Pushing the Envelope of Accuracy and Cost in High-Pressure Electronic Structure Theory: Application to elasticity of silica

SEM of stishovite and coesite between crossed polarizers (Caltech)

Kevin Driver (OSU)Carnegie Institution of Washington

August 30th, 2007

Outline

Motivation and Background� The specific challenge is to understand high pressure phases of silica.� Current state-of-the-art computation methods can not make reliable predictions.� In this talk, we explore the feasibility of applying a highly accurate quantumchemistry method to realistic geophysics problems.

Methodology� Electron exchange and correlation are the essence of material properties.� Density functional theory (DFT) tries to succeed in a clever way.� Quantum Monte Carlo (QMC) is a more accurate and reliable alternative to DFT.

Results� QMC has been found to work for Van der Waals interactions, metals, insulators,and band gaps.� In this work, we apply QMC to high pressure elastic constants of silica.� At a factor of 1200 more in cost, QMC finds agreement with DFT and experiment.

Conclusions� QMC will have a lot to offer to geophysics as computational resources advance.

The Big Picture: Pressure-induced phase transitions in silica up to Earth's core

Mapping out phase details is challenging for both experiment and theory.Experiment has challenges with high pressures, pressure/temperature gradients Theory requires accurate (costly) methods to obtain reliable results.

X-ray Diffraction data: (CaCl2) Tsuchida, 1989; (���������� �� -PbO

2) Dubrovinsky, 1997; (Pa3), Kuwayama 2005

Stishovite CaCl2 α-PbO

2Pyrite-type Coesite

Low-quartz a=b≠c a≠b≠c

1)Quantum Monte Carlo - (nearly) exact many-body method - 100-1000 times more costly than DFT

2)Ab initio: Density Functional Theory (DFT) -nobel Prize in Chemistry 1998 – Kohn, Pople -have to “guess” exchange-correlation energy, N3 scaling

3)Semi-empirical methods (experimental input) - results biased towards experimental input -linear scaling

4)Classical/Empirical Modeling -ignore quantum mechanics -linear scaling

The Computational Scientist's Toolbox

Accurate, slow

Inaccurate, fast

Tools for calculating material properties based on electronic structure

QMC

DFT

Semi-empirical

Empirical/ClassicalPotentials

Number of Atoms Simulated

0.0001 eV

0.1 eV

Qualitative

Topological

10 100 10,000 1,000,000

AccuracyElectronic Structure Methods: Accuracy vs. System Size

To accurately calculate electronic properties of solids, we must know:1) Electrons exhibit exchange (Pauli-exclusion) and 2) Correlated motion (electrons stay out of each other's way)

� The governing electronic equation:(3N dimensional equation; is a many-electron wave function)

Early solution methods:No exchange or correlation: Hartree method (very primitive)Exchange only: Hartree-fock (misses total energies by several eV)

H

� � T � V � � E �

The problem of exchange and correlation of electrons

DFT - A efficient, elegant solution to capture both exchange and correlation� Map the many-body problem onto an independent electron problem, with an effective one-electron potential depending only on the electron density.� Ground state properties are obtained by minimizing E[n(r)].E[n(r)] = T[n(r)] + E

iee[n(r)] + E

XC[n(r)]

� The price we pay: EXC

functional is unknown. Good approximations: LDA, GGA

� Unpredictably, the size of the price paid can be very non-uniform ...�

Hohenberg, Kohn (1964); Kohn, Sham (1965), Nobel Prize (1998).

DFT XC-functionals are unreliable

� DFT works very well in many cases, but can unexpectedly fail.� Predicted properties can be highly dependent on form of the XC-functional.� Quartz/Stishovite: LDA works for structural properties, GGA works for energy.� Errors in volume ~5%; errors in elastic constants ~10%.

Quartz-Stishovite Transition Pressure

7 GPa

Bulk Modulus of Quartz

40 GPa

20 GPa

30 GPa

-3 GPa

Hamann, 1996; Demuth, 1999

A More Accurate (and more costly) Method: Quantum Monte Carlo

What is Quantum Monte Carlo?A theory which solves the Schrödinger equation using Monte Carlo integration.� An alternative to DFT when accuracy is paramount.� Explicit many-body method including correlation and exchange from the outset.

What is Monte Carlo?� An efficient was of solving many-dimensional 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

QMCDiffusion

QMCTrial wavefunction Optimize wavefunction

Project out Ground State

Jastrow factor Orbital Determinant

Correlation Exchange

Error ~1N

Variational Quantum Monte Carlo (VMC)

1) Variational principle evaluated using Monte Carlo integration

E 0 trial H trial

E vmc

� 1M

i � 1

M

E L R i

(assume psi normalized)

Sample configurations {R} according to the probability density function. Metropolis algorithm does this efficiently for us in high dimensional spaces. Evaluate EL for each sampled configuration and average the values.

Probability density function Local Energy

E vmc

� 2 HdR � R 2 E L R dR

1M

Error ~

2) Optimize the wavefunction by minimizing the energy or variance of the energyExtremely important for high accuracy; See C. J. Umrigar, PRL (1988), (2005)

(M = samples*cpu's)

Diffusion Monte Carlo (DMC)

V(x)

�0 x

tτ

� DMC is a stochastic projector method for solving the full, many-body Schrödinger equation. � The Schrödinger equation in imaginary time

describes a combination of diffusionand branching of electron configurations.

� Electron configurations are allowed to propagate in imaginary time until theyare distributed according to the ground-statewavefunction of the system.

� Electron configurations with low potentialenergy proliferate, while those with highpotential energy die.

� After sufficient number of imaginary timesteps (τ), the exact ground-state wavefunction is projected out.x

� In practice, well controlled approximations (fixed node, pseudopotential, blips, finitesize) are used to make calculations tractable. (~95% correlation energy recovered)

A few applications of QMC

QMC of Solid Neon Under Pressure: Difference with Experiment

� Van der Waals bonding within DFT is unreliable (functional dependent). � QMC works for Van der Waals interactions

Drummond, Needs PRL, 2006LDA

DMC

GGA(PBE)

Dif

f of p

ress

ure

with

exp

. pre

ssur

e (G

Pa)

Primitive cell volume (a.u.)

Fractional error still small

120 GPa 0 GPa20 GPa

Silicon Diamond to Beta-tin Transition

Metal-insulator transitionDiamond is a insulatorBeta-tin is a metal

� DFT is unreliable due to dependence on functional choice.� QMC is accurate for insulators and metals under pressure.

QMC:Alfe, et al., PRB (2004)Hennig, et al., PRB (2006)

Experiment:Jamieson, Science (1963)and many others

Bulk Silicon Cohesive Energy

Experiment 4.67 eVDMC 4.68(1)eVGGA 4.59 eVLDA 5.28 eV

Single interstitial defect formation energies

Silicon Band GapWilliamson, et al., PRB (1998)

� QMC is accurate for band gaps, anddefect formation energies.� DFT results are nonuniform.

Leung, PRL (1999); Batista, PRB(2006)

Stishovite to CaCl 2 Phase Transition: Elastic instability

� Transition is driven by instability of the elastic shear modulus (c11

-c12

)

Shieh, Duffy, PRL (2002)Kingma, et al., Nature (1995)

� Elastic and Raman softening predicted by LAPW-DFT (Cohen 1991, 1992).� Tetragonal stishovite transforms to orthorhombic CaCl2-type structure near 50 GPa.

Study of Elastic instabilities are important for understanding phase transformations.! Elasticity of silica has particular geophysical relevance for seismic structure." Stishovite is the simplest silicate to exhibit six-fold coordination and serves as a model for study.

B1g

Ag

cijkl

# 1V

$ 2 E

%'& 2

(

EV

) 12

cijkl

*

ij

+

kl

Strain-energy density relation:

Calculate elastic constants by straining the lattice

, Elastic constants obtained from curvature of energy-strain curve- Double well at 280 Bohr3 indicates elastic instability of stishovite. CaCl2 becoming more stable that stishovite under pressure

Feasibility of Elastic Constants in QMC/ Elasticity is a tough problem for QMC: energy differences ~ 0.005 eV0 Extremely expensive to get accurate error bars for large (100 atom) systems1 Through parallel computation on large supercomputers, it's possible to succeed.

Stishovite DFT(WC)R ' 2 I 354 RStrain the lattice:

For a volume conserving strain:

1500 processor hours

QMC Energy vs. Strain Curves: The “Brute Force” Method

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)

6 QMC at this accuracy level is 1200 times more expensive than DFT.7 QMC error bars must be made much smaller than the strain energy differences.8 VMC error bars decrease twice as fast as DMC error bars.9 Highest pressure curves are most difficult to fit and require smallest error bars(work on high pressure curves is still in progress).

QMC elastic constants compared with DFT and experiment

Shieh, et al., X-ray/Strain, 2002.Brazhkin, et al., Brillouin scattering, 2005.Weidner, et al., Brillouin scattering, 1982.R.E. Cohen, LAPW-DFT, 1992.

Experimentalrange

50±5 pressure0±10 c

11-c

12

: DFT (LAPW-LDA and WC) predicts the elastic constant very accurately.; VMC and DMC agree with DFT. (Error bars at 50 GPa still in progress)< All methods extrapolate into the experimental range of the zero-crossing point.

Concluding Remarks and Future ProspectsConclusions= QMC is a powerful method for calculating electronic properties of materials.> Parallel computing resources enable QMC to be applied to realistic materials.

? This work demonstrates feasibility of calculating high pressure properties with QMC.@ Accomplished ~300 calculations using DFT and QMC; over 3 million cpu hours.A As computational power grows, such calculations will become more routine.B QMC is capable of making a significant impact in geophysics in the years to come.

Future Work:C Elasticity of higher pressure silica phases and other deep Earth minerals.D Correlated sampling techniques to improve efficiency of elasticity calculations.E Use QMC to make more accurate density functionals.

Acknowledgements

GuidenceF Ron Cohen (CIW)G John Wilkins (advisor) (OSU)H Richard Hennig (Cornell)

FundingI NSFJ John Wilkins (DOE)

Computational ResourcesK OSC – IBM opteron (1300 processors)L NERSC – Jacquard, Franklin (20,000 processors) - friendly user programM Carnegie Institution of Washington