How Crystal Field Splitting Works

Quick Answer

Crystal field splitting is the gap that opens between the d orbitals when ligands approach a metal ion. In an octahedral complex the three t2g orbitals drop by 0.4 delta and the two eg orbitals rise by 0.6 delta. Electrons then fill the diagram whichever way costs less: spreading out if the splitting is smaller than the pairing energy, pairing up if it is larger. That single comparison sets the number of unpaired electrons, the crystal field stabilization energy CFSE = (-0.4 x n(t2g) + 0.6 x n(eg)) x delta, the magnetic moment, and the colour. Move the sliders here and watch the electrons rearrange themselves at the crossover.

Why Do Ligands Split The d Orbitals?

Two of the five d orbitals point straight at the six octahedral ligand positions, and the other three point between them. Electrons in the orbitals aimed at a ligand feel more repulsion, so those two rise in energy. The other three fall. The gap between the sets is delta octahedral, written delta o, and every measurable property of the coordination complex follows from it.

• The barycentre is conserved: two orbitals up by 0.6 delta balance three orbitals down by 0.4 delta.
• A tetrahedral field inverts the picture, because there no ligand points at a d orbital directly.
• Delta o is measured in wavenumbers, so a strong field ligand such as cyanide reaches 33000 cm-1 while water sits near 10400.

What Does Delta Octahedral Mean?

Delta octahedral, usually written delta o, is the size of the d orbital splitting in an octahedral field, quoted in wavenumbers. It belongs to the metal and the ligands together, therefore it is looked up rather than derived from first principles. Octahedral vs tetrahedral is the first thing it depends on, because the same ligand set splits a tetrahedral complex only four ninths as far. However, the only number worth using is the one for the geometry you actually have.

• Weak field: iodide and bromide, a few thousand wavenumbers.
• Strong field: cyanide and carbon monoxide, above 30000 wavenumbers.

A Worked Example: Two d6 Complexes

Take the same iron(II) ion in two different coordination complexes. For example, hexaaquairon(II) has delta o near 10400 cm-1, which loses to a 19000 cm-1 pairing energy, therefore the ion stays high spin with four unpaired electrons and a moment of 4.90 BM. Hexacyanoferrate(II) has delta o near 33000 cm-1, which wins, so the electrons pair into t2g and the complex reads 0.00 BM. Nothing about the metal changed; only the ligand did.

• Both presets are on the panel, so you can step between them and watch the arrows rearrange.
• Crystal field theory gets both answers from one comparison, which is what makes it worth learning before the fuller ligand field treatment.

High Spin vs Low Spin: Which One Wins?

Adding a fourth electron forces a decision. It can climb to the upper set at a cost of delta, or it can pair in the lower set at a cost of the pairing energy P. The complex takes the cheaper option, so delta versus P decides the spin state. This simulator costs both arrangements every frame and shows whichever is lower, rather than asserting an answer.

• Weak field, delta below P: high spin, maximum unpaired electrons.
• Strong field, delta above P: low spin, lower set filled first.

Which d Counts Actually Have A Choice?

Only d4 through d7 in an octahedral field. A d3 ion puts one electron in each t2g orbital with nothing to decide, and a d8 ion has t2g full whatever happens. Students lose marks by writing low-spin d3 or high-spin d8 as though those were different species. Drag the d electron slider and the spin state readout goes quiet outside the d4 to d7 window.

How Does The Splitting Change Magnetism?

Unpaired electrons make a complex paramagnetic, and the spin only magnetic moment mu = sqrt(n x (n + 2)) turns that count into a number a magnetic balance can check. For example, high-spin d6 gives four unpaired electrons and 4.90 BM, whereas low-spin d6 gives none at all and reads as diamagnetic. That gap is large, therefore magnetic measurement is the standard way to assign a spin state.

• One unpaired electron: 1.73 BM. Two: 2.83 BM. Five: 5.92 BM.
• Measured moments run a little above spin-only for later first-row metals, because orbital motion contributes as well.

Where Does The Colour Come From?

An electron absorbs a photon and jumps the gap, so the absorbed wavelength is lambda = 10^7 / delta with delta in wavenumbers. A 20000 cm⁻¹ splitting absorbs at 500 nm. What your eye reports is the complement of what was absorbed, which is why that complex looks red rather than blue-green. Both d0 and d10 ions have no d to d transition available, so they are colourless unless something else absorbs.

Where Is This Used Outside An Exam?

A magnetic susceptibility measurement returns a moment, and comparing it against both spin states is how the configuration gets assigned. Haemoglobin is the best-known case: the iron sits high spin when the site is empty and low spin once oxygen binds, and the colour change from dark red to bright red follows the change in splitting. A CFSE calculator is therefore a way to predict what an instrument should read before you run it.

• A predicted 4.90 BM against a measured 5.1 BM is a good match. However, measured values run slightly high, because orbital motion contributes as well.
• The same reasoning ranks ligands into the spectrochemical series from the colours their complexes show.

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