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

Werner Kutzelnigg

Description

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

Sujets

• Fock space
• Université de la Ruhr à Bochum

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

QuantumMechanicsofn-ElectronSystemsin termsoftheCumulantsoftheReducedDensity Matrices WernerKutzelniggandDebashisMukherjee Lehrstuhlfu¨rTheoretischeChemie, Ruhr-Universita¨tBochum D-44780Bochum,Germany,and DepartmentofPhysicalChemistry, IndianAssociationfortheCultivationofScience, Calcutta700032,India November29,2005

Introduction HΨ=EΨ;Don’tsearchforΨ! 1.Ψcontainstoomuchirrelevantinformation. 2.OverlapbetweenexactandanyapproximativeΨ decreasesexponentiallywiththeparticlenumbern. 3.Ψisnotanextensive(additivelyseparable)quantity. FormulatetheenergyE= h Ψ | H | Ψ i / h Ψ | Ψ i andthestationaritycondition δE=0entirelyintermsofadditivelyseparablequantities. Goodcandidates:theone-particledensitymatrix γ=γ 1 andthek-particledensitycumulantsλ k . 1

Excitationoperatorsandreduced densitymatrices. Startfromorthonormalbasis { ψ p } ofspinorbitals. Creationandannihilationoperatorsfortheψ p : a p ;a q =a q † Excitationoperators;W.K.JCP77,3081(1982) a q p =a p a q a r p s q =a q a p a r a s a s p t q u r =a r a q a p a s a t a u ;etc. (1) )2( )3( )4( (particle-number-conservingnormalproductsofcreationandannihilation operators). 2

ConsiderastatedescribedbythewavefunctionΨ: k-particledensitymatrices: Normalization p p γ 1 :γ q = h Ψ | a q | Ψ i γ 2 :γ r p s q = h Ψ | a r p s q | Ψ i γ 3 :γ s p t q u r = h Ψ | a s p t q u r | Ψ i Trγ k =n!/(n − k)! Trγ 1 =n Trγ 2 =n(n − 1) Easilygeneralizedtoensemblestates. 3 )5( )6( )7( ()8 )9( ()01

Hamiltonianinconfigurationspace: nn 1 XX H(1,2,3....n)=h(k)+ k=1k<l=1 r kl FockspaceHamiltonian:;(Einsteinsummationconvention): 1 H=h q p a p q +g r p s q a p r q s 2 p h q = h ψ q | h | ψ p i 1 g r p s q = h ψ r (1)ψ r (2) || ψ p (1)ψ q (2) i r 21 Energyexpectationvalue: 1 E=h q p γ p q +g r p s q γ p r q s 2 )11( 1()2 )31( )41( 51() One-particledensitymatrix:γ=γ 1 ;Naturalorbital(NSO)basis:γ q p =n p δ q p 0 ≤ n p ≤ 1:occupationnumberofthep th NSO 4