幾種常見的優化方法電子結構幾何機構函數穩定點最小點Taylor展開：V(x)=V(xk)+(x-xk)V’(xk)+1/2(x-xk)2V’’(xk)+…..當x是3N個變量的時候，V’(xk)成為3Nx1的向量，而V’’(xk)成為3Nx3N的矩陣，矩陣元如：Hessian1幾種常見的優化方法電子結構函數穩定點Taylor展開：當x一階梯度法a.SteepestdescendentSk=-gk/|gk|directiongradient知道了方向，如何確定步長呢？最常用的是先選擇任意步長l，然后在計算中調節用體系的能量作為外界衡量標準，能量升高了則逐步減小步長。robust,butslow最速下降法2一階梯度法Sk=-gk/|gk|directi最陡下降法（SD）

3最陡下降法（SD）3b.ConjugateGradient（CG)共軛梯度第k步的方向標量UsuallymoreefficientthanSD,alsorobust不需要外界能量等作為衡量量利用了上一步的信息4b.ConjugateGradient（CG)共軛2。二階梯度方法這類方法很多，最簡單的稱為Newton-Raphson方法，而最常用的是Quasi-Newton方法。Quasi-Newton方法：useanapproximationoftheinverseHessian.Formofapproximationdiffersamongmethods牛頓-拉夫遜法BFGSmethodBroyden-Fletcher-Golfarb-ShannoDFPmethodDavidon-Fletcher-Powell52。二階梯度方法這類方法很多，最簡單的稱為Newton-RaMoleculardynamics分子動力學HistoryItwasnotuntil1964thatMDwasusedtostudyarealisticmolecularsystem,inwhichtheatomsinteractedviaaLennard-Jonespotential.Afterthispoint,MDtechniquesdevelopedrapidlytoencompassdiatomicspecies,water(whichisstillthesubjectofcurrentresearchtoday!),smallrigidmolecules,flexiblehydrocarbonsandnowevenmacromoleculessuchasproteinsandDNA.Theseareallexamplesofcontinuousdynamicalsimulations,andthewayinwhichtheatomicmotioniscalculatedisquitedifferentfromthatinimpulsivesimulationscontaininghard-corerepulsions.6Moleculardynamics分子動力學HistWhatcanwedowithMD–CalculateequilibriumconfigurationalpropertiesinasimilarfashiontoMC.–Studytransportproperties(e.g.mean-squareddisplacementanddiffusioncoefficients).–MDintheNVT,NpTandNpHensembles–Theunitedatomapproximation–ConstraintdynamicsandSHAKE–Rigidbodydynamics–MultipletimestepalgorithmsExtendthebasicMDalgorithm7WhatcanwedowithMD–Calcul‘Impulsive’moleculardynamics

1.Dynamicsofperfectly‘hard’particlescanbesolvedexactly,butprocessbecomesinvolvedformanypart(N-bodyproblem).2.Canuseanumericalschemethatadvancesthesystemforwardintimeuntilacollisionoccurs. 3.Velocitiesofcollidingparticles(usuallyapair!)thenrecalculatedandsystemputintomotionagain.4.Simulationproceedsbyfitsandstarts,withameantimebetweencollisionsrelatedtotheaveragekineticenergyoftheparticles.5.Potentiallyveryefficientalgorithm,butcollisionsbetweenparticlesofcomplexshapearenoteasytosolve,andcannotbegeneralisedtocontinuouspotentials. 88Continuoustimemoleculardynamics1.Bycalculatingthederivativeofamacromolecularforcefield,wecanfindtheforcesoneachatomasafunctionofitsposition.2.Requireamethodofevolvingthepositionsoftheparticlesinspaceandtimetoproducea‘true’dynamicaltrajectory.3.StandardtechniqueistosolveNewton’sequationsofmotionnumerically,usingsomefinitedifferencescheme,whichisknownasintegration.4.ThismeansthatweadvancethesystembysomesmalltimestepΔt,recalculatetheforcesandvelocities,andthenrepeattheprocessiteratively.5.ProvidedΔtissmallenough,thisproducesanacceptableapproximatesolutiontothecontinuousequationsofmotion.9ContinuoustimemoleculardynaExampleofintegratorforMDsimulationOneofthemostpopularandwidelyusedintegratorsistheVerletleapfrogmethod:positionsandvelocitiesofparticlesaresuccessively‘leap-frogged’overeachotherusingaccelerationscalculatedfromforcefield.TheVerletschemehastheadvantageofhighprecision(oforderΔt4),whichmeansthatalongertimestepcanbeusedforagivenleveloffluctuations.Themethodalsoenjoysverylowdrift,providedanappropriatetimestepandforcecut-offareused.r(t+Dt)=r(t)+v(t+Dt/2)Dtv(t+Dt/2)=v(t-Dt/2)+a(t+Dt/2)Dt10ExampleofintegratorforMDsOtherintegratorsforMDsimulationsAlthoughtheVerletleapfrogmethodisnotparticularlyfast,thisisrelativelyunimportantbecausethetimerequiredforintegrationisusuallytrivialincomparisontothetimerequiredfortheforcecalculations.Themostimportantconcernforanintegratoristhatitexhibitslowdrift,i.e.thatthetotalenergyfluctuatesaboutsomeconstantvalue.Anecessary(butnotsufficient)conditionforthisisthatitissymplectic.Crudelyspeaking,thismeansthatitshouldbetimereversible(likeNewton’sequations),i.e.ifwereversethemomentaofallparticlesatagiveninstant,thesystemshouldtracebackalongitsprevioustrajectory.11OtherintegratorsforMDsimulOtherintegratorsforMDsimulationsTheVerletmethodissymplectic,butmethodssuchaspredictor-correctorschemesarenot.Non-symplecticmethodsgenerallyhaveproblemswithlongtermenergyconservation.Havingachievedlowdrift,wouldalsoliketheenergyfluctuationsforagiventimesteptobeaslowaspossible.Alwaysdesirabletousethelargesttimesteppossible.Ingeneral,thetrajectoriesproducedbyintegrationwilldivergeexponentiallyfromtheirtruecontinuouspathsduetotheLyapunovinstability.However,thisdoesnotconcernusgreatly,asthethermalsamplingisunaffected?expectationvaluesunchanged.12OtherintegratorsforMDsimulChoosingthecorrecttimestep…1.

Thechoiceoftimestepiscrucial:tooshortandphasespaceissampledinefficiently,toolongandtheenergywillfluctuatewildlyandthesimulationmaybecomecatastrophicallyunstable(“blowup”).2.Theinstabilitiesarecausedbythemotionofatomsbeingextrapolatedintoregionswherethepotentialenergyisprohibitivelyhigh(e.g.atomsoverlapping).3.Agoodruleofthumbisthatwhensimulatinganatomicfluid,thetimestepshouldbecomparabletothemeantimebetweencollisions(about5fsforArat298K).4.Forflexiblemolecules,thetimestepshouldbeanorderofmagnitudelessthantheperiodofthefastestmotion(usuallybondstretching:C—Haround10fssouse1fs).13ChoosingthecorrecttimestepForclassicMD,therecouldbemanytrickstospeedupcalculations,allcenteringaroundreducingtheeffortinvolvedinthecalculationoftheinteratomicforces,asthisisgenerallymuchmoretime-consumingthanintegration.ForexampleTruncatethelong-rangeforces:charge-charge,charge-dipoleLook-uptablesForfirstprinciplesMD,asforcesareevaluatedfromquantummechanics,weareonlyconcernedwiththetime-step.14ForclassicMD,therecouldbeBecausetheinteractionsarecompletelyelasticandpairwiseacting,bothenergyandmomentumareconserved.Therefore,MDnaturallysamplesfromthemicrocanonicalorNVEensemble.Asmentionedpreviously,theNVEensembleisnotveryusefulforstudyingrealsystems.Wewouldliketobeabletosimulatesystemsatconstanttemperatureorconstantpressure.ThesimplestMD,likeverletmethod,isadeterministicsimulationtechniqueforevolvingsystemstoequilibriumbysolvingNewton’slawsnumerically.15BecausetheinteractionsarecMDindifferentthermodynamicensemblesInthislecture,wewilldiscusswaysofusingMDtosamplefromdifferentthermodynamicensembles,whichareidentifiedbytheirconservedquantities.

Canonical(NVT)–Fixednumberofparticles,totalvolumeandtemperature.Requirestheparticlestointeractwithathermostat.

Isobaric-isothermal(NpT)–Fixednumberofparticles,pressureandtemperature.Requiresparticlestointeractwithathermostatandbarostat.

Isobaric-isenthalpic(NpH)–Fixednumberofparticles,pressureandenthalpy.Unusual,butrequiresparticlestointeractwithabarostatonly.16MDindifferentthermodynamicAdvancedapplicationsofMDWewillthenstudysomemoreadvancedMDmethodsthataredesignedspecificallytospeedup,ormakepossible,thesimulationoflargescalemacromolecularsystems.Allthesemethodsshareacommonprinciple:theyfreezeout,ordecouple,thehighfrequencydegreesoffreedom.Thisenablestheuseofalargertimestepwithoutnumericalinstability.Thesemethodsinclude:–Unitedatomapproximation–ConstraintdynamicsandSHAKE–Rigidbodydynamics–Multipletimestepalgorithms17AdvancedapplicationsofMD17RevisionofNVEMDLet’sstartbyrevisinghowtodoNVEMD.Recallthatwecalculatedtheforcesonallatomsfromthederivativeoftheforcefield,thenintegratedthee.o.m.usingafinitedifferenceschemewithsometimestepΔt.Wethenrecalculatedtheforcesontheatoms,andrepeatedtheprocesstogenerateadynamicaltrajectoryintheNVEensemble.Becausethemeankineticenergyisconstant,theaveragekinetictemperatureTKisalsoconstant.However,inthermalequilibrium,weknowthatinstantaneousTKwillfluctuate.IfwewanttosamplefromtheNVTensemble,weshouldkeepthestatisticaltemperatureconstant.18RevisionofNVEMD18ExtendedLagrangiansThereareessentiallytwowaystokeepthestatisticaltemperatureconstant,andthereforesamplefromthetrueNVTensemble.–Stochastically,usinghybridMC/MDmethods–Dynamically,viaanextendedLagrangianWewilldescribethelattermethodinthislectureAnextendedLagrangianissimplyawayofincludingadegreeoffreedomwhichrepresentsthereservoir,andthencarryingoutasimulationonthisextendedsystem.Energycanflowdynamicallybackandforthfromthereservoir,whichhasacertainthermal‘inertia’associatedwithit.AllwehavetodoisaddsometermstoNewton’sequationsofmotionforthesystem.19ExtendedLagrangians19ExtendedLagrangiansThestandardLagrangianLiswrittenasthedifferenceofthekineticandpotentialenergies:Newton’slawsthenfollowbysubstitutingthisintotheEuler-Lagrangeequation:Newton’sequationsandLagrangianformalismareequivalent,butthelatterusesgeneralisedcoordinates...20ExtendedLagrangians..20CanonicalMDSo,ourextendedLagrangianincludesanextracoordinateζ,whichisafrictionalcoefficientthatevolvesintimesoastominimisethedifferencebetweentheinstantaneouskineticandstatisticaltemperatures.Themodifiedequationsofmotionare:TheconservedquantityistheHelmholtzfreeenergy.(modifiedformofNewtonII)21CanonicalMD(modifiedformofCanonicalMDByadjustingthethermostatrelaxationtimetT

(usuallyintherange0.5to2ps)thesimulationwillreachanequilibriumstatewithconstantstatisticaltemperatureTS.TSisnowaparameterofoursystem,asopposedtothemeasuredinstantaneousvalueofTKwhichfluctuatesaccordingtotheamountofthermalenergyinthesystematanyparticulartime.ToohighavalueoftTandenergywillflowveryslowlybetweenthesystemandthereservoir(overdamped).ToolowavalueoftTandtemperaturewilloscillateaboutitsequilibriumvalue(underdamped).ThisistheNosé-Hooverthermostatmethod.22CanonicalMD22CanonicalMDTherearemanyothermethodsforachievingconstanttemperature,butnotallofthemsamplefromthetrueNVTensembleduetoalackofmicroscopicreversibility.Wecallthesepseudo-NVTmethods,andtheyinclude:–BerendsenmethodVelocitiesarerescaleddeterministicallyaftereachstepsothatthesystemisforcedtowardsthedesiredtemperature–GaussianconstraintsMakesthekineticenergyaconstantofthemotionbyminimisingtheleastsquaresdifferencebetweentheNewtonianandconstrainedtrajectoriesThesemethodsareoftenfaster,butonlyconvergeonthetruecanonicalaveragepropertiesasO(1/N).23CanonicalMD23Isothermal-isobaricMDWecanapplytheextendedLagrangianapproachtosimulationsatconstantpressurebysimplyaddingyetanothercoordinatetooursystem.Weuseη,whichisafrictionalcoefficientthatevolvesintimetominimisethedifferencebetweentheinstantaneouspressurep(t),measuredbyavirialexpression,andthepressureofanexternalreservoirpext.TheequationsofmotionforthesystemcanthenbeobtainedbysubstitutingthemodifiedLagrangianintotheEuler-Lagrangeequations.Thesenowincludetworelaxationtimes:oneforthethermostattT,andoneforthebarostattp.24Isothermal-isobaricMD24Isothermal-isobaricMDTheisknownastheNosé-Hoovermethod(Melchionnatype)andtheequationsofmotionare:25Isothermal-isobaricMD25Isothermal-isobaricMD26Isothermal-isobaricMD26Constraintdynamicsfreezethebondstretchingmotionsofthehydrogens(oranyotherbond,inprinciple).Weapplyasetofholonomicconstraintstothesystem,whichar