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

Fig 13.9 Definition of moment of inertia, I

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.

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 = 0 Ic = Ib = Ia Ic = Ib > Ia Ic > Ib > Ia

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)

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

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 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 reflect the population of the initial levels and the strengths of the transition dipole moments

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

Rotational Raman Spectra Fig 13.20 Results of applied electric field When field is parallel to 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 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 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

Alternate intensities 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