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Part III 2302335 Physical Chemistry III

Email: spornthe@hotmail.com

Points and credit: Approximately 20% for quiz & homework 80% final examination

Note *Extra points for good students

Instructor: Assoc. Prof. Dr. Pornthep Sompornpisut

Office hour: Mon. & Tue. 1pm to 2pm + anytime @R1124 MHMK Bld.

- Textbooks : No particular textbook

Thomas Engel “Quantum Chemistry &

Spectroscopy” 2nd edition (Chapter 7, 8,

14 & 16).

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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.

Science Library

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- Bring a Scientific Calculator to the class

- Laptop, notebook, tablet are allowed for the purpose

of the class study only.

Study tools

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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

Main topics

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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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Electromagnetic Radiation : a form of energy whose behavior is described by the properties of both wave and particles.

The dual nature of EM

Wave nature

Particle nature

Behavior: absorption & emissionBehavior: refraction & diffraction

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The oscillations in the electric and magnetic fields are perpendicular to each other, and to the direction of the wave’s propagation

Propagation

EM

electric field magnetic field

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- velocity, - amplitude, - frequency, wavelength, wavenumber - phase angle, etc.

) 2(sin tAA et

Ex. The amplitude of the oscillating electric field at any point along the propagating wave

Max. amplitude

Phase angleFrequency

Properties of electromagnetic wave

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Wavelength () Wavenumber (ṽ)

c

1

c = the speed of light, 3 x 108 m/s

Wavelength & wavenumber

Units: m cm-1

c

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Ex. The wavelength of the sodium D line is 589 nm. What are the frequency and the wavenumber for this line?

1-149

-18

s1009.5m10589

s m103

c

The frequency and wavenumber of the sodium D line are

1-49

cm107.1cm 100

m 1

m10589

11

12

hchc

hE

h = Planck’s constant, 6.6 x 10-34 J sc = the speed of light, 3 x 108 m/sec

Particle properties of electromagnetic radiation

Ex. What is the energy of a photon from the sodium D line at 589 nm.

J1037.3m10589

)s m103( s) J10626.6( 199

-1834

hc

E

The photon energy is

The energy of a photon

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The Electromagnetic Spectrum

Increasing energy

Increasing wavelength

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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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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

Two major categories of spectroscopic techniques

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1) Energy transfer or absorption or emission of photons by an atom or molecule

Undergo transition between energy states

12 EEhvE

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Type of energy transfer

Spectral region

Spectroscopic techniques

absorption -rays Mossbauer spectroscopy

X-rays X-ray absorption spectroscopy

UV/Vis UV/Vis spectroscopyatomic absorption spectroscopy

IR infrared spectroscopyraman spectroscopy

Microwave microwave spectroscopy

Radio wave electron spin resonance spectroscopynuclear magnetic resonance spectroscopy

Examples of Spectroscopic Techniques involving with energy transfer spectroscopy

Continue

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Type of energy transfer

Spectral region

Spectroscopic techniques

emission UV/Vis atomic emission spectroscopy

photoluminescence X-rays X-ray fluorescence

UV/Vis fluorescence spectroscopyphosphorescence spectroscopyatomic fluorescence spectroscopy

chemiluminescence UV/Vis chemiluminescence spectroscopy

Examples of Spectroscopic Techniques involving with energy transfer 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

refraction

diffraction

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Spectral region Type of Interaction

Spectroscopic techniques

X-ray Diffraction X-ray diffraction

UV/Vis refraction refractometry

scattering dynamic light scatteringturbidimetry

dispersion optical rotary dispersion

Examples of Spectroscopic Techniques that do not involve with energy transfer spectroscopy

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Different spectral region : different energy levels of transition

Radio

Microwave

Infrared

Visible

UV

Ene

rgy

(10n

sca

le)

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.

Tkhv Beg

g

N

N /

0

1

0

1

- 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

Random phase, random direction

Incoherent wave Coherent wave

Same phase, same direction

Ex. Lightbulb Ex. Laser

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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:

1cos316

5,

cos4

3,

4

1,

22/1

02

2/10

1

2/10

0

Y

Y

Y

jMjY Providing the notation is used for the preceding

functions.

Assuming the electromagnetic field to lie along the z-axis, and the transition dipole moment takes the form

cosz

dYYd JJz sin,cos, 0

0

0

02

0

0

For the J=0 → J=1 transition,

cos4

3,

4

1,

2/10

12/10

0

YY

0

2

0

2

2

0 0

2

2

0 02/1

2/1

0

00

01

2

0

10

sincos2

3

sincos)2(4

3

sincos4

3

sin4

1coscos

4

3

sin,cos,

d

d

dd

dd

dYYd JJz

0220

2

0

d

For the J=0 → J=1 transition,

dzx

dxxdx

dzxzx

sin

1 ,sin ,cos ,

xz

dzzdzx

xzxdxzxdxx

33

2222

cos3

1

3

1

sin

1sinsinsincos

0

2 sincos dNow consider

Use reduction or substitution method

3

2)11(

3

1cos

3

1sincos 33

0

3

0

2

d

Replace the result into the original integration

From the previous derivation:

0

210 sincos2

3 dz

3

2sincos

0

2

d

For the J=0 → J=1 transition,

3

3

3

2

2

3 10

z

03

310 z

The J=0 → J=1 transition is allowed.

Thus:

For the J=0 → J=2 transition,

4

1, 2/1

00

Y

0

3

2

0 0

3

2

0 02/1

22/1

0

00

02

2

0

20

)sincossincos3(4

5

)sincossincos3(8

5

sin4

1cos1cos3

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5

sin,cos,

d

dd

dd

dYYd JJz

1cos316

5,

2/10

2

Y

0220

2

0

d

0

3 sincos3 d

0

sincos d

Let consider by dividing into two separate terms:

For the J=0 → J=2 transition,

dzx

dxxdx

dzxzx

sin

1 ,sin ,cos ,

xzdzx

xzxdxzxdxx 44333 cos4

3

4

3

sin

1sin3sin3sincos3

Consider

Use the substitution method (similar to the previous one)

Replace x with and integrate from 0 to , we get:

0

3 sincos3 d

0

sincos d

0)1)1((4

3cos

4

3sincos3 44

04

0

3

d

0cos2

1

2

1

sin

1sinsincos 22

xzdz

xxzxdxx

Do the same for

0

sincos d

0)sincossincos3(4

5

0

320

dz

For the J=0 → J=2 transition,

0sincos30

3

d 0sincos0

d

From the previous derivation:

Therefore:

Thus:

020 z

Thus, the J=0 → J=2 transition is forbidden.

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Solution Assuming the electromagnetic field to lie along the zaxis, , and the transition dipole moment takes the form

cosz

03

3

3

cos

2

3sincos

4

3

0

32

0

22

0

10

ddz

dYYd JJz sincos,cos, 0

0

2

0

02

0

0

cos4

3,

4

1,

2/10

12/10

0

YY

For the J=0 → J=1 transition,

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SolutionFor the J=0 → J=2 transition,

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 .

04

1

4

1

8

5

2

cos

4

cos3

8

5sincos1cos3

8

5

0

242

0

22

0

20

ddz

45

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48

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