Rotational Spectroscopy

Applied Spectroscopy

Recommended Reading: 1. Banwell and McCash: Chapter 2

2. Atkins: Chapter 16, sections 4 - 8

Aims In this section you will be introduced to

1) Rotational Energy Levels (term values) for diatomic molecules and linear polyatomic molecules

2) The rigid rotor approximation

3) The effects of centrifugal distortion on the energy levels

4) The Principle Moments of Inertia of a molecule.

5) Definitions of symmetric , spherical and asymmetric top molecules.

6) Experimental methods for measuring the pure rotational spectrum of a molecule

Microwave Spectroscopy - Rotation of Molecules

Microwave Spectroscopy is concerned with transitions between rotational energy levels in molecules.

Molecules can interact with electromagnetic radiation, absorbing or emitting a photon of frequency ω, if they possess an electric dipole moment p, oscillating at the same frequency

Definition Electric Dipole: p = q.d

Most heteronuclear molecules possess a permanent dipole moment

e.g HCl, NO, CO, H2O...

+q -q d

p

H Cl -q +q

p

Gross Selection Rule:

A molecule has a rotational spectrum only if it has a permanent dipole moment.

t

dipo

le m

omen

t p

_ +

+

+

_ +

_

_

Homonuclear molecules (e.g. O2, H2, Cl2, Br2…. do not have a permanent dipole moment and therefore do not have a microwave spectrum!

Rotating molecule

General features of rotating systems

mv

r

Linear velocity v =

distancetime

angular velocity ω = time

radians

v = ω × r Moment of Inertia I = mr2.

A molecule can have three different moments of inertia IA, IB and IC about orthogonal axes a, b and c.

O

ri

R

∑=i

2iirmI

Note how ri is defined, it is the perpendicular distance from axis of rotation

Rigid Diatomic Rotors

IB = Ic, and IA = 0.

C = centre of gravity.

Express I in terms of m1, m2 and r0.

m1 m2

ro

r1 r2 C

( )( )( )3 rmrmI

2 rmrm

1 rrr

222

211

2211

021

⋅+⋅=

⋅=⋅=+

from 20

20

21

21 rμ r mmmm

I ⋅=+⋅=

Derive this expression NOTE:

Units of I = kg.m2.

μ = reduced or effective mass of the molecule

Energy and Angular Momentum

Energy of a body rotating about an axis ‘a’ with constant angular velocity a is 2

AAa I21

E ω=

Angular momentum about axis a is AAA IJ ω=

A body free to rotate about all three axes has a total rotational energy

( ) ( ) ( )2CC2

BB2

AA

2C

2B

2A

2

III

JJJJ

ωωω ++=

++=

2CC

2BB

2AA I

21

I21

I21

E ωωω ++=

And a total angular momentum given by

Recall that J is a vector

Energy Levels of a Rigid Diatomic Rotor

For diatomic molecule IB = Ic = I and IA = 0 so:

I2J

I2J

I2J

I21

I21

E2

c

2c

B

2B2

CC2BB =+=+= ωω

Can find quantum energy levels using Correspondence Principle:

( ) ( )2

222

4

h1JJ1JJJ

π⋅+=⋅+→

1. Rotational energy for a rigid diatomic rotor is quantised

( )I8

h1JJE

2

2

Jπ

⋅+=Units: Joules

J = 0, 1, 2, 3, …

= rotational quantum number

Energy Levels of a Rigid Diatomic Rotor

2. Rotational energy is usually expressed in units of cm-1.

Recall therefore then νν ~chhE ⋅⋅==ch

E~⋅

== εν

εJ gives the allowed energy levels of the rigid rotor (in cm-1).

B can be determined experimentally by spectroscopy

( ) ( ) 1JJIc8

h

hcI24

1JJhch

E

22

2J +=

⋅⋅+=

⋅=

ππεJ cm-1

B - Rotational Constant, units cm-1

( ) 1JJB +⋅=Jε Ic4

Ic8

hB

2 ππ==where

Notation: Rotational Term F(J)

The energy of a rotational state is normally reported as the Rotational Term, F(J), a wavenumber divided by hc

F(J) ! ! J ="Ehc

= B # J J +1( )

Ritz Combination Principle

The wavenumber of any spectral line is the difference between two terms. Two terms T1 and T2 combine to produce a spectral line of wavenumber.

hcE

TT~21

Δ=−=ν

Energy Levels of a Rigid Diatomic Rotor

( ) ..... 3 2, 1, 0, J 1JJB =+⋅=Jε

5

4

3

2 1 0

6

30B

20B

12B

6B 2B 0

42B

F(J) (= εJ)

Can we predict nature of a microwave rotational spectrum?

J

Rotational Transitions in Rigid Diatomic Molecules

Selection Rules: 1. A molecule has a rotational spectrum only if it has a permanent dipole moment.

2. ΔJ = ± 1 +1 = adsorption of photon,

-1 = emission of photon.

J = 5

4

3

2 1 0

Transitions observed in absorption spectrum. J = 0 è J = 1

J = 1è J = 2

1-0J1JJ cm B20B2~ =−=−== == εεΔεν

1-1J2JJ cm B4B2B6~ =−=−== == εεΔεν

In general ( ) 1-1JJ cm 1JB2~ +=+→ν

Spacing between lines in spectrum = 2B

2B 4B

6B 8B

10 B

cm-1

Rotational Spectra of Rigid Diatomic Molecules

Line separation in the rotational spectrum of HCl is ≈ 21.2 cm-1 è B = 10.6 cm-1; IHCl can be found from

Ic8

hB

2π=

Once IHCl is known, then rHCl (the bond length) can be determined from

r r mmmm

I 20

20

21

21 ⋅=+⋅= µ

Need to know mH and mCl, but these are known and tabulated.

Summary so far: 1. Microwave spectroscopy is concerned with transitions between rotational energy levels of molecules

2. General features of rotational systems: I, ω, μ

∑=i

2iirmI 2

020

21

21 rμ r mmmm

I ⋅=+⋅=

3. Energy Levels of a rigid diatomic rotor

( ) 1JJB F(J) +⋅=≡ Jε Ic8

hB

2π=

J = 0, 1, 2, ….; B = rotational constant, units cm-1

4. Selection rules: (1) permanent dipole moment,

(2) ΔJ = ± 1 only 5. Spacing between lines of in rotational spectra of rigid diatomic molecules is constant and equal to 2B cm-1.

Why is Rotational Spectroscopy important?

From pure rotational spectra of molecules we can obtain:

1. bond lengths

2. atomic masses

3. isotopic abundances

4. temperature

Important in Astrophysics: Temperature and composition of interstellar medium

Diatomic molecules found in interstellar gas:

H2, OH, SO, SiO, SiS, NO, NS, HCl, PN, NH, CH+, CH, CN, CO, CS, C2.

S.Taylor, D. Williams, Chemistry in Britain, 1993 pg 680 - 683

Population of Rotational Energy Levels

Boltzmann distribution: NJ+1

NJ=0

= exp ! "EkT

#$%

&'( = exp ! ! Jhc

kT#$%

&'(

Orientation of the angular momentum L is quantised. This results in a (2J+1) degeneracy of the energy level

PopulationJ

PopulationJ=0

! 2J +1( )exp " #EkT

$%&

'() = 2J +1( )exp "

BJ J +1( )hckT

$%&

'()

Spectral Line Intensity ∝ Population

B = 10.6 cm-1,

T = 300K

Example: Rotational Spectrum of HCl

m1 m2

ro

r1 r2 C

Maximum Value of J At what value of J will the intensity be a maximum, at a given temperature?

We can find the value of Jmax by finding the value of J that maximises the population:

Population ! 2J +1( )exp " #EkT

$%&

'() = 2J +1( )exp "

BJ J +1( )hckT

$%&

'()

Differentiate w.r.t. J and set equal to zero for a maximum. This gives

21

hcB2kT

J max −=

must round this value to the nearest integer (J can only be an integer)

Note effect of r0 and m on the rotational constant B.

I ∝ r0 and m and B ∝ 1 / I

Bond lengths and Rotational Constants of some Diatomic Molecules

Effect of Isotope Substitution

From spectra we can obtain:

bond length or atomic weights

and Isotopic Abundances

Effect of Isotope Substitution on Spectra

Atomic weight 12C = 12.0000, 16O = 15.9994

Ic8

hB

2π= 046001.1

'''

'I''I

hc''I8

c'I8

h''B'B 2

2====

µµπ

π

For 12C16O B’ = 1.92118 cm-1. 13C16O B’’ = 1. 83669 cm-1.

13C = 13.0007,

J = 5

4

3

2 1 0

12C16O 13C16O

Non-Rigid Rotor For a non-rigid rotor the bond-length increases as the angular velocity increases Centrifugal Distortion

Ic8

hB

2π=

r

1B

2∝ B decreases as J

increases

For a non-rigid rotor

( ) ( )22 1JJD 1JJB +⋅−+⋅=Jε

D - Centrifugal Distortion constant

(stiffness constant)

units: cm-1.

Like B, D depends on the molecule

rigid rotor non-rigid rotor

J = 5

4

3

2 1 0

( ) ( )22 1JJD 1JJB +⋅−+⋅=Jε( ) 1JJB +⋅=Jε

2B 4B 6B 8B 10 B cm-1 12 B

Non-Rigid Rotor

( ) ( )22 1JJD 1JJB +⋅−+⋅=Jε

( ) ( )31JD 4 1JB2ν~ +⋅−+⋅=−= + J1J εε

The rigid rotor selection rules still apply, i.e. ΔJ = ±1

Typical values: B ~ 1 - 10 cm -1 and D ~ 10-3 – 10-2 cm-1.

D can be related to the

vibrational frequency of the molecule 2vib

3

~B4

Dν

=

Summary

1. Population of energy levels: Boltzmann Distribution degeneracy of rotational states

2. Efect of bond length and mass on rotational constant of diatomic molecule

3. Efect of isotopic substitution atomic weights and isotopic abundances

4. Non-rigid rotor D - centrifugal distortion constant