Question

The characters of \( E, C_2, \sigma_v, \) and \( \sigma' \) symmetry operations, in this order, for valid irreducible representation(s) of the \( C_2v \) point group is/are:

• A. 1, 1, 1, 1
• B. −1, 1, 1, −1
• C. 1, −1, 1, −1
• D. 1, −1, −1, −1

Hint:

The character table of a point group provides essential information on how molecular orbitals transform under symmetry operations.

Answer 1

The Correct Option is A, C

For the \( C_2v \) point group, the symmetry operations \( E, C_2, \sigma_v, \) and \( \sigma' \) have the following characters for the valid irreducible representations.
- \( E \) (identity operation) has a character of 1.
- \( C_2 \) (rotation by 180°) has a character of -1 for some irreducible representations.
- \( \sigma_v \) and \( \sigma' \) (vertical mirror planes) also contribute characters based on the symmetry of the system.
After performing the necessary character table analysis, the correct answer is (D) 1, −1, −1, −1. Final Answer: (D) 1, −1, −1, −1.