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

Transcription

1 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

2 c Copyright by Kevin P. Driver 2011

3 Abstract Quantum mechanics provides an exact description of microscopic matter, but predictions require a solution of the fundamental many-electron Schrödinger equation. Since an exact solution of Schrödinger s equation 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

4 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 trisilicon-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

5 To my family and friends for guidance, help, and love. iv

6 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 education. I would like to thank Cyrus Umrigar, Burkhard Militzer, Hyoungki Park, Amita Wadhera, Mike Fellinger, Jeremy Nicklas, Ken Esler, Neil Drummond, Yaojun Du, Jeongnim 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

7 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 Hazzard, 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, Valerie 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 This work was also made possible by generous computational resources from OSC, NERSC, NCSA, and CCNI. vi

8 Vita April 14, Born New Albany, Indiana, USA B.S., University of Louisville, Louisville, Kentucky Fowler Fellow, Department of Physics, Ohio State University, Columbus, Ohio M.S., Ohio State University, Columbus, Ohio 2005-Present Graduate Research Associate, Department of Physics, Ohio State University, Columbus, Ohio Publications K. P. Driver, R. E. Cohen, Zhigang Wu, B. Militzer, P. López Ríos, 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, (2010). M. Floyd, Y. Zhang, K. P. Driver, Jeff Drucker, P.A. Crozier, and D.J. Smith, Nanometerscale 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

9 Fields of Study Major Field: Physics Studies in quantum Monte Carlo claculations of solids: J. W. Wilkins viii

10 Table of Contents Page Abstract ii Dedication iv Acknowledgments v Vita vii List of Figures xii List of Tables xiv Chapters 1 Introduction Quantum Mechanics Correctly Explains Matter Modeling Matter with Numerical, Quantum Simulations Organization and Summary of Thesis Accomplishments Methods of Solving the Schrödinger Equation The Many Body Problem The Born Oppenheimer Approximation Mean-Field-based Ab Initio Methods The Hartree Approximation: No Exchange, Averaged Correlation The Hartree-Fock Approximation: Explicit Exchange, Averaged Correlation Density Functional Theory Self-interaction Error in DFT Many Body Ab Initio Methods Configuration Interaction Quantum Monte Carlo Coping with DFT Approximations: Benchmarking with Hybrid functionals and QMC Introduction: Approximations and Weaknesses of Density Function Theory Exchange-Correlation Approximations: Categorization of functionals in Jacob s ladder Basis Set Approximations Benchmarking functionals With Hybrid DFT and Quantum Monte Carlo. 38 ix

11 4 Results for Silicon Self-Interstitials Introduction Calculation Details Results Tests for errors in QMC Conclusions Results for Silica Introduction Previous Work and Motivation Computational Methodology Pseudopotential Generation DFT Calculations QMC Calculations Wave-function Construction and Optimization DMC Calculations Results Free Energy Thermal Equations of State and Fit Parameters Phase Stability Thermodynamic Parameters Stishovite Shear Constant Geophysical Implications Conclusions Hybrid DFT Study of Silica Introduction Previous Work Hybrid Calculations of Silica Hybrid B3LYP and PBE0 Calculations of Solids Screened Hybrid (HSE) Calculations of Solids Computational Methodology CRYSTAL Calculations ABINIT Calculations VASP Calculations Results Energy Versus Volume Pressure Versus Volume Equilibrium Quartz and Stishovite Volume from Vinet Fits Equilibrium Quartz and Stishovite Bulk Moduli from Vinet Fits Equilibrium Quartz and Stishovite K 0 from Vinet Fits Enthalpy Versus Pressure Quartz-Stishovite Transition Pressures Conclusions Conclusions Summary x

12 7.2 Future Research Appendices A Error Propagation in QMC Thermodynamic Parameters 169 A.1 Taylor Expansion Method A.2 Monte Carlo Method 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 C.2 Model Periodic Coulomb (MPC) Interaction D Summary of CODES Used in This Work 186 D.1 ABINIT D.2 Quantum ESPRESSO D.3 VASP D.4 CASINO D.5 CHAMP D.6 OPIUM D.7 CRYSTAL D.8 WIEN2K D.9 ELK E Strong and Weak Scaling in the CASINO QMC Code 188 xi

13 List of Figures Figure Page 4.1 Images of single Si interstitial defects Images of Si di-self-interstitial defects Images of Si tri-self-interstitial defects QMC and DFT band gaps of Si QMC and DFT cohesive energy of Si QMC and DFT diamond to β-tin energy difference in Si I1 16-atom formation energy I1 64-atom formation energy I1 diffusion path I2 64-atom formation energy I3 64-atom formation energy DMC time step convergence for Si DMC finite size convergence for X defect DMC formation energy for LDA and GGA pseudopotentials Convergence of Jastrow polynomial order for Si QMC pseudopotential dependence Schematic silica phase diagram Silica energy vs. volume curves Silica equations of state Silica Vinet fit parameter: Zero pressure free energy vs. temperature Silica Vinet fit parameter: Zero pressure volume vs. temperature Silica Vinet fit parameter: Zero pressure bulk modulus vs. temperature Silica Vinet fit parameter: Pressure derivative of the bulk modulus vs. temperature at zero pressure Silica enthalpy curves Silica phase boundaries Thermal pressure of silica Changes in thermal pressure in silica Bulk moduli of silica Pressure derivative of the bulk modulus of silica Thermal expansivity of silica xii

14 5.15 Heat capacity of silica Percentage volume differences of silica Grüneisen ratio of silica Volume dependence of the Grüneisen ratio of silica Anderson-Grüneisen parameter of silica Sound Velocity and Density profile of Earth Bulk Sound Velocity and Density of Silica Energy vs. b/a strain in stishovite Stishovite shear constant softening Gaussian basis set convergence HSE energy versus volume of quartz and stishovite DFT pressure versus volume of quartz DFT pressure versus volume of stishovite DFT zero pressure volumes of silica DFT Bulk Moduli of silica DFT K of silica HSE enthalpy versus pressure of quartz and stishovite DFT quartz-stishovite transition pressures E.1 CASINO VMC weak scaling E.2 CASINO DMC weak scaling E.3 CASINO DMC strong scaling xiii

15 List of Tables Table Page 4.1 Separation of bulk and defect e-n Jastrows Silica thermal equation of state parameters Quartz DFT equation of state parameters Stishovite DFT equation of state parameters xiv

16 Chapter 1 Introduction 1.1 Quantum Mechanics Correctly Explains Matter In the late 17 th 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 20 th 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 Schrödinger, 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

17 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 Schrödinger s wave equation [4], relates the wave properties (wave function) Ψ of a particle to its energy, E, through the action of a Hamiltonian operator, Ĥ. There are two forms of the equation: a time-independent form, ĤΨ = EΨ, (1.1) and a time-dependent form, ĤΨ = i h Ψ t. (1.2) 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 Schrödinger 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 Schrödinger s equation for real materials is to use sophisticated numerical simulations, often requiring massively parallel 2

18 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 s equation 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 computationally 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. However, 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 simulated. 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 materials. 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

19 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

20 mean-field-based Hartree-Fock and density functional approaches to solving the many-body Schrödinger equation. The final section introduces fully many-body approaches to solving Schrödinger s equation, 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 exchangecorrelation functionals and develop strategies to cope with their bias and unreliability. The conceptual systematic improvement of Jacob s ladder of functionals is discussed. Particular 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, diand 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

21 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 Gaussian basis sets, details of finite size error in QMC, scaling in QMC, and summaries of the codes used. 6

22 Chapter 2 Methods of Solving the Schrödinger 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 manyparticle, time-independent Schrödinger equation [6, 10] (Equation 1.1), [ ] N 1 N ĤΨ(R) ˆT + ˆV Ψ(R) = 2 Z i Z j r 2m i + Ψ(R) = EΨ(R), (2.1) i r i r j i=1 which is a 3N-dimensional eigen-problem, where R is the collective coordinate for all N particles r i,...,r N. 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 = m e = h = 4πɛ 0 = 1), and ˆT and ˆV are the kinetic and Coulomb potential energy, respectively. The Coulomb potential term forces Schrödinger s equation to be inseparable for more i>j 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 oneelectron 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

23 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 Schrödinger equation 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) 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. Therefore, 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 Ĥ = i 1 2m i 2 r i + i α Z α r i d α + i>j 1 r i r j α>β Z α Z β, (2.2) d α d β where the terms for electrons of charge -1 at positions r i and ions of charge Z α at positions d α 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

24 2.2 Mean-Field-based Ab Initio Methods 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 Schrödinger equation 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, U el (r) = e dr ρ(r 1 ) r r, (2.3) where and each one-electron equation will yield a single-electron wave-function, ψ i, called an orbital, and an orbital energy. The total electronic charge would be ρ(r) = e i ψ i (r) 2, (2.4) where the sum is over all occupied levels. And, the ion potential is U ion (r) = Z R 1 r R, (2.5) 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(r 1, r 2,..., r N ) = ψ 1 (r 1 )ψ 2 (r 2 )...ψ N (r N ) (2.6) Employing the variational principle and minimizing the expectation value of the Hamiltonian with respect to variations in the wave functions produces a set of one-electron equations, called the Hartree equations: ψ i (r) + U ion (r)ψ i (r) + e 2 j dr ψ j (r ) 2 1 r r ψ i (r) = ɛ i ψ i (r) (2.7) 9

25 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 ψ i 2, 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 The Hartree-Fock Approximation: Explicit Exchange, Averaged Correlation The Hartree equations are a good first attempt at solving the many-body Schrödinger 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 wavefunction 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 meanfield 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: E correlation = E exact,non relativistic E Hartree Fock (2.8) 10

26 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(r 1, r 2,..., r i, r j,..., r N ) = Ψ(r 1, r 2,..., r j, r i,..., r N ) (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: Ψ(r 1, r 2,..., r N ) = ψ 1 (r 1 ) ψ 2 (r 1 ) ψ N (r 1 ) ψ 1 (r 2 ) ψ 2 (r 2 ) ψ N (r 2 ).... ψ 1 (r N ) ψ 2 (r N ) ψ N (r N ) (2.10) The quantum spin variables have been left out for clarity, but they are easily included with the position dependence. Minimizing the expectation value of Ĥ with respect to variations in the one-electron wave-functions results in the one-electron, Hartree-Fock equations: ψ i (r)+u ion (r)ψ i (r)+u el (r)ψ i (r) j δ si s j dr 1 r r ψ j (r )ψ i (r )ψ j (r ) = ɛ i ψ i (r), (2.11) where s i 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

27 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 manybody 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, E tot, of a system is a unique functional of the electron density, n(r). Furthermore, the functional E tot [n(r)] is minimized for the ground state density, n GS (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, E tot [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 Schrödinger equation can be mapped onto the problem of solving an effective single-particle wave equation with an effective potential, V eff. The first step is to write the total energy functional as E tot [n(r)] = T [n(r)] + E e e [n(r)] + E xc [n(r)] + V ext (r)n(r)dr (2.12) where T is the kinetic energy of a noninteracting system, E e e is the electron-electron interaction energy, E xc is the exchange-correlation energy, and V ext represents an external potential including the ions. By minimizing this total energy functional with respect to 12

28 variations in the electron density, subject to the constraint that the number of electrons are fixed, an effective one-particle equation is obtained: where V eff is an effective potential given by } { h2 2m 2 + V eff [n(r)] ψ i (r) = ɛ i ψ i (r), (2.13) V eff [n(r)] = V ext [n(r)] + V e e [n(r)] + V xc [n(r)], (2.14) where V e e is the Hartree potential, V e e (r) = e n(r) r r dr, (2.15) and V xc [n(r)] = δe xc[n(r)]. (2.16) δ[n(r)] Equations 2.13 and 2.14, are known as the Kohn-Sham equations. The quantities ψ i and ɛ 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], E xc [n(r)], is given by E xc = 1 2 dr dr n(r) n(r, r ) r r, (2.17) where, if we introduce a coupling constant λ, which varies from 0 (real-interacting system) to 1 (Kohn-Sham noninteracting system), then n xc (r, r ) = 1 0 dλn λ xc(r, r ) = n x (r, r ) + n c (r, r ) (2.18) is the average over the coupling constant λ of the density at r of the exchange-correlation hole about an electron at r : n λ xc(r, r ) = Ψ α ˆn(r)ˆn(r) Ψ λ n(r) δ(r r ), (2.19) and n x (r, r ) = n α=0 xc (r, r ) is the exchange hole. Here, Ψ α is the correlated ground state wave-function for a system with the same spin densities as the real system but with the 13

29 electron-electron interaction reduced by a factor α. Due to Pauli exclusion and Coulomb repulsion, an exchange-correlation hole forms satisfying the sum rule dr n α xc(r, r ) = 1 (2.20) The main flaw of DFT is that, while the theory is exact, the form of the exchangecorrelation 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 V xc [n(r)], and then solves Eq. (2.13) for the wave functions, ψ i (r), using a standard band theory technique. From the wave functions obtained, one calculates a new charge density: occ. n(r) = ψ i (r) 2. (2.21) This procedure is then repeated until the charge density is converged. i 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

30 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 Interaction approach, which is too expensive for solid calculations, and Quantum Monte Carlo, which is the only many-body method capable of efficiently simulating solids 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 CI Φ = C I Φ I, (2.22) I=0 where C I are the CI expansion coefficients and Φ I are the different configurations of orbitals.if there are N electrons and M basis states, then there are N CI = M! N!(M N)! (2.23) configurations that can be constructed form the orbitals. Setting all C I to 0 except C 0 = 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 electrons. 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

31 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 Schrödinger 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 I = drg(r), (2.24) an importance function, P (R) is introduced such that I = drf(r)p (R), (2.25) where the importance function is a probability density such that P (R) > 0 and 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

32 and computing the sample average: I = lim M [ 1 M ] M f(r m ). (2.26) m=1 However, Monte Carlo only estimates the integral by averaging over a finite sample of points from P(R): I M 1 M M f(r m ), (2.27) m=1 which provides a result with a certain statistical confidence error. estimate of I is σf 2 /M, which is estimated as The variance of the σ f M 1 M(M 1) [ M f(r m ) 1 M m=1 2 M f(r n )], (2.28) n=1 such that the statistical standard deviation decreases with the square root of the number of samples: I I M ± σ f M. (2.29) 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 Schrödinger 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

33 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, Ψ T. In order for VMC to work, Ψ T 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,ψ T and Ψ T must be continuous when the potential is finite and the integrals Ψ T Ψ T and Ψ T ĤΨ T must exist. It is also convient that Ψ T ĤΨ T exist in order to keep the variance of the energy finite. The variational theorem of quantum mechanics states that the expectation value of Ĥ evaluated with any trial wave-function Ψ T is an upper bound on the ground-state energy E 0 : E V = Ψ T (R)ĤΨ T (R)dR Ψ T (R)Ψ T (R)dR (2.30) 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, E L (R) : E V = p(r)e L (R)d(R), (2.31) where p(r) = Ψ 2 T (R) Ψ 2 T (R )d(r ), (2.32) and E L (R) = Ψ 1 T (R)ĤΨ2 T (R). (2.33) The Metropolis algorithm is used generate a set of electron configurations, sometimes called walkers, {(R m : m = 1, M)} from the configuration-space probability density. The 18

34 local energy is evaluated for each walker and the average energy is computed: E V 1 M M E L (R m ), (2.34) m=1 with a statistical error of σ V MC 1 M ( E L(R m ) 2 E L (R m ) 2 ). (2.35) Diffusion Monte Carlo Diffusion Monte Carlo (DMC) is a stochastic, projector-based method that solves the timedependent, many-body Schrödinger equation by allowing the wave-function to decay to the ground state from some initial state in imaginary time. In imaginary time (t it), the Schrödinger equation becomes t Φ(R, t) = (Ĥ E T )Φ(R, t) = ( R + V (R) E T )Φ(R, t), (2.36) where t measures progress in imaginary time, R = (r 1, r 2,..., r N ) is a 3N-dimensional vector (also called a configuration or walker) providing the coordinates of the N electrons, and E T 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 R, t) and time-step, τ : where Φ(R, t + τ) = G(R R, τ)φ(r, t)d(r ), (2.37) G(R R, τ) = R exp( τ(ĥ E T )) R. (2.38) It follows, that the Green s function obeys Equation 2.36, t G(R R, t) = (Ĥ(R) E T )G(R R, t), (2.39) satisfying the initial condition G(R R, 0) = δ(r R ). Using the spectral expansion, exp( τĥ) = i Ψ i exp( τe i ) Ψ i, (2.40) 19

35 the Green s function can be written in a manner which reveals important physics: G(R R, τ) = i Ψ i (R) exp( τ(e i E T ))Ψ i (R ) (2.41) This expression reveals that the critical feature of DMC is that the Green s function operator exp( τ(ĥ E T )) projects out the lowest energy eigenstate Ψ 0 in the limit of infinite time steps (τ ): lim τ Φ(R, t) lim R exp( τ(ĥ E T )) Φ init (2.42) τ = lim G(R R, τ)φ init (R )dr (2.43) τ = lim Ψ i (R) exp( τ(e i E T )) Ψ i Φ init (2.44) τ i = lim τ Ψ 0(R) exp( τ(e 0 E T )) Ψ 0 Φ init (2.45) The last step (the crux of DMC) follows from the fact that E i > E i 1 > E 2 > E 1 > E 0, where E i are excited states and E 0 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 Neglecting the potential term reduces the imaginarytime Schrödinger equation 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 {R i } 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 Ĥ = ˆV + ˆV, where ˆT is the N-electron kinetic energy operator and ˆV 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

36 G(R R, τ) (2πτ) 3N/2 exp [ (R R ) 2 ] exp [ τ [ V (R) + V (R ] ] ) 2E T /2, 2τ (2.46) or G(R R, τ) G D (R R, τ) G B (R R, τ), (2.47) where the first factor, G D (R R, τ) = (2πτ) 3N/2 exp [ (R R ) 2 ], (2.48) 2τ is the Green s function for a diffusion equation, while the second factor, G B (R R, τ) = exp [ τ [ V (R) + V (R ) 2E T ] /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 birthdeath algorithm. Fermion Sign Problem There is one flaw in the DMC method unmentioned up to this point. Thus far, the wavefunction 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

37 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 Ψ T and DMC wave-function Φ, f(r, t) = Ψ T (R)Φ(R, t), (2.50) is purely positive if the nodes of both wave-functions are equal. Substituting into Equation 2.36 gives f t = Rf + R [v D f] + [E L E T ] f, (2.51) where v D (R) = Ψ 1 T (R) RΨ T (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 Schrödinger equation (Equation 2.36) was written in integral form, the importance-sampled imaginary-time Schrödinger equation may be written in integral form: f(r, t + τ) = G(R R, τ)f(r, t)d(r ), (2.53) where G is the modified Green s function for the importance sampled wave-function, which 22

38 amounts to a similar expression for the Green s function as in Equation 2.46, but updated for the importance sampling: G(R R, τ) G D (R R, τ) G B (R R, τ), (2.54) where, again, the first factor, G D (R R, τ) = (2πτ) 3N/2 exp [ (R R τv D (R )) 2 ], (2.55) 2τ is the Green s function for a diffusion equation which now has a new drift term, while the second factor, G B (R R, τ) = exp [ τ [ E L (R) + E L (R ) 2E T ] /2 ], (2.56) is the branching-factor, where the local energy replaced the potential. 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 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 importancesampled mixed distribution, f(r) = Ψ T (R)φ 0 (R), where φ 0 is the ground state for the wave-function expanded in eigenstates,φ i of the Hamiltonian: Φ(R, t) = i c i(t)φ i (t)(r). The fixed-node DMC energy is evaluated using HΨ T = E L Ψ T, where E L is the local energy: E DMC = E 0 = φ 0 Ĥ Ψ T φ 0 Ψ T = f(r)el (R)dR f(r)dr 1 M M E L (R i ). (2.57) 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

39 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: Ψ SJ (R) = exp[j(r)] det[ψ n (r i )] det[ψ n(r j )], (2.58) where exp[j] is the Jastrow factor and det[ψ n (r i )] 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. Unlike 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: J({r i }, {r I }) = N i>j N ions u(r ij ) + I=1 N i=1 N ions χ I (r ii ) + I=1 N f I (r ii, r ji, r ij ), (2.59) i>j where N is the number of electrons, N ions is the number of ions, r ij = r i r j, r ii = r i r I, r i is the position of the electrons, and r I 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

40 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 iterated 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 s equation 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

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

42 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

43 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. Pseudopotentials are generally produced from solving the inverted Schrödinger equation 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

44 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. However, 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

45 Chapter 3 Coping with DFT Approximations: Benchmarking with Hybrid functionals and QMC 3.1 Introduction: Approximations and Weaknesses of Density 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

46 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 Exchange-Correlation Approximations: Categorization of functionals in Jacob s ladder This section discusses a hierarchy of density functional approximations for the exchangecorrelation energy, E xc [15]. The hierarchy is described by a ladder of approximations for 31

47 the exchange-correlation energy as a function of the electron density: E xc [n, n ] = drn(r)ɛ xc ([n, n ] ; r), (3.1) where the integrand nɛ xc is an exchange-correlation energy density, and ɛ xc is an exchangecorrelation 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 = 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 E xc [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).: Exc LDA [n, n ] = drn(r)ɛ unif xc (n (r), n (r)), (3.2) where ɛ uniform xc (n, n ) is the exchange-correlation energy per particle of an electron gas with uniform spin densities n and n. ɛ unif 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), Exc GGA [n, n ] = drn(r)ɛ GGA xc (n (r), n (r), n (r), n (r)), (3.3) 32

48 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, τ σ (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 Exc MGGA [n, n ] = drnɛ MGGA xc (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 improves 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 r around r. The exact DFT expression for exchange 33

49 energy is E x = drn(r)ɛ x (r), (3.5) where and n(r)ɛ x (r) = 1 2 σ dr ρ σ(r, r ) 2 r r, (3.6) ρ sigma (r, r ) = σ α θ(µ ɛ ασ )ψασ(r)ψ ασ (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 noninteraction Kohn-Sham system constructed such that both systems have the same density (See reference [6] for a clear discussion of DFT and hybrid methods): Ĥ λ = ˆT 0 + λ ˆV ee + ˆV λ + ˆV C + ˆV ext, (3.8) where ˆT 0 is the non-interacting kinetic energy operator, V C is the Hartree Coulomb potential (i.e. includes average e-e correlations), V ext is the external (nuclei) potential. V ee is a true many-body operator containing both electron exchange-correlation and the missing kinetic energy of the interacting system, while V xc 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 E xc. In terms of the potential, the adiabatic connection is written as Ψ 1 ˆV ee Ψ 1 = 1 0 ψ λ ˆV ee Ψ λ dλ, (3.9) 34

50 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: ˆVee = ˆV x + ˆV c. Then, for correlation effects, hybrids use the standard DFT approach, but for exchange effects the integral is approximated. 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: E B3LY P xc = E LDA xc + a 0 (E HF x Ex LDA ) + a x (Ex GGA Ex LDA ) + a c (Ec GGA Ec LDA ), (3.10) where a 0 = 0.20, a x = 0.72, and a c = 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: E P BE0 xc = ae HF x + (1 a)e P BE x + E P BE c, (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

51 exchange: E HSE xc = ae HF,SR x + (1 a)e P BE,SR x + E P BE,LR x + E P BE c, (3.12) where E HF,SR x is the short-range Hartree-Fock exchange, E P BE,SR x and E P BE,LR x 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 r = erfc(ωr) r + erf(ωr), (3.13) 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 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: ψ n (r) = k c nk exp(ik r), (3.14) where c nk 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, k max. 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

52 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 Schrödinger euqation: χ(r) = R nl (r)y lm (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,α,n,l,m (r) = 2 n+1 [(2n 1)!!] 1/2 (2π) 1/4 rn 1 exp( αr 2 Y lm (θ, φ)) (3.16) 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

53 contracted function form looks like χ R,k,n,l,m (r) = i u ki χ R,αi,n,l,m(r), (3.17) where the u ki are fixed constants. Only the coefficients to the contracted functions χ 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 Quantum Monte Carlo Progress in computational materials science requires accurate and reliable methods capable of studying large, complex materials. The lower-rung DFT functionals (LDA, GGA, mgga) provide the highest efficiency to accuracy ratio for studying large systems. However, the lower-rung functionals are not always reliable enough to fully trust their predictions. More computationally expensive benchmark accuracy methods test the accuracy of lower-rung functionals. Benchmark methods can help identify the mores accurate lowerrung functional for a given material or property, such that large calculations can be done 38

54 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 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

55 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

56 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 confusion. 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 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): E f = E(defect) N + 1 N E bulk. (4.1) 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

57 (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

58 (a) I3a (b) I3b (c) I3c Figure 4.3: Triple interstitial defects in silicon 43

59 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

60 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

61 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

62 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/SiO 2 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 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 Å) 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

63 detail in Chapter 2. The controlled approximations for silica include statistical Monte Carlo error, numerical grid interpolation ( 5 points/å) of the DFT orbitals [22], finite system size, the number of configurations used, and the DMC time step. approximations are converged to within at least 1 milli-ha/sio 2. All of the controlled 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] 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 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 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

64 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

65 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 ev lower. The mgga predicts T lies highest and lies about 1 ev lower than QMC. 50

66 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 ev lower than QMC. 51

67 Figure 4.9 shows the QMC and DFT energy barriers (migration energies) for the selfdiffusion 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 mev [70], which is similar to GGA results in Figure

68 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

69 The experimental estimates of the diffusion activation energy (formation + migration) for defects in silicon are near 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 ( = 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-selfinterstitial defects, I2 a and Ib 2 in a 64 atom cell. Results again display the expected functional Jacob s ladder trend. QMC predicts the I2 a defect lies 1.2 ev lower in energy than Ib 2. 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

70 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 plus and minus sigma about the mean. QMC predicts the I2 a defect lies 1.2 ev lower in energy than I2 b. 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

71 Figure 4.11 shows QMC and DFT formation energies for the three most stable tri-selfinterstitial defects, I3 a, Ib 3, and Ic 3 in a 64-atom cell. Results again display the Jacob s ladder trend. QMC predicts the stability ordering is I3 a < Ic 3 < Ib 3. GGA predicts the stability ordering I b 3 < Ia 3 < Ic 3, while LDA predicts Ia 3 < Ib 3 < Ic 3. For the Ia 3 and Ic 3 defects, GGA closely agrees with QMC. The improved GGA results may be because the I a 3 and Ic 3 defects are less distorted (smaller coordination number) than the I b 3 defect. LDA energies lie up to 3.5 ev below QMC and GGA lies up to 2 ev below QMC. 56

72 Figure 4.11: Formation energy of the three lowest-energy tri-self-interstitials in silicon (I3 a, I3 b, and Ic 3 ) in a 64-atom cell. The black and white diagonal box on the QMC results respresents plus and minus sigma about the mean. 57

73 Physical Explanation for Results 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

74 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

75 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

76 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

77 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

78 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. Results agree within one-sigma statistical error, indicating negligible dependence on functional choice. 63

79 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 5 th order polynomials. 64

80 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 5 th order polynomials. 65

81 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

82 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

83 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

84 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 ev The di- and tri-interstitials also display the Jacob s ladder trend, but LDA and GGA energies lie ev below QMC. QMC predicts I a 2 and Ia 3 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

85 error, fixed-node error are needed [75]. 70

86 Chapter 5 Results for Silica Silica (SiO 2 ) 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-pbo 2 structure above the core-insulating D 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

87 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 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 CaCl 2 -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 CaCl 2 -structure to an α-pbo 2 -structured polymorph at pressures near the base of the mantle. Figure 5.1 shows a schematic version of the silica pressuretemperature phase diagram. Note the pressure scale is no linear, and the dashed phase boundaries indicate they are not well known. 72

88 Figure 5.1: Schematic version of the silica phase diagram. 73

89 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 CaCl 2 -α-pbo 2 transition is not associated with any global seismic discontinuity, such as D, 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 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

90 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 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/sio 2 accuracy. Converged silica calculations require k-point mesh sizes of 4 4 4, 4 4 6, and for quartz, stishovite/cacl 2, and α-pbo 2, 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 optimizations: 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. Iononly 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 computed 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

91 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 ) = U static (V, T ) T S vib (V, T ), (5.1) where U static is the static internal energy of the crystal lattice and TS vib is the vibrational (phonon) contribution of the thermal atomic motion to the free energy and S vib is the vibrational entropy. The Gibbs free, G = F + P V 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 energy, one must decide how to treat the temperature and volume dependence of the phonon frequencies, ω i of the lattice. Statistical mechanics allows a system s vibrational quantum mechanical energy levels to completely determine the vibrational Helmholtz free energy (F = U static + F vib ) via a partition function, Z: F vib = kt lnz, (5.2) where Z is a sum over all quantum energy levels given by, Z = i exp ( ɛ i ), (5.3) kt where k is the Boltzmann constant, and the ɛ i are the microstate energies: 1 2 hω i, 3 2 hω i, 76

92 5 2 hω i, ect. Therefore, for each mode, there are many energy levels Z i = exp ( 1 hω i 2 all modes kt ) which when combined with Equation 5.2 gives s=0 exp ( s hω i ), (5.4) kt = exp ( 1 hω i 2 kt ) 1 exp ( hω (5.5) i kt ), F vib,i = 1 2 hω i + kt ln(1 exp ( hω i )). (5.6) kt The total expression for the Helmholtz free energy (F = U static + F vib ) in the harmonic state approximation is then all modes F = U static + i=1 all 2 hω 1 modes i + kt i=1 ln(1 exp ( hω i )), (5.7) kt 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 ω i 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, U static, while DFT linear response 77

93 calculations in the QHA provide the much smaller vibrational energy contribution, TS vib. 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 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 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

94 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 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 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/å) 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/sio 2. 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/cacl 2, and 96 atoms (2 2 2) for α-pbo 2. 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, 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

95 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 Parameters, 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, Grüneisen parameters, and the Anderson-Grüneisen parameter. Note that all QMC calculations for silica use orbitals from DFT within the Wu-Cohen (WC) GGA functional approximation 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 ( K). Such small increments allow for the construction of a fine T-V grid for computing thermodynamic functions with finite differences. 80

96 (a) Quartz (b) Stishovite (c) α-pbo 2 Figure 5.2: Computed Ground state (static) QMC free energy as a function of volume for (a) quartz, (b) stishovite and (c) α-pbo 2. 81

97 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/cacl 2 [112, 113], and α-pbo 2 [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 E(V, T ) = E 0 (T ) + 9K 0(T )V 0 (T ) ξ 2 [1 + [ξ(1 x) 1] exp [(1 x)]], (5.8) where E 0 and V 0 are the zero pressure equilibrium energy and volume, respectively, x = (V/V 0 ) 1/3 and ξ = 3 2 (K 0 1)), K 0(T) is the bulk modulus, and K 0 (T) is the pressure derivative of the bulk modulus. The subscript 0 indicates zero pressure. E 0, V 0, K 0, and K 0 are the four fitting parameters. Pressure is obtained analytically as [ ] 3K0 (T )(1 x) P (V, T ) = exp [ξ(1 x)]. (5.9) x 2 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, F 0, volume, V 0, bulk modulus K 0, and pressure derivative of the bulk modulus, K 0 ) 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

98 Figure 5.3: Thermal equations of state of (A) quartz, (B) stishovite and CaCl 2, and (C) α-pbo 2. 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

99 (a) Quartz (b) Stishovite (c) α-pbo 2 Figure 5.4: QMC and WC free energy as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-pbo 2. 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

100 (a) Quartz (b) Stishovite (c) α-pbo 2 Figure 5.5: QMC and WC volume as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-pbo 2. 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

101 (a) Quartz (b) Stishovite (c) α-pbo 2 Figure 5.6: QMC and WC bulk modulus, K 0 as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-pbo 2. 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

102 (a) Quartz (b) Stishovite (c) α-pbo 2 Figure 5.7: Pressure derivative of the bulk modulus, K 0 as a function of temperature at zero pressure for (a) quartz, (b) stishovite and (c) α-pbo 2. 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 K 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

103 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 ) = U static (V, T ) T S vib (V, T ) + P (V, T )V (P, T ). (5.10) At phase equilibrium (i.e. at the transition pressure), P T ), or G phase1 (P T, V 1 (P T )) = G phase2 (P T, V 2 (P T )) F 1 (P T ) + P T V 1 (P T ) = F 2 (P T ) + P T V 2 (P T ), which gives the expression for the transition pressure as P T = [F 2(P T ) F 1 (P T )], [V 2 (P T ) V 1 (P T )] 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. Statistical 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 CaCl 2 -α-pbo 2 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 CaCl 2 and α-pbo 2. 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

104 and the GGA of Perdew, Burke, and Ernzerhof (PBE) [117] matches the QMC result. For the CaCl 2 -α-pbo 2 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 quartzstishovite 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 CaCl 2 -α-pbo 2 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

105 Figure 5.8: Enthalpy difference of the (A) quartz-stishovite, and (B) CaCl 2 -α-pbo 2 transitions. The DMC transition pressures are 6.4(2) GPa and 88(8) GPa for quartz-stishovite and CaCl 2 -α-pbo 2, 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

106 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 CaCl 2 -α-pbo 2 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

107 Table 5.1: Computed QMC thermal equation of state parameters at ambient conditions (300 K, 0 GPa). The units are as follows: F 0 (Ha/SiO 2 ), F (Ha/SiO 2 ), V 0 (Bohr 3 /SiO 2 ), K 0 (GPa), α10 5 (K 1 ). All other quantities are unitless. QMC one-sigma statistical error on F 0 is Ha/SiO 2. Phase quartz stishovite α-pbo 2 Method QMC Exp. QMC Exp. QMC Exp. F F (4) i 0.020(1) (3) V 0 254(2) a (4) d,e 157.1(2) 154.8(1) h K 0 32(6) a (20) d,e (4) h 313(5) K 0 7(1) a 5.7(9) 3.7(6) d,e (1) h 3.43(11) α 3.6(1) b (1) f 1.26(11) 1.2(1) C p /R 1.82(1) b (1) f 1.57(38) 1.69(1) γ 0.57(1) c (1) f,g (1) q 0.40(1) c (1) f,g 2.6(2) (1) δ T 6.27(1) b (1) f,g (5) 6.40(1) a Ref. [111] b Ref. [124] c Ref. [125] d Ref. [112] e Ref. [113] f Ref. [126] g Ref. [127] h Ref. [89] i Ref. [120] 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, F 0 (Ha/SiO 2 ), the Helmholtz free energy difference relative to quartz, F (Ha/SiO 2 ), volume, V 0 (Bohr/SiO 2 ), bulk modulus, K 0, pressure derivative of the bulk modulus, K 0, thermal expansivity,α (K 1 ), heat capacity, C p /R, Grüneisen ratio, γ, volume dependence of the Grüneisen ratio, q, and the Anderson-Grüneisen parameter, δ T, for the quartz, stishovite, and α-pbo 2 phases of silica. QMC generally offers excellent agreement with experiment for each of these parameters. 92

108 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 P th = P (V, T ) P (V, T 0 ) = T T 0 αk T dt, (5.11) where αk T 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-pbo 2. 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

109 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 Figure 5.10: Computed QMC and WC thermal pressure of (a,b) quartz, (c,d) stishovite and (e,f) α-pbo 2 as 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

110 Changes in Thermal Pressure (αk T ) Figure 5.11 shows changes in thermal pressure [109, 128, 129], given by αk T = ( P/ T ) T. Changes in αk T are quite small (note the scale is 10 3 ) as expected for most materials. 95

111 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 Figure 5.11: Computed QMC and WC variation of αk T with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-pbo 2. QMC results are represented by gray shaded curves, indicating one-sigma statistical error. WC results are represented by red dashed lines. 96

112 Bulk Modulus Figure 5.12 shows the calculated temperature and pressure dependencies of the bulk moduli [109] of quartz, stishovite, and α-pbo 2. The bulk modulus is denoted as K T = V ( 2 ) ( ) F P V 2 = V. (5.12) 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-pbo 2 phase. 97

113 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 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-pbo 2. 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

114 Pressure Derivative of the Bulk Modulus Figure 5.13 shows the rate of change in the bulk modulus with respect to pressure [109], denoted as K = ( K T / P ) T. It is an important quantity in many thermodynamic expressions and also occurs as a parameter in most universal equations of state. K is a unitless quantity and shows only slight variation with both temperature and pressure. 99

115 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 Figure 5.13: Computed QMC and WC variation of the pressure derivative of bulk modulus,k, with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α- PbO 2. 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

116 Thermal Expansivity Figure 5.14 shows the computed QMC and WC thermal expansivity [109] for quartz, stishovite, and α-pbo 2. Thermal expansivity is denoted as α = 1 V ( 2 ) ( F 2 ) F / T V V 2 = 1 T V ( ) V T P (5.13) 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 Bohr 3 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-pbo

117 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 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-pbo 2. 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

118 Heat Capacity Figure 5.15 shows the computed QMC and WC heat capacities for quartz, stishovite, and α-pbo 2. Heat capacity [109] may be computed at constant volume or pressure: C v = ( ) U T V (5.14) or C p = ( ) H. (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-pbo 2 phase. 103

119 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 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-pbo 2. 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

120 Change in Volume Figure 5.16 shows the computed QMC and WC volume differences for quartz-stishovite and CaCl 2 -α-pbo 2 at 0 K. The quartz-stishovite volume change is large due to the four to six fold coordination change. The volume change in the CaCl 2 -α-pbo 2 transition is roughly a factor of ten smaller. 105

121 Figure 5.16: Computed QMC and WC percentage volume difference of (A) quartz-stishovite and (B) CaCl 2 -α-pbo 2 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

122 Grüneisen Parameters The Grüneisen ratio [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 constraining 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: γ i = lnω i lnv. (5.16) 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 γ i over the first Brillouin zone leads to a macroscopic (thermodynamic) definition, written as γ = αk TV C V, (5.17) where α is the thermal expansivity, K T is bulk modulus, V is volume, and C V 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) Grüneisen ratios, 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-pbo 2, γ increases with temperature. 107

123 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 Figure 5.17: Computed QMC and WC variation of the Grüneisen ratio with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-pbo 2. 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

124 An additional parameter, q, is used to describe the volume dependence of γ, and is defined as q = lnγ lnv. (5.18) 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

125 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 Figure 5.18: Variation of q with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-pbo 2. 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

126 Anderson-Grüneisen Parameter The Anderson-Grüneisen parameter [109], δ T, which characterizes the relationship between thermal expansivity and pressure is defined as δ T = ( ) lnα lnv T = 1 αk T ( ) KT. (5.19) T P Figure 5.19 shows δ T may initially increase or decrease at low temperatures, and eventually become constant at high temperatures. At all temperatures, δ T shows a strong decrease with pressure. DFT(WC) and QMC predict very similar results in all cases. 111

127 (a) Quartz (b) Quartz (c) Stishovite (d) Stishovite (e) α-pbo 2 (f) α-pbo 2 Figure 5.19: Computed QMC and WC variation of the Anderson-Grüneisen parameter with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-pbo 2. QMC results are represented by black curves and WC results are represented by red dashed lines. 112

128 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

129 (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

130 The bulk sound velocity is defined as V BS = [ KS ρ ] 1 2, (5.20) where ρ is the density, and K S is the adiabatic bulk modulus. The adiabatic bulk modulus is defined as K S = V ( P/ V ) S, where S indicates adiabatic conditions. However, for solids, the adiabatic bulk modulus generally agrees with the isothermal bulk modulus, K T = V ( P/ V ) T within about one percent at room temperature. The two types of bulk moduli are related by K S = K T (1 + αγt ), where α is the thermal expansivity and γ is the Grüneisen ratio. Regarding silica, the important question is whether the CaCl 2 to α-pbo 2 transition is seismically visible. Since phase stability work described above indicates the transition is not associated with the D layer, localized quantities of α-pbo 2 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 D material) measurements [139, 140] at 120 GPa in D, QMC predicts α-pbo 2 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

131 (a) Quartz (b) Quartz Figure 5.21: QMC calculations of the variation in (a) bulk sound velocity and (b) density with pressure for CaCl 2 and α-pbo 2. Profiles are plotted at both room temperature and the mantle-base temperature and compared with the experimental values of perovskite/postperovskite at the base of the mantle. The pairs of lines in the bulk sound velocity plot indicate one-sigma statistical error. 116

132 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 CaCl 2 [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 V = pɛ ii + 1 ( c ijkl 1 ) 2 2 p (2δ ijδ kl δ il δ jk ) ɛ ij ɛ kl, (5.21) where E is the free energy, V is the volume, ɛ ij is the Eulerian strain, c ijkl is the elastic constant tensor, and δ ij is the Kronecker delta. The pressure terms vanish for volume conserving strains leaving E V = 1 2 c ijklɛ ij ɛ kl. (5.22) It s clear from this expression that the elastic constants are second energy derivatives of the energy with respect to strain, given by c ijkl = 1 V 2 E ɛ ij ɛ kl. (5.23) It is the c 11 c 12 shear constant related to the B 1g Raman mode that becomes unstable at the phase transition to CaCl 2. 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

133 c 11 c 12 is given by ɛ = δ δ (5.24) The strain matrix is applied to the matrix of lattice vectors, R, to produce a new set of lattice vectors R corresponding to a structure with the same volume as follows: R = I + δ δ R (5.25) 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

134 (a) Quartz (b) Quartz (c) Stishovite 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

135 Plugging the strain matrix elements into Equation 5.22 gives the expression needed to evaluate c 11 c 12 : where 2 E δ 2 data. c 11 c 12 = 1 2V 2 E δ 2, (5.26) is the curvature of a polynomial fit to the DFT or QMC energy versus strain Figure 5.23 shows the shear softening of c 11 c 12 as a function of pressure at zero temperature. At low pressures c 11 c 12 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 c 11 c 12 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 CaCl 2 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. Accurate 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

136 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 CaCl 2 phase. 121

137 5.6 Geophysical Implications The QMC CaCl 2 -α-pbo 2 boundary indicates that the transition to α-pbo 2, within a twosigma 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 km above the D 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 D layer. Free silica in D, such as in deeply subducted oceanic crust or mantlecore reaction zones, would have the α-pbo 2 structure. However, based on QMC results, the absence of a global seismic anomaly above D suggests that there is little or no free silica in the bulk of the lower mantle. The α-pbo 2 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 transitions 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 CaCl 2 -α-pbo 2 transition pressure is 88(8) GPa. Results show the CaCl 2 -α-pbo 2 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. A number of thermodynamic properties are computed by combining QMC with DFT phonon 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

138 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

139 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

140 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 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

141 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 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 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

142 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 4 Ha/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 4 Ha/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 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

143 Coulomb and Exchange Series Truncation and Additional Convergence Parameters 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. Therefore, 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

144 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 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), G (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 G (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

145 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

146 Figure 6.1: Gaussian basis set convergence 131

147 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/sio 2 accuracy. Converged silica calculations require k-point mesh sizes of for quartz and 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 pseudopotentials for Si and O. Both pseudopotentials are generated using the appropriate exchangecorrelation 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 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/SiO 2 accuracy. Converged silica calculations require k-point mesh sizes of for quartz and for stishovite, and no k-point shift is used in the VASP calculations. 132

148 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 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

149 Figure 6.2: Computed HSE energy versus volume curves for quartz and stishovite. Points are fit with the Vinet equation of state. 134

150 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

151 (a) Quartz LDA (b) Quartz PBE (c) Quartz PW91 and PBEsol (d) Quartz WC (e) Quartz Hybrids Figure 6.3: Computed pressure versus volume curves of quartz using various exchangecorrelation functionals and codes. 136

152 (a) Stishovite LDA (b) Stishovite PBE (c) Stishovite PW91 and PBEsol (d) Stishovite WC (e) Stishovite Hybrids Figure 6.4: Computed pressure versus volume curves of stishovite using various exchangecorrelation functionals and codes. 137

153 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

154 (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

155 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

156 (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

157 6.4.5 Equilibrium Quartz and Stishovite K 0 from Vinet Fits Figures 6.7 (a) and (b) show a comparison of K 0 for quartz and stishovite, respectively; The K 0 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

158 (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

159 Table 6.1 and 6.2 provides zero pressure values of lattice constants (a and c), volume (V 0 ), bulk modulus (K 0 ), and pressure derivative of the bulk modulus (K 0 ) for all of the various DFT functionals and codes. There are two different volumes given in the table: V 0F R corresponds to the fully relaxed, equilibrium DFT geometry and V 0 is the zero pressure volume predicted by the minimum of the Vinet fit to energy versus volume data. V 0F R 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 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

160 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/SiO 2, and the bulk modulus is in units of GPa. Quartz CRYSTAL a c V 0F R V 0 K 0 K 0 LDA PBE PW B3LYP PBE ABINIT a c V 0F R V 0 K 0 K 0 LDA PBE PW PBEsol WCGGA VASP a c V 0F R V 0 K 0 K 0 LDA PBE PW HSE Experiment a (4) 5.7(9) QMC b 254(2) 32(6) 7(1) a Ref. [111] b Ref. [84] 145

161 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/SiO 2, and the bulk modulus is in units of GPa. Stishovite CRYSTAL a c V 0F R V 0 K 0 K 0 LDA PBE PW B3LYP PBE ABINIT a c V 0F R V 0 K 0 K 0 LDA PBE PW PBEsol WCGGA VASP a c V 0F R V 0 K 0 K 0 LDA PBE PW HSE Experiment a (9) 4.45(15) QMC b 159.0(4) 305(20) 3.7(6) a Ref. [87] b Ref. [84] 146

162 Figure 6.8: Computed HSE enthalpy curves of quartz and stishovite as a function of pressure. 147

163 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

164 Figure 6.9: Computed quartz-stishovite transition pressures using various exchangecorrelation 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