The effects of the application of a field to a metal ion, in the formation of a complex, whether examined through the Crystal Field Theory of the Ligand Field Theory, have so far been limited to octahedral complexes. This is because it is easy to visualize the orientation of ligands and orbitals in the octahedral symmetry environment, and the points at the vertices of the octahedron lie on the cartesian axes.

However, metal ion complexes occur in other geometries, the most important being tetrahedral and square-planar. The effects of the ligand field are different for species with these different geometries.

Tetrahedral Coordination compounds

When a metal ion is exposed to an octahedral field, the degeneracy of the d-orbitals is lost, and they split into a lower energy t_2g group, characterized by orbitals whose lobes point between the ligands, and a higher energy e_g group, characterized by orbitals whose lobes point towards the ligands.

In a tetrahedral field, the ligands are sited at opposite corners of the cube which circumscribes the octahedron. Hence, the orbitals which were directed between the ligands in the octahedral orientation are now directed towards the ligands in the tetrahedral orientation. Similarly, those orbitals which were directed towards the ligands in the octahedral orientation are now directed between the ligands in the tetrahedral orientation.

Now, the t₂ set, from the d(xy), d(xz), and d(yz) orbitals, is at higher energy, and the e set, from the d(z²) and d(x²-y²) orbitals, is at lower energy.

The splitting of the energy levels of the d-orbitals in the tetrahedral field is given by the tetrahedral splitting parameter, Δ_t. This has a value of Δ_t = 4/9Δ_o.

The splitting of the d-orbitals in a tetrahedral complex
	The 3 t2 orbitals are at +0.4Δt, and the 2 e orbitals are at -0.6Δt

The fact that the tetrahedral ligand field splitting parameter is smaller than the octahedral ligand field splitting parameter is due to the fact that the relative degree of interaction between the point charge representing the ligand and the orbitals is smaller in the tetrahedral orientation than the octahedral orientation.

The important result is that the smaller value is not enough to stabilize unpaired electrons in the higher energy t₂ orbitals when there are holes in the e orbitals, and hence tetrahedral complexes are always in the weak-field limit.

The calculation of ligand field stabilization energies is the same for tetrahedral compounds as for octahedral compounds. These can be used in discussions of the stability of tetrahedral complexes in the same way as for octahedral complexes, as seen in the discussion of the structure of the inverse spinel Fe₃O₄.

Square-Planar Complexes

Some complexes with six ligands have structures which are not octahedrally symmetric, especially those for Cu²⁺, and are considered to be tetragonally distorted. Other species, such as d⁸ compounds have only four ligands ane a square planar geometry.

These types of complex have a ligand field effect which is different to both the octahedral and tetrahedral cases, though it is useful to use the octahedral case as the basis for discussion of the tetragonal and square planar geometries.

A tetragonal compound can be generated from an octahedral compound by stretching the metal ligand bonds along one of the axes, conveniently chosen as the z-axis.

Generation of a tetragonal distortion

When this happens, the electrostatic repulsion between electrons in the orbitals which have a component in the direction of the z-axis and the ligands situated on the z-axis decreases.

Hence, those orbitals with a z-component are lowered in energy.

Energy levels in a tetragonally distorted complex

If this distortion is continued to its extreme, the ligands in the z-axis can be considered to be not present, and now the d(z²) orbital will be lowest in energy as it has negligible interaction with the ligand field.

The extreme of the tetragonal distortion: the square planar geometry

This kind of distortion occurs when the new geometry has a greater ligand field stabilization energy than the original, octahedral, complex. For example, if the original complex is an octahedral d⁹, t_2g⁶e_g³, complex, the tetragonal distortion will mean that two of the electrons in the e orbitals move to lower energy, and one moves to higher energy, and so overall there is a net reduction in energy, and the distorted environment is more stable. In the d⁸ situation, the extreme of distortion gives full occupation of the lowest lying d(z²), d(xz), d(yz), and d(xy) orbitals and an empty d(x²-y²) orbital, also resulting in an increase in the LFSE.

When the square-planar geometry is adopted, the d-orbitals of the metal now have four different energies, and so three splitting parameters may be defined. If we only consider the total splitting parameter, Δ_sp, which is in fact the sum of the three intermediate splitting parameters, we find that it is actually larger than the octahedral splitting parameter, and Δ_sp = 1.3Δ_o.

The Jahn-Teller effect

The distortion of the octahedral complex to give a tetragonal or even square-planar complex is favourable when the ligand field stabilization energy in the new coordination environment is larger than in the original octahedral environment. This is generally the case when more electrons move to lower energy than move to higher energy.

The tetragonal distortion is an example of the Jahn-Teller effect:

when the ground state electronic configuration of a non-linear compound is degenerate, the compound will distort so that the degeneracy is removed.

In the d⁹ situation discussed above, the octahedral geometry is degenerate as the ninth electron can occupy either of the two orbitals, the d(x²-y²) and the d(z²) which make up the e_g level. When the molecule distorts, the e_g level splits into a lower energy level, the d(z²) orbital, which two electrons occupy, and a higher energy level, the d(x²-y²) orbital, which only one electron occupies, and the degeneracy of the e_g level is lost. There is a net reduction in energy, or an increase in the LFSE, when the degeneracy is broken by the distortion.

Whilst the Jahn-Teller effect will identify a complex as being open to stabilization by distortion if there is the presence of a degeneracy in the ground state electronic configuration of the complex, it does not predict the geometry of the distorted complex.

For example, the tetragonal distortion may occur by either the elongation of the z-axis and compression of the x- and y-axes, or by the compression of the z-axis and elongation of the x- and y-axes. The Jahn-Teller effect cannot say which of these will be the path taken, and the path taken will vary from compound to compound.

Other complexes which have a degenerate ground state may also appear to have an octahedral geometry. The d⁹ [Cu(OH₂)₆]²⁺ ion, for example, appears to be octahedral, when one would predict a tetragonal distortion as being favourable. This can be explained by considering the timescale of the measurement from which the structure is elucidated.

If the tetragonal distortion is fluxional, ie. the extended axis in the tetragonal distortion changes with time, but the timescale of existence of one fluxional form before it is transformed to another is shorter than the timescale of measurement, then only an averaged structure will be seen. When elongation along each of the axes is equally favourable, as is the case in the [Cu(OH₂)₆]²⁺ ion where all of the ligands are the same, then the observed average geometry is the octahedral geometry. This fluxional variation of the distortion from one orientation to another is known as the dynamic Jahn-Teller effect.