PDEs and eletroni struture alulations
Eri CANCES
CERMICS - Eole des Ponts and INRIA
Warwik, January 2009
.
Introdution
2
First-priniple moleular simulation is beoming an essential tool in
Chemistry, Materials Siene, Moleular Biology and Nanosienes
Some objetive fats :
The most ited four artiles in Physis are
1. Kohn & Sham, PR 1965 - Density Funtional Theory (LDA)
2. Hohenberg & Kohn, PR 1964 - Density Funtional Theory
3. Perdew & Zunger, PRB 1981 - Density Funtional Theory (GGA)
4. Ceperley & Alder, PRL 1980 - Quantum Monte Carlo
W. Kohn and J. Pople shared the 1998 Nobel Prie in Chemistry
First-priniple moleular simulation utilizes more than 20% of the
resoures available in sienti omputing enters
Only a handful of Mathematiians are working in this eld
.
Introdution
First-priniple (or
3
ab initio) moleular simulation
In the absene of nulear reations, matter an be desribed as an
assembly of quantum nulei and eletrons interating through the
Coulomb potential : No empirial paramaters !
Atomi units : ~ = 1, me = 1, e = 1,
1
=1
4πε0
Eletrons and nulei
Eletrons : mass me = 1, harge −1,
Nuleus k : mass 1836 ≤ mk ≤ 400 000, harge zk ∈ N∗
.
Introdution
4
Born-Oppenheimer approximation
(lassial nulei, quantum eletrons)
Atomi positions and momenta : ({Rk (t)} , {Pk (t)}) ∈ R3M × R3M


∂Hnuc Pk (t)
dRk


=
(t)
=

 dt
∂Pk
mk


∂Hnuc
dPk


= −∇Rk W (R1(t), · · · , RM (t))
(t) = −

dt
∂Rk
Hnuc({Rk } , {Pk }) =
M
X
|Pk |2
k=1
2 mk
+ W (R1, · · · , RM )
W (R1, · · · , RM ) eetive potential (free of empirial parameters)
.
Introdution
5
Ab initio interatomi potentials and fores
0
nn
W (R1, · · · , RM ) = E{R
+
V
{Rk }
k}
∇Rk W (R1, · · · , RM ) =
nn
V{R
k}
=
ne
V{R
(r)
k}
X
1≤k<l≤M
=−
zk zl
|Rk − Rl |
M
X
k=1
zk
|r − Rk |
Z
R3
nn
ne
(r)
dr
+
∇
V
ρ0{Rk }(r)∇Rk V{R
R
k {Rk }
k}
nulear Coulomb repulsion energy
nulear Coulomb potential
0
0
E{R
and
ρ
{Rk } (r) eletroni ground state energy and density
k}
.
Introdution
6
From now on, the positions {Rk } of the nulei are onsidered as xed
In order to simplify the notation, we set
0
E 0 = E{R
k}
ρ0(r) = ρ0{Rk }(r)
ne
V ne(r) = V{R
(r) = −
k}
M
X
k=1
zk
|r − Rk |
.
Introdution
7
E 0 is the lowest eigenvalue of the eletroni Shrödinger equation
HN Ψ0 = E 0Ψ0
where the eletroni Hamiltonian is given by
HN = −
N
X
1
2
i=1
∆ ri −
N X
M
X
i=1 k=1
zk
+
|ri − Rk |
X
1≤i<j≤N
1
|ri − rj |
and where the eletroni ground state wavefuntion Ψ0 satises
the Pauli priniple
∀i < j,
Ψ0(· · · , rj , · · · , ri, · · · ) = −Ψ0(· · · , ri, · · · , rj , · · · )
the normalization ondition
Z
R3N
|Ψ0(r1, · · · , rN )|2 dr1 · · · drN = 1
.
Introdution
8
ρ0(r) is the density assoiated with Ψ0
The density assoiated with a funtion Ψ(r1, · · · , rN ) satisfying
the Pauli priniple
∀i < j,
Ψ(· · · , rj , · · · , ri, · · · ) = −Ψ(· · · , ri, · · · , rj , · · · )
the normalization ondition
Z
|Ψ(r1, · · · , rN )|2 dr1 · · · drN = 1
R3N
is the funtion ρΨ(r) dened by
ρΨ(r) = N
Z
R3(N −1)
|Ψ(r, r2, · · · , rN )|2 dr2 · · · drN
It holds
ρΨ(r) ≥ 0
and
Z
R3
ρΨ(r) dr = N
.
Introdution
9
Strengths of the model :
It allows to simulate a wide variety of phenomena
It does not ontain any parameter spei to the system
It is extremely aurate
Ionization energy of Helium (Korobov & Yelkhovsky PRL 2001)
alulations : 5 945 262 288 MHz, 5 945 204 223 MHz with RC
exp. : 5 945 204 238 MHz (1997), 5 945 204 356 MHz (1998)
Weaknesses of the model :
The Shrödinger equation HN Ψ = EΨ is a 3N -dimensional PDE
Chemial auray is required
example : atomization energy of water
∆EH2O = EH2O − 2EH − EO
= −76.4389 − 2 × (−0.5) − (−75.0840) = −0.3549 a.u.
.
Introdution
10
Methods for eletroni struture alulations
Hartree−Fock
Wavefunction methods
MPn, CI, CC
MCSCF
Orbital free
Density functional theory
(DFT)
N−body
¨
Schrodinger
equation
Kohn−Sham
Variational MC
Quantum Monte Carlo
Diffusion MC
Reduced Density Matrices
...
.
Introdution
11
Ab initio simulations today
A few atoms (small organi moleules) : spetrosopi auray
A few dozens of rst- or seond-row atoms : hemial auray
Several hundreds / a few thousands of atoms : qualitative results
W.R.( ?) : 11.8 million-atom (1.04×1012 grid points) DFT simulation
(A. Nakano et al., Int. J. High Perform. C. Appl. 2008)
1 - Linear saling algorithms
.
1 - Linear saling algorithms
13
For a system of N non-interating eletrons :
HN =
N
X
hri
i=1
h = − 21 ∆ + V ne
on
L2(R3)
εF
εF
N=5
hφi = εiφi,
E0 =
N
X
i=1
εi ,
Z
R3
φiφj = δij ,
N=6
ε1 < ε2 ≤ ε3 ≤ ... negative eigenvalues of h
1
Ψ0(r1, · · · , rN ) = √ det(φi(rj )),
N!
ρ0(r) =
N
X
i=1
|φi(r)|2
.
1 - Linear saling algorithms
For non-interating eletrons, E 0 =
14
N
X
εi and ρ0(r) =
i=1
N
X
i=1
|φi(r)|2 with

1


− ∆φi + V neφi = εiφi


2



Z
φiφj = δij


3

R





ε1 < ε2 ≤ · · · ≤ εN lowest N eigenvalues of h = − 12 ∆ + V ne
−→
Linear eigenvalue problem
.
1 - Linear saling algorithms
15
Galerkin approximation in atomi orbital basis sets (χµ)1≤µ≤Nb
5
1
2
6
12
4
10
7
3
9
8
11
13
15
S,H =
14
dimension of the approximation spae : Nb

HΦi = εiSΦi





ΦTi SΦj = δij





ε1 < ε2 ≤ · · · ≤ εN
H = [hχµ|h|χν i] ∈ RNb×Nb ,
2 × N ≤ Nb ≤ 5 × N
S = [hχµ|χν i] ∈ RNb×Nb
lowest N eigenvalues of HΦ = εSΦ
.
1 - Linear saling algorithms
16
Assume for simpliity that S = INb (orthonormal basis)
H = [hχµ|h|χν i] symmetri Nb × Nb matrix with Nb ∼ 5N

N

X




E0 =
εi =



i=1




N

X



Φi ΦTi
D=
(HD),
ρ0(r) =
X
Dµ,ν χµ(r)χν (r)
Φi,µχµ(r) =
µ=1
µ,ν
Nb
N X
X
i=1
(Φ1, · · · , ΦN )
i=1

HΦi = εi Φi









ΦTi Φj = δij






 ε ≤ ε ≤ ··· ≤ ε
1
2
N
N
H
for a generi matrix H : algorithmi omplexity in N 3
for a sparse matrix : algorithmi omplexity in N 2
for some hamiltonian matries : algorithmi omplexity in N
.
1 - Linear saling algorithms
17
Fundamental remarks :
1. We do not need to ompute the individual eigenvetors but only
the orthogonal projetor D
2. For insulators and semiondutors, the matrix D is sparse
Formulation in D
H Heaviside funtion
Dopt = H(εF − H),
Dopt = arginf Tr(HD),
D ∈ MS (Nb),
2
D = D,
Occupied ε
Unoccupied (virtual)
F
energy levels
energy levels
εN
ε N+1
ε1
gap
Tr(D) = N
εN b
.
1 - Linear saling algorithms
Some alternatives
18
to brute fore diagonalization
Polynomial or rational fration approximation of H(εF − H) :
FOE, FOP (Goedeker 1999)
Exat penalization methods : DMM (Li, Numes, Vanderbilt 1993)
Domain deomposition methods :
Divide and Conquer (Yang, Lee 1992)
Multilevel Domain Deomposition
(Barrault, Benteux, E.C., Le Bris, Hager, 2007)
Fragment method (Zhao, Meza, Wang 2008)
Linear saling on grids : Garía-Cervera, Lu, Xuan, E (2008)
.
1 - Linear saling algorithms
19
T
Formulation in C : Dopt = CoptCopt
, where Copt is solution to
inf Tr(HCC ),
T
T
C ∈ M(Nb, N ),
C C = IN
(1)
Rotational invariane : if Copt is solution, so is CoptU , for any orthogonal matrix U ∈ U (N )
Loalized orbitals. For an insulator (εN +1 − εN > 0 not too small),
T
where the matrix
there exists a matrix Cloc suh that Dopt = ClocCloc
Cloc is made of almost loally supported vetors
0
Nb
C loc =
0
N
.
1 - Linear saling algorithms
20
For insulators, it sues to solve
inf Tr(HCC ),
T
for matries C of the form
m1
C ∈ M(Nb, N ),
T
C C = IN
(2)
n
H1
n
C1
0
C =
0
N b = (p+1) n/2
mp
N = m 1 + ... + mp
inf
X
p
i=1
Hp
0
Cp
0
H =
Tr
N b = (p+1) n/2
HiCiCiT
,
CiTCi = Imi ,
Ci ∈ M(n, mi),
CiTT Ci+1 = 0,
mi ∈ N,
p
X
i=1
mi = N .
(3)
.
1 - Linear saling algorithms
21
Comparison with LAPACK and DMM (sequential odes)
1e+08
1e+09
LAPACK
DMM
MDD
LAPACK
DMM
MDD
1e+07
1e+08
Memory requirement in Kbytes
CPU Time in seconds
1e+06
100000
10000
1000
1e+06
100000
10000
100
10
100
1e+07
1000
10000
Nb
100000
1e+06
1000
100
1000
10000
Nb
100000
1e+06
Parallel MDD ode (G. Benteux, EDF) :
5 million atom polyethylen hain , STO-3G (17.5 × 106 AO)
solution of the linear subproblem in 60 minutes on 1024 proessors
.
1 - Linear saling algorithms
22
1. Our approah (exat deomposition of the energy, loalization of
the onstraints) provides a general framework for onstruting eient, variational, linear saling, domain deomposition algorithms
2. We have demonstrated the eieny of this general strategy by
proposing a multilevel domain deomposition algorithm (MDD)
performing very well on a benhmark of linear moleules
3. The onvergene properties of the MDD algorithm have been established in a simplied setting (G. Benteux, E.C., W.W. Hager
and C. Le Bris, submitted)
4. Relaxing the orthonormality onstraints on the loalized orbitals
would further inrease the performane of the MDD algorithm
5. The implementation of a 3D version of the MDD algorithm is a
work in progress (G. Benteux, EDF and Cermis, ANR Parmat)
2 - Density Funtional Theory
.
2 - Density Funtional Theory
24
Constrained optimization formulation of the non-interating model
For non-interating eletrons, the ground state energy and density
an be obtained by solving
inf E NI(Φ),
E NI(Φ) =
Φ = (φ1, · · · , φN ) ∈ (H 1(R3))N ,
N
X
i=1
hφi|h|φii =
V ne(r) = −
M
X
k=1
Z
N
X
1
i=1
zk
|r − Rk |
2
R3
|∇φi|2 +
ρΦ(r) =
N
X
i=1
Z
φiφj = δij
Z
ρΦV ne
R3
R3
|φi(r)|2
.
2 - Density Funtional Theory
25
The ase of interating eletrons
In the Kohn-Sham model, the ground state energy and density are
obtained by solving
inf E KS(Φ),
KS
E (Φ) =
Z
N
X
1
i=1
2
R3
Φ = (φ1, · · · , φN ) ∈ (H 1(R3))N ,
2
|∇φi| +
V ne(r) = −
Z
M
X
k=1
R3
ρΦV
ne
zk
|r − Rk |
1
+
2
Exc : exhange-orrelation funtional
Z
R3
φiφj = δij
ρΦ(r)ρΦ(r′)
′
dr
dr
+Exc[ρΦ]
′
|r − r |
R3
Z Z
R3
ρΦ(r) =
N
X
i=1
|φi(r)|2
Hohenberg-Kohn theorem : existene of an exat XC funtional
Z
Xα
Prototypial approximate XC funtional : Exc
[ρ] = −CX
ρ4/3(r) dr
3
.
2 - Density Funtional Theory
26
Kohn-Sham equations ('insulating' ase)

N
X


0

|φiihφi| = 1(−∞,εF](Hρ0 ),
γ
=




i=1







Hρ0 φi = εiφi




Z
ρ0(r) = γ 0(r, r) =
N
X
i=1
φiφj = δij



R3







ε1 < ε2 ≤ · · · ≤ εN lowest N eigenvalues of Hρ0







4

0 1/3
KS
ne
0
−1
 H 0 = − 1 ∆ + V KS
C
ρ
,
V
=
V
+
ρ
⋆
|
·
|
−
0
0
X
ρ
ρ
ρ
2
3
−→
|φi(r)|2
Nonlinear eigenvalue problem
εF
N=5
.
2 - Density Funtional Theory
27
Mathematial and numerial analysis of the models arising in DFT
1. Proof of existene of a solution for neutral and positively harged
systems
R
LDA
(a) for LDA : Exc (ρ) = R3 exc(ρ(r)) dr (Le Bris 1993)
R
GGA
(b) for GGA : Exc (ρ) = R3 exc(ρ(r), ∇ρ(r)) dr but only for two
eletron systems (Anantharaman, E.C., submitted)
2. Constrution and proof of onvergene of some simple SCF algorithms (E.C., Le Bris, 2000-2003). A priori error estimates for
Kohn-Sham LDA (E.C., Chakir, Maday, ongoing work)
3. Thermodynamial limits with Exc(ρ) = 0 : perfet rystal (Catto,
Le Bris, Lions 1998), rystals with loal defets (E.C., Deleurene,
Lewin, 2008)
4. A lot of work remains to be done !
3 - Quantum Monte Carlo
.
3 - Quantum Monte Carlo
29
Spetrum of the N -body Hamiltonian (operating on He =
HN = −
If Z =
N
X
1
i=1
M
X
k=1
2
∆ ri −
N X
M
X
i=1 k=1
zk
+
|ri − Rk |
X
1≤i<j≤N
1
|ri − rj |
=
N
^
L2(R3))
i=1
1
− ∆+V
2
zk ≥ N (neutral or positively harged system),
σess(HN ) = [ΣN , +∞) with ΣN < 0 if N ≥ 2 and Σ1 = 0 ;
HN has an innite sequene of nite multipliity eigenvalues
E 0 = λ1(HN ) ≤ λ2(HN ) ≤ λ3(HN ) ≤ · · · onverging to ΣN
Ground state
Ε 0= λ1(Η N)
Excited states
ΣN
Eigenvalues embedded in
the continuous spectrum
Essential spectrum
.
3 - Quantum Monte Carlo
30
Notations :
Eletroni ground state (supposed to be non-degenerate)
HN Ψ0 = E 0ψ 0
E 0 = λ1(HN )
Let γ = λ2(HN ) − λ1(HN ) > 0
Let ΨI be a trial wavefuntion, relatively lose to Ψ0 for whih the
loal elds
∇ΨI (x)
b(x) =
ΨI (x)
and
(HN ΨI )(x)
1 ∆ΨI (x)
EL(x) =
=−
+ V (x)
ΨI (x)
2 ΨI (x)
are not too diult to ompute (nite sum of Slater determinants)
.
3 - Quantum Monte Carlo
31
Let us onsider the paraboli (imaginary time Shrödinger) equation
and

1
 ∂φ
= −HN φ = ∆φ − V φ
∂t
 φ(0, x) = Ψ (x) 2
I
E(t) =
(4)
(HN ΨI , φ(t))L2
.
(ΨI , φ(t))L2
(5)
One has
0
0 ≤ E(t) − E ≤
(HN ΨI , ΨI )L2 − E
(Ψ0, ΨI )2L2
0
exp(−γt)
Diusion Monte Carlo : simulation of (4)-(5) with probabilisti methods using variane redution tehniques (importane sampling).
.
3 - Quantum Monte Carlo
32
Set f1(t, x) = ΨI (x)φ(t, x). A simple alulation shows that
(HN ΨI , φ(t))L2
=
E(t) =
(ΨI , φ(t))L2
Z
(HN ΨI )(x) φ(t, x) dx
Z
=
ΨI (x)φ(t, x) dx
R3
R3
and that f1 solves
EL(x) f1(t, x) dx
Z
f1(t, x) dx
R3
R3

∂f 1



= ∆f − div (bf ) − ELf
∂t
2


 f (0, x) = Ψ2 (x),
I
with
b(x) =
Z
∇ΨI (x)
ΨI (x)
and
EL(x) =
(HN ΨI )(x)
1 ∆ΨI (x)
=−
+ V (x)
ΨI (x)
2 ΨI (x)
.
3 - Quantum Monte Carlo
33
Interpretation of the equation on f1 in terms of stohasti proess
∂f 1
= ∆f − div (bf ) − ELf
∂t
2
↓
↓
↓
diusion
drift
birth-death
One an then try to approximate E(t) by
Z t
EL (Xs) ds
IE EL(Xt) exp −
Z t 0
E DMC(t) =
IE exp −
EL (Xs) ds
0
where (Xt)t≥0 is the stohasti proess dened by
dXt = b (Xt) dt + dWt
X0 ∼ Ψ2I .
.
3 - Quantum Monte Carlo
34
With B. Jourdain et T. Lelièvre, we have proved the following results
(M3AS 2006) : under some tehnial assumptions,
1. Beause of the singularity of the drift b(x) =
ries dened by
∇ΨI (x)
, the trajetoΨI (x)
dXtx = b (Xtx) dt + dWt
X0x = x
annot ross the nodal surfaes Ψ−1
I (0).
The random variable Xtx has a density p(t, x, y) and the funtion
(x, y)
is symmetri.
7−→
ΨI (x)2p(t, x, y)
3 - Quantum Monte Carlo
.
35
2. It holds
Z
EL(x) f1(t, x) dx
3
E(t) = R Z
f1(t, x) dx
R3
Z
EL(x) f2(t, x) dx
3
E DMC(t) = R Z
f2(t, x) dx
R3
where f1 and f2 are two dierent weak solutions of

∂f 1



= ∆f − div (bf ) − ELf
∂t
2


 f (0, x) = Ψ2 (x),
I
3 - Quantum Monte Carlo
.
36
More preisely
f1(t, x) = ΨI (x) φ(t, x)
f2(t, x) = ΨI (x) φ2(t, x)

∂φ 1



= ∆φ − V φ
∂t
2


 φ(0, x) = Ψ (x),
I

∂φ2 1



= ∆φ2 − V φ2


∂t
2


φ2(0, x) = ΨI (x)






 φ (t, x) = 0 on Ψ−1(0)
2
I
3 - Quantum Monte Carlo
.
37
3. The funtion E DMC(t) onverges when t goes to +∞ to
inf
(
hΨ, HN Ψi,
Ψ∈
N
^
H 1(R3),
i=1
kΨkL2 = 1,
)
Ψ = 0 on Ψ−1
I (0) .
whih is an upper bound of the exat ground state energy
E 0 = inf
(
hΨ, HN Ψi,
Ψ∈
N
^
i=1
H 1(R3),
)
kΨkL2 = 1 .
.
3 - Quantum Monte Carlo
38
This mathematial analysis points out one of the main limitations of
the Diusion Monte Carlo method, well known by its users : using an
importane funtion ΨI introdues a systemati error, exept in the
peuliar ase when the nodal surfaes of Ψ0 and ΨI exatly oinide
Mathematial and numerial hallenges :
1. Analysis and improvement of the existing numerial shemes
(stohasti partile methods)
2. Computation of interatomi fores
3. How to go beyond the xed node approximation ?
.
3 - Quantum Monte Carlo
39
The (Fixed Node) Diusion Monte Carlo method remains to date the
referene method for aurate Quantum Monte Carlo simulations on
large systems
S0 → S2
±