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GATE 2019 CY – Question 62

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. · 2 marks · Numerical answer

For point group D₈, the character-table columns are E, 2C₈, 2C₄, 2C₈³, C₂, 4C₂′, 4C₂″, in that order. Rows are:

A₁: a, 1, 1, 1, 1, 1, 1
A₂: b, 1, 1, 1, 1, h, i
B₁: c, −1, 1, −1, 1, 1, j
B₂: d, −1, 1, −1, 1, −1, 1
E₁: e, √2, 0, −√2, −2, 0, 0
E₂: f, 0, −2, 0, k, 0, 0
E₃: g, −√2, 0, √2, −2, 0, 0

Find a+b+c+d+e+f+g+h+i+j+k.

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Correct answer: 9 to 9

Explanation

Characters under identity equal irrep dimensions: a=b=c=d=1 and e=f=g=2. For A₂, the perpendicular twofold characters are h=i=−1. For B₁, j=−1. In E₂, the principal C₂ rotation gives k=2cos(2π)=2. The requested sum is 4+6−1−1−1+2=9.

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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 →