Chemistry 2. Lecture 6 Vibrational Spectroscopy. Learning outcomes from lecture 5. Be able to draw the wavefunctions for the first few solutions to the Schrödinger equation for the harmonic oscillator
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Learning outcomes from lecture 5
Assumed knowledge
Light behaves like an oscillating electromagnetic field. The electric field interacts with charges. Two charges separated in space represent a dipole moment which can interact with an electric field. Energy can only be taken or added to the electric field in units of hn (photons).
Because light is an oscillating electromagnetic field, it can cause charges to oscillate. If the charge can oscillate in resonancewith the field then energy can be absorbed.
Alternatively, an oscillating charge can emit radiation with frequency in resonance with the original oscillation.
For example, in a TV antenna, the oscillating EM field broadcast by the transmitter causes the electrons in your antenna to oscillate at the same frequency.

+
+
Classical absorption of lightWhat about a molecule as an antenna?
How can we get oscillating charges in a molecule?
n = 0,1,2,…
S.I. units
Need to know!
G(v) & w in cm1
The Quantum Harmonic OscillatorSolution of Schrödinger equation:
But:
Oscillation (vibrational) frequency
G(v) is the “energy” from the bottom of the well and w is the “harmonic frequency”
The Morse Anharmonic Oscillator
all in wavenumber (cm1) units:
This one you DO have to know how to use
In spectroscopy, we tend to use the letter “vee” to indicate the quantum number for vibration. The vibration frequency is indicated by we in cm1. When solving the general quantum mechanical problem we used the letter n, to minimize confusion with “nu”, the vibrational frequency in s1.
Anharmonicity constant
Anharmonic term
Harmonic term
wexe is usually positive, so vibrational levels get closer together
The force constant for the AHO is the same as for the harmonic case:
The Morse energy levelsV (r)
or
r
The level of approximation to use depends on:
i) the information you have
ii) the information you need
D0
But we know about zeropoint energy, therefore slightly less energy is required to break the bond.
De
D0 = De – G(0)
We can estimate the bond dissociation energy from spectroscopic measurements!
Dissociation energyV (r)
This is the energy from the bottom of the well to the dissociation limit.
G(0)
r
(Zeropoint energy)
lower state
molecular dipole
vibrationalcoordinate
Selection rulesselection rule:
Dv=±1
Dv=+1: absorption
Dv=1: emission
Harmonic Oscillator Selection RulesIf an oscillator has only one frequency associated with it, then it can only interact with radiation of that frequency.
Selection rules limit the number of allowed transitions
Much of IR spectroscopy can understood from just these two results of the quantum harmonic oscillator:
E = (v+½)hn and Dv = ±1
Thermal populationAt normal temperatures, only the lowest vibrationalstate(v =0 ) is usually populated, therefore, only the first transition is typically seen.
Transitions arising from v0 are called “hot bands”
(Their intensity is strongly temp. dependent)
v = 1 0
which suggests transitions results of the quantum
might be
0 1
0 2
0 3
What are the new peaks?v=4
v=3
These are almost 1 : 2 : 3
v=2
v=1
v=0
H.O.
There are none!
But!...
Harmonic and anharmonic models are very similar at low energy, so selection rules of AHO converge on HO as the anharmonicity becomes less:
A.H.O.
A.H.O. selection rule:
Dv=±1,±2, ±3
Intensity gets weaker and weaker (typically 10× weaker for each)
A.H.O. selection rule:
Dv= ±1,±2, ±3
Dv = 1 : fundamental
Dv = 2 : first overtone
Dv = 3 : second overtone, etc
Fundamental:
2143 = G(1) – G(0),
Overtone:
4260 = G(2) – G(0)
G(1)G(0) = [(1.5)we–(1.5)2wexe]  [(0.5)we– (0.5)2wexe]
2143= we– 2wexe …(1)
G(2)G(0) = [(2.5)we–(2.5)2wexe]  [(0.5)we – (0.5)2wexe]
4260= 2we– 6wexe …(2)
Two simultaneous equations (simple to solve)
→we= 2169 cm1, andwexe = 13 cm1
V results of the quantum (r)
D0
De
G(0)
r
(Zeropoint energy)
Using the spectra to get information…89,400 cm1 = 1069 kJ/mol
c.f. exp. value: 1080 kJ/mol
Why the difference?
Remember Morse is still an approx. to the true intermolecular potential. Still 2% error is pretty good for just 2 measurements!
The vibrational spectroscopy of polyatomic molecules.
Week 11 homework
6. For a Morse oscillator the observed dissociation energy, D0, is related to the equilibrium vibrational frequency and the anharmonicity by the following expression:
a) ωe2/4ωexe b) [ωe2/4ωexe]G(0) c) (v+½)ωe+(v+½)2ωexe d) (v+½)ωe
7. Which of the following statements about the classical and quantum harmonic oscillator (HO) are true (more than one possible answer here)?
a) The classical HO frequency is continuous, whereas the quantum frequency is discrete.
b) The classical HO has continuous energy levels, whereas the quantum HO levels are discrete.
c) The classical HO depends on the force constant, but the quantum HO does not.
d) The classical HO may have zero energy, but the quantum HO may not.
e) The classical HO does allow the bond to break, whereas the quantum HO does.