Questions: Vibrational Spectroscopy: Theory and Normal Modes

Question 1

A nonlinear molecule contains 4 atoms. How many vibrational normal modes does it have?

• A. 12
• B. 7
• C. 6
• D. 9

Answer: (C)

For a nonlinear molecule with N atoms, the number of vibrational modes is 3N − 6. With N = 4: 3(4) − 6 = 6. The 3N total degrees of freedom minus 3 translational and 3 rotational leaves 6 vibrational. (For a linear molecule it would be 3N − 5 = 7, since linear molecules have only 2 rotational degrees of freedom.)

Question 2

For a centrosymmetric molecule, the mutual exclusion rule states that no vibrational mode can be simultaneously IR-active and Raman-active.

• T. True
• F. False

Answer: True

Centrosymmetric molecules (those with an inversion center, like CO₂ or C₂H₂) have modes that are either symmetric (gerade, 'g') or antisymmetric (ungerade, 'u') with respect to inversion. IR activity requires breaking inversion symmetry (u modes), while Raman activity requires preserving it (g modes). No mode can satisfy both simultaneously, so IR-active and Raman-active sets are mutually exclusive.

Question 3

What physical criterion determines whether a vibrational normal mode is IR-active?

Model answer: A mode is IR-active if the vibration produces a change in the molecular electric dipole moment (∂μ/∂Q ≠ 0, where Q is the normal coordinate). If the dipole moment does not change during the vibration, the mode cannot absorb infrared radiation.

Infrared absorption occurs when the oscillating electric field of IR radiation couples to an oscillating dipole in the molecule. If a vibration is symmetric and produces no dipole change (as in the symmetric stretch of CO₂), there is nothing for the IR field to couple to, and the mode is IR-inactive. This selection rule comes directly from time-dependent perturbation theory applied to the interaction between radiation and molecular dipoles.