Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones
Clusters Containing up to 110 Atoms
David J. Wales*
UniVersity Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, U.K.
Jonathan P. K. Doye
FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
ReceiVed: March 19, 1997; In Final Form: April 29, 1997X
We describe a global optimization technique using “basin-hopping” in which the potential energy surface is
transformed into a collection of interpenetrating staircases. This method has been designed to exploit the
features that recent work suggests must be present in an energy landscape for efficient relaxation to the
global minimum. The transformation associates any point in configuration space with the local minimum
obtained by a geometry optimization started from that point, effectively removing transition state regions
from the problem. However, unlike other methods based upon hypersurface deformation, this transformation
does not change the global minimum. The lowest known structures are located for all Lennard-Jones clusters
up to 110 atoms, including a number that have never been found before in unbiased searches.
I. Introduction
Global optimization is a subject of intense current interest.1
Improved global optimization methods could be of great
economic importance, since improved solutions to traveling
salesman-type problems, the routing of circuitry in a chip, the
active structure of a biomolecule, etc., equate to reduced costs
or improved performance. In chemical physics the interest in
efficient global optimization methods stems from the common
problem of finding the lowest energy configuration of a (macro)-
molecular system. For example, it seems likely that the native
structure of a protein is structurally related to the global
minimum of its potential energy surface (PES). If this global
minimum could be found reliably from the primary amino acid
sequence, this knowledge would provide new insight into the
nature of protein folding and save biochemists many hours in
the laboratory. Unfortunately, this goal is far from being
realized. Instead the development of global optimization
methods has usually concentrated on much simpler systems.
Lennard-Jones (LJ) clusters represent one such test system.
Here the potential is
where and 21/6σare the pair equilibrium well depth and
separation, respectively. We will employ reduced units, i.e.,
)σ)1 throughout. Much of the initial interest in LJ clusters
was motivated by a desire to calculate nucleation rates for noble
gases. However, as a result of the wealth of data generated,
the LJ potential has been used not only for studying global
optimization but also the effects of finite size on phase
transitions such as melting. Through the combined efforts of
many workers, likely candidates for the global minima of LJN
clusters have been found up to N)147.2-16 This represents a
significant achievement since extrapolation of Tsai and Jordan’s
comprehensive enumeration of minima for small LJ clusters17
suggests that the PES of the 147-atom cluster possesses of the
order of 1060 minima.18
Previous studies have revealed that the Mackay icosahedron19
provides the dominant structural motif for LJ clusters in the
size range of 10-150 atoms. Complete icosahedra are possible
at N)13, 55, 147, ... At most intermediate sizes the global
minimum consists of a Mackay icosahedron at the core covered
by a low-energy overlayer. As a consequence of the phase
behavior of LJ clusters, finding these global minima is relatively
easy. Studies have shown that in the region of the solid-liquid
transition the cluster is observed to change back and forth
between a liquid-like form and icosahedral structures.20 As a
result of this “dynamic coexistence,” a method as crude as
molecular dynamics within the melting region coupled with
systematic minimization of configurations generated by the
trajectory is often sufficient to locate the global minimum.21
However, there are a number of sizes at which the global
minimum is not based on an icosahedral structure. These
clusters are illustrated in Figure 1. For LJ38 the lowest energy
structure is a face-centered-cubic (fcc) truncated octahedron,13,14
and for N)75, 76, 77, 102, 103, and 104, geometries based
on Marks’ decahedra22 are lowest in energy.14,15 For these cases,
finding the lowest minimum is much harder because the global
minimum of free energy only becomes associated with the global
potential energy minimum at temperatures well below melting
where the dynamics of structural relaxation are very slow. For
LJ38, the microcanonical temperature for the transition from face-
centered cubic to icosahedral structures has been estimated to
be about 0.12k-1, where kis the Boltzmann constant, and for
XAbstract published in AdVance ACS Abstracts, June 15, 1997.
E)4∑
i<j
[
(
σ
rij
)
12 -
(
σ
rij
)
6
]
Figure 1. Nonicosahedral Lennard-Jones global minima.
5111J. Phys. Chem. A 1997, 101, 5111-5116
S1089-5639(97)00984-5 CCC: $14.00 © 1997 American Chemical Society
