How quantum entanglement can promote [1ex] the...
Transcript of How quantum entanglement can promote [1ex] the...
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
Entanglement and electronic structure Katharina Boguslawski
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”
Entanglement and electronic structure Katharina Boguslawski
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
Entanglement and electronic structure Katharina Boguslawski
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〉
Entanglement and electronic structure Katharina Boguslawski
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?
Entanglement and electronic structure Katharina Boguslawski
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.
Entanglement and electronic structure Katharina Boguslawski
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
Entanglement and electronic structure Katharina Boguslawski
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.
Entanglement and electronic structure Katharina Boguslawski
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
Entanglement and electronic structure Katharina Boguslawski
Reference spin densitiesfrom standard electron correlation methods
Entanglement and electronic structure Katharina Boguslawski
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.
Entanglement and electronic structure Katharina Boguslawski
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.
Entanglement and electronic structure Katharina Boguslawski
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.
Entanglement and electronic structure Katharina Boguslawski
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)
Entanglement and electronic structure Katharina Boguslawski
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
Entanglement and electronic structure Katharina Boguslawski
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
Entanglement and electronic structure Katharina Boguslawski
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 and electronic structure Katharina Boguslawski
Entanglement measuresand
electron correlation effects
Entanglement and electronic structure Katharina Boguslawski
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 and electronic structure Katharina Boguslawski
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 electronic structure Katharina Boguslawski
Entanglement and orbitals—mutual information
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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 electronic structure Katharina Boguslawski
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
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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 electronic structure Katharina Boguslawski
Entanglement and orbitals—a ligated iron nitrosyl complex
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3335
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A’’
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Orbital index
s(1
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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
Entanglement and electronic structure Katharina Boguslawski
Artifacts of small active space calculations
Comparison of DMRG(11,9)[220] to DMRG(13,29)[128,1024,10−5] for [Fe(NO)]2+
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0 2 4 6 8 100
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Orbital index
s(1
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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
Entanglement and electronic structure Katharina Boguslawski
Entanglement measuresand
chemical bonding
Entanglement and electronic structure Katharina Boguslawski
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 electronic structure Katharina Boguslawski
Entanglement and bonding: dissociation of N2
DMRG(10,46)[512,1024,10−5] calculations for r = {2.1, 3.2, 4.4} bohr
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101520
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3211
131421
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4423 46
Ag
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B1g
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Au
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10−2
10−3
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101520
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131421
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4423 46
Ag
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B1g
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Ag
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Au
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0 10 20 30 400
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Orbital index
s(1
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0 10 20 30 400
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Orbital index
s(1
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0 10 20 30 400
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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 electronic structure Katharina Boguslawski
Entanglement and bonding: CxHy
Single-orbital entropies for C2H2,4,6 approaching the dissociation limit:
Mutual Information
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Orbital index
s(1)
#17: 2p⇤�
#3: 2p�
a) 0.8 re
Mutual Information
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Orbital index
s(1)
#17: 2p⇤�
#3: 2p�
b) 1.0 re
Mutual Information
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Orbital index
s(1)
#17: 2p⇤�
#3: 2p�
c) 1.5 re
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Orbital index
s(1)
#17: 2p⇤�
#3: 2p�
d) 2.0 re
1
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Orbital index
s(1)
#9: 2p⇤�
#25: 2p⇤⇡
#22: 2p⇡
#2: 2p�a) 0.8 re
Mutual Information
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Ag
B3u
B2u
B1g
B1u
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B3g
Au
100
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10!2
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Orbital index
s(1)
#9: 2p⇤�
#25: 2p⇤⇡
#22: 2p⇡
#2: 2p�b) 1.0 re
Mutual Information
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Ag
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Orbital index
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#9: 2p⇤�
#25: 2p⇤⇡
#22: 2p⇡
#2: 2p�c) 1.5 re
Mutual Information
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B2u
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Orbital index
s(1)
#9: 2p⇤�
#25: 2p⇤⇡
#22: 2p⇡
#2: 2p�d) 2.0 re
1
Mutual Information
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Ag
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B2u
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Orbital index
s(1
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#16: 2p⇤�
#22/#25: 2p⇤⇡
#8/#11: 2p⇡
#2: 2p�a) 0.8 re
Mutual Information
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Orbital index
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#16: 2p⇤�
#22/#25: 2p⇤⇡
#8/#11: 2p⇡
#2: 2p�b) 1.0 re
Mutual Information
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#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
Entanglement and electronic structure Katharina Boguslawski
Entanglement measuresand
weak interaction
Entanglement and electronic structure Katharina Boguslawski
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.
Entanglement and electronic structure Katharina Boguslawski
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 electronic structure Katharina Boguslawski
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).
Entanglement and electronic structure Katharina Boguslawski
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. . .
Entanglement and electronic structure Katharina Boguslawski