Questions: Molecular Orbital Symmetry Classification

Question 1

Two atomic orbitals are very close in energy but belong to different irreducible representations of the molecule's point group. Can they mix to form molecular orbitals?

• A. Yes — energy proximity is the primary criterion for orbital mixing
• B. No — only orbitals of matching symmetry (same irreducible representation) can mix, regardless of energy
• C. Yes — but the mixing will be weak because of the symmetry mismatch
• D. Only if both orbitals are of σ type

Answer: (B)

Symmetry is a hard constraint, not a preference. The overlap integral between orbitals of different irreducible representations is exactly zero by symmetry — no matter how close in energy they are, there is no interaction. This surprises students who assume that 'similar energy → mixing.' Energy proximity determines the *degree* of mixing when symmetry allows it, but symmetry determines *whether* mixing is allowed at all.

Question 2

In water (C₂ᵥ symmetry), the oxygen 2px orbital belongs to the b₁ irreducible representation. No hydrogen orbital combination shares that symmetry. What type of orbital does the oxygen 2px become in the water MO diagram?

• A. A strongly bonding orbital — it overlaps broadly with the hydrogen 1s orbitals
• B. A strongly antibonding orbital — its mismatched symmetry causes destructive interference
• C. A nonbonding lone pair — it cannot interact with any hydrogen combination
• D. A σ* orbital — it cancels the contribution of the bonding σ orbital

Answer: (C)

When an atomic orbital has no partner of matching symmetry among the other basis orbitals, it cannot mix into any bonding or antibonding MO. It remains as a pure nonbonding orbital — its energy is unchanged from the isolated atom, and it constitutes a lone pair. This is how MO theory predicts the existence and location of lone pairs without any empirical guessing.

Question 3

In a linear molecule, the labels σ, π, and δ are themselves symmetry classifications — σ orbitals are symmetric with respect to rotation about the bond axis, while π orbitals have one nodal plane containing the bond axis.

• T. True
• F. False

Answer: True

Exactly right. σ, π, and δ are not just conventional labels — they encode the orbital's behavior under the symmetry operations of the linear molecule's point group (C∞ᵥ or D∞ₕ). A σ orbital is totally symmetric with respect to rotation; a π orbital changes sign under 180° rotation and is doubly degenerate. For nonlinear molecules, these simple labels are replaced by irreducible representation labels of the appropriate point group, but the logic is identical.

Question 4

Two orbitals can typically be made to interact by adjusting the molecular geometry, even if they currently belong to different irreducible representations.

• T. True
• F. False

Answer: False

Changing geometry changes the point group and can change which irreducible representation each orbital belongs to — so geometry changes can sometimes enable previously forbidden interactions. However, for a fixed geometry with a fixed point group, the symmetry constraint is absolute: orbitals of different irreducible representations have zero overlap and cannot mix. The statement inverts cause and effect: it is the geometry that determines the symmetry labels, not the other way around.

Question 5

Explain why the symmetry matching rule — that only orbitals of the same irreducible representation can mix — allows chemists to predict molecular orbital diagrams without doing any quantum mechanical calculations.

Model answer: Because the overlap integral between orbitals of different irreducible representations is exactly zero, which can be proved from symmetry alone without computing any integrals. Group theory therefore tells you which interactions are forbidden (zero overlap by symmetry) and which are allowed (potentially nonzero overlap). This reduces MO construction to asking 'what symmetry does each atomic orbital transform as?' and grouping orbitals accordingly. The energy ordering within allowed interactions requires calculation, but the pattern of which orbitals interact at all is determined by symmetry alone.

This is the central power of group theory applied to chemistry. Instead of solving the Schrödinger equation for every molecule from scratch, you identify the point group, assign irreducible representations to each atomic orbital, and immediately know the allowed interactions. Orbitals of the same irreducible representation form bonding/antibonding pairs; orbitals with no symmetry-matching partners become nonbonding. The entire MO diagram topology follows from symmetry before any energy calculation begins.