Intelligent Water Drops with Perturbation
Operators for Atomic Cluster Optimization
R.M. Gamot, P.M. Rodger
Centre for Scientific Computing, University of Warwick
r.m.gamot@warwick.ac.uk, p.m.rodger@warwick.ac.uk
B
i
c)
j i
j
d)
j i
j
2
1
3
k
l
l
k
4
B
A
A path measures quality of connectivity 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 velocity; (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 desirable measures.
Figure 1:
Energy
-165
1010
GM configurations from MIWD+GrowEtch
for selected Morse Clusters.
Figure 6:
-170
10 8
10 6 0.0
0
100.5
101.0
101.5
102.0
102.5
103.0
103.5
104.0
-175
0
Iteration Number
5
10
Cluster Number
15
20
Left : Five independent LJ98 test runs
(color lines) (10,000 iterations/run) showing decline
in cluster energy. Right : Cubic Bounding volume
and Grow Etch perturbation operator combination
shows energy decline as tested on LJ38 .
Runs of MIWD alone show improvement as iterations progress. Energies of the final runs for
MIWD+GrowEtch, utilizing spherical bounding
volume for scattering of initial sites, agree exactly
with CCD (Cambridge Cluster Database) results up
to N = 104 atoms. Compactness measures (Fig. 3)
of this study versus CCD results show high-accuracy.
Rotation and translation reveal that chiral clusters were generated. MIWD+GrowEtch achieved
relatively high-success rates for difficult clusters
compared to Basin-Hopping (BH), Basin-Hopping
with Occasional Jumping (BHOJ) and Parallel Fast
Annealing Evoluationary Algorithm (PFAEA) (Table
1).
Cluster MIWD+ BH BHOJ PFAEA
Size
PerOp
38
100%
87% 96%
39%
50%
1%
5%
1%
75
76
20%
5%
10%
4%
77
10%
6%
5%
2%
98
75%
10% 10%
4%
102
5%
3%
16%
9%
103
40%
3%
13%
10%
104
15%
3%
12%
7%
Table 1: Good success rates with all "difficult" LJ
clusters.
Figure 2:
FlowChart of MIWD
End
Initialize static parameters and
dynamic parameter
.
Output best IWDs
Maximum
perturbations
achieved?
No
Probabilistically select next particle, j, from
recently selected particle, i, of current IWD.
Subject the IWDs
to L-BFGS.
Yes
Update velocity of current IWD.
No
Calculate amount of soil removed
from the path atom i to atom j
Maximum
runs
achieved?
Maximum
iterations
achieved?
Update Total Best and
Worst IWDs for this run.
Update soil content of paths
belonging to best and worst IWDs.
No
Identify Best and Worst
IWDs for this iteration.
Yes
No
Yes
Modifications to IWD
1. The probability of choosing a path depends on
amount of soil and the potential energy.
f (soil(i,j))η(i,j)
IW D
P
pi,j =
f (soil(i,j))η(i,j)
1
2+Vtype (ri,j )
a(1−ri,j )
a(1−ri,j )
e
σi,j
ri,j
−2
12
−
σi,j
ri,j
VJ (ri,j ) = VLJ
(ri,j )f
(Ω
i ) f (Ωj )
f (Ωi ) = −exp
2σ 2
0
0.2
0
-1
150
100
0
0
-0.2
-0.5
-0.4
-1
0
-0.5
-0.6
50
0
θ 1j,i
0
50
100
150
100
50
0.5
0.5
-0.5
150
0
50
100
150
θ 1j,i
θ 1i,j
1
θ 1i,j
1
0.6
0.4
0.5
0.5
Attractive
0.2
0
Weak
Attraction
0
-0.2
Repulsive
-0.5
0
0.5
Attractive
Weak
Attraction
0
-0.5
-0.4
-0.5
-0.6
Repulsive
-1
-1
0
θ 1i,j
50
100
150
0
50 100 150
θ 1j,i
150 100 50 0
θ 1i,j
150
100
50
0
θ 1j,i
Orientation interaction, M Vang , for pairs of
◦
◦
angles between 0 to 180 . M Vang for σ = 90 (Column 1) and σ = 30 (Column 2).
Figure 7:
0
0
-100
-10
-200
-20
-300
-30
LJ
MIWD+GrowEtchOr
10
20
30
40
50
MIWD+GrowEtchOr
MIWD+OrDis
MIWD+NeighDis
MIWD+OrGrowEtch
MIWD+GrowEtchNeigh
MIWD+NeighGrowEtch
10
20
30
40
50
Left : Lowest Cluster Energies generated by
MIWd+CombiOp for Janus clusters sizes N = 3 − 50.
Right : Observed basic structures in Janus Clusters.
0.8
8
7
0.7
6
0.6
0.5
0.4
+ exp
6 (θi,j −180)
2σ 2
2
2. An appropriate heuristic undesirability factor,
HUD, is chosen to fit atomic cluster optimization.
2
HU Di,j = η(i, j)−1 + µri,j + β(max(0, ri,j
− D2 ))2
3. Worst iteration agent, TIW, affects the soil content
as well.
IW D soil
soil(i, j) = (1 + ρ)soili,j + ρ
N −1
4. L-BFGS was used as a relaxation algorithm for
IWDs.
Row 1 : Overlayed clusters showing unmatched positions. Row 2 : Rotated and translated
clusters showing matching configurations.
Figure 4:
MIWD+GrowEtchOr
LJ
5
4
3
2
0.3
1
0.2
10
20
30
40
0
50
10
20
Cluster Size
30
40
50
Cluster Size
Left : Average orientation measure of each
Janus particle with its neighbourhing particles. Right
: Compactness of Janus clusters compared to global
optima LJ clusters.
Figure 9:
On Binary LJ and Morse
BINARY LJ : Tested for up to 50 atoms on 6
instances of σBB = 1.05 − 1.30. MIWD+Knead
rediscovered the global minima (GM) for most of the
clusters except for N = 41, 43, 45-49 for σBB = 1.05
and N = 46,47 for σBB = 1.10. MIWD+CutSpliceVar
rediscovered most of the GM except for N = 35-36,
39-50 for σBB = 1.05, N = 46-47, 49-50 for σBB
= 1.10, N = 35 for σBB = 1.15, and N = 30-32
for σBB = 1.30.
Combination of perturbation
operators (CombiOp) in Phase 2 (CutSplice+Knead,
CutSplice+H1L2, CutSplice+H2L1, Knead+H1L2
and Knead+H2L1) were further done. Combinations
were able to arrive at the GM except for N =
41,43,45,46,48,49 for σBB = 1.05 and N = 46 for σBB
= 1.10.
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.
Snapshots from iterations of
MIWD+CombiOp for J100 .
Geometries in a
and b are from Phase 1 while structures obtained
from d to m are Phase 2 results. Geometry in c is
the relaxed configuration resulting from Phase 1.
Figure
10:
×1012
3.5
9
8
3
0
2.5
Energy
Phase 1
Update soil content from
path atom i to atom j and IWD.
2
θi,j
0.4
Yes
No
Completed all
IWDs?
Compactness of clusters MIWD+GrowEtch
versus CCD.
Figure 3:
Phase 2
Apply perturbation operators
to relaxed clusters and
replace current if new
is better.
Assign unique initial
particle site to each IWD.
VLJ (ri,j ) = 4εi,j
0.5
Figure 8:
Initialize dynamic parameters
and
Cluster atoms
reached N?
1
0.6
Cluster Size
Randomly scatter
initial sites on a sphere.
VM = e
1
-50
Yes
η(i, j) =
MIWD+CombiOP was applied on Janus clusters using the LJ potential as the patchy particles model but
where anisotropic attraction and repulsion is modulated by an orientational dependent term M Vang (Fig.
7). Configurations were predicted for cluster sizes
N = 3 − 50 and N = 100. MIWD with GrowEtch
and Patch Orientation Mutation produced the configurations with the lowest energies.
-40
Start
kεVaIW D
On Janus Clusters
MVang
1012
Compactness
i
1014
2
1.5
Compactness
A
1016
MVang
j
Iter80
Iter100
Energy
j i
Iter40
Iter60
Average Cluster Orientation Measure
i
Iter1
Iter20
MVang
b)
LJ38 GO
Phase 1
MVang
a)
-160
1018
Energies
Basic Properties of IWD
On LJ Clusters
Energy
Overview
The Intelligent Water Drops algorithm was modified (MIWD) and adapted to allow it to determine
the most stable configurations, for the first time,
of Lennard-Jones (LJ), Binary LJ (BinLJ), Morse
and Janus Clusters. The algorithm, referred as
MIWD+PerturbOp, is an unbiased type of algorithm where no a priori cluster geometry information and construction were used during initialization. Cluster perturbation operators were applied to clusters generated by MIWD to further
generate lower energies. A limited-memory quasiNewton algorithm, called L-BFGS, was utilized
to further relax clusters to its nearby local minimum.
-50
-100
1
cd e f g h i j k l m
Snapshot
7
6
5
0.5
4
0
-0.5
a
b
c d
e
f
g
h
Snapshot
GM configurations generated from
MIWD+CombiOp for selected Binary LJ Clusters.
Figure
5:
i
j
k
l
m
3
a
b
c d
e
f
g
h
i
j
k
l
m
Snapshot
Left : Energies of Janus configurations in
snapshots a to m. Right : Compactness of configurations in snapshots a to m.
Figure 11:
Acknowledgements
Study is funded by Warwick Chancellor’s Scholarship (formerly WPRS) and Centre for Scientific Computing. Computing facilities are provided by the Centre for
Scientific Computing of the University of Warwick with support from the Science Research Investment Fund. RMT Gamot is also supported by the University of the
Philippines (UP) System under the UP Doctoral Studies Fund.