The Coupled Cluster method for the electronic Schrödinger equation

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




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)


0.001 eV


Magnetic couplings

Atomic cores

1 eV 10 eV0.1 eV0.01 eV

Chemical bonds

CC MP2 DFT




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


0.001 eV


Magnetic couplings

Atomic cores

1 eV 10 eV0.1 eV0.01 eV

Chemical bonds

CC MP2 DFT




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


0.001 eV


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.


0.001 eV


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,...



Ψ0 = Ψ[1, ..,N] :=∧N

i=1ψi(xi , si) =

1√N!

det(ψpi (xj , sj)

)Ni,j=1


2),B = ψ1, . . . , ψN︸︷︷︸

occupied orbitals









Ψ0 = Ψ[1, ..,N] :=∧N

i=1ψi(xi , si) =

1√N!

det(ψpi (xj , sj)

)Ni,j=1


2),B = ψ1, . . . , ψN︸︷︷︸

occupied orbitals









Ψ0 = Ψ[1, ..,N] :=∧N

i=1ψi(xi , si) =

1√N!

det(ψpi (xj , sj)

)Ni,j=1


2),B = ψ1, . . . , ψN︸︷︷︸

occupied orbitals







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 .


X k1 Ψ0 := a†ka1Ψ0

(−1)−pΨk1 = Ψ[k ,2, . . . ,N] = X k

1 Ψ0 = X kj Ψ[1, . . . , . . . ,N] = a†ka1Ψ0


Xµ := X b1,...,bkl1,...,lk

=k∏

i=1

X bili

, 1 ≤ li < li+1 ≤ N , N < bi < bi+1 .



Ψ =

c0 +∑µ∈J

cµXµ

Ψ0 , c0, cµ ∈ R .


X k1 Ψ0 := a†ka1Ψ0

(−1)−pΨk1 = Ψ[k ,2, . . . ,N] = X k

1 Ψ0 = X kj Ψ[1, . . . , . . . ,N] = a†ka1Ψ0


Xµ := X b1,...,bkl1,...,lk

=k∏

i=1

X bili

, 1 ≤ li < li+1 ≤ N , N < bi < bi+1 .



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.


occupied


,






: L2d → L2

d

X a1,..,aki1,..,ik

Ψ[p1, ..,pN ] =

Ψ[ 6i1, .., 6ik ,a1, ..,ak ..,pi , ..]


0 elsewise


Ψ0.


occupied


,






: L2d → L2

d

X a1,..,aki1,..,ik

Ψ[p1, ..,pN ] =

Ψ[ 6i1, .., 6ik ,a1, ..,ak ..,pi , ..]


0 elsewise


Ψ0.


occupied


,






: L2d → L2

d

X a1,..,aki1,..,ik

Ψ[p1, ..,pN ] =

Ψ[ 6i1, .., 6ik ,a1, ..,ak ..,pi , ..]


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 .



Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id

sµXµΨ0

=: (I + S(sd ))Ψ0





Ψd = (I + S(sd ))Ψ0,




Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id

sµXµΨ0 =: (I + S(sd ))Ψ0





Ψd = (I + S(sd ))Ψ0,




Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id






Ψd = (I + S(sd ))Ψ0,




Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id


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 .



Ψd = Ψ0 ⊕L2,F Ψ∗d = Ψ0 +∑µ∈Id


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


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〉.


CC condition


means:

SOLVE



then COMPUTE

E∗ = 〈Ψ0,e−T (td )HeT (td )Ψ0〉.


CC condition


means:

SOLVE



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).



= 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




= 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.


ΦCCSD = eTSDΨ0

= exp(I +∑i1,a1

sa1i1

X a1i1

+∑

i1,i2,a1,a2

sa1,a2i1,i2

X a1,a2i1,i2

)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.


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 ]



e−T AeT = A + [A,T ] +12!

[[A,T ],T ] +13!

[[[A,T ],T ],T ] + . . . =

A +∞∑

k=1

1k !

[A,T ]k .


e−T HeT = H+[H,T ]+12!

[[H,T ],T ]+13!

[[[H,T ],T ],T ]+14!

[H,T ]4


e−TFeT = F + [F ,T ] + [[F ,T ],T ]



e−T AeT = A + [A,T ] +12!

[[A,T ],T ] +13!

[[[A,T ],T ],T ] + . . . =

A +∞∑

k=1

1k !

[A,T ]k .


e−T HeT = H+[H,T ]+12!

[[H,T ],T ]+13!

[[[H,T ],T ],T ]+14!

[H,T ]4


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)



H = F + U ,



[F ,Xµ] = [F ,X a1,...,akl1,...,lk

] = (k∑

j=1



εµ ≥ λN+1 − λN > 0




H = F + U ,



[F ,Xµ] = [F ,X a1,...,akl1,...,lk

] = (k∑

j=1



εµ ≥ λN+1 − λN > 0



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.




∑ν∈Jh

tνXνHe∑ν∈Jh

tνXνΨ0〉 = 0.




tn+1 = tn − A−1f(tn),





∑ν∈Jh

tνXνHe∑ν∈Jh

tνXνΨ0〉 = 0.




tn+1 = tn − A−1f(tn),


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.



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 ′


〈vh, f(uh)〉 = 0 ∀vh ∈ Vh




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 ′


〈vh, f(uh)〉 = 0 ∀vh ∈ 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.






infvh∈Vh

‖uh − vh‖V







infvh∈Vh

‖uh − vh‖V


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∗).


Minimize CC energy


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



Minimize CC energy


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


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.

Documents

The Coupled Cluster method for the electronic Schrödinger equation