1. Materials Process Design and Control Laboratory Using multi-body energy expansions from ab-initio calculations for computation of alloy phase structures Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.eduhttp://mpdc.mae.cornell.edu V. Sundararaghavan and Prof. Nicholas Zabaras

2. Materials Process Design and Control Laboratory PREDICTION OF STABLE STRUCTURES Computational techniques -Exhaustive or heuristic search aided by DFT calculations -Cluster expansion Cu Ca hP6 oP12 Stable Pt clusters (Doye and Wales, New J. Chem., 1998) Stable configurations of adsorbed species

3. Materials Process Design and Control Laboratory Only configurational degrees of freedom Relaxed calculation required but only a few calculations required Periodic lattices, Explores superstructures of parent lattice Configurational and positional degrees of freedom Relaxed DFT calculations are not required Periodicity is not required Requires a large number of cluster energy evaluations Convergence issues Multi-body expansion Materials Process Design and Control Laboratory Comparison with CE Cluster expansion

4. Materials Process Design and Control Laboratory Hybrid cluster expansions Allow positional degrees of freedom in cluster expansions For periodic lattices Cluster expansion for the fixed lattice Pair potentials for local relaxations Materials Process Design and Control Laboratory Geng, Sluiter et al, Phys Rev B 2006

5. Materials Process Design and Control Laboratory Multi-body expansion Total energy Symmetric function Position and species JW Martin - Journal of Physics C, 1975, Empirical potentials (3 body): Murrell-Mottram (Mol. Phys 1990) ∑ = ∑ + ∑ + + …

6. Materials Process Design and Control Laboratory Multi-body expansion Example of calculation of multibody potentials E 1 (X 1 ) = V (1) (X 1 ) E 2 (X 1,X 2 ) = V (2) (X 1,X 2 ) + V (1) (X 1 ) + V (1) (X 2 ) Inversion of potentials Evaluate (ab-initio) energy of several two atom structures to arrive at a functional form of E 2 (X 1,X 2 ) V (2) (X 1,X 2 ) = E 2 (X 1,X 2 ) - (E 1 (X 1 ) + E 1 (X 2 ) ) E 1 (X 2 ) = V (1) (X 2 ) Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004 = Increment in energy due to pair interactions

7. Materials Process Design and Control Laboratory Multi-body expansion Inversion of potentials E L is found from ab-initio energy database, L << M Calculation of energies Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004

8. Materials Process Design and Control Laboratory Fitting energy surfaces To calculate the energy of a 3 body structure (E 3 ), we need to identify E 2 and E 1, values. Two body energy E 2 (X 1,X 2 ) is the energy of an isolated cluster of 2 atoms at positions X 1 and X 2. The database may not contain this energy since the energy values have only been obtained for atoms at locations (x i,y i ) that are different from (X 1,X 2 ) We use interpolation methods for retrieving energy at (X 1,X 2 ) from the database of energies at (x i,y i ). For example, we can use a polynomial interpolation of the form: Interpolation allows us to compute a large number of energies from a well-sampled database

9. Materials Process Design and Control Laboratory Smolyak algorithm Extensively used in statistical mechanics Provides a way to construct interpolation functions based on minimal number of points Uni-variate interpolation Multi-variate interpolation Smolyak interpolation Accuracy the same as tensor product Within logarithmic constant Increasing the order of interpolation increases the number of points sampled

10. Materials Process Design and Control Laboratory Smolyak algorithm: reduction in points For 2D interpolation using Chebyshev nodes Left: Full tensor product interpolation uses 256 points Right: Sparse grid collocation used 45 points to generate interpolant with comparable accuracy For multi-atom systems, sample all combinations of atoms (eg. E(A-A-A), E(A-A-B), E(A-B-B),E(B-B-B) and construct interpolants. Results in multiple orders of magnitude reduction in the number of points to sample

11. Materials Process Design and Control Laboratory CLUSTER REPRESENTATION Specification of clusters of various order by position variables 1 23 4 5 5 1 23 4 a b b a Convex hull technique to represent all atoms in the positive z-direction Use independent coordinates to represent the cluster geometry A point in 6 dimensional space

12. Materials Process Design and Control Laboratory CLUSTER ENERGY COMPUTATIONS Executables Executables –Cluster coordinates –Energy interpolation –Batch input for PWSCF –Read energies from PWSCF –Energy calculation Plane-wave electronic density functional program ‘quantum espresso’ (http://www.pwscf.org) calculations are used to compute energies given the atomic coordinates and lattice parameters.http://www.pwscf.org These calculations employ LDA and use ultra-soft pseudopotentials. Single k-point calculations were used for isolated clusters, the cell size was selected so that the effect of periodic neighbors are negligible. For multi-component systems, a constant energy cutoff equal to cutoff for the hardest atomic potential (e.g. B in B-Fe-Y-Zr) is used.  MP smearing (ismear=1, sigma=0.2) is used for the metallic systems. Computations were performed in parallel on a 64 node quad-processor LINUX cluster

13. Materials Process Design and Control Laboratory LINKING THE MULTIBODY EXPANSION TO OTHER SOFTWARE The multibody expansion software written in C++ Two parts: potential generation & energy computation Energy computation part is the Hamiltonian Molecular dynamics- LAMMPS Multi Body Expansion (MBE) Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is a classical molecular dynamics (MD) code developed by S. Plimpton et. al (Sandia national lab) Directly linked energy computation part in LAMMPS with MBE Useful for molecular dynamics and energy minimization Monte Carlo for Complex Chemical Systems (MCCCS) developed by M. G. Martin, J. I. Siepmann et. al. Available at http://towhee.sourceforge.net/http://towhee.sourceforge.net/ Fortran based code. Linked Towhee and MBE using a library Performs a variety of calculations in all ensembles Monte Carlo- MCCCS Towhee

14. Materials Process Design and Control Laboratory ENERGY SURFACES FOR ISOLATED CLUSTERS b a X Y a = (1.5*X+0.5)*7.5 Bohr b = (1.5*Y+0.5)*7.5 Bohr Platinum isolated cluster energies computed using multi-body potentials

15. Materials Process Design and Control Laboratory COMPUTATION OF CLUSTER ENERGIES Distance between atoms 1-2 (Bohr) Distance between atoms 1-3 (Bohr) Distance between atoms 1-2 (Bohr) Distance between atoms 1-3 (Bohr) (a) (b) The complete potential surface for a 3 Pt cluster. Figure (a) shows computed Platinum three- atom cluster energies, while (b) shows extension of energies using pair potential terms beyond the cutoff.

16. Materials Process Design and Control Laboratory Convergence results for different energy functions Energy of the system scales as n 2 where n is the number of atoms Order of interactions necessary for full convergence: 2 3 body term contribution =0 Energy of the system scales as n 1/2 where n is the number of atoms Order of interactions necessary for full convergence: 4 5 body term contribution =0

17. Materials Process Design and Control Laboratory Convergence results for different energy functions Using pair potentials: Lennard Jones for Helium atoms Order of interactions necessary for full convergence is 2 as expected (since it is a pair potential) 3 body term contribution =0

18. Materials Process Design and Control Laboratory Oscillations in MB energy for complex energy functionals -Energies oscillate around the true energy -Approach: Low pass filtering (convolution operation) that cuts off high frequency oscillations. -Compute the energy at the minima using self consistent field calculation correct energy Energies (E n ) calculated from an n-body expansion EAM potentials: Platinum system

19. Materials Process Design and Control Laboratory Computation of MBE energy filters Weighted MBE + + +.. Is the total energy correlated with structural energies of clusters ?

20. Materials Process Design and Control Laboratory Weighted MB energy a1a1 a2a2 a3a3 True energy

21. Materials Process Design and Control Laboratory Extrapolatory tests on weighted MBE True energy MBE 4 th order Weighted MBE energies once built for a small set of configurations provide accurate energy fit for various different inter-atomic distances within that configuration. 16 atom Au-Cu FCC cluster 4 unit cell, 4 at/cell AuCu 3

22. Materials Process Design and Control Laboratory Selection of order of expansion Weighted 2 nd order MBE Weighted 3rd order MBE Weighted 4th order MBE True energies Weighted MBE expansion coefficients are fitted using 12 atom cluster energies and the results are presented for a 16 atom cluster. Energies may differ but the weighted MBE captures the energy minima within 4 th order expansion. Cohesive energy (Ryd) Test various MBE orders in extrapolatory modes

23. Materials Process Design and Control Laboratory Platinum clusters + + Depth of interpolation 4 120 4 560 4 1820 Number of isolated cluster calculations Coefficients obtained using an 12 atom cluster energies at different lattice parameters 16 atom FCC cluster Actual energy Weighted MBE 4 th order Energy minima 66.577.588.59 -6 -5 -4 -3 -2 0 Lattice parameter (Bohr) Cohesive energy (Ryd)

24. Materials Process Design and Control Laboratory + + Depth of interpolation 4 276 4 2024 4 10626 Number of isolated cluster calculations Actual energy 24 atom FCC cluster Platinum clusters Weighted MBE 4 th order Energy minima 66.577.588.59 -9 -8 -7 -6 -5 -4 -3 -2 0 1 Lattice parameter (Bohr) Cohesive energy (Ryd)

25. Materials Process Design and Control Laboratory + + Depth of interpolation 4 276 4 2024 4 10626 Number of isolated cluster calculations Actual energy Weighted MBE 4 th order A random 24 atom configuration Platinum clusters 66.577.588.59 -9 -8 -7 -6 -5 -4 -3 -2 Lattice parameter (bohr) Cohesive energy (Ryd)

26. Materials Process Design and Control Laboratory Stable phase structures of Au-Cu alloy Super-cell approach For computing stable structures of periodic lattices, a 4x4x4 supercell (216 atoms) is used as an approximation. Weighted MBE is several orders of magnitude faster than a relaxed DFT calculation. Useful for amorphous structures Small cluster calculations are used to compute the weights in the weighted MBE expansion FCC structures are considered here for Au-Cu.

27. Materials Process Design and Control Laboratory Stable phase structures of Au-Cu alloy AuCu 3 cell relaxation 3x3x3 supercell a = 6.62 bohr = 3.50 A Au 3 Cu cell relaxation a = 7.3 bohr = 3.86 A AuCu 3 lattice parameter: 3.76 A Au 3 Cu lattice parameter: 4.04 A

28. Materials Process Design and Control Laboratory Stable phase structures of Au-Cu alloy AuCu 3 cell relaxation 4x4x4 supercell a = 6.71 bohr = 3.55 A Au 3 Cu cell relaxation a = 7.4 bohr = 3.92 A AuCu 3 lattice parameter: 3.76 A Au 3 Cu lattice parameter: 4.04 A

29. Materials Process Design and Control Laboratory G.Kallen,G.Wahnstrom, Quantum treatment of H on a Pt(111) surface, Phys Rev B, 65 (2001) Minimum energy surface of h on Pt(111) Plot of minimum energy in z direction for the primitive cell Highly anharmonic potential energy surface FCC->HCP (55 mev), FCC->TOP (160 mev) H confined to FCC-HCP-FCC valleys APPLICATION TO SURFACE PHENOMENA FCC site (Baskar and Zabaras, 2007)

30. Materials Process Design and Control Laboratory Conclusions MB expansion provides atom position dependent potentials that are used to identify stable structures. Ab-initio database of cluster energies are created and interpolation for various cluster positions are generated using efficient sparse grid interpolation algorithms. Weighted MBE is fast and captures the energy minima within a small order of expansion. Technique is applicable to study stability of amorphous systems, molecules and clusters.