bridging scales through wavefunction...

DNA Conj. Poly. Trans. Met. Outlook Software

Bridging Scales Through Wavefunction Analysis

Felix Plasser

Institute for Theoretical Chemistry, University of Vienna

Excited States Bridging ScalesMarseille, November 7–10, 2016

F. Plasser Wavefunction Analysis 1 / 50


Introduction

Excited state quantum chemistry

I Accurate computations- Computational methods:Semi-emp., TDDFT, CC, ADC, CASSCF, DMRG, CASPT2, MR-CI, ...

- Algorithmic efforts: Linear scaling, resolution-of-the-identity, ...- ParallelizationI Analysis and interpretation of the results- Looking at the orbitals



Introduction

Excited states at increasing length scales

New scales - new problemsI More low-lying excited statesI Sampling of geometries

/ More workI Orbitals of mixed characterI Many interacting configurations

/ Analysis becomes ambiguous and affected by personal bias



Introduction

Excited states at increasing length scales

New scales - new physicsI Small molecules- Molecular orbitals contain all relevant information- nπ∗, ππ∗, . . . statesI Interacting chromophores and extended systems- Excitonic effects- Excited state collectivity and correlation



Introduction

How can we understand excited states for large systems?

/ Tedious analysis work/ Ambiguous results/ Challenging physics, Problems can be solved through systematic wavefunction analysis



DNA

Task:Understand the UV absorption of DNA

I Local excitationsI Delocalized excitations - excitonsI Charge transfer states



DNA

QM/MM calculation

I 4 nucleobases in the QM regionI ADC(2) excitation energies- 20 statesI Sampling of intra- and intermolecular

motions- 300 snapshots

/ How do we analyze 6000 excited states?



DNA

I 16 Possibilities- 4 local transitions- 12 charge transfer transitions

T1→A2 A1→A2 T2→A2 A2→A2

T1→T2 A1→T2 T2→T2 A2→T2

T1→A1 A1→A1 T1→A1 A2→A1

T1→T1 A1→T1 T2→T1 A2→T1

I Where can this information be found?→ Transition density matrix



Transition Density Matrix

1-Electron transition density matrix

D0Iµν = 〈Ψ0| a†µaν |ΨI〉

D0Iµν Matrix representation of the 1TDM

Ψ0,ΨI Ground and excited state wavefunctionsa†µ, aν Creation and annihilation operators

I Connection to physical observables through transition properties- Rigorous meaning- Well-defined independent of the computational method- No explicit dependence on the orbitalsI Approximation: CI vector / response vector



Charge Transfer Numbers

I Summation over squared 1TDM elements- For two nucleobases A and BI Correction for non-orthogonality of the AOs

Charge transfer numbers

ΩAB =1

2

∑µ∈A

∑ν∈B

[(D0IS

)µν

(SD0I

)µν

+D0Iµν

(SD0IS

)µν

]

ΩAA Weight of local excitations on nucleobase AΩAB , A 6= B Amount of charge transfer from A to B

1 FP, H. Lischka JCTC 2012, 8, 2777.2 FP, M. Wormit, A. Dreuw JCP 2014, 141, 024106.



Charge Transfer Numbers

T1→A2 A1→A2 T2→A2 A2→A2

T1→T2 A1→T2 T2→T2 A2→T2

T1→A1 A1→A1 T1→A1 A2→A1

T1→T1 A1→T1 T2→T1 A2→T1

hole

elec

tron

0.00.10.20.30.40.50.60.70.80.9

ΩAB - pseudocolor matrix plots

S1 (4.42 eV) S2 (4.46 eV) S3 (4.56 eV)

S4 (4.74 eV) S5 (4.82 eV) S6 (4.99 eV)



DNA

Additional steps:

I Extract the essential information and classify the states- Charge transfer character- DelocalizationI Do this for all 300 geometries × 20 statesI Decompose the absorption spectrum into different classes of states



DNA

UV absorption spectrum

I Black: full spectrumI Red: CT statesI Blue: deloc. at least 1.5 basesI Green: deloc. at least 2.5 bases

1 FP, A. J. A. Aquino, W. L. Hase, H. Lischka JPCA 2012, 116, 11151.F. Plasser Wavefunction Analysis 15 / 50


Conjugated Polymers

I Poly(para phenylene vinylene)I ADC(2)/SV(P)I Cut into pieces (formally)

→ Same analysis as before

1 A. Panda, FP, A. J. A. Aquino, I. Burghardt, H. Lischka JPCA 2013, 117, 2181.2 S. A. Mewes, J.-M. Mewes, A. Dreuw, FP PCCP 2016, 18,2548.



Exciton Analysis

Wannier excitonsI Hydrogen atom in a boxI Particle-in-a-box statesI Hydrogenic states



Exciton Analysis

Wannier excitons - singlet

11Bu - W(1,1) 21Ag - W(1,2) 21Bu - W(1,3) 31Ag - W(1,4) 71Bu - W(1,5) 101Ag - W(1,6)

41Ag - W(2,1) 31Bu - W(2,2) 81Ag - W(2,3) 91Bu - W(2,4) 111Ag - W(2,5)

101Bu - W(3,1)

Singlet

13Bu - W(1,1) 33Ag - W(1,6)

43Bu - W(1,7)

93Bu - W(2,2)

Triplet

13Ag - W(1,2)

43Ag - W(1,8)

23Ag - W(1,4) 33Bu - W(1,5) 23Bu - W(1,3)

53Ag - W(2,1)

63Ag - W(1,10) 53Bu - W(1,9)



Exciton Analysis

Wannier excitons - triplet

11Bu - W(1,1) 21Ag - W(1,2) 21Bu - W(1,3) 31Ag - W(1,4) 71Bu - W(1,5) 101Ag - W(1,6)

41Ag - W(2,1) 31Bu - W(2,2) 81Ag - W(2,3) 91Bu - W(2,4) 111Ag - W(2,5)

101Bu - W(3,1)

Singlet

13Bu - W(1,1) 33Ag - W(1,6)

43Bu - W(1,7)

93Bu - W(2,2)

Triplet

13Ag - W(1,2)

43Ag - W(1,8)

23Ag - W(1,4) 33Bu - W(1,5) 23Bu - W(1,3)

53Ag - W(2,1)

63Ag - W(1,10) 53Bu - W(1,9)



Conjugated Polymers

Problems with this analysis

I Results depend on fragmentation scheme chosen

I Plots have to be inspected manuallyI Can we do better?



Transition Density Matrix

I Coordinate representation of the 1TDM

1-Electron transition density matrix (1TDM)

γ0I(xh, xe) = n

∫. . .

∫Ψ0(xh, x2, . . . , xn)ΨI(xe, x2, . . . , xn)dx2 . . . dxn

γ0I(xh, xe) Coordinate representation of the 1TDMxh, xe Coordinates of the excitation hole and excited electron

1TDM in second quantization

γ0I(xh, xe) =∑µν

D0Iµνχµ(xh)χν(xe)

D0Iµν Matrix representation of the 1TDM



Exciton Analysis

Exciton analysisI Interpret the 1TDM as the wavefunction χexc of the electron-hole pairI Use as a basis for analysis

Exciton wavefunction

χexc(xh, xe) =∑µν

D0Iµνχµ(xh)χν(xe)

Operator expectation value ⟨O⟩

=〈χexc| O |χexc〉〈χexc|χexc〉

.

1 S. A. Bäppler, FP, M. Wormit, A. Dreuw Phys. Rev. A 2014, 90, 052521.F. Plasser Wavefunction Analysis 23 / 50


Exciton Analysis

Exciton size

Exciton size

dexc2 =

⟨(re − rh)2

⟩I Average separation of the electron and hole quasi-particlesI Static and dynamic charge transfer effectsI No fragment definition requiredI Evaluation through multipole AO integrals

1 S. A. Bäppler, FP, M. Wormit, A. Dreuw Phys. Rev. A 2014, 90, 052521.F. Plasser Wavefunction Analysis 24 / 50


Conjugated Polymers

I Exciton size / excitation energy- 20 singlet and 20 triplet statescompressed into one plot

I Formation of different Wannierexciton bands

I Clustered Frenkel excitonsI Comparison with size of the

molecule



Conjugated Polymers

I Exciton size with increasing sizeof the system

- n = 2, . . . , 8

I TDDFT/CAM-B3LYPI Exciton size quickly levels offI Orbitals stay delocalizedI How is this possible?

Nr.

of

en

tan

gle

dst

ate

s

1.0

2.0

3.0

4.0

Number of phenyl rings2 3 4 5 6 7 8

Co

rre

lati

on

coe

cie

nt

0.0

0.2

0.4

0.6

0.8

1.0

Exc

ito

nsi

ze(Å

)4.0

5.0

6.0

7.02 3 4 5 6 7 8

Singlet Triplet

(b)

(a)

(c)

1 FP JCP 2016, 144, 194107.F. Plasser Wavefunction Analysis 26 / 50


Exciton Analysis

I Quantify correlations between the electron and hole in analogy to Pearson’scorrelation coefficient

Correlation coefficient

Reh =〈rh · re〉 − 〈rh〉 · 〈re〉

σhσe

−1 ≤ Reh ≤ 1

Reh = 0 No correlationReh > 0 Positive correlation - exciton bindingReh < 0 Negative correlation - dynamic repulsion



Conjugated Polymers

I Exciton size and correlationcoefficient

- n = 2, . . . , 8

I Correlation coefficient goes up asexciton size levels off

I Correlation in TDDFT?N

r.o

fe

nta

ng

led

sta

tes

1.0

2.0

3.0

4.0


Co

rre

lati

on

coe

cie

nt

0.0

0.2

0.4

0.6

0.8

1.0

Exc

ito

nsi

ze(Å

)

4.0

5.0

6.0

7.02 3 4 5 6 7 8

Singlet Triplet

(b)

(a)

(c)



Natural Transition Orbitals

Singular value decomposition of the 1TDM

Natural transition orbitals

D0I = U× diag(√

λ1,√λ2, . . .

)×VT

U Hole orbital coefficientsλi Transition amplitudesV Electron orbital coefficients

I Compact representation of the excitation

1 R. L. Martin J. Chem. Phys. 2003, 11, 4775.F. Plasser Wavefunction Analysis 29 / 50


Conjugated Polymers

I NTOs of the S1 stateHole Particle

λ1 = 0.703 λ1 = 0.703

λ2 = 0.192 λ2 = 0.192

λ3 = 0.061 λ3 = 0.061



Conjugated Polymers

I NTOs of the T1 state

Hole Particle

λ1 = 0.573 λ1 = 0.573

λ2 = 0.208 λ2 = 0.208

λ3 = 0.090 λ3 = 0.090

λ4 = 0.045 λ4 = 0.045

I Look similar to S1 but different singular values



Collectivity

I Interpretation of the NTO singular value spectrum within quantuminformation theory

Electron-hole entanglement entropy

SH|E = −∑i

λi log2 λi

Number of entangled states

ZHE = 2SH|E = 1/∏i

λλii

ZHE Number of configurations involved

1 FP J. Chem. Phys. 2016, 144, 194107.F. Plasser Wavefunction Analysis 32 / 50


Conjugated Polymers

I Exciton size, correlationcoefficient, number of entangledstates

I Correlation due tomulticonfigurational characterof the excited state

Nr.

of

en

tan

gle

dst

ate

s

1.0

2.0

3.0

4.0


Co

rre

lati

on

coe

cie

nt

0.0

0.2

0.4

0.6

0.8

1.0

Exc

ito

nsi

ze(Å

)

4.0

5.0

6.0

7.02 3 4 5 6 7 8

Singlet Triplet

(b)

(a)

(c)



Natural Transition Orbitals

Classification of excited states

λ1 = 1 Simple transition between two orbitalsλ1 ≈ . . . ≈ λk ≈ 1/k Collective single-electron excited state∑

i λi 1 Multiple excitation



Conjugated Polymers

I Next step- More molecules- More functionals

S S

S S

S H H

HC H

H H

H

H

H N N

N N

O O

H

H

H

H

(a) (b) (c)

(d) (e) (f)

n

n

n

n

1 S. Kraner, R. Scholz, FP, C. Koerner, K. Leo JCP 2015, 143, 244905.2S. A. Mewes, FP, A. Dreuw, in preparation



Conjugated Polymers

I Plot exciton size /molecular size

I Universal trends amongdifferent conjugatedpolymers

I Strong difference amongfunctionals

- Not only energies butoverall description

- More exchange → strongexciton binding

1S. A. Mewes, FP, A. Dreuw, in preparationF. Plasser Wavefunction Analysis 36 / 50


Conjugated Polymers

I Plot correlation coefficient/ molecular size

I Negative correlation forlocal PBE functional

I Weak positive correlationfor global hybridsB3LYP and PBE0

I Strong positive correlationfor range-separated,M06-2X, and ADC(2)

- ADC(2) ≈ CAM-B3LYP- 100% long-rangeexchange overshoots

1S. A. Mewes, FP, A. Dreuw, in preparationF. Plasser Wavefunction Analysis 37 / 50


Outlook

Transition metal complexes

I Division into central metal (M) andligands (L1, L2, L3)

I 1TDM blocks naturally correspond totypes of states

I MC, MLCT, LMCT, ...

, Automatic assignment, No problems due to mixed orbital

characters

DI0

MLCT

LC

LC

LC

LMCT

LLCT

LLCT

M

M

L1 L2 L3

L1

L2

L3

Electron

Hole

MC

1 FP, A. Dreuw JPCA 2015, 119, 1023.F. Plasser Wavefunction Analysis 39 / 50


Transition metal complex

Example: iridium complex

CASSCF(12/12) ADC(3)State ∆E LC MLCT LLCT ∆E LC MLCT LLCT21A 3.56 0.12 0.74 0.10 3.91 0.50 0.31 0.0731A 3.93 0.05 0.58 0.19 4.35 0.08 0.50 0.2741A 5.36 0.15 0.47 0.20 4.72 0.11 0.36 0.32

I Easy identification of state characterI Comparison between methods possible



Outlook

I Characterization of intramolecular excitations- Automatic assignment of nπ∗, ππ∗, Rydberg etc. character

Proof-of-principle application1

1 FP, B. Thomitzni, S. A. Bäppler et al. JCC 2015, 36, 1609.F. Plasser Wavefunction Analysis 42 / 50


Outlook

Different physics

Unpaired electrons1

Orbital relaxation2

x Electron correlationx Two-electron excitationsx Plasmons

1 FP, H. Pasalic et al. Angew. Chem., Int. Ed. 2013, 52,2581.2 FP, S. A. Bäppler, M. Wormit, A. Dreuw JCP 2014, 141, 024107.



Outlook

Model buildingI Use energies and wavefunction information for model building→ Better transferability→ Property based diabatization

Application to a charge transfer model system1

Application to DNA2

x Application to conjugated polymers

1 FP, H. Lischka JCP 2011, 134, 034309.2 A. A. Voityuk JCP 2014, 140, 244117.



I Charge transfer model system- Ethylene dimer cation- Fragment charge differences(FCD)

I Nonadiabatic couplingscomputed from FCDs

, Property based diabatizationworks

1 FP, H. Lischka JCP 2011, 134, 034309.F. Plasser Wavefunction Analysis 45 / 50


TheoDORE

TheoDORE - Theoretical Density, Orbital Relaxation and Exciton analysis

I Program package for wavefunction analysisI Interfaces to various quantum chemistry programs:

Columbus, Molcas, Turbomole, Orca, GAMESS, Gaussian, ...I Open-source: http://theodore-qc.sourceforge.net

I Analysis functionalitiesI Enhanced post-processing and plotting capabilitiesI Utility functions


http://theodore-qc.sourceforge.net


libwfa

libwfa - An open-source wavefunction analysis tool library1

I State/transition/difference DM analysis methodsI Orbitals + densitiesI Different population analysesI Multipole analysis in coordinate space

I Available in Q-Chem: TDDFT, EOM-CC, ADCI Interface to MOLCAS in progress: CASSCF, CASPT2I Interface to COLUMBUS will come soon...

1https://github.com/libwfa/libwfaF. Plasser Wavefunction Analysis 48 / 50


Conclusions

Analysis of electronic wavefunctions - why?

1 Make things easierI Compact orbital representationsI Automatization

2 Give specific quantitative resultsI Charge transfer characterI DelocalizationI Double excitation character

3 Provide new physical insightI Exciton correlationI Orbital relaxation

There is more to wavefunctions than meets the eye!



Acknowledgements

HeidelbergS. A. MewesM. WormitA. Dreuw

Vienna/Lubbock/TianjinH. Lischka

FrankfurtI. Burghardt


bridging scales through wavefunction...

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