 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-3/49/c/i/solution

In the class order

(E, 8 C_{3}, 3 C_{2}, 6 S_{4}, 6 σ_{d})

, the 15 Cartesian nuclear displacements of methane have characters

(15, 0, - 1, - 1, 3)

. Only nuclei left fixed by an operation contribute to that operation's trace. Subtract translations

T_{2}

T_{1}

Γ_{vib} = A_{1} + E + 2 T_{2} .

1 + 2 + 3 + 3 = 9 = 3 (5) - 6

. There are four frequency families: a nondegenerate symmetric stretch, a doubly degenerate bend, and triply degenerate stretch and bend modes.

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