harmonic oscillator

1. Harmonic Oscillator

2. Harmonic Oscillator

3. Selections rules
= 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.
Permanent Dipole moment
Re
+q -q

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:

4. 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 :
i
f
i
f
fi d 






 ˆ
ˆ 
 
...
2
1
)
( 2
0
2
2
0
0 




















 x
x
x
x
x




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.
Because they are solutions of the
operator H which is Hermitian
0
The higher terms can
be neglected for small
displacements of the
nuclei
...
2
1 2
0
2
2
0
0 




















 

 











 d
x
x
d
x
x
d i
f
i
f
i
f
fi

6. 








 



 d
x
x
i
f
fi
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.
First condition: fi= 0, if ∂/ ∂x = 0 Second condition: 0

 

 d
x i
f
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:
 = frequency
k = spring strength (bond stiffness)
 = reduced mass (~ mass of largest atom)
What Does the Frequency, , Depend On?



k
h
h
E clas
2



 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
11
Alkane
Alkene
Alkyne
-C-H C-C

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

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

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?
N2 = 78.1%
O2 = 20.9%
H20 = 0-2%
Ar + other inert gases = 0.936%
CO2 = 370ppm
CH4 = 1.7 ppm
N20 = 0.35 ppm
O3 = 10^-8
+ other trace gases

17. PY3P05
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.

18. Vibrational modes of CO2

19. PY3P05
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

V  hcDe 1ea(RRe )
 
2
a 
2
2hcDe






1/2

20. PY3P05
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
e
eff
e
e
D
m
a
x
hcx
hc
E
4
~
2
,...
2
,
1
,
0
;
~
2
1
~
2
1
2
max
2






























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