Fast Global Optimization of Difficult Lennard-Jones Clusters.
The minimization of the potential energy function of Lennard-Jones atomic clusters has attracted much theoretical as well as computational research in recent years. One reason for this is the practical importance of discovering low energy configurations of clusters of atoms, in view of applications and extensions to molecular conformation research&semi; another reason of the success of Lennard Jones minimization in the global optimization literature is the fact that this is an extremely easy-to-state problem, yet it poses enormous difficulties for any unbiased global optimization algorithm.In this paper we propose a computational strategy which allowed us to rediscover most putative global optima known in the literature for clusters of up to 80 atoms and for other larger clusters, including the most difficult cluster conformations. The main feature of the proposed approach is the definition of a special purpose local optimization procedure aimed at enlarging the region of attraction of the best atomic configurations. This effect is attained by performing first an optimization of a modified potential function and using the resulting local optimum as a starting point for local optimization of the Lennard Jones potential.Extensive numerical experimentation is presented and discussed, from which it can be immediately inferred that the approach presented in this paper is extremely efficient when applied to the most challenging cluster conformations. Some attempts have also been carried out on larger clusters, which resulted in the discovery of the difficult optimum for the 102 atom cluster and for the very recently discovered new putative optimum for the 98 atom cluster.
Introduction
One of the simplest to describe yet most difficult to solve problems
in computational chemistry is the automatic determination of molecular
conformation. A molecular conformation problem can be described
c
2000 Kluwer Academic Publishers. Printed in the Netherlands.
as that of finding the global minimum of a suitable potential energy
function which depends on relative atom positions. Many models have
been proposed in the literature, ranging from very simple to extremely
like, e.g., the so-called protein folding problem (Neumaier,
1997). While realistic models of atomic interactions take into account
different components of the potential energy function, like pairwise in-
teractions, dihedral angles, torsion, and allow the analysis of composite
molecules in which atom of different kinds interact, it is commonly
recognized in the chemical literature that a fundamental step towards
a better understanding of some molecular conformation problems is
the knowledge of the global minimum of the so called Lennard-Jones
potential energy model; this model is a sufficiently accurate one for
noble gas clusters. Moreover some of the most difficult to find Lennard-Jones
structures, exactly those towards which this paper is oriented,
were found to represent very closely the structure of nickel and gold
clusters (Wales and Scheraga, 1999).
In this model all atoms are considered to be equal and only pairwise
interaction is included in the definition of the potential energy. Let N -
2 be an integer representing the total number of atoms. The Lennard-Jones
(in short L-J) pairwise potential energy function is defined as
follows: if the distance between the centers of a pair of atoms is r, then
their contribution to the total energy is defined to be
r 12
\Gammar 6
(1)
and the L-J potential energy E of the molecule is defined as
represents the coordinates of the center of the i-th
atom and the norm used is the usual Euclidean one. An optimum L-
N g is defined as the solution of the
global optimization problem
Although extremely simplified, this model has attracted research
in chemistry and biology, as it can be effectively considered as a reasonably
accurate model of some clusters of rare gases and as it represents
an important component in most of the potential energy models
used for complex molecular conformation problems and protein folding
(Neumaier, 1997).
From the point of view of numerical optimization methods, this
problem is an excellent test for local and global unconstrained optimization
methods; it is one of the simplest models in the literature of test
problems (see e.g. Floudas and Pardalos, 1999 pag. 188-193), yet one
of the most difficult as it has been conjectured (Hoare, 1979) that the
number of local optimum conformations grows at least exponentially
with N . Many computational approaches can be found in the literature,
ranging from mathematical programming models (Gockenbach et al.,
1997), to genetic methods (Deaven et al., 1996), to Monte Carlo sampling
(Wales and Doye, 1997) and many others. For a recent review of
the state of the art in this subject, the reader may consult (Wales and
Scheraga, 1999).
Unfortunately very few theoretical results are available which could
be used to tune an optimization method for the L-J minimization
problem. One notable exception is the result of (Xue, 1997) where it
is proven that in any global optimum the pairwise interatomic distance
is bounded from below by 0:5: while this result is considered
obvious in the chemical literature, only recently it has been proven
in a formal way. In practice, in all known putative global optima,
no pair of atoms is observed whose distance is less than about 0.95.
In the literature, pairs of atoms which are roughly 1.0 unit apart are
called "near neighbors". Very little is a-priori known on the structure
of the global optima; even a quite reasonable conjecture stating that
the diameter of an optimum N-atoms cluster is O(N 1=3 ) is still open.
Thus, except for extremely small and easy cases, there are no proofs
of global optimality for the putative global optimum configurations
known in the literature (see the L-J page on the Cambridge Clusters
Database at URL http://www.brian.ch.cam.ac.uk). All published
results are aimed at confirming, through experimentation, numerical
results known in the literature or at improving current estimates of the
global optima.
Despite the complexity of the problem, most putative global optima
of micro-clusters (with up to 147 atoms) have been discovered by means
of a very simple and efficient algorithm first proposed in (Northby, 1987)
and further refined in (Xue, 1994), which is based on the idea of starting
local optimization from initial configurations built by placing atoms
in predefined points in space, according to lattice structures which
researchers in chemistry and physics believe are the most common ones
found in nature. However quite a few exceptions to regular lattice-based
structures do exist; these structures are extremely difficult to discover
with general purpose methods. It is to be remarked that new optima
are still being discovered, even though at a slower rate than just a
few years ago. In August 1999, for example, a new configuration for
discovered (Leary and Doye, 1999) and possibly other new
records will appear even in the range N - 147 which has been the most
thoroughly and extensively studied in the last decade. In the chemical
physics literature, it is well known that some "magic numbers" exist,
like 102, for which classical, lattice-based procedures,
are doomed to fail. These numbers correspond to particularly stable
members of particular geometric classes; usually these magic number
clusters correspond to "closed shell" configurations and exhibit a higher
degree of geometric regularity, symmetry, compactness and sphericity
than non-magic number clusters within the same geometric class. Moreover
magic number clusters have a relatively larger number of near
neighbors than non-magic number clusters of the same class.
In this paper we propose a new methodology aimed at discovering
the most difficult structures for Lennard-Jones clusters. Our main aim
is not to introduce a general purpose method, but that of defining a
new strategy for local searches which can be profitably included in any
algorithm which is based on local searches, including the basin-hopping
method (Wales and Doye, 1997), the big-bang method (Leary, 1997),
Leary's descent method (Leary and Doye, 1999) or genetic algorithms
(see (Deaven et al., 1996), (Pullan, 1999)). Our method consists of a
modification of the objective function, in the first phase of the descent,
which enables a local search algorithm to escape from the enormous
number of local optima of the L-J energy landscape; implemented in a
straightforward Multistart-like method, our modification improved by
at least two order of magnitude the number of local searches required to
find the difficult 38 and 75 atom cases and could find the new 98 atom
cluster and the difficult 102 case in less than 10 000 local searches.
In a series of runs the LJ 38 optimum was discovered in 56% of the
local searches performed, an incredible performance if compared with
the best result published so far in which LJ 38 is found in 0.3% of the
attempts (Leary, 1997). In our first attempt to attack the LJ 98 case,
which was discovered only in summer 1999 (Leary and Doye, 1999)
using "millions of local searches" (Anonymous, 1999), we were able to
find the global optimum in less than 10 000 local searches, on average.
The success of the proposed procedure seems to be due to the fact
that the penalty term introduced into the objective function and the
modifications to the potential tend to enlarge the region of attraction
of regular, compact, spherical clusters: these are the characteristics of
the most difficult to find magic number clusters.
2. A new approach to the detection of Lennard-Jones
clusters
2.1. Multistart-like methods
We consider Multistart-like approaches to the problem of globally minimizing
the Lennard-Jones potential function. The pure Multistart method
can be described as follows.
Pure Multistart
1. Generate a point X 2 IR 3N from the uniform distribution over
a sufficiently large box centered at the origin;
2. perform a local search in IR 3N using X as a starting point;
3. if a stopping condition is not satisfied, go back to 1, otherwise
return the local minimum with the lowest function value.
Of course, we can not expect Pure Multistart to be a feasible method
for the solution of the Lennard-Jones problem. Indeed, even though the
difficulty of solving a global optimization problem by Multistart is not
actually related to the number of local minima, but to the measure of
the basin of attraction of the global minimum, the fact that the number
of local minima is exponentially large is a clear indication that Multi-start
may experience great difficulties in solving this kind of problems.
Numerical computations tend to confirm this fact. In Table I we notice
that Pure Multistart (PMS) applied to problems with N 2
fails to detect the putative global optimum for
trials and has very low percentage of successes for many other
values of N .
Therefore, it seems necessary to modify the basic Multistart scheme
in order to be able to solve larger problems. A simple idea is to exploit
the special structure of the Lennard-Jones potential function and
modify the search mechanism accordingly. Looking at the form of the
interaction function (1) we notice that good solutions should
possess some or all of the following characteristics:
atoms should not be too close each other (also recalling the result
in Xue, 1997);
\Gamma the distance between many pairs of atoms should be close to 1.0
(near neighbors), since at 1.0 the Lennard-Jones pair potential
attains its minimum;
\Gamma the optimal configuration should be as spherical and hence compact
as possible;thus the diameter of the cluster should not be too
large while still a minimum near neighbor separation is mantained.
According to these elementary observations, it is possible to substitute
the uniform random generation of points (Step 1 in the Pure
Multistart method) by a generation mechanism which tends to favor
point configuration possessing the above characteristics. As a first attempt
in this direction we substituted the uniform random generation
procedure with the following:
Point Generation Procedure
1. Start with a single atom placed in the origin, i.e. let
2.
2. Generate a random direction d 2 IR 3 and a point X k along
this direction in such a way that its minimum distance r from
every point in X is at least 0:5.
3. If r is greater than a threshold R ? 1 then X k is shifted towards
the origin along direction d, until its distance from at least one
point in X becomes equal to R.
4. Set g. If
and go back to Step 2.
This different point generation procedure slightly improves the performance
of Multistart as it can be seen from Table I where the results
of Multistart equipped with this generation algorithm are displayed
under the heading MMS (Modified Multistart); however even this modified
algorithm soon starts failing in detecting the putative global optima
of moderately large clusters.
2.2. Two phase Multistart
It might be possible to refine further the Point Generation Procedure
in order to produce better starting points, but it is felt that no real
breakthrough can be achieved in this direction. It seems more reasonable
to attack the problem by changing another component of the
Multistart method, i.e. the local search procedure; we are thus led to
search for a local optimization method which avoids as much as possible
being trapped in stationary points of the Lennard-Jones potential
characterized by a high value of the potential energy (2). The idea
is that of performing local searches employing a modified objective
Table

I. Number of successes in 1000 random
trials by Pure Multistart (PMS) and
Modified Multistart (MMS) methods.
14
function which, although related to the Lennard-Jones potential, is in
some sense "biased" towards configurations which satisfy the above
requirements. The local minimum of this modified potential is then
used as a starting point for a local optimization of the Lennard-Jones
potential function. This leads to the following version of the Multistart
method. Let ME(X) be a suitably defined modified potential function.
Two-Phase Multistart
1. Generate a point X 2 IR 3N according to the Point Generation
2. perform a local minimization of the modified potential function
in IR 3N using X as a starting point; let -
X be the local
optimum thus obtained;
3. perform a local optimization of the Lennard-Jones potential (2)
starting from -
4. if a stopping condition is not satisfied, go back to Step 1; otherwise
return the local minimum of E with the lowest function
value.
We notice that, in place of the usual local search of the Pure or
Modified Multistart method, here we have what we call a two-phase
local search: first the function ME is optimized, and then the Lennard-Jones
potential E. We underline that, even if at each iteration two local
searches are started, the computational effort is not doubled: indeed,
the local minimum -
X of ME is typically quite close to a local minimum
of E, so that the computational effort of the second local search is much
lower than that of the first one.
Accordingly we need now to define ME in such a way that the local
minima of this function possess the desired characteristics. In what
follows two classes of functions, among which ME can be chosen, are
introduced. The first class contains functions with the following form
where
are real constants; we note that choosing
coincides with the Lennard-Jones pair potential (1).
In

Figure

1 the case displayed and compared
with the Lennard-Jones pair potential. The parameters p and - have
Lennard-Jones potential
Modified potential

Figure

1. Comparison between Lennard-Jones and Modified potentials.
important effects. By choosing atoms can be moved more freely;
by decreasing p, the effect of the infinite barrier at
prevents atoms from getting too close to each other, is also decreased.
The parameter - has two important effects.
stronger penalty to distances between atoms
greater than 1.0; actually, it also assigns low penalty for pair distances
lower than 1.0, but this is largely overcome by the barrier
effect which, as already remarked, prevents atoms from getting too
close each other.
Global effect : it gives strong penalty to large distances between
atoms, e.g. to the diameter of the molecule.
In order to test the feasibility of this approach, a series of numerical
experiments have been performed by running 10 000 times the algorithm
for As these experiments were carried out on
Pentium II PC's, we did not performed extensive and generalized trials
In Table II the number of two-phase local searches which
led to the putative global optimum are reported. We notice that the
percentage of successes is much higher than the one of the Pure or
Modified Multistart algorithm. In particular, two important cases are
discussed. The first case is which is considered in the literature
a particularly difficult one (Doye et al., 1999a). While most putative
global optima in the range have a so called icosahedral
structure, the putative global optimum for has a FCC (Face
Centered Cubic) structure, and many algorithmic approaches, such as
the lattice search in (Northby, 1987) and (Xue, 1994), biased towards
icosahedral structures, are unable to detect this solution. The new
putative global optimum was first observed only recently in (Pillardy
and Piela, 1995) using a direct approach based on molecular dynamics;
more recently, in (Leary, 1997) the putative global optimum was found
using the "big bang" global optimization algorithm employing on the
average local searches, while for the basin hopping algorithm proposed
in (Wales and Doye, 1997), the expected number of local searches
required to first hit this putative global optimum is 2 000. In the new
approach, choosing 0:3, the expected number of local
searches is reduced to 80, but, with the method described later in this
paper, we were able to obtain the incredible hitting rate of 1.79 local
searches on the average. It is important to notice that the FCC LJ 38
cluster is a truncated octahedron, which is in fact one of the 13 regular
Archimedean solids.
As it can be observed from Table II, although quite successful for
some configurations, our method fails in several cases; most notably
it does not discover, at least in the first 10 000 local searches, the
difficult structure of LJ 75 . This case is the second hard case in the
range and it is much harder than the case (in
order to appreciate the difficulties of both cases see the discussion about
multiple funnel landscapes in (Doye et al., 1999b)). As for 38, the
structure of the putative global optimum is non icosahedral (actually
the structure is a decahedral one). The putative global optimum has
been detected for the first time in (Doye et al., 1995); by employing
the Basin Hopping algorithm the reported expected number of local
searches to first detect this configuration is over 100 000. Thus our
failure in detecting LJ 75 during the first 10 000 local searches was not
a surprise.
2.3. Adding a penalty to the diameter
Instead of persisting with an higher number of local searches, a modification
of (5) was introduced in order to strengthen the global effect.
This lead to the following class of modified functions:
where
where is an underestimate of the diameter of the
cluster. In Figure 2 the case displayed
and compared with the Lennard-Jones pair potential function. We no-
Lennard-Jones potential
Modified potential

Figure

2. Comparison between Lennard-Jones and modified potentials with diameter
penalization.
tice that the penalty term fi(maxf0; r no influence on
pairs of atoms close to each other, but strongly penalizes atoms far
away from each other. Thus, the new term does not affect the local
properties, but strengthens the global ones. The results for this class of
modified functions are reported in Table III. In particular, we note the
following results for the two difficult, non icosahedral cases, obtained
with suitable choices of the parameters.
\Gamma For the expected number of (two-phase) local searches to
first hit the putative global optimum is 10000= 5:46, more than
times faster, in terms of local searches performed, than Big Bang
and 366 faster than Basin Hopping;
the expected number of local searches is 3 333, while
it was 125 000 for the Basin Hopping algorithm: the improvement
factor is thus more than 37.
Given the results of better explanation of the failure
of our first approach can be given, supported by the observation of
the structure of the optimal decahedral structure (see Figure 3) and
icosahedral structure (see Figure 4).
In the best icosahedral structure the number of near neighbors is
higher than what observed in the optimal decahedral structure
(319 pairs). In some sense, the icosahedral structure has better local
properties than the decahedral one. However, this local disadvantage
is compensated by the compactness and sphericity of the decahedral
structure with respect to the icosahedral one: the diameter of the deca-
hedral structure is quite lower than the diameter of the icosahedral one.
Moreover, thanks to the spherical structure, many pairs of atoms in the
decahedral structure have a distance which is equal to the diameter (10
pairs in total, while the icosahedral structure has only 2). In some sense
we can say that the decahedral structure has better global properties
than the icosahedral one. In view of this comparison, it is now possible
to understand the failure for employed. The linear
penalty term -r has, as already remarked, a double effect: a local effect,
rewarding solutions with good local properties (like the icosahedral
structure), and a global effect, rewarding solutions with good global
properties (like the decahedral structure). What appears to happen for
is that the local effect dominates the global one, thus favoring
the icosahedral structure with respect to the decahedral one.
Even though complete computations have been performed only up to
the new approach has been tested for two other difficult cases,
for which the putative global optimum is known to be not icosahedral.

Figure

3. Putative optimum for LJ75
Very recently in (Leary and Doye, 1999) a new, non icosahedral,
putative global optimum for has been detected, displaying
a very compact and spherical structure; it is reported that this
discovery required "millions of local searches" (Anonymous, 1999);
our new approach could detect this solution within local
searches on the average.
In (Wales and Doye, 1997) a non icosahedral putative global optimum
for detected. The new approach was able
to detect this solution within local searches, while the Basin
Hopping algorithm could detect the same solution only 3 times out
of 500 000 local searches.
In other series of experiments with different parameter settings sometimes
better results were found. As a particularly significative instance,
for LJ 38 with parameters 1: the incredible
result of 56% successes was recorded in 1000 random trials. In practice
this means that, with such a parameter setting, the FCC structure of

Figure

4. Icosahedral optimum for LJ75
LJ 38 can be observed after a fraction of a second of CPU time on a
normal personal computer.
2.4. Limits of the proposed approach
It is fair to consider the limits of the proposed approach and possible
ways to overcome them. The main limits of the approach can be seen
from the tables. We notice that for N ? 60 in many cases the putative
global optimum could not be detected. It is not possible to claim that
our new approach is a general one to solve problems for any value of N .
What can be safely assumed is that it is an extremely successful method
in detecting those structures which differ from the icosahedral one; this
might be particularly important as it is believed that when the number
of atoms N is very large, compact, non-icosahedral structures prevail.
For most tested value of N for which the optimal structure is known
to be non icosahedral our method is much
faster (up to two orders of magnitude) than any other approach found
in the literature. It must be again underlined that, in the literature,
these cases are considered by far the most difficult ones. However, the
approach is not able to detect in an effective way some of the optimal
icosahedral structures. In order to detect these optimal structures it
is possible to incorporate the two-phase local search on which the
new approach is based into some of the approaches proposed in the
literature such as the Basin Hopping algorithm: the rationale behind
this is that it appears that the use of our modified potential, two-phase
local optimization actually enlarges the region of attraction of global
optima. In this respect, the choice of imposing a very low penalty on
the diameter should be considered safer, as the effect of this penalty
is usually so strong that only very compact structures are effectively
found (most micro clusters are indeed non compact).
An approach based on a forward procedure followed by a correction
procedure, has been tried, which enabled us to detect all the solutions
which could not be detected by the previous approach. The forward
procedure is already known in the literature and consists of starting the
optimization of LJN from a good configuration of LJN \Gamma1 , adding first a
single atom and then optimizing the overall potential; we implemented
a variant which incorporates the two-phase local search in place of the
regular local optimization. The correction procedure, starting from the
some of the optima found during a two-phase optimization, is based
upon the displacement of a few atoms randomly chosen among those
with highest energy contribution into a different position, followed by
the usual two-phase optimization. Details on these procedures can be
found in (Locatelli and Schoen, 2000), where it is shown that all the
configurations up to 110 atoms can be quite efficiently obtained by
means of these more specialized methods.
Another critic to the proposed approach is the difficulty of choosing
sensible values for the parameters. Again it has to be remarked that
there is no general rule to choose a set of parameters which is sufficiently
good for a whole range of clusters. The main reason for this is
that, as it has already been remarked, cluster structures vary abruptly
around some magic numbers, like However, in an
attempt to find general rules, experiments were performed using, for
the parameter D, a lower bound obtained from a regression analysis of
the diameters of known putative local optima. The results obtained are
very encouraging - details on these experiments will appear elsewhere.
3. Conclusions and further research issues
In a recent survey (Doye, 2000b) global optimization algorithms for
the minimization of the Lennard-Jones potential have been thoroughly
analyzed; the author claims that "Any global optimization method
'worth its salt' should be able to find [. ] the truncated octahedron
at Success for the other non-icosahedral globl minima would
indicate that the method has particular promise". Our method not
only is able to find the two difficult clusters LJ 38 and LJ 75 within a
number of local searches which is roughly two orders of magnitude less
than any previously known method; it is also capable of discovering
the even more difficult cases of LJ 98 and LJ 102 , again within very low
computational times on standard PC hardware. So it can be safely
concluded that the method proposed in this paper performs excitingly
well. An interesting analysis of the reasons for the success of our method
has recently apperared in (Doye, 2000a). However, what we think is
the main result presented in this paper is not an original algorithm,
although a very efficient method has been analyzed and its performance
discussed. The major contribution of this paper is the definition of a
new local search strategy, composed of two phases, the first of which
is built in such a way as to pass over non interesting local minima.
Moreover, this local search promises to be very well suited for general
approaches for the Lennard-Jones and similar problems in molecular
conformation studies; in this paper it was shown how the most difficult
configurations for the Lennard-Jones cluster problem can be discovered
with much greater efficiency by using a simple Multistart algorithm in
which our two-phase local search is used in place of a standard descent
method. Some experiments have already been performed to see if this
two-phase local optimization might be useful when substituted in place
of a standard local search in a more refined method. Our first results
with forward and correct methods are extremely encouraging.
In any case, already from the results presented here it is possible to
infer that the penalties and rewards included in the first phase optimization
succeed in driving the optimization close to very good, compact
clusters, avoiding being trapped in local optima which for a regular local
search method display very large regions of attraction. The structures of
optimal Lennard-Jones clusters are so radically different in some cases
that it seems quite unreasonable to look for general purpose methods
capable of discovering all optima in reasonable computer times. Our
approach greatly reduces the computational effort required to discover
what are commonly accepted as the most difficult configurations. It is
hoped that, when applied to larger clusters, this method will succeed in
finding better putative global optima. Of course, in case of much larger
clusters, the problem arises of efficiently computing the potential as
well as the penalized functions and gradients. Using a naive approach,
these computations require O(N 2 ) distances to be evaluated for each
iteration during local optimization; for large values of N this cost might
be prohibitively large. In order to cope with the curse of dimensionality,
it is planned in the next future to explore the possibility of parallelizing
energy computations; another possibility, which we did not explore up
to now, might be that of using faster approximate potential calcula-
tion, based on approaches similar to the one described in (Hingst and
Phillips, 1999).
Finally, it is important to stress once again that the results in this
paper give a clear demonstration that a carefully chosen, geometry
inspired, penalty function may be dramatically effective in helping to
discover clusters within special geometrical classes. This might prove to
be of great effectiveness when applied to more complex conformation
problems like, e.g., some protein folding models.
4. Appendix: details on the computational experiments
All of the experiments have been performed either on 266Mhz Pentium
II Personal Computers or on a Sun Ultra 5 Workstation. For
local optimization a standard conjugate gradient method was employed
and, in particular, the implementation described in (Gilbert and No-
cedal, 1992) was used with standard parameter settings. For every
choice of the parameters, we ran random 10 000 trials. Experiments
performed with different parameter settings, like those in tables 1 and
2, were conducted using the same seeds for the random generation
mechanism. That is, common random numbers were used for different
experiments: this way, in particular for those instances in which
finding the global optimum is a rare event, a comparison between
the efficiency of different parameter settings becomes more reliable.
The executable code, compiled both for Pentium PC's and for Sun
Ultra Sparc Stations, is freely available for research purpose at URL
http://globopt.dsi.unifi.it/users/schoen.

Acknowledgements

This research was partly supported by projects MOST and COSO of
the Italian Ministry of University and Research (MURST).
The authors would also like to thank an anonymous referee for very
careful reading of the paper and for many comments, which we included
almost verbatim in our paper and greatly enhanced its readability.


--R









Handbook of Test Problems in Local and Global Optimization
Dordrecht: Kluwer Academic Publishers.


'A Performance Analysis of Appel's Algorithm for Performing Pairwise Calculations in a Many Particle System'.














--TR
An Infeasible Point Method for Minimizing the Lennard-Jones Potential
Molecular Modeling of Proteins and Mathematical Prediction of Protein Structure
A Performance Analysis of Appel''s Algorithm for Performing Pairwise Calculations in a Many Particle System
Global Optima of Lennard-Jones Clusters
Minimum Inter-Particle Distance at Global Minimizers of Lennard-Jones Clusters

--CTR
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lennard-jones clusters;global optimization;molecular conformation