Crystal Field Theory — d-Orbital Splitting
Split the d-orbitals with a ligand field and watch spin, magnetism and colour follow.
About this simulation
Crystal field splitting made interactive. Students choose octahedral or tetrahedral geometry, a ligand from the spectrochemical series (I⁻ → CN⁻, which sets the field strength Δ) and the d-electron count, and the energy-level diagram fills the t₂g/eₒ (or e/t₂) sets by the Hund/pairing rules. A strong field (Δ > pairing energy P) forces low-spin, a weak field stays high-spin, and tetrahedral Δt = 4/9 Δo is always high-spin. The number of unpaired electrons, para/diamagnetism, CFSE and even the observed colour (complementary to the absorbed d-d wavelength, λ ≈ 1240/Δ) update live — so [Fe(H₂O)₆]²⁺ (high-spin, paramagnetic) versus [Fe(CN)₆]⁴⁻ (low-spin, diamagnetic) becomes something students build rather than memorise.
How to use it
- Choose 'Geometry' Octahedral, pick 'Ligand' H₂O (aqua) and set 'd-electron count' to 6: the electrons spread out and the top of the stage reads 4 unpaired e⁻ · CFSE = −0.4 Δₒ, then High spin · paramagnetic.
- Keep 6 d-electrons and change the ligand to CN⁻ (cyano): the gap Δₒ widens, all six electrons pair in t₂g and the top now reads 0 unpaired e⁻ · CFSE = −2.4 Δₒ, Low spin · diamagnetic.
- Step the ligand through the spectrochemical series from I⁻ to CN⁻ and watch the gap grow and the colour dot and 'absorbs ~… nm' label change.
- Switch to Tetrahedral and see the order flip (e below t₂) with a much smaller gap Δt, high spin whatever the ligand.
- Count the unpaired arrows in each case and read the configuration printed under the diagram, such as t₂g⁴ eg².
What to notice
- In an octahedral field the d-orbitals split into a lower t₂g set of three and an upper eg set of two; in a tetrahedral field the order is reversed and the gap is only 4/9 as large.
- Strong-field ligands such as CN⁻ make Δₒ larger than the pairing energy, so electrons pair in t₂g first and the complex is low spin.
- Weak-field ligands such as halides and water give a gap smaller than the pairing energy, so electrons occupy all five orbitals singly before pairing.
- The colour seen is complementary to the light absorbed; a larger splitting absorbs shorter wavelengths, and d⁰ and d¹⁰ ions have no d–d transition and are colourless.
The formulas
- CFSE (octahedral) = (−0.4 n(t₂g) + 0.6 n(eg)) Δₒ
- Crystal field stabilisation energy from the numbers of electrons in the lower t₂g and upper eg sets (pairing energy not included).
- Δt = (4/9) Δₒ
- Splitting in a tetrahedral field compared with an octahedral field for the same metal and ligand.
- λ (nm) ≈ 1240 / Δ (eV)
- Wavelength of light absorbed when an electron is promoted across a gap Δ, from E = hc/λ.
Questions
What is crystal field splitting in an octahedral complex?
In an isolated metal ion all five d-orbitals have the same energy. When six ligands approach along the x, y and z axes, the dx²−y² and dz² orbitals, which point at the ligands, are repelled more and rise in energy to form the eg set. The dxy, dyz and dxz orbitals, pointing between ligands, drop to form t₂g. The energy gap is Δₒ.
Why is [Fe(CN)₆]⁴⁻ diamagnetic while [Fe(H₂O)₆]²⁺ is paramagnetic?
Both contain Fe²⁺, a d⁶ ion. CN⁻ is a strong-field ligand, so Δₒ is larger than the pairing energy and all six electrons pair in t₂g (t₂g⁶ eg⁰), leaving no unpaired electrons. Water is a weak-field ligand, so the electrons spread out as t₂g⁴ eg² with four unpaired electrons. Load the two presets to see both diagrams.
What is the spectrochemical series?
The spectrochemical series arranges ligands in order of the splitting they produce: I⁻ < Br⁻ < Cl⁻ < F⁻ < H₂O < NH₃ < en < CN⁻ (with CO stronger still). Ligands at the left are weak-field and tend to give high-spin complexes; those at the right are strong-field and tend to give low-spin complexes. The simulation's Ligand menu lists these eight in this order.
Why are tetrahedral complexes almost always high spin?
In a tetrahedral complex only four ligands surround the metal and none of them point directly at any d-orbital, so the splitting Δt is only about four-ninths of Δₒ. This gap is almost always smaller than the pairing energy, so it is cheaper for electrons to occupy the upper orbitals than to pair. Switch Geometry to Tetrahedral and the complex stays high spin for every ligand.
Why are most transition metal complexes coloured?
When visible light falls on the complex, an electron in the lower d-set absorbs a photon whose energy matches the splitting Δ and jumps to the upper set. The light of that wavelength is removed and we see the complementary colour. If there are no d-electrons, or the d-set is full (d¹⁰, as in Zn²⁺), no such jump is possible and the compound is colourless.