From Electronic Structure Theory to
Simulating Electronic Spectroscopy
Marcel Nooijen
University of Waterloo
Anirban Hazra
Hannah Chang
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.
What is best expansion point for harmonic approximation?
Second order Taylor
series expansion around
the equilibrium geometry
of the excited state.
Accurate 0-0 transition.
Excited state PES
Second Taylor series
expansion around the
equilibrium geometry of
the absorbing state.
Accurate vertical energy.
Ground state PES
Time-dependent picture of spectroscopy
t
 ex (q, t )  e  iH ext Xˆ  0 (q )
Xˆ  0 (q )
P( , T ) 
T
ˆ  (q, t ) eit dt


(
q
)
X
ex
 0
T
 0 (q)
The broad features of the spectrum
are obtained in a short time T.
Accurate PES desired at the equilibrium geometry of the absorbing state.
Ethylene excited state potential energy surfaces
H
CH stretch
H
C
C
H
H
CH2 scissors
C
H
H
C=C stretch
H
C
C
H
H
H
H
H
H
Torsion
C
H
C
H
C
H
Vertical Franck-Condon
•Assume that excited state potential is separable in the normal modes of
the excited states, defined at reference geometry
H ex 

iHarmonic
hˆHarmonic (Qi ) 
hˆHarmonic  n (Qi )  n  n (Qi )

jAnharmonic
K.E.(Q j )  V (Q j )
hˆAnharmonic n (Q j )   n n (Q j )
    i (Qi ) j (Q j )  1 (Q1 )   2 (Q2 )... l (Ql )  m (Qm )...
i
j
Harmonic
Anharmonic
Electronic absorption spectrum of ethylene:
Rydberg + Valence state
Experiment
Geiger and Wittmaack,
Z. Naturforsch, 20A, 628, (1965)
Calculation
Rydberg state spectrum
Total spectrum
Valence state spectrum
Beyond Born-Oppenheimer: Vibronic models
• Short-time dynamics picture:
P( , T ) 
T
ˆ  iHt Xˆ  (q) eit dt


(
q
)
Xe
0
0

ˆ
T
Requires accurate time-dependent wave-packet in Condon-region, for
limited time.
Use multiple-surface model Hamiltonian Ĥ that is accurate near
absorbing state geometry.
E.g. Two-state Hamiltonian with symmetric and asymmetric mode
 E1  hˆ1  k1 qs
Hˆ  
ka qa


ka qa

E2  hˆ2  k2 qs 
Methane Photo-electron spectrum
Fully quadratic vibronic model
Calculating coupling constants in vibronic model
Construct model potential energy matrix in diabatic basis:
i
ij
Vab (q)  Ea ab   Eab
qˆi  12  Eab
qˆi qˆ j
i
i
• 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:
 a (q)   (0)   ca (q)   (q)   (0)

•
Obtain non-diagonal diabatatic Vab (q)
• Use finite differences to obtain linear and quadratic coupling constants.
• Impose Abelian symmetry.
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
1
ˆ
TN   ki qi2   0   ibi†bˆi
2 i
i
1 ˆ ˆ†
qˆi 
(bi  bi )
2
• 2 x 2 Vibronic Hamiltonian (linear coupling)
 E1   0
ˆ
H 
 0
,
 ibˆi†bˆi  11i qˆi

0 
12i qˆi
 


i
†
i

ˆ
ˆ
E2   0  i
12 qˆi
ibi bi  22 qˆi 

• Vibronic eigenstates  

a ,i , j ,...,m
cia, j ,...,m ni , n j ,..., nm
a
• Total number of basis states M i M j ...M m  N a
• Dimension grows very rapidly (but a few million basis states can be
handled easily).
• Efficient implementation: rapid calculation of HC
Ethylene second cationic state: PES slices.
torsion CC=0.0
CH2 rock, CC=0
CC stretch
torsion CC=2.0
CH2 rock, CC=-1.5
Vibronic simulation of second cation state of ethylene.
Simulation includes 4 electronic states and 5 normal modes.
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
M i M j ....M N a 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?
Transformation of Vibronic Hamiltonian
Hamiltonian in second quantization:
ˆ ˆ† ˆj
Hˆ  Eba aˆ †bˆ  Ebjai aˆ †bi
ˆ ˆ†  E a aˆ †bi
ˆ ˆ  E aij aˆ †bi
ˆ ˆ† ˆj †  E a aˆ †bij
ˆ ˆˆ
 Ebai aˆ †bi
bi
b
bij
Electronic states a, b ; a †b a b
Normal mode creation and annihilation operators iˆ† , ˆj
Unitary transformation operator eTˆ †  eTˆ

T  T

G

e
He
Define Hermitian transformed Hamiltonian
Find operator T̂ , such that Ĝ is easier to diagonalize …
Details on transformation
• “Displacement operators” iˆ†  iˆ  pˆ
ˆ ˆ†  bˆ†ai
ˆ ˆ† ˆj †  bˆ†aji
ˆ ˆ)  tbaij (aˆ †bi
ˆˆˆ)
Tˆ  tbai (aˆ †bi
ˆ
ˆ Tˆ
Hˆ  eT He
•

hbai  hbaij  haib  hajib  0
ˆ† j  ˆj †i  Lij
i
“Rotation operators”
ai †   †   †  † 
 i)
U uba ( a †b b†a) ubj
( a bi j b aj
ˆ
ˆ
ˆ Uˆ ;
ˆ Tˆ eUˆ  eUˆ † He
Gˆ  eU eT He

gba  0, a  b,
gbjai  0, except (a  b, i  j )
• After solving non-linear algebraic equations:
ai †  † 
 i
G gaa a †a gai
a ai
....
Adiabatic, Harmonic!?
We are neglecting three-body and higher terms in Ĝ
Eigenstates and eigenvalues of Ĝ :
Gˆ n, a  En ,a n, a
ˆ Tˆ eUˆ n, a  E eTˆ eUˆ n, a
He
n ,a
n,a : original oscillator basis, En ,a harmonic
Transition moments:
ˆ Tˆ eUˆ n, a  n, a eUˆ eTˆ Xˆ 0
0 Xe
Easy calculation of spectrum after transformation.
Akin to Franck-Condon calculation.
# of parameters in Tˆ ,Uˆ scales with # of normal modes
Additional approximations:
• Assume  Aˆ ,Tˆ 


is a two-body operator, if  is two-body.
Reasonable if T̂ is small
a ,i
i
Exact if tb   abt (coordinate transformation)
• Allows recursive expansion of infinite series
1
Gˆ  Hˆ   Hˆ ,Tˆ     Hˆ ,Tˆ  ,Tˆ   ...

2
• 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!! ]
Example: Core-Ionization spectrum ethylene
Exact / Transformed Spectrum
H
H
C*
H
H
C
H
C
H
H
C*
H
Potential along asymmetric CH stretch
Franck-Condon spectrum
Analysis core-ionization spectra
Localized
core-hole Basis l,r:
Delocalized
core-hole Basis g, u:
 Eg  hˆ0  ksˆ

 aˆ


ˆ
 aˆ
Eu  h0  ksˆ 

 E  hˆ0  ksˆ   aˆ

E / 2


ˆ
E / 2
E  h0  ksˆ   aˆ 

In localized picture: Displaced harmonic oscillator basis states
ns , na , l , ns , na , r ,
E  nss  naa
Diagonalize 2x2 Hamiltonian E  nss  naa  E / 2
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
H
*
C
C
H
H
C
*
C
Exact Lanczos / Transformation
Perfect agreement Exact /Transformed
H
Individual states: exact Lanczos
Franck-Condon spectrum
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  ,U and obtain a transformed Hamiltonian:
ˆ
ˆ ˆUˆ
Gˆ  eU ˆ He
• Obtain n, a eUˆ ˆ Xˆ 0 as starting vector Lanczos
• Perform Lanczos with the simplified Hamiltonian Ĝ .
Example: strong Jahn-Teller coupling
Transformation
Active / Lanczos
Full Lanczos vs Full Transform
Full Lanczos vs Partial Transform
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
NSERC
University of Waterloo
NSF