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
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Establishing Quantum MonteCarlo and Hybrid Density
Functional Theory asbenchmarking tools for
complex solids
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor ofPhilosophy in the Graduate School of The Ohio State University
K. P. Driver, R. E. Cohen, Zhigang Wu, B. Militzer, P. Lopez Rıos, M. D. Towler, R. J.Needs, and J. W. Wilkins, Quantum Monte Carlo computations of phase stability, equationsof 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 QMCand DFT under hydrostatic and anisotropic stress, Phys. Rev. B, 82, 014101 (2010).
M. Floyd, Y. Zhang, K. P. Driver, Jeff Drucker, P.A. Crozier, and D.J. Smith, Nanometer-scale composition variations in Ge/Si(100) islands, Appl. Phys. Lett. 82, 1473 (2003).
Y. Zhang, M. Floyd, K. P. Driver, Jeff Drucker, P.A. Crozier, and D.J. Smith, Evolution ofGe/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 BrewsterAngle, Am. J. Phys. 69, 1166 (2001).
vii
Fields of Study
Major Field: Physics
Studies in quantum Monte Carlo claculations of solids: J. W. Wilkins
3 Coping with DFT Approximations: Benchmarking with Hybrid func-tionals and QMC 303.1 Introduction: Approximations and Weaknesses of Density Function Theory 30
Figure 4.1: Single interstitial defects in silicon
(a) I2a (b) I2b
Figure 4.2: Double interstitial defects in silicon
42
(a) I3a (b) I3b (c) I3c
Figure 4.3: Triple interstitial defects in silicon
43
Figure 4.4: QMC and DFT band gaps of Si. Results follow the trend of Jacob’s ladder offunctionals, 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 sigmaabout the mean.
44
Figure 4.5: QMC and DFT cohesive energy of Si. LDA overestimates the energy and GGAimproves DFT agreement with QMC and experiment. The black and white diagonal boxon 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 followthe trend of Jacob’s ladder of functionals, where LDA is least accurate and HSE agrees bestwith QMC and experiment. GGA performs slightly better than mGGA. The black andwhite diagonal box on the QMC result respresents plus and minus sigma about the mean.
46
4.2 Calculation Details
DFT computations are performed within the CPW2000 DFT code [76], which was the
only plane-wave pseudopotential DFT interfaced to work with the QMC code we used
for this project, CHAMP [77]. Calculations use a Ceperely-Alder LDA, norm-conserving
pseudopotential. All silicon calculations use a converged plane-wave energy cutoff of 16 Ha
and Monkhorst-Pack k-point meshes such that total energy converged to tenths of milli-
Ha/SiO2 accuracy. Silicon QMC calculations were done for cell sizes of 8-, 16-, 32-, and
All meshes were shifted to the L-point from the origin by (0.0, 0.5, 0.5), corresponding to
the reduced coordinates in the coordinate system defining the k-point lattice. The defect
structures studied are the most stable candidates as predicted by tight-binding MD [63].
They are further optimized in VASP [78] at the experimental lattice constant of silicon
(5.432 A) until all forces on the atoms are smaller than 10−4 Ha/Bohr.
QMC calculations are performed in the CHAMP [77] code using a Slater-Jastrow type
of wave-function, where the Slater determinant is made of single particle orbitals from a
converged DFT-LDA calculation. The same LDA pseudopotential as used in DFT is used
in QMC calculations. Calculations were repeated with GGA to check for functional bias
and results (shown below) indicated none. The Jastrow used in for this project describes
correlations by including only two-body terms: electron-electron and electron-nuclear with
a total of 12 parameters. The parameters in the Jastrow are optimized by minimizing
the VMC energy [79, 80] over thousands of electron configurations. Several iterations of
computing the VMC energy and re-optimizing the parameters are done until the VMC
energy changes are typically inside of two sigma statistical error.
DMC calculations use the optimized trial wave functions to compute highly accurate
energies. A typical silicon DMC calculation uses 1000 electron configurations, a time step
of 0.1 Ha−1, and several thousand Monte Carlo steps. In order for QMC calculations of
solids to be feasible, a number of approximations [9, 16] are usually implemented that
can be classified into controlled and uncontrolled approximations, which are discussed in
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detail in Chapter 2. The controlled approximations for silica include statistical Monte Carlo
error, numerical grid interpolation (∼5 points/A) of the DFT orbitals [22], finite system
size, the number of configurations used, and the DMC time step. All of the controlled
approximations are converged to within at least 1 milli-Ha/SiO2. Finite size errors are
reduced to less then chemical accuracy by simulating cell sizes up to 64 atoms. Additional
finite size errors are corrected due to insufficient k-point sampling based on a converged
k-point DFT calculation. The uncontrolled approximations include the pseudopotential,
nonlocal evaluation of the pseudopotential, and the fixed node approximation. These errors
are difficult to estimate and generally assumed to be small [28, 81, 19].
4.2.1 Results
Figure 4.7 shows QMC and DFT formation energies for 16-atom bulk and 16(+1)-atom
self-interstitial defects, X, H, and T, while Figure 4.8 shows the QMC and DFT formation
energy for 64-atom bulk and 64(+1)-atom self-interstitial defects, X, H, and T. Results
display the expectation of Perdew’s Jacob’s ladder, with LDA, GGA, mGGA, and HSE
progressively agreeing better with QMC. In both simulation cell sizes, LDA and GGA
calculations generally agree with QMC ordering, but lie 1-1.75 eV below QMC formation
energies. HSE closely agrees with QMC compared to LDA, GGA, and meta-GGA (mGGA)
functionals for 16 atom simulation cell calculations.
QMC results agree well with QMC results of Leung et al. [71]. Leung et al. studied
16-atom and 54-atom cells, while work presented here studied 16-atom and larger 64-atom
cells. Leung et al. predict X and H are degenerate within statistical error in both 16- and
54-atom cells and T lies 0.4-0.5 eV higher in energy, but is within two-sigma statistical
error. Work presented here predicts X and T are degenerate in the 16-atom cell and T lies
about 0.25 eV higher. Finite size errors are expected to be large for the 16-atom cell and,
thus, convergence is not expected. In the 64-atom cell, results agree with Leung etal.. Data
presented here goes beyond Leung et.al. by studying the larger 64-atom cell and obtaining
QMC statistical error that is roughly a factor of 10 smaller. Later in this chapter, several
additional sources or error are also investigated to check results more carefully beyond the
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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.
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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 resultsrespresents plus and minus sigma about the mean. QMC predicts the H defect formationenergy 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 0.25eV below QMC, but predicts T to lie highest. LDA and GGA results predict a similarenergy ordering as QMC, but lie 1-1.75 eV lower. The mGGA predicts T lies highest andlies about 1 eV lower than QMC.
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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 resultsrespresents plus and minus sigma about the mean. QMC predicts the formation energy ofT is 0.6 eV higher than degenerate X and H defects. LDA and GGA predict similar energyordering, but lie 1-1.75 eV lower than QMC.
51
Figure 4.9 shows the QMC and DFT energy barriers (migration energies) for the self-
diffusion path from the X to H to T and back to X. The barriers are the energy required
for a diffusive hope between X, H, and T. The calculations are for 64-atom simulation cells.
X-H and T-X labels correspond to saddle point structures determined from Nudged Elastic
Band (NEB) calculations of diffusion from X to H and T to X, respectively. The QMC
X-to-H diffusion barrier is 355(100) meV and the X-to-T barrier is about 720(100) meV.
There is essentially no barrier for H-to-T diffusion. The X-to-H barrier is similar in QMC
and GGA, while the X-to-T barrier is about 400 meV larger in QMC. Previous LDA and
GGA estimates of the migration energy gave 100-300 meV [70], which is similar to GGA
results in Figure 4.9.
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Figure 4.9: Single interstitial diffusion path in 64 atom cell. QMC(DMC) benchmarksGGA-NEB calculations. The QMC error bar respresents plus and minus sigma about themean value. The lowest barrier from X to H is similar in QMC and DFT. The T defectformation energy and barrier are larger in QMC.
53
The experimental estimates of the diffusion activation energy (formation + migration)
for defects in silicon are near 4.7-4.9 eV [82, 83]. Using the estimates of migration energy
computed above (about 280 meV for GGA and 355 meV for QMC for the lowest path),
indicates that GGA (3.87 + 0.280 = 4.15 eV) underestimates the experimental activation
energy range by about 0.6 eV and the lowest QMC estimate (4.9(1) + 0.3(1) = 5.2(1) eV)
is about 0.3 eV above the experimental range. QMC provides a significant improvement
over GGA. Some of the QMC error could be due to using the DFT(GGA) geometry for the
defects.
Figure 4.10 shows QMC and DFT formation energies for the two lowest energy di-self-
interstitial defects, Ia2 and Ib2 in a 64 atom cell. Results again display the expected functional
Jacob’s ladder trend. QMC predicts the Ia2 defect lies 1.2 eV lower in energy than Ib2. 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.
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Figure 4.10: Formation energy of the two lowest-energy di-self-interstitials in silicon (I2a andI2b) in a 64-atom cell. The black and white diagonal box on the QMC results respresentsplus and minus sigma about the mean. QMC predicts the Ia2 defect lies 1.2 eV lower inenergy than Ib2. LDA and GGA predict a similar energy ordering, but LDA lies up to 3 eVbelow QMC and GGA lies up to 2 eV below QMC.
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Figure 4.11 shows QMC and DFT formation energies for the three most stable tri-self-
interstitial defects, Ia3 , Ib3, and Ic3 in a 64-atom cell. Results again display the Jacob’s ladder
trend. QMC predicts the stability ordering is Ia3 < Ic3 < Ib3. GGA predicts the stability
ordering Ib3 < Ia3 < Ic3, while LDA predicts Ia3 < Ib3 < Ic3. For the Ia3 and Ic3 defects, GGA
closely agrees with QMC. The improved GGA results may be because the Ia3 and Ic3 defects
are less distorted (smaller coordination number) than the Ib3 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 (Ia3 ,Ib3, and Ic3) in a 64-atom cell. The black and white diagonal box on the QMC resultsrespresents 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 intersti-
tial 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.
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Figure 4.12: Convergence of the DMC time step for Si. The QMC error bars represent plusand minus sigma about the mean value. The energy difference on the vertical axis is withrespect to a very small time step energy that has been set to zero. The time step, τ in unitsof 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.
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Figure 4.13: Convergence of DMC finite size error Si X defect. The QMC error barsrepresent plus and minus sigma about the mean value.
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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.
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Figure 4.14: DMC formation energy using LDA and GGA orbitals. The black and whitediagonal box on the QMC results respresents plus and minus sigma about the mean. Re-sults agree within one-sigma statistical error, indicating negligible dependence on functionalchoice.
63
Figure4.15 shows convergence of the Jastrow electron-nuclear and electron-electron poly-
nomial expansion order. Formation energy of the X defect in a 16-atom cell is converged
by 5th order polynomials.
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Figure 4.15: VMC convergence of X-defect formation energy versus Jastrow polynomialorder in 16-atom Si. The QMC error bars represent plus and minus sigma about themean value. The notation MN0 indicates M order electron-nuclear polynomial and N or-der electron-electron polynomial, and no electron-electron-nuclear polynomial. Formationenergy 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 LDAand Hartree-Fock pseudopotentials. The QMC error bars represent plus and minus sigmaabout the mean value. By 32 atoms, when finite-size error is small, it is clear that bothtypes of pseudopotentials produce the same result.
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Table 4.1: VMC and DMC calculations of Si single self-interstitial formation energies. Oneset of calculations uses the same e-n Jastrow for bulk and the defect atom. A second set setof calculations uses independent Jastrow for bulk and defect e-n Jastrows. Results agreewithin one-sigma error, indicating that a single Jastrow is sufficient.
All atoms have same e-n JastrowVMC X H TVMC 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 JastrowsVMC 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
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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 Ia2 and Ia3 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 exper-
iments 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].
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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 de-
scribes calculations of highly accurate ground state QMC plus phonons within the quasi-
harmonic approximation of density functional perturbation theory to obtain benchmark
thermal pressure and equations of state of silica phases up to Earth’s core-mantle bound-
ary [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 D′′ layer, but the absence of a seismic signature suggests
the transition does not contribute significantly to global seismic discontinuities in the lower
mantle. However, the transition could still provide seismic signals from deeply subducted
oceanic crust. Computations also identify the feasibility of QMC to find an accurate shear
elastic constants for stishovite and its geophysically important softening with pressure.
5.1 Introduction
Silica is one of the most widely studied materials across the fields of materials science,
physics, and geology. It plays important roles in many applications, including ceramics,
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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 elec-
tronic 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 min-
erals from the surface through the crust and mantle. However, the abundance of free silica
phases and their role in the structure and dynamics of deep Earth is still unknown.
Free silica phases may form in the Earth as part of subducted slabs [91] or due to
chemical reactions with molten iron [92]. Determination of the phase stability fields and
thermodynamic equations of state are crucial to understand the role of silica in Earth. The
ambient phase, quartz, is a fourfold coordinated, hexagonal structure with nine atoms in
the primitive cell [85]. Compression experiments reveal a number of denser phases. The
mineral coesite, also fourfold coordinated, is stable from 2–7.5 GPa, but is not studied
here due to its large, complex structure, which is a 24 atom monoclinic cell [86]. Further
compression transforms coesite to a much denser, sixfold coordinated phase called stishovite,
stable up to pressures near 50 GPa. Stishovite has a tetragonal primitive cell with six
atoms [87]. In addition to the coesite-stishovite transition, quartz metastably transforms to
stishovite at a slightly lower pressure of about 6 GPa. Near 50 GPa, stishovite undergoes
a ferroelastic transition to a CaCl2-structured polymorph via instability in an elastic shear
constant [88, 93, 94, 95, 96, 97, 98, 99]. This transformation is second order and displacive,
where motion of oxygen atoms under stress reduces the symmetry from tetrahedral to
orthorhombic. Experiments [89, 90, 100] and computations [101, 102, 103] have reported a
further transition of the CaCl2-structure to an α-PbO2-structured polymorph at pressures
near the base of the mantle. Figure 5.1 shows a schematic version of the silica pressure-
temperature phase diagram. Note the pressure scale is no linear, and the dashed phase
boundaries indicate they are not well known.
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Figure 5.1: Schematic version of the silica phase diagram.
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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 D′′, the transition should be
detectable in deeply subducted oceanic crust.
5.3 Computational Methodology
This section discusses the general computational methods and choices made for all silica
calculations. The first section describes pseudopotential generation choices for Si and O
used for all silica calculations. The second section discusses types of silica calculations using
DFT: geometry optimization, wave-function generation, and phonon calculations. The last
section addresses QMC calculations including wave-function optimization and VMC/DMC
choices.
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
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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 calcula-
tions use a converged plane-wave energy cutoff of 100 Ha and Monkhorst-Pack k-point
meshes such that total energy converged to tenths of milli-Ha/SiO2 accuracy. Converged
silica calculations require k-point mesh sizes of 4×4×4, 4×4×6, and 4×4×4 for quartz,
stishovite/CaCl2, and α-PbO2, respectively, and all meshes were shifted from the origin by
(0.5, 0.5, 0.5), corresponding to the reduced coordinates in the coordinate system defining
the k-point lattice.
Geometry Optimization
The ABINIT code allows various types of structural optimizations that are useful for phase
stability and elasticity calculations. This work utilizes several different types of optimiza-
tions: 1) optimization of forces on the ions only, 2) simultaneous optimization of ions and
cell shape at a fixed volume, and 3) full optimization of ions, cell shape, and volume. Ion-
only optimization is used in shear constant calculations of stishovite after an initial full
optimization of the cell. For equations of state of all silica phases, total energies are com-
puted for six or seven cell volumes ranging from roughly 10% expansion to 30% compression
about the fully-optimized equilibrium volume. Constant volume optimization of compressed
and expanded cell geometries relaxes forces on the atoms to less than 10−4 Ha/Bohr.
Wave-function Generation
All QMC calculations start with a trial wave-function that is partially made up of the
Slater determinant of single electron orbitals from corresponding DFT calculations. The
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DFT wave-function is produced from a fixed geometry calculation after all variables are
optimized to produce a converged charge density. The converged charge density is then
used in a non-self-consistent calculation to output the orbitals.
Phonon Free Energy Calculations
There are two commonly used free energies in thermodynamic calculations: 1) Helmholtz
and 2) Gibbs free energies. The Helmholtz free energy is used for the derivation of most
thermodynamic quantities, where volume, V, and temperature, T, are the independent
variables. Gibbs free energy is important for equilibrium studies for determining phase
boundaries, where the convenient independent variables are pressure, P, and T.
The Helmholtz free energy [109] is defined as,
F (V, T ) = Ustatic(V, T )− TSvib(V, T ), (5.1)
where Ustatic is the static internal energy of the crystal lattice and TSvib is the vibrational
(phonon) contribution of the thermal atomic motion to the free energy and Svib is the
vibrational entropy. The Gibbs free, G = F + PV energy is constructed from Helmholtz
energy. Therefore, this section focuses on Helmholtz free energy, while Gibbs free energy
will be discussed in more detail in the phase stability results section below.
Before any thermodynamic properties may be computed from the Helmholtz free en-
ergy, one must decide how to treat the temperature and volume dependence of the phonon
frequencies, ωi of the lattice. Statistical mechanics allows a system’s vibrational quantum
mechanical energy levels to completely determine the vibrational Helmholtz free energy
(F = Ustatic + Fvib) via a partition function, Z:
Fvib = −kT lnZ, (5.2)
where Z is a sum over all quantum energy levels given by,
Z =∑i
exp (−εikT
), (5.3)
where k is the Boltzmann constant, and the εi are the microstate energies: 12 hωi,
32 hωi,
76
52 hωi, ect.
Therefore, for each mode, there are many energy levels
Zi = exp (12hωikT
)all modes∑s=0
exp (−shωikT
), (5.4)
=exp (−1
2hωikT )
1− exp (−hωikT )
, (5.5)
which when combined with Equation 5.2 gives
Fvib,i =12hωi + kT ln(1− exp (
−hωikT
)). (5.6)
The total expression for the Helmholtz free energy (F = Ustatic + Fvib) in the harmonic
state approximation is then
F = Ustatic +all modes∑
i=1
12hωi + kT
all modes∑i=1
ln(1− exp (−hωikT
)), (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 depen-
dence 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 en-
ergies. A typical silica DMC calculation uses 4000 electron configurations, a time step of
0.003 Ha−1, and several thousand Monte Carlo steps. In order for QMC calculations of
solids to be feasible, a number of approximations [9, 16] are usually implemented that can
be classified into controlled and uncontrolled approximations, which are discussed in detail
in Chapter 2. The controlled approximations for silica include statistical Monte Carlo error,
numerical grid interpolation (5 points/A) of the DFT orbitals [22], finite system size, the
number of configurations used, and the DMC time step. All of the controlled approxima-
tions are converged to within at least 1 milli-Ha/SiO2. Finite size errors are reduced by
using a model periodic Coulomb Hamiltonian [24] while simulating cell sizes up to 72 atoms
(2×2×2) for quartz, 162 atoms (3×3×3) for stishovite/CaCl2, and 96 atoms (2×2×2) for
α-PbO2. Additional finite size errors are corrected due to insufficient k-point sampling
based on a converged k-point DFT calculation. The uncontrolled approximations include
the pseudopotential, nonlocal evaluation of the pseudopotential, and the fixed node approx-
imation. These errors are difficult to estimate, but the scheme of Casula [28] minimizes the
nonlocal pseudopotential error and some evidence suggests the fixed node error may be
small [81, 19].
79
5.5 Results
This section presents and discusses all of the results produced from the silica free energy
QMC and DFT calculations: Helmholtz Free energy, Equation of State and Vinet Fit Pa-
rameters, Phase Stability, Thermodynamic parameters, and stishovite shear constant soften-
ing. Thermodynamic parameters computed include the bulk modulus, pressure derivative of
the bulk modulus, thermal expansivity, heat capacity, percent change in volume, Gruneisen
parameters, and the Anderson-Gruneisen 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 com-
puted 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 con-
struction 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 )
ξ2[1 + [ξ(1− x)− 1] exp [(1− x)]] , (5.8)
where E0 and V0 are the zero pressure equilibrium energy and volume, respectively, x =
(V/V0)1/3 and ξ = 32(K ′0 − 1)), K0(T) is the bulk modulus, and K′0(T) is the pressure
derivative of the bulk modulus. The subscript 0 indicates zero pressure. E0, V0, K0, and
K′0 are the four fitting parameters. Pressure is obtained analytically as
P (V, T ) =[
3K0(T )(1− x)x2
]exp [ξ(1− x)] . (5.9)
Figures 5.4, 5.5, 5.6,and 5.7 show the four Vinet fit parameters as a function of tem-
perature. Vinet fits are made for free energy isotherms in increments of 5 K and the zero
pressure fit parameters (free energy, F0, volume, V0, bulk modulus K0, and pressure deriva-
tive of the bulk modulus, K ′0) from each fit form the curves in the plots. The gray shading
of the QMC curves indicates one standard deviation of statistical error from the Monte
Carlo data. The QMC results generally agree well with diamond anvil-cell measurements
at room temperature, as do the corresponding DFT calculations using the GGA functional
of Wu and Cohen (WC) [115].
82
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 setsare near the melting temperature. Gray shaded curves are QMC results, with the shadingindicating one-sigma statistical errors. The dashed lines are DFT results using the WCfunctional. 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 QMCenergy for the static lattice to the WC phonon energy. Vinet fits are made for free energyisotherms in increments of 5 K and the zero pressure free energy parameter from each fitforms the curve in the plot. QMC results are represented by gray shaded curves, indicatingone-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 QMCenergy for the static lattice to the WC phonon energy. Vinet fits are made for free energyisotherms in increments of 5 K and the zero pressure volume parameter from each fit formsthe curve in the plot. QMC results are represented by gray shaded curves, indicatingone-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 pressurefor (a) quartz, (b) stishovite and (c) α-PbO2. The QMC energies are constructed by addingQMC energy for the static lattice to the WC phonon energy. Vinet fits are made for freeenergy isotherms in increments of 5 K and the zero pressure bulk modulus parameter fromeach 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, K′0 as a function of temperature at zeropressure for (a) quartz, (b) stishovite and (c) α-PbO2. The QMC energies are constructedby adding QMC energy for the static lattice to the WC phonon energy. Vinet fits are madefor free energy isotherms in increments of 5 K and the zero pressure K′ parameter fromeach fit forms the curve in the plot. QMC results are represented by gray shaded curves,indicating one-sigma statistical error. WC results are represented by red dashed lines.
87
5.5.3 Phase Stability
A phase transition occurs at the pressure where the Gibbs free energy (or enthalpy at T=0)
of two phases is equal, or, equivalently where the difference in Gibbs free energy changes
sign. The Gibbs free energy is given by
G(V, T ) = Ustatic(V, T )− TSvib(V, T ) + P (V, T )V (P, T ). (5.10)
At phase equilibrium (i.e. at the transition pressure), PT),
Gphase1(PT, V1(PT)) = Gphase2(PT, V2(PT))
or
F1(PT) + PTV1(PT) = F2(PT) + PTV2(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. Sta-
tistical uncertainty in the energy differences determines how well phase boundaries are
constrained. The one-sigma statistical error on the QMC enthalpy difference is 0.5 GPa
for the quartz-stishovite transition and 8 GPa for the CaCl2-α-PbO2 transition. The er-
ror 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 sta-
ble 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 exper-
imental determinations of this transition (Figure 5.9). The experimental variability may be
due to the difficulty in demonstrating rigorous phase transition reversals as well as pressure
and temperature gradients and uncertainties in state-of-the-art experiments.
Figure 5.9a compares QMC and DFT predictions with measurements for the quartz-
stishovite phase boundary. The QMC boundary agrees well with thermodynamic mod-
eling 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 refer-
ence. 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 tran-sitions. The DMC transition pressures are 6.4(2) GPa and 88(8) GPa for quartz-stishoviteand CaCl2-α-PbO2, respectively. Gray shaded curves are QMC results, with the shadingindicating one-sigma statistical errors. The dashed, dot-dashed, and dotted lines are DFTresults using the WC, PBE, and LDA functionals, respectively.
90
Figure 5.9: (a) Computed phase boundary of the quartz-stishovite transition. The grayshaded 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 representshock [118, 119] analysis, while dotted and dash-dash-dot lines represent thermochemicaldata (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 statisticalerrors. 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 blueline are diamond-anvil-cell measurements (Exp.) [89, 90]. The dash-dot-dot line is theboundary inferred from shock data [100]. The vertical light blue bar represents pressuresin the D region. Circles drawn on the geotherm [123] indicate a two-sigma statistical errorin 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 erroron F0 is 0.0002 Ha/SiO2.Phase quartz stishovite α-PbO2
Thermal equations of state facilitate the computation of all desired thermodynamic pa-
rameters. 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, K′0, thermal expansivity,α (K−1),
heat capacity, Cp/R, Gruneisen ratio, γ, volume dependence of the Gruneisen ratio, q, and
the Anderson-Gruneisen parameter, δT , for the quartz, stishovite, and α-PbO2 phases of
silica. QMC generally offers excellent agreement with experiment for each of these param-
eters.
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
Pth = P (V, T )− P (V, T0) =∫ T
T0
αKTdT, (5.11)
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) stishoviteand (e,f) α-PbO2as functions of volume and temperature. Experiments [112, 126] comparefavorably with quartz and stishovite calculations. QMC results are represented by grayshaded curves, indicating one-sigma statistical error. WC results are represented by reddashed 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 pressurefor (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. QMC results are represented by grayshaded curves, indicating one-sigma statistical error. WC results are represented by reddashed lines.
96
Bulk Modulus
Figure 5.12 shows the calculated temperature and pressure dependencies of the bulk mod-
uli [109] of quartz, stishovite, and α-PbO2. The bulk modulus is denoted as
KT = V
(∂2F
∂V 2
)TT
= −V(∂P
∂V
)T
. (5.12)
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 temperatureand pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well withavailable 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 ′ = (∂KT/∂P )T . It is an important quantity in many thermodynamic expres-
sions and also occurs as a parameter in most universal equations of state. K′ is a unitless
quantity and shows only slight variation with both temperature and pressure.
99
(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 bulkmodulus,K ′, 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 arerepresented by gray shaded curves, indicating one-sigma statistical error. WC results arerepresented by red dashed lines.
100
Thermal Expansivity
Figure 5.14 shows the computed QMC and WC thermal expansivity [109] for quartz, stish-
ovite, and α-PbO2. Thermal expansivity is denoted as
α = − 1V
(∂2F
∂T∂V
)/
(∂2F
∂V 2
)T
=1V
(∂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 temperatureand pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well withavailable experimental data [130, 131, 132, 126, 133, 134, 135, 136]. QMC results arerepresented by gray shaded curves, indicating one-sigma statistical error. WC results arerepresented 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
)V
(5.14)
or
Cp =(∂H
∂T
)P
. (5.15)
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 experi-
ment [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 andpressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well with availableexperimental data [130, 126]. QMC results are represented by gray shaded curves, indicatingone-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-stishoviteand (B) CaCl2-α-PbO2 transitions. Gray shaded curves are QMC results, with the shadingindicating one-sigma statistical errors. The dashed lines are DFT results using the WCfunctional.
106
Gruneisen Parameters
The Gruneisen ratio [109], γ, quantifies the relationship between thermal and elastic proper-
ties of a solid. It is a very important parameter to Earth scientists because it sets limitations
for thermoelastic properties of the mantle and core of Earth, the adiabatic temperature
gradient, and the geophysical interpretation of shock (Hugoniot) data. γ is approximately
constant, dimensionless parameter that varies slowly with pressure and temperature [137].
Experimental measurement of γ is difficult, and accurate calculations are useful for con-
straining possible values.
γ has both microscopic and macroscopic definitions. The microscopic definition is based
on the volume dependence of the ith phonon mode of the crystal lattice, and is given by:
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 =αKTV
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) Gruneisen ratios, providing ac-
curate values to help constrain experiments. The computations show reasonable agreement
with quartz data [125] available at low pressures. In general, γ initially decreases with pres-
sure 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 Gruneisen ratio with temperatureand pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. Results agree well withavailable 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) stishoviteand (e,f) α-PbO2. Results agree well with available experimental data [125]. QMC resultsare represented by gray shaded curves, indicating one-sigma statistical error. WC resultsare represented by red dashed lines.
110
Anderson-Gruneisen Parameter
The Anderson-Gruneisen parameter [109], δT, which characterizes the relationship between
thermal expansivity and pressure is defined as
δT =(∂lnα∂lnV
)T
=−1αKT
(∂KT
∂T
)P
. (5.19)
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-Gruneisen parameter withtemperature and pressure for (a,b) quartz, (c,d) stishovite and (e,f) α-PbO2. QMC resultsare 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 discontinuitiescorrespond to major compositional transitions inside Earth [138].
114
The bulk sound velocity is defined as
VBS =[KS
ρ
] 12
, (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 Gruneisen 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 D′′ layer, localized quantities of α-PbO2 may be visible seismically
if the difference in bulk sound velocity and density is large.
Figure 5.21 shows the QMC predicted bulk sound velocity and density as a function
of pressure for room temperature and for a typical lower mantle temperature near the
base. The pairs of lines in the bulk sound velocity plot indicate the one-sigma error bars
on the QMC calculations. With respect to postperovskite, MgSiO3, (the dominate D′′
material) measurements [139, 140] at 120 GPa in D′′, QMC predicts α-PbO2 has 12% lower
density and 67% larger bulk sound velocity, which may provide enough contrast to be seen
seismically if present in appreciable amounts.
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(a) Quartz
(b) Quartz
Figure 5.21: QMC calculations of the variation in (a) bulk sound velocity and (b) densitywith pressure for CaCl2 and α-PbO2. Profiles are plotted at both room temperature and themantle-base temperature and compared with the experimental values of perovskite/post-perovskite at the base of the mantle. The pairs of lines in the bulk sound velocity plotindicate one-sigma statistical error.
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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
∆EV
= −pεii +12
(cijkl −
12p (2δijδkl − δilδjk)
)εijεkl, (5.21)
where E is the free energy, V is the volume, εij 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
∆EV
=12cijklεijεkl. (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 =1V
∂2E
∂εij∂εkl. (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
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c11 − c12 is given by
ε =
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 R′ corresponding to a structure with the same volume as follows:
R′ =
I +
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.
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(a) Quartz (b) Quartz
(c) Stishovite
Figure 5.22: Computed QMC energy versus b/a strain with ion positions optimized by WCat 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 =1
2V∂2E
∂δ2, (5.26)
where ∂2E∂δ2
is the curvature of a polynomial fit to the DFT or QMC energy versus strain
data.
Figure 5.23 shows the shear softening of c11 − c12 as a function of pressure at zero tem-
perature. 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 calcu-
lated results. However, discrepancies can arise in the experimental analysis depending on
the strain model used. Recent Brillouin scattering data [97] agrees well with DMC. Accu-
rate computation of the QMC total energies on a small strain scale is very computationally
expensive, requiring roughly 100-1,000 times more CPU time than a corresponding DFT
calculation. The QMC calculations for this feasibility test require over 3 million CPU hours,
which NERSC made available during alpha and beta testing of their Cray-XT4 (Franklin).
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Figure 5.23: Softening of the c11-c12 shear constant for stishovite with pressure. Downtriangles and circles are the DMC and VMC results, respectively. Diamonds and up trianglesrepresent DFT results within the WC and LDA [93], respectively. Squares represent radialX-ray diffraction data [94] and stars represent Brillouin scattering data [97]. The shearconstant in all methods softens rapidly with increasing pressure and becomes unstable near50 GPa, signaling a transition to the CaCl2 phase.
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5.6 Geophysical Implications
The QMC CaCl2-α-PbO2 boundary indicates that the transition to α-PbO2, within a two-
sigma confidence interval, occurs in the depth range of 2,000-2,650 km (86122 GPa) and
in the temperature range of 2,300-2,600 K in the lower mantle. This places the transition
50650 km above the D′′ layer, a thin boundary surrounding Earth’s core ranging from a
depth of ∼2,700 to 2,900 km [142]. The DFT boundaries all lie within the QMC two-sigma
confidence interval, with PBE placing the transition most near the D′′ layer. Free silica in
D′′, such as in deeply subducted oceanic crust or mantlecore reaction zones, would have the
α-PbO2 structure. However, based on QMC results, the absence of a global seismic anomaly
above D′′ suggests that there is little or no free silica in the bulk of the lower mantle. The
α-PbO2 phase is expected to remain the stable silica phase up to the coremantle boundary.
Bulk sound velocity and density profiles indicate that the transition should be seismically
visible for large enough concentrations.
5.7 Conclusions
This chapter has presented QMC (using WC orbitals) computations of silica equations of
state, phase stability, and elasticity. This work provides highly accurate values for thermal
properties for silica and expands the scope of QMC by studying the complex phase transi-
tions in minerals away from the cubic oxides. The DMC zero temperature quartz-stishovite
transition pressure is 6.4(2) GPa and the QMC zero temperature CaCl2-α-PbO2 transition
pressure is 88(8) GPa. Results show the CaCl2-α-PbO2 transition is not associated with
the global D′′ discontinuity, indicating there is not significant free silica in the bulk lower
mantle. However, the transition should be detectable in deeply subducted oceanic crust.
A number of thermodynamic properties are computed by combining QMC with DFT pho-
non energies. The QMC thermodynamic parameters and their dependence on pressure and
temperature agree well with experimental data. QMC also provides an accurate description
of shear constant softening in stishovite.
Additionally, results document the improved accuracy and reliability of QMC relative
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to DFT. As expected, LDA is the worst for predicting properties based on energy differ-
ences 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 prop-
erties 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 re-
sults 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 func-
tional for the given problem is only identified after comparison with experiment. Reliable
predictive power is needed when desired properties are challenging for experiments to mea-
sure. 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 geome-
tries 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
and PBE0 functionals; ABINIT plane-wave opium-pseudopotential calculations using LDA,
PBE, PW91, PBEsol, and WC functionals; VASP plane-wave PAW-pseudopotential calcu-
lations 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. Pointsare 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 com-
puted 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 func-
tional 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 exchange-correlation 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 exchange-correlation functionals and codes.
137
6.4.3 Equilibrium Quartz and Stishovite Volume from Vinet Fits
Figures 6.5 (a) and (b) show a comparison of the quartz and stishovite zero pressure volumes,
respectively; The volumes are those estimated from the minimum of the Vinet fit to the
energy versus volume data for all of the exchange-correlation functionals and codes used.
The solid horizontal black line indicates the range of experimental data, while the dashed
box (quartz) and line (stishovite) indicates the one-sigma statistical error in the QMC
prediction.
The LDA, PBE, PW91 results are fairly uniform across all types of codes, indicating the
calculations are individually converged and the codes are performing on par with each other.
LDA tends to slightly underestimate the volume; PBE and PW91 tend to overestimate;
PBEsol and WC match experiment very well; The hybrids B3LYP and PBE0 overestimates
the quartz volume, but match experiment for the stishovite volume. The screened hybrid
HSE also slightly over estimates the quartz volume and matches experiment for stishovite.
In general the PBEsol, WC, LDA, and HSE functionals all perform well for the volume.
138
(a) Quartz
(b) Stishovite
Figure 6.5: Zero pressure volumes of (a) quartz and (b) stishovite from Vinet fits of energyversus volume data using various exchange-correlation functionals and codes. The solidhorizontal black line indicates the range of experimental data, while the dashed box or lineindicates 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 stish-
ovite 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 ofenergy versus volume data using various exchange-correlation functionals and codes. Thegray shaded box indicates the range of experimental data, while the dashed box indicatesthe one-sigma statistical error in the QMC prediction.
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6.4.5 Equilibrium Quartz and Stishovite K′0 from Vinet Fits
Figures 6.7 (a) and (b) show a comparison of K′0 for quartz and stishovite, respectively;
The K′0 values are those from the Vinet fits to energy versus volume data. The gray shaded
box indicates the range of experimental data, while the dashed box indicates the one-sigma
statistical error in the QMC prediction.
The LDA, PBE, PW91 results are fairly uniform across all types of codes for quartz,
indicating the calculations are individually converged and the codes are performing on par
with each other. In fact, almost all results for both quartz and stishovite are near the
experimentally measured range of values.
142
(a) Quartz
(b) Stishovite
Figure 6.7: Computed pressure derivative of the bulk modulus for (a) quartz and (b) stisho-vite using various exchange correlation functionals and codes. The gray shaded box indicatesthe range of experimental data, while the dashed box indicates the one-sigma statistical er-ror 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 (K′0) for all of the
various DFT functionals and codes. There are two different volumes given in the table: V0FR
corresponds to the fully relaxed, equilibrium DFT geometry and V0 is the zero pressure
volume predicted by the minimum of the Vinet fit to energy versus volume data. V0FR is the
volume that corresponds with the lattice constants a and c. The DFT values are compared
with experiment and QMC. LDA tends to predict both the structural and elastic constants
in close agreement with experiment. LDA lattice constants are slightly underestimated and
bulk moduli tend to be slightly overestimated. All GGA’s and hybrids tend to overestimate
the lattice constants by varying degrees. Among the GGA’s, PBE and PW91 tend to
perform the worst, and PBEsol and WC improve results. Among hybrids, PBE0, and HSE
offer good agreement with experiment.
6.4.6 Enthalpy Versus Pressure
Figure 6.8 shows the HSE enthalpy curves for quartz and stishovite as one example. Once
the energy is know as a function of volume, the enthalpy as a function of pressure is easily
computed as H = U + PV. Enthalpy is used for analysis of the phase stability. The
quartz-stishovite transition is determined by the crossing of enthalpy curves. For HSE, the
stishovite enthalpy curve becomes more stable after a pressure of 5.9 GPa, marking the
transition pressure.
144
Table 6.1: Computed DFT lattice constants, volume, bulk modulus, and pressure derivativeof the bulk modulus for quartz at zero pressure. Parameters are compared with experimentand QMC. Lattice constants are in units of Bohr, volumes are in units of Bohr/SiO2, andthe bulk modulus is in units of GPa.
Table 6.2: Computed DFT lattice constants, volume, bulk modulus, and pressure derivativeof the bulk modulus for stishovite at zero pressure. Parameters are compared with experi-ment 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.
Figure 6.8: Computed HSE enthalpy curves of quartz and stishovite as a function of pres-sure.
147
6.4.7 Quartz-Stishovite Transition Pressures
Figure 6.9 shows a comparison of the transition pressures for all of the exchange-correlation
functionals and codes used. The gray shaded box indicates the range of experimental data,
while the dashed box indicates the one-sigma statistical error in the QMC prediction.
The first feature to note is that the LDA, PBE, PW91 results are fairly uniform across all
types of codes. This indicates the calculations are individually converged and the codes are
performing on par with each other. The results indicate that the local LDA functional fails
to predict that quartz is the correct ground state in the CRYSTAL and VASP codes, while
the ABINIT code predicts a very small transition pressure. Among the semi-local GGA
functionals, PBE and PW91 agree well with experiments in all codes, while PBEsol and
WC significantly underestimate the transition pressure. Among the hybrids, B3LYP vastly
overestimates the transition pressure, while PBE0 and HSE agree well with experiment and
QMC.
The LDA and GGA results agree with that of Hamann [30]. LDA is expected to perform
worse for the transition pressure because of the stark difference in structure between quartz
and stishovite. GGA improves the energy difference because gradient dependence of the
charge density in the functional estimates the 4- to 6-fold coordination change better. It is
not clear why PBEsol and WC underestimate the transition pressure. Hybrid functionals
likely perform well (save B3LYP) due to description of exact exchange, and correlation
energy should be small in these systems.
148
Figure 6.9: Computed quartz-stishovite transition pressures using various exchange-correlation functionals and codes. The gray shaded box indicates the range of experimentaldata, 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
and hybrid (B3LYP, PBE0, and HSE) functionals. All electron calculations are compared
against projector augmented wave (PAW) pseudopotentials and nonlocal, norm-conserving
pseudopotentials. In addition, results from the CRYSTAL, ABINIT, and VASP codes are
compared.
The LDA functional tends to perform well for structural properties and elastic constants.
The PBE and PW91 GGA functionals tend to overestimate the structural properties and
by 2-3% and underestimate elastic properties by roughly 5%. The PBEsol and WC GGA
functionals significantly improve agreement with experiment. However, among the LDA and
GGA functionals, only PBE and PW91 compute the quartz-stishovite transition pressure
(energy difference) well. LDA is likely fails due to the large coordination change in going
from quartz to stishovite and GGA is expected to improve due to gradient dependence
of the density. The Hybrids tend to improve over PBE and PW91 for structural and
elastic properties. B3LYP significantly overestimates the transition pressure. HSE and
PBE0 results are generally similar and in good agreement with experiment and QMC for
all properties considered. HSE provides the most consistently accurate results and offers a
relatively efficient, yet accurate alternative to QMC.
150
Chapter 7
Conclusions
7.1 Summary
Quantum mechanics provides an exact description of microscopic matter, but an exact
solution of the fully interacting, many-electron Schrodinger equation is intractable. The
Coulomb interaction, responsible for electron exchange-correlation energy, can not be sep-
arated to simplify the many-body problem. This means the wave-function can not legiti-
mately be written as a product of single electron wave-functions, but such a wave-function
is sometimes a good place to start building approximate solutions. In order to accurately
compute, predict, and understand properties of matter, approximate solutions must be
sought.
A reasonable approach to solve the many-body Schrodinger equation is to use a mean-
field-based method in which each electron feels the average potential of all other electrons.
The simplistic Hartree approximation gives an approximate solution for a wave-function
represented by a simple product of one-electron functions. The Hartree-Fock approxima-
tion 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 func-
tional theory (DFT) (Chemistry Nobel Prize 1998) succeeds in exactly mapping the many-
body problem onto an independent electron problem with an effective one-electron potential
depending only on the electron density. However, in practice, the functional for exchange-
correlation must be approximated for real materials. The approximated functionals lead to
151
unreliable results on occasion. Such problems drive development of non-mean-field based
approaches in search of even better accuracy and reliability.
Quantum Monte Carlo (QMC) is a method which abandons the mean-field approach,
and is among the class of methods providing many-body solutions to Schrodinger’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 ex-
pense 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 fail-
ures, 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 Ia2 and Ia3 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 D′′ discontinuity, indicating there is not
significant free silica in the bulk lower mantle. However, the transition should be detectable
in deeply subducted oceanic crust if free silica is at high enough concentrations.
The third project (Chapter 6) investigates whether or not hybrid functionals are capa-
ble of benchmark accuracy for quartz and stishovite silica. Results compare performance of
various exchange-correlation functionals, basis sets, pseudopotentials, and codes, which are
benchmarked against QMC and experiments to determine which are most accurate. Cal-
culations compare local (LDA), semi-local GGA (PBE, PW91, PBEsol, WC), and hybrid
(B3LYP, PBE0, and HSE) functionals. All electron calculations are compared against pro-
jector augmented wave (PAW) pseudopotentials and nonlocal, norm-conserving pseudopo-
tentials. 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 Schrodinger equation is complex and cumbersome to solve.
Standard (LDA, GGA) DFT offers a powerful approximate solution, but functionals occa-
sionally 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 compu-
tationally 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 primi-
tive cells. Its structural phase transitions are found to be consistent with several seismic
discontinuities with increasing depth in the mantle. At a depth of 410 km (12 GPa), the
ambient orthorhombic phase called forsterite transforms to another orthorhombic phase
called wadsleyite. At 520 km (15 GPa) wadsleyite transforms to cubic ringwoodite. At
660 km (25 GPa), ringwoodite transforms to a perovskite structure, and at 2740 km (125
GPa) some evidence indicates a post-perovskite structure forms, consistent with seismic
data from the D” boundary. DFT typically provides lattice constants, bulk moduli, elastic
constants, and sound velocities with in a few per cent of experimental measurements. How-
ever, the transition pressures can vary as much as 50% between LDA and GGA, bracketing
experiment [158]. QMC and hybrid DFT can help constrain the transition pressure.
Preliminary tests on Mg2SiO4 and MgO have revealed a large unexpected inflation in
the variance of the local energies occurs when Mg and O are paired together in a solid.
Investigations into the source of the inflation indicate that the standard numerical orbital
approximation, which only interpolates the first derivatives of the DFT orbitals, is not
accurate enough to converge both energy and variance. The MgO system requires additional
interpolation of the Laplacian of the orbitals in order to also converge the variance. A general
understanding of this potential pitfall for all complex solids still needs to be determined.
156
Capability to interpolate the Laplacian also must be added to the production QMC code,
CASINO.
Lastly, a possible, important new project could be the design and construction of new
exchange correlation functionals based on QMC calculations of solids. The LDA functional
is based on QMC data of the free electron gas. With QMC calculations of large, complex
materials now feasible, the study of exchange-correlation in real systems may allow the
development of new functionals that are even better and more efficient than hybrids. Both
of the projects mentioned here would be important for advancing the field of computational
materials science.
157
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168
Appendix A
Error Propagation in QMCThermodynamic Parameters
QMC computes total ground state energies, which have a statistical error bar. The error bar
propagates in to all quantities computed from the QMC energies. There are two methods
for propagation of error: 1) analytic Taylor series expansion or 2) Monte Carlo.
A.1 Taylor Expansion Method
The Taylor expansion method [159, 160] of propagating error is typically only expanded to
first order. The methods assumes there is a function
x = f(u, v, . . .), (A.1)
with an average value (assuming maximum likelihood)
x = f(u, v, . . .). (A.2)
The uncertainty in the function x is found by considering the spread from combining mea-
surements of ui, Vi, ect, into xi’s.:
xi = f(ui, vi, . . .), (A.3)
which has a variance of
σ2x =
1N
∑(xi − xi)2 (A.4)
169
In order to estimate the error, x is Taylor expanded about the mean values of the
parameters (u, v, . . .) to first order:
xi − x =[∂x
∂u
](ui − u) +
[∂x
∂v
](vi − v) (A.5)
Plugging this expression into the variance (Equation A.4) produces the error propagation
equation:
σ2x = σ2
u
(∂x
∂u
)2
+ σ2v
(∂x
∂v
)2
+ · · ·+ 2σ2uv
(∂x
∂u
)(∂x
∂v
), (A.6)
The rightmost term is zero of the dependent variables are completely independent. Code-
pendent variables are possible in complex systems, requiring higher order mixed terms.
The error propagation can be succinctly written as
σ2x =
∑(∂x
∂u
)(∂x
∂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 GaussianBasis 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 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
O S
61420.0000000 -0.0000200
9199.0000000 -0.0001590
2091.0000000 -0.0008290
590.9000000 -0.0035080
192.3000000 -0.0121560
172
69.3200000 -0.0362610
26.9700000 -0.0829920
11.1000000 -0.1520900
4.6820000 -0.1153310
O S
1.4280000 1.0000000
O S
0.5547000 1.0000000
O S
0.2067000 1.0000000
O P
63.4200000 0.0060440
14.6600000 0.0417990
4.4590000 0.1611430
O P
1.5310000 1.0000000
O P
0.6500000 1.0000000
O P
0.3000000 1.0000000
O D
3.7750000 1.0000000
O D
1.3000000 1.0000000
O D
0.4440000 1.0000000
O F
2.6660000 1.0000000
O 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 0.398056D+00
Si S
513000.0000000 0.178068D-05
76820.0000000 0.138148D-04
17470.0000000 0.730005D-04
4935.0000000 0.307666D-03
1602.0000000 0.112563D-02
574.1000000 0.358435D-02
221.5000000 0.101728D-01
90.5400000 0.237520D-01
38.7400000 0.443483D-01
16.9500000 0.419041D-01
6.4520000 -0.502504D-01
2.8740000 -0.216578D+00
1.2500000 -0.286448D+00
Si S
0.3599000 1.0000000
Si S
0.1699000 1.0000000
Si P
1122.0000000 0.448143D-03
266.0000000 0.381639D-02
85.9200000 0.198105D-01
32.3300000 0.727017D-01
13.3700000 0.189839D+00
5.8000000 0.335672D+00
2.5590000 0.379365D+00
1.1240000 0.201193D+00
Si P
175
1122.0000000 -0.964883D-04
266.0000000 -0.811971D-03
85.9200000 -0.430087D-02
32.3300000 -0.157502D-01
13.3700000 -0.429541D-01
5.8000000 -0.752574D-01
2.5590000 -0.971446D-01
1.1240000 -0.227507D-01
Si P
0.3988000 1.0000000
Si P
0.1533000 1.0000000
Si D
0.3020000 1.0000000
Si D
0.7600000 1.0000000
Si F
0.2120000 1.0000000
Si F
0.5410000 1.0000000
END
176
Appendix C
Details of the Ewald and MPCInteraction 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
are N total charges and the system as a whole is electrically neutral.∑N
i=1 zi = 0 The
electrostatic potential energy of such a system is given by
UCoul =12
N∑i=1
ziφ (ri) , (C.1)
177
where φ (ri) is the electrostatic potential at the position of the ith ion:
φ (ri) =∑~j,~n
′ zjrij + ~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 con-
tribution 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√2/α:
ρGauss = −zi(α/π)3/2exp(−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∑j=1
∑~n
zj(α/π)3/2exp[−α |~r − (~rj + ~nL)|2
](C.4)
The electrostatic potential, φ1, due to ρ1 is computed via Poisson’s equation:
−∇2φ1(r) = 4πρ1(r) (C.5)
or, more conveniently, in Fourier form,
k2φ1(k) = 4πρ1(k). (C.6)
179
Fourier Transforming the charge density ρ1 yields
ρ1(~k) =1V
∫Vd~r exp(−i~k · ~r)ρ1(~r) (C.7)
=1V
∫Vd~r exp(−i~k · ~r)
N∑j=1
∑~n
zj(α/π)3/2exp[−α |~r − (~rj + ~nL)|2
](C.8)
=1V
∫allspace
d~r exp(−i~k · ~r)N∑j=1
zj(α/π)3/2exp[−α |~r − ~rj |2
](C.9)
=1V
N∑j=1
zjexp(−i~k · ~rj)exp(−k2/4α) (C.10)
Inserting ρ1(~k) into equation (6), one obtains
φ1(k) =4πk2
1V
N∑j=1
zjexp(−k2/4α), (C.11)
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) =∑~k 6=0
φ1(k)exp(i~k · ~r) (C.12)
=1V
∑~k 6=0
N∑j=1
4πzjk2
exp[i~k · (~r − ~rj)
]exp(−k2/4α). (C.13)
The following is the contribution of φ1(r) to the potential energy:
180
U1 ≡ 12
∑i
ziφ1(ri) (C.14)
=12
∑~k 6=0
N∑j=1
4πzizjV k2
exp[i~k · (~ri − ~rj)
]exp(−k2/4α)
=V
2
∑k 6=0
4πk2
∣∣∣ρ(~k)∣∣∣2 exp(−k2/4α), (C.15)
where we define
ρ(~k) ≡ 1V
N∑i=1
ziexp(i~k · ~ri). (C.16)
Correction for Self-Interaction
As discussed earlier, equation (14) includes a spurious self-interaction term that results
from the interaction of a point charge with the background charge. This term was included
such that the background charge could be Fourier transformed. The extra term is of the
form (1/2)ziφself (ri), and, in this case, the point charge is sitting at the origin of the
Gaussian distribution. To calculate this term, one must compute the potential energy at
the location of the point charge due to the Gaussian charge distribution. The exact charge
distribution in question is
ρGauss = −zi(α/π)3/2exp(−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
−1r
∂2rφGauss(r)∂r2
= 4πρGauss(r) (C.18)
181
or
−∂2rφGauss(r)
∂r2= 4πrρGauss(r). (C.19)
Integrating by parts once yields
− ∂2rφGauss(r)∂r2
=∫ r
∞4πrρGauss(r) (C.20)
= −2πzi(α/pi)3/2
∫ ∞r
dr2exp(−αr2)
= 2zi(α/π)12 exp(−αr2)
Integrating by parts a second time produces
rφGauss(r) = 2zi(α/π)12
∫ r
0drexp(−αr2) (C.21)
= zierf(√αr)
where,
erf(x) ≡ (2/√π)∫ x
0exp(−u2)du (C.22)
Therefore, the potential due to a Gaussian distribution of charge at any point in space
is given by
φGauss(r) =zirerf(√αr). (C.23)
However, the self-interaction term only depends on the potential at r=0:
φGauss(r = 0) = 2zi(α/π)12 . (C.24)
Therefore, the correction to the potential energy (equation (14)) due to the self-interation
is given by
182
Uself =12
N∑i=1
ziφself (ri)
= (α/π)12
N∑i=1
z2i (C.25)
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 :
φshortrange(r) =zir− zirerf(√αr) (C.26)
=zirerfc(
√αr),
where erfc(x) ≡ 1 − erf(x) is the complementary error function. Therefore, the total
contribution of the screened Coulomb interactions to the potential energy is given by
Ushort range =12
N∑i 6=j
zizjerfc(√αrij) (C.27)
Finally, on obtains the total electrostatic contribution to the potential energy by summing
the three terms (equations (15), (25), and (27)):
183
UCoul =12
∑k 6=0
4πVk2
∣∣∣ρ(~k)∣∣∣2 exp(−k2/4α)
− (α/π)12
N∑i=1
z2i
+12
N∑i 6=j
zizjerfc(√αrij)
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 peri-
odic, 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) =1r
+2π3Ω
rT ·D · r + · · · , (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 1r 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) interac-
tion [24, 163, 164, 23, 21] provides a solution to remove the spurious error to the exchange-
correlation energy. In doing so two rules must be respected: 1) MPC should give the Ewald
interaction for Hartree terms and 2) MPC should be exactly 1r 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:
Hee =∑i>j
f(ri − rj) +∑i
∫WS
[V E(ri − r)− f(ri − r
]n(r)dr, (C.29)
where n is the electronic number density, V E is the Ewald potential, and
f(r) =1rm
(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 exchange-
correlation hole.
The MPC interaction does not completely eliminated the many-body exchange-correlation
finte size error. However, in practice, the finite size error converges much more quickly as
a function of system size than when MPC is not used. So, effectively, using MPC allows
one to do solid calculations with smaller simulations cells than would normally be the case.
This allows a dramatic savings in computational time.
185
Appendix D
Summary of CODES Used in ThisWork
D.1 ABINIT
ABINIT is a free plane-wave, pseudopotential DFT code capable of geometry optimization,
phonon calculations, and a number of other electronic properties. Pseudopotentials are user
created, typically using the norm-conserving, nonlocal variety.
D.2 Quantum ESPRESSO
Quantum ESPRESSO is a free integrated suite of computer codes, including the plane-
wave pseudopotential DFT code PWSCF, for electronic-structure calculations and materials
modeling at the nanoscale. It is based on density-functional theory, plane waves, and
pseudopotentials (both norm-conserving and ultrasoft).
D.3 VASP
VASP is a commercial plane-wave, pseudopotential DFT code capable of geometry opti-
mization, phonon calculations, and a number of other electronic properties. VASP is the
first code capable of doing hybrid, HSE calculations using plane-waves.
186
D.4 CASINO
CASINO is the private code of Richard Need’s Cambridge QMC group. It performs VMC,
DMC with most of the latest and greatest developments in QMC, such as backflow, plane-
wave expansion in the Jastrow, MPC, and lots of useful scripts and a well-developed manual
making it user-friendly.The code has a string user base and requires a small cost and per-
mission to use.
D.5 CHAMP
CHAMP is a private QMC code maintained by Cyrus Umrigar at Cornell University. The
code is typically used as a research code to develop new algorithms and techniques for QMC.
D.6 OPIUM
OPIUM is a free nonlocal, normconserving pseudopotential generator. It produces pseu-
dopotentials 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).
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Appendix E
Strong and Weak Scaling in theCASINO QMC Code
The two most widely-used quantum Monte Carlo methods for continuum electronic struc-
ture 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 sin-
gle configuration of electrons to evaluate expectation values stochastically. In DMC, the
Schrodinger 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 communica-
tion 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.
Error decreases as 1/√NMonte Carlo stepsNconfigurations/procNproc. VMC performance scales
almost perfectly with the square of the number of processors as expected.
The parallel efficiency of DMC is sensitive to the number of configurations used per
188
Figure E.1: Log-Log plot of weak scaling in VMC. The time for each job is scaled to providea result with the same statistical error as the largest processor job. Speedup is the ratio ofscaled parallel time to a scaled serial time. VMC calculations scale almost perfectly withthe 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 oc-
casionally 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 pop-
ulation across processors.
189
Figure E.2: Log-Log plot of weak scaling in DMC. The time for each job is scaled toprovide a result with the same statistical error as the largest processor job. Speedup is theratio of scaled parallel time to a scaled serial time. DMC continuously gains speedup as thenumber 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 toprovide a result with the same statistical error as the largest processor job. Speedup is theratio of scaled parallel time to a scaled serial time. DMC calculations become increasinglyinefficient as the number of processors increase because the total number of configurationsis being held fixed.