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