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GATE 2021 CY – Question 27

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 select

The characters of $E,C_2,\sigma_v,\sigma_v\prime$, in that order, for valid irreducible representations of $C_{2v}$ are ______.

  1. 1, 1, 1, 1
  2. −1, 1, 1, −1
  3. 1, −1, 1, −1
  4. 1, −1, −1, −1

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Correct answer: (A) 1, 1, 1, 1; (C) 1, −1, 1, −1

Explanation

All C2v irreducible representations are one-dimensional, so the identity character must be +1: this rules out B. Since C2=σvσv′, the corresponding signs must multiply consistently. A=(1,1,1,1) and C=(1,−1,1,−1) satisfy the group products. D fails because (−1)(−1)=+1 for the mirrors but gives C2=−1. **A,C**.