Materials Process Design and Control Laboratory

1. ON THE DEVELOPMENT OF WEIGHTED MANY-BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE CRYSTAL STRUCTURES Veera Sundararaghavan1 and Nicholas Zabaras2 1Department of Aerospace Engineering, University of Michigan, Ann Arbor 2Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace EngineeringCornell University Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu Materials Process Design and Control Laboratory

2. The crystal structure prediction problem Predict the stable low-temperature phases of an alloy comprising of elements X,Y. Trial structures Decreasing formation energies True phase structure ? Composition - XY2

3. Ab Initio Structure Prediction • Identify optimum structure through Monte carlo/GA optimization using relaxed ab initio energy calculations (vs) • Identify optimum structures using simplified Hamiltonians – potentials, cluster expansion, multi-body expansion (fitting challenges/ transferability/ accuracy issues)

4. Cluster expansion Ortho-normal and complete set of basis functions are introduced. s is the configuration variable (+/- 1 for binary systems) +1 Basis for M lattice sites is given as: -1 Energy of the lattice (M sites) is given as: +1 For all clusters with number of atoms =K For all cluster sizes Average of energies of all configurations projected onto the basis function Sanchez and de Fontaine, 1981, Sanchez et al, 1984 Physica A Materials Process Design and Control Laboratory

5. ... Cluster expansion

6. Cluster expansion fit • The cluster expansion is able to represent any function E(s) of configurations by an appropriate selection of the values of Ja. • Converges rapidly using relatively compact structures (e.g. short-range pairs or small triplets). • Unknown parameters of the cluster expansion is determined by fitting first-principles energies as shown. Connolly-Williams method, Phys Rev B, 1983

7. Comparison with CE Cluster expansion • Only configurational degrees of freedom • Relaxed calculation required but only a few calculations required • Periodic lattices, Explores superstructures of parent lattice Multi-body expansion • 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* *V. Sundararaghavan and N. Zabaras, "Many-body expansions for computing stable structures", Physical Review B, in review. Materials Process Design and Control Laboratory

8. Multi-body expansion ∑ ∑ ∑ = + + + … Position and species Total energy Symmetric function Materials Process Design and Control Laboratory

9. Multi-body expansion Example of calculation of multi-body potentials E1(X1) = V(1)(X1) E1(X2) = V(1)(X2) E2(X1,X2) = V(2)(X1,X2) + V(1)(X1) + V(1)(X2) Evaluate (ab-initio) energy of several two atom structures to arrive at a functional form of E2(X1,X2) Inversion of potentials V(2)(X1,X2) = E2(X1,X2) - (E1(X1)+ E1(X2)) = Increment in energy due to pair interactions Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004 Materials Process Design and Control Laboratory

10. Multi-body expansion = Increment in energy due to pair interaction = Increment in energy due to trimer interaction

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

12. All potential approximations can be shown to be a special case of multi-body expansion Embedded atom potentials Multi-body expansion Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004

13. Cluster specifiers Specification of clusters of various order by position variables

14. Cluster configurational spaces Symmetry constraints El Er Space of all possible three atom clusters of interest Geometric constraints Corresponds to 9 planes forming a convex hull Fourth order space (6D)

15. Locating a cluster in the configurational space

16. User imposed cut offs 2-atom cluster energy surface Upper cutoff- weak interaction Approximated using lower order (pair) interactions 3-atom cluster energy surface Lower cutoff- unstable configurations Upper cutoff

17. Issues with larger orders of expansion Explosion in number of clusters needed to calculate energies Increase in configurational spaces required for an N-atom cluster

18. EAM potentials: Platinum system Energies (En) calculated from an n-body expansion correct energy Weighted Multi-body expansion • Energies oscillate around the true energy • -Approach: Weight MBE terms. • -Compute the energy at the minima using self consistent field calculation

19. Weighted MBE fit Cluster Energies • The multi body expansion is able to represent energy E of configuration of N atoms by an appropriate selection of the values of coefficients. • Converges rapidly using relatively compact structures (e.g. short-range pairs or small triplets). • Unknown parameters of the expansion is determined by fitting first-principle energies as shown. Structures a

20. Weighted Multi-body expansion (Pt)

21. Convergence test for extrapolatory cases 16 atom Pt Cluster with perturbed atoms

22. Interpolated ab-initio MBE for Pt Calculation of Pt lattice parameter

23. MBE for alloys a-Alumina (Al2O3) system Converges at fourth order. Multi-body expansion for a-Alumina (Al2O3) system using cluster energies computed using the Streitz-Mintmire (SM) model. a-Alumina has a rhombohedral primitive unit cell and is described in space group R-3c (no.167).

24. Ab-initio MBE for alloys – Au-Cu system Cu-Cu-Au Cu-Au-Au For computing stable structures of periodic lattices, a 6x6x6 supercell (864 atoms) is used. Weighted MBE is several orders of magnitude faster than a relaxed DFT calculation. Structure optimization to find the lattice constants for FCC CuAu3system (space group no. 221) using interpolated energies of clusters computed from first principles DFT calculations.

25. Link between MBE and CE CE Global projection operation Sum over all k-atom clusters Fix the k-atom cluster, and find the energy of system for various types of all other atoms Drautz et al, J Phys: Condensed matter, 2004

26. Link between MBE and CE Contribution of N-body potential to K-atom cluster coefficients Sum the N body potential for various positions and types of all other atoms Freeze the k-atom cluster Sum over all orders of expansion > K N = 3, K = 2

27. Because in any calculation only a finite number of clusters can be involved in the CE, there is always an open question of which of the various possible clusters are the most essential ones. Can be identified using MBE Allows one to calculate the energetic contribution of relaxations to the cluster expansion coefficients on a lattice from the many-body potential expansion. Allows consideration of the effect of relaxation, vibrational dofs Advantages

28. Conclusions • MB expansion provides atom position dependent potentials that are used to identify stable phase structures. • Ab-initio database of cluster energies are created and interpolation for various cluster positions are generated using efficient finite element interpolation. • Weighted MBE is fast and captures the energy minima within a small order of expansion. Publication V. Sundararaghavan and N. Zabaras, "Many-body expansions for computing stable structures", Physical Review B, in review. Preprint available for download at http://mpdc.mae.cornell.edu Materials Process Design and Control Laboratory