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•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Graded Orthogonality for Fermionic States Patrick CASSAM-CHENAI Laboratoire J. A. Dieudonne, UMR-6621 du CNRS et de l' Universite de Nice - Sophia-Antipolis Nice, October 2007

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  • grassmann's exterior

  • product

  • functions

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  • left-interior product


Publié le : mardi 19 juin 2012
Lecture(s) : 36
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Source : math.unice.fr
Nombre de pages : 27
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LaboratoireJ.A.Dieudonne´,UMR-6621duCNRS etdelUniversite´deNice-Sophia-Antipolis
Graded Orthogonality for Fermionic States
¨ Patrick CASSAM-CHENAI
Nice, October 2007
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Hone-electron functions, spanned by a set of space of : Hilbert spin-orbitals(ψi)i
Notation-1
n1H ⊗ ∙ ∙ ∙ ⊗ ∧nkH: Hilbert space of partially antisymmetric functions forkdistinguishable groups of electrons of respectivelyn1     nkelectrons
nH: Hilbert space ofn-electron antisymmetric functions, spanned by the Slater determinantal functions(ψi1ψi2∧ ∙ ∙ ∙ ∧ψin)(i1<i2<∙∙∙<in)
∧H Fock space: Fermionic ∧H:=MnH n0
Notation-2
For anyn-particle wave function,Φ :=PcIψi1∧ ∙ ∙ ∙ ∧ψin, I:=(i1<...<in) and any ordered sequence of lengthp,K:= (k1<    < kp)∈ Pn,p, withp∈ {0     n}, 1kjn,
we set, ∙ ∙ ∙(Φ)K∙ ∙ ∙(Φ)K¯∙ ∙ ∙:=ρK,K¯XcI∙ ∙ ∙(ψik1∧ ∙ ∙ ∙ ∧ψipk)∙ ∙ ∙(ψik¯1∧ ∙ ∙ ∙ ∧ψi¯knp)∙ ∙ ∙ I:=(i1<∙∙∙<in) ¯ ¯ ¯ whereK:= (k1<    < knp), complement ofKin (1<2<∙ ∙ ∙< n), ρK,K¯is the sign of the permutation reordering the concatenated sequence ¯ K//Kin increasing order; if the length,|K|, ofKis 0 then, by conven-tion,(Φ)K:= (Φ)= 1, andρ,(1<∙∙∙<n)= 1; note that,(Φ)(1<∙∙∙<n)= Φ.
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which entirely determines the behavior of ann-fermion wave function under the symmetric groupSn is to say, for an. Thatn-fermion single conguration:
The fermionic symmetry is built-in in this exterior algebra because of the following antisymmetry relation between exterior products of 1-particle functions:
X:H7HHΦΨXΨ) = ΦΨ
X(φψ) =φψ=ψφ=−X(ψφ)
Grassmann’s exterior product:
ψ1∧ ∙ ∙ ∙ ∧ψn= (1)|σ|ψσ(1)∧ ∙ ∙ ∙ ∧ψσ(n)
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Conjugate interior products:
Θ∈ ∧npH,Ψ∈ ∧pH,Φ∈ ∧nH,
“left-interior product”hΘ|Ψ-Φi=hΨΘ|Φi
Ψ-Φ =XhΨ|ΦKi ∙ΦK¯ K∈Pn,p
explicit formula
“right-interior product”hΘ|Φ,Ψi=hΘΨ|Φi
ntatavitiy-iocmmtuΨ-Φ = (1)p(np)Φ,Ψ
or for mono-configuration functions (Slater determinants) (ψ1∧∙ ∙ ∙∧ψp)-(φ1∧∙ ∙ ∙∧φn) =XρK,K¯det(hψi|φkji)i,jpφ¯k1∧∙ ∙ ∙∧φk¯np K∈Pn p ,
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Example:
I1[Ψ] ={ω∈ HΦ∈ ∧n1HΦ-Ψ =ω}
Denition:
1-internal space of a wave function
Ψe41={φ1αφ3αφ5αφβ1φ3β+φ1αφ3αφ6αφβ1φβ4+ φ1αφ4αφ6αφβ1φβ3+φ1αφ4αφ5αφ1βφβ+ 4 φ1αφ3αφ5αφβ2φ3β+φ1αφ3αφ6αφβφβ4+ 2 φ1αφ4αφ6αφ2βφ3β+φ1αφ4αφ5αφβ2φβ4+ φ2αφ3αφ5αφ1βφ3β+φ2αφ3αφ6αφβ1φβ+ 4 φ2αφ4αφ6αφ1βφ3β+φ2αφ4αφ5αφβ1φβ4+ φ2αφ3αφ5αφ2βφβ3+φ2αφ3αφ6αφ2βφ4β+ φ2αφ4αφ6αφβ2φβ3+φ2αφ4αφ5αφβ2φβ4}
I1e] =C(ψ1α:=21(φ1α+φ2α) ψ1β:=12(φ1β+φ2β) φ3α φ3β φ4α φ4β φ5α φ6α)
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V⊆ HΨ∈ ∧nVI1[Ψ]V
Smallest one-electron Hilbert-subspace allowing one to express Ψ:
Other characterizations
spanned by the occupied natural spin-One-electron Hilbert-subspace orbitals of Ψ:
Application: compact expressions Ψe21=ψ1αψ1β(φ3αφ5αφβ3+φ3αφ6αφ4β+φ4αφ6αφ3β+φ4αφ5αφβ4)
There exists a basis set of internal spin-orbitals such that for each spin 21orbital there is a spin12orbital with same spatial part.
Application: case of open-shell wave function, eigenfunction ofS2, but where the natural spin12orbitals have are distinct spatial parts from those of the natural spin21ibatrosl
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The casem=nwith no spin constraint is theUHFmethod first developed by Prof. Berthier (and not Pople as found in textbooks).
c(k0)H0∙ ∙ ∙c(kmn)HmncG(k0<∙∙∙<kmn)= 0
H0     Hmn∈ PM,n1G∈ PM,m+1 Xρ(k0),H0∙ ∙ ∙ρ(kmn),Hmnρ(k0<∙∙∙<kmn),G(k0<∙∙∙<kmn) (k0<∙∙∙<kmn)G, {ki}*Hi
Example: Finite basis set of M spin-orbitals
The Complete Active Space Self-Consistent Field Method (CASSCF)
AUCASSCFcalculation withmactive orbitals (and no inactive occu-pied) consists in finding the stationary points of the energy functionnal E(Ψ) =hhΨΨ||HΨΨiiwith the constraint thatdimI1[Ψ]mi.e. within {Ψ∈ ∧nH|∀Φ0    Φmn∈ ∧n1H0-Ψ)∧ ∙ ∙ ∙ ∧mn-Ψ)Ψ = 0}.
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traditional
methods
Variational
spaces
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Strong orthogonality
Or by changing to a basis set representation{φi}i, hφi2  φin1-Ψ1|φi02  φi0n2-Ψ2i= 0i2     in1 i02     i0n2
Since, the (n11)-particle functions,φi2  φin1, span all of(n11)H, and the (n21)-particle functions,φi02  φi0n2, span all of(n21)H, the latter equation is equivalent to orthogonality between any pair of 1-internal functions, that is to say:
Z1Ψ1(τ1 τ2     τn12(τ1 τ02     τ0n2) =hδτ2  δτn1-Ψ1|δτ02  δτ0n2-Ψ2i= 0
Let Ψ1be the wave functions of ann1electron group and Ψ2that of an n2electron group. Ψ1and Ψ2are saidstrongly orthogonalif and only if: τ2     τn1 τ20     τ0n2
I11]⊥ I12]
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dimI2[Ψ] = 10
Ip[Ψ] ={ω∈ ∧pHΦ∈ ∧npHΦ-Ψ =ω}
Denition:
Ψ =ψ1ψ2ψ3ψ4+ψ1ψ2ψ5ψ6
dimI1[Ψ] = 6dim2I1[Ψ] = 15 0 BU Tψψ33ψψ54ψψ35ψψ66 ψ4ψl5inψe4arψd6ependencies
Example:
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