materials process design and control laboratory multibody expansions: an ab- initio based...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


MULTIBODY EXPANSIONS: AN AB-INITIO BASED TRANSFERABLE

POTENTIAL FOR COMPUTATIONAL THERMODYNAMICS

Baskar Ganapathysubramanian and Nicholas ZabarasMaterials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace EngineeringCornell University

Ithaca, NY 14853-3801

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




MATERIAL INTERACTION

Development of new materials for industrial applications

Need to understand and predict behavior

Physical phenomena affecting behavior span several length scales

An accurate descriptor at the lowest scale absolutely essential

- Thermodynamics prediction

- Thermal behavior

- Bulk, surface and isolated interactions

- Extract constitutive relations that can be used to tailor properties





Accurate potential energy surface of interactions between multiple components or surface-molecule or surface-cluster

Long range effects may be critical, affect most stable phase or energetically favorable pathway1,2.

To take into account the quantum effects need an essentially ab-initio approach.

1. P.Nieto, et. al, Science (2006) 312. 86 – 89

2. D. A. Freedman, T.A. Arias, Physical Review Letters, in review.

Need a abinitio level accurate strategy that can model large structures in a computationally tractable way





Complete ab-initio analysis currently infeasible

Strategies to accurately represent interactions

Semi-empirical potentials not accurate enough

Cluster Expansion method1-5.

Fixed lattice: configurational degrees of freedom

Expand energy in converging sequence of cluster energies

1. J. M. Sanchez, F. Ducastelle, D. Gratias, Physica A 128, 334 (1984)

2. J. W. D. Connolly, A. R. Williams, Phys. Rev. B 27,5169--5172 (1983).

3. R. Drutz, R. Singer, M. Fahnle, Phys. Rev. B 67 (2003) 035418

4. M. H. F. Sluiter, Y. Kawazoe, Phys. Rev. B 68 (2003) 085410

5. A. Zunger, NATO Advanced Study Institute on Statics and Dynamics of Alloy Phase Transformations ed P Turchi and A Gonis (New York :Plenum) (1994)




CLUSTER EXPANSION METHODS

Very successfully applied to many systems

Only configurational degrees of freedom

Relaxed calculation required but only a few calculations required

Periodic lattices, Explores superstructures of parent lattice

Issues when alloy phases complex structures

Issues when components have widely varying sizes

Convergence problems when relaxation effects are important1,2.

1. D. de Fontaine, in Solid State Physics, edited by H. Ehrenreich and D. Turnbull, Academic Press, New York, 1994

2. Z.W. Lu, S.H. Wei, A. Zunger, S. Frota-Pessoa, L.G. Ferreira, Phys. Rev. B 44 512--544 (1991).




Allow positional degrees of freedom in cluster expansions

For periodic lattices

Cluster expansion for the fixed lattice

Pair potentials for local relaxations

HYBRID CLUSTER EXPANSION METHODS

1. H.Y. Geng, M.H.F. Sluiter, N.X. Chen, Phys. Rev. B 73, 012202, (2006).

2. R. Drautz, M. Fahnle, J.M. Sanchez, J. Phys.: Condens. Matter 16, 3843, (2004).

3. M. Fahnle, R. Drautz, F. Lechermann, R. Singer, A. Diaz-Ortiz, H. Dosch, Phys. Status Solidi B 242, 1159, (2005).




Total energy 1,2

Symmetric function

Position and species

1. R. Drautz, M. Fahnle, J M Sanchez, J. Phys.: Condens. Matter 16 (2004) 3843–3852

2. J. W. Martin, J. Phys. C, 8 (1975)

MULTIBODY EXPANSION

∑= ∑+ ∑+ + …




Need to find a representation for these functions

Inversion of potentials: Going from energies to potentials, Mobius transform1,2.

EL is found from ab-initio energy database

MULTIBODY EXPANSION

1. N.X. Chen, Phys. Rev. Lett. 64 1193--1195 (1990)

2. N.X. Chen, G.B. Ren, Phys. Rev. B 45, 8177--8180(1992).

All degrees of freedom included No relaxations needed Needs a database of calculations, regression schemes required Periodicity is not required (large cell, one k-point calculation) Can predict energies over several different lattices

Total energy is the sum of energies of higher and higher levels of interaction




MULTIBODY EXPANSION

E0 = V0

E1(X1) = V (1)(X1) + V0

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

(1)(X1) + V (1)(X2) + V0

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) – E0)

1

2

3




All potential approximations can be shown to be a special case of multi-body expansion

Embedded atom potentials

MULTIBODY EXPANSION: LINK TO OTHER HAMILTONIANS




Total energy represented as hierarchical sum of isolated clusters of atoms

- No periodicity

- Fully transferable

- No relaxation necessary

Two issues to be taken care of:

1) How to construct each of these multi body potentials?

2) When to stop the expansion?

∑= ∑+ ∑+ + …

MULTIBODY EXPANSION




MULTIBODY EXPANSIONAs the number of atoms in the n-body potential increases, the dimensionality of the n-body potential increases.

‘Curse of dimensions’ comes into play very quickly

Have to approximate high dimensional surfaces accurately

Cannot utilize a tensor product space!

Come up with intelligent schemes to sample from the hyper-surface

Multi body expansions not a new theory.

One of the standing mathematical problems in representation potential energy surfaces- Roszak & Balasubramanian J. Math Chem (1994)

Techniques devised for representing the PES: but specific to dimension and could not be generalized to higher body interaction

1 2 3

4




TESSELATING HIGH DIMENSIONAL SPACE

First investigations utilized a finite element tessellation and interpolation of the space1.

1. V. Sundararaghavan, N. Zabaras, Phys. Rev. B 77 064101, 2008

Number of elements increased combinatorially as dimensionality increased and also with accuracy

Computationally feasible up to 5 body potentials

Accuracy of 0.1 Ry.

Necessary to incorporate higher orders as well as more accuracy




TESSELATING HIGH DIMENSIONAL SPACEUtilize sparse collocation to interpolate the high dimensional space.

Sparse collocation extensively used to integrate high dimensional functions in statistical mechanics: basic ideas involved in importance sampling

Moving from integration to interpolation non trivial

First ideas based on choosing sparse points on a uniformly sampled grid1.

Sparse tensor product of one-dimensional interpolating functions

1. S.A. Smolyak. Dokl. Akad. Nauk SSSR, 4 240–243, 1963.

Smolyak (1963) came up with a set of rules to construct such products1

Interpolant generated recursively




TESSELATING HIGH DIMENSIONAL SPACE

Theoretical bounds on number of function evaluations M.

1. V. Barthelmann, E. Novak and K. Ritter, Adv. Comput. Math. 12 (2000), 273–288

2. E. Novak, K. Ritter, R. Schmitt and A. Steinbauer, J. Comp. Appl. Math. 112 (1999), 215–228

Depending on the order of the one-dimensional interpolant, construct error estimate of the interpolant1,2

But can improve performance bu incorporating adaptivity




Anisotropic sampling for interpolating functions with steep gradients and other localized phenomena.

Have to detect it on-the-fly.

Utilize piecewise linear interpolating functions: local support

Utilize hierarchical form of basis function: provides natural stopping criterion

Add 2N neighbor points. Scales linearly instead of O(2N)

1. B. Ganapathysubramanian and N. Zabaras, J. Comp. Phys 225 (2007) 652-685

2. X. Ma and N. Zabaras, J. Comp. Phys, under review

ADAPTIVE SPARSE GRID COLLOCATION




Given a user-defined threshold, ε>0.

For points where w > ε, refine the grid to include 2N daughters. Compute the hierarchical surpluses at these new points.

Refine until all w< ε or maximum depth of interpolation is reached

Implementation:

Keep track of uniqueness of new points

Efficient searching and inserting

Parallelizability

Error estimate of the adaptive interpolant





Ability to detect and reconstruct steep gradients





Ene

rgy

Position

•Needs the least number of ab initio calculations to construct the potential,

•Provides capabilities to hierarchically improve the quality of interpolation using the previous interpolant,

•Can be made to adaptively sample the different dimensions to further reduce the computational requirements

•Completely independent of the number of dimensions of the problem.

•Provides a way of constructing fully–transferable ab initio based potentials.





• Plane-wave electronic density functional program ‘quantum espresso’ (http://www.pwscf.org)

•These calculations employ LDA and use ultra-soft pseudopotentials.

• Single k-point calculations were used for isolated clusters.•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.

CONSTRUCTING THE POTENTIALS

Adaptive Sparse Grid Collocation

Framework

N-body potential




∑= ∑+ ∑+ + …

Two issues to be taken care of:

1) How to construct each of these multi body potentials?

2) When to stop the expansion?

1. B. Paulus et. al, Phys. Rev. B 70, 165106 (2004)

2. B. Paulus, Phys Rep 428 (2006)

Work of B.Paulus 1,2 show that the computed energy oscillates between even and odd number of expansion terms, asymptotically converging to the exact energy

Stop the expansion when energy is accurate enough

correct energy

Energies (En) calculated from an n-body

expansion

MULTIBODY EXPANSION




WEIGHTED MULTIBODY EXPANSION

1. V. Sundararaghavan, N. Zabaras, Phys. Rev. B 77 064101, 2008

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

Similar idea to computing the coefficients in the cluster expansion

correct energy

Energies (En) calculated from an n-body expansion




WEIGHTED MULTIBODY EXPANSION ALGORITHM

A. Off-line calculation: Construction the ab-initio based MBE

- Set threshold and maximum depth of interpolation

- Set input file generator to link to first principles software

- Construct adaptive interpolant,

Adaptive sparse grid toolkit

First-principles software

Ab-initio multi-body potentials

B. Energy calculation: Calculating the energy of set of arbitrary M-atom clusters

- Compute the weights for the MBE

- Evaluate energies using the MBE. The energy of isolated L atom clusters are computed by directly interpolating over the multi-body potential

MBE expansion: convert into L atom clusters

N-atom cluster

Interpolate using L-atom potential

Weighted sum gives E




PERFORMANCE OF wMBE

Predict energies of 16-atom Pt clusters using 6-body potentials

True energy

3 order MBE

4 order MBE

5 order MBE

6 order MBE

Beyond 5 body representation energy is accurate to 10-8 Ryd




PERFORMANCE OF wMBE

Predict energies of random 16-atom clusters using 6-body potentials with weights generated from previous case

True energy

3 order MBE

4 order MBE

5 order MBE

6 order MBE

Once constructed, the weighted potentials accurately represent the energy of random configuration of atoms




POTENTIALS FOR PLATINUM

Investigate cluster energies for Platinum

Transition-group metal, applicability in hydrogen adsorption

Current state of art is 4 body potential

Extend to 6 body potential and beyond

Link parameters of adaptive interpolation to physics

ε related to accuracy of ab-initio computation

Two-body potential




ε k

10-2 10

10-3 12

10-4 14

10-5 16


Effect of varying threshold for interpolant

Platinum cluster. Accuracy of abinito computation ~ 0.01-0.1 mRy

2-body space 3-body spaceMaximum error: 5.06x10-3 eV

L2 error: 2.47x10-6 eVPoints: 200

Maximum error: 1.62x10-3 eVL2 error: 5.14x10-7 eV

Points: 8000





Higher order interaction potentials: hyper-surface is highly corrugated

Get reasonable accuracy ~ 0.1mRy with threshold of ε = 10-3

Compute up to 6 body potentials

order dim evals

2 1 ~0.2K

3 3 ~8K

4 6 ~40K

5 9 ~100K

6 12 ~300K




STABILITY STUDIES

One of the standing mathematical problems in representation potential energy surfaces- Roszak & Balasubramanian J. Math Chem (1994)

Prediction of Jahn-Teller distortions, representing effects of non-linear configurations

3-order MBE

3-order wMBE




CLUSTER ENERGIES

Predict energies of N-atom clusters using N-body potentials

Convergence and accuracy check




LARGE CLUSTER ENERGIES

Predict energies of 16-atom clusters using 6-body ab-initio potentials

Beyond 5 body representation energy is accurate to 10-1 eV

True energy

3 order MBE

4 order MBE

5 order MBE

6 order MBE N-body CV score/atom (eV)

3 0.23

4 0.21

5 0.17

6 0.07

Leave-one-out cross validation procedure to check accuracy of weights




LARGE CLUSTER ENERGIES

Predict energies of 128-atom clusters using 5-body ab-initio potentials

Beyond 5 body representation energy is accurate to 10-1 eV

True energy

3 order MBE

4 order MBE

5 order MBE

Computationally effective framework to estimate ab-initio energies of large clusters




LINK TO MD, MC AND THERMODYNAMIC SOFTWARE The atomic potential energy surface (APES) computed from ab-initio techniques

First step towards efficient , quick computation of the PES

1. G.Kallen,G.Wahnstrom, Quantum treatment of H on a Pt(111) surface, Phys Rev B, 65 (2001)

2. S.C.Badescu et al, Energetics and Vibrational states for Hydrogen on Pt(111), PRL 88 (2002)

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)

Confined to fcc-hcp-fcc valleysFCC site

Computational cost

MBE: ~ 10 minutes

DFT: ~ days

From ref 1




1) Represented the energy of a set of atoms as a hierarchical sum of isolated clusters of atoms: The multi body expansion (MBE)

2) Provided a methodology to compute these high dimensional surfaces using sparse grid techniques: Smolyak theorem, adaptive sparse grid methods

3) Possible to Couple the multibody potential framework to several publicly available molecular dynamics and Monte Carlo software

4) Applicability of the MBE to finding the ground state stable configurations

B. Ganapathysubramanian, N. Zabaras, Sparse grid collocation methods for computing ab initio based many-body expansions, Phys Rev B, Under review

V. Sundararaghavan, N. Zabaras, Many-body expansions for computing stable structures of multi-atom systems, Phys. Rev. B 77 064101, 2008

CONCLUSIONS

materials process design and control laboratory multibody expansions: an ab- initio based...

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