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Lesson: Character Tables

Worksheet • 3 Questions

Q1:

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 π‘₯ + 𝑦 2 2 . This means that the 𝐴 1 mode is .

  • Aboth IR and Raman active
  • Bthe only spectroscopically active mode in the molecule
  • CIR active only
  • Dneither IR or Raman active
  • ERaman active only

Q2:

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   π‘₯ + 𝑦 , 𝑧   
A   1 1 βˆ’ 1 1 1 βˆ’ 1 𝑅   
E β€² 2 βˆ’ 1 0 2 βˆ’ 1 0 π‘₯ , 𝑦 π‘₯ βˆ’ 𝑦 , π‘₯ 𝑦  
A οŽ™  1 1 1 βˆ’ 1 βˆ’ 1 βˆ’ 1    
A οŽ™  1 1 βˆ’ 1 βˆ’ 1 βˆ’ 1 1 𝑧  
E β€³ 2 βˆ’ 1 0 βˆ’ 2 1 0 𝑅 , 𝑅   π‘₯ 𝑧 , 𝑦 𝑧

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

  • Aneither IR or Raman active
  • Bthe only spectroscopically active mode in the molecule
  • CIR active only
  • Dboth IR and Raman active
  • ERaman active only
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