This note advertises a simple necessary condition for optimality that any list Nvx(N) of computer-generated putative lowest average pair energies vx(N) of clusters that consist of N monomers has to satisfy whenever the monomers interact with each other through pair forces satisfying Newton’s “action equals re-action.” These can be quite complicated, as, for instance, in the TIP5P model with five-site potential for a rigid tetrahedral-shaped H2O monomer of water, or as simple as the Lennard-Jones single-site potential for the center of an atomic monomer (which is also used for one site of the H2O monomer in the TIP5P model, which in addition has four peripheral sites with Coulomb potentials). The empirical usefulness of the necessary condition is demonstrated by testing a list of publicly available Lennard-Jones cluster data that have been pooled from 17 sources, covering the interval 2 ≤ N ≤ 1610 without gaps. The data point for N = 447 failed this test, meaning the listed 447-particle Lennard-Jones cluster energy was not optimal. To implement this test for optimality in search algorithms for putatively optimal configurations is an easy task. Publishing only the data that pass the test would increase the odds that these are actually optimal, without guaranteeing it, though.

1.
M. W.
Mahoney
and
W. L.
Jorgensen
, “
A five-site model for liquid water and the reproduction of the density anomaly by rigid, nonpolarizable potential functions
,”
J. Chem. Phys.
112
,
8910
8922
(
2000
).
2.
D. J.
Wales
,
J. P. K.
Doye
,
A.
Dullweber
,
M. P.
Hodges
,
F. Y.
Naumkin
,
F.
Calvo
,
J.
Hernández-Rojas
, and
T. F.
Middleton
, The Cambridge cluster database, https://www-wales.ch.cam.ac.uk/CCD.html.
3.
X.
Shao
, The structures of the optimized Lennard-Jones clusters, https://chinfo.nankai.edu.cn/labintroe.html.
4.
L. T.
Wille
and
J.
Vennik
, “
Computational complexity of the ground-state determination of atomic clusters
,”
J. Phys. A: Math. Gen.
18
,
L419
L422
(
1985
).
5.
A. B.
Adib
, “
NP-hardness of the cluster minimization problem revisited
,”
J. Phys. A: Math. Gen.
38
,
8487
8492
(
2005
).
6.
M. K.-H.
Kiessling
, “
A note on classical ground state energies
,”
J. Stat. Phys.
136
,
275
284
(
2009
).
7.
R.
Nerattini
,
J. S.
Brauchart
, and
M. K.-H.
Kiessling
, “
Optimal N-point configurations on the sphere: ‘Magic’ numbers and Smale’s 7th problem
,”
J. Stat. Phys.
157
,
1138
1206
(
2014
).
8.
B. J.
Sutherland
,
S. W.
Olesen
,
H.
Kusumaatmaja
,
J. W. R.
Morgan
, and
D. J.
Wales
, “
Morphological analysis of chiral rod clusters from a coarse-grained single-site chiral potential
,”
Soft Matter
15
,
8147
8155
(
2019
).
9.
S. N.
Fejer
,
D.
Chakrabarti
, and
D. J.
Wales
, “
Emergent complexity from simple anisotropic building blocks: Shells, tubes, and spirals
,”
ACS Nano
4
,
219
228
(
2010
).
10.
J. A.
Northby
, “
Structure and binding of Lennard-Jones clusters: 13 ≤ N ≤ 147
,”
J. Chem. Phys.
87
,
6166
6177
(
1987
).
11.
D. J.
Wales
and
J. P. K.
Doye
, “
Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms
,”
J. Phys. Chem. A
101
,
5111
5116
(
1997
).
12.
R. H.
Leary
, “
Global optima of Lennard-Jones clusters
,”
J. Global Optim.
11
,
35
53
(
1997
).
13.
J. P. K.
Doye
,
M. A.
Miller
, and
D. J.
Wales
, “
The double-funnel energy landscape of the 38-atom Lennard-Jones cluster
,”
J. Chem. Phys.
110
,
6896
6906
(
1999
).
14.
G. L.
Xue
, “
Minimum inter-particle distance at global minimizers of Lennard-Jones clusters
,”
J. Global Optim.
11
,
83
90
(
1997
).
15.
M.
Locatelli
and
F.
Schoen
, “
Global minimization of Lennard-Jones clusters by a two-phase monotonic method
,” in
Optimization and Industry: New Frontiers
,
Applied Optimization Vol. 78
, edited by
P. M.
Pardalos
and
V.
Korotkikh
(
Springer
,
2003
), pp.
221
240
.
16.
T.
Gregor
and
R.
Car
, “
Minimization of the potential energy surface of Lennard-Jones clusters by quantum optimization
,”
Chem. Phys. Lett.
412
,
125
130
(
2005
).
17.
X.
Lai
,
R.
Xu
, and
W.
Huang
, “
Prediction of the lowest energy configuration for Lennard-Jones clusters
,”
Sci. China Chem.
54
,
985
991
(
2011
).
18.
C. L.
Müller
and
I. F.
Sbalzarini
, “
Energy landscapes of atomic clusters as black box optimization benchmarks
,”
Evol. Comput.
20
,
543
573
(
2012
).
19.
M. K.
Cameron
, “
Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network
,”
J. Chem. Phys.
141
,
184113
(
2014
).
20.
M.
Cameron
and
E.
Vanden-Eijnden
, “
Flows in complex networks: Theory, algorithms, and application to Lennard-Jones cluster rearrangement
,”
J. Stat. Phys.
156
,
427
454
(
2014
).
21.
Y.
Forman
and
M.
Cameron
, “
Modeling aggregation processes of Lennard-Jones particles via stochastic networks
,”
J. Stat. Phys.
168
,
408
433
(
2017
).
22.
C.
Barrón-Romero
, “
The oLJ13_N13IC cluster is the global minimum cluster of Lennard Jones potential for 13 particles
,” in
2022 IEEE 3rd International Conference on Electronics, Control, Optimization and Computer Science (ICECOCS)
,
01–02 December 2022
.
24.
K.
Yu
,
X.
Wang
,
L.
Chen
, and
L.
Wang
, “
Unbiased fuzzy global optimization of Lennard-Jones clusters for N ≤ 1000
,”
J. Chem. Phys.
151
,
214105
(
2019
).
25.
J. P. K.
Doye
, Lennard-Jones clusters, http://doye.chem.ox.ac.uk/jon/structures/LJ.html.
26.
T.
James
,
D. J.
Wales
, and
J.
Hernández-Rojas
, “
Global minima for water clusters (H2O)n, n ≤ 21, described by a five-site empirical potential
,”
Chem. Phys. Lett.
415
,
302
307
(
2005
).
27.
Y.
Xiang
,
H.
Jiang
,
W.
Cai
, and
X.
Shao
, “
An effective method based on lattice construction and the genetic algorithm for optimization of large Lennard-Jones clusters
,”
J. Phys. Chem. A
108
,
3586
3592
(
2004
).
28.
X.
Shao
,
Y.
Xiang
, and
W.
Cai
, “
Structural transition from icosahedra to decahedra of large Lennard-Jones clusters
,”
J. Phys. Chem. A
109
,
5193
5197
(
2005
).
29.
L.
Paramonov
and
S. N.
Yaliraki
, “
The directional contact distance of two ellipsoids: Coarse-grained potentials for anisotropic interactions
,”
J. Chem. Phys.
123
,
194111
(
2005
).
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