MULTIBODY EXPANSIONS: AN AB-INITIO BASED TRANSFERABLE POTENTIAL FOR COMPUTATIONAL THERMODYNAMICS

1
MULTIBODY EXPANSIONS AN AB-INITIO BASED
TRANSFERABLE POTENTIAL FOR COMPUTATIONAL
THERMODYNAMICS Baskar Ganapathysubramanian and
Nicholas Zabaras Materials Process Design and
Control Laboratory Sibley School of Mechanical
and Aerospace Engineering Cornell
University Ithaca, NY 14853-3801 zabaras_at_cornell.
edu http//mpdc.mae.cornell.edu
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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
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MATERIAL INTERACTION
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.
Need a abinitio level accurate strategy that can
model large structures in a computationally
tractable way
1. P.Nieto, et. al, Science (2006) 312. 86
89 2. D. A. Freedman, T.A. Arias, Physical
Review Letters, in review.
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MATERIAL INTERACTION
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)
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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).
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HYBRID CLUSTER EXPANSION METHODS

• Allow positional degrees of freedom in cluster
  expansions
• For periodic lattices

Cluster expansion for the fixed lattice
Pair potentials for local relaxations
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).
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MULTIBODY EXPANSION
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Total energy 1,2
Position and species
Symmetric function

• R. Drautz, M. Fahnle, J M Sanchez, J. Phys.
  Condens. Matter 16 (2004) 38433852
• J. W. Martin, J. Phys. C, 8 (1975)

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

• 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

Need to find a representation for these functions
Inversion of potentials Going from energies to
potentials, Mobius transform1,2.
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).
EL is found from ab-initio energy database
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MULTIBODY EXPANSION
E0 V0
E1(X1) V (1)(X1) V0
E2(X1,X2) V (2)(X1,X2) V (1)(X1) V (1)(X2)
V0
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)
E0)
1
3
2
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MULTIBODY EXPANSION LINK TO OTHER HAMILTONIANS

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

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MULTIBODY EXPANSION
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• 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
• How to construct each of these multi body
  potentials?
• When to stop the expansion?

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MULTIBODY EXPANSION
As 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
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TESSELATING HIGH DIMENSIONAL SPACE
First investigations utilized a finite element
tessellation and interpolation of the space1.
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
1. V. Sundararaghavan, N. Zabaras, Phys. Rev. B
77 064101, 2008
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TESSELATING HIGH DIMENSIONAL SPACE
Utilize 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
Smolyak (1963) came up with a set of rules to
construct such products1
Interpolant generated recursively
1. S.A. Smolyak. Dokl. Akad. Nauk SSSR, 4
240243, 1963.
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TESSELATING HIGH DIMENSIONAL SPACE
Theoretical bounds on number of function
evaluations M.
Depending on the order of the one-dimensional
interpolant, construct error estimate of the
interpolant1,2
But can improve performance bu incorporating
adaptivity

• V. Barthelmann, E. Novak and K. Ritter, Adv.
  Comput. Math. 12 (2000), 273288
• E. Novak, K. Ritter, R. Schmitt and A.
  Steinbauer, J. Comp. Appl. Math. 112 (1999),
  215228

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ADAPTIVE SPARSE GRID COLLOCATION
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)

• B. Ganapathysubramanian and N. Zabaras, J. Comp.
  Phys 225 (2007) 652-685
• X. Ma and N. Zabaras, J. Comp. Phys, under review

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ADAPTIVE SPARSE GRID COLLOCATION
Given a user-defined threshold, egt0. For points
where w gt e, refine the grid to include 2N
daughters. Compute the hierarchical surpluses at
these new points. Refine until all wlt e 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
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ADAPTIVE SPARSE GRID COLLOCATION
Ability to detect and reconstruct steep gradients
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ADAPTIVE SPARSE GRID COLLOCATION

• 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 fullytransferable
  ab initio based potentials.

Energy
Position
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CONSTRUCTING THE 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 (ismear1, sigma0.2) is used for the
  metallic systems.

Adaptive Sparse Grid Collocation Framework
N-body potential
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MULTIBODY EXPANSION
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• Two issues to be taken care of
• How to construct each of these multi body
  potentials?
• When to stop the expansion?

Energies (En) calculated from an n-body expansion
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

• B. Paulus et. al, Phys. Rev. B 70, 165106 (2004)
  
• B. Paulus, Phys Rep 428 (2006)

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WEIGHTED MULTIBODY EXPANSION
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
1. V. Sundararaghavan, N. Zabaras, Phys. Rev. B
77 064101, 2008
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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,
First-principles software
Adaptive sparse grid toolkit
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
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PERFORMANCE OF wMBE
Predict energies of 16-atom Pt clusters using
6-body potentials
Beyond 5 body representation energy is accurate
to 10-8 Ryd
True energy
3 order MBE
4 order MBE
5 order MBE
6 order MBE
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PERFORMANCE OF wMBE
Predict energies of random 16-atom clusters using
6-body potentials with weights generated from
previous case
Once constructed, the weighted potentials
accurately represent the energy of random
configuration of atoms
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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 e related to accuracy of ab-initio
computation
Two-body potential
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POTENTIALS FOR PLATINUM
Effect of varying threshold for
interpolant Platinum cluster. Accuracy of abinito
computation 0.01-0.1 mRy
2-body space
3-body space
Maximum error 5.06x10-3 eV L2 error 2.47x10-6
eV Points 200
Maximum error 1.62x10-3 eV L2 error 5.14x10-7
eV Points 8000
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POTENTIALS FOR PLATINUM
Higher order interaction potentials
hyper-surface is highly corrugated Get reasonable
accuracy 0.1mRy with threshold of e
10-3 Compute up to 6 body potentials
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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
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CLUSTER ENERGIES
Predict energies of N-atom clusters using N-body
potentials Convergence and accuracy check
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CLUSTER ENERGIES
Predict energies of N-atom clusters using N-body
potentials Convergence and accuracy check
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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
Leave-one-out cross validation procedure to check
accuracy of weights
3 order MBE
4 order MBE
5 order MBE
6 order MBE
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LARGE CLUSTER ENERGIES
Predict energies of 128-atom clusters using
5-body ab-initio potentials
True energy
3 order MBE
Beyond 5 body representation energy is accurate
to 10-1 eV
4 order MBE
5 order MBE
Computationally effective framework to estimate
ab-initio energies of large clusters
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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
Computational cost MBE 10 minutes DFT days
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-gtHCP (55 meV), FCC-gtTop (160
meV) Confined to fcc-hcp-fcc valleys
FCC site
From ref 1

• G.Kallen,G.Wahnstrom, Quantum treatment of H on a
  Pt(111) surface, Phys Rev B, 65 (2001)
• S.C.Badescu et al, Energetics and Vibrational
  states for Hydrogen on Pt(111), PRL 88 (2002)

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CONCLUSIONS

• Represented the energy of a set of atoms as a
  hierarchical sum of isolated clusters of atoms
  The multi body expansion (MBE)
• Provided a methodology to compute these high
  dimensional surfaces using sparse grid
  techniques Smolyak theorem, adaptive sparse grid
  methods
• Possible to Couple the multibody potential
  framework to several publicly available molecular
  dynamics and Monte Carlo software
• 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