the electronic schrödinger equation, ci and qcdmrg method€¦ · the electronic schrodinger...
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The Electronic Schrodinger equation, CI andQCDMRG method
R. Schneider (TUB Matheon)
John von Neumann Lecture – TU Munich, 2012
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Introduction
Electronic Schrodinger Equation
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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
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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.
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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 ; . . .)
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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.
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Slater determinants
and Full CI method
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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 .
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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〉.
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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〉.
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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
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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〉.
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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.
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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,}
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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)
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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 !!!!
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Binary tensorization,
Fock spaces and second quantization
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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′ .
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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
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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
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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
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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)
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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
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(qunatum chemistry) QC DMRG
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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).
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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?
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QC-DMRG and TT resp, MPS approximations
In courtesy of O Legeza (Hess & Legeza & ..)LiF dissoziation, 1st + 2nd eigenvalue
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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)
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QC-DRMRG and TT approximationsIn courtesy of O LegezaDifferent orderings
0 10 20 30 40 50 600
0.1
0.2
0.3
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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
![Page 30: The Electronic Schrödinger equation, CI and QCDMRG method€¦ · The Electronic Schrodinger equation, CI and¨ QCDMRG method R. Schneider (TUB Matheon) John von Neumann Lecture](https://reader033.vdocuments.us/reader033/viewer/2022052806/605ff5e8789ba269396caea1/html5/thumbnails/30.jpg)
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