Hole-particle characterization of coupled-cluster singles ...
Operators for Atomic Cluster Optimization for Atomic Cluster Optimization R.M.Gamot,P.M.Rodger ......
1
Intelligent Water Drops with Perturbation Operators for Atomic Cluster Optimization R.M. Gamot, P.M. Rodger Centre for Scientific Computing, University of Warwick [email protected], [email protected] Overview The Intelligent Water Drops algorithm was mod- ified (MIWD) and adapted to allow it to deter- mine the most stable configurations, for the first time, of Lennard-Jones (LJ), Binary LJ (BinLJ), Morse and Janus Clusters. The algorithm, re- ferred as MIWD+PerturbOp, is an unbiased type of algorithm where no a priori cluster geome- try information and construction were used dur- ing initialization. Cluster perturbation opera- tors were applied to clusters generated by MIWD to further generate lower energies. A limited- memory quasi-Newton algorithm, called L-BFGS, was utilized to further relax clusters to its nearby local minimum. Basic Properties of IWD a) A B i j i j b) A B i j i j c) i j i j A B m n m n d) Figure 1: A path measures quality of connec- tivity between particles. (a) An IWD gathers soil (brown ellipse) as it flows from particle i to particle j while path(i,j) loses an amount of soil; (b) Soil gathered increases with IWD veloc- ity; (c) An IWD travelling on a path with lesser soil, path(m,n), will gather more soil and higher velocity. (d) The algorithm progressively builds the cluster by choosing the connectivity with de- sirable measures. FlowChart Modifications to IWD 1. The probability of choosing a path depends on amount of soil and the potential energy. p IWD i,j = f (soil(i,j ))η(i,j ) ∑ kV IWD a f (soil(i,j ))η(i,j ) η(i, j )= 1 2+V type (r i,j ) V M = e a(1-r i,j ) ( e a(1-r i,j ) - 2 ) V LJ (r i,j )=4ε i,j ( σ i,j r i,j ) 12 - ( σ i,j r i,j ) 6 V J (r i,j )= V LJ (r i,j )f ( Ω i ) f ( Ω j ) f (Ω i )= -exp θ 2 i,j 2σ 2 + exp (θ i,j -180) 2 2σ 2 2. An appropriate heuristic undesirability factor, HUD, is chosen to fit atomic cluster optimiza- tion. HUD i,j =2+ V type (r i,j )+ μr i,j + β (max(0,r 2 i,j - D 2 )) 2 3. Worst iteration agent, TIW, affects the soil content as well. soil i,j = (1 + ρ)soil i,j + P i,j P i,j = ρ( soil IWD N -1 ) 4. L-BFGS was used as a relaxation algorithm for IWDs. On LJ Clusters Figure 2: Five independent LJ 98 test runs (color lines) (10,000 iterations/run) for Chen bounding volume showing decline in cluster en- ergy. Figure 3: Cubic Bounding volume and Grow Etch perturbation operator combination shows energy decline as tested on LJ 38 . Runs of MIWD alone shows improvement as iterations progress (Fig. 2). Final runs for MIWD+GrowEtch, utilizing spherical bounding volume for scattering of initial sites (Fig. 3), agrees with high-accuracy to (Cambridge Cluster Database) CCD results of up to 104 atoms. Compactness measures (Fig. 4) of this study ver- sus CCD results show high-accuracy. Rotation and translation reveal that chiral clusters were generated (Fig. 5). MIWD+GrowEtch achieved relatively high-success rates for difficult clusters compared to Basin-Hopping with Occasional Jumping (BHOJ)(Table 1). N MIWD+ BHOJ Energy GrowEt 38 100% 96% -173.928426591 75 50% 5% -397.492330983 76 20% 10% -402.894866009 77 10% 5% -409.083517124 98 75% 10% -543.665360771 102 35% 16% -569.363652496 103 40% 13% -575.766130870 104 15% 12% -582.086642068 Table 1: Good success rates with all "difficult" LJ clusters. Figure 4: Compactness of clusters MIWD+GrowEtch versus CCD. Figure 5: Row 1 : Overlayed clusters show- ing unmatched positions. Row 2 : Rotated and translated clusters showing matching configura- tions. On Binary LJ and Morse BINARY LJ : Tested for up to 50 atoms on 6 instances of σ BB =1.05 - 1.30. MIWD+Knead rediscovered the global minima (GM) for most of the clusters except for N = 41,43, 45 -49 for σ BB = 1.05 and N = 47 for σ BB = 1.10. MIWD+CutSpliceVar rediscovered most of the GM except for N = 30-32 for σ BB = 1.30, N = 35 for σ BB = 1.05, 1.15, N = 36, 39-50 for σ BB = 1.05 and N = 47, 49-50 for σ BB = 1.10. Combination of perturbation operators (CombiOp) in Phase 2 (CutSplice+Knead, Cut- Splice+H1L2, CutSplice+H2L1, Knead+H1L2 and Knead+H2L1) were further done. Combina- tions were able to arrive at the GM except for N = 45 for σ BB = 1.05 (Fig. 6). MORSE : Tested for up to 60 atoms on 2 values of interparticle force range (a = 6, 14). MIWD+GrowEtch located the GM for most of the clusters except for N = 47, 55, 57, 58, 60 for a = 14 (Fig. 7). Figure 6: GM configurations generated from MIWD+CombiOp for selected Binary LJ Clus- ters. Figure 7: GM configurations from MIWD+GrowEtch for selected Morse Clusters. On Janus Clusters MIWD+CombiOP was applied on Janus clusters using the LJ potential as the patchy particles model but where anisotropic attraction and re- pulsion is modulated by an orientational depen- dent term MV ang (Fig. 8). Preliminary results were generated for cluster sizes N =3 - 30 (Fig. 9). MIWD with GrowEtch and Patch Orienta- tion Mutation produced the configurations with the lowest energies. 0 25 50 75 100 125 150 175 0 25 50 75 100 125 150 175 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 θ 1i,j θ 1j,i MV ang 0 25 50 75 100 125 150 175 0 25 50 75 100 125 150 175 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 θ 1i,j θ 1j,i MV ang Repulsion Attraction Weak Attraction Figure 8: Orientation measure, MV ang , for pairs of angles between 0 to 180. MV ang for σ = 90(Green) and σ = 30(Red). Figure 9: Lowest Cluster Energies generated by MIWd+CombiOp for Janus clusters sizes N =3 - 30. Figure 10: Observed basic structures in Janus Clusters. Figure 11: Janus cluster configurations with lowest energies. Acknowledgements Study is funded by Warwick Chancellor’s Scholarship (formerly WPRS) and Centre for Scientific Com- puting. Computing facilities are provided by MidPlus Regional Centre of Excellence for Computational Science, Engineering and Mathematics under EPSRC grant EP/K000128/1. RMT Gamot is also sup- ported by the University of the Philippines (UP) System under the UP Doctoral Studies Fund.
Transcript of Operators for Atomic Cluster Optimization for Atomic Cluster Optimization R.M.Gamot,P.M.Rodger ......
Intelligent Water Drops with PerturbationOperators for Atomic Cluster Optimization
R.M. Gamot, P.M. RodgerCentre for Scientific Computing, University of [email protected], [email protected]
OverviewThe Intelligent Water Drops algorithm was mod-ified (MIWD) and adapted to allow it to deter-mine the most stable configurations, for the firsttime, of Lennard-Jones (LJ), Binary LJ (BinLJ),Morse and Janus Clusters. The algorithm, re-ferred as MIWD+PerturbOp, is an unbiased typeof algorithm where no a priori cluster geome-try information and construction were used dur-ing initialization. Cluster perturbation opera-tors were applied to clusters generated by MIWDto further generate lower energies. A limited-memory quasi-Newton algorithm, called L-BFGS,was utilized to further relax clusters to its nearbylocal minimum.
Basic Properties of IWDa)
A B
i j i j
b)
A B
i j i j
c)
i j i j
A B
m n m n
d)
Figure 1: A path measures quality of connec-tivity between particles. (a) An IWD gatherssoil (brown ellipse) as it flows from particle ito particle j while path(i,j) loses an amount ofsoil; (b) Soil gathered increases with IWD veloc-ity; (c) An IWD travelling on a path with lessersoil, path(m,n), will gather more soil and highervelocity. (d) The algorithm progressively buildsthe cluster by choosing the connectivity with de-sirable measures.
FlowChart
Modifications to IWD1. The probability of choosing a path depends onamount of soil and the potential energy.
pIWDi,j
=f(soil(i,j))η(i,j)∑
kV IWDa
f(soil(i,j))η(i,j)
η(i, j) = 12+Vtype(ri,j)
VM = ea(1−ri,j)
(ea(1−ri,j) − 2
)VLJ (ri,j) = 4εi,j
((σi,jri,j
)12−(σi,jri,j
)6)
VJ (ri,j) = VLJ (ri,j)f
(Ωi
)f
(Ωj
)f(Ωi) = −exp
(θ2i,j
2σ2
)+ exp
((θi,j−180)2
2σ2
)2. An appropriate heuristic undesirability factor,HUD, is chosen to fit atomic cluster optimiza-tion.HUDi,j = 2 + Vtype(ri,j) + µri,j+
β(max(0, r2i,j
−D2))2
3. Worst iteration agent, TIW, affects the soilcontent as well.soili,j = (1 + ρ)soili,j + Pi,j
Pi,j = ρ( soilIWD
N−1)
4. L-BFGS was used as a relaxation algorithmfor IWDs.
On LJ Clusters
Figure 2: Five independent LJ98 test runs(color lines) (10,000 iterations/run) for Chenbounding volume showing decline in cluster en-ergy.
Figure 3: Cubic Bounding volume and GrowEtch perturbation operator combination showsenergy decline as tested on LJ38.Runs of MIWD alone shows improvement asiterations progress (Fig. 2). Final runs forMIWD+GrowEtch, utilizing spherical boundingvolume for scattering of initial sites (Fig. 3),agrees with high-accuracy to (Cambridge ClusterDatabase) CCD results of up to 104 atoms.Compactness measures (Fig. 4) of this study ver-sus CCD results show high-accuracy. Rotationand translation reveal that chiral clusters weregenerated (Fig. 5). MIWD+GrowEtch achievedrelatively high-success rates for difficult clusterscompared to Basin-Hopping with OccasionalJumping (BHOJ)(Table 1).
N MIWD+ BHOJ EnergyGrowEt
38 100% 96% -173.92842659175 50% 5% -397.49233098376 20% 10% -402.89486600977 10% 5% -409.08351712498 75% 10% -543.665360771102 35% 16% -569.363652496103 40% 13% -575.766130870104 15% 12% -582.086642068
Table 1: Good success rates with all "difficult"LJ clusters.
Figure 4: Compactness of clustersMIWD+GrowEtch versus CCD.
Figure 5: Row 1 : Overlayed clusters show-ing unmatched positions. Row 2 : Rotated andtranslated clusters showing matching configura-tions.
On Binary LJ and MorseBINARY LJ : Tested for up to 50 atoms on 6instances of σBB = 1.05 − 1.30. MIWD+Kneadrediscovered the global minima (GM) for mostof the clusters except for N = 41,43, 45 -49for σBB = 1.05 and N = 47 for σBB = 1.10.MIWD+CutSpliceVar rediscovered most of theGM except for N = 30-32 for σBB = 1.30, N =35 for σBB = 1.05, 1.15, N = 36, 39-50 for σBB= 1.05 and N = 47, 49-50 for σBB = 1.10.
Combination of perturbation operators(CombiOp) in Phase 2 (CutSplice+Knead, Cut-Splice+H1L2, CutSplice+H2L1, Knead+H1L2and Knead+H2L1) were further done. Combina-tions were able to arrive at the GM except for N= 45 for σBB = 1.05 (Fig. 6).MORSE : Tested for up to 60 atoms on 2values of interparticle force range (a = 6, 14).MIWD+GrowEtch located the GM for most ofthe clusters except for N = 47, 55, 57, 58, 60 fora = 14 (Fig. 7).
Figure 6: GM configurations generated fromMIWD+CombiOp for selected Binary LJ Clus-ters.
Figure 7: GM configurations fromMIWD+GrowEtch for selected Morse Clusters.
On Janus ClustersMIWD+CombiOP was applied on Janus clustersusing the LJ potential as the patchy particlesmodel but where anisotropic attraction and re-pulsion is modulated by an orientational depen-dent term MVang (Fig. 8). Preliminary resultswere generated for cluster sizes N = 3 − 30 (Fig.9). MIWD with GrowEtch and Patch Orienta-tion Mutation produced the configurations withthe lowest energies.
0 25 50 75 100 125 150 175
025
5075
100125
150175
−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
θ1i,j
θ1j,i
MVang
0 25 50 75 100 125 150 1750255075100125150175
−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
θ1i,j
θ1j,i
MVang
Repulsion
Attraction
WeakAttraction
Figure 8: Orientation measure, MVang , forpairs of angles between 0 to 180. MVang forσ = 90(Green) and σ = 30(Red).
Figure 9: Lowest Cluster Energies generated byMIWd+CombiOp for Janus clusters sizes N = 3−30.
Figure 10:Observed basicstructures inJanus Clusters.
Figure 11:Janus clusterconfigurationswith lowestenergies.
AcknowledgementsStudy is funded by Warwick Chancellor’s Scholarship (formerly WPRS) and Centre for Scientific Com-puting. Computing facilities are provided by MidPlus Regional Centre of Excellence for ComputationalScience, Engineering and Mathematics under EPSRC grant EP/K000128/1. RMT Gamot is also sup-ported by the University of the Philippines (UP) System under the UP Doctoral Studies Fund.
