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Observation functional An exact generalization of DFT. Philippe CHOMAZ - GANIL. States, observables, observations Variational principles Generalized mean-Field Hartree-Fock Hierarchies and fluctuations Exact generalized density functional Exact generalized Kohn-Sham Eq. 1. - PowerPoint PPT Presentation
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DFT Philippe CHOMAZ, 2006
: 80
DFT Philippe CHOMAZ, 2006
: 80
Observation functional An exact generalization of DFT
Philippe CHOMAZ - GANIL
States, observables, observations Variational principles Generalized mean-Field Hartree-Fock Hierarchies and fluctuations Exact generalized density functional Exact generalized Kohn-Sham Eq. 1
DFT Philippe CHOMAZ, 2006
: 80
States
Observables
Observation
A) States, Observables and Observations
31
€
ψ Many-body wave functionHilbert or Fock space
€
ˆ O ψ ' = ˆ O ψ
€
< ˆ O >= ψ ˆ O ψ
DFT Philippe CHOMAZ, 2006
: 80
States
Observables
Observation
A) States, Observables and Observations
31
€
ψ Many-body wave functionHilbert or Fock space
€
ˆ D = ψ i pi ψ i
i
∑Density matrixLiouville space
€
ˆ O ψ ' = ˆ O ψ
€
< ˆ O >= ψ ˆ O ψ
€
< ˆ O >= Tr ˆ O ˆ D =<< ˆ O ˆ D >>
Scalar product in matrix space
<A
i
>
<A
j
>D
<a>
DFT Philippe CHOMAZ, 2006
: 80
Static
Dynamics
B) Variational principles
31
€
ψ0 minE[ψ ] = ψ ˆ H ψ
€
i∂t ψ = ˆ H ψ
Schrödinger equation
€
I ψ[ ] = dt ψ i∂
∂t- ˆ H ψ
t0
t1∫
Extremum of the action I
DFT Philippe CHOMAZ, 2006
: 80
Static
Dynamics
B) Variational principles
31
€
ψ0 minE[ψ ] = ψ ˆ H ψ
Zero Temperature minimum energy E
€
E − TS
€
ˆ D 0 minE[ ˆ D ] = Tr ˆ H ˆ D
Finite T minimum free energy
€
S[ ˆ D ] = −Tr ˆ D log ˆ D Entropy
€
i∂t ψ = ˆ H ψ
Schrödinger equation
€
I ψ[ ] = dt ψ i∂
∂t- ˆ H ψ
t0
t1∫
Extremum of the action I Balian and Vénéroni double principle
€
I ˆ B , ˆ D [ ] =< ˆ B >t1- dt tr ˆ B ∂t
ˆ D - ˆ H , ˆ D [ ] /i( )t0
t1∫€
∂tˆ D = ˆ H , ˆ D [ ] /i
Liouville equation
Observables backward from t1
Density forward from t0
€
ˆ B (t)
€
ˆ D (t)
DFT Philippe CHOMAZ, 2006
: 80
Coherent states
Generalized density
Extremum action
C) Generalized mean-field
31
Group transformation
€
ψ Z( ) = ˆ R Z( ) ψ 0( ) = exp i Z. ˆ A ( ) ψ 0( )
€
ˆ R Z( )
€
ˆ A ≡ ˆ A l{ } Lie Algebra
€
Z ≡ Z l{ } Group parameters
€
∂t < ˆ A l >Z= − < ˆ H , ˆ A l[ ] /i >Z
Mean-field <=> Ehrenfest
€
ρl ≡< ˆ A l >Z
DFT Philippe CHOMAZ, 2006
: 80
Coherent states
Generalized density
Extremum action
C) Generalized mean-field
31
Group transformation
€
ψ Z( ) = ˆ R Z( ) ψ 0( ) = exp i Z. ˆ A ( ) ψ 0( )
€
ˆ R Z( )
€
ˆ A ≡ ˆ A l{ } Lie Algebra
€
Z ≡ Z l{ } Group parameters
Maximum entropy trial state
With the constraints
€
ˆ D Z( ) =1
Z0
exp Z. ˆ A ( )
€
< ˆ A l >{ }
Constrained entropy
€
S'= S − Z. ˆ A
Lagrange multipliers
€
Z ≡ Z l{ }
€
∂t < ˆ A l >Z= − < ˆ H , ˆ A l[ ] /i >Z
Mean-field <=> Ehrenfest
€
ˆ B ′ Z ( ) = ′ Z 0 + ′ Z . ˆ A
Trial observables
€
ρl ≡< ˆ A l >Z
DFT Philippe CHOMAZ, 2006
: 80
Lie algebra Observation Trial states
Hamiltonian
Independent particles Mean Field
D) Hartree Fock
31
One-body density
€
ψ Z( ) = exp i ΣijZijˆ c i
+ ˆ c j + h.c.( ) ψ 0( )€
ρ ji =< ˆ c i+ ˆ c j >
€
ˆ A l ≡ ˆ c i+ ˆ c j One-body observables
Thouless theorem (Slaters)
€
ˆ D Z( ) = exp ΣijZijˆ c i
+ ˆ c j + h.c.( ) /Z0
Independent particle state
€
ˆ H = ε ijij
∑ ˆ c i+ˆ c j +
1
4Vijkl
ijkl
∑ ˆ c i+ˆ c j
+ˆ c l ˆ c k
€
ˆ H , ˆ c i+ˆ c j[ ] = εki
ˆ c k+ˆ c j −ε jk
ˆ c i+ˆ c k
k
∑ + 1
2Vkljm
ˆ c k+ ˆ c l
+ˆ c m ˆ c j −Viklmˆ c i
+ˆ c k+ˆ c m ˆ c l
klm
∑
€
ˆ c i+ˆ c j
+ˆ c kˆ c l Z= ρ liρ kj − ρ kiρ lj
€
i∂tρ ij = − < ˆ H , ˆ c j+ ˆ c i[ ] >Z = W, ρ[ ]ij
€
Wij = ε ij + Vikjl ρ lk
kl
∑ = δE[ρ ]/δρ ji
DFT Philippe CHOMAZ, 2006
: 80
Exact dynamics
Hierarchy
Projections <A> Minimum entropy Correlation MF Langevin Mean-Field
E) Hierarchies and fluctuations
31
Close the Lie Algebra including A an H
€
ˆ H , ˆ A l[ ] /i = −Glˆ A , ˆ a ( )
ˆ H , ˆ a m[ ] /i = −gmˆ A , ˆ a ( )
⎧ ⎨ ⎪
⎩ ⎪
€
∂t p ˆ A l f =Gl p ˆ A f ,p ˆ a f( )
∂t p ˆ a m f =gm p ˆ A f ,p ˆ a f( )
⎧ ⎨ ⎪
⎩ ⎪
€
p ˆ a f (p ˆ A f ) = tr ˆ a ˆ D Z(p ˆ A f )( )
€
δ p ˆ a f = p ˆ a f − p ˆ a f (p ˆ A f )
€
∂t p ˆ A l f =Gl p ˆ A f ,p ˆ a f (p ˆ A f )( ) + δGl p ˆ A f ,δ p ˆ a f( )
€
Gl p ˆ A f ,p ˆ a f (p ˆ A f )( ) =p ˆ H , ˆ A l[ ] /i f z≡ Wl p ˆ A f( )
Coupled equations
<A
i
><A
j
>
D(0)
D(t)
<a >
n( )
t( )
<Aj
>
<Ai
>
<A >
DFT Philippe CHOMAZ, 2006
: 80
Exact State Exact Observations
Exact E functional
Min in a subspace
Constrained energy
<=> external field
F) Exact generalized Density functional
31
Or
Generalized density
€
ψ0
€
ˆ D 0 = ψ 0 ψ 0
€
ρl =< ˆ A l >
€
E[ρ] = min< ˆ A >=ρ
< ˆ H >
€
ρl ≡< ˆ A l >= Tr ˆ A l ˆ D =<< ˆ A l ˆ D >>
€
F Z[ ] = min ˆ D < ˆ H − Σl Z l
ˆ A l >
€
ˆ U = −Σl Z lˆ A l
€
F Z[ ] = minρ E[ρ] − Σl Z l ρ l
€
ρl =< ˆ A l >
DFT Philippe CHOMAZ, 2006
: 80
Exact E functional
For a set of observations
Exact ground state E
=> exact densities ρ Variation Equivalent to mean-field Eq.
with Lie algebra including Al , {Al ,
A’m }
G) Exact Generalized Kohn-Sham Eq.
31
Generalized density
€
ρl =< ˆ A l >
€
E[ρ] = min< ˆ A >=ρ
< ˆ H >
€
E0 = minρ E[ρ]
€
0 = ∂ρ lE[ρ]δρ l
l
∑ = W l [ρ]δρ l
l
∑
Exact E and ρ in an external field U=zlAl
DFT Philippe CHOMAZ, 2006
: 80
Remarks
Exact for E
and all observations <Al > =ρl included in E[ρ]
Easy to go from a set of Al to a reduced
set A’l
=> E’[ρ‘]=minρ‘=cst E[ρ]
G) Exact Generalized Kohn-Sham Eq.
31
DFT Philippe CHOMAZ, 2006
: 80
The only information needed is the energy
=> functionals of ρ Local density approximation
Energy density functional
Local densities
matter , kinetic , current
Mean field
H) Density functional theory : LDA
35
€
E = drH (r)∫€
< ˆ H >= E[ ˆ ρ ]
€
hij[ ˆ ρ ] =δE
δρ ji
^
€
H (r) = H [{ρ AB (r)}]
€
ρAB (r) = trδ(r − ˆ r ) ˆ A ̂ ρ ˆ B
€
ραμ =ρσ ατ μ
€
τ μ =ρpσ α τ μ p
€
Jαμ = ρ pσ α τ μ
€
δE = tr ˆ h δ ˆ ρ = dr rstst
∑∫ ˆ h δ ˆ ρ rst
€
δE[ ˆ ρ ] =AB
∑ dr∂H
∂ρ AB
(r)δρ AB (r)∫ =AB
∑ dr∂H
∂ρ AB
(r) tr[δ(r − ˆ r ) ˆ A δ ˆ ρ ˆ B ∫ ]
δE[ ˆ ρ ] =AB
∑ tr[∂H
∂ρ AB
(ˆ r ) ˆ A δ ˆ ρ ˆ B ] ⇒ ˆ h =AB
∑ ˆ B ∂H
∂ρ AB
(ˆ r ) ˆ A
€
ˆ A l ⇒ ˆ B δ(r − ˆ r ) ˆ A
DFT Philippe CHOMAZ, 2006
: 80
H) LDA: Skyrme case
€
ρ(r) = trδ(r − ˆ r ) ˆ ρ
€
ρ3 = trδ ˆ r ˆ τ 3 ˆ ρ = ρ p − ρ n
€
τ(r) = trδ(r − ˆ r )ˆ p ̂ ρ ̂ p
€
τ 3 = trδ ˆ r ˆ τ 3 ˆ p ̂ ρ ̂ p
36
€
J(r) = trδ(r − ˆ r )ˆ p × ˆ σ ̂ ρ
€
J3 = trδ ˆ r ˆ p × ˆ σ ̂ ρ τ 3
€
+...
€
1
€
1
€
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
€
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
€
ˆ h q =1
2mq*
ˆ p 2 + U0q(ˆ r ) + U3q
(ˆ r ) + Ueffq(ˆ r ) + ...
€
ˆ h =AB
∑ ˆ B ∂H
∂ρ AB
(ˆ r ) ˆ A
Standard case few densities Matter isoscalar isovector kinetic isoscalar isovector Spin isoscalar isovector
Energy functional
Mean-field q=(n,p)
DFT Philippe CHOMAZ, 2006
: 80
Standard case few densities
Matter isoscalar isovector kinetic isoscalar isovector Spin isoscalar isovector
Energy functional
Mean-field q=(n,p)
H) LDA: Skyrme case
€
ρ(r) = trδ(r − ˆ r ) ˆ ρ
€
ρ3 = trδ ˆ r ˆ τ 3 ˆ ρ = ρ p − ρ n
€
τ(r) = trδ(r − ˆ r )ˆ p ̂ ρ ̂ p
€
τ 3 = trδ ˆ r ˆ τ 3 ˆ p ̂ ρ ̂ p
36
€
J(r) = trδ(r − ˆ r )ˆ p × ˆ σ ̂ ρ
€
J3 = trδ ˆ r ˆ p × ˆ σ ̂ ρ τ 3
€
+...
Skyrmeparameters
€
1
€
1
€
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
€
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
€
ˆ h q =1
2mq*
ˆ p 2 + U0q(ˆ r ) + U3q
(ˆ r ) + Ueffq(ˆ r ) + ...
DFT Philippe CHOMAZ, 2006
: 80
