9.3 Molecular Orbital Theory for Coordination Compounds

Molecular Orbital Formation

Ligand and Metal Orbital Interactions

Crystal Field Theory and Ligand Field Theory treat metal-ligand bonding as mostly electrostatic or focus on d-orbital splitting alone. Molecular Orbital (MO) Theory goes further: it constructs a full set of bonding, antibonding, and nonbonding orbitals from the overlap of metal and ligand orbitals. This gives you a more complete picture of electron distribution, stability, and spectroscopic behavior in coordination compounds.

The starting point is to identify the orbitals each partner brings to the interaction:

• The metal contributes its valence s, p, and d orbitals. In an isolated metal ion, the five d orbitals ($d_{xy}$, $d_{yz}$, $d_{xz}$, $d_{x^2-y^2}$, $d_{z^2}$) are degenerate (equal in energy).
• The ligands each contribute a lone-pair donor orbital. These individual ligand orbitals are combined into symmetry-adapted linear combinations (SALCs), which are group orbitals that match the symmetry of specific metal orbitals.

Only metal and ligand orbitals of the same symmetry representation can overlap. When they do:

• Bonding MOs form from constructive interference, sitting lower in energy than either parent orbital.
• Antibonding MOs form from destructive interference, sitting higher in energy than either parent orbital.
• Nonbonding MOs have no symmetry match on the other partner, so they stay at roughly the same energy and don't contribute to bond strength.

Orbital Symmetry and Energy Levels

Symmetry is the gatekeeper for orbital mixing. In an octahedral complex, for example, the metal $d_{z^2}$ and $d_{x^2-y^2}$ orbitals (which transform as $e_g$ in the $O_h$ point group) have the right symmetry to overlap with ligand SALCs directed along the bonding axes. The $d_{xy}$, $d_{xz}$, and $d_{yz}$ set ($t_{2g}$) point between the ligands and are nonbonding in a sigma-only model.

Two factors control how far apart the bonding and antibonding MOs end up in energy:

• Extent of overlap: better spatial overlap pushes the bonding orbital lower and the antibonding orbital higher.
• Energy match: when the metal and ligand orbitals are close in energy, mixing is stronger. A large energy mismatch means weaker interaction, even if symmetry allows it.

An MO diagram maps out all of these orbitals by energy and shows where the electrons go. For an octahedral $\sigma$-only complex, you'll typically see six bonding MOs (filled primarily by ligand electrons), a set of nonbonding $t_{2g}$ orbitals, and a set of antibonding $e_g^*$ orbitals. The energy gap between $t_{2g}$ and $e_g^*$ corresponds to $\Delta_o$, the same splitting parameter from Crystal Field Theory, but now it has a clear orbital-overlap origin.

Coordination Geometries

Common Coordination Geometries

• Octahedral complexes have six ligands arranged symmetrically around the metal, forming an eight-faced polyhedron. This is the most common geometry and the one MO theory is most often applied to in detail.
• Tetrahedral complexes have four ligands at the vertices of a tetrahedron. The d-orbital splitting pattern inverts relative to octahedral, and $\Delta_t$ is roughly $\frac{4}{9}\Delta_o$ for the same metal-ligand pair.
• Square planar complexes also have four ligands, but arranged in a plane. This geometry is especially common for $d^8$ metal ions (e.g., $\text{Pt}^{2+}$, $\text{Ni}^{2+}$ with strong-field ligands) because the orbital splitting pattern strongly stabilizes a low-spin $d^8$ configuration.

Types of Bonding in Coordination Compounds

Sigma ($\sigma$) bonding involves head-on overlap between a ligand donor orbital and a metal orbital along the metal-ligand axis. This is the primary bonding interaction in most complexes and the one described by the basic MO diagram.

Pi ($\pi$) bonding involves side-on overlap of metal $t_{2g}$ orbitals with ligand orbitals perpendicular to the bonding axis. This is where MO theory really shows its power over simpler models:

• Pi-donor ligands (e.g., $\text{Cl}^-$, $\text{OH}^-$) have filled p orbitals that donate electron density into the metal $t_{2g}$ set. This raises the energy of the $t_{2g}$ orbitals and decreases $\Delta_o$.
• Pi-acceptor ligands (e.g., CO, $\text{CN}^-$) have empty $\pi^*$ or similar orbitals that accept electron density from the metal $t_{2g}$ set. This lowers the energy of the $t_{2g}$ orbitals and increases $\Delta_o$.

This pi-bonding framework explains why CO and $\text{CN}^-$ sit at the strong-field end of the spectrochemical series while halides sit at the weak-field end. Crystal Field Theory can't account for this.

Delta ($\delta$) bonding arises from face-to-face overlap of d orbitals and is relevant mainly in metal-metal multiple bonds (e.g., the quadruple bond in $[\text{Re}_2\text{Cl}_8]^{2-}$), not in typical metal-ligand interactions.

Factors Affecting Geometry

• Electronic configuration: the number of d electrons and their arrangement in the MO scheme favor certain geometries. For instance, $d^8$ ions with strong-field ligands strongly prefer square planar because of the large stabilization energy gained.
• Ligand field stabilization energy (LFSE): the total stabilization from placing electrons in lower-energy orbitals vs. higher-energy ones. Comparing LFSE across geometries helps predict which one a complex will adopt.
• Steric effects: bulky ligands can prevent a metal from achieving its electronically preferred coordination number, pushing the complex toward lower coordination geometries.

Electronic Configuration and Properties

Spectrochemical Series and Ligand Field Strength

The spectrochemical series ranks ligands by how large a splitting ($\Delta_o$) they produce. A partial ordering:

$\text{I}^- < \text{Br}^- < \text{Cl}^- < \text{OH}^- < \text{H}_2\text{O} < \text{NH}_3 < \text{NO}_2^- < \text{CN}^- < \text{CO}$

Weak-field ligands (left side) are typically pi-donors or have poor orbital overlap with the metal. Strong-field ligands (right side) are typically pi-acceptors. MO theory explains this trend directly through the pi-bonding mechanism described above, which is one of its major advantages.

Spin States in Coordination Complexes

The competition between two energies determines the spin state:

• Pairing energy (P): the energy cost of forcing two electrons into the same orbital.
• Splitting energy ($\Delta_o$): the energy gap between the lower and upper d-orbital sets.

If $\Delta_o < P$, electrons spread out across higher-energy orbitals to avoid pairing, giving a high-spin complex with more unpaired electrons. Weak-field ligands favor this.

If $\Delta_o > P$, electrons fill the lower set first and pair up before occupying higher orbitals, giving a low-spin complex with fewer unpaired electrons. Strong-field ligands favor this.

This distinction matters most for octahedral $d^4$ through $d^7$ configurations, where both high-spin and low-spin arrangements are possible. For $d^1$–$d^3$ and $d^8$–$d^{10}$, there's only one possible arrangement regardless of field strength.

Magnetic and Spectroscopic Properties

Magnetic behavior depends directly on unpaired electrons. Complexes with unpaired electrons are paramagnetic (attracted to a magnetic field), while those with all electrons paired are diamagnetic (weakly repelled). High-spin complexes are more strongly paramagnetic than their low-spin counterparts for the same $d^n$ configuration.

Spectroscopic properties arise from electronic transitions between MOs:

• d-d transitions occur between the lower and upper d-orbital sets (e.g., $t_{2g} \rightarrow e_g^*$ in octahedral complexes). These are typically weak, Laporte-forbidden transitions that produce the characteristic colors of many coordination compounds.
• Charge transfer (CT) transitions involve electron movement between metal-centered and ligand-centered MOs. These are Laporte-allowed and tend to be much more intense than d-d transitions. They can be metal-to-ligand (MLCT) or ligand-to-metal (LMCT), and they often dominate the color of complexes with strong-field or easily oxidized/reduced ligands.

The energy of these transitions, and therefore the color observed, ties directly back to $\Delta_o$ and the overall MO energy level structure.