Download Establishing Quantum Monte Carlo and Hybrid Density Functional

Establishing Quantum Monte
Carlo and Hybrid Density
Functional Theory as
benchmarking tools for
complex solids
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of The Ohio State University
By
Kevin P. Driver, B.S., M.S.
Graduate Program in Physics
The Ohio State University
2011
Dissertation Committee:
John W. Wilkins, Advisor
Richard J. Furnstahl
Ciriyam Jayaprakash
Arthur J. Epstein
c Copyright by
Kevin P. Driver
2011
Abstract
Quantum mechanics provides an exact description of microscopic matter, but predictions
require a solution of the fundamental many-electron 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
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
To my family and friends for guidance, help, and love.
iv
Acknowledgments
The research and underlying educational enlightenment represented by this thesis are most
notably a product of the continuous support and encouragement of my advisor, John
Wilkins. John provided impeccable guidance and maintained a high standard of excellence
in developing my scientific career.
Other than my advisor, a few others deserve specific mention for their critical guidance
and support. Richard Hennig was an excellent mentor and source of scientific inspiration
through out my entire graduate career. My office mate and group partner, William Parker,
provided invaluable amounts of feedback and support throughout my entire graduate career
as well. I am also indebted to the help and guidance of Ronald Cohen, whose training made
significant portions of this research (silica) possible.
There are many other people have taught me or played some role in my scientific 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
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 2008. This work was also made possible by generous computational
resources from OSC, NERSC, NCSA, and CCNI.
vi
Vita
April 14, 1980 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born—New Albany, Indiana, USA
2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S., University of Louisville, Louisville,
Kentucky
2003-2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fowler Fellow, Department of Physics,
Ohio State University, Columbus, Ohio
2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S., Ohio State University, Columbus,
Ohio
2005-Present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Associate, Department of Physics, Ohio State University,
Columbus, Ohio
Publications
K. P. Driver, R. E. Cohen, Zhigang Wu, B. Militzer, P. López Rı́os, M. D. Towler, R. J.
Needs, and J. W. Wilkins, Quantum Monte Carlo computations of phase stability, equations
of state, and elasticity of high-pressure silica, Proc. Natl. Acad. Sci. USA, 107, 9519
(2010).
R. G. Hennig, A. Wadehra, K. P. Driver, W. D. Parker, C. J. Umrigar, and J. W. Wilkins,
Phase transformation in Si from semiconducting diamond to metallic beta-Sn phase in QMC
and DFT under hydrostatic and anisotropic stress, Phys. Rev. B, 82, 014101 (2010).
M. Floyd, Y. Zhang, K. P. Driver, Jeff Drucker, P.A. Crozier, and D.J. Smith, 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
Fields of Study
Major Field: Physics
Studies in quantum Monte Carlo claculations of solids: J. W. Wilkins
viii
Table of Contents
Abstract . . . . . .
Dedication . . . .
Acknowledgments
Vita . . . . . . . .
List of Figures .
List of Tables . .
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Page
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ii
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iv
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v
. vii
. xii
. xiv
Chapters
1 Introduction
1.1 Quantum Mechanics Correctly Explains Matter . . . . . . . . . . . . . . . .
1.2 Modeling Matter with Numerical, Quantum Simulations . . . . . . . . . . .
1.3 Organization and Summary of Thesis Accomplishments . . . . . . . . . . .
1
1
2
4
2 Methods of Solving the Schrödinger Equation
2.1 The Many Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Born Oppenheimer Approximation . . . . . . . . . . . . . . . .
2.2 Mean-Field-based Ab Initio Methods . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Hartree Approximation: No Exchange, Averaged Correlation . .
2.2.2 The Hartree-Fock Approximation: Explicit Exchange, Averaged Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Self-interaction Error in DFT . . . . . . . . . . . . . . . . . . . . . .
2.4 Many Body Ab Initio Methods . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Configuration Interaction . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . .
7
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9
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3 Coping with DFT Approximations: Benchmarking with Hybrid functionals and QMC
3.1 Introduction: Approximations and Weaknesses of Density Function Theory
3.1.1 Exchange-Correlation Approximations: Categorization of functionals
in Jacob’s ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Basis Set Approximations . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Benchmarking functionals With Hybrid DFT and Quantum Monte Carlo .
ix
10
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31
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4 Results for Silicon Self-Interstitials
4.1 Introduction . . . . . . . . . . . . .
4.2 Calculation Details . . . . . . . . .
4.2.1 Results . . . . . . . . . . .
4.2.2 Tests for errors in QMC . .
4.3 Conclusions . . . . . . . . . . . . .
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5 Results for Silica
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Previous Work and Motivation . . . . . . . . . . . . . .
5.3 Computational Methodology . . . . . . . . . . . . . . .
5.3.1 Pseudopotential Generation . . . . . . . . . . . .
5.3.2 DFT Calculations . . . . . . . . . . . . . . . . .
5.4 QMC Calculations . . . . . . . . . . . . . . . . . . . . .
5.4.1 Wave-function Construction and Optimization .
5.4.2 DMC Calculations . . . . . . . . . . . . . . . . .
5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Free Energy . . . . . . . . . . . . . . . . . . . . .
5.5.2 Thermal Equations of State and Fit Parameters
5.5.3 Phase Stability . . . . . . . . . . . . . . . . . . .
5.5.4 Thermodynamic Parameters . . . . . . . . . . .
5.5.5 Stishovite Shear Constant . . . . . . . . . . . . .
5.6 Geophysical Implications . . . . . . . . . . . . . . . . . .
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Hybrid DFT Study of Silica
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Hybrid Calculations of Silica . . . . . . . . . . . . . . . . . . . .
6.2.2 Hybrid B3LYP and PBE0 Calculations of Solids . . . . . . . . .
6.2.3 Screened Hybrid (HSE) Calculations of Solids . . . . . . . . . . .
6.3 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 CRYSTAL Calculations . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 ABINIT Calculations . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 VASP Calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Energy Versus Volume . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Pressure Versus Volume . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Equilibrium Quartz and Stishovite Volume from Vinet Fits . . .
6.4.4 Equilibrium Quartz and Stishovite Bulk Moduli from Vinet Fits
6.4.5 Equilibrium Quartz and Stishovite K00 from Vinet Fits . . . . . .
6.4.6 Enthalpy Versus Pressure . . . . . . . . . . . . . . . . . . . . . .
6.4.7 Quartz-Stishovite Transition Pressures . . . . . . . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Conclusions
151
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
x
7.2
Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
Appendices
A Error Propagation in QMC Thermodynamic Parameters
169
A.1 Taylor Expansion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.2 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B Optimized cc-pVQZ Gaussian Basis Set used for Silica
172
C Details of the Ewald and MPC Interaction in Periodic Calculations
177
C.1 Ewald Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C.2 Model Periodic Coulomb (MPC) Interaction . . . . . . . . . . . . . . . . . . 184
D Summary of CODES Used in
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 . . . . . . . . . . . . .
This Work
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E Strong and Weak Scaling in the CASINO QMC Code
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List of Figures
Figure
Page
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
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 . . . . . . . . . . . . . .
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5.1
5.2
5.3
5.4
5.5
5.6
5.7
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. . . . . . . . . . . . . . . . . . . . . . . . . . .
73
81
83
84
85
86
5.8
5.9
5.10
5.11
5.12
5.13
5.14
xii
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87
90
91
94
96
98
100
102
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
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 . . . . . . . . .
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104
106
108
110
112
114
116
119
121
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
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 K0 of silica. . . . . . . . . . . . . . . . . . . . . .
HSE enthalpy versus pressure of quartz and stishovite
DFT quartz-stishovite transition pressures . . . . . . .
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131
134
136
137
139
141
143
147
149
E.1 CASINO VMC weak scaling . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.2 CASINO DMC weak scaling . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.3 CASINO DMC strong scaling . . . . . . . . . . . . . . . . . . . . . . . . . .
189
190
191
xiii
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List of Tables
Table
Page
4.1
Separation of bulk and defect e-n Jastrows . . . . . . . . . . . . . . . . . . .
68
5.1
Silica thermal equation of state parameters . . . . . . . . . . . . . . . . . .
92
6.1
6.2
Quartz DFT equation of state parameters . . . . . . . . . . . . . . . . . . .
Stishovite DFT equation of state parameters . . . . . . . . . . . . . . . . .
145
146
xiv
Chapter 1
Introduction
1.1 Quantum Mechanics Correctly Explains Matter
In the late 17th century, Isaac Newton’s Principia [1] solidified mathematical physics as the
precise and formal language to describe the nature of the universe. Mathematical theories, in check with experimental observations, allow the logical connection of one fact to
another, giving a comprehensive, disillusioned picture of the universe. Most often, progress
in the scientific conception of the universe comes about from the reciprocal relationship of
experiment and theory: mathematical predictions inspire new experiments and experiments
inspire modifications to mathematical theory. In the case of Newton’s Principia, the mathematical foundation was laid for an entire field of physics known as classical mechanics.
Classical mechanics prevailed as the theory of motion of macroscopic objects for over two
centuries when its inadequacies to describe the microscopic world became apparent in the
atomic era.
A modern, updated theory, quantum mechanics [2, 3], emerged in the early 20th century
as science progressed into the atomic era with the help of many important scientific figures
(Moseley, Thompson, Rutherford, Bohr, Heisenberg, Einstein, Dirac, De Broglie, Millikan,
Stern, Gerlach, Pauli, and 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
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 = ih̄
∂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
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
This thesis also focuses on a highly accurate, stochastic ab initio method called quantum
Monte Carlo (QMC) [8]. QMC explicitly computes the exchange and correlation of electrons
efficiently enough to produce benchmark accuracy for solids [9], but the computational
expense of QMC makes it intractable to replace DFT. One of the important applications
of QMC is for benchmarking the predictions of density functionals. A particular density
functional may fail for a given system, but the failure can be overcome by identifying a
density functional that is capable of describing the system of interest. However, if there is
no experimental data, a highly accurate and reliable method, such as QMC, must be used
to benchmark the functionals.
Part of the aim of this thesis is to investigate whether hybrid DFT functionals can
reliably match the accuracy of QMC and also be used as a benchmarking tool for standard functionals. QMC and hybrid DFT are too computationally expensive to replace the
computational demand standard DFT fulfills for materials science. QMC calculations are
generally heroic computational efforts (millions of CPU hours), hundreds of times more
expensive than standard DFT. Hybrid DFT is about thirty times more expensive than
standard DFT, capable of computing energies of larger systems. Neither QMC or hybrid
calculations are efficient enough to compute atom dynamics in solids, such as forces or
phonons. QMC and hybrid DFT have only been used to publish perhaps a couple dozen
simple solid calculations, compared to tens of thousands of DFT calculations. The main
aim of this thesis is to significantly expand the scope of QMC and hybrid functionals for
solids by applying them to large, complex structures and establish them as a benchmarking
tools for more efficient DFT exchange-correlation functionals.
1.3 Organization and Summary of Thesis Accomplishments
The research presented in this thesis makes use of enormous amounts of knowledge and tools
developed by other researchers in various scientific communities. Chapter 2-3 introduces
the electronic structure methods and concepts that set the foundation for the research in
this thesis. The first sections of chapter 2 discusses the many-body problem and presents
4
mean-field-based Hartree-Fock and density functional approaches to solving the many-body
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
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
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
N
h
i
X
X
Zi Zj 
1
∇2ri +
Ψ(R) = EΨ(R),
ĤΨ(R) T̂ + V̂ Ψ(R) = −
2mi
|ri − rj |
(2.1)
i>j
i=1
which is a 3N-dimensional eigen-problem, where R is the collective coordinate for all N
particles ri ,...,rN . Ψ(R) is a square integrable wave-function, which is anti-symmetric under
exchange of two electrons, obeying the Pauli exclusion principle. The equation is written in
atomic units (e = me = h̄ = 4π0 = 1), and T̂ and V̂ are the kinetic and Coulomb potential
energy, respectively.
The Coulomb potential term forces Schrödinger’s equation to be inseparable for more
than one particle. The simplest possible solution technique of separation of variables is
ruled out, which means the form of the wave-function is not a simple product of 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
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).
2.1.1 The Born Oppenheimer Approximation
A common approximation that all methods discussed in this thesis take advantage of is the
Born-Oppenheimer Approximation [11, 6, 10]. In this approximation, for the purpose of
constructing the Hamiltonian, the nuclei are held in fixed position in order to separate out
the electronic and nuclear degrees of freedom. This approximation is reasonable because
the mass of the nuclei are several thousand times larger than the mass of electrons. 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

X X Zα
X
X
X 1
Zα Zβ 
1
1
∇2 +
+
+
,
Ĥ = −
2mi ri
|r
−
d
|
|r
−
r
|
2
|d
i
α
i
j
α − dβ |
α

i
i
i>j
(2.2)
α>β
where the terms for electrons of charge -1 at positions ri and ions of charge Zα at positions
dα have been separated. This Hamiltonian is still difficult to solve, with still no analytic
solution for more then one electron, but excellent approximations can be made starting with
the Born-Oppenheimer Hamiltonian.
8
2.2 Mean-Field-based Ab Initio Methods
2.2.1 The Hartree Approximation: No Exchange, Averaged Correlation
A common approach for an ab initio treatment of the many-body problem is to break down
the many-electron 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,
Z
el
U (r) = −e
dr0 ρ(r0 )
1
,
|r − r0 |
(2.3)
where and each one-electron equation will yield a single-electron wave-function, ψi , called
an orbital, and an orbital energy. The total electronic charge would be
ρ(r) = −e
X
|ψi (r)|2 ,
(2.4)
i
where the sum is over all occupied levels. And, the ion potential is
U ion (r) = −Z
X
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(r1 , r2 , ..., rN ) = ψ1 (r1 )ψ2 (r2 )...ψN (rN )
(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:

1
− ∇2 ψi (r) + U ion (r)ψi (r) + e2
2

XZ
2
dr0 ψj (r0 )
j
9
1 
ψi (r) = i ψi (r)
r − r0
(2.7)
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
| ψi |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.
2.2.2 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:
Ecorrelation = Eexact,non−relativistic − EHartree−Fock
10
(2.8)
The Pauli exclusion principle requires the wave-function to be antisymmetric under
exchange, such that when two electrons are interchanged the wave-function changes sign:
Ψ(r1 , r2 , ..., ri , rj , ..., rN ) = −Ψ(r1 , r2 , ..., rj , ri , ..., rN )
(2.9)
In the Hartree-Fock method, the indistinguishability and exchange properties of electrons
are included mathematically by representing the wave-function as a Slater determinant of
one-electron orbitals instead of a simple product of orbitals as in the Hartree method. The
determinant is a antisymmetric function of all permutations of one-electron wave-functions:
ψ1 (r1 ) ψ2 (r1 ) · · ·
ψ1 (r2 ) ψ2 (r2 ) · · ·
Ψ(r1 , r2 , ..., rN ) = ..
..
..
.
.
.
ψ1 (rN ) ψ2 (rN ) · · ·
ψN (r1 ) ψN (r2 ) ..
.
ψN (rN ) (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:
X
1
δ si sj
− ∇2 ψi (r)+U ion (r)ψi (r)+U el (r)ψi (r)−
2
j
Z
dr0
1
ψ ∗ (r0 )ψi∗ (r0 )ψj∗ (r0 ) = i ψi (r),
|r − r0 | j
(2.11)
where si represents the spin state. The additional term on the left side compared to the
Hartree equations (Equation 2.7) is known as the exchange term. The exchange term is
only non-zero when considering like spins and causes like-spin electrons to avoid each other.
The exchange term adds considerable complexity to the one-electron equations, making the
Hartree-Fock equations difficult to solve except for special cases.
11
Self-interaction Error Exactly Cancels in Hartree-Fock
Just as in the Hartree Equations, the Hartree-Fock equations have a Hartree potential
(classical Coulomb) term that includes a spurious self interaction of an electron with itself.
However, in the Hartree-Fock equations, the self-interaction energy is exactly cancelled by
the Exchange (Fock) term.
2.3 Density Functional Theory
DFT is currently one of the most successful and popular electronic methods available for
computing properties of real solids. It allows for a great simplification in solving the 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, Etot , of a system is a unique functional of the electron density, n(r).
Furthermore, the functional Etot [n(r)] is minimized for the ground state density, nGS (r). In
short, this affords the possibility of calculating electronic properties based on the electron
density (3 spatial variables), instead of the 3N-variable many-body wave function.
The second part of the theory involves the construction and variational minimization
of the total energy functional, Etot [n(r)], with respect to variations in the electron density.
This part of the theory, developed by Kohn and Sham [14] states that the many-body
Schrödinger equation can be mapped onto the problem of solving an effective single-particle
wave equation with an effective potential, Veff . The first step is to write the total energy
functional as
Z
Etot [n(r)] = T [n(r)] + Ee−e [n(r)] + Exc [n(r)] +
Vext (r)n(r)dr
(2.12)
where T is the kinetic energy of a noninteracting system, Ee−e is the electron-electron
interaction energy, Exc is the exchange-correlation energy, and Vext represents an external
potential including the ions. By minimizing this total energy functional with respect to
12
variations in the electron density, subject to the constraint that the number of electrons are
fixed, an effective one-particle equation is obtained:
h̄2 2
−
∇ + Vef f [n(r)] ψi (r) = i ψi (r),
2m
(2.13)
where Vef f is an effective potential given by
Vef f [n(r)] = Vext [n(r)] + Ve−e [n(r)] + Vxc [n(r)],
(2.14)
where Ve−e is the Hartree potential,
Z
Ve−e (r) = −e
n(r) 0
dr ,
r − r0
(2.15)
and
Vxc [n(r)] =
δExc [n(r)]
.
δ[n(r)]
(2.16)
Equations 2.13 and 2.14, are known as the Kohn-Sham equations. The quantities ψi and
i are auxiliary quantities used to calculate the electron density and total energy, not the
wave function and energy of real electrons.
A formally exact expression for the exchange-correlation energy [15], Exc [n(r)], is given
by
Exc
1
=
2
Z
Z
dr
dr0 n(r)
n̄(r, r0 )
,
|r0 − 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
0
Z
n̄xc (r, r ) =
1
dλnλxc (r, r0 ) = nx (r, r0 ) + n¯c (r, r0 )
(2.18)
0
is the average over the coupling constant λ of the density at r0 of the exchange-correlation
hole about an electron at r :
nλxc (r, r0 ) =
hΨα | n̂(r)n̂(r) | Ψλ i
− δ(r − r0 ),
n(r)
(2.19)
0
and nx (r, r0 ) = nα=0
xc (r, r ) is the exchange hole. Here, Ψα is the correlated ground state
wave-function for a system with the same spin densities as the real system but with the
13
electron-electron interaction reduced by a factor α. Due to Pauli exclusion and Coulomb
repulsion, an exchange-correlation hole forms satisfying the sum rule
Z
dr0 nαxc (r, r0 ) = −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 Vxc [n(r)], and then solves Eq. (2.13) for the wave functions,
ψi (r), using a standard band theory technique. From the wave functions obtained, one
calculates a new charge density:
n(r) =
occ.
X
|ψi (r)|2 .
(2.21)
i
This procedure is then repeated until the charge density is converged.
2.3.1 Self-interaction Error in DFT
Similar to Hartree-Fock, DFT contains a Hartree potential term (Equation 2.15) with a
spurious self-interaction error. In the formal DFT theroy, an exact exchange-correlation
functional potential cancels the self-interaction energy, just as the Fock term cancels the
error in Hartree-Fock. However, in all practical calculations, DFT approximates the exchange and correlation with an approximate exchange-correlation functional. The approximate functional does not likely cancel the self-interaction error in the Hartree term, which
may introduce a significant error in some calculations.
14
2.4 Many Body Ab Initio Methods
Fully many-body methods abandon the mean field approach of reducing the full Schrödinger
equation to a set of one-electron equations and compute exchange and correlation explicitly
using a many-body wave-function. The following section mentions the Configuration 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.
2.4.1 Configuration Interaction
The Configuration Interaction (CI) Method [6] is a general technique of going beyond the
Hartree-Fock Approximation. The Hartree-Fock approximation uses a single Slater determinant to represent the many-electron ground state wave-function. In the CI method, the
many-electron wave-function is written as a linear combination of many slater determinants
representing energetically higher orbitals:
Φ=
N
CI
X
CI Φ I ,
(2.22)
I=0
where CI are the CI expansion coefficients and ΦI are the different configurations of orbitals.If there are N electrons and M basis states, then there are
NCI =
M!
N !(M − N )!
(2.23)
configurations that can be constructed form the orbitals. Setting all CI to 0 except C0 = 1
reduces the CI method to the Hartree-Fock method.
In order to simulate solids, a very large number of determinants is required (perhaps
millions or billions). The computational cost scales exponentially with the number of 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
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
Z
I=
dRg(R),
(2.24)
an “importance function,” P (R) is introduced such that
Z
I=
dRf (R)P (R),
where the importance function is a probability density such that P (R) > 0 and
(2.25)
R
dRP (R) =
1, and f (R) = g(R)/P (R). The mean value theorem from calculus asserts that the exact
vaule of the integral can now be evaluated by sampling infinitely many points from P(R)
16
and computing the sample average:
"
I = lim
M →∞
#
M
1 X
f (Rm ) .
M
(2.26)
m=1
However, Monte Carlo only estimates the integral by averaging over a finite sample of
points from P(R):
IM ≈
M
1 X
f (Rm ),
M
(2.27)
m=1
which provides a result with a certain statistical confidence error. The variance of the
estimate of I is σf2 /M , which is estimated as
"
#2
M
M
X
σf
1
1 X
≈
f (Rm ) −
f (Rn ) ,
M
M (M − 1)
M
m=1
(2.28)
n=1
such that the statistical standard deviation decreases with the square root of the number
of samples:
σf
I ≈ IM ± √ .
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
Variational Monte Carlo
The variational Monte Carlo (VMC) method is the less rigorous of the two QMC methods
discussed in this thesis. VMC provides a less expensive estimate of the total energy and
is used to optimize the trial wave-function, which diffusion Monte Carlo (DMC) uses to
project out the true ground state. VMC is essentially the evaluation of the variational
principle using Monte Carlo integration and a many-body wave-function, ΨT .
In order for VMC to work, ΨT must be a reasonably good approximation to the ground
state. Details of generating a good trial wave-function will be discussed in a later section.
In general,ΨT and ∇ΨT must be continuous when the potential is finite and the integrals
R ∗
R
R
ΨT ΨT and Ψ∗T ĤΨT must exist. It is also convient that Ψ∗T ĤΨT 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 ΨT is an upper bound on the ground-state energy
E0 :
R ∗
Ψ (R)ĤΨT (R)dR
EV = R T∗
ΨT (R)ΨT (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, EL (R) :
Z
EV =
p(R)EL (R)d(R),
(2.31)
Ψ2T (R)
,
Ψ2T (R0 )d(R0 )
(2.32)
where
p(R) = R
and
2
EL (R) = Ψ−1
T (R)ĤΨT (R).
(2.33)
The Metropolis algorithm is used generate a set of electron configurations, sometimes
called walkers, {(Rm : m = 1, M )} from the configuration-space probability density. The
18
local energy is evaluated for each walker and the average energy is computed:
M
1 X
EV ≈
EL (Rm ),
M
(2.34)
m=1
with a statistical error of
r
σV M C ≈
1
(hEL (Rm )2 i − hEL (Rm )i2 ).
M
(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
1
−∂t Φ(R, t) = (Ĥ − ET )Φ(R, t) = (− ∇2R + V (R) − ET )Φ(R, t),
2
(2.36)
where t measures progress in imaginary time, R = (r1 , r2 , ..., rN ) is a 3N-dimensional vector
(also called a configuration or walker) providing the coordinates of the N electrons, and ET
is an energy offset or interaction strength. In order to propagate the walkers in imaginary
time, Equation 2.36 is written in integral form using a Green’s function, G(R ← R0 , t) and
time-step, τ :
Z
G(R ← R0 , τ )Φ(R0 , t)d(R0 ),
(2.37)
G(R ← R0 , τ ) = hR | exp(−τ (Ĥ − ET )) | R0 i.
(2.38)
Φ(R, t + τ ) =
where
It follows, that the Green’s function obeys Equation 2.36,
−∂t G(R ← R0 , t) = (Ĥ(R) − ET )G(R ← R0 , t),
(2.39)
satisfying the initial condition G(R ← R0 , 0) = δ(R − R0 ). Using the spectral expansion,
exp(−τ Ĥ) =
X
| Ψi i exp(−τ Ei )hΨi |,
i
19
(2.40)
the Green’s function can be written in a manner which reveals important physics:
G(R ← R0 , τ ) =
X
Ψi (R) exp(−τ (Ei − ET ))Ψ∗i (R0 )
(2.41)
i
This expression reveals that the critical feature of DMC is that the Green’s function
operator exp(−τ (Ĥ − ET )) projects out the lowest energy eigenstate | Ψ0 i in the limit of
infinite time steps (τ → ∞):
limτ →∞ Φ(R, t)
=
=
=
lim hR | exp(−τ (Ĥ − ET )) | Φinit i
Z
lim
G(R ← R0 , τ )Φinit (R0 )dR0
τ →∞
X
lim
Ψi (R) exp(−τ (Ei − ET ))hΨi | Φinit i
τ →∞
τ →∞
(2.42)
(2.43)
(2.44)
i
lim Ψ0 (R) exp(−τ (E0 − ET ))hΨ0 | Φinit i
τ →∞
(2.45)
The last step (the crux of DMC) follows from the fact that Ei > Ei−1 · · · > E2 > E1 > E0 ,
where Ei are excited states and E0 is the ground state energy. That is, the excited states
are all exponentially damped compared to the ground state and will decay to zero in the
limit of infinite time steps.
Unfortunately, the expression for the exact Green’s function solving Equation 2.36 is
not known except for a few special, simplistic cases. An approximate Green’s function
must be constructed. Some important insight can be gain if, for the moment, the potential
term is neglected in Equation 2.36. Neglecting the potential term reduces the 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 {Ri } to a random
hops to simulate the diffusion process and an ability to undergo a birth-death process to
simulate the rate process, which is sometimes called branching.
Let Ĥ = V̂ +V̂ , where T̂ is the N-electron kinetic energy operator and V̂ is the N-electron
potential energy operator. Then, the Trotter-Suzuki formula [18] for the exponential sum
of operators applied to Equation 2.38leads to an approximate Green’s function for small τ :
20
(R − R0 )2
× exp −τ V (R) + V (R0 ) − 2ET /2 ,
G(R ← R0 , τ ) ≈ (2πτ )3N/2 exp −
2τ
(2.46)
or
G(R ← R0 , τ ) ≈ G˜D (R ← R0 , τ )G˜B (R ← R0 , τ ),
(2.47)
where the first factor,
0
3N/2
GD (R ← R , τ ) = (2πτ )
(R − R0 )2
exp −
,
2τ
(2.48)
is the Green’s function for a diffusion equation, while the second factor,
GB (R ← R0 , τ ) = exp −τ V (R) + V (R0 ) − 2ET /2 ,
(2.49)
is a time-dependent renormalization (re-weighting) of the diffusion Green’s function known
as the branching-factor, which determines the number of walkers that survive in the 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
nodes of the trail function are to that of the exact ground state. It follows that the DMC
energy is always less than or equal to the VMC energy using the same trial wave-function,
and DMC energy is always greater than or equal to the exact ground state. Techniques to
relax the node positions, called backflow, have been developed in an effort to reduce the
fixed-node error in systems [19].
Importance Sampling Transformation
An impotance sampling transformation vastly improves efficiency of the DMC method.
The importance sampling function is a mixed distribution of the trial wave-function ΨT
and DMC wave-function Φ,
f (R, t) = ΨT (R)Φ(R, t),
(2.50)
is purely positive if the nodes of both wave-functions are equal. Substituting into Equation 2.36 gives
−
∂f
1
= − ∇2R f + ∇R · [vD f ] + [EL − ET ] f,
∂t
2
(2.51)
where
vD (R) = Ψ−1
T (R)∇R ΨT (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:
Z
f (R, t + τ ) =
G̃(R ← R0 , τ )f (R0 , t)d(R0 ),
(2.53)
where G̃ is the modified Green’s function for the importance sampled wave-function, which
22
amounts to a similar expression for the Green’s function as in Equation 2.46, but updated
for the importance sampling:
G̃(R ← R0 , τ ) ≈ G̃D (R ← R0 , τ )G̃B (R ← R0 , τ ),
(2.54)
where, again, the first factor,
(R − R0 − τ vD (R0 ))2
,
G̃D (R ← R0 , τ ) = (2πτ )3N/2 exp −
2τ
(2.55)
is the Green’s function for a diffusion equation which now has a new drift term, while the
second factor,
G̃B (R ← R0 , τ ) = exp −τ EL (R) + EL (R0 ) − 2ET /2 ,
(2.56)
is the branching-factor, where the local energy replaced the potential.
The Green’s function, G̃D (R ← R0 , τ ), makes each configuration drift a distance τ v(R0 )
and then diffuse by a random distance drawn from Gaussian noise on τ. Each configuration
is then copied or deleted according to G̃B (R ← R0 , 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) = ΨT (R)φ0 (R), where φ0 is the ground state for the
P
wave-function expanded in eigenstates,φi of the Hamiltonian: Φ(R, t) = i ci (t)φi (t)(R).
The fixed-node DMC energy is evaluated using HΨT = EL ΨT , where EL is the local energy:
EDM C
hφ0 | Ĥ | ΨT i
= E0 =
=
hφ0 | ΨT i
R
M
f (R)EL (R)dR
1 X
R
≈
EL (Ri ).
M
f (R)dR
(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
importance sampling, controls the statistical efficiency, and limits the final accuracy of both
VMC and DMC.
VMC and DMC algorithms repeatedly evaluate the trial-wave-function, which demands
a form that is compact and can be evaluated rapidly. While most quantum chemistry
methods use linear combinations of determinants for the wave-function, they converge slowly
due to their difficulty in describing cusps associated with two electrons coming in close
contact. QMC instead uses a Slater-Jastrow form of the wave-function consisting of a pair
of up and down spin determinants multiplied by a Jastrow correlation factor:
ΨSJ (R) = exp[J(R)] det[ψn (r↑i )] det[ψn (r↓j )],
(2.58)
where exp[J] is the Jastrow factor and det[ψn (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({ri }, {rI }) =
N
X
i>j
u(rij ) +
NX
N
ions X
χI (riI ) +
NX
N
ions X
fI (riI , rjI , rij ),
(2.59)
I=1 i>j
I=1 i=1
where N is the number of electrons, Nions is the number of ions, rij = ri − rj , riI = ri − rI ,
ri is the position of the electrons, and rI is the position of the nucleus I. The functions u, χ,
and f are power expansions with a variety of optimizable coefficients. Reasonable choices
for the form of the functions have been developed by Drummond et al. [20]
Each function comprising the Jastrow has a number of parameters which are optimized
through minimizing either the variance of the local energy, the energy itself, or some com-
24
bination. The first step of the optimization procedure [21] is to generate a set of electron
configurations distributed according to the square of the trial wave-function. The Jastrow
parameters are then passed to an algorithm which adjusts the parameters to minimize the
desired quantity. Following the initial optimization, a set of configurations is generated
using the optimized wave-function and the minimization is performed again. Such an 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
The error can be made as small as one needs for the problem at hand. Typical errors for
cohesive or formation energies are around 0.0001 Ha per formula unit. Elastic constants
may require 0.00004 Ha per formula unit.
Wave-function Form
The Slater-Jastrow form of the VMCM wave-function described in the previous section is
an approximation of the true many-body wave-function. Electron correlation is accounted
for through the fixed form of the Jastrow factor, which has several optimizable parameters.
The exchange is accounted for via a single Slater determinant with single-electron orbitals
from DFT. In principle, there are better forms of the Jastrow, perhaps with more parameter
flexibility, and more than one determinant can be used for the exchange part, as is popular
in quantum chemistry. The fixed wave-function form limits how close VMC can estimate
the true ground state energy. In general, the form used in the codes for this thesis are
extremely good, and often match the results obtained with DMC.
Numerical orbitals
The Slater determinant part of the wave-function uses single-electron orbitals that are
typically produced from a DFT calculation. The orbitals are most often represented in
a plane-wave form. However, the repeated evaluation of plane-wave orbitals is expensive
in QMC because it requires a sum over all plane-waves in the cell. A significant increase
in speed is achieved by re-representing the plane-wave orbitals with a localized polynomial
basis that approximates a grid of plane-wave orbitals. The speed up (typically by a factor of
the number of particles simulated) comes from the fact that while using a localized basis set,
the orbital evaluation has a computational cost independent of the system size. The type of
polynomials that work most efficiently are B-splines. The grid spacing is a approximation
that must be checked for convergence, but typically the so called, natural grid spacing [22]
is well-converged corresponding to about five grid points per angstrom.
26
DMC time-step
The progress of the time-dependent projection algorithm in DMC uses a small, finite, imaginary time step. The size of the time step affects the accuracy of the total energy. Before
a production calculation, the time step must be converged by computing the DMC energy
for a range of increasing small time steps until suitable convergence is found. Typically
convergence in the energy to chemical accuracy or better is the goal.
Finite-Size Errors
Calculations of solids with periodic boundary conditions give the effect of infinitely repeating
the simulation cell in all directions. The finite size of the repeated simulation cell introduces
fictitious errors into the energy in various ways [23]. In fact, there are three types of finite
size errors: 1) The error associated with QMC choosing only a single k-point to evaluate
single particle orbitals of the wave function, 2) Spurious correlation effects due to replication
of electron behavior in mirroring cells, and 3) if the calculation includes a defect, then the
defects can interact in mirroring cells.
The single kpoint approximation can be dealt with a number of ways. The first is to
use so called, twist-averaging, where the DMC energy is computed at several different kpoints and averaged. The k-point error can also simply be estimated from the DFT energy
difference between a converged k-point calculation and one with the equivalent number
of k-points that correspond to the simulation cell with one k-point in each primitive cell.
Alternatively, increasing the size of the simulation cell eventually makes the k-point error
negligible.
The correlation error due to mirroring cells is reduced by modifying the Hamiltonian.
The so called model periodic Coulomb (MPC) interaction [24] alters the Ewald interaction
such that the pure Coulomb interaction is restored within the simulation cell. Alternatively,
a structure factor method is used to provide a correction for both potential and kinetic
energies [25]. A third method [26], which uses the difference in energy of a DFT calculation
of a finite-sized and infinite-sized model of the exchange-correlation functional corrects the
27
finite size error of the total energy.
The defect interaction from periodic cells is estimated either by studying DFT calculations of a range of cells or performing DMC for a range of cell sizes. The error can be made
arbitrarily small with a large enough simulation cell size. Often, the defect interaction error
will cancel when differences in energies are taken.
Walker Population Size
DMC uses a statistical form of the many-body wave-function, which is represented by a
finite number of electron configurations. The number of configurations naturally fluctuate
in the birth-death algorithm for DMC. The total number of walkers must be managed such
that the population of walkers does not diverge or vanish. The control of the population
introduces a bias in the energy, which is made small by using a large number (hundreds to
thousands) of configurations.
Uncontrolled Approximations:
Pseudopotential
QMC is too computationally expensive to simulate all electrons in atoms fora solid. Since
valence electrons are the most important for determining bonding properties, a so called
pseudopotential is used to replace the core electrons with an effective potential. 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
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
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
energies. Material properties are often dependent on the functional, which has generated
a source of skepticism in DFT, especially where there is not reliable experimental data to
compare against.
An important example for this thesis is that of DFT functionals failing compute properties of silica. Silica has simple, closed shell, covalent/ionic bonding [29] and should be ideal
for DFT. However, while LDA provides good results for properties of individual silica polymorphs, it incorrectly predicts stishovite as the stable ground-state structure rather than
quartz. The GGA improves the energy difference between quartz and stishovite [30, 31],
giving the correct ground-state structure, and this discovery was one of the results that
popularized the GGA. However, almost all other properties of silica, such as structures,
equations of state, and elastic constants are often worse within the GGA [30, 31] and other
alternate approximations. Indeed, a drawback of DFT is that there exists no method to
estimate the size of functional bias or to know which functional will succeed at describing
a given property.
The following sections review exchange-correlation approximations and typical basis
sets used with them. A conceptual hierarchy of approximations referred to as “Jacob’s
Ladder” of approximations adds various improvements to the functional approximations.
Functionals, at the highest rungs, called hybrids, should be most accurate, but also most
computationally expensive. One means of improving the reliability of the more efficient
functionals is to benchmark their predictions with the hybrids, or even QMC. Hybrids or
QMC can help identify the best functional for a given system, reducing computational time
and improving the quality of the predictions. One of the goals of this thesis is to test the
ability of hybrids and QMC to benchmark lower-rung DFT functionals for complex solids.
3.1.1 Exchange-Correlation Approximations: Categorization of functionals in Jacob’s ladder
This section discusses a hierarchy of density functional approximations for the exchangecorrelation energy, Exc [15]. The hierarchy is described by a ladder of approximations for
31
the exchange-correlation energy as a function of the electron density:
Z
Exc [n↑ , n↓ ] =
drn(r)xc ([n↑ , n↓ ] ; r),
(3.1)
where the integrand nxc is an exchange-correlation energy density, and xc is an 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 = 0.434 eV). At the lowest rung of the ladder, the energy is
simply determined by the local density, n↑ (r), n↓ (r). The second level incorporates density
gradients ∇n↑ (r) and ∇n↓ (r). Higher rungs incorporate increasingly complex features constructed from the density or the Kohn-Sham orbitals around the volume element dr. Each
new level makes it possible to satisfy additional exact or nearly exact formal properties of
Exc [n↑ , n↓ ]. Higher level rungs are typically more accurate and computationally demanding
the lower rungs. Although, the increase in accuracy is not always true in practice.
Local Density Approximation
The base approximation of Jacob’s ladder is the local density approximation (LDA).:
LDA
Exc
[n↑ , n↓ ]
Z
=
drn(r)unif
xc (n↑ (r), n↓ (r)),
(3.2)
orm
where unif
(n↑ , n↓ ) is the exchange-correlation energy per particle of an electron gas with
xc
uniform spin densities n↑ and n↓ . 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),
GGA
Exc
[n↑ , n↓ ]
Z
=
drn(r)GGA
xc (n↑ (r), n↓ (r), ∇n↑ (r), ∇n↓ (r)),
32
(3.3)
initially offered great improvements over LDA for atomization energies and became a standard method in chemistry. The leading gradient correction for exchange and correlation
is second order for slowly varying densities. Initially, the PW91 [33] GGA functional used
analytic expansions to second order. However, PW91 was found to violate exact properties
of the exchange-correlation holes, which spurred the development of a numerically-defined
GGA parametrized to satisfy exact hole constraints, called PBE [34]. GGA is a semi-local
functional of density since it requires the density in an infinitesimal neighborhood around
r.
Meta-Generalized Gradient Approximation
The meta-GGA (MGGA) functional adds an additional dependence on the Kohn-Sham
kinetic energy densities, τσ (r), beyond GGA. The kinetic energy densities are implicit functionals of the spin densities n↑ (r) and n↓ (r). The MGGA exchange-correlation energy is
given by
M GGA
Exc
[n↑ , n↓ ]
Z
=
GGA
drnM
(n↑ , n↓ , ∇n↑ , ∇n↓ , τ↑ , τ↓ ).
xc
(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 r0 around r. The exact DFT expression for exchange
33
energy is
Z
Ex =
drn(r)x (r),
(3.5)
where
n(r)x (r) = −
1X
2 σ
Z
dr0
|ρσ (r, r0 )|2
,
|r − r0 |
(3.6)
and
∗
ρsigma (r, r0 ) = σα θ(µ − ασ )ψασ
(r)ψασ (r0 )
(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):
Ĥλ = T̂0 + λV̂ee + V̂λ + V̂C + V̂ext ,
(3.8)
where T̂0 is the non-interacting kinetic energy operator, VC is the Hartree Coulomb potential
(i.e. includes average e-e correlations), Vext is the external (nuclei) potential. Vee is a
true many-body operator containing both electron exchange-correlation and the missing
kinetic energy of the interacting system, while Vxc is a local one-particle exchange-correlation
potential of the fictitious, non-interacting Kohn-Sham system.
Becke showed that the lower, non-interacting limit of the associated coupling-constant
integral for the density (Equation 2.18), sometimes called the adiabatic connection, must
mix some exact exchange into Exc . In terms of the potential, the adiabatic connection is
written as
Z
1
hΨ1 | V̂ee | Ψ1 i =
hψλ | V̂ee | Ψλ idλ,
0
34
(3.9)
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: V̂ee = V̂x + V̂c . 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:
B3LY P
LDA
Exc
= Exc
+ a0 (ExHF − ExLDA ) + ax (ExGGA − ExLDA ) + ac (EcGGA − EcLDA ), (3.10)
where a0 = 0.20, ax = 0.72, and ac = 0.81 are three empirical parameters determined in
order to reproduce energies set of molecules as accurately as possible; the GGA exchange
is that of Becke [37] and the GGA correlation is that of Lee, Yang, and Parr (LYP) [38],
and the LDA correlation is the VWN approximation [39] A number of other possible hybrid
functionals exist. Another popular hybrid that performs well is the PBE0 functional:
P BE0
Exc
= aExHF + (1 − a)ExP BE + EcP BE ,
(3.11)
where HF is Hartree-Fock PBE is the DFT functional [34].
Hybrid Screen Exchange functionals (HSE)
The advantage of hybrid functionals is that they tend to significantly improve the quality of
DFT predictions. The disadvantage is they they are much more computationally expensive
due to the fully non-local nature of the exchange. In order to speed up the exact exchange
computation, Hyde, Scuseria and Ernzerhof [40, 41, 42, 43] have exploited the fact that the
range of the exchange interaction decays exponentially in insulators, and algebraically in
metals. A new, approximate hybrid functional, called HSE, applies a screened Coulomb
potential to the exchange interaction in order to screen the long-range part of the HF
35
exchange:
HSE
Exc
= aExHF,SR + (1 − a)ExP BE,SR + ExP BE,LR + EcP BE ,
(3.12)
where ExHF,SR is the short-range Hartree-Fock exchange, ExP BE,SR and ExP BE,LR 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:
erf c(ωr) erf (ωr)
1
=
+
,
r
r
r
(3.13)
where the left term is short-range and the right term is long range, and erf and erfc and
the error and complementary error functions, respectively, and ω = 0.11Bohr−1 [44] is the
screening parameter based on molecular basis tests.
3.1.2 Basis Set Approximations
There are essentially two types of basis sets: Extended and Localized. Most solid-state codes
use plane-waves (extended basis set). Quantum Chemistry codes tend to use localized basis
sets (better for molecules and clusters). Localized, usually Gaussian basis are a zoo sets
must be converged.
Extended Basis Sets
In order to make ab initio calculations possible for realistic systems, the single-particle
orbitals used in DFT and QMC are expanded in a set of pre-defined basis functions [6].
The most convenient basis functions to use for periodic solids are plane waves:
ψn (r) =
X
cnk exp(ik · r),
(3.14)
k
where cnk are the coefficients, and the wave vectors, k, go over the reciprocal lattice vectors
of the super lattice, with k less than the plane wave cutoff, kmax . The plane waves are
completely delocalized and not ascribed to individual atoms.
Plane waves have two advantages: 1) Matrix elements involving plane waves are com36
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) = Rnl (r)Ylm (θ, φ),
(3.15)
where χ belongs to a given atom.
So called Slater-type orbitals choose the radial part as calculated for hydrogen-like
atoms, but these are complicated to evaluate. A more practical basis function is simply
to use Gaussian-type orbitals (GTOs). GTOs are defined as
χ(r) = χR,α,n,l,m (r) = 2n+1
α(2n+1)/4
rn−1 exp(−αr2 Ylm (θ, φ))
[(2n − 1)!!]1/2 (2π)1/4
(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
contracted function form looks like
χR,k,n,l,m (r) =
X
uki χR,αi ,n,l,m (r),
(3.17)
i
where the uki are fixed constants. Only the coefficients to the contracted functions χ̃ 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
efficiently.
QMC is the most obvious choice to benchmark DFT functionals. QMC is the only many
body method that explicitly computes both exchange and correlation to a high level of
accuracy and is able to simulate sizable, solid systems (up to hundreds of atoms). However,
QMC is very expensive and doing more than a few calculations or calculations for a very
large system isn’t always possible with finite computer resources.
Hybrid functionals [42, 41, 43, 34, 45, 46, 47, 48, 49, 50, 51, 52, 53] offer a possible
more efficient alternative benchmark method to QMC. In fact, one of the aims of this thesis
is to test the ability of hybrid functionals to benchmark complex solids. Hybrid methods
use exact (HF) exchange and DFT correlation, which tends to significantly improve their
predictive power over lower-rung DFT functionals. In addition, recent developments of the
screened hybrid functionals (HSE) have made hybrid functional calculations more efficient.
HSE calculations are roughly 30 times more expensive than LDA or GGA, while QMC is
100-1000 times more expensive than LDA and GGA. HSE has already shown great predictive
power for band gaps and solid properties in small, simple systems [54, 55, 56, 44, 42, 41,
57, 40, 43, 58, 59, 60]. In the chapters that follow, the benchmark ability hybrids are tested
for complex solids: silicon defects and high pressure silica phases.
39
Chapter 4
Results for Silicon
Self-Interstitials
4.1 Introduction
Silicon is one of the most important materials in the semiconductor and microelectronics
industry. Due to silicon’s electronic properties, abundance, and cheap cost, silicon wafers are
the basis of all electronics. Fabrication of computer chips and integrated circuits requires
doping silicon wafers to make certain regions n-type or p-type. For example, boron is
commonly used to make p-type regions in silicon. During ion-implantation of dopants, a
large number of silicon self-interstitial defects are formed as the silicon lattice is damaged
by the dopant ions [61, 62]. The silicon interstitials may be made up of a single atom, two
atoms, or three or more. The interstitials may be charged, but in the work shown here all
defects are assumed neutral. During annealing, which is used to repair the damaged lattice
after ion-implantation, the interstitials condense to form larger defects, such as the {311}
planar defect. Evidence suggests that the planar defects limit the performance of electronic
devices. For example, the planar defects facilitate boron transient enhanced diffusion, an
undesirable broadening of the boron doped concentration profiles. This effect limits how
small devices can be made.
Figures 4.1, 4.2,and 4.3 show ball-stick images of the most stable defects as determined
by molecular dynamics simulations [63, 64]. The three lowest energy single-interstitial
defects are named Split-<110> (X), Hexagonal (H), and Tetrahedral (T), named after their
40
geometry. The di- and tri-interstitials are simply name alphabetically based on stability
ordering found by Richie et al. [63].
Direct detection of small interstitials and measurement of their properties is not currently possible [61, 62], though their presence may be inferred indirectly. Numerical simulations offer an aid to experiments to help determine their formation and diffusion properties.
Indeed, the technological importance of silicon defects and diffusion have motivated a number of theoretical studies [65, 66, 63, 67, 68, 69, 70, 71]. However, even among the theoretical
work, including various treatments of exchange-correlation in DFT, and QMC, there is 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 1-1.5 eV larger than GGA.
The activation energy of self-diffusion depends on the formation energy plus the migration
energy. For a simulation of cell size of N atoms, the formation energy for a single interstitial
is computed as follows (di- and tri- are computed with a similar expression):
Ef = E(defect) −
N+1
Ebulk .
N
(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
(a) Bulk
(b) Split-<110>(X)
(c) Hexagonal (H)
Figure 4.1: Single interstitial defects in silicon
(a) I2a
(b) I2b
Figure 4.2: Double interstitial defects in silicon
42
(d) Tetrahedral (T)
(a) I3a
(b) I3b
Figure 4.3: Triple interstitial defects in silicon
43
(c) I3c
Figure 4.4: QMC and DFT band gaps of Si. Results follow the trend of Jacob’s ladder of
functionals, where LDA is least accurate and HSE agrees best with QMC and experiment.
The black and white diagonal box on the QMC result respresents plus and minus sigma
about the mean.
44
Figure 4.5: QMC and DFT cohesive energy of Si. LDA overestimates the energy and GGA
improves DFT agreement with QMC and experiment. The black and white diagonal box
on the QMC result respresents plus and minus sigma about the mean.
45
Figure 4.6: QMC and DFT diamond to β-tin energy difference in Si. Results nearly follow
the trend of Jacob’s ladder of functionals, where LDA is least accurate and HSE agrees best
with QMC and experiment. GGA performs slightly better than mGGA. The black and
white diagonal box on the QMC result respresents plus and minus sigma about the mean.
46
4.2 Calculation Details
DFT computations are performed within the CPW2000 DFT code [76], which was the
only plane-wave pseudopotential DFT interfaced to work with the QMC code we used
for this project, CHAMP [77]. Calculations use a Ceperely-Alder LDA, norm-conserving
pseudopotential. All silicon calculations use a converged plane-wave energy cutoff of 16 Ha
and Monkhorst-Pack k-point meshes such that total energy converged to tenths of milliHa/SiO2 accuracy. Silicon QMC calculations were done for cell sizes of 8-, 16-, 32-, and
64-atoms, requiring 5×5×5, 7×7×7, 7×7×7, and 3×3×3 k-point mesh sizes, respectively.
All meshes were shifted to the L-point from the origin by (0.0, 0.5, 0.5), corresponding to
the reduced coordinates in the coordinate system defining the k-point lattice. The defect
structures studied are the most stable candidates as predicted by tight-binding MD [63].
They are further optimized in VASP [78] at the experimental lattice constant of silicon
(5.432 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
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/SiO2 . Finite size errors are
reduced to less then chemical accuracy by simulating cell sizes up to 64 atoms. Additional
finite size errors are corrected due to insufficient k-point sampling based on a converged
k-point DFT calculation. The uncontrolled approximations include the pseudopotential,
nonlocal evaluation of the pseudopotential, and the fixed node approximation. These errors
are difficult to estimate and generally assumed to be small [28, 81, 19].
4.2.1 Results
Figure 4.7 shows QMC and DFT formation energies for 16-atom bulk and 16(+1)-atom
self-interstitial defects, X, H, and T, while Figure 4.8 shows the QMC and DFT formation
energy for 64-atom bulk and 64(+1)-atom self-interstitial defects, X, H, and T. Results
display the expectation of Perdew’s Jacob’s ladder, with LDA, GGA, mGGA, and HSE
progressively agreeing better with QMC. In both simulation cell sizes, LDA and GGA
calculations generally agree with QMC ordering, but lie 1-1.75 eV below QMC formation
energies. HSE closely agrees with QMC compared to LDA, GGA, and meta-GGA (mGGA)
functionals for 16 atom simulation cell calculations.
QMC results agree well with QMC results of Leung et al. [71]. Leung et al. studied
16-atom and 54-atom cells, while work presented here studied 16-atom and larger 64-atom
cells. Leung et al. predict X and H are degenerate within statistical error in both 16- and
54-atom cells and T lies 0.4-0.5 eV higher in energy, but is within two-sigma statistical
error. Work presented here predicts X and T are degenerate in the 16-atom cell and T lies
about 0.25 eV higher. Finite size errors are expected to be large for the 16-atom cell and,
thus, convergence is not expected. In the 64-atom cell, results agree with Leung etal.. Data
presented here goes beyond Leung et.al. by studying the larger 64-atom cell and obtaining
QMC statistical error that is roughly a factor of 10 smaller. Later in this chapter, several
additional sources or error are also investigated to check results more carefully beyond the
48
work of Leung etal.. The results of Leung et.al. are significantly improved by this work,
but the conclusion remains the same: X and H are degenerate, T is most unstable, GGA
predicts energies roughly 1 eV below QMC, and LDA predicts energies roughly 1.5 eV below
QMC.
49
Figure 4.7: Formation energy of the three lowest-energy single self-interstitials in silicon
(X, H, and T) in a 16-atom cell. The black and white diagonal box on the QMC results
respresents plus and minus sigma about the mean. QMC predicts the H defect formation
energy lies 0.2 eV higher than the X and T defects, whose energies agree within two-sigma.
HSE agrees well with QMC compared to other DFT functional types, lying only about 0.25
eV below QMC, but predicts T to lie highest. LDA and GGA results predict a similar
energy ordering as QMC, but lie 1-1.75 eV lower. The mGGA predicts T lies highest and
lies about 1 eV lower than QMC.
50
Figure 4.8: Formation energy of the three lowest-energy single self-interstitials in silicon
(X, H, and T) in a 64-atom cell. The black and white diagonal box on the QMC results
respresents plus and minus sigma about the mean. QMC predicts the formation energy of
T is 0.6 eV higher than degenerate X and H defects. LDA and GGA predict similar energy
ordering, but lie 1-1.75 eV lower than QMC.
51
Figure 4.9 shows the QMC and DFT energy barriers (migration energies) for the 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 100-300 meV [70], which is similar to GGA
results in Figure 4.9.
52
Figure 4.9: Single interstitial diffusion path in 64 atom cell. QMC(DMC) benchmarks
GGA-NEB calculations. The QMC error bar respresents plus and minus sigma about the
mean value. The lowest barrier from X to H is similar in QMC and DFT. The T defect
formation energy and barrier are larger in QMC.
53
The experimental estimates of the diffusion activation energy (formation + migration)
for defects in silicon are near 4.7-4.9 eV [82, 83]. Using the estimates of migration energy
computed above (about 280 meV for GGA and 355 meV for QMC for the lowest path),
indicates that GGA (3.87 + 0.280 = 4.15 eV) underestimates the experimental activation
energy range by about 0.6 eV and the lowest QMC estimate (4.9(1) + 0.3(1) = 5.2(1) eV)
is about 0.3 eV above the experimental range. QMC provides a significant improvement
over GGA. Some of the QMC error could be due to using the DFT(GGA) geometry for the
defects.
Figure 4.10 shows QMC and DFT formation energies for the two lowest energy di-selfinterstitial defects, I2a and I2b in a 64 atom cell. Results again display the expected functional
Jacob’s ladder trend. QMC predicts the I2a defect lies 1.2 eV lower in energy than I2b . LDA
and GGA predict a similar energy ordering, but LDA lies up to 3 eV below QMC and GGA
lies up to 2 eV below QMC.
54
Figure 4.10: Formation energy of the two lowest-energy di-self-interstitials in silicon (I2a and
I2b) in a 64-atom cell. The black and white diagonal box on the QMC results respresents
plus and minus sigma about the mean. QMC predicts the I2a defect lies 1.2 eV lower in
energy than I2b . LDA and GGA predict a similar energy ordering, but LDA lies up to 3 eV
below QMC and GGA lies up to 2 eV below QMC.
55
Figure 4.11 shows QMC and DFT formation energies for the three most stable tri-selfinterstitial defects, I3a , I3b , and I3c in a 64-atom cell. Results again display the Jacob’s ladder
trend. QMC predicts the stability ordering is I3a < I3c < I3b . GGA predicts the stability
ordering I3b < I3a < I3c , while LDA predicts I3a < I3b < I3c . For the I3a and I3c defects, GGA
closely agrees with QMC. The improved GGA results may be because the I3a and I3c defects
are less distorted (smaller coordination number) than the I3b defect. LDA energies lie up to
3.5 eV below QMC and GGA lies up to 2 eV below QMC.
56
Figure 4.11: Formation energy of the three lowest-energy tri-self-interstitials in silicon (I3a ,
I3b , and I3c ) in a 64-atom cell. The black and white diagonal box on the QMC results
respresents plus and minus sigma about the mean.
57
Physical Explanation for Results
4.2.2 Tests for errors in QMC
Part of the aim of this work is to do an independent check of previous QMC calculations [71]
and examine possible sources of error. Sources of error include DMC time step convergence,
finite size convergence, dependence on exchange-correlation functional, Jastrow polynomial
order, pseudopotential choice, or allowing independent Jastrow correlations for the interstitial atom. Ultimately, results here do not find any significant unexpected error and, thus,
improve the confidence in the Leung etal. results.
Figure 4.12 shows the convergence of the DMC time step τ . The time step is converged
within chemical accuracy by 0.1 Ha−1 .
58
Figure 4.12: Convergence of the DMC time step for Si. The QMC error bars represent plus
and minus sigma about the mean value. The energy difference on the vertical axis is with
respect to a very small time step energy that has been set to zero. The time step, τ in units
of Ha−1 is converge within one-sigma statistical accuracy by 0.1.
59
Figure 4.13 shows the finite size convergence of the simulation cell for LDA, GGA, VMC,
and DMC calculations of the X interstitial. LDA and GGA calculations are converged by
16-atom simulation cell sizes. VMC and DMC converge by the 64-atom simulation cells size,
when neighboring 54 and 64 atom cells agree within one-sigma statistical error. It is also
interesting to note the VMC calculations are in agreement with much more computationally
expensive DMC in this case.
60
Figure 4.13: Convergence of DMC finite size error Si X defect. The QMC error bars
represent plus and minus sigma about the mean value.
61
Figure 4.14 shows a check for DMC energy dependence on choice of functional used to
produce orbitals in DFT. Two identical QMC calculations are performed except for the
exchange correlation functional used to produce the orbitals. In one calculation the orbitals
are produced with LDA and GGA for the other calculation. An LDA pseudopotential is
used in both sets of calculations. Since both sets of calculations agree within one-sigma,
there is negligible dependence on functional choice.
62
Figure 4.14: DMC formation energy using LDA and GGA orbitals. The black and white
diagonal box on the QMC results respresents plus and minus sigma about the mean. Results agree within one-sigma statistical error, indicating negligible dependence on functional
choice.
63
Figure4.15 shows convergence of the Jastrow electron-nuclear and electron-electron polynomial expansion order. Formation energy of the X defect in a 16-atom cell is converged
by 5th order polynomials.
64
Figure 4.15: VMC convergence of X-defect formation energy versus Jastrow polynomial
order in 16-atom Si. The QMC error bars represent plus and minus sigma about the
mean value. The notation MN0 indicates M order electron-nuclear polynomial and N order electron-electron polynomial, and no electron-electron-nuclear polynomial. Formation
energy is converge for 5th order polynomials.
65
Figure 4.16 shows tests of the effect of using a LDA versus Hartree-Fock pseudopotential
in VMC and DMC as a function of simulation cell size. By a cell size of 32 atoms, the X
defect VMC and DMC formation energy using both LDA and Hartree-Fock pseudopotentials
agrees within one sigma. Results indicate choice of pseudopotential does not change the
results.
66
Figure 4.16: DMC and VMC finite size convergence of X-defect formation energy with LDA
and Hartree-Fock pseudopotentials. The QMC error bars represent plus and minus sigma
about the mean value. By 32 atoms, when finite-size error is small, it is clear that both
types of pseudopotentials produce the same result.
67
Table 4.1: VMC and DMC calculations of Si single self-interstitial formation energies. One
set of calculations uses the same e-n Jastrow for bulk and the defect atom. A second set set
of calculations uses independent Jastrow for bulk and defect e-n Jastrows. Results agree
within one-sigma error, indicating that a single Jastrow is sufficient.
All atoms have same e-n Jastrow
VMC X
H
T
VMC 5.44(16) 6.22(15) 4.90(15)
DMC 4.60(7)
5.10(8)
4.15(8)
Defect and bulk atoms have independent e-n Jastrows
VMC 5.55(14) 5.72(14) 4.56(15)
DMC 4.60(7)
4.99(8)
4.19(6)
Table 4.1 shows results of calculations comparing the effects of using independent
electron-nuclear Jastrows for the defect and bulk atoms. Typically, one electron-nuclear
Jastrow with a single set of parameters is used to model correlations for both bulk and
defect atoms. However, one may ask if the correlations are significantly different for the
defect atom, then independent electron-nuclear Jastrows are needed with independently
optimized parameters. Results show that using independent Jastrows does not make a
detectable difference in the VMC or DMC formation energy.
All of the above test for sources of error increase the confidence of our Si interstitial
results. However, it is important to note that other possible sources of error not been
studied here, such as pseudopotential locality approximation and fixed node error, may also
affect results. Further work studying the effect of such sources of error are included in the
work of Parker [75].
4.3 Conclusions
This chapter presents the most accurate results available for QMC and DFT computations of
silicon self-interstitial defect formation energies. For single interstitials, formation energies
and self-diffusion barriers of the three most stable defects were computed. QMC (DMC)
provides a benchmark for various DFT functionals, which follow the expected trend of
Jacob’s ladder: LDA in least agreement with QMC, improved by GGA, mGGA, and HSE
68
in best agreement with QMC. LDA underestimates single interstitial formation energy by
roughly 2 eV, which GGA underestimates the formation energy by about 1.5 eV. The best
QMC results predict the X and H defects are degenerate and more stable than T by about
0.6 eV. Additionally, the lowest path migration energies for GGA and QMC are estimated
to be 280 meV and 355 meV, respectively, for single interstitials. The single interstitial
activation energies (formation + migration) are predicted to be 4.15 eV and 5.2(1) eV in
GGA and QMC, respectively. QMC agrees best with the experimental activation energy
range of 4.7-4.9 eV The di- and tri-interstitials also display the Jacob’s ladder trend, but
LDA and GGA energies lie 2-3.5 eV below QMC. QMC predicts I2a and I3a are the most
stable of the di- and tri-interstitials.
The QMC calculations indicate that DFT is not satisfactory for studying self-interstitial
diffusion in silicon. It could be that more accurate experiments are needed. The experiments are challenging and cannot easily differentiate between interstitials and vacancies,
for example. However, there is also reason to suspect DFT functionals to be inadequate for
silicon defects. First, the self-interaction error could potentially be large in these systems.
In addition, there are a wide range of coordination numbers from 3 (X) to 4 (T) to 6 (H),
compared to the bulk coordination number of 4. The charge in the bulk structure is likely
much more uniform than in the defect structures. LDA is based on the uniform electron gas,
and GGA is based on slowly varying gradients in the density. Therefore, GGA is expected
to estimate the energy difference between different structures with very different interatomic
bonding better than LDA. However, apparently the bulk-defect density difference is still too
stark for GGA to predict the correct energy difference. QMC explicitly computes exchange
and correlation, providing the best estimate of energy differences.
Various tests check of previous QMC calculations of single Interstitials [71] and examine
possible sources of error. Sources of error checked include DMC time step convergence, finite
size convergence, dependence on exchange-correlation functional, Jastrow polynomial order,
pseudopotential choice, or allowing independent Jastrow correlations for the interstitial
atom. Of all possible sources of QMC error checked, none affected results presented outside
of a one-sigma error bar. Further tests of Jastrow optimization, pseudopotential locality
69
error, fixed-node error are needed [75].
70
Chapter 5
Results for Silica
Silica (SiO2 ) is an abundant component of the Earth whose crystalline polymorphs play key
roles in its structure and dynamics. Experiments are often too difficult to probe extreme
conditions that calculations can easily probe. However, DFT calculations may unexpected
fail for silica due to bias of the exchange correlation functional choice. This chapter describes calculations of highly accurate ground state QMC plus phonons within the quasiharmonic approximation of density functional perturbation theory to obtain benchmark
thermal pressure and equations of state of silica phases up to Earth’s core-mantle boundary [84]. The chapter starts with an introduction to the significance of silica and goes over
previous theoretical work and challenges. In the final sections, computational details and
results are discussed. The QMC Results provide the best constrained equations of state and
phase boundaries available for silica. QMC indicates a transition to the most dense α-PbO2
structure above the core-insulating D00 layer, but the absence of a seismic signature suggests
the transition does not contribute significantly to global seismic discontinuities in the lower
mantle. However, the transition could still provide seismic signals from deeply subducted
oceanic crust. Computations also identify the feasibility of QMC to find an accurate shear
elastic constants for stishovite and its geophysically important softening with pressure.
5.1 Introduction
Silica is one of the most widely studied materials across the fields of materials science,
physics, and geology. It plays important roles in many applications, including ceramics,
71
electronics, and glass production. As the simplest of the silicates, silica is also one of the
most ubiquitous geophysically important minerals. It can exist as a free phase in some
portions of the Earth’s mantle. In order to better understand geophysical roles silica plays
in Earth, much focus is placed on improving knowledge of fundamental silica properties.
Studying structural and chemical properties [29] offers insight into the bonding and electronic structure of silica and provides a realistic testbed for theoretical method development.
Furthermore, studies of free silica under compression [85, 86, 87, 88, 89, 90] reveal a rich
variety of structures and properties, which are prototypical for the behavior of Earth minerals from the surface through the crust and mantle. However, the abundance of free silica
phases and their role in the structure and dynamics of deep Earth is still unknown.
Free silica phases may form in the Earth as part of subducted slabs [91] or due to
chemical reactions with molten iron [92]. Determination of the phase stability fields and
thermodynamic equations of state are crucial to understand the role of silica in Earth. The
ambient phase, quartz, is a fourfold coordinated, hexagonal structure with nine atoms in
the primitive cell [85]. Compression experiments reveal a number of denser phases. The
mineral coesite, also fourfold coordinated, is stable from 2–7.5 GPa, but is not studied
here due to its large, complex structure, which is a 24 atom monoclinic cell [86]. Further
compression transforms coesite to a much denser, sixfold coordinated phase called stishovite,
stable up to pressures near 50 GPa. Stishovite has a tetragonal primitive cell with six
atoms [87]. In addition to the coesite-stishovite transition, quartz metastably transforms to
stishovite at a slightly lower pressure of about 6 GPa. Near 50 GPa, stishovite undergoes
a ferroelastic transition to a CaCl2 -structured polymorph via instability in an elastic shear
constant [88, 93, 94, 95, 96, 97, 98, 99]. This transformation is second order and displacive,
where motion of oxygen atoms under stress reduces the symmetry from tetrahedral to
orthorhombic. Experiments [89, 90, 100] and computations [101, 102, 103] have reported a
further transition of the CaCl2 -structure to an α-PbO2 -structured polymorph at pressures
near the base of the mantle. Figure 5.1 shows a schematic version of the silica pressuretemperature phase diagram. Note the pressure scale is no linear, and the dashed phase
boundaries indicate they are not well known.
72
Figure 5.1: Schematic version of the silica phase diagram.
73
5.2 Previous Work and Motivation
The importance of silica as a prototype and potentially key member among lower mantle
minerals has prompted a number of theoretical studies [93, 96, 98, 99, 101, 102, 103, 31,
30, 104] to investigate high pressure behavior of silica. Density functional theory (DFT)
successfully predicts many qualitative features of the phase stability [101, 102, 103, 31,
30], structural [31], and elastic [93, 96, 98, 99, 104] properties of silica, but it fails in
fundamental ways, such as in predicting the correct structure at ambient conditions and/or
accurate elastic stiffness [31, 30]. Work presented here instead uses the quantum Monte
Carlo (QMC) method [9, 16] to compute silica equations of state, phase stability, and
elasticity, documenting improved accuracy and reliability over DFT. This work significantly
expands the scope of QMC by studying the complex phase transitions in minerals away from
the cubic oxides [105, 106]. Furthermore, the QMC results have geophysical implications
for the role of silica in the lower mantle. Though QMC finds the CaCl2 -α-PbO2 transition
is not associated with any global seismic discontinuity, such as D00 , the transition should be
detectable in deeply subducted oceanic crust.
5.3 Computational Methodology
This section discusses the general computational methods and choices made for all silica
calculations. The first section describes pseudopotential generation choices for Si and O
used for all silica calculations. The second section discusses types of silica calculations using
DFT: geometry optimization, wave-function generation, and phonon calculations. The last
section addresses QMC calculations including wave-function optimization and VMC/DMC
choices.
5.3.1 Pseudopotential Generation
In order to improve computational efficiency, pseudopotentials replace core electrons of the
atoms with an effective potential. The Opium code [107] produces optimized nonlocal,
norm-conserving pseudopotentials for Si and O. Both pseudopotentials are generated using
74
the appropriate exchange-correlation functional (LDA, PBE, WC) for DFT calculations.
All QMC calculations use pseudopotentials generated with the WC functional. In all cases,
the silicon potential has a Ne core with equivalent 3s, 3p, and 3d cutoffs of 1.80 a.u. The
oxygen potential has a He core with 2s, 2p, and 3d cutoffs of 1.45, 1.55, and 1.40 a.u.,
respectively.
5.3.2 DFT Calculations
All DFT computations are performed within the ABINIT package [108]. Silica calculations use a converged plane-wave energy cutoff of 100 Ha and Monkhorst-Pack k-point
meshes such that total energy converged to tenths of milli-Ha/SiO2 accuracy. Converged
silica calculations require k-point mesh sizes of 4×4×4, 4×4×6, and 4×4×4 for quartz,
stishovite/CaCl2 , and α-PbO2 , respectively, and all meshes were shifted from the origin by
(0.5, 0.5, 0.5), corresponding to the reduced coordinates in the coordinate system defining
the k-point lattice.
Geometry Optimization
The ABINIT code allows various types of structural optimizations that are useful for phase
stability and elasticity calculations. This work utilizes several different types of 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
DFT wave-function is produced from a fixed geometry calculation after all variables are
optimized to produce a converged charge density. The converged charge density is then
used in a non-self-consistent calculation to output the orbitals.
Phonon Free Energy Calculations
There are two commonly used free energies in thermodynamic calculations: 1) Helmholtz
and 2) Gibbs free energies. The Helmholtz free energy is used for the derivation of most
thermodynamic quantities, where volume, V, and temperature, T, are the independent
variables. Gibbs free energy is important for equilibrium studies for determining phase
boundaries, where the convenient independent variables are pressure, P, and T.
The Helmholtz free energy [109] is defined as,
F (V, T ) = Ustatic (V, T ) − T Svib (V, T ),
(5.1)
where Ustatic is the static internal energy of the crystal lattice and TSvib is the vibrational
(phonon) contribution of the thermal atomic motion to the free energy and Svib is the
vibrational entropy. The Gibbs free, G = F + 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, ωi of the lattice. Statistical mechanics allows a system’s vibrational quantum
mechanical energy levels to completely determine the vibrational Helmholtz free energy
(F = Ustatic + Fvib ) via a partition function, Z:
Fvib = −kT lnZ,
(5.2)
where Z is a sum over all quantum energy levels given by,
Z=
X
exp (
i
−i
),
kT
where k is the Boltzmann constant, and the i are the microstate energies:
76
(5.3)
1
3
2 h̄ωi , 2 h̄ωi ,
5
2 h̄ωi ,
ect.
Therefore, for each mode, there are many energy levels
Zi = exp (
=
1 h̄ωi
)
2 kT
all X
modes
exp (−
s=0
i
exp (− 12 h̄ω
kT )
i
1 − exp ( −h̄ω
kT )
sh̄ωi
),
kT
(5.4)
,
(5.5)
which when combined with Equation 5.2 gives
1
−h̄ωi
Fvib,i = h̄ωi + kT ln(1 − exp (
)).
2
kT
(5.6)
The total expression for the Helmholtz free energy (F = Ustatic + Fvib ) in the harmonic
state approximation is then
F = Ustatic +
all X
modes
i=1
1
h̄ωi + kT
2
all X
modes
ln(1 − exp (
i=1
−h̄ωi
)),
kT
(5.7)
where the second term is the zero-temperature quantum vibrational energy and the third
term is the thermal vibrational energy.
In the pure harmonic approximation, one assumes all the ωi ’s are constant. This choice
causes F to be independent of V, so that all volume derivatives are automatically zero. This
is clearly disastrous for any thermodynamic properties computed with a volume derivative
of F, such as thermal expansivity. In addition, all thermodynamic quantities will not have
a volume dependence, as they should with temperature.
A successful alternative is the so-called quasiharmonic approximation (QHA) [109]. In
the QHA, the phonons frequencies are assumed to depend on volume, but not temperature
(ω = ω(V )). This allows all thermodynamic properties to depend on both T and V using
the harmonic expression in Equation 5.7, effectively giving rise to low order anharmonicity
terms. The phonon frequencies don’t directly depend on T, but the harmonic sum does.
In general, QHA is valid for low temperatures and becomes less valid towards the melting
temperature of materials, where thermal atomic motion is least harmonic-like.
The main novelty of the work presented in this chapter is that QMC (not DFT) is
used to compute the internal energy of the static lattice, Ustatic , while DFT linear response
77
calculations in the QHA provide the much smaller vibrational energy contribution, TSvib .
DFT is a ground-state (zero temperature) method used to compute phonons for each volume
point in the equation of state, corresponding to phonon frequencies that depend on volume,
but not temperature. As a side note, there is also generally an electronic contribution to the
entropy due to thermal excitations in materials with small band gaps or metals. Since silica
is a large band gap insulator, electronic entropy may be ignored for temperature ranges
considered.
The ABINIT code produces phonon free energies by modeled lattice dynamics using the
linear response method [110] within the QHA. Phonon free energies for silica were computed
over a large range of temperatures for each cell volume and added them to ground-state
energies in order to obtain equations of state at various temperatures. Phonon energies were
computed up to the melting temperature in steps of 5 K in order to compute thermodynamic
properties. Converged silica calculations require a plane-wave cutoff energy of 40 Hartree
with matching 4×4×4 q-point and k-point meshes.
5.4 QMC Calculations
The CASINO code [21, 16] facilitates computation of various types of QMC calculations.
The QMC calculations for silica are composed of three major steps: (i) DFT calculation
producing a relaxed crystal geometry and single particle orbitals, (ii) construction of a trial
wave function and optimization within VMC, and (iii) a DMC calculation to determine the
ground-state wave-function accurately. Production of a DFT Slater determinant of orbitals
was discussed above.
5.4.1 Wave-function Construction and Optimization
Construction of the trial QMC wave function is done by multiplying the determinant of
single particle DFT orbitals with a Jastrow correlation factor [9, 16]. As a check for dependence on DFT functional choice, QMC total energies for stishovite were compared using
various functionals for the orbitals and found the energies were equivalent within one-sigma
78
statistical error (tenths of milli-Ha). The Jastrow describes various correlations by including
two-body (electron-electron electron-nuclear), three body (electron-electron-nuclear), and
plane-wave expansion terms, with a total of 44 parameters. Parameters in the Jastrow
are optimized by minimizing the variance of the VMC total energy over several hundred
thousand electron configurations. Several iterations of computing the VMC energy and
re-optimizing the parameters are done until the VMC energy changes are typically inside
of two sigma.
5.4.2 DMC Calculations
DMC calculations use the optimized trial wave functions to compute highly accurate energies. A typical silica DMC calculation uses 4000 electron configurations, a time step of
0.003 Ha−1 , and several thousand Monte Carlo steps. In order for QMC calculations of
solids to be feasible, a number of approximations [9, 16] are usually implemented that can
be classified into controlled and uncontrolled approximations, which are discussed in detail
in Chapter 2. The controlled approximations for silica include statistical Monte Carlo error,
numerical grid interpolation (5 points/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/SiO2 . Finite size errors are reduced by
using a model periodic Coulomb Hamiltonian [24] while simulating cell sizes up to 72 atoms
(2×2×2) for quartz, 162 atoms (3×3×3) for stishovite/CaCl2 , and 96 atoms (2×2×2) for
α-PbO2 . Additional finite size errors are corrected due to insufficient k-point sampling
based on a converged k-point DFT calculation. The uncontrolled approximations include
the pseudopotential, nonlocal evaluation of the pseudopotential, and the fixed node 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
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.
5.5.1 Free Energy
Figure 5.2 shows the zero temperature, Helmholtz free energy versus volume curves computed using QMC. For each phase, the QHA DFT phonon energies are added to the ground
state, static QMC energy curves, producing a set of free energy isotherms for each silica
phase. In this work, an isotherm was fit in temperature increments of 5 K, ranging from 0
K to the melting point of silica (∼2000-4000 K). Such small increments allow for the construction of a fine T-V grid for computing thermodynamic functions with finite differences.
80
(a) Quartz
(b) Stishovite
(c) α-PbO2
Figure 5.2: Computed Ground state (static) QMC free energy as a function of volume for
(a) quartz, (b) stishovite and (c) α-PbO2 .
81
5.5.2 Thermal Equations of State and Fit Parameters
Figure 5.3 shows the computed equations of state compared with experimental data for
quartz [85, 111], stishovite/CaCl2 [112, 113], and α-PbO2 [89, 90]. Thermal equations of
state are computed from the Helmholtz free energy [109]. Pressure is determined from the
expression P = − (∂F/∂V )T . The analytic Vinet [114] equation of state fits isotherms of
the Helmholtz free energies and is defined as
E(V, T ) = E0 (T ) +
9K0 (T )V0 (T )
[1 + [ξ(1 − x) − 1] exp [(1 − x)]] ,
ξ2
(5.8)
where E0 and V0 are the zero pressure equilibrium energy and volume, respectively, x =
(V /V0 )1/3 and ξ =
3
0
2 (K0
− 1)), K0 (T) is the bulk modulus, and K00 (T) is the pressure
derivative of the bulk modulus. The subscript 0 indicates zero pressure. E0 , V0 , K0 , and
K00 are the four fitting parameters. Pressure is obtained analytically as
3K0 (T )(1 − x)
exp [ξ(1 − x)] .
P (V, T ) =
x2
(5.9)
Figures 5.4, 5.5, 5.6,and 5.7 show the four Vinet fit parameters as a function of temperature. Vinet fits are made for free energy isotherms in increments of 5 K and the zero
pressure fit parameters (free energy, F0 , volume, V0 , bulk modulus K0 , and pressure derivative of the bulk modulus, K00 ) from each fit form the curves in the plots. The gray shading
of the QMC curves indicates one standard deviation of statistical error from the Monte
Carlo data. The QMC results generally agree well with diamond anvil-cell measurements
at room temperature, as do the corresponding DFT calculations using the GGA functional
of Wu and Cohen (WC) [115].
82
Figure 5.3: Thermal equations of state of (A) quartz, (B) stishovite and CaCl2 , and (C)
α-PbO2 . The lower sets of curves in each plot are at room temperature and the upper sets
are near the melting temperature. Gray shaded curves are QMC results, with the shading
indicating one-sigma statistical errors. The dashed lines are DFT results using the WC
functional. Symbols represent diamond-anvil-cell measurements (Exp.) [85, 89, 90, 111,
112, 113].
83
(a) Quartz
(b) Stishovite
(c) α-PbO2
Figure 5.4: QMC and WC free energy as a function of temperature at zero pressure for (a)
quartz, (b) stishovite and (c) α-PbO2 . The QMC energies are constructed by adding QMC
energy for the static lattice to the WC phonon energy. Vinet fits are made for free energy
isotherms in increments of 5 K and the zero pressure free energy parameter from each fit
forms the curve in the plot. QMC results are represented by gray shaded curves, indicating
one-sigma statistical error. WC results are represented by red dashed lines.
84
(a) Quartz
(b) Stishovite
(c) α-PbO2
Figure 5.5: QMC and WC volume as a function of temperature at zero pressure for (a)
quartz, (b) stishovite and (c) α-PbO2 . The QMC energies are constructed by adding QMC
energy for the static lattice to the WC phonon energy. Vinet fits are made for free energy
isotherms in increments of 5 K and the zero pressure volume parameter from each fit forms
the curve in the plot. QMC results are represented by gray shaded curves, indicating
one-sigma statistical error. WC results are represented by red dashed lines.
85
(a) Quartz
(b) Stishovite
(c) α-PbO2
Figure 5.6: QMC and WC bulk modulus, K0 as a function of temperature at zero pressure
for (a) quartz, (b) stishovite and (c) α-PbO2 . The QMC energies are constructed by adding
QMC energy for the static lattice to the WC phonon energy. Vinet fits are made for free
energy isotherms in increments of 5 K and the zero pressure bulk modulus parameter from
each fit forms the curve in the plot. QMC results are represented by gray shaded curves,
indicating one-sigma statistical error. WC results are represented by red dashed lines.
86
(a) Quartz
(b) Stishovite
(c) α-PbO2
Figure 5.7: Pressure derivative of the bulk modulus, K00 as a function of temperature at zero
pressure for (a) quartz, (b) stishovite and (c) α-PbO2 . The QMC energies are constructed
by adding QMC energy for the static lattice to the WC phonon energy. Vinet fits are made
for free energy isotherms in increments of 5 K and the zero pressure K0 parameter from
each fit forms the curve in the plot. QMC results are represented by gray shaded curves,
indicating one-sigma statistical error. WC results are represented by red dashed lines.
87
5.5.3 Phase Stability
A phase transition occurs at the pressure where the Gibbs free energy (or enthalpy at T=0)
of two phases is equal, or, equivalently where the difference in Gibbs free energy changes
sign. The Gibbs free energy is given by
G(V, T ) = Ustatic (V, T ) − T Svib (V, T ) + P (V, T )V (P, T ).
(5.10)
At phase equilibrium (i.e. at the transition pressure), PT ),
Gphase1 (PT , V1 (PT )) = Gphase2 (PT , V2 (PT ))
or
F1 (PT ) + PT V1 (PT ) = F2 (PT ) + PT V2 (PT ),
which gives the expression for the transition pressure as
PT =
− [F2 (PT ) − F1 (PT )]
,
[V2 (PT ) − V1 (PT )]
which is equivalent to the so-called common-tangent [116] slope of the two energy versus
volume curves.
Figure 5.8 shows zero temperature phase transitions via changes in the enthalpy. 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 CaCl2 -α-PbO2 transition. The error on the latter is larger because the scale of the enthalpy difference between the quartz
and stishovite phases is about a factor of 10 larger than for CaCl2 and α-PbO2 . In both
transitions, variation in the DFT result with functional approximation is large. For the
metastable quartz-stishovite transition, LDA incorrectly predicts stishovite to be the stable ground state, WC underestimates the quartz-stishovite transition pressure by 4 GPa,
88
and the GGA of Perdew, Burke, and Ernzerhof (PBE) [117] matches the QMC result. For
the CaCl2 -α-PbO2 transition, the same three DFT approximations lie within the statistical
uncertainty of QMC. The variability of the present calculations is less than different experimental determinations of this transition (Figure 5.9). The experimental variability may be
due to the difficulty in demonstrating rigorous phase transition reversals as well as pressure
and temperature gradients and uncertainties in state-of-the-art experiments.
Figure 5.9a compares QMC and DFT predictions with measurements for the 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 CaCl2 -α-PbO2 phase boundary. The WC boundary lies within the QMC statistical
error. Previous DFT work also shows that the LDA boundary lies near the upper range
of the QMC boundary and PBE produces a boundary 10 GPa higher than the LDA [102].
The two diamond-anvil-cell experiments [89, 90] constrain the transition to lie between 65
and 120 GPa near the mantle geotherm (2500 K), while QMC constrains the transition to
105(8) GPa. The boundary inferred from shock data [100] agrees well with QMC and WC.
The boundary slope measured by Dubrovinsky et al. [89] is negative, which is in contrast
to the positive slope inferred by Akins et al. [100]. QMC and WC as well as previous DFT
studies [102, 103] predict a positive slope.
89
Figure 5.8: Enthalpy difference of the (A) quartz-stishovite, and (B) CaCl2 -α-PbO2 transitions. The DMC transition pressures are 6.4(2) GPa and 88(8) GPa for quartz-stishovite
and CaCl2 -α-PbO2 , respectively. Gray shaded curves are QMC results, with the shading
indicating one-sigma statistical errors. The dashed, dot-dashed, and dotted lines are DFT
results using the WC, PBE, and LDA functionals, respectively.
90
Figure 5.9: (a) Computed phase boundary of the quartz-stishovite transition. The gray
shaded curve is the QMC result, with the shading indicating one-sigma statistical errors.
The dashed line is the boundary predicted using WC. The dash-dot and solid lines represent
shock [118, 119] analysis, while dotted and dash-dash-dot lines represent thermochemical
data (Thermo.) [120, 121]. (b) Computed phase boundary of the CaCl2 -α-PbO2 transition.
Gray shaded curves are QMC results, with the shading indicating one-sigma statistical
errors. The dashed, dotted, and dash-dot lines are DFT boundaries using WC, LDA [102],
and PBE [102] functionals, respectively. The dark green shaded region and the solid blue
line are diamond-anvil-cell measurements (Exp.) [89, 90]. The dash-dot-dot line is the
boundary inferred from shock data [100]. The vertical light blue bar represents pressures
in the D region. Circles drawn on the geotherm [123] indicate a two-sigma statistical error
in the QMC boundary.
91
Table 5.1: Computed QMC thermal equation of state parameters at ambient conditions
(300 K, 0 GPa). The units are as follows: F0 (Ha/SiO2 ), ∆F (Ha/SiO2 ), V0 (Bohr3 /SiO2 ),
K0 (GPa), α10−5 (K−1 ). All other quantities are unitless. QMC one-sigma statistical error
on F0 is 0.0002 Ha/SiO2 .
Phase
quartz
stishovite
α-PbO2
Method QMC
Exp.
QMC
Exp.
QMC
Exp.
F0
-35.7946
-35.7764
-35.7689
∆F
0.0182(4) i 0.020(1)
0.0257(3)
a
d,e 157.1(2)
h 157.79
254.32 159.0(4)
V0
254(2)
154.8(1)
a 34
d,e 291–310
h 313(5)
K0
32(6)
305(20)
329(4)
0
a
d,e
h 3.43(11)
K0
7(1)
5.7(9) 3.7(6)
4.29–4.59 4.1(1)
b 3.5
f 1.26(11)
α
3.6(1)
1.2(1)
1.2(1)
b
f 1.57(38)
1.80
Cp /R
1.82(1)
1.71(1)
1.69(1)
c
f,g
γ
0.57(1)
0.57
1.22(1)
1.35-1.33 1.27(1)
c 0.47
f,g 2.6(2)-6.1 2.05(1)
q
0.40(1)
2.22(1)
b 3.3-8.9
f,g 6.6-8.0(5) 6.40(1)
δT
6.27(1)
5.98(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]
5.5.4 Thermodynamic Parameters
Thermal equations of state facilitate the computation of all desired thermodynamic parameters. Table 1 summarizes ambient computed and available experimentally measured
values [89, 111, 112, 113, 124, 125, 126, 127, 120] of the Helmholtz free energy, F0 (Ha/SiO2 ),
the Helmholtz free energy difference relative to quartz, ∆F (Ha/SiO2 ), volume, V0 (Bohr/SiO2 ),
bulk modulus, K0 , pressure derivative of the bulk modulus, K00 , thermal expansivity,α (K−1 ),
heat capacity, Cp /R, Grüneisen ratio, γ, volume dependence of the Grüneisen ratio, q, and
the Anderson-Grüneisen parameter, δT , for the quartz, stishovite, and α-PbO2 phases of
silica. QMC generally offers excellent agreement with experiment for each of these parameters.
92
Thermal Pressure
Thermal pressure is the thermal energy effect on the pressure [109, 128, 129]. It is computed
by taking the volume derivative (P = − (∂F/∂V )T ) of the thermal vibrational energy
term in Equation 5.7. However, through thermodynamic relations thermal pressure can be
written as
Z
T
Pth = P (V, T ) − P (V, T0 ) =
αKT dT,
(5.11)
T0
where αKT is nearly constant at high temperatures for most materials, making the equation
linear in T.
In essence, the expression for thermal pressure provides another form of an equation
of state that can be checked against experimental measurements. From calculations, the
thermal pressure equation of state is simply obtained by computing differences in pressures
between a given isotherm and the ground state isotherm.
Figure 5.10 shows the computed QMC and DFT(WC) variation of thermal pressure with
volume and temperature for quartz, stishovite, and α-PbO2 . QMC results are represented
by gray shaded curves, indicating one-sigma statistical error. WC results are represented by
red dashed lines. All phases show a nearly constant dependence on the volume, but linear
dependence on the temperature, especially at higher temperatures as expected. Quartz and
Stishovite experiments [112, 126] agree well with the predicted thermal pressure equations
of state.
93
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
Figure 5.10: Computed QMC and WC thermal pressure of (a,b) quartz, (c,d) stishovite
and (e,f) α-PbO2 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
Changes in Thermal Pressure (αKT )
Figure 5.11 shows changes in thermal pressure [109, 128, 129], given by αKT = (∂P/∂T )T .
Changes in αKT are quite small (note the scale is 10−3 ) as expected for most materials.
95
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
Figure 5.11: Computed QMC and WC variation of αKT with temperature and pressure
for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2 . QMC results are represented by gray
shaded curves, indicating one-sigma statistical error. WC results are represented by red
dashed lines.
96
Bulk Modulus
Figure 5.12 shows the calculated temperature and pressure dependencies of the bulk moduli [109] of quartz, stishovite, and α-PbO2 . The bulk modulus is denoted as
KT = V
∂2F
∂V 2
= −V
TT
∂P
∂V
.
(5.12)
T
The bulk moduli decrease linearly with temperature and increase linearly with pressure
for all pressure-temperature ranges. Although it is well known that DFT bulk moduli can
vary significantly with choice of functional, results with the WC functional tend to lie only
slightly below QMC. Both DFT and QMC agree with the experimental data for quartz [85]
and stishovite [126] at zero pressure. No measurements are yet available for the elastic
moduli of the α-PbO2 phase.
97
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
Figure 5.12: Computed QMC and WC variation of the bulk modulus with temperature
and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2 . Results agree well with
available experimental data [85, 126]. QMC results are represented by gray shaded curves,
indicating one-sigma statistical error. WC results are represented by red dashed lines.
98
Pressure Derivative of the Bulk Modulus
Figure 5.13 shows the rate of change in the bulk modulus with respect to pressure [109],
denoted as K 0 = (∂KT /∂P )T . It is an important quantity in many thermodynamic expressions and also occurs as a parameter in most universal equations of state. K0 is a unitless
quantity and shows only slight variation with both temperature and pressure.
99
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
Figure 5.13: Computed QMC and WC variation of the pressure derivative of bulk
modulus,K 0 , with temperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) αPbO2 . Results agree well with available experimental data [85, 126]. QMC results are
represented by gray shaded curves, indicating one-sigma statistical error. WC results are
represented by red dashed lines.
100
Thermal Expansivity
Figure 5.14 shows the computed QMC and WC thermal expansivity [109] for quartz, stishovite, and α-PbO2 . Thermal expansivity is denoted as
1
α=−
V
∂2F
∂T ∂V
2 ∂ F
1 ∂V
/
=
∂V 2 T 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 Bohr3 and the
thermal expansivity becomes negative. However, the computations presented here consider
only the α-quartz phase. QMC and DFT also show good agreement with the measured
stishovite expansivity. There have been no expansivity measurements for α-PbO2 .
101
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
Figure 5.14: Computed QMC and WC variation of thermal expansivity with temperature
and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2 . Results agree well with
available experimental data [130, 131, 132, 126, 133, 134, 135, 136]. QMC results are
represented by gray shaded curves, indicating one-sigma statistical error. WC results are
represented by red dashed lines.
102
Heat Capacity
Figure 5.15 shows the computed QMC and WC heat capacities for quartz, stishovite, and
α-PbO2 . Heat capacity [109] may be computed at constant volume or pressure:
Cv =
∂U
∂T
∂H
∂T
(5.14)
V
or
Cp =
.
(5.15)
P
For solids, the two expressions are nearly identical. For all silica phases, the WC results
almost exactly match QMC. The quartz and stishovite results agree well with experiment [126, 130]. There have been no heat capacity measurements for the α-PbO2 phase.
103
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
Figure 5.15: Computed QMC and WC variation of heat capacity with temperature and
pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2 . Results agree well with available
experimental data [130, 126]. QMC results are represented by gray shaded curves, indicating
one-sigma statistical error. WC results are represented by red dashed lines.
104
Change in Volume
Figure 5.16 shows the computed QMC and WC volume differences for quartz-stishovite and
CaCl2 -α-PbO2 at 0 K. The quartz-stishovite volume change is large due to the four to six
fold coordination change. The volume change in the CaCl2 -α-PbO2 transition is roughly a
factor of ten smaller.
105
Figure 5.16: Computed QMC and WC percentage volume difference of (A) quartz-stishovite
and (B) CaCl2 -α-PbO2 transitions. Gray shaded curves are QMC results, with the shading
indicating one-sigma statistical errors. The dashed lines are DFT results using the WC
functional.
106
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:
γi =
∂lnωi
.
∂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 γi over the first Brillouin zone leads to a macroscopic (thermodynamic)
definition, written as
γ=
αKT V
,
CV
(5.17)
where α is the thermal expansivity, KT is bulk modulus, V is volume, and CV is the
heat capacity at constant volume. Both the microscopic and macroscopic definitions are
difficult to analyze experimentally because the former requires knowledge of the phonon
dispersion spectrum and the latter requires measurements of thermodynamic properties at
high temperatures and pressure.
Figure 5.17 shows the computed QMC and DFT(WC) 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-PbO2 , γ increases with temperature.
107
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
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-PbO2 . Results agree well with
available experimental data [125]. QMC results are represented by gray shaded curves,
indicating one-sigma statistical error. WC results are represented by red dashed lines.
108
An additional parameter, q, is used to describe the volume dependence of γ, and is
defined as
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
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
Figure 5.18: Variation of q with temperature and pressure for (a,b) quartz, (c,d) stishovite
and (e,f) α-PbO2 . Results agree well with available experimental data [125]. QMC results
are represented by gray shaded curves, indicating one-sigma statistical error. WC results
are represented by red dashed lines.
110
Anderson-Grüneisen Parameter
The Anderson-Grüneisen parameter [109], δT , which characterizes the relationship between
thermal expansivity and pressure is defined as
δT =
∂lnα
∂lnV
T
−1
=
αKT
∂KT
∂T
.
(5.19)
P
Figure 5.19 shows δT may initially increase or decrease at low temperatures, and eventually
become constant at high temperatures. At all temperatures, δT shows a strong decrease
with pressure. DFT(WC) and QMC predict very similar results in all cases.
111
(a) Quartz
(b) Quartz
(c) Stishovite
(d) Stishovite
(e) α-PbO2
(f) α-PbO2
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-PbO2 . QMC results
are represented by black curves and WC results are represented by red dashed lines.
112
Bulk Sound Velocity and Density
The differences in bulk sound velocity and density are particularly important properties for
phases of minerals because they indicate the strength of corresponding seismic signals. If
the change in bulk sound velocity and density between two phases is large, then there will
be strong discontinuity in seismic data. For example, Figure 5.20 shows the sound velocity
and density profiles for Earth [138]. There are major discontinuities in the profiles as the
phases change form the solid mantle, to the liquid outer core, to the solid Fe inner core.
113
(a) Quartz
Figure 5.20: Profile of the p-wave, α, s-wave, β, and density, ρ, in Earth. The discontinuities
correspond to major compositional transitions inside Earth [138].
114
The bulk sound velocity is defined as
VBS
KS
=
ρ
1
2
,
(5.20)
where ρ is the density, and KS is the adiabatic bulk modulus. The adiabatic bulk modulus
is defined as KS = −V (∂P/∂V )S , where S indicates adiabatic conditions. However, for
solids, the adiabatic bulk modulus generally agrees with the isothermal bulk modulus,
KT = −V (∂P/∂V )T within about one percent at room temperature. The two types of
bulk moduli are related by KS = KT (1 + αγT ), where α is the thermal expansivity and γ
is the Grüneisen ratio.
Regarding silica, the important question is whether the CaCl2 to α-PbO2 transition is
seismically visible. Since phase stability work described above indicates the transition is
not associated with the D00 layer, localized quantities of α-PbO2 may be visible seismically
if the difference in bulk sound velocity and density is large.
Figure 5.21 shows the QMC predicted bulk sound velocity and density as a function
of pressure for room temperature and for a typical lower mantle temperature near the
base. The pairs of lines in the bulk sound velocity plot indicate the one-sigma error bars
on the QMC calculations. With respect to postperovskite, MgSiO3, (the dominate D00
material) measurements [139, 140] at 120 GPa in D00 , QMC predicts α-PbO2 has 12% lower
density and 67% larger bulk sound velocity, which may provide enough contrast to be seen
seismically if present in appreciable amounts.
115
(a) Quartz
(b) Quartz
Figure 5.21: QMC calculations of the variation in (a) bulk sound velocity and (b) density
with pressure for CaCl2 and α-PbO2 . Profiles are plotted at both room temperature and the
mantle-base temperature and compared with the experimental values of perovskite/postperovskite at the base of the mantle. The pairs of lines in the bulk sound velocity plot
indicate one-sigma statistical error.
116
5.5.5 Stishovite Shear Constant
Most information about the deep Earth comes from the study of seismic waves, and elastic
constants determine sound velocities of those waves in the Earth. Much work has been
done using DFT to compute and predict elastic constants for minerals in the Earth [104],
but there is much uncertainty in the predicted elastic constants because different density
functionals predict significantly different values. Here, computations test the feasibility of
using QMC to predict softening of the shear elastic constant, c11-c12, in stishovite, which
drives the ferroelastic phase transition to CaCl2 [88, 93, 99].
While there are many methods of computing elastic constants, the strain-energy density
relation outlined by Barron and Klein [141, 93] is particularly convenient when working
with volume conserving strains. The full expression for strain-energy density is given by
1
∆E
= −pii +
V
2
1
cijkl − p (2δij δkl − δil δjk ) ij kl ,
2
(5.21)
where E is the free energy, V is the volume, ij is the Eulerian strain, cijkl is the elastic
constant tensor, and δij is the Kronecker delta. The pressure terms vanish for volume
conserving strains leaving
1
∆E
= cijkl ij kl .
V
2
(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
cijkl =
1 ∂2E
.
V ∂ij ∂kl
(5.23)
It is the c11 − c12 shear constant related to the B1g Raman mode that becomes unstable
at the phase transition to CaCl2 . The elastic constant is found by computing the energy
versus b/a strain in the tetragonal stishovite lattice for a constant volume and c lattice
parameter. The volume conserving strain matrix applied to the lattice vectors to produce
117
c11 − c12 is given by


 δ 0 0 



=
 0 −δ 0 


0 0 0
(5.24)
The strain matrix is applied to the matrix of lattice vectors, R, to produce a new set of
lattice vectors R0 corresponding to a structure with the same volume as follows:



R =
I

0


 δ 0 0

+
 0 −δ 0

0 0 0


 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
(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
Plugging the strain matrix elements into Equation 5.22 gives the expression needed to
evaluate c11 − c12 :
c11 − c12 =
where
∂2E
∂δ 2
1 ∂2E
,
2V ∂δ 2
(5.26)
is the curvature of a polynomial fit to the DFT or QMC energy versus strain
data.
Figure 5.23 shows the shear softening of c11 − c12 as a function of pressure at zero temperature. At low pressures c11 −c12 is almost constant, but softens as pressure increases and
becomes unstable near 50 GPa. For a well-optimized trial wave function, VMC often comes
close to matching the results of DMC. Due to the large computational cost, VMC computes
c11 − c12 at several pressures and the more accurate DMC checks only the endpoints. The
figure also shows the result of WC and previous LDA computations [93]. Both QMC and
DFT results correctly describe the softening of c11-c12, indicating the zero temperature
transition to CaCl2 near 50 GPa. Radial X-ray diffraction data [94] lies lower than calculated results. However, discrepancies can arise in the experimental analysis depending on
the strain model used. Recent Brillouin scattering data [97] agrees well with DMC. 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
Figure 5.23: Softening of the c11-c12 shear constant for stishovite with pressure. Down
triangles and circles are the DMC and VMC results, respectively. Diamonds and up triangles
represent DFT results within the WC and LDA [93], respectively. Squares represent radial
X-ray diffraction data [94] and stars represent Brillouin scattering data [97]. The shear
constant in all methods softens rapidly with increasing pressure and becomes unstable near
50 GPa, signaling a transition to the CaCl2 phase.
121
5.6 Geophysical Implications
The QMC CaCl2 -α-PbO2 boundary indicates that the transition to α-PbO2 , within a 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
50650 km above the D00 layer, a thin boundary surrounding Earth’s core ranging from a
depth of ∼2,700 to 2,900 km [142]. The DFT boundaries all lie within the QMC two-sigma
confidence interval, with PBE placing the transition most near the D00 layer. Free silica in
D00 , such as in deeply subducted oceanic crust or mantlecore reaction zones, would have the
α-PbO2 structure. However, based on QMC results, the absence of a global seismic anomaly
above D00 suggests that there is little or no free silica in the bulk of the lower mantle. The
α-PbO2 phase is expected to remain the stable silica phase up to the coremantle boundary.
Bulk sound velocity and density profiles indicate that the transition should be seismically
visible for large enough concentrations.
5.7 Conclusions
This chapter has presented QMC (using WC orbitals) computations of silica equations of
state, phase stability, and elasticity. This work provides highly accurate values for thermal
properties for silica and expands the scope of QMC by studying the complex phase 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 CaCl2 -α-PbO2 transition
pressure is 88(8) GPa. Results show the CaCl2 -α-PbO2 transition is not associated with
the global D00 discontinuity, indicating there is not significant free silica in the bulk lower
mantle. However, the transition should be detectable in deeply subducted oceanic crust.
A number of thermodynamic properties are computed by combining QMC with DFT 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
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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.
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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
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the Quantum Chemistry community. However, they are also tend to be computationally
expensive for solids. New generations of hybrid functionals which are screened may be the
best compromise between accuracy and speed, but they remain largely untested.
Silica is one well-known material for which DFT makes unreliable predictions. LDA
generally predicts structural and elastic properties well, while GGA predicts structural
energetics and phase stability better. In fact, LDA fails to predict the lowest pressure, quartz
to stishovite transition, while the GGA functional known as PBE predicts the transition
perfectly. Other GGA functionals do not predict the transition well, however. Chapter 5
showed that QMC can generally be used to predict all properties of silica accurately when
accuracy is paramount. This chapter investigates whether hybrid functionals or perhaps
some newly developed semi-local functionals can afford the same accuracy as QMC for
silica and offer a reliable and computationally inexpensive route to future silica predictions.
Namely, this work studies silica with the following functionals: 1) the local LDA [32, 143]
functional; 2) the semi-local PBE [117], PW91 [33], PBEsol [144], and WC [115, 145]
functionals; and 3) the hybrid B3LYP [146, 147], PBE0 [34, 45, 148, 46], and HSE [40, 41,
42, 43] functionals. The screened hybrid, HSE, is generally found to match the accuracy
of QMC for all properties of silica and the best compromise between standard DFT and
QMC.
6.2 Previous Work
6.2.1 Hybrid Calculations of Silica
There have been small number of studies of certain phases or properties of silica using
the hybrid B3LYP functional [149, 150, 151], all using the CRYSTAL code. Civalleri et
al. [149] studied all-silica zeolite framework stability. Zicovich-Wilson et al. [150] studied
vibrational frequencies of quartz. Ottonello et al. [151] studied vibrational properties of
stishovite. There have been no studies using other hybrid functionals, such as PBE0 or
HSE. B3LYP results typically show B3LYP is superior in performance to LDA and GGA,
and the expectation of this result based on molecular studies is often why B3LYP is chosen.
125
While these studies indicate B3LYP may be a reliable functional for computing properties of
silica, more extensive testing and comparisons need to be done. A thorough comparison of
local, nonlocal, and hybrid functional performance is needed for a wide range of properties
to determine which functional works best.
6.2.2 Hybrid B3LYP and PBE0 Calculations of Solids
The Quantum Chemistry community has a long and prolific history of publishing B3LYP
and PBE0 DFT calculations for molecules and non periodic systems. Indeed, the accuracy
of the B3LYP functional is what convinced most quantum chemists to switch to using DFT
instead of CI. A couple of codes (CRYSTAL and GAUSSIAN) were also developed to study
solids with Gaussian basis sets, but still relatively few solid systems have been studied with
hybrid B3LYP and PBE0 functionals due the large computational expense. Even so, a small
number of periodic solids have been studied with the B3LYP functional [47, 48, 49, 50, 51,
52, 53] and PBE0 [34, 45, 46]. Until recently, all calculations employed Gaussian basis sets.
6.2.3 Screened Hybrid (HSE) Calculations of Solids
With the development of more computationally efficient screened hybrids, plane-wave pseudopotential codes (VASP and PWSCF) have begun to incorporate hybrid functionals into
calculations of periodic solids. There have been roughly a couple dozen published calculations for solids since these developments occurred using the HSE functional [54, 55, 56, 44,
42, 41, 57, 40, 43, 58, 59, 60].
6.3 Computational Methodology
The overall aim of this project is to systematically study and compare the major influences
of ab initio calculations on properties of quartz and stishovite phases of silica and the
quartz-stishovite phase transition. Namely, the aim is to compare performance of various
exchange-correlation functionals, basis sets, pseudopotentials, and codes, and benchmark
against QMC and experiments to determine which are most accurate. Calculations compare
126
local (LDA), semi-local GGA (PBE, PW91, PBEsol, WC), and hybrid (B3LYP, PBE0,
and HSE) functionals. All electron calculations are compared against projector augmented
wave (PAW) pseudopotentials and nonlocal, norm-conserving pseudopotentials. In addition,
results from the CRYSTAL, ABINIT, and VASP codes are compared.
The general protocol for every calculation is to first fully relax the cell shape and volume
until forces are smaller than 10−4 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.
6.3.1 CRYSTAL Calculations
The CRYSTAL [152] code allows for all-electron calculations using Gaussian Basis sets, and
a large variety of exchange-correlation functionals, including LDA, PBE, PW91, B3LYP,
and PBE0 used in this study. CRYSTAL does not allow for plane-wave basis set calculations.
K-point sampling
A k-space sampling converged to at least chemical accuracy for quartz uses Monkhust-Pack
grids with a shrinking factor in reciprocal space and a Gilat shrinking factor of 7. Converged
stishovite calculations use shrinking factors of 9 and 9, respectively.
127
Coulomb and Exchange Series Truncation and Additional Convergence 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
or included in an uncontrolled way [153], then some of the diffuse exponents must be modified or removed. In fact, in the CRYSTAL code, the most diffuse exponent allowed is 0.12.
In the basis sets convergence calculations, the diffuse exponents of each basis set were either
modified as needed and re-optimized using line optimization techniques, or simply removed
if numerical stability could not be achieved otherwise. This aspect makes converging the
basis particularly challenging.
In order to make the basis set optimization and convergence tests efficient, the convergence calculations did not use the production flag option NOBIPOLA, and, in addition,
default options were used for TOLDEP and TOLDEE. However, levshift 6 1 and fmixing
30 were still used, as they increase stability and efficiency of the calculations.
Figure 6.1 shows convergence of the quartz-stishovite energy difference as a function of
increasingly large Gaussian basis sets. The converged LDA, PW91, and B3LYP plane-wave
energies of the quartz-stishovite energy difference provide a benchmark to compare Gaussian
basis set calculations against. The CRYSTAL (LDA, PW91, and B3LYP) quartz-stishovite
energy difference was converged to the plane-wave value by using a series of modified basis
sets that systematically add polarization functions to account for higher angular momentum
orbitals. The series of basis sets tested range from ones including only s and p orbitals:
(3-21GSP, 3-21G, 6-311G), to ones including a single d orbital (cc-pVDZ, 66-21G∗ (Si)/631G∗ (O), 65-111G∗ (Si)/8-411G(O)), to two d orbitals (6-311G∗ ∗), to two d orbitals and
one f orbital (cc-pVTZ), and, finally, ones that include three d orbitals and two f orbitals
(6-311G∗∗ (Si)/cc-pVQZ(O), and cc-pVQZ). Convergence with plane-wave results is reached
when using the cc-pVQZ basis set (See Appendix B for the optimized cc-PVQZ basis set
used).
The convergence of the energy difference is fairly rapid with the addition of extra d and f
polarization functions. There are a couple of important features to notice. The first is that
the 66-21G∗ (Si)/6-31G∗ (O) basis suggested as sufficient by CRYSTAL users and the 65111G∗ (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
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basis sets for silicon and oxygen separately indicate that oxygen is the main component that
requires significant higher angular momentum polarization functions. This is apparent, for
example, when comparing the nearly identical results of (6-311G∗∗ (Si)/cc-pVQZ(O) and
cc-pVQZ).
130
Figure 6.1: Gaussian basis set convergence
131
6.3.2 ABINIT Calculations
The ABINIT package [108] allows for plane-wave pseudopotential calculations using a large
number of different exchange-correlation functionals. In this project, the LDA, PBE, PW91,
PBEsol, and WC functionals are used in ABINIT. At the time of this work, the ABINIT
code was the only plane-wave code to include the PBEsol and WC functionals. All silica
calculations use a converged plane-wave energy cutoff of 100 Ha and Monkhorst-Pack k-point
meshes such that total energy converged to tenths of milli-Ha/SiO2 accuracy. Converged
silica calculations require k-point mesh sizes of 4×4×4 for quartz and 4×4×6 for stishovite,
and all meshes were shifted from the origin by (0.5, 0.5, 0.5), corresponding to the reduced
coordinates in the coordinate system defining the k-point lattice.
The Opium code [107] produces optimized nonlocal, norm-conserving 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.
6.3.3 VASP Calculations
The VASP package [78] allows for plane-wave pseudopotential calculations using a large
number of different exchange-correlation functionals. In this project, the LDA, PBE, PW91,
and HSE functionals are used with VASP. At the time of this work, the VASP code was the
only plane-wave code to include the HSE functional. The VASP code uses preconstructed
projector augmented wave (PAW) pseudopotentials, which are typically highly accurate.
There are no pseudopotentials generated for the HSE functional, so PBE is used as it is
likely the closest match. All silica calculations use a converged plane-wave energy cutoff of 30
Ha and Monkhorst-Pack k-point meshes such that total energy converged to tenths of milliHa/SiO2 accuracy. Converged silica calculations require k-point mesh sizes of 9×9×9 for
quartz and 13×13×13 for stishovite, and no k-point shift is used in the VASP calculations.
132
6.4 Results
The following section presents all of the results comparing performance of various local,
semi-local, and hybrid exchange-correlation functionals, basis sets, pseudopotentials, and
codes: CRYSTAL all-electron, Gaussian basis calculations using LDA, PBE, PW91, B3LYP,
and PBE0 functionals; ABINIT plane-wave opium-pseudopotential calculations using LDA,
PBE, PW91, PBEsol, and WC functionals; VASP plane-wave PAW-pseudopotential calculations using LDA, PBE, PW91, and HSE functionals.
Results show examples of energy vs volume and enthalpy curves for HSE calculations,
as they are fundamental for computing all other properties. Bar plots show comparisons
for the quartz and stishovite zero pressure volume, bulk modulus, pressure derivative of the
bulk modulus, pressure versus volume curves, and quartz-stishovite transition pressure. A
table is presented with values of equilibrium lattice constants, volumes, and bulk moduli.
The main result is that the HSE functional is generally most consistent, outperforming all
other functionals for quartz and stishovite properties, and agreeing well with experiment
and QMC.
6.4.1 Energy Versus Volume
As all other quantities are derived from the energy versus volume data, Figure 6.2 shows
one example of such curve for quartz and stishovite using the HSE functional in a VASP
calculation. The curve shown corresponds to the zero temperature isotherm, fit with the
Vinet equation of state. At zero pressure, the quartz energy is clearly lower, indicating it is
the stable phase at that pressure. To determine phase stability at all pressures, one must
compute the enthalpy.
133
Figure 6.2: Computed HSE energy versus volume curves for quartz and stishovite. Points
are fit with the Vinet equation of state.
134
6.4.2 Pressure Versus Volume
Once the energy is know as a function of volume, the pressure as a function of volume is
computed by taking the derivative of the Vinet energy expression with respect to volume:
P = − (∂F/∂V )T .
Figure 6.3 and Figure 6.4 compare all pressure versus volume equations of state computed with various functionals and codes for quartz and stishovite. Experimental results
are plotted as points on each figure as symbols. The curve produced with the PBEsol functional seems to best match experimental data for quartz, while HSE result best matches
experimental data for stishovite.
135
(a) Quartz LDA
(b) Quartz PBE
(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
(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
6.4.3 Equilibrium Quartz and Stishovite Volume from Vinet Fits
Figures 6.5 (a) and (b) show a comparison of the quartz and stishovite zero pressure volumes,
respectively; The volumes are those estimated from the minimum of the Vinet fit to the
energy versus volume data for all of the exchange-correlation functionals and codes used.
The solid horizontal black line indicates the range of experimental data, while the dashed
box (quartz) and line (stishovite) indicates the one-sigma statistical error in the QMC
prediction.
The LDA, PBE, PW91 results are fairly uniform across all types of codes, indicating the
calculations are individually converged and the codes are performing on par with each other.
LDA tends to slightly underestimate the volume; PBE and PW91 tend to overestimate;
PBEsol and WC match experiment very well; The hybrids B3LYP and PBE0 overestimates
the quartz volume, but match experiment for the stishovite volume. The screened hybrid
HSE also slightly over estimates the quartz volume and matches experiment for stishovite.
In general the PBEsol, WC, LDA, and HSE functionals all perform well for the volume.
138
(a) Quartz
(b) Stishovite
Figure 6.5: Zero pressure volumes of (a) quartz and (b) stishovite from Vinet fits of energy
versus volume data using various exchange-correlation functionals and codes. The solid
horizontal black line indicates the range of experimental data, while the dashed box or line
indicates the one-sigma statistical error in the QMC prediction.
139
6.4.4 Equilibrium Quartz and Stishovite Bulk Moduli from Vinet Fits
Figures 6.6 (a) and (b) show a comparison of the quartz and stishovite zero pressure bulk
moduli, respectively; The bulk moduli are those computed from the curvature around the
minimum of the Vinet fit of the energy versus volume data for all of the exchange-correlation
functionals and codes used. The gray shaded box indicates the range of experimental data,
while the dashed box indicates the one-sigma statistical error in the QMC prediction.
The LDA, PBE, PW91 results are fairly uniform across all types of codes for quartz,
indicating the calculations are individually converged and the codes are performing on par
with each other. However, the LDA, PBE, PW91 results for stishovite are less uniform
across all types of codes for the bulk moduli. The all-electron CRYSTAL LDA, PBE and
PW91 bulk moduli tend to be about 7-10% larger than the plane-wave pseudopotential
results. The CRYSTAL LDA result agrees with the all-electron LAPW result of Cohen
et al. [156]. However, the LAPW LDA and PBE result of Zupan et al. [157] agrees with
the ABINIT and VASP results. LDA tends to predict the bulk modulus in agreement with
experiment; PBE, PW91, PBEsol, and WC tend to underestimate; The hybrids B3LYP
significantly underestimates the quartz bulk modulus and slightly over estimates the stishovite bulk modulus. PBE0 underestimates the quartz bulk modulus, and overestimates for
stishovite. The screened hybrid HSE slightly underestimates the quartz bulk modulus and
matches experiment for stishovite. In general the LDA and HSE functionals perform well
for the bulk modulus.
140
(a) Quartz
(b) Stishovite
Figure 6.6: Zero pressure bulk moduli of (a) quartz and (b) stishovite from Vinet fits of
energy versus volume data using various exchange-correlation functionals and codes. The
gray shaded box indicates the range of experimental data, while the dashed box indicates
the one-sigma statistical error in the QMC prediction.
141
6.4.5 Equilibrium Quartz and Stishovite K00 from Vinet Fits
Figures 6.7 (a) and (b) show a comparison of K00 for quartz and stishovite, respectively;
The K00 values are those from the Vinet fits to energy versus volume data. The gray shaded
box indicates the range of experimental data, while the dashed box indicates the one-sigma
statistical error in the QMC prediction.
The LDA, PBE, PW91 results are fairly uniform across all types of codes for quartz,
indicating the calculations are individually converged and the codes are performing on par
with each other. In fact, almost all results for both quartz and stishovite are near the
experimentally measured range of values.
142
(a) Quartz
(b) Stishovite
Figure 6.7: Computed pressure derivative of the bulk modulus for (a) quartz and (b) stishovite using various exchange correlation functionals and codes. The gray shaded box indicates
the range of experimental data, while the dashed box indicates the one-sigma statistical error in the QMC prediction.
143
Table 6.1 and 6.2 provides zero pressure values of lattice constants (a and c), volume
(V0 ), bulk modulus (K0 ), and pressure derivative of the bulk modulus (K00 ) for all of the
various DFT functionals and codes. There are two different volumes given in the table: V0F R
corresponds to the fully relaxed, equilibrium DFT geometry and V0 is the zero pressure
volume predicted by the minimum of the Vinet fit to energy versus volume data. V0F 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.
6.4.6 Enthalpy Versus Pressure
Figure 6.8 shows the HSE enthalpy curves for quartz and stishovite as one example. Once
the energy is know as a function of volume, the enthalpy as a function of pressure is easily
computed as H = U + PV. Enthalpy is used for analysis of the phase stability. The
quartz-stishovite transition is determined by the crossing of enthalpy curves. For HSE, the
stishovite enthalpy curve becomes more stable after a pressure of 5.9 GPa, marking the
transition pressure.
144
Table 6.1: Computed DFT lattice constants, volume, bulk modulus, and pressure derivative
of the bulk modulus for quartz at zero pressure. Parameters are compared with experiment
and QMC. Lattice constants are in units of Bohr, volumes are in units of Bohr/SiO2 , and
the bulk modulus is in units of GPa.
Quartz
CRYSTAL
a
c
V0F R
V0
K0
K00
LDA
9.16 10.10 244.50 245.34 37.58 6.39
PBE
9.47 10.40 268.90 268.85 32.96 5.81
PW91
9.47 10.39 268.99 268.87 31.47 6.00
B3LYP
9.53 10.44 273.73 272.94 27.68 6.87
PBE0
9.42 10.33 264.43 262.93 31.69 6.35
ABINIT
a
c
V0F R
V0
K0
K00
LDA
9.17 10.12 245.44 245.96 35.16 5.63
PBE
9.49 10.42 271.10 270.13 32.59 4.54
PW91
9.48 10.40 269.93 270.68 31.37 5.20
PBEsol
9.34 10.28 259.13 259.16 31.65 5.74
WCGGA
9.38 10.30 261.65 261.54 29.18 5.89
VASP
a
c
V0F R
V0
K0
K00
LDA
9.28 10.21 253.90 250.31 33.07 5.93
PBE
9.53 10.43 273.69 273.39 30.63 5.26
PW91
9.56 10.47 276.29 273.85 29.19 5.44
HSE
9.46 10.37 267.77 267.10 32.42 5.06
a
Experiment
9.29 10.22 254.30 254.32 34(4) 5.7(9)
QMCb
—
—
—
254(2) 32(6) 7(1)
a Ref. [111]
b Ref. [84]
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Table 6.2: Computed DFT lattice constants, volume, bulk modulus, and pressure derivative
of the bulk modulus for stishovite at zero pressure. Parameters are compared with experiment and QMC. Lattice constants are in units of Bohr, volumes are in units of Bohr/SiO2 ,
and the bulk modulus is in units of GPa.
Stishovite
CRYSTAL
a
c
V0F R
V0
K0
K00
LDA
7.84 5.03 154.87 154.95
326.09
4.26
PBE
7.99 5.09 162.65 162.72
281.33
4.86
PW91
7.98 5.09 162.08 162.13
284.75
4.87
B3LYP
7.94 5.06 159.53 159.38
326.36
4.05
PBE0
7.88 5.03 156.28 156.09
343.80
4.19
ABINIT
a
c
V0F R
V0
K0
K00
LDA
7.83 5.02 154.12 154.45
305.04
3.96
PBE
8.01 5.10 163.61 164.48
253.19
4.41
PW91
7.99 5.09 162.55 163.45
255.81
4.43
PBEsol
7.92 5.07 159.03 159.27
288.19
4.00
WCGGA
7.92 5.07 159.17 159.38
288.82
4.00
VASP
a
c
V0F R
V0
K0
K00
LDA
7.85 5.04 155.47 155.59
306.23
4.78
PBE
8.00 5.09 162.71 162.94
251.57
6.40
PW91
7.99 5.09 162.41 162.59
255.82
6.31
HSE
7.89 5.04 156.78 157.33
311.29
4.01
a
Experiment
7.90 5.04 157.29 157.29
300(9)
4.45(15)
QMCb
—
—
—
159.0(4) 305(20) 3.7(6)
a Ref. [87]
b Ref. [84]
146
Figure 6.8: Computed HSE enthalpy curves of quartz and stishovite as a function of pressure.
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6.4.7 Quartz-Stishovite Transition Pressures
Figure 6.9 shows a comparison of the transition pressures for all of the exchange-correlation
functionals and codes used. The gray shaded box indicates the range of experimental data,
while the dashed box indicates the one-sigma statistical error in the QMC prediction.
The first feature to note is that the LDA, PBE, PW91 results are fairly uniform across all
types of codes. This indicates the calculations are individually converged and the codes are
performing on par with each other. The results indicate that the local LDA functional fails
to predict that quartz is the correct ground state in the CRYSTAL and VASP codes, while
the ABINIT code predicts a very small transition pressure. Among the semi-local GGA
functionals, PBE and PW91 agree well with experiments in all codes, while PBEsol and
WC significantly underestimate the transition pressure. Among the hybrids, B3LYP vastly
overestimates the transition pressure, while PBE0 and HSE agree well with experiment and
QMC.
The LDA and GGA results agree with that of Hamann [30]. LDA is expected to perform
worse for the transition pressure because of the stark difference in structure between quartz
and stishovite. GGA improves the energy difference because gradient dependence of the
charge density in the functional estimates the 4- to 6-fold coordination change better. It is
not clear why PBEsol and WC underestimate the transition pressure. Hybrid functionals
likely perform well (save B3LYP) due to description of exact exchange, and correlation
energy should be small in these systems.
148
Figure 6.9: Computed quartz-stishovite transition pressures using various 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
6.5 Conclusions
Given that QMC is very expensive and standard DFT is inconsistent due to functional bias,
this work has investigated reliability of hybrid functionals for silica. The results compare
performance of various exchange-correlation functionals, basis sets, pseudopotentials, and
codes, which are benchmarked against QMC and experiments to determine which are most
accurate. Calculations compare local (LDA), semi-local GGA (PBE, PW91, PBEsol, WC),
and hybrid (B3LYP, PBE0, and HSE) functionals. All electron calculations are compared
against projector augmented wave (PAW) pseudopotentials and nonlocal, norm-conserving
pseudopotentials. In addition, results from the CRYSTAL, ABINIT, and VASP codes are
compared.
The LDA functional tends to perform well for structural properties and elastic constants.
The PBE and PW91 GGA functionals tend to overestimate the structural properties and
by 2-3% and underestimate elastic properties by roughly 5%. The PBEsol and WC GGA
functionals significantly improve agreement with experiment. However, among the LDA and
GGA functionals, only PBE and PW91 compute the quartz-stishovite transition pressure
(energy difference) well. LDA is likely fails due to the large coordination change in going
from quartz to stishovite and GGA is expected to improve due to gradient dependence
of the density. The Hybrids tend to improve over PBE and PW91 for structural and
elastic properties. B3LYP significantly overestimates the transition pressure. HSE and
PBE0 results are generally similar and in good agreement with experiment and QMC for
all properties considered. HSE provides the most consistently accurate results and offers a
relatively efficient, yet accurate alternative to QMC.
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Chapter 7
Conclusions
7.1 Summary
Quantum mechanics provides an exact description of microscopic matter, but an exact
solution of the fully interacting, many-electron Schrödinger equation is intractable. The
Coulomb interaction, responsible for electron exchange-correlation energy, can not be separated to simplify the many-body problem. This means the wave-function can not legitimately be written as a product of single electron wave-functions, but such a wave-function
is sometimes a good place to start building approximate solutions. In order to accurately
compute, predict, and understand properties of matter, approximate solutions must be
sought.
A reasonable approach to solve the many-body Schrödinger equation is to use a meanfield-based method in which each electron feels the average potential of all other electrons.
The simplistic Hartree approximation gives an approximate solution for a wave-function
represented by a simple product of one-electron functions. The Hartree-Fock approximation improves on the Hartree approximation by representing the wave-function as a single
Slater determinant, which forces it to obey the Pauli principle. Therefore, Hartree-Fock
correctly computes exchange, but ignores correlation of electrons. Amazingly, density functional theory (DFT) (Chemistry Nobel Prize 1998) succeeds in exactly mapping the manybody problem onto an independent electron problem with an effective one-electron potential
depending only on the electron density. However, in practice, the functional for exchangecorrelation must be approximated for real materials. The approximated functionals lead to
151
unreliable results on occasion. Such problems drive development of non-mean-field based
approaches in search of even better accuracy and reliability.
Quantum Monte Carlo (QMC) is a method which abandons the mean-field approach,
and is among the class of methods providing many-body solutions to Schrödinger’s equation.
QMC is advantageous because it is the only many body technique capable of efficiently and
accurately computing energies of large system sizes (solids). QMC achieves efficiency over
other many body techniques by using stochastic methods to explicitly compute exchange
and correlation with a many-body wave-function. The input trial wave-function is a product
of a Slater determinant and a Jastrow correlation factor. Two common QMC algorithms
are variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). The VMC method
is efficient and provides an upper bound on the ground-state energy using the variational
principle applied to a trial form of the many-body wave-function. However, VMC usually
cannot give the required chemical accuracy. DMC further evolves a statistical representation
of the many-body wave-function and projects out the ground-state solution consistent with
the nodes of the trial wave-function. The fixed-node approximation is the only essential
approximation, though other approximations are made to increase efficiency.
Although QMC is a highly accurate and reliable method, excessive computational expense prevents its prolific use beyond a benchmarking tool for spot-checking DFT results.
Additionally, QMC calculations for solids are currently intimately tied with DFT, as QMC
is too expensive to optimize structural geometries, and DFT provides orbitals for a trial
wave-function. Standard DFT (LDA, GGA) is not always reliable and has well known failures, such as when computing band gaps or some silica phase transitions. A computational
method with both the efficiency of DFT and accuracy of QMC is needed.
In effort to ratchet up DFT functional quality, new-generations of functionals have been
developed conceptualized as rungs of “Jacob’s Ladder.” Each higher rung represents a new
conceptual improvement: LDA, GGA, meta-GGA, and hybrids. In practice, higher-rung
functionals do not always provide an improved result, but, nonetheless, efforts are still being
made to fine tune functional quality as its still the most reasonable solution to the many
body problem. Most recently, the screen hybrid functional, HSE, has shown particular
152
promise. HSE computes band gaps and solid properties extremely well compared with
QMC. Gaussian basis sets are sometimes used with hybrid DFT functional calculations of
solids and particular care must be taken to converge the basis set to plane-wave accuracy.
Convergence studies show silica calculations need a cc-PVQZ basis set to reach plane-wave
accuracy. Published literature indicates that DFT(HSE) is capable of offering benchmark
accuracy calculations along with QMC.
Prior to work in this thesis, almost all QMC and hybrid-HSE DFT calculations have
been limited to small, cubic systems. This thesis presents three large-scale applications,
which are used to expand the scope of QMC and hybrid DFT methods and establish their
usefulness as benchmarking tools for complex solids.
The first project (Chapter 4) provides the most accurate results for silicon self-interstitial
defect formation energies using QMC and DFT. Both formation energies and self-diffusion
barriers are computed for the three most stable single interstitial defects. Experimental
measurement of the formation energies is very challenging. QMC (DMC) and HSE provide
a benchmark for various DFT functionals, which follow the expected trend of “Jacob’s
ladder”: LDA in least agreement with QMC, followed by GGA, mGGA, and HSE in best
agreement with QMC. LDA underestimates the QMC single interstitial formation energy
by roughly 2 eV, while GGA underestimates the formation energy by about 1.5 eV. The
best QMC results predict the X and H defects are degenerate and more stable than T by
about 0.6 eV. The di- and tri-interstitials also display the “Jacob’s ladder” trend, but LDA
and GGA energies lie 2-3.5 eV below QMC. QMC predicts I2a and I3a are the most stable
of the di- and tri-interstitials.
In addition, various tests were performed to check possible sources of error in the QMC
interstitial calculations. Sources of error checked include DMC time step convergence, finite
size convergence, dependence on exchange-correlation functional, Jastrow polynomial order,
pseudopotential choice, and allowing independent Jastrow correlations for the interstitial
atom. Of all possible sources of QMC error checked, none affected results presented outside
of a one-sigma error bar of chemical accuracy.
The second project (Chapter 5) uses QMC combined with DFT phonon computations
153
to compute silica equations of state, phase stability, and elasticity. This work provides
highly accurate values for thermal properties for silica and expands the scope of QMC
by studying the complex phase transitions in minerals away from the cubic oxides. QMC
benchmark calculations are needed because a number of discrepancies between experimental
data and DFT results are documented and silica is an important material to many fields
of science. First, results document feasibility of QMC for computing thermodynamic and
elastic properties of complex minerals. Secondly, results document improved accuracy and
reliability of QMC relative to standard DFT functionals. The efficiency of standard DFT
functionals combined with the ability of QMC to benchmark their performance makes a
powerful tool for predicting and understanding materials physics that is challenging for
experiment to uncover. The main geophysical implication of the results is that the CaCl2 α-PbO2 transition is not associated with the global D00 discontinuity, indicating there is not
significant free silica in the bulk lower mantle. However, the transition should be detectable
in deeply subducted oceanic crust if free silica is at high enough concentrations.
The third project (Chapter 6) investigates whether or not hybrid functionals are capable of benchmark accuracy for quartz and stishovite silica. Results compare performance of
various exchange-correlation functionals, basis sets, pseudopotentials, and codes, which are
benchmarked against QMC and experiments to determine which are most accurate. Calculations compare local (LDA), semi-local GGA (PBE, PW91, PBEsol, WC), and hybrid
(B3LYP, PBE0, and HSE) functionals. All electron calculations are compared against projector augmented wave (PAW) pseudopotentials and nonlocal, norm-conserving pseudopotentials. In addition, results from the CRYSTAL, ABINIT, and VASP codes are compared.
Silica DFT results indicate that choice of basis set, pseudopotential, or code do not
make much difference as long as calculations are converged. Choice of exchange-correlation
functional has the most influence on the predicted result. The LDA functional typically
predicts structural properties and elastic constants within 1-2% or better of experiment and
QMC. The PBE and PW91 GGA functionals tend to overestimate the structural properties
and by 2-3% and underestimate elastic properties by roughly 5%. The PBEsol and WCGGA
functionals tend to improve agreement with experiment over LDA/PBE/PW91, but not for
154
the transition pressure. In fact, among the LDA and GGA functionals, only PBE and
PW91 predict the quartz-stishovite transition pressure in close agreement with experiment.
LDA, WCGGA, and PBEsol underestimate the transition pressure by more than 70%. The
hybrid functionals (B3LYP, PBE0, HSE) tend to improve over PBE and PW91 for structural
and elastic properties. However, B3LYP significantly overestimates the transition pressure.
HSE and PBE0 results are generally similar and in good agreement with experiment and
QMC for all properties considered. HSE provides the most consistently accurate results for
structural and elastic properties and the transition pressure. Of all the functionals studied,
HSE demonstrates consistent benchmark accuracy and is a more efficient alternative to
QMC. HSE is more expensive than standard DFT by a factor of 30, but more efficient than
QMC by a factor of at least 3.
In summary, the many body Schrödinger equation is complex and cumbersome to solve.
Standard (LDA, GGA) DFT offers a powerful approximate solution, but functionals occasionally fail causing DFT to be unreliable. Often, a DFT failure can be fixed by simply
identifying which DFT functional best describes the system under study. Identifying the
best functional for the job is a challenging task, particularly if there is no experimental
measurement to compare against. Higher accuracy methods, which are vastly more computationally expensive, can be used to benchmark DFT functionals and identify those which
work best for a material when experiment is lacking. If no DFT functional can perform
adequately, then it is important to show more rigorous methods are capable of handling the
task.
QMC is a well-known, high accuracy alternative to DFT, but QMC is too expensive to
replace DFT. Hybrid DFT functionals appear to be a good compromise between QMC and
standard DFT. Not many large scale computations have been done to test the feasibility
or benchmark capability of either QMC or hybrid DFT for complex materials. Each of the
three applications presented in this thesis expands the scope of QMC and hybrid DFT to
large, scale complex materials. Results verify the benchmark accuracy of both QMC and the
HSE hybrid DFT functional for silicon defects and high pressure silica phases. Standard
DFT is still the most efficient and useful for general computation. However, this thesis
155
shows that QMC and hybrid calculations can aid and evaluate shortcomings associated
the exchange-correlation potential in DFT by offering a route to benchmark and improve
reliability of DFT predictions. As next generation computers appear, QMC and hybrid
DFT are bound to have an increasingly large impact on computational materials science.
7.2 Future Research
In future projects, it is important to continuing expanding the scope of QMC to even
more complex materials in order to identify pitfalls with the QMC algorithms and continue
pushing its development to get the most out next generation computers. On materials of
particular importance in geophysics is magnesium silicate. Magnesium silicate is one of
the most abundant minerals in Earth’s mantle. This is a ternary oxide with large primitive cells. Its structural phase transitions are found to be consistent with several seismic
discontinuities with increasing depth in the mantle. At a depth of 410 km (12 GPa), the
ambient orthorhombic phase called forsterite transforms to another orthorhombic phase
called wadsleyite. At 520 km (15 GPa) wadsleyite transforms to cubic ringwoodite. At
660 km (25 GPa), ringwoodite transforms to a perovskite structure, and at 2740 km (125
GPa) some evidence indicates a post-perovskite structure forms, consistent with seismic
data from the D” boundary. DFT typically provides lattice constants, bulk moduli, elastic
constants, and sound velocities with in a few per cent of experimental measurements. However, the transition pressures can vary as much as 50% between LDA and GGA, bracketing
experiment [158]. QMC and hybrid DFT can help constrain the transition pressure.
Preliminary tests on Mg2 SiO4 and MgO have revealed a large unexpected inflation in
the variance of the local energies occurs when Mg and O are paired together in a solid.
Investigations into the source of the inflation indicate that the standard numerical orbital
approximation, which only interpolates the first derivatives of the DFT orbitals, is not
accurate enough to converge both energy and variance. The MgO system requires additional
interpolation of the Laplacian of the orbitals in order to also converge the variance. A general
understanding of this potential pitfall for all complex solids still needs to be determined.
156
Capability to interpolate the Laplacian also must be added to the production QMC code,
CASINO.
Lastly, a possible, important new project could be the design and construction of new
exchange correlation functionals based on QMC calculations of solids. The LDA functional
is based on QMC data of the free electron gas. With QMC calculations of large, complex
materials now feasible, the study of exchange-correlation in real systems may allow the
development of new functionals that are even better and more efficient than hybrids. Both
of the projects mentioned here would be important for advancing the field of computational
materials science.
157
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168
Appendix A
Error Propagation in QMC
Thermodynamic Parameters
QMC computes total ground state energies, which have a statistical error bar. The error bar
propagates in to all quantities computed from the QMC energies. There are two methods
for propagation of error: 1) analytic Taylor series expansion or 2) Monte Carlo.
A.1 Taylor Expansion Method
The Taylor expansion method [159, 160] of propagating error is typically only expanded to
first order. The methods assumes there is a function
x = f (u, v, . . .),
(A.1)
with an average value (assuming maximum likelihood)
x̄ = f (ū, v̄, . . .).
(A.2)
The uncertainty in the function x is found by considering the spread from combining measurements of ui , Vi , ect, into xi ’s.:
xi = f (ui , vi , . . .),
(A.3)
which has a variance of
σx2 =
1 X
(xi − x̄i )2
N
169
(A.4)
In order to estimate the error, x is Taylor expanded about the mean values of the
parameters (u, v, . . .) to first order:
∂x
∂x
(ui − ū) +
(vi − v̄)
xi − x̄ =
∂u
∂v
(A.5)
Plugging this expression into the variance (Equation A.4) produces the error propagation
equation:
σx2
=
σu2
∂x
∂u
2
+
σv2
∂x
∂v
2
+ ··· +
2
2σuv
∂x
∂u
∂x
∂v
,
(A.6)
The rightmost term is zero of the dependent variables are completely independent. Codependent variables are possible in complex systems, requiring higher order mixed terms.
The error propagation can be succinctly written as
σx2 =
X ∂x ∂x ∂u
∂v
Cov[u, v],
(A.7)
where Cov[] is the covariance matrix.
A.2 Monte Carlo Method
A more powerful method, but one which involves some additional simulation, is the Monte
Carlo method of error propagation [161]. The advantage of the Monte Carlo method is that
all nonlinear terms the Taylor expansion leaves out of the propagation formula are included.
Thus, codependent variables are taken fully into account.
The Monte Carlo method is conceptually simple. One repeatedly adds random Gaussian
noise proportional to the statistical error to the actual data and proceeds with the analysis
to compute the desired property. After such a procedure is completed many times, the
standard deviation in the results provides is the desired propagated error bar.
As an example, imagine there are a set of QMC energies Ei , each with a statistical error
of σi . Imagine that an equation of state is fit, weighted appropriately to the error bars and
then, say the heat capacity is computed from the equation of state. In order to determine the
error propagated into the heat capacity from the initial set of energies one can use a Monte
Carlo simulation. The simulation allows Gaussian fluctuations of the original energies with
170
a standard deviation that corresponds to the size of the one-sigma statistical error for each
energy. Then the fit is performed and the heat capacity is computed. Independent iterations
of the same procedure are repeated in a loop several hundred or thousands of times. Then
the standard deviation of those hundreds or thousands of computed heat capacities is the
propagated error bar on the heat capacity. The number of iterations should be checked for
convergence.
171
Appendix B
Optimized cc-pVQZ Gaussian
Basis Set used for Silica
cc-pVQZ
EMSL
Basis Set Exchange Library 1/24/11
BASIS SET: (12s,6p,3d,2f,1g) -> [5s,4p,3d,2f,1g]
O
O
S
61420.0000000
0.0000900
9199.0000000
0.0006980
2091.0000000
0.0036640
590.9000000
0.0152180
192.3000000
0.0524230
69.3200000
0.1459210
26.9700000
0.3052580
11.1000000
0.3985080
4.6820000
0.2169800
S
61420.0000000
-0.0000200
9199.0000000
-0.0001590
2091.0000000
-0.0008290
590.9000000
-0.0035080
192.3000000
-0.0121560
172
O
69.3200000
-0.0362610
26.9700000
-0.0829920
11.1000000
-0.1520900
4.6820000
-0.1153310
S
1.4280000
O
S
0.5547000
O
O
63.4200000
0.0060440
14.6600000
0.0417990
4.4590000
0.1611430
P
1.0000000
F
2.6660000
O
1.0000000
D
0.4440000
O
1.0000000
D
1.3000000
O
1.0000000
D
3.7750000
O
1.0000000
P
0.3000000
O
1.0000000
P
0.6500000
O
1.0000000
P
1.5310000
O
1.0000000
S
0.2067000
O
1.0000000
1.0000000
F
173
0.8590000
1.0000000
BASIS SET: (16s,11p,3d,2f,1g) -> [6s,5p,3d,2f,1g]
Si
S
513000.0000000
0.260920D-04
76820.0000000
0.202905D-03
17470.0000000
0.106715D-02
4935.0000000
0.450597D-02
1602.0000000
0.162359D-01
574.1000000
0.508913D-01
221.5000000
0.135155D+00
90.5400000
0.281292D+00
38.7400000
0.385336D+00
16.9500000
0.245651D+00
6.4520000
0.343145D-01
2.8740000
-0.334884D-02
1.2500000
0.187625D-02
Si
S
513000.0000000
-0.694880D-05
76820.0000000
-0.539641D-04
17470.0000000
-0.284716D-03
4935.0000000
-0.120203D-02
1602.0000000
-0.438397D-02
574.1000000
-0.139776D-01
221.5000000
-0.393516D-01
90.5400000
-0.914283D-01
38.7400000
-0.165609D+00
16.9500000
-0.152505D+00
6.4520000
0.168524D+00
2.8740000
0.569284D+00
174
1.2500000
Si
0.398056D+00
S
513000.0000000
0.178068D-05
76820.0000000
0.138148D-04
17470.0000000
0.730005D-04
4935.0000000
0.307666D-03
1602.0000000
0.112563D-02
574.1000000
0.358435D-02
221.5000000
0.101728D-01
90.5400000
0.237520D-01
38.7400000
0.443483D-01
16.9500000
0.419041D-01
6.4520000
-0.502504D-01
2.8740000
-0.216578D+00
1.2500000
-0.286448D+00
Si
S
0.3599000
Si
S
0.1699000
Si
Si
1.0000000
1.0000000
P
1122.0000000
0.448143D-03
266.0000000
0.381639D-02
85.9200000
0.198105D-01
32.3300000
0.727017D-01
13.3700000
0.189839D+00
5.8000000
0.335672D+00
2.5590000
0.379365D+00
1.1240000
0.201193D+00
P
175
Si
1122.0000000
-0.964883D-04
266.0000000
-0.811971D-03
85.9200000
-0.430087D-02
32.3300000
-0.157502D-01
13.3700000
-0.429541D-01
5.8000000
-0.752574D-01
2.5590000
-0.971446D-01
1.1240000
-0.227507D-01
P
0.3988000
Si
P
0.1533000
Si
1.0000000
F
0.2120000
Si
1.0000000
D
0.7600000
Si
1.0000000
D
0.3020000
Si
1.0000000
1.0000000
F
0.5410000
1.0000000
END
176
Appendix C
Details of the Ewald and MPC
Interaction in Periodic
Calculations
C.1 Ewald Interaction
In solid-state simulations of real materials, it is not computationally feasible to use a clutster
with millions of atoms to approximate a bulk structure. Instead, one usually uses a supercell
method, in which a cluster containing a small number of atoms is artificially repeated
through out space via periodic boundry conditions. One difficultly that arises in such a
method is finding a convenient way to sum up all the contributions to the electrostatic
potential energy restulting from the coulomb interaction of the charges in the supercell
with their periodic images. A popular scheme for performing this sum is known as the
Ewald method [162].
To demonstrate the Ewald method, consider an ideal system of positive and negative
charges in a cube of side length L subject to periodic boundary conditions. Suppose there
PN
are N total charges and the system as a whole is electrically neutral.
i=1 zi = 0 The
electrostatic potential energy of such a system is given by
UCoul =
N
1 X
zi φ (ri ) ,
2
i=1
177
(C.1)
where φ (ri ) is the electrostatic potential at the position of the ith ion:
φ (ri ) =
X0
~j,~
n
zj
rij + ~nL
(C.2)
The prime denotes that summation is done over all periodic images, n, and over all particles,
j, except j=i when n=0. Unfortunately, φ (ri ) as written is only conditionally convergent
sum.
Ewald found that he could imporve the convergence of the sum by recasting the charge
density into a different form. In equation (2) the charge density is implicitly cast as a sum
of delta-functions. Alternatively, if one thinks of the distribution as screened point charges
among a compensating background charge, then equation (2) can be broken into two nicely
converging summations: one over Fourier space and one over real space. The problem with
the delta-function distribution is that the unscreened coulomb interaction decays slowy as
1/r, and, hence, the interactions are long-range. By assumimg that every point charge, zi ,
is surrounded by a diffuse charge distribution of opposite sign and magnitude, the Coulomb
interaction is much shorter range. In this situation, the contribution of a point charge to
the electrostatic potential is due only to the fraction of charge that is not screened. The
rate at which the Coulomb interaction decays depends on the functional form chosen for
the screening charge distribution.
Our goal is to find the potential due to a set of point charges, not screened charges.
One corrects for the screening charge by adding in a compensating background charge
distribution. The one catch is that one would like the background charge distribution to be
a smoothly varying function in space. In general, when computing the electrostatic potential
energy at a ion site, i, the contribution of the charge zi to the energy is not included beacuse
it would be a unphysical self interaction. This means the screening charge and background
charge around ion i should not contribute to the energy at site i either. However, for
convenience, the background charge around ion i is included when calculating the energy
at ion i such that the background charge distribution is a continuous and smoothly varying
function. A correction for the self-interaction of ion i with it’s background charge will be
178
computed later. The advantage to this method is that the background distribution can be
represented by a rapidly converging Fourier series.
In what follows, three individual terms will be computed to evaluate the Coulomb contribution to the electrostatic potential energy. First, the contribution to the Coulomb
energy due to the background charge is computed in Fourier space, then the correction for
the self-interaction, and finally the contribution due to the screening charges in real-space.
All equations will be in Gaussian units to make the notation compact. The compensating
background charge around an ion, i, is chosen to be a Gaussian distribution with width
p
2/α:
ρGauss = −zi (α/π)3/2 exp(−αr2 ),
(C.3)
where α is parameter adjusted for computational efficiency.
Fourier Part
Choosing the compensating background charge distribution as Gaussian around an ion,
i, means that the electrostatic potential at a point ri is due to a periodic sum, ρ1 , of
Gaussians:
ρ1 (r) =
N X
X
h
i
zj (α/π)3/2 exp −α |~r − (~rj + ~nL)|2
(C.4)
j=1 ~
n
The electrostatic potential, φ1 , due to ρ1 is computed via Poisson’s equation:
−∇2 φ1 (r) = 4πρ1 (r)
(C.5)
or, more conveniently, in Fourier form,
k 2 φ1 (k) = 4πρ1 (k).
179
(C.6)
Fourier Transforming the charge density ρ1 yields
ρ1 (~k) =
1
=
V
Z
d~r
exp(−i~k · ~r)
V
1
=
V
1
V
Z
exp(−i~k · ~r)ρ1 (~r)
d~r
(C.7)
V
N X
X
h
i
zj (α/π)3/2 exp −α |~r − (~rj + ~nL)|2
(C.8)
j=1 ~
n
Z
d~r
exp(−i~k · ~r)
allspace
N
X
h
i
zj (α/π)3/2 exp −α |~r − ~rj |2
(C.9)
j=1
=
N
1 X
zj exp(−i~k · ~rj )exp(−k 2 /4α)
V
(C.10)
j=1
Inserting ρ1 (~k) into equation (6), one obtains
N
4π 1 X
φ1 (k) = 2
zj exp(−k 2 /4α),
k V
(C.11)
j=1
which is not defined for k = 0. Assumeing the k=0 term is negligible describes a periodic
system embedded in a medium with infinite dielectric constant.
In order to compute the potential energy due to φ1 (k) using equation (1), one must
compute φ1 (r) by Fourier Transforming equation (11):
φ1 (r) =
X
φ1 (k)exp(i~k · ~r)
(C.12)
~k6=0
=
N
h
i
1 X X 4πzj
~k · (~r − ~rj ) exp(−k 2 /4α).
exp
i
V
k2
~k6=0 j=1
The following is the contribution of φ1 (r) to the potential energy:
180
(C.13)
U1 ≡
1X
zi φ1 (ri )
2
(C.14)
i
=
=
N
1 X X 4πzi zj
2
~k6=0 j=1
V k2
h
i
exp i~k · (~
ri − ~rj ) exp(−k 2 /4α)
V X 4π ~ 2
ρ(k) exp(−k 2 /4α),
2
k2
(C.15)
k6=0
where we define
N
1 X
ρ(~k) ≡
zi exp(i~k · ~ri ).
V
(C.16)
i=1
Correction for Self-Interaction
As discussed earlier, equation (14) includes a spurious self-interaction term that results
from the interaction of a point charge with the background charge. This term was included
such that the background charge could be Fourier transformed. The extra term is of the
form (1/2)zi φself (ri ), and, in this case, the point charge is sitting at the origin of the
Gaussian distribution. To calculate this term, one must compute the potential energy at
the location of the point charge due to the Gaussian charge distribution. The exact charge
distribution in question is
ρGauss = −zi (α/π)3/2 exp(−αr2 ).
(C.17)
Using Poisson’s equation, one can compute the electrostatic potential of this charge
distribution. Assuming spherical symmetry of the Gaussian distribution, Poisson’s equation
can be written as
−1 ∂ 2 rφGauss (r)
= 4πρGauss (r)
r
∂r2
181
(C.18)
or
−
∂ 2 rφGauss (r)
= 4πrρGauss (r).
∂r2
(C.19)
Integrating by parts once yields
∂ 2 rφGauss (r)
−
∂r2
Z
r
4πrρGauss (r)
Z
3/2
= −2πzi (α/pi)
(C.20)
=
∞
∞
dr2 exp(−αr2 )
r
1
= 2zi (α/π) 2 exp(−αr2 )
Integrating by parts a second time produces
rφGauss (r) = 2zi (α/π)
1
2
Z
√
= zi erf ( αr)
r
drexp(−αr2 )
(C.21)
0
where,
√
Z
erf (x) ≡ (2/ π)
x
exp(−u2 )du
(C.22)
0
Therefore, the potential due to a Gaussian distribution of charge at any point in space
is given by
φGauss (r) =
√
zi
erf ( αr).
r
(C.23)
However, the self-interaction term only depends on the potential at r=0:
1
φGauss (r = 0) = 2zi (α/π) 2 .
(C.24)
Therefore, the correction to the potential energy (equation (14)) due to the self-interation
is given by
182
N
Uself
=
1X
zi φself (ri )
2
i=1
1
= (α/π) 2
N
X
zi2
(C.25)
i=1
It is worth noting that this correction term is a constant provided all charges are fixed in
opsition.
Real Space Sum
Recall the point charges are screened by Gaussian charge distributions,opposite in charge
and equal in magnitude. In this final section, the electrostatic energy due to the screened
point charges must be computed. Using equation (23),one can write the (now short range)
electrostatic potential due to a point charge zi surrounded by a Gaussian charge distribution
with net charge −zi :
√
zi zi
− erf ( αr)
r
r
√
zi
erf c( αr),
r
φshortrange (r) =
=
(C.26)
where erf c(x) ≡ 1 − erf (x) is the complementary error function. Therefore, the total
contribution of the screened Coulomb interactions to the potential energy is given by
N
Ushort
range
√
1X
zi zj erf c( αrij )
=
2
(C.27)
i6=j
Finally, on obtains the total electrostatic contribution to the potential energy by summing
the three terms (equations (15), (25), and (27)):
183
UCoul =
1 X 4πV
2
k2
~ 2
ρ(
k)
exp(−k 2 /4α)
k6=0
1
− (α/π) 2
N
X
zi2
i=1
+
N
1X
2
√
zi zj erf c( αrij )
i6=j
C.2 Model Periodic Coulomb (MPC) Interaction
In QMC, finite size errors arise fom the fact that a preiodically repeated finite simulation
cell is used to model an infinite solid. For the exchange correlation energy, the periodicity
introduces a spurious contribution because electron correlations are also forced to be periodic, which each electron interacts with peridodic images of it’s exchange correlation hole.
Another way of thinking about the error is that electron correlation in each periodically
repeated cell is the same, which is unphysical, and causes the electron interactions to be
unphysical.
The Ewald sum models the interaction of periodically repeated electrons. Exapnding
the Ewald interaction gives
vEwald (r) =
1 2π T
+
r · D · r + ··· ,
r 3Ω
(C.28)
where Ω is the volume of te simulation cell, and D is a tensor that depends on the shape
of the cell (cubic = identity). The deviations from
1
r
are what make the Ewald interaction
periodic, but are responsible for spurious contributions to the exchange-correlation energy.
A modification to the Hamiltonian called the Model Periodic Coulomb (MPC) interaction [24, 163, 164, 23, 21] provides a solution to remove the spurious error to the exchangecorrelation energy. In doing so two rules must be respected: 1) MPC should give the Ewald
interaction for Hartree terms and 2) MPC should be exactly
1
r
for the interaction with the
exchange correlation hole. The solution of Poisson’s equation for a periodic array of charges
only obeys the second rule in the limit of an infinitly sized cell.
184
The Model Periodic Coulomb interaction replaces the Ewald interaction in order to
satisfy both rules:
Ĥee =
X
i>j
f (ri − rj ) +
XZ
i
E
V (ri − r) − f (ri − r n(r)dr,
(C.29)
WS
where n is the electronic number density, V E is the Ewald potential, and
f (r) =
1
rm
(C.30)
is a cutoff Coulomb function within a minimum image convention, which corresponds to
reducing the vector r into the Wigner-Seitz (WS) cell of the simulation cell by removal of
lattice vectors, leaving rm . The full MPC Hamiltonian consists of the sum of the Hartree
energy computed with the Ewald interaction and the exchange-correlation energy computed
with the cutoff function to prevent electrons to interact with mirror images of their exchangecorrelation hole.
The MPC interaction does not completely eliminated the many-body exchange-correlation
finte size error. However, in practice, the finite size error converges much more quickly as
a function of system size than when MPC is not used. So, effectively, using MPC allows
one to do solid calculations with smaller simulations cells than would normally be the case.
This allows a dramatic savings in computational time.
185
Appendix D
Summary of CODES Used in This
Work
D.1 ABINIT
ABINIT is a free plane-wave, pseudopotential DFT code capable of geometry optimization,
phonon calculations, and a number of other electronic properties. Pseudopotentials are user
created, typically using the norm-conserving, nonlocal variety.
D.2 Quantum ESPRESSO
Quantum ESPRESSO is a free integrated suite of computer codes, including the planewave pseudopotential DFT code PWSCF, for electronic-structure calculations and materials
modeling at the nanoscale. It is based on density-functional theory, plane waves, and
pseudopotentials (both norm-conserving and ultrasoft).
D.3 VASP
VASP is a commercial plane-wave, pseudopotential DFT code capable of geometry optimization, phonon calculations, and a number of other electronic properties. VASP is the
first code capable of doing hybrid, HSE calculations using plane-waves.
186
D.4 CASINO
CASINO is the private code of Richard Need’s Cambridge QMC group. It performs VMC,
DMC with most of the latest and greatest developments in QMC, such as backflow, planewave expansion in the Jastrow, MPC, and lots of useful scripts and a well-developed manual
making it user-friendly.The code has a string user base and requires a small cost and permission to use.
D.5 CHAMP
CHAMP is a private QMC code maintained by Cyrus Umrigar at Cornell University. The
code is typically used as a research code to develop new algorithms and techniques for QMC.
D.6 OPIUM
OPIUM is a free nonlocal, normconserving pseudopotential generator. It produces pseudopotentials that interface with most of the codes listed here and more.
D.7 CRYSTAL
CRYSTAL is a commercial DFT code which uses Gaussian basis sets with or without a
pseudopotential (effective core potential). The code is maintained by people at a variety of
European institutions.
D.8 WIEN2K
WIEN2k is a commerical FLAPW code.
D.9 ELK
ELK is a free FLAPW code (essentially a free version of WIEN2K).
187
Appendix E
Strong and Weak Scaling in the
CASINO QMC Code
The two most widely-used quantum Monte Carlo methods for continuum electronic structure calculations are: variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC).
The VMC method computes the electronic energies by applying the variational principle
to a given functional form of the many-body wave-function. Each processor uses a single configuration of electrons to evaluate expectation values stochastically. In DMC, the
Schrödinger equation is written as a diffusion equation and the many-body wavefunction
takes on a statistical form, represented as several electronic configurations per processor.
The configurations diffuse, branch, or vanish until distributed according to the ground-state
charge density.
Parallel efficiency in VMC is only sensitive to a small amount of inter-node communication due to a final step of accumulating and averaging energies computed on each processor.
Since only one configuration is used per processor, the size of the problem per node is always
fixed. Therefore, only study of weak scaling is possible.
Figure 1 shows VMC pays a negligible penalty in parallel efficiency due to inter-node
communication. The speedup is shown relative to a serial run, where the time of each job
is scaled to provide equivalent statistical error in the energy as the largest processor job.
p
Error decreases as 1/ NMonte Carlo steps Nconfigurations/proc Nproc . VMC performance scales
almost perfectly with the square of the number of processors as expected.
The parallel efficiency of DMC is sensitive to the number of configurations used per
188
Figure E.1: Log-Log plot of weak scaling in VMC. The time for each job is scaled to provide
a result with the same statistical error as the largest processor job. Speedup is the ratio of
scaled parallel time to a scaled serial time. VMC calculations scale almost perfectly with
the square of the number of processors.
processor. As the number configurations increase or decrease on each processor due to
the branching algorithm, the populations on each processor may become uneven and must
occasionally be rebalanced. Processors with the fewest number of configurations will occasionally have to wait on other processors to finish cycling through a larger number of
configurations. The efficiency can be indefinitely improved by increasing the number of
configurations per processor. However, in practice the number of configurations used is
limited by available memory. DMC also has a small amount of inter-node communication
for accumulation and averaging of energies as in VMC.
Figure 2 shows weak scaling in DMC, where the number of configurations per processor
is held fixed. This means the total number of configurations increases with the number of
processors, allowing DMC efficiency to improve as the number of processors is increased.
However, scaling is not perfect due to time required for rebalancing the configuration population across processors.
189
Figure E.2: Log-Log plot of weak scaling in DMC. The time for each job is scaled to
provide a result with the same statistical error as the largest processor job. Speedup is the
ratio of scaled parallel time to a scaled serial time. DMC continuously gains speedup as the
number of processors increases because the total number of configurations is also increasing.
Efficiency is imperfect due to time required to rebalance configurations across processors.
Figure 3 shows strong scaling in DMC, where the total number of configurations is fixed.
Efficiency decreases with processor size because the number of configurations per processor
is decreasing, causing increased time for rebalancing the configurations. Figure 2 and 3
together show why it is important to use a large number of configurations to maximize the
speedup of parallel computing in DMC.
190
Figure E.3: Log-Log plot of strong scaling in DMC. The time for each job is scaled to
provide a result with the same statistical error as the largest processor job. Speedup is the
ratio of scaled parallel time to a scaled serial time. DMC calculations become increasingly
inefficient as the number of processors increase because the total number of configurations
is being held fixed.
191