LAB: FRANCK-CONDON PRINCIPLE
I2 Vibronic Absorption
A UV/VIS absorption experiment typically involves the transition of an electron from a
molecular ground state (So) to the first excited singlet state (S1). Once excited, the molecule will
relax at some later time, possibly giving off a photon in the form of optical emission.
It might be reasonable to predict that the optical absorption of a molecule undergoing an So → S1
transition would occur at the same energy as the emission S1 → So. In practice, however, one sees
a shift in energy (or inverse wavelength) between the absorption and emission spectra. This is due
to the contribution of the vibrational states that accompany the electronic transitions.
A molecule at room temperature normally begins in its ground electronic state (So) and lowest
vibrational state (v=0).
1) A ground-state molecule can absorb a photon and undergo a transition to its excited S1
state. But rather than being limited to a pure electronic transition, it can also absorb
photons having the additional energy needed to reach one of the molecule’s upper-tier
vibrational levels (v’ ). This coupling of an electronic transition and the simultaneous
transition to a new vibrational state is called a vibronic transition.
2) Once a molecule is in this excited vibronic state, it immediately relaxes back down to the
lowest vibrational level in the S1 state, releasing energy in the form of heat.
3) After a delay the electron relaxes back down to its ground electronic state, So. But similar
to absorption, this relaxation to So can involve the lower vibrational levels (v) rather than
exclusively undergoing emission back to the very lowest So v=0 state.
The result of this 3-step process is that the photons absorbed by the molecule are typically higher
in energy than the photons that are emitted by the molecule due to the involvement of the
vibrational levels.
Vibrational Relaxation
v’
So
Absorption
?
Emission
Absorption
S1
v
Emission
0-0’
?
← Transition Energy
Wavelength →
This discussion explains why the absorption and emission bands are split apart. But what is still
missing is an explanation for why the maxima for the two peaks are shifted away from the 0-0’
or 0-1’ transitions. To answer this question we must look at the impact of the Franck-Condon
Principle on the wavefunction transition moment.
Gentry, 2015
OPTICAL SELECTION RULES FOR VIBRATIONAL TRANSITIONS
Pure Vibrational Transition: In a typical FTIR experiment, the instrument monitors the
transition between different vibrational states in a single potential energy well. Under these
conditions the intensity of an optical transition from one vibrational state to another is controlled
by the transition moment.
(1)
where ψ i and ψ f are the wavefunctions for the initial and final
states respectively, and µ is the molecular dipole moment. So long as
the system stays within the same potential energy well the only time
that the integral in (1) is non-zero is if the vibrational transition
obeys the selection rule ∆v = ±1 where v is the vibrational quantum
number. For all other transitions, e.g. v = 0→2, the transition
moment goes to zero and the transition will not be visible in an
absorption experiment.
Single Potential Well
Potential Energy, V(r)
µif = ∫ψ *f ⋅ µ ⋅ψ i dτ
3 v
2
1
∆v =+1
0
re
Internuclear Distance, r
Two Potential Wells
Excited Electronic
State, E ’
v’
Energy
Vibronic Transition: This ∆v = ±1 selection rule breaks down,
however, if the vibrational transition is coupled to an electronic
transition. The process of moving an electron from one molecular
orbital to another causes a change in the electron distribution around
the nuclei. This change in electron distribution, in turn causes a
change in the bond strength, and thus a change in the bond distance,
the bond dissociation energy, and the vibrational frequency.
Ground Electronic
State, E
v
re
re’
Internuclear Distance
FRANCK-CONDON PRINCIPLE FOR VIBRONIC TRANSITIONS
This shift in equilibrium bond length and the timing for when it occurs are very important when
it comes to how vibrational wavefunctions interact with one another.
Franck-Condon Principle
The Franck-Condon Principle states that an optical vibronic
transition is essentially instantaneous compared to the relatively
slow response of the nuclei shifting to new equilibrium positions.
2) a much slower step that occurs at a later time when the nuclei
shift to their new equilibrium positions.
Franck-Condon Factors
1) Absorption
1) very fast optical absorption to the excited electronic state
while the nuclei remain in their ground-state positions; and
Energy
This means that a vibronic transition occurs in two discrete steps:
2) Shift in Nuclei
re
re’
Internuclear Distance
Page 2 of 10
Franck-Condon Factor
The sideways translation in the potential energy curves that occurs between two different
vibronic states leads to optical absorbance values that are very different in strength than those
seen with pure vibrational transitions. The Franck-Condon Principle says that the optical
absorption (or emission) for a vibronic transition will be proportional to the square of the overlap
integral of the two vibrational wavefunctions. This squared integral is called the Franck-Condon
Factor (FCF). The total optical strength is then given by the FCF times a constant electronic
interaction term.
2
FCF =  ∫ψ *f ⋅ψ i dτ 
vib
(2)
The overlap integral is strongly affected by the offset between the equilibrium bond lengths for
the ground-state and excited-state potential energy wells. This can be seen in the figure below
which looks at the overlap between v=1 and v’=4 states for wavefunctions that are either
centered at the same re position or with one function offset to the side from the other. In the first
case the overlaps cancel and FCF = 0. In the second case the offset in positions means that the
overlaps no longer cancel and there is a sizeable FCF interaction.
Two wavefunctions
centered at
same r position
One wavefunction
offset (centered at
different r position
v1
v1
v4
v4
offset
v1* v4
∫ (v1*v4) = 0
overlaps
cancel out
v1* v4
overlaps
do not
cancel
∫ (v1*v4) > 0
The result of the offset in vibronic wavefunctions is that :
- There will be a large number of v→v’ transitions that all have sizeable absorbance
values.
- The maximum absorbance will occur for the transition that has the largest overlap
between the ground-state v=0 wavefunction and the offset excited-state v’
wavefunction. This can occur at a relatively large v’ value.
Franck-Condon Factors
Page 3 of 10
Approximation Technique to Locate the Maximum Franck-Condon Overlap
An approximate way of finding the excited quantum state having the maximum overlap with the
ground state v=0 state is shown below. This quantum level should correspond to where the edge
of the excited-state Morse potential curve has the same internuclear separation distance as the
center of the ground-state well. That is to say, one can draw a vertical line from the center of the
bottom ground-state potential well and look for where it intersects the upper excited-state
potential curve. The excited vibrational quantum level that matches the energy of the
intersection point should correspond to the observed maximum in an absorption spectrum, v’max.
Absorption Experiment
FranckCondon
intersection
point
Excited Electronic
State
vmax’
Energy
v’max
0’
v’
0
0-vmax’
Ground Electronic
State
optical
transition
strong
overlap
0’
v
little
overlap
0
0-0’
0
← Transition Energy
re
re’ Internuclear Distance
Wavelength →
MORSE POTENTIAL
A Franck-Condon analysis requires one to be able to calculate the vibrational energies for the
different electronic states. The potential energy surface for molecular vibrations can be
described using the Morse potential.
(
1 − e −α ( r − re )
)
2
(3)
De is the dissociation energy. It describes the difference in energy
from the very bottom of the potential energy well up to the
dissociation plateau. re is the equilibrium bond length, i.e. the
internuclear distance corresponding to the bottom of the potential
energy well. α is a factor that controls the width of the well and is
dependent on the vibrational frequency and the reduced mass of the
molecule.
Franck-Condon Factors
Potential Energy, V(r)
V ( r ) = De
Morse
potential
3
2
1
0 v
De
re
Internuclear Distance, r
Page 4 of 10
Given that a molecular vibration is controlled by the Morse potential, the energies for the various
vibrational quantum levels can then be determined.
1
ɶ vib = ɶve  v + 1  − ɶve χ
ɶ  v + 
E


2
2


2
v = 0,1,2,…
(4)
When the energy equation is written this way with the use of the ~ headers it is based on
spectroscopic wavenumbers (cm-1) rather than pure SI-unit energy.
The transition energy, ∆Eɶ v −v ' , from a ground-state vibrational state ( v) to an excited
electronic/vibrational state ( v’ ) depends on the vibrational energies of the initial and final states
(Evib and E’vib) as well as on the electronic transition energy, Te’. Te is measured from the bottom
of the ground electronic potential well to the bottom of the excited-state electronic potential well.
∆Eɶ v −v ' = Tɶe ' + Eɶ 'vib  − Eɶ vib
(5)
The inter-relationships between diatomic molecular constants and the constants used in Eqn’s 3
and 4 can be shown to give the following equations:
Eɶ vib , ɶve , Dɶ e , and Tɶe
µ =
ma ⋅ mb
ma + mb
α = ɶve π
ɶ =
χ
Franck-Condon Factors
ɶve
ɶe
4D
2µ c
ɶe
hD
given in units of cm-1
ma and mb are mass of atoms a and b in kg
α calculated in units of meters-1
c is the speed of light, in cm/s
h is Planck’s constant, in J·s
ɶ = anharmonicity constant, with
χ
dimensionless units in this format
Page 5 of 10
Experimental
In this experiment you will experimentally measure the vibronic absorption spectrum for Iodine,
I2(g). You will then compare your experimental result to calculated predictions based on the
Franck-Condon Principle. If you already measured the spectrum of I2 earlier in the semester you
can skip over the experimental section.
To measure the spectrum, put a few crystals of I2 in a cuvette and then seal it with a cap. You
will then need to warm the iodine so that it sublimes to vapor. This can be done with a heat gun.
It helps to warm both the cuvette and the sample holder in the spectrometer otherwise the large
thermal mass of the sample holder will cool the cuvette before you have a chance to record the
spectrum.
Use the Unicam UV/Vis spectrometer to record the absorbance spectrum. Do a quick spectral
run to observe where the primary absorption peak is located. This can be done by setting the
instrument to scan speed = “Intelliscan”, data interval = normal, and bandwidth = 1nm. Try a
wavelength range of 350 – 700 nm first.
Once you have a spectrum, record a second, higher-quality spectrum over the active absorption
region for your sample. You will want to record the entire span of the absorption band. For this
scan, set the instrument so that the data interval is set to the highest resolution and the bandwidth
to 0.2 nm. Export the data as an ASCII batch file when done.
•
Include a copy of the spectrum in your report as well as report the wavelength location of the
maximum absorbance. Do this even if you included the spectrum in your earlier lab report.
Calculations
We will analyze the data using two different techniques. We will first use Excel to calculate the
ground-state and excited-state Morse potentials for I2. We will use these curves along with the
Franck-Condon Approximation technique to estimate the energy, wavelength, and quantum
number for the location of maximum absorbance. Secondly, we will use a Java application1 to
explore the individual Franck-Condon Factors.
These calculations require that you know the vibronic constants for Iodine.
Literature Values2 for I2(g)
Term
Symbol
1
Ground State, X
Excited State, B
3
Σ
+
g
Π 0 +u
Dɶ e
(cm-1)
(Å)*
Tɶe
(cm-1)
18713.
vɶe
(cm-1)
214.502
2.6663
0
5168.7
125.697
3.0247
15769.01
re
* Note that the equation for alpha (α) for use in Eqn (3) will give a result in m-1. This will need to
be converted to Angstroms-1 (Å-1) if the units are going to cancel with re in the equation for α.
Franck-Condon Factors
Page 6 of 10
A) Excel Analysis
•
Use the equation for the Morse potential (Eqn. 3) to calculate the potential energies V(r)
for both the ground-state and excited state vibrations. Do this over a range of
r = 2.2 to 6.0Å. Use increments of 0.01 Å for your data table, but insert an additional
row in the middle so as to include the ground state equilibrium distance r = re = 2.6663Å
point in your calculations.
•
Add the excited-state electronic energy, Te’ to the excited-state vibrational curve to get
the total energy for the excited vibronic system. E’tot = E’vib+Te’
•
Plot the ground-state and excited-state energy curves on the same graph showing energy
on the y axis and internuclear separation distance (r) on the x axis. Limit your y-axis
scale to a maximum of 30,000 cm-1 so that you can more easily see the differences in the
potential wells for the two states. Your graph should resemble the figure that shows the
two Morse potentials on page 2.
To double check your calculations, for r = 4.0 Å,
Ground X state: Evib ≈ 14113cm-1, Excited B State: E’vib+Te’ ≈ 19149cm-1
Use Franck-Condon Approximation Technique to Locate Maximum Wavefunction
Overlap
•
On your graph of energy vs. internuclear distance, draw a straight vertical line on your
graph starting from the center re position of the lower potential energy curve and
proceeding upwards until it intersects the upper energy curve. Include this graph and line
in your report.
•
Use your table of calculated values to actually determine the values of E’vib+Te’ and E’vib
at that intersection point of r = re,ground state .
•
Use Eqn. 4 to determine the quantum number corresponding to that excited vibrational
state. [The easiest way to do this may be to try various values of quantum number v’
until the quantum energy E’vib(v) approximately matches your observed intersection E’vib
value. Alternatively you could use the Solver add-in for Excel.]
•
Using Eqn. 5, what is the energy of transition ( ∆Eɶ v −v ' , in cm-1) for that point? [Do not
forget to account for the zero-point energy of the ground-state vibration and the
electronic transition energy.]
•
What is the wavelength of light corresponding to that 0-v’max transition?
•
By contrast, what is the energy and the wavelength for the 0-0’ transition?
•
What is the shift in wavelength (in nm) from the 0-0’ origin to the 0-v’max peak?
•
Report these results in a table in your report and compare the transition wavelengths to
the observed locations and absorbances seen in your experimental spectra.
Franck-Condon Factors
Page 7 of 10
B) Java Analysis
S.W. North and his colleagues at Texas A&M University have prepared a handy Java program
that plots and analyzes Franck-Condon factors. 1
•
Go to our class website and do a right-click “Save Target as…” to download the
“Java.FCIntensity.jar” file onto your computer. If you are using one of the departmental
computers you may need to have me do this for you (with my administrator access).
•
Start the program by double clicking on the icon.
•
You should see two panels appear. The main panel on the left is where you enter the
molecular constants. In order to model your absorption results you will want to put the
ground-state values into the boxes marked “Initial State”. The excited-state constants are
entered in the boxes for “Final State”. You will also need to tell the program what the
starting vibrational constant is. This should be v=0. (Why?) The final state should be a
range of v’ = 0 to 60. Lastly you will need to insure that the masses for the two nuclei in
the diatomic model are entered correctly.
•
Click on “Calculate”. You should now see a plot of the two energy curves at the top of
the panel and the relative intensities for all of the different 0-v’ transitions in the center of
the panel. Estimate the wavelength corresponding to the peak maximum? How do your
experimental peak location and peak width compare to this calculated spectrum?
•
The “Wavefunction” panel on the right allows you to take a closer look at specific
transitions based on the molecular constants that you entered on the other panel. In the
center of the Wavefunction panel you can enter a starting quantum number (v=0) and an
excited-state final quantum number (v’ ). When you click on the “Plot” button at the
bottom, the program will show you the two wavefunctions (offset in space according to re
and re’ ). The graph at the bottom shows the result of multiplying one wavefunction
times the other wavefunction. At the very bottom of the panel you will see boxes that
report the wavelength and the integrated Franck-Condon Factor for that transition.
•
Using the Wavefunction panel, repeatedly try various values of v’ until you find the
maximum value for the Franck-Condon Factor. As you change the value of v’ note how
the overlap changes between the two wavefunctions. Once you have found the v’ value
that gives the largest FCF, report the wavelength, and quantum number for the transition.
The wavelength should match the location of the peak maximum on the left panel.
•
The main panel on the left has a File menu tab that allows you to print the two panels.
Your lab report should include a print of the calculated spectrum on the left and the
wavefunction transition on the right showing the maximum overlap.
•
When you are done with the analysis of the maximum overlap, use the Wave Function
panel to determine the FCF and wavelength for the 0-0’ transition. Note the difference in
how the peaks do or do not overlap as compared to the maximum value.
Franck-Condon Factors
Page 8 of 10
Questions
Comment on the comparison of your experimental data to the two calculated results (Excel and
Java). How closely do the locations of the peak maxima agree? Discuss the advantages and
shortcomings of the different approaches.
How far in wavelength (nm) did the peak maxima shift from the original 0-0’ transition? What
would your absorption spectrum have looked like if the system limited itself to the ∆v = ±1.
What did you observe with the Java program when you compared the wavefunctions and their
overlaps for the 0-0’ versus 0-v’max transitions?
Observation of Hot Bands in Experimental Spectra
If you take a closer look at your experimental spectrum in the range of 530-590nm, you will see
the emergence of a second set of peaks, and eventually a third set, appearing at longer
wavelengths. Normally molecules start in their lowest vibrational level because there is not
enough thermal energy available to populate higher vibrations. In the case of Iodine, however,
the vibrational energy gap is small enough that now there is enough thermal energy to generate
some small amount of additional population in the next two higher vibrational levels. Vibronic
transitions starting from these higher vibrational levels are called “hot bands” .
0-v’
Iodine ( I2 )
v’
Energy
Absorbance
S1
1-v’
0-v’
1-v’ hot band
2-v’
v
So
530
540
550
560
570
Wavelength (nm)
580
590
- Plot your high-resolution UV/Vis data in the range of 530-590nm.
- Use the Boltzmann distribution (beginning of first semester Thermodynamics) to
calculate the population probability ratio of the v=1 versus v=0 vibrational levels
in the ground electronic state in I2. Report your results in terms of the Boltzmann
ratio, P1 / P0 = e − ( E1 − E0 )/ kT . Comment on the amount of population seen in the
higher vibrational levels for the I2 molecule and how this affects your ability to
observe hot bands.
- Given that you have shown that I2 has a sizeable population distribution in the v=1,
return to the Java program and on the main panel on the left change the starting
quantum number from v=0 to v=1. [You may get an error message that will
require you to limit your range of excited-state quantum numbers. You might
recall that there is an upper limit v’ level, beyond which the bond breaks rather
than vibrates.]
Franck-Condon Factors
Page 9 of 10
- Similarly change the starting wavelength on the Wavefunction panel on the right to
find the maximum wavelengths and quantum numbers for the two hot bands.
- Lab-report question: Is the observed long-wavelength shoulder in your experimental
spectrum generally consistent with the predicted locations and populations of the
hot bands?
References
1
Gucchese, G., Lucchese, R.R., North, S.W.; J. Chem. Ed. 2010, 87, 345.
2
Bother, J.R., Tainter, C.J.; J Phys Chem Lab 2006, 10, 55.
Franck-Condon Factors
Page 10 of 10