Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Using concentration-compactness to solve some
quantum chemistry models
A. Anantharaman and E. Cancès
CERMICS - INRIA
CIMPA-UNESCO-EGYPT School - February 2nd, 2009
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Plan
Background - Density Functional Theory (DFT)
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Plan
Background - Density Functional Theory (DFT)
Models of interest and results
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Plan
Background - Density Functional Theory (DFT)
Models of interest and results
Sketch of the proofs
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Computation of electronic structures and DFT
Electronic structures : atoms, molecules, solids,...
What atoms, molecules can exist, and with which properties ?
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Computation of electronic structures and DFT
Electronic structures : atoms, molecules, solids,...
What atoms, molecules can exist, and with which properties ?
What are the groundstate energies and corresponding electronic
densities ?
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Computation of electronic structures and DFT
Electronic structures : atoms, molecules, solids,...
What atoms, molecules can exist, and with which properties ?
What are the groundstate energies and corresponding electronic
densities ?
What are the bond lengths and angles ?
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Computation of electronic structures and DFT
Electronic structures : atoms, molecules, solids,...
What atoms, molecules can exist, and with which properties ?
What are the groundstate energies and corresponding electronic
densities ?
What are the bond lengths and angles ?
How much energy is necessary to ionize the systems, to break
bonds ?
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Figure: Some DFT computation (I. Dabo - CERMICS)
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Wavefunction approach
Consider a system with N electrons (charge −1, position
ri ∈ R3 ) and
P
M nuclei (charge Zk ∈ N, position Rk ∈ R3 , Z = Zk ). The
Hamiltonian of the system is defined by
b =T
b +V
bee +
H
N
X
V (ri )
i=1
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Wavefunction approach
Consider a system with N electrons (charge −1, position
ri ∈ R3 ) and
P
M nuclei (charge Zk ∈ N, position Rk ∈ R3 , Z = Zk ). The
Hamiltonian of the system is defined by
b =T
b +V
bee +
H
N
X
V (ri )
i=1
where
b = PN − 1 ∇2 is the kinetic energy of the electrons
T
i=1
2 ri
PN P 1 1
b
Vee = i=1 j6=i 2 |ri −rj | is the electron-electron repulsion
P
Zk
V (r) = − M
k=1 |r−Rk | is the electron-nucleus attraction
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Wavefunction approach
Consider a system with N electrons (charge −1, position
ri ∈ R3 ) and
P
M nuclei (charge Zk ∈ N, position Rk ∈ R3 , Z = Zk ). The
Hamiltonian of the system is defined by
b =T
b +V
bee +
H
N
X
V (ri )
i=1
where
b = PN − 1 ∇2 is the kinetic energy of the electrons
T
i=1
2 ri
PN P 1 1
b
Vee = i=1 j6=i 2 |ri −rj | is the electron-electron repulsion
P
Zk
V (r) = − M
k=1 |r−Rk | is the electron-nucleus attraction
Born-Oppenheimer approximation : nuclei have fixed positions.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
The N electrons are represented using a wavefunction ψ with
following properties :
ψ = ψ(r1 , σ1 , r2 , σ2 , · · · rN , σN ) where the σi are the spin
variables with values {|↑i, |↓i}
hψ|ψi = 1 : ψ is normalized
ψ(· · · , ri , σi , · · · , rj , σj , · · · ) = −ψ(· · · , rj , σj , · · · , ri , σi , · · · ) : ψ
is antisymmetric
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
The N electrons are represented using a wavefunction ψ with
following properties :
ψ = ψ(r1 , σ1 , r2 , σ2 , · · · rN , σN ) where the σi are the spin
variables with values {|↑i, |↓i}
hψ|ψi = 1 : ψ is normalized
ψ(· · · , ri , σi , · · · , rj , σj , · · · ) = −ψ(· · · , rj , σj , · · · , ri , σi , · · · ) : ψ
is antisymmetric
With any wavefunction ψ can be associated the electronic density ρψ
XZ
X
···
ρψ (r) = N
|ψ(r, σ1 , r2 , σ2 , · · · , rN , σN )|2 dr2 · · · drN
σ1
so that
σN
Z
ρψ (r)dr = N
R3
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
The evolution of the system is given by the Schrödinger equation
ih d
b
H|ψi
=
|ψi
2π dt
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
The evolution of the system is given by the Schrödinger equation
ih d
b
H|ψi
=
|ψi
2π dt
b
We define the energy functional E (ψ) = hψ|H|ψi.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
The evolution of the system is given by the Schrödinger equation
ih d
b
H|ψi
=
|ψi
2π dt
b
We define the energy functional E (ψ) = hψ|H|ψi.
The groundstate (E0 , ψ0 ) is the minimum of E on all possible
wavefunctions, satisfying :
b 0 = E0 ψ0
Hψ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
The evolution of the system is given by the Schrödinger equation
ih d
b
H|ψi
=
|ψi
2π dt
b
We define the energy functional E (ψ) = hψ|H|ψi.
The groundstate (E0 , ψ0 ) is the minimum of E on all possible
wavefunctions, satisfying :
b 0 = E0 ψ0
Hψ
Similarly, stationary states are found solving the eigenvalue problems
b n = En ψn
Hψ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
The evolution of the system is given by the Schrödinger equation
ih d
b
H|ψi
=
|ψi
2π dt
b
We define the energy functional E (ψ) = hψ|H|ψi.
The groundstate (E0 , ψ0 ) is the minimum of E on all possible
wavefunctions, satisfying :
b 0 = E0 ψ0
Hψ
Similarly, stationary states are found solving the eigenvalue problems
b n = En ψn
Hψ
Wavefunctions are never used for large electron number N. On the
contrary, working with the sole density would be convenient.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Density Functional Theory
Hohenberg-Kohn theorem (1964) :
Theorem
There exists a functional F (ρ) of the electronic density ρ, such that
the groundstate energy E0 and the corresponding density ρ0 for N
electrons in presence of the external potential V is given by
Z
E0 = min F (ρ) +
ρ(r)V (r)dr
ρ
R3
Rwhere the minimum is taken over all positive ρ such that
R3 p(r)dr = N.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
Density Functional Theory
Hohenberg-Kohn theorem (1964) :
Theorem
There exists a functional F (ρ) of the electronic density ρ, such that
the groundstate energy E0 and the corresponding density ρ0 for N
electrons in presence of the external potential V is given by
Z
E0 = min F (ρ) +
ρ(r)V (r)dr
ρ
R3
Rwhere the minimum is taken over all positive ρ such that
R3 p(r)dr = N.
F is universal (does not depend on the potential V ). The problem is
to find approximations of F .
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
DFT : Thomas-Fermi model
Thomas and Fermi introduced the following approximation for F :
F
TF
3
(ρ) =
10
3
π2
2/3 Z
5/3
ρ
R3
1
(r)dr +
2
Z
R3
Z
R3
ρ(r) ρ(r0 )
dr dr0
|r − r0 |
This model is not adequate for complicated structures. Generally
speaking, approximations of F that are explicit functions of ρ are too
crude.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
DFT : Kohn-Sham models
In 1965, Kohn and Sham introduced a scheme to calculate the biggest
part of F exactly. It writes :
F (ρ) = TKS (ρ) + J(ρ) + Exc (ρ)
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
DFT : Kohn-Sham models
In 1965, Kohn and Sham introduced a scheme to calculate the biggest
part of F exactly. It writes :
F (ρ) = TKS (ρ) + J(ρ) + Exc (ρ)
where
TKS is an approximation of the kinetic energy which is computed
using a small wavefunctions set
R R
ρ(r0 )
0
J(ρ) = 12 R3 R3 ρ(r)
|r−r0 | dr dr is the classical electron-electron
repulsion
Exc is called the exchange-correlation energy
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Computation of electronic structures
Wavefunction approach - Schrödinger equations
DFT
Kohn-Sham models
DFT : Kohn-Sham models
In 1965, Kohn and Sham introduced a scheme to calculate the biggest
part of F exactly. It writes :
F (ρ) = TKS (ρ) + J(ρ) + Exc (ρ)
where
TKS is an approximation of the kinetic energy which is computed
using a small wavefunctions set
R R
ρ(r0 )
0
J(ρ) = 12 R3 R3 ρ(r)
|r−r0 | dr dr is the classical electron-electron
repulsion
Exc is called the exchange-correlation energy
The modeling effort concerns Exc .
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
Studied models and results
We restrict ourselves to spinless models : there is an even number of
elecrons N = 2Np which go by pairs .
The Kohn-Sham classical model then writes
INRKS
= inf
X
Np Z
i=1
R3
2
Z
|∇ϕi | +
R3
A. Anantharaman and E. Cancès
ρΦ V + J(ρΦ ) + Exc (ρΦ ), Φ ∈ WNp
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
Studied models and results
We restrict ourselves to spinless models : there is an even number of
elecrons N = 2Np which go by pairs .
The Kohn-Sham classical model then writes
INRKS
= inf
X
Np Z
i=1
R3
2
Z
|∇ϕi | +
ρΦ V + J(ρΦ ) + Exc (ρΦ ), Φ ∈ WNp
R3
with
WNp
= Φ = (ϕ1 , · · · , ϕNp ) |
and
ρΦ (r) = 2
1
3
Z
ϕi ∈ H (R ),
ϕi ϕj = δij
R3
Np
X
|ϕi (r)|2
i=1
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
We also considered an extended Kohn-Sham model where the
minimization is taken on bounded self-adjoint operators :
Z
REKS
IN
= Tr (−∆γ) +
ργ V + J(ργ ) + Exc (ργ ), γ ∈ KNp
R3
where
KNp = γ ∈ S(L2 (R3 )) | 0 ≤ γ ≤ 1, Tr (γ) = Np , Tr (−∆γ) < ∞
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
We also considered an extended Kohn-Sham model where the
minimization is taken on bounded self-adjoint operators :
Z
REKS
IN
= Tr (−∆γ) +
ργ V + J(ργ ) + Exc (ργ ), γ ∈ KNp
R3
where
KNp = γ ∈ S(L2 (R3 )) | 0 ≤ γ ≤ 1, Tr (γ) = Np , Tr (−∆γ) < ∞
P+∞
γ ∈ KNp writes γ = i=1
ni |φi ihφi | with φi ∈ H 1 (R3 ),
Z
φi φj = δij ,
ni ∈ [0, 1],
R3
and ργ (r) = 2
+∞
X
ni = Np ,
i=1
+∞
X
+∞
X
ni k∇φi k2L2 (R3 ) < ∞.
i=1
ni |φi (r)|2
i=1
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
If Exc was exact, then we would have INRKS = INREKS = E0 .
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
If Exc was exact, then we would have INRKS = INREKS = E0 .
In computations, two approximations are commonly used :
Local Density Approximation (LDA)
R
R
always
− R3
Exc (ρ) = R3 g (ρ(r)) dr =
3
4
3 1/3
ρ(r)4/3
π
Generalized
(GGA)
p
R Gradient Approximation
Exc (ρ) = R3 h(ρ(r), 12 |∇ ρ(r)|2 ) dr
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
If Exc was exact, then we would have INRKS = INREKS = E0 .
In computations, two approximations are commonly used :
Local Density Approximation (LDA)
R
R
always
− R3
Exc (ρ) = R3 g (ρ(r)) dr =
3
4
3 1/3
ρ(r)4/3
π
Generalized
(GGA)
p
R Gradient Approximation
Exc (ρ) = R3 h(ρ(r), 12 |∇ ρ(r)|2 ) dr
Our goal : finding conditions on the functions g (LDA) and h (GGA)
so that the minimization problems be well posed.
The existence of a minimizer for the RKS-LDA case was proved by Le
Bris in 1993.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
Under some conditions on g and h, we proved the existence of a
minimizer for :
the REKS-LDA case
the RKS-GGA case with two electrons
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
Under some conditions on g and h, we proved the existence of a
minimizer for :
the REKS-LDA case
the RKS-GGA case with two electrons
The difficulties lie in the following points :
generally speaking :
- lack of compactness (minimization on R3 )
- nonconvexity with respect to ρ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
Under some conditions on g and h, we proved the existence of a
minimizer for :
the REKS-LDA case
the RKS-GGA case with two electrons
The difficulties lie in the following points :
generally speaking :
- lack of compactness (minimization on R3 )
- nonconvexity with respect to ρ
according to the model considered :
- REKS-LDA : concentration-compactness on operators
- RKS-GGA : a priori non elliptic, quasilinear Euler-Lagrange
equations
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
LDA setting
Recall Exc (ρ) =
R
R3
g (ρ(r))
Suppose that g (ρ) is a C 1 function from R+ to R, twice differentiable
and such that :
g (0) = 0
g0 ≤ 0
∃0 < β− ≤ β+ <
3
∃1 ≤ α <
2
s.t.
2
3
s.t.
sup
ρ∈R+
lim sup
ρ→0+
A. Anantharaman and E. Cancès
|g 0 (ρ)|
<∞
ρβ− + ρβ+
g (ρ)
<0
ρα
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
Theorem (Extended KS-LDA model)
Assume that Z ≥ N = 2Np (neutral or positively charged system) and
that the function g satisfies the previous conditions. Then the
extended Kohn-Sham LDA model with Exc has a minimizer γ0 .
Besides, γ0 satisfies the self-consistent field equation
γ0 = χ(−∞,F ) (Hργ0 ) + δ
for some F ≤ 0, where
1
Hργ0 = − ∆ + V + ργ0 ? |r|−1 + g 0 (ργ0 ),
2
where χ(−∞,F ) is the characteristic function of the range (−∞, F )
and where δ ∈ S(L2 (R3 )) is such that 0 ≤ δ ≤ 1 and
Ran(δ) = Ker(Hργ0 − F ).
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
GGA setting
Recall Exc (ρ) =
R
R3
p
h(ρ(r), 12 |∇ ρ(r)|2 ) dr
Suppose that the h(ρ, κ) is a C 1 function from R+ × R+ to R, twice
differentiable w.r.t κ, and such that
h(0, κ) = 0, ∀κ ∈ R+
∂h
≤0
∂ρ
∃0 < β− ≤ β+ <
2
3
s.t.
A. Anantharaman and E. Cancès
sup
(ρ,κ)∈R+ ×R+
∂h
(ρ, κ)
∂ρ
<∞
ρβ− + ρβ+
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
∃1 ≤ α <
3
2
s.t.
h(ρ, κ)
<0
ρα
(ρ,κ)→(0+ ,0+ )
∃0 < a ≤ b < ∞ s.t.
∀(ρ, κ) ∈ R+ × R+ ,
LDA and GGA models
Results
lim sup
∀(ρ, κ) ∈ R+ × R+ ,
1+
a≤1+
∂h
∂2h
(ρ, κ) + 2κ 2 (ρ, κ) ≥ 0.
∂κ
∂κ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
∂h
(ρ, κ) ≤ b
∂κ
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
LDA and GGA models
Results
Theorem (Extended KS-GGA model for two electron systems)
Assume that Z ≥ N = 2Np = 2 (neutral or positively charged system
with two electrons) and that the function h satisfies the previous
conditions. Then the Kohn-Sham GGA model has a minimizer φ
satisfying the Euler equation
∂h
1
2
1+
(ρφ , |∇φ| ) ∇φ
− div
2
∂κ
∂h
−1
2
+ V + ρφ ? |r| +
(ρφ , |∇φ| ) φ = φ
∂ρ
for some < 0. φ can be chosen non-negative and (, φ) is the lowest
eigenpair of the self-adjoint operator
1
∂h
∂h
2
− div
1+
(ρφ , |∇φ| ) ∇· + V + ρφ ? |r|−1 + (ρφ , |∇φ|2 ).
2
∂κ
∂ρ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Sketch of the proof for the GGA case
In the GGA two electron setting, the energy writes
Z
Z
Z
|∇u|2 +
ρu V + J(ρu ) +
h(ρu , |∇u|2 )
E (u) =
R3
R3
R3
with
ρu = 2|u|2
and we consider the minimizing problem
Z
1
3
I1 = inf E (u), u ∈ H (R ),
|u| = 1
2
R3
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
As usual, we embed this problem in the family of problems
Z
1
3
2
Iλ = inf E (φ), u ∈ H (R ),
|u| = λ
R3
and introduce the problem at infinity
Z
∞
Iλ = inf E ∞ (u), u ∈ H 1 (R3 ),
2
|u| = λ
R3
where
∞
Z
2
Z
|∇u| + J(ρu ) +
E (u) =
R3
A. Anantharaman and E. Cancès
h(ρu , |∇u|2 )
R3
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
The Euler equation associated with Iλ writes
1
∂h
2
− div
1+
(ρu , |∇u| ) ∇u
2
∂κ
∂h
2
−1
(ρu , |∇u| ) u = θu
+ V + ρu ? |r| +
∂ρ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
The Euler equation associated with Iλ writes
1
∂h
2
− div
1+
(ρu , |∇u| ) ∇u
2
∂κ
∂h
2
−1
(ρu , |∇u| ) u = θu
+ V + ρu ? |r| +
∂ρ
Conditions on h entail :
ellipticity of the Euler equation
convexity of the energy w.r.t ∇u
Iλ > −∞, λ → Iλ continuous, boundedness of minimizing
sequences,...
nice spectral properties for the operator
1
∂h
∂h
2
− div
(ρu , |∇u| ) ∇· +V +ρu ?|r|−1 + (ρu , |∇u|2 )
1+
2
∂κ
∂ρ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
We show the following useful lemmas :
Lemma
∀ λ > 0, −∞ < Iλ ≤ Iλ∞ < 0
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
We show the following useful lemmas :
Lemma
∀ λ > 0, −∞ < Iλ ≤ Iλ∞ < 0
Lemma
∞
∀ 0 < µ < λ, Iλ ≤ Iµ + Iλ−µ
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
We show the following useful lemmas :
Lemma
∀ λ > 0, −∞ < Iλ ≤ Iλ∞ < 0
Lemma
∞
∀ 0 < µ < λ, Iλ ≤ Iµ + Iλ−µ
Lemma
Let 0 ≤ µ ≤ 1 and let (un )n∈N be a minimizing sequence for Iµ (resp.
for Iµ∞ ) which converges to some u ∈ H 1 (R3 ) weakly in H 1 (R3 ).
Assume that kuk2L2 = µ. Then u is a minimizer for Iµ (resp. for Iµ∞ ).
Therefore we only have to prove that the minimizing sequences are
compact.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Let un be an Ekeland sequence for I1 , which means
Z
1
3
∀n ∈ N, un ∈ H (R ) and
un2 = 1
R3
lim E (un ) = I1
n→+∞
lim E 0 (un ) + θn un = 0
n→+∞
in H −1 (R3 )
.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Let un be an Ekeland sequence for I1 , which means
Z
1
3
∀n ∈ N, un ∈ H (R ) and
un2 = 1
R3
lim E (un ) = I1
n→+∞
lim E 0 (un ) + θn un = 0
n→+∞
in H −1 (R3 )
We can assume un ≥ 0. un satisfies an equation of the kind
−div (an ∇un ) + Wn un + θn un → 0
in H −1 (R3 )
where an et Wn depend on un and ∇un through h.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Let un be an Ekeland sequence for I1 , which means
Z
1
3
∀n ∈ N, un ∈ H (R ) and
un2 = 1
R3
lim E (un ) = I1
n→+∞
lim E 0 (un ) + θn un = 0
n→+∞
in H −1 (R3 )
We can assume un ≥ 0. un satisfies an equation of the kind
−div (an ∇un ) + Wn un + θn un → 0
in H −1 (R3 )
where an et Wn depend on un and ∇un through h.
If un vanishes, then I1 ≥ 0 which is impossible.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Let un be an Ekeland sequence for I1 , which means
Z
1
3
∀n ∈ N, un ∈ H (R ) and
un2 = 1
R3
lim E (un ) = I1
n→+∞
lim E 0 (un ) + θn un = 0
n→+∞
in H −1 (R3 )
We can assume un ≥ 0. un satisfies an equation of the kind
−div (an ∇un ) + Wn un + θn un → 0
in H −1 (R3 )
where an et Wn depend on un and ∇un through h.
If un vanishes, then I1 ≥ 0 which is impossible.
If dichotomy happens, then un breaks into two bits u1,n (mass
δ1 ) and u2,n (mass δ2 = 1 − δ1 ) and we have I1 = Iδ1 + Iδ∞
.
2
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
These bits can also break into two bits and so on, so that we get a
binary tree.
k corresponds to some mass δ k .
Each uj,n
j
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
All the bits ui,n satisfy quasilinear equations of the kind
−div (ai,n ∇ui,n ) + Wi,n ui,n + θn ui,n → 0
in H −1 (R3 )
in which ai,n et Wi,n depend on ui,n et ∇ui,n through h.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
All the bits ui,n satisfy quasilinear equations of the kind
−div (ai,n ∇ui,n ) + Wi,n ui,n + θn ui,n → 0
in H −1 (R3 )
in which ai,n et Wi,n depend on ui,n et ∇ui,n through h.
We pass to the limit thanks to H-convergence and obtain an
infinity of equations
−div (ai ∇ui ) + Wi ui + θui = 0
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
All the bits ui,n satisfy quasilinear equations of the kind
−div (ai,n ∇ui,n ) + Wi,n ui,n + θn ui,n → 0
in H −1 (R3 )
in which ai,n et Wi,n depend on ui,n et ∇ui,n through h.
We pass to the limit thanks to H-convergence and obtain an
infinity of equations
−div (ai ∇ui ) + Wi ui + θui = 0
Due to spectral properties of −div (ai ∇·) + Wi , we have θ > 0
and this implies that there are a finite number of bits.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
All the bits ui,n satisfy quasilinear equations of the kind
−div (ai,n ∇ui,n ) + Wi,n ui,n + θn ui,n → 0
in H −1 (R3 )
in which ai,n et Wi,n depend on ui,n et ∇ui,n through h.
We pass to the limit thanks to H-convergence and obtain an
infinity of equations
−div (ai ∇ui ) + Wi ui + θui = 0
Due to spectral properties of −div (ai ∇·) + Wi , we have θ > 0
and this implies that there are a finite number of bits.
Consequently the dichotomy tree has a finite number (say P) of
compact leaves ui which areR minimizersPof the problems Iδ1 and
Iδ∞
for i ∈ J1, PK with δi = R3 ui2 and Pi=1 δi = 1.
i
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Suppose there are only two bits u1 and u2 , solutions of
−div (a1 ∇u1 ) + W1 u1 + θu1 = 0
and
−div (a2 ∇u2 ) + W2 u2 + θu2 = 0
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Suppose there are only two bits u1 and u2 , solutions of
−div (a1 ∇u1 ) + W1 u1 + θu1 = 0
and
−div (a2 ∇u2 ) + W2 u2 + θu2 = 0
It is easy to see that u1 and u2 have exponential decay at infinity.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Suppose there are only two bits u1 and u2 , solutions of
−div (a1 ∇u1 ) + W1 u1 + θu1 = 0
and
−div (a2 ∇u2 ) + W2 u2 + θu2 = 0
It is easy to see that u1 and u2 have exponential decay at infinity.
Thus we can easily compute the energies of
wt (r) = u1 (r) + u2 (r − te1 ), vt (r) = kwt kwt2 3 and obtain
L (R )
E (vt ) < E (u1 ) + E ∞ (u2 )
for t large enough.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Suppose there are only two bits u1 and u2 , solutions of
−div (a1 ∇u1 ) + W1 u1 + θu1 = 0
and
−div (a2 ∇u2 ) + W2 u2 + θu2 = 0
It is easy to see that u1 and u2 have exponential decay at infinity.
Thus we can easily compute the energies of
wt (r) = u1 (r) + u2 (r − te1 ), vt (r) = kwt kwt2 3 and obtain
L (R )
E (vt ) < E (u1 ) + E ∞ (u2 )
for t large enough.
Then I1 < Iδ1 + Iδ∞
= I1 , which is a contradiction.
2
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Suppose there are only two bits u1 and u2 , solutions of
−div (a1 ∇u1 ) + W1 u1 + θu1 = 0
and
−div (a2 ∇u2 ) + W2 u2 + θu2 = 0
It is easy to see that u1 and u2 have exponential decay at infinity.
Thus we can easily compute the energies of
wt (r) = u1 (r) + u2 (r − te1 ), vt (r) = kwt kwt2 3 and obtain
L (R )
E (vt ) < E (u1 ) + E ∞ (u2 )
for t large enough.
Then I1 < Iδ1 + Iδ∞
= I1 , which is a contradiction.
2
We have a relatively compact minimizing sequence un , which
therefore converges to a minimizer.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models
Computation of electronic structures and DFT
Studied models and results
Sketch of the proof for the GGA case
Conclusions
Conclusions
The LDA setting is now comprehensively treated.
Many things are yet to be done in the GGA setting : spin, N > 2
electrons.
The conditions on g and h are satisfied by some functionals
commonly used in computations.
A. Anantharaman and E. Cancès
LDA and GGA Kohn-Sham models