RaymondAtta-Fynn

RaymondAtta-Fynn(UniversityofTexasatArlington)NSFSummerSchoolonDisorderedMaterialsModeling

Summer2019attafynn@uta.edu

IntroductiontoMolecularDynamicsSimulations

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RaymondAtta-Fynn

=WhatismolecularDynamics?=Numericalsolutionstoequationsofmotion= IngredientsofMolecularDynamicsSimulations

=Constantenergymoleculardynamics=Constanttemperaturemoleculardynamics

=OtherConsiderations=Summary

Outline

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= Moleculardynamicsprovidesameansfordynamicalquantities[i.e.time-dependentquantities]andthermalsquantitiesofasystemtobereadilycomputed.

= Examplesofsuchpropertiesare:(i) therateofdiffusion (howfastorslowatomsmovewhensubjectedtothermaleffects)(ii) thermalconductivityofamaterial(howgoodorbadamaterialconductsheat)(iii) Phasetransformationsinmaterials (howmaterialschange,forexample,theirsoftnessor

hardnesswhensubjecttotemperatureandpressure)

WhatisMolecularDynamics?

Whatismoleculardynamics?𝑀𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟𝑑𝑦𝑛𝑎𝑚𝑖𝑐𝑠𝑖𝑠𝑡ℎ𝑒𝑡𝑖𝑚𝑒𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑜𝑓𝑎𝑠𝑦𝑠𝑡𝑒𝑚𝑜𝑓𝑎𝑡𝑜𝑚𝑠𝑑𝑒𝑠𝑐𝑟𝑖𝑏𝑒𝑑𝑏𝑦𝑁𝑒𝑤𝑡𝑜𝑛H𝑠𝑠𝑒𝑐𝑜𝑛𝑑𝑙𝑎𝑤𝑜𝑓𝑚𝑜𝑡𝑖𝑜𝑛𝑢𝑛𝑑𝑒𝑟𝑡ℎ𝑒𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑐𝑒𝑜𝑓𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒(𝑎𝑛𝑑𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒)

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WhatisMolecularDynamics?

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Customizedmoleculardynamicssimulationscanbeemployedtoaccessdifferentenergeticstates ofasystemormaterial.

Thisappliestocaseswherethesystemcanbetrappedinoneenergystateforalongtime[seecartoon;𝜉 denotesastateofthesystem and𝐹(𝜉) denotesenergyofthatstate]

RaymondAtta-Fynn

= Newton’ssecondlawofmotion: Thenetforce �⃗� actingonobjectofmass𝑚 isproportionaltotheobject’sacceleration �⃗�:

�⃗� ∝ �⃗� ⇒ �⃗� = 𝑚�⃗�

= Since�⃗� istherateofchangeofvelocity �⃗� withtime 𝑡 [thatis, �⃗� = 𝑑�⃗� 𝑑𝑡⁄ ]and�⃗� istherateofchangeofdisplacement𝑟 with𝑡 [thatis,�⃗� = 𝑑𝑟 𝑑𝑡⁄ ],Newton’ssecondlawcanberestatedat:

�⃗� = 𝑚𝑑�⃗�𝑑𝑡and�⃗� =

𝑑𝑟𝑑𝑡

WhatisMolecularDynamics?

Newton′ssecondlawemployedinmoleculardynamics

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= Considerasystemof𝑁 atomsofmasses{𝑚X,𝑚Y,….,𝑚Z} withatomicpositions{𝑟X,𝑟Y,….,𝑟Z},velocities{�⃗�X,�⃗�Y,….,�⃗�Z} undertheactionforces{�⃗�X,�⃗�Y,….,�⃗�Z} atagiveninstantintime𝑡.

= PerNewton’ssecondlaw,thetimeevolutionofthesystemisgovernedbythepairofcoupledequations:

𝑑�⃗�\(𝑡)𝑑𝑡

=�⃗�\𝑚\

𝑑𝑟\(𝑡)𝑑𝑡

= �⃗�\

[1]

where𝑖 = 1, 2,… ,𝑁 labelstheatoms

= Letusexamineafewstandardnumericalalgorithms tosolvingthesystemofequationsin1 above.

WhatisMolecularDynamics?

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= Beforewedelveintowidelyemployednumericalalgorithmsinmoleculardynamicssimulation,letusexamineafewheuristics.

= First,wenotefollowingaboutnumericalapproximationtofirstderivatives:𝑑�⃗�\(𝑡) 𝑑𝑡⁄ ≈ �⃗�\ 𝑡 + ∆𝑡 − �⃗�\ 𝑡 ∆𝑡⁄𝑑𝑟\(𝑡) 𝑑𝑡⁄ ≈ 𝑟\ 𝑡 + ∆𝑡 − 𝑟\ 𝑡 ∆𝑡⁄

whereitisimplicitlyassumedthat∆𝑡 issmallenough.= Pertheaboveapproximations,asimplesolutiontothepairofNewton’sequationsare:𝑑�⃗�\𝑑𝑡

=�⃗�\𝑚\

⇒ �⃗�\ 𝑡 + ∆𝑡 ≈ �⃗�\ 𝑡 + �⃗�\ 𝑡 ∆𝑡 where�⃗�\ 𝑡 =�⃗�\ 𝑡𝑚\

𝑑𝑟\𝑑𝑡

= �⃗�\ ⇒ 𝑟\ 𝑡 + ∆𝑡 ≈ 𝑟\ 𝑡 + �⃗�\ 𝑡 ∆𝑡

where𝑖 = 1, 2,… ,𝑁 and∆𝑡 isknownasthetimestep.= Letusexamineafewstandardnumericalapproaches

WhatisMolecularDynamics?

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Verletmethod:Theequationsofmotionare

𝑟\ 𝑡 + ∆𝑡 = 2𝑟\ 𝑡 − 𝑟\ 𝑡 − ∆𝑡 + �⃗�\ 𝑡 ∆𝑡 Ywhere�⃗�\ 𝑡 =�⃗�\ 𝑡𝑚\

�⃗�\ 𝑡 =𝑟\ 𝑡 + ∆𝑡 − 𝑟\ 𝑡 − ∆𝑡

2∆𝑡

= AdvantagesoftheVerlet method:(i)Straightforwardtoimplement;(ii)thepropagationoftheatomicpositionsdoesnotexplicitlyrequiretheatomicvelocities[thusonecanignorethevelocitiesunlesstheyarereallyneeded]

= DisadvantagesoftheVerlet method:(i)Itisnon-selfstarting;(ii)thevelocitiesaccumulatesignificantround-offerrors(especiallyfordynamicsspanninglongtimescales)

Numericalsolutionstoequationsofmotion

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Leapfrogmethod:Theequationsofmotionare

�⃗�\ 𝑡 +∆𝑡2

= �⃗�\ 𝑡 −∆𝑡2

+ �⃗�\ 𝑡 ∆𝑡where�⃗�\ 𝑡 =�⃗�\ 𝑡𝑚\

𝑟\ 𝑡 + ∆𝑡 = 𝑟\ 𝑡 + �⃗�\ 𝑡 +∆𝑡2

∆𝑡

Thevelocityattime𝑡 isgivenby�⃗�\ 𝑡 = �⃗�\ 𝑡 − ∆𝑡 2⁄ + �⃗�\ 𝑡 + ∆𝑡 2⁄ 2⁄

= AdvantagesoftheLeapfrogmethod:(i)Itistimereversible;(ii)itissymplectic (i.e.itconservestheenergyofthesystem)

= DisadvantageoftheLeapfrogmethod:Thevelocitiesandpositionsarecomputedatdifferenttimes,thatis,theyleapfrogeachotherbyhalfatimestep[∆𝑡 2⁄ ]

Numericalsolutionstoequationsofmotion

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VelocityVerlet method:Theequationsofmotionare

𝑟\ 𝑡 + ∆𝑡 = 𝑟\ 𝑡 + �⃗�\ 𝑡 ∆𝑡 +12�⃗�\ 𝑡 ∆𝑡 Ywhere�⃗�\ 𝑡 =

�⃗�\ 𝑡𝑚\

�⃗�\ 𝑡 + ∆𝑡 = �⃗�\ 𝑡 +12�⃗�\ 𝑡 + �⃗�\ 𝑡 + ∆𝑡 ∆𝑡

= AdvantagesofthevelocityVerlet method:(i)Itistimereversible;(ii)itissymplectic (i.e.itconservestheenergyofthesystem)

= Thisalgorithmisverypopular;itisveryreliableandaccurate.WewillemployitintheLabsessions.

Numericalsolutionstoequationsofmotion

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Thefollowingarethekeyingredientsrequiredtosuccessfullyexecuteamoleculardynamics:

(1)Atomisticmodel

(2)Interactionpotential(orpotentialenergy)

(3)Atomicforces

(4)Basicmoleculardynamicsobservables:kineticenergy,totalenergy,temperature,andpressure.

(5)Thermodynamicensembles:microcanonicalandcanonicalensembles

(6)Theergodichypothesis

(7)Methodforsolvingtheequationsofmotion[alreadydiscussedinthreepreviousslides]

IngredientsofMolecularDynamicsSimulations

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Ingredient1:Atomisticmodel:= Comprisesasetofatoms,whichrepresentativeofthesystemofinterest,withwell-defined

positions(orcoordinates){𝑟X,𝑟Y,….,𝑟Z}.

= Normally,arandomsetofatomicpositionsarechosensubjecttogeometricconstraintsandotherconstraintssuchasthecorrectatomicdensity(i.e.thenumberofatomspervolume).

IngredientsofMolecularDynamicsSimulations

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Atomisticmodelofachargedmetalionimmersedinliquidwater.

RaymondAtta-Fynn

Ingredient2:Potentialenergy(orinteractionpotential):= Thepotentialenergy ofasystemof𝑁 atoms,denotedby 𝑈,tellsushowtheatomsinteract

witheachorpositionthemselvesrelativetoeachother.

= Thepotentialenergydependsontheatomicpositions:𝑈 ≡ 𝑈(𝑟X,𝑟Y,….,𝑟Z) .

= 𝑈 usuallyhasawell-definedanalytic/mathematicalrepresentation.

Example:thepotentialenergy duetothebondbetweentwoatoms𝑖 and𝑗 locatedatpositions𝑟\ = (𝑥\, 𝑦\, 𝑧\) and𝑟l = (𝑥l, 𝑦l , 𝑧l) respectively,mayberepresentedas

𝑈(𝑟\,𝑟l) =12𝑘 𝑟\ − 𝑟l

Y =12𝑘 (𝑥l − 𝑥\)Y+(𝑦l − 𝑦\)Y+(𝑧l − 𝑧\)Y

where𝑘 isaconstant.

IngredientsofMolecularDynamicsSimulations

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Ingredient3:Atomicforces:= Theatomicforces(neededtosolvetheequationsofmotion)areobtainedfromthe

potentialenergy𝑈 ≡ 𝑈(𝑟X,𝑟Y,….,𝑟Z).= Theforce �⃗�\ actingonatom𝑖 isgivenbythenegativegradientofthepotentialenergy𝑈

withrespecttotheatom’sposition 𝑟\:

�⃗�\ = −𝛻\𝑈 𝑟X,𝑟Y,….,𝑟Z = −𝜕𝑈𝜕𝑟\

= −𝜕𝑈𝜕𝑥\

, −𝜕𝑈𝜕𝑦\

, −𝜕𝑈𝜕𝑧\

= Example:ifthepotentialenergy ofa2-atomsystemisgivenby

𝑈(𝑟\,𝑟l) =12𝑘 𝑟\ − 𝑟l

Y =12𝑘 (𝑥l − 𝑥\)Y+(𝑦l − 𝑦\)Y+(𝑧l − 𝑧\)Y

where𝑘 isaconstant,thenforce�⃗�\ onatom𝑖 isgivenby

�⃗�\ = −𝜕𝑈𝜕𝑥\

,−𝜕𝑈𝜕𝑦\

,−𝜕𝑈𝜕𝑧\

= 𝑘(𝑥l − 𝑥\), 𝑘(𝑦l − 𝑦\), 𝑘(𝑧l − 𝑧\)

IngredientsofMolecularDynamicsSimulations

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RaymondAtta-Fynn

(4)Basicmoleculardynamicsobservables:

(a)Thetotalkineticenergy(i.e.theenergyduetoatomicmotion)is

𝐾𝐸 =12r 𝑚\𝑣\Y

Z

\sXwhere𝑚\ and𝑣\ = �⃗�\ arethemassandspeedofatom𝑖.

(b)Thetotalenergy 𝐸 ofasystematanyinstantisthesumofthekineticenergy𝐾𝐸 andthepotentialenergy𝑈:

𝐸 = 𝐾𝐸 + 𝑈 𝑟X,𝑟Y,….,𝑟Z =12r 𝑚\𝑣\Y

Z

\sX+ 𝑈(𝑟X ,𝑟Y,….,𝑟Z)

IngredientsofMolecularDynamicsSimulations

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Ingredient4:Basicmoleculardynamicsobservables:(c)Thetemperature 𝑇 ofthesystematanyinstantisgivenbytheequipartitiontheorem:

12𝑁u𝑘v𝑇 = 𝐾𝐸 ⇒ 𝑇 =

2𝐾𝐸𝑁u𝑘v

=∑ 𝑚\𝑣\YZ\sX𝑁u𝑘v𝑇

where𝑁u isthedegreesoffreedom(usually𝑁u = 3𝑁 − 3)and𝑘v istheBoltzmannconstant.

(d)Thepressure𝑃 ofthesystematanyinstantisgivenbyvirialequation

𝑃𝑉 = 𝑁𝑘v𝑇 −13r �⃗�\ { 𝑟\

Z

\sX⇒ 𝑃 =

𝑁𝑘v𝑇𝑉

−13𝑉

r �⃗�\ { 𝑟\Z

\sX

where denotes averageoverallatomsinthesystem.

IngredientsofMolecularDynamicsSimulations

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Ingredient4:Basicmoleculardynamicsobservables:(c)Thetemperature 𝑇 ofthesystematanyinstantisgivenby:

12𝑁u𝑘v𝑇 = 𝐾𝐸 ⇒ 𝑇 =

2𝐾𝐸𝑁u𝑘v

=∑ 𝑚\𝑣\YZ\sX𝑁u𝑘v𝑇

where𝑁u isthedegreesoffreedom(usually𝑁u = 3𝑁 − 3)and𝑘v istheBoltzmannconstant.

(d)Thepressure𝑃 ofthesystematanyinstantisgivenbyvirialequation

𝑃𝑉 = 𝑁𝑘v𝑇 −13r �⃗�\ { 𝑟\

Z

\sX⇒ 𝑃 =

𝑁𝑘v𝑇𝑉

−13𝑉

r �⃗�\ { 𝑟\Z

\sX

where denotes averageoverallatomsinthesystem.

IngredientsofMolecularDynamicsSimulations

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Ingredient5:Thermodynamicensembles:Anensemble,withinthecontextofstatisticalthermodynamics,denotesanassemblyofidenticalcopiesofasystemindifferentstates.Wewillconsideronlytwoofsuchensembles.

(a)Microcanonicalor(N,V,E)ensemble:Inthisensemble,copiesofthesystemeachhavethesamenumberofparticles𝑁,thesamevolume𝑉,andthesameenergy 𝐸.Buteachcopycanbeinadifferentstate characterizedbytemperature𝑇 andpressure𝑃.

Example: anmicrocanonicalensemblecomprising6copies;eachcopyhasthesamevaluesof𝑁,𝑉,and𝐸 [notethat𝑁 = 4].However,eachsystemcanbeindifferent𝑇 and𝑃 states.

IngredientsofMolecularDynamicsSimulations

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Ingredient5:Thermodynamicensembles:(b)Canonicalor(N,V,T)ensemble:Inthisensemble,copiesofthesystemeachhavethesamenumberofparticles𝑁,thesamevolume𝑉,andthesystemisconnectedtoaheatbath(orthermostat)tokeeptemperature 𝑇 constant.Buteachcopycanbeinadifferentstatecharacterizedbytheenergy𝐸 andpressure𝑃.

Example: acanonicalensemblecomprising3copies;eachcopyhasthesamevaluesof𝑁,𝑉,and𝑇 [connectedtoheatbath].However,eachsystemcanbeindifferent𝐸 and𝑃 states.

IngredientsofMolecularDynamicsSimulations

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heatbath𝑇

heatbath𝑇

heatbath𝑇

RaymondAtta-Fynn

Ingredient6:ErgodicHypothesis:Ofteninmoleculardynamicssimulationsoneisinterestedinsomedynamicalproperty𝐴.

Perstatisticalthermodynamics,arepresentativevalueofthequantity𝐴 istheaveragevalue𝐴 ,obtainedbyaveraging𝐴 isoveralargeensemble(i.e.overseveralcopies;seefig.below).

Practically,computing 𝐴 isnotfeasible.Luckily,theergodichypothesis providesasolution.

IngredientsofMolecularDynamicsSimulations

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………… . 𝐴 isthevalueof𝐴 averagedoveralargeensemble

ErgodicHypothesis:thetimeaveragedvalueof𝐴overalongtimespaninonlyasinglecopyofthesystem,denotedby �̅� isequaltotheemsembleaverage 𝐴 ,thatis,

𝐴 = �̅�.

RaymondAtta-Fynn

TemperatureControlinNVTsimulation:(i)Velocityscalingmethod

= Scalevelocitiestomatchthetargettemperature[easytoimplement]= Efficient,butdoesnotcorrectlydescribedynamics

(ii)Nose-Hooverthermostat= FictitiousdegreeoffreedomisaddedbutcorrectlydescribesNVTdynamics=Cancausetemperaturetofluctuate;notsoeasytoimplement

(iii)Berendsenthermostat= Scalesvelocitiestomatchtemperature=Goodcompromisebetweenthe(i)and(ii);okaytouseinsimulations

OtherConsiderations

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RaymondAtta-Fynn

Initialvelocities:Maxwell-Boltzmannvelocitydistribution= TheMaxwell-Boltzmann(MB)probabilityoffindingaparticlewithspeedvx atagiven

temperatureT

= Sameforvy and vz

= Generaterandominitialatomicvelocitiesusingthe

= Usetheequipartitiontheoremtoscaleeachinitialvelocity byafactor𝛾 sothatthekineticenergymatchesthetemperatureT :

OtherConsiderations

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1/221( ) exp /

2 2x x BB

mP v mv k Tk Tπ

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

2 2 23

( )B

x y z

k Tm v v v

γ =+ +

RaymondAtta-Fynn

Choosingthetimemoleculardynamicsstep∆𝑡

= ∆𝑡 cannotbechosentobetoobigotherenergyconservationwillbedestroyed

= Typically,∆𝑡 ≤ 4fs isemployed[fs denotes“femto-seconds” i.e.1fs = 10Xf].

= Itisfairlycommontouse∆𝑡 = 1fs inpractice

= Ifthecomputationalcostisverycheaporifhighlyaccuratedynamicalresultsarerequired,thenchoose∆𝑡 assmallaspossible.E.g.∆𝑡 = 0.1fs isconsideredsmall

OtherConsiderations

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MolecularDynamicsinAction

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Moleculardynamicssimulationsofachargedmetalioninteractingwithwatermoleculesatroomtemperature(300K).

RaymondAtta-Fynn

Summary

6/4/2019 NSFSummerSchool:Introduction toMolecularDynamics 25Calculate results and finish

Set initial conditions and)( 0tir )( 0tiv

Get the forces )( ii rF

Use forces to olve the equations of motionnumerically over a short step

)()( ttt ii Δ+→ rr)()( ttt ii Δ+→ vv

tΔ

ttt Δ+=

Is ?maxtt >

If NVT ensemble, then use thermostat to control temperature T