i ) direct minimization in hartree fock and dft - ii ... · i ) direct minimization in hartree fock...
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I ) Direct minimization in Hartree Fock and
DFT - II) Converence of Coupled Cluster
Reinhold Schneider, MATHEON TU Berlin
Computational methods in many electron systems : MP2
Workshop Göteborg 2009
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Ab initio computation
Reliable computation of atomisitic molecular phenomena will
play an important role in modern material science, chemistry,
(molecular biology)
Ab initio computation is based on first principles of quantum
mechanics
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Electronic Structure CalculationElectronic Schrödinger equa-
tion N′ stationary nonrelativistic elec-
trons + Born Oppenheimer approxima-
tion
HΨ = EΨ
The Hamilton operator
H = −12
∑i
∆i −N′∑i
K∑ν=1
Zν|xi − aν |
+12
N′∑i 6=j
1|xi − xj |
acts on anti-symmetric wave functions Ψ ∈ H1((R3 × ±12)
N′),Ψ(x1, s1, . . . , xN′ , sN′) ∈ R , (xi , si) ∈ R3 × ±1
2 .
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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GoalsOutput: ground-state energy
E0 = min〈Ψ,Ψ〉=1〈HΨ,Ψ〉 , Ψ = argmin〈Ψ,Ψ〉=1〈HΨ,Ψ〉
most quantities for molecules (chemistry) and
crystals (solid state physics) can be derived from E0, e.g.
atomic forces, molecular geometry, bonding and ionization
energies etc.
these quantities are (small) differences E0,a − E0,b
excited states are required for opto-electronic effects
accuracy is limited due to neglecting relativistic and
non-Born-Openheimer effects.. is beyond the present presentation
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Basic Problem - Curse of dimensions
linear eigenvalue problem, but extremely high-dimensional
+ anti-symmetry constraints + lack of regularity.
traditional approximation methods (FEM, Fourier series,
polynomials, MRA etc.): approximation error in R1: . n−s,
s- regularity , R3N′ : . n−s3N′ , (s < 5
2 ) with n DOFs
For large systems N ′ >> 1 ( N ′ > 1) the electronic Schrödinger
equation seems to be intractable! But 70 years of impressive
progress has been awarded by the Nobel price 1998 in
Chemistry: Kohn, Pople
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Tensor product approx. - separation of variables
Approximation by sums of anti-symmetric tensor products:
Ψ =∞∑
k=1
ck Ψk
Ψk (x1, s1; . . . ; xN′ , sN′) = ϕ1,k ∧ . . . ∧ ϕN′,k =1√N ′!
det(ϕi,k (xj , sj))
with ϕi,k ∈ ϕj : j = 1, . . ., w.l.o. generality
〈ϕi , ϕj〉 =∑
s=± 12
∫R3ϕi(x, s)ϕj(x, s)dx = δi,j .
A Slater determinant: Ψk is an (anti-symmetric) product of N ′
orthonormal functions ϕi , called spin orbital functions
ϕi : R3 × ±12 → R , i = 1, . . . ,N,
For present applications it is sufficient to consider real valued
functions Ψ, ϕ .
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Spin functions and spatial orbitals
Example of Slater determinant (N ′ = 2)
Ψ[φ1, φ2]((x, s1; y, s2) =1√2
(φ1(x, s1)φ2(y, s2)− φ2(x, s1)φ1(y, s2)) .
spin functions χ and spatial orbital functions φ:
ϕ(x, s) = φα(x)χα(s) + φβ(x)χβ(s)
with spin functions χα(+12) = 1, χα(−1
2) = 0, χβ(s) = 1− χα(s)
Closed shell RHF (Restricted Hartree Fock) :
φα,i = φβ,i = φi , i = 1, . . . ,N =N ′
2.
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Hartree-Fock- (HF) Approximation
Ground state energy E0 = min〈Hψ,ψ〉 : 〈ψ,ψ〉 = 1
approximation of ψ by a single Slater determinant
ΨSLΦ (x1, s1, . . . , xN′ , sN′) :=
1√N ′!
det(φi(xj , sj))
Closed Shell Restricted HF (RHF): N := N′2 electron pairs
minimization of the functional J HF (Φ)
Φ 7→ J HF (Φ) :=DHΨSL
Φ ,ΨSLΦ
E=
NXi=1
„Z `|∇φi (x)|2 + 2Vcore(x)|φi (x)|2+
+NX
j=1
ZR3
"|φj (y)|2|φi (x)|2
‖x − y‖−
12|φi (x)φi (y)||φj (x)φj (y)|
‖x − y‖
#dy´dx
1Aw.r.t. orthogonality constraints
Φ = (φi)Ni=1 ∈
(H1(R3)
)N and⟨φi , φj
⟩= δi,j
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Kohn-Sham model
Theorem (Kohn-Hohenberg)
The ground state energy E0 is a functional of the electron
density n.
12E0 ≈ EKS = infJ KS(Φ) : 〈φi , φj〉 = δij
minimization of the Kohn Sham energy functional J KS(Φ)
J KS(Φ) =
(Z12
NXi=1
|∇φi |2 +
ZnVcore +
12
Z Zn(x)n(y)
|x − y |dx dy − Exc(n)
)
φi ∈ H1(R3), electron density n(x) :=∑N
i=1 |φi(x)|2
Exc(n) exchange-correlation-energy (not known explicitly)
e.g. LDA (local density approximation) n 7→ Exc(n) : R→ R
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Φ := (φ1, . . . , φN) ∈ (H1(R3))N = V N = V
Gelfand triple V := H1(R3) ⊆ L2(R3) ⊆ H−1(R3) = V ′
〈ΦT Ψ〉 := (〈φi , ψj〉)i,j ∈ RN×N
scalar product 〈〈Φ,Ψ〉〉 := tr〈ΦT Ψ〉 =∑N
i=1〈φi , ψi〉 ∈ R
AΦ := (Aφ1, . . . ,AφN), A : V → V ′
Simplified Problem: minimize
J SCF (Φ) :=N∑
i=1
〈Aφi , φi〉 = tr〈ΦTAΦ〉 = 〈〈Φ,AΦ〉〉
w.r.t. to orthogonality constraints 〈ΦT Φ〉 = I. I.e. finding the
invariant subspace for the first N eigenfunctions.
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Geometry of the admissible setDefinition (Stiefel and Grassman manifolds)
Stiefel manifold VV ,N := V := Φ = (φi)Ni=1|φi ∈ V , 〈φi , φj〉 = δi,j
Grassmann manifold is a quotient manifold
GV ,N := G := VV ,N/∼, Φ∼Φ⇔ Φ = ΦU , U ∈ U(N)
(identify ONB spanning the same subspace span Φ)
Density matrix operator projects onto span Φ := spanφi,
DΦ :=N∑
i=1
〈φi , ·〉φi
There is a one-to-one correspondence between
[Φ] ∈ G ! DΦ (density matrix operator)Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Tangent space
Theorem (Edelman, Arias, Smith (98); Blauert, Neelov,
Rohwedder, S. (08))
tangent space T[Φ]G = δΨ ∈ V N |〈(δΨ)T Φ〉 = 0 ∈ RN×N
T[Φ]G = (δψi)Ni=1 : δψi ∈ V , δψi ⊥ span φi : i = 1, . . . ,N
(I −DΦ) : V N → T[Φ]G, is an orthogonal projection onto the
tangent space T[Φ]G
tangent space TΦS = T[Φ]G + ΦA : AT = −A
= Θ ∈ V N : 〈ΘT Φ〉 = −〈ΦT Θ〉
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Definition (Gradient ∇JKS(Ψ) = FKSΨ Ψ )
Kohn-Sham Fock operator F KSn = F KS
Φ : V → V ′ defined by
F KSΦ ϕ(x)=− 1
2 ∆ϕ(x) + Vcore(x)ϕ(x)+R n(y)|x−y|dyϕ(x) + vxc(n)(x)ϕ(x)
Unitary invariance: JKS(Φ) = JKS(ΦU)
Theorem (Necessary 12t order conditions)
If [Ψ] = argmin J (Φ) : [Φ] ∈ G ∈ V N(V Nh ) then
〈〈F[Ψ]Ψ, δΦ〉〉 = 0∀ δΦ ∈ T[Ψ]G ⊂ V N(V Nh )
〈〈(I −DΨ)F[Ψ]Ψ, δΦ〉〉 = 0∀ δΦ ∈ V N(V Nh )
F KS[Ψ]ψ
(k)i −
N∑j=1
ψ(k)j λji = 0 ∀i = 1, . . . ,N SCF-iteration Cances -LeBris (01)
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Projected Preconditioned Gradient Step Algorithm
Algorithm.1 Guess initial Φ(0) = (ϕ1, . . . , ϕN) ∈ V
2 repeat
1 compute Λ(k) by λ(k)ij = 〈ϕi ,F KS
[Φ(k)]ϕ
(k)j 〉
2 ϕ(k+1)i = ϕ
(k)i − B−1
k (F KS[Φ(k)]
ϕ(k)i −
∑Nj=1 ϕ
(k)j λ
(k)ji ) with
appropriate preconditioner Bk
3 project Φ(k+1) onto Vglob: orthonormalize Φ(k+1) to Φ(k+1)
until convergence reached
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Projection onto the Stiefel manifold
Projection P : V N → S,(resp. G), PΦ =: Φ ∈ V, s.t.
span φi : i = 1, . . . ,N = span φi : i = 1, . . . ,N
Löwdin transformation Φ = L−1Φ where LLT = 〈ΦT Φ〉
Diagonalization of Λ(n+1) = 〈ΦTF[Φ]Φ〉 = (〈ϕi ,F[Φ]ϕi〉)Ni=1,
yields the first N eigenvalues λ(n)1 ≤ . . . ≤ λ(n)
N of F[Φ].
LemmaAt the minimizer [Ψ], if F[Ψ] has a spectral gap λN < ΛN+1, then
F[Ψ] is V -elliptic on T[Ψ]
〈φ,F[Ψ]φ〉 ∼ ‖φ‖2V ∀φ ⊥ span ψi ; i = 1, . . . ,N .
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Comment on direct minimization
The algorithm computes an ON basis of the invariant subspace
of FΨ corresponding to the N lowest eigenvalues.
everything is valid if V := Vh is a finite dimensional
subspace (Galerkin approximation)
improvement by subspace acceleration: e.g. DIIS
gradient directed→ convergence with Armijo line search
B : V → V ′, ‖φ‖2B = 〈φ, φ〉B := 〈Bφ, φ〉 ∼ ‖φ‖2H1
e.g.: B ≈ −12 ∆ + C, e.g. multigrid or convolution by FFT
for a fixed operator A it is a block PINVIT iteration cf. e.g.
(Bramble & Knyazev et al.) – use for SCF Iteration
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Convergence results
Definitionmeasure of the error between subspaces spanned by
Φ = (ϕ1, . . . , ϕN),Ψ
‖(I −DΨ)Φ‖2B :=N∑
i=1
‖φi −DΨφi‖2B
Theorem ( Blauert, Neelov, Rohwedder, S. (08))
If Φ(0) ∈ Uδ(Ψ), and 〈〈(J ′′(Ψ)− Λ)Φ,Φ〉〉 ≥ γ‖Φ‖2H1N for all
Φ ∈ T[Ψ]G, then there exists χ < 1 such that
‖(I −DΨ)Φ(n+1)‖B ≤ χ · ‖(I −DΨ)Φ(n)‖B
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Energy error
Theorem ( Blauert, Neelov, Rohwedder, S. (08))
Let J ∈ C2(V N ,R), then
J (Φ(n))− J (Ψ) +R2 =
2|〈〈(Ψ− Φ(n)), (A[Φ(n)]Φ(n) − Φ(n)Λ(n))〉〉| . ‖(I −DΨ)Φ(n)‖2V N
where R2 = O(‖(I −DΨ)Φ(n)‖2V N ).
Corresponding results for a priori and a posterriori estimates for
the Galerkin solution together with S. Schwinger and
W.Hackbusch (08).
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Energy error
Constrained optimization problem (revisited):
u = argminJ(v) : G(v) = 0
Lagrangian L(x) := L(u,Λ) = J(u)− ΛG(u) (x = (u,Λ) ∈ X )
Theorem (Rannacher et al. )
If L′(x)y = 0 ∀y ∈ X and L′(xn)xn = 0,
L(x)− L(xn) =12
L′(xn)(x − xn) +O(‖x − xn‖3X )
J(u)− J(un) =12[J ′(un)(u − un)− ΛnG′(un)(u − un)
−(Λ− Λn)G(un)]
Here L(Φ,Λ) = J (Φ) + trΛ(〈ΦT ,Φ〉 − I)Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Orbital based functional
L(Φ,Λ) = J HF (Φ) + trΛ(〈ΦT ,Φ〉 − I)
error measure on G :
‖[Φ]− [Ψ]‖ := infU ∈ U(N) : ‖Φ−ΨU‖V N
Theorem
J (Ψh)− J (Ψ) +R3(ΦU − Φh) =
2〈〈(ΨU − Φh),AΨh Ψh −ΨhΛ〉〉 ∀Φh ∈ Vh
But U = argminU∈U(N)R3(ΦU − Φh), R3 = O(‖Ψh −ΨU‖3V N ) is
unknown.
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Interpolation estimates
For local basis functions like Finite Elements or wavelets
(Interpolation estimates)
Let Ωk := suppψk ⊂ Ωk , Vh := spanψk,
hk ∼ diamΩk ∼ diamΩk , then there ex. an operator
Ph : V → Vh reproducing polynomials such thtat
‖u − Phu‖L2(Ωk ) . h2k‖u‖H2(Ωk )
e.g. Ph (Clermont, Scott-Zhang quasi) interpolation operatorLemma (H2 regularity)
The minimizer Φ = (ϕi) of J HF ,J SCF is in (H2(R3))N . This is
assumed to hold also for J KS
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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A posteriori error estimates
Let R := (Ri)Ni=1 := (F[Ψh] − Λh)Φh ∈ V N be the residual.
The local residuals rk , ρk are defined by
r2i,k :=
N∑i=1
‖Ri‖2L2(Ωk ) , ρ2i,k := h−1/2
k
N∑i=1
‖[∂nϕh,i ]‖2L2(ek )
where [∂nϕh,i ]|ek denotes the jump of the normal derivatives
across the edges in Ωk if ψk 6∈ C1.
The error estimator depends on the representation Φ ∈ V
of [Φ] ∈ G.
it allows an individual discretization of ϕi
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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A posteriori error estimates
Theorem (S. &Schwinger )
If Φ ∈ G, Φ ∈ V N , Φi ∈ V Nh be the minimizers of
J = J HF ,J KS,J SCF , then
J (Φh)− J (Φ) .
∑i,k
(r2i,k + ρ2
i,k )h4k
1/2
‖Φ‖(H2)N
it reveals the eigenvalue error estimator of Larsen
The H2-regularity assumption simplifies the proof, but may
not be required, see (Heuveline & Rannacher).
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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BigDFT Project
T. Deutsch (CEA Grenoble) http://inac.cea.fr/L_Sim/BigDFT/index.html
S. Goedecker (Uni Basel)
X. Gonze (UC Louvain)
R. Schneider (U Kiel→ TU Berlin)
Aims:
implementing an electronic structure calculation program
with wavelet bases
usable on massively parallel computer
embed this code in the existing program package ABINIT
(www.abinit.org)
linear scaling code with respect to the number of electrons
of the systemReinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Scaling Function ϕ and Wavelet ψ
0 0.5 1 1.5 2 2.5 3−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4p = 2
0 1 2 3 4 5 6 7 8 9−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2p = 5
Figure: Daubechies scaling functions ϕ with p = 2 and 5.
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
2p = 2
−4 −3 −2 −1 0 1 2 3 4 5
−1
−0.5
0
0.5
1
p = 5
Figure: Daubechies wavelets ψ with p = 2 and 5.Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Adaptivity
two computational regions: coarse and fine region
(sufficient with pseudopotentials)
fine region: scaling functions and wavelets (8 basis
functions per grid point)
coarse region: only scaling functions (1 basis function per
grid point)
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Adaptivity
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Adaptivity
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Adaptivity
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Numerical results
example: C19H22N2O (N = 55)
,
Figure:
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Numerical results
Convergence history for the direct minimization scheme and
together with DIIS acceleration
0 5 10 15 20 25 30 35 40 45 5010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
iter
abso
lute
err
or
steepest descent
hgrid = 0.3
hgrid = 0.45
hgrid = 0.7
, 0 5 10 15 20 25 30 35 40 45 5010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
iter
abso
lute
erro
r
history size for DIIS = 6
hgrid = 0.3
hgrid = 0.45
hgrid = 0.7
Figure: convergence
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Accuracy Depending on hgrid
convergence rate: O(h14grid )
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Wavelets vs. Plane Waves (Degrees of Freedom)
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Wavelets vs. Plane Waves (Runtime)
10−4
10−3
10−2
10−1
100
0
500
1000
1500
2000
2500
3000
3500
4000
4500
abs. prec. error
sec.
Runtimes for Cinchonidine
Abinit
CPMD
BigDFT
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Computing Times
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Summary
Tensor product approximation is the key to treat the high dimensional problem
Hartree Fock is a rank one anti-symmetric tensor product approximation
In theory, Density Functional Theory (DFT) can provide the exact ground state
energy and density
Due to the unknown exchange correlation potential there remains an
unavoidable modelling error
The high dimensional (d = 3N) linear eigenvalue problem is reduced by HF and
DFT to a (system of) low dimensional (d = 3) nonlinear eigenvalue type
problems
relatively large systems can be treated if one accepts the modeling error,
complexity is O(N2 dim Vh) ∼ O(N3)→ O(N) for large systems.
for more accurate computations one has to consider the original highdimensional
problem
Quantum Monte Carlo Methods or Wave Function - Post Hartree Fock Methods
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Full CI Configuration Interaction Method
Approximation space for (spin) orbitals (xj , sj)→ ϕ(xj , sj)
Xh := span ϕi : i = 1, . . . ,N ⊂ H1(R3×±12) , 〈ϕi , ϕj〉 = δi,j ,
Full CI (for benchmark computations ≤ N = 18) is a
Galerkin method w.r.t. the subspace
VFCI =N∧
i=1
Xh = spanΨSL = Ψ[ν1, ..νN ] =1√N!
det(ϕνi (xj , sj))Ni,j=1
Galerkin ansatz: Ψ = c0Ψ0 +∑
ν∈J cνΨν
H = (〈Ψν′ ,HΨν〉) , Hc = Ec ,but dim Vh =
NN
!
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FCI Method - canonical orbitals
Let Ψ0 = Ψ[1, ..,N] = 1√N!
det(ϕi(xj , sj))Ni,j=1, be a reference
These first N orbital functions ϕi are called occupied orbitals
the others are called unoccupied orbitalsϕ1, . . . , ϕN , ϕN+1, . . . , ϕN
they are eigen functions of F : X → X ′,X := H1(R3 × ±12)
〈Fϕi − λiϕi , φh〉 = 0 ∀φh ∈ Xh , f pr := 〈Fϕr , ϕp〉 = δr ,p , r ,p,≤ N
Lemma
The Fock operator F := Fh =∑N
k=1 Fk : VFCI → VFCI admits
FΨ[ν1, . . . , νN ] =
(N∑
i=1
λνi
)Ψ[ν1, . . . , νN ]
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Some remarks about choie of ϕ
H = F + U , U is a two particle operator
FCI exponentially scaling wr.t. to N, one has to confine to a small subspace
Usually N ∼ 10N but one hat to resolve the e-e cusp
F Fock operator for (HF or KS) , precomputation
H. Yserentant has shown the existence of F where the eigen-functions provide a
sparse grid basis (to avoid curse of dimensions)→ adaptive schemes (Griebel et
al., Flad, Rohwedder & S)
ϕk can be choosen optimally (MRSCF), not eigen fuctions of a single operator,
existence proved by Friesecke, Levin but still exonential complexity of FCI
Localized basis functions reduced complexity of matrix elements→ linear
scaling (?!)
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Second quantization
Second quantization: annihilation operators:
ajΨ[j ,1, . . . ,N] := Ψ[1, . . . ,N]
and := 0 if j not apparent in Ψ[. . .].
The adjoint of ab is a creation operator v
a†bΨ[1, . . . ,N] = Ψ[b,1, . . . ,N] = (−1)NΨ[1, . . . ,N,b]
Theorem (Slater-Condon Rules)H : V → V resp. H : VFCI → VFCI reads as (basis dependent)
H = F + U =∑p,q
f pr ar a
†p +
∑p,q,r ,s
upqrs ar asa†qa†p
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Excitation operators
Single excitation operator , Let Ψ0 = Ψ[1, . . . ,N] be a reference
determinant then e.g.
X k1 Ψ0 := a†ka1Ψ0
(−1)−pΨk1 = Ψ[k ,2, . . . ,N] = X k
1 Ψ0 = X kj Ψ[1, . . . , . . . ,N] = a†ka1Ψ0
higher excitation operators
Xµ := X b1,...,bkl1,...,lk
=k∏
i=1
X bili
, 1 ≤ li < li+1 ≤ N , N < bi < bi+1 .
A CI solution Ψh = c0Ψ0 +∑
µ∈JhcµΨµ can be written by
Ψh =
c0 +∑µ∈Jh
cµXµ
Ψ0 , c0, cµ ∈ R , Jh ⊂ J .
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Coupled Cluster Method - Exponential-Ansatz
Theorem (S. 06, (Coester & Kümmel ∼ 1959))Let Ψ0 be a reference Slater determinant, e.g. Ψ0 = ΨHF and
Ψ ∈ VFCI , V, satisfying
〈Ψ,Ψ0〉 = 1 intermediate normalization .
Then there exists an unique excitation operator(T1 - single-, T2 - double- , . . . excitation operators)
T =N∑
i=1
Ti =∑µ∈J
tµXµ
such that
Ψ = eT Ψ0
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CC Energy and Projected Coupled Cluster Method
Let Ψ ∈ VFCI satisfying HΨ := HhΨ = E0Ψ, then, due to the
Slater Condon rules and 〈Ψ,Ψ0〉 = 1
E = 〈Ψ0,HΨ〉 = 〈Ψ0,HeT Ψ0〉 = 〈Ψ0,H(I + T1+T2 +12
T 21 )Ψ0〉
The Projected Coupled Cluster Method applies the ansatz
Ψh = eTh Ψ0 , Th := T :=l∑
k=1
Tk =∑µ∈Jh
tµXµ , 0 6= µ ∈ Jh ⊂ J
Ψµ ∈ subspace of VFCI , CCSD T = T1 + T2 = T (t) and
(Galerkin) projection onto subspace.
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Projected Coupled Cluster Method
Let T =∑l
k=1 Tk =∑
µ∈JhtµXµ , 0 6= µ ∈ Jh ⊂ J using
0 = 〈Ψ0, (H − E)Ψ〉 = 〈Ψ0, (H − E(th)eT (th)Ψ0〉
The unlinked projected Coupled Cluster formulation
0 = 〈Ψµ, (H − E(th))eT (th)Ψ0〉 =: gµ(t) , t = (tν)ν∈Jh , µ, ν ∈ Jh
The linked projected Coupled Cluster formulation consists in0 = 〈Ψµ,e−T HeT Ψ0〉 =: fµ(t) , t = (tν)ν∈Jh , µ, ν ∈ Jh
These are L = ]Jh << N nonlinear equations for L unknown
excitation amplitudes tµ.Theorem ( Kümmel ∼ 1959, S. 09)The (projected) CC Method is size consistent!:
HAB = HA + HB ⇒ ECCAB = ECC
A + ECCB .
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Baker-Campell-Hausdorff expansion
We recall the Baker-Campell-Hausdorff formula
e−T AeT = A + [A,T ] +12!
[[A,T ],T ] +13!
[[[A,T ],T ],T ] + . . . =
A +∞∑
k=1
1k !
[A,T ]k .
For Ψ ∈ Vh the above series terminates,
e−T HeT = H+[H,T ]+12!
[[H,T ],T ]+13!
[[[H,T ],T ],T ]+14!
[H,T ]4
e.g. for a single particle operator e.g. F there holds
e−TFeT = F + [F ,T ] + [[F ,T ],T ]
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Iteration method to solve CC amplitude equations
0 = fµ(t) = 〈Ψµ,e−T HeT Ψ0〉
= 〈Ψµ, [F ,Xµ]Ψ0〉+4∑
k=0
1k !〈Ψµ, [U,T ]k Ψ0〉
The nonlinear amplitude equation f(t) = 0 is solved byAlgorithm (quasi Newton-scheme)
1 Choose t0, e.g. t0 = 0.
2 Compute
tn+1 = tn − A−1f(tn),
where A = diag (εµ)µ∈J > 0.
−εµ < λN − λN+1 < 0 (Bach-Lieb-Sololev) the matrix A > 0Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Analysis of the Coupled Cluster Method
We consider the projected CC as an approximation of the full CI
solution!
If h→ 0, thenM→∞ and max εµ →∞! We need estimates
uniformly w.r.t. h,N
Definition
LetM := dimVFCI dimensional parameter space V = RM
equipped with the norm
‖t‖2V := ‖∑µ∈J
εµtµΨµ‖2L2((R3×± 12)N )
=∑µ∈J
εµ|tµ|2
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Analysis of the CC Methods - Lemmas
Lemma (S.06)There holds
‖t‖V ∼ ‖T Ψ0‖H1((R3×± 12)N ) ∼ ‖T Ψ0‖V .
Lemma (S.06)
For t ∈ `2(J ), the operator T :=∑
ν∈J tνXν maps
‖T Ψ‖L2 . ‖t‖`2‖Ψ‖L2 ∀Ψ ∈ VFCI ⊂N∧
i=1
L2(R3 × ±12)
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Analysis of CC Method - Lemmas
Lemma (S.06)
For t ∈ V, the operator T :=∑
ν∈J tνXν maps
‖T Ψ‖H1 . ‖t‖V‖Ψ‖H1 ∀Ψ ∈ VFCI
Corollary (S06)
The function f : V → V ′ is differentiable at t ∈ V with the
Frechet derivative f′[t] : V → V ′ given by
(f′[t])ν,µ = 〈Ψν ,e−T [H,Xµ]eT Ψ0〉
= ενδν,µ + 〈Ψν ,e−T [U,Xµ]eT Ψ0〉
All Frechet derivatives t 7→ f (k)[t] : V → V ′, are Lipschitz
continuous. In particular f(5) ≡ 0.
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Convergence of the Coupled Cluster MethodLemma
Let Ψh = eTh Ψ0 where
‖t− th‖V . infv∈R]Jh
‖t− vh‖V .
then
‖Ψ−Ψh‖H1 . infv∈RL‖Ψ− e
Pµ∈Jh
vµXµΨ0‖H1 .
DefinitionA function g : is called strongly monotone at t if
〈g(t)− g(t′), (t− t′〉 ≥ γ‖t− t′‖2V
for some γ > 0 and all ‖t′ − t‖V < δ.
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Local existence and quasi-optimal convergence
Strict monotonicity of f is not known yet! Let T (t) :=∑
µ tµXµ
we consider g : V → V , g(t)ν := 〈Ψν , (H − E(t)eT (t)Ψ0〉 .Theorem (S. 2008)
Let E be a simple EV. If ‖Ψ−Ψ0‖V < δ sufficiently small, and
Jh excitation complete, then
1 for E = E(th) := 〈Ψ0,HeT (th)Ψ0〉, there holds
〈g(th),v〉 = 0 , ∀v ∈ Vh ⇔ 〈f(th),v〉 = 0 , ∀v ∈ Vh
2 g is strongly monontone at t ∀‖t‖V ≤ δ′
3 there ex. th ∈ Vh with 〈g(th),v〉 = 〈f(th),v〉 = 0, ∀v ∈ Vh,
‖t− th‖V . infv∈Vh‖t− vh‖V .
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Convergence of the Coupled Cluster EnergiesTheorem (S. 06 a priori estimate)
The error in the energy |J(t)− J(th)| can be estimated by
|E − Eh| . ‖t− th‖V‖a− ah‖V + (‖t− th‖V )2
. infuh∈Vh
‖t− uh‖V‖a− ah‖V
+( infuh∈Vh
‖t− uh‖V )2.
|E − Eh| . infuh∈Vh
‖t− uh‖V‖a− ah‖V
+( infuh∈Vh
‖t− uh‖V )2
. ‖t− th‖V‖a− ah‖V
. infuh∈Vh
‖t− uh‖V infbh∈V
‖a− bh‖V
All constants involved above are uniform w.r.t. N →∞.
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Numerical examples – CCSD versus CISD
The relative difference in the correlation energy between CI and CC for several
molecules in bonding configuration is plotted over the total number of electrons N and
the number of valence electrons.
All computations were performed with MOLPRO
The lack of size consistency suggests a behavior√
N
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2
0.25
0.3
0.35
rel.
corr
elat
ion
ener
gy d
iffer
ence
nuclear charge0 5 10 15 20 25 30 35 40
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
rel.
corr
elat
ion
ener
gy d
iffer
ence
number of valence electrons
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster
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Conclusions - CC is a most powerful wave function
method
If the reference Ψ0 is sufficiently close to the exact solution
Ψ, granted by the results of Yserentant, then we have
quasipotimal convergence w.r.t the wave function and
super-optimal convergence w.r.t. the energy
+ size consistency
CCSD and CCSD(T) are standard, CCSDT; CCSDTQ etc.
only for extremely accurate computations
not good for multi-configurational problems, e.g. (near-)
degenerate ground state ( where RHF is rather bad)
Reinhold Schneider, MATHEON TU Berlin I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster