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Electronic Spectroscopy
Born-Oppenheimer Principle
When we were looking at bonding, the wavefunction that we used had no dependence on the
internuclear distance. We used such wavefunctions without justification and were implicitly
assuming that the Born-Oppenheimer principle was valid.
The Born-Oppenheimer principle states that the we can find the wavefunction for the
electrons in molecule without considering the motion of the nuclei.
Reasoning behind the Born-Oppenheimer principle
- The nuclei are much, much heavier than the electrons.
- Thus the electrons respond much, much quicker to changes than the nuclei.
- Thus we can separate the electronic motion from the nuclear motion.
- Such a separation implies that we can write the total wavefunction of the system as the
product of an electronic wavefunction and a nuclear wavefunction.
- The electron wavefunctions would be the wavefunctions we used in VB or MO theory.
- The nuclear wavefunction would the harmonic oscillator wavefunctions as we used in
the vibrational spectroscopy section.
molecule  r, R   electronic  r  nuclear  R 
In the analysis of the electronic structure of a molecule, one does not need to account for the
motion of the nuclei. (We still need to include the distance of the nuclei.)
Sometimes the nuclear motion is important and creates what is called vibronic coupling that
is magnetic effect similar to spin-orbit coupling. We will not discuss vibronic coupling
further.
Franck-Condon Principle
Introduction
Electronic transitions occur when electrons change their arrangement in an atom or molecule.
The Franck-Condon principle states that electronic transitions in molecules occur before
nuclei have time to adjust.
One of the consequences of the Franck-Condon principle is that transitions between
electronic states is most likely to occur when the nuclei are “stationary”.
Nuclei are “stationary” where the vibrational wavefunction has the highest probability
density
1.) For ground vibrational state, maximum probability occurs at equilibrium bond
distance.
2.) For excited vibrational state, maximum probability occurs at turning points.
- Turning points are positions where the motion of particle reverses itself.
- For oscillators, the turning points are at the boundary of the potential.
- I.e., turning point is where kinetic energy is zero and energy of particle is all
potential energy.
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Spectral Line Intensities
Spectral line intensities are proportional to the Einstein absorption coefficient. I  B
However the Einstein absorption coefficient is related to the transition dipole matrix element.
2
f ˆ i
I
6 0 2
**Generally, the greater the value of the transition dipole matrix element, f ˆ i , the more
intense the transition, f  i.**
Franck-Condon Factors
Spectral line intensities of electronic transitions depend on the transition dipole matrix
element.
However wavefunctions of final and initial states have an electronic portion and a vibrational
portion. I.e., i   ie  i
- This separation occurs
Also the dipole operator has an electronic portion and a nuclear portion.
   e rj   ZK e R K
j
K

- sum of j electrons
j

- sum of K nuclei
K
Calculate the transition dipole matrix element.


f ˆ i   ef ,  f ˆ  ie ,  i   ef ,  f   erj   ZK eR K   ie ,  i
K
 j



  ef  f   erj   ZK eR K   i  ie
K
 j

  ef
 er
j
 ie  f  i   fe  ie  f
j
Z
K
eR K  i
K
The electronic states are orthogonal, that is,  ef  ie  0
f ˆ i  ef
 er
j
ie f i
j
Note: Why aren’t vibrational wavefunctions are orthogonal?
- The wavefunctions come from “different molecules”.
- Geometry of excited state is different from the geometry of the ground state.
 f i
2
are known as Franck-Condon factors (FCF)
- Intensity of spectral line related to the overlap of vibrational wavefunctions.
- Ideally FCF can be calculated analytically, but practically they are either taken from
experiment or calculated numerically.
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E
S1
S0
R
Note in figure above, the greatest overlap of vibrational wavefunctions is between the
 = 0 of the electronic ground state and the  = 2 or the first excited singlet state.
- The FCF,  f  2 i  0
2
is the largest.
Energy Flow in Electronic Excited States
Optical Transitions
Fluorescence – emission between electronic states with the same spin state.
Decay of excited state singlet state to ground state (singlet state, usually)
Transition occurs quickly, lifetime of excited state is short,  10-9 – 10-5 sec.
Phosphorescence – emission between electronic states with different spin states.
Decay of triplet state to ground state
Example of a spin-forbidden transition.
- Ideally S = 0 in electronic spectroscopy
- Because of spin-orbit coupling, S = 0 as a selection rule is broken.
Transition occurs slower, lifetime of triplet state is longer than excited singlet state
-  10-4 – 101 sec.
Radiationless Transitions
Vibrational Relaxation
Vibrationally excited molecule can transfer energy to other molecules via collisions.
Intersystem Crossing (ISC)
When a molecule in a specific spin state and vibronic state has a transition to a new
vibronic state in a different spin state with the same energy.
Internal Conversion (IC)
When a molecule in a specific spin state and vibronic state has a transition to a new
vibronic state in a same spin state with the same energy.
For ISC and IC, the molecules with a specific arrangement of electrons and nuclei have
multiple vibronic wavefunctions to describe state.
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Jablonski Diagram
Jablonski diagram illustrates energy flow in electronic excited states.
S0
S1
IC
ab
sor
pti
on
flu
ore
s ce
nc
e
ISC
T0
VR
ISC
VR
nce
s ce
e
r
o
sph
pho
VR
Disassociation of Vibronic States
Two possibilities for the disassociation of vibronic states will be considered.
1.) Increasing vibrational level leads to dissociation.
- Potential energy approaches constant value as R increases.
- Thus vibronic states have a practical limit as to the number of their vibrational states.
energy continuum
dissociation limit
quantized states
- If energy is absorbed into molecule such that the transition is above the dissociation
limit, the molecule dissociates. The energy of the molecule is somewhere in the
continuum.
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2.) Excitation to unbound electronic state leads to dissociation.
- Worth emphasizing is that not all electronic states are bound states.
Examples of unbound electronic states
1 
g for He2.
3
 g for H2.
E
g
3
1
+
g+
R
Note lowest energy of unbound state is at R = .
- Sometimes molecules can dissociate because of ISC or IC to an unbound state.
E
crossing from bound state to
unbound state may occur
R