High Temperature High Pressure Properties of Silica From Quantum Monte Carlo

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1 High Temperature High Pressure Properties of Silica From Quantum Monte Carlo K.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) Physics of silica compression 2) Quantum Monte Carlo (QMC) as a benchmarking tool for first principles lattice dynamics. 3) Computing silica thermal EoS using QMC for static lattice and DFT for vibrations. 4) Compare QMC thermodynamic and elastic properties with DFT and experiment.

2 High-Pressure Silica Physics Simplest of Earth's silicates; free phases may exist at base of mantle. Complex series of phase transitions. Quartz to stishovite is a four fold to six fold coordination change. Stishovite undergoes a ferroelastic transition to a CaCl structured phase. 2 CaCl2 structure transforms to a PbO2 structure, which is stable to the core.

3 Motivation for QMC Computation 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. DFT generally great, but can fail: LDA predicts stishovite as ambient ground state. GGA's may improve transition, but may worsen other properties. Goals of this work: Explore feasibility of using QMC with DFPT for high pressure/temperature properties of silica. Thermal equations of state Thermodynamic properties of silica Elastic constants

4 Density Functional Theory and Quantum Monte Carlo DFT WCGGA functional and potentials ABINIT, Opium PPs 9/6/12 atom cells 1)Single particle theory in effective potential. 2)Choose XC functional and pseudopotential. 3)Relax crystal structures. 4)Compute energy and wavefunction. DFT Vol/BulkMod at T=300 K, P= 0 GPa phase quartz stishovite apbo2 V0 QMC 254(2) 159.0(4) 154.8(1) V0 WCGGA V0 Experiment N/A K0 QMC 32(6) 305(20) 329(4) K0 WCGGA K0 Experiment N/A DFT(WCGGA) V too high, K too low QMC generally improves QMC 1)Explicit many body method. 2)Use DFT's relaxed crystal structures. 3)Optimize DFT wavefunction (fixed nodes). 4)Compute energy stochastically. Finite size errors < 30 mev MPC, 72/162/96 atom simulation cells CASINO, B spline basis, time step Jastrow with 2 & 3 body, PW (44 parameters)

5 Computing P V T Equations of State Compute quasi harmonic Helmholtz free energy QMC Static Energy vs Volume F QHA=E static V F vibration V, T Compute static lattice energy with QMC Dominant energy contribution Most accurate method available for solids CASINO code T=0 K Transition Pressure (GPa) Quartz Stishovite CaCl2 apbo2 Experiment QMC WCGGA 6 to 7 ~90 6.3(0.14) 88(8) Quartz Phonon Dispersion (P=0 GPa) Frequency (cm 1) Compute vibrational free energy with DFT Currently too costly for QMC Vibrational energy is small Typically well described in DFT ABINIT, Linear Response, Quasi harmonic Dispersion data from Burkel, et al. Physica B, , pp (1999).

6 Thermal Equations of State P= Quartz F QHA V T Thermal EoS determines fundamental thermodynamic parameters and phase relations. QMC improves agreement with experiment in each phase: quartz, stishovite, PbO2. Stishovite Only small number of measurements for PbO2, making QMC most accurate available. PbO2 Gray shading indicates one standard deviation of statistical error.

7 Enthalpy Difference (Ha/SiO2) Vquartz stish % Volume Difference VCaCl2 PbO2 H PbO2 CaCl2 (Ha/SiO2) Hstish quartz(ha/sio2) Enthalpy Difference and Volume Difference Pressure (GPa) Pressure (GPa) Enthalpy differences and errors determine equilibrium phase relations. CaCl PbO enthalpy change is not measurable; phases are not quenchable to zero pressure. 2 2 CaCl2 & PbO2 enthalpy curves lie very close together compared to quartz stishovite. Volume change in quartz stishovite is 20 times larger than in CaCl2 PbO2.

8 Silica Phase Boundaries Equilibrium phase boundaries computed from Gibbs free energies. Temperature (K) Metastable quartz stishovite transition measured with thermocal or shock. QMC agrees well. Temperature (K) QMC CaCl2 PbO2 transition lies between two measurements. DFT(WCGGA) boundary 4 GPa too low for quartz stishovite and within statistical error of QMC for CaCl2 PbO2 Pressure (GPa)

9 c11 c12 (GPa) Stishovite Shear Modulus c ij = 1 2 E V 2 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 strain energy relation (brute force 1000 CPU cost of DFT) All methods roughly agree, with the shear modulus going unstable around 50 GPa.

10 Bulk Modulus K = V P V T QMC and DFT(WCGGA) temperature and pressure dependence of K. Bulk Modulus (GPa) K decreases linearly with T K increases linearly with P QMC and DFT generally agree. (T dependence comes from DFT) Temperature (K)

11 Thermal Expansivity Thermal Expansivity (10 5K 1) = 1 V V T P QMC and DFT(WCGGA) temperature and pressure dependence of. QMC shows best agreement for quartz. Experimentally, quartz appears at 846 K we only consider quartz. QMC and DFT show good agreement with stishovite measurements. Temperature (K) PbO2 curves are the best available.

12 Heat Capacity C p= H T P QMC and DFT(WCGGA) temperature and pressure dependence of Cp. Cp/R QMC and DFT results are nearly identical. Good agreement with experiment for quartz and stishovite. Temperature (K) QMC PbO2 curves are best available.

13 Grüneisen Ratio =V P U V = KT V CV change in P due to increase in energy density Grüneisen ratio quantifies relation between thermal and elastic properties. Close agreement with experiment for quartz. QMC stishovite and PbO2 curves are the best available. Pressure (GPa)

14 Conclusions QMC with DFT phonons is a route to benchmark solid ab initio high P/T calculations. QMC gives excellent agreement with experiment for thermal equations of state and thermodynamic properties. QMC gives accurate phase transformation boundaries for quartz-stishovite and CaCl2 PbO2 transitions. QMC computation of shear modulus softening in stishovite confirms the stishovite CaCl2 transition is near 50 GPa at low temperatures. Publication is written, few details remain to be worked out.

15 Backup Slides 1)DMC time step convergence 2)DMC finite size convergence 3)Shear modulus strain energy technique 4)DMC and VMC strain energy curves 5)Statistical error propagation 6)Disordered Silica

16 DMC Time Step Convergence Time step of a.u. is converged to within 30 mev.

17 DMC Finite Size Convergence

18 Calculate elastic constants by straining the lattice Stishovite DFT(WC) Strain energy density relation: Strain the lattice: R ' =[ I ] R For a volume conserving strain: E 1 = c ijkl ij kl V processor hours c ijkl = 1 2 E V 2 Elastic constants obtained from curvature of energy strain curve Double well at 280 Bohr3 indicates elastic instability of stishovite CaCl becoming more stable that stishovite under pressure 2 Feasibility of Elastic Constants in QMC Elasticity is a tough problem for QMC: energy differences ~ ev Extremely expensive to get accurate error bars for large (100 atom) systems Through parallel computation on large supercomputers, it's possible to succeed.

19 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 small for 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).

20 Statistical Error Propagation 1) Propagation of error equation from Taylor expanding a function about the mean values of its parmeters: F = i, j F F Ci C j Cov [i, j ] P = i 2) Propagation with Monte Carlo Pi Ei E i actual data set Allow random Gaussian fluctuations on QMC energies Ei with stdv Ei Fit Vinet Equation to new set of energies and compute property Allow random Gaussian fluctuations on QMC energies Ei with stdv Ei Allow random Gaussian fluctuations on QMC energies Ei with stdv Ei Fit Vinet Equation to new set of energies and compute property Fit Vinet Equation to new set of energies and compute property standard deviation of synthetic data sets gives uncertainty in property......

21 Does silica undergo order disorder transition at high T? Stishovite Stishovite and post stishovite phases can be described as distorted hcp arrays of oxygen ions with octahedral intersticies half filled by silicon ions. Suggests AB alloy in which silicons can be disordered. Goal: get a rough estimate of disorder temperature without doing any significant additional calculations. Approach: 1) Stishovite is unkinked, PbO2 is maximally kinked. Guess a transition structure and compute it's energy in DFT. PbO2 2)Assume we can estimate the enthalpy of a disordered silica phase by weighting three ordered phases (stish, PbO2, and the transition structure) according to their Boltzmann probabilities and including a configurational entropy term. 3) Compute entropy of disordered phase with lattice gas model?