Bridging the gapfrom QM to interatomic potentials potential based directly on detailed QM data high...
Transcript of Bridging the gapfrom QM to interatomic potentials potential based directly on detailed QM data high...
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Bridging the gap:
Gaussian Approximation Potential
Albert Bartók-Pártay
ESDG 2009
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Gábor Csányi
Risi Kondor
ESDG 2009
Bridging the GAP
Albert Bartok-Partay
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● from QM to interatomic potentials● potential based directly on detailed QM data● high dimensional fit (Gaussian Processes)● atomic neighbourhoods: bispectrum● the first GAP for carbon● other uses: defining the local energy
Bridging the GAP
Albert Bartok-Partay ESDG 2009
Outline
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● quantum mechanics is the 'ultimate truth'● expensive to solve● sequence of approximations:
● Full CI● QMC● DFT-LDA● tight binding
● interatomic potentials
Bridging the GAP
Albert Bartok-Partay ESDG 2009
From QM...
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● energy is sum of atomic energies● atomic energy depends on neighbouring atoms● electronic problem is not solved
● cluster expansion of total energy
● EAM expansion
Bridging the GAP
Albert Bartok-Partay ESDG 2009
...to interatomic potentials
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● How is an interatomic potential generated?
● empirical, analytic formula
● choose target properties (even forces)
● fit free parameters to reproduce properties
● hope that the formula remains reasonably valid everywhere in the configurational space
● The GAP way:● no fixed formula
● search in the space of smooth functions
● identify target configurations
● fit to arbitrary precision QM data
● extend target set if needed
Bridging the GAP
Albert Bartok-Partay ESDG 2009
Generating potentials
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● energy is sum of atomic energies
● is not practical
● matrix is complete but not invariant to permutations
● symmetric polynomials are also complete but not invariant to rotation
● : all terms, chosen the rotationally invariant ones● our solution
● atomic energy is a functional of atomic density● express atomic density in rotationally invariant terms
GAP
Albert Bartok-Partay ESDG 2009
GAPBridging the GAP
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● how to choose the target configurations?
● an optimal way to interpolate many-dimensional functions
● magic trick: finding which fitting points are optimal for reproducing a very large data set
● calculate accurate forces and energies (DFT)
● perform fit using sum of derivatives of (forces) and sum of (energy) as the target function
Albert Bartok-Partay ESDG 2009
GAP: function fittingBridging the GAP
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● Two examples:
● 1D periodic function ● 2D object
● 1D:
● 2D: project on the Riemann-sphere then express it in spherical harmonics basis:
Albert Bartok-Partay ESDG 2009
Invariant representationsBridging the GAP
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● Power spectrum:● 1D:
● 2D: ● incomplete representation, phase information lost
Albert Bartok-Partay ESDG 2009
Invariant representationsBridging the GAP
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● Bispectrum (almost complete):● 1D:
● 2D:
Albert Bartok-Partay ESDG 2009
Invariant representationsBridging the GAP
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● project on a 4D sphere
●
● and are the same as the 3D polar coordinates
● express in 4D spherical harmonics basis: the Wigner D-matrices
● bispectrum is analogous to 3D case
● 4D CG-coefficients are direct products of 3D CG-coefficients
Albert Bartok-Partay ESDG 2009
Invariant representations:3D objects
Bridging the GAP
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target function: sin xfit points between 3 and 7random noise on fit pointsexpectation value and variance predicted
Albert Bartok-Partay ESDG 2009
GP: a simple interpolationBridging the GAP
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● Hot MD of interesting systems● surfaces● interstitial● vacancy● quenched liquid
● energies and forces of configuration samples with DFT
Albert Bartok-Partay ESDG 2009
Target configurationsBridging the GAP
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● GP gives 0 as an answer when unsure
● add core repulsion to correct for very close atoms● EAM formula● fitted to high pressure DFT
results
● add dispersion term for long-range interactions
Albert Bartok-Partay ESDG 2009
'Baseline' potentialBridging the GAP
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● target configurations● bulk phases● transition from diamond to graphite● 111 and 100 surfaces● vacancy● interstitial● amorphous carbon
Albert Bartok-Partay ESDG 2009
GAP for carbonBridging the GAP
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● force correlation DFT vs (REBO, GAP)
● = 0.26 eV/A
● 100 teaching points● 2.5 A cutoff
Albert Bartok-Partay ESDG 2009
GAP testBridging the GAP
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● diamond to graphite transition along arbitrary reaction coordinate
● energy from DFT, REBO and GAP
Albert Bartok-Partay ESDG 2009
GAP testBridging the GAP
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● how to obtain local energy from DFT● surface energy● visualisation
● best fit potential: use local energy● equivalent to chemical potential of an atom
● replacing atoms to infinity
Albert Bartok-Partay ESDG 2009
Defining local energiesBridging the GAP
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● restricted part of configurational space● use force and energy information only along a
minimisation from 'gas' phase or● blow up a configuration
● use it as post-processing tool● visualise 'hot' atoms
Albert Bartok-Partay ESDG 2009
Approximate local energiesBridging the GAP
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● interpolation in high-dimensional space via GP● target is quantum mechanical data● extendable method● carbon potential that captures sp2-sp3
transition● new approach in defining local energies
Albert Bartok-Partay ESDG 2009
SummaryBridging the GAP
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● more than one atom type:
● electrostatics: subtract Coulomb energy
Bridging the GAP
ESDG 2009
Albert Bartok-Partay ESDG 2009
Bridging the GAP