A neural network potential for searching the atomic structures of pure and mixed nanoparticles. Application to ZnMg nanoalloys with an eye on their anticorrosive properties

ContentslistsavailableatScienceDirect

Acta Materialia

journalhomepage:www.elsevier.com/locate/actamat

Full length article

A neural network potential for searching the atomic structures of pure and mixed nanoparticles. Application to ZnMg nanoalloys with an eye on their anticorrosive properties

P. Álvarez-Zapatero

^∗

, A. Vega, A. Aguado

Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, E–47011 Valladolid, Spain

a rt i c l e i nf o

Article history:

Received 2 March 2021 Revised 13 September 2021 Accepted 17 September 2021 Available online 24 September 2021 Keywords:

Atomistic simulation Artificial neural networks Density functional theory Magnesium alloy Corrosion

a b s t r a c t

The accurate description of the potential energy landscape of moderate-sized nanoparticles is a formidabletask,butofparamountimportanceifoneaimstocharacterize,inarealisticway,theirphys- icalandchemical properties.Wepresent hereaNeuralNetworkpotentialabletopredictstructuresof pureandmixednanoparticleswithanerrorinenergyandforcesoftheorderofchemicalaccuracyas comparedwiththevaluesprovidedbythetheoreticalmethodusedinthetrainingprocess,inourcase the densityfunctional theory.The neuralnetworkisintegratedintoabasin-hopping algorithmwhich dynamicallyfeedsthe trainingprocess.Themain ingredientsofthe neuralnetworkalgorithmas well astheprotocolusedforitsimplementationandtrainingaredetailed,withparticularemphasisonthose aspectsthatmakeitsoefficientandtransferable.Asafirsttest,wehaveappliedittothedetermination oftheglobalminimumstructuresofZnMgnanoalloyswithupto52atomsand stoichiometriescorre- spondingtoMgZn₂andMg₂Zn₁₁,ofspecialinterestinthecontextofanticorrosivecoatings.Wepresent and discussthestructural properties,chemical order,stabilityand pertinent electronicindicators, and weextractsomeconclusionsonfundamentalaspectsthatmaybeattherootsofthegoodperformance ofZnMgnanoalloysasprotectivecoatings.Finally,wecommentonthestepforwardthatthepresented machinelearningapproachconstitutes,bothinthefactthatitallowstoaccuratelyexplorethepotential energysurfaceofsystemsthatothermethodologiescannot,andthatitopensnewprospectsforavariety ofproblemsinMaterialsScience.

© 2021TheAuthor(s).PublishedbyElsevierLtdonbehalfofActaMaterialiaInc.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Structure determinationinsmallnanoparticlesremains asone ofthefundamentalproblemsinclusterresearch.Onone hand,an accurateknowledgeofaclusterstructureisaprerequisiteforade- tailed understandingofits severalphysico-chemicalproperties,of broadinterestinmanynanotechnologyapplications; ontheother hand,therearenodirectexperimentalprobesofstructureforfree- standingclusters,onlyindirectprobessuchasphotoemissionspec- tra,vibrationalspectra,electrondiffractionpatterns,etc.[1–5],that havetobe comparedtotheoreticalpredictions beforeobtaininga definitivestructuralassessment[6,7].Tomakethingsworse,athe- oretical determinationoftheglobalminimum(GM)structureofa

∗Corresponding author.

E-mail address: [email protected] (P. Álvarez-Zapatero).

cluster requiresan exhaustivecomputational samplingofthe po- tentialenergysurface.Althoughseveralglobaloptimizationmeth- ods, such as genetic [8,9] or Basin Hopping [10–13] algorithms, have been proposed to efficiently tackle this problem, they tend tobecomputationallyimpracticaltosampletheabinitiopotential energysurfaceofallbutthesmallestnanoparticles.Awell estab- lishedpracticalsolutiontothisproblemconsistsofperformingan exhaustive samplingof the approximate potential energy surface providedby anempirical potential(EP) toquicklydeterminetrial structures that are then reoptimized at a first-principles density functionaltheory(DFT)level,andhascometobeknownasEP-DFT approach[14].It introduces newproblems,however:inorderfor thisapproachtobe successful, theEP modelmustprovidea rea- sonably accurate andtransferable account of atomicinteractions.

In recentyears,neural network (NN) potentials haveemerged as a particulartype ofempiricalpotentials whichmayreach an un- precedentednumericalaccuracy,socombiningglobaloptimization

https://doi.org/10.1016/j.actamat.2021.117341

1359-6454/© 2021 The Author(s). Published by Elsevier Ltd on behalf of Acta Materialia Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

withNNmethodsmaybeanoptimalsolutionforthestructurede- terminationproblemin clusters,takingintoaccount thecapabili- tiesofpresentlyavailablecomputers[15–21].

Machine Learningtoolsfallunderseveralcategoriesdepending onthetype ofproblemtheyare intendedtotackleandhowthey are implemented. In thiswork, we focus onthe so calledsuper- vised learning techniques, in which the computer is fed with a set of inputsand the desiredoutputs, so that it can find a logi- cal rule that connectsboth. Inpractical terms,the computerwill findthefunctionalrelationshipbetweenthenumericalinputsand targets through a regression analysis, thus obtaining a continu- ousfunction.Wehavechosentheneuralnetworkapproachasthe particularmethodtoperformtherequiredregression.NeuralNet- worksarecomputationalalgorithmsinspiredinthebiologicalneu- ral networks [22]. They are formed by an ensemble of intercon- nectedunits(ornodes)calledartificialneurons,eachofwhichre- ceivesasignalthatpropagatestothesubsequentneuronsthrough a nonlinearactivation function.AllNeuralNetworkshaveacom- mon general structure: an input layer of nodeswhere the signal iscreated,followedbyoneormorelayersofnodes(calledhidden layers)throughwhichthe signalevolvesinanon-linearway,and lastly one output layer where the signal dies providing the user with a result. We will be concerned withthe simplestNN type:

the feed-forward NeuralNetwork[23–28],wherethe information evolves onlyin one direction, fromtheinput layer to theoutput layerpassingthrougheachhiddenlayeronce.Eachnodeinagiven layerisconnectedtoallnodesofthesubsequentlayerviaanon- linearmatrixrelationship,whichmustbeoptimizedtogive accu- rateoutput dataduringa trainingstage.In ourcase, thetraining involvesfittingabinitiodataincludingbindingenergiesandatomic forcescalculatedonalargeanddiversedatapoolofclusterstruc- tures.Despiteitssimplicity,theNNenclosesanoutstandingpoten- tiality.AccordingtotheUniversalApproximationTheorem[29,30], anycontinuousmultidimensionalfunctioncanbeapproximatedto anydesireddegree ofaccuracyby increasingthe numberofneu- ronsinasinglehiddenlayerofafeed-forward neuralnetwork.In practice, theaccuracyof theNNininterpolating thedatais nev- ertheless limited by the finite size of the input data set and by thephysicalappropriatenessoftheinputdescriptorschosenbythe user.

The structural andelectronic propertiesof metallic nanoparti- cles in thesize rangeup to severalhundred atoms generallyex- hibita markednon-monotonousdependencewithsize, toalarge extent duetoquantum-confinementeffects. Thisfact complicates thetrainingprocess,sinceunveilingsuchcomplexsize-dependent behaviorsisdifficultwithoutanexplicitrepresentationoftheelec- tronicdegreesoffreedom.Thus,developingaNeuralNetworkpo- tentialtoaccuratelydescribethepotentialenergysurfaceofsmall nanostructures ismore challengingthanfortheir bulk crystalline counterparts, where the absence of such non-monotonous elec- tronic and geometric behaviors plus the dramatic reduction of structuralrelaxationpossibilities,allowsforasimplerandsuccess- ful training.So far,considerableefforts havebeendevoted tode- velopNNtechniquesforbulksystems[31–35]butnotasmuchfor smallnanostructures[36,37].Besides,inmostcaseswherenanos- tructures are the goal, the employed approaches in the training stage extrapolatefromthebulklimit[38,39].Onthecontrary,our NeuralNetworkpotentialimplementationisspecificallytailoredto train and test finite nanostructures comprising a wide range of sizes and chemical bondings, for both homoatomic clusters and nano-alloyswithupto3differentchemicalelements.

Inthiswork,wepresentbothourcomputerimplementationof the neural network approach for clusters, and its particular ap- plication to Zn-Mg nanoalloys withup to 52 atoms. Specifically, we focusontwocompositions,MgZn₂ andMg₂Zn₁₁,becausethey havebeenfoundtobeoptimalforuseassacrificialcorrosionpro-

tective coatings [40–42]. We have previously reported the struc- turesofsmallZnxMg_20−xnanoalloysinthewholerangeofcompo- sitions[43],aswell asthe structuresofequiatomic nanoalloysas a functionof size[44], andanalyzedtheir electronic structure in ordertoprovidesome preliminaryclues aboutthe enhancedcor- rosionprotectioncapabilitiesofthisalloy.Weneedtogetaccurate structuralmodels ofMgZn₂ andMg₂Zn₁₁ beforeembarkingintoa dedicatedstudyoftheirreactivitywithexternalre-agentssuchas oxygen,chlorine orwater,thisbeingthemain motivationforthe presentstudy.

2. Theneuralnetworkapproachandprotocolforthestructural search

TheuseofNeuralNetworkscanpotentially overcomethelim- itations and shortcomings of usual empirical potentials. When properlytrained,NeuralNetworkpotentials canbe used toaccu- rately describe atomic interactions inarbitrary materials, sothey becomeanidealalternativeinthosesystemswhereavailable em- piricalpotentialsare not ableto capturetheessential features of thetrue potential energysurface. Theirmain advantagerelies on the flexibility offered by the machine learning technique: while classical interatomic potentialshave a limitednumber of param- eters tobe tuned on a definite simplefunctional expression that itself does not account for all of the important geometrical and electroniceffects,intheneuralnetworkapproachonecan,inprin- ciple,fiteveryfunctionalformdescribingtherealpotentialenergy surface[29,30].In order to take full advantage of thisflexibility, wehavewrittenaNeuralNetworkFortran90codespecificallytai- lored to deal with cluster systems. As detailed in the following, the codeis fed withseparate trainingandtesting datasets gen- erated by an external code. Preferably, andfor thesake ofaccu- racy, thosedata sets arehere generatedata first-principleslevel oftheory,althoughotheroptionswouldofcoursebepossible.The code produces as output a NN potential providing both energies andanalyticatomicforces,thatcanbelaterusedbyotherexternal codessuchasglobaloptimizationormoleculardynamicsroutines.

Atpresent,itcantrainbothhomoatomicandheteroatomicnanos- tructuresofarbitrarysizeandwithuptothreedifferentchemical species,though inthissection we will describeits particular ap- plicationtoZn-Mgbinarynanoalloys.

2.1. Inputandoutputlayers.Structuraldescriptors

OurparticularNNimplementationforclustersisbased onthe Behler-Parrinellomethod[45–47].Inthismethod,thetotalcluster energyiswrittenasasumofatomiccontributions:

E=

N

i

E_i, (1)

whereN isthetotalnumberofatomsinthecluster.Eachatomic energyforms the output layer of a separate feed-forward Neural Networkdealingwiththat particular atom.The energyof atomi isexpressedintermsofitslocalatomicenvironment,describedby adequatesymmetryfunctionsthatformtheinputlayerofthecor- respondingneuralnetwork.Althougheachatomisprocessedsepa- rately,atomsofthesamekindaredescribedbythesameNN.This way,onehastotrainasinglefeed-forwardNeuralNetworkforev- ery chemical speciespresent inthe system, whose separate out- puts,E_i,areauxiliaryquantitiesusedtorecoverthetotalenergyof thesystem.

The additivity assumption implied by Eq.(1) is to be consid- ered an approximation (in the sense that it can not be demon- stratedstartingfromfundamentalquantum-mechanicallawssuch asSchrödingerequation),whoseaccuracymaybejudgedaposteri- oribythequalityofthetrainingfit.Wewillshowbelowthatitis

accurate enoughforthemetallic nanoalloysdealtwithinthispa- per.Its mostimportantcomputational advantageis that itallows to generatea singleneuralnetwork potential todescribe nanoal- loys of arbitrarysize andcomposition [45], asthe NNis dealing only withone particular atom at agiven time. Machine learning modelsavoidingtheadditivityassumptionandbasedonglobalde- scriptors[48–50]needtobefittedseparatelyforeachcurrentsys- temsizeorcomposition,whichweconsideranundesirablefeature asourgoalistostudyabroadrangeofsizesorcompositions.

Choosinganappropriatesetofstructuraldescriptorsforthein- putlayerisoneofthekeyfactorstosuccessfullytraintheNNand obtainanaccuratepotential.Thewisertheuserisinprovidingthe rightphysical descriptors(thosethat arestatisticallydominantin determiningtheclusterenergy),thefasterthetrainingprocesswill be, thelower theirreducibleerrorofthefit,andfewer inputde- scriptorswillbeneeded.Inourwork,thesymmetryfunctionsare separatelytailoredforeachchemicalelement,andareconstructed taking into account the local atomic andchemical environments of theatom we are describing(an atom-centredapproach) up to some cutoff radius. So in first place, we have to define a cutoff function fcaroundatomi,whichdefinesthesizeofitsatomicen- vironmentandtheradialextensionofthesymmetryfunctions:

fc

(

^ri j

)

=



₁

2



cos

π

^ri j

rc



+1



, ri j≤ rc

0, ri j>rc,

(2)

which smoothly decays to zero in value and slope at the cutoff radius rc.The userhas complete freedom inchoosing therc val- ues,andcanjudgetheappropriatenessofthechosencutoffsbythe quality ofthefit.Inparticular,forfinitenanoscalesystemswhich areourtargetinthispaper,thecutoffscanalwaysbechosenlong enoughsoastoprovidestructuralinformationaboutthecomplete set of atoms in the environment of a reference atom i. In bulk systemswithsignificantlong-rangeinteractions (electrostatic,van der Waals,etc.), however,theenergyofanatom iisnotexpected to depend only onits local environment up to a predefinedcut- off distance,so our NNenergy expresion shouldbe complemented with separate long-range energy terms [51]. We emphasize that ourcode,intheversionpresentedinthispaper,isspecificallytai- loredtodealwithfinite-sizeatomicsystems.

To describe theradial arrangement ofatoms inthe surround- ingsofatomi,radialsymmetry functionsaredefinedasasumof productsofGaussianstimesthecutoff function:

G^r_i=

j

e^{−η(rs−ri j)2}fc

(

^ri j

)

, (3)

where

η

^{isawidthparameterthatdeterminestheradialextension,}

andrsashiftingparameterthatdisplacestheGaussianstoimprove sensitivityatspecificradii.Forhomoatomicclusters,thesumruns overalltheneighborswithin thecutoff spherearoundatomiand theG^r_i functionsincorporatepurelygeometricalinformation.Inthe case of binarynanoalloys (formed by A andB chemical species), separateradialfunctionsaredefinedforA− A,A− BandB− Bdis- tancedistributionsinordertoincorporatechemicalorderinginfor- mationaswell.

Describingjustthe radial distributionofthe atomsisnot suf- ficientto obtainasuitable fingerprintofthe atomicenvironment.

Adescriptionoftheangulardistributionoftheneighborsofatom i is accomplished by employing the following angular symmetry functions:

G^θ_i =2^{(1−ζ )}

j,k=j

(

¹+

λ

^cos

θ

i jk

)

^{ζe−η(r2i j+r2ik+r2jk)f}c

(

^ri j

)

^fc

(

^rik

)

^fc

(

^rjk

)

,

(4) where

θ

i jk istheangleconformedbytheatomictriplet(i;j;k),

η

determines againthewidthofthegaussianfunctions,and

λ

=±1

isusedtoinverttheshapeofthecosinefunctionforanimproved sensitivity at different values of

θ

i jk. When dealing with binary nanoalloys,separateangularfunctionsareemployedtodistinguish thechemicalnatureofthetwoneighborsofatomiineachtriplet (i; j; k). Therefore, there are three different versions of each G^θ_i function,asthe(j;k)neighborsofagiventripletcanbeofAA,AB orBBtypes.

All the parameters appearing in these functions, as well as thetotal numberof symmetryfunctions employed,must be cus- tomized beforehand to accurately characterize the cluster struc- tures.Usingtoofewofthesefunctionswouldresultinan incom- pletedescription ofthestructuralandchemicalenvironmentand, thus, to a poorrelationship betweendescriptors andtargets. On theotherhand,thesetofsymmetryfunctionsshouldalsobekept assmallaspossibletoincreasethecomputationalefficiencyofthe trainingandtestingstages.Oneshouldinparticularavoidthatdif- ferentsymmetryfunctionsarelinearlyrelated.

Itis interesting tocompare thestructuralinformation content of our localdescriptors to that of global descriptors such as the Coulombmatrix[48–50],whichessentially compiles thepairdis- tances r_{i j} between all atoms in the molecule. Therefore, in the limitof longenoughcutoffs, thestructuralinformationcontained inourradial descriptorsforatomiisessentiallythesameasthat intheithrowoftheCoulombmatrix.However,inglobaldescriptor methods[48–50] it isthewhole Coulombmatrixthat isfed into themachinelearningcodeina singlestep,whileinourlocalap- proachwearefeeding,sotosay,eachrowofthatmatrixatatime.

Theconsequenceisthat,whiletheCoulombmatriximplicitlycon- tainscompleteangularinformation,wehavetoseparatelyaddthat informationthroughthe angularlocaldescriptors,whichparallels theinclusion of3-body termsinacluster expansion.Wedemon- stratebelowthattheaccuracyofourfittingisofthesamequality asthat obtainedwithrecentmethodsbasedontheCoulombma- trixdescriptor[50],evenifourtrainingsetcontainsa widevari- etyofsystemsizesandcompositionsandthusrepresentsamuch morestringenttransferability test.Thisfinding demonstratesthat 2-bodyand3-bodyinteractions,togetherwiththenon-linearityof theNN,sufficetofaithfullyrepresentthemetallicnanoalloyscon- sideredinthispaper.Thisisaveryinterestingconclusionbecause withthe local method we can fit a single potential forarbitrary sizesandcompositions,asexplainedabove.Anotheradvantageof thelocaldescriptorsisthattheyareeasilygeneralizedtodealwith chemicalorderinnanoalloys,bysimplyreplicatingthedescriptors foreachpossiblepairortripletofatomsinthenanoalloys.

2.2. Architectureandtrainingoftheneuralnetwork

Boththenumberofhiddenlayersandthenumberofnodesin eachlayeraresetby theuser,andtogether definethesizeofthe neuralnetwork.Allofthesignal valuesinagivenlayerkarecol- lected inavector array X_k.The signalsevolvefroma givenlayer tothenextoneinafeed-forwardway,accordingtotheequation:

X_k+1=

σ (

^XkW_k,k+1+B_k+1

)

, (5) whereW_k,k+1isthematrixofweightsthatconnectsthenodesbe- tween layers k and k+1, B_k+1 is a biasterm and

σ

^{is the non-}

linearactivationfunctionwhichbuildsupthesignalsofthek+1 layer.Allmatricesandvectorarraysare real-valued,andonlythe signalintheinputlayermustbeprovidedbytheuser.

The activation function employed in our work is the Swish function [52],

σ

(^x)=₁₊^x_e−x, exceptfor the output layer where a linearfunction wasused.Swish belongstothe familyofrectifier functions,whosedistinguishingfeatureistobeunbounded,asop- posedtomanypreviousimplementationswhichreliedonbounded sigmoidfunctions.Wefoundrectifierstoproduceamoreaccurate

fitting to the ab initio data in the training stage (see below), as comparedtosigmoidssuchasthehyperbolictangentfunction(see Table S1 intheESI whichcompares theperformance of different activation functions). Our observation is in line with the known superiority in performance of Swish over sigmoids in regression problems[52–54].

Theoptimalvaluesfortheseveralweightmatricesandbiasar- rays areobtainedbytrainingtheNeuralNetworktomatchtheab initio valuesofenergies andforces calculatedon alarge dataset ofclusterstructures.Thisisdonebyminimizing,inaleast-squares sense, thevalue ofthefollowingcost function(orobjectivefunc- tion):



= 1 2M

M

k



E_k^DFT− E_k N_k



²

+

Nk



l

3 α

_FDFT k,lα − Fk,lα

3N_k



²



, (6)

whereM isthetrainingsetsize(inourcasethenumberofcluster structures includedinthetrainingset), N_kthesizeofclusterk in the datapool, and

^{a scalingfactor (set to0.05) used toobtain}

balanced relative errorsinenergies andforces.The NNenergyof structure k,E_k,isgivenby thesumoftheindividual atomiccon- tributionsE_k,iasstatedbytheBehler-Parrinelloapproach(Eq.(1)).

TheNNforceonatoml ofstructurekisthenobtainedas:

F_k,lα=−

∂

^Ek

∂

^rk,lα =−

Nk



i

∂

^Ek,i

∂

^rk,lα =−

Nk



i Ms



s

∂

^Ek,i

∂

^Gi,s

∂

^Gi,s

∂

^rk,lα, (7) where M_s is the number of symmetry functions associated with atom l ofcluster k, and

α

^{denotes the cartesian direction. Thus,}

theforceonatoml dependson

∂

^Ek,i

∂

^Gi,s,thatisthederivativeofthe output layervalue withrespect tothesymmetry functionsinthe input layer,andon

∂

^Gi,s

∂

^rk,lα,thatisthederivativesofthesymmetry functionswithrespecttothecartesiancoordinates.Bothtermsare obtainedanalytically,whichmakesourcodecomputationallyvery efficient.Full detailsabouttheanalytic forcesareprovidedinthe ESI.Noticethattheatomicforcesarenotexplicitlyprovidedbythe output layeroftheNN(asthey canbeanalyticallyobtainedfrom the atomicenergies intheoutput layer),yetthey areincluded in thecostfunctiontoexplicitlytraintheNNtowardsaccurateforces aswell.

The minimization of the cost function  ^{is afforded by em-} ploying the Nadam algorithm[55,56],a gradientdescent method thatexploitsthefirst-andsecond-ordermomentsofthegradients toimprovecomputationalperformanceandstability,basedonthe reliable Adam [57] algorithm.The gradients of ^{with respectto} theweightsandbiasesappearinginEq.(5),neededbyNadam,are calculatedwiththebackpropagationalgorithm[58].An important technical pointhereis therateatwhich thecost function isup- dated,asitsignificantlyimpactstheefficiencyandaccuracyofthe minimization problem. InNNjargon,the evaluationofthewhole datasetonceiscalledanepoch.Intheso-calledbatchgradientde- scent, the weights and biasesare updated after one epoch, pro- ducing arelativelysmooth errorsurface,butbeingveryslowand requiringa lotofmemoryforlarge datasets.In stochasticgradi- ent descent,one updatesthevariables afterevaluatingeach indi- vidual inthe trainingset.Beingthe fastestupdaterate, it results in anoisygradient, butnoticethat some noiseis not necessarily harmfulsinceithelpsthealgorithminescapingfromshallowlocal minima. A good compromise is the mini-batch gradient descent, whichupdatestheweightsbasedonasmallsubsetofindividuals (32, 64or128are typicalchoicesinparallel computations).After oneepoch,thesesubsets(ormini-batches)arerandomlyreselected fromthe wholedata set.Thiswaythegradients are morerobust comparedtostochasticgradientdescent whilehavingsome white noise.Thisistheparticularimplementationusedinourwork,with

amini-batchsizeof64individuals.Other technicaldetailsareex- plainedintheESI.

A final important point concerning the training stage is the stoppingcriterion, whichshouldtake intoconsideration thebias- variancetradeoff problem [59,60].Ourcoderandomlydivides the datasetprovidedbytheuserintotwoparts:90%ofthestructures goto thetrainingset onwhich thecost function ^isevaluated;

theremaining 10%isused asa testingset,containingdatawhich arenotexplicitlyusedtotraintheNN.Thetrainingprocessiscon- tinued until the cost function erroron the test set is minimal, a procedureknownasearlystoppingmethod.Thisstrategyaims to minimize both the bias andvariance errors, so that both under- andover-fittingproblemsareavoided.Thefinalerrorofthistrain- ingstrategywillmostlybeofanirreducibletype,i.e.onethatcan not be significantly reduced by increasing the complexity of the NNofbyrunningtheminimization foralongertime. Infact,the irreducibleerrorfaithfullyrepresentstheinherentnoiseofthefit- ting,due for example to a sparse data set or, more importantly, to the non-existenceof a deterministic relationship between in- puts andtargets.It can onlybe reduced by theuser (notby the machine)byprovidingmoreexplicitdataormoreappropriate de- scriptorsintheinputlayer.

2.3. Symmetryissues

In a recent report, Chmiela et al. [49] have developed the sGDMLmodel,agradient-domainmachinelearningforce-fieldthat makesexplicituseofsymmetriesinordertoreducethesizeofthe trainingsetandalsotoimprovetheaccuracyandefficiencyofcal- culations performedonhigh-symmetry molecules.In ourcurrent implementationoftheneuralnetwork, wehavenotexplicitlyex- ploitedsymmetryinordertoimprovetheefficiencyoftheenergy calculations.Itjustturnsoutthatthegreatmajorityofgeometries sampled during a global optimization run on Zn-Mg nanoalloys have no rotational symmetries (i.e. most of them have C₁ point group),socomputational savingswouldnotbesubstantialforthe particulartargetsysteminthispaper.Formoresymmetricsystems, the point-group symmetry could be exploitedby feeding the in- putlayeroftheNNwithjustoneatomfromeachsymmetryorbit, thusreducingtheexplicitnumberofatomicenergycalculations.In effect,twoatomsequivalentbysymmetrymusthaveequalcontri- butionstothetotalenergy,andtheirforcevectorsmustberelated byagroupoperation.Thisimprovementmightbeimplementedin futureversionsifconsideredworthwhile.

Notwithstanding this computational efficiency issue, our code satisfies many symmetry properties by construction, and so ac- counts for all relevant effects associated to symmetry, as the sGDML model does. For example, homogeneity and isotropy of spaceimply that the total energyof an isolated system must be invariant against global translations and rotations of the system.

Thisisensured by construction inourcode,asthe localdescrip- tors depend only on pair distances ortriplet angles, both quan- titiesbeing independent onthe coordinate system. Similarly, our NNforce-fieldisconservativebyconstruction,i.e.itwillsatisfythe conservation ofthe total energy ofan isolated systemduring its time evolutionifusedinMDsimulations, becausetheexactcon- sistency relations betweenenergies andforces isanalytically en- sured(seeprevious section).Finally,thephysicaleffects ofpoint- groupsymmetrywillbecapturedalsobyconstructionofourlocal descriptors:ineffect,twoatomsthatare equivalentby symmetry willhaveexactlythesamesetoflocaldescriptors,andsothecode willassign exactly thesame atomicenergyto both ofthem,and forceswhicharerelatedbythecorrespondinggroupoperation.

The authors of the sGDML model take advantage of permutation-inversion symmetries to reduce the size of the data set and thus optimize the cost of the training process. In

Fig. 1. Flow diagram summarizing our training protocol. See the main text for full details about each different step of the process.

effect,permutationsofidenticalatomsdefinephysicallyequivalent but different local minima on the PES, andthere is no point in explicitlysamplingtwo equivalentbasins. Similarityofisomersis quantified through their euclidean distance, properly minimized with respect topermutation-inversion operations, whichrequires the use of bi-partite and multi-partitematching techniques [49]. We havealsoconsidered similarityofisomers whenbuildingour trainingdataset,withthesamegoalofdefiningatrainingsetthat doesnotcontainduplicateortoosimilarstructures.Wehavedone this witha less elaborate, home-built,similarity indicator,which isinvariant topermutation-inversionoperationsthusavoidingthe complications inherent to the euclidean distance measure. This similarityindicatorisdescribedintheESI.

Aflowdiagramsummarizingthemajormilestonesofourtrain- ingprotocolisofferedinFig.1.

2.4. ApplicationtoZn-Mgnanoalloys

We employ atotal of70symmetry functions(bothradial and angular) intheinput layers ofthetwodifferentNeuralNetworks that deal with Mg and Zn atoms, respectively (details of these functionsareprovidedintheESI).Thisnumberofsymmetryfunc- tionswasoptimizedthroughacarefulcorrelationanalysisinorder to reduce the number of redundant descriptors, that is, we dis- carded thosewhichinourinitialdescriptorsethadalargecorre- lationwithdescriptorsalreadyconsidered.

WehavegeneratedtwodifferentNNpotentials,inordertopro- videalargeramountofplausiblelow-energystructures,i.e.toen- hancestructuraldiversity.BothNeuralNetworkshavethreehidden layers,butdifferentarchitectures:70× 40× 20× 10× 1(amount- ingto3881adjustableparameters)inthefirstone,and70× 45× 30× 20× 1(5216parameters)inthesecondone.Thosearchitec- tureswere selectedsincethey providesufficientaccuracywithout resultinginover-fitting.

WehaveimplementedourNNpotentialsintoamoduleofthe freelyavailable GMINcode[61].We usethisprogramtodynami- callyenlargethesizeofourdatasetduringaniterativeandauto- matedtrainingprocessinvolvingBasinHopping(BH)runs[10,11]. In each iteration, the currently fittedNN force field samples the potential energysurface ofthe nanoalloythrough BH runs, com- prising 35000steps andincluding bothrandom andswapmoves (thosethat exchange thechemical identity ofa randomlychosen Zn-Mg pair of atoms). We then feed the data set with the new structures that the NN predicts, which involves single-pointDFT calculationsonthosenewstructures, andstart anewtrainingon the enlarged dataset. During the initial iterations, i.e. when the NNisnot yetsufficientlytrained,we foundthatiteasily tendsto explore areas ofthe potential energy surfacenot covered by the initial data set. In other words, the NN extrapolates, thus failing to provide reliable structures. With relatively short BH runs one can easilycheck the qualityof thecurrentNNpotential andfind newrelevant structures tofeed thedata set.This loopis contin- ueduntilextrapolationisnolongerobservedandtheNNproduces reasonablestructures. Notice that globaloptimizationrunsare to bepreferredoverother optionssuchasshortmoleculardynamics runs, asourultimategoalisto usetheNNforcefield inaglobal optimization aimed to locate the global minimum structure of a cluster ornanoalloy,andforthat we needglobalaccuracy inthe wholePES,ratherthan extensivesamplingoflocalregions ofthe PES.

The initial dataset wasbuilt upon the structures provided in previous works on Zn-Mgnanoalloys [43,44].It was dynamically enlargedasexplainedaboveuptoafinalsizeof49000nanoparti- clesin thesize rangebetween4and100 atoms, containingboth pure Zn and Mg, and (mostly) Zn-Mg nanoalloys of all possible compositionratiosandwithdifferentchemicalorderings.Thefinal NeuralNetworkpotentialswerefittedto reproducebothenergies andforcesonthishuge,DFT-quality,dataset.

With the two resulting NN potentials, we performed a Basin Hoppingglobaloptimizationsearch forZn-Mgnanoalloysofsizes rangingfrom6to52atomsandstoichiometriescorresponding to Zn₂Mg and Zn₁₁Mg₂. We performed 200000 BH steps for each nanoalloyforeachofthetwodifferentpotentials.BHmovescom- prisedeitherarandomchangeoftheatomiccoordinatesoraswap move, the latter amounting to a 20% of the total BH moves. If no lower energystructure isidentified during10000consecutive steps, the cluster structure is re-seeded to a random one in or- dertoenhancethesampling.Atotalamountofaround100struc- tures (the most stableones) wassaved fromboth potentials af- ter removing duplicates inthe twolists. The resultinglistof 100 structures wasthen relaxedatthe Kohn-Sham densityfunctional theory (KS-DFT) level by using the SIESTA code [62]. Exchange- correlationeffectsweretreatedwithinthegeneralizedgradientap- proximationofPerdew,BurkeandErnzerhof(PBE)[63],andnorm conservingpseudopotentials[64,65]includingnon-linearcorecor- rections[66]wereusedtorepresentcore-valenceinteractions.The valenceactivespaceincludes3d¹⁰4s² electronsforZn,and3s²for Mg. The size ofthe basis setwas double-zeta plus two polariza- tionorbitals(DZP2).The clusterswere placedinacubicsupercell of30˚Aofside.ThereliabilityofthisSIESTAsetuphasbeendemon- strated in our previous works (the single-point ab initio calcula- tions needed duringthe trainingstage were performedwith the samesetup). The structures were relaxedwithout geometrycon- straintsuntilthe forceoneach atomwassmallerthan0.01 eV/˚A.

Finally,forthebeststructureobtainedafterDFTrelaxation,anex- haustive homotopic search was performed with the Neural Net- work potential in order to better sample the chemical ordering.

ThiswasaccomplishedthroughaBH searchof200000stepsand withonly swap moves. The best configurations found were then relaxedagainwithSIESTA.

Table 1

Mean absolute errors in cohesive energy E coh (in meV/atom) and atomic forces (in eV/ ˚A) on the train and test sets for the two Neu- ral Network potentials.

NN with 3881 parameters NN with 5216 parameters Train set Test set Train set Test set

E coh 10.80 11.73 8.63 9.97

Force 0.14 0.14 0.14 0.15

3. Resultsanddiscussion

3.1. Qualityperformanceoftheneuralnetwork

Table1showsthequalityofthefitofourtwoNeuralNetwork potentials. Specifically, the table displays mean absolute errors (MAE)incohesiveenergy(bindingenergyperatom)andforcesin both trainingand testing sets. The cohesive energy of a nanoal- loywithN atomsisdefinedasE_coh(^N)=(^E(^Mg)^NMg+E(^Zn)^NZn− E_N)/N, where E_N is the energy of the nanoalloy formed by N_Mg andN_Zn MgandZnatoms, whoseatomicenergies areE(^Mg)^and E(^Zn)^,respectively.

The energetic MAE of both NN’s is around 10 meV/atom ≈ 1 kJ/mol,anerrorsizeusuallyconsideredasrepresentativeofchem- ical accuracy. This means that both NN’sreproduce PBE energies within an errorthatis alreadysmaller thantheinherent errorof PBE functional in reproducing experimental cohesive energies of metals [67], which suggeststhe NNpotentials are asaccurate as onemaywishthemtobe(thereisnopointinrequiringaNNfit- ting error significantly smaller than the inherent error of the ab initio referencedata) and, inparticular, justifiesthe additivityas- sumption (Eq. (1)). The MAE of the forces, around 0.15 eV/˚A, is alsoverysmallaswehavecheckedthatforcesofthat magnitude areinduced byatomicdisplacementsofaround0.04 ˚A,sothege- ometriespredictedbytheNNwillbeveryaccurate.Weaddition- allynoticethatthesecondNN,whilecontainingaround35%more adjustable parameters thanthe firstone, improvesthe MAEby a mere1%,soweconcludethatbothNN’saresimilarlyaccurate,and thattheerrorcannolongerbesubstantiallyreducedbyincreasing thecomplexityoftheneuronalconnections.Webelieveitissafeto concludethattheremainingerrorismostlyofanirreducibletype.

Consideringthedynamicaladjustmentofthetrainingsetsizeem- ployedinourwork(seeprevioussection),theirreducibleerrorcan not be significantly improved by increasing thesize of thetrain- ing set. Itcould only be improvedby providing additionalphysi- cal descriptors inthe input layer. Right now, theNN istrying to correlate the energy contributionof a given atom with the local geometric andchemicalenvironments ofthatatom. However,the stability ofmetal clustersis well known to be additionallyinflu- enced byelectronic shelleffects (associatedwiththediscrete na- tureoftheelectroniceigenvaluespectrumoffinitesystems),which aremoredependentonglobalpropertiessuchastheclustershape, thanonlocalatomicdescriptors.Itisplausiblethatincludingsuch globaldescriptorsexplicitlyintheinputlayerwouldfurtherdimin- ish thefittingerror. Nevertheless,ourresultsshow that localde- scriptorssufficeforthisparticularnanoalloy.Asanadditionalaccu- racycheck,wedemonstrateintheESIthatourNeuralNetworkpo- tentialclearlyoutperformstheempiricalCoulomb-correctedGupta potential developed in a previous work [44] in proposing better candidatestructuresforDFTreoptimization.

3.2. PutativeglobalminimumstructuresofZnMgnanoalloyswith nominalcompositionsZn₂MgandZn₁₁Mg₂

The putative GM structures located in this work are shown in Figs. 2, 3 and4. In general, their skeletal geometries coincide

Fig. 2. Putative GM structures and approximate point group symmetries of Zn 2 Mg nanoalloys with N = 6 − 27 atoms. Brown and golden spheres represent Zn and Mg atoms, respectively. For some clusters, we add in brackets the point group of the corresponding homo-atomic cluster in order to better appreciate the symmetry of the atomic skeleton.

Fig. 3. Putative GM structures and approximate point group symmetries of Zn 2 Mg nanoalloys with N = 30 − 51 atoms. Rest of the caption as in Fig. 2 .

Fig. 4. Putative GM structures and approximate point group symmetries of Zn 11 Mg 2

nanoalloys with N = 13 − 52 atoms. Rest of the caption as in Fig. 2 .

with those of the pure Zn atomic clusters [7]. Zn₄Mg₂ is con- formedby2tetrahedralunitssharingoneedge.TheGMstructure ofZn₆Mg₃ isatri-cappedtrigonalprism(TTP),andall nanoalloys withN=12− 15atomsareobtainedbyaddingatomstothatTTP unit.ThefirststructurewithaninternalcoreatomisZn₁₂Mg₆,and isbasedonadistorted13-atomdecahedronwithadatomscapping itssquare facets,thedistortions increasingthecoordinationnum- ber of some adatoms. The GMstructure ofZn₁₄Mg₇ is based on aC₃ twistedpyramidwithadanglingatomattachedtooneofits corners,anexoticstructurealsofound forZn₂₁ [7].Fromthissize on,theGMstructures tendtobemoreamorphous-likeduetothe size-mismatchbetweenthetwoatomicspecies,althoughmanyof them locally display a recognizabletendency towards decahedral packing (with the orientations chosen in the figures, decahedral unitscan be visually appreciated inclusters with30,36, 39and 51 atoms, for example). Notable exceptions to thisgeneral trend are Zn₁₈Mg₉ andZn₃₃Mg₆, which are poly-tetrahedral structures

composed of icosahedral and Frank-Kasper polyhedral units. The number of coreatomsobviously increaseswith cluster size: two coreatomsfirstappearatsizeN=26,andthebiggerclusterscon- sideredherehave8coreatoms.

Nextwedescribethechemicalorderingtrends.Therearethree importantfactors thatconjointly determinethetendencytowards segregationormixinginZn-Mgnanoalloys:(1)thebulk cohesive energyofMgis12%largerthanthatofZn.Assumingthattherel- ative strength of Mg-Mg and Zn-Zn bonds is maintained in the nanoalloys,thisfactorwouldtendtomaximizethenumberofMg- Mg bonds, and so wouldfavor Mg@Zn segregation; (2) however, interatomic distances in bulk Mgare around 30% longer than in bulkZn.Inordertominimizebondstrain,thesmallelementtends tosegregatetotheclustercore,sothisfactoralonewouldfavorthe opposite Zn@Mg segregation. The observation that energetic and steric effectsoppose each other alreadysuggeststhat segregation trends are not clear; (3) finally, charge transfer effects introduce an ionic bondingcomponentthat tends tomaximize the number of Zn-Mg bonds and thus promotes mixing. In orderto quantify the degree of mixing, we haveevaluated the following indicator foreachoftheGMstructures:

p_mix=NAB− N_AB^m

N_AB^M − N_AB^m, (8)

whereN_AB isthenumberofZn-Mgbonds.N^M_ABandN^m_ABarerespec- tively the maximum and minimumvalues that N_AB can take for thatparticularfrozennuclearskeleton.Thesetwolastnumbersare obtainedfromswap-onlyBH runswithoutallowing forstructural relaxation.Theparameterthusdefinedisnormalizedbetweenzero andone,withavalue p_mix=1indicatingthemaximumdegreeof mixing that a given nuclear skeleton allows; the value p_mix=0, onthecontrary,isassociatedwithleft-rightsegregatedstructures, displaying an interface separatingpure zinc frompure Mg sides.

The calculation of this indicator demonstrates that most of the GMstructuresfoundfortheZn₂Mgcomposition areindeedmaxi- mallymixed,andinanycaseallofthemhave p_mix>0.9.Regard- ing the Zn₁₁Mg₂ composition, all GM structures have p_mix>0.8. Thus, concerning chemical order, the dominantbuilding rule for ZnMgnanoalloysistomaximize mixing.Thisisachieved, asseen in the figures, by preferentially placing the minority component Mgatomsinclustersiteswithahighercoordinationnumber,but atthesametimeavoidingasmuchaspossibledirectMg-Mgcon- tacts. More in detail, we have checked that the shell of ZnMg nanoalloys tends to be well triangulated, each surfacesite being connected toeither5, 6or7surfacesites. Withveryfew excep- tions, all of the5-fold sitesare occupied by Znatoms, while Mg atoms occupy the 6-fold and 7-fold sites. This is a strong trend commontoallclustersconsideredinthispaper,althoughissome- timesincompatiblewiththeglobal2:1compositionratio.Forex- ample,wehaveobservedthataMgatommayoccupya5-foldsite in the shell ofZn2Mg nanoalloys whenoccupation of anyofthe remaining6-foldor7-foldsiteswouldimplythecreationofaMg- Mgbond.

The main secondary factor that competes with the maximal mixing rule seems to be steric in nature. In effect, the internal core sites tend to be under a high compressive stress for most oftheatomicpackingsusuallyobservedinsmallclusters, andoc- cupying them with the biggerMg atoms would result in a less denseatomicpackingbecauseofthesizemismatcheffect,thusde- creasingtheclusterstability.Additionally,thecoordinationvolume around internalsitesissmallerforZn-richer compositions.There- fore,thefirstMgcoreatomfortheZn₁₁Mg₂nanoalloysisnotob- served untilsize N=52. Thisis clearlythe reasonwhy Zn₁₁Mg₂ nanoalloys donotachievea p_mix=1value,eveniftheMgatoms consistently occupy well separated surface sites with the maxi- mumpossiblesurfacecoordination:themixingcouldonlyincrease

if more Mg atoms were at core sites. The same trend operates forthe2:1compositionalthoughisrelativelylessimportantthere.

Tosummarize,thereisaslightMg-enrichment(beyondthenom- inalcomposition)ofthesurfaceinZn-Mgnanoalloys,enforced by thesizemismatcheffect,thatcompeteswiththemaximalmixing trend. The two factors together consistently explain the detailed chemicalorderingtrendsinthesenanoalloys.

BulkMgZn₂ crystallizesinone oftheAB₂ Lavesphases,which are notorious by a number of reasons. The three simplest Laves phases, occurring for MgCu₂ (cubic C15 phase), MgZn₂ (hexago- nal C14 phase) and MgNi₂ (hexagonal C36 phase), are tetrahe- drallyclose-packedstructureswheretheintersticesareexclusively tetrahedral. They are all obtained by packing Frank-Kaspercoor- dination polyhedra [68], and differjust in the stacking sequence of those polyhedral units. The coordination number (CN) of the smallerZnatomis12,anditscoordinationpolyhedronisanicosa- hedronformedby6Znand6Mgatoms;CN=16forthelargerMg atom,anditscoordinationpolyhedronisaZ16Frank-Kasperpoly- hedronformedby12Znand4Mgatoms.Giventheimportanceof Lavesphasesinmanymaterialsprocessesaninparticularincorro- sionproblems[68],itisinteresting toanalyzetowhatextentthe smallclusters studiedin thispaper havedeveloped thebulk-like propertiesoftheLavesphases.

Tostartwith,wehaveexplicitlycheckedthat p_mix=1forboth cubicandhexagonalbulkLavesphases,soatleastconcerningthis global tendency towards a maximally mixed chemical ordering, eventhesmallestMgZn₂ nanoalloysarealreadybulk-like.Another similaritybetweennano- andbulk alloysis thelarger totalcoor- dination number of Mg atoms ascompared to Zn atoms, asex- plainedintheprevious paragraphs.Inparticular,thetotalcoordi- nationnumbersofcoreatomsarealreadyclosetobulkvalues:on average,wefindCN=11forZncoreatomsandCN=15forMgcore atoms in most cases, although there are a few examples where thebulk CN valuesare strictlyrecovered. Thevery smallclusters hereconsidered are thus slightly lesscompactly packed ascom- paredtothebulklimit.Wealsoobservethatthefirstcoordination layer around internal atoms tends to contain similar numbers of Zn andMgneighborsaround Zn, buta substantially largernum- berofZn atomsaroundMg. Concerningthecoordinationpolyhe- drasurroundinginternalatoms,thesetendtobedifferenttothose inthe bulk Lavesphase, mainlybecause mostofthe clustersare stillbasedondistorteddecahedralpacking,whichisdifferentand lessdensethan thepoly-tetrahedralpacking realizedinthe bulk.

Nevertheless,at leastforthe special Zn₁₈Mg₉ andZn₃₃Mg₆ clus- ters, which are based on poly-tetrahedral packing,the coordina- tionpolyhedraareindeeddistortedversionsoftheirbulkcounter- parts(seeFig.5),thedistortionsbeingexpectedduetothesurface roundingandrelaxationeffects.Otherdistinguishingstructuralfea- turesofthebulk phasesareofalongerspatialrangeandcannot berealizedinsmallclusters.Forexample,thereisnopossibledis- tinction between C14, C15 and C36 phases as there are not yet enough polyhedral unitsto define a stacking sequence. Similarly, thecompositionallayeringobservedinthebulkishardtoappreci- ateintheseverysmallsystems,althoughatleastMg₁₅Zn₃₀shows clearsignaturesofalternatingstacked Zn/Mgsheets.Insummary, we conclude that there are significant similarities in the short- rangestructuraldescriptorsofclusterandbulksystems.

3.3. ElectronicpropertiesofZn₂Mgnanoalloys

We have evaluated the vertical ionization potential (IP) and electron affinity (EA) ofall GM structures through a −SCFcal- culation,i.e. by explicitlycalculating the cation oranion state at theoptimalgeometryoftheneutralclusterandtakingthecorre- spondingtotalenergydifferences. Wehaveadditionallycomputed thefundamentalgap,definedasE_GAP=(IP-EA),whichistwice the

Fig. 5. The top row compares the coordination polyhedron of the internal Zn atom in Zn 18 Mg ⁹ with a perfect Z12 polyhedron, i.e. an icosahedron. The bottom row shows the corresponding comparison between the coordination polyhedron of the internal Mg atom in Zn 18 Mg 9 and a perfect Z16 Frank-Kasper polyhedron.

chemicalhardness.Thesethreequantitiesaretakenasindicatorsof electronicstability:largegapvaluesidentifythoseclustersthatare more stable against both oxidation and reduction processes and, assuch,displayanenhancedelectronicstability;alsoalargeIPor a low EA are indicativeof a particularlystableconfiguration due to the reluctance to releasing or accepting one electron, respec- tively. According to a simple Jellium picture of delocalised elec- trons[69,70],itistheelectronicstabilitythatdrivesthethermody- namicstabilityofthecluster,andclustersdisplayinganelectronic shellclosingarethusexpectedtobeparticularlystable.Thespher- ical Jellium model predicts these electronic shell closings to oc- curformetallic clusterswithNe=8,20,34,40,58,68− 70,92,· · · electrons. Jellium models allowing for ellipsoidaldeformationsof the confining ionic potential predict additional (sub-shell) clos- ings due to the splitting of angular momentum multiplets in a non-spherical potential. Now, considering that Zn and Mg atoms arebothdivalent,thenumberofdelocalizedelectronsincreasesin steps of 6along the Zn₂Mgseries, andin steps of26 along the Zn₁₁Mg₂ series,sononeoftheNevaluesassociatedwithaspheri- calelectronicshellclosureoccursinourclustersample.Moreover, the non negligible ionic contribution to bonding associated with charge transfer causes Zn-Mgnanoalloys to slightly deviate from apristine jelliumpicture,asshowninourpreviouswork[44].All takentogether,it canmake ithard todiscernmarked features in thesizeevolutionoftheelectronicindicators.

As usual insmallmetalclusters, theIP displaysagloballyde- creasingtrend asafunction ofsize(meanwhile theEAshowsan increasing envelope),whichissteeperforsmallersizes.Theslope oftheseindicatorsallowstoidentifytwodifferentsizerangescon- cerning theelectronic properties.The ionizationpotential, forex- ample, steeply decreases fromaround 6.5 eV to5 eV inthe N= 6− 30 size interval, andthen stabilizes by decreasing ata much slowerpace.Inourpreviousstudiesonpurezincclusters[6,7],we analyzed the evolution of metallicityand found that while most clusterswithN≥ 25− 30alreadydisplay typically metallicbond-

Fig. 6. Electronic stability indicators of Zn 2 Mg nanoalloys. Standardized vertical ionization energy, electron affinity and fundamental gap as a function of cluster size N (lower scale) or the number of electrons N e (upper scale) are shown.

ing,forsmallerclustersmetallicityisstillpoorlydeveloped.Inthis sensetheZn-Mgnanoalloysbehavesimilarlytothepurezincclus- ters. In the regions with a globally steep slope, it is difficult to identifylocalfeaturesinthoseelectronicindicators,soforoptimal visualizationwehavedecidedtoremovetheirglobalsizeevolution by fitting the data to a cubic polynomial function, and subtract- ing from the data series the resulting fit; then, we have shifted the mean value of the trendless data to zero and divided them bythestandarddeviationofthedataset.Theresulting“standard- ized” dataaredimensionlessandareshownforZn₂Mgnanoalloys inFig.6,whiletherawdataareprovidedinFig.S2intheESI.

Althoughnoneofthe clustershasthenumberofelectrons re- quiredforasphericalshellclosing,theresultscanbeshowntoac- commodatequitewelltojelliumexpectationsbyfocusingonthose clustersizesthatbracketanelectronshellclosing.TheEA,tostart with, is consistently higher than average just before (and lower than average justafter) the mainelectronic shellclosings, which explainsallofthesuddendropsinthatcurve.Forexample,theEA isveryhighforZn₁₂Mg₆ andlowforZn₁₄Mg₇,thesetwoclusters bracketing theshell closing occurringforNe=40 electrons. Sim- ilarly, one would expect substantial IP drops after each electron shell closing, butthose drops only occur in clusters witha well developed metallicity, specifically at cluster pairs bracketing the Ne=58,70,92shellclosings.Meanwhile,theIPdropatNe=40is veryweak,anduponcrossingtheNe=20shellclosingtheIPeven increases.Thispeculiarbehaviorhasbeenexplainedinourprevi- ousworksonpurezincclusters[6,7]:itisrelatedtothepresence of low-coordinated adatoms and to the coexistence of insulating andmetallicbondingcontributionswithinasinglecluster,features that obviously depart from a jellium picture of delocalized elec- trons. Zn₁₄Mg₇, for example, displays a low-coordinated adatom attached to the corner ofa 20-atom twisted pyramid. The pyra- miditself is asuperatom with40 valenceelectrons, butthe two additional electrons in Zn₁₄Mg₇ form a lone pairon the adatom ratherthancontributingtothejelliumseaofdelocalizedelectrons [69,70].Thisway,theclusteravoidstheoccupationofveryunsta- bledelocalizedorbitalsandkeepsahighIPvaluebeyondtheelec- tronicshellclosing.

Thefundamentalgapresultsfromthedelicatebalancebetween the IP andEAvalues. The gapshould be locally maximumat an

Fig. 7. Size dependent stabilities of Zn 2 Mg nanoalloys as a function of the total number of atoms N (lower scale) or the total number of electrons N e (upper scale).

The upper graph shows the cohesive energies as standardized dimensionless quan- tities (see Fig. S3 in the ESI for the absolute cohesive energy values). The middle graph shows the energy cost of evaporating (or dissociating) a Zn 2 Mg trimer. Fi- nally, the lower graph displays the second-order energy differences.

electronicshellclosing,butitsbehavioraroundtheshellclosingis notclearinadvance.IntheN≥ 25metallicregion,thedropinthe EA upon crossinga shell closingtends to damp the effectofthe corresponding dropintheIP,sothefundamentalgapiscompara- tively featurelessinthatregion.AroundNe=58,forexample,the dropintheIPisstrongerthanthedropintheEA,soaweakdrop isobservedinthegap;around Ne=92,onthecontrary,thedrop intheEAisstrongersoweobserveanincrease inthegap.Inthe smallsizeregime withN≤ 25,theelectronlocalizationeffectson theIP,describedinthepreviousparagraph,explainthesteepestin- creasesinthegap,occurringpreciselyattheNe=20andNe=40 shellclosings.

Thereisasecondarymagicnumber,predictedbyellipsoidaljel- lium modelsandobservedin manymetal clusters, thatisexactly realized in theZn₂Mg series.This isthe 15-atom nanoalloy con- taining 30 electrons, which according to the deformable jellium models should be a marked magicnumber induced by a prolate shape.Ineffect,thestructureofZn₁₀Mg₅containstwostackedTTP unitsresultinginastrongprolatedeformation,anditdisplaysalo- calmaximuminboththeIPandthegap.

In conclusion, the main variations inthe electronic indicators of Zn₂Mg nanoalloys conform to general expectations aboutsys- tems with delocalized electrons, evenif noneof theclusters has therightnumberofelectronstoproduceasphericalshellclosing.

Some care hasto be takenfor afew clusters duetoelectron lo- calization effects (inlone pairorbitals),asthisaffects theproper countingofdelocalizedelectronsinthesystem.

3.4. StabilitiesofZn₂Mgnanoalloys

WeanalyzethesizedependenceofclusterstabilitiesforZn₂Mg nanoalloysthrough3differentindicators,showninFig.7:theco- hesiveenergyE_coh(definedpreviously)providesanabsolutemea- sure for the cluster global stability, as it quantifies the total in- ternalenergycontent storedinthechemical bonds;theevapora- tion energy Eevap is here defined as the energy required to dis- sociate a Zn2Mg formula unit, so it quantifies the stability with respect toaparticularfragmentationchannel;finally,thesecond- orderenergydifferenceisdefinedas2(^N)=E_N−3+E_N+3− 2E_N=

Eevap(^N)− Eevap(^N+1)^{. Both Eevap and} 2 provide more “local”

stability measures in comparing the energy of a cluster of size N with its adjacent neighbors at N− 3 and N+3. In particular, 2 estimatesthedifferencebetweentheevaporationratesoftwo neighboringcluster sizesinan evaporativeensemble,sothisindi- catorcorrelates withthecluster populations determinedby mass spectrometryonfree-standingclusterbeams.There areno exper- imental studiesonZn₂Mgnanoalloys asfarasweknow,andob- viouslythereisnoevidencethatevaporationofaZn₂Mgtrimeris thedominantfragmentationchannel, yetwehavefound ituseful to considertheseindicators todiscusscluster stabilityata theo- reticallevel.Our2(^N)^{valuesareexpectedtocorrelatewithtrue} abundances in an evaporative ensembleof Zn2Mgnanoalloys as- suming that trimer evaporation is the dominant channel. Notice thatthethreeindicatorsproviderelatedbutdifferentinformation:

inparticular,themorestableclusters(asdeterminedbythecohe- siveenergy)donotneedtocoincidewiththemoreabundantsizes (asdeterminedby2).

Aswiththeelectronicpropertiesdiscussedintheprevioussec- tion, theanalysis ofstabilities is slightlycomplicated by the fact thattheNe scaleisquitesparse alongtheZn₂Mgseries,andalso dueto the re-entrance of localized molecular orbital states right afterthemainelectronic shellclosings.Asshowninourprevious worksonzincclusters[6,7],occupationofthoselocalizedorbitals amounts to a temporary departure fromthe usual jellium filling pattern, which slightly delays the opening of a new electronic shell. The main effect on stabilities is that clusters with Ne+2 electrons(withNeasphericalshellclosing)tendtobemorestable than clusters with Ne− 2 electrons. With these caveats in mind, the stabilities in Fig. 7 can be seen to correlate quite well with electronic shell closing effects. The clusters with a higher-than- averagecohesiveenergyarepreciselythoseclosertothespherical shellclosings.Ineffect,thestandardizedcohesive energydisplays localmaximaat(orsubstantial dropsafter) Ne=18,36,42,60,72 and 90electrons, all of them two electrons away froman exact electronicshell closing.The stabilitydropisspeciallypronounced afterNe=40,90,producingdeepstabilityminimaforclusterswith 48and102electrons.

Someoftheclusterswithanenhancedstabilitywouldalsodis- play an enhanced population in an evaporative ensemble where dissociation of a Zn₂Mg trimer is the dominant fragmentation channel.Asanexample,theclusterwith72valenceelectronsper- sistsasalocalmaximumintheevaporationenergyand2 indi- cators.Forsomeother clusters,however,ahighstabilitydoesnot correlatewithan enhanced abundance.The cluster with36elec- trons, forexample, ismore stable thanthe cluster with 42elec- tronsbutsignificantly lessabundant;electron localizationon the danglingatom of Zn₁₄Mg₇ equips thecluster witha highevapo- rationenergyevenifithastwoadditionalelectrons ontop ofan electronic shell closing,producinga marked maximumin 2 for Ne=42electrons.Analogously, thepopulationofclusterswith90 and 96 electrons is comparable due to the very low stability at Ne=102 electrons. As a final example, the maximum in the co- hesive energy for Ne=60 shifts to Ne=54 in the 2 indicator:

here, itisthe verylow stabilityofthecluster withNe=48elec- tronsthat resultsin a very high evaporationenergy forNe=54, sothatthecascadeoftrimerevaporationeventswouldbestalled there maximizing the population ofZn₁₈Mg₉. Electronic shell ef- fects are thus the dominant factor determining the stabilities of Zn₂Mgnanoalloys.IntheESIwefurtheranalyzepossiblerelation- shipsbetweenstabilityandstructuralproperties,anddemonstrate thatthemorestableclustersarealsomorecompactlypacked.

AlthoughthediscussionofEevap and2 isinitselfinteresting, weemphasizethat itisrestrictedtooneparticularfragmentation channel,enforcedbytheconsiderationofasinglefixedstoichiom- etryinourwork.Forexample,evenifZn₁₄Mg₇displaysahighre-