Crystal field theory: splitting, spin states, and color
See how ligands split a metal's d orbitals into energy levels, why complexes are colored, and how high-spin and low-spin states arise.
Quick answer
Crystal field theory explains the properties of transition-metal complexes by treating ligands as point charges whose electric fields split the metal's five d orbitals into higher- and lower-energy sets.
Example: [Fe(CN)6]3- is low-spin with one unpaired electron because CN- creates a large splitting, while [FeF6]3- is high-spin with five unpaired electrons because F- creates a small one.
What you will learn
- Draw the octahedral splitting pattern and label the t2g and eg sets
- Predict high-spin versus low-spin configurations for d4-d7 metals
- Compare octahedral and tetrahedral splitting in size and orbital order
- Explain complex colors as d-d electron transitions
- Calculate crystal field stabilization energies for simple cases
- Use the spin-only formula to connect magnetism with unpaired electrons
The core idea: ligands split the d orbitals
Crystal field theory starts with a simple picture: treat each ligand as a point of negative charge approaching the metal ion, and ask how that charge affects the metal's five d orbitals. The d electrons are repelled by the ligand charges, so their energies rise, but not equally.
In an octahedral complex, ligands sit on the x, y, and z axes. The dz2 and dx2-y2 orbitals point directly at the ligands and feel the strongest repulsion, so they rise in energy. The dxy, dxz, and dyz orbitals point between the axes and feel less repulsion, so they sit lower. The five d orbitals split into a lower set of three and an upper set of two.
Octahedral splitting
The lower set is labeled t2g and the upper set eg. The energy gap between them is the octahedral splitting energy, Δo (sometimes written 10 Dq). Relative to the unsplit average energy, called the barycenter, each t2g orbital drops by 0.4 Δo and each eg orbital rises by 0.6 Δo, so the total energy is conserved.
The size of Δo depends on the ligand, the metal, and its oxidation state. Strong-field ligands like CN- and CO produce large splittings; weak-field ligands like I- and F- produce small ones. Higher oxidation states and heavier 4d or 5d metals also give larger splittings.
Filling the split orbitals: spin states
Electrons fill the split orbitals following Hund's rule and the Aufbau principle, but now there is a competition. Placing an electron in the upper eg set costs Δo, while pairing it in the lower t2g set costs the pairing energy P.
When Δo is smaller than P, electrons occupy the upper set before pairing: the high-spin state with maximum unpaired electrons. When Δo exceeds P, electrons pair in the lower set first: the low-spin state. Only d4 through d7 configurations can go either way; d1-d3, d8, d9, and d10 have a single possible arrangement.
Compare Δo with the pairing energy P. If Δo < P, the complex is high-spin; if Δo > P, it is low-spin. This choice exists only for d4-d7 in octahedral geometry.
Tetrahedral splitting
In a tetrahedral complex the ligands approach between the axes, so the splitting pattern inverts: the e set (dz2, dx2-y2) sits lower and the t2 set sits higher. Fewer ligands and a less direct approach make the gap much smaller, roughly Δt = 4/9 Δo.
Because the gap is so small, it essentially never exceeds the pairing energy, and tetrahedral complexes are always high-spin in practice. The inverted order also means the stabilization energies are smaller, one reason octahedral complexes are far more common.
The spectrochemical series
The spectrochemical series ranks ligands by the splitting they produce:
I- < Br- < S2- < Cl- < NO3- < F- < OH- < H2O < NH3 < en < CN- ~ CO
Pi-donor ligands such as halides push electron density onto the metal and shrink the gap, while pi-acceptor ligands such as CO and CN- pull density away and widen it. The metal matters too: for a given ligand, Δo increases down a group and with higher oxidation state, which is why Fe(III) complexes split more than Fe(II) ones.
Why complexes are colored
A complex is colored when a d electron absorbs visible light and jumps from the lower set to the upper set, a d-d transition. The absorbed wavelength matches Δo, and the color observed is the complement of the color absorbed.
[Ti(H2O)6]3+, with a single d electron, absorbs yellow-green light for the t2g to eg jump and appears purple. Complexes with no possible d-d transition, such as d0 Sc3+ or d10 Zn2+, are colorless, while some intense colors, like the purple of MnO4-, come from charge-transfer transitions rather than d-d jumps.
Crystal field stabilization energy
The crystal field stabilization energy (CFSE) measures how much the splitting stabilizes a given electron configuration relative to the unsplit ion. Each t2g electron contributes -0.4 Δo and each eg electron +0.6 Δo, with an extra pairing-energy penalty for each pair beyond the high-spin arrangement.
For example, high-spin d6 (t2g4 eg2) has CFSE = 4(-0.4 Δo) + 2(0.6 Δo) = -0.4 Δo, while low-spin d6 (t2g6) has CFSE = 6(-0.4 Δo) + 2P = -2.4 Δo + 2P. The larger stabilization of the low-spin form helps explain why strong-field d6 complexes are so favorable.
Magnetism and the spin-only formula
Unpaired electrons make a complex paramagnetic, attracted into a magnetic field, and the strength of that attraction reveals how many there are. The spin-only formula gives the magnetic moment from the number of unpaired electrons n:
μ = √(n(n+2)) Bohr magnetons
High-spin Fe(III), d5 with five unpaired electrons, gives √(35) ≈ 5.92 BM, matching experiment closely. The formula ignores orbital contributions, so it works best for first-row metals where the orbital angular momentum is largely quenched, but it remains the standard quick check of a spin-state assignment.
Variables and units
| Symbol | Name | Unit | Meaning |
|---|---|---|---|
| octahedral splitting energy | kJ/mol | Energy gap between the t2g and eg sets in an octahedral complex | |
| tetrahedral splitting energy | kJ/mol | Energy gap between the e and t2 sets in a tetrahedral complex, about 4/9 of Δo | |
| pairing energy | kJ/mol | Energy cost of placing two electrons in the same orbital instead of separate ones | |
| crystal field stabilization energy | kJ/mol | Net stabilization of an electron configuration relative to the unsplit d orbitals | |
| spin-only magnetic moment | Bohr magnetons | Approximate magnetic moment calculated from unpaired electrons only | |
| number of unpaired electrons | dimensionless | Count of electrons with no partner in the split d orbitals |
Common mistakes
The same metal with different ligands gives different colors because the ligands set the splitting energy. [Co(H2O)6]2+ is pink while [CoCl4]2- is deep blue, though both contain cobalt(II).
Low spin requires the splitting to exceed the pairing energy, and it also depends on the metal and oxidation state. Configurations d1-d3 and d8-d10 have only one possible arrangement no matter the ligand.
Tetrahedral splitting is only about 4/9 of the octahedral value and the geometry can never produce a large gap, so tetrahedral complexes are essentially always high-spin.
A spherical shell of charge would raise them equally, but real ligands approach from specific directions. Orbitals pointing at the ligands rise while the others sink, keeping the average energy at the barycenter.
Real-world applications
Cobalt blue and chrome green pigments get their colors from crystal field splitting in cobalt and chromium complexes
Ruby's red color comes from trace Cr3+ ions whose d orbitals are split by the surrounding oxide lattice
The blood-red thiocyanate test for iron(III) works because SCN- forms the intensely colored complex [Fe(SCN)]2+
Measured magnetic moments reveal the number of unpaired electrons and confirm high-spin or low-spin assignments
Gemstone colors, such as emerald's green from Cr3+ or V3+, are explained by d-d transitions in a crystal field
Frequently asked questions
Why does CN- produce a much larger splitting than F-?
Cyanide is a pi-acceptor ligand: it pulls electron density from the metal's t2g orbitals into its own empty pi orbitals, which stabilizes the t2g set and widens the gap. Fluoride is a pi-donor that pushes electron density back onto the metal, raising the t2g set and shrinking the gap. The difference in pi bonding, not just sigma donation, sets their positions in the spectrochemical series.
Why are 4d and 5d complexes almost always low-spin?
The 4d and 5d orbitals are larger and more diffuse than 3d orbitals, so they overlap with ligand orbitals more effectively and experience a stronger field. The resulting splitting is typically so large that it always exceeds the pairing energy, making low-spin configurations the rule for second- and third-row transition metals.
Can any d-electron count be high- or low-spin?
No. Only d4 through d7 configurations in an octahedral field can be either high- or low-spin. Configurations d1-d3 must place each electron in a separate t2g orbital, and d8-d10 fill the orbitals in only one possible way regardless of the splitting size.
Why is [Ti(H2O)6]3+ purple?
The single d electron of Ti3+ sits in the t2g set and can absorb yellow-green light to jump to the eg set. Removing yellow-green from white light leaves the complementary color, purple, which is what you observe. One absorption band and one complementary color is the simplest possible case of complex color.
What exactly is the pairing energy P?
Pairing energy is the energy cost of placing two electrons in the same orbital: increased electron-electron repulsion minus the exchange stabilization lost by pairing. When the splitting energy exceeds this cost, electrons pair up in the lower set and the complex goes low-spin; when it does not, they spread out and stay high-spin.
Is crystal field theory the complete explanation?
It is a useful model, not the full story. Crystal field theory treats ligands as point charges and ignores covalency, yet metal-ligand bonds have real covalent character. Ligand field theory extends the model by allowing orbital mixing, which explains trends like the pi-donor and pi-acceptor effects in the spectrochemical series.