 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/28/a/solution

For the character of a localized orbital basis, take the trace of the transformation matrix: both permutation of centres and signs of atomic orbitals matter. The two metal centres lie on the principal axis and remain fixed under rotation about

z

. At either metal the

(p_{x}, p_{y})

R (θ) = (cos θ sin θ - sin θ cos θ) .

The full four-dimensional group representation is block diagonal with two such blocks. Consequently

χ (R) = 4 cos θ

, and for the

6 0^{\circ}

rotation of the D6h molecular point group, $χ (C_{6}) = 4 cos (π /3) = 2$ .

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