The Electronic Schrodinger equation, CI andQCDMRG method

R. Schneider (TUB Matheon)

John von Neumann Lecture – TU Munich, 2012

Introduction

Electronic Schrodinger Equation

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 determines e.g.

B bonding energies,reactivity

B ionization energiesB conductivity,B in a wider sense

molecular geometry,-dynamics,...

of atoms, molecules, solidsetc.

Electronic Schrodinger equationN nonrelativistic electrons +

Born Oppenheimer approxi-

mationHΨ = EΨ

The Hamilton operator in atomic units

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 (Pauli principle)

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

Ψ(. . . ; xi , si ; . . . ; xj , sj ; . . .) = −Ψ(. . . ; xj , sj ; . . . ; xi , si ; . . .)

Variational formulationInput: position aν of the ν’s atom, nuclear charge Zν , number ofelectrons N.

Output: We are mainly interested in the ground-state energy,i.e. the lowest eigenvalue in the configuration spaceΨ ∈ V = H1(R3 × {±1

2})N ∩

∧Ni=1 L2(R3 × {±1

2}) .E0 = min〈Ψ,Ψ〉=1〈HΨ,Ψ〉 , Ψ = argmin〈Ψ,Ψ〉=1〈HΨ,Ψ〉

Basic Problem - Curse of dimensionalityI linear eigenvalue problem, but extremely high-dimensionalI + anti-symmetry constraints + lack of regularity.I traditional approximation methods (FEM, Fourier series, polynomials, MRA etc.):

approximation error in R1: . n−s , s- regularity , R3N′ : . n−s3N′ , (s < 5

2 ) with nDOFs i

I Nondeterministic methods: Quantum Monte Carlo methods have problems withfermions

For large systems N′ >> 1 ( N′ > 1) the electronic Schrodinger equation seems to be

intractable! But 70 years of impressive progress has been awarded by the Nobel price

1998 in Chemistry: Kohn, Pople.

Slater determinants

and Full CI method

Anti-symmetric tensor products - Slater determinantsApproximation 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 valuedfunctions Ψ, ϕ .

Single and two particle operatorsAbbreviations:

a) Single particle operators A = h =∑N′

i=1 hi =∑N′i=1(1

2∆i +∑M

j=1−Zj|xi−Rj |

) =∑N′

i=1(12∆i + Vcore(xi)), hi = hj ,

〈i |h|j〉 := 〈ϕi ,hjϕj〉

a) Two particle operators G =∑N′

i∑N′

j>i1

|xi−xj |, H = h + G

〈a,b|i , j〉 :=∑

s,s′=± 12

∫ ∫ϕ∗a(x, s)ϕ∗b(x′, s′)ϕi(x, s)ϕj(x′, s′)

|x− x′|dxdx′

〈a,b||i , j〉 := 〈a,b|i , j〉 − 〈a,b|j , i〉.

Single and two particle operatorsAbbreviations:

a) Single particle operators A = h =∑N′

i=1 hi =∑N′i=1(1

2∆i +∑M

j=1−Zj|xi−Rj |

) =∑N′

i=1(12∆i + Vcore(xi)), hi = hj ,

〈i |h|j〉 := 〈ϕi ,hjϕj〉

a) Two particle operators G =∑N′

i∑N′

j>i1

|xi−xj |, H = h + G

〈a,b|i , j〉 :=∑

s,s′=± 12

∫ ∫ϕ∗a(x, s)ϕ∗b(x′, s′)ϕi(x, s)ϕj(x′, s′)

|x− x′|dxdx′

〈a,b||i , j〉 := 〈a,b|i , j〉 − 〈a,b|j , i〉.

Slater-Condon Rules

Single particle operators: Ψ1 = ΨSL = Ψ[. . . , νi , νj , . . .]

1) Ψ1 = Ψ2 =: Ψ Ψ1 = Ψ[. . . , νi , νj , . . .]

Ψ2 = Ψ[. . . , νi , νj , . . .] 〈Ψ, hΨ〉 =∑N

l=1〈l|h|l〉2) Ψ2 = X a

j Ψ1 Ψ1 = Ψ[. . . , νi , νj , . . .]

Ψ2 = Ψ[. . . , νi , νa, . . .] 〈Ψ2, hΨ1〉 = 〈a|h|j〉3) Ψ1 = X a,b

i,j Ψ2 Ψ1 = Ψ[. . . , νi , νj , . . .]

or higher excitations Ψ2 = Ψ[. . . , νa, νb, . . .] 〈Ψ2, hΨ1〉 = 0

Proof *exercise Hint: Leibniz formula +

〈Ψ2, hl Ψ1〉 =

∫ϕa(xl )(hlϕi )(xl ) ·

∫ϕb(xk )ϕj (xk )

∫ϕm(x1)ϕm(x1) . . . = 〈a|h|l〉δb,j

Slater-Condon Rules

Slater-Condon Rules for two particle operators:1) Ψ1 = Ψ2 Ψ1 = Ψ[. . . , νi , νj , . . .]

Ψ2 = Ψ[. . . , νi , νj , . . .] 〈Ψ1,GΨ1〉 = 12∑N

i=1∑N

j=1〈ij||ij〉2) Ψ2 = X a

j Ψ1 Ψ1 = Ψ[. . . , νi , νj , . . .]

Ψ2 = Ψ[. . . , νi , νa, . . .] 〈Ψ2,GΨ1〉 =∑N

i=1〈j i||ai〉3) Ψ1 = X a,b

i,j Ψ2 Ψ1 = Ψ[. . . , νi , νj , . . .]

Ψ2 = Ψ[. . . , νa, νb, . . .] 〈Ψ2,GΨ1〉 = 〈ij||ab〉3) Ψ1 = X a,b,c

i,j,l Ψ2 Ψ1 = Ψ[. . . , νi , νj , νl . . .]

or higher excitations Ψ2 = Ψ[. . . , νa, νb, νc . . .] 〈Ψ2,GΨ1〉 = 0

with

〈a, b|i, j〉 :=∑

s,s′=± 12

∫ ∫ϕ∗a (x, s)ϕ∗b (x′, s′)ϕi (x, s)ϕj (x′, s′)

|x− x′|dxdx′

〈a, b||i, j〉 := 〈a, b|i, j〉 − 〈a, b|j, i〉.

Spin functions and spatial orbitalsSpin orbital function: ϕ(x, s) ∈ R, (C), x ∈ R3 s = ±1

2 ,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)

ϕα(x, s) = φα(x)χα(s)

UHF (Unrestricted Hartree Fock, single spin state orbitals) :

φα,i , φβ,j , N ′ = Nα + Nβ .

Closed shell RHF (Restricted Hartree Fock) :

φα,i = φβ,i = φi , i = 1, . . . ,N =N ′

2.

CI Configuration Interaction MethodApproximation 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 ofthe discretized single particle operator (e.g Fock operator)

F := Fh =N∑

k=1

Fk : VFCI → VFCI ,

〈Fϕi − λiϕi , φh〉 = 0 ∀φh ∈ Xh

Full CI (for benchmark computations ≤ N = 18) is aGalerkin 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 ∈ XhThe first N eigenfunctions ϕi are called occupied orbitals theothers 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

)The matrix coefficients of H = (〈Ψν′ ,HΨν〉) can be computedby Slater-Condon-rules. (sparse matrix)

Configuration Interaction Method

Assumption: For any ϕ ∈ H1 there exist ϕh ∈ Xh such that‖ϕh−ϕ‖H1 → 0, if h→ 0 (roughly: limh→0Xh = H1(R3×{±1

2}))

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 CImethod is infeasible for large N or N !!!!

Binary tensorization,

Fock spaces and second quantization

Configuration SpaceConsider a (complete) orthonormal basis {ϕi : i = 1, . . . ,d}(e.g. by MCSCF)

Xh := span {ϕi : i = 1, . . . ,d} ⊂ X = H1(R3,±12

)

ONB of antisymmetric functions by Slater determinants

ΨSL[k1, . . . , kN ](x1, s1; . . . ; xN , sN) :=1√N!

det(ϕki (xj , sj))Ni,j=1

VNFCI =

N∧i=1

Xh = span{ΨSL = Ψ[k1, . . . , kN ] : k1 < . . . < kN ≤ d} ⊂ V

Curse of dimensionality dimVNFCI =

(dN

)!

For a discrete operator H : VNFCI =: V → V ′

HΨ =∑ν′,ν

〈Ψν′ ,HΨν〉cνΨν′ =∑ν′

(Hc)ν′Ψν′ .

Fock spaceLet Ψµ := ΨSL[ϕk1 , . . . , ϕkN ] = Ψ[k1, . . . , kN ] basis Slater det.Labeling of indices µ ∈ I by an binary string of length d

e.g.: µ = (0,0,1,1,0, . . .) =:d−1∑i=0

µi2i , µi = 0,1 ,

I µi = 1 means ϕi is (occupied) in Ψ[. . .].I µi = 0 means ϕi is absend (not occupied) in Ψ[. . .].

(discrete) Fock space Fd is of dimFd = 2d , (K := C,R)

Fd :=d⊕

N=0

VNFCI = {Ψ : Ψ =

∑µ

cµΨµ}

Fd ' {c : µ 7→ c(µ0, . . . , µd−1) = cµ ∈ K , µi = 0,1 } =d⊗

i=1

K2

This is a basis depent formalism⇒ : Second Quantization

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

Discrete annihilation and creation operators

A :=

(0 10 0

), AT =

(0 01 0

)In order to obtain the correct phase factor, we define

S :=

(1 00 −1

),

and the discrete annihilation operator

ap ' Ap := S ⊗ . . .⊗ S ⊗ A(p) ⊗ I ⊗ . . .⊗ I

where A(p) means that A appears on the p-th position in theproduct.The creation operator

a†p ' ATp := S ⊗ . . .⊗ S ⊗ AT

(p) ⊗ I ⊗ . . .⊗ I

Schrodinger operatorOne and two electron integrals

hpq := 〈ϕq, (

−12

∆− Vcore)ϕp〉 , p,q, r , s = 1, . . . ,d ,

gp,qr ,s :=

12〈ϕr (x, s1)ϕs(y, s2),

ϕp(x, s1)ϕq(y, s2)

|x− y|〉

Theorem (Slater -Condon, (HRS (2010)))The Galerkin matrix H of electronic Schrodinger Hamiltonian issparse and can be represented by

H =d∑

p,q=1

hqpAT

p Aq +d∑

p,q,r ,s=1

gp,qr ,s AT

r ATs ApAq .

Modification, treating spin explicitely, (spin symmetries andother symmetries could be enfored (Legeza et al. ))

Fd =d⊗

i=1

K4 , (K = C,R)

Particle number operator and Schrodinger eqn.

P :=d∑

p=1

a†pap , ' P :=d∑

p=1

ATp Ap .

The space of N-particle states is given by

VN := {c ∈d⊗

i=1

K2 : Pc = Nc} .

Variational formulation of the Schrodinger equation

c = (c(µ)) = argmin{〈Hc,c〉 : 〈c,c〉 = 1 , Pc− Nc = 0} .

Approximate

c ∈d⊗

i=1

K2 ≈ cε ∈Mr (Tr )

in hierarachical tensor format (e.g. MPS) by means of e.g.(M)ALS⇒ QC DMRG

(qunatum chemistry) QC DMRG

Separation into (two) subsystems Ai ,Bi

FAi = generated by {ϕ1 . . . ϕi},FBi = generated by {ϕi+1 . . . ϕd}

SVD factorization of Uxi+1x1,...,xi

(E. Schmidt). ri is a measure forthe entanglement of the two subsystems Ai ,Bi , i = 1, . . . ,d − 1.If Ai ,Bi are independent→ ri = 1 - size consistency.

Working assumption: ri moderate, and truncation error small!Deficiencies:

1. representation depends on the ordering of the basisfunctions (variables)

2. representation depends on the choice basis functions(molecular, natural, localized orbitals)

3. scaling O(r2).

DiscussionConsequences having an MPS approximation:

1. represents a FCI wave function2. permeates the full FCI space3. complexity (storage) : 2

∑di1 ri−1ri = O(r2d)

4. electron density and reduced density matrices can becomputed⇒ gradients, forces, ...

5. subspaces, mixed states, (e.g. multiple eigenvalues,degeneracy and near degeneracy) can be computed aswell

6. size consistency and black box (yes and no)I cannnot (will not) answer the open questions:

1. for which systems the working assumption holds?2. are there better tensor formats (e.g. HT or tensor networks)?

(of course ri < rc usually ri << rc )

Instead we consider the questions1. is the TT or MPS parametrization stable? (yes) can we

remove rendundancy completely? (yes)2. how to compute the MPS approximation of the FCI wave

function?

QC-DMRG and TT resp, MPS approximations

In courtesy of O Legeza (Hess & Legeza & ..)LiF dissoziation, 1st + 2nd eigenvalue

QC-DMRG and TT resp. MPS approximations

In courtesy of O Legeza Example transition metals: copper

oxide cluster N = 18,d = 2× 43 (Legza & Reiher & )

3 (1.936) 14 (1.988)13 (0.070) 25 (1.958)

34 (1.942)26 (0.028) 35 (0.550)

QC-DRMRG and TT approximationsIn courtesy of O LegezaDifferent orderings

0 10 20 30 40 50 600

0.1

0.2

0.3

Site index

Site en

tropy 01

020304

5 10 15 20 25 30 35 40 45 50 550

0.2

0.4

0.6

Left block length

Block

entrop

y 01020304

5 10 15 20 25 30 35 40 45 50 55−0.6

−0.4

−0.2

0

Left block length

Mutua

l inform

ation

01020304

0 10 20 30 40 50 600

0.1

0.2

0.3

Site index

Site ent

ropy

01020304

5 10 15 20 25 30 35 40 45 50 550

0.2

0.4

0.6

Left block length

Block en

tropy 01

020304

5 10 15 20 25 30 35 40 45 50 55−0.6

−0.4

−0.2

0

Left block length

Mutual

informa

tion 01020304

QC-DMRG and TT, resp. MPS approximations

Copper Cluster (-perox-) before and after reordering

0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

Orbital index

Orb

ital e

ntro

py (a)

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Left block length

Blo

ck e

ntro

py (b)

5 10 15 20 25 30 35 40

−1

−0.5

0

Left block length

Mut

ual i

nfor

mat

ion

(c) 010203040506

0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

Orbital index

Orb

ital e

ntro

py (a)

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Left block length

Blo

ck e

ntro

py (b)

5 10 15 20 25 30 35 40−1.5

−1

−0.5

0

Left block length

Mut

ual i

nfor

mat

ion

(c) 010203040506

In courtesy of O Legeza