Z. Phys. D 40, 194–197 (1997)
ZEITSCHRIFT
FÜR PHYSIK D
c Springer-Verlag 1997
Surveying a potential energy surface by eigenvector-following
Applications to global optimisation and the structural transformations of clusters
J.P.K. Doye, D.J. Wales
University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, UK
Received: 4 July 1996
Abstract. We have developed a method to search potential
energy surfaces which avoids some of the difficulties associated with trapping in local minima. Steps are directly taken
between minima using eigenvector-following. Exploration
of this space by low temperature Metropolis Monte Carlo
is a useful global optimisation tool. This method successfully finds the lowest energy icosahedral minima of LennardJones clusters from random starting configurations, but cannot find the global minimum in a reasonable time for difficult
cases such as the 38-atom Lennard-Jones cluster where the
face-centred-cubic truncated octahedron is lowest in energy.
However, by performing searches at higher temperatures,
we have found a pathway between the truncated octahedron
and the lowest energy icosahedral minima. Such a pathway
may be illustrative of some of the structural transformations
that are observed for supported metal clusters by electron
microscopy.
PACS: 02.60.Pn; 36.40.-c; 61.46.+w
1 Introduction
Most methods used to search a potential energy surface
(PES) take steps either in configuration or phase space and
are hindered by the vast number of minima which generally exist on a multidimensional PES. Trajectories on a PES
involve both vibrational motion about the minima and rearrangements between minima that involve crossing barriers.
At high temperatures, the time scales for these two types of
motion are of the same order. However, at low temperatures,
the time scale for interwell motion is much longer than for
intrawell motion, and this can lead to the system becoming
trapped in local minima. This effect makes ergodicity hard
to achieve and gives rise to problems for global optimisation strategies such as simulated annealing [1]. A great deal
of work has been done to try to reduce the negative effects
of trapping. For example, the jump-walking method tries to
improve the ergodicity of low temperature simulations by
allowing jumps to configurations sampled at higher temperatures [2], and the taboo search method attempts to increase
the rate of interwell motion by forbidding steps that go into
regions of configuration space that have recently been visited
[3].
An alternative strategy is to remove the intrawell motion
altogether by taking steps directly between minima using
eigenvector-following [4]. Here we illustrate some of the
advantages of this method by application to Lennard-Jones
(LJ) clusters.
2 Method
The eigenvector-following technique can find transition
states on a PES by maximising the energy along a chosen
eigenvector of the Hessian whilst simultaneously minimising the energy along all other eigenvectors. Once a transition
state is located, the minima it connects can be found by taking small initial steps away from the stationary point along
the eigenvector of the Hessian corresponding to the unique
negative eigenvalue (in both the positive and negative directions) and then performing minimisations. The details of our
implementation of the eigenvector-following method can be
found elsewhere [5].
The essential points of our PES-searching method are as
follows. First, we find a minimum from which to start our
search. From this minimum, we can search for a transition
state along any of the eigenvectors of the Hessian in both the
positive and negative directions, giving 6N −12 possibilities.
We make a random choice of which eigenvector to follow,
but weight this choice towards those with lower eigenvalues,
because a search along an eigenvector with a low eigenvalue
generally converges to a transition state more rapidly. Once
a transition state is found, we find the minima it connects.
Eigenvector-following occasionally finds a transition state
unconnected to the original minimum, and so we check that
the original minimum is at one end of the rearrangement
pathway.
Having found a connected minimum, a decision whether
to accept the step must be taken. Here, we choose a Metropolis step criterion, which accepts moves with a probability of
min[1, exp(−∆E/kT )], where ∆E is the change in energy
associated with the step [6]. This choice simply results in
195
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a
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-37
Energy / ε
-38
-39
-40
-41
-42
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-44
0
1
2
3
4
5
6
S /σ
7
8
9
10
b
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Energy / ε
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-168
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0
5
10
15
S /σ
20
25
30
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c
Energy / ε
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record is kept of each transition state search, so that if the
same step is attempted the search does not need to be repeated.
The categories provided by Kunz and Berry for analysing
the topography of a PES [7, 8] are particularly useful for
understanding the behaviour of our PES-searching method.
These authors defined a basin as a set of minima on the PES
which belong to monotonic sequences of minima leading
down to the lowest energy minimum in the basin. (This
definition overlaps with the concept of a funnel that is used
in the protein folding literature [9, 10].) The global minimum
is at the bottom of the primary basin. Basins have a similar
role in this ‘minima space’ to that played by minima in
configuration space. However, there are far fewer basins than
minima on the PES. The time scale for interbasin flow is
much slower than for intrabasin flow between minima [7,
8]. Therefore, secondary basins (basins that do not end at
the global minimum) can act as traps.
If a Monte Carlo walk in the minima space is performed
at zero temperature, the system is ‘quenched’ to the bottom
of a basin. If there are a small number of basins on the
PES, we might expect that a quench from a random configuration might lead to the global minimum, especially as
it is likely that the primary basin would have the largest
catchment area. This suggests that these quenches may be a
useful global optimisation tool. If a low rather than a zero
temperature is used, the method would be able to escape
from shallow basins, whilst not significantly compromising
the favourability of downhill moves.
We investigate the properties of our PES-searching
method for LJ clusters, which have been much studied. The
LJ potential has the form
"
#
X σ 12 σ 6
−
,
(1)
V = 4
rij
rij
i<j
where rij is the distance between atoms i and j. We use and σ as the units of energy and distance.
-270
3 Results
-275
-280
0
5
10
15
S /σ
20
25
30
Fig. 1a–c. Reaction profiles of typical downhill pathways from a random
minimum to the lowest energy icosahedral minimum for a LJ13 , b LJ38 and
c LJ55 . Stationary points are denoted by diamonds. S is the integrated path
length along the reaction pathway [5]
a Monte Carlo simulation of this discrete system of minima. There are many other possible step criteria that could
be used. For example, the probability of acceptance could
be related to the transition rate as calculated using RRKM
theory, and so could be used to simulate the dynamics of the
real cluster (assuming the density of states of the minimum
and transition state could be calculated accurately). Once the
decision to accept the step has been made the whole process
is repeated, and a new transition state search is started. A
Our PES-searching method has been applied to LJ13 , LJ38
and LJ55 . For the 13- and 55-atom clusters the global minima
are Mackay icoshedra [11]. However, it has been recently
shown that the global minimum for the 38-atom cluster is a
face-centred-cubic (fcc) truncated octahedron [12, 13]. This
cluster therefore represents a relatively difficult test for a
global optimisation method.
The energy profiles for typical low temperature quenches
performed from random starting configurations are shown
in Fig. 1 for the three clusters. The quenches for the 13atom cluster typically find the icosahedral global minimum
in three or four accepted steps. This is not surprising given
the relative simplicity of the PES; the number of minima
on the surface is probably of the order of 1500 [14, 15].
Furthermore, a comprehensive survey of the topography of
this PES has shown that none of the minima are more than
four steps away from the global minimum [15].
The quenches for LJ55 also find the global minimum,
although the pathway and the number of steps is inevitably
196
1
2
6
7
11
12
3
8
13
4
5
9
10
14
15
Fig. 2. Minima on the LJ38 pathway from the truncated octahedron to the lowest energy icosahedral minimum. The minima are numbered by their position
along the pathway
larger because of the increased system size. Still it is a remarkable feature of this PES that the global minimum is so
easy to find considering there are probably of the order of
1021 minima on the PES [16]. In the last part of the pathway
shown in Fig. 1c the cluster finds the Mackay icosahedron
in two steps from a low energy liquid-like minimum, rather
than by passing through defective Mackay icosahedra as was
observed for some of the other pathways.
For LJ38 the system never finds the fcc global minimum
but instead the quenches always take the system down into
the basin of icosahedral minima. The lowest energy icosahedral minimum is a C5v structure which has only been
recently reported [17]. Although this represents a failure of
our method as a global optimisation tool, it provides us with
some very interesting information about the topography of
the PES. It seems that the basin containing the icosahedral
minima is much more easily accessed from higher energy
minima than the basin leading down to the fcc minima. This
is probably a result of the greater structural similarity between the icosahedral structures and the liquid. It has been
shown that simple liquids have significant polytetrahedral
character (the structure is a tessellation of all space by tetrahedra with atoms at the vertices) [18–20], and the icosahedral structures have the greatest degree of polytetrahedral
character of all the ordered types of structure that clusters
adopt.
One difference between the quench pathway for LJ38
and that of the other two clusters is that in the last stages
of the pathway the system has to search through the low
energy icosahedral isomers. The bottom of the icosahedral
basin is flatter than for LJ13 and LJ55 , both of which have
global minima that are significantly lower in energy than
the next lowest energy minimum, making the last stages of
global optimisation easier. For comparison, the energy gaps
between the lowest energy icosahedral minima are 2.644 ,
0.118 and 2.855 for LJ13 , LJ38 and LJ55 , respectively.
The PES of LJ75 has some similarities to LJ38 in that it
has a non-icosahedral global minimum which is very hard to
find by global optimisation [13]. For this cluster the global
minimum is a Marks’ decahedron [21]. However, model calculations on this system suggest that the icosahedral structures become the free energy global minimum at low temperature because the large number of these minima gives
them a higher configurational entropy [13].
Similarly, it may be that the 38-atom truncated octahedra is the free energy global minimum only at low temperature. If this is the case, we would expect a high temperature
Monte Carlo walk in minima space starting from the truncated octahedra to escape from the fcc basin and pass into
the icosahedral basin, and this is indeed what we observed.
The length of the initial pathway between the two lowest
energy structures produced by this method was about 100 σ.
However, by performing quenches on either side of the barrier, the pathway was significantly shortened. The resulting
pathway is illustrated in Figs. 2 and 3.
Initially the truncated octahedron undergoes a series of
rearrangements that take it to the high energy hexagonal
close-packed minimum 6. This structure then collapses to
a polytetrahedral structure typical of the liquid-like state,
which becomes more and more ordered until it reaches the
decahedral structure 11. After two localised rearrangements,
the decahedral minimum 13 is then converted to an icosahe-
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pensive, involving repeated calculation and diagonalisation
of the Hessian matrix. Furthermore, our method does not
offer a solution to difficult cases where the basin associated
with the global minimum is not the most accessible.
However, the real advantage of this method is that it can
give detailed information about the topography of the PES,
such as rearrangement pathways for relaxation to the global
minimum and structural transformations. Hence, we obtain
further insight into the behaviour of the system, especially
the dynamics [23].
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Energy / ε
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9
45
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10
3
11 12
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1
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0
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20
25
We are grateful to the the Engineering and Physical Sciences Research
Council (J.P.K.D.) and the Royal Society (D.J.W.) for financial support.
30
S /σ
Fig. 3. Reaction profile of a pathway on the LJ38 PES from the truncated
octahedron to the lowest energy icosahedral minimum. Stationary points are
denoted by diamonds. The minima are numbered by their position along
the pathway
dral minimum by a π/5 rotation of one half of the structure
about the decahedral axis.
In electron microscopy studies of metal clusters supported on a surface, transformations are often observed between icosahedral, decahedral and fcc structures [22]. The
pathway we have found here, although for a cluster much
smaller than those observed experimentally, is probably
fairly typical of the complex series of rearrangements and
high energy barriers that need to be overcome to move between ordered structures with different morphologies.
4 Conclusion
We have illustrated some of the properties of a PES-searching
method that takes steps directly between minima, particularly focussing on its potential for global optimisation. Although it is quite successful our method is probably less
efficient than techniques such as genetic algorithms [17],
because the transition state searches are computationally ex-
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