hedin equations and kohn–s ham potentialetsf.polytechnique.fr/system/files/slides.pdffunctions, in...

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



Hedin equations



Conclusions

Goals and overview

2 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions

Goals and overview


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



Hedin equations



Conclusions

Goals and overview


e−β(H−µN)]

.


Derivation of Hedin equations.

2 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions

Goals and overview


e−β(H−µN)]

.



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



Hedin equations



Conclusions

Goals and overview


e−β(H−µN)]

.



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



Hedin equations



Conclusions

References

3 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions

References


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



Hedin equations



Conclusions

References



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



Hedin equations



Conclusions

References




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



Hedin equations



Conclusions

References





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



Hedin equations



Conclusions

References






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



Hedin equations



Conclusions

References






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



Hedin equations



Conclusions

The system: the electron gas

4 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions



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



Hedin equations



Conclusions



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


†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



Hedin equations



Conclusions



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


†σ′ (x

′)e2

|x− x′| ψσ′ (x′)ψσ(x)



(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



Hedin equations



Conclusions



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


†σ′ (x

′)e2

|x− x′| ψσ′ (x′)ψσ(x)



(k)σ1...σN (x1...xN )


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



Hedin equations



Conclusions



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


†σ′ (x

′)e2

|x− x′| ψσ′ (x′)ψσ(x)



(k)σ1...σN (x1...xN )



′, τ ′)〉ρFundamental quantity in many body theory:

4 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions



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


†σ′ (x

′)e2

|x− x′| ψσ′ (x′)ψσ(x)



(k)σ1...σN (x1...xN )




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



Hedin equations



Conclusions



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


†σ′ (x

′)e2

|x− x′| ψσ′ (x′)ψσ(x)



(k)σ1...σN (x1...xN )





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



Hedin equations



Conclusions



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


†σ′ (x

′)e2

|x− x′| ψσ′ (x′)ψσ(x)



(k)σ1...σN (x1...xN )





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



Hedin equations



Conclusions

Hedin equations and DFT

5 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions


G11′ = G011′ + G

012Σ

∗22′G2′1′

5 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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′


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



Hedin equations



Conclusions


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′



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



Hedin equations



Conclusions


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′




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



Hedin equations



Conclusions


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′




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



Hedin equations



Conclusions


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′




u(x)↓

G(1, 1+)

⇐⇒


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



Hedin equations



Conclusions


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′




u(x)↓

G(1, 1+)

⇐⇒




↑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



Hedin equations



Conclusions


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′




u(x)↓

G(1, 1+)

⇐⇒




↑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



Hedin equations



Conclusions


6 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


DFT:φj(x)

6 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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



Hedin equations



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



Hedin equations



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



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

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



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

tN, ..., t

N

)

→ 〈φ| A |ψ〉 =∫


→∫

dq|q〉〈q| =∫

dp|p〉〈p| = 1

7 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

tN, ..., t

N

)

→ 〈φ| A |ψ〉 =∫


→∫

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



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

tN, ..., t

N

)

→ 〈φ| A |ψ〉 =∫


→∫

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



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

tN, ..., t

N

)

→ 〈φ| A |ψ〉 =∫


→∫

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′))

]


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



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

tN, ..., t

N

)

→ 〈φ| A |ψ〉 =∫


→∫

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′))

]



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



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

tN, ..., t

N

)

→ 〈φ| A |ψ〉 =∫


→∫

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′))

]



i ψi or ψ†i ψ

†j ψj ψi:


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



Hedin equations



Conclusions


Z(T, V, µ) = Tr[

e−β(H−µN)]

it = ~β←−−−−→ 〈x, t|x0, 0〉 = 〈x|e−i~Ht|x0〉


(

tN, ..., t

N

)

→ 〈φ| A |ψ〉 =∫


→∫

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′))

]



i ψi or ψ†i ψ

†j ψj ψi:


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


d3x ψσ(x)ψσ(x)|ψ〉〈ψ|

7 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=

8 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x


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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x



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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x



Z = limN→∞

∫

... 〈−ψ|e− 1~

~βNK ... e−

1~

~βNK ... e−

1~

~βNK |ψ〉

↑ ↑ ↑ ↑1F− 1F−

(

1 =

∫

∏


−∫

dx ψσ(x)ψσ(x)|ψ〉〈ψ|

)

8 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x



Z = limN→∞

∫

... 〈−ψ|e− 1~

~βNK ... e−

1~

~βNK ... e−

1~

~βNK |ψ〉

↑ ↑ ↑ ↑1F− 1F−

(

1 =

∫

∏


−∫


)

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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x



Z = limN→∞

∫

... 〈−ψ|e− 1~

~βNK ... e−

1~

~βNK ... e−

1~

~βNK |ψ〉

↑ ↑ ↑ ↑1F− 1F−

(

1 =

∫

∏


−∫


)

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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x



Z = limN→∞

∫

... 〈−ψ|e− 1~

~βNK ... e−

1~

~βNK ... e−

1~

~βNK |ψ〉

↑ ↑ ↑ ↑1F− 1F−

(

1 =

∫

∏


−∫


)

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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x



Z = limN→∞

∫

... 〈−ψ|e− 1~

~βNK ... e−

1~

~βNK ... e−

1~

~βNK |ψ〉

↑ ↑ ↑ ↑1F− 1F−

(

1 =

∫

∏


−∫


)

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



Hedin equations



Conclusions


Z = Tr[

e−β(H−µN)]

=↑

Tr[A] =

∫

∏


−∫


Z =

∫∏

σ,x



Z = limN→∞

∫

... 〈−ψ|e− 1~

~βNK ... e−

1~

~βNK ... e−

1~

~βNK |ψ〉

↑ ↑ ↑ ↑1F− 1F−

(

1 =

∫

∏


−∫


)

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



Hedin equations



Conclusions

Path–integral form of Z

9 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


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



Hedin equations



Conclusions


Z =

∫

D[ψσ(x, τ)


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



Hedin equations



Conclusions


Z =

∫

D[ψσ(x, τ)


1

~S[ψ,ψ

]

S[ψ, ψ,ϕ] =

∫

dx ψσ(x)

(

~∂

∂τ−

~2

2m∇

2+ u(x) − µ

)

ψσ(x) +

+1

2

∫



↑

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



Hedin equations



Conclusions


Z =

∫

D[ψσ(x, τ)


1

~S[ψ,ψ

]

S[ψ, ψ,ϕ] =

∫

dx ψσ(x)

(

~∂

∂τ−

~2

2m∇

2+ u(x) − µ

)

ψσ(x) +

+1

2

∫



↑


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



Hedin equations



Conclusions


Z =

∫

D[ψσ(x, τ)


1

~S[ψ,ψ

]

S[ψ, ψ,ϕ] =

∫

dx ψσ(x)

(

~∂

∂τ−

~2

2m∇

2+ u(x) − µ

)

ψσ(x) +

+1

2

∫



↑


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



Hedin equations



Conclusions


Z =

∫

D[ψσ(x, τ)


1

~S[ψ,ψ

]

S[ψ, ψ,ϕ] =

∫

dx ψσ(x)

(

~∂

∂τ−

~2

2m∇

2+ u(x) − µ

)

ψσ(x) +

+1

2

∫



↑


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



Hedin equations



Conclusions


Z =

∫

D[ψσ(x, τ)


1

~S[ψ,ψ

]

S[ψ, ψ,ϕ] =

∫

dx ψσ(x)

(

~∂

∂τ−

~2

2m∇

2+ u(x) − µ

)

ψσ(x) +

+1

2

∫



↑


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



Hedin equations



Conclusions

Two–points functions and generating functionals

10 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions


−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



Hedin equations



Conclusions


−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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


U(x;x′) = e2

~〈ϕ(x)ϕ(x′)〉Z = xx′


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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


U(x;x′) = e2

~〈ϕ(x)ϕ(x′)〉Z = xx′


Z [η,η,ρ] =

∫



↓Ssorg =

∫

dx[


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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


U(x;x′) = e2

~〈ϕ(x)ϕ(x′)〉Z = xx′


Z [η,η,ρ] =

∫



↓Ssorg =

∫

dx[


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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


U(x;x′) = e2

~〈ϕ(x)ϕ(x′)〉Z = xx′


Z [η,η,ρ] =

∫



↓Ssorg =

∫

dx[


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] = e−1~W[η,η,ρ]

10 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


U(x;x′) = e2

~〈ϕ(x)ϕ(x′)〉Z = xx′


Z [η,η,ρ] =

∫



↓Ssorg =

∫

dx[


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] = 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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


U(x;x′) = e2

~〈ϕ(x)ϕ(x′)〉Z = xx′


Z [η,η,ρ] =

∫



↓Ssorg =

∫

dx[


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′)




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



Hedin equations



Conclusions


−Gσσ′ (x;x′) = 〈ψσ(x)ψσ′ (x′)〉Z = xx′


U(x;x′) = e2

~〈ϕ(x)ϕ(x′)〉Z = xx′


Z [η,η,ρ] =

∫



↓Ssorg =

∫

dx[


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′)




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



Hedin equations



Conclusions

Self–Energy and Proper Polarization

11 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Properties of the Legendre Transform:

δ(2)Γ[ψcl, ψcl,ϕcl]

δψclµ (1)δψclν (2)= ~G−1

µν (1, 2)


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



Hedin equations



Conclusions





µν (1, 2) = ~

[

G0−1σσ′ (1, 1′)− Σ∗

σσ′ (1, 1′)]


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



Hedin equations



Conclusions





µν (1, 2) = ~

[

G0−1σσ′ (1, 1′)− Σ∗

σσ′ (1, 1′)]


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



Hedin equations



Conclusions





µν (1, 2) = ~

[

G0−1σσ′ (1, 1′)− Σ∗

σσ′ (1, 1′)]


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



Hedin equations



Conclusions

Schwinger–Dyson equations → Hedin equations

Z [η,η,ρ] =

∫



12 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Z [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



Hedin equations



Conclusions


Z [η,η,ρ] =

∫





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



Hedin equations



Conclusions


Z [η,η,ρ] =

∫






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



Hedin equations



Conclusions


Z [η,η,ρ] =

∫






Sold −→ Sold +

∫


)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



Hedin equations



Conclusions


Z [η,η,ρ] =

∫






Sold −→ Sold +

∫


)ψσ(x) + ησ(x)

]

↑k(x, τ) = ~

∂∂τ− ~

2

2m∇2 + u(x)− µ

[

k(x) +δWδρ(x)

]δW

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



Hedin equations



Conclusions


Z [η,η,ρ] =

∫






Sold −→ Sold +

∫


)ψσ(x) + ησ(x)

]

↑k(x, τ) = ~

∂∂τ− ~

2

2m∇2 + u(x)− µ

[

k(x) +δWδρ(x)

]δW

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



Hedin equations



Conclusions

Ward Identities → Hedin equations

13 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions


[

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



Hedin equations



Conclusions


[

k(1) +δWδρ(1)

]δWδησ(1)

− ~δ(2)W

δρ(1)δησ(1)+ ησ(1) = 0


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



Hedin equations



Conclusions


[

k(1) +δWδρ(1)

]δWδησ(1)

− ~δ(2)W

δρ(1)δησ(1)+ ησ(1) = 0


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)


(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



Hedin equations



Conclusions


[

k(1) +δWδρ(1)

]δWδησ(1)

− ~δ(2)W

δρ(1)δησ(1)+ ησ(1) = 0


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)


(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



Hedin equations



Conclusions


[

k(1) +δWδρ(1)

]δWδησ(1)

− ~δ(2)W

δρ(1)δησ(1)+ ησ(1) = 0


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)


(0)

ψψ= −~Σ∗:

Σ∗σρ(1, 2)Gρσ′ (2, 1′) +

δ

δρ(1)Gσσ′ (1, 1′)− i e

~ϕcl(1)Gσσ′ (1, 1′) = 0


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



Hedin equations



Conclusions


[

k(1) +δWδρ(1)

]δWδησ(1)

− ~δ(2)W

δρ(1)δησ(1)+ ησ(1) = 0


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)


(0)

ψψ= −~Σ∗:

Σ∗σρ(1, 2)Gρσ′ (2, 1′) +

δ

δρ(1)Gσσ′ (1, 1′)− i e

~ϕcl(1)Gσσ′ (1, 1′) = 0


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



Hedin equations



Conclusions

Hedin equations

14 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions

Hedin equations


′):

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



Hedin equations



Conclusions

Hedin equations


′):

Σ∗σσ′ (1, 1

′) =1

~δσσ′δ(1− 1′)UH(1) + Σ∗

xcσσ′(1, 1′)


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



Hedin equations



Conclusions

Hedin equations


′):

Σ∗σσ′ (1, 1

′) =1

~δσσ′δ(1− 1′)UH(1) + Σ∗

xcσσ′(1, 1′)


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



Hedin equations



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



Hedin equations



Conclusions

Hedin equations


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



Hedin equations



Conclusions

Hedin equations


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



Hedin equations



Conclusions

Hedin equations


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



Hedin equations



Conclusions

Hedin equations


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



Hedin equations



Conclusions

Hedin equations: approximations

16 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Hartree–Fock approximation:

Γ1′

1≈

1′

1

16 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions



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



Hedin equations



Conclusions



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



Hedin equations



Conclusions

Density Functional Theory at finite temperature

17 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions


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



Hedin equations



Conclusions


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



Hedin equations



Conclusions



[

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



Hedin equations



Conclusions



[

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



Hedin equations



Conclusions



[

ρ(


)]


Mermin theorem (1965):

17 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions



[

ρ(


)]



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



Hedin equations



Conclusions



[

ρ(


)]



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



Hedin equations



Conclusions



[

ρ(


)]



n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(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



Hedin equations



Conclusions



[

ρ(


)]



n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]


∣

∣

∣

n=n∗= 0


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



Hedin equations



Conclusions



[

ρ(


)]



n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]


∣

∣

∣

n=n∗= 0


d3x n(x)u(x)


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



Hedin equations



Conclusions



[

ρ(


)]



n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]


∣

∣

∣

n=n∗= 0


d3x n(x)u(x)


EH [n(x)] = e2

2

∫



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



Hedin equations



Conclusions



[

ρ(


)]



n(x)1−1!−−−→ u(x)→ H → ρ→ Ω[n(x)]


∣

∣

∣

n=n∗= 0


d3x n(x)u(x)


EH [n(x)] = e2

2

∫



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



Hedin equations



Conclusions

Density Functional Theory: functional approach

18 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



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



Hedin equations



Conclusions


e−1~W[θ] =

∫


1~[S+

∫


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



Hedin equations



Conclusions


e−1~W[θ] =

∫


1~[S+

∫



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



Hedin equations



Conclusions


e−1~W[θ] =

∫


1~[S+

∫



n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]

δθ(x)


Γ [n] =W [θ]−∫


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



Hedin equations



Conclusions


e−1~W[θ] =

∫


1~[S+

∫



n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]

δθ(x)


Γ [n] =W [θ]−∫



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



Hedin equations



Conclusions


e−1~W[θ] =

∫


1~[S+

∫



n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]

δθ(x)


Γ [n] =W [θ]−∫



θ(x) = − δΓ [n]

δn(x)


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



Hedin equations



Conclusions


e−1~W[θ] =

∫


1~[S+

∫



n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]

δθ(x)


Γ [n] =W [θ]−∫



θ(x) = − δΓ [n]

δn(x)


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



Hedin equations



Conclusions


e−1~W[θ] =

∫


1~[S+

∫



n(x) = 〈ψ(x)ψ(x)〉θ =δW [θ]

δθ(x)


Γ [n] =W [θ]−∫



θ(x) = − δΓ [n]

δn(x)


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



Hedin equations



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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form

19 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form↓


(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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form↓





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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form↓





Z =

∫

D [ϕ] e− 1

~

[

∫

dxϕ(x)(− 18π


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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form↓





Z =

∫

D [ϕ] e− 1

~

[

∫

dxϕ(x)(− 18π


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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form↓





Z =

∫

D [ϕ] e− 1

~

[

∫

dxϕ(x)(− 18π


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∫


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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form↓





Z =

∫

D [ϕ] e− 1

~

[

∫

dxϕ(x)(− 18π


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∫



∑


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



Hedin equations



Conclusions


Z =

∫

D [ϕ]D[ψ]D [ψ] e

− 1~

∫

dx[

ϕ(x)(− 18π


)ψ(x)

]

︸︷︷︸

quadratic form↓





Z =

∫

D [ϕ] e− 1

~

[

∫

dxϕ(x)(− 18π


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∫



∑


n · uxc + Fxc)

Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


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



Hedin equations



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∫


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



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−G−1H

(x, y)

20 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−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



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−G−1H

(x, y)→ nH(x) 6= n(x)↓


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



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−G−1H

(x, y)→ nH(x) 6= n(x)↓


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∫


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



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−G−1H

(x, y)→ nH(x) 6= n(x)↓


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∫


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



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−G−1H

(x, y)→ nH(x) 6= n(x)↓


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∫


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



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−G−1H

(x, y)→ nH(x) 6= n(x)↓


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∫


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



Hedin equations



Conclusions


Ω [n(x)] = −EH [n(x)]− 1

βln

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+]

e− 1

~

[

−ie∫


1~

(k(x)+ieϕH(x)


]]

︸︷︷︸

−G−1H

(x, y)→ nH(x) 6= n(x)↓


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∫


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



Hedin equations



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



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

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



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

x

n ·uxc−1

βln

∫

D [ϕ] e− 1

~

[

...

]


Ω [n(x)] =∑

i


∫

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



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

x

n ·uxc−1

βln

∫

D [ϕ] e− 1

~

[

...

]


Ω [n(x)] =∑

i


∫

x


A comparison gives:

e−βFxc =

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+~∑∞n=2

1n (

ie~)n Tr

[GKSδϕ

]n]


Moreover it is possible to prove:

21 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

x

n ·uxc−1

βln

∫

D [ϕ] e− 1

~

[

...

]


Ω [n(x)] =∑

i


∫

x


A comparison gives:

e−βFxc =

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+~∑∞n=2

1n (

ie~)n Tr

[GKSδϕ

]n]


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



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

x

n ·uxc−1

βln

∫

D [ϕ] e− 1

~

[

...

]


Ω [n(x)] =∑

i


∫

x


A comparison gives:

e−βFxc =

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+~∑∞n=2

1n (

ie~)n Tr

[GKSδϕ

]n]



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



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

x

n ·uxc−1

βln

∫

D [ϕ] e− 1

~

[

...

]


Ω [n(x)] =∑

i


∫

x


A comparison gives:

e−βFxc =

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+~∑∞n=2

1n (

ie~)n Tr

[GKSδϕ

]n]




The Sham–Schlüter equation holds

21 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

x

n ·uxc−1

βln

∫

D [ϕ] e− 1

~

[

...

]


Ω [n(x)] =∑

i


∫

x


A comparison gives:

e−βFxc =

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+~∑∞n=2

1n (

ie~)n Tr

[GKSδϕ

]n]




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



Hedin equations



Conclusions


Ω [n(x)] =∑

i


∫

x

n ·uxc−1

βln

∫

D [ϕ] e− 1

~

[

...

]


Ω [n(x)] =∑

i


∫

x


A comparison gives:

e−βFxc =

∫

D [ϕ] e− 1

~

[∫

dxϕ(x)(− 18π

∇2)ϕ(x)+~∑∞n=2

1n (

ie~)n Tr

[GKSδϕ

]n]




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



Hedin equations



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



Hedin equations



Conclusions



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



Hedin equations



Conclusions



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



Hedin equations



Conclusions




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



Hedin equations



Conclusions





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



Hedin equations



Conclusions






=⇒ 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



Hedin equations



Conclusions







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



Hedin equations



Conclusions







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



Hedin equations



Conclusions








Outlooks:

22 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions








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



Hedin equations



Conclusions









contribution). TDDFT in a path integral approach.

22 / 24

HEDIN EQUATIONS

AND KOHN–SHAM

POTENTIAL

IN THE

PATH–INTEGRAL

FORMALISM

Marco Vanzini

General overview



Hedin equations



Conclusions









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



Hedin equations



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



Hedin equations



Conclusions


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



Hedin equations



Conclusions



f (θ1, ..., θn) = c0 +n∑

k=1

n∑

i1...ik=1


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



Hedin equations



Conclusions



f (θ1, ..., θn) = c0 +n∑

k=1

n∑

i1...ik=1



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



Hedin equations



Conclusions



f (θ1, ..., θn) = c0 +n∑

k=1

n∑

i1...ik=1




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



Hedin equations



Conclusions



f (θ1, ..., θn) = c0 +n∑

k=1

n∑

i1...ik=1




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



Hedin equations



Conclusions



f (θ1, ..., θn) = c0 +n∑

k=1

n∑

i1...ik=1




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



Hedin equations



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



Hedin equations



Conclusions



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



Hedin equations



Conclusions




∫

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



Hedin equations



Conclusions




∫

R

dk√2πa

e−k2

2a−ikx


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



Hedin equations



Conclusions




∫

R

dk√2πa

e−k2

2a−ikx


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



Hedin equations



Conclusions




∫

R

dk√2πa

e−k2

2a−ikx


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



Hedin equations



Conclusions




∫

R

dk√2πa

e−k2

2a−ikx


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)]↑


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



Hedin equations



Conclusions




∫

R

dk√2πa

e−k2

2a−ikx


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)]↑


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



Hedin equations



Conclusions




∫

R

dk√2πa

e−k2

2a−ikx


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)]↑


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



Hedin equations



Conclusions




∫

R

dk√2πa

e−k2

2a−ikx


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)]↑


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

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hedin equations and kohn–s ham potentialetsf.polytechnique.fr/system/files/slides.pdffunctions, in...

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