A point at (1,3,5) is transformed by O 1 to (-1,-3,5), then by O 2 to (-1,-3,-5). The symmetry operators O 1, O 2, respectively, are .

$\hat C_2(x)$ and $\hat\sigma(xy)$

$\hat\sigma(xy)$ and $\hat C_2(z)$

GATE 2022 CY – Question 23

Physical Chemistry · Group Theory: Symmetry elements and symmetry operations. Basic postulates of groups and group multiplication tables. Classes, reducible and irreducible representations of symmetry elements. Classification and labelling of molecular point groups. Symmetry based selection rules and their application to electronic and vibrational spectroscopy. Internal coordinates and symmetry labelling of vibrational modes; symmetry adapted linear combination of atomic orbitals (LCAO-MO); construction of hybrid orbitals using molecular point group symmetry. · 1 mark · Multiple choice

A point at $(1,3,5)$ is transformed by $\hat O_1$ to $(-1,-3,5)$, then by $\hat O_2$ to $(-1,-3,-5)$. The symmetry operators $\hat O_1,\hat O_2$, respectively, are ______.

$\hat C_2(x)$ and $\hat\sigma(xy)$
$\hat C_2(z)$ and $\hat\sigma(xy)$
$\hat\sigma(xy)$ and $\hat C_2(z)$
$\hat S_1$ and $\hat S_2$

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Correct answer: (B) $\hat C_2(z)$ and $\hat\sigma(xy)$

Explanation

A 180° rotation about z changes(x, y, z) to(−x,−y, z), matching the first step. Reflection in the xy plane changes only z’s sign, matching the second step.

Therefore **C2(z),σ(xy), B**.