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 Dh group given below, what is the number of one-dimensional irreducible representations?

DhE2C3C𝜎h2S3𝜎v
A111111𝑥+𝑦,𝑧
A111111𝑅
E210210𝑥,𝑦𝑥𝑦,𝑥𝑦
A111111
A111111𝑧
E210210𝑅,𝑅𝑥𝑧,𝑦𝑧

Q2:

Fill in the blank: The character table for a C3v molecule, such as PCl3, lists three symmetry labels: 𝐴, 𝐴, and 𝐸. The 𝐴 mode transforms 𝑧, 𝑧, and 𝑥+𝑦. This means that the 𝐴 mode is .

  • Aneither IR nor Raman active
  • BIR active only
  • Cboth IR and Raman active
  • Dthe only spectroscopically active mode in the molecule
  • ERaman active only

Q3:

Fill in the blank: The character table for a D3h ion, such as CO32, lists six symmetry labels: 𝐴, 𝐴, 𝐸, 𝐴, 𝐴, and 𝐸. The 𝐴 mode transforms 𝑅 only. This means that the 𝐴 mode is .

  • Athe only spectroscopically active mode in the molecule
  • Bboth IR and Raman active
  • CIR active only
  • DRaman active only
  • Eneither IR nor Raman active

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