Crystal Field Splitting Visualiser

What is a crystal field splitting diagram?

A crystal field splitting diagram shows how the five d orbitals of a transition metal ion, which share one energy in the free ion, separate into sets of different energy when ligands surround it. In an octahedral complex the three orbitals that point between the ligands, d_xy, d_xz and d_yz, fall to −0.4Δo and are called t_2g. The two that point straight at the ligands, d_z² and d_x²−y², rise to +0.6Δo and are called e_g. The gap between the two sets is the octahedral splitting, Δ_o, and their average, the barycentre, stays where it was: 3 × (−0.4) + 2 × (+0.6) = 0.

The d electrons then fill the levels, and the crystal field stabilisation energy, CFSE, is the energy they gain against the free ion:

CFSE = (−0.4x + 0.6y)Δo + mP

Here x is the number of electrons in t_2g, y the number in e_g, P the pairing energy and m the number of electron pairs beyond those the free ion already has. A negative CFSE is a stabilisation. The visualiser above draws the split levels to scale for octahedral, tetrahedral and square planar complexes, fills them with electrons, decides between high and low spin, and reads out the CFSE, the unpaired electrons and the spin-only magnetic moment. The theory goes back to Hans Bethe’s 1929 paper on ions in crystals and to John Van Vleck’s work on their magnetism in 1932.

Octahedral, tetrahedral and square planar splitting

Which orbitals rise depends on where the ligands sit, since an electron in an orbital pointing at a ligand is pushed up by it:

• Octahedral, six ligands on the x, y and z axes: t_2g at −0.4Δ_o and e_g at +0.6Δ_o.
• Tetrahedral, four ligands at alternate corners of a cube: the pattern turns over, with e (d_z², d_x²−y²) at −0.6Δ_t and t₂ (d_xy, d_xz, d_yz) at +0.4Δ_t. For the same metal, ligands and distance, Δt = (4/9)Δo: there are four ligands instead of six, and none points straight at any orbital. The labels lose their g because a tetrahedron has no centre of symmetry.
• Square planar, four ligands on the x and y axes: d_x²−y² points at all four and rises far above the rest. In the point-charge model of Krishnamurthy and Schaap (1969), d_xz and d_yz sit at −0.514Δ_o, d_z² at −0.428Δ_o, d_xy at +0.228Δ_o and d_x²−y² at +1.228Δ_o. The gap from d_xy up to d_x²−y², which the visualiser calls Δ₁, is exactly Δ_o, and the whole spread is 1.742Δ_o.

The shapes are the ones the VSEPR Molecular Geometry tool builds in three dimensions, and the Atomic Orbitals Visualiser shows which d orbital lobes point at the ligands and which point between them.

High spin or low spin: the splitting against the pairing energy

For d¹ to d³ in an octahedron the electrons go into t_2g one to an orbital. The fourth has a choice. It can go up into e_g at a cost of Δ_o, or pair with an electron already in t_2g at a cost of the pairing energy P, the extra repulsion of two electrons sharing one orbital together with the exchange energy they lose. If Δo < P it goes up and the complex is high spin; if Δo > P it pairs and the complex is low spin. The same comparison settles d⁵, d⁶ and d⁷, so in an octahedron the choice exists for d⁴ to d⁷ only. From d⁸ on, t_2g is full and the rest must go into e_g whatever Δ_o is.

The pairing term counts only the pairs beyond those the free ion has under Hund’s rule. High-spin d⁶, t_2g⁴e_g², has one pair, like the free ion, so its CFSE is simply −0.4Δo. Low-spin d⁶, t_2g⁶, has three pairs, so it is −2.4Δo + 2P. The two differ by 2(Δo − P), which changes sign exactly where Δ_o equals P, and low-spin d⁶, with t_2g full, has the largest CFSE of any octahedral arrangement. In a tetrahedron the choice exists for d³ to d⁶ and turns on Δ_t against P.

Using the visualiser

Choose a complex from the list, or set your own geometry, d electron count, Δ_o and P. A complex sets all four from measured data: Δ_o from its absorption spectrum, as tabulated by Shriver, Atkins and Langford (1994), and, where its d count leaves a choice of spin, the pairing energy of the free metal ion, from Miessler, Fischer and Tarr (2014). The Weaker ligand and Stronger ligand buttons step the same metal ion along the spectrochemical series, and moving any slider takes you back to your own values.

The d electron count of an ion is its group number less its charge: iron is in group 8, so Fe²⁺ is d⁶. The Periodic Table gives the group of every metal. The Electron arrangement menu can force either limit, so you can read the CFSE of an arrangement that is not the lowest, and the readouts then say how far above the lowest it sits. With the diagram focused, the left and right arrow keys change the d count, the up and down keys move Δ_o by 500 cm⁻¹, and Page Up and Page Down move it by 5000 cm⁻¹.

Reading the diagram and the readouts

Each orbital is a short bar at its energy, and each electron a half arrow, up or down for its spin. On a wide screen the free ion appears on the left, five bars at the barycentre filled by Hund’s rule. The coloured bracket is the splitting, and in the tetrahedral view a dashed one shows the Δ_o the same ligands would give in an octahedron. Under the diagram, bars set the splitting against P.

• Spin state is high spin, low spin, intermediate spin, or one arrangement where the d count leaves no choice.
• CFSE is in units of the geometry’s own splitting, Δ_t for a tetrahedron, and CFSE with pairing adds the P term. In wavenumbers and Per mole put numbers to it, at 11.96 J/mol for each cm⁻¹.
• Spin-only moment is μ = √(n(n + 2)) μB for n unpaired electrons.
• Other spin state is how far the alternative lies above the arrangement drawn.

The plot under the diagram draws the energy of each possible arrangement against Δ_o as a straight line. Low spin starts higher, by its extra pairing energy, and falls faster, so the lines cross where the spin changes, and the dot marks the arrangement drawn.

Worked example: [Fe(H₂O)₆]²⁺ and [Fe(CN)₆]⁴⁻

The visualiser opens on hexaaquairon(II), with Δ_o = 10,400 cm⁻¹ and the free Fe²⁺ pairing energy P = 17,600 cm⁻¹:

• Iron is in group 8, so Fe²⁺ has 8 − 2 = 6 d electrons.
• Δ_o is less than P, so the fifth and sixth electrons go up into e_g rather than pair: t_2g⁴e_g², with four unpaired electrons.
• CFSE = 4 × (−0.4) + 2 × (+0.6) = −0.4Δo, with no P term, since high-spin d⁶ has one pair, as the free ion does.
• In numbers, −0.4 × 10,400 = −4160 cm⁻¹, which is −49.76 kJ/mol.
• Low spin would come to −2.4 × 10,400 + 2 × 17,600 = +10,240 cm⁻¹, which is 14,400 cm⁻¹ higher, 2 × (17,600 − 10,400).
• The spin-only moment is √(4 × 6) = 4.899 μB. High-spin octahedral iron(II) measures 5.1 to 5.7 μB, a little more, because orbital motion adds to the spin.

Now press Stronger ligand. Cyanide gives Δ_o = 32,800 cm⁻¹, well above P, and all six electrons pair in t_2g:

• CFSE = −2.4Δo + 2P = −2.4 × 32,800 + 2 × 17,600 = −43,520 cm⁻¹, or −520.6 kJ/mol.
• High spin would sit 30,400 cm⁻¹ higher, and with no unpaired electrons [Fe(CN)₆]⁴⁻ is diamagnetic.

The spectrochemical series

Ligands can be put in order of the splitting they give. Ryutaro Tsuchida set the series out in 1938 from the absorption spectra of cobalt(III) complexes, and a common short form of it is:

I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < bipy < phen < NO₂⁻ < CN⁻ < CO

The weak end is the halides and other π donors, whose filled p orbitals push t_2g up and close the gap. The strong end is π acceptors such as CN⁻ and CO, which draw t_2g down and widen it. The point-charge picture cannot explain that order, since in it neutral CO could never outrank an anion. The measured values in the list follow it: for chromium(III), Δ_o rises from 13,000 cm⁻¹ with Cl⁻ to 17,400 with H₂O, 21,500 with NH₃ and 26,600 with CN⁻.

The metal matters as well. Δ_o grows with the charge on the ion, from 10,400 to 14,300 cm⁻¹ for hexaaqua iron going from Fe²⁺ to Fe³⁺ and from 8500 to 21,000 cm⁻¹ for manganese. It grows down a group too: 22,900 cm⁻¹ for [Co(NH₃)₆]³⁺, 34,000 for [Rh(NH₃)₆]³⁺ and 41,000 for [Ir(NH₃)₆]³⁺. Together with their smaller pairing energies, that is why 4d and 5d complexes are nearly always low spin. Cobalt(III) shows the series deciding the spin: [CoF₆]³⁻, at 13,000 cm⁻¹, is high spin against P = 21,000 cm⁻¹, while [Co(NH₃)₆]³⁺, only 1900 cm⁻¹ above P, is low spin and diamagnetic.

Why tetrahedral complexes are high spin

With Δt = (4/9)Δo, a tetrahedral splitting rarely exceeds the pairing energy, so tetrahedral complexes are almost always high spin. The cobalt(II) halides in the list show it: Δ_t is 2700 cm⁻¹ for [CoI₄]²⁻, 2900 for [CoBr₄]²⁻ and 3300 for [CoCl₄]²⁻, in spectrochemical order, against 22,500 cm⁻¹ for the pairing energy of free Co²⁺. For [CoCl₄]²⁻ the visualiser sets the Δ_o slider to 7425 cm⁻¹, the 9/4 × 3300 that the 4/9 rule implies.

Cobalt(II) is d⁷, e⁴t₂³ in a tetrahedron, with three unpaired electrons and a CFSE of 4 × (−0.6) + 3 × (+0.4) = −1.2Δt. That is −3960 cm⁻¹ for the chloride, or −0.5333Δ_o. Its spin-only moment is 3.873 μB, and tetrahedral cobalt(II) complexes measure 4.2 to 4.8 μB.

Square planar d⁸ complexes

In a square planar field d_x²−y² points straight at all four ligands and sits Δ₁ above d_xy, which the point-charge model makes equal to Δ_o. Eight d electrons fill the four lower orbitals exactly, so once Δ₁ passes P the last two pair in d_xy and d_x²−y² stays empty. The CFSE is then 4 × (−0.514) + 2 × (−0.428) + 2 × (+0.228) = −2.456Δo, plus P for the one pair beyond the free ion’s three. Every electron is paired, which is why [Ni(CN)₄]²⁻, [PdCl₄]²⁻, [PtCl₄]²⁻ and [AuCl₄]⁻ are diamagnetic. Nearly all 4d and 5d d⁸ complexes with four ligands are square planar, while nickel(II) with chloride is tetrahedral instead: [NiCl₄]²⁻ has two unpaired electrons.

With d⁴, d⁵ or d⁶ in the square planar field, the unevenly spread levels give an intermediate spin over a range of Δ_o. For d⁵ at P = 17,600 cm⁻¹, five unpaired electrons give way to three at Δ_o = 10,100 cm⁻¹, and three give way to one at 23,720 cm⁻¹.

Magnetic moments: spin-only and measured

Unpaired electrons make a complex paramagnetic, and counting them by a magnetic measurement is how high and low spin are told apart in the laboratory. The spin-only moment μ = √(n(n + 2)) μB gives 1.732, 2.828, 3.873, 4.899 and 5.916 μB for one to five unpaired electrons. Measured moments are often higher, because the electrons’ orbital motion adds to their spin: octahedral high-spin cobalt(II) complexes such as [Co(H₂O)₆]²⁺ measure 4.3 to 5.2 μB against a spin-only 3.873. A moment near zero means every electron is paired, as in [Co(NH₃)₆]³⁺ and [Fe(CN)₆]⁴⁻.

What this model leaves out

• Covalent bonding. Crystal field theory treats the ligands as point charges, which cannot explain why neutral CO splits the d orbitals more than F⁻ does. Ligand field and molecular orbital theory can, by letting the ligand orbitals mix with the metal’s, and the Molecular Orbital Diagram Visualiser builds that kind of diagram for small molecules. The measured Δ_o values already contain those effects.
• A smaller pairing energy in the complex. The pairing energies are free-ion values. In a complex the d electrons spread onto the ligands, the nephelauxetic effect, and P falls, so a complex close to its crossover can be low spin where the free-ion P predicts high spin.
• Jahn-Teller distortion. High-spin d⁴, low-spin d⁷ and d⁹ ions have an odd number of electrons in e_g, and their complexes distort, usually by lengthening two opposite bonds, which splits e_g further. [Mn(H₂O)₆]³⁺ and copper(II) complexes do this.
• Absorption energies. For a single d electron the absorption band sits at Δ_o itself, so [Ti(H₂O)₆]³⁺, at 20,300 cm⁻¹, absorbs near 493 nm and looks violet. With more electrons, repulsion between them shifts and splits the bands, and Tanabe-Sugano diagrams are needed to read Δ_o from a spectrum.
• The order of the lower square planar levels. It is a result of the model. Point charges put d_xy above d_z², while other models and real ligands can swap them or move d_z² lower still, as Börgel, Campbell and Ritter (2016) review. For d⁸ the order makes no difference, because the four lower orbitals are full.
• The choice of geometry. The visualiser draws the geometry you pick, while a real complex settles its own through CFSE, the size of its ligands and more.

Common mistakes

• Counting d electrons from the neutral atom. Iron is 3d⁶4s², but Fe²⁺ is 3d⁶ and Fe³⁺ is 3d⁵, because an ion loses its 4s electrons first. The Electron Configuration Calculator writes out the ion’s configuration.
• Using the octahedral pattern for a tetrahedron. The levels turn over, with e below t₂, and the splitting is Δ_t, about 4/9 of Δ_o.
• Charging P for every pair. Only pairs beyond the free ion’s count, so low-spin d⁶ is −2.4Δ_o + 2P, not + 3P. Some books add every pair instead; the comparison between spin states comes out the same either way, since the free ion’s pairs appear on both sides.
• Looking for a spin choice in d³ or d⁸. In an octahedron only d⁴ to d⁷ have one, and in a tetrahedron only d³ to d⁶.
• Dropping the sign. A CFSE of −0.4Δ_o means 0.4Δ_o below the free ion. Tables that list it as 0.4 are giving the size of the stabilisation, not a destabilisation.