Worksheet: Character Tables

In this worksheet, we will practice writing and interpreting character tables listing the irreducible representations for common molecular point groups.

Q1:

Using the character table for the D h group given below, what is the number of one-dimensional irreducible representations?

D h E 2 C 3 C 𝜎 h 2 S 3 𝜎 v
A 1 1 1 1 1 1 𝑥 + 𝑦 , 𝑧
A 1 1 1 1 1 1 𝑅
E 2 1 0 2 1 0 𝑥 , 𝑦 𝑥 𝑦 , 𝑥 𝑦
A 1 1 1 1 1 1
A 1 1 1 1 1 1 𝑧
E 2 1 0 2 1 0 𝑅 , 𝑅 𝑥 𝑧 , 𝑦 𝑧

Q2:

Fill in the blank: The character table for a C 3 v molecule, such as P C l 3 , lists three symmetry labels: 𝐴 , 𝐴 , and 𝐸 . The 𝐴 mode transforms 𝑧 , 𝑧 , and 𝑥 + 𝑦 . This means that the 𝐴 mode is .

  • Aboth IR and Raman active
  • BRaman active only
  • Cneither IR nor Raman active
  • Dthe only spectroscopically active mode in the molecule
  • EIR active only

Q3:

Fill in the blank: The character table for a D 3 h ion, such as C O 3 2 , lists six symmetry labels: 𝐴 , 𝐴 , 𝐸 , 𝐴 , 𝐴 , and 𝐸 . The 𝐴 mode transforms 𝑅 only. This means that the 𝐴 mode is .

  • Aneither IR nor Raman active
  • BIR active only
  • CRaman active only
  • Dthe only spectroscopically active mode in the molecule
  • Eboth IR and Raman active

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