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ON THE DEVELOPMENT OF WEIGHTED MANY-BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE CRYSTAL STRUCTURE PowerPoint Presentation
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ON THE DEVELOPMENT OF WEIGHTED MANY-BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE CRYSTAL STRUCTURE - PowerPoint PPT Presentation


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

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slide1
ON THE DEVELOPMENT OF WEIGHTED MANY-BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE CRYSTAL STRUCTURES

V. 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

the crystal structure prediction problem
The crystal structure prediction problem

Given elements A, B, C, …

predict the stable low-temperature phases

Energy

Local minima

Global minimum

True structure

Atomic positions

ab initio structure prediction
Ab Initio Structure Prediction
  • Directly optimize ab initio Hamiltonian with Monte Carlo, genetic algorithms, etc. (too slow)
  • Simplified Hamiltonians – potentials, cluster expansion (fitting challenges, limited transferability/accuracy)

Obtain a manageable list of likely candidate structures for structure calculation

slide4

Cluster expansion

  • 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 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

slide6

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

slide7

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

Materials Process Design and Control Laboratory

multi body expansion
Multi-body expansion

=

+

+

+ …

Position and species

Total energy

Symmetric function

Materials Process Design and Control Laboratory

multi body expansion1
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

slide10

Multi-body expansion

= Increment in energy due to pair interaction

= Increment in energy due to trimer interaction

multi body expansion2
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

slide12
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

slide13

Cluster specifiers

Specification of clusters of various order by position variables

slide14

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)

slide16

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

slide17

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

slide18

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
slide19

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

slide21

Convergence test for extrapolatory cases

16 atom Pt Cluster with perturbed atoms

slide22

Interpolated ab-initio MBE for Pt

Calculation of Pt lattice parameter

slide23

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).

slide24

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.

slide25

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