Talk1: Wavefunctionanalysistoolsforenergyandelectron transfer · F. Plasser Wavefunction analysis...

Intro Density Matrices DNA Conj. Poly.

Talk 1:Wavefunction analysis tools for energy and electron

transfer

Felix Plasser

Institute for Theoretical Chemistry, University of Vienna

Helsinki, 18 December 2017

F. Plasser Wavefunction analysis tools 1 / 50


Introduction

I Excitation energy transfer? Where is the excitation- Localization on which chromophore- Delocalization

I Electron transfer- Charge transfer excited-states- Partial charge transfer

I Formal and practical questions



Goals

GoalsI Formal definition of excited-state localization and charge transfer using- Ground-state wavefunction Ψ0- Excited-state wavefunction ΨI

I Development of practical analysis methods



Outline

1 Introduction

2 Density Matrices

3 DNA

4 Conjugated Polymers



DNA

Photophysics of interacting nucleobases? What happens after DNA is excited by

UV lightI Energy transfer1

I Electron transfer and exciplex formation2

Starting point: UV absorptionI Localized/delocalized excitationsI Charge transfer states

1 D. Onidas, T. Gustavsson, E. Lazzarotto, D. Markovitsi PCCP 2007, 9, 5143.2 T. Takaya, C. Su, K. De La Harpe, C. E. Crespo-H., B. Kohler PNAS 2008, 105, 10285.



DNA

Polyadenine (single stranded)I Time-dependent density functional theory→ Excitation energiesI Multiscale QM/MM calculation- 8 nuclebases in the QM regionI GPU-based Terachem codeI 100 MD snapshots × 60 excited states

/ How do we analyze 6000 excited states?

1 J. J. Nogueira, FP, L. González Chem. Sci. 2017, 8, 5682.F. Plasser Wavefunction analysis tools 7 / 50


DNA

Leading configurationsI S1 state

H-2 → L+1 (-0.70)H-2 → L (-0.47)H-2 → L+10 (0.29)

I S2 stateH-2 → L+10 (-0.45)H-2 → L+1 (-0.40)H-2 → L+2 (0.35)

/ Tedious work/ Possible ambiguities

Canonical orbitals

HOMO L (LUMO)

H-2 L+1

L+2

L+10



Motivation

I Description of electronic excitations- Kohn-Sham orbitals- TDDFT response vector

? Physical meaning



Many-electron wavefunctions

Starting point:

Time-independent Schrödinger Equation

ĤΨ0(x1, x2, . . .) = E0Ψ0(x1, x2, . . .)

ĤΨI(x1, x2, . . .) = EIΨI(x1, x2, . . .)

I How to describe the many-electron function Ψ0(x1, x2, . . .)?I How to discuss changes between Ψ0(x1, x2, . . .) and ΨI(x1, x2, . . .)?

, Complexity reduction through reduced density matrices



Density Matrices

Integrate out all the coordinates except for one

1-Electron reduced density matrix (1DM)

γ(x, x′) = n

∫. . .

∫Ψ(x, x2, . . . , xn)Ψ(x

′, x2, . . . , xn)dx2 . . . dxn

1DM in second quantization

Dµν = 〈Ψ| â†µâν |Ψ〉

γ(x, x′) =∑µν

Dµνχµ(x)χν(x′)

γ(x, x′) Coordinate representation of the 1DMDpq Matrix representation of the 1DM

χµ, χν Atomic orbitalsâ†µ, âν Creation and annihilation operators



Density Matrix

Physical meaning

Expectation value of a 1-electron operator

〈Ψ| Ô1 |Ψ〉 = n∫. . .

∫Ψ(x1, x2, . . . , xn)Ô1Ψ(x1, x2, . . . , xn)dx1dx2 . . . dxn

Dipole moment, angular momentum, kinetic energy, ...

Using the 1DM

〈Ψ| Ô1 |Ψ〉 =∑µν

Dµν

∫χµ(x1)Ô1χν(x1)dx1

I Many-electron integral reduced to a summation over 1-electron integralsI All wavefunction-specific information contained in the 1DM→ Logical starting point for wavefunction analysis



Electron Density

Electron Density

ρ(x) = γ(x, x)

ρ(x) = n

∫. . .

∫Ψ(x, x2, . . . , xn)

2dx2 . . . dxn

I Description of the overall electron distribution

Adenine

Ground state nπ∗ state ππ∗(Lb) state ππ∗(La) state

/ Not really helpful ...



Difference Density Matrices

Subtract the density matrices of the ground state and excited state

1-Electron difference density matrix (1DDM)

∆0I = DII −D00

Adenine

nπ∗ state ππ∗(Lb) state ππ∗(La) state

I Physical meaning: changes in one-electron properties during theexcitation

/ Still not very intuitive



Attachment/detachment analysis

Natural difference orbitals

DII −D00 = W × diag (κ1, κ2, . . .)×WT

W Natural difference orbital coefficientsκi < 0 Detachment eigenvalues, diκi > 0 Attachment eigenvalues, ai

I Weighted sum over all positive (negative) eigenvalues leads to theattachment (detachment) densities1

1 M. Head-Gordon, A. M. Grana, D. Maurice, C. A. White J. Chem. Phys. 1995, 99, 14261.F. Plasser Wavefunction analysis tools 16 / 50


Attachment/Detachment Densities

Adenine, ADC(2)/cc-pVDZ

↑ ↑ ↑

nπ∗ state ππ∗(Lb) state ππ∗(La) state



Transition Density Matrices

Comparison of two wavefunctions Ψ0 and ΨI

1-Electron transition density matrix (1TDM)

γ0I(xh, xe) = n

∫. . .

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

1TDM in second quantization

D0Iµν = 〈Ψ0| â†µâν |ΨI〉

γ0I(xh, xe) =∑µν

D0Iµνχµ(xh)χν(xe)

γ0I(xh, xe) Coordinate representation of the 1TDMD0Iµν Matrix representation of the 1TDM

xh, xe Coordinates of the excitation hole and excited electron



Transition Density Matrices

Electron/hole picture

Ψ0(x1, x2, . . . , xn) ΨI(x1, x2, . . . , xn)

I “Subtract” the ground state Ψ0(x1, x2, . . . , xn)

Fermi vacuum γ0I(xh, xe)

! Still two effective particlesF. Plasser Wavefunction analysis tools 19 / 50


Transition Density Matrix

Physical meaning

Transition property of a 1-electron operator

〈Ψ0| Ô1 |ΨI〉 =∑µν

D0Iµν 〈χµ| Ô1 |χν〉

I Transition moments, oscillator strengthI Spin-orbit coupling (mean field approximation)

I Interaction with lightI Coulomb coupling with other chromophores




TDDFTI No wavefunctions → no 1TDMs→ Assignment problem of TDDFTI But: 1TDM can be reconstructed from the oscillator strength1

→ Use the following form in terms of the excitation Xia and de-excitation Yiaamplitudes2

1TDM

γ0I(xh, xe) =

occ∑i

virt∑a

[Xiaφi(xh)φa(xe) + Yiaφa(xh)φi(xe)]

1M. E. Casida (1995). In D. P. Chong (Ed.), Recent advances in density functional methods,Part I (pp. 155–192).

2 S. A. Mewes, F. Plasser, A. Dreuw JCP 2015, 143, 171101.F. Plasser Wavefunction analysis tools 21 / 50


Transition Density

Transition Density

ρ0I(x) = γ0I(x, x)

Adenine

nπ∗ state ππ∗(Lb) state ππ∗(La) stateI Physical meaning: transition moments→ Interaction with light→ Coulomb coupling for energy transfer/ No intuitive interpretation/ Disappears for charge transfer states



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 excitationI Independent description of hole and electronI Important for large systems, large basis sets etc.

1 R. L. Martin J. Chem. Phys. 2003, 11, 4775.F. Plasser Wavefunction analysis tools 23 / 50


DNA

Leading configurationsI S1 state

H-2 → L+1 (-0.70)H-2 → L (-0.47)H-2 → L+10 (0.29)

I S2 stateH-2 → L+10 (-0.45)H-2 → L+1 (-0.40)H-2 → L+2 (0.35)

? Are all these basesinvolved

? Are all these orbitalsinvolved

Canonical orbitals

HOMO L (LUMO)

H-2 L+1

L+2

L+10




I S1 state- Locally excited state (La)- Only one important configuration

Hole Electron

90%




I S2 state- Locally excited state (Lb)- Two important configurations

77%

17%




I S5 state- Delocalized state- One configuration for every involved base

39%

23%

23%




I S10 state- Charge transfer state

77%

10%



DNA

I Excited states in multichromophoric systems

Locally excited

state

Charge-transfer

stateFrenkel exciton

(coupled local

excitations)

Where the excitation comes from - "hole"

Where the excitation goes to - "electron"

Charge resonance

state

(coupled CT states)

I Connection between electron and hole decisive! 2-dimensional picture



Quantitative Description

I 2-dimensional population analysis of the1TDM

I Consider individual adenine moleculesA1, A2, A3, A4, ...

I Locally excited contributions (diagonal)I CT contributions (off-diagonal)

I Charge transfer numbers

Transition density matrix

A1

A1

A2 A3 A4

A2

A3

A4

Hole

Electron

CT

CT

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



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

I Result: small matrix- All local and CT contributions

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




S1 S2 S5 S10

1 2 3 4 5 6 7 812345678

1 2 3 4 5 6 7 812345678

1 2 3 4 5 6 7 812345678

1 2 3 4 5 6 7 812345678

localized localized delocalized charge transferA4 A4 A1-A4 A4→ A3

, Information about localization and CT encoded in a small matrix→ Use for statistical analysis



Statistical Analysis

Charge transfer character

CT = Ω−1∑B 6=A

ΩAB Ω =∑A,B

ΩAB

CT=0 Locally excited state or Frenkel excitonCT=1 Charge transfer or charge resonance state




I Delocalization length? How many fragments contribute to the excitation→ Count the number of non-vanishing ΩAB values




I Counting non-vanishing values

{ni} = {1

k,

1

k, . . . ,

1

k︸︷︷︸k

, 0 . . . , 0}∑i

ni = 1

? What is k

Participation ratio

PR =1∑i n

2i

I Insert

PR =1∑ki=1

1k2

=1

k 1k2= k




Delocalization Length

DL =Ω2∑

A

(∑B

ΩAB+ΩBA2

)2DL=1 Locally excited state (only one molecule involved)DL>1 Delocalized exciton or charge transfer state




S1 S2 S5 S10

1 2 3 4 5 6 7 812345678

1 2 3 4 5 6 7 812345678

1 2 3 4 5 6 7 812345678

1 2 3 4 5 6 7 812345678

localized localized delocalized charge transferA4 A4 A1-A4 A4→ A3

CT = 0.13 CT = 0.16 CT = 0.05 CT = 0.65DL = 1.27 DL = 1.31 DL = 3.46 DL = 2.30

I Always mixed character- Qualitative analysis does not tell the whole truth



Polyadenine

Delocalization length (DL)

I Decomposition of the spectrum- Analysis of 6000 excited states

I Main contribution: DL=2- Nearest neighbor interactionsI Additionally: DL=1, DL=3I No significant contributions > 4



Polyadenine

Charge transfer (CT)

I Local and Frenkel exciton states(CT< 0.1)

→ 51% of the spectral intensity→ CT admixture for remaining

states

I States with significant CTcharacter (CT> 0.3)

→ Low intensity, high energies



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) 2

1Bu - W(1,3) 31Ag - W(1,4) 7

1Bu - W(1,5) 101Ag - W(1,6)

41Ag - W(2,1) 31Bu - W(2,2) 8

1Ag - W(2,3) 91Bu - W(2,4) 11

1Ag - 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) 2

3Bu - 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) 2

1Bu - W(1,3) 31Ag - W(1,4) 7

1Bu - W(1,5) 101Ag - W(1,6)

41Ag - W(2,1) 31Bu - W(2,2) 8

1Ag - W(2,3) 91Bu - W(2,4) 11

1Ag - 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) 2

3Bu - 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?




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


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


Operator expectation value 〈Ô〉

=〈χexc| Ô |χexc〉〈χexc|χexc〉

→ Evaluate using analytic integration techniques.

1 S. A. Bäppler, FP, M. Wormit, A. Dreuw Phys. Rev. A 2014, 90, 052521.F. Plasser Wavefunction analysis tools 48 / 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 tools 49 / 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


IntroductionDensity MatricesDNAConjugated Polymers

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