Paper: CSIR-NET December 2011 · Subject: Physical Chemistry · Chapter: Group Theory · Topic: Character Tables · Marks: 2 · Difficulty: Hard
The character table of the $\mathrm{C_{2v}}$ point group is given below: The two functions $\varphi=p_1+2p_2+2p_3+p_4$ and $\varphi=p_1-2p_2-2p_3+p_4$ (where $p_k$ is the $p$-orbital on the $k^{\text{th}}$ atom of cis-butadiene and $\sigma_v$ is the molecular plane) belong to
(a)$A_1$ and $A_2$ respectively
(b)Both $\mathrm{A}_{2}$
(c)Both $\mathrm{B_{2}}$
(d)$B_1$ and $B_2$ respectively.
Answer
Answer: C ✓ checked by 4AB · confidence high
Explanation
With σ_v as the molecular plane every p orbital changes sign under σ_v (χ = −1) and under the in-plane C₂, which also swaps p₁↔p₄, p₂↔p₃ (χ = −1 for both symmetric combinations); the perpendicular σ_v′ swaps the orbitals without a sign change (χ = +1). Characters (1, −1, −1, 1) are B₂ for both functions.