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 LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y





Using multi-body energy expansions from ab-initio

calculations for computation of alloy phase structures

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

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

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

V. Sundararaghavan and Prof. Nicholas Zabaras

http://mpdc.mae.cornell.edu/

http://mpdc.mae.cornell.edu/




PREDICTION OF STABLE STRUCTURES

Computational techniques -Exhaustive or heuristic search aided by DFT calculations-Cluster expansion

CuCa

hP6oP12

Stable Pt clusters (Doye

and Wales, New J. Chem., 1998)

Stable configurations of adsorbed species




• 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




Comparison with CE

Cluster expansion




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




Geng, Sluiter et al, Phys Rev B 2006








Total energy

Symmetric function

Position and species

JW Martin - Journal of Physics C, 1975, Empirical potentials (3 body): Murrell-Mottram (Mol. Phys 1990)

∑= ∑+ ∑+ + …





Example of calculation of multibody potentialsExample of calculation of multibody potentials

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








Inversion of potentialsInversion of potentials

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

Calculation of energiesCalculation of energies

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







Fitting energy surfaces

To calculate the energy of a 3 body structure (E3), we need to identify E2 and E1, values.

• Two body energy E2(X1,X2) is the energy of an isolated cluster of 2 atoms at positions X1 and X2.

• The database may not contain this energy since the energy values have only been obtained for atoms at locations (xi,yi) that are different from (X1,X2)

• We use interpolation methods for retrieving energy at (X1,X2) from the database of energies at (xi,yi) . For example, we can use a polynomial interpolation of the form:

2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5x 10

6

Order of expansion

Num

ber o

f add

ition

al c

alcu

latio

ns

Interpolation allows us to compute a large number of energies from a well-sampled

database




Smolyak algorithm

Extensively used in statistical mechanics

Provides a way to construct interpolation functions based on minimal number of points

( ) ( )i

i i

i i

xx X

U f a f x

1

0 11

, 1,

0, ,

( ) ( ) ( )( )d

i i id

iiq d q d

i q

U U U i i i

A f A f f

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




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







CLUSTER REPRESENTATION

Specification of clusters of various order by position variables

1 2 3

4 5

5

1 2 3

4

a

bba

• Convex hull technique to represent all atoms in the positive z-direction

• Use independent coordinates to represent the cluster geometryA point in 6 dimensional

space







CLUSTER ENERGY COMPUTATIONS

• Executables Executables –Cluster coordinatesCluster coordinates–Energy interpolationEnergy interpolation–Batch input for PWSCFBatch input for PWSCF–Read energies from Read energies from

PWSCFPWSCF–Energy calculationEnergy 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. •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




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/

Fortran based code. Linked Towhee and MBE using a library

Performs a variety of calculations in all ensembles

Monte Carlo- MCCCS Towhee




3 4 5 6 7 8 9 10 11 12-104.8

-104.7

-104.6

-104.5

-104.4

-104.3

-104.2

Interatomic distance (Bohr)

En

erg

y (

Ry

d)




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

4 5 6 7 8 9 10 11

4

5

6

7

8

9

10

11

-157

-156.9

-156.8

-156.7

-156.6

-156.5

-156.4

-156.3

Platinum isolated cluster energies computed using multi-body potentials







COMPUTATION OF CLUSTER ENERGIES

4 5 6 7 8 9 10 11

4

5

6

7

8

9

10

11

-157

-156.9

-156.8

-156.7

-156.6

-156.5

-156.4

-156.3

4 5 6 7 8 9 10 11

4

5

6

7

8

9

10

11

Distance between atoms 1-2 (Bohr)

Dis

tan

ce b

etw

ee

n a

tom

s 1

-3 (

Bo

hr)

Distance between atoms 1-2 (Bohr)

Dis

tan

ce b

etw

ee

n a

tom

s 1

-3 (

Bo

hr)

(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.







Convergence results for different energy functions

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

10

20

30

40

50

60

70

80

90

Cont

ribut

ion

to to

tal e

nerg

y in

%

value of n in n-body interation term

convergence result

Energy of the system scales as n2 where n is the number of atoms

Order of interactions necessary for full convergence: 2

3 body term contribution

=0

1 1.5 2 2.5 3 3.5 4 4.5 50

50

100

150

200

250

Con

tribu

tion

to to

tal e

nerg

y in

%


convergence result

Energy of the system scales as n1/2 where n is the number of atoms

Order of interactions necessary for full convergence: 4

5 body term contribution =0




1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

20

40

60

80

100

120

Co

ntr

ibu

tio

n t

o t

ota

l e

ne

rgy

in

%


convergence result




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







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 (En) calculated from an n-body expansion

EAM potentials: Platinum system







Computation of MBE energy filters

Weighted MBE

+

+

+ ..

Is the total energy correlated with

structural energies of clusters ?







Weighted MB energy

11 2( , ,.., )M ME X X X

21 2( , ,.., )M ME X X X

31 2( , ,.., )M ME X X X

a1

a2

a31 2 2 1 2

3 1 2 4 1 2

( , ,.., ) 0.1484 ( , ,.., )

0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M

M M

E X X X E X X X

E X X X E X X X

6 6.5 7 7.5 8 8.5 9-8.2

-8

-7.8

-7.6

-7.4

-7.2

-7

-6.8

Lattice Parameter (Bohr)

Co

he

siv

e E

ne

rgy

(R

yd

)

MBE energyTrue energy

2 3 4

-20

-15

-10

-5

0

Truncation order

Co

he

siv

e E

ne

rgy

(R

yd

)

True energy




4 4.5 5 5.5 6 6.5 7-10

0

10

20

30

40

50

60

70

80


Co

he

siv

e E

ne

rgy

(R

yd

)


Extrapolatory tests on weighted MBE

True energy

MBE 4th order

1 2 2 1 2

3 1 2 4 1 2

( , ,.., ) 0.1484 ( , ,.., )

0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M

M M

E X X X E X X X

E X X X E X X X

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

AuCu3

6 6.5 7 7.5 8 8.5 9-8.2

-8

-7.8

-7.6

-7.4

-7.2

-7

-6.8


Coh

esiv

e E

nerg

y (R

yd)








Selection of order of expansion

Weighted 2nd order MBE

Weighted 3rd order MBE

Weighted 4th order MBE

True energies

True energies

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 4th order expansion.

Coh

esi

ve e

ne

rgy

(Ryd

)

Coh

esi

ve e

ne

rgy

(Ryd

)

Coh

esi

ve e

ne

rgy

(Ryd

)

Test various MBE orders in extrapolatory modes







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

1 2 2 1 2

3 1 2 4 1 2

( , ,.., ) 0.5884 ( , ,.., )

0.3014 ( , ,.., ) 0.0353 ( , ,.., ).M M M

M M

E X X X E X X X

E X X X E X X X

Actual energy

Weighted MBE 4th order

Energy minima

6 6.5 7 7.5 8 8.5 9-6

-5

-4

-3

-2

-1

0

Lattice parameter (Bohr)

Coh

esi

ve e

ne

rgy

(Ryd

)







+

+


4 276

4 2024

4 10626


Actual energy

24 atom FCC cluster

1 2 2 1 2

3 1 2 4 1 2

( , ,.., ) 0.5884 ( , ,.., )

0.3014 ( , ,.., ) 0.0353 ( , ,.., ).M M M

M M

E X X X E X X X

E X X X E X X X

Platinum clusters


Energy minima

6 6.5 7 7.5 8 8.5 9-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1


Coh

esi

ve e

ne

rgy

(Ryd

)







+

+


4 276

4 2024

4 10626


Actual energy


1 2 2 1 2

3 1 2 4 1 2

( , ,.., ) 0.5884 ( , ,.., )

0.3014 ( , ,.., ) 0.0353 ( , ,.., ).M M M

M M

E X X X E X X X

E X X X E X X X

A random 24 atom configuration

Platinum clusters

6 6.5 7 7.5 8 8.5 9-9

-8

-7

-6

-5

-4

-3

-2

-1

Lattice parameter (bohr)

Coh

esi

ve e

ne

rgy

(Ryd

)







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.




6 6.5 7 7.5 8 8.5 9-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Lattice parameter (A)

Coh

esiv

e en

ergy

(Ryd

/ato

m)

6 6.5 7 7.5 8 8.5 9-0.46

-0.44

-0.42

-0.4

-0.38

-0.36

-0.34

-0.32

-0.3

-0.28

-0.26


Co

he

siv

e e

ne

rgy

(R

yd

/ato

m)





AuCu3 cell relaxation

3x3x3 supercell

a = 6.62 bohr = 3.50 A

Au3Cu cell relaxation

a = 7.3 bohr = 3.86 A

1 2 2 1 2

3 1 2 4 1 2

( , ,.., ) 0.1484 ( , ,.., )

0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M

M M

E X X X E X X X

E X X X E X X X

AuCu3 lattice parameter: 3.76 A Au3Cu lattice parameter: 4.04 A




6 6.5 7 7.5 8 8.5 9-0.19

-0.18

-0.17

-0.16

-0.15

-0.14

-0.13

-0.12

-0.11


Coh

esiv

e en

ergy

(R

y/at

om)





AuCu3 cell relaxation

4x4x4 supercell

a = 6.71 bohr = 3.55 A

Au3Cu cell relaxation

6 6.5 7 7.5 8 8.5 9-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25


Coh

esiv

e E

nerg

y (R

yd/a

t)

a = 7.4 bohr = 3.92 A

AuCu3 lattice parameter: 3.76 A Au3Cu lattice parameter: 4.04 A

1 2 2 1 2

3 1 2 4 1 2

( , ,.., ) 0.1484 ( , ,.., )

0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M

M M

E X X X E X X X

E X X X E X X X




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)







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.

Documents

Materials Process Design and Control Laboratory Using multi-body energy expansions from ab-initio calculations for computation of alloy phase structures