Materials Process Design and Control Laboratory ON THE DEVELOPMENT OF WEIGHTED MANY- BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y





ON THE DEVELOPMENT OF WEIGHTED MANY-BODY EXPANSIONS USING AB-INITIO

CALCULATIONS FOR PREDICTING STABLE CRYSTAL STRUCTURES

1Department of Aerospace Engineering,University of Michigan, Ann Arbor

2Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

Cornell University

Email: [email protected]: http://mpdc.mae.cornell.edu

Veera Sundararaghavan1 and Nicholas Zabaras2




The crystal structure prediction The crystal structure prediction problemproblem

Predict the stable low-temperature phases of an alloy comprising of

elements X,Y.

Decreasing form

ation energies

Composition - XY2

Trial structures

True phase structure ?




Ab Initio Structure PredictionAb Initio Structure Prediction

• Identify optimum structure through Identify optimum structure through Monte carlo/GA optimization using Monte carlo/GA optimization using relaxed ab initio energy calculations relaxed ab initio energy calculations

(vs)(vs)

• Identify optimum structures using Identify optimum structures using simplified Hamiltonianssimplified Hamiltonians – potentials, – potentials, cluster expansion, multi-body expansion cluster expansion, multi-body expansion (fitting challenges/ transferability/ (fitting challenges/ transferability/ accuracy issues)accuracy issues)




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




Sanchez and de Fontaine, 1981, Sanchez et al, 1984 Physica A

Cluster expansion

-1

+1

+1






Cluster expansionCluster expansion

...




Cluster expansion fitCluster 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




• Only configurational degrees of freedomOnly configurational degrees of freedom• Relaxed calculation required but only a few calculations Relaxed calculation required but only a few calculations

required required • Periodic lattices, Explores superstructures of parent latticePeriodic lattices, Explores superstructures of parent lattice

• Configurational and positional degrees of freedomConfigurational and positional degrees of freedom• Relaxed DFT calculations are not requiredRelaxed 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 expansionMulti-body expansion




Comparison with CEComparison with CE

Cluster expansionCluster expansion

*V. Sundararaghavan and N. Zabaras, "Many-body expansions for computing stable structures", Physical Review B, in review.







Multi-body expansion

Total energy

Symmetric function

Position and species

∑= ∑+ ∑+ + …








Example of calculation of multi-body Example of calculation of multi-body potentialspotentials

E1(X1) = V (1)(X1)

E2(X1,X2) = V (2)(X1,X2) + V

(1)(X1) + V (1)(X2)

Inversion of potentials

Evaluate (ab-initio) energy of several two atom structures to arrive at a

functional form of E2(X1,X2) V (2)(X1,X2) = E2(X1,X2) - (E1(X1) + E1(X2) )

E1(X2) = V (1)(X2)

Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004

= Increment in energy due to pair interactions




= Increment in energy due to pair interaction

= Increment in energy due to trimer interaction









Inversion of potentials (Mobius formula)Inversion of potentials (Mobius formula)

EL is found from ab-initio energy database, L << M

Calculation of energiesCalculation of energies





• All potential approximations can be shown to be a special case of multi-body expansion– Embedded atom potentials






Specification of clusters of various order by position variables

Cluster specifiers




Cluster configurational spaces

Space of all possible three atom clusters of interest

Geometric constraints

Symmetry constraints

Corresponds to 9 planes forming a convex hull

El Er

Fourth order space (6D)




Locating a cluster in the configurational space




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




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




-Energies oscillate around the true energy

-Approach: Weight MBE terms.

-Compute the energy at the minima using self consistent field calculation

correct energy

Energies (En) calculated from an n-body expansion

EAM potentials: Platinum system

Weighted Multi-body expansion




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




Weighted Multi-body expansion (Pt)




Convergence test for extrapolatory cases

16 atom Pt Cluster with perturbed atoms




Interpolated ab-initio MBE for Pt

Calculation of Pt lattice parameter




MBE for alloys

Multi-body expansion for -Alumina (Al2O3) system using cluster energies computed using the Streitz-Mintmire (SM) model. -Alumina

has a rhombohedral primitive unit cell and is describedin space group R-3c (no.167).

Converges at fourth order.

-Alumina (Al2O3) system




Ab-initio MBE for alloys – Au-Cu systemCu-Cu-Au Cu-Au-Au

Structure optimization to find the lattice constants for FCC CuAu3 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.




Sum over all k-atom clusters

Fix the k-atom cluster, and find the energy of system for various types of all other atoms

Link between MBE and CE

CE Global projection operation

Drautz et al, J Phys: Condensed matter, 2004




Freeze the k-atom cluster Sum the N body potential for various positions and types of all other atoms

Contribution of N-body potential to K-atom cluster coefficients

Sum over all orders of expansion > K

N = 3, K = 2

Link between MBE and CE




Advantages

• 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







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

Documents

Materials Process Design and Control Laboratory ON THE DEVELOPMENT OF WEIGHTED MANY- BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE