Questions: Molecular Orbital Theory for Transition Metal Complexes

Question 1

In the MO diagram of an octahedral complex with sigma-only ligands, which metal-based orbitals become the HOMO for a d⁶ low-spin configuration?

• A. The bonding eg MOs
• B. The nonbonding t₂g MOs
• C. The antibonding eg* MOs
• D. The metal 4s-based a₁g MO

Answer: (B)

With sigma-only ligands, the metal t₂g orbitals (d_xy, d_xz, d_yz) point between the ligands and have no sigma-bonding ligand counterpart — they remain nonbonding. In a d⁶ low-spin complex, all six electrons fill the t₂g set completely. These are the highest occupied molecular orbitals, making them the HOMO. The bonding eg MOs are lower in energy (filled by ligand electrons), while the antibonding eg* MOs are the LUMO — the 'Δ_oct' gap in MO theory is the energy difference between t₂g and eg*.

Question 2

The crystal field splitting parameter Δ_oct in MO theory corresponds to the energy gap between the nonbonding t₂g orbitals and the antibonding eg* orbitals, not between bonding sets.

• T. True
• F. False

Answer: True

This is a critical conceptual point. In CFT, Δ_oct appears to be between two sets of d-orbitals. In MO theory, the full picture reveals that the 'eg' orbitals involved in the splitting are actually antibonding MOs formed from metal eg-symmetry orbitals mixing with ligand sigma-donor SALCs. The t₂g orbitals are nonbonding (with sigma-only ligands) or modified by pi interactions. The gap between these two sets — t₂g (nonbonding or pi-modified) and eg* (antibonding) — is what CFT calls Δ_oct. This explains why stronger sigma-donors produce larger Δ: they push the eg* antibonding orbitals higher.

Question 3

In the MO diagram of [Co(CO)₆]³⁺, the t₂g orbitals are significantly stabilized compared to a sigma-only complex. This stabilization results from pi-back-bonding into CO's empty π* orbitals.

• T. True
• F. False

Answer: True

CO has empty π* orbitals with the correct symmetry (t₂g in Oh) to overlap with the filled metal t₂g orbitals. This interaction creates bonding and antibonding combinations: the bonding combination lowers the energy of the metal-based t₂g electrons (stabilizing them), while the antibonding combination raises CO-based orbitals. The net effect is that the occupied t₂g level drops in energy, increasing the gap to eg* and producing a very large Δ_oct. This is the MO basis for CO being the strongest common field ligand.

Question 4

Construct the qualitative MO energy level diagram for an octahedral ML₆ complex with pi-donor ligands and explain how it differs from the sigma-only case.

Model answer: Start with the sigma-only case: six ligand sigma SALCs interact with metal orbitals of matching symmetry (a₁g with s, t₁u with p, eg with d_z²/d_x²−y²) to form bonding and antibonding pairs. The t₂g metal orbitals are nonbonding. Now add pi-donor interactions: the ligand pi-donor orbitals (filled) have t₂g symmetry and interact with the metal t₂g orbitals. Since the ligand pi-orbitals are filled, this produces a bonding MO (mostly ligand character, lower energy) and an antibonding MO (mostly metal character, higher energy). The metal-based t₂g level is pushed UP in energy, decreasing the gap to eg* — hence Δ_oct shrinks. This is the MO explanation for why pi-donors are weak-field ligands.

For pi-acceptors, the effect reverses: the ligand pi* orbitals are empty and higher in energy than the metal t₂g. The interaction produces a bonding MO (mostly metal character, lowered) and antibonding MO (mostly ligand character, raised). The metal-based t₂g drops in energy, increasing Δ_oct. The full MO diagram thus provides a unified picture of the entire spectrochemical series.