Objectives

I vocabulary :

Schrodinger’s equation , wavefunction, etc. . .

I what is the problem ? many body problem

I first steps toward resolution :

Hartree-Fock , Density Functional Theory , . . .

I what ”we” do : Range Separated Hybrid

I what ”I” do : Random Phase Approximation

What is it all about ?

- this system is described by a wavefunction |Ψ(xN ,RM)〉eigenfunction of the Schrodinger’s equation

Schrodinger’s equation

H|Ψ(xN ,RM)〉= E (xN ,RM)|Ψ(xN ,RM)〉H = Tn + Te + Vne + Vnn + Vee

Born-Oppenheimer : RM are fixed

Helec|Ψelec(xN)〉= Eelec(xN)|Ψelec(xN)〉Helec = Te + Vne + Vee

|Ψelec(xN)〉 is a function ofthe Hilbert space L2(XN ,C)

|Ψelec(xN)〉 is anantisymmetric function

What is it all about ?

- a molecule is a collection of M nuclei and N electronsnuclei : charges Z1→M , masses M1→M , positions R1→M ∈ R3

electrons : spins s1→N ∈ S = {−12 ,

12}, positions r1→N ∈ R3

i.e. coordinates x1→N ∈ X = R3 ⊗ S

- this system is described by a wavefunction |Ψ(xN ,RM)〉eigenfunction of the Schrodinger’s equation

Schrodinger’s equation

H|Ψ(xN ,RM)〉= E (xN ,RM)|Ψ(xN ,RM)〉H = Tn + Te + Vne + Vnn + Vee

Born-Oppenheimer : RM are fixed

Helec|Ψelec(xN)〉= Eelec(xN)|Ψelec(xN)〉Helec = Te + Vne + Vee

|Ψelec(xN)〉 is a function ofthe Hilbert space L2(XN ,C)

|Ψelec(xN)〉 is anantisymmetric function

What is it all about ?

- a molecule is a collection of M nuclei and N electronsnuclei : charges Z1→M , masses M1→M , positions R1→M ∈ R3

electrons : spins s1→N ∈ S = {−12 ,

12}, positions r1→N ∈ R3

i.e. coordinates x1→N ∈ X = R3 ⊗ S

- this system is described by a wavefunction |Ψ(xN ,RM)〉eigenfunction of the Schrodinger’s equation

Schrodinger’s equation

H|Ψ(xN ,RM)〉= E (xN ,RM)|Ψ(xN ,RM)〉H = Tn + Te + Vne + Vnn + Vee

Born-Oppenheimer : RM are fixed

Helec|Ψelec(xN)〉= Eelec(xN)|Ψelec(xN)〉Helec = Te + Vne + Vee

|Ψelec(xN)〉 is a function ofthe Hilbert space L2(XN ,C)

|Ψelec(xN)〉 is anantisymmetric function

What is it all about ?

- a molecule is a collection of M nuclei and N electronsnuclei : charges Z1→M , masses M1→M , positions R1→M ∈ R3

electrons : spins s1→N ∈ S = {−12 ,

12}, positions r1→N ∈ R3

i.e. coordinates x1→N ∈ X = R3 ⊗ S

- this system is described by a wavefunction |Ψ(xN ,RM)〉eigenfunction of the Schrodinger’s equation

Schrodinger’s equation

H|Ψ(xN ,RM)〉= E (xN ,RM)|Ψ(xN ,RM)〉H = Tn + Te + Vne + Vnn + Vee

Born-Oppenheimer : RM are fixed

Helec|Ψelec(xN)〉= Eelec(xN)|Ψelec(xN)〉Helec = Te + Vne + Vee

|Ψelec(xN)〉 is a function ofthe Hilbert space L2(XN ,C)

|Ψelec(xN)〉 is anantisymmetric function

What is it all about ?

- a molecule is a collection of M nuclei and N electronsnuclei : charges Z1→M , masses M1→M , positions R1→M ∈ R3

electrons : spins s1→N ∈ S = {−12 ,

12}, positions r1→N ∈ R3

i.e. coordinates x1→N ∈ X = R3 ⊗ S

- this system is described by a wavefunction |Ψ(xN ,RM)〉eigenfunction of the Schrodinger’s equation

Schrodinger’s equation

H|Ψ(xN ,RM)〉= E (xN ,RM)|Ψ(xN ,RM)〉H = Tn + Te + Vne + Vnn + Vee

Born-Oppenheimer : RM are fixed

Helec|Ψelec(xN)〉= Eelec(xN)|Ψelec(xN)〉Helec = Te + Vne + Vee

|Ψelec(xN)〉 is a function ofthe Hilbert space L2(XN ,C)

|Ψelec(xN)〉 is anantisymmetric function

What is it all about ?

- a molecule is a collection of M nuclei and N electronsnuclei : charges Z1→M , masses M1→M , positions R1→M ∈ R3

electrons : spins s1→N ∈ S = {−12 ,

12}, positions r1→N ∈ R3

i.e. coordinates x1→N ∈ X = R3 ⊗ S

- this system is described by a wavefunction |Ψ(xN ,RM)〉eigenfunction of the Schrodinger’s equation

Schrodinger’s equation

H|Ψ(xN ,RM)〉= E (xN ,RM)|Ψ(xN ,RM)〉H = Tn + Te + Vne + Vnn + Vee

Born-Oppenheimer : RM are fixed

Helec|Ψelec(xN)〉= Eelec(xN)|Ψelec(xN)〉Helec = Te + Vne + Vee

|Ψelec(xN)〉 is a function ofthe Hilbert space L2(XN ,C)

|Ψelec(xN)〉 is anantisymmetric function

Hamiltonian

Helec = Te + Vne + Vee =∑i

hi +∑ij

gij

N body with instanteneous Coulombic interaction(literally) infinitely complex problem

THE approximation : mean field

N-body problem → superposition of N 1-body problemsinteraction of each pair → sum of interactions of 1e− with the mean field

Helec =∑i

hi +∑i

vmeani

What do we gain ?wavefunction factorisable

What do we loose ?the correlation between the electrons

fluctuating repulsion/ correlation hole missing

Hamiltonian

Helec = Te + Vne + Vee =∑i

hi +∑ij

gij

N body with instanteneous Coulombic interaction(literally) infinitely complex problem

THE approximation : mean field

N-body problem → superposition of N 1-body problemsinteraction of each pair → sum of interactions of 1e− with the mean field

Helec =∑i

hi +∑i

vmeani

What do we gain ?wavefunction factorisable

What do we loose ?the correlation between the electrons

fluctuating repulsion/ correlation hole missing

Hamiltonian

Helec = Te + Vne + Vee =∑i

hi +∑ij

gij

N body with instanteneous Coulombic interaction(literally) infinitely complex problem

THE approximation : mean field

N-body problem → superposition of N 1-body problemsinteraction of each pair → sum of interactions of 1e− with the mean field

Helec =∑i

hi +∑i

vmeani

What do we gain ?wavefunction factorisable

What do we loose ?the correlation between the electrons

fluctuating repulsion/ correlation hole missing

Hamiltonian

Helec = Te + Vne + Vee =∑i

hi +∑ij

gij

N body with instanteneous Coulombic interaction(literally) infinitely complex problem

THE approximation : mean field

N-body problem → superposition of N 1-body problemsinteraction of each pair → sum of interactions of 1e− with the mean field

Helec =∑i

hi +∑i

vmeani

What do we gain ?wavefunction factorisable

What do we loose ?the correlation between the electrons

fluctuating repulsion/ correlation hole missing

Remember ? ”Hilbert space”one particuleN particulesantisymmetric

L2(X,C)L2(XN ,C)A2(XN ,C)

set of spin-orbitals {|ψi 〉}∞set of all Hartree products {|ψaψb . . . ψc〉}∞set of all Slater determinants {ΨA}∞

Hartree (Hartree product)

|ΨH(x1→N)〉= |ψ1(x1) . . . ψn(xn)〉totally uncorrelatedFermi principle : no

Hartree-Fock (monodeterminant)

|ΨHF(x1→N)〉 ∝

∣∣∣∣∣∣∣∣ψ1(x1) ψ2(x1) ···

ψ1(x2). . .

... ψn(xn)

∣∣∣∣∣∣∣∣uncorrelated for e− of different spinFermi : yes

〈ψiψj |gij |ψiψj〉 =

∫∫ψ∗i (x1)ψi (x1) gij ψ

∗j (x2)ψj(x2) Hartree term EH

〈ψiψj |gij |ψjψi 〉 =

∫∫ψ∗i (x1)ψj(x1) gij ψ

∗j (x2)ψi (x2) Exchange term Ex

Remember ? ”Hilbert space”one particuleN particulesantisymmetric

L2(X,C)L2(XN ,C)A2(XN ,C)

set of spin-orbitals {|ψi 〉}∞set of all Hartree products {|ψaψb . . . ψc〉}∞set of all Slater determinants {ΨA}∞

Hartree (Hartree product)

|ΨH(x1→N)〉= |ψ1(x1) . . . ψn(xn)〉totally uncorrelatedFermi principle : no

Hartree-Fock (monodeterminant)

|ΨHF(x1→N)〉 ∝

∣∣∣∣∣∣∣∣ψ1(x1) ψ2(x1) ···

ψ1(x2). . .

... ψn(xn)

∣∣∣∣∣∣∣∣uncorrelated for e− of different spinFermi : yes

〈ψiψj |gij |ψiψj〉 =

∫∫ψ∗i (x1)ψi (x1) gij ψ

∗j (x2)ψj(x2) Hartree term EH

〈ψiψj |gij |ψjψi 〉 =

∫∫ψ∗i (x1)ψj(x1) gij ψ

∗j (x2)ψi (x2) Exchange term Ex

Remember ? ”Hilbert space”one particuleN particulesantisymmetric

L2(X,C)L2(XN ,C)A2(XN ,C)

set of spin-orbitals {|ψi 〉}∞set of all Hartree products {|ψaψb . . . ψc〉}∞set of all Slater determinants {ΨA}∞

Hartree (Hartree product)

|ΨH(x1→N)〉= |ψ1(x1) . . . ψn(xn)〉totally uncorrelatedFermi principle : no

Hartree-Fock (monodeterminant)

|ΨHF(x1→N)〉 ∝

∣∣∣∣∣∣∣∣ψ1(x1) ψ2(x1) ···

ψ1(x2). . .

... ψn(xn)

∣∣∣∣∣∣∣∣uncorrelated for e− of different spinFermi : yes

〈ψiψj |gij |ψiψj〉 =

∫∫ψ∗i (x1)ψi (x1) gij ψ

∗j (x2)ψj(x2) Hartree term EH

〈ψiψj |gij |ψjψi 〉 =

∫∫ψ∗i (x1)ψj(x1) gij ψ

∗j (x2)ψi (x2) Exchange term Ex

Remember ? ”Hilbert space”one particuleN particulesantisymmetric

L2(X,C)L2(XN ,C)A2(XN ,C)

set of spin-orbitals {|ψi 〉}∞set of all Hartree products {|ψaψb . . . ψc〉}∞set of all Slater determinants {ΨA}∞

Hartree (Hartree product)

|ΨH(x1→N)〉= |ψ1(x1) . . . ψn(xn)〉totally uncorrelatedFermi principle : no

Hartree-Fock (monodeterminant)

|ΨHF(x1→N)〉 ∝

∣∣∣∣∣∣∣∣ψ1(x1) ψ2(x1) ···

ψ1(x2). . .

... ψn(xn)

∣∣∣∣∣∣∣∣uncorrelated for e− of different spinFermi : yes

〈ψiψj |gij |ψiψj〉 =

∫∫ψ∗i (x1)ψi (x1) gij ψ

∗j (x2)ψj(x2) Hartree term EH

〈ψiψj |gij |ψjψi 〉 =

∫∫ψ∗i (x1)ψj(x1) gij ψ

∗j (x2)ψi (x2) Exchange term Ex

Density Functional Theory|Ψ(xN)〉 : 4N coordinates

Hohenberg-Kohn : density n(r) is enough !

E0 = minn∈N

minΨ→n

{〈Ψ| Te + Vee |Ψ〉+

∫n(r)vext(r)

}= min

n∈N

{minΨ→n〈Ψ| Te + Vee |Ψ〉+

∫n(r)vext(r)

}= min

n∈N

{F [n] +

∫n(r)vext(r)

}

F [n] = minΨ→n〈Ψ| Te + Vee |Ψ〉= Te [n] + Vee [n] is the universal functionnal

E0 = minn∈N

{Ts [n] + EH [n] + Exc [n] +

∫n(r)vext(r)

}

Exc [n] is the exchange-correlation functionnal

This is where we talk about DFAs(LDA,GGA,PBE,PBE0,. . .)

Density Functional Theory|Ψ(xN)〉 : 4N coordinates

Hohenberg-Kohn : density n(r) is enough !

E0 = minn∈N

minΨ→n

{〈Ψ| Te + Vee |Ψ〉+

∫n(r)vext(r)

}= min

n∈N

{minΨ→n〈Ψ| Te + Vee |Ψ〉+

∫n(r)vext(r)

}= min

n∈N

{F [n] +

∫n(r)vext(r)

}

F [n] = minΨ→n〈Ψ| Te + Vee |Ψ〉= Te [n] + Vee [n] is the universal functionnal

E0 = minn∈N

{Ts [n] + EH [n] + Exc [n] +

∫n(r)vext(r)

}

Exc [n] is the exchange-correlation functionnal

This is where we talk about DFAs(LDA,GGA,PBE,PBE0,. . .)

Density Functional Theory|Ψ(xN)〉 : 4N coordinates

Hohenberg-Kohn : density n(r) is enough !

E0 = minn∈N

minΨ→n

{〈Ψ| Te + Vee |Ψ〉+

∫n(r)vext(r)

}= min

n∈N

{minΨ→n〈Ψ| Te + Vee |Ψ〉+

∫n(r)vext(r)

}= min

n∈N

{F [n] +

∫n(r)vext(r)

}

F [n] = minΨ→n〈Ψ| Te + Vee |Ψ〉= Te [n] + Vee [n] is the universal functionnal

E0 = minn∈N

{Ts [n] + EH [n] + Exc [n] +

∫n(r)vext(r)

}

Exc [n] is the exchange-correlation functionnal

This is where we talk about DFAs(LDA,GGA,PBE,PBE0,. . .)

Objectives

I vocabulary :

Schrodinger’s equation , wavefunction, etc. . .

I what is the problem ? many body problem

I first steps toward resolution :

Hartree-Fock , Density Functional Theory , . . .

I what ”we” do : Range Separated Hybrid

I what ”I” do : Random Phase Approximation

Range Separated Hybrid1

r= v lr

ee(r) + v sree(r)

vSR

ee

vLR

ee

vee

F[n] = minΨ→n〈Ψ| Te + Vee |Ψ〉= min

Ψ→n〈Ψ| Te + V lr

ee |Ψ〉+ E srHxc[n]

and :

E0 = minΨ→N

{〈Ψ| Te + V lr

ee |Ψ〉+ E srHxc[nΨ] +

∫nΨ(r)vext(r)

}

RSH scheme :

ERSH = minΦ

{〈Φ| Te + Vne + V lr

ee |Φ〉+ E srHxc[nΦ]

}total energy : E = ERSH + E lr

c

Range Separated Hybrid1

r= v lr

ee(r) + v sree(r)

vSR

ee

vLR

ee

vee

F[n] = minΨ→n〈Ψ| Te + Vee |Ψ〉= min

Ψ→n〈Ψ| Te + V lr

ee |Ψ〉+ E srHxc[n]

and :

E0 = minΨ→N

{〈Ψ| Te + V lr

ee |Ψ〉+ E srHxc[nΨ] +

∫nΨ(r)vext(r)

}

RSH scheme :

ERSH = minΦ

{〈Φ| Te + Vne + V lr

ee |Φ〉+ E srHxc[nΦ]

}total energy : E = ERSH + E lr

c

Range Separated Hybrid1

r= v lr

ee(r) + v sree(r)

vSR

ee

vLR

ee

vee

F[n] = minΨ→n〈Ψ| Te + Vee |Ψ〉= min

Ψ→n〈Ψ| Te + V lr

ee |Ψ〉+ E srHxc[n]

and :

E0 = minΨ→N

{〈Ψ| Te + V lr

ee |Ψ〉+ E srHxc[nΨ] +

∫nΨ(r)vext(r)

}

RSH scheme :

ERSH = minΦ

{〈Φ| Te + Vne + V lr

ee |Φ〉+ E srHxc[nΦ]

}total energy : E = ERSH + E lr

c

Random Phase Approximation

I Dispersion- mutual dynamical polarization of electron cloud

- fluctuations of the electronic density correlating

- need a good description of the long-range correlations

I RPA- treating the long-range

- exchange in the response function

- polarizability yields the correct long-range behaviour (C6/R6)

I Performances- average, due to the quality of the short-range correlation- considerably better in a range separation context

The origin/The way it was thought

In a gaz of electron : organized oscillations

(LR collective behavior) and (SR screened interaction)

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H = Hparticules

+ Hchamp

+ Hinteraction particules/champ

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HRPA = Hparticules

+ Hoscillations

+ Hsrinteraction particules/particules

contribution from particules in phase with the oscillation

the rest, which have random phases , are zeroed out

The origin/The way it was thought

In a gaz of electron : organized oscillations

(LR collective behavior) and (SR screened interaction)

����������

����

����

������

H = Hparticules

+ Hchamp

+ Hinteraction particules/champ

����������

����

����

������

HRPA = Hparticules

+ Hoscillations

+ Hsrinteraction particules/particules

contribution from particules in phase with the oscillation

the rest, which have random phases , are zeroed out

EAC-FDTc =

1

2

∫ 1

0dα

∫1,2;1′,2′

∫ ∞−∞

−dω2πi

w(1, 2; 1′, 2′)[χα(1, 2; 1′, 2′;ω)

− χ0(1, 2; 1′, 2′;ω)]

EAC-FDTc

(∫ 1

0

dα et

∫ ∞

−∞−dω2πi

)

Formalisme "densité de corrélation"(∫dα numérique

) Formalisme "matrice diélectrique"(∫

dω numérique

)

Formalisme de "plasmon"

Formalisme "rCCD"

∫dω analytique

∫dα analytique

∫dω analytique

∫dα analytique