Ewald summation technique for interaction site mod

arXiv:physics/9901031v1[physics.comp-ph]19Jan1999EwaldsummationtechniqueforinteractionsitemodelsofpolarfluidsIgorP.OmelyanInstituteforCondensedMatterPhysics,NationalUkrainianAcademyofSciences1SvientsitskySt.,UA-290011Lviv,Ukraine∗AbstractAcomputeradaptedfluctuationformulaforthecalculationofthewavevec-tor-andfrequency-dependentdielectricpermittivityforinteractionsitemod-elsofpolarfluidswithintheEwaldsummationtechniqueisproposedandappliedtomoleculardynamicssimulationsoftheTIP4Pwater.TheformulaisanalyzedandoptimalparametersoftheEwaldmethodareidentified.Acomparisonoftheobtainedresultswiththoseevaluatedwithinthereactionfieldapproachismade.Keywords:Computersimulation;Ewaldtechnique;DielectricpropertiesPACSnumbers:61.20.Ja;77.22.-d;24.60.-k∗E-mail:[e-mail]11MotivationInordertoachieveamacroscopicbehaviourforinvestigatedquantitiesincom-puterexperimentbasedontheobservationoffinitesystems,itisnecessarytoreducetheinfluenceofsurfaceeffectstoaminimum.Thisisespeciallyimportantforpolarsystemswiththelong-rangenatureofinteractions.Excludingthesurfaceeffectsinsimulationscanbeperformedwithineitherthereactionfield(RF)[1–5]orEwaldsummation[6–10]techniques.Nowanequivalencebetweenthesetechniqueshasbeenestablishedformodelsofpointdipolesandpropercalculationscanbemadewithineithermethod[11,12].Theexplicitconsiderationofafinite-sizemediumleadtocomputeradaptedfluctuationformulas[11–17]whichallowonetocalculateboundaryfreevaluesforthedielectricconstantonthebasisofdipolemomentfluc-tuationsobtainedinsimulations.TheseformulasdifferconsiderablywithrespecttothoseknownfromthetheoryofmacroscopicsystemseveniftheEwaldmethodisused[11].Detailsofthesummationmustbetakenintoaccountexplicitlyinordertoobtaincorrectvaluesforthebulkdielectricconstant.Previously[18–20],thestandardRFofpointdipoles(PDRF)[3]hasbeenappliedtoinvestigatemorerealistic,interactionsitemodels(ISMs)[21]ofpolarfluids.ThePDRF,however,beingexactforpointdipolemodels,maynotbenecessarilyapplicablytointerpretsimulationresultsforarbitrarysystems[5].Recently,ithasbeenshownbyactualcalculationsforaMCYwatermodelthatuncertaintiesforthedielectricquantitiesaresignificantifthePDRFisusedincomputersimulationsofISMsandanalternativescheme,theinteractionsitereactionfield(ISRF)geometry,hasbeenproposed[22].Atthesametime,thereisnotsuchanapproachconcerningtheentirewavevectorandfrequencydependenceforthedielectricpermittivityofISMswithintheEwaldgeometry.Themainattentionofprevioussimulations[23–31]wasdirectedtostudythedielectricpropertiesinthestaticlimitoratzeroandsmallwavevectorvalues.Moreover,themacroscopicfluctuationformulashavebeenusedinthesimulationresultswithouttakingintoaccountdetailsoftheEwaldsummation.InthepresentpaperweapplytheEwaldtechniquefortreatingCoulombinter-actionsinISMs.Thepaperisorganizedasfollows.Afluctuationformulasuitable2forthecalculationofthewavevector-andfrequency-dependentdielectricconstantisderivedinSec.2andoptimalvaluesoftheEwaldparametersaredeterminedthere.TheresultsofmoleculardynamicssimulationsoftheTIP4PwaterfortimecorrelationfunctionsrelatedtodielectricpolarizationarepresentedinSec.3.TheseresultsarecomparedwiththosecomputedwithintheISRFgeometry.ConcludingremarksaregiveninSec.4.2EwaldsummationforISMsConsiderapolarfluidwithNmoleculescomposedofMinteractionsiteswhichareconfinedinavolumeV.ThemicroscopicelectricfieldcreatedbythemoleculesatpointrandtimetcanbepresentedasˆE(r,t)=ZVD(r−r′)ˆQ(r′,t)dr′,whereˆQ(r,t)=PNi=1PMa=1qaδ(r−rai(t))isthemicroscopicchargedensity,rai(t)andqadenotethepositionandcharge,respectively,ofsiteawithinthemoleculeiandD(ρ)=−∇1/ρistheoperatoroftheCoulombinteractions.Obviously,thefieldˆE(r,t)forinfinitesystems(N,V→∞)cannotberepro-ducedexactlyincomputerexperimentwhichdealswithafinite,asarule,cubicvolumeV=L3,whereListhelengthofthesimulationboxedge.However,usingthelatticesummation,amacroscopicbehaviourcanbeachievedconsideringtheinteractionsbetweensiteswithinthebasiccellaswellasaninfinitelatticeofitsperiodicimages(theperiodicboundaryconvention).Thiscanbeinterpretedasaneffectiveinteractionwhichinvolvesonlythesitesinthebasiccellandcharac-terizedbyamodifiedoperatorD(ρ)=PnD(ρ+nL),wherethesummationisextendedoverallvectorsnwithintegercomponents.Itismoreconvenienttorep-resentthelatticesuminaform,proposedbyEwaldandKornfeld(EK)[6],namely,D(ρ)=D1(ρ)+D2(ρ),whereD1(ρ)=X0≤|n|≤ND(ρ+nL)nerfc(η|ρ+nL|+2η√π|ρ+nL|exp(−η2|ρ+nL|2)o(1)isasuminrealcoordinatespace,whileD2(ρ)=1VX0|k|≤kmaxD(k)exp(−k2/4η2+ik·ρ)(2)3correspondstosummationoverwavevectorsk=2πn/LofthereciprocallatticespaceandD(k)=Zdre−ik·ρD(ρ)=−4πik/k2isthespatialFouriertransformofD(ρ).Fortheidealizedsummations(N→∞,kmax→∞),thetotalsumof(1)and(2)isindependentontheparameterη.ThemainadvantageoftheEKrepresentationliesinthefactthatvaluesforηcanbefoundinsuchawaythatthebothsums,D1andD2,convergeveryquicklyandmaybetruncatedafterafinitenumberofterms.Iftheparameterηischosensufficientlylarge,wecanrestrictourselvestoasingleterm(N=0)intherealspacesum,correspondingtothebasiccelltowhichtoroidalboundaryconditionsareapplied,and,additionally,tothesphericaltruncation|ρ|≤R,where