1. Harmonic Oscillator

2. Harmonic Oscillator

3. Permanent Dipole moment
Selections rules
Permanent Dipole moment
An electric dipole consists of two electric charges q and -q separated by a distance R. This arrangement of charges is represented by a vector, the electric dipole moment  with a magnitude: Re
+q
-q 
= Re q
Unit: Debye, 1D = 3.33×10-30Cm
When the molecule is at its equilibrium position, the dipole moment is called “permanent dipole moment” 0.

4. Electric dipole moment operator
Selections rules
Electric dipole moment operator
 The probability for a vibrational transition to occur, i.e. the intensity of the different lines in the IR spectrum, is given by the transition dipole moment fi between an initial vibrational state i and a vibrational final state f :
The electric dipole moment operator depends on the location of all electrons and nuclei, so its varies with the modification in the intermolecular distance “x”. 0 is the permanent dipole moment for the molecule in the equilibrium position Re

5. The two states i and f are orthogonal.
The higher terms can be neglected for small displacements of the nuclei
The two states i and f are orthogonal.
Because they are solutions of the operator H which is Hermitian

6. Second condition:
First condition: fi= 0, if ∂/ ∂x = 0
In order to have a vibrational transition visible in IR spectroscopy: the electric dipole moment of the molecule must change when the atoms are displaced relative to one another. Such vibrations are “ infrared active”. It is valid for polyatomic molecules.
By introducing the wavefunctions of the initial state i and final state f , which are the solutions of the SE for an harmonic oscillator, the following selection rules is obtained:
 = ±1

7. Note 1: Vibrations in homonuclear diatomic molecules do not create a variation of   not possible to study them with IR spectroscopy.
Note 2: A molecule without a permanent dipole moment can be studied, because what is required is a variation of  with the displacement. This variation can start from 0.

8. IR Stretching Frequencies of two bonded atoms:
What Does the Frequency, , Depend On?
= frequency
k = spring strength (bond stiffness)
 = reduced mass (~ mass of largest atom)
is directly proportional to the strength of the bonding between the two atoms (  k)
is inversely proportional to the reduced mass of the two atoms (v  1/) 51

9. Stretching Frequencies
Frequency decreases with increasing atomic weight.
Frequency increases with increasing bond energy. 52

10. IR spectroscopy is an important tool in structural determination of unknown compound

11. IR Spectra: Functional Grps
Alkane
C-C
-C-H
Alkene
Alkyne 11 11

12. IR: Aromatic Compounds
(Subsituted benzene “teeth”)
C≡C 12 12

13. IR: Alcohols and Amines
O-H broadens with Hydrogen bonding
CH3CH2OH
C-O
N-H broadens with Hydrogen bonding
Amines similar to OH 13 13

14. CO2, A greenhouse gas ?

15. Electromagnetic Spectrum
Over 99% of solar radiation is in the UV, visible, and near infrared bands
Over 99% of radiation emitted by Earth and the atmosphere is in the thermal IR band (4 -50 µm)
Near Infrared
Thermal Infrared

16. What are the Major Greenhouse Gases?
Ar + other inert gases = 0.936%
CO2 = 370ppm
CH4 = 1.7 ppm
N20 = 0.35 ppm
O3 = 10^-8
+ other trace gases

17. Molecular vibrations
The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations.
Transition occur for v = ±1
This potential does not apply to energies close to dissociation energy.
In fact, parabolic potential does not allow molecular dissociation.
Therefore more consider anharmonic oscillator.
PY3P05

18. Vibrational modes of CO2

19. Anharmonic oscillator
A molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations.
At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit.
Must therefore use a asymmetric potential. E.g., The Morse potential:
where De is the depth of the potential minimum and
PY3P05

20. Anharmonic oscillator
The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels:
where xe is the anharmonicity constant:
The second term in the expression for E increases
with v => levels converge at high quantum numbers.
The number of vibrational levels for a Morse
oscillator is finite:
v = 0, 1, 2, …, vmax
PY3P05

21. Energy Levels: Basic Ideas
Basic Global Warming: The C02 dance …
About 15 micron radiation