3、周期性边界条件

1.7ModelSystemandInteractionPotentialInmostofthiscourse,themicroscopicofasystemmaybespecifiedintermsofthepositionandmomentaofaconstituentsetofparticles.Inthiscasetherapidmotionoftheelectronshavebeenaveragedout.)()(),(qppqVKHHamiltonianKineticenergyPotentialenergyNiiimpK122/iiijiijijkkjijiivvvV),,(),()(321rrrrrrAnalysisofthePotentialEnergyiiijiijijkkjijiivvvV),,(),()(321rrrrrrExternalpotentialPairpotentialabout90%ThreebodycontributionInFCCcrystal,upto10%Effectivepairpotential),()(21jiiiijeffivvVrrr)()(),(2rvrvvijjieffrr1.7.1Effectivepairpotentialforsphericalmolecules•1.7.1.1Hard-spherepotentialForthepurposesofinvestigatinggeneralpropertiesofliquidsandforcomparisonwiththeory,highlyidealizedpairpotentialsmaybeofvalue.Inthissection,Iwillintroducehard-sphere,square-well,YukawaandLennard-Jonespotentials,etc.)(0)()(HSrrrvisthediameterofhardspheresr)(rv1.7.1.2Square-Wellpotential)(0)()()(SWrrrrvr)(rvSWpotentialisthesimplestoneincludingtheattractiveforcesandcanbeappliedtoinertgasesandsomenon-polarsubstances,etc.1.7.1.3Yukawapotentialrrrrrv/]/)(exp[)(Y01234-1.5-1.0-0.50.00.51.01.52.0)(rv/rWhen,itcanbeusedtomodelArreasonablywell.8.1Yukwapotentialcanalsobeusedtomodelplasma,colloidalparticles,andsomeelectricalinteractions.1.7.1.4Lennard-Jonespotential612)/()/(4)(rrrvLJ0.30.40.50.60.70.8-150-100-50050100150v(r)/k(K)r(nm)Forargon:K120/knm34.0CodesforcalculatingthetotalpotentialV=0.0DO100I=1,N-1RXI=RX(I)RYI=RY(I)RZI=RZ(I)DO100J=I+1,NRXIJ=RXI-RX(J)RYIJ=RYI-RY(J)RZIJ=RZI-RZ(J)RIJSQ=RXIJ**2+RYIJ**2+RZIJ**2SR2=SIGSQ/RIJSQSR6=SR2*SR2*SR2SR12=SR6**2V=V+SR12-SR6100CONTINUEV=V*4.0*EPSLONThecoordinatevectorsofLJatomsarestoredinthreearraysRX(I),RY(I),RZ(I)2ijrCalculate)(jixx6)/(ijr1.7.1.5PotentialsforionsAsimpleapproachtoconstructpotentialsforionsistosupplementoneoftheabovepairpotentialswiththeCoulombcharge-chargeinteraction:ijrjiXrezzrvrv024)()(WhereXmaybeHS,SW,LJ.Thepopularoneis:ijrjiijrrezzrrv024)(1.7.2potentialformacromoleculesTheenergy,V,isafunctionoftheatomicpositions,R,ofalltheatomsinthesystem,theseareusuallyexpressedintermofCartesiancoordinates.Thevalueoftheenergyiscalculatedasasumofinternal,orbondedterms,whichdescribethebonds,anglesandbondrotationsinamolecule,andasumofexternalornonbondedterms.Thesetermsaccountforinteractionsbetweennonbondedatomsoratomsseparatedby3ormorecovalentbonds.bondednonbonded)(VVVRBondedpotentialBondangle-bendBond-strentchTorsion(rotate-along-bond)Angle-bendpotentialThedeviationofanglesfromtheirreferencevaluesisfrequentlydescribedusingaHooke’slaworharmonicpotential:anglesKV20)(2)(Bond-stretchpotentialpairslllKlV2,120)()(TorsionpotentialTorsionalpotentialsarealmostalwaysexpressedasacosineseriesexpansion.)cos(12)(4,1nKVpairs‘barrier’heightDetermineswherethetorsionanglepassesthroughitsminimumvalue.Thenumberofminimumpointsinthefunctionasthebondisrotatedthrough2.2/12)/(*mtt1.8ReducedUnitsWhyusereducedunit?Avoidsthepossibleembarrassmentofconductingessentiallyduplicatesimulations.Andtherearealsotechnicaladvantagesintheuseofreducedunitsduetothesimulationboxisinthemagnitudeofmolecularscale.Density3*Temperature/*TkTBEnergy/*UUPressure/*3PPForce/f*fTorque/*Surfacetension/*2TimeReducedunit----continueDiffusioncoefficient21/2*(/)DDmViscosity21/2*/()mThermalconductivity21/2*(/)(/)BkmSIUnits:W/(mK)Pasm2/sTestofunit(Forexample,viscosity):21/22221/2m[*]Pas(Jkg)mNms1(NmNs/m)Reducedunit----continueItshouldbepointedthatallthereducedunitsintheprevioustwoslidesarebasedonLennard-Jonesinteractionpotential.Forhardspherefluid,thereducedunitsareobtainedusingkBTtoreplace.Thereducedunitsforotherpropertiessuchaschemicalpotentialandheatcapacitycanbededucedfromtheunitsgiveninthissection.Especially,forelectrolytesolution,wecanuse20/4BrBelkT2*/QQeSurfacechargedensity:1.9SimulationBoxandItsBoundaryConditionsComputersimulationsareusuallyperformedonasmallnumberofmolecules,10N10,000.ThetimetakenforadoubleloopusedtoevaluatetheforcesandpotentialenergyisproportionaltoN2.Whetherornotthecubeissurroundedbyacontainingwall,moleculesonthesurfacewillexperiencequitedifferentforcesfrommoleculesinthebulk.Itisessentialtoproposepropermethodstoovercometheproblemofsurfaceeffects.1.9.1SimulationboxxyzCubeHexagonalprismxyzExample:DNAsimulation1.9.1Simulationbox---continueTruncatedoctahedronRhombicdodecahedron1.9.2PeriodicboundaryconditionBAHDGFECBAHDGFECInacubicbox,thecutoffdistanceissetequaltoL/2.MinimumimageconventionAE•Asideviewofthebox(b)AtopviewoftheboxBDCAEHFGSimulationofmoleculesinslit-likepore1.9.3ComputercodeforperiodicboundariesHowdowehandleperiodicboundariesandtheminimumimageconventioninasimulationprogram?Letusassume,initially,theNmoleculesinthesimulationliewithacubicboxofsideBOXL,withtheoriginatitscenter,i.e.,allcoordinatelieintherange(-BOXL/2,BOXL/2).Afterthemoleculeshavebeenmoved,wemusttestthepositionimmediatelyu