Establishing Quantum Monte Carlo and Hybrid Density Functional Theory as Benchmarking Tools for Complex Solids

Transcription

1 Establishing Quantum Monte Carlo and Hybrid Density Functional Theory as Benchmarking Tools for Complex Solids Kevin P. Driver, Ph.D. Defense, February 1, 011 Outline 1) Introduction: Quantum mechanics describes matter but is difficult to solve. ) Computational Methods: Density functional theory(dft) and quantum Monte Carlo (QMC) provide accurate and computationally affordable solutions to large quantum mechanical systems. 3) Accuracy: Expensive QMC and hybrid DFT provide benchmarks for less expensive DFT methods: Tests of DFT's accuracy provides a route to deal with DFT weaknesses. 4) Applications: Silicon self-interstitial defects and silica are difficult and important real world problems that simultaneously exhibit important technical tests for ab initio methods. 5) Conclusions: In addition to QMC, hybrid DFT(HSE) is a benchmarking standard for DFT. Results establish the feasibility of QMC and HSE to study and benchmark complex materials significantly beyond the scope of previous calculations.

2 Quantum Mechanics Describes Matter But No Exact Solution Quantum Mechanics is a non-deterministic theory that relates wave properties of matter to its energy: A particle is described by a wavefunction (prob amp) r, t in states of discrete energy, E with a probability of being found at point r, at time t of r, t Mathematical description is given by Schrödinger's Wave Equation: N where H is the full Hamiltonian operator: = H 1 mi =E H i i=1 is a N particle (many body) wavefunction. kinetic energy N Zi Z j i j r i r j potential Coulomb energy Coulomb term describes each electron interacts with all others in a system: many-body problem The coulomb term is inseparable, which means simple analytic solutions are not possible. Additionally, each electron has 3 dimensions; total number of dimensions for solids is large! mn >> me: allows separation of electron and nuclear degrees of freedom (Born-Oppenheimer)

3 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 r 1, r = r, r 1 ) Electron Correlation Interactions (Coulomb correlation/hund #) Coulomb interactions cause electrons to stay out of each others way. e e In solids, there are Avogadro's number of electrons interacting within essentially continuous bands of quantum states. Is there a clever way to proceed?

4 Computational Methods: Numerical (Approximate) Solutions to Schrödinger's Equation

5 Hartree Fock: Explicit Exchange, Averaged Correlation Method: consider each electron separately interacting with the average repulsion (mean field potential) of all others; allows Schrödinger's equation to be broken down into a set of one electron equations. Pros: Slater determinant wavefunction form obeys Pauli exclusion: exchange is exactly computed. Cons: Correlation is only accounted for by the average effect of repulsion; also includes self interaction. Mean field potential: U = e dr ' el n r ' r r ' 1 r 1 r 1 electron example: r, r = 1 n r = e i r 1 r r i i r are one electron wavefunctions = 1 r 1 r r 1 1 r includes permutation operator, P1 The electron potential terms in the Hamiltonian become orbital dependent operators: V HF =[ J K ] j r 1 = dr i r r r 1 * j r 1 dr i r i r 1 r r 1 j r Coulomb operator Hartree term V H Guess form of i r construct Self consistent Field solution converges VHF exchange operator V X =T V V H ion HF solve new i r H =E

6 Density Functional Theory: Approximate Exchange and Correlation =E Tot =E unique [ n r ] Hohenberg Kohn: GS H GS GS GS There is a 1 to 1 mapping from psi to n; ngs(r) uniquely determines the GS psi and E; it's the true ngs. Real system Electrons fully interact density, n Energy, E e Kohn Sham Exact mapping e e Fictitious model system Electrons treated independently; interact via Veff same density, if Veff is exact. same energy, E, but functional of n(r) 0 e R, V E TOT [ n]=t [ n] E ion [ n] E H [ n] E XC [ n] known form real unknown e e e e r, V i non interacting KE eff EXC includes all the information that is not known: exchange correlation energy and T T0 Minimizing ETOT[n] wrt density variation gives effective one particle Kohn Sham eqns: 1 H i =[ V eff r ] i r = i r where V eff r =V ion V H V XC DFT is exact for universal V, but it is unknown; VXC must be approximated. XC Guess i r construct n r = e i r Self consistent solution converges n i construct V eff new i r solve H i r = i r

7 Quantum Monte Carlo (QMC): Explicit Exchange and Correlation 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 1 falls off increasingly slow with the dimension of the problem. Error ~ Statistical error from Monte Carlo is independent of dimension. N What is Quantum Monte Carlo? Solution to the Schrödinger equation (SE) using Monte Carlo integration with a many body wavefunction. An alternative to DFT when accuracy is paramount. Explicit many body method including correlation and exchange from the outset. QMC comes in two forms: 1) Variational MC and ) more accurate Diffusion MC Basic outline of a DMC calculation: DFT Trial wavefunction Variational QMC Optimize wavefunction = Jastrow factor Orbital Determinant Correlation Exchange Diffusion QMC Project out Ground State

8 VMC: provides best solution for a fixed wavefunction form. Compute energy via variational principle evaluated using Monte Carlo integration. Sample configurations {R} according to the probability density function (Metropolis algorithm). Evaluate E for each sampled configuration and average the values. L E vmc = R PDF [ ] H R R dr 1 N MC local energy E L N MC E L Ri ± i=1 1 N MC E L E L The VMC Slater Jastrow wavefunction form: R =e J R D R where J(R) is the Jastrow correlation factor and D(R) is a Slater determinant J(R) is a polynomial expansion in particle separation: body (e n, e e) and 3 body (e e n) terms D(R) is made up of single particle electron orbitals (from DFT calculations). Jastrow parameters are optimized by minimizing the energy or variance of the energy. Iterate VMC and VMC optimization until energy change is below desired statistical error. equilibration loop statistics accumulation loop VMC Algorithm random configuration accept trial move reject update e positions accept trial move reject update e positions compute EL optimize Jastrow

9 DMC: projects best solution for fixed wavefunction nodes V(x) DMC transforms Schrödinger's equation to imaginary time diffusion/rate equation ( it): i t t =H =[ 1 V R ] Recast in terms of a Green's function propagator with time step that projects out ground state relative to damped excited states. N 0 x x E DMC = 1 N step N config set up configurations N step N config j=1 k=1 GS = lim N exp H initial 0 j=1 is represented statistically by a population of e configurations which, in each time step, undergo a random diffusion and E L R k, j potential dependent rate(birth/death) processes. DMC the Green's function is the PDF; Possible to sample negative regions. Nodes are fixed to ensure positive PDF. Allow e 's to diffuse for small time steps obeying fixed node and weighted Metropolis. compute rate process probability accumulate averages over all configs increment time step

10 Accuracy Thesis broadly asks two general questions: 1) How accurate are our methods? Efficiency accuracy trade-off. ) Can we solve some important, practical problems?

11 Strengths, Weaknesses, and Approximations in DFT Strength: DFT works great for many materials (thousands of calculations published). Weaknesses: In general, DFT cannot do: 1) Excited states ) Band Gaps 3) van der Waals attractions 4) Energy differences between structures with vastly different interatomic bonding. Source of DFT's weaknesses: el 1) Self interaction error in mean field approach: U = e dr ' n r ' r r ' ) Approximations to exchange-correlation functional (LDA, GGA, meta-gga, Hybrids) Functional performance is sometimes unreliable; There is skepticism amongst the zoo. Jacob's ladder classification systematically includes more and more physical information. Higher rungs of the ladder are not always better. Must find ways to determine the accuracy of these methods. Benchmark methods are needed to ultimately determine the appropriate functional for the job. 3) Pseudopotentials: Construct from atomic calculations; compare tests with all electron before use. ik r Basis sets: Planewaves are easy to converge for solids, but not molecules n r = c nk e k Gaussians are troublesome to converge, but efficient for molecules (and hybrid functionals).

12 Jacob's Ladder Approximation of XC functionals E XC [ n]= dr n r xc [ n r ] where xc is the exchange correlation energy per particle nonlocal nonlocal n, semilocal orbitals semilocal local Hybrid functionals are times more expensive than lower-rung (local, semilocal) functionals. HSE hybrid functional has performed very well for band gaps and properties of simple solids. Performance of this ladder of functionals has not been studied for large, complex solids. QMC benchmarks help determine which functional is most accurate for a given system/property.

13 Strengths, Weaknesses, and approximations in QMC Strengths: Accurate and reliable because XC is computed explicitly. Weaknesses: Computational expense requires approximations and xDFT CPU time 1) Statistical (VMC & DMC) error ~1/sqrt(N) Tunable approximations Increase number of samples (MC steps). J R ) Fixed VMC wavefunction form limits ability to estimate E of GS. R =e D R Expand J in more terms/parameters until energy converges; converge DFT orbitals. 3) Interpolation of PW orbitals (VMC & DMC) PW orbitals are grid interpolated for factor N speed up. Decrease grid size for better accuracy. 4) DMC time-step (DMC) determines accuracy at which wavefunction is sampled Reduce until energy converges 5) Finite-size (VMC & DMC) periodic simulation cell causes systematic error in energy 1 k-point/cell in QMC: use DFT k-point correction or larger supercells in QMC Self-interaction caused by Ewald summation. Fixed by restoring e-e interaction in cell (MPC). 6) Finite population size (DMC) causes bias in energy Increase configuration sample size 1) Pseudopotential necessary for solids Non-tunable approximations Compare results with different pseudopotentials;use DFT all-electron correction ) Pseudopotential locality calculate nonlocal channels with variational lattice DMC 3) Fixed-node apply backflow transformation to electron coordinates, or try different orbitals.

14 Applications 1) Silicon defects ) QMC study of high pressure silica phases 3) Hybrid functional study of silica Semiconductors: Silicon Defects Geophysics: High pressure silica

15 Application 1: Silicon defects affect semiconductor device performance Silicon chips require photographic and chemical processing. Ion implantation induced interstitials precipitate silicon self interstitial defects. Annealing causes defects to coalesce into extended defects. Extended defects broaden dopant profiles, limiting device size. Direct experimental detection is not possible; Watch diffusion of radio isotope profiles to estimate activation energy. Need accurate simulations to understand Si defect formation and migration energies. Defect Evolution (Tight Binding MD/DFT) Single Interstitials Tri Interstitials Anneal Di Interstitials Anneal

16 DFT and QMC Study Silicon Self Interstitial Defect Formation Energy 16 atom Supercell at L point Defect Type LDA lies roughly ev below QMC; GGA 1.5 ev below QMC. Likely due to self-interaction plus inability of LDA and GGA to compute energy differences between distorted structures. X and H defect formation energies are degenerate and T has highest by about 0.5 ev in 64-atom cell. Observe systematic improvement of functionals, consistent with Jacob's ladder. QMC results agree with less accurate results of Leung et al.

17 DFT and QMC Self Diffusion Barrier (Migration Energy) QMC and GGA results for self-diffusion from X to H to T to X. Lowest energy barrier from X to H is similar in both QMC and GGA (~300 mev). X to T barrier is twice as large in QMC as in GGA. Activation energy for diffusion is formation + migration energy Experiments estimate ev QMC predicts 5.(1) ev GGA predicts 4.15 ev

18 Di and Tri interstitials LDA lies roughly 3 ev below QMC; GGA ev below QMC. Observe systematic improvement of functionals, consistent with Jacob's ladder. A I is most stable di-interstitial and I3A is most stable tri-interstitital. GGA improves for less distorted defect.

19 Tunable QMC approximation tests DMC time step convergence Finite size convergence Bulk Si, L point X defect, L point DMC time step converged by 0.1 Ha-1 Finite size errors converged by 64-atom cell VMC energy converged by Jastrow polynomial order of 550. (order 5 in e-n expansion, order 5 in e-e expansion, no e-e-n expansion) VMC, X defect, 16 atom cell

20 Non Tunable QMC approximation tests DMC Formation energy does not vary with choice of DFT orbitals. DMC and VMC energies are equivalent with HF and LDA pseudopotentials by 3-atom supercell

21 Application Compression of silica Application: QMC and:dft Study High Pressure Silica Simplest of Earth's silicates; ubiquitous component of Earth; Complex phase transitions with pressure. Quartz to stishovite is a four fold to six fold coordination change. nd Stishovite undergoes a order transition to a CaCl structured phase. CaCl structure transforms to a PbO structure, which is stable to the core. Goals of this work: Explore feasibility of using QMC for high pressure/temperature properties of silica. Compute thermal equations of state and phase diagram (Combine QMC with DFT phonons). Compute thermodynamic properties of silica.

22 Compute static energies of silica phases at several volumes using QMC QMC Static Energy vs Volume Static energy is dominant crystal energy QMC is most accurate method for solids DFT 1)Choose XC functional and pseudopotential. )Relax crystal structures. 3)Compute energy and wavefunction. DFT QMC 1)Use DFT's relaxed crystal structures. )Optimize DFT wavefunction. 3)Compute energy stochastically, fit E(V) T=0 K Transition Pressure (GPa) Quartz Stishovite CaCl apbo Experiment QMC DFT(WC) 6 to 7 ~90 6.3(0.14) 88(8).1 86

23 Compute energy due to thermal vibrations (phonons) using DFT 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 Phonons computed within Quasi harmonic approx. = V Solve equations of motion for frequencies using small displacements. Dispersion data from Burkel, et al. Physica B, 63 64, pp (1999). Phonons computed from Helmholtz vibrational energy: F =E static V F vibration V, T = ktln Z 1 modes Z = exp i / kt i Implicit T dependence because of V dependence modes ℏ i V kt ln 1 exp ℏ i V / kt i=1 i=1 static energy F QHA = E 0 V zero point E 1 thermal vib E is force constant matrix; sum times phase factor gives dynamical matrix K F QHA=E 0 K u Phonon frequencies are eigenvalues of dynamical matrix DFT End of the day: F, T =F QMC T =0 F, T Static vib

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

25 Silica Phase Boundaries G=F PV G 1 P T, V 1 =G P T,V Gibbs FE equal at equilibrium F 1 P T P T V 1 P T =F P T P T V P T Equilibrium phase boundaries computed from Gibbs free energies. Metastable quartz stishovite transition measured with thermocal or shock. QMC agrees well. QMC CaCl PbO transition lies between two measurements. No free silica at base of mantle. DFT(WCGGA) boundary 4 GPa too low for quartz stishovite and within statistical error of QMC for CaCl PbO

26 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) PbO curves are the best available.

27 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 PbO curves are best available.

28 ... And many more thermodynamic properties Room temperature comparison to experiment for all computed thermodynamic quantities: Agreement with QMC is very good for all computed quantities

29 First QMC Calculation of 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 1500 processor hours c ijkl = 1 E V Elastic constants obtained from curvature of energy strain curve Double well at 80 Bohr3 indicates elastic instability of stishovite CaCl becoming more stable that stishovite under pressure 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.

30 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) ) Run QMC on thousands of processors for many 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 100 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).

31 c11 c1 (GPa) Stishovite Shear Modulus c ij = 1 E V Shear constant: 3 million CPU hours (1 summer). Total QMC silica project: 10 million CPU hours. Stishovite to CaCl 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.

32 Application 3: Hybrid DFT Study of Silica Quartz Stishovite Transition Hybrid functionals are potentially as accurate as QMC (particularly HSE) Hybrid functionals have not been well tested for large, complex systems due to expense associated with exact exchange. HSE improves efficiency by screening exchange. Hybrid functionals typically require localized (Gaussian) basis sets; difficult to converge

33 QMC and hybrid comparison of zero pressure volume Procedure Compute E at several volumes at zero temperature. Use Vinet EoS to fit E(V) for quartz and Stishovite All following properties are then computed from E(V) Compare dependence on codes, pseudopotentials, basis set, and XC functional. CRYSTAL: all-electron, Gaussian basis set ABINIT: norm-conserving, nonlocal pseudopotential (based on atomic calc), planewaves VASP: PAW pseudopotential (based on all-electron solid calc), planewaves GGA's: PBE, PW91, PBEsol, WC Hybrids: B3LYP, PBE0, HSE LDA better than GGAs for structural properties; HSE performs as well as LDA. No Jacob's Ladder trend in performance

34 QMC and hybrid comparison of zero bulk modulus K = V P V T =V F V LDA better than GGAs for elastic properties HSE performs better for stishovite than quartz. No Jacob's Ladder trend in performance

35 QMC and hybrid comparison of quartz stishovite transition pressures GGAs better than LDA for energy differences for two systems with very different structures. PBE, PW91, HSE, and PBE0 all predict the correct transition pressure. Overall conclusion: LDA is better than GGA for structural and elastic properties. GGA is better than LDA for energy differences HSE is good for all properties and the best compromise of accuracy and speed for benchmarks.

36 Conclusions Quantum mechanics describes matter, but its equations are difficult to solve for matter. Mean-field-based approaches (HF, DFT) provide manageable single-particle equations. QMC offers stochastic solution to the full many-body Schrödinger equation.-vmc uses variational principle for fixed many-body wavefunction form -DMC projects the ground state using a diffusing/branching, statistical wavefunction DFT is efficient, but it's accuracy is limited mainly by the XC functional. QMC is expensive, but computes XC explicitly at a high accuracy level. A benchmark method that compromises accuracy and speed is needed to validate XC functionals Jacob's ladder of functionals indicates hybrid functionals should be most accurate QMC benchmarks of silicon interstitial defects indicate LDA and GGA are not adequate, but HSE performs well. QMC benchmark calculations of silica use DFT phonons to produce QMC-based thermal equations of state and thermodynamic properties and the first elastic constant. LDA, PBE, WC are not adequate to study high pressure phases. QMC provides reliable geophysical implications. Hybrid studies of the quartz-stishovite silica transition indicates the HSE functional performs very well for all silica properties. HSE is a less expensive compromise for benchmark calculations of complex materials.

37 Backup Slides

38 Harmonic approximation to free energy F =E TS= ktln Z Helmholtz Free Energy: Partition function: Z = exp i / kt i 1 3 Assume microstates are harmonic: i = ℏ i, ℏ i, 5 ℏ i,... Partition function for harmonic quantum states: modes Z = exp i=1 F =E 0 static internal 1 1 ℏ i / kt modes s=0 modes i=1 modes exp s ℏ i / kt = i=1 modes ℏ i kt zero point i=1 exp 1 ℏ i / kt 1 exp ℏ i / kt ln 1 exp ℏ i / kt harmonic vibrational contribution (thermal)

39 Quasi harmonic adds anharmonicity the easy way Assume = V is a function of volume only F QHA=E 0 V 1 modes i=1 modes ℏ i V kt i=1 ln 1 exp ℏ i V / kt Notice the thermal term now has both T and V dependence T dependence comes from the partition function/definition of F V dependence comes from frequency By assuming a frequency volume dependence, we add in low order anharmonicity terms Since Fth=Fth(T,V), T and V dependence of all thermodynamic functions can be computed. (Thermal expansion and gruneisen are no longer zero) In DFT: Compute phonons for small displacements at various volumes of a structure Phonons are computed at T=0 (frequencies are T independent) When will this breakdown?

40 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 ) 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......

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42 QMC Locality Approximation Pseudopotentials separate core and valence electrons; effective potential reproduces core. The pseudopotential is made separately for each angular momentum state; must be nonlocal. DMC requires pseudopotentials for efficiency (Z6.5 scaling without, Z3 scaling using Pps) In DMC, nonlocal pseudopotentials create an addition sign problem if left untreated: H eff =K V loc V nloc result of Vnloc on trial wfn Schrodinger's equation becomes f R, = 1 m v f f d The locality approximation assumes H T T E T f V DMC DMC V T T f DMC = T in the last term. Without this approximation, the elements V R can cause G (R,R') to change sign. R e B Going beyond the locality approximation Use variational, lattice regularized DMC (Casula) Silica QMC calculations used this correction scheme. nloc