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GATE 2020 CY – Question 25

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 pyramidal AB₃ molecule has C₃ᵥ character-table rows (E,2C₃,3σᵥ): A₁ = (1,1,1), bases z and (x²+y²,z²); A₂ = (1,1,−1), base Rz; E = (2,−1,0), bases (x,y),(Rx,Ry),(x²−y²,xy),(xz,yz). Its total reducible representation is Γ = (12,0,2). Which gives all normal Raman-active modes?

  1. $A_1+A_2+2E$
  2. $3E$
  3. $3A_1+A_2+E$
  4. $2A_1+2E$

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Correct answer: (D) $2A_1+2E$

Explanation

Reduction gives Γ₃N = 3A₁ + A₂ + 4E. Translations transform as A₁+E and rotations as A₂+E. Subtracting both yields Γvib = 2A₁+2E, six vibrational degrees of freedom. A₁ and E have quadratic basis functions in the character table, so all these modes are Raman active. Select D.

More 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. questions

All 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. questions →