Tetrahedral d-Orbital Splitting

Introduction

Four ligands at the corners of a tetrahedron approach between the Cartesian axes, unlike six octahedral ligands that approach along them. The direction change reverses which d-orbital group is raised most. Tetrahedral splitting is also usually smaller than comparable octahedral splitting, so electron pairing is less often favoured before the upper group is occupied.

Core explanation

The d xy, d xz and d yz orbitals have lobes between axes. In a tetrahedral ligand arrangement, those directions interact more strongly with the four ligands, so these orbitals form the higher t₂ set in the simple CFT picture. The d x²−y² and d z² orbitals point along axes, farther from the ligand approach directions, and form the lower e set. The names resemble octahedral labels but the ordering is opposite: octahedral t₂g lower/e g upper, tetrahedral e lower/t₂ upper. The g subscript is not used in the tetrahedral labels because tetrahedral symmetry lacks the relevant inversion center.

The tetrahedral energy gap is Δ tet. For the same metal, oxidation state and ligand type, it is generally smaller than the octahedral gap. A simple point-charge estimate often quotes Δ tet ≈ 4/9 Δ oct for idealized matched circumstances; real values depend on bond lengths and covalency, so do not treat the ratio as a universal measured constant. Fewer ligands and different approach directions contribute to the smaller separation.

Because Δ tet is commonly below the electron-pairing energy, electrons usually occupy the higher t₂ orbitals singly before pairing extensively in the lower e orbitals. Tetrahedral transition-metal complexes are therefore usually high spin. This is a tendency, not a mathematical impossibility of low-spin tetrahedral systems. State actual ligand and metal conditions when making a precise prediction.

For tetrahedral d⁵, fill the two lower e orbitals singly, then the three upper t₂ orbitals singly. The result has five unpaired electrons in the ideal high-spin picture. For d⁶, the sixth electron pairs in a lower e orbital, leaving four unpaired. Counting boxes is safer than memorising every dⁿ unpaired number. The metal's d count must first come from its oxidation state, not from the four ligands.

Tetrahedral compounds may be coloured when an excitation between split d levels absorbs visible light. Their d–d absorption intensity can differ from octahedral complexes because tetrahedral structures lack a center of inversion, relaxing some selection restrictions. However, charge-transfer transitions and other factors can also dominate. The direction and size of splitting alone do not uniquely predict perceived colour.

The ideal diagram applies to a regular tetrahedron. Unequal ligands or distortion split the e or t₂ groups further. A four-coordinate complex may also be square planar, with a very different energy pattern. Before drawing a tetrahedral diagram, confirm geometry from the problem or suitable chemical evidence.

Step-by-step reasoning

1. Verify that the four-coordinate complex is tetrahedral. 2. Calculate the metal's oxidation state and d-electron count. 3. Draw two lower e boxes and three upper t₂ boxes. 4. Compare Δ tet with pairing energy; high spin is the usual starting assumption. 5. Fill orbitals and count unpaired electrons.

Visual explanation

Draw two lower boxes labelled e and three upper boxes labelled t₂. Sketch four ligand arrows approaching between axes. Place the octahedral three-lower/two-upper diagram beside it so the reversed group ordering is obvious.

Real-world analogy

If obstacles move from the main roads to diagonal paths, travellers who use the diagonals face more interference. Tetrahedral ligands approach between axes, making the between-axis orbitals relatively higher in the basic repulsion picture.

Real-world example

Tetrahedral [CoCl₄]²⁻ is a useful classroom comparison with octahedral cobalt complexes. Changing coordination and ligands alters the d-level pattern and can produce a visible colour change without necessarily changing cobalt's +2 state.

Why?

Why are tetrahedral complexes often high spin? Their d-orbital splitting is usually smaller than the pairing energy, so placing electrons singly in higher orbitals can cost less than pairing them prematurely below.

Common misconception

“The tetrahedral diagram is the octahedral diagram with four ligands removed but the same orbital ordering.” The different approach directions reverse the lower and upper groups in the ideal CFT model.

Worked example

Assume a tetrahedral d⁵ ion. Put one electron in each lower e orbital, then one in each upper t₂ orbital. All five electrons remain unpaired in the ordinary high-spin arrangement. If the same d⁵ count were placed in a strong octahedral field, pairing could produce a different unpaired count; geometry and splitting conditions matter.

Quick check

1. Which orbital set is lower in an ideal tetrahedral field? Answer: The two-orbital e set.

Exam focus

Write e lower and t₂ upper for tetrahedral CFT. Use “usually high spin” rather than “always.” Only compare Δ tet and Δ oct under specified matched conditions.

Advanced insight

The often-quoted 4/9 ratio is derived within a simple electrostatic model for comparable arrangements. Real metal–ligand covalency and bond distances modify measured gaps, so the ratio is a teaching estimate rather than an exact identity.

Summary

Tetrahedral ligand approach raises the between-axis t₂ orbitals above the axis-directed e orbitals. Its splitting is usually smaller than octahedral splitting, making high-spin occupancy common. Geometry must be known before using this diagram.

Practice questions

1. How many orbitals are in the tetrahedral lower e set? Answer: Two. 2. Which set is upper in an ideal tetrahedral field? Answer: t₂, containing three orbitals. 3. How many unpaired electrons does ideal high-spin tetrahedral d⁵ have? Answer: Five. 4. Is Δ tet exactly 4/9 Δ oct for every real pair of complexes? Answer: No. That is an idealized matched-case estimate.