Questions: Group Theory and Vibrational Mode Classification

Question 1

After applying the reduction formula to water (C₂ᵥ), a student finds all 3 vibrational modes belong to symmetry species that transform as x, y, or z in the character table. What is the correct conclusion?

• A. All 3 modes are Raman active only, because C₂ᵥ symmetry prohibits IR activity
• B. All 3 modes are IR active, because they transform as linear functions matching the dipole moment components
• C. None of the modes are spectroscopically active, because water is too simple to show IR absorption
• D. The modes are IR active but the mutual exclusion rule prevents any Raman activity in C₂ᵥ

Answer: (B)

IR activity requires that the vibrational mode transform as x, y, or z — the components of the dipole moment vector. If all 3 modes satisfy this, all 3 are IR active. The mutual exclusion rule (option D) applies only to centrosymmetric molecules with an inversion center, like CO₂. Water (C₂ᵥ) has no inversion center, so there is no mutual exclusion — some modes may be both IR and Raman active simultaneously. This is read directly from the character table without any wavefunction calculation.

Question 2

CO₂ is a linear, centrosymmetric molecule belonging to a point group with an inversion center. What does the mutual exclusion rule predict about its vibrational spectra?

• A. All modes are both IR and Raman active, because the molecule's high symmetry makes every mode allowed
• B. No mode can be simultaneously IR and Raman active — each vibrational mode is exclusively one or the other (or inactive in both)
• C. All modes are IR active and none are Raman active, because linear molecules cannot change polarizability
• D. Symmetry-based selection rules do not apply to linear molecules, so all modes must be calculated numerically

Answer: (B)

The mutual exclusion rule states that for centrosymmetric molecules (those with an inversion center), no normal mode can be both IR and Raman active. IR-active modes must be antisymmetric under inversion (they change the dipole); Raman-active modes must be symmetric under inversion (they change the polarizability). These are mutually exclusive symmetry requirements. CO₂'s symmetric stretch is Raman active and IR inactive; its asymmetric stretch is IR active and Raman inactive. The two spectra are complementary, giving non-overlapping but jointly complete information.

Question 3

To determine whether a vibrational mode of a molecule is IR active, one should solve the Schrödinger equation to calculate the vibrational wavefunction and its dipole moment integral.

• T. True
• F. False

Answer: False

This is precisely what group theory allows you to bypass. By determining the molecule's point group, constructing Γ_total from atomic displacement coordinates, subtracting Γ_trans and Γ_rot, and decomposing what remains using the reduction formula, you identify which irreducible representations each vibrational mode belongs to. You then look up whether those representations appear in the x, y, or z column of the character table. No wavefunctions are computed — the result follows entirely from the molecular geometry and the character table.

Question 4

Translations and rotations must be subtracted from Γ_total before identifying vibrational modes because they contribute real, nonzero characters to the reducible representation even though they are not vibrations.

• T. True
• F. False

Answer: True

The reducible representation Γ_total is constructed from all 3N Cartesian displacement coordinates, which describe every possible motion of the molecule — including pure translations (the whole molecule moving in x, y, z) and rotations. These non-vibrational motions transform as specific irreducible representations explicitly listed in the character table. Subtracting Γ_trans and Γ_rot is mandatory to isolate Γ_vib. Forgetting this step produces a count of vibrational modes that is too high by 5 (linear molecules) or 6 (nonlinear molecules) — the single most common procedural error in this analysis.

Question 5

Explain in conceptual terms why IR activity requires a change in dipole moment and how group theory predicts which vibrational modes will produce such a change.

Model answer: For a vibration to absorb IR radiation, its electric field must couple to a changing dipole in the molecule. A changing dipole is a vector quantity with x, y, z components — so the vibrational motion must transform under the molecule's symmetry operations the same way that a linear translation in x, y, or z does. Group theory assigns each vibrational mode to an irreducible representation; the character table shows which representations transform as x, y, or z. Any mode assigned to one of those representations changes the dipole moment when atoms move along that mode, making it IR active. The prediction requires only symmetry arguments — no computation of the actual dipole integral.

The same logic applies to Raman activity (requiring transformation as a quadratic function like x², xy, etc., because Raman involves the polarizability tensor). Understanding that selection rules are symmetry properties — not energetic calculations — is the central payoff of group theory in spectroscopy.