REVISTA MEXICANA DE FÍSICA'¡7 SW'LEl\IENTO 1. 54-5H MARZO 2001

Minimum-energy configurations of metallic c1usters obtained by simulatedannealing molecular dynamics using an Extended-Hückel Hamiltonian

Luis RincónDepartamento de Química, Facultad dc CienciasUnú'ersidad de Los Andes. Mérida-5 J(JJ. Venezuela

Recibido el 20 de febrero de 2000: aceptado el JO de ao;"ildc 2000

ScmicllIpirical simulated annealing molecular dynamics method using a liclitiolls Lagrangian has heen developed fm the study of struclUralanJ e1eclronic propcrlies of micTO-and nano-c1usters. As an applicalion uf the prescnl schel11c. we study the slructure of Nar¡ c1uslers in Iherange of 11 == 2-100. and compared the present calculation with some ah-in/lio molle! calclllatioll.

KI'.\'w(m!s: ClllS!ers; slructure; stability; molecular dynamics

Se desarrolló un lllétodo de Dinámica Molecular-Recocido Simulado usando un Lagrangiano licticio para estudiar las propiedades elec-trónicas y estructurales de mino- y nano-agregados. Como una aplicación del presente esquema. se estudió la estructura de agregados dcNa" CII el rango entre" = 2-100. Yse compararon los resultados con algunos cákulos ah-il/ilio modelo.

f)('scri¡lton'.\: Aglomcrados: estructura: estabilidad; dinámica molecular

PACS: 61.46.+w; .16.40.-c; 71.15.Pd

1. lntroduclion

J\10lcclllar dynamics is an importanl 1001 for the s{udy 01'lime derendent amI ensemhle properlies 01' molecular sys-lems [1. :n 1I is also uscd for geomelry optimization ol' dil'-llcult cases in simulated ~nnealing [3-5]. as an alg:orithm fors<lmpling Ihe conl1guralion space [6]. Hov.'ever, the computa-Ilonal eost 01'ah-il/itio molecular dynamics (molecular dyna-mics WlthoUI ti priori knowledge 01' the pOlential energy sur-facc) is prohihilive. even with the recent hrcakthrollgh 01' IheCar-Parrincl1o Illcthod [71. In Ihis melhod, the l'ull calculalion01' Ihe 'I,/avefunction al each time in Ihe dynamics simulationis ahandoned amI instcad a ficlilious c1assical propagation 01"lhe: waveflllKlion paramclers is performed. just in lhe samcway as Ihe nuclei are propagalcd. A molecular dynamics al-gorithm can he lIsed to drive the system through time. amIthe wave-fllllclions forces are constrained lo maintain thc ort-hogonality hetween Ihe molecular orbitals at each lime. ThcCar-Parrinello algorithm has made rnany applications and dc- .vclopmelll possihle [81; most of Ihcm for the study ol' periodicsyslems ,\.'hcre density functional thcory has been employedtogether with r1anc ''':ave hasis sets.Let us mention that. thismClhod can also lIscd in conjunction with traditional Gaus.sian oasis set and electronic slructure mcthods 01' quantumchemistry. For instance see Ihe work hy Carter and collaoo-ralors [(JI.

Despile Ihe advances descrioed ahovc, the ab-il/itio mo-lecular dynamil:s silllulatinns has heen limitcd. in the hest ca-ses. (o cluslas 01"maximun 01' ahout 50 atoms. and for largesystems. the majority 01"molecular dynamics simulalions areperformcd using cmpirical potentials [91. which are chosen tolllodcl a silllplilled (analytical) represenlation ol' the interac-(ions lhat in l1wny c¡¡ses cannot descrihe adequalely quantllJ1lcrfccls sllch as polarisalion. chemical reactions or excited sla-

le processes. In spile of Ihis. they have Ihe advantage lhat Ihccllcrgy and forces upon aloms of a system can he evaluatcdmosl rapidly than in lirsl-principle calclllations.

There is ¡¡ largc class 01' prohlcms thal involves morea10ms than (hose in Ihe present Iimils ol' first-principle tc-chniques. and also demands a quanlum mechanical Ircal-men!, that empirical pOlenlial cannot provide. For lhis kindnI' prohlcllls. cakulalions of molecular dynamics with appro-ximated molecular oroilals schernes (tight-hinding [10-121or scmi-cmpirical [131) have hecn reporled in the literalU-re. Since in thesc mcthods. a minimal hasis ser is uscd amithe Hamiltonian malrix elemcnts are rarameterized. a largcnllmher of alollls can he tacklcd with the aClual computerscapahilitics.

Tite use of molecular dynamics together with a simulatedanncaling to perfonn geometry optimizalion. has al so opened!lew possihilities over Iraditional methods. based on gradienttcchniqlles that are often faceu wilh the problem of local mi.nima. Thl1s. when the nl1mher of degrccs ol' freedom ge(s lar-ger. lhe dimensionalily 01"lhe potential energy surl'ace increa-ses and the 11UlIloer of local minima grows exponcntially 1141.The siml1la(cd annealing has heell introduced lo remed)' (hissituation. The idea here is sampling the energy surl'ace (using;¡ suitahle deterministic algorithm) subjecled to a control va-riahle (lhe (emperalure). The comhination of molecular dyna-mies ,\'ith simulated annealing is a rohust rncthod for f1ndinglhe glohal lllinima on complicated energy hypersurfaces.

This paper reports thc illlplclllentation of a moleculardynamics scheme llsing a Car~Parrinello algorithm for anExtended lHickel Iype Hamillonian. As an application. Ihisl1lethod was lIscd. in conjullction with a simlllaled annca.ling algorithm. to determine the equilibrium slructure amiIhe cleclronic properties 01"sodilllll clusters Nan. wilh 11 =2, ,1, G, 8, 10,20, ,,0, 100. Le!', menllon lh"l lhe "Ibli ag-

~-- -- -------~------------------------------------------

MINJMUM~ENERGY CONFIGURATIONS OF METALLlC CLUSTER SOBTAINEO BY SJMULATED ... 55

2. Total energy ca1culation

whcrc fj.EEIl is thc Extended Hückcl hinding cnergy.Givcn N alkali-l11ctal aIOI11S.with a hasis set of one va-

riational H]J orhital per alom, the Extcnded Hückel hindingcllcrgy is calculated as.

gregales are <!mong lhe octtcr known rnicroclustcrs, 001h cx-pcrimcntally and thcoretically. In Seco 2 wc descrihe lhe totalcncrgy cnlculation. Secrioo 3 prcscnts lhe simulatcd annca-ling molecular dynamics algorilhm and Seco 4 shows lhe rc-sults. (7)1 (")do = 'OC¡ = - Qo,

1.1,., (

- R ,,1 (1') ,.- dI'.0 r

E,o' = ¿ {4¿f(E¡)C¡¡CJ,(lsiIHllsj)

i<j k

where (fU) is lhe valencc charge on atom i, ohtained fromMulliken population analysis. Thus, lhe total energy can hercwrittcn as a sum of pairwise ínlcractions,

+ ER,,,(ij) }. (9)

where (lo is the Bahr radius and ( the sp orbital exponent.This refonnulal;on of Ihe EXJended Hüekel method le-

ads to a significant improvcmcnt of the potenlial energy cur-ves 01' diatomic and small organic molecules, specialty af-tcr optimizalion 01' the J\, and c5 parameters for a given classof eompound. As is pointed out hy Caro el al. [12J, due tolhe nonorthogonal nature 01' the basis sct, the orbilal ener-gies are shifted. As a conscqucnce of it, when it is used tocalculate the forccs on molecular dynamics algorithm thismodel has a very important drawhack. respcct the traditionaltight-hinding mcthod in which lhe hasis functions are assu-med orthogonal. Por cxample. in Na3 the linear configurationís prcfcred over the triangular one (which is thc most stahlc).Onc wjJY 10 o\'crcome this limitaríon is solving the lraditionaleigenvaluc equalion. (ll - EI)C = O, in lhe same way as isdone in the compleJe neglcet of diffrential overiap (CNDO)melhod, or like in lhe Hückel tllclhod. This point is essentialfor the success 01' lhe Extended Hückel molecular dynamics.

f-or lhe repulsivc potcntial we usc a pairwise potential 01'Ihe form [10-121,

E = "' E ( .. ) = "' (3r¡(i)r¡(j) cxp( -21";JO (8)({"p L UI']) 1) L . ', .

l<j i<i 1)

1\, ami Ó are posilive elllpirical parameters satisfying 0.75 <f,', < 1.25 and 0.35 A-l < ó < 0.35 A-l, and do being ;11CSUIll of alomic radii deflned hy,

(I)

(2)

(3)

N

= 2¿f(E,)E, - NEO,i=l

N

1>¡ = LCkiSPklk=l

The Extended Hüekel model is lhe simplesl quantulll chcllli-cal Illclhod that cOllsiders all lhe valcncc clcclrons [15-171. Itis \Vell known that in lhe original forOl Ihis mcthod is nnl ahlelo optimizc gcomclry correctly as it lack rcpulsivc clcctros-latie inlcraClions. Ho\\'cvcr. Ihis dcficicncy can he ovcrcomchy introducing an clectrostatic corrcction lcrm 116, 1¡J. Thus,Ihe total cncrgy is calculalcd as.

wherc E() is rhe 8p-state ionization energy, E¡ is the molecu-lar orhitals cncrgy, f( E;) is the Fermi dislrihuJion at T = O Kand Ef~'1I is the c1assical Extended IHickel (ot<11energy (asllm over Ihe occupied molecular orhital cllergics). Ir eachmolecular orhital of the c1usler, 1>i' is expanded as a linearcomhinalion of the .••¡>~:-atomic hyhrid orhilals,

Ihe eigenvalues (E¡) and cigenvectors (Cki) are ohtained assolutions 01' lhe generalized eigenvalue equation,

Here, S is the overlap matrix helwcen .<;¡Hype orhilals,and H is the Extended~Hückel Hamiltonian matrix, whosedi<1gonalized matrix elements are equal to ea (lhe valence-slate ionization energy), and the non-diagonal (hopping) cle-ment, ¡¡ji are given hy !l8J.

(H - ES)C = O. (4)

In this cxpresion we have introduce four empirical para-mcters, o, 1", J and IJ, which are chosen to reproduce lhehinding encrgy and hond distance 01' diatomic alkali mo-lccules. For lhe prescnt calculalions, values of (} = O.Gi,1 + ". = 1.75, J = 0.13 and {3 = 1.0 were used. Thcsevalues render an equilihríum distance of 5.4 a.u., a dissocia-tion energy 01' O. 75 eV ¡¡mi a vihraclional frequcncy 01' 220.0cm-l. for Nal.

3, Molecular dynamics

with rij equal to the disrance hetween atoms i and j. (1 con s-[ant. and l\ ij given hy (17]:

1,,; = 1 +" cxp[-J(rij - do)], (6)

In the Car-Parrincllo molecular dynamics algorilhm with ap-proximaled (or empirical) Hamiltonian {10-12]. lhe wavefunction eigcnvcctors (Ck¡) are combined with lhe nucle<1rcoordinares (R¡) into onc system undergoíng a joint c1assicaldynamics. Since the wave function coefficients are massless.we ha ve to givc thclll a fictitious mass Ji.

Ret'. Me.\". Fí.,. 47 SI (2001) 54-58

56 LUIS RINCÓN

Thc Lagrangian governing the equations 01'motion of thenuclei ane!clectrons is given hy,

L = ~IIL ('IC,,)' + ~ML (dRi)'_ dt _ dii)' i

- Etot({RJ; (cid). (10)

hy Ihe fklitious Lagrangian using a smaJl fictitious electronicmass.

Thc equalions of mOlion ror lhe variables gencrated hythe ficlitious Lagrangian are,

(1 1)

for lhe nuclei.Using a pairwise potcntial ror the tolal energy IEq. (9)J,

the force aCling on atom j are evaluated according lo lheHcllman-Feynllwn lheorcm (no Pulay corrections appear),and is obtailled to he,

Noticc that the total encrgy ohtained from Extended-Hückel Hamiltonian represcnts the potential energy lerm.Thc firts Iwo tcnns represent lhe kinclic energy 01' elec-trnns variahles ami nuclci. Let's cmphasizc that, as written,the elcctronic kinetic lerm is fictitious, since eleclron kineliccncrgy cOl1l1olhe ohtained in lhis classical like manner. Cal'and Parrinello suggestcd thal lhe lruc dynamics of lhe nuclearsyslcm (Born-Oppcnhaimer dynamics) can he approximated

I

for the elcctronic variahles, and

.JUd2R¡ = _ DEtotdt' DRi •

(12 )

( 13)

F¡ is a many-hody force sincc the electronic cigenvectorstakc in lo account the cnvironmcnt of lhe interacting aloll1s.Thc derivatives 01' lhe total energy with respecl to Ihe molc-cular orhital cocrticients (lhe fktitious rorces) are,

DEtot=4'CH.DC- 0)1.: 1)

Ik j-#í

The orthonormalily condition,

L CikC'il - Si! =.0,

(14 )

( 15)

near their ground state and follow Ihe nuc1e<lfmotionadiahatically. this condition is only ohlained in the ¡i-mit 01'infinitesimal! small¡l. However. it is found Ihata very slllal1 liclitiolls mass requires very smaJl timeslcp. Thus, in él compromise hi:lween 11 and time SICp,ror a tipical 11 valucs 01'0.10 a.m.u. and a time step of0.10!-" in Ihe Na1 cluster Icatls lo a conservation 01'thecllcrgy on the order 01' 10-5 hartree ovcr a pcriod ofn.12 ps (In" sICp).

must hi: preservcd during thc rropagation. This conslrainthClwl'ell Ihc cod'Hciellls force to perform a Gram-Schmidlortogonalization aner the electronic coordenales arc updatesal each timc step. The sequellce in which the orhitals are or-togonalized (top dO\\'1lor hOtl0I1111p)does not affect the par-ticlcs trajcctorics. In general. other ortogonalization methodsproduce similar results [11]. rvtost 01'authors use the 5;HAKEIllelhod in which the orthogonalily is includcd as a constrainlllsing a Lagrangc 1ll11ltiplierin the equation 01"motion [191.

Thc SOllllioll01'the equations 01'1110tionis easily obtained\\'ith lhe standard molecular dynamics rolltincs [1,21. I1rieny,(he procedure is as I"ollows:

1. Dctlne the illitial values for alomic coordinates antlsolve the corrcsponding Extended Hückel Hamiltoniallrol' the illilial molecular orbital coeftkienls ami cncrgy.

'1 Assign random nuclear vclociries corresponding lo agivcn iniliallemperaturc (or kinetic energy), and assignzero vc10cities 10 the molecular orhital cocfficienls.Choosc <1timeslcp ror the simulation and a ficliousIllass, 11, rOl" the coefticients. The llctitious mass is li-miled hy the requircmenl lhal we wanllo reproduce thenuclear dynamics. Wc require lhat lhe electrons remaill

3. Calculale the cncrgy, ETot' ami forces, DE/DR¡ anJDE¡DC,¡.

4. Update the posilion 01' the atoll1s and thcir velociliesaccording the Verlet algorilhm.

5. Update the values 01"lhe wavefunclion cocflkients ac-cording the Verlet algorithm. Pcrfonn a Gram-Schmidtorlogonalization on Ihe molecular orbitals. Calculalelhe vclocilies ror the slCp.

Ú. Calculare ¡he instanrancolls lempcraturc (from kinetici:nergy) amI perform the velocity llIoJifications Ihat isrCLJuircdhy lhe tcmperaturc schedule oí"the simulatcdanllcaling.

7. Rcturn to slep (3) rOl"as many stcps 01"simulatcd anne-a1ing rcquired.

4. Results 01'simulated annealing molecular dy-nanlics

Gcollletry oplirnil.atioll involves Ihe minimi/ation 01'the Ex-tended Hückel lotal energy. E'ot. which is a fUIlCtiol1of Ihe

Re•..Mex. Fú. 47 SI (2(XlI) 54~5X

t\.t1NIMUM-ENERGY CONFIGURATIONS OF METALLlC CLUSTERS OBTAINED BY SIMULATED. 57

posilion 01' Ihe cluster particles. Thc simulatcd anncaling mo-lecular dynamics method is a scquencc of molecular dyna-mies propagation 01' nuclci and coefficicnls in whieh Ihe ins-tanlancous tcmperalurc nI' each SICP is sealing aecording to:t lemperature schedule. Initially the systern is allowcd, for along lime, 10 follo\V a dynamics consistent with a high lem-peralurc. \Vilhin this period energy harriers can he slInnolln-ted, and Ihe syslell1 is not conlined lo any single calchmcnlrL'gion. Then, :l slow cooling is performed, which guaranleereaching the glohal minimulll. In simulaled annealing mole-cular dynamics. lhe propagalion is merely an cfficienl mean01"sampling the potenlial energ)' surface, this is so hecallschere \Ve are not interested in rcprodllcing any physical realIrajectory oí" nuclei.

Sincc the potential cl1ergy surfacc is quitc nal in Illany01' lhe glohal minimulll slructure 01' alkali metal, in order topre\"ent dissociatinl1 nI' the cluster at very high lemperalll-re. ;¡ lllaxil11l1n critica! temperature. which has 10 he passedslowly in the cnoling shedlllc 01' the sil1lulaled annealing. isused. rhe simulated anncaling molecular dynamics in theseexalllples COllsisl in going from 1 K to the critical lempera-Ime (TJ in lOO steps. next Ihe systern is equilihrated al thecrilical lempcralure for ahoul 107_10li sleps ami then coo-led frollllhal lemperature to 1 K over 107 sleps ami from 1 100.01 K over lOfi addilional steps. Small clustcrs are cOl1lparedwilh Ihe results 01"l\lanin el al. [20J ami Bonaci,-Koutecky£'raI.1211. They wcrc ohtaincJ with a!J-illirio mclhod. and forour ellccl \Ve shall consider <lS'exact'.

Tahle 1 shows the average valllc nf the ncarcsl-neighllorhond dislance (JI) ,Ihe smallcr inleratolllic distance n .

/111 Illlli'

Ihe \"ari •.IIlcc 01' Ihc Ilcarcst Ilcighhor distancc ~ ¡{'III' the avc-rage nUlllher 01' ncaresl llcighor atOJ1lSDo/I/

I1lalllllhe hinding

ellcrgy pcr atom 01' sodium [3E/II. For the slllall cluster l/h.

illirio \'allles are givcn in parcnthcsis ['201, The range 01'C!lIS-ter size Ihal \Ve llave studied here. 2 to I(JO :1l01llS, is tooslllalJ lo alJo\V a good extrapolatioll 01" the aSYllltolic heha-vim nI' the material hulk. Na bulk slructure have a hcc pac-king. wilh S neighbor per Na alOm, ami inleratomie distallce01"1.02'2 ,UI. It is I"ound thal 1'01'small cluslers ('2 < 1/. < 8),these optimizatiolls yield qualitatively lile salllc cluster sha-pes thal more solisticalcd Icchniqlles. Small cluskl"s are verys~/[nllletric: Na,¡ is rhomboid, U'.!h symlllL'try. Nali planar.J):~/¡ sYlllllletry. anJ Na!'j is Ictrahedral. For 11 > S, no silll~pIel' point group can be assigned to the cluster. howc\"Cl". 110thcubil' amI letrahC'dral suh-lIllilS are obsL'r\'C'd. It is ;,Iso ohser-\"L'dthat lhe Ilraresl neighhol" distallcc lends to inerrase willl

l. D. FrenkeJ and B. Smil. UIlr!l'I"s(wu/mg ¡\foll'clI/(/r Sil/ll//arltm:

/-¡l/m i\lgorithlll.\" ro AI,lIicari()lJs, (Academil: i'res<;. J (96).

"2. De. Rap;¡porL TI//' Arl (~r,\foll'cul(// !>Y"l/llliC".\ Sil/l/f/l/l/olI,

(Cllllhridge Uni\"cr"iIY i'rcss, Carnhridgl'. (1)95)..~. S. Kirl\patril:l\. C.D. Ciclatt, JI'. ami ~1.P. VCl'chi • .\'('/1'1I("1' 22n

(llJX.1) 671

T/\BLE 1. Propcrtics of sodium clusters. Valucs in parenthcsis co-rrcspond to Rcf. 20. Distances in alomic units (0.52 Á) and energiesin cv.

Cluster Rltfl Rmin D.Rnn D.Hnn BE("Na! 5.3 5.3 0.00 1.00 0.37

(55) (5.5) (O (lO) (0.45)

Na, 6.0 5.6 0.67 2.50 0.60(60) (5.5) (0.69) (0.61)

Na¡; 6.0 5.9 0.18 3.33 068(6.0) (5.8) (0.20) (0.73)

Na!l 6.0 5.9 (U19 4.00 0.73((>0) 16.1) (0.05) (0.86)

NalO 6. 1 6.0 0.05 4.80 0.77

Na:w 6.3 6.3 (W2 5.50 0.86

Na.',n (J.S 6.5 0.01 6.90 0.88

NalOo 6Jl 6.6 0.01 7.44 0.91

cluster sil.e. in all cases lhe values arc smaller than Ihe bulkinteralomic distancc. Thc variance of neares! neighbor dis-lance slrongly decreases wilh cluster size. Thc variation 011l!le average nllmher 01' Ileighbors shows thal the structures nI'small melal clusters diller from lhe highly syrnrnetrical bu!kstructure. and Ihercfore it is no! eorrect lo build these clustersfrom blllk unil cells, as is onen assumed in model calcula-liolls. The calclllaled binding encrgy lcnds to converge morequiekly Ihan Ihe 11u Illher 01' l1eighhors.

These preliminary lesls have shown that our melhodworks correclly ami produce physical reasonahle rcsulls. Thcadvantage 01"the current mode! over lhe tradilional olles isit greal speed that makes possihle its use in molecular dyna-mies invonving a \Vide range 01' atoms Ilumhcr and severa!Il10llsand slcps. 1\ IInal COlllt11enl, wc think lhat this Illelhodis very promising rOl' Iarge-scalc molecular dynamics ca1cll-laliolls SillCC il has several advantages compared to other ap-pro;¡ches: (a) lhe slllall sit.e 01' lhe basis. (h) easy evalualionnI' alOmie forces. and (e) lhe Hamiltonian allows a consislentdescriplion from slllalllllolccules lo the malcrial bulk.

Acknowlcdgmcnl

This rcscan:h \Vas suported hy CONICIT lhrough lhe COIll-putalional Calalysis Projecl (Grant G-97(00667).

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-;- R Cal' :tnl! ~ 1, PalTincllo, ['''-,",l' Rl'l~ Ll'tI. 55 ( 1985 2471).

Rel'. Mex. /-II. 47 SI (2001) 5-l--5X

58 LUIS RINCÓN

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RCI'.Mex. Fú. 47 SI (2001) 54-58