From Electronic Structure Theory toSimulating Electronic Spectroscopy

Marcel Nooijen Anirban HazraK. R. Shamasundar Hannah Chang

University of Waterloo Princeton University

What is Calculated?

8.738.668.798.711B3u

------8.491B1g

8.218.408.458.271Ag

7.897.957.997.971B2g

7.867.857.947.921B1g

7.968.407.987.781B1u

7.137.177.287.211B3u

MRD-CI [Petrongolo]

CASPT2 [Roos]

EOM-CCSD[Bartlett]

STEOM[Nooijen]

State

Vertical transitions, e.g. ethylene:

V

V

R

R

R

R

R

What is measured?

Ethylene: Ethylene: Two lowest statesTwo lowest states

Hudson et al.

Beyond vertical excitation energies for “sizeable” molecules:

Some type of “Harmonic” approximation to potential energy surfaces is required.

• Franck-Condon Approach: single surface.

• Non-adiabatic vibronic models: multiple surfaces.

Ground state PES

At equilibrium geometry of the absorbing state?

True excited state PES

At equilibrium geometry of the excited state?

What is best expansion point for harmonic approximation?

0 ( )qΦ

0ˆ ( )X qΦ

t0

ˆ( , ) ( )exiH tex q t e X q−Φ = Φ

0ˆ( , ) ( ) ( , )

Ti t

exT

P T q X q t e dtωω ω−

= Φ Φ∫

The broad features of the spectrum are obtained in a short time T.

Time-dependent picture of spectroscopy

Excited State PES may not have minimum?? Harmonic Franck-Condon ??

Going beyond the strict harmonic approximation in vertical FC

The treatment of double well potentials:

Potential energy surfaces of ethylene cationic states along torsion mode.

General 1-d Vertical Franck-Condon:

ˆ ˆˆ ( ) ( )

ˆ ˆ( ) ( ) ( ) ( )

ex harm i gen ji harmonic j general

harm n i n n i gen n j n n j

H h q h q

h q q h q qφ ω φ ψ ε ψ

∈ ∈

= +

= =

∑ ∑% %

% % % %

Introduce intermediate harmonic basis for trouble modes j

( ) ( ) ( ) ( )n j k j k j n jk

q q q qψ φ φ ψ=∑% % % %

Calculate FC factors in intermediate, fully harmonic basis.Transform FC factors to true basis states and obtain spectrum.

( ) ,n j nqφ ω%

General 1-d potentials, but decoupled in excited state normal modes.

How to calculate general Franck-Condon factors ?

0 1 0 2 0 1 2( ) ( )... ( ) ( ) ( )... ( )M n m z Mq q q q q qφ φ φ φ ψ φ% % %

Ethylene lowest Ethylene lowest RydbergRydberg state PES along torsion modestate PES along torsion mode

Harmonic approximation at ground state equilibrium geometry

Actual potential from electronic structure calculation

Ethylene UV absorption spectrum

Ethylene Ethylene RydbergRydberg state:state:Vertical FranckVertical Franck--Condon spectrumCondon spectrum

Symmetric modes only.Harmonic vertical FC.

Torsion mode includedGeneral vertical FC.

Ethylene Valence state:Ethylene Valence state:Vertical FranckVertical Franck--Condon spectrumCondon spectrum

Symmetric modes only.Harmonic vertical FC.

Torsion mode included.General vertical FC.

Ethylene: Ethylene: Two lowest statesTwo lowest states

Experiment

General VFC calculationSTEOM-CC electronic structure

Summary Franck-Condon

• Adiabatic Harmonic Franck-Condon:- Optimize excited state geometries and frequencies. - Suitable for 0-0 transitions (if they exist !!).- Less appropriate for excitation spectra.

• General 1-d Vertical Franck-Condon:- No geometry optimization of excited states.- Requires full 1-d potentials for normal modes with negative force

constants at ground state geometry.- Applicable only, if no degeneracies at ground state geometry.

Electronic structure calculations determine computational costs. The FC calculations themselves take negligible computer time.

Beyond Born-Oppenheimer: Vibronic models

• Short-time dynamics picture:

Requires accurate time-dependent wave-packet in Condon-region, for limited time.

Use multiple-surface model Hamiltonian that is accurate near absorbing state geometry.

E.g. Two-state Hamiltonian with symmetric and asymmetric mode

ˆ0 0

ˆ ˆ( , ) ( ) ( )T

iHt i t

T

P T q Xe X q e dtωω ω −

−

= Φ Φ∫

H

1 11

2 22

ˆˆ

ˆharm

harm

s s a a

a a s s

E h k q k qH

k q E h k q

+ + = + +

Comparison of model FC and vibronic calculations

O=C=O symmetric and asymmetric stretch included.

Methane Photo-electron spectrumFully quadratic vibronic model

Calculating coupling constants in vibronic model

• Calculate geometry and harmonic normal modes of absorbing state.• Calculate "excited" states in set of displaced geometries along normal modes.• From adiabatic states construct set of diabatic states :

Minimize off-diagonal overlap:

• Obtain non-diagonal diabatatic

• Use finite differences to obtain linear and quadratic coupling constants. • Impose Abelian symmetry.

12ˆ ˆ ˆ( ) i ij

ab a ab ab i ab i ji i

V E E q E q qδ= + +∑ ∑qr

( ) ( ) ( ) ( ) ( )a acµ λ λ µλ

Φ ∆ Ψ = ∆ Ψ ∆ Ψ∑q 0 q q 0r rr r r

Construct model potential energy matrix in diabatic basis:

( )abV q∆

Advantages of vibronic model in diabatic basis

• "Minimize" non-adiabatic off-diagonal couplings.• Generate smooth Taylor series expansion for diabatic matrix.

The adiabatic surfaces can be very complicated.• No fitting required; No group theory.• Fully automated / routine procedure.

• Model Potential Energy Surfaces Franck-Condon models.

• Solve for vibronic eigenstates and spectra in second quantization. Lanczos Procedure:

- Cederbaum, Domcke, Köppel, 1980's - Stanton, Sattelmeyer: coupling constants from EOMCC calculations.- Nooijen:

Automated extraction of coupling constants in diabatic basis.Highly efficient Lanczos for many electronic states.

Vibronic calculation in Second Quantization

• 2 x 2 Vibronic Hamiltonian (linear coupling)

• Vibronic eigenstates

• Total number of basis states • Dimension grows very rapidly (but a few million basis states can be

handled easily).• Efficient implementation: rapid calculation of HC

†1 ˆ ˆˆ ( )2i i iq b b= + 2 †

01 ˆˆ2N i i i i i

i iT k q b bε ω+ → +∑ ∑

,

†1 0 11 12

†2 0 12 22

ˆ ˆ ˆ ˆ0ˆˆ ˆ0 ˆ ˆ

i ii i i i i

i ii i i i i i

E b b q qH

E q b b q

ε ω λ λε λ ω λ

+ + = + + +

∑

...i j m aM M M N⋅

, ,...,, , ,...,

, ,...,ai j m i j m a

a i j mc n n nΨ = ∑

torsion CC=0.0

torsion CC=2.0

CC stretch

CH2 rock, CC=0

CH2 rock, CC=-1.5

Ethylene second cationic state: PES slices.

C C

H

HH

H

VibronicVibronic simulation of second simulation of second cationcation state of ethylene.state of ethylene.Simulation includes 4 electronic states and 5 normal modes.Simulation includes 4 electronic states and 5 normal modes.

Model includes up to Model includes up to quarticquartic coupling constants.coupling constants.

Experimental spectrum (Holland, Shaw, Hayes, Shpinkova, Rennie, Karlsson, Baltzer, Wannberg,Chem. Phys. 219, p91, 1997)

Simulated Spectrum

Third and fourth ionized states of ethylene

Simulation

Experiment

Circular Dichroism Spectrum of dimethyloxirane

Comparison of experiment and vertical excitation simulation:

Inclusion of FC factors: PES’s along most important normal mode2, ,Im[ ] 0el e mag e

if if f inµ µ⋅r r%

Agreement between experiment and FC calculation after smalladjustments of vertical excitation energies (~ 0.2 eV)

Remaining discrepancies absolute intensities: vibronic coupling?

Summary

• Vibronic models are a convenient tool to simulate electronic spectra.• Coupling constants that define the vibronic model can routinely be obtained

from electronic structure calculations & diabatization procedure.

• Full Lanczos diagonalizations can be very expensive. Hard to converge.

• Vibronic model defines electronic surfaces: Can be used in (vertical and adiabatic) Franck-Condon calculations. No geometry optimizations; No surface scans.

• Other possibilities to use vibronic models (in the future):Investigate short-time photochemical processes.Resonance Raman processes and other spectroscopies (CD).Studies of transition metal chemistry.Coupling constants from Amsterdam Density Functional program.

Anirban Hazra Hannah Chang Alexander Auer

NSERCUniversity of WaterlooNSF