Phenomenology 1DM 1TDM/1DDM Exciton TheoDORE C
Lecture 3:A closer look into electronic wavefunctions
Felix Plasser
SHARC Workshop
Vienna, October 3rd–7th, 2016
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Phenomenology 1DM 1TDM/1DDM Exciton TheoDORE C
Excited State Phenomenology
I Intramolecular excitationsI Valence states: ππ∗, nπ∗, πσ∗
I Rydberg statesI Intermolecular excitations
I Charge transfer statesI Excitonic states
I One-electron excited states (single excitations)I Two-electron excited states (double excitations)
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Valence states
Valence statesBonding / non-bonding orbital → antibonding orbital
I Excitation within the valence spaceI Bonding → antibonding orbital: ππ∗, πσ∗, σσ∗
I Lone pair → antibonding orbital: nπ∗
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Valence states
ππ∗ statesI π-orbital: nodal plane in the
molecular planeI Typical UV absorbing states
↑
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Valence states
nπ∗ statesI Usually "dark" ↑
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Valence states
πσ∗, σσ∗ statesI Can lead to dissociation of the
molecule↑
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Rydberg states
Rydberg statesI Excitations into diffuse orbitalsI Molecular ion circled by an
electron→ Similar to "Rydberg series" of the
hydrogen atom
Diffuse s-orbital
↑
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Rydberg states
Diffuse p-orbitals
↑ ↑ ↑
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Rydberg states
Rydberg states - in practice
↑ ↑ ↑
Cutoff 0.04 Cutoff 0.03 Cutoff 0.02
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Charge Transfer States
Charge Transfer StatesI Electron transfer between two
chromophoresI Approximate energy
ECT ≈ IP + EA− 1
R
I Strong dependence on theintermolecular separation!
↑
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Excitons
ExcitonsI Excited states in larger systemsI Coupled local excitations- many orbitals involvedI Alternative viewpoint:
electron-hole pair- two-body wavefunctionχexc(xh, xe)
↑ (58%)
↑ (21%)
↑ (9%)
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Wavefunction Analysis
How to move from a computation to a phenomenological description?
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Many-electron wavefunctions
Starting point:
Time-independent Schrödinger Equation
HΨ0(x1, x2, . . .) = E0Ψ0(x1, x2, . . .)
HΨI(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, . . .)?→ Systematic 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| a†µaν |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 orbitalsa†µ, aν Creation and annihilation operators
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Density Matrix
Physical meaning
Expectation value of a 1-electron operator
〈Ψ| O1 |Ψ〉 = n
∫. . .
∫Ψ(x1, x2, . . . , xn)O1Ψ(x1, x2, . . . , xn)dx1dx2 . . . dxn
Dipole moment, angular momentum, kinetic energy, ...
Using the 1DM
〈Ψ| O1 |Ψ〉 =∑µν
Dµν
∫χµ(x1)O1χν(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) stateI Not really helpful ...
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Population Analysis
Next step: Divide the electron density into contributions of different atoms
I There is no unique way to do so!
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Population Analysis
Mulliken population of atom A
n(A) =∑µ∈A
(DS)µµ
S AO-overlap matrix
I Summation over diagonal 1DM elements+ Correction for non-orthonormality of the AOs
Löwdin population of atom A
n(A) =∑µ∈A
(S1/2DS1/2)µµ
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Population Analysis
Many other options existI Atoms in molecules decomposition of the density1
I Hirshfeld chargesComparison with atomic densities
I Natural population analysis2
I Fitting to electrostatic potentialI ...
1 R. F. W. Bader Acc. Chem. Res. 1985, 18,9.2 A. Reed, R. B. Weinstock, F. Weinhold J. Chem. Phys. 1985, 83, 735.
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Population Analysis
I Different states of AdenineI ADC(2)/cc-pVDZI Mulliken charges of the N-atoms
N1 N3 N7 N9
gr. st. -0.28 -0.27 -0.25 -0.17nπ∗ -0.09 -0.23 -0.23 -0.17
ππ∗(Lb) -0.21 -0.21 -0.23 -0.16ππ∗(La) -0.28 -0.27 -0.27 -0.15
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Natural Orbitals
Diagonalization of the 1DM
Natural orbitals
D = T× diag (n1, n2, . . .)×TT
T Natural orbital coefficientsni Occupation numbers
I Closed shell orbitals (ni = 2)I Open shell / active orbitals (0 < ni < 2)I Unoccupied orbitals (ni = 0)
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Natural orbitals
NaII S3 state at 7.0 Å separationI Closed-shell stateI Na+ + I−
Na I
ni = 0.00
ni = 2.00
ni = 2.00
ni = 2.00
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Natural orbitals
NaII S0 state at 7.0 Å separationI Biradical stateI Na ·+ · II Only small covalent interaction
Hint: Use a smaller cutoff value (0.03)
Na I
ni = 0.88
ni = 1.12
ni = 2.00
ni = 2.00
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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| a†µaν |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 Matrix
Physical meaning
Transition property of a 1-electron operator
〈Ψ0| O1 |ΨI〉 =∑µν
D0Iµν 〈χµ| O1 |χν〉
I Transition moments, oscillator strength
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Transition Density
Transition Density
ρ0I(x) = γ0I(x, x)
Adenine
nπ∗ state ππ∗(Lb) state ππ∗(La) state
I Physical meaning: transition moments→ Interaction with lightI But: no intuitive interpretation
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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 Important for large basis sets
1 R. L. Martin J. Chem. Phys. 2003, 11, 4775.F. Plasser Electronic Wavefunctions 31 / 54
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Natural Transition Orbitals
I Adenine: ADC(3)/aug-cc-pVDZI S4 stateI Canonical orbitals, 5 dominant transitions
LUMO+6 LUMO+11 LUMO+13 LUMO+10 LUMO+14↑-0.39 ↑-0.36 ↑+0.21 ↑-0.20 ↑-0.13
I Low energy virtual orbitals are very diffuse (quasi-continuum)
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Natural Transition Orbitals
I Individual orbitals do not have a physical meaningI Canonical orbitals can be misleadingI Decisive: 1TDM γ0I(xh, xe)
I Optimal representation through natural transition orbitals
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Natural Transition Orbitals
I Adenine: ADC(3)/aug-cc-pVDZI Natural transition orbitalsI One dominant transition- Compact valence stateI Diffuse orbitals were only
artifacts!
↑ 97%
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Natural Transition Orbitals
Linear combination of the orbitals
= −0.39× −0.26×
+0.21× −0.20×
−0.13×
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Statistical Analysis
Analysis of the distribution of the hole/electron in space
Hole density
ρh(xh) =
∫γ0I(xh, xe)dxe
(Excess) electron density
ρe(xe) =
∫γ0I(xh, xe)dxh
I Analysis of statistical moments- Centroids- Root-mean-square size σh, σe
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Statistical Analysis
I Adenine, ADC(3)/aug-cc-pVDZI Statistical moments
∆E(eV) Osc. Str. σh(σh,z) σe CharacterS1 5.23 0.138 2.09(0.75) 2.27 ππ∗
S2 5.41 0.142 2.17(0.73) 2.39 ππ∗
S3 5.53 0.010 2.19(0.75) 4.11 RydbergS4 5.62 0.002 1.98(0.43) 2.23 nπ∗
S5 5.90 0.000 2.20(0.75) 4.73 Rydberg
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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 the excitationprocess
I But: 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 Electronic Wavefunctions 39 / 54
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Attachment/Detachment Densities
Adenine, ADC(2)/cc-pVDZ
↑ ↑ ↑
nπ∗ state ππ∗(Lb) state ππ∗(La) state
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Natural Difference Orbitals
I Adenine, ADC(2)/cc-pVDZ, nπ∗ stateI Comparison of transition and difference DMs
NTOs NDOs
λ1 = 0.84 a1 = 0.94 a2 = 0.07 a3 = 0.06
λ1 = 0.84 d1 = 0.93 d2 = 0.07 d3 = 0.06
I Main transition the sameI But additional contributions in the 1DDMI Orbital relaxation
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Exciton Analysis
Main ideaI 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)
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Exciton Analysis
Charge transfer numbers
ΩAB =
∫A
dxh
∫B
dxeχexc(xh, xe)
A Fragment where the hole is localizedB Fragment where the electron is localized
ΩAB Probability
I Pseudocolor matrix plot of the ΩAB matrix
rh
re
re
rh
χexc
(a) (b) re
rh
(c)
+
_
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Phenomenology 1DM 1TDM/1DDM Exciton TheoDORE C
Exciton Analysis
I Example: Poly(para phenylene vinylene)I ADC(2)/SV(P)
1 S. A. Mewes, J.-M. Mewes, A. Dreuw, F. Plasser PCCP 2016, 18,2548.F. Plasser Electronic Wavefunctions 45 / 54
Phenomenology 1DM 1TDM/1DDM Exciton TheoDORE C
Exciton Analysis
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)
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Exciton Analysis
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)
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TheoDORE
TheoDORE - Theoretical Density, Orbital Relaxation and Exciton analysis
I Program package for wavefunction analysisI Interfaces to a number of quantum chemistry programs: Molcas, Turbomole,
Columbus, Orca, GAMESS, ...I Open-source: http://theodore-qc.sourceforge.net
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TheoDORE
Main scriptsI theoinp
Input generationI analyze_sden.py
I Analysis of state/difference density matricesI Population analysisI Attachment/detachment analysis
I analyze_tden.pyI Analysis of transition density matricesI Natural transition orbitalsI Electron-hole correlation plots
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libwfa
libwfa - A wavefunction analysis library
I State/transition/difference DM analysis methodsI Orbitals + densitiesI Different population analysesI Also tools in coordinate spaceI Available in Q-Chem: TDDFT, EOM-CC, ADCI Interface to MOLCAS in progress: CASSCF, CASPT2
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Conclusions
Analysis of electronic wavefunctions - why?
1 Make things easierI Compact orbital representationsI Automatization
2 Give specific quantitative resultsI Charge transfer characterI DelocalizationI Spatial extentI Double excitation character
3 Provide new physical insightI Orbital relaxationI Exciton correlation
There is more to wavefunctions than meets the eye!
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Literature
Density matricesI A. Szabo and N. Ostlund "Modern Quantum Chemistry" 1989,
McGraw-Hill, Inc.I Other textbooks ...
Basic analysis techniquesI F. Plasser, M. Wormit, A. Dreuw J. Chem. Phys. 2014, 141, 024106.I F. Plasser, S. A. Bäppler, M. Wormit, A. Dreuw J. Chem. Phys. 2014, 141,
024107.
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