MO Diagrams for Tetrahedral and Square-Planar Complexes

Introduction

Four-coordinate complexes may be tetrahedral or square planar, and a single “four ligands” label cannot determine their orbital diagram. In an ideal tetrahedron, four radial σ donors form A₁+T₂ combinations and the metal d e pair is comparatively nonbonding. In an ideal square plane, donor combinations include A₁g, B₁g and E u; the metal d(x²−y²) orbital strongly faces all four ligands and often sits highest.

Core explanation

In T d symmetry, the four ligand σ orbitals decompose as A₁+T₂, whose dimensions 1+3 match the four donors. The metal s orbital has A₁ symmetry. Metal p orbitals and the three d(xy), d(xz), d(yz) functions have T₂ symmetry, allowing interactions with ligand T₂ combinations. The two d(z²) and d(x²−y²) functions have E symmetry and no radial σ SALC partner in this restricted construction, so they are approximately σ-nonbonding. The metal-rich t₂-derived levels are pushed up by σ mixing, giving lower e and upper t₂ d-like sets—the reverse of the octahedral orbital-energy ordering.

Tetrahedral splitting Δ t is commonly smaller than analogous octahedral Δₒ, often estimated near 4Δₒ/9 in a simple same-metal, same-ligand comparison. The four-ligand geometry gives fewer interactions and no ligand exactly on a conventional d-orbital axis in the idealised orientation picture. Consequently, pairing cost usually exceeds the saving from occupying lower e early, and familiar tetrahedral transition-metal complexes are high spin. A d⁸ tetrahedron has e⁴t₂⁴ and two unpaired electrons.

In D₄h square-planar symmetry, the four radial σ donors combine as A₁g+B₁g+E u, with dimensions 1+1+2=4. The B₁g ligand pattern can interact strongly with metal d(x²−y²), also B₁g. Its antibonding metal-rich descendant is typically the highest d-like level. Metal d(z²), an A₁g function, also has symmetry-allowed σ interaction; metal p x/p y can match E u. Other metal d functions, such as d(xy) and d(xz)/d(yz), are not all equivalent and their detailed order can depend on π bonding and metal–ligand distances. A one-size-fits-all ladder for every square-planar ligand set is therefore unsafe.

For many square-planar d⁸ complexes, the four lower d-like levels each hold a pair and the high d(x²−y²)-like level stays empty. The simple prediction is diamagnetism. Pd²⁺ and Pt²⁺, with extended 4d and 5d orbitals, frequently adopt this arrangement. Ni²⁺ d⁸ may be square planar with strong-field ligands such as CN⁻ but tetrahedral with some weaker-field ligands such as Cl⁻. Both descriptions retain d⁸ and four ligands; geometry and ligand bonding change the orbital energies.

The two geometries have different inversion symmetry. D₄h has a centre of inversion, so parity labels and a Laporte restriction apply to ideal electric-dipole d–d transitions. T d lacks inversion, making parity mixing more permissive; tetrahedral d–d bands can be relatively intense. Band intensity is not an exact geometry test by itself because spin rules, covalency and charge transfer also matter.

Changing geometry changes total energy beyond d-level CFSE. Ligand–ligand repulsion, bond strength, solvation, pairing energy and entropy all contribute to whether tetrahedral or square-planar coordination is observed. The MO diagrams explain why the geometries can differ magnetically and spectroscopically, not a universal rule that one geometry is always favoured for a given d count.

Step-by-step reasoning

Identify T d or D₄h geometry. Decompose four σ donors into A₁+T₂ for T d or A₁g+B₁g+E u for D₄h and match metal functions. Rank the d-like levels qualitatively, paying attention to tetrahedral e versus square-planar high d(x²−y²). Fill the formal d count and predict unpaired electrons, then check spectra or structure before choosing between geometries.

Visual explanation

Draw a tetrahedron and a square plane side by side. Under the tetrahedron, show a lower twofold e line and upper threefold t₂ line. Under the square, show four lower d-like positions and a separated high d(x²−y²) line, marking its lobes toward four in-plane donors.

Real-world analogy

The same four speakers arranged around a listener in a tetrahedron or flat square produce different directional interference patterns. Matching sound patterns change with geometry; orbital overlap patterns likewise change even with identical numbers of ligands.

Real-world example

[NiCl₄]²⁻ and [Ni(CN)₄]²⁻ both have Ni²⁺ d⁸. The chloride complex is commonly tetrahedral and paramagnetic with two unpaired electrons; the cyanide complex is square planar and diamagnetic. Their contrasting response supports geometry-specific MO diagrams.

Why?

Why is the tetrahedral e pair approximately σ-nonbonding? Four radial σ donors form only A₁ and T₂ SALCs, with no E combination to match the metal d(z²)/d(x²−y²) pair in ideal T d symmetry.

Common misconception

“Four-coordinate d⁸ always gives two unpaired electrons.” That is the usual high-spin tetrahedral filling; square-planar d⁸ often pairs all eight electrons below a high empty d(x²−y²)-like orbital.

Worked example

For [Ni(CN)₄]²⁻, four CN⁻ ligands total −4 and the complex is −2, so Ni is +2 and d⁸. In the square-planar model, eight electrons occupy four lower d-like orbitals as four pairs, leaving the high B₁g d(x²−y²)-like level empty: n=0. For tetrahedral [NiCl₄]²⁻, the same d⁸ count gives e⁴t₂⁴ and n=2. Magnetic data can distinguish the assignments.

Quick check

1. What are the four σ-donor SALC symmetries in ideal T d? Answer: A₁ plus T₂, onefold plus threefold for four total combinations.

Exam focus

Use geometry-specific symmetry labels and avoid applying octahedral −0.4/+0.6 coefficients to four-coordinate complexes. Distinguish tetrahedral magnetic predictions from square-planar ones for d⁸.

Advanced insight

Metal p and d functions can share the same T₂ symmetry in a tetrahedron, so symmetry permits mixing among more than one metal basis set and ligand T₂ combinations. Energy matching determines how much each contributes to the actual molecular orbitals.

Summary

Tetrahedral σ donors form A₁+T₂, leaving metal e d orbitals relatively low; square-planar σ donors form A₁g+B₁g+E u, with strong B₁g interaction raising d(x²−y²). Geometry can switch d⁸ magnetism.

Practice questions

1. Why does a tetrahedral d⁸ ion usually have two unpaired electrons? Answer: Its high-spin e⁴t₂⁴ filling has two singly occupied t₂ orbitals after the e pair is full. 2. Which D₄h ligand SALC can interact with metal d(x²−y²)? Answer: B₁g, because d(x²−y²) has B₁g symmetry in the ideal square plane. 3. Why can tetrahedral d–d bands be stronger than ideal centrosymmetric square-planar d–d bands? Answer: T d lacks an inversion centre, so the strict g-to-g Laporte parity restriction of a centrosymmetric D₄h skeleton does not apply in the same way. 4. Which square-planar metal d orbital is commonly highest? Answer: d(x²−y²), because it points directly at all four in-plane ligands.