Crystal-field and ligand-field descriptions

Read the occupation picture before the algebra

On a first reading, follow which levels are lower, how many electrons occupy them and how many remain unpaired. The pairing-cost and orbital-mixing calculations are optional deeper passes. These simplified models organize comparisons; they do not replace a spectrum or an electronic-structure calculation for a named complex.

Crystal-field splitting starts with a restricted model

Crystal-field theory treats the ligand environment electrostatically and compares d orbitals with different directions relative to the donors. In an ideal octahedron, the d_xy,d_xz,d_yz set is lower and d_z²,d_x²−y² higher in the usual crystal-field description. The lower set has three orbitals and the upper set two. Δ_o denotes their separation, not the entire absolute bonding energy.

Relative to the five-orbital weighted average, the lower energies are −0.4Δ_o and the upper +0.6Δ_o. The sum 3(−0.4)+2(+0.6)=0 preserves that reference. In an ideal tetrahedral field the ordering is reversed: two lower orbitals at −0.6Δ_t and three upper at +0.4Δ_t, again with zero weighted sum.

The often used Δ_t≈(4/9)Δ_o relation is a comparison within a restricted comparable-ligand electrostatic model. Different real distances, ligand identities and covalency can change it. A splitting diagram also omits other orbitals and interactions; its energy zero is not the energy of free separated reagents.

Pairing competes with occupying the upper set

For a selected octahedral d⁶ count, a high-spin occupation is (2, 1, 1) in the lower set and (1, 1) above: four unpaired electrons and one pair. The low-spin occupation is (2, 2, 2) below and (0, 0) above: no unpaired electrons and three pairs. Spin choice follows an energy competition, rather than the d count alone.

In an invented comparison, take a constant pair cost P=15000 cm⁻¹ per pair. The high-spin crystal-field contribution is −0.4Δ_o and the low-spin contribution −2.4Δ_o. The low-minus-high total difference is −2Δ_o+2P. For Δ_o=10000 cm⁻¹ it is +10000 cm⁻¹, favouring high spin in this toy; for Δ_o=20000 cm⁻¹ it is −10000 cm⁻¹, favouring low spin.

These spectroscopic energy units represent energy divided by hc. The stipulated pair cost, common reference and occupations are necessary assumptions. Actual electron repulsion, covalency, distortion and spin–orbit coupling can require a more complete model. Calling a ligand strong field is a contextual comparison, not a guarantee of every metal’s spin state.

Keep tetrahedral occupation and its reference separate

For a selected tetrahedral d⁸ model, filling the two lower and three upper orbitals gives (2, 2;2, 1, 1), with two unpaired electrons. Stipulate a comparable octahedral splitting 18000 cm⁻¹ and use the restricted 4/9 relation: Δ_t=8000 cm⁻¹. The crystal-field contribution is 4(−0.6Δ_t)+4(+0.4Δ_t)=−0.8Δ_t=−6400 cm⁻¹.

That is the splitting contribution relative to the barycentre, rather than the complex’s full formation energy. It does not prove that every coordination-number 4 d⁸ entity is tetrahedral: a square-planar field has a different splitting pattern, and actual geometry needs its own evidence.

Ligand-field language can include covalent mixing

Ligand-field and molecular-orbital descriptions include interactions between metal and ligand orbital combinations of compatible symmetry. σ donation, π donation and π acceptance can alter energies and metal/ligand character. The electrostatic crystal-field picture by itself cannot fully account for those covalent interactions or a universal ligand ordering.

As a restricted original example, choose orthonormal metal and ligand basis functions with Hamiltonian rows (0, 1) and (1,−2), in invented eV units. Its eigenvalues are −1−√2≈−2.41421 and −1+√2≈ 0.414214 eV. Normalized coefficients may be (0.382683,−0.923880) for the lower combination and (0.923880, 0.382683) for the upper. Signs depend on the chosen basis phases.

Squaring the metal coefficients gives metal weights 0.146447 and 0.853553, with complementary ligand weights. These are model basis weights, not oxidation states, measured charges or a complete molecular population. The two-state calculation illustrates mixing and level separation; it has no real ligand geometry, overlap matrix, additional orbitals or many-electron correlation.

Worked examples

Try it yourself

1. Keep P=15000 cm⁻¹ but set Δ_o=25000 cm⁻¹ in the same octahedral d⁶ toy. Find the low-minus-high difference and the favoured state.

   Show the answer and reasoning

   −20000 cm⁻¹; low spin in this toy.

   2(P−Δ_o)=2(15000−25000)=−20000 under the stated constant pair-cost and common-reference assumptions.

2. Use the same selected tetrahedral d⁸ occupation, but stipulate Δ_t=9000 cm⁻¹. Find its splitting contribution and unpaired count.

   Show the answer and reasoning

   −7200 cm⁻¹; two unpaired electrons.

   Four lower and four upper electrons give −0.8Δ_t; the selected occupation has two singly occupied upper orbitals.

3. Can Δ_t=4Δ_o/9 be transferred without qualification to any two real complexes?

   Show the answer and reasoning

   No.

   It is a restricted comparison; ligand identity, geometry, distances and other interactions matter.

4. Does model metal weight 0.146447 mean metal oxidation state +0.146447?

   Show the answer and reasoning

   No.

   Basis weights and formal oxidation-state assignments are different descriptions.

What to carry forward

• State the splitting pattern and its energy reference.
• Count pairs as well as unpaired electrons.
• Compare pairing cost and splitting under explicit assumptions.
• Do not infer unique geometry from a d count.
• Orbital mixing adds physics beyond an electrostatic field.