energy minimization of mixed argon-xenon microclusters using a genetic algorithm

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Energy Minimization of MixedArgon]Xenon Microclusters Using aGenetic Algorithm

W. J. PULLAN*Department of Mathematics and Computing, Central Queensland University, Rockhampton,Queensland, 4702, Australia

Received 10 July 1996; accepted 25 September 1996

ABSTRACT: This article describes a parallel real-coded genetic algorithmimplemented to find global minimum energy structures of microclusters ofnon-bonded argon and xenon atoms. Using appropriate genetic operators, thegenetic algorithm was able to find minimum energy structures for microclustersof two to twenty atoms, in all possible combinations of argon and xenon.Q 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 1096]1111, 1997

Keywords: global optimization; genetic algorithm; molecular conformation

Introduction

tomic microclusters are currently an activeA field of theoretical and experimental re-search. Microclusters are aggregates of atoms ormolecules, sufficiently small so that a significantproportion of these units are present on the surfaceof the microcluster. Theoretical investigations ofmicroclusters address the following optimizationproblem:

Given N particles, interacting with two-bodycentral forces, find the configuration in three-di-mensional Euclidean space for which the totalpotential energy attains its global minimum.

*Present address: University of Colorado at Boulder, De-partment of Computer Science, Campus Box 430, Boulder, CO80309-0430.

Correspondence to: W. J. Pullan

Simplifications normally used are that many-bodyand angle dependent interactions are ignored,quantum effects are not taken into account and allparticles are assumed to be spherical and the samesize. Using these simplifications, the potential en-ergy of an N atom microcluster can be written as:

Ny1 N

Ž .V s v rÝ Ý i jis1 jsiq1

where r is the Euclidean distance between atomsi jŽ .i and j, and v r is the pair potential betweeni j

atoms i and j. From a molecular point of view, thepotential energy of a microcluster can be describedby a hypersurface, normally referred to as a poten-

Žtial energy surface, of high dimensionality 3N y 6,where N is the number of atoms in the microclus-

.ter . The geometry of the microcluster’s most sta-ble conformation is the geometry corresponding to

Q 1997 by John Wiley & Sons, Inc. CCC 0192-8651 / 97 / 081096-16

MIXED ARGON ]XENON MICROCLUSTERS

the global minimum of the potential energy sur-face. Even with the simplifying assumptions, theminimization of the potential energy of a micro-cluster is very difficult to achieve as it is a noncon-vex optimization problem involving numerous lo-cal minima. In fact, Hendrickson1 has shown thateven simple versions of the problem are NP-com-plete; that is, the expected time to find the globalminimum increases exponentially with the dimen-sion of the problem.

This article describes a parallel real-coded ge-Ž .netic algorithm GA which has been applied to

the problem of finding minimum energy configu-rations of microclusters containing a mixture ofargon and xenon atoms. The remainder of thissection describes pure and mixed microclustersand presents a review of optimization methodsthat have been used to find their minimum energyconfigurations. A later section describes the imple-mentation of the computational method used inthis study. The results are then presented anddiscussed. Finally, the conclusions are presented inthe final section.

PURE MICROCLUSTERS

As pure microclusters are composed of a singletype of atom, the scaled Lennard]Jones pair poten-tial is normally used to investigate their proper-ties. This pair potential is shown in Figure 1 anddefined by

12 6Ž . Ž . Ž .v r s 1rr y 2 1rr

FIGURE 1. Scaled Lennard ]Jones pair potential,( ) ( )12 ( )6v r = 1 / r y 2 1 / r .

where r is the distance between the atoms. Thefirst term is an approximation for the repulsive

Žcomponent for which no rigorously derived ex-.pression exists , whereas the second attraction term

can be rigorously derived for spherical, chemicallysaturated atoms.

An extensive review of the theory and structureof microclusters is presented in Hoare2 where thequestion of the number, and types, of local minimapossible on the potential energy surface for a mi-crocluster is addressed. In terms of the energyfunction, the only quantitative statement that canbe made is that, for a pure N atom microcluster

Ž .with v r as the pair potential, there exists a weaklower bound given by:

Ž .yN N y 1 r2

That is, a simplex in N y 1 dimensions in whichŽ .all atoms are nearest neighbors distance 1 apart ,

gives the lowest energy possible. Liao3 has shownŽ .that, for an N atom microcluster with v r as the

pair potential, a global minimum exists and thecorresponding configuration is contained within asphere, centered at the origin, and of radius:

'Ž .N y 1 N y 1

Ž .For smaller microclusters, with v r as the pairpotential, globally minimum configurations are:

B Two atoms—separated by 1 unit.B Three atoms—an equilateral triangle, each

side 1 unit.B Four atoms—a tetrahedron, each side 1 unit.B Five atoms—a triangular bipyramid, slightly

contracted along the symmetry and dis-tended in the symmetry plane.

B Six atoms—the regular octahedron withslightly contracted sides.

B Seven atoms—part of a regular icosahedronwith slightly distended edges and contractedaxial distances.

For more than seven atoms, minimum energy con-figurations are based on icosahedral lattice struc-tures.4

Determining the number of distinct local min-Ž .ima which includes the global minimum is not

straightforward for microclusters due to the sym-metries involved. Minima associated with struc-tures which differ only in labeling of atoms mustfirst be eliminated. In addition, minima that areassociated with structures which are equivalent

JOURNAL OF COMPUTATIONAL CHEMISTRY 1097

PULLAN

under proper symmetry operations in three di-mensions must also be eliminated. Finally, thereare some pairs of structures which must be elimi-nated as they share the same potential energy and

Žthe same distances between atoms to within atom.numbering but are not superimposable under

normal symmetry operations. The total number ofŽ .distinct local minima, m N , for microclusters con-

sisting of 2 to 13 atoms is 1, 1, 1, 1, 2, 4, 8, 18, 57,145, 366, and 988, respectively, and is approxi-mated by 2:

Ž . Ž 2 .m N s exp y2.5176 q 0.3572 N q 0.0286N

OPTIMIZATION OF PURE MICROCLUSTERS

Lattice-Based Methods

Hoare et al.2, 5 ] 7 developed a general growthalgorithm and used it to generate large numbers ofstable structures, mainly in the range N F 55.These were compared to find the lowest energystructures, which in turn became candidates forthe globally minimum structures. Hoare and Pal5

observed that, while what they termed as the‘‘icosahedral growth sequence’’ did not, in general,produce minimal structures, icosahedral subunitsdid appear regularly in relaxed configurationsgenerated by other sequences.

Using results from the Hoare and Pal study,Northby 4 produced what is currently the mostsuccessful algorithm for minimizing Lennard]Jones microclusters. At a high level of abstraction,Northby’s algorithm proceeds as follows:

B For N atoms, define the set S of all possibleŽicosahedral shell lattices with inner shells

.densely packed .B Define a potential function for optimizing

Žthis discrete problem i.e., optimizing where.each atom must remain on a lattice vertex .

Northby used a ‘‘square well’’ potential ofthe form:

0 0 F r F 0.8Ž .v r s y1 0.8 - r - 1.2½ 0 1.2 F r F `

B For each lattice in the set S, perform a lat-tice-based search optimization to identifyminimum energy lattice-based configura-tions.

B Using the Lennard]Jones potential function,optimize all geometrically distinct structuresusing a gradient search method.

Clearly, the critical assumption in Northby’salgorithm is that a well-defined set of lattice struc-tures contains at least one initial cluster configura-tion which relaxes to the cluster configuration cor-responding to the global minimum of the potentialenergy.

Simulated Annealing8 Ž .Wille applied simulated annealing SA to

Lennard]Jones microclusters for 4 F N F 25. ForŽ .small microcluster sizes 4 F N F 13 , one atom

was kept in a fixed central position, to eliminatetranslational degrees of freedom, and for the larger

Ž .ones 14 F N F 25 13 atoms were kept in a centralicosahedral environment during annealing. As iscommon with the use of SA, the difficult part isthe determination of the cooling schedule. A cool-ing rate that is too slow is time consuming, while afast cooling rate will lead to minima other than theglobal minima. To analyze the influence of thecooling schedule on the final state, Wille per-formed a number of simulations on 13-atom mi-croclusters making the number of possible steps

Žper atom at each temperature equal to K 1 y.log T , where K s 200 and 2500 with the tempera-

ture reduction factor being 0.9. In the first case, 3of 30 runs produced the isosahedral structure,whereas in the second case 4 of 5 runs gave thisresult. This indicates how the likelihood of detect-ing the global minimum increases with decreasingcooling rate.

Xue9 implemented a two-level parallel versionof simulated annealing for use with an icosahedrallattice-based search technique which had increasedcomputational efficiency over that used by

Ž .Northby. Xue’s basic nonparallel two-level simu-lated annealing algorithm uses a local optimizer todetermine the local minimum for the catchmentbasin within which each generated point lies. Newpoints are moved to by comparing the local min-ima of the associated catchment basins rather thanthe values associated with the actual points. Whenthe temperature gets low, two-level simulated an-nealing still accepts moves that lead to worse func-

Žtion values, but these moves all lead to better or.the current local minima. That is, the two-level

simulated annealing algorithm can easily climb upa hill at any temperature, but is very unlikely to

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move into the catchment basin of a worse localminima when the temperature is low.

Genetic Algorithms10 Ž .Hartke used a genetic algorithm GA to find

global minimum energy structures for four atoms.An encoding method, using just four parametersto specify the four-atom configuration, allowed astandard GA to find the global minimum energy.Mestres and Scuseria11 were able to find globalminimum energy configurations for Lennard]Jonesmicroclusters of up to 13 atoms. More recently,Deaven and Ho13 used a real-coded GA with tai-lored genetic operators to find the global mini-mum energy for fullerene structures up to C and60Gregurick et al.12 used a modified deterministicrstochastic genetic algorithm to find all currentlyaccepted global minima for Lennard]Jones atomicclusters in the range 2, . . . , 29. However, thismethod required a seed growth method for gener-ating initial structures for all clusters containingmore than 20 atoms. More recently, Deaven et al.,19

using a real-coded GA with innovative phenotypegenetic crossover operators, were able to find allglobal minimum energies for Lennard]Jones pureatomic clusters in the range $2, . . . , 100$ includingnew minima for 38, 65, 69, 76, 88 and 98 atomicclusters.

MIXED MICROCLUSTERS

While all the studies mentioned above havefocused on pure microclusters, investigations havealso been performed on microclusters containing amixture of argon and xenon atoms. For these mi-croclusters, the scaled Lennard]Jones pair poten-tial is not appropriate as it does not account for thedifferences in pairwise interaction between differ-ent atom types. For mixed argon]xenon microclus-ters, the following form of the Lennard]Jones pairpotential is normally used:

12 6Ž . Ž . Ž .v r s 4e srr y srr

where r is the distance between atoms and e ands are parameters whose values depend on thetypes of interacting atoms. The appropriate valuesof e and s 14, along with the associated r andminŽ .v r for argon and xenon atoms, are tabulatedmin

in Table I, whereas the corresponding curves forŽ .v r are shown in Figure 2.

Within this study, the following notation is usedto define mixed microclusters of argon and xenon

TABLE I.Lennard – Jones Parameters for Argon – XenonPair Potentials.

˚ ˚( ) ( ) ( ) ( )Interaction s A « kJ / mol r A v rmin min

Ar ]Ar 3.40 1.0000 3.82 y1.00Ar ]Xe 3.65 1.4800 4.10 y1.48Xe ]Xe 4.10 1.8525 4.60 y1.85

atoms:

B Ar Xe : a mixed microcluster containing ii jargon and j xenon atoms.

B Ar Xe : the set of microclustersN y n nAr Xe , . . . , Ar Xe .0 N N 0

B Ar Xe N : the set of microclustersNy n n iF N F jAr Xe , . . . , Ar Xe .iyn n jyn n

Intuitively, finding the minimum energy struc-ture of a mixed microcluster would appear to be atleast as difficult an optimization problem as that offinding the minimum energy structure for the cor-responding pure microcluster. For example, com-pare the pure Ar Xe microcluster with the mixed5 0Ar Xe microcluster. As shown by Hoare2, the3 2Ar Xe microcluster has a single energy minima,5 0corresponding to a triangular bipyramid structure,slightly contracted along the symmetry axis anddistended in the symmetry plane. This configura-tion can be constructed from the optimal Ar Xe5 0

Ž .microcluster a tetrahedron by adding the fifthargon atom at the apex of a tetrahedron based on a

FIGURE 2. Argon ]xenon Lennard ]Jones pair( ) [( )12 ( )6]potentials, v r = 4e s / r y s / r , where e and s

are defined in Table I for each possible pair of atoms.


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face of the optimal Ar Xe microcluster and per-4 0forming local optimization. Although this opera-tion may be performed with all three faces of theAr Xe tetrahedron, the resulting structures are5 0identical under symmetry operations and all resultin the same energy minima for Ar Xe . Similarly,5 0locally optimal Ar Xe microclusters can be con-3 2structed by either adding an argon atom to theoptimal Ar Xe microcluster, or by adding a xenon2 2atom to the optimal Ar Xe microcluster. Using3 1these operations, four triangular bipyramid struc-tures may be constructed as follows:

1. An argon atom positioned at the apex of a� 4tetrahedron based on an Ar, Ar, Xe face of

the optimal Ar Xe microcluster.2 2

2. An argon atom positioned at the apex of a� 4tetrahedron based on an Ar, Xe, Xe face of


3. A xenon atom positioned at the apex of a� 4tetrahedron based on an Ar, Ar, Xe face of


4. A xenon atom positioned at the apex of a� 4tetrahedron based on an Ar, Ar, Ar face of


Under symmetry operations, structures 1 and 3above are identical, leaving three unique struc-tures corresponding to three distinct local minima.Extending this argument, it would be reasonableto conjecture that the number of local minima forAr Xe is bounded below by the number ofNy n nlocal minima for Ar Xe , for all N.N 0

OPTIMIZATION OF MIXED MICROCLUSTERS

Navon et al.14 and Robertson et al.15 investi-gated Ar Xe , Ar Xe , and Ar Xe mi-7yn n 13yn n 19yn ncroclusters using a combination of simulated an-nealing and conjugate gradient descent. The ap-proach used by Navon and Robertson was, for amicrocluster containing N atoms, to start with theknown structure of the Ar Xe microcluster andN 0randomly substitute the appropriate number of Aratoms with Xe atoms to form an initial structure,S , for optimization. Using a local optimizer, a0minimum energy, E , is located near S . A new0 0structure, S , is then generated from S by ran-1 0domly interchanging the position of one Ar atomwith that of one Xe atom. Using a local optimizer,a minimum energy, E , is located near S . If E -1 1 1E , then S becomes the new starting structure. If0 1E G E , then S is accepted with Boltzmann prob-1 0 1

ability as the new starting structure. This processis repeated for a fixed number of steps. The methodclearly relies on the assumption that a mixed mi-crocluster has an underlying structure which isbased on that for the corresponding pure micro-cluster. The Ar Xe , Ar Xe , and Ar Xe7yn n 13yn n 19yn nmicroclusters investigated by Navon et al.14 werechosen because, as their pure forms, Ar Xe ,7 0Ar Xe , and Ar Xe , respectively, are particu-13 0 19 0larly stable, this assumption was more likely to bevalid. However, for some of the microclusters in-vestigated in this study, the assumption was foundto be invalid and the optimization method justdescribed would be unlikely to find all minimumenergy structures.

Computational Method

The computational method used in this study tooptimize mixed microclusters was a parallel real-coded genetic algorithm. GAs are based on theprocess of natural selection where the proportionof ‘‘fitter’’ individuals in a population tends toincrease at each generation. GAs operate on anencoded version of the parameters of a problemand use genetic operators such as proportional

Ž .selection based on a fitness factor , random muta-tion, and crossover to generate a new populationfrom an existing population. A detailed descrip-tion of GAs may be found in Goldberg.16

Ž .Standard or binary coded GAs are efficientwhen applied to problems whose parameters canbe encoded as short, low-order schema, which arerelevant to the underlying problem and relativelyunrelated to schema over other fixed positions.16

Unfortunately, optimization of microclusters is dif-ficult for standard GAs because:

B Atomic positions are real valued parameters.Using a binary coded GA requires that thesepositions must be discretized so they can becoded as binary integers. To obtain an accu-racy of three digits after the decimal place,for a 30-atom microcluster, requires a binarysolution vector of length 1260 bits. This inturn generates a search space of approxi-mately 10400 elements over which a binarycoded GA can be expected to perform poorly.An additional problem is that discrete binarycoded GAs sample a continuous surface atpoints defined by an arbitrary grid and theglobal minimum may, in fact, be unobtain-

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able, even when a deterministic local opti-Žmizer is used to ‘‘fine tune’’ solutions Fig.

.3a shows just such a situation . It is interest-ing to note that, using a real-coded GA inconjunction with a local optimizer is, in ef-fect, a variable discretization process wherethe discrete value is the minimum associatedwith the catchment basin within which theparameter lies, and the discretization inter-

Ž .val at that point is the width of this catch-Ž .ment basin Fig. 3b . For a real-coded GA,

the global minimum is always attainable.B ŽIn evaluating the potential energy and hence

.the fitness factor of a microcluster, the posi-tion of each atom is only meaningful in thecontext of the positions of all other atoms.That is, schema over different positions arenot unrelated, so standard GAs can be ex-pected to perform poorly when applied tothe microcluster problem. However, asshown by Deaven and Ho,13 if genetic opera-tors are developed which operate in thethree-dimensional space in which the micro-clusters exist and are able to use buildingblocks within the configuration, then a GA

FIGURE 3. GA parameter mappings to objective( )function. a Binary parameter values and corresponding

( )locally optimized values, and b real-coded parameters( )mapping to the complete surface and locally optimizedvalues.

can be successfully used on the microclusterproblem.

The parallel real-coded GA described in thisarticle contains many of the basic features of theDeaven and Ho GA,13, 19 in particular the use ofgeometric crossover operators. The following defi-nitions are used: g— generation number;p—processor number; P —current population forggeneration g ; P —the subset of P going to pro-p g gcessor p; and T —temporary population generatedgfrom P . Thus, the algorithm obtained is:g

( )Master Slave p

g s 0Initialize Initialize

Next:g s g q 1SelectStartAll WaitFor P For P1 g p g

Crossover CrossoverMutate Mutate

if finishedstop all processes

elsego to Next Go to Next

Details of the elements of this algorithm are:

B Initialize creates population T of N atomgmicroclusters by randomly placing each atomwithin a cube, where the size of the cube isdependent on N. These microclusters arethen locally optimized using a BFGS opti-mizer.

B Select chooses microclusters from T for in-gclusion in P . Selection is based on micro-gcluster energy with the proviso that only onemicrocluster will be chosen with a given

Ž .energy plus or minus a small delta . At thecompletion of Select, T is cleared.g

B StartAll sends a message to all slave proces-sors to start processing, P . The subsets,p gP , are chosen such that the processing loadp gis evenly distributed over all available pro-cessors.

B Wait waits for the start message from themaster processor.

B Crossover applies crossover genetic opera-tors to selected microclusters from P top g


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generate new microclusters for inclusion inT . Two crossover operators for generatinggnew microclusters were used, with equalprobability. The first, x , randomly rotated1the coordinate system around each axis andstrictly divided both parent microclustersalong a plane parallel to, and a random dis-tance above, the xy-plane, into two volumeelements. The second, x , randomly rotated2the coordinate system around each axis andused atoms within a quadrant or an octant ofthe parent microclusters to construct volumeelements. In both cases, new microclustersare constructed by exchanging correspond-ing volume elements between parents.

B Mutate performs BFGS local minimization,with a performance-related early terminationoption, on all new microclusters. In addition,with probability, p , one of two equallymlikely mutation operators was applied. Ei-ther a combination of BFGS local minimiza-tions and APSE17 searches was used to per-form a localized random search within thestructural vicinity of the new microcluster,or randomly chosen argon and xenon atomswere interchanged and a BFGS local mini-mization was performed. In both cases, thebest microcluster found was added to T .g

Although no formal investigations were per-formed to investigate the effect of changing globalparameters for this algorithm, it appeared that bestresults were obtained with small constant-sized

Ž .populations 8]12 microclusters , a low mutationŽ .probability , 0.01 , and a high crossover proba-

Ž .bility , 0.8 distributed equally between x and1x .2

Crossover operators such as x and the quad-1rant version of x interchange large volumes be-2tween current microclusters. Fine tuning of themicrocluster was provided by the octant version ofx , the use of BFGS and APSE optimization, and2swapping of Ar and Xe atoms within the muta-tion operator.

An experimental and analytical evaluation ofthe efficiency of these geometric operators, com-pared to the normal random genetic crossoveroperators, is presented in Pullan.18

Results

The GA successfully located all currently ac-cepted global minima14 for Ar Xe and7yn n

Ar Xe microclusters while the results for13yn n

Ar Xe were generally improved. In addition,19yn n

minima for the remaining microclusters inAr Xe N were determined by runningNy n n 2 F N F 20

the GA for 50 generations, at which point all mini-mum population energies had stabilized. The low-est energies, E, and the substitution energy, D E, asthe Ar to Xe substitutions were performed arepresented in Appendix A. The energy, E, repre-sents an upper bound on the energy of the globalminima for these systems as there is no guaranteethat a GA will find the global minima. However, itshould be noted that this GA, in addition to re-peating the results of Navon et al.,14 has previ-ously located the currently accepted global min-ima for all Ar Xe N microclusters.18

N 0 2 F N F 80

The energy values obtained and correspondingmicrocluster structures are discussed in the follow-ing section.

The effectiveness of the GA in optimizing mixedmicroclusters is shown in Figure 4 where the mini-mum energy in the current population of micro-clusters, as a function of generation, is plotted forthe optimization of the Ar Xe microcluster. This10 10

pattern, with an initial sharp decrease in energyfollowed by a slow approach to the global mini-mum, was observed for all microclusters.

The number of generations required to find thecurrently accepted global minima for Ar XeN 0

N , as a function of the number of atoms in2 F N F 80

the microcluster, is shown in Figure 5. A least-

FIGURE 4. Minimum population energies whileoptimizing Ar Xe starting from randomly generated,10 10locally optimized, initial configurations.

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FIGURE 5. Generations required to optimizemicroclusters Ar Xe N . Each point representsN 0 2F NF 80the number of generations required for a singleoptimization of each cluster.

squares fit of the data gives:

G s 0.0065N 2 y 0.0970N q 4.028

where G is the number of generations required.However, as each datapoint is the result of a singleoptimization run for a cluster, and as there ap-pears to be a new trend developing for N G 74,this fit may be inappropriate for higher N.

The multiprocessor scalability of the GA isshown in Figure 6 for the Ar Xe microcluster,40 0

FIGURE 6. Decrease in the elapsed time to optimizeAr Xe as the number of computer processors is40 0increased. The elapsed time for a single processor istaken as unity and each point represents the number ofgenerations required for a single optimization of thiscluster.

Žoptimized using from 1 to 10 processors takingthe elapsed time for the single processor case as

.unity . A least-squares fit of this data gives:

E s 0.0105P 2 y 0.2022 P q 1.113

where P is the number of processors and E is theratio of elapsed time for P processors to the elapsedtime for a single processor.

Inefficiencies arise from normal message han-dling in the multiprocessor implementation andalso when there is a mismatch between the popu-lation size and the number of processors. That is,the required crossoverrmutation operations can-not be equally spread across available processors.

Discussion

As observed in Navon et al.,14 in discussing theresults obtained, it is important to keep in mindthe relative energy minima of the three

Ž .Lennard]Jones pair potentials Table I , and theinteratomic distances at which they occur. The

Ž .strongest attractive interaction lowest minima isthe Xe]Xe interation, whereas the weakest isAr]Ar. Navon et al.14 suggested that, as a crudemodel, if there are more argon than xenon atoms,then the number of Ar]Xe nearest neighborsshould be maximized and the number of Ar]Arnearest neighbors minimized. Alternatively, ifthere are more xenon than argon atoms then thenumber of Xe]Xe nearest neighbors should bemaximized. Clearly, this model only takes intoaccount the relative Lennard]Jones energy minimaand not the interatomic distances at which theyoccur.

An alternative perspective, which does incorpo-rate the different interatomic distances at whichthe energy minima occur, is to consider the num-ber of atom pairs whose separation is within a

Ž .small d of the optimum r , Table I for thatmin˚atom pair. For d s 0.2 A, N , the count of nearA A

optimal Ar]Ar pairings, N , the count of nearA Xoptimal Ar]Xe pairings, and N , the count ofX Xnear optimal Xe]Xe pairings, are shown in Ap-pendx A for all optimized Ar Xe NNy n n 2 F N F 20microclusters. Although the value of d is some-what arbitrary, it is reasonable to assume that ifthe distance between two atoms is greater than 0.2A of the optimal distance for that pair, then theoptimization process has not focused on optimiz-ing the relationship between these two atoms. Ifwe denote the minimum value of the Ar]Ar pair


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potential by V , that for Ar y Xe by V , andA A A Xfor Xe y Xe by V , then the optimization prob-X Xlem is approximated by:

Ž .min N V q N V q N VA A A A A X A X X X X X

Optimization studies of pure microclusters2

have shown that the maximum possible values ofN , N , and N , for 2]13 atoms, are 1, 3, 6, 9,A A A X X X12, 16, 19, 23, 27, 31, 36, and 42, respectively. Usingthese results, inspection of the actual counts pre-sented in Appendix A shows that, when there is arelatively large number of argon atoms, N isA Xmaximized. As a secondary effect, N is mini-A A

Ž .mized which is equivalent to maximizing N .X XOnly when there are considerably more xenonthan argon atoms does the focus move to maximiz-ing N . Intuitively, the initial maximizing ofX XN can be understood in terms of the relativeA Xenergy minima of the different pair potentials. Forexample, if there are only two xenon atoms, thenthe maximum value of N is one. In contrast, ifX Xthe two Xe atoms are separated, then the possiblevalues for N have a much greater range. EvenA Xthough the Xe]Xe pair has a lower potential en-ergy minimum than the Ar]Xe, the number ofpossible Ar y Xe pairs more than compensates.This effect is clearly seen in the optimized Ar Xe5 2microcluster shown in Figure 7, where the twoxenon atoms were placed at the apex positions,thus maximizing N .A X

Analysis of the values obtained for the substitu-tion energy, D E, in Appendix A, shows that, for allAr Xe N microclusters, the D E valuesNy n n 2 F N F 9are generally higher for greater N and slowlydecrease as Xe atoms are substituted for Ar atoms.

FIGURE 7. Optimized structure, Ar Xe .5 2

( )FIGURE 8. Substitution energies DE for Ar Xe ,7y n nAr Xe , and Ar Xe as xenon atoms are13y n n 19 y n nsubstituted for argon atoms.

However, for all Ar Xe N micro-Ny n n 10 F N F 20clusters, whereas substitution of the first seven Aratoms follows this pattern, the D E values aremore erratic as the remaining substitutions areperformed. Figure 8 shows the behavior of D Eobtained in this study, for all Ar Xe ,7yn nAr Xe , and Ar Xe microclusters. The13yn n 19yn nsubstitution energies shown here follow the samepattern as those found in Novon et al.14

To determine the reasons for the observed be-havior of D E, three microclusters were furtherinvestigated.

B Microcluster Ar Xe : Figure 9 shows the10yn nvariation in D E as xenon atoms replace ar-

( )FIGURE 9. Substitution energies DE for Ar Xe10y n nas xenon atoms are substituted for argon atoms.

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gon atoms in the Ar Xe microclusters.10yn nThe first xenon atom is placed at the centerof the partially constructed Ar Xe icosahe-10 0dron and, up until the sixth xenon atom,substitute argon atoms in the shell of thepartial icosahedron. The structure changeswhen the sixth xenon is substituted in thatthe xenon atom at the center of the icosahe-

Ždron is replaced by an argon atom Figs. 10.and 11 . This corresponds to the first increase

in D E shown in Figure 9. As more xenonatoms are added, they are placed in theicosahedral shell which gradually envelopesthe central argon atom. The second increasein D E occurs when the eighth xenon atom is

Ž .added Figs. 12 and 13 . The rectangular baseof four xenon atoms beneath the apex xenonatom in the optimized Ar Xe microcluster3 7

˚ ˚has sides of length 5.83 A and 4.71 A. Whenthe eighth xenon atom is added, the remain-ing rectangular base of four xenon atoms is

˚able to contract to a square of side 4.91 Awhich is closer to the Xe]Xe r of 4.60.minThis results in a considerable increase inN .A X

B Microcluster Ar Xe : Figure 14 shows the12yn nvariation in D E as xenon atoms replace ar-gon atoms in the Ar Xe microclusters.12yn nThe pattern followed was that the first xenonatoms substituted argon atoms in the partialicosahedral shell. The last xenon atom re-placed the argon atom at the center of the

Ž .icosahedron Figs. 15 and 16 . The subse-

FIGURE 10. Minimum energy structure for Ar Xe .5 5





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quent relaxation and adjustment of the mi-crocluster resulted in N increasing fromX X25 to 36, giving the large increase in D Eshown in Figure 14.

B Microcluster Ar Xe : Figure 17 shows the17yn nvariation in D E as xenon atoms replace ar-gon atoms in Ar Xe microclusters. The17ynincrease in D E when the twelfth xenon atomis added results from a major change in

Ž .structure between Ar Xe Fig. 18 and6 11Ž .Ar Xe Fig. 19 . At this point, the structure5 12

changes from a central xenon atom with allother argon and xenon atoms in an enclosingshell to two interpenetrating icosahedronscentered on two argon atoms. This created asignificant increase in N . The next twoA Xmajor increases in D E occur when the four-


VOL. 18, NO. 81106




teenth and sixteenth xenon atoms are added.As in the Ar Xe to Ar Xe , these allowed a3 7 2 8rectangular base of xenon atoms to trans-form to a square whose sides were closer tor for Xe y Xe, thus increasing N .min X X

Conclusions

In this article, an investigation designed to de-termine the energy-optimized structures ofLennard]Jones microclusters containing a mix ofargon and xenon atoms has been presented. A GA,whose genetic operators operated in the problem



space, was able to find these structures efficiently.In common with all GAs, it explored the searchdomain in parallel using all microclusters in thecurrent population as starting points. The crossoverand mutation genetic operators functioned at anumber of levels when generating new microclus-ters. At a high level, crossover operators such asx and the hemisphere version of x interchanged1 2large volumes between current microclusters. Finetuning of the microcluster was provided by thequadrant and octant version of x , the use of2BFGS and APSE optimization, and the swappingof Ar and Xe atoms within the mutation operator.


PULLAN

An important feature of the GA is that, as italways starts from randomly generated structuresand has no bias toward any particular structure, itwill perform a more complete search of the do-main of possible structures. Alternative methods,which start from a particular structure, are likelyto have an inherent bias toward that structure andare less able to find different structures that mayhave lower energy.

The energies, substitution energies, and atompair counts have been tabulated for all Ar XeNy n nN microclusters. In addition, further inves-2 F N F 20tigation of the optimized structures for Ar Xe ,10yn nAr Xe and Ar Xe microclusters was per-12yn n 17yn nformed to determine the reasons for the observedchanges in substitution energy.

Acknowledgments

The author thanks Graham Wood and theanonymous referees for their many critical andconstructive comments while preparing this paper.

Appendix A: Microcluster Properties

The following table contains:

B The optimized energies for all Ar XeNy n nN microclusters.2 F N F 20

B The substitution energy, D E, as Xe atomsare substituted for Ar atoms in each micro-cluster. D E is tabulated as an absolute value.

B The counts of Ar]Ar, Ar]Xe, and Xe]Xeatom pairings where the distance between

˚the atoms is within 0.2 A of r for theminrespective atom types.

n E D E N N NA A A X X X

Ar Xe2yn n

0 y1.0000 1 0 01 y1.4800 0.4800 0 1 02 y1.8525 0.3725 0 0 1

Ar Xe3yn n

0 y3.0000 3 0 01 y3.9600 0.9600 1 2 02 y4.8125 0.8525 0 2 13 y5.5575 0.7450 0 0 3

Ar Xe4yn n

0 y6.0000 6 0 01 y7.4400 1.4400 3 3 0


2 y8.7725 1.3325 1 4 13 y9.9975 1.2250 0 3 34 y11.1150 1.1175 0 0 6

Ar Xe5yn n

0 y9.1039 9 0 01 y11.0176 1.9137 5 4 02 y12.8290 1.8114 2 6 13 y14.5408 1.7118 0 6 34 y15.6858 1.1450 0 3 65 y16.8649 1.1791 0 0 9

Ar Xe6yn n

0 y12.7121 12 0 01 y14.7512 2.0391 8 4 02 y16.9602 2.2090 3 8 13 y18.7090 1.7488 1 8 34 y20.5018 1.7928 0 8 45 y21.9889 1.4871 0 4 86 y23.5491 1.5602 0 0 12

Ar Xe7yn n

0 y16.5054 15 0 01 y19.2622 2.7568 10 6 02 y22.0179 2.7557 5 9 13 y23.7468 1.7289 3 10 24 y25.5027 1.7559 1 10 45 y27.2248 1.7221 0 8 76 y28.8997 1.6749 0 4 127 y30.5762 1.6765 0 0 15

Ar Xe8yn n

0 y19.8215 19 0 01 y23.0058 3.1843 12 7 02 y25.8141 2.8083 7 11 13 y28.0031 2.1890 4 12 24 y30.1491 2.1460 2 11 55 y31.9569 1.8078 0 11 76 y33.6846 1.7277 0 7 127 y35.3818 1.6972 0 3 168 y36.7193 1.3375 0 0 19

Ar Xe9yn n

0 y24.1134 23 0 01 y27.5466 3.4332 15 8 02 y30.3876 2.8410 10 12 13 y32.9050 2.5174 6 14 34 y35.1329 2.2279 4 13 65 y37.3987 2.2658 2 12 96 y39.1971 1.7984 1 10 127 y41.0267 1.8296 0 8 158 y42.8423 1.8156 0 4 199 y44.6700 1.8277 0 0 23

VOL. 18, NO. 81108



Ar Xe10yn n

0 y28.4225 27 0 01 y31.8734 3.4509 12 9 02 y34.7706 2.8972 7 13 13 y37.4314 2.6608 5 15 34 y40.2013 2.7699 4 16 65 y42.2426 2.0413 2 15 96 y44.5857 2.3431 3 15 97 y46.8097 2.2240 2 8 118 y49.8288 3.0191 1 12 129 y51.9091 2.0803 0 9 1810 y52.6528 0.7437 0 0 27

Ar Xe11yn n

0 y32.7660 31 0 01 y36.1896 3.4236 19 10 02 y39.1815 2.9919 14 14 13 y42.1373 2.9558 10 17 34 y45.1324 2.9951 4 18 65 y47.6269 2.4945 0 18 96 y50.5539 2.9270 6 18 87 y53.3974 2.8435 4 16 108 y56.9554 3.5580 0 16 89 y58.6126 1.6572 0 12 1610 y60.2662 1.6536 0 9 2011 y60.6990 0.4328 0 0 31

Ar Xe12yn n

0 y37.9676 36 0 01 y41.2283 3.2607 30 6 02 y44.5236 3.2953 24 12 03 y47.6411 3.1175 19 16 14 y50.4576 2.8165 15 19 25 y53.4130 2.9554 11 22 36 y56.3567 2.9437 8 18 77 y59.3386 2.9819 5 20 108 y62.1998 2.8612 3 18 129 y64.4855 2.2857 1 16 1710 y66.1764 1.6909 0 13 2111 y67.8740 1.6976 0 9 2512 y70.3350 2.4610 0 0 36

Ar Xe13yn n

0 y44.3268 39 0 01 y47.6968 3.3700 36 6 02 y51.1213 3.4245 19 16 13 y54.5957 3.4744 24 18 04 y57.8543 3.2586 19 22 15 y60.7363 2.8820 17 20 56 y63.7979 3.0616 12 24 67 y66.5918 2.7939 10 22 108 y69.0209 2.4291 3 23 13


9 y71.6010 2.5801 5 20 1710 y74.0185 2.4175 2 20 2011 y76.2746 2.2561 1 11 2512 y78.7008 2.4262 0 1 3613 y82.1154 3.4146 0 0 39

Ar Xe14yn n

0 y47.8452 45 0 01 y51.6528 3.8076 38 7 02 y55.2653 3.6125 32 12 13 y58.8911 3.6258 27 15 34 y62.3267 3.4356 21 21 35 y65.5107 3.1840 16 25 46 y68.6972 3.1865 12 27 67 y71.5258 2.8286 10 25 108 y73.9865 2.4607 7 25 139 y76.5749 2.5884 5 23 1710 y79.0019 2.4270 3 19 2111 y81.2590 2.2571 1 14 2512 y83.7123 2.4533 0 7 3613 y87.1603 3.4480 0 3 4214 y88.6800 1.5197 0 0 45

Ar Xe15yn n

0 y52.3227 49 0 01 y56.3670 4.0443 30 14 02 y60.1992 3.8322 33 14 13 y63.9532 3.7540 28 17 34 y67.6800 3.7268 24 20 45 y71.1313 3.4513 17 26 46 y74.3084 3.1771 11 30 57 y77.1841 2.8757 11 28 108 y79.6808 2.4967 0 32 129 y82.2536 2.5728 5 26 1710 y84.7267 2.4731 4 23 2111 y86.9356 2.2089 1 17 2512 y89.5321 2.5965 1 10 3513 y92.9570 3.4249 1 6 4114 y94.9799 2.0229 0 4 4415 y97.0191 2.0392 0 0 48

Ar Xe16yn n

0 y56.8158 53 0 01 y61.8505 5.0347 33 14 02 y65.1729 3.3224 29 18 03 y68.8390 3.6661 27 19 34 y72.5794 3.7404 25 22 35 y76.1583 3.5789 19 28 36 y79.6076 3.4493 13 31 77 y82.6623 3.0547 10 31 108 y85.1669 2.3458 8 29 149 y87.7763 2.7682 4 30 1610 y90.2711 2.4948 2 26 19


PULLAN


11 y92.8687 2.5976 1 23 2412 y95.5958 2.7271 1 24 1613 y98.5781 2.9823 0 9 4014 y101.0880 2.5101 0 8 4315 y103.2260 2.1382 0 4 4716 y105.3870 2.1609 0 0 51

Ar Xe17yn n

0 y61.3180 57 0 01 y66.7034 5.3854 36 14 02 y70.2609 3.5575 31 19 13 y73.9456 3.6847 26 22 34 y77.6197 3.6741 28 25 35 y81.2377 3.6180 22 31 36 y84.6540 3.4163 17 30 77 y88.1712 3.5172 10 34 108 y90.8381 2.6669 8 33 119 y94.0120 3.1739 5 32 1610 y96.4962 2.4842 3 29 2011 y99.1802 2.6840 2 26 2312 y103.0480 3.8679 3 30 2213 y104.8350 1.7873 2 22 2714 y108.2390 3.4039 1 21 3215 y109.4510 1.2115 0 8 45

16 y111.6100 2.1595 0 4 4917 y113.7890 2.1788 0 0 54

Ar Xe18yn n

0 y66.5310 61 0 01 y71.6344 5.1034 42 14 02 y75.4300 3.7956 43 16 03 y79.6966 4.2666 38 22 14 y83.7028 4.0062 32 26 35 y87.5703 3.8675 26 30 56 y91.2575 3.6872 20 36 57 y94.1626 2.0879 17 37 78 y97.2132 3.8678 14 38 99 y99.9746 2.7614 11 38 1210 y102.9080 2.9334 9 34 1611 y105.6790 2.7706 7 31 2012 y108.2810 2.6019 0 26 3113 y111.0780 2.7979 2 29 2714 y113.5770 2.4987 1 25 3215 y116.1860 2.6091 2 12 4316 y118.2490 2.0625 1 12 4717 y120.4410 2.1925 0 3 5318 y123.4810 3.0398 0 0 560 y72.6620 67 0 01 y77.2985 4.6365 59 8 0

2 y81.9088 5.6103 51 16 03 y86.3286 4.4198 44 22 1


Ar Xe19yn n

4 y90.4325 4.1039 38 26 35 y94.3813 3.9488 32 30 56 y98.1058 3.7245 26 36 57 y101.8270 3.7215 20 42 58 y104.5980 2.7704 17 42 89 y107.5650 2.9672 15 38 1210 y110.2860 2.7211 13 35 1611 y113.0830 2.7974 11 38 1712 y115.7640 2.6805 9 35 2213 y118.4020 2.6382 7 34 2614 y120.8970 2.4949 6 27 3215 y123.9530 3.0556 5 17 4316 y126.7740 2.8215 3 15 4717 y129.3820 2.6074 1 12 4918 y132.1260 2.7448 0 7 5119 y134.9560 2.8297 0 0 65

Ar Xe20yn n

0 y77.1794 70 0 01 y81.8554 4.6760 63 8 02 y86.5218 4.6664 53 16 13 y91.1201 4.5983 45 24 14 y95.1858 4.0657 39 28 35 y99.2034 4.0176 35 34 2

6 y102.8620 3.6584 27 38 57 y106.6060 3.7443 21 44 58 y109.8870 3.2808 17 45 89 y113.0180 3.1308 14 44 910 y115.9530 2.9353 12 41 1411 y118.8340 2.8805 11 38 1912 y121.5280 2.6944 9 36 2213 y124.2350 2.7075 7 34 2514 y126.7900 2.5545 5 30 3015 y129.8970 3.1072 5 18 4216 y132.8010 2.9044 3 17 4517 y135.4560 2.6540 1 15 4818 y138.1490 2.6934 0 9 5219 y141.0400 2.8911 0 2 6220 y143.3990 2.3591 0 0 62

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energy minimization of mixed argon-xenon microclusters using a genetic algorithm

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