Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 10 / b

Use the stated local axes: $z$ is radial along O–Ru, $x$ is tangential within the Ru–O plane, and $y$ is perpendicular to that plane. The oxygen's three $s{p}^2$ hybrids make the in-plane sigma framework. Its remaining $p_y$ orbital is perpendicular to the plane and can overlap with one $d_{yz}$ orbital on each Ru.

Choose the local orbital phases so their O-facing lobes agree. Of the three metal orbitals, onlyhas the appropriate combination to interact with oxygen $p_y$. The other two independent combinations have coefficients summing to zero and are nonbonding with respect to the central oxygen. Mixing $D$ with $p_y$ gives one lower pi-bonding orbital and one upper pi-antibonding orbital. Each Ru also retains its $d_{xy}$ and $d_{xz}$ orbitals as nonbonding with respect to this central-oxo pi interaction. Thus the complete low-energy diagram contains one bonding level, eight metal-nonbonding levels, and one antibonding level. The eight consist of two $d_{yz}$ combinations, three $d_{xy}$ and three $d_{xz}$ orbitals. They need not be exactly degenerate once other ligands are considered.

For the neutral formula as printed, oxide contributes $-2$ and six acetates contribute $-6$. The Ru oxidation numbers therefore sum to $+8$, giving sixteen metal d electrons overall, formally two Ru(III) and one Ru(II), possibly delocalized. Including the oxygen pi pair gives eighteen electrons, filling the bonding orbital and all eight nonbonding orbitals, leaving pi* empty. If the often encountered singly charged cation were intended instead, there would be seventeen electrons and one singly occupied nonbonding level. This charge distinction changes occupancy, not the orbital connectivity.

The three $d_{xz}$ orbitals are in the Ru–O plane and can overlap directly along the Ru–Ru edges. Their locally rotated lobes have nonzero edge-directed overlap, producing metal–metal bonding and antibonding combinations; a simple triangle model resolves the three orbitals into a singlet and a doublet. The phases and ordering depend on the chosen local basis, so the important feature is the complete set of three combinations.

In the occupancy obtained in (i), this $d_{xz}$ set contains six electrons. All three combinations are filled: the stabilization of the bonding combinations is offset by occupation of their antibonding partners. Equivalently, the sum of the three eigenvalues of a pure off-diagonal interaction matrix is zero. Hence there is no net Ru–Ru bond order from this filled set in the stated model. Overlap alone is not evidence of a net metal–metal bond. This conclusion neglects higher-order mixing and the interactions explicitly excluded from the qualitative diagram.