G∗∗), to two d orbitals and

one f orbital (cc-pVTZ), and, ﬁnally, ones that include three d orbitals and two f orbitals

(6-311G∗∗(Si)/cc-pVQZ(O), and cc-pVQZ). Convergence with plane-wave results is reached

when using the cc-pVQZ basis set (See Appendix B for the optimized cc-PVQZ basis set

used).

The convergence of the energy diﬀerence is fairly rapid with the addition of extra d and f

polarization functions. There are a couple of important features to notice. The ﬁrst is that

the 66-21G∗(Si)/6-31G∗(O) basis suggested as suﬃcient by CRYSTAL users and the 65-

111G∗(Si)/8-411G(O) basis, which has been typically cited [154, 155] and used in published

CRYSTAL silica calculations, do not appear to suﬃcient for chemical accuracy convergence

of the quartz-stishovite energy diﬀerence. Secondly, use of a mixture of smaller and larger

129 basis sets for silicon and oxygen separately indicate that oxygen is the main component that requires signiﬁcant higher angular momentum polarization functions. This is apparent, for example, when comparing the nearly identical results of (6-311G∗∗(Si)/cc-pVQZ(O) and cc-pVQZ).

130 Figure 6.1: Gaussian basis set convergence

131 6.3.2 ABINIT Calculations

The ABINIT package [108] allows for plane-wave pseudopotential calculations using a large number of diﬀerent exchange-correlation functionals. In this project, the LDA, PBE, PW91,

PBEsol, and WC functionals are used in ABINIT. At the time of this work, the ABINIT code was the only plane-wave code to include the PBEsol and WC functionals. All silica calculations use a converged plane-wave energy cutoﬀ of 100 Ha and Monkhorst-Pack k-point meshes such that total energy converged to tenths of milli-Ha/SiO2 accuracy. Converged silica calculations require k-point mesh sizes of 4×4×4 for quartz and 4×4×6 for stishovite,

and all meshes were shifted from the origin by (0.5, 0.5, 0.5), corresponding to the reduced

coordinates in the coordinate system deﬁning the k-point lattice.

The Opium code [107] produces optimized nonlocal, norm-conserving pseudopoten-

tials for Si and O. Both pseudopotentials are generated using the appropriate exchange-

correlation functional (LDA, PBE, PW91, PBEsol, and WC) for DFT calculations. All

QMC calculations use pseudopotentials generated with the WC functional. In all cases, the

silicon potential has a Ne core with equivalent 3s, 3p, and 3d cutoﬀs of 1.80 a.u. The oxygen

potential has a He core with 2s, 2p, and 3d cutoﬀs of 1.45, 1.55, and 1.40 a.u., respectively.

6.3.3 VASP Calculations

The VASP package [78] allows for plane-wave pseudopotential calculations using a large

number of diﬀerent exchange-correlation functionals. In this project, the LDA, PBE, PW91,

and HSE functionals are used with VASP. At the time of this work, the VASP code was the

only plane-wave code to include the HSE functional. The VASP code uses preconstructed

projector augmented wave (PAW) pseudopotentials, which are typically highly accurate.

There are no pseudopotentials generated for the HSE functional, so PBE is used as it is

likely the closest match. All silica calculations use a converged plane-wave energy cutoﬀ of 30

Ha and Monkhorst-Pack k-point meshes such that total energy converged to tenths of milli-

Ha/SiO2 accuracy. Converged silica calculations require k-point mesh sizes of 9×9×9 for quartz and 13×13×13 for stishovite, and no k-point shift is used in the VASP calculations.

132 6.4 Results

The following section presents all of the results comparing performance of various local, semi-local, and hybrid exchange-correlation functionals, basis sets, pseudopotentials, and codes: CRYSTAL all-electron, Gaussian basis calculations using LDA, PBE, PW91, B3LYP, and PBE0 functionals; ABINIT plane-wave opium-pseudopotential calculations using LDA,

PBE, PW91, PBEsol, and WC functionals; VASP plane-wave PAW-pseudopotential calculations using LDA, PBE, PW91, and HSE functionals.

Results show examples of energy vs volume and enthalpy curves for HSE calculations, as they are fundamental for computing all other properties. Bar plots show comparisons for the quartz and stishovite zero pressure volume, bulk modulus, pressure derivative of the bulk modulus, pressure versus volume curves, and quartz-stishovite transition pressure. A table is presented with values of equilibrium lattice constants, volumes, and bulk moduli.

The main result is that the HSE functional is generally most consistent, outperforming all other functionals for quartz and stishovite properties, and agreeing well with experiment and QMC.

6.4.1 Energy Versus Volume

As all other quantities are derived from the energy versus volume data, Figure 6.2 shows one example of such curve for quartz and stishovite using the HSE functional in a VASP calculation. The curve shown corresponds to the zero temperature isotherm, ﬁt with the

Vinet equation of state. At zero pressure, the quartz energy is clearly lower, indicating it is the stable phase at that pressure. To determine phase stability at all pressures, one must compute the enthalpy.

133 Figure 6.2: Computed HSE energy versus volume curves for quartz and stishovite. Points are ﬁt with the Vinet equation of state.

134 6.4.2 Pressure Versus Volume

Once the energy is know as a function of volume, the pressure as a function of volume is computed by taking the derivative of the Vinet energy expression with respect to volume:

P = − (∂F/∂V )T. Figure 6.3 and Figure 6.4 compare all pressure versus volume equations of state computed with various functionals and codes for quartz and stishovite. Experimental results are plotted as points on each ﬁgure as symbols. The curve produced with the PBEsol functional seems to best match experimental data for quartz, while HSE result best matches experimental data for stishovite.

135 (a) Quartz LDA (b) Quartz PBE

(e) Quartz Hybrids

Figure 6.3: Computed pressure versus volume curves of quartz using various exchange- correlation functionals and codes.

136 (a) Stishovite LDA (b) Stishovite PBE

(e) Stishovite Hybrids

Figure 6.4: Computed pressure versus volume curves of stishovite using various exchange- correlation functionals and codes.

137 6.4.3 Equilibrium Quartz and Stishovite Volume from Vinet Fits

Figures 6.5 (a) and (b) show a comparison of the quartz and stishovite zero pressure volumes, respectively; The volumes are those estimated from the minimum of the Vinet ﬁt to the energy versus volume data for all of the exchange-correlation functionals and codes used.

The solid horizontal black line indicates the range of experimental data, while the dashed box (quartz) and line (stishovite) indicates the one-sigma statistical error in the QMC prediction.

The LDA, PBE, PW91 results are fairly uniform across all types of codes, indicating the calculations are individually converged and the codes are performing on par with each other.

LDA tends to slightly underestimate the volume; PBE and PW91 tend to overestimate;

PBEsol and WC match experiment very well; The hybrids B3LYP and PBE0 overestimates the quartz volume, but match experiment for the stishovite volume. The screened hybrid

HSE also slightly over estimates the quartz volume and matches experiment for stishovite.

In general the PBEsol, WC, LDA, and HSE functionals all perform well for the volume.

138 (a) Quartz

(b) Stishovite

Figure 6.5: Zero pressure volumes of (a) quartz and (b) stishovite from Vinet ﬁts of energy versus volume data using various exchange-correlation functionals and codes. The solid horizontal black line indicates the range of experimental data, while the dashed box or line indicates the one-sigma statistical error in the QMC prediction.

139 6.4.4 Equilibrium Quartz and Stishovite Bulk Moduli from Vinet Fits

Figures 6.6 (a) and (b) show a comparison of the quartz and stishovite zero pressure bulk moduli, respectively; The bulk moduli are those computed from the curvature around the minimum of the Vinet ﬁt of the energy versus volume data for all of the exchange-correlation functionals and codes used. The gray shaded box indicates the range of experimental data, while the dashed box indicates the one-sigma statistical error in the QMC prediction.

The LDA, PBE, PW91 results are fairly uniform across all types of codes for quartz, indicating the calculations are individually converged and the codes are performing on par with each other. However, the LDA, PBE, PW91 results for stishovite are less uniform across all types of codes for the bulk moduli. The all-electron CRYSTAL LDA, PBE and

PW91 bulk moduli tend to be about 7-10% larger than the plane-wave pseudopotential results. The CRYSTAL LDA result agrees with the all-electron LAPW result of Cohen et al. [156]. However, the LAPW LDA and PBE result of Zupan et al. [157] agrees with the ABINIT and VASP results. LDA tends to predict the bulk modulus in agreement with experiment; PBE, PW91, PBEsol, and WC tend to underestimate; The hybrids B3LYP signiﬁcantly underestimates the quartz bulk modulus and slightly over estimates the stishovite bulk modulus. PBE0 underestimates the quartz bulk modulus, and overestimates for stishovite. The screened hybrid HSE slightly underestimates the quartz bulk modulus and matches experiment for stishovite. In general the LDA and HSE functionals perform well for the bulk modulus.

140 (a) Quartz

(b) Stishovite

Figure 6.6: Zero pressure bulk moduli of (a) quartz and (b) stishovite from Vinet ﬁts of energy versus volume data using various exchange-correlation functionals and codes. The gray shaded box indicates the range of experimental data, while the dashed box indicates the one-sigma statistical error in the QMC prediction.

141 0 6.4.5 Equilibrium Quartz and Stishovite K0 from Vinet Fits

0 Figures 6.7 (a) and (b) show a comparison of K0 for quartz and stishovite, respectively; 0 The K0 values are those from the Vinet ﬁts to energy versus volume data. The gray shaded box indicates the range of experimental data, while the dashed box indicates the one-sigma statistical error in the QMC prediction.

The LDA, PBE, PW91 results are fairly uniform across all types of codes for quartz, indicating the calculations are individually converged and the codes are performing on par with each other. In fact, almost all results for both quartz and stishovite are near the experimentally measured range of values.

142 (a) Quartz

(b) Stishovite

Figure 6.7: Computed pressure derivative of the bulk modulus for (a) quartz and (b) stishovite using various exchange correlation functionals and codes. The gray shaded box indicates the range of experimental data, while the dashed box indicates the one-sigma statistical error in the QMC prediction.

143 Table 6.1 and 6.2 provides zero pressure values of lattice constants (a and c), volume

0 (V0), bulk modulus (K0), and pressure derivative of the bulk modulus (K0) for all of the

various DFT functionals and codes. There are two diﬀerent volumes given in the table: V0FR

corresponds to the fully relaxed, equilibrium DFT geometry and V0 is the zero pressure

volume predicted by the minimum of the Vinet ﬁt to energy versus volume data. V0FR is the volume that corresponds with the lattice constants a and c. The DFT values are compared

with experiment and QMC. LDA tends to predict both the structural and elastic constants

in close agreement with experiment. LDA lattice constants are slightly underestimated and

bulk moduli tend to be slightly overestimated. All GGA’s and hybrids tend to overestimate

the lattice constants by varying degrees. Among the GGA’s, PBE and PW91 tend to

perform the worst, and PBEsol and WC improve results. Among hybrids, PBE0, and HSE

oﬀer good agreement with experiment.

6.4.6 Enthalpy Versus Pressure

Figure 6.8 shows the HSE enthalpy curves for quartz and stishovite as one example. Once

the energy is know as a function of volume, the enthalpy as a function of pressure is easily

computed as H = U + PV. Enthalpy is used for analysis of the phase stability. The

quartz-stishovite transition is determined by the crossing of enthalpy curves. For HSE, the

stishovite enthalpy curve becomes more stable after a pressure of 5.9 GPa, marking the

transition pressure.

144 Table 6.1: Computed DFT lattice constants, volume, bulk modulus, and pressure derivative of the bulk modulus for quartz at zero pressure. Parameters are compared with experiment and QMC. Lattice constants are in units of Bohr, volumes are in units of Bohr/SiO2, and the bulk modulus is in units of GPa. Quartz 0 CRYSTAL a c V0FR V0 K0 K0 LDA 9.16 10.10 244.50 245.34 37.58 6.39 PBE 9.47 10.40 268.90 268.85 32.96 5.81 PW91 9.47 10.39 268.99 268.87 31.47 6.00 B3LYP 9.53 10.44 273.73 272.94 27.68 6.87 PBE0 9.42 10.33 264.43 262.93 31.69 6.35 0 ABINIT a c V0FR V0 K0 K0 LDA 9.17 10.12 245.44 245.96 35.16 5.63 PBE 9.49 10.42 271.10 270.13 32.59 4.54 PW91 9.48 10.40 269.93 270.68 31.37 5.20 PBEsol 9.34 10.28 259.13 259.16 31.65 5.74 WCGGA 9.38 10.30 261.65 261.54 29.18 5.89 0 VASP a c V0FR V0 K0 K0 LDA 9.28 10.21 253.90 250.31 33.07 5.93 PBE 9.53 10.43 273.69 273.39 30.63 5.26 PW91 9.56 10.47 276.29 273.85 29.19 5.44 HSE 9.46 10.37 267.77 267.10 32.42 5.06 Experimenta 9.29 10.22 254.30 254.32 34(4) 5.7(9) QMCb — — — 254(2) 32(6) 7(1) aRef. [111] bRef. [84]

145 Table 6.2: Computed DFT lattice constants, volume, bulk modulus, and pressure derivative of the bulk modulus for stishovite at zero pressure. Parameters are compared with experiment and QMC. Lattice constants are in units of Bohr, volumes are in units of Bohr/SiO2, and the bulk modulus is in units of GPa. Stishovite 0 CRYSTAL a c V0FR V0 K0 K0 LDA 7.84 5.03 154.87 154.95 326.09 4.26 PBE 7.99 5.09 162.65 162.72 281.33 4.86 PW91 7.98 5.09 162.08 162.13 284.75 4.87 B3LYP 7.94 5.06 159.53 159.38 326.36 4.05 PBE0 7.88 5.03 156.28 156.09 343.80 4.19 0 ABINIT a c V0FR V0 K0 K0 LDA 7.83 5.02 154.12 154.45 305.04 3.96 PBE 8.01 5.10 163.61 164.48 253.19 4.41 PW91 7.99 5.09 162.55 163.45 255.81 4.43 PBEsol 7.92 5.07 159.03 159.27 288.19 4.00 WCGGA 7.92 5.07 159.17 159.38 288.82 4.00 0 VASP a c V0FR V0 K0 K0 LDA 7.85 5.04 155.47 155.59 306.23 4.78 PBE 8.00 5.09 162.71 162.94 251.57 6.40 PW91 7.99 5.09 162.41 162.59 255.82 6.31 HSE 7.89 5.04 156.78 157.33 311.29 4.01 Experimenta 7.90 5.04 157.29 157.29 300(9) 4.45(15) QMCb — — — 159.0(4) 305(20) 3.7(6) aRef. [87] bRef. [84]

146 Figure 6.8: Computed HSE enthalpy curves of quartz and stishovite as a function of pressure.

147 6.4.7 Quartz-Stishovite Transition Pressures

Figure 6.9 shows a comparison of the transition pressures for all of the exchange-correlation functionals and codes used. The gray shaded box indicates the range of experimental data, while the dashed box indicates the one-sigma statistical error in the QMC prediction.

The ﬁrst feature to note is that the LDA, PBE, PW91 results are fairly uniform across all types of codes. This indicates the calculations are individually converged and the codes are performing on par with each other. The results indicate that the local LDA functional fails to predict that quartz is the correct ground state in the CRYSTAL and VASP codes, while the ABINIT code predicts a very small transition pressure. Among the semi-local GGA functionals, PBE and PW91 agree well with experiments in all codes, while PBEsol and

WC signiﬁcantly underestimate the transition pressure. Among the hybrids, B3LYP vastly overestimates the transition pressure, while PBE0 and HSE agree well with experiment and

QMC.

The LDA and GGA results agree with that of Hamann [30]. LDA is expected to perform worse for the transition pressure because of the stark diﬀerence in structure between quartz and stishovite. GGA improves the energy diﬀerence because gradient dependence of the charge density in the functional estimates the 4- to 6-fold coordination change better. It is not clear why PBEsol and WC underestimate the transition pressure. Hybrid functionals likely perform well (save B3LYP) due to description of exact exchange, and correlation energy should be small in these systems.

148 Figure 6.9: Computed quartz-stishovite transition pressures using various exchange- correlation functionals and codes. The gray shaded box indicates the range of experimental data, while the dashed box indicates the one-sigma statistical error in the QMC prediction.

149 6.5 Conclusions

Given that QMC is very expensive and standard DFT is inconsistent due to functional bias, this work has investigated reliability of hybrid functionals for silica. The results compare performance of various exchange-correlation functionals, basis sets, pseudopotentials, and codes, which are benchmarked against QMC and experiments to determine which are most accurate. Calculations compare local (LDA), semi-local GGA (PBE, PW91, PBEsol, WC), and hybrid (B3LYP, PBE0, and HSE) functionals. All electron calculations are compared against projector augmented wave (PAW) pseudopotentials and nonlocal, norm-conserving pseudopotentials. In addition, results from the CRYSTAL, ABINIT, and VASP codes are compared.

The LDA functional tends to perform well for structural properties and elastic constants.

The PBE and PW91 GGA functionals tend to overestimate the structural properties and by 2-3% and underestimate elastic properties by roughly 5%. The PBEsol and WC GGA functionals signiﬁcantly improve agreement with experiment. However, among the LDA and

GGA functionals, only PBE and PW91 compute the quartz-stishovite transition pressure

(energy diﬀerence) well. LDA is likely fails due to the large coordination change in going from quartz to stishovite and GGA is expected to improve due to gradient dependence of the density. The Hybrids tend to improve over PBE and PW91 for structural and elastic properties. B3LYP signiﬁcantly overestimates the transition pressure. HSE and

PBE0 results are generally similar and in good agreement with experiment and QMC for all properties considered. HSE provides the most consistently accurate results and oﬀers a relatively eﬃcient, yet accurate alternative to QMC.

150 Chapter 7 Conclusions

7.1 Summary

Quantum mechanics provides an exact description of microscopic matter, but an exact solution of the fully interacting, many-electron Schr¨odingerequation is intractable. The

Coulomb interaction, responsible for electron exchange-correlation energy, can not be separated to simplify the many-body problem. This means the wave-function can not legiti- mately be written as a product of single electron wave-functions, but such a wave-function is sometimes a good place to start building approximate solutions. In order to accurately compute, predict, and understand properties of matter, approximate solutions must be sought.

A reasonable approach to solve the many-body Schr¨odingerequation is to use a mean-

ﬁeld-based method in which each electron feels the average potential of all other electrons.

The simplistic Hartree approximation gives an approximate solution for a wave-function represented by a simple product of one-electron functions. The Hartree-Fock approximation improves on the Hartree approximation by representing the wave-function as a single

Slater determinant, which forces it to obey the Pauli principle. Therefore, Hartree-Fock correctly computes exchange, but ignores correlation of electrons. Amazingly, density functional theory (DFT) (Chemistry Nobel Prize 1998) succeeds in exactly mapping the many- body problem onto an independent electron problem with an eﬀective one-electron potential depending only on the electron density. However, in practice, the functional for exchange- correlation must be approximated for real materials. The approximated functionals lead to

151 unreliable results on occasion. Such problems drive development of non-mean-ﬁeld based approaches in search of even better accuracy and reliability.

Quantum Monte Carlo (QMC) is a method which abandons the mean-ﬁeld approach, and is among the class of methods providing many-body solutions to Schr¨odinger’s equation.

QMC is advantageous because it is the only many body technique capable of efficiently and accurately computing energies of large system sizes (solids). QMC achieves efficiency over other many body techniques by using stochastic methods to explicitly compute exchange and correlation with a many-body wave-function. The input trial wave-function is a product of a Slater determinant and a Jastrow correlation factor. Two common QMC algorithms are variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). The VMC method is efficient and provides an upper bound on the ground-state energy using the variational principle applied to a trial form of the many-body wave-function. However, VMC usually cannot give the required chemical accuracy. DMC further evolves a statistical representation of the many-body wave-function and projects out the ground-state solution consistent with the nodes of the trial wave-function. The fixed-node approximation is the only essential approximation, though other approximations are made to increase efficiency.

Although QMC is a highly accurate and reliable method, excessive computational expense prevents its proliﬁc use beyond a benchmarking tool for spot-checking DFT results.

Additionally, QMC calculations for solids are currently intimately tied with DFT, as QMC is too expensive to optimize structural geometries, and DFT provides orbitals for a trial wave-function. Standard DFT (LDA, GGA) is not always reliable and has well known failures, such as when computing band gaps or some silica phase transitions. A computational method with both the eﬃciency of DFT and accuracy of QMC is needed.

In effort to ratchet up DFT functional quality, new-generations of functionals have been developed conceptualized as rungs of “Jacob’s Ladder.” Each higher rung represents a new conceptual improvement: LDA, GGA, meta-GGA, and hybrids. In practice, higher-rung functionals do not always provide an improved result, but, nonetheless, efforts are still being made to fine tune functional quality as its still the most reasonable solution to the many body problem. Most recently, the screen hybrid functional, HSE, has shown particular

152 promise. HSE computes band gaps and solid properties extremely well compared with

QMC. Gaussian basis sets are sometimes used with hybrid DFT functional calculations of solids and particular care must be taken to converge the basis set to plane-wave accuracy.

Convergence studies show silica calculations need a cc-PVQZ basis set to reach plane-wave accuracy. Published literature indicates that DFT(HSE) is capable of oﬀering benchmark accuracy calculations along with QMC.

Prior to work in this thesis, almost all QMC and hybrid-HSE DFT calculations have been limited to small, cubic systems. This thesis presents three large-scale applications, which are used to expand the scope of QMC and hybrid DFT methods and establish their usefulness as benchmarking tools for complex solids.

The ﬁrst project (Chapter 4) provides the most accurate results for silicon self-interstitial defect formation energies using QMC and DFT. Both formation energies and self-diﬀusion barriers are computed for the three most stable single interstitial defects. Experimental measurement of the formation energies is very challenging. QMC (DMC) and HSE provide a benchmark for various DFT functionals, which follow the expected trend of “Jacob’s ladder”: LDA in least agreement with QMC, followed by GGA, mGGA, and HSE in best agreement with QMC. LDA underestimates the QMC single interstitial formation energy by roughly 2 eV, while GGA underestimates the formation energy by about 1.5 eV. The best QMC results predict the X and H defects are degenerate and more stable than T by about 0.6 eV. The di- and tri-interstitials also display the “Jacob’s ladder” trend, but LDA

a a and GGA energies lie 2-3.5 eV below QMC. QMC predicts I2 and I3 are the most stable of the di- and tri-interstitials.

In addition, various tests were performed to check possible sources of error in the QMC

interstitial calculations. Sources of error checked include DMC time step convergence, ﬁnite

size convergence, dependence on exchange-correlation functional, Jastrow polynomial order,

pseudopotential choice, and allowing independent Jastrow correlations for the interstitial

atom. Of all possible sources of QMC error checked, none aﬀected results presented outside

of a one-sigma error bar of chemical accuracy.

The second project (Chapter 5) uses QMC combined with DFT phonon computations

153 to compute silica equations of state, phase stability, and elasticity. This work provides highly accurate values for thermal properties for silica and expands the scope of QMC by studying the complex phase transitions in minerals away from the cubic oxides. QMC benchmark calculations are needed because a number of discrepancies between experimental data and DFT results are documented and silica is an important material to many ﬁelds of science. First, results document feasibility of QMC for computing thermodynamic and elastic properties of complex minerals. Secondly, results document improved accuracy and reliability of QMC relative to standard DFT functionals. The eﬃciency of standard DFT functionals combined with the ability of QMC to benchmark their performance makes a powerful tool for predicting and understanding materials physics that is challenging for experiment to uncover. The main geophysical implication of the results is that the CaCl2-

00 α-PbO2 transition is not associated with the global D discontinuity, indicating there is not signiﬁcant free silica in the bulk lower mantle. However, the transition should be detectable in deeply subducted oceanic crust if free silica is at high enough concentrations.

The third project (Chapter 6) investigates whether or not hybrid functionals are capable of benchmark accuracy for quartz and stishovite silica. Results compare performance of various exchange-correlation functionals, basis sets, pseudopotentials, and codes, which are benchmarked against QMC and experiments to determine which are most accurate. Cal- culations compare local (LDA), semi-local GGA (PBE, PW91, PBEsol, WC), and hybrid

(B3LYP, PBE0, and HSE) functionals. All electron calculations are compared against projector augmented wave (PAW) pseudopotentials and nonlocal, norm-conserving pseudopotentials. In addition, results from the CRYSTAL, ABINIT, and VASP codes are compared.

Silica DFT results indicate that choice of basis set, pseudopotential, or code do not make much diﬀerence as long as calculations are converged. Choice of exchange-correlation functional has the most inﬂuence on the predicted result. The LDA functional typically predicts structural properties and elastic constants within 1-2% or better of experiment and

QMC. The PBE and PW91 GGA functionals tend to overestimate the structural properties and by 2-3% and underestimate elastic properties by roughly 5%. The PBEsol and WCGGA functionals tend to improve agreement with experiment over LDA/PBE/PW91, but not for

154 the transition pressure. In fact, among the LDA and GGA functionals, only PBE and

PW91 predict the quartz-stishovite transition pressure in close agreement with experiment.

LDA, WCGGA, and PBEsol underestimate the transition pressure by more than 70%. The hybrid functionals (B3LYP, PBE0, HSE) tend to improve over PBE and PW91 for structural and elastic properties. However, B3LYP signiﬁcantly overestimates the transition pressure.

HSE and PBE0 results are generally similar and in good agreement with experiment and

QMC for all properties considered. HSE provides the most consistently accurate results for structural and elastic properties and the transition pressure. Of all the functionals studied,

HSE demonstrates consistent benchmark accuracy and is a more eﬃcient alternative to

QMC. HSE is more expensive than standard DFT by a factor of 30, but more eﬃcient than

QMC by a factor of at least 3.

In summary, the many body Schr¨odingerequation is complex and cumbersome to solve.

Standard (LDA, GGA) DFT oﬀers a powerful approximate solution, but functionals occasionally fail causing DFT to be unreliable. Often, a DFT failure can be ﬁxed by simply identifying which DFT functional best describes the system under study. Identifying the best functional for the job is a challenging task, particularly if there is no experimental measurement to compare against. Higher accuracy methods, which are vastly more computationally expensive, can be used to benchmark DFT functionals and identify those which work best for a material when experiment is lacking. If no DFT functional can perform adequately, then it is important to show more rigorous methods are capable of handling the task.

QMC is a well-known, high accuracy alternative to DFT, but QMC is too expensive to replace DFT. Hybrid DFT functionals appear to be a good compromise between QMC and standard DFT. Not many large scale computations have been done to test the feasibility or benchmark capability of either QMC or hybrid DFT for complex materials. Each of the three applications presented in this thesis expands the scope of QMC and hybrid DFT to large, scale complex materials. Results verify the benchmark accuracy of both QMC and the

HSE hybrid DFT functional for silicon defects and high pressure silica phases. Standard

DFT is still the most eﬃcient and useful for general computation. However, this thesis

155 shows that QMC and hybrid calculations can aid and evaluate shortcomings associated the exchange-correlation potential in DFT by oﬀering a route to benchmark and improve reliability of DFT predictions. As next generation computers appear, QMC and hybrid

DFT are bound to have an increasingly large impact on computational materials science.

7.2 Future Research

In future projects, it is important to continuing expanding the scope of QMC to even more complex materials in order to identify pitfalls with the QMC algorithms and continue pushing its development to get the most out next generation computers. On materials of particular importance in geophysics is magnesium silicate. Magnesium silicate is one of the most abundant minerals in Earth’s mantle. This is a ternary oxide with large primitive cells. Its structural phase transitions are found to be consistent with several seismic discontinuities with increasing depth in the mantle. At a depth of 410 km (12 GPa), the ambient orthorhombic phase called forsterite transforms to another orthorhombic phase called wadsleyite. At 520 km (15 GPa) wadsleyite transforms to cubic ringwoodite. At

660 km (25 GPa), ringwoodite transforms to a perovskite structure, and at 2740 km (125

GPa) some evidence indicates a post-perovskite structure forms, consistent with seismic data from the D” boundary. DFT typically provides lattice constants, bulk moduli, elastic constants, and sound velocities with in a few per cent of experimental measurements. How- ever, the transition pressures can vary as much as 50% between LDA and GGA, bracketing experiment [158]. QMC and hybrid DFT can help constrain the transition pressure.

Preliminary tests on Mg2SiO4 and MgO have revealed a large unexpected inﬂation in the variance of the local energies occurs when Mg and O are paired together in a solid.

Investigations into the source of the inﬂation indicate that the standard numerical orbital approximation, which only interpolates the ﬁrst derivatives of the DFT orbitals, is not accurate enough to converge both energy and variance. The MgO system requires additional interpolation of the Laplacian of the orbitals in order to also converge the variance. A general understanding of this potential pitfall for all complex solids still needs to be determined.

156 Capability to interpolate the Laplacian also must be added to the production QMC code,

CASINO.

Lastly, a possible, important new project could be the design and construction of new exchange correlation functionals based on QMC calculations of solids. The LDA functional is based on QMC data of the free electron gas. With QMC calculations of large, complex materials now feasible, the study of exchange-correlation in real systems may allow the development of new functionals that are even better and more eﬃcient than hybrids. Both of the projects mentioned here would be important for advancing the ﬁeld of computational materials science.

157 Bibliography

[1] I. Newton. Philosophiæ Naturalis Principia Mathematica (1687).

[2] P. A. M. Dirac. The Principles of Quantum Mechanics. Claredon Press, Oxford, 1rst edition, 1930.

[3] R. Shankar. Principles of Quantum Mechanics. Kluwer Academic, New York, New York, 2nd edition, 1994.

[4] E. Schr¨odinger.An undulatory theory of the mechanics of atoms and molecules. Phys Rev., 28:1049, 1926.

[5] R. M. Martin. Electronic Structure. Cambridge University Press, Cambridge, United Kingdom, 2004.

[6] M. Springborg. Methods of Electronic Structure Calculations: From molecules to solids. Wiley, New York, New York, 2000.

[7] D. S. Sholl and J. A. Steckel. Density Functional Theory: A Practical Introduction. Wiley, Hoboken, New Jersey, 2009.

[8] M. P. Nightingale and C. J. Umrigar. Quantum Monte Carlo Methods in Physics and Chemistry. Kluwer Academic, Proceedings of the NATO Advanced Study Institute on Quantum Monte Carlo Methods in Physics and Chemistry, Ithaca, NY, 1999.

[9] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal. Quantum Monte Carlo Simulations of Solids. Rev. Mod. Phys., 73:33, 2001.

[10] D. Young. Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems. Wiley Interscience, New York, New York, 2001.

[11] M. Born and R. Oppenheimer. Zur Quantentheorie der Molekeln. Annalen der Physik, 389:457, 1927.

[12] N. Ashcroft and D. Mermin. Solid State Physics. Saunders College, Philadelphia, PA, 1976.

[13] P. Hohenberg and W. Kohn. Self consistent equations including exchange and correlation eﬀects. Phys. Rev., 136, 1964. 158 [14] W. Kohn L. J. Sham. Inhomogeneous electron gas. Phys. Rev., 140, 1965.

[15] J. P. Perdew and Karla Schmidt. Jacob’s ladder of density functional approximations for the exchange-correlation energy. In V. Van Doren, C. Van Alsenoy, and P. Geer- lings, editors, Density Functional Theory and Its Applications to Mateials, pages 1–20. American Institute of Physics, 2001.

[16] R. J. Needs, M. D. Towler, N. D. Drummond, and P. L´opez-R´ıos. Continuum variational and diﬀusion Monte Carlo calculations. J. Phys-Conds Mat., 22:023201, 2010.

[17] N. A. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equation of state calculation by fast computing machines. J. Chem. Phys., 21:1087, 1953.

[18] H. F. Trotter. On the product of semi-groups of operators. P AM MATH SOC, 10:545, 1959.

[19] P. L´opez-R´ıos,A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs. Inhomoge- neous backﬂow transformations in quantum Monte Carlo. Phys. Rev. E, 74:066701, 2006.

[20] N. D. Drummond, M. D. Towler, and R. J. Needs. Jastrow correlation factor for atoms, molecules, and solids. Phys. Rev. B, 70:235119, 2004.

[21] R. J. Needs, M. D. Towler, N. D. Drummond, and P. L´opez-R´ıos. CASINO version 2.1 User Manual, University of Cambridge, Cambridge (2007).

[22] D. Alf`eand M. J. Gillan. Eﬃcient localized basis set for quantum Monte Carlo calculations on condensed matter. Phys. Rev. B, 70:161101(R), 2004.

[23] N. D. Drummond, R. J. Needs, A. Sorouri, and W. M. C. Foulkes. Finite-size errors in continuum quantum monte carlo calculations. Phys. Rev. B, 78:125106, 2008.

[24] L. M. Fraser, W. M. C. Foulkes, G. Rajagopal, R. J. Needs, S. D. Kenny, and A. J. Williamson. Finite-size eﬀects and Coulomb interactions in quantum Monte Carlo calculations for homogeneous systems with periodic boundary conditions. Phys. Rev. B, 53:1814, 1996.

[25] S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holtzman. Finite-Size Error in Many-Body Systems with Long-Range Interactions. Phys. Rev. Lett., 97:076404, 2006.

[26] H. Kwee, S. Zhang, and H. Krakauer. Finite-Size correction in many-body electronic structure calculations. Phys. Rev. Lett., 100:126404, 2008.

[27] M. Pozzo and D. Alf`e. Structural properties and enthalpy of formation of magnesium hydride from quantum Monte Carlo calculations. Phys. Rev. B, 77:104103, 2008.

[28] M. Casula. Beyond the locality approximation in the standard diﬀusion Monte Carlo method. Phys. Rev. B, 74:161102(R), 2006.

159 [29] R. E. Cohen. Bonding and electronic structure of minerals. In K. Wright and C. R. A. Catlow, editors, Microscopic Properties and Processes in Minerals, page 201. Kluwer, The Netherlands, 1999.

[30] D. R. Hamann. Generalized gradient theory for silica phase transitions. Phys. Rev. Lett., 76:660, 1996.

[31] Th. Demuth, Y. Jeanvoine, J. Hafner, and J. G. Angy´an.Polymorphism´ in silica studied in the local density and generalized-gradient approximations. J. Phys.-Condens. Mat., 11:3833, 1999.

[32] D. M. Ceperley and B. J. Alder. Ground State of the Electron Gas by a Stochastic Method. Phys. Rev. Lett., 45:566, 1980.

[33] J. P. Perdew. Electronic structure of solids ’91. page 11. Akademie Verlag, Berlin, 1991.

[34] J. Perdew and M. Enzerhof. Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys., 105:9982, 1996.

[35] A. D. Becke. A new mixing of Hartree-Fock and local density-functional theories. J. Chem. Phys., 98:1372, 1993.

[36] A. D. Becke. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys., 98:5648, 1993.

[37] A. D. Becke. Density-functional exchange-energy approximation iwth correct asymp- totic behavior. Phys. Rev. A, 38:3098, 1988.

[38] C. Lee, W. Yang, and R. G. Parr. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B, 37:785, 1988.

[39] S. H. Vosko, L. Wilk, and M. Nusair. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys., 59:1200, 1980.

[40] J. Heyd, G. E. Scuseria, and M. Ernzerhof. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys., 118:8207, 2003.

[41] J. Heyd and G. E. Scuseria. Eﬃcient hybrid density functional calculations in solids: Assessment of the Heyd-Scuseria-Ernzerhof screened Coulomb hybrid functional. J. Chem. Phys., 121:1187, 2004.

[42] J. Heyd, J. E. Peralta, G. E. Scuseria, and R. L. Martin. Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhor screened hybrid functional. J. Chem. Phys., 123:174101, 2005.

[43] J. Heyd, J. E. Peralta, G. E. Scuseria, and R. L. Martin. Erratum: ”Hybrid functionals based on a screened Coulomb potential“ [J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys., 124:219906(E), 2006.

160 [44] A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys., 125:224106, 2006. [45] C. Adamo and V. Barone. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys., 110:6158, 1999. [46] M. Ernzerhof and G. E. Scuseria. Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional. J. Chem. Phys., 110:5029, 1999. [47] J. Muscat, A. Wander, and N. M. Harrison. On the prediction of band gaps from hybrid functional theory. Chem. Phys. Lett., 342:397, 2001. [48] N. Wilson and J. Muscat. The calculation of structural, elastic and phase stability properties of minerals using first principles techniques: A comparison of the HF, DFT and hybrid functional treatments of exchange and correlation. Molc. Sim., 28:903, 2002. [49] A. Lichanot. Hartree-Fock and density functional calculations of the elastic constants of alkaline-earth oxides: comparison with experiment. J. Phys.: Condens. Matter, 116:543, 2000. [50] C. Zhao-Ying and Y. Jin-long. The B3LYP hybrid density functional study on solids. Front. Phys. China, 3:339, 2006. [51] K. Doll. Electronic structure of GdN, and the influence of exact exchange. J. Phys.: Condens. Matter, 20:075214, 2008. [52] Y. Noel, M. Catti, Ph. D’Arco, and R. Dovesi. The vibrational frequencies of foresterite Mg2SiO4: an all-electon ab initio study with the CRYSTAL code. Phys. Chem. Minerals, 33:383, 2006. [53] F. Marinelli and A. Lichanot. Elastic constants and electronic structure of alkaline- earth chalcogenides. Performances of various hamiltonians. Chem. Phys. Lett., 367:430, 2003. [54] E. R. Batista, J. Heyd, R. G. Hennig, B. P. Uberuaga, R. L. Martin, G. E. Scuseria, C. J. Umrigar, and J. W. Wilkins. Comparison of screened hybrid density functional theory to diffusion Monte Carlo in calculations of total energies of silicon phases and defects. Phys. Rev. B, 74:121102(R), 2006. [55] T. M. Henderson, A. F. Izmaylov, G. Scuseria, and A. Savin. The importance of middle-range Hartree-Fock type exchange for hybrid density functionals. J. Chem. Phys., 127:221103, 2007. [56] J. E. Peralta, J. Heyd, and G. E. Scuseria. Spin-orbit splittings and energy band gaps calculated with the Heyd-Scuseria-Ernzerhof screened hybrid functional. Phys. Rev. B, 74:073101, 2006. [57] E. N. Brothers, A. F. Izmaylov, J. O. Normand, V. Barone, and G. E. Scuseria. Accurate solid-state band gaps via screened hybrid electronic structure claculations. J. Chem. Phys., 129:011102, 2008. 161 [58] O. A. Vydrov, J. Heyd, A. V. Krukau, and G. E. Scuseria. Importance of short- range versus long-range Hartree-Fock exchange for the performance of hybrid density functionals. J. Chem. Phys., 125:074106, 2006.

[59] I. C. Gerber and J. G. Angy´an.Range´ separated hybrid density functional with long- range Hartree-Fock exchange applied to solids. J. Chem. Phys., 127:054101, 2007.

[60] J. L. F. Da Silva and M. V. Ganduglia-Pirovano and J. Sauer and V. Bayer and G. Kresse. Hybrid functionals applied to rare-earth oxides: The example of ceria. Phys. Rev. B, 75:045121, 2007.

[61] P. M. Fahey, P. B. Griﬃn, and J. D. Plummer. Point defects and dopant diﬀusion in silicon. Phys. Rev. Lett, 61:289, 1989.

[62] D. J. Eaglesham, P. A. Stolk, H. J. Grossmann, and J. M. Poate. Implantation and transient B diﬀusion in Si: The source of the interstitials. Appl. Phys. Lett., 65:2305, 1994.

[63] D. A. Richie, J. Kim, S. A. Barr, K. R. A. Hazzard, R. G. Hennig, and J. W. Wilkins. Complexity of small silicon self-interstitial defects. Phys. Rev. Lett, 92:45501, 2004.

[64] L. A. Marqu´es,L. Pelaz, P. Castrillo, and J. Barbolla. Molecular dynamics studies of the conﬁgurational and energetic properties of the silicon self-interstitial. Phys. Rev. B, 71:085204, 1989.

[65] Y. Bar-Yam and J. D. Joannopoulos. Silicon self-interstitial migration: Multiple paths and charge states. Phys. Rev. B, 30:1844, 1984.

[66] P. E. Bl¨ochl, E. Smargiassi, R. Car, D. B. Laks, W. Andreoni, , and S. T. Pantelides. First-principles calculations of self-diﬀusion constants in silicon. Phys. Rev. Lett., 70:2435, 1993.

[67] Y. A. Du, R. G. Hennig, and J. W. Wilkins. Diﬀusion mechanisms for silicon di- interstitials. Phys. Rev. B, 73:245203, 2006.

[68] Y. A. Du, S. A. Barr, K. R. A. Hazzard, T. J. Lenosky, R. G. Hennig, and J. W. Wilkins. Fast diﬀusion mechanism of silicon tri-interstitial defects. Phys. Rev. B, 72:241306(R), 2005.

[69] Y. A. Du, R. G. Hennig, T. J. Lenosky, and J. W. Wilkins. From compact point defects to extended structures in silicon. Eur. Phys. J. B, 57:229, 2007.

[70] R. J. Needs. First-principles calculations of self-interstitial defect structures and diffusion paths in silicon. J. Phys: Condens. Matt., 11:10437, 1999.

[71] W. K. Leung, R. J. Needs, G. Rajagopal, S. Itoh, and S. Ihara. Calculations of Silicon Self-Interstitial Defects. Phys. Rev. Lett., 83:2351, 1999.

[72] A. J. Williamson, R. Q. Hood, R. J. Needs, and G. Rajagopal. Diﬀusion quantum Monte Carlo calculations of the excited states of silicon. Phys. Rev. B, 57:12140, 1998.

162 [73] R. G. Hennig, A. Wadehra, K. P. Driver, W. D. Parker, C. J. Umrigar, and J. W. Wilkins. Phase transformation in si from semiconducting diamond to metallic β- sn phase in qmc and dft under hydrostatic and anisotropic stress. Phys. Rev. B, 82:014101, 2010.

[74] W. D. Parker, J. W. Wilkins, and R. G. Hennig. Accuracy of Quantum Monte Carlo Methods for Point Defects in Solids. Phys. Status Solidi B, 00:00, published online 16 August, 2010.

[75] W. D. Parker, Ph.D. Thesis, Ohio State University, 2010.

[76] Private communication with Jos´eLu´ısMartins http://bohr.inesc-mn.pt/∼jlm/.

[77] C. J. Umrigar. http://www.tc.cornell.edu/∼cyrus/champ.html.

[78] G. Kresse and J. Furthm¨uller. Eﬃcient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 54:11169, 1996.

[79] C. J. Umrigar and C. Filippi. Energy and Variance Optimization of Many-Body Wave Functions. Phys. Rev. Lett., 94:150201, 2005.

[80] J. Toulouse and C. J. Umrigar. Optimization of quantum Monte Carlo wave functions by energy minimization. J. Chem. Phys., 126:084102, 2007.

[81] C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig. Alleviation of the Fermion-Sign Problem by Optimization of Many-Body Wave Functions. Phys. Rev. Lett., 98:110201, 2007.

[82] H. Bracht, E. E. Haller, and R. Clark-Phelps. Silicon self-diﬀusion in isotope het- erostructures. Phys. Rev. Lett., 81:393, 1998.

[83] A. Ural, P. B. Griﬃn, and J. D. Plummer. Self-diﬀusion in silicon: Similarity between the properties of native point defects. Phys. Rev. Lett., 83:3454, 1999.

[84] K. P. Driver, R. E. Cohen, Z. Wu, B. Militzer, P. L´opez R´ıos,M. D. Towler, R. J. Needs, and J. W. Wilkins. Quantum Monte Carlo computations of phase stability, equations of state, and elasticity of high-pressure silica. Proc. Natl. Acad. Sci. U S A, 107:9519, 2010.

[85] L. Levien, C. T. Prewitt, and D. J. Weidner. Structure and elastic properties of quartz at pressure. Am. Mineral, 65:920, 1980.

[86] L. Levien and C. T. Prewitt. High-pressure crystal structure and compressibility of coesite. Am. Mineral, 66:324, 1981.

[87] N. L. Ross, J. F. Shu, R. M. Hazen, and T. Gasparik. High-pressure crystal chemistry of stishovite. Am. Mineral., 75:739, 1990.

[88] K. J. Kingma, R. E. Cohen, R. J. Hemley, and H. K.Mao. Transformation of stishovite to a denser phase at lower-mantle pressures. Nature, 374:243, 1995.

163 [89] L. S. Dubrovinsky, N. A. Dubrovinskaia, S. K. Saxena, F. Tutti, S. Rekhi, T. Le Bihan, G. Shen, and J. Hu. Pressure-induced transformations of cristobalite. Chem. Phys. Lett., 333:264, 2001.

[90] M. Murakami, K. Hirose, S. Ono, and Y. Ohishi. Stability of CaCl2-type and αPbO2- type SiO2 at high pressure and temperature determined by in-situ X-ray measurements. Geophys. Res. Lett., 30:1207, 2003.

[91] S. E. Kesson, J. D. FitzGerald, and J. M. G. Shelley. Mineral chemistry and density of subducted basaltic crust at lower mantle pressures. Nature, 372:767, 1994.

[92] E. Knittle and R. Jeanloz. Earth’s core-mantle boundary: Results of experiments at high pressures and temperatures. Science, 251:1483, 1991.

[93] R. E. Cohen. First-principles predictions of elasticity and phase transitions in high pressure SiO2 and geophysical implications. In Y. Syono and M. H. Manghnani, editors, High-Pressure Research: Application to Earth and Planetary Sciences, page 425. Terra Scientiﬁc, Tokyo and Am. Geophys. Union, Washington, D.C., 1992.

[94] S. R. Shieh and T. S. Duﬀy. Strength and elasticity of sio2 across the stishovite-cacl2- type strutural phase boundary. Phys. Rev. Lett., 89:255507, 2002.

[95] V. V. Brazhkin, L. E. McNeil, M. Grimsditch, N. A. Bendeliani, T. I. Dyuzheva, and L. M. Lityagina. Elastic constants of stishovite up to its amorphization temperature. J. Phys.: Condens. Matter, 17:1869, 2005.

[96] B. B. Karki, L. Stixrude, and J. Crain. Ab initio elasticity of three high-pressure polymorphs of silica. Geophys. Res. Lett., 24:3269, 1997.

[97] F. Jiang, G. D. Gwanmesia, T. I. Dyuzheva, and T. S. Duﬀy. Elasticity of stishovite and acoustic mode softening under high pressure by Brillouin scattering. Phys. Earth. Planet. In., 172:235, 2009.

[98] A. Togo, F. Oba, and I. Tanaka. First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures. Phys. Rev. B, 78:134106, 2008.

[99] C. Lee and X. Gonze. SiO2 stishovite under high pressure: Dielectric and dynamical properties and the ferroelastic phase transition. Phys. Rev. B, 56:7321, 1997.

[100] J. A. Akins and T. J. Ahrens. Dynamic compression of sio2: A new interpretation. Geophys. Res. Lett., 29:1394, 2002.

[101] B. B. Karki, M. C. Warren, L. Stixrude, G. J. Ackland, and J. Crain. Ab initio studies of high-pressure structural transformations in silica. Phys. Rev. B, 55:3465, 1997.

[102] T. Tsuchiya, R. Caracas, and J. Tsuchiya. First principles determination of the phase boundaries of high-pressure polymorphs of silica. Geophys. Res. Lett., 31:11610, 2004.

[103] A. R. Oganov, M. J. Gillan, and G. D. Price. Structural stability of silica at high pressures and temperatures. Phys. Rev. B, 71:064104, 2005.

164 [104] B. B. Karki, L. Stixrude, and R. M. Wentzcovitch. High-pressure elastic properties of major materials of Earth’s mantle from ﬁrst principles. Rev. Geophys., 39:507, 2001.

[105] D. Alf`e,M. Alfredsson, J. Brodholt, M. J. Gillan, M. D. Towler, and R. J. Needs. Quantum Monte Carlo calculations of the structural properties and the B1-B2 phase transition of MgO. Phys. Rev. B, 72:014114, 2005.

[106] J. Kolorenˇcand L. Mitas. Quantum Monte Carlo calculations of structural properties of FeO under pressure. Phys. Rev. Lett., 101:185502, 2005.

[107] For OPIUM pseudopotential generation programs, see http://opium.sourceforge.net.

[108] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.-Y. Raty, and D. C. Allan. First-principles computation of material properties: the ABINIT software project. Comp. Mater. Sci., 25:478, 2002.

[109] O. L. Anderson. Equations of State of Solids for Geophysics and Ceramic Sceince. Oxford Univ Press, New York, New York, 1995.

[110] X. Gonze and C. Lee. Dynamical matricies, Born eﬀective charges, dielectric permit- tivity tensors, and interatomic force constants from density functional perturbation theory. Phys. Rev. B, 55:10355, 1997.

[111] R. M. Hazen, L. W. Finger, R. J. Hemley, and H. K. Mao. High pressure crystal chemistry and amorphization of α-quartz. Solid State Commun., 72:507, 1989.

[112] D. Andrault, G. Fiquet, F. Guyot, and M. Hanﬂand. Pressure-induced Landau-type transition in stishovite. Science, 282:720, 1998.

[113] D. Andrault, J. L. Mosenfelder R. J. Angel, and T. Le Bihan. Equation of state of stishovite to lower mantle pressures. Am. Mineral., 88:301, 2003.

[114] P. Vinet, J. R. Smith, J. Ferrante, and J. H. Rose. Temperature eﬀects on the universal equation of state of solids. Phys. Rev. B, 35:1945, 1987.

[115] Z. Wu and R. E. Cohen. More accurate generalize gradient approximation for solids. Phys. Rev. B, 73:235116, 2006.

[116] L. E. Reichl. A Modern Course in Statistical Physics. Wiley-Interscience, New York, New York, 1998.

[117] J. P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple. Phys. Rev. Lett., 77:3865, 1996.

[118] T. J. Ahrens and V. G. Gregson Jr. Shock compression of crustal rocks: Data for quartz, calcite, and plagioclase rocks. J. Geophys. Res., 69:4839, 1964.

[119] J. C. Beottger. New model for the shock-induced α-quartz → stishovite phase transition in silica. J. Appl. Phys., 72:5500, 1992.

165 [120] M. Akaogi and A. Navrotsky. The quartz-coesite-stishovite transformations: new calorimetric measurements and calculation of phase diagrams. Phys. Earth Planet. Inter., 36:124, 1984. [121] J. L. Holm and O. J. Kleppa. Thermodynamics of polymorphic transformations in silica. Thermal properties from 5 to 1070 K and pressure-temperature stability fields for coesite and stishovite. Geochemica et Cosmochimica Acta, 31:2289, 1967. [122] S. N. Lou, T Çaˇgin,A Strachan, W. A. Goddard III, and T. J. Ahrens. Molecular dynamics modeling of stishovite. Earth Planet. Sc. Lett., 202:157, 2002. [123] R. Beohler. High-pressure experiments and the phase diagram of lower mantle and core materials. Rev. Geophys., 38:221, 2000. [124] Y. Sumino and O. L. Anderson. Elastic constants of minerals. In R. S. Carmichael, editor, CRC Handbook of Physical Properties of Rocks, vol. III, pages 39–138. CRC Press, Boca Raton, FL, 1984. [125] R. Boehler. Adiabats of quartz, coestite, olivine, and magnesium oxide to 50 kbar and 1000 K, and the adiabatic gradient in Earth’s mantle. J. Geophys. Res., 87:5501, 1982. [126] Y. Nisihiara, K. Nakayama, E. Takahashi, I. Iguchi, and K. Funakoshi. P-V-T equation of state of stishovite to the mantle transition zone conditions. Phys. Chem. Miner., 31:660, 2005. [127] S. N. Lou, J. L.Mosenfelder, P. D. Asimow, and T. J. Ahrens. Direct shock wave loading of stishovite to 235 GPa: Implications for perovskite stability relative to an oxide assemblage at lower mantle conditions. Geophys. Res. Lett., 29:1691, 2002. [128] R. E. Cohen and O. Gülseren.Thermal equation of state of tantalum. Phys. Rev. B, 63:224101, 2001. [129] X. Sha and R. E. Cohen. Lattice dynamics and thermodynamics of bcc iron under pressure: First-principles linear response study. Phys. Rev. B, 73:104303, 2006. [130] T. H. K. Barron. Thermal expansion, Grüneisenfunctions and static lattice properties of quartz. J. Phys. C: Solid State, 15:4311, 1982. [131] K. Kihara. An X-ray study of the temperature dependence of the quartz structure. Eur. J. Mineral., 2:63, 1990. [132] M. A. Carpenter, E. K. H. Salje, A. Graeme-Barber, B. Wruck, M. T. Dove, and K. S. Knight. Calibration of excess thermodynamic properties and elastic constant variations associated with the alpha ↔ beta phase transition in quartz. Am. Mineral., 83:2, 1998. [133] J. Liu, J. Zhang, L. Flesch, B. Li, D. J. Weidner, and R. C. Liebermann. Thermal equation of state of stishovite. Phys. Earth Planet. In., 112:257, 1999. [134] H. Ito, K. Kawada, and S. Akimoto. Thermal expansion of stishovite. Phys. Earth Planet. In., 8:277, 1974. 166 [135] J. S. Weaver, T. Takahashi, and W. A. Basset. Thermal expansion of stishovite (abstr.). EOS T Am. Geophys. Un., 54:475, 1973.

[136] A. M. Doroshev, V. M. Galkin, and G. N. Kuznetsov. Thermal expansion of stishovite at 98- 420 K. Geokhimiya, 10:1463, 1987. English translation: Geochem. Int. 25, 92, (1988).

[137] L. Voˇcadlo,J. P. Poirer, and G. D. Price. Gr¨uneisenparameters and isothermal equations of state. Am. Mineral., 85:390, 2000.

[138] A. M. Dziewonskia and D. L. Anderson. Preliminary reference Earth model. Phys. Earth Planet. Inter., 25:297, 1981.

[139] M. Murakami, K. Hirose, K. Kawamura, N. Sata, and Y. Ohishi. Post-perovskite phase transition in MgSiO3. Science, 304:855, 2004. [140] A. R. Hutko, T. Lay, J. Revenaugh, and E. J. Garnero. Anticorrelated seismic velocity anamalies from post-perovskite in the lowermost mantle. Science, 320:1070, 2008.

[141] T. H. K. Barron and M. L. Klein. Second-order elastic constants of a solid under stress. Proc. Phys. Soc., 85:523, 1965.

[142] I. Sidorin, M. Gurnis, and D. V. Helmberger. Evidence for a ubiquitous seismic discontinuity at the base of the mantle. Science, 286:1326, 1999.

[143] J. Perdew and A. Zunger. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B, 23:5048, 1981.

[144] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett., 100:136406, 2008.

[145] F. Tran, R. Laxkowski, P. Blaha, and K. Schwarz. Performance on molecules, surfaces, and solids of the wu-cohen gga exchange-correlation energy functional. Phys. Rev. B, 75:115131, 2007.

[146] K. Kim and K. D. Jordan. Comparison of density functional and MP2 calculations on the water monomer and dimer. J. Phys. Chem., 98:10089, 1994.

[147] P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch. Ab Initio calculation of vibrational absorption and circular dichroism spectra using density functional force ﬁelds. J. Phys. Chem., 98:11624, 1994.

[148] M. Ernzerhof, J. P. Perdew, and K. Burke. Coupling-constant dependence of atomization energies. Int. J. Quantum Chem., 64:285, 1997.

[149] B. Civalleri, C. M. Zicovich-Wilson, P. Ugliengo, V. R. Saunders, and R. Dovesi. A periodic ab initio study of the structure and relative stability of silica polymorphs. Chem. Phys. Lett., 292:394, 1998.

167 [150] C. M. Zicovich-Wilson, F. Pascale, C. Roetti, V. R. Saunders, R. Orlando, and R. Dovesi. Calculation of the vibration frequencies of α-quartz: The eﬀect of Hamil- tonian and basis set. J. Comput. Chem., 25:1873, 2004.

[151] G. Ottonello, M. Vetuschi Zuccolini, and B. Civalleri. Thermo-chemical and thermo- physical properties of stishovite: An ab-initio all-electron investigation. CALPHAD, 33:457, 2009.

[152] R. Dovesi, V.R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-Wilson, F. Pas- cale, B. Civalleri, K. Doll, N. M. Harrison, I. J. Bush, Ph. D’Arco, and M. Llunell. CRYSTAL06 User’s Manual, University of Torino, Torino, (2006).

[153] A. Gr¨uneich and B. A. Heb. Choosing GTO basis sets for HF calculations. Theor. Chem. Acc., 100:253, 1998.

[154] P. D’Arco, G. Sandrone, R. Dovesi, R Orlando, and V. R. Saunders. A quantum mechanical study of the perovskite structure type of MgSiO3. Phys. Chem. Minerals, 20:407, 1993.

[155] R. Dovesi, C. Roetti, C. Freyria-Fava, M. Prencipe, and V. R. Saunders. On the elastic properties of lithium, sodium and potassium oxide. An ab initio study. Chem. Phys., 156:11, 1991.

[156] R. E. Cohen. Bonding and elasticity of stishovite SiO2 at high pressure: Linearized augmented plane wave calculations. Am. Mineral, 76:733, 1991.

[157] A. Zupan, P. Blaha, K. Schwarz, and J. P. Perdew. Pressure-induced phase transitions in solid Si, SiO2, and Fe: Performance of local-spin-density and generalized-gradient- approximation density functionals. Phys. Rev. B, 58:11266, 1998.

[158] Y. Yu and R. M. Wentzcovitch. α ↔ β ↔ γ phase transformations in Mg2SiO4 in Earth’s transition zone. Submitted to Earth Planet. Sci. Lett., 00:00, 2007. http://www.vlab.msi.umn.edu/reports/allpublications/ﬁles/77 EPSL-ABG.pdf.

[159] A. C. Melissinos. Experiments in Modern Physics. Academic Press, Boston, 1966.

[160] O. A. Asbjornsen. Error in the propagation of Error formula. AiChE Journal, 32:332, 1986.

[161] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes. Cambridge University Press, Cambridge, 1986.

[162] D. Frenkel and B. Smit. Understanding Molecular Simulation: From Algotithms to Applications. Academic Press, 1996.

[163] P. R. C. Kent, R. Q. Hood, A. J. Williamson, and R. J. Needs. Finite-size errors in quantum many-body simulations of extended systems. Phys. Rev. B, 59:1917, 1999.

[164] A. J. Williamson, G. Rajagopal, R. J. Needs, L. M. Fraser, W. M. C. Foulkes, Y. Wang, and M.-Y. Chou. Elimination of Coulomb ﬁnite-size eﬀects in quantum many-body simulations. Phys. Rev. B, 55:R4851, 1997.

168 Appendix A Error Propagation in QMC Thermodynamic Parameters

QMC computes total ground state energies, which have a statistical error bar. The error bar propagates in to all quantities computed from the QMC energies. There are two methods for propagation of error: 1) analytic Taylor series expansion or 2) Monte Carlo.

A.1 Taylor Expansion Method

The Taylor expansion method [159, 160] of propagating error is typically only expanded to

ﬁrst order. The methods assumes there is a function

x = f(u, v, . . .), (A.1)

with an average value (assuming maximum likelihood)

x¯ = f(¯u, v,¯ . . .). (A.2)

The uncertainty in the function x is found by considering the spread from combining measurements of ui, Vi, ect, into xi’s.:

xi = f(ui, vi,...), (A.3)

which has a variance of 1 X σ2 = (x − x¯ )2 (A.4) x N i i

169 In order to estimate the error, x is Taylor expanded about the mean values of the

parameters (u, v, ...) to ﬁrst order:

∂x ∂x x − x¯ = (u − u¯) + (v − v¯) (A.5) i ∂u i ∂v i

Plugging this expression into the variance (Equation A.4) produces the error propagation equation: ∂x2 ∂x2 ∂x ∂x σ2 = σ2 + σ2 + ··· + 2σ2 , (A.6) x u ∂u v ∂v uv ∂u ∂v The rightmost term is zero of the dependent variables are completely independent. Code- pendent variables are possible in complex systems, requiring higher order mixed terms.

The error propagation can be succinctly written as

X ∂x ∂x σ2 = Cov[u, v], (A.7) x ∂u ∂v where Cov[] is the covariance matrix.

A.2 Monte Carlo Method

A more powerful method, but one which involves some additional simulation, is the Monte

Carlo method of error propagation [161]. The advantage of the Monte Carlo method is that all nonlinear terms the Taylor expansion leaves out of the propagation formula are included.

Thus, codependent variables are taken fully into account.

The Monte Carlo method is conceptually simple. One repeatedly adds random Gaussian noise proportional to the statistical error to the actual data and proceeds with the analysis to compute the desired property. After such a procedure is completed many times, the standard deviation in the results provides is the desired propagated error bar.

As an example, imagine there are a set of QMC energies Ei, each with a statistical error of σi. Imagine that an equation of state is ﬁt, weighted appropriately to the error bars and then, say the heat capacity is computed from the equation of state. In order to determine the error propagated into the heat capacity from the initial set of energies one can use a Monte

Carlo simulation. The simulation allows Gaussian ﬂuctuations of the original energies with

170 a standard deviation that corresponds to the size of the one-sigma statistical error for each energy. Then the ﬁt is performed and the heat capacity is computed. Independent iterations of the same procedure are repeated in a loop several hundred or thousands of times. Then the standard deviation of those hundreds or thousands of computed heat capacities is the propagated error bar on the heat capacity. The number of iterations should be checked for convergence.

171 Appendix B Optimized cc-pVQZ Gaussian Basis Set used for Silica

cc-pVQZ EMSL Basis Set Exchange Library 1/24/11

BASIS SET: (12s,6p,3d,2f,1g) -> [5s,4p,3d,2f,1g]

61420.0000000 0.0000900

9199.0000000 0.0006980

2091.0000000 0.0036640

590.9000000 0.0152180

192.3000000 0.0524230

69.3200000 0.1459210

26.9700000 0.3052580

11.1000000 0.3985080

4.6820000 0.2169800

61420.0000000 -0.0000200

9199.0000000 -0.0001590

2091.0000000 -0.0008290

590.9000000 -0.0035080

192.3000000 -0.0121560

172 69.3200000 -0.0362610

26.9700000 -0.0829920

11.1000000 -0.1520900

4.6820000 -0.1153310

1.4280000 1.0000000

0.5547000 1.0000000

0.2067000 1.0000000

63.4200000 0.0060440

14.6600000 0.0417990

4.4590000 0.1611430

1.5310000 1.0000000

0.6500000 1.0000000

0.3000000 1.0000000

3.7750000 1.0000000

1.3000000 1.0000000

0.4440000 1.0000000

2.6660000 1.0000000

173 0.8590000 1.0000000

BASIS SET: (16s,11p,3d,2f,1g) -> [6s,5p,3d,2f,1g]

Si S

513000.0000000 0.260920D-04

76820.0000000 0.202905D-03

17470.0000000 0.106715D-02

4935.0000000 0.450597D-02

1602.0000000 0.162359D-01

574.1000000 0.508913D-01

221.5000000 0.135155D+00

90.5400000 0.281292D+00

38.7400000 0.385336D+00

16.9500000 0.245651D+00

6.4520000 0.343145D-01

2.8740000 -0.334884D-02

1.2500000 0.187625D-02

Si S

513000.0000000 -0.694880D-05

76820.0000000 -0.539641D-04

17470.0000000 -0.284716D-03

4935.0000000 -0.120203D-02

1602.0000000 -0.438397D-02

574.1000000 -0.139776D-01

221.5000000 -0.393516D-01

90.5400000 -0.914283D-01

38.7400000 -0.165609D+00

16.9500000 -0.152505D+00

6.4520000 0.168524D+00

2.8740000 0.569284D+00

174 1.2500000 0.398056D+00

Si S

513000.0000000 0.178068D-05

76820.0000000 0.138148D-04

17470.0000000 0.730005D-04

4935.0000000 0.307666D-03

1602.0000000 0.112563D-02

574.1000000 0.358435D-02

221.5000000 0.101728D-01

90.5400000 0.237520D-01

38.7400000 0.443483D-01

16.9500000 0.419041D-01

6.4520000 -0.502504D-01

2.8740000 -0.216578D+00

1.2500000 -0.286448D+00

Si S

0.3599000 1.0000000

Si S

0.1699000 1.0000000

Si P

1122.0000000 0.448143D-03

266.0000000 0.381639D-02

85.9200000 0.198105D-01

32.3300000 0.727017D-01

13.3700000 0.189839D+00

5.8000000 0.335672D+00

2.5590000 0.379365D+00

1.1240000 0.201193D+00

Si P

175 1122.0000000 -0.964883D-04

266.0000000 -0.811971D-03

85.9200000 -0.430087D-02

32.3300000 -0.157502D-01

13.3700000 -0.429541D-01

5.8000000 -0.752574D-01

2.5590000 -0.971446D-01

1.1240000 -0.227507D-01

Si P

0.3988000 1.0000000

Si P

0.1533000 1.0000000

Si D

0.3020000 1.0000000

Si D

0.7600000 1.0000000

Si F

0.2120000 1.0000000

Si F

0.5410000 1.0000000

END

176 Appendix C Details of the Ewald and MPC Interaction in Periodic Calculations

C.1 Ewald Interaction

In solid-state simulations of real materials, it is not computationally feasible to use a clutster with millions of atoms to approximate a bulk structure. Instead, one usually uses a supercell method, in which a cluster containing a small number of atoms is artificially repeated through out space via periodic boundry conditions. One difficultly that arises in such a method is finding a convenient way to sum up all the contributions to the electrostatic potential energy restulting from the coulomb interaction of the charges in the supercell with their periodic images. A popular scheme for performing this sum is known as the

Ewald method [162].

To demonstrate the Ewald method, consider an ideal system of positive and negative charges in a cube of side length L subject to periodic boundary conditions. Suppose there PN are N total charges and the system as a whole is electrically neutral. i=1 zi = 0 The electrostatic potential energy of such a system is given by

N 1 X U = z φ (r ) , (C.1) Coul 2 i i i=1

177 th where φ (ri) is the electrostatic potential at the position of the i ion:

X0 zj φ (ri) = (C.2) rij + ~nL ~j,~n The prime denotes that summation is done over all periodic images, n, and over all particles,

j, except j=i when n=0. Unfortunately, φ (ri) as written is only conditionally convergent sum.

Ewald found that he could imporve the convergence of the sum by recasting the charge density into a diﬀerent form. In equation (2) the charge density is implicitly cast as a sum of delta-functions. Alternatively, if one thinks of the distribution as screened point charges among a compensating background charge, then equation (2) can be broken into two nicely converging summations: one over Fourier space and one over real space. The problem with the delta-function distribution is that the unscreened coulomb interaction decays slowy as

1/r, and, hence, the interactions are long-range. By assumimg that every point charge, zi, is surrounded by a diﬀuse charge distribution of opposite sign and magnitude, the Coulomb interaction is much shorter range. In this situation, the contribution of a point charge to the electrostatic potential is due only to the fraction of charge that is not screened. The rate at which the Coulomb interaction decays depends on the functional form chosen for the screening charge distribution.

Our goal is to ﬁnd the potential due to a set of point charges, not screened charges.

One corrects for the screening charge by adding in a compensating background charge distribution. The one catch is that one would like the background charge distribution to be a smoothly varying function in space. In general, when computing the electrostatic potential energy at a ion site, i, the contribution of the charge zi to the energy is not included beacuse it would be a unphysical self interaction. This means the screening charge and background charge around ion i should not contribute to the energy at site i either. However, for convenience, the background charge around ion i is included when calculating the energy at ion i such that the background charge distribution is a continuous and smoothly varying function. A correction for the self-interaction of ion i with it’s background charge will be

178 computed later. The advantage to this method is that the background distribution can be represented by a rapidly converging Fourier series.

In what follows, three individual terms will be computed to evaluate the Coulomb contribution to the electrostatic potential energy. First, the contribution to the Coulomb energy due to the background charge is computed in Fourier space, then the correction for the self-interaction, and ﬁnally the contribution due to the screening charges in real-space.

All equations will be in Gaussian units to make the notation compact. The compensating background charge around an ion, i, is chosen to be a Gaussian distribution with width p2/α:

3/2 2 ρGauss = −zi(α/π) exp(−αr ), (C.3)

where α is parameter adjusted for computational eﬃciency.

Fourier Part

Choosing the compensating background charge distribution as Gaussian around an ion,

i, means that the electrostatic potential at a point ri is due to a periodic sum, ρ1, of Gaussians:

N X X 3/2 h 2i ρ1(r) = zj(α/π) exp −α |~r − (~rj + ~nL)| (C.4) j=1 ~n

The electrostatic potential, φ1, due to ρ1 is computed via Poisson’s equation:

2 −∇ φ1(r) = 4πρ1(r) (C.5)

or, more conveniently, in Fourier form,

2 k φ1(k) = 4πρ1(k). (C.6)

179 Fourier Transforming the charge density ρ1 yields

1 Z ρ1(~k) = d~r exp(−i~k · ~r)ρ1(~r) (C.7) V V

N 1 Z X X h i = d~r exp(−i~k · ~r) z (α/π)3/2exp −α |~r − (~r + ~nL)|2 (C.8) V j j V j=1 ~n

N 1 Z X h i = d~r exp(−i~k · ~r) z (α/π)3/2exp −α |~r − ~r |2 (C.9) V j j allspace j=1

N 1 X = z exp(−i~k · ~r )exp(−k2/4α) (C.10) V j j j=1

Inserting ρ1(~k) into equation (6), one obtains

N 4π 1 X φ (k) = z exp(−k2/4α), (C.11) 1 k2 V j j=1 which is not deﬁned for k = 0. Assumeing the k=0 term is negligible describes a periodic system embedded in a medium with inﬁnite dielectric constant.

In order to compute the potential energy due to φ1(k) using equation (1), one must compute φ1(r) by Fourier Transforming equation (11):

X φ1(r) = φ1(k)exp(i~k · ~r) (C.12) ~k6=0

N 1 X X 4πzj h i = exp i~k · (~r − ~r ) exp(−k2/4α). (C.13) V k2 j ~k6=0 j=1

The following is the contribution of φ1(r) to the potential energy:

180 1 X U ≡ z φ (r ) (C.14) 1 2 i 1 i i N 1 X X 4πzizj h i = exp i~k · (~r − ~r ) exp(−k2/4α) 2 V k2 i j ~k6=0 j=1 V X 4π 2 = ρ(~k) exp(−k2/4α), (C.15) 2 k2 k6=0

where we deﬁne

N 1 X ρ(~k) ≡ z exp(i~k · ~r ). (C.16) V i i i=1

Correction for Self-Interaction

As discussed earlier, equation (14) includes a spurious self-interaction term that results from the interaction of a point charge with the background charge. This term was included such that the background charge could be Fourier transformed. The extra term is of the form (1/2)ziφself (ri), and, in this case, the point charge is sitting at the origin of the Gaussian distribution. To calculate this term, one must compute the potential energy at the location of the point charge due to the Gaussian charge distribution. The exact charge distribution in question is

3/2 2 ρGauss = −zi(α/π) exp(−αr ). (C.17)

Using Poisson’s equation, one can compute the electrostatic potential of this charge distribution. Assuming spherical symmetry of the Gaussian distribution, Poisson’s equation can be written as

−1 ∂2rφ (r) Gauss = 4πρ (r) (C.18) r ∂r2 Gauss

181 or ∂2rφ (r) − Gauss = 4πrρ (r). (C.19) ∂r2 Gauss Integrating by parts once yields

2 Z r ∂ rφGauss(r) − 2 = 4πrρGauss(r) (C.20) ∂r ∞ Z ∞ 3/2 2 2 = −2πzi(α/pi) dr exp(−αr ) r 1 2 = 2zi(α/π) 2 exp(−αr )

Integrating by parts a second time produces

Z r 1 2 rφGauss(r) = 2zi(α/π) 2 drexp(−αr ) (C.21) 0 √ = zierf( αr) where, √ Z x erf(x) ≡ (2/ π) exp(−u2)du (C.22) 0 Therefore, the potential due to a Gaussian distribution of charge at any point in space is given by

z √ φ (r) = i erf( αr). (C.23) Gauss r However, the self-interaction term only depends on the potential at r=0:

1 φGauss(r = 0) = 2zi(α/π) 2 . (C.24)

Therefore, the correction to the potential energy (equation (14)) due to the self-interation is given by

182 N 1 X U = z φ (r ) self 2 i self i i=1 N 1 X 2 = (α/π) 2 zi (C.25) i=1 It is worth noting that this correction term is a constant provided all charges are ﬁxed in opsition.

Real Space Sum

Recall the point charges are screened by Gaussian charge distributions,opposite in charge and equal in magnitude. In this ﬁnal section, the electrostatic energy due to the screened point charges must be computed. Using equation (23),one can write the (now short range) electrostatic potential due to a point charge zi surrounded by a Gaussian charge distribution with net charge −zi :

z z √ φ (r) = i − i erf( αr) (C.26) shortrange r r z √ = i erfc( αr), r where erfc(x) ≡ 1 − erf(x) is the complementary error function. Therefore, the total contribution of the screened Coulomb interactions to the potential energy is given by

N 1 X √ U = z z erfc( αr ) (C.27) short range 2 i j ij i6=j

Finally, on obtains the total electrostatic contribution to the potential energy by summing the three terms (equations (15), (25), and (27)):

183 1 X 4πV 2 U = ρ(~k) exp(−k2/4α) Coul 2 k2 k6=0 N 1 X 2 − (α/π) 2 zi i=1 N 1 X √ + z z erfc( αr ) 2 i j ij i6=j

C.2 Model Periodic Coulomb (MPC) Interaction

In QMC, finite size errors arise fom the fact that a preiodically repeated finite simulation cell is used to model an infinite solid. For the exchange correlation energy, the periodicity introduces a spurious contribution because electron correlations are also forced to be periodic, which each electron interacts with peridodic images of it’s exchange correlation hole.

Another way of thinking about the error is that electron correlation in each periodically repeated cell is the same, which is unphysical, and causes the electron interactions to be unphysical.

The Ewald sum models the interaction of periodically repeated electrons. Exapnding the Ewald interaction gives

1 2π v (r) = + rT · D · r + ··· , (C.28) Ewald r 3Ω where Ω is the volume of te simulation cell, and D is a tensor that depends on the shape

1 of the cell (cubic = identity). The deviations from r are what make the Ewald interaction periodic, but are responsible for spurious contributions to the exchange-correlation energy.

A modiﬁcation to the Hamiltonian called the Model Periodic Coulomb (MPC) interaction [24, 163, 164, 23, 21] provides a solution to remove the spurious error to the exchange- correlation energy. In doing so two rules must be respected: 1) MPC should give the Ewald

1 interaction for Hartree terms and 2) MPC should be exactly r for the interaction with the exchange correlation hole. The solution of Poisson’s equation for a periodic array of charges only obeys the second rule in the limit of an inﬁnitly sized cell.

184 The Model Periodic Coulomb interaction replaces the Ewald interaction in order to satisfy both rules:

Z X X E Hˆee = f(ri − rj) + V (ri − r) − f(ri − r n(r)dr, (C.29) i>j i WS where n is the electronic number density, V E is the Ewald potential, and

1 f(r) = (C.30) rm is a cutoﬀ Coulomb function within a minimum image convention, which corresponds to reducing the vector r into the Wigner-Seitz (WS) cell of the simulation cell by removal of lattice vectors, leaving rm. The full MPC Hamiltonian consists of the sum of the Hartree energy computed with the Ewald interaction and the exchange-correlation energy computed with the cutoﬀ function to prevent electrons to interact with mirror images of their exchange- correlation hole.

The MPC interaction does not completely eliminated the many-body exchange-correlation

finte size error. However, in practice, the finite size error converges much more quickly as a function of system size than when MPC is not used. So, effectively, using MPC allows one to do solid calculations with smaller simulations cells than would normally be the case.

This allows a dramatic savings in computational time.

185 Appendix D Summary of CODES Used in This Work

D.1 ABINIT

ABINIT is a free plane-wave, pseudopotential DFT code capable of geometry optimization, phonon calculations, and a number of other electronic properties. Pseudopotentials are user created, typically using the norm-conserving, nonlocal variety.

D.2 Quantum ESPRESSO

Quantum ESPRESSO is a free integrated suite of computer codes, including the plane- wave pseudopotential DFT code PWSCF, for electronic-structure calculations and materials modeling at the nanoscale. It is based on density-functional theory, plane waves, and pseudopotentials (both norm-conserving and ultrasoft).

D.3 VASP

VASP is a commercial plane-wave, pseudopotential DFT code capable of geometry optimization, phonon calculations, and a number of other electronic properties. VASP is the

ﬁrst code capable of doing hybrid, HSE calculations using plane-waves.

186 D.4 CASINO

CASINO is the private code of Richard Need’s Cambridge QMC group. It performs VMC,

DMC with most of the latest and greatest developments in QMC, such as backﬂow, plane- wave expansion in the Jastrow, MPC, and lots of useful scripts and a well-developed manual making it user-friendly.The code has a string user base and requires a small cost and per- mission to use.

D.5 CHAMP

CHAMP is a private QMC code maintained by Cyrus Umrigar at Cornell University. The code is typically used as a research code to develop new algorithms and techniques for QMC.

D.6 OPIUM

OPIUM is a free nonlocal, normconserving pseudopotential generator. It produces pseudopotentials that interface with most of the codes listed here and more.

D.7 CRYSTAL

CRYSTAL is a commercial DFT code which uses Gaussian basis sets with or without a pseudopotential (eﬀective core potential). The code is maintained by people at a variety of

European institutions.

D.8 WIEN2K

WIEN2k is a commerical FLAPW code.

D.9 ELK

ELK is a free FLAPW code (essentially a free version of WIEN2K).

187 Appendix E Strong and Weak Scaling in the CASINO QMC Code

The two most widely-used quantum Monte Carlo methods for continuum electronic structure calculations are: variational Monte Carlo (VMC) and diﬀusion Monte Carlo (DMC).

The VMC method computes the electronic energies by applying the variational principle to a given functional form of the many-body wave-function. Each processor uses a single conﬁguration of electrons to evaluate expectation values stochastically. In DMC, the

Schrödingerequation is written as a diffusion equation and the many-body wavefunction takes on a statistical form, represented as several electronic configurations per processor.

The conﬁgurations diﬀuse, branch, or vanish until distributed according to the ground-state charge density.

Parallel eﬃciency in VMC is only sensitive to a small amount of inter-node communication due to a ﬁnal step of accumulating and averaging energies computed on each processor.

Since only one conﬁguration is used per processor, the size of the problem per node is always

ﬁxed. Therefore, only study of weak scaling is possible.

Figure 1 shows VMC pays a negligible penalty in parallel eﬃciency due to inter-node communication. The speedup is shown relative to a serial run, where the time of each job is scaled to provide equivalent statistical error in the energy as the largest processor job. p Error decreases as 1/ NMonte Carlo stepsNconﬁgurations/procNproc. VMC performance scales almost perfectly with the square of the number of processors as expected.

The parallel eﬃciency of DMC is sensitive to the number of conﬁgurations used per

188 Figure E.1: Log-Log plot of weak scaling in VMC. The time for each job is scaled to provide a result with the same statistical error as the largest processor job. Speedup is the ratio of scaled parallel time to a scaled serial time. VMC calculations scale almost perfectly with the square of the number of processors.

processor. As the number configurations increase or decrease on each processor due to the branching algorithm, the populations on each processor may become uneven and must occasionally be rebalanced. Processors with the fewest number of configurations will occasionally have to wait on other processors to finish cycling through a larger number of configurations. The efficiency can be indefinitely improved by increasing the number of configurations per processor. However, in practice the number of configurations used is limited by available memory. DMC also has a small amount of inter-node communication for accumulation and averaging of energies as in VMC.

Figure 2 shows weak scaling in DMC, where the number of configurations per processor is held fixed. This means the total number of configurations increases with the number of processors, allowing DMC efficiency to improve as the number of processors is increased.

However, scaling is not perfect due to time required for rebalancing the conﬁguration population across processors.

189 Figure E.2: Log-Log plot of weak scaling in DMC. The time for each job is scaled to provide a result with the same statistical error as the largest processor job. Speedup is the ratio of scaled parallel time to a scaled serial time. DMC continuously gains speedup as the number of processors increases because the total number of configurations is also increasing. Efficiency is imperfect due to time required to rebalance configurations across processors.

Figure 3 shows strong scaling in DMC, where the total number of conﬁgurations is ﬁxed.

Efficiency decreases with processor size because the number of configurations per processor is decreasing, causing increased time for rebalancing the configurations. Figure 2 and 3 together show why it is important to use a large number of configurations to maximize the speedup of parallel computing in DMC.

190 Figure E.3: Log-Log plot of strong scaling in DMC. The time for each job is scaled to provide a result with the same statistical error as the largest processor job. Speedup is the ratio of scaled parallel time to a scaled serial time. DMC calculations become increasingly inefficient as the number of processors increase because the total number of configurations is being held fixed.

191