Analysis of Floquet formulation of time-dependent

arXiv:cond-mat/0611428v1[cond-mat.mtrl-sci]16Nov2006AnalysisofFloquetformulationoftime-dependentdensity-functionaltheoryPrasanjitSamal∗andManojK.Harbola†DepartmentofPhysics,IndianInstituteofTechnology,KanpurU.P.208016,IndiaAbstractFloquetformulationoftime-dependentdensity-functionaltheoryisrevisitedinlightofitsrecentcriticism[MaitraandBurke,Chem.Phys.Lett.359(2002),237].ItisshownthatFloquettheoryiswellfoundedanditscriticismhasoverlookedimportantpointsofboththeRunge-GrossformalismandFloquetformulationitself.WesubstantiateouranalysisbyexamplessimilartothatconsideredbyMaitraandBurke.∗Electronicaddress:Fax:[phone];E-mail:[e-mail]†Electronicaddress:Fax:[phone];E-mail:[e-mail]1Densityfunctionaltheory(DFT)isawellestablishedtheoryfortheground-statesofCoulombicsystems.Itisconceptuallysimpleandpracticallyusefulinvariousbranchesofphysics,chemistryandmaterialscience.BasedontheworkofHohenbergandKohn(HK)[1]andKohnandSham(KS)[2],DFThasbecomeapracticaltoolfortheelectronicstruc-turecalculationofatoms,moleculesandsolids[3].Time-independentdensityfunctionalformalismisalsobeingextendedtodealwithexcited-states[4,5,6,7,8,9,10,11].Ontheotherhand,time-dependentphenomenaarenotaccessiblethroughtraditionalDFT.Time-dependentdensityfunctionaltheory(TDDFT)[12,13,14]isthegeneralizationoftheground-stateDFTtotime-dependentproblems.ThedevelopmentofTDDFTisrelativelyrecent.ThecentralresultofmodernTDDFTisasetoftime-dependentKohn-Shamequationswhicharestructurallysimilartothetime-dependentHartree-Fockequationsbutincludeinprincipleexactlyallthemany-bodyeffectsthroughalocaltime-dependentexchange-correlationpotential.TDDFTallowsaccesstolotsofinterestingandimportantphenomenawhichcan’tbeeasilydealtwithinstaticDFT.Ithasbecomepopularforstudyingatomsandmoleculesinintenselaserfields[15],calculatingtheexcitationspectraandoscillatorstrengths[16,17],frequencydependentpolarizabilities,hyperpolarizabilities[18]andopticalresponseofsolids,etc.[19].TDDFTisespeciallyveryusefulforquantumcontrolproblems[20].Thisisbecauseforintenselaserfieldswherecorrelationeffectsarequitecrucial,TDDFTappearstobetheonlywayofstudyingthequantumdynamicsofamany-electronsystem.InitialstepstowardstherigorousfoundationofTDDFTweretakenbyDebandGhosh[12]andbyBartolotti[13].TheyformulatedandexploredHKandKStypetheoremsforthetime-dependentdensity.Eachofthesederivations,however,wasrestrictedtoaparticularclassofallowabletime-dependentpotentials(topotentialsperiodicintime).Becauseoftheperiodicityofthepotentialintime,DebandGhoshdevelopedTDDFTwiththeFloquetformulation.FurtherinrecenttimesTelnovandChu[21,22,23,24,25]havedevelopedapproachesforthenonperturbativetreatmentofthestrongfieldprocesses.ThesearetheapproacheswhichinvolvethetransformationoftheTDDFTequationsintoanequivalenttime-independentgeneralizedFloquetmatrixeigenvalueproblem.Avariational2perturbationmethodbasedontheFloquettheoryhasalsobeendevelopedandapplied[26]tostudyopticalpropertiesofatomswithinTDDFT.ThisperturbativeFloquetapproachissuitableforthetreatmentofweak-fieldprocesses.Ageneralproofofdensitytopotentialmapping(ρ(~r,t)→v(~r,t))foratime-dependentdensitywasgivenbyRungeandGross[14].RungeandGross(RG)[14]formallyestablishedtheTDDFTbyshowingthatforagiveninitialstatetheevolvingdensityρ(~r,t)uniquelydeterminesthecorrespondingtime-dependentpotentialv(~r,t).TwofeaturesofRGformalismarenoteworthy:onethatthetime-dependentwavefunctionforagivendensityhasfunctionaldependenceonboththedensityaswellastheinitialwavefunctionΨ0.SecondlytheformalismisvalidforpotentialsthatareTaylorseriesexpandableattheinitialtimet=t0.Thismakestheformalismappli-cabletosuddenlyswitched-onpotentialsbutnottotheadiabaticallyswitched-onpotentials.LikeinthecaseofstaticDFT,inTDDFTalsoaninteractingmany-electronsystemismappedontoafictitiousnon-interactingKohn-Shamsystemwiththesametime-dependentdensityastheinteractingsystem.InFloquetformulation[12,13]theKohn-Shamsystemisdevelopedintermsoftheperiodictime-dependentdensityonly.Ontheotherhand,inthetheRGformalismtheKohn-Shamwavefunctionisafunctionalofboththetime-dependentdensityandaninitialstate[27,28].TheuniquenessoftheKSsystemfollowsfromtheone-to-oneρ(~r,t)→v(~r,t)mapping.However,inarecentwork[29]byMaitraandBurkeithasbeensuggestedthataKSsystemwithintheFloquetformalismisnotunique.Thenon-uniquenessoftheKSsystemhasbeenattributedtoitsfunctionaldependenceontheinitialstate.Further,theysuggestthattheHohenberg-Kohnliketheorem[12]maynotexistinFloquettheoryofTDDFTbecausethe”groundFloquetstate”cannotbedefinedproperly.Thisappearstohavecreatedsomedoubts[30]abouttheFloquettheoryinTDDFT.ItisthepurposeofthispapertoanalyzetheFloquettheoryofTDDFTandshowthatitisperfectlyvalid.ThepresentanalysisalsobringsoutsubtledifferencesbetweentheRGandFloquettheoryandshowsthetwoformulationstobedistinctandfoundedontwodifferentprinciples.AssuchFloquettheoryofTDDFTcannotbethoughtofasaparticularcaseoftheRGtheoryandtheRGargumentscan’tbeappliedtoFloquetformulation.WebeginwithashortdescriptionofthefoundationsofFloquetandRGtheoryinTDDFT.3Foratime-dependent(T