Global geometry optimization of silicon clusters using the space-fixed geneticalgorithmMasao Iwamatsu

Citation: The Journal of Chemical Physics 112, 10976 (2000); doi: 10.1063/1.481737 View online: http://dx.doi.org/10.1063/1.481737 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/112/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The structure of 4-methylphenol and its water cluster revealed by rotationally resolved UV spectroscopy using agenetic algorithm approach J. Chem. Phys. 123, 044304 (2005); 10.1063/1.1961615 Global geometry optimization of silicon clusters described by three empirical potentials J. Chem. Phys. 119, 1442 (2003); 10.1063/1.1581849 Global optimization analysis of water clusters ( H 2 O ) n (11n13) through a genetic evolutionary approach J. Chem. Phys. 116, 8327 (2002); 10.1063/1.1471240 A study of genetic algorithm approaches to global geometry optimization of aromatic hydrocarbon microclusters J. Chem. Phys. 108, 2208 (1998); 10.1063/1.475601 Global geometry optimization of atomic clusters using a modified genetic algorithm in spacefixed coordinates J. Chem. Phys. 105, 4700 (1996); 10.1063/1.472311

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JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 24 22 JUNE 2000

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Global geometry optimization of silicon clusters using the space-fixedgenetic algorithm

Masao Iwamatsua)

Department of Computer Engineering, Hiroshima City University, Hiroshima 731-3194, Japanand Department of Physics, National Central University, Chung-li 320, Taiwan, Republic of China

~Received 20 April 1999; accepted 30 March 2000!

The space-fixed genetic algorithm originally proposed by Niesse and Mayne@J. Chem. Phys.105,4700~1996!# is modified and used to study the lowest energy structure of small silicon clusters byemploying empirical interatomic potentials. In this new space-fixed genetic algorithm, agradient-free simplex method, rather than the conventional gradient-driven conjugate gradientminimization employed by Niesse and Mayne, is selected by virtue of its flexibility and applicabilityto any form of interatomic potentials for which the calculation of derivatives is difficult. Using twoempirical three-body potentials, we calculated the ground state structure up to Si15 successfullyusing this new genetic algorithm based on the simplex method. The effect of angular dependentthree-body potentials on the cluster structures is examined and compared with the experimentalresults. © 2000 American Institute of Physics.@S0021-9606~00!51324-5#

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I. INTRODUCTION

It is well known that technological problems can leadproblems of global optimization. Hence, an improved glooptimization method is highly desirable. In chemistry aphysics, many theoretical studies have been devoted togoal of global optimization; most of these studies havecused on the search for the lowest energy structure of atoand molecular clusters. The main difficulty associated wglobal optimization of clusters is the exponential increasethe search space, which results from the increasing sizthe clusters.1–3 In fact, it has already been proven that tdetermination of the ground state of clusters, which intereven under two-body central force, belongs to the classNP- ~non-deterministic polynomial!-hard problems3 forwhich no known algorithm is guaranteed to find the globminimum in polynomial time. Recently, interest has beshown in the use of the genetic algorithm~GA!4 to solve theproblem of global optimization. GA is a global optimizatiomethod based on several metaphors from Darwinian biolcal evolution. The method is based on the principle of svival of the fittest, considered in the sense that candidsolutions ~clusters! in the population compete with eacother for survival and produce offspring.

It has been proven4 that the GA approach is more powerful than the traditional simulated annealing approa5

based on the Monte Carlo method. In particular, the Gusing real-valued Cartesian variables6 and appropriate genetic operations is much simpler and more efficient thantraditional GA based on binary coding.4 Using real valuecoding, Zeiri6 obtained the geometry of ArnH2 clusters andfound that the performance of the GA is superior to thatthe traditional simulated annealing. Greguricket al. pro-

a!Address after April 1, 2000: Department of Information and CompuEngineering, Kisarazu National College of Technology, Chiba 292-00Japan.

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posed the deterministic/stochastic genetic algorithm~DS-GA!, in which they combined the global GA method bason binary coding and local optimization by using a conjuggradient method.7 This strategy, which combines global anlocal searches, is also called the ‘‘Lamarckian algorithm4

Subsequent to these efforts, Niesse and Mayne8 integratedthe real coding of Zeiri and the hybrid algorithm of Gregrick et al. and proposed thespace-fixed genetic algorithm.They8 showed that this method is more than ten times fasin CPU time than the DS-GA of Greguricket al.7 They re-cently extended their algorithm in order to calculate mocomplex aromatic hydrocarbon micro clusters,9 and con-cluded that although the genetic algorithm is powerful, itessential to combine a global search method based onwith a local search method such as a conjugate gradmethod. A simpler but more intuitive and geometric GA prposed by Deaven and Ho10 was successfully used to predithe ground state structure of C60 buckyball.

Because of their technological importance, silicon cluters have been studied by many researchers and by vamethods. The GA has already been applied to predictground state structure of Si4 and Si10 by means of standardbinary coding by Hartke.11 In addition, the space-fixed GAhas been used by Niesse and Mayne12 to predict the clusterstructure of Sin (n53 – 10) by means of the empirical potential presented by Bolding and Andersen.13 Recently, Gonget al.14,15 proposed a new empirical three body potentbased on the popular empirical potential proposed by Singer and Weber,16 which has been used over the yearsmany researchers to calculate various properties of silicincluding the structure of clusters.17 The Gong potential14

contains not only the original three-body potential of Stiinger and Weber,16 which reflects the tetrahedral bond angu;109°, but an additional factor that takes into accountpreferred bond angleu;60° observed in theab initio mo-lecular dynamic simulation for the liquid state. Using th

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10977J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Global geometry optimization of Si clusters

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model potential, they14,15 calculated the cluster structure oSi by using traditional simulated annealing.

In this paper, we simplify the space-fixed genetic algrithm of Niesse and Mayne8 and calculate the lowest-energstructure of silicon clusters Sin for 3<n<15 by using theempirical three-body potential of Stillinger and Weber16 aswell as that of Gonget al.14 in order to test the ability of thisgenetic algorithm. Although we use the real-coding methof Zeiri6 and four genetic operations proposed by NiesseMayne,8 we employ a simpler gradient-free simplex metho5

for local optimization.

II. ALGORITHM

Our algorithm is schematically shown in Fig. 1. We prpare a population that consists ofN clusters~individuals! Xi

( i 51,2,3,. . . ,N) as the first generation. In our calculatiowe fix the population of clusters asN510. Each individualXi is represented by agenethat consists ofm53n26 realvalues that represent the coordinates of each atom ofcluster, which consists ofn atoms. Initially, the coordinateare randomly chosen. We found it useful to choose an inrandom cluster smaller than that of the true ground sstructure; then the algorithm can successfully find the grostate. A similar philosophy was used by Leary2 in his ‘‘bigbang’’ algorithm.

A simplex minimization,5 rather than the conventionaconjugate gradient minimization,8 is performed for each in-dividual in order to place each cluster at its local minimuThe initial simplex with (m11) vertices for the individualcluster characterized by m real coordinates x0

5(a1 ,a2 , . . . ,am) is made fromx0 itself and m pointsxi( i 51,2,. . . ,m), whose coordinates are given byxi

5(a1 ,a2 , . . . ,a i1d, . . . ,am), whered is chosen to be theeffective size of an atom (51 in our unit in III!. We haveused the routineamoeba5 and the tolerance on the potentivalue 1026. Therefore, our method is derivative-free, aoffers much wider applicability to various forms of inteatomic potentials. This simplex local minimization is termnated when the cluster starts to dissociate.

For each individualXi ( i 51,2,. . . ,N), the locally mini-mized total potential energyF i5F(Xi) is obtained. Amongthe population, the bestM50.8N ~80% of population! indi-viduals are used as parents that are allowed to producenext generation. In order to choose the parents, the cadates are characterized by the fitnessf i ( i 51,2,. . . ,M ),which is used as the probability of choosing that individuas a parent. This fitness is calculated from the expressio

f i5FiY (i 51

M

Fi , ~2.1!

and the intermediate quantityFi is calculated from the lineascaling of locally minimized total potential energyF i , de-fined as

Fi5~Fmax2F i !/~Fmax2Fmin!, i 51,2,. . . ,M , ~2.2!

where the sum in~2.1! is taken of the candidates for thparents, andFmax andFmin are the maximum energy and thminimum energy, respectively, among theM candidates for

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the parents. Many different kinds of scaling to define tfitness of the candidates have been proposed,4,8,10 though weselected the simplest, linear scaling, in order to avoid adtional fitting parameters.

The next generation~children! is obtained from the par-ents of the current generation by choosing one geneticeration. The parent~s! is/are selected according to the proability proportional to their fitness by the roulette whemethod.4,8,12 We followed the procedures outlined by Zeir6

and Niesse and Mayne,12 and used a mating process that wchosen equally randomly from four genetic operations,

FIG. 1. Flow chart of genetic algorithm used. The algorithm is similarthat proposed by Zeiri~Ref. 6!, or those proposed by Niesse and May~Ref. 12!.

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10978 J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Masao Iwamatsu

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~1! Inversion,~2! Arithmetic mean,~3! Geometric mean,~4! m-points crossover.

The details of these genetic operations can be found inwork of Niesse and Mayne8,12 and are schematically showin Fig. 2.

Among these operations, the inversion produceschild from one parent; the arithmetic mean and the geomemean, respectively, produce one child from two parents;the m-point crossover produces two children from two paents. Although these four genetic operations are seleequally randomly, a slight modification, giving the arithmemean and them-point crossover operations more weight, improves the convergence. The child thus produced was imdiately relaxed to the local minimum by the simplex methoand incorporated into the new generation~Lamarckism4!. Inorder to maintain the size of the population constant,current member~s! of lowest fitness is/are automatically replaced by the new child~children!. Thus, in our algorithm,we accept at most two new children with each generatand the evolution occurs gradually. The iteration is termnated when the 40% (0.4N) of the current population has thsame lowest energy.

We have selected this derivative-free simplex methrather than the faster conjugate gradient method for lominimization because~i! the calculation of the derivative fothe three-body potential is tedious, and~ii ! there seem to bemore metastable structures for three-body potentials thantwo-body potentials; therefore, it is difficult to reach thlowest-energy structure using the conjugate gradient metNotably, we cannot find the lowest minimum for Sin (n>5) with this space-fixed genetic algorithm combined wthe conjugate gradient local minimization routinefrprmn,5

though it is much faster than the simplex minimization. Tlowest energy of 100 independent runs for Si5 using the con-jugate gradient method is21.1377, with a pyramidal shapwhose base is a parallelogram, while that using the simpmethod is21.1504, with a shape that is a trigonal bipyramwhen we use the Gong14 potential~III !. We have confirmedthat the latter is also obtainable from the conjugate gradmethod, by inserting the final structure of the cluster~trigo-nal bipyramid! obtained from the simplex method into thconjugate gradient program as the initial structure. Basedthis result, we conclude that the simplex method, althouslow, is more efficient and stable than the conjugate gradmethod if we start from a completely random structure. Tresult can be anticipated from the result of Feustonet al.,17

who discovered several metastable structures even for silclusters as small as Si3 using the Stillinger-Weberpotential.16 The conjugate gradient method would be eastrapped by one of those metastable minima. An advantagthe simplex method, which allows barrier crossing, has bpointed out by other authors.4

III. RESULTS AND DISCUSSIONS

We carry out the calculation by using the StillingerWeber~SW! potential16 as well as the modified SW potentia

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proposed by Gong.14 The total potential energyF for then-atom cluster bound by those potentials is written as

F~1,2,. . . ,n!5(i , j

n

v2~ i , j !1 (i , j ,k

n

v3~ i , j ,k!, ~3.1!

where the two-body potentialv2 is given by the followinganalytic functions:

v2~ i , j !5A~Bri j2p2r i j

2q!exp@~r i j 2a!21#, ur i j u,a, ~3.2!

wherer i j is the distance betweeni and j atoms. The three-body potentialv3 is given by

v3~ i , j ,k!5h~r j i ,r ki!1h~r k j ,r i j !1h~r ik ,r jk!, ~3.3!

with

h~r j i ,r ki!5l exp~g~~r i j 2a!21

1~r ki2a!21!!~cosu j ik11/3!2,

ur i j u,a, ur kiu,a, ~3.4!

whereu j ik is the angle made by three atomsj 2 i 2k. Thevalues of the seven parametersA,B,a,p,q,l, and g aregiven by16

A57.049 556 277, B50.602 224 558 4,

FIG. 2. Four genetic operations used in our algorithm, which were defioriginally by Niesse and Mayne~Ref. 12!.

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10979J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Global geometry optimization of Si clusters

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a51.8, p54, q50,

l521.0, g51.20.

The energy and the length are measured in units oe(52.17 eV! ands(52.0951 Å!, respectively. This analyticapotential of Stillinger and Weber has been used by mresearchers.17

Recently, inspired by the results of the bond-angle dtribution of bulk liquid obtained fromab initio moleculardynamics simulations with local-density approximati~LDA !, Gong14 improved the three-body termv3 of the em-pirical SW potential. This was accomplished by multiplyinthe SW three-body potentialv3 by an extra factor, followedby an attempt to reproduce the preferred bond angle;60°observed in LDA simulation.14 Specifically, Gong modifiedthe three-body potential~3.4! as

h~r j i ,r ki!5l exp~g~~r i j 2a!211~r ki2a!21!!~cosu j ik

11/3!2@~cosu j ik1c0!21c1#, ~3.5!

and the new values of parametersc0 , c1 , andl are given by

l525.0, c0520.5, c150.45. ~3.6!

The angular parts of these two three-body potentialscompared in Fig. 3. As indicated in Fig. 3, we can anticipthat the clusters bound by the Gong potential will have copact structures, while those bound by the SW potential whave an open structure with a bond angle;109°.

In order to visualize the combined effect of the threbody potential of Stillinger and Weber~SW!16 and the im-provement of Gong,14 we have calculated the lowest enerstructure of~the! silicon cluster Sin (n53,4,. . . ,15) boundby

FIG. 3. The angular part of the three-body potentialh(r j i ,r ki)/l ofStillinger–Weber~3.4! ~solid curve! and that of Gong~3.5! ~dashed curve!,where r j i 5r ki5a. Note the deep minimum atu;109° of the Stillinger–Weber potential.

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~1! Two-bodyv2 potential of Stillinger and Weber,16

~2! Full three-body potentialv21v3 of Stillinger andWeber,16 and

~3! Improved three-body potentialv21v3 of Gong,14

using our genetic algorithm.In Fig. 4~a!, we show the lowest energy in each gene

tion versus the number of the generation~the number of ge-

FIG. 4. ~a! The lowest energy of Si11 clusters among the population plotteas a function of the number of generations~mating operations! during atypical run of the genetic algorithm program.~b! The lowest-energy clusterstructure corresponding to the generations at~1!–~5! of ~a!.

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10980 J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Masao Iwamatsu

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netic mating operations that have been performed! for theSi11 cluster. As has been reported by previoresearchers,8–10 the curve looks like a staircase. The evoltionary improvement occurs occasionally, and the algoriteventually finds the global minimum. This is in marked cotrast to the traditional simulated annealing method, wherthe improvement occurs gradually. Clearly, global searchcomplished via genetic operation, along with fine-tuningmeans of local minimization, are co-factors of the succesthis algorithm. Figure 4~b! shows the intermediate clusteduring the evolution of Fig. 4~a!. We observe that the clusteevolves quickly to the nearly optimal structure, but it takerather long period of fine-tuning to reach the true groustate.

Table I, for comparison, shows the lowest energyatom E5F/n, the cluster structure, and the point grosymmetry for small Si clusters (Sin , n53 – 8) determinedfrom our genetic algorithm by using three types of inteatomic potentials. The results are the best of 100 indepenruns. Our genetic algorithm can successfully reproducestructures and energies of all silicon clusters bound byStillinger–Weber potential, predicted by means of the trational simulated annealing technique and based on thelecular dynamics simulation.17 Unfortunately, the values othe binding energies of silicon clusters bound by the Gopotential are not published;14,15 the ground state structurewe found, however, seem in marked accord with thoseported by Gonget al. Figure 5 shows a comparison of thlowest-energy structures of small Si clusters (Sin , n54 –8) bound by three kinds of potentials predicted by ournetic algorithm.

We find that the structures predicted from the Gongtential are virtually the same as those predicted fromtwo-body potentialv2 . It can be seen from the comparisoof the angular part of the SW and the Gong potentials in F3 that the angular part of the three-body term of the Go

TABLE I. Structures, their point groups, and their potential energiesatom (F/n) in units of e~52.17 eV! of small silicon clusters Sin (n53 – 8) calculated from the space-fixed genetic algorithm using three tof interatomic potentials: the two-bodyv2 potential ~left!, the three-bodypotential of Stillinger and Weber~SW! ~Ref. 16! ~middle!, and the three-body potential of Gong~Ref. 14!.

v2 of SWa v21v3 of SWa v21v3 of Gongb

Si3 Equilateral triangle Equilateral triangle Equilateral triangle(D3h , 21.0000! (D3h , 20.6827! (D3h , 20.8058!

Si4 Tetrahedron Square Tetrahedron(Td , 21.5000! (D4h , 20.9387! (Td , 21.0004!

Si5 Regular trigonal Pentagon Compressed trigonabipyramid bipyramid(D3h , 21.8000! (D5h , 20.9996! (D3h , 21.1504!

Si6 Compressed octahedron Trigonal prism Octahedron(D4h , 22.0478! (D3h , 21.0906! (Oh , 21.2783!

Si7 Pentagonal bipyramid Chair Pentagonal bipyram(D5h , 22.2845! (C3v , 21.1788! (D5h , 21.3749!

Si8 Unicapped distorted Cube Unicapped distortedpentagonal bipyramid pentagonal bipyrami(C2v , 22.3853! (Oh , 21.3223! (C2v , 21.4418!

aReference 16.bReference 14.

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potential is much weaker than that of the SW potentTherefore, the Gong potential is more isotropic than the Spotential, and closer to the isotropic two-body pair potenv2 . Then the geometric structure of clusters bound byGong potential becomes as compact as those bound puby the two-body potentialv2 .

As expected, it can be seen that the full SW potentialcontrast to the compact structure predicted by the ppotentialv2 and the Gong potential, predicts more open adistorted structures. The angular dependence of the thbody potential of Stillinger and Weber is too rigid to keethe bond-angleu fixed at the tetrahedral angle (u;109°) forclusters. Therefore, the surface of the cluster bound by

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FIG. 5. Optimal structure of small silicon clusters Sin (n54 – 8) calculatedfrom the space-fixed genetic algorithm using~a!–~e!: the pair potentialv2 ofthe Stillinger–Weber potential~left!, ~f!–~j!: the Stillinger–Weber three-body potentialv21v3 ~middle!, and~k!–~o!: the Gong three-body potentiav21v3 ~right!.

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10981J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Global geometry optimization of Si clusters

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TABLE II. Structures, their point groups, and their potential energies per atom (F/n) in units ofe~52.17 eV!of medium-sized silicon clusters Sin (n59 – 15) calculated from the space-fixed genetic algorithm using thtypes of interatomic potentials: the two-bodyv2 potential~left!, the three-body potential of Stillinger and Webe~Ref. 16! ~middle!, and the three-body potential of Gong~Ref. 14!.

v2 of SW v21v3 of SW v21v3 of Gong

Si9 Bicapped Edge-capped Tricapped trigonalpentagonal bipyramid cube prism(C2v , 22.5528! (C2v , 21.3271! (D3h , 21.5111!

Si10 Tricapped Symmetric stacking of Bicapped tetragonalpentagonal bipyramid two regular pentagons antiprism(C3v , 22.6937! (D5h , 21.3797! (C4v , 21.5649!

Si11 Bicapped tetragonal Smallest fourfold Icosahedron withantiprism with one center coordinated structure one vertex missing(C4v , 22.8409! (C2v , 21.3829! (C5v , 21.5800!

Si12 Icosahedron with Four identical Regular icosahedronone vertex missing pentagons with a vacant center(C5v , 22.9808! (C1v , 21.4178! (Ih , 21.6661!

Si13 Regular icosahedron One fourfold and 12 threefold Vacant distorted icoscoordinated atoms hedron with one cap

(Ih , 23.2085! (C4v , 21.4234! (C2v , 21.6245!Si14 Unicapped Six identical pentagons Bicapped hexagonal

icosahedron and three squares antiprism(C3v , 23.1936! (C2v , 21.4455! (C6v , 21.6496!

Si15 Bicapped Three fourfold and 12 threefold Ninecapped trigonalicosahedron coordinated atoms prism(C2v , 23.2456! (C1v , 21.4410! (C3h , 21.6464!

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SW potential is covered mostly by the square (u590°) or bythe regular pentagon (u5108°), while those bound byv2

and the Gong potential are mainly covered by the equilattriangle (u560°).

Table II and Fig. 6 show a comparison of the loweenergy structures of medium-sized Si clusters (Sin , n59 –15). In this case the structures predicted from the ppotentialv2 and those from the Gong potential becometally different. For example, the surface of clusters boundthe Gong potential are covered mostly by triangles asquares~Fig. 6!. Further, atoms try to occupy the surfacand the inner atom is missing. This difference can be cleseen for the Si12 regular icosahedron with a vacant cen~Fig. 6! calculated from the Gong potential, while the reguicosahedron with one center appears for the Si13 from thev2

potential. The full SW potential predicts more open and dtorted structures even for medium-sized clusters, as showFig. 6.

The total potential energyF is displayed in Fig. 7 asfunctions of the number of atomsn in clusters, which seemto indicate the exceptional stability of this Si12 regular icosa-hedron calculated from the Gong potential and that ofSi13 calculated from thev2 potential.

The cluster structures calculated from the Gong14 poten-tial are close to those calculated from accurateab initio18,19

or tight-binding ~TB!20,21 theories, and seem more realistthan those calculated from the Stillinger–Weber potentHowever, small discrepancies remain. For example, theabinitio and TB theories predicted an open triangle for Si3 ,18,20

while we predicted an equilateral triangle as the lowestergy structure. Further, theab initio and TB theories pre-dicted a flat rhombus for Si4 ,18,20 while our genetic algo-rithm predicted a tetrahedron as the ground state struc

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Gong14 has asserted that these two structures havealmostthesame energy, though he did not specify which is truly lowThe compressed trigonal bipyramid for Si5 from ab initioand TB theory19,20agrees with our prediction using the Gonpotential. The face-capped trigonal bipyramid for Si6 ,18,20

however, does not agree with our prediction of an octadron. The pentagonal bipyramid for Si7 predicted from theab initio19 and the TB calculation20 agrees with our resulusing the Gong potential. The distorted bicapped octahedfor Si8 ,19,20 is slightly different from our distorted unicappepentagonal bipyramid~Fig. 5, Table I! and from the tran-scapped octahedron predicted by Gong.14 Raghavachari andRohlfing19 predicted a tricapped octahedron for Si9 whileOrdejon et al.20 predicted a distorted tricapped trigonprism. Recent calculation based on the local density futional method25 has predicted a similar tricapped trigonprism structure. Our result using the Gong potential for9agrees with the results of Ordejo´n et al.20 and Ho et al.,25

respectively. The tetracapped trigonal prism for Si10 from abinitio19 and the TB theory,20 however, does not agree witour results; our genetic algorithm predicted the bicappedtragonal antiprism as the ground state structure. AgGong14 has noted that these two structures, the tetracaptrigonal prism and the bicapped tetragonal antiprism, halmost the same energy. Of the two, based on our genalgorithm, the bicapped tetragonal antiprism seems to belower energy structure. Our regular icosahedron with acant center for Si12 agrees with the result reported by Bahand Ramakrishna21 calculated from the TB theory.20

Although current literature has evinced a continuointerest18–21 in the theoretical prediction of the structuresmall silicon clusters, conclusive experimental evidencescarce.22–25 However, comparison of experimental resu

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10982 J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Masao Iwamatsu

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FIG. 6. Optimal structure of medium-sized silicon clusters Sin (n59 –15). ~a!–~g!: the pair potentialv2 of the Stillinger–Weber potential~left!,~h!–~n!: the Stillinger–Weber three-body potentialv21v3 ~middle!, and~o!–~u!: the Gong three-body potentialv21v3 ~right!.

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obtained from mass spectroscopy combined with thehanced Raman spectroscopy22 and infrared spectroscopy,23

and those results obtained fromab initio quantum chemicalcalculation, suggest that Si4 is a planar rhombus, Si6 is adistorted~compressed! octahedron, and Si7 is a pentagonalbipyramid. The last two structures are similar to the clusstructure predicted by the Gong potential. Regarding larclusters, at present it is difficult to draw any direct concsions concerning their structure. In general, the Gong potial predicts fairly compact and symmetrical structuresmedium-sized clusters, while the recent calculation usinglocal density functional method25 suggests open distortestructures. It has been asserted that such open and diststructures are more consistent with the results of ion mobexperiments.25

Indirect evidence drawn from the chemical reactionclusters with other molecules is that the Si6 and Si10 areexceptionally stable24 among other clusters because the dsolution of larger cluster occurs via the release of Si6 andSi10 clusters, rather than by means of a single atom Si. This also some evidence that the 13 atom cluster Si13 is stable.This conclusion was inferred from the reactivity of chargsilicon clusters Sin

1 with oxygen (O2) and with heavy water(D2O).24 In order to check the stability of clusters andlook at the possibility of a shell structure, we plotted tsecond energy difference15

D2~n!5F~n11!1F~n21!22F~n!, ~3.7!

shown in Fig. 8. A similar result has been reported for Sin forn.10.15 The distinct peaks of theD2 curve in Fig. 8 clearlysuggest the stability of Si10 as well as that of Si12 among

FIG. 7. The total potential energyF as a function of the number of atomsnfor three kinds of interatomic potentials.

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10983J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Global geometry optimization of Si clusters

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other clusters. Altogether, these results seem to suggesthe empirical potential of Gong14,15 is more favorable thanthat of the popular Stillinger–Weber potential.16,17

IV. CONCLUSION

In conclusion, we have tested the space-fixed genalgorithm originally proposed by Zeiri6 and by Niesse andMayne8 by calculating the ground state structure of silicclusters. We have used the empirical three-body interatopotential proposed by Stillinger and Weber16 as well as thatproposed by Gong.14 Even for these three-body potentialthe space-fixed genetic algorithm successfully found all ttative ground state structures, all of which seemed reasable.

In our algorithm, we used the simplex local minimiztion method rather than any conventional gradient-drivencal minimization methods such as the conjugate gradmethod. We found that the simplex method is more stathan the conjugate gradient method, due to the fact thatlatter may be easily trapped by metastable minima. Althouour investigation required slight mitigation of the efficien

FIG. 8. The second energy differenceD2(n) for silicon clusters as functionsof the number of atomsn for three kinds of interatomic potentials.

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

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of the algorithm, the preliminary results for the gallium anthe sodium clusters in supercooled liquid states indicatethe algorithm based on the simplex method is stable, evenpathological interatomic potential with damped oscillatitail resulting from the Friedel oscillation.26 This space-fixedgenetic algorithm is simple to use; fine tuning of multiparaeters is not necessary. We believe that this algorithmpromising for the determination of the lowest-energy struture of clusters. In particular, it will be suited to parallcomputation.

ACKNOWLEDGMENTS

The author is grateful for the financial support and hopitality extended to him by the Department of Physics, Ntional Central University~Republic of China!, where thiswork was partially conducted. The author is particularly idebted to Professor S. K. Lai, who offered warm hospitalHe is grateful to Messrs H. Nishikawa, A. Naka, andIwata, for their assistance in data analysis.

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