MO Description of Tetrahedral and Square-Planar Complexes

The tetrahedral MO picture explains the geometric preference and magnetism of numerous biological and industrial metal centers: high-spin tetrahedral Fe(II)/Fe(III) and Co(II) sites in zinc-finger-like and rubredoxin-type proteins, tetrahedral Zn(II) in carbonic anhydrase and hundreds of hydrolase active sites (Zn²⁺ is d¹⁰, so there is no CFSE preference either way, and tetrahedral geometry is chosen by sterics/charge, illustrating the d⁰/d¹⁰ limiting case directly), and the systematic weak-field tendency of transition-metal halide and oxide tetrahedral complexes used as catalysts and pigments (cobalt blue glass, [CoCl4]²⁻-doped materials). The square-planar picture is the electronic backbone of essentially all d⁸ homogeneous catalysis: Wilkinson's catalyst RhCl(PPh3)3 (technically 5-coordinate but built from a square-planar-like d⁸ Rh(I) fragment), Vaska's complex IrCl(CO)(PPh3)2 used to teach and study oxidative addition, Pd(0)/Pd(II) cross-coupling catalytic cycles (Suzuki, Negishi, Stille couplings all cycle a square-planar d⁸/d¹⁰ Pd center), Ni(II) cross-coupling catalysts, and cisplatin [PtCl2(NH3)2], whose square-planar geometry (Pt(II), d⁸) is essential both to its DNA-crosslinking mechanism and to the cis/trans isomer distinction that makes only the cis isomer an active anticancer drug. Square-planar d⁸ Ni(II), Pd(II), and Pt(II) complexes are also textbook cases for spectrochemical/magnetochemical characterization: diamagnetic square-planar versus paramagnetic tetrahedral Ni(II) complexes are routinely distinguished by simple magnetic susceptibility (Evans method NMR or SQUID magnetometry) as an undergraduate laboratory exercise, directly testing the predictions of this MO framework. The 18-electron/16-electron counting logic derived here underlies modern computational catalyst design: density-functional-theory screening of candidate cross-coupling or hydrogenation catalysts routinely checks whether a proposed square-planar d⁸ intermediate is genuinely "coordinatively unsaturated" (16e, with an accessible empty axial b1g*/dz²-adjacent pocket) versus saturated — a computational shortcut that is really a restatement of the orbital-symmetry argument built above. The same Ligand Field Theory: Sigma-Bonding MO Diagram (Octahedral) and Pi-Bonding and Rationalizing the Spectrochemical Series machinery, extended geometry-by-geometry as in this article, remains the standard first-pass tool for predicting spin state, color, and reactivity of any new coordination complex before more expensive computation is applied.

Ask the tutor: How does adding π-donor or π-acceptor character to the ligands quantitatively shift the tetrahedral e set and the square-planar eg/b2g set, and how would you estimate the resulting change in Δt or Δsp from spectroscopic data?

Every ligand-field MO diagram is built the same way regardless of geometry. First, place the metal at the origin, and generate a set of ligand group orbitals (LGOs, or SALCs) from the σ-donor orbital on each ligand, using the group theory of the point group. Second, find the irreducible representations spanned by the metal's valence s, p, and d orbitals in that point group. Third, match symmetry labels: any metal orbital and any ligand SALC of the same irreducible representation can mix; metal orbitals with no ligand-SALC partner remain strictly nonbonding. The bonding/antibonding energy gap for each matched pair scales with the square of their overlap integral, divided by the energy separation of the unmixed orbitals — exactly the two-level perturbation result you already used to build the octahedral diagram.

Octahedral baseline (Oh, six ligands along ±x, ±y, ±z). The six σ-SALCs span a1g + t1u + eg. Metal s (a1g), metal p (t1u), and two of the five d orbitals — dz² and dx²−y² together spanning eg — find symmetry partners and go strongly bonding/antibonding. The remaining three d orbitals, dxy, dxz, dyz, span t2g and have zero net overlap with the σ-SALCs (their lobes point between the ligands, at the octahedron edges) so t2g is strictly σ-nonbonding. Filling in electrons: bonding a1g, t1u, eg(σ) orbitals hold the 12 ligand-donated electrons; the metal d electrons occupy t2g (nonbonding) and eg (σ-antibonding), the two sets you already know as "t2g below eg," separated by Δo.

Tetrahedral (Td, four ligands at alternate cube corners). Now project the metal d orbitals onto the same coordinate system, but ligands sit along the four body diagonals of a cube circumscribing the metal, not along the axes. Character-table analysis of Td shows the five d orbitals split as e (dz², dx²−y²) and t2 (dxy, dxz, dyz) — note the swap: it is now the e pair that has no symmetry partner among the four ligand σ-SALCs (which span a1 + t2), so e is nonbonding, while t2 becomes σ-antibonding because it matches the ligand t2 SALC. This is the geometric heart of the whole article: the same real d orbitals (dz², dx²−y², dxy, dxz, dyz) get sorted into "points-at-ligand" (antibonding) versus "points-between-ligand" (nonbonding) roles that are exchanged between Oh and Td.

Square planar (D4h, four ligands in the xy-plane along ±x, ±y). This can be built directly by group theory, or — more intuitively for a first pass — by taking the Oh diagram and removing the two axial ligands along ±z, watching each Oh level split as its axial character is stripped away. The four σ-SALCs (from ligands in the xy-plane) span a1g + b1g + eu. Metal dx²−y² (b1g) keeps its full σ*-antibonding character (it still points straight at all four ligands). Metal dz² (a1g symmetry in D4h) loses its σ-interaction with the (now absent) axial ligands and drops sharply in energy, becoming close to nonbonding (it retains a small "donut" overlap with the equatorial ligands, so it is weakly stabilized but not zero). dxy (b2g) remains σ-nonbonding as in Oh. dxz, dyz (eg) were π-interacting with axial ligands in Oh; removing those ligands drops them close to nonbonding too, though residual in-plane π-interactions with the four remaining ligands can perturb them. The net d-orbital ordering that results — and that is confirmed by both photoelectron spectroscopy and electronic spectra of square-planar d⁸ complexes — is

with a large gap opening only above dx²−y², separating four low-lying, largely nonbonding/weakly-bonding orbitals from one strongly antibonding orbital.

The unifying quantitative statement behind every ligand-field diagram, in any geometry, is a two-level (or few-level) perturbation-theory mixing formula applied orbital-by-orbital:

where:

• $E_d$, $E_L$ are the unperturbed energies of the metal d orbital and the ligand SALC of matching symmetry,
• $H_{dL}$ is the resonance (overlap-derived) interaction integral between them,
• $E_+$ is the antibonding combination (metal-d-dominated, pushed up), $E_-$ the bonding combination (ligand-dominated, pushed down).

Reading: whenever a metal d orbital and a ligand SALC share an irreducible representation, they push each other apart by an amount that grows with $H_{dL}^2$ (i.e., with the square of the overlap) and shrinks as the unperturbed energies $E_d, E_L$ diverge. Orbitals with no symmetry-matched SALC partner have $H_{dL}=0$ and stay at $E_d$ — strictly nonbonding, regardless of geometry. The geometry-dependence enters entirely through which metal d orbitals have a symmetry match, and through the geometric prefactor in $H_{dL}$ set by orbital overlap. For d-orbital/ligand-σ overlap specifically, the key relative-magnitude result is:

where:

• $\Delta_t$ is the tetrahedral splitting (t2 − e) for a given metal and ligand set,
• $\Delta_o$ is the octahedral splitting (eg − t2g) for the same* metal–ligand pair at comparable metal–ligand distance,
• the factor 4/9 combines a coordination-number factor (4 ligands vs 6, i.e. 4/6 = 2/3) and a geometric-overlap factor (2/3, since Td metal–ligand σ-overlap projects less efficiently onto each d lobe than the Oh arrangement) multiplied together.

Reading: because Δt is always smaller than Δo for comparable ligands, essentially all tetrahedral complexes are high-spin — Δt rarely exceeds the mean pairing energy P, even for strong-field ligands, so you almost never need to worry about a low-spin tetrahedral case in a first analysis. The square-planar energetics can be written as a sum of the Oh splitting plus explicit axial-ligand-removal corrections:

where:

• $\Delta_{oh}|_{d_{x^2-y^2}}$ is the energy dx²−y² already sat above dz² conceptually within Oh's degenerate eg,
• $\delta_{ax}(d_{z^2})$ is the extra stabilization dz² receives when the two axial σ-bonds are removed (dz² loses its σ*-antibonding character toward the axial ligands and drops toward nonbonding).

Reading: the square-planar gap above dx²−y² is larger than Δo for the parent octahedral complex, because dx²−y² keeps all its Oh antibonding character while dz² additionally falls in energy — both effects widen, rather than narrow, the highest one-orbital gap. This is the electronic reason low-spin d⁸ square-planar complexes are so robust.

Example 1 (foundational): assign symmetry labels and count nonbonding electrons for high-spin [CoCl4]²⁻ (d⁷, tetrahedral). [CoCl4]²⁻ is a well-known deep-blue tetrahedral complex. Co²⁺ is d⁷. Using the Td diagram: e (dz², dx²−y²) nonbonding, t2 (dxy, dxz, dyz) σ(+π)-antibonding, with Δt small (Cl⁻ is a weak-to-moderate field ligand and Δt is intrinsically reduced by 4/9 anyway), so the configuration is always high-spin:

Filling: e gets 4 electrons (2 each, both orbitals filled, no unpaired spins forced yet since e can hold up to 4), t2 gets the remaining 3 electrons, one in each of the three t2 orbitals (Hund's rule, since Δt ≪ P). Unpaired electron count: e is fully paired (4 electrons in 2 orbitals ⇒ 0 unpaired), t2 has 3 electrons in 3 orbitals, all parallel-spin ⇒ 3 unpaired. Total unpaired spins = 3, giving a ⁴A2 ground term and a magnetic moment

in good agreement with the experimentally observed moment for [CoCl4]²⁻ (typically reported near 4.4–4.8 μB once orbital contribution from the ⁴A2/⁴T1 mixing is included, but the spin-only value from the e⁴t2³ configuration correctly predicts three unpaired electrons, the qualitative and dominant feature). Check against the limiting case: if Δt were somehow large (it never is for real ligands), one might imagine a "low-spin" e⁴t2³ alternative pairing electrons within t2 before populating a third orbital singly — but since Δt ≈ (4/9)Δo is always far below the pairing energy P for Cl⁻ (a weak/medium field ligand), the high-spin configuration above is essentially the only physically realized case, consistent with the empirical rule that virtually no tetrahedral complex is low-spin.

Example 2 (applied): rationalize why [Ni(CN)4]²⁻ is diamagnetic square planar while [NiCl4]²⁻ is paramagnetic tetrahedral, both d⁸ Ni²⁺. Ni²⁺ is d⁸ in both. In [NiCl4]²⁻ (Td, weak-field Cl⁻, small Δt): configuration e⁴t2⁴, filling e fully (4 e⁻, paired) then t2 with 4 electrons across 3 orbitals — by Hund's rule, one t2 orbital gets 2 electrons (paired) and the other two get 1 electron each (parallel), giving 2 unpaired electrons overall. This predicts

matching the experimentally observed ~3.2–3.4 μB for tetrahedral Ni(II) (spin-only value plus modest orbital contribution) — consistent with the known paramagnetism of [NiCl4]²⁻. In [Ni(CN)4]²⁻ (D4h, strong-field CN⁻, large Δsp), the level ordering from the square-planar derivation places dxz, dyz (eg), dxy (b2g), and dz² (a1g) all well below the strongly antibonding dx²−y² (b1g). Because CN⁻ is such a strong-field π-acceptor ligand, the gap $\Delta_{sp}$ is large — large enough that pairing all 8 electrons into the 4 lower orbitals (2 each) costs far less energy than promoting one electron into b1g. Configuration:

All electrons paired ⇒ diamagnetic, exactly as observed for [Ni(CN)4]²⁻. The qualitative rule this illustrates: strong-field ligands favor square-planar low-spin d⁸ geometry over tetrahedral high-spin d⁸ geometry, because only the square-planar orbital pattern offers a large enough single gap to make full pairing worthwhile — Td's Δt is architecturally too small (by the 4/9 argument) to ever force pairing for a d⁸ ion regardless of ligand strength, while D4h's $\Delta_{sp} = \Delta_o + \delta_{ax}$ is architecturally larger than even Δo and easily exceeds P for strong donors like CN⁻. Limiting-case check: for a weak-field ligand in principle capable of square-planar coordination (rare, but conceptually), $\Delta_{sp}$ would shrink toward P, and one could imagine a high-spin square-planar d⁸ case — but in practice essentially all structurally characterized square-planar d⁸ complexes involve mid-to-strong field ligands (CN⁻, phosphines, N-heterocycles, deprotonated amides) precisely because weak-field ligands cannot supply the crystal-field stabilization needed to overcome the steric/electronic preference for tetrahedral geometry with 4-coordinate d⁸ in the first place — this is why weak-field d⁸ complexes (e.g., with Cl⁻ or Br⁻) are observed as tetrahedral, not high-spin square planar.

Example 3 (exam/competition level): use orbital symmetry to explain the electron count and unusual air-stability of Vaska's complex, trans-[IrCl(CO)(PPh3)2], and predict the geometric change upon oxidative addition of H2. Vaska's complex is square-planar Ir(I), d⁸, 16-electron. Apply the square-planar MO analysis: with strong-field π-acceptor ligands (CO, PPh3) present, dxz and dyz (eg) are further stabilized by back-bonding into CO/PPh3 π orbitals (the same π-acceptor stabilization mechanism used in the octahedral π-bonding treatment, now acting on the eg pair specifically because those are the orbitals with the correct symmetry to overlap axial-direction-analogous π combinations of the in-plane π-acceptor ligands). dxy (b2g) is π-nonbonding (no ligand π-SALC of b2g symmetry available from a simple set of in-plane σ/π donors along the standard axes, generically), a1g (dz²) is weakly stabilized/nonbonding, and b1g (dx²−y²) remains strongly σ-antibonding and empty. Total d-electron count: 8, all paired in the 4 lower orbitals — diamagnetic, exactly as observed (Vaska's complex is EPR-silent and diamagnetic). Electron count: Ir(I) d⁸ + 4 two-electron ligand donors (Cl⁻, CO, 2×PPh3) = 8 + 8 = 16 valence electrons, a stable "16e square-planar d⁸" configuration by the same logic as Example 2. Oxidative addition of H2 across the empty axial positions (above and below the square plane) converts Ir(I) d⁸ square planar 16e into Ir(III) d⁶ octahedral 18e: the two new Ir–H bonds use the previously nonbonding/weakly-antibonding dz² orbital (which now must re-engage in σ-bonding along z, exactly reversing the "axial ligand removal" argument of the derivation above) and promote electron density into what becomes, upon re-forming an Oh-like ligand field, the t2g/eg pattern of an octahedral d⁶ ion. Because oxidative addition simultaneously (i) adds 2 ligands (going from 4-coordinate to 6-coordinate), (ii) formally removes 2 electrons from the metal in oxidation-state bookkeeping (Ir(I)→Ir(III)) while adding 2 new 2-electron ligand (H⁻, H⁻) donations, the electron count is conserved at 18 (as expected for the resulting octahedral d⁶ Ir(III) product), and this product is now t2g⁶eg⁰ low-spin octahedral, diamagnetic, and substitution-inert — matching the well-known reactivity of Vaska's complex as a classic oxidative-addition substrate. This example demonstrates the predictive power of the orbital-symmetry-plus-electron-counting method across a geometry change: the square-planar 16e picture correctly predicts an empty, reactive dx²−y²/dz²-derived axial "pocket" available for incoming σ-donor ligands, and the resulting product's stability is correctly read off the octahedral t2g⁶eg⁰ closed-shell pattern once the geometry has changed.

The MO/ligand-field treatment of non-octahedral transition-metal complexes did not spring from a single named "square-planar theorem"; it is the joint product of ligand field theory's founders (notably the systematic angular-overlap and symmetry-based extensions worked out through the 1950s–60s by workers building on Bethe's original crystal-field group-theory treatment) applied piece-by-piece to Td and D4h point groups as four-coordinate organometallic and coordination chemistry matured. The single individual most associated with turning this into a rigorous, general symmetry-based bonding method — applicable to any point group, not just Oh — is Carl J. Ballhausen, whose systematic MO treatments of transition-metal complexes in the 1950s–60s established the character-table-driven SALC method used throughout this article.