Principle of Rotational and Vibrational Spectroscopy, Selection Rule for Application in Diatomic Molecules

1. Introduction

Spectroscopy is the study of the interaction of electromagnetic radiation with matter.

• Rotational spectroscopy deals with transitions between quantized rotational energy levels of molecules.
• Vibrational spectroscopy deals with transitions between quantized vibrational energy levels.

These techniques provide information about bond lengths, bond strengths, and molecular structures, particularly in diatomic molecules.

2. Principle of Rotational Spectroscopy

2.1 Concept

Molecules rotate about their center of mass, and these rotations have discrete energy levels. For a rigid diatomic molecule, the rotational energy is given by:

\( E_J = \frac{h^2}{8\pi^2 I} J(J+1) \)

• \( J \) = rotational quantum number (0, 1, 2, …)
• \( I = \mu r^2 \) = moment of inertia
• \( \mu = \frac{m_1 m_2}{m_1 + m_2} \) = reduced mass
• \( r \) = bond length
• \( h \) = Planck's constant

2.2 Principle

When a molecule absorbs microwave radiation, it undergoes rotational transitions. The energy difference between consecutive levels is:

\( \Delta E = 2B(J + 1)h c \)

where \( B = \frac{h}{8\pi^2 c I} \) is the rotational constant.

2.3 Selection Rule

• \( \Delta J = \pm 1 \) for absorption or emission.
• The molecule must have a permanent dipole moment.
• Examples: HCl, CO, NO -> active; H2, N2, O2 -> inactive.

3. Principle of Vibrational Spectroscopy

3.1 Concept

Atoms in a molecule vibrate about their equilibrium positions. For a harmonic oscillator, the vibrational energy is given by:

\( E_v = \left(v + \frac{1}{2}\right)h\nu_0 \)

• \( v \) = vibrational quantum number (0, 1, 2, …)
• \( \nu_0 = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}} \)
• \( k \) = force constant of the bond

3.2 Principle

A vibrational transition occurs when IR radiation of frequency equal to the difference between two vibrational levels is absorbed.

3.3 Selection Rule

• Harmonic oscillator: \( \Delta v = \pm 1 \)
• Anharmonic oscillator: \( \Delta v = \pm 1, \pm 2, \pm 3, \dots \) (overtones)
• Vibration must cause a change in dipole moment.

4. Combined Rotational–Vibrational Spectroscopy

When both transitions occur simultaneously, rotational fine structure appears on the vibrational spectrum. Diatomic molecules show P-branch (\( \Delta J = -1 \)) and R-branch (\( \Delta J = +1 \)), but no Q-branch (\( \Delta J = 0 \)).

5. Applications

• Determination of bond length from rotational spectra.
• Calculation of force constant from vibrational spectra.
• Identification of molecules in astronomy and atmospheric studies.
• Structural analysis of gaseous diatomic species.

6. Summary Table