Quantum Monte Carlo Equations of State of - and-Mg2SiO

4

K. P. Driver and J. W. Wilkins;  Funding: DOE, NSF; Computation: NERSC, OSC, CCNI

Outline:1) Quantum Monte Carlo (QMC) as a benchmarking tool for first principles lattice dynamics. 2) Computing Mg

2SiO

4 PVT-EoS using QMC for static lattice and DFT for vibrations.

3) Compare QMC thermodynamic properties with DFT and experiment.

DFT generally great, but can fail:●LDA underpredicts quartz-stishovite and forsterite-wadsleyite transition by 6 GPa.●GGA's may improve transition, but may worsen other properties.

●Extreme materials research demands pushing the frontier of first-principle simulations.●DFT is current workhorse, but exchange-correlation (XC) functionals leave room for doubt.●QMC avoids XC approximation, but is too expensive replace DFT anytime soon. ●Use QMC as a benchmark tool to spot-check important results.

Introduction and Motivation

Tem

pera

ture

 (K)

0

2000

1000

0 25

Pressure (GPa)6

Figure from:Yu et al., Earth Planet. Sci. Lett. 273, 115 (2008)Recent DFT work:Li et al., J. Geophys. Res. 112, B05206 (2007)Wu et al., J. Geophys. Res. 112, B12202 (2007)

 DFT(LDA)

 DFT(GGA)

12

Goals of this work:●Explore feasibility of using QMC with DFPT for high pressure/temperature phase transitions.

●Check QMC/DFPT performance for first ternary solid oxide Mg

2SiO

4. (Number of QMC calculations on

solid oxides is tiny: (MgO, NiO, MnO, FeO, SiO2)

to Mg2SiO4 

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.

Density Functional Theory and Quantum Monte Carlo

●WCGGA functional and potentials ●small­core Mg potential (Opium code)●28/56 atom cells

●MPC, 112 atom simulation cells, ~1.5 mH FSE●B­spline basis, 0.005 time step●Jastrow with 2 & 3­body, PW (109 paramaters)●~4 million CPU hours(NERSC, CCNI, OSC)

Experiment  123­129 289­292QMC 136(14) 309(1)WCGGA  112 297.8LDA  126.4 289.5

Experiment  160­175 535­539QMC T=0 K 156(12) 544(3.4)WCGGA T=0K 159.5 545.9LDA  166.7 541.35

Forsterite Bulk Modulus (GPa) Volume (Ang3)

Wadsleyite

DFT

Properties at T=300 K, P= 0 GPa

Computing P­V­T Equations of State

FQHA=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 quasi­harmonic Helmholtz free energy

Dispersion data from Lam et al., Am. Mineral 75, 109 (1990)

 T=0 K Transition Pressure (GPa)Experiment 8, 12QMC 10.5 (2.3)WCGGA 8.6PBE 12.6LDA 6.5

QMC Static Forsterite and Wadsleyite EoS

Freq

uenc

y (c

m­1)

Forsterite Phonon Dispersion (P=0 GPa)

Forsterite and Wadsleyite EoS

●QMC equations of state still need work.●QMC volumes and curvature are not yet correct – off by several per cent.●Must reconsider finite size errors, pseudopotential, and DFT structures.●WC­GGA and LDA calculations agree well with experiment.

P=− ∂FQHA

∂V T

Compute pressure from free energy

        Forsterite        Wadsleyite (T=0 only)

LDA ­ Li et al., J. Geophys. Res. 112, B05206 (2007) and Wu et al., J. Geophys. Res. 112, B12202 (2007)

Bulk Modulus Temperature and Pressure Dependence

P=0 GPa

K=−V ∂P∂V

T

●QMC modulus variation with temperature agrees with experiment.●Small QMC variation with pressure reflects nearly constant P(V) slope.●DFT(WC) modulus falls too rapidly with temperature.●DFT(WC) modulus increases with pressure as expected.

Thermal Expansivity and Heat Capacity

●QMC thermal expansivity agrees well with experiment.●DFT(WC) expansivity is too large above room temperature.●Thermal expansivity decreases by 20­50% from P=0 to P=20 GPa.●QMC and DFT heat capacity both agree well with experiment.●Heat Capacity shows only slight variation with pressure.

=1V ∂V

∂T P

C p= ∂H∂T

P

Conclusions

●QMC with DFT phonons is a route to benchmark solid ab initio high P/T calculations.

●QMC solid calculations may be cumbersome and expensive with current resources.(How many DFT calculations did it take to make DFT what it is today?)

●Current QMC Mg2SiO

4 equation of state requires more work/thought to improve.

●QMC thermodynamic property variation with temperature is good despite the EoS.

●QMC thermodynamic property variation with pressure is poorly described due to poor quality of current EoS.