Genetic Algorithms for the Geometry Optimization of Atomic and Molecular Clusters

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JOURNAL OFCOMPUTATIONAL AND THEORETICAL NANOSCIENCE

Vol 1 117ndash131 2004

Copyright copy 2004 American Scientific PublishersAll rights reservedPrinted in the United States of America

CONTENTS

1 Introduction 1172 Genetic Algorithm as Global Optimization Method 119

21 General Genetic Algorithm An Overview 11922 Cluster Geometry Optimization by GA History

and Development of the Methodologies 12023 Implementation of Genetic Algorithm in Cluster

Geometry Optimization 12124 Other Competitive Global Optimization

Algorithms 1223 Noble Gas Lennard-Jones Clusters by Genetic

Algorithm 12231 Small LJ Clusters with n 30 12332 Medium-Size LJ Clusters with 30 n 150 12333 Large LJ Clusters with n 150 124

Genetic Algorithms for the GeometryOptimization of Atomic and Molecular Clusters

Jijun Zhaoa and Rui-Hua Xieb

aDepartment of Physics and Astronomy University of North Carolina at Chapel Hill North Carolina 27599 USAbDepartment of Chemistry Queenrsquos University Kingston Ontario K7L 3N6 Canada

Clusters aggregates of a few to thousands of atoms or molecules have been intensively studied duringthe past 20 years due to their importance in physics chemistry biochemistry and material science as wellas their potential applications as building blocks in nanoscience and nanotechnology Determination of theground-state geometries of clusters is a well-known nondeterministic polynomial-time hard problem be-cause of the numerous structural isomers on the potential energy surface To overcome this difficulty andto explore the magic number clusters with high stability genetic algorithms were introduced and developedin the global optimization of cluster geometry In this article we review the methodological developments ofgenetic algorithms in cluster science and present their broad applications for the geometry optimization ofatomic and molecular clusters In detail benchmark studies on the noble gas clusters modeled by theLennard-Jones potential based on different global optimization approaches are summarized and dis-cussed Applications of the genetic algorithm to the realistic clusters such as metal clusters semiconduc-tor clusters alloy and compound clusters and molecular clusters are presented The virtues limitation andfuture improvement of the genetic algorithm for the geometry optimization of clusters are remarked

Keywords Cluster global optimization local minimization genetic algorithm potential energy surfacemany-body potential simulated annealing molecular dynamics ab initio DFT tight-bindingLennard-Jones cluster Monte Carlo mutation selection crossover semiconductor metalalloy water carbon fullerene silicon gold platinum rhodium zirconium titanium palladiumnickel aluminum sodium potassium cobalt copper chromium vanadium lanthanumcerium praseodymium

4 Application in the Realistic Clusters 12441 Metal Clusters 12442 Semiconductor Clusters 12643 Alloy and Compound Clusters 12744 Molecular Clusters 12745 Applications in Other Related Fields 128

5 Summary and Outlook 128Acknowledgment 129References 129

1 INTRODUCTION

Clusters are aggregates of a few to thousands of identicalor different atomsmolecules and their length scales rangefrom a few angstroms to tens of nanometers In experimentsthey are usually generated from mass-selective clusterbeams and can be isolated in a number of media such as inertmatrices and colloidal suspensions or they can be deposited

J Comput Theor Nanosci Vol 1 No 2 2004 copy 2004 by American Scientific Publishers 1546-198X200401117131$1700+25 doi101166jctn2004010 117

Author to whom correspondence should be addressed

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on the surface of substrates The relative abundance ioniza-tion potentials photoelectron spectra magnetic momentsand optical adsorption of these clusters in different formshave been measured by various of experimental means

As the intermediate state between microscopic atomsand macroscopic condensed matters the structures andphysical properties of clusters exhibit many interesting size-dependent phenomena such as ldquomagic numberrdquo in the rela-tive abundance associated with electron or geometry shellsmetal-nonmetal transition enhanced magnetic momentsred- or blue-shift of optical gap and so on1ndash8 From thetechnological point of view clusters can be considered assuperatoms9 and serve as building blocks in nanoscienceand nanotechnology such as cluster-deposited film cluster-assembled solid and cluster-based nanoelectronics10ndash16

During the past 20 years the structures and properties ofatomic and molecular clusters have been intensively stud-ied from many aspects In particular it is fundamentallyimportant to understand the evolution behavior of the ma-terials as a function of size from isolated atoms or smallmolecules to nanoparticles and to bulk condensed matters

Since the physical and chemical properties of clusterssensitively depend on the cluster geometry it is critical toknow the lowest-energy atomic structures of clusters inorder to further understand the clusters and to design thecluster-based materials and devices The determination ofground-state geometry of a cluster with n atoms can beconsidered as a global optimization problem in the 3n-dimension configuration space constituted by the atomiccoordinates As the cluster size n increases the number ofstructural isomers will increase too Correspondingly the

multi-dimension potential energy surface (PES) of thecluster will become very complicated For example it wasestimated that the total number M of minimal energy struc-tures for a Lennard-Jones (LJ) cluster scales exponentiallywith the cluster size n as given below17

M exp[25176 03572 n 00286 n2] (1)

Indeed the global geometry optimization of LJ clusterswas proven to be a nondeterministic polynomial-time hard(NP-hard) problem18 For the true atomic or molecularclusters other than LJ ones the realistic potential energylandscape could be even more complicated Thus the de-termination for the ground-state geometry of clusters isone of the most fundamental and challenging problems incluster science On the other hand many cluster propertiesare difficult to measure directly Their spectroscopic andmass spectrometric data are usually interpreted in terms oftheoretical models Hence theory and computational toolsplay important roles in cluster science It is also note-worthy that the ground-state geometries of clusters dis-cussed in this paper are all for the zero-temperature caseIn reality the structures of clusters may change because ofthe temperatures If there are several metastable structuralisomers on the PES with comparable energy the clusterstructures may fluctuate among these isomers under a finitetemperature As the cluster temperature further increasesand approaches the melting point the cluster will undergophase transition and become liquid-like

For larger clusters with hundreds to thousands of atoms ormolecules ab initio electronic structure calculations are

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118 J Comput Theor Nanosci Vol 1 No 2 2004

Dr (John) Rui-Hua Xie is currently working at the National Institute of Standards and Technology(NIST) Gaithersburg Maryland His research interest and expertise include computational nanoscience(nanoelectronics nanostructured materials nanocrystals and quantum dots) cluster physics molecularphysics quantum computing quantum optics quantum theory quantum chemistry and electronic spec-troscopy such as NMR UV-vis and Raman Dr Xie received his bachelorrsquos degree in theoretical physicsfrom Wuhan University Hubei China in 1991 and his PhD in theoretical physics from NanjingUniversity Jiangsu China in 1996 From 1997 to 1998 he was a postdoctoral fellow at the University ofToronto Canada During the period from 1998 to 2000 he moved to Germany as an Alexander vonHumboldt fellow working at the Max-Planck-Institut fuumlr Stroumlmungsforschung in the beautiful universitytown of Goumlttingen Before joining the Quantum Process Group at NIST in 2001 he came to the famous

town of Kingston the first capital of Canada working at Queenrsquos University He has contributed over 80 peer-reviewed journal articlesand several review articleschapters in journals books and encyclopedias and presented over 10 invited seminarscolloquiums

Dr Jijun Zhao was born and educated in Jiangsu China in 1973 He received his Bachelor degree inphysics from Nanjing University China in 1992 and also his PhD in condensed matter physics fromNanjing University in 1996 From 1997 to 1998 he was a postdoctoral fellow in International Centre forTheoretical Physics (ICTP) Italy From 1998 to 2002 he became a research associate and then a researchassistant professor in Department of Physics and Astronomy University of North Carolina at Chapel HillUSA He is currently a senior research associate in Institute for Shock Physics Washington StateUniversity USA His major research field is in computational materials science with special interest fornanostructures (nanotubes nanowires etc) nanoelectronic devices atomic clusters and cluster-based ma-terials high pressure physics and molecular crystals He has contributed over 80 refereed journal papersand two book chapters in this field

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computationally expensive This led to the utilization ofempirical potentials19ndash27 for interatomic or intermolecularinteractions in the simulations of clusters However no mat-ter which method one may use one of the main objectives incomputational cluster science is to find the lowest-energyconfiguration of atoms or molecules or ions for a given clus-ter size This corresponds to the lowest potential energy iethe global minimum on the potential energy hypersurfaceThus the structural relaxation of clusters plays a key role inachieving this goal Molecular dynamics (MD)28 is one ofthe most commonly adopted methods This method is ableto simulate the structural and dynamic properties of clustersOn the other hand gradient-driven numerical minimizationmethods29 such as steepest descent quasi-Newton andconjugate gradient methods can be employed to directlydetermine the local minima on the PES of clusters How-ever neither MD nor numerical minimization is a globaloptimization method The optimized structures from thesemethods critically depend on the choice of starting config-urations If there are a number of local minima on the clus-ter PES separated by high-energy barriers it is almostimpossible to locate the global minimum of clusters Toovercome this difficulty some global optimization meth-ods such as simulated annealing (SA)30 31 and genetic al-gorithm (GA)31ndash33 have been developed The philosophiesof these methods are to mimic the processes in nature TheSA method models an annealing process while the GAmethod models a natural selection and evolution processBoth methods have achieved remarkable success in solv-ing many global optimization problems

The SA method which was first introduced by Kirk-patrick Gelatt and Vecchi30 simulates the natural anneal-ing process A substance is initially heated to a high tem-perature above its melting points and then cooled graduallyto approach its crystalline state as the global minimumProvided that the cooling schedule is slow enough and thetraveled trajectory in the phase space is long enough theglobal minimum of the system can be found In previousworks the SA simulations based on either Monte-Carlo34

or MD28 methods have been widely applied to search thelowest-energy structures of small clusters However inpractice if the cluster size further increases (for examplen 20 50) andor the PES is described by some realisticfunctions (for example many-body potentials densityfunctional or tight-binding Hamiltonian) the PES wouldbecome so complicated that the search procedure wouldlead to a formidable simulation in order to escape fromthose numerous local minima Moreover the SA procedurecan hardly overcome energy barriers that are higher thanthe initial kinetic energy of the system

For the global optimization of cluster geometry it is es-sential to develop some other optimization methods thatcan sample the PES more efficiently and that can hop fromone region of the PES to another region more easily Onepromising tool satisfied for such a requirement is the GAmethod This method is inspired by Darwinian evolution

theory where only the fittest individual can survive Inother words the essential idea of the GA procedure is toallow a population of many individual candidates to evolveunder a given selection rule that maximizes the fitness TheGA method was first developed by Holland32 and furthergeneralized by Goldberg33 Since 1993 there have beenseveral pioneering works on the structures of clusters fromthe GA optimization35ndash39 Even in its early stage the GAdemonstrated an impressive efficiency in searching theglobal minima of different systems ranging from the noblegas clusters to the semiconductor clusters An outstandingexample is the direct location of the global minimum struc-ture of C60 starting from a random configuration byDeaven and Ho39 Because of the strong directional bondsin carbon clusters and the consequent high energy barriersbetween different isomers the standard SA simulation pro-cedures failed to yield the buckyball structure of C60 fromunbiased starting configurations39 Motivated by the suc-cess of these pioneering works there have been increasinginterests in the applications of GA in the cluster geometryoptimization as well as in developing the GA methodologyin nanoscience and related fields For example the numberof published research papers on the GA study of clusters in-creased from five per year in 1996 to about 20 per year in2002 based on our incomplete search from the Web ofScience Such a rapid expansion of this field encouraged usto write this article for reviewing the recent progress on theapplication of genetic algorithm in clusters while the fieldis still in a highly active phase of development This articleis organized as follows In Section 2 we illustrate how toimplement the GA procedure into the global optimizationof cluster geometry and we review the methodological de-velopment as well as the other related global optimizationapproaches Then the applications of GA for example inthe noble gas clusters and the other clusters are reviewed inSection 3 and 4 respectively Finally we end this reviewgiving a brief summary and outlook in Section 5

2 GENETIC ALGORITHM AS GLOBALOPTIMIZATION METHOD

21 General Genetic Algorithm An Overview

Similar to all of the other optimization methods the ge-netic algorithm first needs to define a finite set of optimiza-tion parameters pi and the cost function f f [pi]Then the genetic algorithm will search for the minimumcost on the M-dimensional configuration space constitutedby the optimization parameters pi p1 p2 pMThe optimization parameters can be chosen in either binaryor continuous mode while the latter option is obviouslyapplicable to a much wider range of systems and problemsIn the practical applications of genetic algorithms it is crit-ical to define suitable optimization parameters and the costfunction according to the specific characteristics of the sys-tem that one wants to study

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The genetic algorithm procedure starts with an initialpopulation that has totally Np individuals The individualsin the initial population are usually generated randomly bychoosing a set of initial values of the optimization parame-ters pi for each of them It is possible to generate a muchlarger number of the initial population members and dis-card most of the high-cost members via natural selectionWith the initial population that is created two individuals(ldquoparentsrdquo) from the population are then selected to pro-duce the new offspring (ldquomatingrdquo) A variety of selectionrule can be used in pairing the individuals in the populationfor example random pairing pairing by the sequence in thepopulation weighted pairing by the cost etc After the twoparents are chosen one or more offspring (ldquochildrdquo) can becreated in the mating (or crossover) process If the geneticalgorithm optimization is performed in binary mode themating operation can be rather straightforward eg mix-ing the binary codes of the two parents by some given reg-ulations On the other hand in the continuous genetic al-gorithm there is more flexibility in defining a mating(crossover) operation We will illustrate a practical way ofmating for cluster geometry optimization in the next part

In the practical global optimization the genetic algo-rithm can converge very quickly into one region of theconfiguration space of optimization parameters whichmay not be the region of global minimum To avoid beingtrapped in the local minima we have to introduce somestochastic variation (ldquomutationrdquo) in the optimization para-meters pi so that the other areas of the configurationspace can be also reached However too much mutationmay destroy the existing ldquogoodrdquo parameters and lower theefficiency of the genetic algorithm Thus the mutation isusually restricted by a mutation rate Pmu that is only thePmu percentage of the optimization parameters will bechanged Similar to the natural process mating operationrealizes the inheritance and combination of ldquogenesrdquo fromthe parents while mutation introduces variation in ldquogenesrdquoto achieve the possible stronger offspring Both are essen-tial to the global optimization After the mating and muta-tion the cost function of the offspring in the next genera-tion is evaluated Natural selection based on the value ofcost function can be applied to both the parents andthe children so that only the fittest individuals can survivein the population The procedure of pairing mating muta-tion and selection will be iterated until the convergence isachieved or a maximum number of iteration is exceeded

22 Cluster Geometry Optimization by GAHistory and Development of theMethodologies

In those earlier efforts of applying the genetic algorithm tocluster geometry optimization for example Hartke onSi435 Xiao and Williams on small molecular clusters ofbenzene naphthalene and anthracene37 the cluster geome-

tries were binary encoded This is obviously inconvenientand hard to use As a major breakthrough Zeiri used thereal Cartesian coordinates of the atoms as the natural genesand converted the conventionally binary genetic algorithminto continuous genetic algorithm38

Two other significant steps toward the broad applica-tions of GA to cluster optimization have been made byDeaven Ho and co-workers39 40 First Deaven and Hointroduced an efficient ldquocut and splicerdquo crossover opera-tion39 40 As shown in Fig 1 random planes that passthrough the center of the two parent clusters are chosenAfterward a new child cluster is assembled from the twohalves (upper half in A and down half in B) of the parentclusters A B This operation allows the child cluster to in-herit about half of the (geometry) characteristics from itsparents similar to the natural mating process The secondimportant contribution in their work is the application of alocal gradient-driven minimization to structures of thechild cluster right after each crossover Such a local mini-mization is proven to significantly improve the efficiencyof the GA global minimization

In recent years several further efforts have been madeto improve the GA and to apply to the clusters with morecomplicated potential energy surface Some noticeableachievements in this area include modified deterministicstochastic genetic algorithm by Gregurick et al41 space-fixed modified genetic algorithm by Niesse and Mayne42 43

modified genetic algorithm by Wolf and Landman44 sym-biotic algorithm by Michaelian45 predictor algorithm byZacharias et al46 phonotype algorithm by Hartke47 single-parent evolution algorithm by Rate et al48 neural networkassisted genetic algorithm by Lemes et al49 hierarchicalgreedy algorithm by Krivov50 modified genetic algorithm

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120 J Comput Theor Nanosci Vol 1 No 2 2004

Figure 1 Illustrations of the ldquocut and splicerdquo crossover operation intro-duced by Deaven and Ho39 The two halves of ldquoparentrdquo clusters A and Bare spliced into a new ldquochildrdquo cluster A1-B2

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with two new evolutionary operators (annihilator and his-tory operator) by Guimaratildees et al51 improved genetic al-gorithm with a self-guiding search strategy by Kabredeand Hentschke52 etc We will not discuss the details of themin this article Some of the progress on genetic algorithm incluster science can be found in recent reviews53 54

23 Implementation of Genetic Algorithmin Cluster Geometry Optimization

Since the GA method is independently developed in clus-ter science by a number of groups the detailed operationsof doing GA simulations may differ from one to the otherHere we briefly illustrate a practical procedure that is usedin our own works55ndash68 The corresponding genetic algo-rithm code developed by us is also available upon requestBasically it is very close to the scheme introduced byDeaven and Ho39 40 and is also similar to those used bymost other researchers As shown in Fig 2 our scheme ofGA optimizations can be in general divided into three es-sential steps crossover mutation and selection

At the first step an initial population which contains anumber of different cluster geometries is generated at ran-dom To ensure a global optimization the number Np ofthe individuals in the population typically increases withthe cluster size Then two individuals in the population areselected as parent clusters to generate a new child clusterwhich is known as crossover (or mating) operation Thepossibility of being chosen as a parent is identical for allthe candidates in population The crossover operation iscomposed by cutting each parent via a randomly chosenplane passing through its mass center and then assemblingthe child from the upper and down halves of the two re-

spective parents (see Fig 1 for an illustration) This is thepopular ldquocut and splicerdquo crossover operation introduced byDeaven and Ho39 If the generated child does not containthe right number of atoms or the resulting bond lengths areunacceptably small the above mating process will be re-peated until a proper child is yielded

At the end of crossover we can perform a mutation onthis new generated child with certain possibility Pmu It isnoteworthy that the current mutation rate is for the entirecluster instead of the optimization parameters (ie Carte-sian coordinates) as stated in Section 21 Such a mutationoperation can be carried out by applying a number of ran-dom displacements (eg 100 steps per atom with walklength 005 Aring per step) on the Cartesian coordinates ofeach atom in the cluster We found that the mutation stageis crucial to obtain the global minima if the configurationspace is sufficiently complicated

The child cluster generated from crossover and muta-tion operations will be locally optimized by either a MDquenching or a numeric minimization like BFGS69 In thecase of MD relaxation the cluster is quenched from a rel-atively high temperature to a low temperature within a rel-atively small number of MD steps Both the BFGS mini-mization and MD quenching are expected to relax thecluster to a reasonable local minimum nearby This ldquolocalrdquoquenching strategy is able to obtain the lowest energyminima corresponding to a considerable portion of thePES such that the system will not be trapped in the near-est minima from the starting configuration that might havea high energy As a consequence numerous minima struc-tures with relatively higher energies can be skipped in thesearch process

After a locally stable ldquochildrdquo cluster is obtained we per-form the natural selection during which the child is eitheraccepted into the population or discarded according to itsenergy and geometry To preserve the diversity of the pop-ulation we first compare the coordination numbers of eachatom in the child to all those for the individuals inside thepopulation If the child has the same configuration as anyexisting individual it will be either discarded or replacethis individual according to its energy If the child has alower energy than one of its parents and has geometry dif-ferent from all of the existing individuals in the populationit will replace its high-energy parent and enter the popula-tion Therefore the population always contains individualsthat have different atomic structures In this scheme theGA optimization not only determines the ground statestructure but also collects the other locally stable isomersof the cluster As we shall discuss below the availability ofthe other structural isomers is very important for an empir-icalquantum combination study

In practice it was found that the efficiency of the GAoptimization sensitively depends on the number Np of indi-viduals in the population and the mutation probability PmuFor example Fig 3 presents the convergence of the lowest-energy structures for a Si16 clusters using different Np and

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child

mutation

crossover

parent parent parent

local optimization (empirical or TB)

parent parent parent

parent

input for DFT minimization

initiation

weaker

crossover

selection

parent

discard parent

stronger

Figure 2 Flow chart for an empiricalquantum combination geneticalgorithm optimization

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Pmu In the GA optimizations Menon and Subbaswamyrsquosnonorthogonal tight-binding total energy model70 and thelimited memory BFGS minimization method69 are usedFor a given cluster size there are some optimal choices forNp and Pmu that approach the lowest-energy structuresfastest

In order to improve the accuracy of the geometry globaloptimization Zhao and coworkers have further proposeda GA-based empiricalquantum combination scheme58 andapplied it to study a variety of atomic clusters58ndash60 65ndash67 Theessential idea is to use the GA optimization at either empiri-cal or semiempirical (eg tight-binding model57 70 71) levelfor generating a number of structural isomer candidates ofthe cluster Then accurate density function theory (DFT)calculations are performed to locally optimize these isomerstructures and finally determine the lowest-energy structureof the cluster at the accuracy level of DFT A similar empir-icalquantum combination method was also used by otherssuch as Ho et al for silicon clusters72 Tang et al for Si8H8

clusters73 and Garzoacuten et al for gold clusters74

24 Other Competitive Global OptimizationAlgorithms

Besides the GA method many other competitive globaloptimization algorithms have also been developed for theseNP-hard problems such as cluster geometry optimizationor protein folding Some major achievements in theseunbiased global optimization methods are Monte-Carlowith minimization75 J-walking76 Monte-Carlo growth

method77 78 generalized simulated annealing79 80 confor-mational space annealing81 82 basin-hopping83ndash86 quan-tum annealing87 parallel tempering algorithm88 smartwalking annealing89 multicanonical jump walk anneal-ing90 quantum thermal annealing91 random tunnelingalgorithm92 fast annealing evolutionary algorithm93ndash95

quantum path minimization96 etc Some details can befound in other review articles97 98 Among these achieve-ments it is worthy to mention that Wales and Doye intro-duced a basin-hopping algorithm83 in the sprite of Monte-Carlo with minimization75 By transforming the PES intoa collection of interpenetrating staircases83 the basin-hopping algorithm is found to be very robust in clustergeometry optimizations demonstrating transferability be-tween atomic and molecular clusters and yielding usefulresults for biomolecules In Table 1 we summarize someof these previous works that utilized the LJ clusters asbenchmark systems The performance on the LJ clustersbetween the GA method and some of those competitive ap-proaches will be compared in the next section

3 NOBLE GAS LENNARD-JONESCLUSTERS BY GENETIC ALGORITHM

Although the LJ potential is analytically simple it is stillwidely used by researchers in many areas There has beena vast amount of literature on the clusters of noble gasatoms modeled by the LJ pair potential Almost everyglobal optimization method in existence has been tested onthe LJ clusters as benchmark system In the 6ndash12 LJ pairpotential

(2)

the reduced units j 1 e 1 are conventionally usedWithin a LJ pair potential it is well-known that the geome-tries of the cluster are dominated by the packing effectIn general the LJ cluster tends to adopt the icosahedralcompact structures as called ldquoMackay icosahedronrdquo99 (seeFig 4) where the multi-shell structures are complete atparticular sizes for example n 13 55 147 309 etc(so-called geometry magic number) However for some ofthose clusters with sizes between the complete icosahedranonicosahedral close-packed structures were found83 84 98

Therefore such clusters with non-icosahedral global min-ima are more critical to test the performance of the globaloptimization methods

Table 1 summarizes the previous studies of LJ clustersusing genetic algorithm and other competitive global opti-mization methods A complete collection of energies andCartesian coordinates for those global minima structuresup to n 150 can be found in the Cambridge ClusterDatabase100

E r rij ij

i j

412 6

e j j ( ) ( )[ ]sum

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122 J Comput Theor Nanosci Vol 1 No 2 2004

Figure 3 The evolution of lowest-energy of a Si16 cluster within ge-netic algorithm optimization with different choices of numbers of parentsin population Np and mutation probability Pmu It is clear that Np 8 andPmu 03 lead to the best optimization efficiency for the current case Inthe GA optimizations Menon and Subbaswamyrsquos nonorthogonal tight-binding total energy model70 and the limited memory BFGS minimiza-tion method69 are used The present global minimum structures (shownon the right) found for Si16 within the nonorthogonal tight-binding modelis similar to the metastable C2h isomer predicted by Ho72

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31 Small LJ Clusters with n 30

Small LJ clusters have been studied as a benchmark forGA optimization in those earlier works Zeiri has studiedthe lowest energy structures of very small LJ clusters (n 4 10) using genetic algorithm101 Excellent agreementwas obtained between the GA results with exact classicaldata102 He has also shown that the genetic algorithmbased method converges to the global minima much morerapidly than both the simulated annealing and quantum

annealing Similarly for LJ Arn clusters with n 30 thedeterministicstochastic genetic algorithm optimization con-verges much more quickly to the global minimum than therandom search procedures41 Moreover Chaudhury andBhattacharyya demonstrated that GA can locate criticalpoints on the multi-dimensional potential energy surfaceusing Arn (n 4 30) clusters as a test system104

32 Medium-Size LJ Clusters with 30 n 150

The medium-sized LJ clusters with n 30 150 havebeen intensively studied by different groups using the GAmethod40 42 44 45 47 103 In this size range non-icosa-hedral structures (see Fig 4) were found as global minimaat several specific sizes (n 38 75 77 98 102 104)Thus they can be considered as the ldquohurdlesrdquo for thoseglobal optimization methods As shown in Fig 4 a fcctruncated octahedron was found as a global minimumstructure of the 38-atom LJ cluster It is particularly inter-esting as the only small LJ cluster with fcc lowest-energystructures during the size range of n 30 150 In previ-ous GA studies the fcc truncated octahedron was firstfound by Deaven et al and then successfully reproducedby other workers42 44 45 47 103 At n 75 77 and 102 104 the global minima structures of LJ clusters are basedon Marks decahedra (see Fig 4) These global minimawere systematically summarized by Wales and Doye intheir basis-hopping study of LJ clusters83 It is noteworthythat Deavenrsquos original GA optimization of LJ clustersfailed to produce those decahedra40 Wolf and Landmanimproved the methodology of GA and were able to repro-duce all of these global minima of LJ clusters with up to100 atoms44 Later Hartke47 and Barron et al103 have per-

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Table 1 Summary of previous works on LJ clusters using genetic algorithm and related global optimization methods

Size Method Authors

4 10 genetic algorithm Zeiri101

2 29 deterministicstochastic genetic algorithm Gregurick et al41

4 30 genetic algorithm Chaudhury et al104

4 55 space fixed modified genetic algorithm Niesse and Mayne42

6 18 23 38 55 symbiotic algorithm based on GA Michaelian45

2 100 genetic algorithm Deaven et al40

2 100 genetic algorithm Pullan105

10 100 modified genetic algorithm Wolf and Landman44

13 147 genetic algorithm Barron et al103

10 150 phenotype algorithm Hartke47

148 309 genetic algorithm Romero et al106

75 77 98 500 1000 hierarchchical greedy algorithm Krivov50

2 110 basin-hopping algorithm Wales and Doye83 84

19 30 38 significant structures basin hopping White and Mayne85

5 100 self-consistent basin-to-deformed-basis mapping Pillardy86

2 100 random tunneling algorithm Jiang et al92

2 13 fast annealing evolutionary algorithm Cai and Shao93

2 74 fast annealing evolutionary algorithm Cai et al94

2 116 parallel fast annealing evolutionary algorithm Cai et al95

5 30 multocanonical jump walk annealing Xu and Berne90

201 conformational space annealing Lee et al82

11 100 quantum path minimization Liu and Berne96

Figure 4 (Upper) Global minima structures of LJ clusters with com-plete Mackay icosahedron 13-atoms 55-atoms 147-atoms (Lower)Typical non-icosahedron global minima structures of LJ cluster 38-atoms truncated fcc octahedron 75-atoms Marks decahedron 98-atomstetrahedron-based packing All the structures are taken from CambridgeCluster Database100

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formed independent GA global optimization of LJ clusterswith up to 150 atoms respectively Both of the studieshave successfully obtained the global minima for n 75 77 and 102 104 However for the 98-atom LJ cluster atetrahedral global minimum was found by Leary and Doyeusing the basis-hopping method83 On the PES this globalminimum is at the bottom of a narrow tunnel so that it isdifficult to find Indeed within the best of our knowledgenone of the current GA optimizations have produced thisstructure Thus determination of global minimum of the98-atom LJ cluster is still a challenge for the genetic algo-rithm in clusters science

Besides the genetic algorithm and the basis-hoppingmethods it would be interesting to examine the perfor-mance of other global-optimization approaches in this sizerange Pillard et al have proposed a self-consistent basin-to-deformed-basin mapping method and tested it in the LJclusters with up to 100 atoms86 Although most of theglobal minima including n 38 are reproduced theirmethod failed in yielding the global minimum for n 75 77 and 98 for which the second-to-the-lowest-energystructures were found Jiang Cai and Shao have put for-ward two global optimization methods and applied them toLJ clusters92ndash95 The global optimization using randomtunneling algorithm has successfully produced all theglobal minima of LJ clusters with up to 10092 while an-other work done with fast annealing evolutionary algo-rithm has extended the successful range to n 11695 Liuand Berne proposed a quantum path minimization methodand found that this new method is able to locate all theglobal minima of LJ clusters of size up to n 100 withexception at n 76 77 9896 Very recently Lee and co-workers applied the conformational space annealing methodto the LJ clusters with up to 201 atoms and reproduced allknown global energy minima82

33 Large LJ Clusters with n 150

So far there are only few reports on the LJ clusters withmore than 150 atoms using genetic algorithm50 106 As anextension of their previous study on LJ clusters with up to147 atoms103 Remero and co-workers proposed a geneticalgorithm combined with a stochastic search procedure onicosahedral lattices to perform global optimization of LJclusters with n 148 309106 Icosahedral and Marksdecahedral structures were found and a new morphologycalled FD was predicted For the larger LJ clusters with upto 500ndash1000 atoms a hierarchical greedy algorithm hasbeen developed by Krivov and applied to search the globalminima of the clusters50

4 APPLICATION IN THE REALISTICCLUSTERS

In this section we review the current progress on the GAworks done for the true atomic and molecular cluster such

as metal clusters semiconductor clusters ionic compoundclusters and molecular clusters Certainly the PESs of allthese clusters would be much more complicated than thenoble gas clusters modeled by LJ potentials making thedetermination of the global minimum structures of clustersmore challenging Therefore most of the current studiesare still limited to relatively smaller clusters ie n 50

41 Metal Clusters

Metal clusters have attracted tremendous attention due totheir interesting atomic and electronic structures Althoughthe cohesion in the metal clusters should be dominated by theeffect of valence electrons bonding empirical many-bodypotentials in the sprite of ldquoembedded-atomrdquo are still widelyused to account for the interatomic interactions in the metalclusters Among these empirical potentials the Gupta-likepotential19 is the most popular one in previous GA studies

(3)

The second attraction part of the potential can be associ-ated with the tight-binding model within the second mo-ment approximation20 which essentially describes the bandcharacter of the metallic bond Other than empirical poten-tials semiempirical tight-binding models and extended-Huumlckel method have also been incorporated into the GA ap-proach for a global search of lowest-energy structures of themetal clusters Moreover empirical GA simulation has beenused to provide a number of structural isomer candidateswhich are then further optimized by more accurate DFT cal-culations In Table 2 we summarize the total energy meth-ods and the size range of metal clusters reported in previousworks The detailed descriptions are presented below

411 Noble Metal Clusters

Noble metal clusters especially gold clusters have receivedincreasing attention as building blocks in the nanoscalematerials and devices It is particularly interesting to inves-tigate the noble metal clusters at the magic sizes such as38 and 55 where the complete fcc or icosahedral close-packing structures might form (see Fig 4 in Section 3) Forsmall Aun (n 2 20) clusters Wang and co-workershave carried out a comprehensive GA search for their low-est-energy structures and have extensively studied theelectronic properties using the empiricalquantum combi-nation method65 Using the same method they have re-cently investigated the lowest-energy structures of the 20-atoms noble metal clusters (Cu20 Ag20 and Au20)66 After

E B pr

r

A qr

r

ij

j ii

ij

j i

exp

exp 2

0

2

0

1 2

1

1

ne

ne

sumsum

sum

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fitting a semiempirical tight-binding model from bulk anddimer properties of silver Zhao et al have studied thestructural and electronic properties of silver clusters up to20 atoms57

In the serial works reported by Garzoacuten and co-workers74 107 108 the lowest-energy structures of Aun

clusters were obtained from the empirical GA optimiza-tions using Gupta potentials These structures were furtherconfirmed by the DFT relaxation In general disordered orlow-symmetry structures were found to be global minimastructures which are different from the LJ clusters Similarresults were also obtained by Li et al in their empirical GAsimulations56

412 Simple Metal Clusters

The structures of monovalent (Na K Rb Cs) and polyva-lent (Pb) clusters with the number of atoms up to 56 havebeen systematically studied by using Gupta potentials109

Both the genetic algorithm and the basin-hopping methodhave been used in the global minimization and excellentagreement was found in the global minima from bothschemes109 Loyd110 used the GA to optimize the geome-try of aluminum clusters with 21 55 atoms bound by theMurrell-Mottram (MM) 2 3 body potential111 The MMpotential is based on a many-body expansion of the poten-tial energy truncated up to the three-body term The poten-tial parameters in the MM potentials are derived by fittingexperimental data such as phonon frequencies elastic con-stants vacency energy etc for the solid phase

413 Divalent Metal Clusters

Size-dependent evolution of metallic behaviors of the diva-lent metal clusters is an interesting issue in cluster sciencesince the very small clusters are bound by van der Waalsinteraction similar to helium Zhao and co-workers haveemployed the combination technique of empirical GAglobal search and DFT local minimization to obtain thelowest-energy structures of Cd58 Be59 Zn67 clusters up to20 atoms The size-dependent electronic properties of thoseclusters and the nonmetal-metal transitions were then stud-ied based on the optimized structures58 59 67 For the largerZn and Cd clusters at ldquomagicrdquo sizes of 13 38 55 75 and147 atoms disordered structures or structures of very lowsymmetry were found as the lowest-energy structures fromempirical GA optimizations using Gupta potentials113

Hartke and co-workers have studied Hgn (n 7 14)clusters via a GA global geometry optimization within aquantum mechanical-empirical hybrid model114

414 Transition Metal Clusters

As for the transition metal clusters the size-dependentmagnetism of small clusters is particularly interestingCombined techniques of empirical GA optimization usingGupta potential and spin-polarized calculations withintight-binding approximations have been used to investi-gate the structures and magnetic moments of Cr13115 Rhn

(n 9 13 15 17 19 in Ref [116] 4 n 26 in Ref[117]) and Vn (n 3 19)118 clusters As well as the 3d

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Table 2 Summary of the cluster size range and total energy methods in previous GA studies of metal clusters ldquoGuptapotential DFTrdquo denotes that first principles calculated result have been further applied to the local stable candidatestructures from GA optimizations based on empirical potentials

Element Size Total energy method Authors

Au 220 Gupta potential DFT Wang et al65

Au 3855 Gupta potential Li et al56

Au 38 55 75 Gupta potential DFT Garzoacuten et al74

Au Ni Ag 675 Gupta potential Michaelian et al107

Au 12212 Gupta potential DFT Garzoacuten et al108

Cu Ag Au 20 Gupta potential Wang et al66

Ag 221 tight-binding Zhao et al57

Na K Rb Cs Pb 356 Gupta potential Lai et al109

Al 2155 Murrell-Mottram potential Lloyd et al110

Be 221 Gupta potential DFT Wang et al59

Sr 263 Size-dependent Gupta potential Wang et al112

Cd 221 Gupta potential DFT Zhao58

Zn 220 Gupta potential DFT Wang et al67

Zn Cd 13147 Gupta potential DFT Michaelian et al113

Hg 714 Quantum-empirical hybrid model Hartke et al114

Cr 13 Gupta potential Luo et al115

V 319 Gupta potential Sun et al118

Rh 919 Gupta potential Sun et al116

Rh 426 Gupta potential Granja et al117

La Ce Pr 320 Moumlbius inversion pair potential Luo and Wang119

Ni 213 extended-Huumlckel Curotto et al120

Fe 226 density functional tight-binding Parvanova et al121

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and 4d transition metals Luo and Wang have extended theGA optimization to the geometries of rare-earth metalclusters ie Lan Cen and Prn (n 3 20) using theMoumlbius inversion pair potential119

Besides empirical potentials semiempirical extended-Huumlckel method and tight-binding models have been uti-lized in the GA optimization of transition metal clustersFor example Curotto obtained the minima and isomers ofNin (n 2 13) clusters based on the extended-Huumlckeltotal energy model120 DFT-based tight-binding (DFTB)calculations121 have been performed to search for theground-state candidate structures of iron clusters up to 26atoms using a single-parent GA procedure48

42 Semiconductor Clusters

421 Silicon Clusters

Compared to noble gas clusters and metal clusters semi-conductor clusters might have more complicated PESs dueto the intrinsic nature of the covalent bonding Thereforesilicon clusters up to 20 atoms have been frequently usedas prototypes of the clusters with rather complicated PESlandscape in the GA global optimizations35 36 43 46 48 49

62 72 122ndash126 In Hartkersquos pioneering work Si435 and Si10

36

clusters described by an empirical Bolding-Anderson po-tential22 were used as model systems for testing the per-formance strengths and possible limitations of the GAmethod Niesse and Mayne applied space-fixed modifiedgenetic algorithm (SFMGA) approach to search the globalminima for silicon clusters43 using the same empirical po-tential22 Later this SFMGA was modified and used tostudy the lowest-energy structures of Sin with n 15where the interatomic interaction was described by empir-ical Stillinger-Weber (SW) type potential21 and a revisedform proposed by Gong23

For the larger Sin clusters up to n 20 Ho et al em-ployed a combined technique of tight-binding GA globaloptimization and LDA local minimization to explore thecluster ground state structures72 A single-parent evolu-tion algorithm similar to the GA was introduced to op-timize Si clusters in which the Si-Si bonding was de-scribed by DFT-based tight-binding method48 Zachariaset al introduced a novel hybrid approach of GA and SA(namely predictor algorithm) and applied it to study thegeometry of Si20 cluster46 A modified GA optimizationmethod called neural-network-assisted genetic algo-rithm49 was tested in determining the ground stategeometries of Sin (n 10 15) based on a tightbindingtotal energy method124 Recently the same group hasextended their study to larger silicon clusters with 16to 21 atoms125 Using a nonorthogonal tight-bindingmodel70 Wang et al have obtained the ground statestructures of medium-sized Sin (n 7 21) clustersand exploited their melting behavior by Monte Carlosimulations62

Above empirical or tight-binding levels global geom-etry optimization of small silicon clusters Sin (n 4 5 710) using DFT calculations has been achieved byHartke123 while Bazterra et al employ the GA method toproduce the local minima of 14 isomers of Si9 and furtherrelax them at the DFT level126

422 Hydrogenated Silicon Clusters

The GA global optimization of hydrogenated silicon clus-ters SinHm has been carried out by several groups73 127ndash129

Using tight-binding models Chakraborti et al127 deter-mined the ground-state structures of SinH (n 1 4 8)and Tang et al performed a global structural optimizationfor Si8H8 cluster73 At semiempirical AM1 level Ge andHead performed the global geometry optimization of fullyand partially H-passivated Si clusters Si10Hm (m 4 812 16 20) and Si10H20128 Very recently they have furtherimproved the accuracy of their simulations at the ab initiolevel by reparametrizing AM1 with the aid of the ab initioresults129 As an example of their new strategy the ab ini-tio global minima for Si6H2 and Si6H6 were determined

423 Germanium Clusters

Compared to silicon clusters less work has been done ongermanium clusters by using the GA procedure Wang andco-workers have systematically studied the Gen (n 2 25) clusters60 61 68 using the combination technique of theGA simulation with a tight-binding model71 and a DFTminimization Based on the global minima structures ob-tained the melting behaviors of the germanium clustersare simulated and their electronic properties such as ion-ization potentials energy gaps and polarizabilities as afunction of cluster size have been discussed

424 Carbon Clusters

The structures of carbon clusters are of particular inter-est8 130 131 because of the discovery of novel cage-like C60

and related fullerenes In a precursory work using the GAprocedure and a tight-binding total energy model Deavenand Ho have obtained the fullerene cluster structures up toC60

39 Later several other groups have also optimized thecarbon cluster geometries132ndash134 using the Brenner bond-order potential24 Hobday and Smith studied Cn(n 3 60) and predicted a structural transition from planar poly-cyclic to fullerenes around n 20132 In contrast Zhanget al found bow-like configurations for most Cn (n 20 30) clusters with exception at C28 (fullerene structure)133

In order to discuss the formation mechanism of odd-num-bered fullerene-related species Kong et al used the GAoptimization to find the lowest-energy structures of odd-numbered Cn with n 51 59134

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43 Alloy and Compound Clusters

431 Bimetallic Alloy Clusters

Bimetallic clusters present a number of structures andphases that are different from those of corresponding puremetals The lowest-energy structures of the clusters shoulddepend both on the cluster size and on the concentration ofthe micro-alloys The GA has also been employed in theglobal optimization of alloy (eg Co-Cu Cu-Au Ni-AlPt-Pd) clusters63 64 135ndash138 while Gupta potentials wereused to model the interatomic interactions in these worksFor instance Wang et al optimized the Co-Cu bimetallicclusters with 18ndash20 atoms and simulated their melting be-havior63 Using the same technique they have explored thelowest-energy structures of Co18mCum with all possibleconcentrations (0 m 18) and then calculated the mag-netic moments of these clusters within a parameterizedtightbinding Hamiltonian64 Recently Johnston and co-workers studied the structural properties of a varietiesof nanoalloy clusters up to about 55 atoms including Pt-Pd135 Cu-Au136 137 Ni-Al138

432 Ionic and Covalent Compound Clusters

Using an improved GA with a self-guiding search strat-egy Kabrede and Hentschke have searched for the globalminima structures of the neutral sodium chloride clustersthat consist of 2 to 100 ions52 The empirical Born-Mayerrepulsive potential plus a long-range electrostatic Coulombterm were used to account for the interactions betweenNa and Cl ions Similar Coulomb Born-Mayer po-tentials were used for optimizing the geometries of stoi-chiometric and non-stoichiometric MgO clusters byRoberts and Johnston139 Flikkema and Bromley devel-oped a new interatomic potential for nanoscale silica140

and applied it to the GA simulation of (SiO2)7 clustersBy incorporating the GA with DFTB model the struc-tural and electronic properties of small titanium carbo-hedrene clusters TimCn with m 7 8 and n 10 14have been investigated141 Employing a GA-based searchstrategy Tomasulo and Ramakrishna have carried out aglobal search for the lowest-energy structures of small(AlP)n (n 6) clusters using the DFT with the local den-sity approximation142

44 Molecular Clusters

441 Water Clusters

In molecular clusters the constituting molecules (eg H2OC6H6 etc) are weakly bound by hydrogen bonding andor van der Waals interaction Among molecular clusterswater clusters are of particular interest51 143ndash148 Assumingrigid molecules several intermolecular potentials such asthe transferable intermolecular potentials (TIP)25 and sim-ple point chargeextended (SPCE) model26 have been used

in the GA global optimization of water clusters51 143ndash145

In an earlier work by Niesse and Mayne143 the water clustersup to (H2O)13 were globally optimized using a modifiedGA approach The transferable intermolecular potentials3 point (TIP3P)25 was employed to model the interactionbetween H2O molecules Later based on the same TIP3PGuimaraes et al determined the water clusters (H2O)n withn 11 1351 With the aid of two new evolutionary op-erators namely annihilator and history operators new globalminimum energy structures were found for all (H2O)1113

water clusters As well as TIP3P another three-centermodel potential SPCE was adopted by Qian to explorethe global potential energy minima of water clusters(H2O)n up to n 14144 while the more realistic TIP4P(4 points) potentials were used by Hartke145 RecentlyKabrede and Hentschke have computed apparent globalminimum structures of (H2O)n (n 25) using all threeavailable empirical potentials (SPCE TIP3P TIP4P)146

Hartke has performed global geometry optimization ofpure neutral water clusters in the size range of n 2 30using the TTM2-F potential148 Above the empirical levelthe global geometry optimization of water pentamer andhexamer clusters were performed using a NEMO modelpotential and further improved by minimizing the geome-tries using ab initio local second-order Moller-Plesset per-turbation theory (MP2)147

Ion-containing in H2O clusters can be viewed as a mi-croscopic model of solvation Ions of either alkali metalsor halogens inside H2O clusters have been studied TIP4P-like empirical potentials can still be used to describe theintermolecular interaction and the interaction between H2Omolecules and alkali-metalhalogen ions In previous worksthe global minima structures of Cl1(Br1 I1)(H2O)n

(n 1 6)149 Na(H2O)n (n 4 25)150 M(H2O)n

(n 4 22 M Na K Cs)151 clusters were studied usingGA optimization

442 Clusters of Aromatic HydrocarbonMolecules

In an earlier work by Xiao and Williams a computer pro-gram GAME (genetic algorithm for minimization of energy)was developed to obtain the global minimum structures ofbenzene naphthalene and anthracene molecular clusters upto tetramer37 Later Pullan determined the global minimumenergy structures of (C6H6)n (n 2 15) clusters152 Usingthe space-fixed modified GA technique White et al studiedthe benzene (C6H6)n (n 2 13) naphthalene (C10H8)n

(n 2 6) and anthracene (C14H10)n (n 2 6) clus-ters153 They found that the GA method is superior to theSA and comparable to the basis-hopping technique83

443 C60 Clusters

In addition to the clusters composed by small moleculesthe structure of (C60)n clusters have been studied up to n

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25 by Luo et al55 and up to n 57 by Zhang et al154 Bothworks used the Pacheco-Ramalho intermolecular poten-tial27 Luo found a crossover from icosahedral to decahe-dral at n 17 and (C60)13 as a stable magic numbered clus-ter while Zhang further found n 38 55 as the moststable species for the larger (C60)n clusters

45 Applications in Other Related Fields

As a robust global minimization method GA has been ap-plied to other related fields of computational materials sci-ence and nanoscience In this paper we only intend to pre-sent some highlights on the recent progress in this aspect

451 Nanoscience

The applications of GA to the other problems in compu-tational nanoscience include supported clusters155 156 clus-ter magnetism157 nanoparticle-assemblies158 and ultrathinmetal nanowires159 Using the noble gas atoms by LJ poten-tials as model systems for both surface and supported clus-ters Miyazaki and Inoue simulated the island and layerstructures of the deposited atoms155 Using the surface-embedded-atom method Zhang et al have considered amore realistic system by adatom clusters up to 39 atoms onfcc (111) surfaces of different metals (Ag Pt Pd Ni Al)156

Oda et al applied a Monte Carlo method incorporated withthe GA approach to simulate the magnetization and suscep-tibility of randomly generated spin clusters and a-HQNNorganic radical crystal within Ising model157 Stucke andCrespi used the GA procedure for optimizing the packingdensity of self-assembled multicomponent crystals of nano-particles and predicted several new crystalline structures158

Wang and coworkers have extended the GA to the struc-tural optimization of one-dimensional ultrathin metal nano-wires159 In these studies periodic boundary conditions areapplied along the wire axis while the mating is modifiedfrom the Deaven-Ho ldquocut and splicerdquo operation for clus-ters39 Helical multi-shell cylindrical structures were foundfor Au Zr Ti Rh Pt metal nanowires in agreement withexperimental observation on Au and Pt nanowires160

452 Computational Materials Science

The GA approach was also employed in determination ofcrystal structures as well as the computer-aid materials de-sign For example Woodley et al have used the GAmethod to generate plausible crystal structures of inorganiccrystals including binary oxides and ternary oxides161

Bazterra et al developed a new computational scheme tomodel structures of organic crystals using a modifiedGA162 Based on the knowledge of the geometry of the in-dividual molecules the crystal structures of benzenenaphthalene and anthracene were predicted with an empir-ical intermolecular potential162 Recently DFT-based cal-culations in conjunction with an evolutionary algorithm

were used to search for the ldquosuper alloyrdquo materials withspecific properties163

5 SUMMARY AND OUTLOOK

From the above discussions it is obvious that introducingthe genetic algorithm into cluster science has led to a sig-nificant breakthrough in the global optimization of clus-ters For the global optimization of those relatively simplersystems like LJ clusters genetic algorithm has proven it-self to be an efficient way for solving the NP-hard prob-lem However it is not quite clear whether the genetic al-gorithm is able to find the global minima of the clusterswith a rather complicated potential energy surface up tomedium and large size For instance it is still a challengeto find out the global minima structures of silicon clusterswith the cluster size over 20ndash30 atoms In those clustersthe specific operation of encoding mating and mutationshould be more carefully considered according to thenature of the chemical bonding in the cluster to furtherimprove the efficiency Although the ldquocut and splicerdquocrossover operation introduced by Deaven and Ho hasachieved a great success for LJ clusters carbon clustersmetal clusters etc there might be still some other possibleefficient ways of doing crossover For example graph the-ory might be used to define a new way of crossover in thecovalent bonding clusters with a network structure

Among other global optimization approaches it seemsthat the basin-hopping method83 97 is a good alternative ofthe genetic algorithm Therefore it would be more reliableif researchers can perform a global optimization by usingboth techniques and compare the computational results ob-tained from them In addition fast annealing evolutionaryalgorithm95 and conformational space annealing method82

also demonstrated impressive success in the optimizationof LJ clusters On the other hand combining the virtues ofGA with other global optimization methods like SA mightalso lead to a good way for further improving the globaloptimization An example along this direction is the pre-dictor algorithm46

One advantage of the GA approach is that it is an idealalgorithm for implementing on a parallel computer152

First the evolution of total energy and local minimizationof each individual in a population are totally independentand can be simultaneously done on different processes Itis also possible to simultaneously run several genetic algo-rithms on each parallel processor and allow the migrationbetween the populations of each of the simulations

Up to now most GA optimizations are based on empiri-cal potentials The empiricalquantum combination tech-niques58ndash60 65ndash67 72 74 used by several research groups aresignificant improvements over the accuracy of computa-tional simulations However it is not guaranteed that theempirical potentials or tight-binding model used in thecombination study (usually fitted for the bulk phase prop-erties) can exactly reproduce the true potential energy sur-

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face of clusters dominated by quantum chemical bondingIn particular for many clusters there are size-dependentchanges of the characteristics of chemical bonding Be-cause of the mismatch between the empirical and quantumpotential energy surfaces the empiricalsemiempirical GAsearch of structural isomers may still miss some of the im-portant candidates To avoid this problem it is desirable todirectly incorporate the GA procedure with a DFT-basedtotal energy method Furthermore the order-N technique164

of electronic structures calculations developed in the re-cent years can be used to further increase the durable sys-tem size in the study

Acknowledgments We would like to thank Prof G HWang Dr J L Wang Dr Y H Luo and Prof B L Wangfor collaborations and Prof J P Lu and Dr J Kohanofffor valuable suggestions JZ would like to acknowledgethe financial support from the University Research Councilof the University of North Carolina at Chapel Hill theOffice of Naval Research Grant No N00014-98-1-0597and the NASA Ames Research Center

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239 (1997)79 C Tsallis and D A Stariolo Physica A 233 395 (1996)80 Y Xiang D Y Sun and X G Gong J Phys Chem A 104 2746

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1222 (1997)82 J Lee I H Lee and J Lee Phys Rev Lett 91 080201 (2003)83 D J Wales and J P K Doye J Phys Chem A 101 5111 (1997)84 J P K Doye Phys Rev E 62 8753 (2000)85 R P White and H R Mayne Chem Phys Lett 289 463 (1998)86 J Pillardy A Liwo and H A Scheraga J Phys Chem A 103

9370 (1999)87 A B Finnila M A Gomez C Sebenik C Stenson and J D Doll

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(2002)93 W Cai and X Shao J Comput Chem 23 428 (2002)94 W Cai Y Feng X Shao and Z Pan J Mol Str (Theochem) 579

229 (2002)95 W Cai H Jiang and X Shao J Chem Inf Comput Sci 42 1099

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Received 29 January 2004 RevisedAccepted 9 February 2004