1. Rotational and Vibrational Spectra
Spectroscopy 1:
Rotational and Vibrational Spectra
CHAPTER 16

2. Set up expressions for the energy levels of molecules
Then apply selection rules and population considerations to infer the form of the spectra
Rotational energy levels:
Derive expressions for their values
Interpret rotational spectra in terms of molecular dimensions
Consider selection rules w.r.t. nuclear spin and Pauli exclusion principle
Vibrational energy levels:
Use harmonic oscillator model with modifications
Polyatomic vibrational levels

3. Pure Rotational Spectra
Rotational energy levels:
Derive expressions for their values
Interpret rotational spectra in terms of molecular dimensions
Consider selection rules w.r.t. nuclear spin and Pauli exclusion principle

4. Fig 13.9 Definition of moment of inertia, I

5. Rotational properties of the molecule can
be expressed in terms of the moments of inertia about the three perpendicular axes set in the molecule.
Labeled as Ia, Ib, Ic
Assigned so that Ic ≥ Ib ≥ Ia
e.g., For linear molecules, Ic = Ib, Ia = 0.

6. Fig 13.10 An asymmetric rotor (most molecules)
Ic > Ib > Ia

7. Fig 13.11 Classification of rigid rotors (i.e., no distortion)
Ic = Ib, Ia = 0
Ic = Ib = Ia
Ic = Ib > Ia
Ic > Ib > Ia

8. Fig 13.12 Rotational levels of a linear or spherical
rotor
where:
the rotational quantum number
J = 0, 1, 2, 3, ...
Normally expressed in terms of the rotational constant, B:
Rotational term in cm-1:
F(J) = BJ(J+1)

9. Fig 13.16 The effect of rotation on a molecule
Including the centrifugal
distortion constant, DJ:
F(J) = BJ(J+1) – DJJ2(J+1)2

10. Fig 13.17 Rotating polar molecule appears as an
oscillating dipole that can be stirred by the em field
Gross selection rule:
In order to give a pure
rotational spectrum, a
molecule must have a
permanent dipole
Specific selection rule:
ΔJ = ± MJ = 0, ±1

11. Fig 13. 18 When a photon is absorbed by a molecule, angular
Fig When a photon is absorbed by a molecule, angular momentum is conserved

12. Fig 13.14 Significance of quantum number MJ
Laboratory axis

13. Fig 13.19 Rotational energy levels of a linear rotor
For the allowed transition
J+1 ← J:
v = 2B(J+1)
with J = 0, 1, 2,...
Relative intensities reflect
the population of the initial levels and the strengths of the transition dipole moments

14. Rotational Raman Spectra
Involves the inelastic scattering of a photon
Photon may lose energy (Stokes)
Photon may gain energy (anti-Stokes)
Photon may not change energy (Rayleigh)
Gross selection rule:
Molecule must be anisotropically polarizable
Specific selection rule:
Linear rotors: ΔJ = 0, ±2
Symmetric rotors: ΔJ = 0, ±1, ±2

15. Rotational Raman Spectra
Fig Results of applied electric field
When field is parallel
to molecular axis
When field is perpendicular
to molecular axis

16. Distortion induced in a molecule by an applied electric field
Distortion returns to its initial value after 180°
i.e., twice a full revolution
Hence: ΔJ = 0, ±2

17. Fig 13. 21 Rotational energy levels of a linear rotor and
Fig Rotational energy levels of a linear rotor and the transitions allowed by ΔJ = 0, ±2
J+2 ← J:
v = vi - 2B(2J+3)
with J = 0, 1, 2,...
J-2 ← J:
v = vi + 2B(2J-1)
with J = 2, 3, 4, ...

18. Nuclear Spin Statistics
From Pauli principle: if two identical spin nuclei are
exchanged the overall wavefunction must remain
unchanged
Number of ways of achieving odd J
Number of ways of achieving even J
= (I+1)/I for half-integral spins
= I/(I+1) for integral spins
e.g., for H-H or F-F, both atoms have same nuclear spin = ½
∴ Populations between odd J and even J are 3 : 1

19. Alternate intensities
Fig Rotational Raman spectrum of a diatomic molecule with two identical spin-1/2 nuclei
Alternate intensities
is the result of
nuclear statistics