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Exploring deep Earth minerals with accurate theoryK.P. Driver, R.E. Cohen, Z. Wu, B. Militzer, P. Lopez Rios, M. Towler, R. Needs, and J.W. Wilkins Funding: NSF, DOE; Computation: NCAR, TeraGrid, NCSA, NERSC, OSC, CCNI
Outline:1) Probing minerals in deep Earth (with focus on silica, SiO
2)
2) Simulation methods predict minerals; First-principles (ab initio) methods are best3) Introduction to many-body problem in solids; Quantum Monte Carlo succeeds4) Computation of QMC silica phase diagram and thermodynamic properties
crust: (18 mi, P<5 GPa, T<1000 K), mantle: (1800 mi, P<135GPa, T<2700 K),core: (4000 mi, <300 GPa, T~7000 K (>surface of sun))
What materials comprise Earth?●Seismic waves directly probe Earth's interior.●Diamond anvils and computers simulations infer.
Take away from this talk:An appreciation for firstprinciples simulationsand their usefulness for studying Earth minerals.
Earth 4.5 billion years old 1 GPa = 145,000 lbs/in2
Nuclear Bombsgo here
“restart core spin” = save the world
What would you do for a klondike bar?
(2003)
Compression of silica
●Simplest of Earth's silicates; ubiquitous component of Earth.●Complex series of phase transitions with increasing pressure.●Quartz to stishovite is a fourfold to sixfold coordination change.●Stishovite undergoes a 2nd order transition to a CaCl
2structured phase.
●CaCl2structure transforms to a PbO
2structure, which is stable to the core.
1)Ab initio: Quantum Monte Carlo - (nearly) exact many-body method - 100-1000 times more costly than DFT, N2
2)Ab initio: Density Functional Theory (DFT) -nobel prize in Chemistry 1998 – Kohn, Pople -mean field theory, N3 scaling
3)Semi-empirical methods (experimental input) -results biased towards experimental input -compute time scales linearly with N
atoms
4)Classical/Empirical Modeling -ignore quantum mechanics -compute time scales linearly with N
atoms
Quantum, slow (refine structural data)
Classical, fast (search for structures)
Tools for calculating material properties based on electronic structure:
Zhong, OSU
Hierarchy of Simulation Techniques
1) Electron Exchange Interactions (Fermi Correlation/Hund #1)●the interaction of electrons via the Pauli exclusion principle●largely responsible for the shape/volume of matter
2) Electron Correlation Interactions (Coulomb correlation/Hund #2)●Coulomb interactions cause electrons to stay out of each others way.
e eIn solids, there are Avagadro's number of electrons interacting within essentially continuous bands of quantum states.Is there a clever way to proceed?
DFT cleverly maps the manybody problem onto a singleparticle problem while keeping,the theory exact. In practice, we don't know functionals for exchange and correlation exactly.
DFT's long term illness:
Many-body electron interactions required for accuracy
Fermion's obey Pauli
● DFT works very well in many cases, but can unexpectedly fail.● Predicted properties can be highly dependent on form of the XCfunctional.● Quartz/Stishovite: LDA works for structural properties, GGA works for energy.● DFT errors in volume ~5%; errors in elastic constants ~10%.
(Uninterested people can take a nap here and still pass the quiz at the end)
DFT XC-functionals can be unreliable
What is Quantum Monte Carlo?
What is Quantum Monte Carlo?●A stochastic theory which solves the Schrödinger equation using Monte Carlo integration.●Uses a statistical representation of the wave function explicitly including manybody effects.●1001000 times more computationally expensive than DFT.
What is Monte Carlo?● An efficient way of solving manydimensional integrals (mean value theorem).● Evaluation: Randomly sample the integrand and average the sampled values.Why use Monte Carlo?● Conventional integration methods (e.g. Simpson's rule) use a mesh of points and error in the result falls off increasingly slow with the dimension of the problem.● Statistical error from Monte Carlo is independent of dimension.
DFTVariational
MCDiffusion
MCTrial wave function Optimize wave function
Project out Ground State
=Correlation functionOrbital Determinant
Correlation Exchange
Error ~ 1
N
(Pauli happy!)
What is Quantum Monte Carlo?
What is Quantum Monte Carlo?●A stochastic theory which solves the Schrödinger equation using Monte Carlo integration.●Uses a statistical representation of the wave function explicitly including manybody effects.●1001000 times more computationally expensive than DFT.
What is Monte Carlo?● An efficient way of solving manydimensional integrals (mean value theorem).● Evaluation: Randomly sample the integrand and average the sampled values.Why use Monte Carlo?● Conventional integration methods (e.g. Simpson's rule) use a mesh of points and error in the result falls off increasingly slow with the dimension of the problem.● Statistical error from Monte Carlo is independent of dimension.
DFTVariational
MCTrial wave function Optimize wave function
Project out Ground StateCorrelation Exchange
Error ~ 1
N
(Pauli happy!)
=Correlation functionOrbital Determinant
Use QMC to compute phase diagram and properties of silica
Goals of this work:●Explore feasibility of using QMC for high pressure/temperature properties of silica.●Compute thermal equations of state and phase diagram.●Compute thermodynamic properties of silica.
QMC1)Explicit manybody method.2)Use DFT's relaxed crystal structures.3)Optimize DFT wavefunction (fixed nodes).4)Compute energy stochastically.
DFT1)Singleparticle theory in effective potential.2)Choose XCfunctional and pseudopotential.3)Relax crystal structures.4)Compute energy and wavefunction.
Compute total energies of silica phases at several volumes (pressures)
DFT
QMC Static Energy vs Volume
T=0 K Transition Pressure (GPa)
CaCl2aPbO2Experiment 6 to 7 ~90 QMC 6.3(0.14) 88(8)DFT(WC) 2.1 86
QuartzStishovite
Add in energy due to thermal vibrations (temperature dependence)
F=E staticV F vibrationV ,T
Compute static lattice energy with QMC●Dominant energy contribution●Most accurate method available for solids●CASINO code
Compute vibrational free energy with DFT●Currently too costly for QMC●Vibrational energy is small●Typically well described in DFT●ABINIT, Linear Response, Quasiharmonic
●Compute Helmholtz free energy
Dispersion data from Burkel, et al. Physica B, 263264, pp412415 (1999).Fr
eque
ncy
(cm
1)
Quartz Phonon Dispersion (P=0 GPa)
For anyone who is currently zoned out, this is a good time to wake up!
The theoretical background part is over ... onto the results, whichwill be on the quiz.
x2
A subtle transition slide to the results section
Thermal Equations of State
● Thermal EoS determines fundamentalthermodynamic parameters and phase relations.
● QMC improves agreement with experiment in each phase: quartz, stishovite, PbO
2.
●Only small number of measurementsfor PbO
2, making QMC most accurate
available.
●QMC gives internal estimate of error. Gray shading indicates one standard deviation of statistical error.
P=− ∂F∂V
T
Quartz
Stishovite
PbO2
Silica Phase Boundaries
Pressure (GPa)
Tem
pera
ture
(K)
Tem
pera
ture
(K) ●Equilibrium phase boundaries
computed from Gibbs free energies.
●Metastable quartzstishovitetransition measured with thermocalor shock. QMC agrees well.
●QMC CaCl2PbO
2 transition lies
between two measurements.
●DFT(WCGGA) boundary 4 GPa toolow for quartzstishovite and withinstatistical error of QMC for CaCl
2PbO
2
G=FPV G1PT ,V 1=G2PT ,V 2 at equilbrium
Thermal Expansivity
Temperature (K)
Ther
mal
Exp
ansiv
ity (1
05K
1)
●QMC and DFT(WCGGA) temperature and pressure dependence of .
●QMC shows best agreement for quartz.
●Experimentally, quartz appears at846 K – we only consider quartz.
●QMC and DFT show good agreement with stishovite measurements.
●PbO2 curves are the best available.
=1V ∂V
∂T P
Heat Capacity
Temperature (K)
C p/R
●QMC and DFT(WCGGA) temperature and pressure dependence of C
p.
●QMC and DFT results are nearly identical.
●Good agreement with experiment forquartz and stishovite.
●QMC PbO2 curves are best available.
C p= ∂H∂T
P
Conclusions
●Highly accurate first-principles calculations can be used to compute properties of minerals deep inside of Earth.
●Accurate description of many-body effects known as exchange and correlation are critical for successful prediction.
●QMC is the most accurate method available for computing materials properties, whichexplicitly includes many-body electron interactions.
●QMC has provided highly accurate phase boundaries and thermodynamical properties of silica phases up to the Earth's core.
Backup Slides1)DMC time step convergence2)DMC finite size convergence3)Shear modulus strainenergy technique4)DMC and VMC strainenergy curves5) Shear Modulus vs Pressure6)Statistical error propagation7)Silica enthalpy difference and volume difference8)Silica bulk moduli9)VMC10)DMC
DMC Time Step Convergence
●Time step of 0.003 a.u. is converged to within 30 meV.
DMC Finite Size Convergence
cijkl=1V
∂2 E∂2
EV
=12
cijklijkl
Strainenergy density relation:
● Elastic constants obtained from curvature of energystrain curve● Double well at 280 Bohr3 indicates elastic instability of stishovite● CaCl
2 becoming more stable that stishovite under pressure
Feasibility of Elastic Constants in QMC● Elasticity is a tough problem for QMC: energy differences ~ 0.005 eV● Extremely expensive to get accurate error bars for large (100 atom) systems● Through parallel computation on large supercomputers, it's possible to succeed.
Stishovite DFT(WC)R '=[ I]RStrain the lattice:
For a volume conserving strain:
1500 processor hours
Calculate elastic constants by straining the lattice
1) Take optimized input structures from DFT(WC) (we can't do forces in QMC yet)2) Run QMC on thousands of processors for a few days until error bars are sufficiently smallfor each structure.
VMC (500,000 hrs) DMC (additional 1.3 million hrs)
● QMC at this accuracy level is 1200 times more expensive than DFT.● QMC error bars must be made much smaller than the strain energy differences.● VMC error bars decrease twice as fast as DMC error bars.● Highest pressure curves are most difficult to fit and require smallest error bars(work on high pressure curves is still in progress).
QMC Energy vs. Strain Curves: The “Brute Force” Method
Stishovite Shear Modulusc 11
c12
(GPa
)
Pressure (GPa)
●Stishovite to CaCl2 transition is driven by instability in the elastic shear modulus.
●VMC modulus computed at several pressures and DMC checks at endpoints.●Shear modulus computed from strainenergy relation (brute force 1000 CPU cost of DFT)●All methods roughly agree, with the shear modulus going unstable around 50 GPa.
cij=1V
∂2 E∂2
F=∑i , j
∂F∂C i
∂F∂C j
Cov [i , j ]
Statistical Error Propagation
1) Propagation of error equation from Taylor expanding a function about the mean values of its parmeters:
P=∑i
P i
E i
E i
Allow random Gaussian fluctuations on QMC energies E
i with stdv
Ei
Fit Vinet Equation tonew set of energiesand compute property
actual data set
...Fit Vinet Equation tonew set of energiesand compute property
Fit Vinet Equation tonew set of energiesand compute property
standard deviation of synthetic data sets gives uncertainty in property
Allow random Gaussian fluctuations on QMC energies E
i with stdv
Ei
Allow random Gaussian fluctuations on QMC energies E
i with stdv
Ei
...
2) Propagation with Monte Carlo
Enthalpy Difference and Volume DifferenceEnthalpy Difference (Ha/SiO
2) % Volume Difference
●Enthalpy differences and errors determine equilibrium phase relations.●CaCl
2PbO
2 enthalpy change is not measurable; phases are not quenchable to zero pressure.
●CaCl2 &
PbO
2 enthalpy curves lie very close together compared to quartzstishovite.
●Volume change in quartzstishovite is 20 times larger than in CaCl2PbO
2.
HP
bO2
CaCl
2 (H
a/Si
O2)
Hsti
shq
uartz
(Ha/
SiO
2)
V
CaCl
2P
bO2
V
quar
tzs
tish
Pressure (GPa) Pressure (GPa)
Bulk Modulus
Temperature (K)
Bulk
Mod
ulus
(GPa
)
●QMC and DFT(WCGGA) temperature and pressure dependence of K.
●K decreases linearly with T
●K increases linearly with P
●QMC and DFT generally agree.(T dependence comes from DFT)
K=−V ∂P∂V
T
1) Variational principle evaluated using Monte Carlo integration
E 0⟨trial∣H∣trial⟩
E vmc≈1M∑i=1
M
E L Ri
(assume psi normalized)
● Sample configurations {R} according to the probability density function.● Metropolis algorithm does this efficiently for us in high dimensional spaces.● Evaluate E
L for each sampled configuration and average the values.
Probability density function Local Energy
E vmc=∫∣∣2[ H
]dR=∫∣R∣2 E L RdR
1
MError ~
2) Optimize the wavefunction by minimizing the energy or variance of the energy● Extremely important for high accuracy; See C. J. Umrigar, PRL (1988), (2005)
(M = samples*cpu's)
Variational Quantum Monte Carlo (VMC)
τ
●DMC is a stochastic projector method for solving the full, manybody Schrödinger equation.
● The Schrödinger equation in imaginary timedescribes a combination of diffusionand branching of electron configurations.
● Electron configurations are allowed to propagate in imaginary time until theyare distributed according to the groundstatewavefunction of the system.
● Electron configurations with low potentialenergy proliferate, while those with highpotential energy die.
● After sufficient number of imaginary timesteps (τ), the exact groundstate wavefunction is projected out.
V(x)
0x
t
x
Diffusion Quantum Monte Carlo (DMC)