 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b/section-a/29/b/solution

The C2v molecular point group has four one-dimensional irreducible representations. In the operation order used above, the character table rows are

A_{1} = (1, 1, 1, 1)

A_{2} = (1, 1, - 1, - 1)

B_{1} = (1, - 1, 1, - 1)

and

B_{2} = (1, - 1, - 1, 1)

χ_{Γ} = (6, 0, 2, 0)

give

m_{A_{1}} = \frac{6 + 2}{4} = 2, m_{A_{2}} = \frac{6 - 2}{4} = 1, m_{B_{1}} = 2, m_{B_{2}} = 1.

Hence $Γ = 2 A_{1} \oplus A_{2} \oplus 2 B_{1} \oplus B_{2}$ , whose dimensions sum to six. The labels

B_{1}, B_{2}

refer to the source axes; swapping

x

and

y

swaps these names.

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