Ab initio thermodynamics of solids under the quasiharmonic approach

Transcription

1 MolEng Thermodynamics Analytical EOS DFT EEC Conclusions Ack Bib Ab initio thermodynamics of solids under the quasiharmonic approach Víctor Luaña ( ) & Alberto Otero-de-la-Roza ( ) ( ) Departamento de Química Física y Analítica, Universidad de Oviedo ( ) National Institute of NanoTechnology, Edmonton, Alberta, Canada European school on Theoretical Solid State Chemistry ZCAM, Zaragoza, May 12 16, 2014 VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

2 Content 1 Molecular engineering tools 2 ab initio Thermodynamics of Crystals 3 Analytical Equations of State 4 DFT calculation of the electronic structure 5 Empirical Energy Corrections 6 Conclusions 7 Main programmers in the Oviedo s Quantum Chemmistry Group 8 Bibliography VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

3 The need for a molecular engineering tool Columbus, Ohio (May 24, 2011) Chemical Abstracts Service (CAS) records the 60 millionth substance in CAS REGISTRY, a potential antiviral. Some new entries are added everyday. The properties of most substances will remain unkonwn unless there is some previous hint of their importance. (January 1, 2009) The Cambridge Structural Database (CSD) records the crystal structure of different compounds (mostly organic and organometallic). (April 2009) The Inorganic Crystal Structure Database (ICSD) contains structures. The structure of most compounds will remain unknown, including the effect of p and T conditions. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

4 The Standard Model of Material Science The thermodynamic properties of a solid under a given pressure (p) and temperature (T) are determined by the general Gibbs energy: G(p, T, x) = E(V, x) + pv(x) + A vem(v, T, x) (1) where x represents the internal geometry of the crystal, V is the cell volume, E is the electronic energy. The Helmholtz energy A vem term includes the contribution from lattice vibrations, free electrons in metals, magnons in paramagnetic materials, etc. E(V, x) and p = de/dv is the source of the static approach, quite useful for a first analysis of the high pressure solid properties. Three steps are fundamental in the full thermal treatment: Fit an analytical Equation of State (EOS) to the E(V, x) discrete set of data. Objectives: (1) determine the equilibrium properties (V 0, E 0, B 0, B 0,...); (2) interpolate; (3) derive the curve; and perhaps (4) extrapolate. Scale the data if agreement with the experiment is critical. Determine thermal effects (including zero point energy). VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

5 Practical calculation of thermodynamic properties under QHA HA (harmonic approximation): Given an equilibrium configuration of the crystal (i.e. zero forces: qe = 0, where q i = M i(r i R 0 i ) for atom i) each atom moves in an harmonic potential (K = q qe). QHA (quasiharmonic approximation): The force constant matrix depends on the crystal volume: K(V). At difference from HA, the QHA solid expands when the temperature is raised. Density Functional Perturbation Theory [1] produces analitically the ab initio K matrix. Frequencies and vibrational modes (phonons) are obtained by diagonalizing K. Sampling the frequencies on the first Brillouin zone produces the phonon-density of States (φ-dos). Vibrational free energy in QHA: A vib (V, x, T) = 0 [ ω ( )] g(ω; V, x) 2 + kbt ln 1 e ω/k BT dω. (2) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

6 Frequency (cm -1 ) MgO E (Ry) Γ X Γ L X W L LDA PBE DOS (1/cm -1 ) V (bohr 3 ) 0.05 LDA 0.04 PBE at V PBE at V exp Frequency (cm -1 ) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

7 The Murnaghan EOS Murnaghan historical paper [2] is based on the principle of conservation of mass combined with Hooke s law for an inifinitesimal variation of stress in the solid. An equivalent result, however, can be simply obtained by assumming a linear variation of the bulk modulus with pressure, B(p) = B 0 + B 0p, to integrate B = V( p/ V) T: which can be inverted to ( ) 1/B V(p) = V B 0 0 p (3) B 0 p(v) = B0 B 0 [ ( ) B ] V (4) V The hydrostatic work equation, dw = pdv, can then be used to integrate: [ ] E = E 0 + B0V (V 0/V) B 0 B 0 B B0V0 B 0 1 (5) for the volume dependent energy under null temperature conditions. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

8 The Birch-Murnaghan EOS family Origin: Taylor expansion of the strain energy E = [ ] (V r/v) 2/3 1 Eulerian strain: f = 1 { 2 Conditions: lim V; E; p = de dp ; B = V f 0 dv dv ; B = db dp ;... Results: V r = V 0; c 0 = E 0; c 1 = 0; c 2 = 9 V0B0; 2 c 3 = 9 2 V0B0(B 0 4); c 4 = 3 8 V0B0{9[B0B 0 + (B 0) 2 ] 63B };... BM2: Let x = (V/V 0). n c kf k. k=0 } = { V 0; E 0; 0; B 0; B 0;... } E = E B0V0(x 2/3 1) 2, p = 3 2 B0 ( x 7/3 x 5/3 ). (6) BM3: E = E V0B0 (x 2/3 1) 2 x 7/3 {x 1/3 (B 0 4) x(b 0 6)},... (7) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

9 Non-linear fitting and convergence of results Test system: Rock-salt phase of MgO. Calculations: QUANTUM ESPRESSO [3] LDA using plane-waves and norm-conserving pseudopotentials. Very strict conditions: cutoff energy (80 Ry), k-mesh (shifted Monkhorst-Pack grid) and q-mesh (6 6 6). Data points: linear grid of 129 points in the volume range bohr 3. Equilibrium properties of the rock-salt phase of MgO. EOS V 0 (bohr 3 ) B 0 (GPa) B 0 B 0 (GPa 1 ) Murn Vinet BM ,023 6 PT ,090 4 BM ,025 4 PT ,016 2 The dispersion of fitting parameters is very large (17 GPa for B 0). The five parameter EOS (BM4 and PT4) are more in agreement, but we can do better with our excellent input data. Converging BM3 and BM4 is easy, BM5 is difficult, who wants to try BM10? VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

10 BM as a polynomial strain family The big question: If the BM family comes from assuming a polynomial E(f ) equation, why do not use a linear fit for the polynomial coefficients? Is the result of linear and nonlinear fittings identical? model fit V 0 E 0 B 0 B 0 B 0 BM3 nonlin BM3 lin BM4 nonlin BM4 lin BM12 lin Linear BM fittings can be done precise and efficiently to any order required by the data Linear fittings can be generalized to many strain definitions. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

11 BM linear fitting procedure 1 Get the eulerian strain f = ((V/V r) 2/3 1)/2. V r is a reference volume. 2 Linear fitting to the E(f ) = n 0 ckf k polynomial. 3 Get minimum of E(f ) to determine the equilibrium properties. 4 Get the derivatives E nf d n E/df n. 5 Determine f nv = d n f /dv n. For the eulerian strain: f (n+1)v = (3n + 2)f nv/(3v), f 1V = (1/3V)(V r/v) 2/3, n = 1, 2, 3,... (8) 6 Obtain E nv = d n E/dV n : 7 Finally E 1V = E 1f f 1V; E 2V = f 2 1VE 2f + f 2VE 1f ; (10) E 3V = f 3 1VE 3f + 3f 1Vf 2VE 2f + f 3VE 1f ; (11) E 4V = f 4 1VE 4f + 6f 2 1Vf 2VE 3f + (4f 1Vf 3V + 3f 2 3V)E 2f + f 4VE 1f ;... (12) (9) p = E 1V; B = VE 2V; B p = VE3V E 2V 1; B = E 3 2V [V(E 4VE 2V E 2 3V) + E 3VE 2V];... (13) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

12 An indefinite number of polynomial strain families strain EOS f f (n+1)v f 1V s 1 ( ) eulerian BM x 2/3 1 (3n+2)sf nv x 2/ V 3V 1 ( ) lagrangian Thomson x 2/3 1 (3n 2)sf nv x2/ V 3V 1 natural PT 3 ln x 1 nsfnv 1 3V V (1 f ) 4 infinitesimal Bardeen 1 x 1/3 (3n+1)sf nv 1 3V r 3V x 3 x 3 x 1/3 (3n 1)sf nv 1 3V V V 0 3V 1 0 x = V/V r, η = x 1/3. BM: Birch-Murnaghan [2, 4, 5]; Thomson [6]; PT: Poirier-Tarantola [7]; Bardeen [8]. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

13 Average of polynomials and predicted error bars Idea: Produce a better polynomial by averaging several strain polynomials according to their efficiency. The square sum of residuals: S i = k [Ek E(Vk)]2. w i = [N i/n i][s i/s min], where N i is the number of data, n i is the degree of the polynomial, and S min = min i S i. A normalized probability distribution is defined as P i = e w i /( k e w k ) (method 1), or P i = e w2 i /( k e w2 k ) (method 2). The P i are used to: (1) average the polynomial coefficients, thus defining a average polynomial; and (2) average the equilibrium properties of the individual polynomials, determining in this way the probability distribution of the properties (mean, standard deviation, skew, kurtosis, etc). Use bootstrap sampling when data is noisy. This method can be used with arbitrary strain polynomial families. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

14 Do the different average strain families converge and agree? B (GPa) (GPa -1 ) B order n BM PT Lag Inf V (bohr 3 ) B B (GPa) (GPa -1 ) B BM average up to order n V (bohr 3 ) B VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

15 The results are stable (within the predicted error bars) for the many experiments that we have done: (1) decreasing the 129 volume points to 65, 33, 17, or increasing them to 257, 1025, 2049; (2) moving the grid points; (3) changing the volume range up to 1/2 or 2 times;... The different strain families agree. For a good set of data, the error bars are small, orders of magnitude better than the errors currently assumed by using BM3, BM4 or Vinet. Equilibrium properties of MgO EOS V 0 (bohr 3 ) B 0 (GPa) B 0 B 0 (GPa 1 ) BM BM Vinet avg14bm (83) (20) (58) (85) Contrary to the current opinion, the average of strain polynomials produces a significative value for B 0, B 0, and B 0. In the case of B 0 the error bar is larger than the predicted value. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

16 Resampling error bars The resampling method (aka Monte Carlo statistics) is a standard technique for (a) estimating the precision of sample statistics by using subsets of available data (jackknifing) or drawing randomly with replacement from a set of data points (bootstrapping); and (b) validating models by using random subsets (bootstrapping, cross validation). MgO: BM6 fitting to a resampling of the 129 data points. sample points V 0 (bohr 3 ) B 0 (GPa) B 0 B 0 (GPa 1 ) (36) (78) (13) (17) (34) (82) (12) (15) (34) (75) (11) (15) (34) (75) (11) (15) (34) (75) (11) (15) avg14bm (83) (20) (58) (85) The error bars of the strain polynomials average method are conservative. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

17 The solid state physics approach: Density Functional Theory Much is due to the seminal work of J.C. Slater and coworkers (Xα and APW methods, for instance), Wigner-Seitz (cellular method), Philips-Kleinman (pseudopotentials),... However, nowadays is common to start from the Hohenberg-Kohn theorems (1964): HK1: The ground state electron density, ρ 0(r 1), of a bound system of interacting electrons in some external potential, V ext, determines this potential uniquely. In consequence, the ground state electron density is sufficient to construct the full Hamilton operator and hence to calculate, in principle, any ground state property of the system. In other words, any ground state property can be expressed in terms of the ground state electron density ρ 0. Tipically V ext represents the Coulomb potential of a set of nuclei. The theorem can be extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states. HK2: The functional E for the ground state energy is minimized by the ground state electron density ρ 0. I.e.: E[ρ] E[ρ 0] for every trial electron density ρ. This enables calculating ρ 0 variationally. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

18 The Kohn-Sham method DFT is, in principle, an orbital-free method: minimize the energy E[ρ] = T[ρ]+ V 1 ρ(r)ρ(r ) ext(r)ρ(r)dr+ drdr +E xc[ρ] = T[ρ]+E n[ρ]+e R 3 2 R 6 r r H [ρ]+e xc[ρ] (14) by varying ρ over all densities containing N electrons. However, T[ρ] an essential component of the total energy (T[ρ] = E[ρ] by the virial theorem), is unknown, and HK1 demands that the electron density comes from a well behaved wave function (N-representability). Most DFT calculations follow the Kohn-Sham route: ρ(r) = λ nocc λ ψ λ 2, where the spinorbitals are the result of solving, self-consistently, the KS equations: { 1 } 2 2 r + ˆV n(r) + ˆV H(r, [ρ]) + ˆV xc[ρ] ψ λ (r) = ɛ λ ψ λ (r) (15) ˆV n: nuclear electrostatic potential (or sum of pseudopotentials of the chemical kernels); R3 ρ(r ˆV )dr H(r) = : Coulomb potential due to the (possibly valence) electron r r density. ˆV xc: Exchange and correlation term. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

19 There is no systematic way of improving the xc functional but J. P. Perdew has popularized the Jacob s ladder metaphor. First rung (LDA/LSDA) [Local (spin) density approach]: Based on the Quantum Monte Carlo solution of an homogeneous electron gas. Ex: VWN91. Jacob s ladder by Shalom de Safed ( ) Second rung (GGA) [Generalized gradient approach]: Determined by the value of the electron density and its gradient on each point. Ex: PW91, PBE,... Third rung (meta-gga): Depends on the local density, gradient, and kinetic energy. Ex: TPSS,... Fourth rung, Hybrid functionals: Mix of several kinds. Typically, the Hartree-Fock exchange (exact exchange) is mixed with GGA. In other words, the functional depends on the occupied orbitals. Ex: B3LYP,... Fifth rung, Double hybrid functionals: Depends on the virtual and occupied orbitals. Ex: XYG3,... Increasing cost: L(S)DA GGA < meta-gga <<? hybrid <<< double-hybrid. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

20 DFT calculations on MgO Problems that prevent true accuracy: Choice of xc: source of uncertainty in DFT. Known trends in solids: LDA overbinds, PBE underbinds. No systematic ways of improving the results. Wait for a hypothetical perfect functional? Fitting of the E(V) and A(V; T) curves. Solved!!!. V (bohr 3 ) Speziale (2001) Li (2006) Tange (2009) LDA p (GPa) PBE Fiquet (1999) Sinogeikin (2000) Dubrovinsky (1997) T (K) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

21 Empirical Energy Corrections (EEC) The systematic deviations in the exchange-correlation functionals can be corrected by using one or two accurate experimental results: ambient conditions volume (V 0 exp) and bulk modulus (B 0 exp). Kunc and Syassen s observation: p/b 0 vs. V/V 0 is transferable, V 0 and B 0 being the static equilibrium volume and bulk modulus, respectively. ( ) Ẽ sta(v) E sta V V 0 V exp = (16) B exp B 0 The bpscal EEC is: Ẽ sta(v) = E sta(v 0) + BexpVexp B 0V 0 ( ) ] [E sta V V0 E sta(v 0) V exp (17) The experimental V exp and B exp extrapolated to static conditions are missing, and are treated as parameters to fit the experimental data. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

22 Does the EEC really works? V (bohr 3 ) Speziale (2001) Li (2006) Tange (2009) LDA PBE Fiquet (1999) Sinogeikin (2000) Dubrovinsky (1997) p (GPa) T (K) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

23 B s (GPa) p (GPa) Anderson (1990) Sinogeikin (2000) T (K) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

24 Any thermodynamic property can be obtained (MgO) The three steps: (1) energy fitting, (2) full QHA, and (3) empirical energy corrections allow us to predict thermodynamic properties with unprecedented accuracy. Room (0 GPa and K) (0 GPa and 1000 K) PBEcorr expt. PBEcorr expt. V (bohr 3 ) (19) (84) F vib (kj/mol) (47) (59) S (J/mol K) (88) (56) p th (GPa) 2.624(15) (67) 4.96 B T (GPa) (50) (20) 141 B S (GPa) (50) (20) α ( K) (97) (68) 4.47 C v (J/mol K) (50) (28) C p (J/mol K) (57) (55) B T 4.173(43) 4.53(12) B T (GPa 1 ) (28) (24) γ (24) (11) 1.54 VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

25 MgO as a high pt pressure scale Magnesium oxide has been proposed as pressure scale for diffraction experiments at extreme pressure and temperature. This requires a very precise knowledge of the p(v, T) EOS This study Tange(2009), Vinet Tange(2009), BM3 Wu(2008) Speziale(2001) p (GPa) K, T m o =3125 K VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

26 Performance of the method for other crystals V (bohr 3 ) V (bohr 3 ) Occelli (2003) Dewaele (2008) Dewaele (2004) Chijioke (2005) Akahama (2006) p (GPa) Reeber (1996) C Langelaan (1999) T (K) Al VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

27 Availability Any method able to provide static E(V) data and phonon DOS as a function of volume can be used. Among the ab initio open source electronic structure codes: Quantum Espresso, abinit, siesta,... The codes for fitting and for doing the complete analysis described here are open source and publicly available: GIBBS & gibbs2: The quasi-harmonic model and robust fitting techniques. Comput. Phys. Commun. 158 (2004) 57; 182 (2011) 1708; The technique and results have also been described in: Equations of State in Solids. Fitting theoretical data, possibly including noise and jumps. Comput. Theor. Chem. 975 (2011) 111. Rigorous treatment of first-principles data for quasiharmonic thermodynamics of solids: the case of MgO. Phys. Rev. B 84 (2011) Equations of state and thermodynamics of solids using empirical corrections in the quasiharmonic approximation Phys. Rev. B 84 (2011) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

28 Conclusions The conventional procedure of fitting nonlinearly an analytical low order EOS (BM3, Vinet,...) introduces a significant error in the equilibrium properties and in the derivatives calculated from E(V) theoretical data. The average of strain polynomial method: is efficient, numerically stable, can be extended to any order required by the data, and provides an estimation of the accuracy, in the form of error bars, that can be translated to any thermodynamic property. An empirical energy correction (EEC) can be used to remove the systematic error in DFT calculations due, mainly, to the defects in the exchange-correlation functional. The quasiharmonic approximation, improved with the EEC and a careful fit of the E(V) and A(V) curves, can predict the thermodynamic properties of a crystal in a very wide range of pressures and temperatures. Genetic algorithms and other methods of random sampling can be used to predict new metastable phases and conformations of solids. Chemical bonding properties in a solid do depend heavily on the pressure and, to a lesser extent, the temperature conditions. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

29 MolEng Thermodynamics Analytical EOS DFT EEC Conclusions Ack Bib Main Programmers in the Oviedo s QCG group Oviedo s Quantum Chemistry group around VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

30 This is a tribute to our friend and colleague Dr. Miguel Álvarez Blanco, whose untimely death left a deep hole in our hearts yet not in our brains. We miss him but he has not left us and never will. Dr. Miguel Álvarez Blanco ( ), Dr. Aurora Costales Castro, Miguel y Jaime Álvarez Costales. VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31

31 References I S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Modern Physics 73 (2001) F. D. Murnaghan, The compressibility of media under extreme pressures, Proc. Natl. Acad. Sci. USA 30 (1944) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, R. M. Wentzcovitch, QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Phys.-Condens. Matter 21 (2009) F. Birch, Finite elastic strain of cubic crystals, Phys. Rev. 71 (1947) F. Birch, Finite strain isotherm and velocities for single-crystal and polycrystalline NaCl at high pressures and 300 K, J. Geophys. Res. 83 (1978) L. Thomson, On the fourth order anharmonic equation of state of solids, J. Phys. Chem. Solids 31 (1970) J.-P. Poirier, A. Tarantola, A logarithmic equation of state, Phys. Earth Planet. Int. 109 (1998) 1 8. J. Bardeen, Compressibilities of the alkali metals, J. Chem. Phys. 6 (1938) VLC & AOR & DMC () Thermodynamics of Solids ZCAM, Zaragoza / 31