PHYSICSSEPI[MBER isso

TODAY

COMPUTATIONAL MATERIALS SCIENCE

COMPUTATIONAL MATERIALSSCIENCE: THE ERA OF

APPLIED QUANTUMMECHANICS

For manycenturies,mate-rials were discovered,

mined, and processedin alargely serendipitousway.However, the characteriza-tion of theatom andthepro-gress madein x-ray diffrac-tion duringtheearly yearsofthis century starteda questfor a theory of materialsinterms of their atomic constituents.Later decadessawscientistsdevelopingmanyqualitativeandsemi-quantita-tive modelsthat explainedthe principles of atomic cohe-sion andthe basicpropertiesof semiconductors,metals,andsalts. Consideringtheir simplicity, someofthemodelsweresurprisinglyaccurateandledto remarkableprogress.However, for most materialsof current interest, the in-teratomic interactions are intricate enough to requirefairly elaboratemodels.Fortunately,we are entering anerain which high-performancecomputingis coming intoits own, allowing true predictive simulationsof complexmaterialsto be madefrom informationon their individualatoms.

Methods for computing the propertiesof materialscanbe divided into two classes:thosethatdo not useanyempirically or experimentallyderived quantities, andthosethat do. The former are often called ab initio, orfirst-principles methods,while the latter are called em-pirical or semi-empirical.The ab initio methodsare par-ticularly useful in predicting thepropertiesof new mate-rials or newcomplexmaterialstructures,andfor predict-ing trends across a wide range of materials. Thesemi-empiricalmethods excel at interpolating and ex-trapolatingfrom knownproperties.This article focusesontheab initio methods,whichretain their predictivepowerevenwhen experimentaldataare scarceor unavailable.

Methodologyof ab initio calculationsAs is well known, the binding in moleculesand solids isdueto the Coulombforcesbetweenelectronsandnuclei.The exact solution of the full, many-body Schrodingerequationdescribinga material is, of course,impossible,but one can make surprisingly accurateapproximationsof a system’sgroundstate,andsuch approximationsarewidely usedin condensed-matterphysics. Theseapproxi-mations are basedon density functional theory (DFT),1which wasdevelopedat the University of California, SanDiego, by Walter Kohn, PierreHohenberg,andLu Sham(and for which Kohn shared the 1998 Nobel Prize inchemistrywith John Pople,a pioneerin quantumchem-

istry.) The impact of DFT onphysics, however, was cer-tainly not smaller than itsimpact on chemistry. DFTprovesthat theground-stateenergyof anM-electronsys-temis afunction only of theelectrondensityp(r). In DFT,theelectronsarerepresentedby one-bodywavefunctions

iQ, which satisfy the Schrodinger-like equations (inRydbergs):

Thepropertiesof newandartificiallystructuredmaterialscanbepredictedandexplainedentirelyby computations,using

atomicnumbersasthe only input.

JerzyBernholc

KV2+VN(r)+12p(r’)dr +/¾c[P(r)]Ja~~(r) =

i=1,. . . , M.

The first term representsthe kinetic energy;the secondis the potentialdueto all nuclei; the third is the classicalelectron—electronrepulsion potential; andthe fourth, theso-calledexchangeandcorrelationpotential, accountsforthe Pauli exclusion principle andspin effects. The exactform of the exchange—correlationterm is unknown,but alocal approximation, in which the exchange—correlationpotential of a homogeneouselectrongasof density p(r) isused at each point, has proved highly successful.DFTgenerallypredictslatticeconstants,atomicpositions,elas-tic propertiesandphononfrequencieswith errors smallerthan a few percent.For example,my researchgroup atNorthCarolinaStateUniversity computedthe radialdis-tribution function of the then new solid C60 (seefigure 1)six months beforethe first neutron scatteringdata, yettheresultofthe computationagreedalmostperfectlywithexperiment.

2In fact,the theoreticalresultswereusedforthe initial calibrationof the experimentalsetup.

Density functional theorypredictionsof cohesiveen-ergies usedto be lessaccurate,but including terms de-pendenton the gradientof the electrondensityhassub-stantially improved the agreementwith experimentand

with high-level quantumchemistrycalculations.3

Large-scalecomputationsBecausethe electronic structure calculationsdescribedaboveare computationally demanding,progressin thefield dependsin perhapsequal measureon advancesintheoretical methodsand on advancesin computertech-nology. For simple materials,suchassilicon with only twoatomsin a periodically repeatedunit cell, the computa-tional effort requiredhasbecomeso modestnow that thecalculationscanbe carriedout on any contemporaryper-sonalcomputer.However,understandingthe growth andpropertiesof silicon devicesrequirescalculatingthe char-acteristicsof silicon surfaces.Becausesilicon is a cova-lently bondedmaterial, the creationof a surfaceleadsto

30 SEPTEMBER 1999 PHYSICS TODAY © 1999 AmericanInstitute of Physics,S-0031-9228-9909-020-1

FIGURE 1. CALCULATEDelectrondistributionin C60 at1000 K, obtainedfromquantummoleculardynamicssimulations.

2Yellow, green,andbluedenoteregionsofsuccessivelygreaterelectrondensity.Theatomicstructureof C

60 (background)consistsof five- andsix-memberednngsarrangedin theshapeofasoccerball.

broken bonds.This processis energeticallyunfavorable,so some atoms will move to rebond, forming a recon-structedsurfacewith less symmetry. (Seethe article byJohn J. Boland and John H. Weaver, PHYsIcs TODAY,August 1998, page34.) Dependingon the complexity ofthe resulting structure,thenumberof atomsN that onemust considerin the unit cell will rangefrom 16 to 400.Becausethesizeof thecomputationaleffortgrowsroughlyasN

2, andasymptoticallyevenasN3, it is no wonderthatprogressin computationalmaterialsphysicsis closelytiedto progressin methodologyandcomputers.

The largestab initio calculationsusually make useof functions called “pseudopotentials,”which replace thenuclearpotentialandthechemicallyinertcoreelectronswithan effective potential, so that only valence electronsareexplicitly included in the calculations.(Seethearticle byMarvin L. Cohen,PHYsIcsTODAY, July 1979,page40; alsoseeref. 4.) The pseudopotentialsare derivedfrom atomiccalculationsthat useatomic numbersas the only input.Becausepseudowavefunctionsaresmooth andnodeless,planewavescan beusedasa basisset.This offers threemajor advantages:> Planewavesdo not dependon theatomic positions,sousing them makesthe resultsmore precise.> The accuracyof the result is determinedby a singleparameter,the highest kinetic energy of the waves in-cludedin the calculations.> The kinetic energy(— V2) is diagonalin Fourier space,whereasthe potential is diagonalin real space.

Thetransformationbetweenthetwo spacesoccursvia

the well-known fastFourier transform (FFT)algorithm, which worksvery quickly on vector su-percomputersandmodernworkstations.However,be-causethe FFT is a globaloperation,its performanceslows down on massivelyparallel computers.For alargeproblem,thenumberof plane waves can be 50000 or more,andonemustuse iterative diagonaliza-tion methods that mostlywork with the occupiedsubspace.A particularlyef-fective approachwas firstdevelopedby Roberto Carand Michele Parrinello(then at the InternationalSchool for AdvancedStud-ies in Trieste), who com-bined the solution of theelectronic structure prob-lem with molecular dy-

namicsfor the atoms.5Another approachis to solveitera-tively for theelectronicwavefunctions,computetheforces,and movethe atomsby a large step.6 In both casestheatoms follow Newton’sequationsof motionwith ab initiointeratomicforces. Thesemethodsarecalled ab initio, orquantummolecular, dynamics.

Ab initio calculationshave long beenuseful in ex-plaining experimentalresults and providing unique in-sights. The recent advancesmake it possibleto predictthepropertiesof materialswith complexatomicarrange-ments,whosestudywould havebeenprohibitively expen-sivejust a few yearsago. The examplesbelow illustratetherole that accurateandquantitativelyreliablecalcula-tions can play in modern condensedmatterphysics andmaterialsscience,while alsohighlighting the advantagesof collaboration and close interaction betweentheoristsandexperimenters.

Solid C36Someof themostexcitingnewmaterialsdiscoveredin thelast decadeare the fullerenes. Solid C

60, once it wasproducedin quantity, was shown to havea number ofremarkableproperties,including superconductivityafterintercalationwith alkalimetalatoms.In fact,its transitiontemperaturesT~ are approaching40 K, secondonly tohigh-Taoxides. The relativelyhigh T~ of C60 is due to thestrong electron—phononinteraction in curved fullerenes.This has stimulatedMarvin Cohen, Steven Louie, andtheir coworkersat theUniversity of California, Berkeley,to examinefullereneswith evengreatercurvatures.

7In-

SEPTEMBER 1999 PHYSICSTODAY 31

FIGURE 2. ATOMIC STRUCTUREandsurfaceof constantchargeden-sity of rhombohedralC36. Blue spheresrepresentcarbonatoms.Thecovalent-likebondingbetweenthe C36 units is clearlyvisible. It ispredictedthat this new solid will form from individual moleculesandwill exhibit high-temperaturesuperconductivity.(From ref. 7.)

deed, they showedthat a solid formedout of C36 shouldhaveevenmore remarkableproperties.Unlike the mole-cules in the C60 solid, the individual C36 moleculescrosslink (see figure 2). The cohesiveenergyof solid C36shouldbe comparableto thatof solid C60, andthe densityof electronicstatesat theFermi level shouldalsobehigh,as is necessaryfor high T~ accordingto theBardeen—Coo-per—Schrieffertheory of superconductivityAnd the elec-tron—phononcoupling? The calculationsshow that it isevenstrongerthanin C60, so that solid C36 is anexcellentcandidatefor a superconductingsolid with a high Tc.

How aboutan experimentalconfirmation?Growinga“desiguer” solid is not easyThe C36moleculesform in thegasphase,but Alex Seattle’sgroup(alsoat Berkeley)wasableto depositthem on a substrateandmeasuresomeoftheir properties.Indeed,the individual moleculesturn outto bemore reactivethan C60 moleculesandseemto formdimersandtrimers.But asolid sampleof C36 goodenoughfor the delicatemeasurementsof superconductingbehav-ior remainsto be made.

SiliconLet us now turn to the technologically important caseofsilicon. Although mostphysiciststhink of devicesasbeingmadeof a single crystal, polycrystalline silicon is usedextensivelyin microelectronics.Whenpolycrystallmesili-con is doped,a siguificantnumberof dopant atoms seg-regateto the grain boundariesand becomeelectricallyinactive. What are the driving forces for that behavior,andcanit be prevented?

Considerthe caseof a substitutionaldonor, arsenic.Experimentsindicate that its segregationenergyis sub-stantial,about0.5 eV. How is thatpossible?We knowfromelementarychemistry that arsenichasthe propensitytobe either trivalent or pentavalent.It is the perfection ofcrystalline silicon that forces it to assumea four-foldcoordinationandthusbecomea shallowdonor. However,

at a grain boundarythe coordination constraintbecomesweaker.Thereis more roomfor relaxation,andarsenicatomscanassumetheir preferredthree-fold coordination. Wouldn’t that behavior breakbondsandmakesomesilicon atomsveryunhappy?Not if arsenicatomsactin pairs, effectivelybreak-ing one bond and moving away from eachother.Indeed,calculationsby a joint theoretical—experi-mentalteamheadedby SokratesPantelidesofVan-derbilt University and StephenPennycookof OakRidgeNational Laboratoryfound dimers andevenchainsof dimers to be energeticallyfavorable.

8Infact, dimer chains, shown in figure 3, were some-what preferred.

The story doesnot end there. Two yearslaterthe sameteam succeededin imaging the arsenicatoms at a grainboundaryalbeit a more complexboundarythan the one examinedtheoreticallyAr-senic atoms were indeedin the dimer configura-tions, but the dimers were isolated. Follow-up cal-culations,althoughcarriedout for anidealizedsys-tem, provided the explanationin terms of kineticconstraints,the atomic geometryat the more com-plex boundaryandentropiceffectsat thefairly highdoping temperature.

MagnetismIn theearly daysof quantummechanics,the Heis-enberg and Ising models of maguetismprovidedimportant conceptualmodelsfor the entire field ofphasetransitions. However, magueticphenomenaare complex, and there are still many unsolvedproblems—especiallyat the microscopiclevel—be-

cause spin effects are so subtle. Harnessingmagueticphenomenato thefullest extentpossibleis alsoimportantfor many applications,most notably in high-densitycom-puter storage.

In DFT it haslong beenpossibleto carry out spin-polarizedcalculations,treatingthespin-up andspin-downelectrons separatelyFor largely homogeneoussystems,where there is a single preferred spin direction, thattreatmenthasworkedwell and has led to a numberofsurprises.It is now well acceptedthat maguetismis aneffect of undercoordination, in which some atoms havefewerneighborsthanthemaximumnumberof bondstheycan form. This is becausein a covalent bond a spin-upelectronpairs up with a spin-downelectron,leadingto nonet spin. Nevertheless,it took yearsto arrive at a satis-factory descriptionof low-coordinatedsystems.Pioneeringcalculationsby Arthur Freemanand his coworkers atNorthwesternUniversity haveshownthat surfacelayerscanpossessmagueticmomentsenhancedfar beyondthosein the bulk, andthat thin films of nonmagueticmetalscan becomemaguetic.

In disorderedor low-symmetry systems,evenmorecomplicatedspin arrangementsarepossible,in which thevarious spins are not aligued along a single preferreddirection. How do we know which arrangementis ener-getically favored?The usual procedureis to computetheenergiesofeachcompetingstructureto identify thegroundstate.This canbe tedious.But thereis a betterway Ulfvon Barth and Lars Hedin at the University of Lunddevised a generalizedspin-density theory in which thewavefunctions are two-component spinors \P(r) =

[i~1(r),i~2(r)],where i~1 and i~2 are complexfunctions.This

approachallows the spin quantizationaxis to vary con-tinuously with position, but the total energy needsto beminimized without any constraintson thespin direction.Very recently Car andhis coworkersat the Institute forNumerical Researchin Lausanne,developeda generali-

32 SEPTEMBER 1999 PHYSICSTODAY

zation of the Car—Parrinelloprocedurethat simultane-ously optimizes both the spin directions andthe atomiccoordinates.9Thefirst applicationsto small iron structuresfound both collinear and noncollinearground states,de-pendingon the clustersize. (Seefigure 4.) Very interest-ingly there are energybarriers separatingthe collinearandnoncollinearstructures.Suchbarriersdo not exist inthe classicalHeisenbergmodel of a spin system.

This examplealsohighlights theemergenceof clustersasanewandexcitingclassof materials.Atoms in clustersareundercoordinatedcomparedto thosein bulk, resultingin different structures, larger magueticmoments, andenhancedreactivity patternsin catalysis. Clustersmaybecomethebuilding blocksof a hostof newmaterials,buttheir structuresandproperties,which can vary dramati-cally with cluster size,must first be well understood.

New methodsAlthough materialscomputationshavealreadyadvancedquite far, the quest for faster and bigger calculationscontinuesunabated.This hasstimulatedinterestin “realspace”methods,which avoid the useof planewaves thatextendthroughoutthe entire systemby having the vastmajority of operationsbe local, in “real space.”Working

FIGURE3. GRAIN BOUNDARY. This two-dimensionalplotshowstheelectrondensity at a silicon grainboundarysaturatedby achainof arsenicatoms.Thearsenicatomsareshownin red, the silicon atomsin blue. Thecolor scheme(inorderof decreasingelectrondensity)is red,yellow, green,lightblue, anddeepblue. (Fromref. 8.)

in real spacehasseveraladvantages.It allows oneto useadvancedmathematicaltechniques that automaticallyseparatethevariouslengthscalespresentin theproblem,substantiallyacceleratingconvergence.(Seethe box be-low.) Also, parallelizationbecomesmuch easier,becauseeachprocessorcan be assigueda given region of space.Finally real-spacemethodsmay reducescaling of thecomputationaleffort with systemsizeto the orderof thenumber of atomsN ratherthanthe current0(N3).

The easiestway to exploit the multiscaleaspectsofreal-spacecalculations is by using methodsbased ontechniquesof applied mathematics:multigrids, wavelets,andfinite elements.Of thosethree,the grid-basedmeth-ods10haveprogressedthe mostin applications,andI willgive a few exampleshere.

For clusters,grid-basedmethodsoffer a naturalad-vantageby making it easyto implementboundarycondi-tions.JamesChelikowskyandhisgroupat theUniversityofMinnesotahavebeautifully exploitedthatadvantageindeterminingstructuralproperties,polarizabilities,opticalenergygaps,andexcitationspectra.11It is much easiertocalculateopticalgapsinclustersthanin thebulk, becausein clusters evencorrectionsto density functional theorycan be determinedby computing the DFT total energydifferences.Excitation spectraare siguificantlymore dif-ficult to work with, butagainthe real-spacemethodologyand the small size of the clusters allow simplification.Usingexpressionsderivedfrom time-dependentlocal den-sity theory, Chelikowskyandhis colleaguesobtainedex-cellentresults for the spectraandpolarizabilities of so-dium andsilicon clusters.Theagreementwith experimentwasalmostas good asin high-quality quantumchemistrycalculations,which aremuchmore expensiveto do.

Therehavealso beenasubstantialnumberof large-scalesolid-stateapplicationsof real-spacemethods.In myown group,the multigrid approachis now the methodofchoice for studies of large systems.We haveused it todeterminelarge surfacereconstructionsandtheir opticalspectra,and to investigatedoping, surfacemelting, and

MultiscaleCalculationson aGridNA ultiscalephenomenaoccurin manyareasof physics.In.LVleach casewe can use the sameconceptualframework:Consideran iterative solution of a differential equationon agrid. Although the iterationsquickly reduceerrorcomponentsthatareoscillatoryon thescaleof thegrid, thelong-wavelengthcomponentsconvergeonly slowly. However,theapproximatesolutioncanbetransferredto acoarsenedgrid, wheretheerroris now oscillatory, and iteratedthere. After a convergenceslowdownis observed,a still coarsergrid is used. -

Up to this point, thecoarseningprocedureis veryreminis-centof therenormalization-groupapproach,which focusesonthe large-scalefeaturesof theproblem.However,in the multi-grid method,developedby themathematicianAchi Brandtofthe WeizmannInstitute, onerecursivelytransfersthe coarse-grid solutionsbackto finer grids.18 This introduceshigh-fre-quencyerrors,of course,but theyareeasilyremovedby afewiterations.Thusweobtainamachine-precisionsolutionin 0(1<)operations,whereK is thenumberof points on the finegrid.

Actualcalculationsrequirehigh accuracyandthushigh-or-

I. Global (fine) Basis

II. Medium ~ Multigrids

III. Coarse*

derdiscretizations,becausethecohesiveenergyis averysmallfraction of the ground-stateenergy.Consider,as anexample,theone-dimensionalPoissonequation,‘V(x) = f(x).By Taylor-expanding(13(x), we obtainanapproximatediscretizationbasedon centralfinite differences:

[F(x~ - h) - 213(x~) + F(x~ + h)]/h2 — f(x,) + 0(h2).However,expandingboth 13(x)andf(x), andusingthefactthat(1Y’(x) is equal to f(x) at convergence,leadsto amoreaccurateexpressionthatstill involves only the nearestneighbors:

[l2F(x~ — h) — 2413(x,) + l2’P(x~ +

= f(x~ - h) + lQf(x~) + f(x~ + h) + 0(h4).

The short rangeof this “Mehrstellen”discretizationis particu-larly importantfor parallelizationand0(N) methods.

SEPTEMBER 1999 PHYSICSTODAY 33

P-B

3.72 au

b 2.88P-B

3.66 au

A2.55 /.tB

2.58P-B

nanotubes.Quantum molecular dynamics, which mustconservetotal energyto a high degreeof accuracyworksvery well andallows for largetime steps.The advantageof multigrids becomesparticularly apparentin systemswhere convergenceis difficult to attain becauseof suchobstaclesaschargesloshingor multiple lengthscales.Theuseof parallelizationhasalso beenvery effective in ourwork, and the computational speedon the massivelyparallelCray T3E computerhasscaledlinearly with thenumberof processors—upto 1024, themaximumnumberto which we had access.

As an example of a specific application, considernanotubes,which are the strongestmaterials known.Early classicalmoleculardynamicssimulationsindicatedthat they had exceptional strength, a suggestionthatneededto be confirmed.However,becausecurrentlymadenanotubesare so very short, it is extremelydifficult tomeasuretheir strengths.With no experimentaldataavail-able at the time, ab initio simulationsto determinethemechanismsof breakageseemedto be the next bestapproach.But how weretheyto beperformed,if eventheclassicalsimulationshadproved computationallyexpen-sive?Well, therewas atrick—namelyto simulatea tubeat ahigh fixed dilation andhigh temperature.Indeed,alongquantummoleculardynamicscomputerrun identifiedthe key transformationthat leadsto either breakageorplasticbehavior,dependingon thetemperatureandstrainrate.’2The predictedstrengthis thehighestby far of any

FIGURE4. ATOMIC AND MAGNETIC STRUCTURESof Fe3and

Fe,clusters.In the isomersa andc the spinsarenoncollinear.Thecorrespondingcollinearstructuresareb andd. Magneticmomentsareoftenmuchlargerin clustersthanin bulk, a factthatcouldbeimportant for applications.(From ref. 9.)

materialto date,andtherewill be no shortageof excitingapplicationswhen nanotubescan be grown in sufficientquantity andquality

OvercomingsizelimitsWe now turn to methodsthatseekto overcomethelimitson the sizeof the systemsthat can be studiedab initio.There is an urgent, growing need to include more andmoreatomsin simulations—newspace-agematerialsarebecomingmore complex or consist of multiple layers of“simple” materialsthat haveto be modeledtogether.

Computer desiguersare doing their part, but thestandarddensityfunctionalmethodsdescribedabovepre-sentaproblem.ConsiderasystemofN atoms.AsN grows,the number of electronsit containsgrows as 0(N). In asolid, the wavefunctionof eachelectroncan extendoverthe whole solid, which meansthat computingonewave-function will take at least 0(N) operations.Hence, thecomputationsmust scaleat leastas0(N

2). Furthermore,the individual wavefunctionsmust be orthogonalto eachother, and the processof orthogonalizationor diagonali-zationscalesas 0(N3).

Recentlya numberof ingenious methodshavebeenproposedfor evaluating the ground-stateenergyin 0(N)operations.How is thatpossible?The main ideain mostof the approachesis to useabasisof localized,variation-ally optimized functions. Each function is confined to agivenregionof space,but thevariousconfinementregionsoverlapso that thereis little loss of generalityBecausethematrix elementsbetweenfunctionslocalizedin regionssufficiently far awaywill vanish, the numberof non-zeroelementsbecomes0(N).Nevertheless,the resultsarestillquite accuratewith only a few functions per atom andwith localizationregionsspanningonly afewbondlengths.Recentreviewshavecomparedthevariousaspectsof theproposed0(N) methods.’3

The’ DFT-based0(N) methodsare still being devel-oped,but the localized functions are turning out to beusefulevenapartfrom facilitatingvery-large-scalecalcu-lations.’4The optimizationshownin figure 5, for example,whichwasstartedfrom randomnumbers,ledto a chemi-cally very intuitive result: afunction stronglylocalizedona carbon—carbonbond. That exampleprovidessomein-sight into and justification for expectingthe eventualsuccessof 0(N)-like methods.

Chemistshavelong known that good approximationsofthecohesiveenergiesofcomplicatedstructurescanoftenbe obtainedby summingup the known“strengths”of theindividual bonds. Using the quantum-mechanicalbasisfunctions that automaticallylocalize in bondingregionsshouldleadto accurateresultswhile keepingthenumberofbasis functionsto a minimum. The strict localizationalsomakesit moreefficient to do large-scaleparallelization,andit is likely that DFT calculationsincluding more than athousandatomswill appearsoon.Such calculationsshouldhavea significant impact on our understandingof compli-catedsolids, largebiomolecules,andcomplexliqulds.

BeyonddensityfunctionaltheoryDensity functional theoryhasknown drawbacks.For ex-ample, it does not do well in describingpropertiesthatinvolve excitedstates,suchasbandgapsof semiconductorsandinsulators.That limitation is not surprising though,

a

c

34 SEPTEMBER 1999 PHYsicsTODAY

becauseDFT is a variational approachfor obtainingthetotal energyof only thegroundstate.To obtaintheenergiesof the excitedstates,we needto usemany-bodytheoryand properly take into accountthe nonlocal interactionbetweenelectrons.The simplest working theory, the so-called GW approximation, developed by Hedin andadaptedfor usewith semiconductorsby StevenLouie andMark Hybertsen,” leads to bandgapsin excellentagree-mentwith experiment.’6However, the computationalex-penseis siguificantly greaterthan that of DFT calcula-tions, andfurther simplifications would be desirable.

Although theDFT ground-stateenergies,andin par-ticular thedifferencesbetweentheenergiesof alternativeground-statestructures,arequite accurate,theerrors inunfavorablecasesare often still too large, becausetheexchange—correlationpotential is not known exactly Ifgreateraccuracyis required, a sophisticatedmany-bodytreatmentis againnecessary.The mostsuccessfulmeth-odology is basedon the quantumMonte Carlo approach,in which the full many-electronequationsare solved bystatistical sampling.’7 Although it is substantiallymoreexpensivethan density functional theory the quantumMonte Carloapproachis likely to be themethodof choicefor high-accuracybenchmarkcalculationsin small andmedium-sizedsystems.

FutureoutlookProgressin materialssimulationis acceleratingwhile alsobecomingmoreimportant.Therapid increasein thespeedof computersis likely to continue unabatedfor at leastthe next decade,and possibly siguificantly longer. Thealgorithmic progress,stimulatedby the interest and ad-vancesin the field, is evenmore robust. As someof theexamplesaboveillustrate, it is alreadypossibleto predictmanypropertiesreliably andin somecasesthe theoreticalinput hasstimulatedimportant experimentaldiscoveries.As the capabilities for performing realistic simulationsincrease,it might becomepossibleto routinely “desigu”on a computer, at least in part, a new material withdesirable properties.The candidatematerialswould beexaminedtheoreticallyandonly the mostpromisingoneswould be investigatedexperimentally

Materialsgrowth andprocessingis anotherfield thatwould benefit siguificantly from reliable simulations.Al-thoughthe industrially relevantprocessesareoftenrather

FIGURE5. OPTIMIZED ELECTRON-DENSITYFUNCTION for acarbonnanotube,calculatedby Jean-LucFattebertandhiscoworkersat North CarolinaStateUniversity. Notethatalthoughthe allowedlocalizationregionextendsover 6 A(shadedin blue),this functionis largelyconfinedto onecarbon-carbonbond. Theplotting planeis alongthe surfaceofthenanotube.Thecalculationsusedanon-orthogonalorbitalmultigrid generalizationof a methodproposedby Giulia GalliandMicheleParrinello.The optimizedlocalizedfunctionshavealreadyenabledFattebertandMarcoBuongiornoNardelli to makeefficient ab initio calculationsof quantumtransportin nanotubes.

complicated,materialssimulationsarecoming to be rec-oguized as an important componentof industrial R&D.Atomistic-simulationsoftwareis beingmarketedto indus-try, andmajorindustrial firms suchasMotorola andDowChemicalare building up their materials-theorydepart-ments. The US government,which has long been thedriving force in high-performancecomputing, hasmadematerialssimulation one of its main areasof emphasis.In view of the current interest in it and its rate ofprogress, the field of materialssimulation is likely toexperiencerapidgrowth and a commensurateexpansionin job opportunities.

Thiswork wassupportedin part by theOffice ofNavalResearch,theNationalScienceFoundation, the NationalAeronauticsandSpaceAdministration,andthe Departmentof Energy.

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