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The Coupled Cluster method for theelectronic Schrodinger equation
Reinhold Schneider and Thorsten RohwedderTU Berlin
TU Munich 2012
Matheon_en_big.pdf
Acknowledgement: Thanks toDr. H.-J. Flad (MATHEON; TU Berlin),Prof. W. Hackbusch (Max Plank Institute MIS Leipzig)Prof. H. Yserentant (TUB)
Overview:
I. Introduction - the electronic Schrodinger equation
II. The (projected) Coupled Cluster method
IIII. The continuous Coupled Cluster method
I.Introduction
Quantum mechanics
Goal: Calculation of physical and chemicalproperties on a microscopic (atomic)
length scale.
B E.g. atoms, molecules, clusters, solidsB E.g. chemical behaviour, bonding
energies, ionization energies,conduction properties, essentialmaterial properties
Electronic structure calculationReduction of the problem to the computation of an electronicwave function Ψ for given fixed nuclei.
The electronic Schrodinger equation, describes the stationarynonrelativistic behaviour of system of N electrons in anelectrical field
HΨ = EΨ.Electronic structure determinates e.g.
B bonding energies,reactivity
B ionization energiesB conductivity,B in a wider sense
molecular geometry,-dynamics,...
of atoms, molecules, solidsetc.
The stationary electronic Schrodinger equation(variational formulation)
Find antisymmetric wave function Ψ ∈ H1 and eigenvalue E ∈ Rsuch that
〈Φ, HΨ〉 = E 〈Φ,Ψ〉 for all Φ ∈ H1.
Energy scales (1Eh ≈ 27, 2114eV Hartree)
0.001 eV
Intermolecular interactions
Magnetic couplings
Atomic cores
1 eV 10 eV0.1 eV0.01 eV
Chemical bonds
CC MP2 DFT
The stationary electronic Schrodinger equation(variational formulation)
Find antisymmetric wave function Ψ ∈ H1 and eigenvalue E ∈ Rsuch that
〈Φ, HΨ〉 = E 〈Φ,Ψ〉 for all Φ ∈ H1.
B the wave function Ψ is antisymmetric (Pauli principle),
Ψ((x1, s1), . . . , (xi , si), . . . , (xj , sj), . . . , (xN , sN))
= −Ψ((x1, s1), . . . , (xj , sj), . . . , (xi , si), . . . , (xN , sN)).
B N-fermion space:
Ψ ∈ L2 :=∧N
i=1L2(R3 × ±1
2)
Energy scales (1Eh ≈ 27, 2114eV Hartree)
0.001 eV
Intermolecular interactions
Magnetic couplings
Atomic cores
1 eV 10 eV0.1 eV0.01 eV
Chemical bonds
CC MP2 DFT
The stationary electronic Schrodinger equation(variational formulation)
Find antisymmetric wave function Ψ ∈ H1 and eigenvalue E ∈ Rsuch that
〈Φ, HΨ〉 = E 〈Φ,Ψ〉 for all Φ ∈ H1.
B
H : H1(R3N × ±12N) → H−1(R3N × ±1
2N)
is the weak Hamiltonian, defined via
HX = −12
N∑i=1
∆i +12
N∑i=1
N∑j=1j 6=i
1|xi − xj |
−N∑
i=1
M∑k=1
Zk
|xi − Rk |.
B H1 := H1(R3N × ±12
N) ∩ L2
Energy scales (1Eh ≈ 27, 2114eV Hartree)
0.001 eV
Intermolecular interactions
Magnetic couplings
Atomic cores
1 eV 10 eV0.1 eV0.01 eV
Chemical bonds
CC MP2 DFT
The stationary electronic Schrodinger equation(variational formulation)
Find antisymmetric wave function Ψ ∈ H1 and eigenvalue E ∈ Rsuch that
〈Φ, HΨ〉 = E 〈Φ,Ψ〉 for all Φ ∈ H1.
B
H : H1(R3N × ±12N) → H−1(R3N × ±1
2N)
is the weak Hamiltonian, defined via
HX = −12
N∑i=1
∆i +12
N∑i=1
N∑j=1j 6=i
1|xi − xj |
−N∑
i=1
M∑k=1
Zk
|xi − Rk |.
B H1 := H1(R3N × ±12
N) ∩ L2
Energy scales (1Eh ≈ 27, 2114eV Hartree)
0.001 eV
Intermolecular interactions
Magnetic couplings
Atomic cores
1 eV 10 eV0.1 eV0.01 eV
Chemical bonds
CC MP2 DFT
The stationary electronic Schrodinger equation(variational formulation), ground state problem
Find antisymmetric wave function Ψ ∈ H1 and eigenvalue E∗ ∈ Rsuch that
〈Φ, HΨ〉 = E∗〈Φ,Ψ〉 for all Φ ∈ H1.
and such that E∗ is the lowest eigenvalue of H.
Energy scales (1Eh ≈ 27, 2114eV Hartree)
0.001 eV
Intermolecular interactions
Magnetic couplings
Atomic cores
1 eV 10 eV0.1 eV0.01 eV
Chemical bonds
CC MP2 DFT
II.
The (projected)
Coupled Cluster method
|Ψd〉 = eTd |Ψ0〉
Projected CC = approximation to fixed Galerkin/”full CI” scheme
Starting point: One-particle basis
B = ψ1, . . . , ψd,
antisymmetric tensor basis (Slater determinants)
Bd = Ψµ = Ψ[p1, ..,pN ], 1≤pi<pi+1≤d,
Ψ[p1, ..,pN ] :=∧N
i=1ψpi =
1√N!
det(ψpi (xj , sj)
)Ni,j=1.
CC is approximation of Galerkin (full CI) solution Ψd , solving
〈Ψµ,HΨd〉 = E 〈Ψµ,Ψd〉 for all Ψµ ∈ Bd .
(an extremely high-dimensional problem, mostly unsolvable inpractice)
Ansatz space and reference determinantHartree-Fock (or DFT) calculation
gives(a) a (quite good) rank-1 approximation of eigenfunction Ψ,
Ψ0 = Ψ[1, ..,N] :=∧N
i=1ψi(xi , si) =
1√N!
det(ψpi (xj , sj)
)Ni,j=1
(b) one-particle basis B of L2d (R3 × ±1
2),B = ψ1, . . . , ψN︸ ︷︷ ︸
occupied orbitals
, ψN+1, . . . , ψd︸ ︷︷ ︸virtual orbitals
occ ⊥ virt in L2 and w.r.t. inner product F ∼ H1
tensor basis Bd = Ψ[p1, ..,pN ], 1≤pi<pi+1≤d of L2d
⇓Post-Hartree-Fock calculation
CI (Galerkin) calculation Coupled Cluster calculation
Accuracy, size consistency,...
Ansatz space and reference determinantHartree-Fock (or DFT) calculation
gives(a) a (quite good) rank-1 approximation of eigenfunction Ψ,
Ψ0 = Ψ[1, ..,N] :=∧N
i=1ψi(xi , si) =
1√N!
det(ψpi (xj , sj)
)Ni,j=1
(b) one-particle basis B of L2d (R3 × ±1
2),B = ψ1, . . . , ψN︸ ︷︷ ︸
occupied orbitals
, ψN+1, . . . , ψd︸ ︷︷ ︸virtual orbitals
occ ⊥ virt in L2 and w.r.t. inner product F ∼ H1
tensor basis Bd = Ψ[p1, ..,pN ], 1≤pi<pi+1≤d of L2d
⇓Post-Hartree-Fock calculation
CI (Galerkin) calculation Coupled Cluster calculation
Accuracy, size consistency,...
Ansatz space and reference determinantHartree-Fock (or DFT) calculation
gives(a) a (quite good) rank-1 approximation of eigenfunction Ψ,
Ψ0 = Ψ[1, ..,N] :=∧N
i=1ψi(xi , si) =
1√N!
det(ψpi (xj , sj)
)Ni,j=1
(b) one-particle basis B of L2d (R3 × ±1
2),B = ψ1, . . . , ψN︸ ︷︷ ︸
occupied orbitals
, ψN+1, . . . , ψd︸ ︷︷ ︸virtual orbitals
occ ⊥ virt in L2 and w.r.t. inner product F ∼ H1
tensor basis Bd = Ψ[p1, ..,pN ], 1≤pi<pi+1≤d of L2d
⇓Post-Hartree-Fock calculation
CI (Galerkin) calculation Coupled Cluster calculation
Accuracy, size consistency,...
Ansatz space and reference determinantHartree-Fock (or DFT) calculation
gives(a) a (quite good) rank-1 approximation of eigenfunction Ψ,
Ψ0 = Ψ[1, ..,N] :=∧N
i=1ψi(xi , si) =
1√N!
det(ψpi (xj , sj)
)Ni,j=1
(b) one-particle basis B of L2d (R3 × ±1
2),B = ψ1, . . . , ψN︸ ︷︷ ︸
occupied orbitals
, ψN+1, . . . , ψd︸ ︷︷ ︸virtual orbitals
occ ⊥ virt in L2 and w.r.t. inner product F ∼ H1
tensor basis Bd = Ψ[p1, ..,pN ], 1≤pi<pi+1≤d of L2d
⇓Post-Hartree-Fock calculation
CI (Galerkin) calculation Coupled Cluster calculation
Accuracy, size consistency,...
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
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
Excitation operatorsSingle excitation operator , Let Ψ0 = Ψ[1, . . . ,N] be a referencedeterminant 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 Ψ = c0Ψ0 +∑
µ∈J cµΨµ can be written by
Ψ =
c0 +∑µ∈J
cµXµ
Ψ0 , c0, cµ ∈ R .
Excitation operatorsSingle excitation operator , Let Ψ0 = Ψ[1, . . . ,N] be a referencedeterminant 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 Ψ = c0Ψ0 +∑
µ∈J cµΨµ can be written by
Ψ =
c0 +∑µ∈J
cµXµ
Ψ0 , c0, cµ ∈ R .
Excitation operatorsSingle excitation operator , Let Ψ0 = Ψ[1, . . . ,N] be a referencedeterminant 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 Ψ = c0Ψ0 +∑
µ∈J cµΨµ can be written by
Ψ =
c0 +∑µ∈J
cµXµ
Ψ0 , c0, cµ ∈ R .
Coupled Cluster Method - Exponential-ansatzTheorem (S. 06)Let Ψ0 be a reference Slater determinant, e.g. Ψ0 = ΨHF andΨ ∈ VFCI , V, satisfying
〈Ψ,Ψ0〉 = 1 intermediate normalization .
Then there exists an excitation operator(T1 - single-, T2 - double- , . . . excitation operators)
T =N∑
i=1
Ti =∑µ∈J
tµXµ such that
Ψ = eT Ψ0 = Πµ(I + tµXµ)Ψ0 .
Key observations: for analytic functions :
f (T ) =N∑
k=0
akT k since [Xµ,Xν ] = 0 , X 2µ = 0 , T N = 0 .
Reformulation of the Galerkin ansatzB One-particle basis B = ψ1, ..., ψN︸ ︷︷ ︸
occupied
, ψN+1, ..., ψd︸ ︷︷ ︸virtual
,
tensor basis Bd = Ψ[p1, ..,pN ], 1 ≤ p1 < ... < pN ≤ d.
B Replacement of occupied by virtual orbitals in reference Ψ0,
Ψ[1, . . . , i1, .., ik , ..,N]“excitation”−→ Ψµ = Ψ[1, ., 6i1, .., 6ik , ..,a1, ..,ak ],
gives Bd = Ψ0 ∪ Ψµ | µ ∈ Id.
B Reformulation: Excitation operator X a1,..,aki1,..,ik
: L2d → L2
d
X a1,..,aki1,..,ik
Ψ[p1, ..,pN ] =
Ψ[ 6i1, .., 6ik ,a1, ..,ak ..,pi , ..]
if i1, .., ik ∈ ind(Ψ)and a1, ..,ak /∈ ind(Ψ)
0 elsewise
With this, Ψ[i1, .., , iN−k ,a1, ..,ak ] = X a1,...,aki1,...,ik
Ψ0.
Reformulation of the Galerkin ansatzB One-particle basis B = ψ1, ..., ψN︸ ︷︷ ︸
occupied
, ψN+1, ..., ψd︸ ︷︷ ︸virtual
,
tensor basis Bd = Ψ[p1, ..,pN ], 1 ≤ p1 < ... < pN ≤ d.
B Replacement of occupied by virtual orbitals in reference Ψ0,
Ψ[1, . . . , i1, .., ik , ..,N]“excitation”−→ Ψµ = Ψ[1, ., 6i1, .., 6ik , ..,a1, ..,ak ],
gives Bd = Ψ0 ∪ Ψµ | µ ∈ Id.
B Reformulation: Excitation operator X a1,..,aki1,..,ik
: L2d → L2
d
X a1,..,aki1,..,ik
Ψ[p1, ..,pN ] =
Ψ[ 6i1, .., 6ik ,a1, ..,ak ..,pi , ..]
if i1, .., ik ∈ ind(Ψ)and a1, ..,ak /∈ ind(Ψ)
0 elsewise
With this, Ψ[i1, .., , iN−k ,a1, ..,ak ] = X a1,...,aki1,...,ik
Ψ0.
Reformulation of the Galerkin ansatzB One-particle basis B = ψ1, ..., ψN︸ ︷︷ ︸
occupied
, ψN+1, ..., ψd︸ ︷︷ ︸virtual
,
tensor basis Bd = Ψ[p1, ..,pN ], 1 ≤ p1 < ... < pN ≤ d.
B Replacement of occupied by virtual orbitals in reference Ψ0,
Ψ[1, . . . , i1, .., ik , ..,N]“excitation”−→ Ψµ = Ψ[1, ., 6i1, .., 6ik , ..,a1, ..,ak ],
gives Bd = Ψ0 ∪ Ψµ | µ ∈ Id.
B Reformulation: Excitation operator X a1,..,aki1,..,ik
: L2d → L2
d
X a1,..,aki1,..,ik
Ψ[p1, ..,pN ] =
Ψ[ 6i1, .., 6ik ,a1, ..,ak ..,pi , ..]
if i1, .., ik ∈ ind(Ψ)and a1, ..,ak /∈ ind(Ψ)
0 elsewise
With this, Ψ[i1, .., , iN−k ,a1, ..,ak ] = X a1,...,aki1,...,ik
Ψ0.
Reformulation of the Galerkin ansatzB One-particle basis B = ψ1, ..., ψN︸ ︷︷ ︸
occupied
, ψN+1, ..., ψd︸ ︷︷ ︸virtual
,
tensor basis Bd = Ψ[p1, ..,pN ], 1 ≤ p1 < ... < pN ≤ d.
B Replacement of occupied by virtual orbitals in reference Ψ0,
Ψ[1, . . . , i1, .., ik , ..,N]“excitation”−→ Ψµ = Ψ[1, ., 6i1, .., 6ik , ..,a1, ..,ak ],
gives Bd = Ψ0 ∪ Ψµ | µ ∈ Id.
B Reformulation: Excitation operator X a1,..,aki1,..,ik
: L2d → L2
d
X a1,..,aki1,..,ik
Ψ[p1, ..,pN ] =
Ψ[ 6i1, .., 6ik ,a1, ..,ak ..,pi , ..]
if i1, .., ik ∈ ind(Ψ)and a1, ..,ak /∈ ind(Ψ)
0 elsewiseWith this, Bd = Ψ0 ∪ XµΨ0 | µ ∈ Id.
Cluster operator/Coupled-Cluster ansatz
B Galerkin solution Ψd is expressed by excitations,
Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id
sµΨµ
=: (I + S(sd ))Ψ0
B Reformulated Galerkin ansatz:
Linear Parametrisation for Ψd = Ψ0 + Ψ∗d :
Find cluster operator S = S(sd ) =∑
µ∈IdsµXµ such that
Ψd = (I + S(sd ))Ψ0,
〈Φd , H(I + S(sd ))Ψ0〉 = E∗〈Φd , (I + S(sd ))Ψ0〉 ∀ Φd ∈ H1d .
Cluster operator/Coupled-Cluster ansatz
B Galerkin solution Ψd is expressed by excitations,
Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id
sµXµΨ0
=: (I + S(sd ))Ψ0
B Reformulated Galerkin ansatz:
Linear Parametrisation for Ψd = Ψ0 + Ψ∗d :
Find cluster operator S = S(sd ) =∑
µ∈IdsµXµ such that
Ψd = (I + S(sd ))Ψ0,
〈Φd , H(I + S(sd ))Ψ0〉 = E∗〈Φd , (I + S(sd ))Ψ0〉 ∀ Φd ∈ H1d .
Cluster operator/Coupled-Cluster ansatz
B Galerkin solution Ψd is expressed by excitations,
Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id
sµXµΨ0 =: (I + S(sd ))Ψ0
B Reformulated Galerkin ansatz:
Linear Parametrisation for Ψd = Ψ0 + Ψ∗d :
Find cluster operator S = S(sd ) =∑
µ∈IdsµXµ such that
Ψd = (I + S(sd ))Ψ0,
〈Φd , H(I + S(sd ))Ψ0〉 = E∗〈Φd , (I + S(sd ))Ψ0〉 ∀ Φd ∈ H1d .
Cluster operator/Coupled-Cluster ansatz
B Galerkin solution Ψd is expressed by excitations,
Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id
sµXµΨ0 =: (I + S(sd ))Ψ0
B Reformulated Galerkin ansatz:
Linear Parametrisation for Ψd = Ψ0 + Ψ∗d :
Find cluster operator S = S(sd ) =∑
µ∈IdsµXµ such that
Ψd = (I + S(sd ))Ψ0,
〈Φd , H(I + S(sd ))Ψ0〉 = E∗〈Φd , (I + S(sd ))Ψ0〉 ∀ Φd ∈ H1d .
Cluster operator/Coupled-Cluster ansatz
B Galerkin solution Ψd is expressed by excitations,
Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id
sµXµΨ0 =: (I + S(sd ))Ψ0
B Coupled-Cluster-Ansatz:
Nonlinear Parametrisation for Ψd = Ψ0 + Ψ∗d :
Find cluster operator T = T (td ) =∑
µ∈IdtµXµ such that
Ψd = eT (td )Ψ0,
〈Φd , HeT (td )Ψ0〉 = E∗〈Φd ,eT (td )Ψ0〉 ∀ Φd ∈ H1d .
Cluster operator/Coupled-Cluster ansatz
B Galerkin solution Ψd is expressed by excitations,
Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id
sµXµΨ0 =: (I + S(sd ))Ψ0
B Coupled-Cluster-Ansatz:
Nonlinear Parametrisation for Ψd = Ψ0 + Ψ∗d :
Find cluster operator T = T (td ) =∑
µ∈IdtµXµ such that
Ψd = eT (td )Ψ0,
〈Φd ,e−T (td )HeT (td )Ψ0〉 = E∗〈Φd ,Ψ0〉 ∀ Φd ∈ H1d .
CC Energy and Projected Coupled Cluster MethodLet Ψ ∈ VFCI satisfying HΨ := HhΨ = E0Ψ, then, due to theSlater Condon rules and 〈Ψ,Ψ0〉 = 1
E = 〈Ψ0,HΨ〉 = 〈Ψ0,H(I + T1+T2 +12
T 21 )Ψ0〉
Variants: (probably better but not computable)I unitary CC:
Ψ = e12 (T−T∗)Ψ0 ,
I variational CC
Ψ = argmin〈eTψ0,HeT Ψ0〉
I general CC (Noijens conjecture)
Ψ = e(∑
i,j,p,q,r,s t ji a†i aj +tp,q
r,s a†r a†paqas
)Ψ0 .
CC Energy and Projected Coupled Cluster MethodLet Ψ ∈ VFCI satisfying HΨ := HhΨ = E0Ψ, then, due to theSlater Condon rules and 〈Ψ,Ψ0〉 = 1
E = 〈Ψ0,HΨ〉 = 〈Ψ0,H(I + T1+T2 +12
T 21 )Ψ0〉
Variants: (probably better but not computable)I unitary CC:
Ψ = e12 (T−T∗)Ψ0 ,
I variational CC
Ψ = argmin〈eTψ0,HeT Ψ0〉
I general CC (Noijens conjecture)
Ψ = e(∑
i,j,p,q,r,s t ji a†i aj +tp,q
r,s a†r a†paqas
)Ψ0 .
Projected Coupled Cluster MethodLet T =
∑lk=1 Tk =
∑µ∈Jh
tµXµ , 0 6= µ ∈ Jh ⊂ J using0 = 〈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 in
0 = 〈Ψµ,e−T HeT Ψ0〉 =: fµ(t) , t = (tν)ν∈Jh , µ, ν ∈ Jh
These are L = ]Jh << N nonlinear equations for L unknownexcitation amplitudes tµ.TheoremThe CC Method is size consistent!:
HAB = HA + HB ⇒ ECCAB = ECC
A + ECCB .
e−(TA+TB)(HA + HB)eTa+TB = e−TAHAeTA + e−TB HBeTB
Projected Coupled Cluster MethodLet T =
∑lk=1 Tk =
∑µ∈Jh
tµXµ , 0 6= µ ∈ Jh ⊂ J using0 = 〈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 in
0 = 〈Ψµ,e−T HeT Ψ0〉 =: fµ(t) , t = (tν)ν∈Jh , µ, ν ∈ Jh
These are L = ]Jh << N nonlinear equations for L unknownexcitation amplitudes tµ.TheoremThe CC Method is size consistent!:
HAB = HA + HB ⇒ ECCAB = ECC
A + ECCB .
e−(TA+TB)(HA + HB)eTa+TB = e−TAHAeTA + e−TB HBeTB
The (discrete) CC equations
CC condition
〈Φd ,e−T (td )HeT (td )Ψ0〉 = E∗〈Φd ,Ψ0〉 ∀ Φd ∈ H1d .
means:
SOLVE
f (t) = (〈Ψµ,e−T (td )HeT (td )Ψ0〉)µ∈Id = 0,
for td = (tµ)µ∈Id , e.g. by quasi-Newton method,
then COMPUTE
E∗ = 〈Ψ0,e−T (td )HeT (td )Ψ0〉.
The (discrete) CC equations
CC condition
〈Φd ,e−T (td )HeT (td )Ψ0〉 = E∗〈Φd ,Ψ0〉 ∀ Φd ∈ H1d .
means:
SOLVE
f (t) = (〈Ψµ,e−T (td )HeT (td )Ψ0〉)µ∈Id = 0,
for td = (tµ)µ∈Id , e.g. by quasi-Newton method,
then COMPUTE
E∗ = 〈Ψ0,e−T (td )HeT (td )Ψ0〉.
The (discrete) CC equations
CC condition
〈Φd ,e−T (td )HeT (td )Ψ0〉 = E∗〈Φd ,Ψ0〉 ∀ Φd ∈ H1d .
means:
SOLVE
f (t) = (〈Ψµ,e−T (td )HeT (td )Ψ0〉)µ∈Id = 0,
for td = (tµ)µ∈Id , e.g. by quasi-Newton method,
then COMPUTE
E∗ = 〈Ψ0,e−T (td )HeT (td )Ψ0〉.
Basis reductionDiscrete one-particle basis B = χ1, ..., χN , χN+1, ...., χD+1.
Write (full Galerkin, “full CI”) solution ΨFCI asΨFCI = (I + Tfull CI)Ψ0
= Ψ0 +∑i1,a1
sa1i1
X a1i1
Ψ0 +∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
Ψ0
+ . . .+∑
i1,..,iN ,a1,..,aN
sa1,..,aNi1,..,iN
X a1,..,aNi1,..,iN
Ψ0.
Truncation according to excitation level, e.g.:I CISD (single/double):
ΦCISD = (I + TSD)Ψ0
= (I +∑i1,a1
sa1i1
X a1i1
+∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
)Ψ0
Truncated CC is not equivalent to corresponding CI truncation(but superior due to favourable properties).
Basis reductionDiscrete one-particle basis B = χ1, ..., χN , χN+1, ...., χD+1.
Write (full Galerkin, “full CI”) solution ΨFCI asΨFCI = (I + Tfull CI)Ψ0
= Ψ0 +∑i1,a1
sa1i1
X a1i1
Ψ0 +∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
Ψ0
+ . . .+∑
i1,..,iN ,a1,..,aN
sa1,..,aNi1,..,iN
X a1,..,aNi1,..,iN
Ψ0.
Truncation according to excitation level, e.g.:I CCSD (single/double):
ΦCCSD = eTSDΨ0
= exp(I +∑i1,a1
sa1i1
X a1i1
+∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
)Ψ0
Truncated CC is not equivalent to corresponding CI truncation(but superior due to favourable properties).
Basis reductionDiscrete one-particle basis B = χ1, ..., χN , χN+1, ...., χD+1.
Write (full Galerkin, “full CI”) solution ΨFCI asΨFCI = (I + Tfull CI)Ψ0
= Ψ0 +∑i1,a1
sa1i1
X a1i1
Ψ0 +∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
Ψ0
+ . . .+∑
i1,..,iN ,a1,..,aN
sa1,..,aNi1,..,iN
X a1,..,aNi1,..,iN
Ψ0.
Truncation according to excitation level, e.g.:I CCSD (single/double):
ΦCCSD = eTSDΨ0
= exp(I +∑i1,a1
sa1i1
X a1i1
+∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
)Ψ0
Truncated CC is not equivalent to corresponding CI truncation(but superior due to favourable properties).
Basis reductionDiscrete one-particle basis B = χ1, ..., χN , χN+1, ...., χD+1.
Write (full Galerkin, “full CI”) solution ΨFCI asΨFCI = (I + Tfull CI)Ψ0
= Ψ0 +∑i1,a1
sa1i1
X a1i1
Ψ0 +∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
Ψ0
+ . . .+∑
i1,..,iN ,a1,..,aN
sa1,..,aNi1,..,iN
X a1,..,aNi1,..,iN
Ψ0.
Truncation according to excitation level, e.g.:I CCSD (single/double):
ΦCCSD = eTSDΨ0
= exp(I +∑i1,a1
sa1i1
X a1i1
+∑
i1,i2,a1,a2
sa1,a2i1,i2
X a1,a2i1,i2
)Ψ0
Evaluation of the CC function: BCH-formula, operator algebra,Second quantization, Wick’s theorem, anticommutation laws.
The (linked) CCSD equations
E(t) = 〈Ψ0,HΨ0〉 +∑IA
fIAtAI +
1
4
∑IJAB
〈IJ‖AB〉tABIJ +
1
2
∑IJAB
〈IJ‖AB〉tAI tB
J ,
f (t)AI = fIA +
∑C
fAC tCI −
∑K
fKI tAK +
∑KC
〈KA‖CI〉tKC +
∑KC
fKC tACIK +
1
2
∑KCD
〈KA‖CD〉tCDKI
−1
2
∑KLC
〈KL‖CI〉tCAKL −
∑KC
fKC tCI tA
K −∑KLC
〈KL‖CI〉tCK tA
L +∑KCD
〈KA‖CD〉tCK tD
I
−∑
KLCD
〈KL‖CD〉tCK tD
I tAL +
∑KLCD
〈KL‖CD〉tKC tDA
LI −1
2
∑KLCD
〈KL‖CD〉tCDKI tA
L −1
2
∑KLCD
〈KL‖CD〉tCAKL tD
I
f (t)ABIJ = 〈IJ‖AB〉 +
∑C
(fBC tAC
IJ − fAC tBCIJ)−∑
K
(fKJ tAB
IK − fKI tABJK)
+1
2
∑KL
〈KL‖IJ〉tABKL
+1
2
∑CD
〈AB‖CD〉tCDIJ + P(IJ)P(AB)
∑KC
〈KB‖CJ〉tACIK + P(IJ)
∑C
〈AB‖CJ〉tCI − P(AB)
∑K
〈KB‖IJ〉tKA
+1
2P(IJ)P(AB)
∑KLCD
〈KL‖CD〉tACIK tDB
LJ +1
4
∑KLCD
〈KL‖CD〉tCDIJ tAB
KL +1
2P(AB)
∑KLCD
〈KL‖CD〉tACIJ tBD
KL
−1
2P(IJ)
∑KLCD
〈KL‖CD〉tABIK tCD
JL +1
2P(AB)
∑KL
〈KL‖IJ〉tAK tB
L +1
2P(IJ)
∑CD
〈AB‖CD〉tCI tD
J
− P(IJ)P(AB)∑KC
〈KB‖IC〉tAK tC
J + P(AB)∑KC
fKC tAK tBC
IJ + P(IJ)∑KC
fKC tCI tAB
JK − P(IJ)∑KLC
〈KL‖CI〉tCK tAB
LJ
+ P(AB)∑KCD
〈KA‖CD〉tCK tDB
IJ + P(IJ)P(AB)∑KCD
〈AK‖DC〉tDI tBC
JK + P(IJ)P(AB)∑KLC
〈KL‖IC〉tAL tBC
JK
+1
2P(IJ)
∑KLC
〈KL‖CJ〉tCI tAB
KL −1
2P(AB)
∑KCD
〈KB‖CD〉tAK tCD
IJ +1
2P(IJ)P(AB)
∑KLC
〈KB‖CD〉tCI tA
K tDJ
Baker-Campell-Hausdorff expansion
Solving f(th) = 0 we recall the Baker-Campell-Hausdorffformula
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, exercise**
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 ]
Baker-Campell-Hausdorff expansion
Solving f(th) = 0 we recall the Baker-Campell-Hausdorffformula
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, exercise**
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 ]
Baker-Campell-Hausdorff expansion
Solving f(th) = 0 we recall the Baker-Campell-Hausdorffformula
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, exercise**
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 ]
Iteration method to solve CC amplitude equations
We decompose the (discretized) Hamiltonian
H = F + U ,
F - Fock operator, U - fluctuation potential.
LemmaThere holds for MOs ( discrete eigenfunctions of F)
[F ,Xµ] = [F ,X a1,...,akl1,...,lk
] = (k∑
j=1
(λaj − λlj ))Xµ =: εµXµ .
and [[F ,Xµ],Xµ] = 0 together with
εµ ≥ λN+1 − λN > 0
(due to Bach-Lieb-Solojev)
Iteration method to solve CC amplitude equations
We decompose the (discretized) Hamiltonian
H = F + U ,
F - Fock operator, U - fluctuation potential.
LemmaThere holds for MOs ( discrete eigenfunctions of F)
[F ,Xµ] = [F ,X a1,...,akl1,...,lk
] = (k∑
j=1
(λaj − λlj ))Xµ =: εµXµ .
and [[F ,Xµ],Xµ] = 0 together with
εµ ≥ λN+1 − λN > 0
(due to Bach-Lieb-Solojev)
Iteration method to solve CC amplitude equations
We decompose the (discretized) Hamiltonian
H = F + U ,
F - Fock operator, U - fluctuation potential.
LemmaThere holds for MOs ( discrete eigenfunctions of F)
[F ,Xµ] = [F ,X a1,...,akl1,...,lk
] = (k∑
j=1
(λaj − λlj ))Xµ =: εµXµ .
and [[F ,Xµ],Xµ] = 0 together with
εµ ≥ λN+1 − λN > 0
(due to Bach-Lieb-Solojev)
Iteration method to solve CC amplitude equations
The amplitude function t 7→ f(t) = (fµ(t))µ∈Jh = 0
fµ(t) = 〈Ψµ,e−T HeT Ψ0〉 = 〈Ψµ,e−
∑ν∈Jh
tνXνHe∑ν∈Jh
tνXνΨ0〉 = 0.
The nonlinear amplitude equation f(t) = 0 is solved by
Algorithm (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.
Iteration method to solve CC amplitude equations
The amplitude function t 7→ f(t) = (fµ(t))µ∈Jh = 0
fµ(t) = 〈Ψµ,e−T HeT Ψ0〉 = 〈Ψµ,e−
∑ν∈Jh
tνXνHe∑ν∈Jh
tνXνΨ0〉 = 0.
The nonlinear amplitude equation f(t) = 0 is solved by
Algorithm (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.
Iteration method to solve CC amplitude equations
The amplitude function t 7→ f(t) = (fµ(t))µ∈Jh = 0
fµ(t) = 〈Ψµ,e−T HeT Ψ0〉 = 〈Ψµ,e−
∑ν∈Jh
tνXνHe∑ν∈Jh
tνXνΨ0〉 = 0.
The nonlinear amplitude equation f(t) = 0 is solved by
Algorithm (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.
Coupled Cluster...
....in practice:
B CC ansatzes introduced ∼ 1960 (Coester, Kummel)
B CC is nowadays standardly used in commercial quantumchemistry codes
B CCSD(T): often yields chemical accuracy, comparable topractical experiments
....from the point of view of numerical analysis:
B Schneider, 2009: First error estimates for reducedequations w.r.t. Galerkin solution Ψd .
B Problem: Does not admit for estimates, convergencestatements, error estimators w.r.t. full solution Ψ.
Coupled Cluster...
....in practice:
B CC ansatzes introduced ∼ 1960 (Coester, Kummel)
B CC is nowadays standardly used in commercial quantumchemistry codes
B CCSD(T): often yields chemical accuracy, comparable topractical experiments
....from the point of view of numerical analysis:
B Schneider, 2009: First error estimates for reducedequations w.r.t. Galerkin solution Ψd .
B Problem: Does not admit for estimates, convergencestatements, error estimators w.r.t. full solution Ψ.
III.
Analysis of the
Coupled Cluster methodS: & Th. Rohwedder (Dissertation 2010)
|Ψ〉 = eT |Ψ0〉
Globalization to continuous Coupled Cluster method
i.e analogeous reformulation of the continuous equation
HΨ = EΨ
to continuous Coupled Cluster equation
〈Ψµ,e−T (t∗)HeT (t∗)Ψ0〉 = E 〈Ψµ,Ψ0〉 ∀µ ∈M
for Ψ = eT (t∗)Ψ0.
Now formulated in continuous basis sets,
B = ψ1, ..., ψN︸ ︷︷ ︸occupied
∪ψa|a ∈ virt︸ ︷︷ ︸virtual
, B = Ψµ|µ ∈ I.
with analogous definition of cluster operator
T (t) : L2 → L2, T (t) =∑µ∈I
tµXµ
and suitable reference determinant
Ψ0 = Ψ[1, ..,N] :=∧N
i=1ψi(xi , si).
Globalization to continuous Coupled Cluster method
i.e analogeous reformulation of the continuous equation
HΨ = EΨ
to continuous Coupled Cluster equation
〈Ψµ,e−T (t∗)HeT (t∗)Ψ0〉 = E 〈Ψµ,Ψ0〉 ∀µ ∈M
for Ψ = eT (t∗)Ψ0. Now formulated in continuous basis sets,
B = ψ1, ..., ψN︸ ︷︷ ︸occupied
∪ψa|a ∈ virt︸ ︷︷ ︸virtual
, B = Ψµ|µ ∈ I.
with analogous definition of cluster operator
T (t) : L2 → L2, T (t) =∑µ∈I
tµXµ
and suitable reference determinant
Ψ0 = Ψ[1, ..,N] :=∧N
i=1ψi(xi , si).
Main problem, assumption on the basis
Main problem:
H1-continuity of cluster operator T and L2-adjoint T † have to beestablished!(to make 〈Ψµ,e−T (t∗)HeT (t∗)Ψ0〉 well-defined)
Assumption:
There holds
〈FχI , χA〉 = 〈χI , χA〉 = 0 for all I ∈ occ, A ∈ virt.
for a symmetric operator
F : H1(R3 × ±12)→ H−1(R3 × ±1
2),
spectrally equivalent to the H1(R3 × ±12)-norm.
(e.g. Fock operator, if HF ground state exists.)
Main problem, assumption on the basis
Main problem:
H1-continuity of cluster operator T and L2-adjoint T † have to beestablished!(to make 〈Ψµ,e−T (t∗)HeT (t∗)Ψ0〉 well-defined)
Assumption:
There holds
〈FχI , χA〉 = 〈χI , χA〉 = 0 for all I ∈ occ, A ∈ virt.
for a symmetric operator
F : H1(R3 × ±12)→ H−1(R3 × ±1
2),
spectrally equivalent to the H1(R3 × ±12)-norm.
(e.g. Fock operator, if HF ground state exists.)
Continuity of the cluster operator
Theorem (S., 2009; R., 2010)For any Ψ∗ =
∑α∈M∗ tαΨα ∈ H1, T = T (t) and T † its
L2-adjoint,
‖T‖H1→H1 ∼ ‖Ψ∗‖H1 , ‖T †‖H1→H1 ≤ ‖Ψ∗‖H1 .
Sketch of proof:
B Reduction to L2-orthogonal basis set,
B projection on Fi -orthonormal basis sets, F ∼ H1.
B Estimation with `1 . `2-estimate (Schneider 2009).
The continuous Coupled Cluster equations
Theorems (S., 2009; R., 2010)The eigenvalue equation
〈Ψµ, (H − E∗)Ψ〉 = 0, ∀µ ∈ I,
holds for Ψ = Ψ0 + Ψ∗ ∈ H1, E∗ ∈ R iff the Coupled Clusterequations
〈Ψµ,e−T (t∗)HeT (t∗)Ψ0〉 = 0, ∀µ ∈ I∗,〈Ψ0,e−T (t∗)HeT (t∗)Ψ0〉 = E∗,
hold for Ψ = eT (t∗)Ψ0, T (t∗) =∑
µ∈I∗ t∗µXµ, ‖t∗µ‖V <∞.
Coefficient vector t∗ ∈ V is solution of CC root equation,
f (t∗) = 0 ∈ V′for CC function
f : V → V′, f (t) :=(〈Ψµ,e−T (t)HeT (t)Ψ0 〉
)µ∈I∗ .
The continuous Coupled Cluster equations
Theorems (S., 2009; R., 2010)The eigenvalue equation
〈Ψµ, (H − E∗)Ψ〉 = 0, ∀µ ∈ I,
holds for Ψ = Ψ0 + Ψ∗ ∈ H1, E∗ ∈ R iff the Coupled Clusterequations
〈Ψµ,e−T (t∗)HeT (t∗)Ψ0〉 = 0, ∀µ ∈ I∗,〈Ψ0,e−T (t∗)HeT (t∗)Ψ0〉 = E∗,
hold for Ψ = eT (t∗)Ψ0, T (t∗) =∑
µ∈I∗ t∗µXµ, ‖t∗µ‖V <∞.
Coefficient vector t∗ ∈ V is solution of CC root equation,
f (t∗) = 0 ∈ V′for CC function
f : V → V′, f (t) :=(〈Ψµ,e−T (t)HeT (t)Ψ0 〉
)µ∈I∗ .
Local strong monotonicity of the CC function
Theorem (S., 2009; R., 2010)If E∗ < σess(h) is simple and Ψ0 close enough to Ψ, then f islocally strongly monotone at the solution t∗, i.e. there areγ, δ > 0 such that
〈f (s)− f (t), s − t〉 ≥ γ · ‖s − t‖2V
holds for s, t ∈ V with ‖s − t∗‖V, ‖t − t∗‖V < δ.
Sketch of proof:
I Local Lipschitz continuity from continuity of T
I 〈Φ, (H − E∗)Φ〉 ≥ γ′||Φ||21 on Ψ0⊥from Garding estimate for h, perturbation argument
I estimate remaining perturbations
Local strong monotonicity of the CC function
Theorem (S., 2009; R., 2010)If E∗ < σess(h) is simple and Ψ0 close enough to Ψ, then f islocally strongly monotone at the solution t∗, i.e. there areγ, δ > 0 such that
〈f (s)− f (t), s − t〉 ≥ γ · ‖s − t‖2V
holds for s, t ∈ V with ‖s − t∗‖V, ‖t − t∗‖V < δ.
Sketch of proof:
I Local Lipschitz continuity from continuity of T
I 〈Φ, (H − E∗)Φ〉 ≥ γ′||Φ||21 on Ψ0⊥from Garding estimate for h, perturbation argument
I estimate remaining perturbations
(Abstract) Galerkin Scheme
H - Hilbert space, V is a (reflexive) Banach space, V ′ its dual
V ⊂ H ⊂ V ′ ,
e.g. H := L2,V = H1 = u : ‖u‖2H1 := 〈u, (I −∆)u〉 <∞,
Vh ⊂ V : h < 0 be a dense family of finite dimensionalsubspaces. f : V → V ′ and u ∈ V where f(u) = 0 ∈ V ′
Definition (Galerkin scheme)An approximate solution uh ∈ Vh is obtained by the Galerkinscheme solving
〈vh, f(uh)〉 = 0 ∀vh ∈ Vh
i.e. the residual f(uh) ⊥ Vh is perpendicular to Vh.
(Abstract) Galerkin Scheme
H - Hilbert space, V is a (reflexive) Banach space, V ′ its dual
V ⊂ H ⊂ V ′ ,
e.g. H := L2,V = H1 = u : ‖u‖2H1 := 〈u, (I −∆)u〉 <∞,Vh ⊂ V : h < 0 be a dense family of finite dimensionalsubspaces. f : V → V ′ and u ∈ V where f(u) = 0 ∈ V ′
Definition (Galerkin scheme)An approximate solution uh ∈ Vh is obtained by the Galerkinscheme solving
〈vh, f(uh)〉 = 0 ∀vh ∈ Vh
i.e. the residual f(uh) ⊥ Vh is perpendicular to Vh.
(Abstract) Galerkin Scheme
H - Hilbert space, V is a (reflexive) Banach space, V ′ its dual
V ⊂ H ⊂ V ′ ,
e.g. H := L2,V = H1 = u : ‖u‖2H1 := 〈u, (I −∆)u〉 <∞,Vh ⊂ V : h < 0 be a dense family of finite dimensionalsubspaces. f : V → V ′ and u ∈ V where f(u) = 0 ∈ V ′
Definition (Galerkin scheme)An approximate solution uh ∈ Vh is obtained by the Galerkinscheme solving
〈vh, f(uh)〉 = 0 ∀vh ∈ Vh
i.e. the residual f(uh) ⊥ Vh is perpendicular to Vh.
Abstract Convergence Analysis
DefinitionA function f is called (locally) strongly monotone at u if
〈f(u)− f(u′), (u− u′〉 ≥ γ‖u− u′‖2V
for some γ > 0 and all ‖u′ − u‖V < δ.
ExampleLet A := f ′(u) : V → V ′ (linear) with
〈v,Au〉 ≤ L‖u‖V‖v‖V and
〈u,Au〉 ≥ γ‖u‖2V i.e. ReA > 0,
then f is Lipschitz continuous and strongly monoton.
Abstract Convergence Analysis
DefinitionA function f is called (locally) strongly monotone at u if
〈f(u)− f(u′), (u− u′〉 ≥ γ‖u− u′‖2V
for some γ > 0 and all ‖u′ − u‖V < δ.
ExampleLet A := f ′(u) : V → V ′ (linear) with
〈v,Au〉 ≤ L‖u‖V‖v‖V and
〈u,Au〉 ≥ γ‖u‖2V i.e. ReA > 0,
then f is Lipschitz continuous and strongly monoton.
Quasi-Optimal Convergence
Theorem (standard result)Let f be Lipschitz continuous and strongly monotone, theGalerkin scheme admits a (unique) solution uh ∈ Vh, C > 0satisfying ∀h < h0 the estimates
‖u− uh‖V ≤Lγ‖f(uh)‖V ′ , ‖uh‖V ≤ C‖u‖V ′
together with the quasi-optimal error estimate
‖u− uh‖V ≤Lγ
infvh∈Vh
‖uh − vh‖V
Example (CI-method)If ‖Ψ−Ψ0‖V < δ sufficiently small and E0 = 〈Ψ0,HΨ0〉, then(t) 7→ hν (t) := 〈Ψν , (H − E0)(I+T (t))Ψ0〉 is strongly monotone.
Quasi-Optimal Convergence
Theorem (standard result)Let f be Lipschitz continuous and strongly monotone, theGalerkin scheme admits a (unique) solution uh ∈ Vh, C > 0satisfying ∀h < h0 the estimates
‖u− uh‖V ≤Lγ‖f(uh)‖V ′ , ‖uh‖V ≤ C‖u‖V ′
together with the quasi-optimal error estimate
‖u− uh‖V ≤Lγ
infvh∈Vh
‖uh − vh‖V
Example (CI-method)If ‖Ψ−Ψ0‖V < δ sufficiently small and E0 = 〈Ψ0,HΨ0〉, then(t) 7→ hν (t) := 〈Ψν , (H − E0)(I+T (t))Ψ0〉 is strongly monotone.
Quasi-Optimal Convergence
Theorem (standard result)Let f be Lipschitz continuous and strongly monotone, theGalerkin scheme admits a (unique) solution uh ∈ Vh, C > 0satisfying ∀h < h0 the estimates
‖u− uh‖V ≤Lγ‖f(uh)‖V ′ , ‖uh‖V ≤ C‖u‖V ′
together with the quasi-optimal error estimate
‖u− uh‖V ≤Lγ
infvh∈Vh
‖uh − vh‖V
Example (CI-method)If ‖Ψ−Ψ0‖V < δ sufficiently small and E0 = 〈Ψ0,HΨ0〉, then(t) 7→ hν (t) := 〈Ψν , (H − E0)(I+T (t))Ψ0〉 is strongly monotone.
Local existence and quasi-optimal convergence
Let T (t) :=∑
µ tµXµwe 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, andJh 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‖ ≤ δ′
3. there ex. th ∈ Vh with 〈g(th),v〉 = 〈f(th),v〉 = 0, ∀v ∈ Vh,‖t− th‖V . inf
v∈Vh‖t− vh‖V .
Existence and uniqueness; quasi-optimality
Theorem (S., 2009; R., 2010)(i) Under assumptions as above, the solution t∗ is unique in
the neighbourhood Bδ(t∗).(ii) For closed subspaces Vd for which
d(t∗,Vd ) := minv∈Vd ‖t∗ − v‖V is sufficiently small,
〈f (td ), vd〉 = 0 for all vd ∈ Vd
admits a solution td in Bδ,d := Vd ∩ Bδ(t∗) which is uniqueon Bδ,d and fulfils the quasi-optimality estimate
‖td − t∗‖V ≤ Lγ
d(t∗,Vd ).
Sketch of proof:I Uniqueness from strong monotonicityI Ex. of discrete solutions uses lemma based on Browder’s fixed
point theorem
Error estimators (following Rannacher et al.)Lagrangian approach:
Minimize CC energy
E(t) = 〈Ψ0,e−T (t)HeT (t)Ψ0〉,under side condition f (t) = 0:
L(t , z) = E(t) + 〈f (t), z〉
Lemma (S., 2009; R., 2010)Monotonicity⇒ First order condition
L′(t∗, z∗) =
〈E ′(t∗), s〉 − 〈Df (t∗)s, z∗〉
〈f (t∗), s〉
= 0 for all s ∈ V.
has unique dual solution (Lagrangian multiplier) z∗ ∈ V. and
‖zd − z∗‖V . maxd(Vd , t∗), d(Vd , z∗).
Error estimators (following Rannacher et al.)Lagrangian approach:
Minimize CC energy
E(t) = 〈Ψ0,e−T (t)HeT (t)Ψ0〉,under side condition f (t) = 0:
L(t , z) = E(t) + 〈f (t), z〉
Lemma (S., 2009; R., 2010)Monotonicity⇒ First order condition
L′(t∗, z∗) =
〈E ′(t∗), s〉 − 〈Df (t∗)s, z∗〉
〈f (t∗), s〉
= 0 for all s ∈ V.
has unique dual solution (Lagrangian multiplier) z∗ ∈ V.
and
‖zd − z∗‖V . maxd(Vd , t∗), d(Vd , z∗).
Error estimators (following Rannacher et al.)Lagrangian approach:
Minimize CC energy
E(t) = 〈Ψ0,e−T (t)HeT (t)Ψ0〉,under side condition f (t) = 0:
L(t , z) = E(t) + 〈f (t), z〉
Lemma (S., 2009; R., 2010)Monotonicity⇒ Discrete first order condition
L′(td , zd ) =
〈E ′(td ), sd〉 − 〈Df (td )sd , zd〉
〈f (td ), sd〉
= 0 for all sd ∈ Vd
has unique dual solution (Lagrangian multiplier) zd ∈ V, and
‖zd − z∗‖V . maxd(Vd , t∗), d(Vd , z∗).
Dual weighted residual approach
Theorem (Becker/Rannacher, 2001)
Let (t∗, z∗) ∈ V2 and (td , zd ) ∈ V2d be the solutions of the
Lagrange equations for a thrice differentiable functional L, anddenote
ρ(td ) := 〈f (td ), ·〉V ρ∗(td , zd ) := 〈E ′(td ), ·〉V − 〈Df (td )·, zd〉V.
Then there holds
E(t∗)− E(td ) =12ρ(td )(z∗ − vd ) +
12ρ∗(td , zd )(t∗ − wd ) + R3
d
for all vd ,wd in Vd , where
R3d = O(max‖t∗ − td‖, ‖z∗ − zd‖3).
Error estimators for the CC equation
Theorem (S., 2009; R., 2010)
(i) For maxd(Vd , t∗), d(Vd , z∗) sufficiently good, under theabove assumptions, there holds
|E(t∗)− E(td )| ≤ ‖td − t∗‖V(
c1 ‖td − t∗‖V + c2 ‖zd − z∗‖V),
|E(t∗)− E(td )| .(
d(Vd , t∗) + d(Vd , z∗))2
for the solutions (t∗, z∗), (td , zd ) of the continuous/discreteCoupled Cluster equations and corr. dual solutions.
(ii) For Ψ = Ψ0 + Ψ∗ = eT (t∗)Ψ0, Ψz∗ := Ψ0 + Ψz∗ := eT (z∗)Ψ0,there holds
|E(t∗)− E(td )| .(
infΦ∈H1
d,⊥
‖Φ−Ψ∗‖H1 + infΦ∈H1
d,⊥
‖Φ−Ψz∗‖H1
)2.
Error estimators for the CC equation
Theorem (S., 2009; R., 2010)
(i) For maxd(Vd , t∗), d(Vd , z∗) sufficiently good, under theabove assumptions, there holds
|E(t∗)− E(td )| ≤ ‖td − t∗‖V(
c1 ‖td − t∗‖V + c2 ‖zd − z∗‖V),
|E(t∗)− E(td )| .(
d(Vd , t∗) + d(Vd , z∗))2
for the solutions (t∗, z∗), (td , zd ) of the continuous/discreteCoupled Cluster equations and corr. dual solutions.
(ii) For Ψ = Ψ0 + Ψ∗ = eT (t∗)Ψ0, Ψz∗ := Ψ0 + Ψz∗ := eT (z∗)Ψ0,there holds
|E(t∗)− E(td )| .(
infΦ∈H1
d,⊥
‖Φ−Ψ∗‖H1 + infΦ∈H1
d,⊥
‖Φ−Ψz∗‖H1
)2.
Comparison with Jastrow factor ansatzExample (– Quantum Monte Carlo Methods)Let us consider the Jastrow factor ansatz:
Ψ(x) ≈ F (x)Ψ0(x)
Ψ0(x) - reference (determinant), F - multiplication operator1. Linear ansatz:
F (x) =∑N
i f1(xi) +∑N
i>j f2(xi ,xj) + f3 . . .2. exponential ansatz : ( Krotzschek, ... )
F (x) = e[∑N
i f1(xi )+∑N
i>j f2(xi ,xj )+f3...] . ANOVA approx. is sizecons. only for the exponential ansatz 2)
3. In Coupled Cluster and Perturbation Theory
Ψ(x) = FΨ0(x) , CC : F = eT
is an operator. (In principle this is an exact ansatz - nofixed node error.)
Quantum Monte Carlo Methods (QMC)
Ψ(x) ≈ F (x)Φ(x) = F (x)Ψ0(x)e12∑N
i>j ‖xi−xj‖χ
I Φ(x) = Ψ0(x)e12∑N
i>j ‖xi−xj‖χ -reference, Ψ0 = ΨSL[1, . . . ,N]
I f1/2 := e12∑N
i>j ‖xi−xj‖ (e-e cusp) (f1/2 - e.g. Klopper in CC)I F - unknown Jastrow factor ( Ceperly, Umrigar, . . . )
Schrodinger eqn. ⇒ EVP for F ⇒ Fokker Planck eqn. t →∞
∂
∂tF =
12(∆F +∇ log |Φ|2 · ∇F
)−(∆Φ
Φ− Vcore + E0
)F → 0 .
Dirichlet boundary conditions F |∂Ω = 0, ∂Ω := x : Ψ0(x) = 0.
(Ito Calculus)⇐⇒ Stochastic differential equation (SDE)⇒ MC
(Small) systematic error: fixed node approximation (Cances &
Jourdan & Lelievre) - but accuracy comparable with CCSD!
Notes
I Projected CC is a compromise making the exponentialansatz computable
I it is more a perturbational approach for improving areference solution Ψ0.
I Analysis lays base for goal-oriented error estimators forCC, for example in combination with extrapolation schemes
I Analysis is only local, but it shows
I importance of quality of reference determinant Ψ0I importance of gap infσ(h)\E∗ − E∗
These do not only enter in convergence estimates foralgorithms (and reflect in practical experience), but alsoenter in quasi-optimality estimates.
Summary
I Schrodinger equation = high dimensional eigenvalueproblem with additional antisymmetry constraint
I Reformulation of linear Galerkin ansatz by nonlinear(projected) Coupled Cluster ansatz gives practical method
I Formulation in infinite dimensional spaces givescontinuous CC ansatz, equivalent to electronicSchrodinger equation
I Local existence/uniqueness statements for CC ansatz
I Error estimators for energy
I
Thank youfor your attention.
References:
H. Yserentant, Regularity and Approximability of Electronic Wave Functions,Lecture Notes in Mathematics series, Springer-Verlag, 2010.
T. Helgaker, P. Jørgensen, J. Olsen, Molecular Electronic-Structure Theory,John Wiley & Sons, 2000.
R. Schneider, Analysis of the projected coupled cluster method in electronicstructure calculation, Num. Math. 113, 3, p. 433, 2009.
R. Becker, R. Rannacher, An optimal control approach to error estimationand mesh adaptation in finite element methods,
Acta Numerica 2000 (A. Iserlet, ed.), p. 1, Cambridge University Press, 2001.
Th. Rohwedder, An analysis for some methods and algorithms of Quantum Chemistry.PhD thesis, 2010.