Calculate absorption profiles

1
Calculate absorption profiles?
Experimental spectrum
?
Calculated spectrum for candidate molecule
?
2
Calculate absorption profiles?
Experimental spectrum
?
Calculated spectrum for candidate molecule
Assignment
?
3
Calculate absorption profiles?
Experimental spectrum
?
Calculated spectrum for candidate molecule
Assignment
?
This sort of assignment can often provide clear
picture of the nature of vibronic states
4
Vibronic coupling and the computational modelling
of electronic spectra

• The Born-Oppenheimer approximation and the
  adiabatic potential energy surface
• The Franck-Condon approximation
• Breakdown of the simple Born-Oppenheimer picture
  - Vibronic coupling
• Diabatic potential energy surfaces and a
  simplified model of spectroscopy
• Complicated adiabatic surfaces and couplings
• The model Hamiltonian approach of Köppel, Domcke
  and Cederbaum (KDC)
• Application to NO2 molecule (photoelectron and
  electronic spectra)

Lecture I, Frontiers in Spectroscopy, Columbus,
1/2007
5
The molecular Hamiltonian
H Te Tn Vne Vee
Vnn
Electrons and nuclei treated on an equal
footing Qualitatively inconsistent with our
models and thinking about molecules in
chemistry Does not lend itself to eigenstates
described as electronic, vibrational and
rotational Where is the notion of molecular
structure?

• O
• O O

Example Ozone
H ? Te ? Tn ?? Vne
1/2 ?? Vee 1/2 ?? Vnn
-
i ?
? i
i j ?
?
Where is the unique oxygen? What is the bond
length?
6
The two major paradigms in chemistry
H Te Tn Vne Vee
Vnn

7
The two major paradigms in chemistry
H Te Tn Vne Vee
Vnn
1. The separation of electronic and nuclear
motions (adiabatic approximation) Electrons move
much faster than nuclei, due to (relatively) huge
nuclear mass H Te Tn
Vne Vee Vnn

8
The two major paradigms in chemistry
H Te Tn Vne Vee
Vnn
1. The separation of electronic and nuclear
motions (adiabatic approximation) Electrons move
much faster than nuclei, due to (relatively) huge
nuclear mass H Te Tn
Vne Vee Vnn
Nuclear coordinates are parameters Electronic
coordinates are variables

9
The two major paradigms in chemistry
H Te Tn Vne Vee
Vnn
1. The separation of electronic and nuclear
motions (adiabatic approximation) Electrons move
much faster than nuclei, due to (relatively) huge
nuclear mass H Te Tn
Vne Vee Vnn
Nuclear coordinates are parameters
Constant
Electronic coordinates are variables

10
The two major paradigms in chemistry
H Te Tn Vne Vee
Vnn
1. The separation of electronic and nuclear
motions (adiabatic approximation) Electrons move
much faster than nuclei, due to (relatively) huge
nuclear mass H Te Tn
Vne Vee Vnn
Nuclear coordinates are parameters
Constant
Electronic coordinates are variables

Depend on nuclear coordinates (molecular
geometry)
11
The two major paradigms in chemistry
H Te Tn Vne Vee
Vnn
1. The separation of electronic and nuclear
motions (adiabatic approximation) Electrons move
much faster than nuclei, due to (relatively) huge
nuclear mass H Te Tn
Vne Vee Vnn
Nuclear coordinates are parameters
Constant
Electronic coordinates are variables

Depend on nuclear coordinates (molecular
geometry)
Helec(r) Te(r) Vne(rR)
Vee(r) Vnn
12
The two major paradigms in chemistry
H Te Tn Vne Vee
Vnn
1. The separation of electronic and nuclear
motions (adiabatic approximation) Electrons move
much faster than nuclei, due to (relatively) huge
nuclear mass H Te Tn
Vne Vee Vnn
Nuclear coordinates are parameters
Constant
Electronic coordinates are variables

Depend on nuclear coordinates (molecular
geometry)
Helec(r) Te(r) Vne(rR)
Vee(r) Vnn
The electronic Hamiltonian
13
Helec(r) Te(r) Vne(rR)
Vee(r) Vnn Helec(r) ? Te(r)
Vne(rR) ?? Vee(r) Vnn
? Hi ?? Hij
Vnn
14
Helec(r) Te(r) Vne(rR)
Vee(r) Vnn Helec(r) ? Te(r)
Vne(rR) ?? Vee(r) Vnn
? Hi ?? Hij
Vnn
The one electron Electron-electron
Hamiltonian repulsion
15
A reminder from quantum mechanics if
H ? hi then
? ? ?i
Each term must involve independent coordinates
Separability condition
16
Helec(r) Te(r) Vne(rR)
Vee(r) Vnn Helec(r) ? Te(r)
Vne(rR) ?? Vee(r) Vnn
? Hi ?? Hij
Vnn 2. The neglect of electron correlation -
the independent particle approximation ?
Hi ?vieff Vnn
17
Eigenvalues of electronic Hamiltonian
Second eigenvalue
x
x

E
Lowest eigenvalue
r
18
Eigenvalues of electronic Hamiltonian
x
x
x
x x
x

x
x x
x
x
x x
x x x
x x
x x
E
r
19
Adiabatic potential energy surfaces
x
x
x
x x
x

x
x x
x
x
x x
x x x
x x
x x
E
r
20
Adiabatic potential energy surfaces
x
x
x
x x
x

x
x x
x
x
x x
x x x
x x
x x
E
Equilibrium bond length (re)
r
21
Adiabatic potential energy surfaces
x
x
x
x x
x

x
x x
x
x
x x
x x x
x x
x x
E
Dissociation energy (De)
Equilibrium bond length (re)
r
22
Adiabatic potential energy surfaces
x
x
x
x x
x

x
x x
x
x
x x
x x x
x x
x x
E
Vertical excitation energy
Dissociation energy (De)
Equilibrium bond length (re)
r
23
The Born-Oppenheimer Separation
H(r,R) Helec(rR) Hnuc(R)
Hnuc(R) Tnuc(R) V(R)
? ?elec ?nuc
? ?elec ?vib?rot E
Eelec Evib Erot
Adiabatic potential energy surface
(PES)
Note ?elec diagonalizes electronic
Hamiltonian ?nuc diagonalizes
nuclear Hamiltonian B.O. approximation
assumes that ? diagonalizes full Hamiltonian
24
Adiabatic potential energy surfaces
x
x
x
x x
x

x
x x
x
x
x x
x x x
x x
x x
E
Vertical excitation energy
Dissociation energy (De)
2B0
Zero-point Energy (ZPE)
Equilibrium bond length (re)
r
25
H Te Tn Vne Vee
Vnn
Qualitatively inconsistent with our models and
paradigms in chemist Does not lend itself to
eigenstates described as electronic,
vibrational and rotational Where is the
structure?
Born-Oppenheimer approximation
Independent electron approximation
26

• The crude
  Born-Oppenheimer approximation
• and
  electronic spectroscopy
• Ignore R dependence of electronic wavefuction
• ?ve ?e(rR0)
  ?v(R)
• -Spectroscopic transition moments in dipole
  approximation
• M lt?ve ? ?ve gt
• M lt ?e(rR0) ?v(R) ?n ?e
  ?e(rR0) ?v(R)gt
• lt ?e(rR0) ?e ?e(rR0) gt lt
  ?v(R) ?v(R) gt gt
• lt ?v(R) ?n ?v(R) gt lt
  ?e(rR0) ?e(rR0) gt

Reference geometry
27

• The crude
  Born-Oppenheimer approximation
• and
  electronic spectroscopy
• Ignore R dependence of electronic wavefuction
• ?ve ?e(rR0)
  ?v(R)
• -Spectroscopic transition moments in dipole
  approximation
• M lt?ve ? ?ve gt
• M lt ?e(rR0) ?v(R) ?n ?e
  ?e(rR0) ?v(R)gt
• lt ?e(rR0) ?e ?e(rR0) gt lt
  ?v(R) ?v(R) gt gt
• lt ?v(R) ?n ?v(R) gt lt
  ?e(rR0) ?e(rR0) gt

Reference geometry
0
28

• The crude
  Born-Oppenheimer approximation
• and
  electronic spectroscopy
• Ignore R dependence of electronic wavefuction
• ?ve ?e(rR0)
  ?v(R)
• -Spectroscopic transition moments in dipole
  approximation
• M lt?ve ? ?ve gt
• M lt ?e(rR0) ?v(R) ?n ?e
  ?e(rR0) ?v(R)gt
• lt ?e(rR0) ?e ?e(rR0) gt lt
  ?v(R) ?v(R) gt gt

Reference geometry
Electronic transition moment
Vibrational overlap integral
The Franck-Condon approximation
29
-Spectroscopic transition probabilities
A lt?ve ? ?ve gt2 A
lt ?e(rR0) ?e ?e(rR0) gt2 lt
?v(R) ?v(R) gt2
Me(r,R0)2
FCF
30
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31
What is required in the simulation of electronic
spectra?
The simplest case The Franck-Condon approximation
Applicable to most transitions from one
isolated electronic state to another,
particularly in low-energy part of the spectrum.
Assumptions
?(r,R) ?(rR) ?(R)
Vibrational Wavefunction calculated from Veff(R)
Vibronic Electronic
Wavefunction Wavefunction
lt ?(r,R) ? ?(r,R) gt lt ?(r,R0) ?
?(r,R0) gt Mij
Dipole operator Reference
geometry
32
Vibronic level positions Electronic energy
difference (adiabatic)
33
Vibronic level positions Electronic energy
difference - ZPE of ground state
34
Vibronic level positions Electronic energy
difference - ZPE of ground state vibrational
energy in final state
35
Vibronic level positions Electronic energy
difference ZPE of ground state vibrational
energy in final state
Origin
36
Vibronic level positions Electronic energy
difference ZPE of ground state vibrational
energy in final state
Origin
Intensities
Dipole approximation Intensity ? lt
?(r,R) ? ?(r,R)gt2


? Me(r,R0)2 Fk


37
Vibronic level positions Electronic energy
difference ZPE of ground state vibrational
energy in final state

Origin
Intensities
Dipole approximation Intensity ? lt
?(r,R) ? ?(r,R)gt2


? Me(r,R0)2 Fk


Origin
38
With several modes

Origin
1 2 3
0 1 2 3 4
5 6
Origin
39
To do a Franck-Condon simulation, we
need Quantum chemical calculation of ground
state geometry and force field
EASY - CAN BE DONE WITH ANY METHOD
Quantum chemical calculation of excited
state geometry and force field MORE
CARE REQUIRED WITH RESPECT TO CHOICE OF METHOD
(CIS, RPA, CIS(D),
MCSCF, CASPT2, EOM-CC, MRCI)
(balance is important here) Quantum
chemical calculation of transition dipole
moment NOT SO HARD - ACCURACY USUALLY
NOT VERY IMPORTANT
40
Franck-Condon simulations account
for Progressions in totally symmetric
vibrations (provides a measure of the geometry
change due to excitation) Even-quantum
transitions in nonsymmetric vibrations (shows up
only if there is an appreciable force constant
change) but do not account for Final states
that are not totally symmetric (due to vibronic
coupling) Spectroscopic manifestations of
non-adiabaticity (BO breakdown) (effects of
conical intersections, avoided crossings
etc.) Low-lying states usually heavily affected
by vibronic coupling!
41
A common class of problems Polyatomic molecules
with close-lying states of different
symmetry Example NO2 X2A1 - Ground state (?
? 134 degrees) A2B2 - Excited state (? ? 102
degrees) Three vibrational modes ?1 (a1) -
symmetric stretch (modulates gap between the
states)) ?2 (a1) - symmetric stretch (modulates
gap between the states) ?3 (b2) - symmetric
stretch (directly couples the two states)
?
(adiabatic separation - 1.15 eV)
?
42
The adiabatic potential energy surface(s) of NO2
One dimensional representations
N
O
O
43
2B2
Stretch coordinate
2A1
Anion geometry
2B2
Bend coordinate
2A1
44
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45
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46
The adiabatic potential energy surface(s) of NO2
Two dimensional representations
N
O
O
47
q2
2A1 minimum
2B2 minimum
q1
48
q2
2A1 minimum
2B2 minimum
q1
49
q2
Conical intersection seam
Seam minimum
q1
50
q2
Conical intersection seam
Seam minimum
q1
51
Upper adiabatic sheet (cone state)
States of different symmetry can interconvert on
same adiabatic surface! (pseudorotation)
Rapidly changinging wavefunction
2B2 state
Lower adiabatic sheet
2A1 state
52
Top view
2B2 minimum
Vertex of cone state
53
Obviously, adiabatic potential energy surfaces in
these cases are exceedingly complicated --
mathematically they are nondifferentiable along
the conical intersection seam, and very far from
the parabolic ideal Moreover the adiabatic
states that diagonalize the electronic
Hamiltonian vary rapidly near the conical
intersection the off-diagonal matrix elements
of the nuclear kinetic energy operator are no
longer negligible. Equivalently, they are poor
approximations to eigenfunctions of the nuclear
kinetic energy operator -- wavefunction is
varying rapidly in the region of an
intersection (recall diatomic curves) This
means that the Born-Oppenheimer approximation is
not very good here, and our simple idea of the
separable wavefunction, the Franck-Condon approxim
ation, and all that goes with this, will no
longer give us a good model for understanding
spectra ?ve ? ?e(rR0)
?v(R)
54
Going beyond the crude Born-Oppenheimer
approximation The adiabatic approximation ?ve
i ? c?i ?e?(rR0) ? c?i ??(R) The
crude Born-Huang expansion
?vei ? c??i ?e?(rR0) ??(R) (A
formally exact expansion for the vibronic
wavefunction) The adiabatic approximation -
which is the Born-Oppenheimer approximation to
most quantum chemists allows for variation of the
electronic wavefunction with R, but is difficult
to apply to spectroscopy and separability of
vibrational and electronic parts The crude
Born-Huang expansion provides a convenient
formalism for vibronic coupling, and we will use
it in the following.
?
?
Wavefunctions calculated from
quantum chemistry (R dependence carried by
coefficients)
??
55
Diabaticity and quasi-diabaticity The
electronic states appearing in the crude
Born-Huang expansion are strictly and rigorously
diabatic (they diagonalize the nuclear kinetic
energy operator - for obvious reasons) and the
treatment of vibronic coupling we are about to
study is exact in principle. However
implementation of a theory based strictly on the
CBH expansion is clearly a
daunting and terrible prospect -- so many
electronic states have to be
considered (think even about dissociation of a
diatomic) The (usually quite) slow variation
of electronic wavefunctions with R can be viewed
as a consequence of a mixing-in of many functions
in the frozen basis as the nuclei move, and Is
NOT due exclusively (or even substantially) to
mixing with those states that are strongly
coupled vibronically. Hence, we can make an
approximation in the formalism that accounts for
this.
56
Diabaticity and quadi-diabaticity The
electronic states appearing in the crude
Born-Huang expansion are strictly and rigorously
diabatic (they diagonalize the nuclear kinetic
energy operator - for obvious reasons) and the
treatment of vibronic coupling we are about to
study is exact in principle. However
implementation of a theory based strictly on the
CBH expansion is clearly a
daunting and terrible prospect -- so many
electronic states have to be
considered (think even about dissociation of a
diatomic) The (usually quite) slow variation
of electronic wavefunctions with R can be viewed
as a consequence of a mixing-in of many functions
in the frozen basis as the nuclei move, and Is
NOT due exclusively (or even substantially) to
mixing with those states that are strongly
coupled vibronically. Hence, we can make an
approximation in the formalism that accounts for
this.
57
The vibronic Hamiltonian in pictures
Diabatic electronic basis
H(R)
Adiabatic electronic basis
H(R)
TN V
Diabatic wavefunctions do not change with nuclear
displacement, necessitating enormous
configuration expansions. Consider the case
where a few states are strongly coupled
vibronically, and relatively isolated from the
rest.
58
The vibronic Hamiltonian in pictures
H(R)
Diabatic electronic basis
Block diagonalize potential energy
Quasidiabatic electronic basis
H(R)
TN V
Mixing of the orthogonal complement with the
coupled diabatic states (via the
block diagonalization) allows for the smooth
variation of the wavefunction associated
with nuclei following. The resultant weak
nuclear coupling with other states is small
and it is reasonable to neglect it. Which leads
to the KDC model Hamiltonian
59
The vibronic Hamiltonian in pictures
Quasidiabatic electronic basis
H(R)
T V
The fundamental approximation
?
T1
V11
V12
HKDC(R)
T2
V12
V22
60
T1
V11
V12
HKDC(R)
T2
V12
V22

• Some important points
• If ?1 and ?2 have different symmetries, V12
  vanishes unless the nuclear displacement
• corresponding to R transforms as ?1 x ?2.
• Due to the quasidiabatization procedure, the
  diagonal blocks of the potential are the
• adiabatic potential energy surfaces for all R
  such that ?R ? ?1 x ?2.
• 3. The electronic subblocks of the KDC
  vibronic Hamiltonian are further projected onto a
  
• harmonic oscillator basis the complete basis is
  of the direct-product type seen in the
• CBH expansion except that the electronic terms
  now refer to the quasidiabatic states.
• 4. Use of a common set of dimensionless
  normal coordinates (with a common origin)
• facilitates the calculations. Usually these are
  those of the absorbing (initial) state, but need
• not be.

61

• Vibronic effects on potential energy surfaces
• A model potential for a two coupled-mode, two
  state system
• ?1 0.4 eV (323
  cm-1) symmetric
• ?2 0.1 eV (807
  cm-1) non-symmetric
• Parameters
• - Vertical energy gap between the states
• - Linear coupling constant between the two
  states
• ?A - Slope of diabatic potential of state A along
  q1 at q1q20
• ?B - Slope of diabatic potential of state A along
  q1 at q1q20
• KDC Hamiltonian corresponding to model (in
  quasidiabatic basis)
• T1 0 ?A q1
  1/2 ?1 q12 ?2 q22 ? q2
• H ( ) (
  
  )

(Model due to Köppel, Domcke and Cederbaum)
TN V
62
Vibronic effects on potential energy
surfaces Diagonalization of potential energy
(V) gives adiabatic potential energy surfaces

? 0 eV

q2
Note that the associated diagonal basis does not
necessarily block-diagonalize the Hamiltonian
63
? 0.05 eV

q2
64
? 0.10 eV

q2
65
? 0.15 eV

q2
66
? 0.20 eV

q2
67
Vibronic effects on potential energy
surfaces Example ? 0.5 eV ?1 0.2 eV ?2
-0.2 eV ? variable
Diabatic surfaces
A state
B state

68
? 0.2 eV
Pseudorotation transition state
Lowest adiabatic surface (what we calculate in
quantum chemistry)
Equivalent minima

Conical Intersection
69
Lower adiabatic surface
Upper adiabatic surface
Conical intersections
Note that complicated (but realistic) adiabatic
surfaces arise from an extremely simple model
potential

70
More visualization
TOP VIEW
PERSPECTIVE VIEW
71
Lowest adiabatic surface with different coupling
strengths
kk
?0.05 eV
?0.15 eV
?0.10 eV

?0.25 eV
?0.30eV
?0.20 eV
72
Lowest adiabatic surface with different coupling
strengths
kk
?0.05 eV
?0.15 eV
?0.10 eV

?0.25 eV
?0.30eV
?0.20 eV
73
In this case, both states are minima on the
potential energy surface. But note that two
different electronic states lie on the
same potential energy surface!!! Far too
infrequently thought about in quantum chemistry.
Real world example 2A1 and 2B2 states of NO2
B state
A state
74
An astrophysical example Propadienylidene

1A1
1B1

1A2
1A1

Coupled by modes of b2 symmetry
75
Weak vibronic coupling between dark state and
bright state
2A2 state (minimum)
2B1 state (minimum)
Conical intersection
Lowest adiabatic potential sheet
76
Franck-Condon simulation
77
Franck-Condon simulation
78
Vibronic simulation
79
Vibronic effects on vibrational energy
levels Diagonalization of complete Hamiltonian
(TV) gives vibronic energy levels Model
potential again ? 0.20 eV ? 0.5 eV ?1
0.2 eV ?2 -0.2 eV Harmonic A state
frequencies from diabatic potential 806 cm-1
(?s) 323 cm-1 (?a) Harmonic A state frequencies
from adiabatic potential 806 cm-1 (?a) 253 cm-1
(?a) Exact vibronic levels below 1050 cm-1 250
cm-1 (?n), 500 cm-1 ( 2?n), 752 cm-1 ( 3?n), 802
cm -1 ( ?s),1004 cm-1 ( 4?n), 1043 cm-1 ( ?s?n)

80
Vibronic effects on vibrational energy
levels Diagonalization of complete Hamiltonian
(TV) gives vibronic energy levels Model
potential again ? 0.20 eV ? 0.5 eV ?1
0.2 eV ?2 -0.2 eV Harmonic A state
frequencies from diabatic potential 806 cm-1
(?s) 323 cm-1 (?a) Harmonic A state frequencies
from adiabatic potential 806 cm-1 (?a) 107i
cm-1 (?a) Exact vibronic levels below 1050
cm-1 0, 121, 327, 590, 720, 866, 888, 1046
(symmetric levels) 19, 213, 453, 723, 736, 931,
1045 (nonsymmetric levels)

81
Vibronic effects on potential energy
surfaces Diagonalization of potential energy
(V) gives adiabatic potential energy surfaces

2?n
?n
3?n
?s
82
Add some slides here with wavefunctions,
discussion of nodes, etc.
83
Wavefunctions and stationary state
energies Eigenstates of system obtained by
diagonalizing Hamiltonian Given by

Vibrational basis functions
? ?A ? ci ?i ?B ? ci ?i
Diabatic electronic states

84
Wavefunctions and stationary state
energies Eigenstates of system obtained by
diagonalizing Hamiltonian Given by

Vibrational basis functions
? ?A ? ci ?i ?B ? ci ?i

true only if ?0 ? ?A ? ci ?I
or ? ?B ? ci ?i Vibrational level
of electronic state A
Vibrational level of electronic state
A Electronic states are coupled by the
off-diagonal matrix element Breakdown
of the Born-Oppenheimer Approximation
Diabatic electronic states

85
Add more slides here with wavefunctions,
densities, projection onto diabatic states, etc.
86
The Calculation of Electronic Spectra including
Vibronic Coupling
Energies given by
Eigenvalues of model Hamiltonian
Relative intensities given by
lt ?(r,R) ? ?(r,R)gt2
Ground state
Final state
87
Model system Adiabatic perspective
Corresponding Hamiltonian very complicated
Potential energy matrix is diagonal but not
simple (discontinuities) kinetic energy matrix
is clearly not diagonal transition dipole moment
very sensitive wrt geometry
88
Model system Adiabatic perspective
Corresponding Hamiltonian very complicated
Potential energy matrix is diagonal but not
simple (discontinuities) kinetic energy matrix
is clearly not diagonal transition dipole moment
very sensitive wrt geometry
Green arrow - transition to bright state
89
Model system Adiabatic perspective
Corresponding Hamiltonian very complicated
Potential energy matrix is diagonal but not
simple (discontinuities) kinetic energy matrix
is clearly not diagonal transition dipole moment
very sensitive wrt geometry
Red arrow - transition to dark state Green
arrow - transition to bright state
90
An aside Traditional quantum chemistry
assumes TA 0
VA 0 H

0 TB 0 VB
Vibrational energy levels calculated from the
Schrödinger equations
(TA VA) ? Evib ?
(TB VB) ? Evib ? and total
(vibronic energies) given by
Eev(A) (VA)min Evib
Eev(B)
(VB)min Evib
)
)
Adiabatic potential energy surfaces
)
)
91
Diabatic perspective (KDC Hamiltonian)
conceptually (and computationally a much simpler
approach
T1 0 ?A q1 1/2
?1 q12 ?2 q22 ? q2 H (
) (

) 0 T2
? q2 ?
?Bq1 1/2 ?1 q12 ?2 q22

• Treatment
• 1. Assume initial state not coupled to final
  states (not necessary, but a simple place to
  start)
• 2. Assume transition moments between diabatic
  states are constant
• 3. Diagonalize Hamiltonian (Lanczos recursion is
  best choice)

? ?0 ?000
lt?0? ?Agt MA
lt?0? ?Bgt MB
? cA0 ?A ?000 cB0 ?B ?000 ? cAi ?A ?i
cBi ?B ?ii
92
T1 0 ?A q1 1/2
?1 q12 ?2 q22 ? q2 H (
) (

) 0 T2
? q2 ?
?Bq1 1/2 ?1 q12 ?2 q22

• 4. Stick spectra given by
• Basis set and symmetry considerations
• Direct product basis
• ?A ?00, ?A ?01, ?A ?02
• ?B ?00, ?B ?01, ?B ?02
• Symmetry of vibronic level
• ?ve ?v x ?e

cA0 cB0 2 ?(E - ?)
93
T1 0 ?A q1 1/2
?1 q12 ?2 q22 ? q2 H (
) (

) 0 T2
? q2 ?
?Bq1 1/2 ?1 q12 ?2 q22

• 4. Stick spectra given by
• Basis set and symmetry considerations
• Direct product basis
• ?A ?00, ?A ?01, ?A ?02
• ?B ?00, ?B ?01, ?B ?02
• Symmetry of vibronic level
• ?ve ?v x ?e

Makes entire contribution to intensity - only ONE
element of eigenvector matters
cA0 cB0 2 ?(E - ?)
94
Appearance of eigenvectors - pictorial view
?S
?A
?N
?S
?B
?N
Franck-Condon Vibronic
coupling
(weaker)
vibronically allowed level
95
Appearance of eigenvectors - pictorial view
?S
?A
?N
?S
?B
?N
Franck-Condon Vibronic
coupling
(stronger)
vibronically allowed level
96
Franck-Condon
Vibronic coupling
97

• Application to NO2 photoelectron spectra The
  nuts and bolts
• Choose a quantum-chemical method

98

• Application to NO2 photoelectron spectra The
  nuts and bolts
• Choose a quantum-chemical method
  EOMIP-CCSD with
• 
  cc-pVDZ basis

99

• Application to NO2 photoelectron spectra The
  nuts and bolts
• Choose a quantum-chemical method
  EOMIP-CCSD with
• 
  cc-pVDZ basis
• 2. Choose a reference state and coordinate
  system

100

• Application to NO2 photoelectron spectra The
  nuts and bolts
• Choose a quantum-chemical method
  EOMIP-CCSD with
• 
  cc-pVDZ basis
• Choose a reference state and coordinate system
  Anion and its
• 
  
  DNCs

101

• Application to NO2 photoelectron spectra The
  nuts and bolts
• Choose a quantum-chemical method
  EOMIP-CCSD with
• 
  cc-pVDZ basis
• Choose a reference state and coordinate system
  Anion and its
• 
  
  DNCs
• Quantum-chemical calculations begin!
• A. Optimize geometry and get DNCs for NO2-
  
• no2 anion - optimized geometry
• O
• N 1 R
• O 2 R 1 A
• R 1.253566124038471

Input for geometry optimization
102
Forces are in hartree/bohr and hartree/radian.
Parameter values are in Angstroms and
degrees. -----------------------------------------
--------------------------------- Parameter
dV/dR Step Rold
Rnew ---------------------------------------------
----------------------------- R
0.0000008145 -0.0000005378 1.2629668229
1.2629662851 A 0.0000013123
-0.0001124564 115.9925199928 115.9924075364 ----
--------------------------------------------------
-------------------- Minimum force
0.000000815 / RMS force 0.000001092
Output from geometry optimization
no2 anion - optimized geometry O N 1 R O 2 R 1
A R 1.262966285109282 A
115.992407536378849 CRAPS(CALCCCSD,BASISPVDZ,V
IBEXACT CHARGE-1,CC_CONV9,LINEQ_CONV9,MEM1000
00000)
Input for harmonic Frequency calcualtion
103
Normal Coordinate Analysis --------------------
--------------------------------------------
Irreducible Harmonic Infrared
Type Representation Frequency
Intensity --------------------------------------
--------------------------
(cm-1) (km/mol) -----------------------
-----------------------------------------
A1 810.6092 8.4024
VIBRATION B2 1389.0607
469.7518 VIBRATION A1
1406.5167 18.3785 VIBRATION
--------------------------------------------------
--------------
Normal Coordinates A1
B2 A1
810.61 1389.06
1406.52 VIBRATION
VIBRATION VIBRATION O
0.000 0.6121 0.1953 0.0000 0.3689 -0.2305
0.0000 0.3540 -0.3377 N 0.000 0.0000
-0.4176 0.0000 -0.7884 0.0000 0.0000
0.0000 0.7219 O 0.000-0.6121 0.1953
0.0000 0.3689 0.2305 0.0000 -0.3540 -0.3377
Output from frequency calculation
T A 0 ?A1 q1
?A2 q2 1/2 ?1 q12 ?2 q22 ?3 q32
? q3 H ( ) (


) 0 TB
? q3
? ?B1q1 ?B2q1 1/2 ?1 q12 ?2 q22 ?3
q32
B. Get gradients and energies of neutral states
at anion geometry
104
no2 anion - optimized geometry O N 1 R O 2 R 1
A R 1.262966285109282 A
115.992407536378849 CRAPS(CALCCCSD,BASISPVDZ,D
ERIV_LEV1,RESRAMANON,EXCITEEOMIP CHARGE-1,CC_C
ONV9,LINEQ_CONV9,MEM100000000) excite 1 1 1
0 12 0 1.00
Input for energy and gradient calculation (require
s FCMFINAL)
Converged eigenvalue 5.642856285778000E-002
a.u. Total EOMIP-CCSD electronic energy
-204.56410335862603 a.u.
EOMIP-CCSD eval and Final state total energy
Gradient vector in normal coordinate
representation --------------------------
-------------------------------- i W(I)
dE/dQ(i) dE/dq dE/dq
dE/dQ(i)/w(i)
(cm-1) (eV) (relative)
--------------------------------------------------
-------- 7 810.61 0.0155393214 1314.383
0.162964 0.0000004126 8 1389.06
0.0000000000 0.000 0.000000
0.0000000000 9 1406.52 -0.0441068964
-2832.237 -0.351156 0.0000019159
--------------------------------------------------
--------
EOMIP-CCSD eval and Final state total energy
105
T A 0 ?A1 q1
?A2 q2 1/2 ?1 q12 ?2 q22 ?3 q32
? q3 H ( ) (


) 0 TB
? q3
? ?B1q1 ?B2q2 1/2 ?1 q12 ?2 q22 ?3
q32
106
no2 anion - optimized geometry O N 1 R O 2 R 1
A R 1.262966285109282 A
115.992407536378849 CRAPS(CALCCCSD,BASISPVDZ,D
ERIV_LEV1,RESRAMANON,EXCITEEOMIP CHARGE-1,CC_C
ONV9,LINEQ_CONV9,MEM100000000) excite 1 1 1
0 10 0 1.00
Input for energy and gradient calculation (require
s FCMFINAL)
EOMIP-CCSD eval and Final state total energy
Converged eigenvalue 9.307240503891839E-002
a.u. Total EOMIP-CCSD electronic energy
-204.52745951644488 a.u.
Gradient vector in normal coordinate
representation --------------------------
-------------------------------- i W(I)
dE/dQ(i) dE/dq dE/dq
dE/dQ(i)/w(i)
(cm-1) (eV) (relative)
--------------------------------------------------
-------- 7 810.61 -0.0240429216 -2033.654
-0.252143 0.0000009878 8 1389.06
0.0000000000 0.000 0.000000
0.0000000000 9 1406.52 0.0109852129
705.394 0.087459 0.0000001188 ----------------
------------------------------------------
EOMIP-CCSD eval and Final state total energy
107
T A 0 ?A1 q1
?A2 q2 1/2 ?1 q12 ?2 q22 ?3 q32
? q3 H ( ) (


) 0 TB
? q3
? ?B1q1 ?B2q2 1/2 ?1 q12 ?2 q22 ?3
q32
108
What do we do about this????? Hamiltonian Is in
diabatic basis, and q.c. calculates adiabatic
energies
T A 0 ?A1 q1
?A2 q2 1/2 ?1 q12 ?2 q22 ?3 q32
? q3 H ( ) (


) 0 TB
? q3
? ?B1q1 ?B2q1 1/2 ?1 q12 ?2 q22 ?3
q32
109
What do we do about this????? Hamiltonian Is in
diabatic basis, and q.c. calculates adiabatic
energies
T A 0 ?A1 q1
?A2 q2 1/2 ?1 q12 ?2 q22 ?3 q32
? q3 H ( ) (


) 0 TB
? q3
? ?B1q1 ?B2q1 1/2 ?1 q12 ?2 q22 ?3
q32
transform model Hamiltonian back to adiabatic
basis!
T A TAB ?A1 q1
?A2 q2 1/2 f1 q12 f2 q22 fA3 q32
0 H ( ) (


) TAB TB
0 ?
?B1q1 ?B2q1 1/2 f1 q12 f2 q22 fB3 q32
110
Now, it is fairly obvious that
?A1 ?A1 ?A2 ?A2
?B1 ?B1
?B2 ?B2 f1 f1
f2 f2
T is complicated like crap and it can be
shown that fA3 ?3 - 2 ?2/ ? O(?4) fB3
?3 2 ?2/ ? O(?4) which allows us to
calculate ? from the adiabatic coupling mode
force constants in the upper and lower states via
the simple formula
? 1/2 (fA3 - fB3) ?1/2
111
This equation is exact within the assumptions of
the quasidiabatic model, and the centrality of
the upper and lower force constants provide a
check on the assumptions implicit in the
model. An important point to practitioners is
that the upper and lower state adiabatic force
constants for the coupling mode must be evaluated
at the reference state geometry (not the
respective minima) and it must be evaluated in
dimensionless normal coordinates, which takes a
bit of betting used to and is not entirely
trivial (Sarah) For NO2, we find
fA3 0.293 eV
fB3 0.049 eV along
with
? 0.037 eV
?3 0.172 eV We find ?
0.091 eV
112
T A 0 ?A1 q1
?A2 q2 1/2 ?1 q12 ?2 q22 ?3 q32
? q3 H ( ) (


) 0 TB
? q3
? ?B1q1 ?B2q2 1/2 ?1 q12 ?2 q22 ?3
q32
And we are done. The Hamiltonian is completely
parametrized and we can calculate spectra now.
Note that this (quite simple and simplest) KDC
Hamiltonian has but eight parameters, but
contains most of the important physics. Lets
see how it does.