 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2024/paper-1/a5/a/ii/solution

Let

d_{μ, A}

and

d_{μ, B}

be corresponding real 4d orbitals on the upper and lower metals, with identical global-axis phases. Their symmetry-adapted linear combinations are

ϕ_{μ, \pm} = \frac{d _{μ, A} \pm d _{μ, B}}{2},

with the normalization shown in the negligible-overlap approximation; including overlap changes the denominator but not the symmetry. Inversion exchanges the centres and each local d orbital has even intrinsic parity, so the plus combinations are gerade and the minus combinations ungerade.

metal orbital d_{z^{2}} (d_{x z}, d_{yz}) d_{x y} d_{x^{2} - y^{2}} ϕ_{+} a_{1 g} e_{g} b_{2 g} b_{1 g} ϕ_{-} a_{2 u} e_{u} b_{1 u} b_{2 u} Mo-Mo bonding a_{1 g} (σ) e_{u} (π) b_{2 g} (δ) b_{1 g} (δ) Mo-Mo antibonding a_{2 u} (σ^{*}) e_{g} (π^{*}) b_{1 u} (δ^{*}) b_{2 u} (δ^{*})

The reversal of bonding phase for

d_{x z}, d_{yz}

is important: the facing lobes have opposite signs when the two atomic orbitals share the same global phase convention, so their difference is bonding. Each

e

irrep is two-dimensional. Thus

Γ_{10 d} = A_{1 g} + A_{2 u} + B_{1 g} + B_{2 u} + B_{2 g} + B_{1 u} + E_{g} + E_{u},

with total dimension ten. Rotating

x, y

4 5^{\circ}

interchanges the names of the two delta families; all labels here retain the chloride-aligned axes fixed above.

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