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 1 , 𝐴 2 , and 𝐸 . The 𝐴 1 mode transforms 𝑧 , 𝑧 2 , and 𝑥 + 𝑦 2 2 . This means that the 𝐴 1 mode is .

  • ARaman active only
  • BIR active only
  • Cneither IR or Raman active
  • Dboth IR and Raman active
  • Ethe only spectroscopically active mode in the molecule

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 1 , 𝐴 2 , 𝐸 , 𝐴 1 , 𝐴 2 , and 𝐸 . The 𝐴 2 mode transforms 𝑅 𝑧 only. This means that the 𝐴 2 mode is .

  • ARaman active only
  • BIR active only
  • Cboth IR and Raman active
  • Dneither IR or Raman active
  • Ethe only spectroscopically active mode in the molecule

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.