From Electronic Structure Theory toSimulating Electronic Spectroscopy

Marcel Nooijen Anirban Hazra Hannah Chang

University of Waterloo Princeton University

Beyond vertical excitation energies for “sizeable” molecules:

(A) Franck-Condon Approach:

Single surface (Born-Oppenheimer).

Decoupled in normal modes.

Computationally efficient.

(B) Non-adiabatic vibronic models:

Multiple coupled surfaces.

Coupling across normal modes.

Accurate but expensive.

Ground state PES

Second Taylor series expansion around the

equilibrium geometry of the absorbing state.

Accurate vertical energy.

Excited state PES

Second order Taylor series expansion around the equilibrium geometry

of the excited state.Accurate 0-0 transition.

What is best expansion point for harmonic approximation?

0 ( )q

0ˆ ( )X q

t0

ˆ( , ) ( )exiH tex q t e X q

0ˆ( , ) ( ) ( , )

Ti t

ex

T

P T q X q t e dt

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

Time-dependent picture of spectroscopy

Accurate PES desired at the equilibrium geometry of the absorbing state.

Ethylene excited state potential energy surfacesEthylene excited state potential energy surfaces

Torsion

CH stretch C=C stretch

CH2 scissors C C

H

HH

H

C C

H

HH

H

C C

H

HH

H

C C

H

HH

H

Vertical Franck-CondonVertical Franck-Condon

•Assume that excited state potential is separable in the normal modes of the excited states, defined at reference geometry

1 1 2 2

ˆ ( ) K.E.( ) ( )

ˆ ˆ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )... ( ) ( )...

Harmonicex i j ji Harmonic j Anharmonic

Harmonic n i n n i Anharmonic n j n n j

i i j j l l m mi j

H h Q Q V Q

h Q Q h Q Q

Q Q Q Q Q Q

Harmonic Anharmonic

Experiment

Calculation

Electronic absorption spectrum of ethylene: Rydberg + Valence state

Geiger and Wittmaack,Z. Naturforsch, 20A, 628, (1965)

Rydberg state spectrumRydberg state spectrum Valence state spectrum

Total spectrum

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

ˆ

0 0ˆ ˆ( , ) ( ) ( )

TiHt i t

T

P T q Xe X q e dt

H

1 1 1

2 2 2

ˆˆ

ˆs a a

a a s

E h k q k qH

k q E h k q

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 q q

( ) ( ) ( ) ( ) ( )a ac

q 0 q q 0

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 potential energy 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. 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 ( )2

i i iq b b 2 †0

1 ˆˆ2N i i i i i

i i

T k q b b

,

†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

...i j m aM M M N

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

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

a i j m

c n n n

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.

Vibronic simulation of second cation state of ethylene.Vibronic simulation of second cation 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 quartic coupling constants.Model includes up to quartic coupling constants.

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

Issues with current vibronic model + Lanczos scheme

• Computational cost and memory requirements scale very rapidly with number of normal modes included in calculation.

• For efficiency in matrix-vector multiplications we keep

basis states

(no restriction to a maximum excitation level)

• Vibronic calculation requires judicious selection of important modes, which is error prone and time-consuming.

What can be done?

....i j NM M M a

Transformation of Vibronic Hamiltonian

Hamiltonian in second quantization:

Electronic states ;Normal mode creation and annihilation operators

Unitary transformation operator

Define Hermitian transformed Hamiltonian

Find operator , such that is easier to diagonalize …

† † †

† † † † † † †

ˆ ˆˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

a aib bj

ai a aij ab bi b bij

H E a b E a bi j

E a bi E a bi E a bi j E a bij

†a b a b ,a b†ˆ ˆ,i j

G e HeT T

†ˆ ˆT Te e

T G

Details on transformation

• “Displacement operators”

• “Rotation operators”

• After solving non-linear algebraic equations: Adiabatic, Harmonic!?

† † † † † † †ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ( ) ( )ai aijb bT t a bi b ai t a bi j b aji

ˆ ˆˆ ˆ 0T T ai aij b bb b ai ajiH e He h h h h

( ) ( )† † † † † †U u a b b a u a bi j b aj iba

bjai

†ˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ ;

0, , 0, ( , )

U T T U U U

a aib bj

G e e He e e He

g a b g except a b i j

....† † †G g a a g a ai iaa

aiai

†ˆ ˆ ˆi i p

† †ˆ ˆiji j j i L

We are neglecting three-body and higher terms in

Eigenstates and eigenvalues of :

: original oscillator basis, harmonic

Transition moments:

Easy calculation of spectrum after transformation. Akin to Franck-Condon calculation.

# of parameters in scales with # of normal modes

,

ˆ ˆ ˆ ˆ

,

ˆ , ,

ˆ , ,

n a

T U T Un a

G a E a

He e a E e e a

n n

n n

ˆ ˆ ˆ ˆˆ ˆ0 , , 0T U U TXe e a a e e X n n

G

G

,an ,aEn

ˆ ˆ,T U

Additional approximations:

• Assume is a two-body operator, if is two-body. Reasonable if is small Exact if (coordinate transformation)

• Allows recursive expansion of infinite series

• Method is “exact” for adiabatic harmonic surfaces (including displacements & Duschinsky rotations).

• Unitary expansion does not terminate, but preserves norm of wave function.

• [ Transformation with pure excitation operators can yield unnormalizable wfn’s Similarity transform changes eigenvalues Hamiltonian!! ]

ˆ ˆ,A T

1ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , , ...2

G H H T H T T

T

A

,a i ib abt t

Example: Core-Ionization spectrum ethylene

C C*

H

H

H

H

C* C

H

H

H

H

Exact / Transformed Spectrum

Franck-Condon spectrum

Potential along asymmetric CH stretch

Analysis core-ionization spectra

0

0

ˆ ˆ ˆ

ˆˆ ˆ

g

u

E h ks a

a E h ks

Delocalized core-hole Basis g, u:

Localized core-hole Basis l,r:

0

0

ˆ ˆ ˆ / 2

ˆ ˆ ˆ/ 2

E h ks a E

E E h ks a

In localized picture: Displaced harmonic oscillator basis states

, , , , , ,s a s a s s a an n l n n r E n n

Diagonalize 2x2 Hamiltonian / 2s s a aE n n E Same frequencies as in ground state (very different from FC)

Transformation works in delocalized picture. Slightly different frequencies.Symmetric mode coupling is the same in the two states.

Core Ionization Spectrum acetylene

Individual states: exact Lanczos

Franck-Condon spectrumExact Lanczos / Transformation

C*C HH

*C C HH

Perfect agreement Exact /Transformed

Benzene Core ionization Transformation with 6 electronic states, 11 normal modes

‘Breakdown’ of transformation method for strongly coupled E1u mode at 1057 cm-1

Combined Transform – Lanczos approach

• Identify "active" normal modes that would lead to large / troublesome transformation amplitudes.

• Transform to zero all ‘off-diagonal’ operators that do not involve purely active modes.

• Solve for and obtain a transformed Hamiltonian:

• Obtain as starting vector Lanczos

• Perform Lanczos with the simplified Hamiltonian .

ˆ ˆˆ ˆˆ ˆU UG e He

, U

ˆ ˆ ˆ, 0Ua e X n

G

Example: strong Jahn-Teller coupling

Full Lanczos vs Partial Transform

Full Lanczos vs Full TransformTransformation Active / Lanczos

Summary

• Vibronic approach is a powerful 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. Efficient.

• “Transform & Diagonalize” approach is interesting possibility to extend vibronic approach to larger systems.

Anirban Hazra Hannah Chang Alexander Auer

NSERCUniversity of WaterlooNSF