spectroscopy 1: rotational and vibrational spectra chapter 16
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Spectroscopy 1:Spectroscopy 1:Rotational and Vibrational SpectraRotational and Vibrational Spectra
CHAPTER 16CHAPTER 16
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
Pure Rotational Spectra
Rotational energy levels:Rotational energy levels:
Derive expressions for their valuesDerive expressions for their values
Interpret rotational spectra in terms of Interpret rotational spectra in terms of molecular dimensionsmolecular dimensions
Consider selection rules w.r.t. nuclear spin Consider selection rules w.r.t. nuclear spin and Pauli exclusion principleand Pauli exclusion principle
Fig 13.9 Definition of moment of inertia, I
i
2iirmI
• Rotational properties of the molecule canbe 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.
Fig 13.10 An asymmetric rotor (most molecules)
Ic > Ib > Ia
Fig 13.11 Classification of rigid rotors (i.e., no distortion)
Ic = Ib = Ia
Ic = Ib > Ia
Ic > Ib > Ia
Ic = Ib, Ia = 0
Fig 13.12 Rotational levels of a linear or spherical
rotor
I2)1J(JE
2
J
where:
the rotational quantum number
J = 0, 1, 2, 3, ...
Normally expressed in terms of the rotational constant, B:
)1J(hcBJEJ
F(J) = BJ(J+1)
Rotational term in cm-1:
cI4B
Fig 13.16 The effect of rotation on a molecule
F(J) = BJ(J+1) – DJJ2(J+1)2
Including the centrifugaldistortion constant, DJ:
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 = ±1 MJ = 0, ±1
Fig 13.18 When a photon is absorbed by a molecule, angular momentum is conserved
Fig 13.14 Significance of quantum number MJ
Laboratory axis
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 reflectthe population of the initial levels and the strengths of
the transition dipole moments
kTE
JJ
J
eNgN
• 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
Rotational Raman Spectra
Rotational Raman Spectra
Fig 13.20 Results of applied electric field
When field is parallelto molecular axis
When field is perpendicular
to molecular axis
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
Fig 13.21 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, ...
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 JNumber 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
Fig 13.23 Rotational Raman spectrum of a diatomic molecule with two identical spin-1/2 nuclei
Alternate intensities
is the result of
nuclear statistics