Crystal Field Theory

Crystal Field Theory (CFT) takes a completely different starting point from VBT. It treats the metal–ligand bond as purely electrostatic (ionic), arising from the attraction and repulsion between the metal centre and the ligands, and makes no reference to orbital overlap or covalency at all. Ligands are modelled as:

• point negative charges, if they are anions, or
• point dipoles, if they are neutral molecules (with the negative end of the dipole facing the metal).

Why the d orbitals split at all

In an isolated gaseous metal atom or ion, all five $d$ orbitals are degenerate — they have exactly the same energy. This degeneracy survives even if the metal is surrounded by a perfectly spherically symmetric field of negative charge, because such a field pushes all five orbitals up in energy by the same amount.

Real ligands, however, do not surround the metal symmetrically in space — they approach from specific directions fixed by the geometry of the complex. This makes the electric field around the metal asymmetric, and it is this asymmetry that lifts the degeneracy of the $d$ orbitals, splitting them into sets of different energy. The exact pattern of splitting depends on the geometry of the ligand arrangement, so octahedral, tetrahedral, and other geometries each split the $d$ orbitals differently.

(a) Splitting in an octahedral field

In an octahedral coordination entity, six ligands approach the metal symmetrically along the $\pm x$, $\pm y$, $\pm z$ axes. The repulsion between the electrons (or negative charge) on the ligands and the electrons in the metal's $d$ orbitals is not the same for every $d$ orbital — it depends on how directly that orbital points toward an approaching ligand:

• The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals point directly along the axes, straight at the incoming ligands, so they experience the greatest repulsion and are raised in energy. Together these form the higher-energy $e_g$ set.
• The $d_{xy}$, $d_{yz}$, and $d_{xz}$ orbitals point between the axes, away from the ligands, so they experience comparatively less repulsion and are lowered in energy relative to the average. Together these form the lower-energy $t_{2g}$ set.

This splitting of the originally degenerate $d$ orbitals into the $t_{2g}$ (lower) and $e_g$ (upper) sets is called crystal field splitting, and the energy gap between the two sets is denoted $\Delta_o$ (the subscript $o$ stands for octahedral). Relative to the hypothetical "barycentre" — the average energy the $d$ orbitals would have in a spherical field — the $e_g$ set is raised by $\tfrac{3}{5}\Delta_o$ and the $t_{2g}$ set is lowered by $\tfrac{2}{5}\Delta_o$.

The spectrochemical series

The size of $\Delta_o$ is not fixed — it depends on both the nature of the ligand and the charge on the metal ion. Some ligands produce a strong field, giving a large splitting; others produce a weak field, giving only a small splitting. Ranking common ligands by the size of the splitting they produce (determined experimentally, from how strongly complexes absorb light) gives the spectrochemical series — an ordering of ligands purely by field strength, running from the weakest field-producing ligands to the strongest.

The ligands, arranged in order of increasing field strength, form the series:

Halide ions and other simple anions sit toward the weak-field end, while ligands such as ethylenediamine, cyanide, and carbon monoxide sit toward the strong-field end.

Filling electrons into the split orbitals — weak field vs. strong field

For $d^1$, $d^2$, and $d^3$ octahedral entities, there is no ambiguity: the electrons simply occupy the lower-energy $t_{2g}$ orbitals singly, following Hund's rule, since there are three $t_{2g}$ orbitals available.

The interesting case arises at $d^4$: the fourth electron has two possible homes, and which one it actually takes depends on a competition between two energies —

• $\Delta_o$, the crystal field splitting energy separating $t_{2g}$ from $e_g$, and
• $P$, the pairing energy — the energy cost of forcing two electrons to share a single orbital.

This gives two distinct outcomes:

1. If $\Delta_o < P$: pairing costs more than promotion, so the fourth electron avoids pairing and instead enters an $e_g$ orbital, giving the configuration $t_{2g}^{3}e_g^{1}$. Ligands that produce this outcome (i.e. $\Delta_o < P$) are called weak field ligands, and the resulting complexes are called high spin complexes (maximum number of unpaired electrons).
2. If $\Delta_o > P$: promotion to $e_g$ costs more than pairing, so it becomes energetically favourable for the fourth electron to pair up within the $t_{2g}$ set instead, giving the configuration $t_{2g}^{4}e_g^{0}$. Ligands that produce this outcome are called strong field ligands, and the resulting complexes are called low spin complexes (minimum number of unpaired electrons, i.e. maximum pairing).

Calculations further show that, across the range $d^4$ to $d^7$, coordination entities are generally more stable under a strong crystal field (low spin arrangement) than under a weak one.

(b) Splitting in a tetrahedral field

Tetrahedral coordination entities show the same underlying idea — asymmetric approach of the ligands lifts the degeneracy of the $d$ orbitals — but with the pattern inverted relative to the octahedral case: …