A molecular dynamics primer.docx

A molecular dynamics primer.docx

《A molecular dynamics primer.docx》由会员分享，可在线阅读，更多相关《A molecular dynamics primer.docx（65页珍藏版）》请在冰点文库上搜索。

Amoleculardynamicsprimer

Amoleculardynamicsprimer

FurioErcolessi

UniversityofUdine,Italy

WWW:

http:

//www.fisica.uniud.it/~ercolessi/

Contents

∙Contents

∙Introduction

oTheroleofcomputerexperiments

▪Butisittheoryorexperiment?

oWhatismoleculardynamics?

oHistoricalnotes

oToday'sroleofmoleculardynamics

oLimitations

▪Useofclassicalforces

▪Realismofforces

▪Timeandsizelimitations

∙Thebasicmachinery

oModelingthephysicalsystem

▪TheLennard-Jonespotential

▪Potentialtruncationandlong-rangecorrections

oPeriodicboundaryconditions

▪Theminimumimagecriterion

▪Geometrieswithsurfaces

oTimeintegrationalgorithm

▪TheVerletalgorithm

▪Predictor-correctoralgorithm

∙Running,measuring,analyzing

oStartingasimulation

▪Startingfromscratch

▪Continuingasimulation

oControllingthesystem

oEquilibration

oLookingattheatoms

oSimplestatisticalquantitiestomeasure

▪Potentialenergy

▪Kineticenergy

▪Totalenergy

▪Temperature

▪Thecaloriccurve

▪Meansquaredisplacement

▪Pressure

oMeasuringthemeltingtemperature

oRealspacecorrelations

oReciprocalspacecorrelations

oDynamicalanalysis

oAnnealingandquenching:

MDasanoptimizationtool

oOtherstatisticalensembles

∙Interatomicpotentials

oTheBorn-Oppenheimerapproximation

oThedesignofpotentials

oTheproblemswithtwo-bodypotentials

oMany-bodypotentialsformetals

oMany-bodypotentialsforsemiconductors

▪TheStillinger-Weberpotential

▪TheTersoffpotential

oLong-rangeforces

oButdowereallyneedpotentials?

oFittingtoabinitiodataby``forcematching''

∙Choosinghardwareandsoftware

oWhatkindofcomputer?

▪Selectingacomputerformoleculardynamics

▪Storagespace

oWhatlanguage?

▪Highlevellanguages

▪FortranorC?

∙References

Introduction

Theroleofcomputerexperiments

 Computerexperimentsplayaveryimportantroleinsciencetoday.Inthepast,physicalscienceswerecharacterizedbyaninterplaybetweenexperimentandtheory.Inexperiment,asystemissubjectedtomeasurements,andresults,expressedinnumericform,areobtained.Intheory,amodelofthesystemisconstructed,usuallyintheformofasetofmathematicalequations.Themodelisthenvalidatedbyitsabilitytodescribethesystembehaviorinafewselectedcases,simpleenoughtoallowasolutiontobecomputedfromtheequations.Inmanycases,thisimpliesaconsiderableamountofsimplificationinordertoeliminateallthecomplexitiesinvariablyassociatedwithrealworldproblems,andmaketheproblemsolvable.

Inthepast,theoreticalmodelscouldbeeasilytestedonlyinafewsimple``specialcircumstances''.So,forinstance,incondensedmatterphysicsamodelforintermolecularforcesinaspecificmaterialcouldbeverifiedinadiatomicmolecule,orinaperfect,infinitecrystal.Eventhen,approximationswereoftenrequiredtocarryoutthecalculation.Unfortunately,manyphysicalproblemsofextremeinterest（bothacademicandpractical）falloutsidetherealmofthese``specialcircumstances''.Amongthem,onecouldmentionthephysicsandchemistryofdefects,surfaces,clustersofatoms,organicmolecules,involvingalargeamountofdegreesoffreedom;anaccuratetreatmentoftemperatureeffects,includinganharmonicitiesandphasetransitions;disorderedsystemsingeneral,wheresymmetryisofnohelptosimplifythetreatment;andsoon.

Theadventofhighspeedcomputers--whichstartedtobeusedinthe50s--alteredthepicturebyinsertinganewelementrightinbetweenexperimentandtheory:

thecomputerexperiment.Inacomputerexperiment,amodelisstillprovidedbytheorists,butthecalculationsarecarriedoutbythemachinebyfollowinga``recipe''（thealgorithm,implementedinasuitableprogramminglanguage）.Inthisway,complexitycanbeintroduced（stillwithcaution!

）andmorerealisticsystemscanbeinvestigated,openingaroadtowardsabetterunderstandingofrealexperiments.

Needlesstosay,thedevelopmentofcomputerexperimentsalteredsubstantiallythetraditionalrelationshipbetweentheoryandexperiment.Ononeside,computersimulationsincreasedthedemandforaccuracyofthemodels.Forinstance,amoleculardynamicssimulationallowstoevaluatethemeltingtemperatureofamaterial,modeledbymeansofacertaininteractionlaw.Thisisadifficulttestforthetheoreticalmodeltopass--andatestwhichhasnotbeenavailableinthepast.Therefore,simulation``bringstolife''themodels,disclosingcriticalareasandprovidingsuggestionstoimprovethem.

Ontheotherside,simulationcanoftencomeveryclosetoexperimentalconditions,totheextentthatcomputerresultscansometimesbecompareddirectlywithexperimentalresults.Whenthishappens,simulationbecomesanextremelypowerfultoolnotonlytounderstandandinterprettheexperimentsatthemicroscopiclevel,butalsotostudyregionswhicharenotaccessibleexperimentally,orwhichwouldimplyveryexpensiveexperiments,suchasunderextremelyhighpressure.

Lastbutnotleast,computersimulationsallowthoughtexperiments--thingswhicharejustimpossibletodoinreality,butwhoseoutcomegreatlyincreasesourunderstandingofphenomena--toberealized.Fantasyandcreativityareimportantqualitiesforthecomputersimulator!

Butisittheoryorexperiment?

Simulationisseensometimesastheory,sometimesasexperiment.Ononeside,wearestilldealingwithmodels,notwiththe``realthing'':

thissuggeststoclassifysimulationasbelongingtotheoreticalmethodswithouthesitation.Ontheotherside,theprocedureofverifyingamodelbycomputersimulationresemblesanexperimentquiteclosely:

weperformarun,andthenanalyzetheresultsinprettymuchthesamewayasexperimentalphysicistsdo.Sohowshouldweclassifysimulation?

Thereisnosharpanswertothisquestion:

bothsidesrepresentlegitimatepointofviews,andthisiswhatmakescomputationalscienceabranchonitsown.Thereishoweveranotherimportantconsideration.

Theoryistraditionallybasedonthereductionisticapproach:

wedealwithcomplexitybyreducingasystemtosimplersubsystems,continuinguntilthesubsystemsaresimpleenoughtoberepresentedwithsolvablemodels.Whenwelookatsimulationassimplyapracticaltoolto``verifyandtest''modelsinsituationswhicharetoocomplextohandleanalytically（forexample,whencomputingthephasediagramofasubstancemodeledbyacertainforcelaw）,weareimplicitlyassumingthatthemodelrepresentsthe``theorylevel''wheretheinterestisfocused.

Butitisimportanttorealizethatsimulationmayplayamoreimportantandinterestingrole.Wecanconsideritnotasanaidtoreductionismbut--tosomeextent--asanalternativetoit.Simulationincreasesthethresholdofcomplexitywhichseparates``solvable''and``unsolvable''models.Wecantakeadvantageofthisthresholdshiftandmoveuponelevelinourdescriptionofphysicalsystems.Thankstothepresenceofsimulation,wedonotneedtoworkwithmodelsassimpleasthoseusedinthepast.Thisgivesusanadditionaldegreeoffreedomtoexploreandopensentirelynewpossibilities.

Oneexampleofthispointofviewisrepresentedbyinteratomicpotentials.Inthepast,interactionswereobtainedbytwo-bodypotentialswithsimpleanalyticalform,suchasMorseorLennard-Jones.Today,themostaccuratepotentialscontainmany-bodytermsandaredeterminednumericallybyreproducingascloselyaspossibleforcespredictedbyfirst-principlemethods（thisisdiscussedin4.8）.Wehavethusmoveduponelevelinthedegreeofreductionismcontainedinthepotential,nowlimitedonlytotheselectionofitsanalyticalform.Theadvantageisofcourseamuchbetterrealism,whichinturnallowsinvestigationofphysicsproblemwhichrequirealevelofaccuracyinthemodelthatwasnotachievedbefore.Thesenewpotentialscouldnotexistwithoutsimulation:

simulationisnotonlyalinkbetweenexperimentandtheory,itisalsoapowefultooltopropelprogressinnewdirections.

Thereaderinterestedinthese``philosophical''aspectsofcomputationalsciencecanfindaveryenjoyablediscussioninchapter1ofref.[6].

Whatismoleculardynamics?

 Wecallmoleculardynamics（MD）acomputersimulationtechniquewherethetimeevolutionofasetofinteractingatomsisfollowedbyintegratingtheirequationsofmotion.

Inmoleculardynamicswefollowthelawsofclassicalmechanics,andmostnotablyNewton'slaw:

（1）

foreachatomiinasystemconstitutedbyNatoms.Here,miistheatommass,

itsacceleration,and

theforceactinguponit,duetotheinteractionswithotheratoms.Therefore,incontrastwiththeMonteCarlomethod,moleculardynamicsisadeterministictechnique:

givenaninitialsetofpositionsandvelocities,thesubsequenttimeevolutionisinprinciple

completelydetermined.Inmorepictorialterms,atomswill``move''intothecomputer,bumpingintoeachother,wanderingaround（ifthesystemisfluid）,oscillatinginwavesinconcertwiththeirneighbors,perhapsevaporatingawayfromthesystemifthereisafreesurface,andsoon,inawayprettysimilartowhatatomsinarealsubstancewoulddo.

Thecomputercalculatesatrajectoryina6N-dimensionalphasespace（3Npositionsand3Nmomenta）.However,suchtrajectoryisusuallynotparticularlyrelevantbyitself.Moleculardynamicsisastatisticalmechanicsmethod.LikeMonteCarlo,itisawaytoobtainasetofconfigurationsdistributedaccordingtosomestatisticaldistributionfunction,orstatisticalensemble.Anexampleisthemicrocanon