Question

The correct electronic configuration of central metal ion in ferrocene with \(z\)-axis passing through the centre of two \(\mathrm{C_5H_5^-}\) rings and the metal ion is:

• A. \(d_{xy}^2 = d_{x^2-y^2}^2 < d_{z^2}^2 < d_{xz}^0 = d_{yz}^0\)
• B. \(d_{xz}^2 = d_{yz}^2 < d_{z^2}^2 < d_{x^2-y^2}^0 = d_{xy}^0\)
• C. \(d_{xy}^2 = d_{xz}^2 = d_{yz}^2 < d_{x^2-y^2}^0 = d_{z^2}^0\)
• D. \(d_{xy}^2 < d_{xz}^2 = d_{yz}^2 < d_{x^2-y^2}^0 < d_{z^2}^0\)

Hint:

Rank the d-orbitals by overlap with the Cp ring \(\pi\) system: the in-plane pair is lowest (nonbonding), \(d_{z^2}\) is next, and the tilted pair is highest; fill \(d^6\) from the bottom, fully paired since ferrocene is diamagnetic.

Answer 1

The Correct Option is A

Step 1: Set up the coordinate system and point group.
In ferrocene, \(\mathrm{Fe(C_5H_5)_2}\), the two cyclopentadienyl rings sandwich the iron centre. With the \(z\)-axis through the centres of both rings, the effective symmetry seen by the metal \(d\)-orbitals is \(D_{5d}\) or \(D_{5h}\); either way the \(d\)-orbitals split into three sets by overlap with the ring \(\pi\) orbitals: \(a_{1g}\) (\(d_{z^2}\)), \(e_{1g}\) (\(d_{xz},d_{yz}\)), and \(e_{2g}\) (\(d_{xy},d_{x^2-y^2}\)).

Step 2: Rank the three sets by overlap with the ring \(\pi\) system.
\(d_{xy}\) and \(d_{x^2-y^2}\), the \(e_{2g}\) pair, point between the ring carbons in the \(xy\)-plane and overlap poorly with the ring \(\pi\) orbitals, staying essentially nonbonding and lowest in energy. \(d_{z^2}\) (\(a_{1g}\)) points straight along the sandwich axis at both rings and has a small \(\sigma\)-type overlap, sitting slightly higher. \(d_{xz}\) and \(d_{yz}\) (\(e_{1g}\)) have the strongest directional overlap with the ring \(\pi\) orbitals and rise highest. So the order is \(e_{2g} < a_{1g} < e_{1g}\).

Step 3: Fill in the electrons.
Ferrocene is \(\mathrm{Fe^{2+}}\), \(d^6\), and diamagnetic (an 18-electron, low-spin complex), so all 6 electrons pair up in the lowest available orbitals: 4 electrons fill the doubly degenerate \(e_{2g}\) pair (\(d_{xy}^2,d_{x^2-y^2}^2\)) and the remaining 2 fill \(a_{1g}\) (\(d_{z^2}^2\)), leaving \(e_{1g}\) empty.

Step 4: Write and match the configuration.
\[ d_{xy}^2 = d_{x^2-y^2}^2 < d_{z^2}^2 < d_{xz}^0 = d_{yz}^0 \]
This is option (A). Option (B) swaps the order of the \(e_{1g}\) and \(e_{2g}\) sets; option (C) wrongly places \(d_{z^2}\) as degenerate with the empty set; option (D) breaks the required degeneracy within each \(e\) pair.

Final Answer:
The configuration is \(d_{xy}^2 = d_{x^2-y^2}^2 < d_{z^2}^2 < d_{xz}^0 = d_{yz}^0\), option (A). \[ \boxed{\text{(A)}} \]