Exploring deep Earth minerals with accurate theory

Transcription

1 Exploring deep Earth minerals with accurate theory 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) Probing minerals in deep Earth (with focus on silica, SiO2) 2) Simulation methods predict minerals; First-principles (ab initio) methods are best 3) Introduction to many-body problem in solids; Quantum Monte Carlo succeeds 4) Computation of QMC silica phase diagram and thermodynamic properties Earth 4.5 billion years old crust: (18 mi, P<5 GPa, T<1000 K), 1 GPa = 145,000 lbs/in2 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 first principles simulations and their usefulness for studying Earth minerals.

2 Nuclear Bombs go here restart core spin = save the world What would you do for a klondike bar? (2003)

3 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 four fold to six fold coordination change. nd Stishovite undergoes a 2 order transition to a CaCl2 structured phase. CaCl2 structure transforms to a PbO2 structure, which is stable to the core.

4 Hierarchy of Simulation Techniques Tools for calculating material properties based on electronic structure: Quantum, slow 1)Ab initio: Quantum Monte Carlo - (nearly) exact many-body method (refine structural data) 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 Natoms Classical, fast (search for structures) 4)Classical/Empirical Modeling -ignore quantum mechanics -compute time scales linearly with Natoms Zhong, OSU

5 Many-body electron interactions required for accuracy 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 Fermion's obey Pauli 2) Electron Correlation Interactions (Coulomb correlation/hund #2) Coulomb interactions cause electrons to stay out of each others way. e e DFT's long term illness: In 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 many body problem onto a single particle problem while keeping, the theory exact. In practice, we don't know functionals for exchange and correlation exactly.

6 DFT XC-functionals can be 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. 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)

7 What is Quantum Monte Carlo? What is Monte Carlo? An efficient way 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. 1 Error ~ Statistical error from Monte Carlo is independent of dimension. N 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 many body effects times more computationally expensive than DFT. DFT Trial wave function Variational Optimize wave function MC = Correlation function Orbital Determinant Correlation Exchange (Pauli happy!) Diffusion MC Project out Ground State

8 What is Quantum Monte Carlo? What is Monte Carlo? An efficient way 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. 1 Error ~ Statistical error from Monte Carlo is independent of dimension. N 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 many body effects times more computationally expensive than DFT. DFT Trial wave function Variational Optimize wave function MC = Correlation function Orbital Determinant Correlation Exchange (Pauli happy!) Project out Ground State

9 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.

10 Compute total energies of silica phases at several volumes (pressures) QMC Static Energy vs Volume DFT 1)Single particle theory in effective potential. 2)Choose XC functional and pseudopotential. 3)Relax crystal structures. 4)Compute energy and wavefunction. DFT QMC 1)Explicit many body method. 2)Use DFT's relaxed crystal structures. 3)Optimize DFT wavefunction (fixed nodes). 4)Compute energy stochastically. T=0 K Transition Pressure (GPa) Quartz Stishovite CaCl2 apbo2 Experiment QMC DFT(WC) 6 to 7 ~90 6.3(0.14) 88(8)

11 Add in energy due to thermal vibrations (temperature dependence) Compute Helmholtz free energy F =E static V F vibration V, T 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 Quartz Phonon Dispersion (P=0 GPa) Frequency (cm 1) Compute static lattice energy with QMC Dominant energy contribution Most accurate method available for solids CASINO code Dispersion data from Burkel, et al. Physica B, , pp (1999).

12 A subtle transition slide to the results section For anyone who is currently zoned out, this is a good time to wake up! x2 The theoretical background part is over... onto the results, which will be on the quiz.

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

14 Silica Phase Boundaries G=F PV G 1 P T, V 1 =G 2 P T,V 2 at equilbrium 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)

15 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.

16 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.

17 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, which explicitly includes many-body electron interactions. QMC has provided highly accurate phase boundaries and thermodynamical properties of silica phases up to the Earth's core.

18 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) Shear Modulus vs Pressure 6)Statistical error propagation 7)Silica enthalpy difference and volume difference 8)Silica bulk moduli 9)VMC 10)DMC

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

20 DMC Finite Size Convergence

21 Calculate elastic constants by straining the lattice Strain energy density relation: Stishovite DFT(WC) 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.

22 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).

23 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.

24 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......

25 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.

26 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)

27 Variational Quantum Monte Carlo (VMC) 1) Variational principle evaluated using Monte Carlo integration E 0 trial H trial E vmc = 2 [ ] H (assume psi normalized) dr= R E L R dr 2 Probability density function Local Energy Sample configurations {R} according to the probability density function. Metropolis algorithm does this efficiently for us in high dimensional spaces. Evaluate E for each sampled configuration and average the values. L E vmc 1 M M E L Ri i=1 Error ~ 1 M (M = samples*cpu's) 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)

28 Diffusion Quantum Monte Carlo (DMC) DMC is a stochastic projector method for solving the full, many body Schrödinger equation. The Schrödinger equation in imaginary time describes a combination of diffusion and branching of electron configurations. V(x) Electron configurations are allowed to propagate in imaginary time until they are distributed according to the ground state wavefunction of the system. τ t Electron configurations with low potential energy proliferate, while those with high potential energy die. x 0 x After sufficient number of imaginary time steps (τ), the exact ground state wave function is projected out.