d-Orbital Splitting in Lower Symmetries

Introduction

Few coordination compounds are perfectly octahedral. Unequal ligands, Jahn–Teller effects or a different coordination number lower the symmetry and split levels that were degenerate in the ideal diagram. A useful prediction begins with ligand directions and orbital shapes, not a memorised five-line order. Tetragonal, square-planar and five-coordinate geometries each provide a different directional field.

Core explanation

Start with an octahedron and lengthen its two axial z bonds while retaining four shorter equatorial bonds. This tetragonal elongation weakens interactions involving z. The octahedral e g pair separates: d(z²) interacts less strongly with distant axial ligands and usually falls relative to d(x²−y²), which still points at four equatorial ligands. The t₂g triplet also separates, with d(xz) and d(yz) generally affected differently from d(xy). Compression along z reverses the axial-versus-equatorial emphasis, though exact order can depend on bonding details.

A square-planar environment can be viewed qualitatively as an extreme removal of the two axial ligands. Four ligands on ±x and ±y strongly destabilise d(x²−y²), commonly making it the highest d-like orbital. Other orbital positions depend on σ and π interactions and cannot always be captured by one universal sequence copied from an ideal point-charge figure. For d⁸ Pd(II) or Pt(II), eight electrons can pair in four lower levels and leave the high d(x²−y²)-like level empty.

Five-coordinate complexes offer two common limiting geometries. A trigonal bipyramid has three equatorial ligands in a plane and two axial ligands; a square pyramid has four equatorial ligands and one axial. Their orbital symmetries differ: trigonal-bipyramidal positions are related by a threefold rotation when all ligands are equivalent, while square-pyramidal positions have a fourfold axis in the ideal case. The metal d orbitals therefore form different symmetry sets and different energies. A real five-coordinate complex may lie between these shapes or distort dynamically.

For a trigonal bipyramid, d(z²) points toward axial ligands and has density around its equator, so its energy reflects both sets; d(x²−y²) and d(xy) interact with equatorial ligands, while d(xz) and d(yz) sample other directions. The detailed energy ranking depends on axial versus equatorial ligand strength and bond length. For a square pyramid, d(x²−y²) strongly faces the four basal ligands; d(z²) interacts with the single apical ligand and the equatorial belt. One cannot simply remove a box from the octahedral diagram and keep the old Δₒ coefficients.

Lower symmetry has spectroscopic consequences. Degenerate electronic states can split into multiple components, making bands broaden or multiply. Polarised absorption or EPR can reveal anisotropy; crystallography can directly show unequal bond lengths. The splitting pattern should be tested against such observations. A CFT picture remains a starting approximation, especially when ligand covalency differs around the coordination sphere.

The electron count still matters. A d⁹ octahedral ion with uneven e g occupancy has a strong reason to distort tetragonally, while a completely filled d¹⁰ shell lacks that particular first-order orbital benefit. A four-coordinate d⁸ ion may choose square planar over tetrahedral depending on total bonding and pairing energies. Geometry is both an input to and sometimes an outcome of the electronic structure problem.

Step-by-step reasoning

Locate ligand positions and compare axial with equatorial distances and identities. Sketch d-orbital lobes relative to those directions to rank direct σ repulsion. Use the geometry’s symmetry to group orbitals that remain equivalent, then fill electrons and test predicted bond lengths or spectra. Avoid numerical CFSE formulas derived for a different point group.

Visual explanation

Draw an octahedron with two long z-axis bonds, then remove those axial ligands to form a square plane. Beside them draw a trigonal bipyramid with three equatorial and two axial ligands and a square pyramid with four basal and one apical ligand. Mark d(x²−y²) as strongly directed at square basal ligands.

Real-world analogy

A room with six equally bright lamps has one pattern of illumination. Dimming two ceiling lamps, removing them entirely or replacing the layout with five lamps changes which directions are brightest. Orbital energies similarly respond to both number and direction of ligands.

Real-world example

Cu(II) aqua environments often have two long axial bonds and four shorter equatorial bonds. Their d⁹ electronic occupancy motivates tetragonal elongation, and the resulting lower symmetry can be seen in structural distances and electronic or EPR spectra.

Why?

Why does d(x²−y²) remain high in a square plane? Its four lobes point toward the four in-plane ligand directions. Strong σ-antibonding interaction raises its energy even after the axial ligands have been removed.

Common misconception

“Every five-coordinate complex has the same d-level order.” Trigonal bipyramids and square pyramids have different ligand directions, and unequal axial/equatorial ligands alter their energy ranking further.

Worked example

An ideal octahedral d⁹ Cu²⁺ ion has t₂g⁶e g³. On elongation along z, suppose d(z²) moves below d(x²−y²) by a separation δ while their mean is unchanged. Two e g electrons occupy the lower d(z²)-like level and one the higher d(x²−y²)-like level. Their sum falls by δ/2 relative to equal e g energies, providing an electronic driving force for distortion if it exceeds the structural cost.

Quick check

1. Which orbital typically becomes highest in a square-planar σ-donor field? Answer: d(x²−y²), because it faces all four in-plane ligands.

Exam focus

State geometry and ligand directions before ranking orbitals. Use “typically” for detailed lower-symmetry orders and show the physical reason for the levels most confidently assigned.

Advanced insight

Many five-coordinate species move along a low-energy structural path between square-pyramidal and trigonal-bipyramidal shapes, sometimes called Berry pseudorotation in appropriate molecules. Dynamic exchange can average sites on a spectroscopic timescale.

Summary

Lower symmetry splits formerly degenerate d sets. Tetragonal elongation separates axial and equatorial orbital interactions; square-planar and five-coordinate geometries require their own diagrams and structural evidence.

Practice questions

1. In a tetragonally elongated octahedron, why does d(z²) generally fall relative to d(x²−y²)? Answer: The two axial ligands move farther away, weakening their interaction with d(z²), while four equatorial ligands still strongly face d(x²−y²). 2. Why is an octahedral CFSE formula unsafe for a square pyramid? Answer: One axial ligand is absent and symmetry is lower, so the degeneracies and orbital energies underlying −0.4Δₒ/+0.6Δₒ no longer apply. 3. Give one measurement that can test a proposed tetragonal distortion. Answer: Crystallography can reveal two axial metal–ligand distances different from the four equatorial ones; EPR or optical spectra can also show lower-symmetry effects. 4. What distinguishes ideal trigonal-bipyramidal from square-pyramidal coordination? Answer: The former has three equatorial and two axial ligand positions; the latter has four basal and one apical position.