Paper: CSIR-NET December 2019 · Subject: Physical Chemistry · Chapter: Group Theory · Topic: Character Tables · Marks: 2 · Difficulty: Hard
The following is the character table for the $\mathrm{C}_{2 \mathrm{v}}$ point group
$C_{2 v}$
E
$\mathrm{C}_{\mathrm{2}}$
$\sigma_{v}$
$\sigma_{v}{ }^{\prime}$
$\mathrm{A}_{\mathrm{1}}$
1
1
1
1
$\mathrm{A}_{\mathrm{2}}$
1
1
-1
-1
$\mathrm{B}_{\mathrm{1}}$
1
-1
1
-1
$\mathrm{B}_{\mathrm{2}}$
1
-1
-1
1
There are two functions $\mathrm{f}_{1}$ and $\mathrm{f}_{2}$ belonging to $\mathrm{A}_{2}$ and $\mathrm{B}_{1}$ representations, respectively. The correct option for the product of two functions $\mathrm{f}_{1}$ abd $\mathrm{f}_{2}$ and the integral $\int \mathrm{f}_{1} \mathrm{f}_{2} \mathrm{dt}$ is
(a)The product belongs to $\mathrm{A}_{2}$ representation and the integral is non-zero
(b)The product belongs to $\mathrm{B}_{2}$ representation and the integral is zero
(c)The product belongs to $\mathrm{A}_{1}$ representation and the integral is zero
(d)The product belongs to $\mathrm{B}_{1}$ representation and the integral is non-zero
Answer
Answer: B ✓ checked by 4AB · confidence high
Explanation
A₂ × B₁ = B₂ in C₂v; the integral of a function that is not totally symmetric vanishes.