One and two particle density matrices for single determinant HF wavefunctions. (1) = φ 2. )β(1) ( ) ) + β(1)β * β. (1)ρ RHF
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1 One and two partcle densty matrces for sngle determnant HF wavefunctons One partcle densty matrx Gven the Hartree-Fock wavefuncton ψ (,,3,!, = Âϕ (ϕ (ϕ (3!ϕ ( 3 The electronc energy s ψ H ψ = ϕ ( f ( ϕ ( + ϕ ( ϕ ( g(, ( P ϕ ( ϕ ( = < The one electron contrbuton may be wrtten as * ' ( f( ( = d ( f( ( ( = = ϕ ϕ τ ϕ ϕ ' from whch we deduce ' * ' (, = ( ( = γ ϕ ϕ and we see that the natural orbtals of a sngle determnant wavefuncton are smply the occuped spn orbtals of the system. If we are dealng wth a RHF wavefuncton these spn orbtals may be parttoned nto pars of doubly occuped spatal orbtals and so ϕ ( = φ (! r (, ϕ ( = φ (! r β(, ϕ 3 ( = φ (! r (,!, ϕ ( = φ / (! r β( and the one partcle densty matrces becomes / γ RHF (, ' = φ φ * ' ( * ( ' + β(β * ( ' and so the electron densty may be wrtten as ρ RHF ( = γ RHF (, = ( * (ρ RHF ( + β(β * β (ρ RHF where ρ RHF and ρ RHF ( r! are the denstes of & wave-functon these are dentcal β electrons and for the RHF James F Harrson Mchgan State Unversty December 3, 07
2 ρ RHF = ρ RHF / ( r! = φ ( r! φ * ( r! and snce we have ρ ξ RHF ( r! dv = /, where = or ρrhf ( dτ ( = ξ β If however we have an unrestrcted HF wavefuncton ψ UHF (,,3,!, = Â ( χ (χ (!χ ( φ β( +φ β( +!φ β β( + β Where { χ } and { } β = ρ UHF and ρ UHF φ = are the & (! r are not the same β spatal orbtals, then the spn denstes ρ UHF ( r! = χ ( r! χ * & ρ UHF ( r! = φ ( r! φ * ( r! and at a gven pont r! there s an mbalance n the number of & β electrons. One often defnes a spn densty (rather than a spn densty dfference as Q( r! = ρ UHF ρ UHF ( r! whch quantfes ths mbalance as a functon of! r. ote that ( Q( r! dv = ρ UHF ( r! ρ β UHF ( r! dv = β Two partcle densty matrces We may rewrte the two-electron contrbuton to the energy of a sngle determnant wavefuncton as β ϕ ϕ g P ϕ ϕ = dτ g ϕ ϕ P ϕ ϕ < < ( ( (,( ( ( (, (, ( (( ( ( that allows us to wrte the two-partcle densty matrx as James F Harrson Mchgan State Unversty December 3, 07
3 (,, ϕ ( (( ϕ P ϕ( ϕ( < Γ = In ths specal case of a sngle determnant wavefuncton can re-wrte Γ n terms of γ as follows. Frst the summaton s changed so that t's unrestrcted Γ (,, = ( (( ( ( = ( (( ( ( ϕ ϕ P ϕ ϕ ϕ ϕ P ϕ ϕ <, then Γ = = (,, ϕ ( (( ϕ P ϕ( ϕ( γ(, γ(, γ(, γ(,, ( Beng able to express Γ n terms of γ s a property of the Hartree-Fock wavefuncton and reflects the lmted electron correlaton n such a wavefuncton. For example for totally uncorrelated moton of the electrons we expect ψ (,,!, ψ * (,,!, dτ (3,!, = ψ (,,3,!, ψ * (,,3,!, dτ (,3,!, ( ( ψ (,,3,!, ψ * (,,3,!, dτ (,3,!, or, n terms of the densty matrces γ(, γ(, Γ uncorr (,, = ( Rewrtng the relatonshp between Γ and γ we express the effect of electron correlaton due to antsymmetry as τ (, where γ(, γ(, Γ (,, = τ(, Γ (,, = τ(, uncorr ( γ(, γ(, τ (, = γ(, γ(, The analyss of the two-electron densty matrx s a bt more complcated. Usng the general form for a sngle determnant wave-functon James F Harrson Mchgan State Unversty December 3, 07 3
4 Γ (,, = (, (, (, (, γ γ γ γ ( and the defnton of γ we may wrte for the UHF functon Γ UHF (,, = P ; ( * (( * ( + P ;ββ ( * (β(β * ( + P ββ ; β(β * (( * ( + P β ;β (β * (β( * ( + P β ;β β( * ((β * ( + P ββ ;ββ β(β * (β(β * ( where P ; = ρ ρ ( r! ϕ ϕ * ϕ ( r! ϕ * ( r! = P ;ββ (! r,! r = ρ (! r ρ β (! r P ββ ; (! r,! r = ρ β (! r ρ (! r P β ;β (! r,! r = β = ϕ (! r φ * (! r φ (! r ϕ * (! r P β ;β (! r,! r = and β = φ (! r ϕ * (! r ϕ (! r φ * (! r P ββ ;ββ = ρ β ρ β ( r! β β φ φ * φ ( r! φ * ( r! = ote that P β ;β = ( P β ;β * These components of Γ (,, have the followng nterpretaton: James F Harrson Mchgan State Unversty December 3, 07 4
5 P ; (! r,! r represents the probablty that an electron wth spn wll be at the termnus of r! n the volume element dv whle smultaneously another electron, also wth spn wll be at the termnus of r! n the volume element dv. P ;ββ ( r! represents the probablty that an electron wth spn wll be at the termnus of r n the volume element dv whle smultaneously another electron, wth β spn wll be at the termnus of r! n the volume element dv. P ββ ; ( r! and P ββ ;ββ have smlar nterpretatons. However ts not clear how one should nterpret the terms P β ;β and P β ;β. ote the normalzaton of these probabltes P ; = ( P ββ,ββ = β ( β P ;ββ = P ββ ; = β β P β ;β = ϕ φ * φ ( r! ϕ * ( r! = Snce the & βorbtals are not egenfunctons of the same Fock operator they need not be orthogonal and so Δ = ϕ * (! r φ (! r dv and therefore P β ;β = In a smlar fashon β Δ Δ = * = P β ;β = * Δ Δ = β = β β = = Δ Δ James F Harrson Mchgan State Unversty December 3, 07 5
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