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
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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
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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
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Outline
1 Introduction
2 Density Matrices
3 DNA
4 Conjugated Polymers
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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.
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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
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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
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Motivation
I Description of electronic excitations- Kohn-Sham orbitals- TDDFT response vector
? Physical meaning
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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
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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| ↵âν |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 orbitals↵, âν Creation and annihilation operators
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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
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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 ...
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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
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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
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Attachment/Detachment Densities
Adenine, ADC(2)/cc-pVDZ
↑ ↑ ↑
nπ∗ state ππ∗(Lb) state ππ∗(La) state
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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| ↵âν |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
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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
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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
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Transition Density Matrix
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
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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
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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
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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
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Natural Transition Orbitals
I S1 state- Locally excited state (La)- Only one important configuration
Hole Electron
90%
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Natural Transition Orbitals
I S2 state- Locally excited state (Lb)- Two important configurations
77%
17%
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Natural Transition Orbitals
I S5 state- Delocalized state- One configuration for every involved base
39%
23%
23%
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Natural Transition Orbitals
I S10 state- Charge transfer state
77%
10%
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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
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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.
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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.
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Charge Transfer Numbers
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
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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
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Statistical Analysis
I Delocalization length? How many fragments contribute to the excitation→ Count the number of non-vanishing ΩAB values
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Statistical Analysis
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
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Statistical Analysis
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
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Charge Transfer Numbers
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
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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
1 J. J. Nogueira, FP, L. González Chem. Sci. 2017, 8, 5682.F. Plasser Wavefunction analysis tools 39 / 50
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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
1 J. J. Nogueira, FP, L. González Chem. Sci. 2017, 8, 5682.F. Plasser Wavefunction analysis tools 40 / 50
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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.
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Exciton Analysis
Wannier excitonsI Hydrogen atom in a boxI Particle-in-a-box statesI Hydrogenic states
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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)
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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)
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Conjugated Polymers
Problems with this analysis
I Results depend on fragmentation scheme chosen
I Plots have to be inspected manuallyI Can we do better?
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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
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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 〈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
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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
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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
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IntroductionDensity MatricesDNAConjugated Polymers
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