Materials Process Design and Control Laboratory ON THE DEVELOPMENT OF WEIGHTED MANY- BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE CRYSTAL STRUCTURES 1 Department of Aerospace Engineering, University of Michigan, Ann Arbor 2 Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering Cornell University URL: V. Sundararaghavan 1 and Nicholas Zabaras 2

Materials Process Design and Control Laboratory The crystal structure prediction problem Given elements A, B, C, … predict the stable low-temperature phases Energy Atomic positions Local minima Global minimum True structure

Materials Process Design and Control Laboratory Ab Initio Structure Prediction Directly optimize ab initio Hamiltonian with Monte Carlo, genetic algorithms, etc. (too slow) Directly optimize ab initio Hamiltonian with Monte Carlo, genetic algorithms, etc. (too slow) Simplified Hamiltonians – potentials, cluster expansion (fitting challenges, limited transferability/accuracy) Simplified Hamiltonians – potentials, cluster expansion (fitting challenges, limited transferability/accuracy) Obtain a manageable list of likely candidate structures for structure calculation

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

Materials Process Design and Control Laboratory For binary system... Cluster expansion

Materials Process Design and Control Laboratory Cluster expansion fit The cluster expansion is able to represent any function E(  ) of configuration  by an appropriate selection of the values of J . 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

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

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

Materials Process Design and Control Laboratory Multi-body expansion Example of calculation of multi-body 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

Materials Process Design and Control Laboratory = Increment in energy due to pair interaction = Increment in energy due to trimer interaction Multi-body expansion

Materials Process Design and Control Laboratory Multi-body expansion Inversion of potentials (Mobius formula) E L 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 All potential approximations can be shown to be a special case of multi-body expansion –Embedded atom potentials Multi-body expansion Fahnle et al., 2004

Materials Process Design and Control Laboratory Specification of clusters of various order by position variables Cluster specifiers

Materials Process Design and Control Laboratory Cluster configurational spaces Space of all possible three atom clusters of interest Geometric constraints Symmetry constraints Corresponds to 9 planes forming a convex hull ElEl ErEr Fourth order space (6D)

Materials Process Design and Control Laboratory Locating a cluster in the configurational space

Materials Process Design and Control Laboratory User imposed cut offs Lower cutoff- unstable configurations Upper cutoff- weak interaction 3-atom cluster energy surface 2-atom cluster energy surface Approximated using lower order (pair) interactions Upper cutoff

Materials Process Design and Control Laboratory 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

Materials Process Design and Control Laboratory -Energies oscillate around the true energy -Approach: Weight MBE terms. -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 Weighted Multi-body expansion

Materials Process Design and Control Laboratory Weighted MBE fit 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. Cluster Energies  Structures

Materials Process Design and Control Laboratory Weighted Multi-body expansion (Pt)

Materials Process Design and Control Laboratory Convergence test for extrapolatory cases 16 atom Pt Cluster with perturbed atoms

Materials Process Design and Control Laboratory Interpolated ab-initio MBE for Pt Calculation of Pt lattice parameter

Materials Process Design and Control Laboratory MBE for alloys Multi-body expansion for  -Alumina (Al 2 O 3 ) system using cluster energies computed using the Streitz-Mintmire (SM) model.  -Alumina has a rhombohedral primitive unit cell and is described in space group R-3c (no.167). Converges at fourth order.  -Alumina (Al 2 O 3 ) system

Materials Process Design and Control Laboratory Ab-initio MBE for alloys – Au-Cu system Cu-Cu-Au Cu-Au-Au Structure optimization to find the lattice constants for FCC CuAu 3 system (space group no. 221) using interpolated energies of clusters computed from first principles DFT calculations. 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.

Materials Process Design and Control Laboratory 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