Rotational and Vibrational Levelsof Molecules

Lecture 23

www.physics.uoguelph.ca/~pgarrett/Teaching.html

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Review of L-21

• Beer-Lambert law

• Transmittance

• Absorbance

• Extinction coefficient

xCx eIeII 00−− ==

)(

)(

0 λλ

I

IT =

xI

IA 4343.0

)(

)(log 0 =��

�

����

�=λλ

4343.0=

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Rotations and vibrations

• We have examined electronic transitions in molecules– But they can also rotate and vibrate

– ex. O2

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Rotations and vibrations

• Become more complicated for more complex molecules– ex. H2O rotational modes

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Rotations and vibrations

• Become more complicated for more complex molecules– ex. H2O vibrational modes

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Vibrations

• Just like the electrons, molecular motion is governed by quantum mechanics– Energies due to rotation and vibration are quantized

• Molecular vibrations– Chemical bond acts like a spring and can display SHM– Have an effective spring constant k for the bond involved and

effective mass meff

– Angular frequency

– Energy of vibration

– ½�ω comes from quantum mechanics and represents zero-point energy

effm

k=

( ) ( )hfvvEv 21

21 +=+= �

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Vibrations

• Vibrational energy

• Vibrational quantum number v = 0,1,2,3,… • The zero point energy ½�ω implies molecule never stops

vibrating, even when its in the v = 0 state!– Zero point energy cannot be harvested or extracted– Still exists at absolute zero

• All molecules are then in v = 0 state

• Energy levels are equally spaced with separation �ω• Obey selection rule ∆v = ±1 if no accompanying electronic

transition– Otherwise can be anything

( ) ( )hfvvEv 21

21 +=+= �

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Molecular vibrations

• For diatomic molecule with mass M1 and M2, effective mass meff can take the simple form

• Energy scale for molecular vibrations is much less than for electronic excitations

• Excitation energies correspond to IR region of the spectrum– Typical wavelengths are 2 – 50 µm = 2000 – 50000 nm for organic

molecules

• Vibrational levels are built on electronic states – each electronic state will host the whole range of vibrational states

21

21

MM

MMmeff +

=

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Vibrational excitation and de-excitation

v

3 2 10

π electronic state n = 2

.

.

.

v

3 2 10

.

.

.

IR radiation

visible radiation

At normal temperatures, most of the

molecules will be in the v = 0

state

Probability distribution for which v state is

populated during the ∆n

transitionvisible radiation

IR radiationFundamental IR transition

π electronic state n = 1

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Molecular rotations

• In quantum mechanics, the rigid rotor has energy levels

where ℑ is the moment of inertia (PHY1080), J is the angular momentum, J = 0,1,2,3,…

• The quantity is called the rotational parameter

• Moment of inertia, hence rotational parameter, can be different for each rotation axis

• Excitation energies correspond to the microwave region

• Energy scale for rotations << vibrations– Each vibrational level has rotational bands built on it

• Selection rule ∆J = ±1

( )12

2

+ℑ

= JJEJ

�

ℑ2

2�

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Rotational levels

J

3 2 10

vibrationalstate v = 1

.

.

.

J

3 2 10

.

.

.

IR radiation

At normal temperatures,

molecules will have a distribution

amongst the J states

Two types of transitions, J

increasing, and Jdecreasing, populated

during the ∆v transition

microwave radiation

vibrationalstate v = 1

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Vibrational-rotational IR spectrum

• HCl

1→→→→22→→→→3

3→→→→4

4→→→→5

5→→→→6

6→→→→7

7→→→→8

8→→→→9

9→→→→10

10→→→→11

11→→→→12

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Energy levels

• Taking rotations, vibrations, and electronic excitation into account

• If the measuring instrument has very good resolution, it is possible to see the discrete transitions

• Complex molecules may have many vibrational modes, rotational modes, etc. The combination of these different modes leads to a “smearing” of the discrete spectrum (temp. effects too) so that broad bumps appear rather than discrete lines

( ) ( )122

2

21

2

22

,,

,,

+ℑ

+++=

++=

JJvm

hnE

EEEE

eJvn

JvnJvn

��

�

Ring molecule

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Water absorption spectrum

IR Radiowave