Crystal structure prediction

Transcription

1 Crystal structure prediction Klaus Doll Institute for Mathematical Physics, TU Braunschweig Max Planck Institute for Solid State Research, Stuttgart MSSC2011, Turin, September 2011

2 Motivation structure prediction for solids: here: global search on potential surface; minima correspond to stable/metastable structures method: simulated annealing with ab initio energies in all the steps examples: bulk: LiF, PbS ionic clusters: BN CaC2 GeF2 covalent mixed ionic-covalent lone pair LiF, MgF2 model potential good for ionic bond, less so for other bond types NEW: ab initio energies, less biased with respect to the bond type

3 Strategies perform local optimisation on known structures (e.g. from databases) genetic/evolutionary algorithm: - start from ensemble of structures (first generation), relax, reject worst ones - next generation: combination of structures (e.g. combine slabs from two parent structures), permutation of atoms, change lattice parameter ; relax for solids, ab initio: A. R. Oganov, C. W. Glass, J. Chem. Phys. 124, (2006) random structure search: guess random structure, relax with gradients C. J. Pickard and R. J. Needs, Phys. Rev. Lett. 97, (2006) here: simulated annealing

4 Method, part I I) simulated annealing with G42 code 0) guess structure -> compute energy 1) create new structure (atoms moved or exchanged, lattice constant changed ), compute new energy 2) accept new structure according to probability exp (-ΔE / kt) (Metropolis algorithm) reduce temperature during the run; stop after ~ steps followed by quench (~ steps at T=0) J. C. Schön and M. Jansen, Angew. Chem., Int. Ed. Engl. 35, 1286 (1996)

5 Step 1 Step 2543 Step Step 2544 (2 atoms exchanged)

6 typically: 4 formula units used (8 atoms for BN, 12 for CaC 2) periodic boundary conditions employed, no symmetry (space group P1) => some restriction in the global search: structures with 1,2,4 formula units in the primitive cell can be found, but not with e.g. 6,8 formula units

7 Method, part II II) local optimisation: analytical gradients (usually implemented in ab initio codes, here: CRYSTAL09) atom positions: K. Doll, V. R. Saunders, N. M. Harrison, Int. J. Quant. Chem. 82, 1 (2001) unit cell: K. Doll, R. Dovesi, R. Orlando, Theor. Chem. Acc. 112, 394 (2004) before the local optimisation after the local optimisation

8 Method, part III III) symmetry-analysis with KPLOT (R. Hundt, Bonn) idealise the structure and search for the symmetry operators, identify the space group P1 KPLOT: P63/mmc as a whole: ~ runs required new run: new initialisaton of the random number generator => new initial geometry, new moves... further analysis

9 New: ab initio energies in all the stages ab initio energy (CRYSTAL09): - during simulated annealing in most cases: HF (restricted/unrestricted) HF+dispersion [S. Grimme, J. Comp. Chem. 27, 1787 (2006)] easiest to achieve SCF converge for insulators due to large gaps (SCF convergence for random structures required!) B3LYP slower; LDA for metals - local optimisation: routine task, various density functionals used

10 Ab initio energies in all the stages: difficulties ab initio: HΨ=EΨ model potential: e.g. Born-Mayer: E= i j qi q j cij Aij exp r ij /ϱ 6 rij r ij main problem: CPU time initial estimate: years of CPU time required for a single run; ~100 runs required centuries! model potential: seconds/minutes use less accurate ab initio parameters (integral selection, SCF convergence, k-points ) + shorter simulated annealing run now: ~ 1 week for a single run on 1 CPU

11 Why ab initio? model potentials good for ionic systems, less for covalent or metallic systems ab initio calculations more generally applicable, also if type of bond unknown important to test whether ab initio landscape explorations find the same minima as calculations based on model potentials

12 Simulated annealing: choice of parameters I) unit cell: 4 Li, 4 F atoms, initial volume ~3 times larger than estimated volume (using atomic or ionic radii) II) move classes: 1) move atom (70% of suggested moves) 2) exchange 2 atoms (10%) 3) change lattice constant (20%) restriction: minimum distance of atoms= ~0.7 * sum of atomic or ionic radii (radius from databases, charge from Mulliken population) III) ~ simulated annealing steps sufficient, followed by ~ quench steps

13 LiF: proof of principle NaCl 5-5 Pnma (62) Wurtzite zinc blende Pc (7) Cmc21 (36) NiAs Confirmation of the calculations with model potential model potential: J. C. Schön and M. Jansen, Comput. Mater. Sci. 4, 43 (1995) ab initio: K. Doll, J. C. Schön, M. Jansen, Phys. Chem. Chem. Phys. 9, 6128 (2007) experiment (new LiBr, LiCl structure: wurtzite): Y. Liebold Ribeiro, D. Fischer, M. Jansen, Angew. Chemie, 120, 4500 (2008) A. Bach, D. Fischer, M. Jansen, Z. Anorg. Allg. Chem. 635, 2406 (2009)

14 Energetics structure type rock salt zincblende 5-5 wurtzite NiAs LiF(I) LiF(II) LiF(III) relative energy, per 1 formula unit (ev) HF LDA total: 70 runs, 38 give good structure number of times found (local optimisation may end up in different structure with different functional) HF LDA

15 PbS NaCl TlI 4.5 GPa (FeB) CsCl 25 GPa

16 Negative pressure NaCl (rhombohedral distortion) GeTe D. Zagorac, K. Doll, J. C. Schön, M. Jansen, Phys. Rev. B 84, (2011)

17 BN structures found: wurtzite, zinc blende, hexagonal BN., no rock salt B N _ P63/mmc (194) space group P42/mnm : R3m (160) P6m2 (187) different view: β-beo (isoelectronic with BN, but never before suggested for BN) K. Doll, J. C. Schön, M. Jansen, Phys. Rev. B 78, (2008)

18 SrAl2 like structure: isoelectronic: Sr 2 + ignore (Al)2 2 B - N

19 pressure driven phase transition: enthalpy required use p= E ; V H=E+pV

20 layered structures become unfavorable at high pressure (due to their low coordination number) synthesis of new β - BeO phase: not feasible by just applying pressure

21 Convergence for initial structure: gap: LDA ~0.1 ev; B3LYP ~ 0.5 ev; HF ~6 ev

22 Results for BN: convergence: Hartree-Fock quickest! (large gap helps convergence) B3LYP possible, but more time consuming (factor 4) -> most important to have many runs; only rough position of minima required, local optimisation on any level possible several hundred runs with Hartree-Fock, ~ 15 % give good structure 10 runs with B3LYP K. Doll, J. C. Schön, M. Jansen, Phys. Rev. B 78, (2008)

23 CaC2 Experimentally known (at low temperature): CaC2 -I (space group I4/mmm) Energy : Eh CaC2 -II (space group C2/c) CaC2 -III (space group C2/m) Energy : Eh Energy : Eh Exp.: M. Knapp and U. Ruschewitz, Chem. Eur. J. 7, 874 (2001)

24 New modifications predicted, most interesting ones: CaC2 -V (space group Immm) Energy : Eh lower in energy than known structures _ CaC2 VII (space group R3m) Energy : Eh high pressure modification A. Kulkarni, K. Doll, J. C. Schön, M. Jansen, J. Phys. Chem. B 114, (2010) Now: experimental evidence for high pressure modification in BaC2 (G. J. Vajenine, I. Efthimopoulos, E. Stavrou, K. Kunc, K. Syassen, St. Liebig, U. Ruschewitz, M. Hanfland)

25 GeF2 Structures found, and corresponding building units a) c,d) e) b)

26 GeF2: predicted structures 6 energetically lowest ones: P P43212 (TeO2) or P41212 Pnma Pbcm P21/c Pnma K. Doll, M. Jansen, Angew. Chem. Int. Ed. 50, 4627 (2011)

27 (LiF)n Clusters n=1 n=3 n=4 up to (LiF)8 n=2 Li F

28 Tree graph from threshold run Threshold run: start from one structure accept all moves up to a certain value, then quench -> possible transition to new structure after quench K. Doll, J. C. Schön, M. Jansen, J. Chem. Phys. 133, (2010)

29 Conclusion+Acknowledgment NEW: structure prediction based on simulated annealing and ab initio energies in all the steps feasible LiF: proof of principle K. Doll, J. C. Schön, M. Jansen, Phys. Chem. Chem. Phys. 9, 6128 (2007) PbS: possible rhombohedral distortion D. Zagorac, K. Doll, J. C. Schön, M. Jansen, Phys. Rev. B 84, (2011) BN: example for a covalent system, new modifications (synthesis?) K. Doll, J. C. Schön, M. Jansen, Phys. Rev. B 78, (2008) CaC2: mixed covalent/ionic system, new modifications A. Kulkarni, K. Doll, J. C. Schön, M. Jansen, J. Phys. Chem. B 114, (2010) GeF2: chain like+3d-structures K. Doll, M. Jansen, Angew. Chem. Int. Ed. 50, 4627 (2011) clusters: LiF: K. Doll, J. C. Schön, M. Jansen, J. Chem. Phys. 133, (2010) MgF2: S. Neelamraju, J. C. Schön, K. Doll, M. Jansen recent reviews: J. C. Schön, K. Doll, M. Jansen, Phys. Status Solidi B 247, 23 (2010) M. Jansen, K. Doll, J. C. Schön, Acta Cryst. A 66, 518 (2010)