How quantum entanglement can promote [1ex] the ...cassam/Workshop13/Presentations/boguslawski.pdf · How quantum entanglement can promote the understanding of electronic structures

How quantum entanglement can promote

the understanding of

electronic structures of molecules

Katharina Boguslawski

McMaster University, Department of Chemistry, Canada

http://www.chemistry.mcmaster.ca/ayers/

November 2013

Entanglement and electronic structure Katharina Boguslawski

Introduction

The electronic structure problem

Conventional and unconventional methods

A benchmark example: spin densities

Entanglement and electronic structure

Electron correlation

Bond orders

Weak interaction


Introduction

Electronic structure of molecules in quantum chemistry

↙ ↘density-functional theory (DFT)

↓central quantity: electron density ρ(r)

↓Kohn–Sham formalism:

E[ρ] = Ts[ρ] + Vext[ρ] + J[ρ] + Exc[ρ]

⊕ computationally feasible for largemolecules

exact exchange–correlation (xc)density functional not known

approximations can fail

ab initio wave function methods↓

Hartree–Fock theory (singledeterminant Φ0)

↓Account for electron correlation byincluding excitations Ψ =

∑I CIΦI

↙ ↘exponential cost:(F)CI, CASSCF,CC, . . .

⊕ highly accurate size limited

polynomial cost:DMRG,AP1roG, . . .

⊕ highly accurate⊕ “size unlimited”


Parameterization of the wave function

|Ψ〉 =∑

ni n2···nN

Cn1n2···nN |n1〉 ⊗ |n2〉 ⊗ · · · ⊗ |nN〉

Restricted sum over basis states with a certain excitation pattern

Configuration Interaction ansatz

|CI〉 =(

1 +∑µ

Cµτµ)|HF〉

numerous specialized selection/restriction protocols


Parameterization of the wave function

Find better parameterization schemes for the electronic wave function

|Ψ〉 =∑

n1n2···nN

Cn1n2···nN |n1〉 ⊗ |n2〉 ⊗ · · · ⊗ |nN〉

Coupled Cluster ansatz

|CC〉 = exp(∑

µ

tµτµ)|HF〉

MPS representation

|MPS〉 =∑{n}

An1 . . .Anl−1Ψnl nl+1Anl+2 . . .AnL |n1 . . . nL〉


Motivation

DFT instrumental in the study of transition metal complexes(structures(!), energies, spectroscopic signatures)But: open-shell systems remain a challenge, e.g.,

relative energetical ordering of closely lying states of different spin Sspin density distributions of compounds containing noninnocentligands

⇒ accurate reference calculations are necessary, e.g., CASSCF/CASPT2,MRCI are well-establishedexamples:M. Radon, K. Pierloot, J. Phys. Chem. A 2008, 112, 11824.

M. Radon, E. Broclawik, J. Chem. Theory Comput. 2007, 3, 728.

B. O. Roos, V. Veryazov, J. Conradie, P. R. Taylor, A. Ghosh, J. Phys. Chem. B 2008, 112, 14099.

But what if the system size is too large?


Analysis of spin density distributions


Most critical test case: noninnocent iron nitrosyl complexes

(a) Fe(salen)(NO) conformation a

(b) Fe(salen)(NO) conformation b

(c) Fe(porphyrin)(NO)

transition metal nitrosyl complexes have acomplicated electronic structure

standard functionals might yield’reasonable’ spin state splittings — butspin densities can still be wrongJ. Conradie, A. Ghosh, J. Phys. Chem. B2007, 111, 12621.

(see also review on spin-DFT:Ch. R. Jacob, M. Reiher, IJQC, 2012, 112,3661.)

systematic comparison of DFT spindensities with CASSCF:K. Boguslawski, C. R. Jacob, M. Reiher, J.Chem. Theory Comput. 2011, 7, 2740;see also work by K. Pierloot et al.


DFT spin densities: the salen-complex example

(a) OLYP (b) OPBE (c) BP86 (d) BLYP

(e) TPSS (f) TPSSh (g) M06-L (h) B3LYP

Only for high-spin complexes similar spin densities are obtained

⇒ [Fe(NO)]2+ moiety determines the spin density


The model system for accurate reference calculations

zyx

Fe

N

O

�

dpcdpc

dpcdpc

Structure:

Four point charges of −0.5 e model a square-planarligand field (ddp = 1.131 A)

⇒ Similar differences in DFT spin densities as present forlarger iron nitrosyl complexes

Advantage of the small system size:Standard correlation methods (CASSCF, . . .) can be efficiently employed

Study convergence of the spin density w.r.t. the size of the active orbitalspace

K. Boguslawski, C. R. Jacob, M. Reiher, J. Chem. Theory Comput. 2011, 7, 2740.


DFT spin densities

(a) OLYP (b) OPBE (c) BP86 (d) BLYP

(e) TPSS (f) TPSSh (g) M06-L (h) B3LYP

Spin density isosurface plots Spin density difference plots w.r.t. OLYP

⇒ Similar differences as found for the large iron nitrosyl complexes


Reference spin densitiesfrom standard electron correlation methods


CASSCF calculations: oscillating spin densities

CAS(7,7) CAS(11,9) CAS(11,11) CAS(11,12) CAS(11,14)

CAS(11,13) CAS(13,13) CAS(13,14) CAS(13,15) CAS(13,16)

stable CAS with allimportant orbitalsdifficult to obtain

⇒ Reference spindensities for verylarge CAS required

⇒ Apply DMRGalgorithm !

K. Boguslawski, K. H. Marti,O. Legeza, M. Reiher, J. Chem.Theory Comput. 2012, 8, 1970.


Spin densities forlarge active spaces


DMRG spin densities — measures of convergence

Three convergence criteria:

Qualitative convergence measure: spin density difference plots

Quantitative convergence measure:

∆abs =

∫|ρs1(r)− ρs2(r)|dr < 0.005

∆sq =

√∫|ρs1(r)− ρs2(r)|2dr < 0.001

Quantitative convergence measure: quantum fidelity Fm1,m2

Fm1,m2 = |〈Ψ(m1)|Ψ(m2)〉|2

⇒ Reconstructed CI expansion of the DMRG wave function indicatesimportant configurationsK. Boguslawski, K. H. Marti, M. Reiher, J. Chem. Phys. 2011, 134, 224101.


DMRG spin densities for large active spaces

3 different active spaces in DMRG calculations:CAS(13,20), CAS(13,24) and CAS(13,29)

Orbital basis: natural orbitals from a CAS(11,14) calculation, Dunning’scc-pVTZ basis set

CAS(13,24) and CAS(13,29): the fifth Fe double-d-shell orbital (dx2−y2 ) isincluded

Orbital ordering was optimized for each CAS in order to keep the numberof DMRG active-system states m as small as possibleG. Barcza, O. Legeza, K. H. Marti, M. Reiher, Phys. Rev. A 2011, 83, 012508.O. Legeza, J. Solyom, Phys. Rev. B 2003, 68, 195116.

K. Boguslawski, K. H. Marti, O. Legeza, M. Reiher, J. Chem. Theory Comput. 2012, 8, 1970.


DMRG spin densities for large active spaces

∆abs and ∆sq for DMRG(13,y)[m] calculations w.r.t. DMRG(13,29)[2048] reference

Method ∆abs ∆sq Method ∆abs ∆sq

DMRG(13,20)[128] 0.030642 0.008660 DMRG(13,29)[128] 0.032171 0.010677DMRG(13,20)[256] 0.020088 0.004930 DMRG(13,29)[256] 0.026005 0.006790DMRG(13,20)[512] 0.016415 0.003564 DMRG(13,29)[512] 0.010826 0.003406DMRG(13,20)[1024] 0.015028 0.003162 DMRG(13,29)[1024] 0.003381 0.000975DMRG(13,20)[2048] 0.014528 0.003028

DMRG(13,20;128) DMRG(13,20;256) DMRG(13,20;512) DMRG(13,20;1024) DMRG(13,20;2048)

DMRG(13,29;128) DMRG(13,29;256) DMRG(13,29;512) DMRG(13,29;1024) DMRG(13,29;2048)


Importance of empty ligand orbitals

Some important Slater determinants with large CI weights from DMRG(13,29)[m] Upper part: Slaterdeterminants containing an occupied dx2−y2 -double-shell orbital (marked in bold face). Bottom part:Configurations with occupied ligand orbitals (marked in bold face). 2: doubly occupied; a: α-electron; b:β-electron; 0: empty.

CI weightSlater determinant m = 128 m = 1024

b2b222a0a0000000 0000000 a 00000 0.003 252 0.003 991bb2222aa00000000 0000000 a 00000 −0.003 226 −0.003 611222220ab00000000 0000000 a 00000 −0.002 762 −0.003 328ba2222ab00000000 0000000 a 00000 0.002 573 0.003 022b2a222a0b0000000 0000000 a 00000 −0.002 487 −0.003 017202222ab00000000 0000000 a 00000 0.002 405 0.002 716b222a2a0b0000000 0000000 0 0000a 0.010 360 0.011 55822b2a2a0a0000000 0000000 0 b0000 0.009 849 0.011 36622b2a2a0b0000000 0000000 0 a0000 −0.009 532 −0.011 457b2222aab00000000 0000000 0 0000a −0.009 490 −0.010 991a2222baa00000000 0000000 0 0000b −0.009 014 −0.010 017b2b222a0a0000000 0000000 0 0a000 0.008 820 0.010 327


Assessment of CASSCF spin densities

CAS(11,11) CAS(11,12) CAS(11,13) CAS(11,14)

CAS(13,13) CAS(13,14) CAS(13,15) CAS(13,16)

⇒ CASSCF spin densities oscillate around DMRG reference distribution


Outlook: DMRG Spin Densities for (Full) Complexes

Spin density of [Fe(NO)salen] complex with DMRG(13,43)[2048]

K. Boguslawski, O Legeza, M. Reiher, in preparation


Entanglement measuresand

electron correlation effects


Electron correlation effects

In quantum chemistry, electron correlation effects are usually divided into3 contributions:

nondynamic (account for degeneracies, required for correctdissociation)static (strong, account for degeneracies)dynamic (weak, account for electron cusp)

Diagnostic tools to characterize single- and multireference correlationeffects:

C0 coefficientT1, D1 and D2 diagnostics for SR-CCConcepts from quantum information theory (von Neumann entropy) interms of one- or two-particle reduced density matricesdistribution of effectively unpaired electrons (radical character)

Different approach: Employ knowledge from many-particle densitymatrices to measure interaction among orbitals directly


Entanglement measures

Single-orbital entropy

s(1)i =∑α

ωα,i lnωα,i

ωα,i are the eigenvalues of the one-orbital reduced density matrixρi = TrE |Ψ(ni ,E)〉〈Ψ(ni ,E)| (of orbital i)

Measures the entanglement of orbital i with the environment EO. Legeza, J. Solyom, Phys. Rev. B 2003, 68, 195116.

Mutual information

Iij =12

(s(2)i,j − s(1)i − s(1)j )(1− δij )

s(2)i,j is the two-orbital entropy determined from the eigenvalues of the two-orbitalreduced density matrix ρi,j = TrE |Ψ(ni ,nj ,E)〉〈Ψ(ni ,nj ,E)|Measures the interaction of orbitals i and j embedded in the environment EJ. Rissler, R.M. Noack, S.R. White, Chem. Phys. 2006, 323, 519.

Applied to increase convergence and optimize orbital ordering

⇒ Can we draw further insights from a quantum chemical point of view?


Entanglement and orbitals—mutual information

1

3

5

7

9

11

1315

17

19

21

23

25

27

29

2

4

6

8

10

12

1416

18

20

22

24

26

28

A

100

10−1

10−2

10−3

Three blocks of orbitals⇒ high entanglement⇒ medium entanglement⇒ weak entanglement

Strong interaction for pair correlations:

(d , π∗)⇐⇒ (d , π∗)∗

π ⇐⇒ π∗

σMetal ⇐⇒ σLigand

⇒ Important for static and nondynamiccorrelation (⇐⇒ chemical intuition ofconstructing a CAS)

zyx

Fe

N

O

�

dpcdpc

dpcdpc

4 point charges in xy -plane at dpc = 1.133 A

Natural orbital basis: CAS(11,14)SCF/cc-pVTZ

DMRG(13,29) with DBSS (mmin = 128,mmax = 1024)


Entanglement and orbitals—single orbital entropy

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1

)

Three blocks of orbitals⇒ large single orbital entropy⇒ medium single orbital entropy⇒ (very) small single orbitalentropy

Configurations with occupiedorbitals belonging to the thirdblock have small CI coefficients

⇒ Important for dynamic correlation

Orbital index 29 28 27 26 25 24 23s(1) 0.0188 0.0041 0.0044 0.0119 0.0164 0.0051 0.0019

Orbital index 22 21 20 19 18 17 16s(1) 0.0042 0.0068 0.0058 0.0046 0.0019 0.0055 0.0019

K. Boguslawski, P. Tecmer, O. Legeza, M. Reiher, J. Phys. Chem. Lett. 2012, 3, 3129.


Entanglement and orbitals—a ligated iron nitrosyl complex

1

3

5

7

9

11

13

15

1719

21

23

25

27

29

31

3335

2

4

6

8

10

12

14

161820

22

24

26

28

30

32

34

A’

A’’

100

10−1

10−2

10−3

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1

)

O

NFe O

N

N

O

BP86/TZP/Cs

Single orbital entropies and mutual information for aDMRG(21,35)[128,1024,10−5] calculation

Three groups of orbitals classified by their combinedIi,j and s(1)i contribution (nondynamic, static anddynamic)

Increasing dynamic and decreasing static electroncorrelation effects


Artifacts of small active space calculations

Comparison of DMRG(11,9)[220] to DMRG(13,29)[128,1024,10−5] for [Fe(NO)]2+

1

3

5

7

9

2

4

6

8

A

100

10−1

10−2

10−3

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1

)

1

3

5

7

9

11

1315

17

19

21

23

25

27

29

2

4

6

8

10

12

1416

18

20

22

24

26

28

A

100

10−1

10−2

10−3

Iij and s(1)i are overestimated forsmall active spaces

⇒ Too large entanglement betweenorbitals

⇒ Missing dynamic correlation iscaptured in an artificial way



chemical bonding


The chemical bond

The chemical bond is a basic concept in chemistry

⇒ understanding of bonding can serve as a guide to synthesis

The chemical bond is not a quantum chemical observable!

Quantum chemistry uses a variety of analysis tools for a qualitativeunderstanding of electronic wave functions

Extract local quantities from quantum states:

Local spin concept (study spin–spin interactions)

(Effective) bond order

Entanglement-based approaches:

Orbital communication theory (OCT) by R. Nalewaiski et al.Extract bond orders from entanglement analysis


Entanglement and bonding: dissociation of N2

DMRG(10,46)[512,1024,10−5] calculations for r = {2.1, 3.2, 4.4} bohr

46

8

1

3

28

26

24

31

12

101520

19

17

33

35

37

39

41

43

2245

57

9

2

29

27

25

30

3211

131421

16

18

34

36

38

40

42

4423 46

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10−1

10−2

10−3

46

8

1

3

28

26

24

31

12

101520

19

17

33

35

37

39

41

43

2245

57

9

2

29

27

25

30

3211

131421

16

18

34

36

38

40

42

4423 46

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10−1

10−2

10−3

46

8

1

3

28

26

24

31

12

101520

19

17

33

35

37

39

41

43

2245

57

9

2

29

27

25

30

3211

131421

16

18

34

36

38

40

42

4423 46

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10−1

10−2

10−3

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

0.3

Orbital index

s(1

)

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Orbital index

s(1

)

0 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2

1.4

Orbital index

s(1

)

Resolve bond-breaking processes of individual σ-, π-, and δ-bonds inmulti-bonded centersExtract formal bond order from s(1)i diagram in the dissociation limit


Entanglement and bonding: CxHy

Single-orbital entropies for C2H2,4,6 approaching the dissociation limit:

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

Au

Bu

Bg

100

10!1

10!2

10!3

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1)

#17: 2p⇤�

#3: 2p�

a) 0.8 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

Au

Bu

Bg

100

10!1

10!2

10!3

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1)

#17: 2p⇤�

#3: 2p�

b) 1.0 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

Au

Bu

Bg

100

10!1

10!2

10!3

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1)

#17: 2p⇤�

#3: 2p�

c) 1.5 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

Au

Bu

Bg

100

10!1

10!2

10!3

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1)

#17: 2p⇤�

#3: 2p�

d) 2.0 re

1

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Orbital index

s(1)

#9: 2p⇤�

#25: 2p⇤⇡

#22: 2p⇡

#2: 2p�a) 0.8 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Orbital index

s(1)

#9: 2p⇤�

#25: 2p⇤⇡

#22: 2p⇡

#2: 2p�b) 1.0 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Orbital index

s(1)

#9: 2p⇤�

#25: 2p⇤⇡

#22: 2p⇡

#2: 2p�c) 1.5 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Orbital index

s(1)

#9: 2p⇤�

#25: 2p⇤⇡

#22: 2p⇡

#2: 2p�d) 2.0 re

1

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Orbital index

s(1

)

#16: 2p⇤�

#22/#25: 2p⇤⇡

#8/#11: 2p⇡

#2: 2p�a) 0.8 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Orbital index

s(1

)

#16: 2p⇤�

#22/#25: 2p⇤⇡

#8/#11: 2p⇡

#2: 2p�b) 1.0 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Orbital index

s(1

)

#16: 2p⇤�

#22/#25: 2p⇤⇡

#8/#11: 2p⇡

#2: 2p�c) 1.5 re

Mutual Information

1

3

5

7

9

11

1315

17

19

21

23

25

27

2

4

6

8

10

121416

18

20

22

24

26 28

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10!1

10!2

10!3

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Orbital index

s(1

)

#16: 2p⇤�

#22/#25: 2p⇤⇡

#8/#11: 2p⇡

#2: 2p�d) 2.0 re

1

H3C–CH3

single bondσ

H2C=CH2

double bondσ, π

HC≡CHtriple bondσ, π, π



weak interaction


Mysterious spin-state splitting of CUO

CUONg4 (Ng = Ne, Ar)

U

C

Ng NgNg Ng

O

J. Li, B. E. Bursten, B. Liang and L. Andrews, Science,2002, 295, 2242–2245.

Experimental anticipation:The spin state of CUO changes(1Σ+ → 3Φ) if Ng matrix ischanged from Ne to ArSo far, quantum chemistry wasunable to resolve this problem!

spin state splitting requiresmultireference methodsHartree–Fock and DFTincorrectly predicts theground state of the CUO

P. Tecmer, H. van Lingen, A. S. P. Gomes and L. Visscher, J. Chem.Phys., 2012, 137, 084308.


DMRG spin state splittings and potential energy surfaces

(i) CUONe4 (ii) CUOAr4

(a) Potential energy surfaces

(i) CUONe4 (ii) CUOAr4

(b) Spin state splittings

P. Tecmer, K. Boguslawski, O. Legeza and M. Reiher, Phys. Chem. Chem. Phys., 2013 (DOI: 10.1039/C3CP53975J).


Entanglement and weak interaction: CUONg4, Ng = Ne, ArDMRG(38,36)[2048,512,2048,10−5] calculations at r = re

1

3

15

23

25

33

4

6

810

12

17

19

21

26

28

30

35

2

14

16

24

32

34

5

79

11

13

18

20

22

27

29

3136

A1

B1

B2

A2

10−1

10−2

10−3

10−4

10−5

(i) 1Σ+

1

3

15

23

25

33

4

6

810

12

17

19

21

26

28

30

35

2

14

16

24

32

34

5

79

11

13

18

20

22

27

29

3136

A1

B1

B2

A2

10−1

10−2

10−3

10−4

10−5

(ii) 3Φ(v)

1

3

15

23

25

33

4

6

810

12

17

19

21

26

28

30

35

2

14

16

24

32

34

5

79

11

13

18

20

22

27

29

3136

A1

B1

B2

A2

10−1

10−2

10−3

10−4

10−5

(iii) 3Φ(a)

(a) CUONe4

1

3

15

23

25

33

4

6

810

12

17

19

21

26

28

30

35

2

14

16

24

32

34

5

79

11

13

18

20

22

27

29

3136

A1

B1

B2

A2

10−1

10−2

10−3

10−4

10−5

(i) 1Σ+

1

3

15

23

25

33

4

6

810

12

17

19

21

26

28

30

35

2

14

16

24

32

34

5

79

11

13

18

20

22

27

29

3136

A1

B1

B2

A2

10−1

10−2

10−3

10−4

10−5

(ii) 3Φ(v)

1

3

15

23

25

33

4

6

810

12

17

19

21

26

28

30

35

2

14

16

24

32

34

5

79

11

13

18

20

22

27

29

3136

A1

B1

B2

A2

10−1

10−2

10−3

10−4

10−5

(iii) 3Φ(a)

(b) CUOAr4

Mutual information Decay of mutual information

Entanglement measures elucidate different interactionstrengths/complexation energies for CUO embedded in a Ng4

surroundingP. Tecmer, K. Boguslawski, O. Legeza and M. Reiher, Phys. Chem. Chem. Phys., 2013 (DOI: 10.1039/C3CP53975J).


Conclusion and Outlook

DMRG or tensor network ansatze can be applied to transition metal,(lanthanide) and actinide compounds where large active spaces arerequired

But: need to define an active space for practical calculations⇒ Use geminal-based wave function forms where no active spaces needto be defined

Entanglement measures are a useful tool to interpret electronicstructures:

Dissect electron correlation effectsAnalysis of chemical bondingResolving weak interactions. . .


Acknowledgments

Markus Reiher

Ors Legeza

Paweł Tecmer

Christoph R. Jacob

Konrad H. Marti

Matthieu Mottet

Paul W. Ayers

Thank you for listening!


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How quantum entanglement can promote [1ex] the ...cassam/Workshop13/Presentations/boguslawski.pdf · How quantum entanglement can promote the understanding of electronic structures