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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)
����������
����
����
������
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
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