Time dependent density functional theory calculati

arXiv:0801.1388v1[physics.atm-clus]9Jan2008TimedependentdensityfunctionaltheorycalculationofvanderWaalscoefficientC6ofalkali-metalatomsLi,Na,K,alkalidimersLi2,Na2,K2andsodiumclustersNanArupBanerjeeaandJochenAutschbachb,(a)LaserPhysicsApplicationDivision,RajaRamannaCentreforAdvancedTechnologyIndore452013,India(b)DepartmentofChemistry,StateUniversityofNewYorkatBuffalo,Buffalo,NewYork14260-3000AbstractInthispaperweemployall-electrontimedependentdensityfunctionaltheory(TDDFT)tocalculatethelongrangedipole-dipoledispersioncoefficient(vanderWaalscoefficient)C6ofalkali-metalatomsLi,Na,K,alkali-metalatomdimersLi2,Na2,K2andsodiumclusterscontainingevennumberofatomsrangingfrom2to20atoms.ThedispersioncoefficientsareobtainedviaCasimir-Polderexpressionwhichrelatesittothefrequencydependentlinearpolarizabiltyatimaginaryfrequencies.ThefrequencydependentpolarizabilitiesarecalculatedbyemployingTDDFT–basedcompletesum-over-statesexpressionsfortheatoms,anddirectTDDFTlinearresponsetheoryfortheclosedshelldimersandclusters.1I.INTRODUCTIONThecontributionofthelong-rangevanderWaalsforce(dispersionforce)totheinterac-tionbetweentwomany-electronsystemsisquitesignificant.Thisforceplaysanimportantroleinthedescriptionmanyphysicalandchemicalphenomenasuchasadhesion,surfacetension,physicaladsorptionetc.Physically,thislongrangeforcearisesfromthecorrelationbetweentheelectrondensityfluctuationsatwidelyseparatedspatiallocationssuchthattheelectronsbelongingtothedifferentmoleculesaredistinguishable.Forsuchalargesepara-tionamathematicalexpressionforthepotentialcorrespondingtothelongrangedispersionforceisobtainedbyemployingperturbationtheoryforthecalculationofthesecond-orderchangeinenergyduetocoulombinteractionbetweentwochargedistributions.Thefirsttermintheperturbativeexpansionoftheinteractionpotential(afterorientationalaverageshavebeenperformed)decaysas−C6/R6,whereRistheintermoleculardistanceandthevanderWaalscoefficientC6describesthedipole-dipoleinteractionbetweenthetwopolarizablesystems.ThecalculationofthiscoefficientcanbeperformedbyusingtheCasimir-Polderexpression[1,2]whichrelatesittothedynamicdipolepolarizabilityatimaginaryfrequen-cies.Thedynamicpolarizabilitywhichdescribestheresponseofanatomormoleculetoaweak,timedependentexternalelectricfieldhasbeenwellstudied.Thereexistanum-berofwavefunction–basedab-initiomethodsforcalculatingthisquantityatvaryinglevelsofsophisticationtakingintoaccountelectroncorrelation.Alternatively,timedependentdensityfunctionaltheory(TDDFT)representsanefficienttoolforfirst–principlestheoreti-calcalculationsofdynamicpolarizabilityofatomsandmolecules,typicallyatsignificantlylowercomputationalcostascorrelatedwavefunction–basedmethods.TDDFToftenyieldsanaccuracysimilartowavefunctionbasedcorrelatedmethodsaslongaslong–rangechargetransferexcitationsarenotoverlyimportantforthepropertyunderconsideration.Recently,TDDFThasbeenappliedtocalculatevanderWaalscoefficientsofvarietyofatomsandsmallmolecules[3]extensivesetofpolycyclicaromatichydrocarbons[4,5],C60andC70molecules[4,6]andsmallsizedsodiumclustersupto20atoms[7,8].TheresultsofthesecalculationsarequiteencouragingandclearlydemonstratethatTDDFTbasedmethodyieldsresultsforC6whichareveryclosetoothertheoreticalandexperimentaldata(whereavailable).Thissuccesshasmotivatedustofurtherapplyall-electronTDDFTbasedmethodtocalculatethecoefficientC6tovarietyofsystemscontainingalkali-metalatoms.2Inthispaper,wecalculateC6forfollowinginteractions(i)alkaliatom-alkaliatom(Li,Na,andK),(ii)alkaliatom-alkaliatomdimer(Li2,Na2,andK2),and(iii)alkaliatom-sodiumcluster(Nan,wheren≤20).Thechoiceofalkali-metalatomsandtheirmoleculesforourcalculationsarealsomotivatedbytherecentdevelopmentoflasercoolingandtrappingoftheseatoms[9].Theseadvanceshaverekindledtheinterestintheknowledgeoflongrangeforcesbetweenalkaliatomsandclustersthereofastheseforcesplayanimportantroleinthepropertiesofcoldgasesofatomsandmolecules.Moreover,theresultsofthecalculationsofC6forsystems(i)and(ii)giveagoodopportunitytoassesstheaccuracyofTDDFTtocalculateC6foralkaliatomsanddiaomicsasforthesesystemsaccuratetheoreticaldataobtainedbycorrelatedwave-functionbasedmethodsexistintheliterature.Moreover,thecalculationsofC6forthealkaliatom–NaninteractionsismotivatedbyanexperimentperformedbyKresinandco-workers[10].Kresinetal.measuredtheintegralscatteringcrosssectioninlowenergycollisionsbetweenabeamofsodiumclustersandvapoursofalkali-metalatomsLi,Na,andK.Forlowenergycollisions,theintegralscatteringcrosssectiondependsonthevanderWaalscoefficientC6.TheexperimentalresultsforintegralscatteringcrosssectionmatchedquitewellwiththetheoreticalpredictionswhichwereobtainedbyemployingtheLondondispersionformulaforC6.TheLondondispersionformulaassumesthatallthestrengthofdipoletransitionisconcentratedinasinglepeaklocatedataneffectivefrequencyandtheformulainvolvesonlytwoparameters.(seeEq.(3)below).WewillcompareourTDDFTbasedresultswiththenumbersusedinRef.[10]toreproducetheirexperimentaldata.Wenotethat,tothebestofourknowledge,nofirst–principlestheoreticaldataforC6betweenalkali-metalatomandsodiumclustershavesofarbeenre