Part III
2302335 Physical Chemistry III
Instructor: Assoc. Prof. Dr. Pornthep Sompornpisut
Office hour: Mon. & Tue. 1pm to 2pm + anytime @R1124 MHMK Bld.
Email: spornthe@hotmail.com
Points and credit:
Approximately 20% for quiz & homework
80% final examination
Note *Extra points for good students
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Study materials
- PP lectures: Download from my facebook
Send your name and student ID to my email, I will send
you an invited message for joining the group. You can later
add Facebook ID of your friends into the group.
- Textbooks : No particular textbook
Thomas Engel “Quantum Chemistry &
Spectroscopy” 2nd edition (Chapter 7, 8,
14 & 16).
Science Library
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Study tools
- Bring a Scientific Calculator to the class
- Laptop, notebook, tablet are allowed for the purpose
of the class study only.
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Main topics
1.
An Introduction to Spectroscopy
2.
Vibrational Spectroscopy: Harmonic oscillator
model treated by classical vs quantum mechanics
3.
Rotational Spectroscopy : Rigid rotor model
treated by classical vs quantum
4.
Electronic Spectroscopy: electronic transition
5
An Introduction to Spectroscopy
Outlines
•
Electromagnetic radiation: the dual nature of EM
•
Properties of electromagnetic waves and particles
•
Electromagnetic Spectrum
•
Spectroscopic techniques and two major
categories
•
The relationship between electronic, vibrational,
rotational state energies
•
Spontaneous emission vs Stimulated emission
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An Introduction to Spectroscopy
- Spectroscopy are tools that chemists have to elucidate
chemical structure, bonding, properties and reactivity of
the molecules.
- In most spectroscopies, atoms or molecules absorb
electromagnetic radiation and undergo transitions
between allowed quantum states.
What if molecules had a continuous energy spectrum?
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The dual nature of EM
Electromagnetic Radiation : a form of energy whose
behavior is described by the properties of both wave and
particles.
Particle
nature
Behavior: absorption
& emission
Wave
nature
Behavior: refraction &
diffraction
8
Propagation
EM
electric field
magnetic field
The oscillations in the electric and magnetic fields are
perpendicular to each other, and to the direction of the
wave’s propagation
9
Properties of electromagnetic wave
- velocity,
- amplitude,
- frequency, wavelength,
wavenumber
- phase angle, etc.
Ex. The amplitude of the oscillating electric field at any point
along the propagating wave
Max. amplitude
At  Ae (sin 2 t   )
Frequency
Phase angle
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Wavelength & wavenumber
Wavelength ()

Units:
c

Wavenumber (ṽ)
 
m
c = the speed of light, 3 x 108 m/s
1



c
cm-1
11
Ex. The wavelength of the sodium D line is 589 nm. What
are the frequency and the wavenumber for this line?
The frequency and wavenumber of the sodium D line are
3 108 m s -1
14 -1
 

5
.
09

10
s
9
 589 10 m
c
1
1
1 m
4
-1
  


1
.
7

10
cm
 589 10 9 m 100 cm
12
Particle properties of electromagnetic radiation
The energy of a photon
E  h 
hc

 hc
h = Planck’s constant, 6.6 x 10-34 J s
c = the speed of light, 3 x 108 m/sec
Ex. What is the energy of a photon from the sodium D line
at 589 nm.
The photon energy is
(6.626 10 34 J s) (3 108 m s -1 )
19
E


3
.
37

10
J
9

589 10 m
hc
13
The Electromagnetic Spectrum
 Increasing energy
Increasing wavelength 
14
Types of Atomic & Molecular Transitions
• -rays: nuclear
• X-rays: core-level electrons
• Ultraviolet (UV): valence electrons
• Visible (Vis): valence electrons
• Infrared (IR): molecular vibrations
• Microwave: molecular rotations, X-band electron spin
• Radio waves: nuclear spin, electron spin
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Two major categories of spectroscopic techniques
1) Energy transfer or absorption or emission of photons by
an atom or molecule
2) Electromagnetic radiation undergoes a change in
amplitude, phase angle, polarization, or direction of
propagation
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1) Energy transfer or absorption or emission of photons by
an atom or molecule
Undergo transition between energy states
E  hv  E2  E1
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Examples of Spectroscopic Techniques involving with
energy transfer spectroscopy
Type of
energy
transfer
Spectral
region
Spectroscopic techniques
absorption
-rays
Mossbauer spectroscopy
X-rays
X-ray absorption spectroscopy
UV/Vis
UV/Vis spectroscopy
atomic absorption spectroscopy
IR
infrared spectroscopy
raman spectroscopy
Microwave
microwave spectroscopy
Radio wave
electron spin resonance spectroscopy
nuclear magnetic resonance spectroscopy
Continue 
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Examples of Spectroscopic Techniques involving with
energy transfer spectroscopy
Type of energy
transfer
Spectral
region
Spectroscopic techniques
emission
UV/Vis
atomic emission spectroscopy
photoluminescence
X-rays
X-ray fluorescence
UV/Vis
fluorescence spectroscopy
phosphorescence spectroscopy
atomic fluorescence spectroscopy
chemiluminescence UV/Vis
chemiluminescence spectroscopy
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Two major categories of spectroscopic techniques
2) Electromagnetic radiation undergoes a change in
amplitude, phase angle, polarization, or direction of
propagation as a result of
• refraction,
• reflection,
• scattering,
• diffraction,
• or dispersion
diffraction
refraction
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Examples of Spectroscopic Techniques that do not involve
with energy transfer spectroscopy
Spectral region
Type of
Spectroscopic techniques
Interaction
X-ray
Diffraction
X-ray diffraction
UV/Vis
refraction
refractometry
scattering
dynamic light scattering
turbidimetry
dispersion
optical rotary dispersion
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Different spectral region : different energy levels of transition
Visible
Infrared
Microwave
Radio
Energy
(10n scale)
UV
EUV required for electronic
transition is larger than Evib
required for transition from one
vibrational state to another
vibrational state.
Eelec >> Evib >> Erot
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The relationship between electronic, vibrational, rotational
state energies
• Each electronic state will
have a group of vibrational
(and rotational) states.
• Vibrational transition takes a
lot of energy more than
rotational transition.
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Pure electronic transition & the electronic transition couples
with the vibrational transition
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Transition from the ground to the first
excited vibrational state.
N1 g1  hv / k BT

e
N0 g0
- N1/N0 is very low.
- All the molecules in a macroscopic sample are in their
ground vibrational state (n=0) at room temperature
(even at 1000K).
- only the n = 0  n = 1 transition is observed in
vibrational spectroscopy
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Spontaneous emission vs Stimulated emission
Incoherent wave
Random phase,
random direction
Ex. Lightbulb
Coherent wave
Same phase,
same direction
Ex. Laser
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28
29
30
Molecular motion
Translation
Vibration
Rotation
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Example: Using the following total energy eigenfunctions for
the three-dimensional rigid rotor, show that the J=0 → J=1
transition is allowed, and that the J=0 → J=2 transition is
forbidden:
Providing the notation Y jM is used for the preceding functions.
j
Y  ,   
0
0
1
4 1/ 2
1/ 2
 3 
0
Y1  ,    

 4 
cos 
1/ 2
 5 
Y20  ,    

 16 
3 cos
2
  1
Assuming the electromagnetic field to lie along the zaxis, z   cos and the transition dipole moment takes the
2

form
J0
0
0
 z    d  YJ  ,  cos  Y0  ,  sin d
0
0
For the J=0 → J=1 transition,
Y  ,   
0
0
2

0
0
2

1/ 2
 3 
Y  ,    

 4 
1
cos 
0
1
4 
1/ 2
0
0



 10


d

Y

,

cos

Y
z
J  0  ,   sin d
  J 1
1/ 2
 3 
   d  

4 
0
0


3
2
4
0
0

(2 )  cos  sin d
2
0

1
4 
1/ 2
sin d

2
d

cos
   sin d
4
3
cos  cos  
3
2

cos
 sin d

2 0
2
 d   0  2  0
0
2
For the J=0 → J=1 transition,

Now consider
2
cos
  sin d
0
Use reduction or substitution method
x   , z  cos x,
dz
1
  sin x, dx  
dz
dx
sin x
1 

2
2
2
2
cos
x
sin
xdx

z
sin
xdx

z
sin
x

dz


z
dz






 sin x 
1
1
  z 3   cos 3 x
3
3
Replace the result into the original integration


1
2
 1
2
3 
3
3
0 cos  sin d   3 cos   0   3 (1  1 )  3
For the J=0 → J=1 transition,
From the previous derivation:

3
2

cos
 sin d

2 0
 10
z

10
z
3 2
3

 
2 3
3

2
cos
  sin d 
0
Thus:
3
 
0
3
10
z
The J=0 → J=1 transition is allowed.
2
3
For the J=0 → J=2 transition,
Y00  ,   
2

0
0
2

1/ 2
 5 
Y20  ,    

 16 
1
4 
1/ 2
3 cos   1
 z20    d  YJ0 2  ,  cos  YJ00  ,  sin d
1/ 2
 5 
   d  

16 
0
0

5
8
2

0
0
3 cos
2
  1cos  
1
4 
1/ 2
sin d
3
 sin   cos  sin  )d
cos
3
(

d
 
2

5
3
 sin   cos  sin  )d
cos
3
(


4 0
 d   0  2  0
0
Let consider by dividing into two separate terms:


0
0
3
3
cos
 sin d

 cos sin d
2
Consider


0
0
For the J=0 → J=2 transition,
 cos sin d
3
3
cos
 sin d

Use the substitution method (similar to the previous one)
dz
1
x   , z  cos x,
  sin x, dx  
dz
dx
sin x
1 
3 4
3

3
3
3
4
3
cos
x
sin
xdx

3
z
sin
xdx

3
z
sin
x

dz


z


cos
x





4
4
 sin x 
Replace x with  and integrate from 0 to , we get:


3
4
3
cos

sin

d



cos

0
4
3


0
3
  ((1) 4  14 )  0
4

Do the same for
 cos sin d
0
1 
1 2
1

2
cos
x
sin
xdx

z
sin
x

dz


z


cos
x0




2
2
 sin x 
For the J=0 → J=2 transition,
Thus:


0
0
3
3
cos
 sin d  0

 cos sin d  0
From the previous derivation:
 z20

5
3

(
3
cos
 sin   cos  sin  )d  0

4 0
Therefore:
 z20  0
Thus, the J=0 → J=2 transition is forbidden.
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40
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Solution Assuming the electromagnetic field to lie along
the zaxis, z   cos,  and the transition dipole moment
takes the form
2
2
0
0
 zJ 0    d  YJ0  ,  cos   ,  Y00 cos  sin d
For the J=0 → J=1 transition,
Y  ,   
0
0
 
10
z
3
4
2
1
4 
1/ 2
2
 d  cos
0
0
1/ 2
2
 3 
Y  ,    

 4 
cos 
0
1
 sin d 
 3  cos  

2 
3
3
 
 
 0
 3
3
0
43
Solution
For the J=0 → J=2 transition,
 z20  
5
8
2
2
0
0
 
 5  3 cos 4  cos 2  
 5  1 1
2
3 cos   1 cos  sin d 

  0

 

8 
4
2  0
8  4 4 
 d  

The preceding calculations show that the J=0 → J=1
transition is allowed and that the J=0 → J=2 transition is
forbidden. You can also show that is also zero unless
MJ=0 .
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