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
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
=