Franck-Condon Principle

Report
Chemistry 2
Lecture 10
Vibronic Spectroscopy
Learning outcomes from lecture 9
• Be able to qualitatively explain the origin of the Stokes and antiStokes line in the Raman experiment
• Be able to predict the Raman activity of normal modes by working
out whether the polarizability changes along the vibration
• Be able to use the rule of mutual exclusion to identify molecules
with a centre of inversion (centre of symmetry)
Assumed knowledge
Excitations in the visible and ultraviolet correspond to excitations of
electrons between orbitals. There are an infinite number of
different electronic states of atoms and molecules.
Which electronic transitions are allowed?





The allowed transitions are associated with electronic vibration
giving rise to an oscillating dipole
Electronic spectroscopy of diatomics
• For the same reason that we started our examination of IR spectroscopy
with diatomic molecules (for simplicity), so too will we start electronic
spectroscopy with diatomics.
• Some revision:
– there are an infinite number of different electronic states of atoms
and molecules
– changing the electron distribution will change the forces on the
atoms, and therefore change the potential energy, including k, we,
wexe, De, D0, etc
Depicting other electronic states
Excited Electronic States
1. Unbound
2. Bound
Ground Electronic State
There is an infinite number of excited
states, so we only draw the ones
relevant to the problem at hand.
Notice the different shape
potential energy curves including
different bond lengths…
Ladders upon ladders…
Each electronic state has its own
set of vibrational states.
De’
we’
Note that each electronic state
has its own set of vibrational
parameters, including:
- bond length, re
- dissociation energy, De
- vibrational frequency, we
De”
we”
Notice: single prime (’) = upper state
double prime (”) = lower state
re” re’
The Born-Oppenheimer Approximation
The total wavefunction for a molecule is a function of both
nuclear and electronic coordinates:
(r1…rn, R1…Rn)
where the electron coordinates are denoted, ri , and the
nuclear coordinates, Ri.
The Born-Oppenheimer approximation uses the fact the nuclei, being much
heavier than the electrons, move ~1000x more slowly than the electrons.
This suggests that we can separate the wavefunction into two components:
(r1…rn, R1…Rn) = elec (r1…rn; Ri) x vib(R1…Rn)
Total wavefunction = Electronic wavefunction at ×
each geometry
Nuclear wavefunction
The Born-Oppenheimer Approximation
(r1…rn, R1…Rn) = elec (r1…rn; Ri) x vib(R1…Rn)
Total wavefunction = Electronic wavefunction at × Nuclear wavefunction
each geometry
The B-O Approximation allows us to think about (and calculate) the motion
of the electrons and nuclei separately. The total wavefunction is
constructed by holding the nuclei at a fixed distance, then calculating the
electronic wavefunction at that distance. Then we choose a new distance,
recalculate the electronic part, and so on, until the whole potential energy
surface is calculated.
While the B-O approximation does break down, particularly for some
excited electronic states, the implications for the way that we interpret
electronic spectroscopy are enormous!
Spectroscopic implications of the B-O approx.
1. The total energy of
the molecule is the
sum of electronic
and vibrational
energies:
Etot = Eelec + Evib
Evib
Eelec
Etot
Spectroscopic implications of the B-O approx.
• In the IR spectroscopy lectures we introduced the concept of
a transition dipole moment:
μ 21    (ri , Ri )μˆ 1 (rj , R j )drdR  0
*
2
|2
transition
dipole
moment
upper state
wavefunction
lower state integrate
dipole
moment wavefunction over all
operator
coords.
|1
using the B-O approximation:
μ 21   2*vib ( Ri ) 2*elec (ri )μˆ  1elec (rj )1vib ( R j )drdR
  2*vib ( Ri )μ( R) 1vib ( R j )dR
2. The transition moment is a smooth function of the nuclear coordinates.
Spectroscopic implications of the B-O approx.
μ 21   
*vib
2
( Ri )
 
*vib
2
( Ri )μ( R)  ( R j )dR
|2
( Ri )  ( R j )dR
|1
 μ0  
*vib
2
*elec
2
elec
vib
ˆ
(ri )μ  1 (rj )1 ( R j )drdR
vib
1
vib
1
2. The transition moment is a smooth function of the nuclear coordinates. If
it is constant then we may take it outside the integral and we are left with a
vibrational overlap integral. This is known as the Franck-Condon
approximation.
3. The transition moment is derived only from the electronic term. A
consequence of this is that the vibrational quantum numbers, v, do not
constrain the transition (no Dv selection rule).
Electronic Absorption
There are no vibrational selection
rules, so any Dv is possible.
But, there is a distinct favouritism
for certain Dv. Why is this?
Franck-Condon Principle (classical idea)
“Most probable bond length for a
molecule in the ground electronic
state is at the equilibrium bond
length, re.”
Energy
Classical interpretation:
0
1
2
3
R
4
5
The Franck-Condon Principle
states that as electrons move
very much faster than nuclei, the
nuclei as effectively stationary
during an electronic transition.
Energy
Franck-Condon Principle (classical idea)
In the ground state, the
molecule is most likely in v=0.
0
1
2
3
R
4
5
•The Franck-Condon Principle
states that as electrons move
very much faster than nuclei, the
nuclei as effectively stationary
during an electronic transition.
The electron excitation is
effectively instantaneous; the
nuclei do not have a chance to
move. The transition is
represented by a VERTICAL ARROW
on the diagram (R does not
change).
Energy
Franck-Condon Principle (classical idea)
0
1
2
3
R
4
5
•The Franck-Condon Principle
states that as electrons move
very much faster than nuclei, the
nuclei as effectively stationary
during an electronic transition.
The most likely place to find an
oscillating object is at its turning
point (where it slows down and
reverses). So the most likely
transition is to a turning point on
the excited state.
Energy
Franck-Condon Principle (classical idea)
0
1
2
3
R
4
5
Quantum (mathematical) description of FC
principle
*vib
vib
μ 21  μ 0  2 ( Ri ) 1 ( R j )dR
approximately constant
with geometry
Franck-Condon (FC)
factor
μ21 = constant × FC factor
FC factors are not as restrictive as IR selection rules (Dv=1). As a result there are
many more vibrational transitions in electronic spectroscopy.
FC factors, however, do determine the intensity.
Franck-Condon Principle (quantum idea)
In the ground state, what is the most likely position to find the nuclei?
3
v
Prob   2
2
1
Max. probability at Re
(0)
0
v=0
0
1
2
R
3
4
Franck-Condon Factors
If electronic excitation is much faster
than nuclei move, then wavefunction
cannot change. The most likely
transition is the one that has most
overlap with the excited state
wavefunction.
2
1
v’ = 0
0
1
2
Wave number
3
4
v” = 0
Look at this more closely…
Negative overlap in middle
Positive overlap at edges
overall very small overlap
Negative overlap to left, postive overlap
to right
overall zero overlap
• Excellent overlap
everywhere
Franck-Condon
Factors
1
2
0
3
Wave number
4
Franck-Condon
Factors
v=10
Note: analogy with classical
picture of FC principle!
v. poor v=0 overlap
Electronic Absorption
There are no vibrational selection
rules, so any Dv is possible.
Relative vibrational intensities
come from the FC factor
μ21 = constant × FC factor
Absorption spectrum of binaphthyl
•Example of real spectra showing FC profile
16
17
15
18
19
20
21
22
14
23
13
24
12
25
11
26
27
28
10
9
(3) (4) (5)
30100
6
7
30200
8
30300
30400
30500
30600
-1
Wave number (cm )
30700
30800
30900
Absorption spectrum of CFCl
= CCl2 peaks
(0,n,0)
(0,n,1)
(0,n,2)
(1,n,0)
3
3
2
(0,n,0)
(0,n,1)
(0,n,2)
5
5
4
17000
18000
}
10
(0,0,1) hot bands
19000
20000
-1
Wave number (cm )
21000
Unbound states (1)
If the excited state is dissociative, e.g. a
p* state, then there are no vibrational
states and the absorption spectrum is
broad and diffuse.
Unbound states (2)
Even if the excited state is bound, it is
possible to access a range of vibrations,
right into the dissociative continuum.
Then the spectrum is structured for low
energy and diffuse at higher energy.
Some real examples…
HI
A purely dissociative state
leads to a diffuse spectrum.
Some real examples…
I2
The dissociation limit observed in
the spectrum!
0.25
I2
Absorbance
0.20
0.15
0.10
0.05
0.00
16000
18000
20000
-1
Wave number (cm )
22000
Analyzing the spectrum…
All transitions are (in principle) possible. There is
no Dv selection rule
Vibrational structure
0.25
I2
Absorbance
0.20
0.15
0.10
0.05
0.00
16000
18000
20000
-1
Wave number (cm )
22000
Analysing the spectrum…
v”
0
v’
25
cm-1
18327.8
0
26
18405.4
0
27
18480.9
0
28
18555.6
0
29
18626.8
0
30
18706.3
0
31
18780.0
0
32
18846.6
0
33
18911.5
0
34
18973.9
0
35
19037.5
(

(

2
1
1
G( v)  v  2 we  v  2 we xe
How would you solve this?
(you have too much data!)
1. Take various combinations of v’ and
solve for we and wexe simultaneously.
Average the answers.
2. Fit the equation to your data (using XL
or some other program).
Analyzing the spectrum…
Etot  Eelec  Evib
2
Etot  Eelec  G( v)  Eelec  (v  1 2 we  (v  1 2  we xe
v”
0
v’
25
cm-1
18327.8
0
26
18405.4
0
27
18480.9
0
28
18555.6
0
29
18626.8
0
30
18706.3
0
31
18780.0
0
32
18846.6
0
33
18911.5
0
34
18973.9
10
0
35
19037.5
Dissociation energy = 19950 cm-1
20000
-1
Wave number (cm )
19500
19000
18500
Eelec = 15,667 cm-1
18000
we = 129.30 cm-1
wexe = 0.976 cm-1
17500
17000
20
30
40
50
v'
60
70
Learning outcomes
• Be able to draw the potential energy curves for excited electronic
states in diatomics that are bound and unbound
• Be able to explain the vibrational fine structure on the bands in
electronic spectroscopy for bound excited states in terms of the
classical Franck-Condon model
• Be able to explain the appearance of the band in electronic
spectroscopy for unbound excited states
The take home message from this lecture is to understand the
(classical) Franck-Condon Principle
Next lecture
• The vibrational spectroscopy of polyatomic molecules.
Week 12 homework
• Vibrational spectroscopy worksheet in tutorials
• Practice problems at the end of lecture notes
• Play with the “IR Tutor” in the 3rd floor computer lab and with the
online simulations:
http://assign3.chem.usyd.edu.au/spectroscopy/index.php
Practice Questions
1. Which of the following molecular parameters are likely to change when a molecule is
electronically excited?
(a) ωe (b) ωexe (c) μ (d) De (e) k
2.
Consider the four sketches below, each depicting an electronic transition in a diatomic
molecule. Note that more than one answer may be possible
(a) Which depicts a transition to a dissociative state?
(b) Which depicts a transition in a molecule that has a larger bond length in the excited
state?
(c) Which would show the largest intensity in the 0-0 transition?
(d) Which represents molecules that can dissociate after electronic excitation?
(e) Which represents the states of a molecule for which the v”=0 → v’=3 transition is
strongest?

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