I ) Direct minimization in Hartree Fock and DFT - II ... · I ) Direct minimization in Hartree Fock...

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

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


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 .


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


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


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 .


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.


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


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


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.


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

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〉


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)


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


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 .


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


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


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).


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

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.


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


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


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).


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

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

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)


Adaptivity


Numerical results

example: C19H22N2O (N = 55)

,

Figure:


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


Accuracy Depending on hgrid

convergence rate: O(h14grid )


Wavelets vs. Plane Waves (Degrees of Freedom)


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


Computing Times


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


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

!


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 ]


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 (?!)


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 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 .


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


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.


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 .


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 ]


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

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


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)


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.


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.


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 .


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 →∞.


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


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)


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