lecture 3: a closer look into electronic wavefunctions

Phenomenology 1DM 1TDM/1DDM Exciton TheoDORE C

Lecture 3:A closer look into electronic wavefunctions

Felix Plasser

SHARC Workshop

Vienna, October 3rd–7th, 2016

F. Plasser Electronic Wavefunctions 1 / 54


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)



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∗



Valence states

ππ∗ statesI π-orbital: nodal plane in the

molecular planeI Typical UV absorbing states

↑



Valence states

nπ∗ statesI Usually "dark" ↑



Valence states

πσ∗, σσ∗ statesI Can lead to dissociation of the

molecule↑



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

↑



Rydberg states

Diffuse p-orbitals

↑ ↑ ↑



Rydberg states

Rydberg states - in practice

↑ ↑ ↑

Cutoff 0.04 Cutoff 0.03 Cutoff 0.02



Charge Transfer States

Charge Transfer StatesI Electron transfer between two

chromophoresI Approximate energy

ECT ≈ IP + EA− 1

R

I Strong dependence on theintermolecular separation!

↑



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%)



Wavefunction Analysis

How to move from a computation to a phenomenological description?



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



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



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



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 ...



Population Analysis

Next step: Divide the electron density into contributions of different atoms

I There is no unique way to do so!



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



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.



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



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)



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



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



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



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



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



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



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)




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




I Adenine: ADC(3)/aug-cc-pVDZI Natural transition orbitalsI One dominant transition- Compact valence stateI Diffuse orbitals were only

artifacts!

↑ 97%




Linear combination of the orbitals

= −0.39× −0.26×

+0.21× −0.20×

−0.13×



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



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



Difference Density Matrices

Subtract the density matrices of the ground state and excited state

1-Electron difference density matrix (1DDM)

∆0I = DII −D00

Adenine


I Physical meaning: changes in one-electron properties during the excitationprocess

I But: 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 Electronic Wavefunctions 39 / 54


Attachment/Detachment Densities

Adenine, ADC(2)/cc-pVDZ

↑ ↑ ↑




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



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)



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)

+

_



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


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)



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


http://theodore-qc.sourceforge.net


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



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



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!



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.


lecture 3: a closer look into electronic wavefunctions

Documents