SurfaceScience411(1998)186–202ThesurfaceenergyofmetalsL.Vitosa,A.V.Rubana,H.L.Skrivera,*,J.Kolla´rbaCenterforAtomic-scaleMaterialsPhysicsandDepartmentofPhysics,TechnicalUniversityofDenmark,DK-2800Lyngby,DenmarkbResearchInstituteforSolidStatePhysics,H-1525Budapest,P.O.Box49,HungaryReceived3November1997;acceptedforpublication2May1998AbstractWehaveuseddensityfunctionaltheorytoestablishadatabaseofsurfaceenergiesforlowindexsurfacesof60metalsintheperiodictable.Thedatamaybeusedasaconsistentstartingpointformodelsofsurfacesciencephenomena.Theaccuracyofthedatabaseisestablishedinacomparisonwithotherdensityfunctionaltheoryresultsandthecalculatedsurfaceenergyanisotropiesareappliedinadeterminationoftheequilibriumshapeofnano-crystalsofFe,Cu,Mo,Ta,PtandPb.©1998ElsevierScienceB.V.Allrightsreserved.Keywords:Abinitioquantumchemicalmethodsandcalculations;Densityfunctionalcalculations;Green’sfunctionmethods;Highindexsinglecrystalsurfaces;Lowindexsinglecrystalsurfaces;Metals;Singlecrystalsurfaces;Surfaceenergy1.IntroductionThesurfaceenergycdefinedasthesurfaceexcessfreeenergyperunitareaofaparticularcrystalfacetisoneofthebasicquantitiesinsurfacephysics.Itdeterminestheequilibriumshapeofmezoscopiccrystals,itplaysanimportantroleinfaceting,roughening,andcrystalgrowthphen-omena,andmaybeusedtoestimatesurfacesegre-gationinbinaryalloys.Mostoftheexperimentalsurfaceenergydata[1,2]stemsfromsurfaceten-sionmeasurementsintheliquidphaseextrapolatedtozerotemperature.Althoughthesedataatpre-sentformthemostcomprehensiveexperimentalsourceofsurfaceenergiestheyincludeuncertain-tiesofunknownmagnitudeandcorrespondtoanisotropiccrystal.Hence,theydonotyieldinforma-*Correspondingauthor.Tel.:(+45)45882488;fax:(+45)45932399;e-mail:skriver@fysik.dtu.dktionastothesurfaceenergyofaparticularsurfacefacet.ExceptfortheclassicmeasurementsonPbandIn[3,4]therearetoourknowledgenodirectexperimentaldeterminationsoftheanisotropyinthesurfaceenergyofsolids.Atheoreticaldetermi-nationofthesurfaceenergyisthereforeofvitalimportance.Duringthelastdecadetherehavebeenmanycalculationsofthesurfaceenergyofmetalseitherfromfirst-principles[5–7]orbysemi-empiricalmethods[8].Thelatterareofcoursecomputation-allyhighlyefficientandinmanycasesprovideagooddescriptionoftheenergeticsofsurfaces.Hence,theyhavebeenusedwithgreatsuccesstostudyandunderstandtrends.Incontrast,mostfirst-principlesmethodsarecomputationallydemandingandhavetypicallybeenusedonlyforparticularcases,focusingonafewelementsoronaspecialapplicationforagivenmetalsurface.However,recentlyMethfesseletal.[5]haveused0039-6028/98/$seefrontmatter©1998ElsevierScienceB.V.Allrightsreserved.PII:S0039-6028(98)00363-XL.Vitosetal./SurfaceScience411(1998)186–202187thefullpotentiallinearmuffin-tinorbitals(LMTO)methodtoinvestigatethetrendsinthesurfaceenergy,workfunction,andsurfacerelaxationinthebccandfcc4dtransitionmetals.Concurrently,Skriverandco-workers[6,7,9,10]usedtheGreen’sfunctionLMTOmethod[11]intheatomic-sphereapproximation(ASA)tocalculatethesurfaceenergyandworkfunctionofmostoftheelementalmetalsincludingthelightactinides.Inallofthesefirst-principlescalculationslittleattentionhasbeenpaidtothedependenceofthesurfaceenergyontheorientationofthesurfacefacetsandhencethereisatpresentnocomprehensivefirst-principlesdatabaseforsurfaceenergyanisotopies.Itistheaimofthepresentpapertofillthisgap.Ourstartingpointisthewell-establishedfactthatdensityfunctionaltheory(DFT),eitherinthelocaldensityapproximation(LDA)orthegeneral-izedgradientapproximation(GGA),yieldsgroundstatepropertiesofmetalsincloseagreementwithexperimentalobservations.Hence,byusingtheLDAandGGAforallmetalsintheperiodictableweshouldbeabletoformaconsis-tentandaccuratedatabaseofsurfaceenergiesincludingtheanisotropiesforwhichtherearefewexperimentalobservations.GiventheLDAortheGGAweneedtobeabletosolvetheone-electronproblemaccuratelyandefficientlyifwearetocovermostoftheperiodictableforamultitudeofsurfacefacets.Tothisendweapplytherecentlydevelopedfullchargedensity(FCD)LMTOmethod[12,13]whichwasshowntohaveanaccuracycomparablewiththatofthefullpotentialmethods.Toestablishtheaccuracyforthepresentpurposewecompareoursurfaceenergiesforthe4dtransitionmetalswiththefullpotentialcalcula-tions[5]andfindcloseagreementbetweenthetwosetsofresults.WethereforeexpectourcalculateddatabaseofsurfaceenergiestoreflectthetrueDFTresult.2.ComputationalmethodAfulldescriptionoftheFCDmethodmaybefoundinRefs.[13–15].Here,weoutlinetheimpor-tantnumericaldetailsandestablishtheaccuracyoftheFCDapproachthroughacomparisonwithotherfirst-principlesLDAcalculations.Wefurthercalculatethesurfaceenergiesofthe4dmetalsintheLDAaswellastheGGAassumingaclosepackedfcc(111)surfaceandcomparetheresultswiththeisotropicexperimentalresults[2].2.1.FullchargedensityschemeforsurfacesTheFCDmethodisbasedondensityfunctionaltheory[16].TheKohn–Shamone-electronequa-tionsaresolvedbymeansoftheextremelyefficienttight-bindingLMTOmethodintheASA[17–20].Thecompletenon-sphericallysymmetricchargedensityn(r)isconstructedfromtheoutputofaself-consistentGreen’sfunctionLMTO-ASAcal-culationandnormalizedwithinspace-fillingnon-overlappingcellscenteredaroundeachatomicsite.Thetotaldensitysoconstructediscontinuousandcontinuouslydifferentiableinallspace[15].ItwasshownbyAndersenetal.[21]thatthefullchargedensityobtainedonthebasisofasphericallysymmetricself-consistentcalculationisclosetothedensityobtainedusingafullpotentialmethodevenforopenstructuressuchasdiamond.Therefore,weexpectthetotalenergyfunctionalevaluatedfromthefullchargedensityderivedbyanLMTO-ASAcalculationtobeaverygoodapproximationtothetotalenergyofthesystem.TheHartreeandexchange-correlationpartsoftheFCDenergyfunctionalEFCD[n]arecalculatedexactlyfromthechargedensityn(r)usingtheLDAorGGAexchange-correlationenergyfunctionals,whilethekineticenergyisgivenbytheKohn–ShamkineticenergyobtainedfromtheLMTO-ASAequationscorrectedbyatermduetothenon-sphericallysymmetricpartofthechargedensityneglectedintheself-consistencyprocedure.Thiscorrectionisevaluatedintermsofagradientexpansionaroundthesphericallysymmetricchargedensityusingthefunctionalformforthekineticenergyofnoninteractingparticles[16],asdescribedinRef.[13].Thereby,theFCDmethodretainsmostofthesimplicityandthecomputa-tionalefficiencyoftheLMTO-ASAmethod,butattainsanaccuracycomparablewiththatofthefullpotentialmethodsasdemonstratedinthesuccessfulcalculationsofthebulkgroundstateproperties,includingtheshearelasticconstants,forthe4dtransitionmetalsandtheequilibrium188L.Vitosetal./SurfaceScience411(1998)186–202volumesofthea-phasesofthelight5fmetals[12,13,22].IntheapplicationoftheFCDmethodtosurfaceenergycalculationstheone-electronequationsaresolvedbymeansofthesurfaceGreen’sfunctiontechniquedevelopedbySkriverandRosengaard[11].Themethodcorrectlyaccountsforthesemi-infinitenatureofasurfacebymeansoftheprinci-pallayertechnique[23].TheDysonequationdescribingtherelaxationoftheelectronicstructureofthesurfaceregionissetupandsolvedwithintheatomicsphereapproximation,takingintoaccounttheelectrostaticmono-anddipolecontri-butionstothesphericallysymmetricone-electronpotential[11].Inthepresentimplementationtherelaxationofthesurfaceatomicpositionsisneglected.Accordingtothefirst-principlesworksbyFeibelmanetal.[24,25]andbyMansfieldetal.[26]theeffectofrelaxationonthecalculatedsurfaceenergyofaparticularcrystalfacetmayvaryfrom2to5%dependingontheroughness.Thesemi-empiricalresultsbyRodriguezetal.[8]showthatsurfacerelaxationtypicallyaffectstheanisotropybylessthan2%andthereforetheneglectofrelaxationsinthepresentworkhaslittleeffectontheaccuracyofthecalculateddatabase.Sincethesurfaceone-electronpotentialincludestheelectrostaticdipolecontribution,theKohn–ShamkineticenergyobtainedwithintheASAimplicitlycontainsthemaineffectofthesemi-infinitesurface[11].ForclosepackedsurfacestheASAkineticenergyisthereforeagoodapproxima-tiontotheexactkineticenergy.InthesurfaceFCDcalculationsthenonsphericalcorrectiontotheASAsurfacekineticenergyisestimatedbymeansofasecond-ordergradientexpansionofthekineticenergyfunctional[16].Theactualcalcula-tionsshowthatthiscorrectionbecomesimportantonlyforopensurfacesandthatinmostcasesonlydensitiesoflowgradientsareinvolved.Hence,althoughadensitygradientexpansiondoesnotingeneralworkforthecompletekineticenergywefindthatitworkswellforthecorrectiontotheASAkineticenergy.Atzerotemperaturethesurfaceexcessfreeenergymaybecalculatedasthedifferencec=EFCD2D(Na+Nv)−NaEFCD3D(1)betweenthetotalenergyofNaatomsplusNvemptyspheresinthesurfaceregion(2D)andNtimesthetotalenergyperatominthebulk(3D).aThisapproachispurelysemi-infiniteanddoesnotrelyonaslaborsupercellgeometry.Further,toreducenumericalerrorsthetotalenergyofthebulkEFCDisevaluatedbymeansofaGreen’sfunction3DtechniquesimilartothatusedinthesurfaceFCDcalculations,i.e.,allone-electronquantitiesareobtainedbycontourintegralsofGreen’sfunctionsratherthanbyaconventionalHamiltonianeigenvaluetechnique.2.2.NumericaldetailsInallcalculationstheoneelectronequationsweresolvedwithinthescalar-relativisticandfrozen-coreapproximations.IntheLMTObasissetweincludedspanddorbitalsinthecaseofthesimplemetals,andspdandforbitalsinthecaseoftransitionmetalsandlightactinidesincludingFrandRa.ForInthe4d,forTlandPbthe5d,andforthelightactinidesthe6pstatesweretreatedasbandstatesusingasecondenergypanel.Thevalenceelectronsweretreatedself-consistentlyinthelocaldensityapproximationwiththePerdew–Wangparametrization[27]oftheresultsofCeperleyandAlder[28]forexchangeandcorrelationandtoobtainthetotalenergyinthegeneralizedgradientapproximationtheFCDfunc-tionalincludedtheexchange-correlationfunctionalgivenbyPerdewetal.[29].TheDysonequationforthebulkandsurfacecalculationswassolvedfor16complexenergypointsdistributedexponentiallyonasemicircularcontour.Thek-pointsamplinginthebulkcalcula-tionswasperformedonauniformgrid,whileinthesurfacecalculationsspecialk-pointswereused[30].Thenumberofk-pointsintheirreduciblewedgeofthe3Dand2DBrillouinzonesforeachstructureandsurfaceorientationareshowninTable1.Thehcpstructurehastwo(101:0)surfacesdependingonthefirstandinthetable(101:interlayerdistanced−10,0)d−10=(ǰ3/6)aArefers:tothesurfacewith−while10(1010)denotesthesurfacewithAd−10=2d.InthecalculationsBofatomicBthenumberslayersAwerechosensuchthattherelativeerrorinthesurfaceenergycausedbythedifferencebetweentheenergyofthedeepestlyingatomicL.Vitosetal./SurfaceScience411(1998)186–2021Table1Numberofk-vectorsintheirreduciblewedgeofthethreedimensional(3D)andthetwodimension(2D)Brillouinzonesandnumberofatomic(N)andvacuum(N)layersusedintheself-consistentcalculations.Aistheareaofthe2Dunitcellandaandcareav2DlatticeparametersStructurefccNIBZ−3Dk505Surface(111)(100)(110)(110)(100)(211)(310)(111)(0001):0)(101A:0)(101B(001)(110)(100)(100)(110)A2D2DBrillouinzonehexagonalsquarerectangular-Prectangular-Csquarerectangular-Prectangular-Chexagonalhexagonalrectangular-Prectangular-Psquarerectangular-Crectangular-Psquarerectangular-PNIBZ−2Dk45363612812845453636Na4466688848844446Nv2232344424422423ǰ3/4a21/2a2ǰ2/2a2ǰ2/2a2a2ǰ3/2a2ǰ10/2a2ǰ3a2ǰ3/2a2ǰ8/3a2ǰ8/3a2a2ǰ2/2acaca2ǰ2a2bcc506hcp225bct594sc455layerandthebulkenergywaslessthan1%.Theactualnumbersofatomic(N)andvacuumlayersa(N)usedareshowninthelasttwocolumnsvofTable1.Theelectronicchargedensitywasrepresentedinanone-centerexpansionintermsofsphericalharmonicsincludingtermsuptol=10andthemaxnormalizationofthechargedensitywasensuredbythetechniquedescribedinSectionIIAofRef.[14].TheFCDenergyfunctionalwaseval-uatedbymeansoftheshapefunctiontechniqueusingalinearradialmeshbetweentheinscribedandcircumscribedspheres.TheconventionalMadelungtermswerecalculateduptol=8.maxForthenearestneighborcellinteractionweusedthetechniquedescribedinRefs.[14,31]takingl=30forthemultipolemoments.max2.3.Comparisonwithfull-potentialLDAcalculationsToestablishtheaccuracyofthepresentmethodformetalsurfaceswemaycompareourcalculatedsurfaceenergieswithotherfirst-principlesresults.Toourknowledge,theonlysystematiccalculationofthesurfaceenergyofatransitionmetalseriesandofthecorrespondingsurfaceenergyanisot-ropyiscarriedoutbyMethfesseletal.[5]intheirfullpotential(FP)LMTOstudyofthe4dmetals.Fortheclosepackedfcc(111)surfacesthecompar-isonbetweentheresultsobtainedbythesetwomethodshavebeenpresentedinRef.[14],wherewefounda10%meandeviationinthesurfaceenergyoverthe4dseries.Thepresentimplementa-tionoftheFCDmethod,usingthesamebasissetandthesameexchange-correlationfunctionalasthefullpotentialcalculation,givesameandevia-tionfortheclosedpackedsurfacessimilartothatfoundinRef.[14].TheaccuracyoftheorientationdependenceofthesurfaceenergymaybeassessedfromFig.1wherewecomparetheFCDanisotropyratiosc/candc/cforthe4dtransitionmetals100111110111assumingthefccstructureforthewholeserieswiththoseobtainedintheFP-LMTOcalculationsbyMethfesseletal.[5].Wenotethatc/c100111andc/cinthenearestneighborbrokenbond110111modelare1.15and1.22,respectivelyandthatthefirst-principlesresultsshowninthefiguredeviatesomewhatfromthesevalues.Inparticular,the190L.Vitosetal./SurfaceScience411(1998)186–202Fig.1.ComparisonoftheFCDsurfaceenergyanisotropiesforthelowindexfccsurfacesofthe4delementswiththefull-potentialslabcalculationbyMethfesseletal.[5].Thethindashedlinesindicatethevaluesobtainedinthenearestneighborbrokenbondmodel.NotethattheresultsshowninthefigureareLDAresultscalculatedforcomparisononly.Theyareapproximately1%largerthanthecorrespondingGGAresultslistedinTable6.elementsMo–Pdexhibitanincreasedcareducedc100/c111butmodel.This110/cmay111relativetothebrokenbondbeexplainedintermsofthepair-potentialexpansion,tobeintroducedinSection4.Inthisexpansionthenext-nearestneighborpoten-tialsarenegativewhich,coupledwiththefactthatthenumberofbrokennext-nearestneighborbondsforthe(100)andthe(110)surfacesis2and4,respectively,leadstothedeviationsfromthenear-estneighborbrokenbondmodelobservedinFig.1forMo–Pd.Inthecomparisononeshouldnotethatthefullpotentialcalculationsrelyonaslabsupercellgeometrywith7metalliclayerswhichmaybeinsufficientfortheopensurfaces1,arenon-relativ-1Toobtainconvergenceweuseinthepresentcalculationsfrom4to8metalliclayers,seeTable1,whichhavethecorrectboundaryconditionatthevacuumandbulkinterfaces.Asim-ilarconvergenceintermsofthenumberofmetalliclayersinaslabsupercellmethodneedsapproximately8to16metalliclayer.istic,andallowtheatomicpositionsatthesurfacetorelaxwhilethepresentGreen’sfunctiontech-niquedescribesasinglesemi-infinitesurface,isscalarrelativistic,anddoesnotincluderelaxationofatomicpositions.Further,thefullpotentialcalculationsutilizeatriplespdbasiswhilethepresentcalculationsutilizeasinglebasisbutincludetheforbitalsaswell.Finally,thefullpotentialmethodmakesuseofthePerdew–Zungerparametrization[32]oftheresultsofCeperleyandAlder[28]fortheexchange-correlationfunctionalwhilethepresentFCDmethod,forthiscompari-sononly,usesthePerdew–Wangparametrization[27]ofthesameresults.Withthesedifferencesinmindtheagreementbetweenthetwocalculationsisverygood,themeandeviationsforcc/cforthefccsurfacesoverthe100/c4d111andperiodbeing110111around5%and6%,respectively.ThelargestdeviationbetweenthetwosetsofcalculationsisfoundinMowhereananomaloussurfacerelaxationcausesasubstantialreductioninthesurfaceenergyofthe(100)and(110)facetsinthefullpotentialcalculations.However,Moformsinthebccstructureandforthisthetwomethodsyieldsc/cvalues1%.Fortheother100bcc110thatagreewithinmetalNbouranisotropyissubstantiallylowerthanthatobtainedinthefullpotentialLMTOcalculationbutinthiscasethefullpotentiallinearaugmentedplanewave(LAPW)calculationbyWeinertetal.[33]givesc/c=1.07whichcompareswellwithof100ourvalue1.06110andwiththesimilarlylowanisotropywefindforVandTa(seeSection3.2).2.4.LDAversusGGAThemostsignificantassumptionintheFCDcalculationsistheapproximatefunctionalformfortheexchangeandcorrelationenergy.Inmostdensityfunctionalcalculationssomeformoflocaldensityapproximationisusedwithgreatsuccesstodescribegroundstatepropertiesofawiderangeofbulkandsurfacesystems.ForsurfacestheaccuracyoftheLDAwasstudiedwithintheself-consistentjelliumsurfacemodelbyPerdewetal.[34],andgoodagreementwasfoundwiththeL.Vitosetal./SurfaceScience411(1998)186–202191experimentalresult2,especiallyforslowlyvaryingdensityprofiles.Itssuccesswasascribedtoacancellationbetweentheerrorsintheexchangeandcorrelationenergies.ThegradientcorrectiontotheLDA[29,34]representsanimportantimprovementforthecorrelationpart,butitunder-estimatestheexchangeenergy,andasaconse-quenceitgivessurfaceenergieswhichare7–16%lowerthantheLDAvaluesforjelliumand16–29%lowerthantheexperimentalresults[34].OnthesegroundsonemaythereforeprefertheLDAresultsinthedatabase.However,sincethestudybyPerdewetal.ofthevalidityoftheLDAandGGAforsurfacesisbasedonthejelliummodelwhichatbestisonlyapplicabletosimplemetalsoneneedstocomparesurfaceenergiesobtainedforalargerrangeofrealmetalsbeforejudgingtherelativemeritsofthetwoapproximationstoden-sityfunctionaltheory.BeforecomparingLDAandGGAresultswemustaddressthequestionoftheatomicvolumeusedinthecalculationsofthesurfaceenergy.InthelowerpanelofFig.2weshowthetotalenergyversusWigner–Seitzradiusforabulkatom(EFCD)andtheenergyperatomoftheatomsinthe3Dsurfaceregion(EFCD(N)/N)offccThedifferencebetween2D(111)Ru.thesetwoenergiesisthesurfaceenergy,asdefinedinEq.(1),andthisisplottedintheupperpanel.InspectionofthesefiguresshowsthatthecalculatedsurfaceenergiesdependsensitivelyonthevolumeoftheunderlyingbulklatticeandthereforeitisimportantthatintheLDA-GGAcomparisonthecalculationsareperformedattheproperequilibriumvolumes.KeepingthevolumefixedattheLDAvaluewefindthatthemeandeviationbetweentheLDAandGGAresultsalongthe4dseriesincludingRbandSrisabout10%.RepeatingtheGGAcalcula-tionsattheGGAequilibriumvolumesthemeandeviationdecreasesto6%.ItistheresultsobtainedattheproperLDAandGGAvolumeswhichareplottedinFig.3.Asonemaysee,theeffectoftheGGAovertheLDAisrathersmallforthesimple2The‘‘experimental’’valuesusedinthecomparisonwiththejelliumsurfaceenergyofRef.[34]wereobtainedfromtheexperimentalvaluesforthecorrespondingsimplemetal[1]dividedbythecorrugationfactorof1.2asdefinedinRef.[35].Fig.2.Upperpanel:surfaceenergyversusWigner–SeitzradiusforaRufcc(111)surface.Lowerpanel:totalenergyperatomversusWigner–Seitzradiusforbulk(EFCD)andforinthesurfaceregion(EFCD3Dtheatoms2D(N)/N)forfccRu.Fig.3.ComparisonoftheFCD-LDAandFCD-GGAsurfaceenergiesforthefcc(111)surfaceofthe4dmetalsincludingRbandSr.metals,e.g.forRbandSrthedeviationis5and3%,respectively,whichismuchlowerthanthe7–16%foundinthejelliumcalculations,butit192L.Vitosetal./SurfaceScience411(1998)186–202increasestoapproximately20%inPdandAgattheendofthe4dseries.Inthecomparisonbetweenthetheoreticalfcc(111)surfaceenergiesplottedinFig.3andtheisotropicvaluesderivedfromexperiment[2]wefindthatonaveragetheLDAresultsare8%largerandtheGGAresults7%lowerthantheexperimen-talresults.Therefore,sincetheGGAyieldsmar-ginallybettersurfaceenergiesandsubstantiallybetteratomicequilibriumvolumesrelativetoexperimentsallthesurfaceenergiesfromthepre-sentworkpresentedinTables2–8havebeenobtainedbymeansofthemostrecentGGAexchange-correlationfunctional[29]atthetheoret-icalGGAequilibriumvolumesdeterminedinaseriesofbulkcalculationsusingthesameGreen’sfunctiontechniqueasinthesurfacecalculations.FinallywenotethattheeffectofthegradientcorrectiontotheLDAincreaseswiththeroughnessandasaconsequencereducesthesurfaceenergyanisotropy.Theeffectisfoundtobeabout1%forc/c,c/cinthefccmetalsandfor100111110111c/cinthebccmetals.1001103.ResultsanddiscussionThepresentsurfaceenergyresultsareshowninTables2–8ineVatom−1aswellasinJm−2.Forcomparisonwealsopresenttheavailablefullpotentialresultsandtwosetsofexperimentallyderivedvalues.AllthefullpotentialcalculationsshowninthetablesemployanLDAexchange-correlationfunctional.AlmostallFCDcalcula-tionshavebeenperformedinthemeasuredlowtemperatureequilibriumcrystalstructuresbutinfewcaseswherethea-phasehasalowersymmetrythanbodycenteredtetragonal(bct)weconsiderinsteadahigh-pressurephaseoraclosepackedfcc(111)surface.Inthetablesthesestructuresareindicatedbyanasterisk.Forthehcpmetalswehaveassumedanidealc/aratiowiththeexceptionTable2SurfaceenergiesforthemonovalentspmetalscalculatedbytheFCDmethodintheGGA.Forcomparisonwehaveincludedtheavailablefullpotentialandexperimentalresults.Thecalculatedequilibriumlatticeconstantsaarelistedinthesecondcolumn˚))Structure(a(ALibcc(3.431)bcc(4.197)bcc(5.300)bcc(5.714)bcc(6.2)bcc(6.320)Surface(110)(100)(111)(110)(100)(111)(110)(100)(111)(110)(100)(111)(110)(100)(111)(110)(100)(111)FCD(eVatom−1)0.20.3830.7500.1970.2900.5460.1670.2490.4620.1500.2290.4170.1420.2280.3900.1220.2020.346FCD(Jm−2)0.5560.5220.5900.2530.20.2870.1350.1420.1520.1040.1120.1180.0820.0930.0920.0690.0810.080FP(Jm−2)0.545a0.506a0.623aExperiment(Jm−2)0.522b,0.525cNa0.261b,0.260cK0.145b,0.130cRb0.117b,0.110cCs0.095b,0.095cFraPseudopotential,Ref.[36].bExperimental,Ref.[1].cExperimental,Ref.[2].L.Vitosetal./SurfaceScience411(1998)186–202193Table3SurfaceenergiesforthedivalentspmetalscalculatedbytheFCDmethodintheGGA.Forcomparisonwehaveincludedtheavailablefullpotentialandexperimentalresults.Thecalculatedequilibriumlatticeconstantsaarelistedinthesecondcolumn˚))Structure(a(ACafcc(5.624)fcc(6.169)bcc(5.2)bcc(5.372)bcc(4.757)fcc(5.697)hcp(2.236)hcp(3.196)hcp(2.684,c/a=1.86)hcp(3.061,c/a=1.)hcp*(3.528)Surface(111)(100)(110)(111)(100)(110)(110)(100)(111)(110)(100)(111)(110)(100)(111)(111)(100)(110)(0001):0)(101A:0)(101B(0001):0)(101A:0)(101B(0001)(0001)(0001)FCD(eVatom−1)0.4840.5350.8110.4400.4840.7250.40.6161.1990.3770.5151.0100.4840.6531.2820.4230.4840.7210.4951.0831.6260.4370.8141.0720.3850.3000.111FCD(Jm−2)0.5670.5420.5820.4280.4080.4320.3760.3530.3970.2960.2860.3240.4850.4630.5240.4820.4780.5031.8342.1263.1920.7920.7821.0300.90.5930.1651.924c,2.1d0.450bFP(Jm−2)Experiment(Jm−2)0.502a,0.490bSr0.419a,0.410bBa0.380a,0.370bRaEuYb0.500bBe1.628a,2.700bMg0.1e0.785a,0.760bZnCdHg0.993a,0.990b0.762a,0.740b0.605a,0.575baExperimental,Ref.[1].bExperimental,Ref.[2].cFullpotentialLAPW,Ref.[24].dPseudopotential,Ref.[37].ePseudopotential,Ref.[38].ofZnandCd.Forthesetwoelementsandforthebctstructuresweusedtheexperimental[52]c/aratioslistedinthetables.ThetheoreticalGGAequilibriumlatticecon-stantscorrespondingtotheatomicvolumesusedinthesurfacecalculationsareshowninthesecondcolumnintheTables2–8.ThemeandeviationsbetweentheselatticeconstantsobtainedforthelowtemperatureequilibriumcrystalstructuresandtheexperimentalvaluesofRef.[53]is2.4%inthecaseofthesimplemetalsand1.4%,1.8%,and2.1%inthecaseofthe3d,4d,and5dtransitionmetals,respectively.Inthedeterminationoftheequilibriumvolumeofthetransitionmetalsweincludedthe3p,4p,and5pstatesinasecondenergypanel,i.e.treatedthemassemicorestates.194L.Vitosetal./SurfaceScience411(1998)186–202Table4SurfaceenergiesforthespmetalsfromGroupIIIA–VIAcalculatedbytheFCDmethodintheGGA.Forcomparisonwehaveincludedtheavailablefullpotentialandexperimentalresults.Thecalculatedequilibriumlatticeconstantsaarelistedinthesecondcolumn˚))Structure(a(AAlfcc(4.049)bct*(3.018,c/a=1.58)bct(3.352,c/a=1.52)hcp(3.714)bct(3.187,c/a=1.83)fcc(5.113)sc*(3.102)sc*(3.257)sc(3.349)Surface(111)(100)(110)(001)(110)(100)(001)(110)(100)(0001):0)(101A:0)(101B(001)(110)(100)(111)(100)(110)(100)(110)(100)(110)(100)(110)FCD(eVatom−1)0.5310.60.9190.3760.5070.6950.3420.4220.6320.2210.4940.5290.3870.5090.7160.2260.3070.5130.3650.5600.3560.5070.3060.370FCD(Jm−2)1.1991.3471.2710.6610.7970.7730.4880.5600.5920.2970.3520.3770.6110.6200.6160.3210.3770.4450.6080.6590.5370.5410.4370.3730.496d0.592d0.597b,0.535c0.4b,0.490cFP(Jm−2)0.939a1.081a1.090aExperiment(Jm−2)1.143b,1.160cGa0.881b,1.100cIn0.700b,0.675cTl0.602b,0.575cSn0.709b,0.675cPb0.593b,0.600cSbBiPoaPseudopotential,Ref.[39].bExperimental,Ref.[1].cExperimental,Ref.[2].dPseudopotential,Ref.[26].However,theinclusionofsemicore3p,4p,or5pstatesinthesurfacepartofthecalculationsaffectsthesurfaceenergyofthetransitionmetalsbylessthan2%andthereforethesestateswereconsideredcorestatesinthesurfacecalculations.Afurtherapproximationinthepresentcalculationsistheneglectoftherelaxationoftheatomicpositionswhichmayleadtoerrorsofuptoafewpercent.Asaresultweestimatethecombinederrorsinthepresentsurfaceenergiestobe2–5%dependingonthesurfaceroughnessand2%inthesurfaceenergyanisotropy,bothrelativetoanexactdensityfunc-tional,LDAorGGA,calculation.3.1.ThespmetalsTheFCDsurfaceenergiesofthespmetalsforanumberoflow-indexsurfacesarepresentedinTables2–4.Sincethea-structureofGaisortho-rhombicwith8atomsintheunitcellthecalcula-tionshavebeenperformedforthehigh-pressurebctphase.Furthermore,thea-structureofSbandBimaybeconsideredaslightlydistortedsimplecubicstructureandthesemetalshavethereforebeentreatedinthescstructure,whileSnhasbeencalculatedinthebctphaseofmetallicwhitetin.Thesurfaceenergyanisotropiesofthemonova-L.Vitosetal./SurfaceScience411(1998)186–202195Table5Surfaceenergiesforthe3dmetalscalculatedbytheFCDmethodintheGGA.Forcomparisonwehaveincludedtheavailablefullpotentialandexperimentalresults.Thecalculatedequilibriumlatticeconstantsaarelistedinthesecondcolumn˚))Structure(a(ASchcp(3.300)hcp(2.945)bcc(3.021)Surface(0001):0)(101A:0)(101B(0001):0)(101A:0)(101B(110)(100)(211)(310)(111)(110)(100)(211)(310)(111)(111)(110)(100)(211)(310)(111)(0001):0)(101A:0)(101B(111)(100)(110)(111)(100)(110)FCD(eVatom−1)1.0801.6942.0111.2342.2242.4351.3121.7252.4022.9213.4941.2582.0202.4203.0303.6261.0430.9781.2651.8042.1532.6940.9611.9822.4760.6950.9691.3370.7070.9061.323FCD(Jm−2)1.8341.5261.8122.6322.5162.7543.2583.0283.4433.2443.5413.5053.9793.23.7754.1233.1002.4302.2222.52.3932.7332.7753.0353.7912.0112.4262.3681.9522.1662.2371.94e1.802f2.194bFP(Jm−2)Experiment(Jm−2)1.275aTi1.9c,2.100aV2.622c,2.550a3.18dCrbcc*(2.852)2.354c,2.300aMnFefcc*(3.529)bcc(3.001)1.543c,1.600a2.417c,2.475aCohcp(2.532)fcc(3.578)fcc(3.661)2.522c,2.550aNi2.380c,2.450aCu1.790c,1.825aaExperimental,Ref.[2].bFullpotentialLAPW,Ref.[40].cExperimental,Ref.[1].dFullpotentialLAPW.Intheoriginalpaper[41],theauthorsobtainedthefollowingresults:#5.1Jm−2fortheW(100)surface,and#3.4Jm−2fortheV(100)surface.Theresultsquotedbyuscontainacorrectionthatappearsinthecaseofthin-filmtotal-energycalculationaswaspointedoutbyBoettger[42].eFullpotentialLMTO,Ref.[43].fModifiedAPW,Ref.[44].lentmetals,withtheexceptionofLi,areingoodagreementwiththeresultsobtainedwithinthejelliummodelbyPerdew[54].ThepresentGGAcalculationsyieldforNa,K,Rb,andCsc/c100110intherangefrom1.04to1.13comparedwith1.14forthejelliummodel.TheFCDcalculationforLigivesc/c=0.94,whichislowerthanthecorre-100110spondingvalueobtainedinthejelliummodel.On196L.Vitosetal./SurfaceScience411(1998)186–202Table6Surfaceenergiesforthe4dmetalscalculatedbytheFCDmethodintheGGA.Forcomparisonwehaveincludedtheavailablefullpotentialandexperimentalresults.Thecalculatedequilibriumlatticeconstantsaarelistedinthesecondcolumn˚))Structure(a(AYhcp(3.638)hcp(3.248)bcc(3.338)Surface(0001):0)(101A:0)(101B(0001):0)(101A:0)(101B(110)(100)(211)(310)(111)(110)(100)(211)(310)(111)(0001):0)(101A:(1010)B(0001):0)(101A:(1010)B(111)(100)(110)(111)(100)(110)(111)(100)(110)FCD(eVatom−1)1.0771.6762.0591.2882.2692.5921.3201.9872.4103.1453.6681.5342.4102.7383.6014.0681.5273.0403.31.5743.2013.6691.0021.3101.9190.8241.1521.5590.5530.6530.953FCD(Jm−2)1.5061.2431.5272.2602.1112.4112.6852.8582.8292.8613.0453.4543.8373.6003.6263.7403.6913.74.93.9284.23.8562.4722.7992.91.9202.3262.2251.1721.2001.2383.0g,4.3g2.044b,1.729cFP(Jm−2)Experiment(Jm−2)1.125aZr1.909d,2.000aNb2.36e,2.9f2.86e,3.1f2.655d,2.700aMobcc(3.173)3.14e3.52e2.907d,3.000aTchcp(2.767)hcp(2.723)fcc(3.873)fcc(3.985)fcc(4.179)3.150aRu3.043d,3.050aRh2.53e2.81e,2.65h,2.592i2.88e1.e1.86e,2.3f,2.130j1.97e,2.5f1.21e1.21e,1.3f,1.27k1.26e1.4f2.659d,2.700aPd2.003d,2.050aAg1.246d,1.250aaExperimental,Ref.[2].bPseudopotential,Ref.[45].cFullpotentialLAPW,Ref.[40].dExperimental,Ref.[1].eFullpotentialLMTO,Ref.[5].fFullpotentialLAPW,Ref.[33].gPseudopotential,Ref.[46].iFullpotentialLAPW,Ref.[25].hPseudopotential,Ref.[47].jPseudopotential,Ref.[48].kFullpotentialLAPW,Ref.[49].theotherhandourresultsforLiareinverygoodagreementwiththepseudopotentialcalculationbyKokkoetal.[36].ForthedivalentfccandbccmetalswefindthatthesurfaceenergyofthesecondmostclosepackedsurfaceisconsistentlylowerthanthatofthemostL.Vitosetal./SurfaceScience411(1998)186–202197Table7Surfaceenergiesforthe5dmetalscalculatedbytheFCDmethodintheGGA.Forcomparisonwehaveincludedtheavailablefullpotentialandexperimentalresults.Thecalculatedequilibriumlatticeconstantsaarelistedinthesecondcolumn˚))Structure(a(ALahcp*(3.873)hcp(3.566)hcp(3.237)bcc(3.354)Surface(0001):0)(101A:0)(101B(0001):0)(101A:0)(101B(0001):0)(101A:0)(101B(110)(100)(211)(310)(111)(110)(100)(211)(310)(111)(0001):0)(101A:0)(101B(0001):0)(101A:0)(101B(111)(100)(110)(111)(100)(110)(111)(100)(110)FCD(eVatom−1)0.9091.3981.6901.1021.8452.0931.4002.4712.21.5312.1742.7993.4854.2011.8062.9553.2614.3384.9161.7813..7701.8693.8744.5951.2251.7722.4281.0041.3782.0090.6110.51.321FCD(Jm−2)1.1210.9151.1061.6041.4241.6162.4722.3142.7093.0843.0973.2563.1393.4554.0054.6354.1774.3034.4524.2144.6285.9854.5665.0215.9552.9713.7223.6062.2992.7342.8191.2831.6271.7002.067d4.78cFP(Jm−2)Experiment(Jm−2)1.020aLu1.225aHf2.193b,2.150aTa2.902b,3.150aWbcc(3.196)3.265b,3.675aRehcp(2.797)hcp(2.752)fcc(3.907)fcc(4.019)fcc(4.198)3.626b,3.600aOs3.439b,3.450aIr3.048b,3.000aPt2.4b,2.475aAu1.04e1.506b,1.500aaExperimental,Ref.[2].bExperimental,Ref.[1].cFullpotentialLAPW;seefootnotedofTable5.dPseudopotential,Ref.[50].ePseudopotential,Ref.[51].closepackedsurface,theratiobeingapproximately0.96.FromTables2and3wefindthatformostofthemonovalentanddivalentmetals,wherethetwosetsofexperimentallyderiveddataareclose,therelativedifferencebetweentheexperimentallyderivedandthetheoreticalresultsissmall.TheexceptionishcpBewheretheexperimentallyderivedvaluesdifferconsiderablyamongeachother.However,inthiscasethepresentresultforthe(0001)surfaceisincompleteagreementwith198L.Vitosetal./SurfaceScience411(1998)186–202Table8Surfaceenergiesforthe5fmetalscalculatedbytheFCDmethodintheGGA.Forcomparisonwehaveincludedtheavailableexperimentalresults.Thecalculatedequilibriumlatticeconstantsaarelistedinthesecondcolumn˚))Structure(a(AAcfcc(5.786)fcc(5.188)bct(3.986,c/a=0.82)fcc*(4.784)UNpPufcc*(4.634)fcc*(4.580)fcc*(4.513)Surface(111)(100)(110)(111)(100)(110)(110)(100)(001)(111)(111)(111)(111)FCD(eVatom−1)0.7860.71.0061.0731.2331.7221.82.0752.6381.4241.3671.2521.104FCD(Jm−2)0.8680.7320.6811.4761.4681.4502.9022.5842.6612.3022.3562.2082.0072.000a1.939b,1.900a1.500aExperiment(Jm−2)ThPaaExperimental,Ref.[2].bExperimental,Ref.[1].thefirst-principlesfullpotentialcalculations[24,37].ThecalculatedsurfaceenergyanisotropiesforthespmetalsfromgroupsIIIA–VIAshowninTable6arehigherthanthoseofthemono-anddivalentmetals,especiallyforfccPbwhereaparticularlystrongfacetdependenceisfound.TheanisotropyforPbisstudiedexperimentallyatdifferenttemperaturesbyHeyraudandMetois[3,4]whofindthatatT=473Kc/c#1.06110111whichissubstantiallylowerthanourresultc/c=1.39validatT=0K.Todeterminethe110111surfaceenergyanisotropyHeyraudandMetoisreversedtheso-calledWulffconstruction[55]assumingthesurfaceenergyanisotropytobepro-portionaltothedistancefromthecentertotheperimeterofthecrystallite.However,thisapproachneglectsentropyeffectsandcannotyieldsurfaceenergiesforfacetswhichhavesuchahighanisotropythattheyarenotpresentintheequilib-riumshapeatlowtemperature.ForPb(110)thecalculatedanisotropyof1.39iswellbeyond1.22whichisthelimitfortheexistenceofa(110)facetatzerotemperatureandthereforetheexperimentalvalueof1.06forPb(110)is,atbest,alowerboundary.OurresultsforfccAlareslightlyhigherthanthoseobtainedbyScho¨chlinetal.[39]whoappliedaslabapproachinconjunctionwiththepseudo-potentialmethod.However,Scho¨chlinetal.usedWignerexchangewhichforsimplemetalsgives13–16%lowersurfaceenergies[11]thantheCeperley–Alderexchange-correlationfunctional[28].WenotethatthelatterfunctionalinthecaseofAlgivessurfaceenergiesthatdifferfromtheGGAvaluesbylessthan2%.3.2.ThetransitionmetalsTheFCDsurfaceenergiesofthe3d,4d,and5dtransitionmetalsforanumberoflow-indexsur-facesarepresentedinTables5–7.Inthecalcula-tionsbccFe,hcpCo,andfccNiaretreatedasferromagnets,i.e.withaspin-polarizedexchange-correlationfunctional[27],whileallothermetalsL.Vitosetal./SurfaceScience411(1998)186–202199aretreatedasparamagnets.ForMnweconsideronlytheclosepackedfcc(111)surface.FirstwecompareourFCDresultswiththeotherfullpotentialvaluesandfromTables5–7weobservethatingeneraltheagreementisverygood.However,forTi(0001)andZr(0001)theLAPWcalculationofRef.[40]givessomewhatlowersur-faceenergiesthanthepresentmethodbutthesurfacerelaxations,whichlowersthesurfaceenergycomparedwiththeunrelaxedresults,foundinRef.[40]aresubstantiallylargerthantheexperi-mentalvaluesandthereforetheLAPWsurfaceenergiesmaybetoolow.InthecaseofRu(0001)theonlyavailablepseudopotentialcalculation[46]liststwoverydifferentvaluesforthesurfaceenergy,4.3and3.0Jm−2,obtainedbydifferentbasissets.Theincreaseinthesurfaceenergyofthetrans-itionmetalswithincreasingroughnessmaybequalitativelyunderstoodintermsofthebrokenbondsmodel.Itmayevenbedescribedsemi-quantitativelybymeansofamomentexpansionofthestatedensityaswasdoneinthepioneeringworkbyCyrot-Lackmann[56].AsweshallseeinSection3.3,theinteractionsinatransitionmetalarequiteaccuratelydescribedbyonlypairwiseinteratomicpotentials[57,58]andthereforethemaincontributiontothesurfaceenergiesofthesemetalscomesfromthebrokenbondsatthesemi-infinitesurface.Sincethenumberofsuchbrokenbondsincreasesasthesurfacebecomesmoreopenweexpectasimilartrendinthesurfaceenergies.Thistrendisindeedexhibitedbythepresentresults,whenexpressedineVatom−1,inperfectagreementwithpreviousstudies[5,6].Owingtoasimultaneousincreaseinsurfacearea,however,thisbehaviorcannotbeseeninthesurfaceenergiesexpressedinJm−2.Intheearlyhcptransitionmetalsthesurfaceenergyexhibitsaweakorientationdependenceandfortheseelementswefindthatthe(101:0)facetsaremorestablethanthemostclosepacked(0001)facets.Furthermore,thesurfaceenergiesofthe(101:0)facetsofthehcpmetalsfromthemiddleoftheBseriesofthe(101:areusually15–30%largerthanthose0)facets.Hence,forthesemetalsthe(101:0)facetAshouldnotbeobservedexperimen-tally,orBonlyinaverysmallfractioncomparedwiththe(101:0)ofthecubictransitionAand(0001)facets.TheanisotropymetalswillbeanalyzedinSection4.3.3.ThelightactinidesTheFCDsurfaceenergiesofthelightactinidesforafewlow-indexsurfacesarepresentedinTable8.Sincethea-structureofUandNpisorthorhombicwith4and8atomsperunitcell,respectively,andthatofPuismonoclinicwith16atomsperunitcellweconsiderhereonlytheclosepackedfcc(111)surfaces.Forthe5fmetalstheagreementbetweentheexperimentallyderivedandthetheoreticalvaluesisverygood.Theonlypriorfirst-principlescalcula-tionoftheclosepackedfcc(111)surfacesofthelightactinideshasbeenperformedbyKolla´retal.[7]usinganearlierversionoftheFCDmethod.ThepresentsurfaceenergiesofU,Np,andPuarelowerthanthoseofRef.[7]asaresultoftheneglectofthehigherlchargedistributioninthepreviouswork.WenotethattheeffectoftheGGAforthesesurfaceenergiesisabout5%comparedwiththeLDAvalues.TheanisotropiesofAcandThareveryclosetothosefoundintheearlytransitionmetals.InthecaseofbctPa,wherethefelectronsalreadyplayanimportantrole,themostclosepacked(110)facetappearstobetheleaststablefacet.4.TheequilibriumshapeofthecrystalsAsanexampleoftheapplicationofthedatabaseweusethefirst-principlessurfaceenergiestodeter-minetheequilibriumshapeofcrystalsofnanome-tersize.Thisistheshapethatminimizesthetotalsurfaceenergyatconstantvolume,andatT=0Kitiscompletelydeterminedbytheanisotropyofthesurfaceenergy[57].Hence,inthecaseofweakanisotropytheequilibriumcrystalshapeisasphere,whileinthecaseofstronganisotropyitmaybeacomplicatedpolyhedron.Theequilibriumshapemaybedeterminedbytheso-calledWulffconstruction[55],whichassumesacompleteknowledgeoftheorientationdependenceofthesurfaceenergyc(n).Inthe200L.Vitosetal./SurfaceScience411(1998)186–202presentapplicationoftheWulffconstructionforthelow-indexsurfacesweusethefirst-principlesresultsfromtheTables4–7,andforthehigh-indexsurfacesweuseaclusterexpansionofthetotalenergybasedonthelow-indexfirst-principlesresults.Inaclusterexpansiononeexpressesthetotalenergyofagivencrystalasalinearcombina-tionofone-site,two-site,andhigherinteratomicpotentials[58].Itturnsoutthatfortransitionmetalsurfacesoneneedsonlytoincludepairwiseinteractionstodescribethehigher-indexsurfaceswithareasonableaccuracy[57].WethereforeapplytheexpansionNsc#∑nV(2)(2)ssswherenisthenumberofatomsinthecoordina-stionshells,V(2)thepairpotentialsofthisshell,sandNthenumberofshellsconsideredinthesexpansion.InTable9wecompareforanumberoffccandbccmetalstheresultsobtainedbytheclusterexpansionwiththecorrespondingfirst-principlesresults.Inthesecalculationsweincludetwonearestneighborinteractionsforthefcctransitionmetals,i.e.Nfcc=2,andfournearestneighborinteractionssforthebcctransitionmetals,i.e.Nbcc=4,andfindsthattherelativeerrorinthesurfaceenergiesofthenexthigher-indexsurfacesobtainedusingtheclusterexpansionisapproximately2.3%.Thisrela-tivehighaccuracyoftheclusterexpansionobservedinthetransitionmetalsmaybebecausethecoefficientsofsometwo-andmulti-siteinter-actionsareproportionaltothecoefficientsofthenearestneighourpairwiseinteractions,andthere-foreimplicitlyincludedinourexpansion.The12.1%relativeerrorforPbshowsthatforsimplemetalsthezeroth-orderorvolumeterm[58],neglectedinEq.(2),shouldbetakenintoaccount.ThepolarplotsofthesurfaceenergiesofFe,Cu,Mo,Ta,PtandPbareshowninFig.4.ThealmostsphericalshapeofabccTaclusterreflectstheweakanisotropyinthismetal.Incontrast,bccFig.4.ThecalculatedequilibriumshapeofbccTaandMoclustersinthe(001)plane,offccPtandCuclustersinthe(110)plane,andofbccFeandfccPbclustersinthe(001)and(110)planes,respectively.Thedashedlinesindicatethedirectionsforwhichfirst-principlescalculationswereperformed.Thethinlinesdenotetheresultsoftheclusterexpansionandtheheavylinesthetheoreticalequilibriumcrystalshapes.Table9Comparisonofthesurfaceenergies(ineVatom−1units)calculatedusingtheFCDmethodandtheclusterexpansion,Eq.(2)MethodFCDClusterexpansionPercentageerrorPt2.0092.0471.9Cu1.3231.3773.9Pb0.5130.45812.1Ta4.2014.1361.6Mo4.0684.1271.4Fe2.6942.6133.1L.Vitosetal./SurfaceScience411(1998)186–202201Moexhibitsastrongeranisotropyandformsinanearlycubicshapewithtruncatedcorners.TheequilibriumshapeofPtincludesonlythe(111)and(100)facets.OwingtotherelativelowvalueofcinCuasmallfractionofthe(110)facetisformed.110TheparticularsquareshapeofbccFeclustersisaconsequenceoftheirferromagneticgroundstate.Thusthemagneticcontributiontothesurfaceenergy[10]ofthe(100)Fesurfacelowersthesurfaceenergyofthisfacetrelativetothatofthemostclosepacked(110)surface.Asaresult,intheequilibriumshapetheareaofanindividual(100)facetishigherthanthatofa(110)facet.Wenotethatinaparamagneticcalcu-lationtheanisotropyisreversedandasaresulttheequilibriumshapecorrespondstothatshowninFig.4butrotatedby45°.InrecentTEMexperiments[59]onefindsashapeandorientationoffacetsincompleteagreementwiththepredictioninFig.4.AsalastexampleweshowtheequilibriumshapeoffccPbwhichonaccountofthestronganisot-ropyformsinaslightlydistortedhexagonalshapewhereonlythe(111)and(001)facetsarepresent.Averysimilarequilibriumshapewithoutthe(110)facetwasobservedexperimentallybyHeyraudandMetois[3,4]atTabove473K,cf.thediscussioninSection3.1.5.ConclusionsWehavecalculatedadatabaseoflowindexsurfaceenergiesfor60metalswhichmaybeusedasastartingpointfortheunderstandingofawiderangeofsurfacephenomenaincludingfaceting,roughening,crystalgrowth,surfacesegregation,andtheequilibriumshapeofmezoscopiccrystals.Wehavetakengreatcareinmaintainingahighdegreeofaccuracyandconsistencyinourcompu-tationalproceduresminimizingpossiblenumericalsourcesoferror.Thecomparisonwithavailablefullpotentialcalculationsshowthatourdatabaseisaccurate.ThecomparisonwithsurfaceenergiesderivedfromexperimentsshowthattheGGAyieldsresultwhicharemarginallybetterthantheLDA.Finally,asanexamplethedatabaseisusedtopredicttheequilibriumshapesofnano-crystalsofafewselectedmetallicelements.AcknowledgementsTheCenterforAtomic-scaleMaterialsPhysicsissponsoredbytheDanishNationalResearchFoundation.PartofthisworkwassupportedbytheresearchprojectOTKA016740and23390oftheHungarianScientificResearchFund.References[1]W.R.Tyson,W.A.Miller,Surf.Sci.62(1977)267.[2]F.R.deBoer,R.Boom,W.C.M.Mattens,A.R.Miedema,A.K.Niessen,CohesioninMetals,North-Holland,Amsterdam,1988.[3]J.C.Heyraud,J.J.Metois,Surf.Sci.128(1983)334.[4]J.C.Heyraud,J.J.Metois,Surf.Sci.177(1986)213.[5]M.Methfessel,D.Henning,M.Scheffler,Phys.Rev.B46(1992)4816.[6]H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