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HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
HEDIN EQUATIONS AND KOHN–SHAM POTENTIALIN THE PATH–INTEGRAL FORMALISM
Marco Vanzini
Supervisor: Prof. Luca G. Molinari
Co–Supervisor: Prof. Giovanni Onida
Co–Supervisor: Dr. Guido Fratesi
Milan, July 17th 2014
1 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Goals and overview
2 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Goals and overview
Path integral formulation for Z = Tr[
e−β(H−µN)]
.
2 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Goals and overview
Path integral formulation for Z = Tr[
e−β(H−µN)]
.
Diagrammatic theory: G, U , Σ∗, Π∗, Γ.
2 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Goals and overview
Path integral formulation for Z = Tr[
e−β(H−µN)]
.
Diagrammatic theory: G, U , Σ∗, Π∗, Γ.
Derivation of Hedin equations.
2 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Goals and overview
Path integral formulation for Z = Tr[
e−β(H−µN)]
.
Diagrammatic theory: G, U , Σ∗, Π∗, Γ.
Derivation of Hedin equations.
Riformulation of density functional theory at finite temperature (Mermin1965) in a path integral approach.
2 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Goals and overview
Path integral formulation for Z = Tr[
e−β(H−µN)]
.
Diagrammatic theory: G, U , Σ∗, Π∗, Γ.
Derivation of Hedin equations.
Riformulation of density functional theory at finite temperature (Mermin1965) in a path integral approach.
Functional expression for Fxc[n(x)].
2 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
Hedin equations and the path integral:
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
Hedin equations and the path integral:
C. Itzykson e J. B. Zuber, Quantum Field Theory, McGraw–Hill (1980).
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
Hedin equations and the path integral:
C. Itzykson e J. B. Zuber, Quantum Field Theory, McGraw–Hill (1980).
L. Hedin, New Method for Calculating the One–Particle Green’s Function withApplication to the Electron–Gas Problem, Phys. Rev. 139 (3A) A796–A823(1965).
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
Hedin equations and the path integral:
C. Itzykson e J. B. Zuber, Quantum Field Theory, McGraw–Hill (1980).
L. Hedin, New Method for Calculating the One–Particle Green’s Function withApplication to the Electron–Gas Problem, Phys. Rev. 139 (3A) A796–A823(1965).
L. G. Molinari, An introduction to functional methods for many–body Greenfunctions, in CECAM workshop “Green’s function methods: the nextgeneration”, Toulouse, June 2013.
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
Hedin equations and the path integral:
C. Itzykson e J. B. Zuber, Quantum Field Theory, McGraw–Hill (1980).
L. Hedin, New Method for Calculating the One–Particle Green’s Function withApplication to the Electron–Gas Problem, Phys. Rev. 139 (3A) A796–A823(1965).
L. G. Molinari, An introduction to functional methods for many–body Greenfunctions, in CECAM workshop “Green’s function methods: the nextgeneration”, Toulouse, June 2013.
Thermal density functional theory and the path integral:
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
Hedin equations and the path integral:
C. Itzykson e J. B. Zuber, Quantum Field Theory, McGraw–Hill (1980).
L. Hedin, New Method for Calculating the One–Particle Green’s Function withApplication to the Electron–Gas Problem, Phys. Rev. 139 (3A) A796–A823(1965).
L. G. Molinari, An introduction to functional methods for many–body Greenfunctions, in CECAM workshop “Green’s function methods: the nextgeneration”, Toulouse, June 2013.
Thermal density functional theory and the path integral:
R. Fukuda, T. Kotani, Y. Suzuki, S. Yokojima, Density Functional Theorythrough Legendre Transformation, Prog. Theor. Phys. 92 (4), 833 (1994).
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
References
Hedin equations and the path integral:
C. Itzykson e J. B. Zuber, Quantum Field Theory, McGraw–Hill (1980).
L. Hedin, New Method for Calculating the One–Particle Green’s Function withApplication to the Electron–Gas Problem, Phys. Rev. 139 (3A) A796–A823(1965).
L. G. Molinari, An introduction to functional methods for many–body Greenfunctions, in CECAM workshop “Green’s function methods: the nextgeneration”, Toulouse, June 2013.
Thermal density functional theory and the path integral:
R. Fukuda, T. Kotani, Y. Suzuki, S. Yokojima, Density Functional Theorythrough Legendre Transformation, Prog. Theor. Phys. 92 (4), 833 (1994).
M. Valiev, G. W. Fernando, Path–integral analysis of the exchange–correlationenergy and potential in density–functional theory: Unpolarized systems, Phys.Rev. B 54, 7765 (1996).
3 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
Associated Schrödinger equation:
Hφ(k)σ1...σN (x1...xN ) = Ekφ
(k)σ1...σN (x1...xN )
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
Associated Schrödinger equation:
Hφ(k)σ1...σN (x1...xN ) = Ekφ
(k)σ1...σN (x1...xN )
One–particle thermal Green’s function:
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
Associated Schrödinger equation:
Hφ(k)σ1...σN (x1...xN ) = Ekφ
(k)σ1...σN (x1...xN )
One–particle thermal Green’s function:
Gσσ′ (x, τ ;x′, τ ′) := −〈T ψσ(x, τ)ψ†σ′ (x
′, τ ′)〉ρ
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
Associated Schrödinger equation:
Hφ(k)σ1...σN (x1...xN ) = Ekφ
(k)σ1...σN (x1...xN )
One–particle thermal Green’s function:
Gσσ′ (x, τ ;x′, τ ′) := −〈T ψσ(x, τ)ψ†σ′ (x
′, τ ′)〉ρFundamental quantity in many body theory:
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
Associated Schrödinger equation:
Hφ(k)σ1...σN (x1...xN ) = Ekφ
(k)σ1...σN (x1...xN )
One–particle thermal Green’s function:
Gσσ′ (x, τ ;x′, τ ′) := −〈T ψσ(x, τ)ψ†σ′ (x
′, τ ′)〉ρFundamental quantity in many body theory:
Thermodynamic equilibrium expectation value of any one–particle operator,e.g. density:
n(x) = Gσσ(x, τ ;x, τ+)
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
Associated Schrödinger equation:
Hφ(k)σ1...σN (x1...xN ) = Ekφ
(k)σ1...σN (x1...xN )
One–particle thermal Green’s function:
Gσσ′ (x, τ ;x′, τ ′) := −〈T ψσ(x, τ)ψ†σ′ (x
′, τ ′)〉ρFundamental quantity in many body theory:
Thermodynamic equilibrium expectation value of any one–particle operator,e.g. density:
n(x) = Gσσ(x, τ ;x, τ+)
Total energy expectation value: Migdal–Galitskii formula;
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The system: the electron gas Hamiltonian: N interacting electrons in an external potential u(x) (atoms,
molecules, solids, ...):
H =N∑
i=1
(p2i
2m+ u(xi)
)
+1
2
N∑
i,j=1(i 6=j)
e2
|xi − xj |=
=
∫
V
d3x ψ†σ(x)
(
− ~2
2m∇2 + u(x)
)
ψσ(x)+
+1
2
∫
V
d3x d3x′ ψ†σ(x)ψ
†σ′ (x
′)e2
|x− x′| ψσ′ (x′)ψσ(x)
Associated Schrödinger equation:
Hφ(k)σ1...σN (x1...xN ) = Ekφ
(k)σ1...σN (x1...xN )
One–particle thermal Green’s function:
Gσσ′ (x, τ ;x′, τ ′) := −〈T ψσ(x, τ)ψ†σ′ (x
′, τ ′)〉ρFundamental quantity in many body theory:
Thermodynamic equilibrium expectation value of any one–particle operator,e.g. density:
n(x) = Gσσ(x, τ ;x, τ+)
Total energy expectation value: Migdal–Galitskii formula; Excitation energies: Lehmann representation;
4 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
Density functional theory (Hohenberg–Kohn, 1964; Mermin, 1965;Kohn–Sham, 1965): ground–state / Gibbs state:
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
Density functional theory (Hohenberg–Kohn, 1964; Mermin, 1965;Kohn–Sham, 1965): ground–state / Gibbs state:
Many Body System:N interacting electronsin an external potential
u(x)↓
G(1, 1+)
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
Density functional theory (Hohenberg–Kohn, 1964; Mermin, 1965;Kohn–Sham, 1965): ground–state / Gibbs state:
Many Body System:N interacting electronsin an external potential
u(x)↓
G(1, 1+)
⇐⇒
Kohn–Sham System:N free electronsin an effective potentialuKS(x)↓GKS(1, 1+)
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
Density functional theory (Hohenberg–Kohn, 1964; Mermin, 1965;Kohn–Sham, 1965): ground–state / Gibbs state:
Many Body System:N interacting electronsin an external potential
u(x)↓
G(1, 1+)
⇐⇒
Kohn–Sham System:N free electronsin an effective potentialuKS(x)↓GKS(1, 1+)= n(x) =
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
Density functional theory (Hohenberg–Kohn, 1964; Mermin, 1965;Kohn–Sham, 1965): ground–state / Gibbs state:
Many Body System:N interacting electronsin an external potential
u(x)↓
G(1, 1+)
⇐⇒
Kohn–Sham System:N free electronsin an effective potentialuKS(x)↓GKS(1, 1+)= n(x) =
Kohn–Sham equation− ~2
2m∇2φj(x) + uKS(x)φj(x) = ǫjφj(x)
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
Density functional theory (Hohenberg–Kohn, 1964; Mermin, 1965;Kohn–Sham, 1965): ground–state / Gibbs state:
Many Body System:N interacting electronsin an external potential
u(x)↓
G(1, 1+)
⇐⇒
Kohn–Sham System:N free electronsin an effective potentialuKS(x)↓GKS(1, 1+)= n(x) =
Kohn–Sham equation− ~2
2m∇2φj(x) + uKS(x)φj(x) = ǫjφj(x)
↑uKS [n(x)] ≡ u(x) + uH [n(x)] + uxc [n(x)]
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT Many–body–perturbation–theory: Hedin equations (1965):
G11′ = G011′ + G
012Σ
∗22′G2′1′
Σ∗11′ = 1
~δ11′U
012G22+ − 1
~G12Γ1′23U31
U11′ = U011′ + U
012Π
∗22′U2′1′
Π∗11′ = 1
~G12Γ2′21′G2′1+
Γ123 = δ12δ23 + Γ5′4′3G5′5
δΣ∗xc21
δG45
G44′
5 integro–differential equations for the 5 quantities G, U , Σ∗, Π∗, Γ
Density functional theory (Hohenberg–Kohn, 1964; Mermin, 1965;Kohn–Sham, 1965): ground–state / Gibbs state:
Many Body System:N interacting electronsin an external potential
u(x)↓
G(1, 1+)
⇐⇒
Kohn–Sham System:N free electronsin an effective potentialuKS(x)↓GKS(1, 1+)= n(x) =
Kohn–Sham equation− ~2
2m∇2φj(x) + uKS(x)φj(x) = ǫjφj(x)
↑uKS [n(x)] ≡ u(x) + uH [n(x)] + uxc [n(x)]
↓n(x) = 2
∑
j nj |φj(x)|2
5 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT
6 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT
DFT:φj(x)
6 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT
DFT:φj(x)
MBPT:ψ(x), ψ†(y) = δ(x− y)
6 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations and DFT
DFT:φj(x)
MBPT:ψ(x), ψ†(y) = δ(x− y)
︸ ︷︷ ︸
Both theories find a natural and elegant riformulation in the functional integralformalism, for T = 0 and for T 6= 0 as well.
6 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ):
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
→∫
dq|q〉〈q| =∫
dp|p〉〈p| = 1
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
→∫
dq|q〉〈q| =∫
dp|p〉〈p| = 1
→ K(p) |p〉 = K(p) |p〉, V (q) |q〉 = V (q) |q〉
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
→∫
dq|q〉〈q| =∫
dp|p〉〈p| = 1
→ K(p) |p〉 = K(p) |p〉, V (q) |q〉 = V (q) |q〉↓
〈x, t|x0, 0〉 =
∫
x(0)=x0x(t)=x
D[x(t′)]D[p(t
′)]e
i~
∫ t0 dt
′[
p(t′)·∂x(t′)∂t′
−H(p(t′),x(t′))
]
Feynman phase–space path integral (QM)
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
→∫
dq|q〉〈q| =∫
dp|p〉〈p| = 1
→ K(p) |p〉 = K(p) |p〉, V (q) |q〉 = V (q) |q〉↓
〈x, t|x0, 0〉 =
∫
x(0)=x0x(t)=x
D[x(t′)]D[p(t
′)]e
i~
∫ t0 dt
′[
p(t′)·∂x(t′)∂t′
−H(p(t′),x(t′))
]
Feynman phase–space path integral (QM)
In the second quantization formalism, H − µN is written in terms ofnormal–ordered creation and annihilation operators, ψ†
i ψi or ψ†i ψ
†j ψj ψi:
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
→∫
dq|q〉〈q| =∫
dp|p〉〈p| = 1
→ K(p) |p〉 = K(p) |p〉, V (q) |q〉 = V (q) |q〉↓
〈x, t|x0, 0〉 =
∫
x(0)=x0x(t)=x
D[x(t′)]D[p(t
′)]e
i~
∫ t0 dt
′[
p(t′)·∂x(t′)∂t′
−H(p(t′),x(t′))
]
Feynman phase–space path integral (QM)
In the second quantization formalism, H − µN is written in terms ofnormal–ordered creation and annihilation operators, ψ†
i ψi or ψ†i ψ
†j ψj ψi:
Coherent States: eigenstates of the annihilation operator:
ψσ(x)|ψ〉 = ψσ(x)|ψ〉
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
→∫
dq|q〉〈q| =∫
dp|p〉〈p| = 1
→ K(p) |p〉 = K(p) |p〉, V (q) |q〉 = V (q) |q〉↓
〈x, t|x0, 0〉 =
∫
x(0)=x0x(t)=x
D[x(t′)]D[p(t
′)]e
i~
∫ t0 dt
′[
p(t′)·∂x(t′)∂t′
−H(p(t′),x(t′))
]
Feynman phase–space path integral (QM)
In the second quantization formalism, H − µN is written in terms ofnormal–ordered creation and annihilation operators, ψ†
i ψi or ψ†i ψ
†j ψj ψi:
Coherent States: eigenstates of the annihilation operator:
ψσ(x)|ψ〉 = ψσ(x)|ψ〉 ψσ(x) must be a Grassmann number:
ψσ(x)ψρ(y) = −ψρ(y)ψσ(x)
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integral: coherent statesGrand canonical ensemble (T, V, µ): partition function Z:
Z(T, V, µ) = Tr[
e−β(H−µN)]
it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉
transition amplitude (QM)→ t →
(
tN, ..., t
N
)
→ 〈φ| A |ψ〉 =∫
dqdq′φ∗(q) 〈q| A |q′〉ψ(q′)
→∫
dq|q〉〈q| =∫
dp|p〉〈p| = 1
→ K(p) |p〉 = K(p) |p〉, V (q) |q〉 = V (q) |q〉↓
〈x, t|x0, 0〉 =
∫
x(0)=x0x(t)=x
D[x(t′)]D[p(t
′)]e
i~
∫ t0 dt
′[
p(t′)·∂x(t′)∂t′
−H(p(t′),x(t′))
]
Feynman phase–space path integral (QM)
In the second quantization formalism, H − µN is written in terms ofnormal–ordered creation and annihilation operators, ψ†
i ψi or ψ†i ψ
†j ψj ψi:
Coherent States: eigenstates of the annihilation operator:
ψσ(x)|ψ〉 = ψσ(x)|ψ〉 ψσ(x) must be a Grassmann number:
ψσ(x)ψρ(y) = −ψρ(y)ψσ(x)
Tr[A] =
∫
∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|A|ψ〉
1F− =
∫
∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)|ψ〉〈ψ|
7 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
Z = limN→∞
∫
... 〈−ψ|e− 1~
~βNK ... e−
1~
~βNK ... e−
1~
~βNK |ψ〉
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
Z = limN→∞
∫
... 〈−ψ|e− 1~
~βNK ... e−
1~
~βNK ... e−
1~
~βNK |ψ〉
↑ ↑ ↑ ↑1F− 1F−
(
1 =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)|ψ〉〈ψ|
)
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
Z = limN→∞
∫
... 〈−ψ|e− 1~
~βNK ... e−
1~
~βNK ... e−
1~
~βNK |ψ〉
↑ ↑ ↑ ↑1F− 1F−
(
1 =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)|ψ〉〈ψ|
)
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−
∑
k
∫
d3x ψ(k)σ (x)ψ
(k)σ (x)·
·N−1∏
j=0
〈ψ(j+1)|e− 1~
~βNK |ψ(j)〉
∣∣∣∣∣∣ψ(0)=−ψ(N)
ψ(0)=−ψ(N)
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
Z = limN→∞
∫
... 〈−ψ|e− 1~
~βNK ... e−
1~
~βNK ... e−
1~
~βNK |ψ〉
↑ ↑ ↑ ↑1F− 1F−
(
1 =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)|ψ〉〈ψ|
)
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−
∑
k
∫
d3x ψ(k)σ (x)ψ
(k)σ (x)·
·N−1∏
j=0
〈ψ(j+1)|e− 1~
~βNK |ψ(j)〉
∣∣∣∣∣∣ψ(0)=−ψ(N)
ψ(0)=−ψ(N)↑
ψσ(x)|ψ(j)〉 = ψ(j)σ (x)|ψ(j)〉
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
Z = limN→∞
∫
... 〈−ψ|e− 1~
~βNK ... e−
1~
~βNK ... e−
1~
~βNK |ψ〉
↑ ↑ ↑ ↑1F− 1F−
(
1 =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)|ψ〉〈ψ|
)
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−
∑
k
∫
d3x ψ(k)σ (x)ψ
(k)σ (x)·
·N−1∏
j=0
〈ψ(j+1)|e− 1~
~βNK |ψ(j)〉
∣∣∣∣∣∣ψ(0)=−ψ(N)
ψ(0)=−ψ(N)↑
ψσ(x)|ψ(j)〉 = ψ(j)σ (x)|ψ(j)〉
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−∑
k
∫
d3x
ψ(k+1)σ (x)·
e
·[
ψ(k+1)σ (x)−ψ
(k)σ (x)
]
+ 1~
~βN
(H−µN
)[ψ
(k+1)σ (x),ψ
(k)σ (x)]
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
Z = limN→∞
∫
... 〈−ψ|e− 1~
~βNK ... e−
1~
~βNK ... e−
1~
~βNK |ψ〉
↑ ↑ ↑ ↑1F− 1F−
(
1 =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)|ψ〉〈ψ|
)
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−
∑
k
∫
d3x ψ(k)σ (x)ψ
(k)σ (x)·
·N−1∏
j=0
〈ψ(j+1)|e− 1~
~βNK |ψ(j)〉
∣∣∣∣∣∣ψ(0)=−ψ(N)
ψ(0)=−ψ(N)↑
ψσ(x)|ψ(j)〉 = ψ(j)σ (x)|ψ(j)〉
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−∑
k
∫
d3x
ψ(k+1)σ (x)·
e
·[
ψ(k+1)σ (x)−ψ
(k)σ (x)
]
+ 1~
~βN
(H−µN
)[ψ
(k+1)σ (x),ψ
(k)σ (x)]
Last step: letting N approach∞:
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Construction of the path–integralUsing coherent states to get a functional expression for Z:
Z = Tr[
e−β(H−µN)]
=↑
Tr[A] =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)〈−ψ|A|ψ〉
Z =
∫∏
σ,x
dψσ(x)dψσ(x) e−∫
d3x ψσ(x)ψσ(x)〈−ψ|e− 1~(~β)K |ψ〉
Z = limN→∞
∫
... 〈−ψ|e− 1~
~βNK ... e−
1~
~βNK ... e−
1~
~βNK |ψ〉
↑ ↑ ↑ ↑1F− 1F−
(
1 =
∫
∏
σ,xdψσ(x)dψσ(x) e
−∫
dx ψσ(x)ψσ(x)|ψ〉〈ψ|
)
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−
∑
k
∫
d3x ψ(k)σ (x)ψ
(k)σ (x)·
·N−1∏
j=0
〈ψ(j+1)|e− 1~
~βNK |ψ(j)〉
∣∣∣∣∣∣ψ(0)=−ψ(N)
ψ(0)=−ψ(N)↑
ψσ(x)|ψ(j)〉 = ψ(j)σ (x)|ψ(j)〉
Z =
N∏
k=1
∫∏
σ,x
dψ(k)σ (x)dψ
(k)σ (x)
e−∑
k
∫
d3x
ψ(k+1)σ (x)·
e
·[
ψ(k+1)σ (x)−ψ
(k)σ (x)
]
+ 1~
~βN
(H−µN
)[ψ
(k+1)σ (x),ψ
(k)σ (x)]
Last step: letting N approach∞: continuum limit: ψ(k)σ (x)→ ψσ(x, τ)...
8 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
Z =
∫
D[ψσ(x, τ)
]D [ψσ(x, τ)] exp−
1
~S[ψ,ψ
]
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
Z =
∫
D[ψσ(x, τ)
]D [ψσ(x, τ)] exp−
1
~S[ψ,ψ
]
S[ψ, ψ,ϕ] =
∫
dx ψσ(x)
(
~∂
∂τ−
~2
2m∇
2+ u(x) − µ
)
ψσ(x) +
+1
2
∫
dxdy ψσ(x)ψρ(y)e2
|x − y|ψρ(y)ψσ(x)
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
Z =
∫
D[ψσ(x, τ)
]D [ψσ(x, τ)] exp−
1
~S[ψ,ψ
]
S[ψ, ψ,ϕ] =
∫
dx ψσ(x)
(
~∂
∂τ−
~2
2m∇
2+ u(x) − µ
)
ψσ(x) +
+1
2
∫
dxdy ψσ(x)ψρ(y)e2
|x − y|ψρ(y)ψσ(x)
↑
The very last step:Hubbard–Stratonovich transformation:from a four–fermion–fields interactionto a two–fermion–fields plusan auxiliary–boson–field one (∼ QED):
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
Z =
∫
D[ψσ(x, τ)
]D [ψσ(x, τ)] exp−
1
~S[ψ,ψ
]
S[ψ, ψ,ϕ] =
∫
dx ψσ(x)
(
~∂
∂τ−
~2
2m∇
2+ u(x) − µ
)
ψσ(x) +
+1
2
∫
dxdy ψσ(x)ψρ(y)e2
|x − y|ψρ(y)ψσ(x)
↑
The very last step:Hubbard–Stratonovich transformation:from a four–fermion–fields interactionto a two–fermion–fields plusan auxiliary–boson–field one (∼ QED):
e−12~
n·U0n =∫
D[ϕ]e−e2
2~ϕ·U
−10 ϕ− ie
~ϕ·n
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
Z =
∫
D[ψσ(x, τ)
]D [ψσ(x, τ)] exp−
1
~S[ψ,ψ
]
S[ψ, ψ,ϕ] =
∫
dx ψσ(x)
(
~∂
∂τ−
~2
2m∇
2+ u(x) − µ
)
ψσ(x) +
+1
2
∫
dxdy ψσ(x)ψρ(y)e2
|x − y|ψρ(y)ψσ(x)
↑
The very last step:Hubbard–Stratonovich transformation:from a four–fermion–fields interactionto a two–fermion–fields plusan auxiliary–boson–field one (∼ QED):
e−12~
n·U0n =∫
D[ϕ]e−e2
2~ϕ·U
−10 ϕ− ie
~ϕ·n =
ψ
ψ
ϕ
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
Z =
∫
D[ψσ(x, τ)
]D [ψσ(x, τ)] exp−
1
~S[ψ,ψ
]
S[ψ, ψ,ϕ] =
∫
dx ψσ(x)
(
~∂
∂τ−
~2
2m∇
2+ u(x) − µ
)
ψσ(x) +
+1
2
∫
dxdy ψσ(x)ψρ(y)e2
|x − y|ψρ(y)ψσ(x)
↑
The very last step:Hubbard–Stratonovich transformation:from a four–fermion–fields interactionto a two–fermion–fields plusan auxiliary–boson–field one (∼ QED):
e−12~
n·U0n =∫
D[ϕ]e−e2
2~ϕ·U
−10 ϕ− ie
~ϕ·n =
ψ
ψ
ϕ
Z =
∫
D[
ψ]
D [ψ]D [ϕ] exp−1
~S[
ψ,ψ,ϕ]
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Path–integral form of Z
Z =
∫
D[ψσ(x, τ)
]D [ψσ(x, τ)] exp−
1
~S[ψ,ψ
]
S[ψ, ψ,ϕ] =
∫
dx ψσ(x)
(
~∂
∂τ−
~2
2m∇
2+ u(x) − µ
)
ψσ(x) +
+1
2
∫
dxdy ψσ(x)ψρ(y)e2
|x − y|ψρ(y)ψσ(x)
↑
The very last step:Hubbard–Stratonovich transformation:from a four–fermion–fields interactionto a two–fermion–fields plusan auxiliary–boson–field one (∼ QED):
e−12~
n·U0n =∫
D[ϕ]e−e2
2~ϕ·U
−10 ϕ− ie
~ϕ·n =
ψ
ψ
ϕ
Z =
∫
D[
ψ]
D [ψ]D [ϕ] exp−1
~S[
ψ,ψ,ϕ]
S[ψ, ψ,ϕ] =
∫
dx ψσ(x)
(
~∂
∂τ−
~2
2m∇
2+ u(x) − µ
)
ψσ(x)+
+
∫
dx ϕ(x)
(
−1
8π∇
2)
ϕ(x) + ie
∫
dx ψσ(x)ϕ(x)ψσ(x)
9 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
↓Ssorg =
∫
dx[
ησ(x)ψσ(x) + ψσ(x)ησ(x) + ieρ(x)ϕ(x)]
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
↓Ssorg =
∫
dx[
ησ(x)ψσ(x) + ψσ(x)ησ(x) + ieρ(x)ϕ(x)]
Gσσ′ (x; x′) = −
~2
Z
δ(2)Z[η,η,ρ]
δησ1 (1)δησ1′(1′)
U(x; x′) = −
~
Z
δ(2)Z[η,η,ρ]
δρ(1)δρ(1′)
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
↓Ssorg =
∫
dx[
ησ(x)ψσ(x) + ψσ(x)ησ(x) + ieρ(x)ϕ(x)]
Gσσ′ (x; x′) = −
~2
Z
δ(2)Z[η,η,ρ]
δησ1 (1)δησ1′(1′)
U(x; x′) = −
~
Z
δ(2)Z[η,η,ρ]
δρ(1)δρ(1′)
Z[η,η,ρ] is the generator of the proper Green’s functions.
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
↓Ssorg =
∫
dx[
ησ(x)ψσ(x) + ψσ(x)ησ(x) + ieρ(x)ϕ(x)]
Gσσ′ (x; x′) = −
~2
Z
δ(2)Z[η,η,ρ]
δησ1 (1)δησ1′(1′)
U(x; x′) = −
~
Z
δ(2)Z[η,η,ρ]
δρ(1)δρ(1′)
Z[η,η,ρ] is the generator of the proper Green’s functions.
Z[η,η,ρ] = e−1~W[η,η,ρ]
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
↓Ssorg =
∫
dx[
ησ(x)ψσ(x) + ψσ(x)ησ(x) + ieρ(x)ϕ(x)]
Gσσ′ (x; x′) = −
~2
Z
δ(2)Z[η,η,ρ]
δησ1 (1)δησ1′(1′)
U(x; x′) = −
~
Z
δ(2)Z[η,η,ρ]
δρ(1)δρ(1′)
Z[η,η,ρ] is the generator of the proper Green’s functions.
Z[η,η,ρ] = e−1~W[η,η,ρ] −→W[η,η,ρ] is the generator of the
connected Green’s functions.
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
↓Ssorg =
∫
dx[
ησ(x)ψσ(x) + ψσ(x)ησ(x) + ieρ(x)ϕ(x)]
Gσσ′ (x; x′) = −
~2
Z
δ(2)Z[η,η,ρ]
δησ1 (1)δησ1′(1′)
U(x; x′) = −
~
Z
δ(2)Z[η,η,ρ]
δρ(1)δρ(1′)
Z[η,η,ρ] is the generator of the proper Green’s functions.
Z[η,η,ρ] = e−1~W[η,η,ρ] −→W[η,η,ρ] is the generator of the
connected Green’s functions.
Γ[ψc,ψc,ϕc] =W[η,η,ρ]−∑
σ
∫dx
[η ·ψc + ψc · η + ieρ ·ϕc
],
with ϕc(x) ≡ 〈ϕ(x)〉Z , ..., that is performing a Legendre transform onW
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Two–points functions and generating functionals Electronic Propagator:
−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′
Dressed Interaction (∼ boson propagator):
U(x;x′) = e2
~〈ϕ(x)ϕ(x′)〉Z = xx′
Linear coupling with external sources ησ(x), ησ(x), ρ(x) (Schwinger, ‘50):fields become derivatives...
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
↓Ssorg =
∫
dx[
ησ(x)ψσ(x) + ψσ(x)ησ(x) + ieρ(x)ϕ(x)]
Gσσ′ (x; x′) = −
~2
Z
δ(2)Z[η,η,ρ]
δησ1 (1)δησ1′(1′)
U(x; x′) = −
~
Z
δ(2)Z[η,η,ρ]
δρ(1)δρ(1′)
Z[η,η,ρ] is the generator of the proper Green’s functions.
Z[η,η,ρ] = e−1~W[η,η,ρ] −→W[η,η,ρ] is the generator of the
connected Green’s functions.
Γ[ψc,ψc,ϕc] =W[η,η,ρ]−∑
σ
∫dx
[η ·ψc + ψc · η + ieρ ·ϕc
],
with ϕc(x) ≡ 〈ϕ(x)〉Z , ..., that is performing a Legendre transform onW−→ Γ[ψc,ψc,ϕc] is the generator of the 1PI Green’s functions.
10 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Self–Energy and Proper Polarization
11 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Self–Energy and Proper Polarization
Properties of the Legendre Transform:
δ(2)Γ[ψcl, ψcl,ϕcl]
δψclµ (1)δψclν (2)= ~G−1
µν (1, 2)
δ(2)Γ[ψcl, ψcl,ϕcl]
δϕcl(1)δϕcl(2)= e2U−1(1, 2)
11 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Self–Energy and Proper Polarization
Properties of the Legendre Transform:
δ(2)Γ[ψcl, ψcl,ϕcl]
δψclµ (1)δψclν (2)= ~G−1
µν (1, 2) = ~
[
G0−1σσ′ (1, 1′)− Σ∗
σσ′ (1, 1′)]
δ(2)Γ[ψcl, ψcl,ϕcl]
δϕcl(1)δϕcl(2)= e2U−1(1, 2) = e2
[
U−10 (1, 1′)−Π∗(1, 1′)
]
11 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Self–Energy and Proper Polarization
Properties of the Legendre Transform:
δ(2)Γ[ψcl, ψcl,ϕcl]
δψclµ (1)δψclν (2)= ~G−1
µν (1, 2) = ~
[
G0−1σσ′ (1, 1′)− Σ∗
σσ′ (1, 1′)]
δ(2)Γ[ψcl, ψcl,ϕcl]
δϕcl(1)δϕcl(2)= e2U−1(1, 2) = e2
[
U−10 (1, 1′)−Π∗(1, 1′)
]
︸ ︷︷ ︸
G = G0 + G0Σ∗G U = U0 + U0Π∗UDyson equations
11 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Self–Energy and Proper Polarization
Properties of the Legendre Transform:
δ(2)Γ[ψcl, ψcl,ϕcl]
δψclµ (1)δψclν (2)= ~G−1
µν (1, 2) = ~
[
G0−1σσ′ (1, 1′)− Σ∗
σσ′ (1, 1′)]
δ(2)Γ[ψcl, ψcl,ϕcl]
δϕcl(1)δϕcl(2)= e2U−1(1, 2) = e2
[
U−10 (1, 1′)−Π∗(1, 1′)
]
︸ ︷︷ ︸
G = G0 + G0Σ∗G U = U0 + U0Π∗UDyson equations
= + Σ∗ = + Π∗
11 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Schwinger–Dyson equations → Hedin equations
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
12 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Schwinger–Dyson equations → Hedin equations
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
Infinitesimal shift of the fields −→ Z [η,η,ρ] doesn’t change!−→ Schwinger–Dyson equations (’50):
equations of motion forthe generating functionals.
12 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Schwinger–Dyson equations → Hedin equations
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
Infinitesimal shift of the fields −→ Z [η,η,ρ] doesn’t change!−→ Schwinger–Dyson equations (’50):
equations of motion forthe generating functionals.
Shift of the field ψσ(x): ψσ(x)→ ψσ(x) + δψσ(x)
12 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Schwinger–Dyson equations → Hedin equations
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
Infinitesimal shift of the fields −→ Z [η,η,ρ] doesn’t change!−→ Schwinger–Dyson equations (’50):
equations of motion forthe generating functionals.
Shift of the field ψσ(x): ψσ(x)→ ψσ(x) + δψσ(x)
Sold −→ Sold +
∫
dx δψσ(x)[(k(x) + ieϕ(x)
)ψσ(x) + ησ(x)
]
↑k(x, τ) = ~
∂∂τ− ~
2
2m∇2 + u(x)− µ
12 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Schwinger–Dyson equations → Hedin equations
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
Infinitesimal shift of the fields −→ Z [η,η,ρ] doesn’t change!−→ Schwinger–Dyson equations (’50):
equations of motion forthe generating functionals.
Shift of the field ψσ(x): ψσ(x)→ ψσ(x) + δψσ(x)
Sold −→ Sold +
∫
dx δψσ(x)[(k(x) + ieϕ(x)
)ψσ(x) + ησ(x)
]
↑k(x, τ) = ~
∂∂τ− ~
2
2m∇2 + u(x)− µ
[
k(x) +δWδρ(x)
]δW
δησ(x)− ~
δ(2)Wδρ(x)δησ(x)
+ ησ(x) = 0
12 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Schwinger–Dyson equations → Hedin equations
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
Infinitesimal shift of the fields −→ Z [η,η,ρ] doesn’t change!−→ Schwinger–Dyson equations (’50):
equations of motion forthe generating functionals.
Shift of the field ψσ(x): ψσ(x)→ ψσ(x) + δψσ(x)
Sold −→ Sold +
∫
dx δψσ(x)[(k(x) + ieϕ(x)
)ψσ(x) + ησ(x)
]
↑k(x, τ) = ~
∂∂τ− ~
2
2m∇2 + u(x)− µ
[
k(x) +δWδρ(x)
]δW
δησ(x)− ~
δ(2)Wδρ(x)δησ(x)
+ ησ(x) = 0
Shift of the field ϕ(x): ϕ(x)→ ϕ(x) + δϕ(x)
12 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Schwinger–Dyson equations → Hedin equations
Z [η,η,ρ] =
∫
D[ψ]D [ψ]D [ϕ] e−
1~S+Ssorg [η,η,ρ]
Infinitesimal shift of the fields −→ Z [η,η,ρ] doesn’t change!−→ Schwinger–Dyson equations (’50):
equations of motion forthe generating functionals.
Shift of the field ψσ(x): ψσ(x)→ ψσ(x) + δψσ(x)
Sold −→ Sold +
∫
dx δψσ(x)[(k(x) + ieϕ(x)
)ψσ(x) + ησ(x)
]
↑k(x, τ) = ~
∂∂τ− ~
2
2m∇2 + u(x)− µ
[
k(x) +δWδρ(x)
]δW
δησ(x)− ~
δ(2)Wδρ(x)δησ(x)
+ ησ(x) = 0
Shift of the field ϕ(x): ϕ(x)→ ϕ(x) + δϕ(x)
− 1
4πie∇2 δW
δρ(x)− ie~ δ(2)W
δησ(x)δησ(x+)+ ieρ(x) = 0
12 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations
13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations Begin with Schwinger–Dyson equation:
[
k(1) +δWδρ(1)
]δWδησ(1)
− ~δ(2)W
δρ(1)δησ(1)+ ησ(1) = 0
13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations Begin with Schwinger–Dyson equation:
[
k(1) +δWδρ(1)
]δWδησ(1)
− ~δ(2)W
δρ(1)δησ(1)+ ησ(1) = 0
Derivative w.r.t. external source (G11′ = −~W(2)η1η1′
with sources to 0):
[
k(1) + ieϕcl(1)
]
Gσσ′ (1, 1′)− ~δ
δρ(1)Gσσ′ (1, 1′) = −~δσσ′δ(1− 1′)
13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations Begin with Schwinger–Dyson equation:
[
k(1) +δWδρ(1)
]δWδησ(1)
− ~δ(2)W
δρ(1)δησ(1)+ ησ(1) = 0
Derivative w.r.t. external source (G11′ = −~W(2)η1η1′
with sources to 0):
[
k(1) + ieϕcl(1)
]
Gσσ′ (1, 1′)− ~δ
δρ(1)Gσσ′ (1, 1′) = −~δσσ′δ(1− 1′)
Using k(1)δ(1− 1′) = − δ(2)Γ(0)
δψcl(1)δψcl(1′)and Γψψ − Γ
(0)
ψψ= −~Σ∗:
13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations Begin with Schwinger–Dyson equation:
[
k(1) +δWδρ(1)
]δWδησ(1)
− ~δ(2)W
δρ(1)δησ(1)+ ησ(1) = 0
Derivative w.r.t. external source (G11′ = −~W(2)η1η1′
with sources to 0):
[
k(1) + ieϕcl(1)
]
Gσσ′ (1, 1′)− ~δ
δρ(1)Gσσ′ (1, 1′) = −~δσσ′δ(1− 1′)
Using k(1)δ(1− 1′) = − δ(2)Γ(0)
δψcl(1)δψcl(1′)and Γψψ − Γ
(0)
ψψ= −~Σ∗:
Σ∗σρ(1, 2)Gρσ′ (2, 1′) +
δ
δρ(1)Gσσ′ (1, 1′)− i e
~ϕcl(1)Gσσ′ (1, 1′) = 0
13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations Begin with Schwinger–Dyson equation:
[
k(1) +δWδρ(1)
]δWδησ(1)
− ~δ(2)W
δρ(1)δησ(1)+ ησ(1) = 0
Derivative w.r.t. external source (G11′ = −~W(2)η1η1′
with sources to 0):
[
k(1) + ieϕcl(1)
]
Gσσ′ (1, 1′)− ~δ
δρ(1)Gσσ′ (1, 1′) = −~δσσ′δ(1− 1′)
Using k(1)δ(1− 1′) = − δ(2)Γ(0)
δψcl(1)δψcl(1′)and Γψψ − Γ
(0)
ψψ= −~Σ∗:
Σ∗σρ(1, 2)Gρσ′ (2, 1′) +
δ
δρ(1)Gσσ′ (1, 1′)− i e
~ϕcl(1)Gσσ′ (1, 1′) = 0
Apply functional inverse (againW(2)η1η2
Γ(2)
ψ2ψ3= −δ13):
13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations Begin with Schwinger–Dyson equation:
[
k(1) +δWδρ(1)
]δWδησ(1)
− ~δ(2)W
δρ(1)δησ(1)+ ησ(1) = 0
Derivative w.r.t. external source (G11′ = −~W(2)η1η1′
with sources to 0):
[
k(1) + ieϕcl(1)
]
Gσσ′ (1, 1′)− ~δ
δρ(1)Gσσ′ (1, 1′) = −~δσσ′δ(1− 1′)
Using k(1)δ(1− 1′) = − δ(2)Γ(0)
δψcl(1)δψcl(1′)and Γψψ − Γ
(0)
ψψ= −~Σ∗:
Σ∗σρ(1, 2)Gρσ′ (2, 1′) +
δ
δρ(1)Gσσ′ (1, 1′)− i e
~ϕcl(1)Gσσ′ (1, 1′) = 0
Apply functional inverse (againW(2)η1η2
Γ(2)
ψ2ψ3= −δ13):
Σ∗σσ′(1, 1
′) = ie
~ϕcl(1)δσσ′δ(1− 1′)+
− 1
~Gσρ(1, 2)
δϕcl(3)
δρ(1)
δ(3)Γ
δϕcl(3)δψclρ (2)δψclσ′ (1′)
13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Ward Identities → Hedin equations Begin with Schwinger–Dyson equation:
[
k(1) +δWδρ(1)
]δWδησ(1)
− ~δ(2)W
δρ(1)δησ(1)+ ησ(1) = 0
Derivative w.r.t. external source (G11′ = −~W(2)η1η1′
with sources to 0):
[
k(1) + ieϕcl(1)
]
Gσσ′ (1, 1′)− ~δ
δρ(1)Gσσ′ (1, 1′) = −~δσσ′δ(1− 1′)
Using k(1)δ(1− 1′) = − δ(2)Γ(0)
δψcl(1)δψcl(1′)and Γψψ − Γ
(0)
ψψ= −~Σ∗:
Σ∗σρ(1, 2)Gρσ′ (2, 1′) +
δ
δρ(1)Gσσ′ (1, 1′)− i e
~ϕcl(1)Gσσ′ (1, 1′) = 0
Apply functional inverse (againW(2)η1η2
Γ(2)
ψ2ψ3= −δ13):
Σ∗σσ′(1, 1
′) = ie
~ϕcl(1)δσσ′δ(1− 1′)+
− 1
~Gσρ(1, 2)
δϕcl(3)
δρ(1)
δ(3)Γ
δϕcl(3)δψclρ (2)δψclσ′ (1′)
Hedin equation for Σ∗!13 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
14 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper self–energy Σ∗σσ′ (1, 1
′):
Σ∗σσ′ (1, 1
′) =1
~δσσ′δ(1− 1′)UH(1) + Σ∗
xcσσ′(1, 1′)
14 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper self–energy Σ∗σσ′ (1, 1
′):
Σ∗σσ′ (1, 1
′) =1
~δσσ′δ(1− 1′)UH(1) + Σ∗
xcσσ′(1, 1′)
Hartree insertion (tadpole):
1
~UH(1) =
1
~
∫
d2 U0(1, 2)Gρρ(2, 2+)
14 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper self–energy Σ∗σσ′ (1, 1
′):
Σ∗σσ′ (1, 1
′) =1
~δσσ′δ(1− 1′)UH(1) + Σ∗
xcσσ′(1, 1′)
Hartree insertion (tadpole):
1
~UH(1) =
1
~
∫
d2 U0(1, 2)Gρρ(2, 2+)
Exchange–correlation term:
Σ∗xcσσ′
(1, 1′) = −
1
~
∫
d2d3 Gσρ(1, 2)U(3, 1)Γσ′ρ(1′, 2, 3)
14 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper self–energy Σ∗σσ′ (1, 1
′):
Σ∗σσ′ (1, 1
′) =1
~δσσ′δ(1− 1′)UH(1) + Σ∗
xcσσ′(1, 1′)
Hartree insertion (tadpole):
1
~UH(1) =
1
~
∫
d2 U0(1, 2)Gρρ(2, 2+)
Exchange–correlation term:
Σ∗xcσσ′
(1, 1′) = −
1
~
∫
d2d3 Gσρ(1, 2)U(3, 1)Γσ′ρ(1′, 2, 3)
1′
Σ∗(1; 1′)
1
=1 ≡ 1′
2+
Γ
1′
1
32
14 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper polarization Π∗(1, 1′):
15 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper polarization Π∗(1, 1′):
Π∗(1, 1′) =1
~Gσµ(1, 2)Gνσ(2′, 1+)Γνµ(2
′, 2, 1′)
15 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper polarization Π∗(1, 1′):
Π∗(1, 1′) =1
~Gσµ(1, 2)Gνσ(2′, 1+)Γνµ(2
′, 2, 1′)
Π∗(1, 1′) = Γ1′ 1
2′
2
15 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper polarization Π∗(1, 1′):
Π∗(1, 1′) =1
~Gσµ(1, 2)Gνσ(2′, 1+)Γνµ(2
′, 2, 1′)
Π∗(1, 1′) = Γ1′ 1
2′
2
Hedin equation for the dressed vertex Γµν(1, 2, 3):
15 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations
Hedin equation for the proper polarization Π∗(1, 1′):
Π∗(1, 1′) =1
~Gσµ(1, 2)Gνσ(2′, 1+)Γνµ(2
′, 2, 1′)
Π∗(1, 1′) = Γ1′ 1
2′
2
Hedin equation for the dressed vertex Γµν(1, 2, 3):
Γµν(1, 2, 3) =1 ≡ 2 ≡ 3
+ ΓδΣ∗xcδG
1
23 4′
5′
4
5
15 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations: approximations
16 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations: approximations
Hartree–Fock approximation:
Γ1′
1≈
1′
1
16 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations: approximations
Hartree–Fock approximation:
Γ1′
1≈
1′
1
Random–Phase approximation:
Γ1′
1≈
1′
1
16 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Hedin equations: approximations
Hartree–Fock approximation:
Γ1′
1≈
1′
1
Random–Phase approximation:
Γ1′
1≈
1′
1
GW approximation:
Γ1′
1≈
1′
1
16 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperature
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]
Minimization condition: δΩ[n(x)]δn(x)
∣
∣
∣
n=n∗= 0
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]
Minimization condition: δΩ[n(x)]δn(x)
∣
∣
∣
n=n∗= 0
Mermin decomposition: Ω[n(x)] = F [n(x)] +∫
d3x n(x)u(x)
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]
Minimization condition: δΩ[n(x)]δn(x)
∣
∣
∣
n=n∗= 0
Mermin decomposition: Ω[n(x)] = F [n(x)] +∫
d3x n(x)u(x)
↓ Kohn–Sham approach: F [n(x)] = Ffree + EH + Fxc:
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]
Minimization condition: δΩ[n(x)]δn(x)
∣
∣
∣
n=n∗= 0
Mermin decomposition: Ω[n(x)] = F [n(x)] +∫
d3x n(x)u(x)
↓ Kohn–Sham approach: F [n(x)] = Ffree + EH + Fxc:
EH [n(x)] = e2
2
∫
d3xd3yn(x)n(y)|x−y|
“Hartree mean field”.
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]
Minimization condition: δΩ[n(x)]δn(x)
∣
∣
∣
n=n∗= 0
Mermin decomposition: Ω[n(x)] = F [n(x)] +∫
d3x n(x)u(x)
↓ Kohn–Sham approach: F [n(x)] = Ffree + EH + Fxc:
EH [n(x)] = e2
2
∫
d3xd3yn(x)n(y)|x−y|
“Hartree mean field”.
uxc(x) ≡δFxc[n(x)]δn(x)
exchange–correlation potential.
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory at finite temperatureGeneralization for T 6= 0 of Hohenberg–Kohn theorems (1964):
From the energy E[ψ] = 〈ψ|H|ψ〉→ to the thermodynamic potential Ω[ρ] = Tr
[
ρ(
H − µN + 1βln ρ
)]
From the minimum condition for the energy E[ψGS ] < E[ψ]→ to the minimization of the potential Ω[ρgc] < Ω[ρ]
Mermin theorem (1965):
n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]
Minimization condition: δΩ[n(x)]δn(x)
∣
∣
∣
n=n∗= 0
Mermin decomposition: Ω[n(x)] = F [n(x)] +∫
d3x n(x)u(x)
↓ Kohn–Sham approach: F [n(x)] = Ffree + EH + Fxc:
EH [n(x)] = e2
2
∫
d3xd3yn(x)n(y)|x−y|
“Hartree mean field”.
uxc(x) ≡δFxc[n(x)]δn(x)
exchange–correlation potential.
Ω =∑
i
niǫi − TSs − µN − EH [n(x)]−∫
d3x n(x)uxc(x) + Fxc[n(x)]
(
n(x) = n∗(x))
17 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approach
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
Thermal expectation value of the density:
n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]
δθ(x)
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
Thermal expectation value of the density:
n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]
δθ(x)
Legendre Transformation: from an external–source–based description toa density–based description:
Γ [n] =W [θ]−∫
dx θ(x)ψσ(x)ψσ(x)
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
Thermal expectation value of the density:
n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]
δθ(x)
Legendre Transformation: from an external–source–based description toa density–based description:
Γ [n] =W [θ]−∫
dx θ(x)ψσ(x)ψσ(x)
Equation of motion for Γ:
θ(x) = − δΓ [n]
δn(x)
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
Thermal expectation value of the density:
n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]
δθ(x)
Legendre Transformation: from an external–source–based description toa density–based description:
Γ [n] =W [θ]−∫
dx θ(x)ψσ(x)ψσ(x)
Equation of motion for Γ:
θ(x) = − δΓ [n]
δn(x)
But our actual system corresponds to θ(x) = 0 (remember Z = e−βΩ):
δΓ [n]
δn(x)
∣∣∣∣θ=0
= 0
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
Thermal expectation value of the density:
n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]
δθ(x)
Legendre Transformation: from an external–source–based description toa density–based description:
Γ [n] =W [θ]−∫
dx θ(x)ψσ(x)ψσ(x)
Equation of motion for Γ:
θ(x) = − δΓ [n]
δn(x)
But our actual system corresponds to θ(x) = 0 (remember Z = e−βΩ):
δΓ [n]
δn(x)
∣∣∣∣θ=0
= 0 =⇒ δΩ [n]
δn(x)
∣∣∣∣n=n∗
= 0 DFT!
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
Thermal expectation value of the density:
n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]
δθ(x)
Legendre Transformation: from an external–source–based description toa density–based description:
Γ [n] =W [θ]−∫
dx θ(x)ψσ(x)ψσ(x)
Equation of motion for Γ:
θ(x) = − δΓ [n]
δn(x)
But our actual system corresponds to θ(x) = 0 (remember Z = e−βΩ):
δΓ [n]
δn(x)
∣∣∣∣θ=0
= 0 =⇒ δΩ [n]
δn(x)
∣∣∣∣n=n∗
= 0 DFT!
Moreover it is possible to prove that:
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Density Functional Theory: functional approachExternal source linearly coupled to the density (vs. coupling source/field):
e−1~W[θ] =
∫
D [ϕ]D[ψ]D [ψ] e−
1~[S+
∫
dx θ(x)ψσ(x)ψσ(x)]
Thermal expectation value of the density:
n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]
δθ(x)
Legendre Transformation: from an external–source–based description toa density–based description:
Γ [n] =W [θ]−∫
dx θ(x)ψσ(x)ψσ(x)
Equation of motion for Γ:
θ(x) = − δΓ [n]
δn(x)
But our actual system corresponds to θ(x) = 0 (remember Z = e−βΩ):
δΓ [n]
δn(x)
∣∣∣∣θ=0
= 0 =⇒ δΩ [n]
δn(x)
∣∣∣∣n=n∗
= 0 DFT!
Moreover it is possible to prove that: If the source is real, n∗ is a minimum (W is concave, Γ is convex). The Mermin decomposition holds: Ω[n(x)] = F [n(x)] +
∫
d3x n(x)u(x).
18 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
(gaussian Berezin–integral:∫∏
k dθkdθke−θiAijθj = detA =
= eln detA = eTr lnA)
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
(gaussian Berezin–integral:∫∏
k dθkdθke−θiAijθj = detA =
= eln detA = eTr lnA)
Z =
∫
D [ϕ] e− 1
~
[
∫
dxϕ(x)(− 18π
∇2)ϕ(x)−~Tr ln[
1~
(k(x)+ieϕ(x)
)δ(x,y)
]]
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
(gaussian Berezin–integral:∫∏
k dθkdθke−θiAijθj = detA =
= eln detA = eTr lnA)
Z =
∫
D [ϕ] e− 1
~
[
∫
dxϕ(x)(− 18π
∇2)ϕ(x)−~Tr ln[
1~
(k(x)+ieϕ(x)
)δ(x,y)
]]
↑ϕ(x) = ϕH(x) + ϕ′(x)
ϕH(x, τ) = −ie
∫
d3yn(y, τ)
|x − y|=
1
ieuH(x) =
1
ie
δEH [n(x)]
δn(x)Hartree field
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
(gaussian Berezin–integral:∫∏
k dθkdθke−θiAijθj = detA =
= eln detA = eTr lnA)
Z =
∫
D [ϕ] e− 1
~
[
∫
dxϕ(x)(− 18π
∇2)ϕ(x)−~Tr ln[
1~
(k(x)+ieϕ(x)
)δ(x,y)
]]
↑ϕ(x) = ϕH(x) + ϕ′(x)
ϕH(x, τ) = −ie
∫
d3yn(y, τ)
|x − y|=
1
ieuH(x) =
1
ie
δEH [n(x)]
δn(x)Hartree field
Z = eβEH [n(x)]
∫
D [ϕ] e−1~[∫
dxϕ(x)(− 18π
∇2)ϕ(x)−ie∫
dxϕ(x)n(x)−~Tr ln[...]]
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
(gaussian Berezin–integral:∫∏
k dθkdθke−θiAijθj = detA =
= eln detA = eTr lnA)
Z =
∫
D [ϕ] e− 1
~
[
∫
dxϕ(x)(− 18π
∇2)ϕ(x)−~Tr ln[
1~
(k(x)+ieϕ(x)
)δ(x,y)
]]
↑ϕ(x) = ϕH(x) + ϕ′(x)
ϕH(x, τ) = −ie
∫
d3yn(y, τ)
|x − y|=
1
ieuH(x) =
1
ie
δEH [n(x)]
δn(x)Hartree field
Z = eβEH [n(x)]
∫
D [ϕ] e−1~[∫
dxϕ(x)(− 18π
∇2)ϕ(x)−ie∫
dxϕ(x)n(x)−~Tr ln[...]]
With this shift, a first separation occurs (Z = e−βΩ):(remember Kohn–Sham formula: Ω =
∑
niǫi − TSs − µN − EH −∫
n · uxc + Fxc)
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
(gaussian Berezin–integral:∫∏
k dθkdθke−θiAijθj = detA =
= eln detA = eTr lnA)
Z =
∫
D [ϕ] e− 1
~
[
∫
dxϕ(x)(− 18π
∇2)ϕ(x)−~Tr ln[
1~
(k(x)+ieϕ(x)
)δ(x,y)
]]
↑ϕ(x) = ϕH(x) + ϕ′(x)
ϕH(x, τ) = −ie
∫
d3yn(y, τ)
|x − y|=
1
ieuH(x) =
1
ie
δEH [n(x)]
δn(x)Hartree field
Z = eβEH [n(x)]
∫
D [ϕ] e−1~[∫
dxϕ(x)(− 18π
∇2)ϕ(x)−ie∫
dxϕ(x)n(x)−~Tr ln[...]]
With this shift, a first separation occurs (Z = e−βΩ):(remember Kohn–Sham formula: Ω =
∑
niǫi − TSs − µN − EH −∫
n · uxc + Fxc)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)+ieϕ(x)
)δ(x,y)
]]
19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integralGo back to the path integral form of the grand–canonical partition function:
Z =
∫
D [ϕ]D[ψ]D [ψ] e
− 1~
∫
dx[
ϕ(x)(− 18π
∇2)ϕ(x)+ψ(x)(k(x)+ieϕ(x)
)ψ(x)
]
︸ ︷︷ ︸
quadratic form↓
Integrate over fermion fields:obtain an effective bosonic theory!
(gaussian Berezin–integral:∫∏
k dθkdθke−θiAijθj = detA =
= eln detA = eTr lnA)
Z =
∫
D [ϕ] e− 1
~
[
∫
dxϕ(x)(− 18π
∇2)ϕ(x)−~Tr ln[
1~
(k(x)+ieϕ(x)
)δ(x,y)
]]
↑ϕ(x) = ϕH(x) + ϕ′(x)
ϕH(x, τ) = −ie
∫
d3yn(y, τ)
|x − y|=
1
ieuH(x) =
1
ie
δEH [n(x)]
δn(x)Hartree field
Z = eβEH [n(x)]
∫
D [ϕ] e−1~[∫
dxϕ(x)(− 18π
∇2)ϕ(x)−ie∫
dxϕ(x)n(x)−~Tr ln[...]]
With this shift, a first separation occurs (Z = e−βΩ):(remember Kohn–Sham formula: Ω =
∑
niǫi − TSs − µN − EH −∫
n · uxc + Fxc)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)+ieϕ(x)
)δ(x,y)
]]
Price for EH : more difficult interaction term!19 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)→ nH(x) 6= n(x)↓
Solution: add and remove anew field ϕxc(x) withthe following property:
GKS(x, x+) = G(x, x+)
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)→ nH(x) 6= n(x)↓
Solution: add and remove anew field ϕxc(x) withthe following property:
GKS(x, x+) = G(x, x+)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[−G−1
KS(x,y)+ie
(ϕ(x)−ϕxc(x)
)δ(x,y)
]]
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)→ nH(x) 6= n(x)↓
Solution: add and remove anew field ϕxc(x) withthe following property:
GKS(x, x+) = G(x, x+)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[−G−1
KS(x,y)+ie
(ϕ(x)−ϕxc(x)
)δ(x,y)
]]
︸ ︷︷ ︸
δϕ(x)
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)→ nH(x) 6= n(x)↓
Solution: add and remove anew field ϕxc(x) withthe following property:
GKS(x, x+) = G(x, x+)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[−G−1
KS(x,y)+ie
(ϕ(x)−ϕxc(x)
)δ(x,y)
]]
︸ ︷︷ ︸
δϕ(x)
Ω [n(x)] = −EH [n(x)]− 1
βTr ln(−G−1
KS)−∫
d3x n(x)uxc(x)+
− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)→ nH(x) 6= n(x)↓
Solution: add and remove anew field ϕxc(x) withthe following property:
GKS(x, x+) = G(x, x+)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[−G−1
KS(x,y)+ie
(ϕ(x)−ϕxc(x)
)δ(x,y)
]]
︸ ︷︷ ︸
δϕ(x)
Ω [n(x)] = −EH [n(x)]− 1
βTr ln(−G−1
KS)−∫
d3x n(x)uxc(x)+
− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
The very last step:
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)→ nH(x) 6= n(x)↓
Solution: add and remove anew field ϕxc(x) withthe following property:
GKS(x, x+) = G(x, x+)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[−G−1
KS(x,y)+ie
(ϕ(x)−ϕxc(x)
)δ(x,y)
]]
︸ ︷︷ ︸
δϕ(x)
Ω [n(x)] = −EH [n(x)]− 1
βTr ln(−G−1
KS)−∫
d3x n(x)uxc(x)+
− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
The very last step: − 1βTr ln(−G−1
KS) =∑
i niǫi − TSs − µN
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Kohn–Sham Decomposition & path–integral
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[
1~
(k(x)+ieϕH(x)
)δ(x,y)+ieϕ(x)δ(x,y)
]]
︸ ︷︷ ︸
−G−1H
(x, y)→ nH(x) 6= n(x)↓
Solution: add and remove anew field ϕxc(x) withthe following property:
GKS(x, x+) = G(x, x+)
Ω [n(x)] = −EH [n(x)]− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+]
e− 1
~
[
−ie∫
dxϕ(x)n(x)−~Tr ln[−G−1
KS(x,y)+ie
(ϕ(x)−ϕxc(x)
)δ(x,y)
]]
︸ ︷︷ ︸
δϕ(x)
Ω [n(x)] = −EH [n(x)]− 1
βTr ln(−G−1
KS)−∫
d3x n(x)uxc(x)+
− 1
βln
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
The very last step: − 1βTr ln(−G−1
KS) =∑
i niǫi − TSs − µN Price: really difficult interaction term!
20 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
A comparison gives:
e−βFxc =
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
Exact form of the exchange–correlation free energy!
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
A comparison gives:
e−βFxc =
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
Exact form of the exchange–correlation free energy!
Moreover it is possible to prove:
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
A comparison gives:
e−βFxc =
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
Exact form of the exchange–correlation free energy!
Moreover it is possible to prove: The cluster–decomposition–theorem holds: Fxc is made up of only fully connected
vacuum diagrams.
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
A comparison gives:
e−βFxc =
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
Exact form of the exchange–correlation free energy!
Moreover it is possible to prove: The cluster–decomposition–theorem holds: Fxc is made up of only fully connected
vacuum diagrams. Fxc and uxc are not independent quantities: uxc(x) = δFxc/δn(x)
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
A comparison gives:
e−βFxc =
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
Exact form of the exchange–correlation free energy!
Moreover it is possible to prove: The cluster–decomposition–theorem holds: Fxc is made up of only fully connected
vacuum diagrams. Fxc and uxc are not independent quantities: uxc(x) = δFxc/δn(x)
The Sham–Schlüter equation holds
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
A comparison gives:
e−βFxc =
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
Exact form of the exchange–correlation free energy!
Moreover it is possible to prove: The cluster–decomposition–theorem holds: Fxc is made up of only fully connected
vacuum diagrams. Fxc and uxc are not independent quantities: uxc(x) = δFxc/δn(x)
The Sham–Schlüter equation holds Non–trivial first–order expansion: exchange “Fock” contribution:
F(1)xc ≡ Ex = −
e2
2
∫
d3xd
3yn(x,y)n(y,x)
|x − y|=
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Exchange–Correlation Free Energy On the one hand: Path integral expression for Ω:
Ω [n(x)] =∑
i
niǫi−TSs−µN−EH [n]−
∫
x
n ·uxc−1
βln
∫
D [ϕ] e− 1
~
[
...
]
On the other hand: Kohn–Sham expression for Ω:
Ω [n(x)] =∑
i
niǫi − TSs − µN − EH [n] −
∫
x
n · uxc + Fxc [n(x)]
A comparison gives:
e−βFxc =
∫
D [ϕ] e− 1
~
[∫
dxϕ(x)(− 18π
∇2)ϕ(x)+~∑∞n=2
1n (
ie~)n Tr
[GKSδϕ
]n]
Exact form of the exchange–correlation free energy!
Moreover it is possible to prove: The cluster–decomposition–theorem holds: Fxc is made up of only fully connected
vacuum diagrams. Fxc and uxc are not independent quantities: uxc(x) = δFxc/δn(x)
The Sham–Schlüter equation holds Non–trivial first–order expansion: exchange “Fock” contribution:
F(1)xc ≡ Ex = −
e2
2
∫
d3xd
3yn(x,y)n(y,x)
|x − y|=
Besides: besides RPA as far as the interaction is concerned,besides One–Loop–Expansion as far as the integral is concerned...
21 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing.
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory.
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations.
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework.
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework. Change/Shift of variables =⇒ DFT in the Kohn–Sham picture:
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework. Change/Shift of variables =⇒ DFT in the Kohn–Sham picture:
Extension of articles in the literature to finite values of temperature.
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework. Change/Shift of variables =⇒ DFT in the Kohn–Sham picture:
Extension of articles in the literature to finite values of temperature. Explicit expression for the exchange–correlation free energy Fxc.
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework. Change/Shift of variables =⇒ DFT in the Kohn–Sham picture:
Extension of articles in the literature to finite values of temperature. Explicit expression for the exchange–correlation free energy Fxc.
Outlooks:
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework. Change/Shift of variables =⇒ DFT in the Kohn–Sham picture:
Extension of articles in the literature to finite values of temperature. Explicit expression for the exchange–correlation free energy Fxc.
Outlooks: Expansion of Fxc besides the first non–trivial term (exchange
contribution).
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework. Change/Shift of variables =⇒ DFT in the Kohn–Sham picture:
Extension of articles in the literature to finite values of temperature. Explicit expression for the exchange–correlation free energy Fxc.
Outlooks: Expansion of Fxc besides the first non–trivial term (exchange
contribution). TDDFT in a path integral approach.
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Conclusions and outlook
One single elegant formalism to describe the two most powerful tools forstudying many–body–systems: the coherent–state–path–integral:
Fields and sources are handled on the same footing. A source–based description =⇒ generating functionals:
many–body–perturbation–theory. Invariance of the integral under an infinitesimal shift of the integration
variable =⇒ Hedin equations. A picture based on the classic density nc(x) (Legendre transformation)
=⇒ DFT in the Hohenberg–Kohn framework. Change/Shift of variables =⇒ DFT in the Kohn–Sham picture:
Extension of articles in the literature to finite values of temperature. Explicit expression for the exchange–correlation free energy Fxc.
Outlooks: Expansion of Fxc besides the first non–trivial term (exchange
contribution). TDDFT in a path integral approach.
Thank you for your attention
22 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Grassmann numbersA set of n anticommuting generators θ1, ..., θn: θiθj = −θjθi
23 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Grassmann numbersA set of n anticommuting generators θ1, ..., θn: θiθj = −θjθi
Together with the identity 1, they form a 2n–dimensional algebra:
f (θ1, ..., θn) = c0 +n∑
k=1
n∑
i1...ik=1
ci1...ikθi1 ...θik
23 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Grassmann numbersA set of n anticommuting generators θ1, ..., θn: θiθj = −θjθi
Together with the identity 1, they form a 2n–dimensional algebra:
f (θ1, ..., θn) = c0 +n∑
k=1
n∑
i1...ik=1
ci1...ikθi1 ...θik
For example: ex = 1 + x+ 12x2 + ..., but eθ = 1 + θ.
23 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Grassmann numbersA set of n anticommuting generators θ1, ..., θn: θiθj = −θjθi
Together with the identity 1, they form a 2n–dimensional algebra:
f (θ1, ..., θn) = c0 +n∑
k=1
n∑
i1...ik=1
ci1...ikθi1 ...θik
For example: ex = 1 + x+ 12x2 + ..., but eθ = 1 + θ.
Clifford algebra: introduce a derivative: ∂∂θi
θj = δij
23 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Grassmann numbersA set of n anticommuting generators θ1, ..., θn: θiθj = −θjθi
Together with the identity 1, they form a 2n–dimensional algebra:
f (θ1, ..., θn) = c0 +n∑
k=1
n∑
i1...ik=1
ci1...ikθi1 ...θik
For example: ex = 1 + x+ 12x2 + ..., but eθ = 1 + θ.
Clifford algebra: introduce a derivative: ∂∂θi
θj = δij
[∂
∂θi, θj
]
+
= δij
[∂
∂θi,∂
∂θj
]
+
= 0
23 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Grassmann numbersA set of n anticommuting generators θ1, ..., θn: θiθj = −θjθi
Together with the identity 1, they form a 2n–dimensional algebra:
f (θ1, ..., θn) = c0 +n∑
k=1
n∑
i1...ik=1
ci1...ikθi1 ...θik
For example: ex = 1 + x+ 12x2 + ..., but eθ = 1 + θ.
Clifford algebra: introduce a derivative: ∂∂θi
θj = δij
[∂
∂θi, θj
]
+
= δij
[∂
∂θi,∂
∂θj
]
+
= 0
Introduce an integral (Berezin), with two main properties: linearity andinvariance under a shift of the integration variable:
∫
dθ = 0
∫
dθ θ = 1
23 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
Grassmann numbersA set of n anticommuting generators θ1, ..., θn: θiθj = −θjθi
Together with the identity 1, they form a 2n–dimensional algebra:
f (θ1, ..., θn) = c0 +n∑
k=1
n∑
i1...ik=1
ci1...ikθi1 ...θik
For example: ex = 1 + x+ 12x2 + ..., but eθ = 1 + θ.
Clifford algebra: introduce a derivative: ∂∂θi
θj = δij
[∂
∂θi, θj
]
+
= δij
[∂
∂θi,∂
∂θj
]
+
= 0
Introduce an integral (Berezin), with two main properties: linearity andinvariance under a shift of the integration variable:
∫
dθ = 0
∫
dθ θ = 1
For example:N∏
i=1
∫
dθidθie−θiAijθj = detA
(while in the bosonic case one has∏Ni=1
∫
R
dz∗i dzi2πi
e−z∗i Aijzj = 1
detA)
23 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
e−12
∫
d1d2n(1) 1~U0(1,2)n(2) =
=
∫
D[ϕ]e−12
∫
d1 e~ϕ(1)
(
− ~
4πe2∇2)
e~ϕ(1)−i
∫
d1 e~ϕ(1)n(1)
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
e−12
∫
d1d2n(1) 1~U0(1,2)n(2) =
=
∫
D[ϕ]e−12
∫
d1 e~ϕ(1)
(
− ~
4πe2∇2)
e~ϕ(1)−i
∫
d1 e~ϕ(1)n(1)
=
∫
D[ϕ]e− 1~
∫
d1[ϕ(1)(− 18π
∇2)ϕ(1)+ieϕ(1)ψ(1)ψ(1)]
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
e−12
∫
d1d2n(1) 1~U0(1,2)n(2) =
=
∫
D[ϕ]e−12
∫
d1 e~ϕ(1)
(
− ~
4πe2∇2)
e~ϕ(1)−i
∫
d1 e~ϕ(1)n(1)
=
∫
D[ϕ]e− 1~
∫
d1[ϕ(1)(− 18π
∇2)ϕ(1)+ieϕ(1)ψ(1)ψ(1)]↑
new interaction term (QED)
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
e−12
∫
d1d2n(1) 1~U0(1,2)n(2) =
=
∫
D[ϕ]e−12
∫
d1 e~ϕ(1)
(
− ~
4πe2∇2)
e~ϕ(1)−i
∫
d1 e~ϕ(1)n(1)
=
∫
D[ϕ]e− 1~
∫
d1[ϕ(1)(− 18π
∇2)ϕ(1)+ieϕ(1)ψ(1)ψ(1)]↑
new interaction term (QED)
In this way no more than just quadratic forms!
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
e−12
∫
d1d2n(1) 1~U0(1,2)n(2) =
=
∫
D[ϕ]e−12
∫
d1 e~ϕ(1)
(
− ~
4πe2∇2)
e~ϕ(1)−i
∫
d1 e~ϕ(1)n(1)
=
∫
D[ϕ]e− 1~
∫
d1[ϕ(1)(− 18π
∇2)ϕ(1)+ieϕ(1)ψ(1)ψ(1)]↑
new interaction term (QED)
In this way no more than just quadratic forms! Price: a new bosonic field appear...
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
e−12
∫
d1d2n(1) 1~U0(1,2)n(2) =
=
∫
D[ϕ]e−12
∫
d1 e~ϕ(1)
(
− ~
4πe2∇2)
e~ϕ(1)−i
∫
d1 e~ϕ(1)n(1)
=
∫
D[ϕ]e− 1~
∫
d1[ϕ(1)(− 18π
∇2)ϕ(1)+ieϕ(1)ψ(1)ψ(1)]↑
new interaction term (QED)
In this way no more than just quadratic forms! Price: a new bosonic field appear... Lint = ieϕ(x, τ)
∑
σ ψσ(x, τ)ψσ(x, τ)
LQED = ecAµ(x, t)∑
ab ψa(x, t)γµabψb(x, t)
24 / 24
HEDIN EQUATIONS
AND KOHN–SHAM
POTENTIAL
IN THE
PATH–INTEGRAL
FORMALISM
Marco Vanzini
General overview
Many–particle–system
Construction of thepath–integral
Hedin equations
Density FunctionalTheory
DFT & path–integral
Conclusions
The Hubbard–Stratonovich transformation
From a four–fermions interaction to a two–fermions plus an auxiliary bosonfield one.
Numerical analogous: e−a2x2 =
∫
R
dk√2πa
e−k2
2a−ikx
The same with fields: using Poisson equation ∇2U0(x) = −4πe2δ(4)(x):
e−12
∫
d1d2n(1) 1~U0(1,2)n(2) =
=
∫
D[ϕ]e−12
∫
d1 e~ϕ(1)
(
− ~
4πe2∇2)
e~ϕ(1)−i
∫
d1 e~ϕ(1)n(1)
=
∫
D[ϕ]e− 1~
∫
d1[ϕ(1)(− 18π
∇2)ϕ(1)+ieϕ(1)ψ(1)ψ(1)]↑
new interaction term (QED)
In this way no more than just quadratic forms! Price: a new bosonic field appear... Lint = ieϕ(x, τ)
∑
σ ψσ(x, τ)ψσ(x, τ)
LQED = ecAµ(x, t)∑
ab ψa(x, t)γµabψb(x, t)
Lem = − 116πFµνF
µν = 18π
[
|E|2 − |B|2]
= 18π [∇ϕ]2
24 / 24
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