Optimized energy landscape exploration for nanosciences using ab-initio based methods

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1 Optimized energy landscape exploration for nanosciences using ab-initio based methods Eduardo Machado-Charry Nanosciences Foundation & Laboratoire de simulation atomistique (L Sim), SP2M, UMR-E CEA-Grenoble GDR Modélisation des Matériaux Marseille, février 21-22, 2013

2 Energy Landscapes Generic problem How to explore the space of variables of a high dimensional cost function? ART nouveau : Zero temperature method Samples the local minima only, going through common saddle point, generating a continuous trajectory. Barkema et al, PRL 1996 and Malek et al, PRB 2000 Marinica et al, PRB 83, (2002) Disordered systems, polymers and proteins

3 Activation: Leave the harmonic basin SP er GS1 Activation GS2 Random e R deformation (global, local, guess) Inflection point Follow e R until an λ become negative

4 Activation: Converging to a saddle point e SP GS2 Activation GS1 Push the configuration up along the direction e λ Convergence Zero total force & negative eigenvalue λ

5 Some detail of the step q i+1 = q i + ê λ δ λ 1 [H 3N 3N ] O(N 3 ) Lanczos method = L n n Typically, 15 < n < 20 = Still costly n force evaluations

6 Relax to a new local minimum SP GS2 Activation Relaxation GS1 Push the configuration slightly over the saddle point Kick: Relaxation (FIRE) Metropolis test Warning: The path is not the MEP

Characteristics Minimum Saddle Not sensitive

Not biased toward pre-determined mechanisms

7 Characteristics Minimum Saddle Not sensitive to the real space complexity of activated jump nor the height of the activation energy barrier, which can be very high. Not biased toward pre-determined mechanisms Seems to sample all classes of events (ergodic) Events are reversible

8 some details of an event (a) E (ev) (b) 10 1 log ( f (ev/å) ) (c) λ 1 (ev/å 2 ) force evaluations (d) 10 3 Log (force eval.) force evaluations log(steps) log(steps)

9 ab initio : BigDFT* Wavelets: localised in real and fourier space Systematic Placed only where are needed Low and high resolution Pseudopotentials Efficiency of the order of 90%, up to thousands of processors. Hybrid architectures (GPU) * J. Chem. Phys. 129, (2008)

10 Improving convergence Direct Inversion in the Iterative Subspace (DIIS) Proposed by Pulay 1980 for finding minima configurations Speed up the convergence when the systems is near to the SP Energy saddle point shoulder Minimum Coordinate PROS A remarkable improvement in the efficiency CONS detects all critical points including (shoulders)

11 DIIS Use your memory x = c i x i g = c i g(x i ) ci = 1 Minimize x new x 2 with respect to c i where x new = x + H 1 g By solving ( ) ( ) A 1 c = 1 0 λ ( ) 0 1 A = a ij = g T i g j and λ is the Lagrangian multiplier Warning: We use a diagonal approximation to H 1

12 Lanczos + DIIS (a) E (ev) (b) 10 1 log ( f (ev/å) ) (c) λ 1 (ev/å 2 ) force evaluations (d) 10 3 Log (force eval.) force evaluations log(steps) log(steps)

13 Statistic over events Amorphous silicon (SW semiempirical potential) Steps after minimum in λ Attempted events New events f s f s force evaluation per successful event

14 Comparison between algorithms Algo. ART nouveau Improved dimer a b GSM c System V Si C 20 SiC C 6H 10 PHBH/H 2O VO x/sio 2 BC Bulk Isol.e Surf. Isol. Sol. Isol. Pot. DFT DFT DFT DFT QM/MM DFT Method PBE LDA PBE B3LYP AM1 B3LYP DOF f f s A/S 79/78 434/ / a J. Chem. Phys 123, (2005) b J. Chem. Phys. 128, (2009) c J. Chem. Phys. 130, (2009)

15 Clusters : C (a) Tm=0.5 ev (b) Tm=0.9 ev Energy (ev) Events eV 1.04 ev 0.0 ev 0.89 ev 5.36 ev open cages closed cages Accepted event

16 Bulk: Vacancy diffusion in silicon bulk barrier split vacancy structure Energy a JT i JT j a Displacement = JT a i JT j a Energy (ev) α β α α γ γ β α α β JT i a FFCD α JT i a SVC α δ γ γ γ γ = JT a i = JT a i a JT i b JT i 0.2 = JT a i JT j b Events

17 Graphene/SiC Growth : LEED pattern

Surface: 4H-SiC (000 1) (2 2) C 7 6 (a) (b) (c) (d) (e) (f) (g) 5 Energy (ev) 4 3 2 1 0 0 4 8 12 0 4 8 12 16 0 4 8 0 4 8 0

18 Surface: 4H-SiC (000 1) (2 2) C 7 6 (a) (b) (c) (d) (e) (f) (g) 5 Energy (ev) Events T4 T4 H3 T4 T4 H3 H3 H3 H3 H3 Energy (ev) E 0 E 2 E 3 E Cumulative displacement (Å)

19 Final remarks Lanczos + DIIS improve the convergence to saddle points by a factor of 3 A tool for energy landscape exploration in material science simulations using ab initio forces Not a black box: Requires a interaction with the user QM/MM (SW potentials) L. Karim U. Montréal Kinetic-ART N. U. Montréal

20 People involved Pascal Luigi Damien Normand Pochet Genovese Caliste Mousseau Support Nanosciences Fondation: Muscade project Canada: NSERC-CRSNG, FRNT Québec, CRC Computer time: GENCI (Grant 6323), Tera-100 from CEA-DAM, and Calcul Québec

21 People involved Pascal Luigi Damien Normand Pochet Genovese Caliste Mousseau Thanks for your attention!! Look for J. Chem. Phys. 135, (2011) J. At. Mol. Opt. 2012, (2012) Sim/BigDFT/ mousseau/

22 The Lanczos method H x 0 > = a o x 0 > +b 1 x 1 > H x 1 > = a 1 x 1 > +b 1 x 0 > +b 2 x 2 > H x 2 > = a 2 x 2 > +b 2 x 1 > +b 3 x 3 >. H x l 1 > = a l 1 x l 1 > +b l 1 x l 2 > +b 1 x 1 > H x 1 > = a1 x l > +b l x k 1 > (1) a 0 b 1 b 1 a 1 b 2 L l =... λ 1[H] λ 1 [L l ](l 3N). b l 1 a l 1 b l

23 H[q 0 ]u by the finite difference on the forces by a Taylor expansion H[q 0 ]u = f(qo + δ lu) f(q 0 ) δ l + O(δ 2 l ) (2) H[q 0 ]u = f(qo + δ lu) f(qo δ l u) 2δ l + O(δ 3 l ) (3)

24 Metropolis test p accept = min[e (E fin E ini /K B T), 1] (4) T ia a Metropolis temperature. Vibrational entropic contributions are not included Fictitious temperature It is possible to thermalize the system at this temperature

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