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Quantum Mechanics of n Electron Systems in terms of the Cumulants of the Reduced Density

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52 pages
Quantum Mechanics of n-Electron Systems in terms of the Cumulants of the Reduced Density Matrices Werner Kutzelnigg and Debashis Mukherjee Lehrstuhl fur Theoretische Chemie, Ruhr-Universitat Bochum D-44780 Bochum, Germany, and Department of Physical Chemistry, Indian Association for the Cultivation of Science, Calcutta 700 032, India November 29, 2005

  • rs pq

  • fock space

  • reduced density

  • ruhr-universitat bochum

  • overlap between exact

  • excitation operators


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QuantumMechanicsofn-ElectronSystemsintermsoftheCumulantsoftheReducedDensityMatricesWernerKutzelniggandDebashisMukherjeeLehrstuhlfu¨rTheoretischeChemie,Ruhr-Universita¨tBochumD-44780Bochum,Germany,andDepartmentofPhysicalChemistry,IndianAssociationfortheCultivationofScience,Calcutta700032,IndiaNovember29,2005
IntroductionHΨ=EΨ;Don’tsearchforΨ!1.Ψcontainstoomuchirrelevantinformation.2.OverlapbetweenexactandanyapproximativeΨdecreasesexponentiallywiththeparticlenumbern.3.Ψisnotanextensive(additivelyseparable)quantity.FormulatetheenergyE=hΨ|H|Ψi/hΨ|ΨiandthestationarityconditionδE=0entirelyintermsofadditivelyseparablequantities.Goodcandidates:theone-particledensitymatrixγ=γ1andthek-particledensitycumulantsλk.1
Excitationoperatorsandreduceddensitymatrices.Startfromorthonormalbasis{ψp}ofspinorbitals.Creationandannihilationoperatorsfortheψp:ap;aq=aqExcitationoperators;W.K.JCP77,3081(1982)aqp=apaqarpsq=aqaparasasptqur=araqapasatau;etc.(1))2()3()4((particle-number-conservingnormalproductsofcreationandannihilationoperators).2
ConsiderastatedescribedbythewavefunctionΨ:k-particledensitymatrices:Normalizationppγ1:γq=hΨ|aq|Ψiγ2:γrpsq=hΨ|arpsq|Ψiγ3:γsptqur=hΨ|asptqur|ΨiTrγk=n!/(nk)!Trγ1=nTrγ2=n(n1)Easilygeneralizedtoensemblestates.3)5()6()7(()8)9(()01
Hamiltonianinconfigurationspace:nn1XXH(1,2,3....n)=h(k)+k=1k<l=1rklFockspaceHamiltonian:;(Einsteinsummationconvention):1H=hqpapq+grpsqaprqs2phq=hψq|h|ψpi1grpsq=hψr(1)ψr(2)||ψp(1)ψq(2)ir21Energyexpectationvalue:1E=hqpγpq+grpsqγprqs2)11(1()2)31()41(51()One-particledensitymatrix:γ=γ1;Naturalorbital(NSO)basis:γqp=npδqp0np1:occupationnumberofthepthNSO4