wave function methods for the electronic schrödinger equation · schrödinger equation reinhold...

Wave function methods for the electronic

Schrödinger equation

Reinhold Schneider, MATHEON TU Berlin

Zürich 2008

Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

DFG Reseach Center Matheon:

Mathematics in Key Technologies

A7: Numerical Discretization Methods in Quantum Chemistry

DFG Priority Program:

Modern and Universal First-Principle-Methods for Many-Electron Systems

in Chemistry and Physics

DFG Priority Program:

Extraction of Essential Information from Complex Systems


Basic model - electronic Schrödinger equationElectronic Schrödinger equa-

tion N nonrelativistic electrons + Born

Oppenheimer approximation

HΨ = EΨ

The Hamilton operator

H = −12

∑i

∆i −N∑i

K∑ν=1

Zν|xi − aν |

+12

N∑i,j

1|xi − xj |

acts on anti-symmetric wave functions Ψ ∈ H1(R3 × {±12})

N ,

Ψ(x1, s1, . . . , xN , sN) ∈ R , xi = (xi , si) ∈ R3 × {±12} .


Goals

To go beyond the accuracy of Density Functional Models one

tries to approximate the eigenfuntion Ψ, of HΨ = EΨ.

high precision: Correlation energy. Ecorr := EHF − E0

dynamic correlation: (closed shell R) HF is a good

approximation, but the resolution of the electron- electron

cusp hempers good convergence

static correlation: a single Slater deteminant cannot provide

a sufficiently good reference approximation, due e.g. open

shells, requires multi configurational models

excited statesdegenerate ground states


CI Configuration Interaction Method

Approximation space for (spin) orbitals (xj , sj)→ ϕ(xj , sj)

Xh := span {ϕi : i = 1, . . . ,N} , 〈ϕi , ϕj〉 = δi,j

E.g. Canonical orbitals ϕi , i = 1, . . . ,N are eigenfunctions of

the discretized Fock operator F := Fh =∑N

k=1 Fk : VFCI → VFCI

〈Fϕi − λiϕi , φh〉 = 0 ∀φh ∈ Xh

Full CI (for benchmark computations ≤ N = 18) is a

Galerkin method w.r.t. the subspace

VFCI =N∧

i=1

Xh = span{ΨSL = Ψ[ν1, ..νN ] =1

N!det(ϕνi (xj , sj))N

i,j=1,}


CI Configuration Interaction Method

N (discrete) eigenfunction ϕi , 〈Fϕi − λiϕi , φh〉 = 0 ∀φh ∈ Xh

The first N eigenfunctions ϕi are called occupied orbitals the

others are called unoccupied orbitals (traditionally)

ϕ1, . . . , ϕN , ϕN+1, . . . , ϕN

Galerkin ansatz: Ψ = c0Ψ0 +∑

ν∈J cνΨν

H = (〈Ψν′ ,HΨν〉) , Hc = Ec , dim Vh =

NN

Theorem (Brillouins theorem)

Let Ψ0 = ΨHF and Ψai = ΨS then

〈Ψai ,HΨ0〉 = 0


Configuration Interaction MethodAssumption: For any ϕ ∈ H1 there exist ϕh ∈ Xh such that

‖ϕh−ϕ‖H1 → 0, if h→ 0 (roughly: limh→0Xh = H1(R3×{±12}))

TheoremLet E0 be a single eigenvalue and HΨ = E0Ψ and E0,h,

Ψh ∈ Vh ⊂ VFCI be the Galerkin solution, then for h < h0 hold

‖Ψ−Ψh‖V ≤ c infφh∈Vh

‖Ψ− φh‖V

E0,h − E0 ≤ C infφh∈Vh

‖Ψ− φh‖2V .

Since dim Vh = O(NN), (curse of dimension), the full CI

method is infeasible for large N or N !!!!Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Multi-Configuration Self Consistent Field Method -

MCSCF Method

Optimization of the orbital basis functions ϕi , N ≤ M << N

Yh := span {ϕi : i = 1, . . . ,M} ⊂ Xh := span {ϕi : i = 1, . . . ,N},

VMCSCF =N∧

i=1

Yh = span{ΨSL = Ψ[ν1, ..νN ] : ϕν ∈ Yh}

For Ψ = c0Ψ0 +∑

ν∈J cνΨν and E = 〈Ψ,HΨ〉 = J MCSCF (c,Φ)

E0 ≈ min{J MCSCF (c,Φ ) : ‖c‖ = 1,Φ ∈ SM}

unknowns c,Φ := (ϕi)i=1,...,M ∈ S ⊂ V M on Stiefel manifold S

Exisitence results: Friesecke, Lewin(03)Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Conclusions

MCSCF can be solved with 2nd order optimization

methods, e.g. trust region Newton methods

Instead of FCI one can use a Complete Active Space

MCSCF is for multi-configurational problems, ( where RHF

is rather bad) static correlation

Due to the e-e cusp, M ∼ ε−1/2 is large!

FCI and MCSCF methods are scaling exponentially with N

O(

M

N

) = O(eN)!!!

restriction to single-double excitations etc. are not size

consistent (see below)Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Rayleigh-Schrödinger perturbation theory

H = H0 + U,

Solution for H0 is known

H0ψ0 = E0ψ0 .

Adiabatic perturbation H(λ) = H0 + λU.

Assumption: E0 is a simple eigenvalue of H0 and λ→ E(λ),

λ→ ψ(λ) are analytic in {λ ∈ C : |λ| ≤ 1 + ε} for some ε > 0.

E(λ) =∞∑

k=0

Ekλk , ψ(λ) =

∞∑k=0

ψkλk



Intermediate normalization ψ = ψ(1)

〈ψ,ψ0〉 = 1 ⇔ 〈ψl , ψ0〉 = 0 , l > 0.

The projection P0 onto the orthogonal complement of ψ0 is

defined by P0u = I − 〈ψ0,u〉ψ0.

Inserting the ansatz into the Schrödinger equation

∞∑

k=0

(H0 − E0)ψkλk = −

∞∑k=0

λUψkλk +

∞∑k=1

(k∑

l=1

Elψk−l)λk .

for all |λ| ≤ 1 + ε, + sorting w.r.t. to powers of λ gives

(H0 − E0)ψk = −Uψk−1 +k∑

l=1

Elψk−l , k = 1, . . . .



Testing against ψ0 gives

0 = 〈ψ0, (H0 − E0)ψk 〉 = 〈ψ0, (−Uψk−1 +k∑

l=1

Elψk−l)〉 =

〈ψ0,−Uψk−1〉+ Ek .

Ek = 〈ψ0,Uψk−1〉

ψk = P0(H0 − E0)−1P0(−Uψk−1 +k∑

l=1

Elψk−l).

For the computation of the energy contribution Ek we need all

ψl up to l = k − 1.Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Møller-Plesset Perturbation theory

In the sequel we restrict ourselves finite dimensional subspace

to Vh = VFCI !

H = F + U =N∑

i=1

Fi + (∑i<j

1|ri − rj |

−N∑

i=1

(Ji − Ki))

where F is the Fock operator and U is called the fluctuation

potential.

Proposition

Let Ψ = ΨSL = Ψ[ν1, . . . , νN ] be a Slater determinant of

canonical orbitals, then

FΨ = εΨ , ε =N∑

i=1

λνi , E0 =N∑

i=1

λi .


Let Ψ0 be the HF wave function, the first order energy

contribution is

E1 = −〈Ψ0,UΨ0〉 = −12

N∑i,j=1

〈ij ||ij〉

Hartree-Fock energy EMP1 = E0 + E1 = EHF .

Ψ1 = −P0(F − E0)−1P0UΨ0

and the second order contribution to the energy MP2:

E2 = 〈Ψ0,UΨ1〉 = 〈Ψ0,UP0(F − E0)−1P0UΨ0〉,

EMP2 = E0 + E1 + E2 = EHF + E2 .

Higher order contributions can be computed, e.g. MP3, MP4

etc. too. The convergence of the expansion is not guaranteed.Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Second quantization

Second quantization: annihilation operators:

ajΨ[j ,1, . . . ,N] = Ψ[1, . . . ,N]

and = 0 if j not apparent in Ψ[. . .].

sign-normalization: j appears in the first place in Ψ[j ,1, . . . ,N].

The adjoint of ab is a creation operator a†b

a†bΨ[1, . . . ,N] = Ψ[b,1, . . . ,N] = (−1)NΨ[1, . . . ,N,b]

Lemma

akal = −alak , a†ka†l = −a†l a†k , a†kal + ala

†k = δk .l


Excitation operators

Single excitation operator e.g. X k1

(−1)−pΨk1 = Ψ[k ,2, . . . ,N] = X k

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

higher excitation operator

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 = (I + T )Ψ0 T = T1 + T2 + T3 + . . .


Coupled Cluster Method - Exponential-ansatz

Theorem (S. 06)Let Ψ0 be a reference Slater determinant, e.g. Ψ0 = ΨHF and

Ψ ∈ VFCI , or 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


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, 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 ]


Coupled Cluster energy

Ψ ∈ V or VFCI = VFCI = span{Ψν : ν ∈ J }

〈Φ, (H − E0)Ψ〉 = 0∀Φ ∈ V,VFCI

due to Slater Condon rules and normalization 〈Ψ,Ψ0〉 = 1

E = 〈Ψ0,HΨ〉

= E〈Ψ0,H(I + T +12

T 2 + . . .)Ψ0〉

= 〈Ψ0,H(I + T1+T2 + T3 + . . .+12

T 21 + . . .)Ψ0〉

Proposition

E = 〈Ψ0,HΨ〉 = 〈Ψ0,H(I + T1+T2 +12

T 21 )Ψ0〉


Projected Coupled Cluster Methodamplitude equations

〈Ψµ,e−T HeT Ψ0〉 = 〈Ψµ,e−T HΨ〉 = E〈Ψµ,e−T Ψ〉 = 0∀µ ∈ J

The Projected Coupled Cluster Method consists in the ansatz

T =l∑

k=1

Tk =∑µ∈Jh

tµXµ , 0 6= µ ∈ Jh ⊂ J , i.e. Ψµ ∈ Vh ⊂ VFCI

e.g. CCSD T = T1 + T2 = T (t) satisfying

0 = 〈Ψµ,e−T HeT Ψ0〉 =: fµ(t) , t = (tν)ν∈Jh , µ, ν ∈ Jh

These are L = ]Jh << dimVFCI nonlinear equations for L

unknown excitation amplitudes tµ.Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Iteration method to solve CC amplitude equations

Quasi-Newton method to solve CC amplitude equations

We decompose the (discretized) Hamiltonian

H = F + U ,

F - Fock operator, U - fluctuation potential. There holds

[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

(Bach-Lieb-Solojev)Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Iteration method to solve CC amplitude equations

The amplitude function t 7→ f(t) = (fµ(t))µ∈Jh must be zero

fµ(t) = 〈Ψµ,e−T HeT Ψ0〉 = 〈Ψµ, [F ,Xµ]Ψ0〉+〈Ψµ, [U,T ]Ψ0〉 = 0.

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.

The Coupled Cluster Method is size consistent!:

HAB = HA+HB , e−(TA+TB)(HA+HB)eTa+TB = e−TAHAeTA+e−TB HBeTB ⇒ ECCAB = ECC

A +ECCB .


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



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})



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 Method

A function f : is called strictly monotone at t if

〈f(t)− f(t′), (t− t′〉 ≥ γ‖t− t′‖2V

for some γ > 0 and all ‖t′ − t‖V < δ.

Theorem (S.06 (a priori estimate))

If f(t) = 0 and f is strictly monotone at t, then th, resp.

Ψh = eTh Ψ0 satisfy

‖t− th‖V . infv∈R]Jh

‖t− vh‖V .

‖Ψ−Ψh‖H1 . infv∈RL‖Ψ− e

Pµ∈Jh

vµXµΨ0‖H1 .


Convergence of the CC Method - Duality estimates

Energy functional

J(t) := E(t) := 〈Ψ0,H(1 + T2 +12

T 21 )Ψ0〉

where t solves the amplitude equations

(f(t))ν = 〈Ψν ,e−T HeT Ψ0〉 = 0 , ∀ν ∈ J .

Let us further consider the Lagrange functional

L(t,a) := J(t)− 〈f(t),a〉 , t ∈ V ′ , a ∈ V .

and its stationary points

Lt[t,a](r,b) := J ′[t]r− 〈f′[t]r,a〉 = 0 , for all a ∈ V .

and La[t,a](r,b) = 〈f(t),b〉 = 0 for all b ∈ V .Reinhold Schneider, MATHEON TU Berlin Wave function methods for the electronic Schrödinger equation

Duality estimates for the Coupled Cluster Method

The dual solution a satisfies

f′[t]>a = −(J ′[t]) ∈ V ′.

Its Galerkin approximation is given by ah = (aµ)µ∈Jh ∈ Vh,

〈f′[th]>ah′ ,vh〉 = −〈(J ′[th]),vh〉 , vh ∈ Vh .

The discrete primal solution th = (tν)ν∈Jh ∈ Vh solves

〈f[th],bh〉 = 0 for all bh ∈ Jh .

We define the corresponding residual r, r∗ ∈ V ′

(r(th))µ =

(fµ) , µ 6∈ Jh

0 , µ ∈ Jh

together with the dual residual

(r∗(th,ah))µ =

(f′[th]>ah − (J ′[t]h)µ∈J , µ 6∈ Jh

0 , otherwise


Duality estimates for the Coupled Cluster Method

The derivatives J ′[t] and f ′[t] can be explicitly computed

(J ′[t]) =

〈Ψ0,HXµeT Ψ0〉 µ ∈ J1 single

〈Ψ0,UΨµ〉 , µ ∈ J2 double

0 , otherwise

where T =∑

tνXν , and(f ′[t])µ,ν = εµδµ,ν + 〈Ψµ,eT [U,Xν ]eT Ψ0〉, µ, ν ∈ J .

Lemma (dual weighted residual, Rannacher)

Let x := (t,a) ∈ V × V, xh := (th,ah) ∈ Vh × Vh and

eh = x − xh. L′(x) = 0, L′(xh)yh = 0 ∀yh ∈ Vh × Vh Then

L(x)− L(xh) = L′[xh](x − yh) +R3 , ∀yh ∈ Vh × Vh

where the remainder term

R3 = 12

∫ 10 L(3)[xh + seh](eh,eh,eh)s(s − 1)ds is depends

cubically on the error eh.


Convergence of the Coupled Cluster Energies

Theorem (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 infbh∈V

‖a− bh‖V +

+( infuh∈Vh

‖t− uh‖V )2.

|E − Eh| . ‖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 →∞.


Conclusions

Improvement by adding the e-e-cusp singularity explicitely,

r1,2, f1,2 methods (Kutzelnigg-Klopper ... )

CCSD and CCSD(T) are standard

CCSDT; CCSDTQ etc. only for extremely accurate

computations

CC is the most powerful tool for computing dynamical

correlation

not good for multi-configurational problems, ( where RHF is

rather bad)

How to do Multi Reference Coupled Cluster ???


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