Crystal Field Theory and Magnetism

01 / UNDERSTAND & APPLY

1. What the model explains, and what its energy zero means

Crystal field theory asks how the spatial arrangement of surrounding ligands changes the relative energies of a metal ion’s five d orbitals. Its elementary form represents ligands electrostatically, using charges or dipoles, and does not explicitly build covalent metal–ligand molecular orbitals. It is useful for organizing electron occupancy, magnetic behaviour and some spectral trends, but it is not a complete theory of bonding.

The five d orbitals are degenerate for a spherically symmetric reference field. Arranging ligands in particular directions removes some of that degeneracy. Orbitals directed more strongly toward the approaching negative donor regions experience greater repulsion in the electrostatic model. Their energies rise relative to orbitals oriented more between those directions.

The reference called the barycentre is the degeneracy-weighted average of the split orbital energies. Setting it to zero does not mean the metal and ligands have no interaction, nor that a complex with zero crystal-field stabilization has no bonds. It removes a common reference contribution so that the extra effect of directional splitting can be isolated.

A negative signed crystal-field energy means stabilization relative to that chosen unsplit reference. Some questions instead report the positive magnitude of stabilization. State the convention before comparing answers. Pairing energy, electrostatic attraction, covalency, ligand strain and solvation contribute to real stability, so a crystal-field number alone is not the total formation enthalpy or a universal formation-constant predictor.

02 / UNDERSTAND & APPLY

2. Octahedral splitting and the −0.4/+0.6 factors

Choose an octahedron with ligand directions along ±x, ±y and ±z. The d(x²−y²) and d(z²) orbitals have substantial density along these axes and form the higher-energy e_g set. The d(xy), d(xz) and d(yz) orbitals form the lower-energy t₂g set. The subscripts label symmetry; the important elementary facts are the membership, the degeneracies and the energy ordering.

Let the separation between the two sets be Δₒ, always a positive gap. If the lower set lies at energy a relative to the barycentre, the upper set lies at a+Δₒ. Because three orbitals are lower and two are upper, the weighted sum must be 3a+2(a+Δₒ)=0. Therefore a=−2Δₒ/5, and the upper energy is +3Δₒ/5.

Each electron placed in a t₂g orbital contributes −0.4Δₒ to the signed orbital energy; each electron in e_g contributes +0.6Δₒ. The gap is Δₒ, not 0.6Δₒ, because the gap spans both offsets. The zero line is also not halfway between the two sets: there are three lower orbitals but only two upper orbitals.

For a d-electron occupancy t₂gᵃe_gᵇ, add contributions from all a+b electrons. This arithmetic counts orbital-energy contributions only. It does not yet account for the difference in electron-pairing costs between possible occupancies.

03 / UNDERSTAND & APPLY

3. Determine the ion before filling orbitals

The metal’s d count belongs to its oxidation state, not to the overall charge on the coordination entity. Compute that oxidation state from ligand charges first. Neutral ammonia and water do not change the charge sum, whereas chlorido and cyanido ligands each contribute −1 in the usual ionic bookkeeping.

For common transition-metal ions, a convenient formal d count is the group number minus the oxidation state. Thus iron(II) is d⁶, iron(III) is d⁵, cobalt(III) is d⁶, nickel(II) is d⁸ and copper(II) is d⁹. This is the standard ionic configuration convention for the present complexes, not a literal measurement of localized charge in a covalent bond.

When deriving the ionic configuration from a neutral atom, remove outer ns electrons before the (n−1)d electrons for these ordinary transition-metal cations. Do not retain a neutral atom’s 4s² electrons in Fe²⁺ merely because 4s appeared before 3d in a mnemonic filling order.

Once the d count is correct, draw the orbital levels for the actual geometry. A d⁸ ion in an octahedron and a d⁸ ion in a square plane need not have the same unpaired-electron count. Metal oxidation state and electron count are necessary information, but neither alone determines a magnetic answer.

04 / UNDERSTAND & APPLY

4. Pair downstairs or occupy an upstairs orbital?

Within a set of degenerate orbitals, occupy each orbital singly with parallel spins before pairing. The additional decision is whether it costs less to pair in a lower orbital or occupy a higher orbital. In the elementary model, P is an effective pairing cost and Δₒ is the promotion cost between the two orbital sets.

When Δₒ is smaller than P, occupying the higher set can cost less than extra pairing: the high-spin arrangement is favoured. When Δₒ is larger than P, extra pairing in the lower set can be favoured: the low-spin arrangement results. Near a real crossover, more detailed electronic and thermodynamic effects matter; a single fixed P is an instructional simplification.

For ideal octahedral complexes, the ordinary distinct high-spin/low-spin choices occur for d⁴ through d⁷. The occupancies are d⁴: t₂g³e_g¹ versus t₂g⁴; d⁵: t₂g³e_g² versus t₂g⁵; d⁶: t₂g⁴e_g² versus t₂g⁶; and d⁷: t₂g⁵e_g² versus t₂g⁶e_g¹. The first occupancy in each pair is high spin.

The corresponding unpaired-electron counts are 4 versus 2 for d⁴, 5 versus 1 for d⁵, 4 versus 0 for d⁶, and 3 versus 1 for d⁷. Low spin does not mean that every electron is paired: low-spin d⁵ and d⁷ each retain one unpaired electron. High spin does not mean there are no pairs: high-spin d⁶ contains one pair.

For octahedral d¹, d² and d³, the lower three orbitals accommodate the electrons singly, giving one, two and three unpaired electrons. For d⁸, the lower set is full and the two upper orbitals are singly occupied, giving two unpaired electrons in the ideal model. d⁹ has one unpaired electron and d¹⁰ has none. Applying a strong-field label cannot magically pair ideal octahedral d⁸ into a square-planar occupancy.

05 / UNDERSTAND & APPLY

5. Keep pairing references explicit

Two common energy expressions look different because they choose different pairing references. One writes the total model energy as E_CF plus N_pair P, where N_pair counts all doubly occupied d orbitals. Another subtracts the pairing already present in the unsplit free-ion reference and adds only the number of extra pairs. Both can give the same energy differences if used consistently.

For d⁶, the high-spin octahedral state has t₂g⁴e_g², orbital energy −0.4Δₒ and one pair. The low-spin state has t₂g⁶, orbital energy −2.4Δₒ and three pairs. The difference E_LS−E_HS is therefore −2Δₒ+2P. If Δₒ>P, the low-spin state lies lower in this simple model.

Relative to the unsplit high-spin d⁶ reference, which already has one pair, the pairing-corrected stabilization expressions are −0.4Δₒ for high spin and −2.4Δₒ+2P for low spin. Adding only one P to the latter would miss one of the two additional pairs. Always count actual boxes rather than memorizing a table with an unexplained convention.

The word CFSE is used with slightly different pairing conventions in different educational sources. Here E_CF explicitly means the orbital splitting contribution alone. Whenever a question asks for a pairing-inclusive value, specify the reference and show the pair count. Comparisons between spin states should use the same metal, geometry model and energy parameters.

06 / UNDERSTAND & APPLY

6. Other geometries require other energy ladders

In a tetrahedral arrangement, the ligands lie between the Cartesian axes rather than directly along them. The two e orbitals, d(z²) and d(x²−y²), lie lower, while the three t₂ orbitals, d(xy), d(xz) and d(yz), lie higher. The g subscripts used for the centrosymmetric octahedral labels are absent in tetrahedral symmetry.

Let the tetrahedral separation be Δₜ. Two lower and three upper orbitals give offsets −0.6Δₜ and +0.4Δₜ, because 2(−0.6)+3(+0.4)=0. Thus E_CF=[−0.6n(e)+0.4n(t₂)]Δₜ. The degeneracy pattern and ordering are reversed relative to an octahedron; simply reusing the octahedral −0.4/+0.6 formula is wrong.

The ideal point-charge comparison gives Δₜ approximately 4Δₒ/9 for corresponding metal, ligand and distance assumptions. It is not a rule for comparing unrelated real complexes with different bond lengths. Tetrahedral splittings are commonly small enough that ordinary first-row examples are high spin. Treat this as the standard regime for these problems rather than a claim that low-spin tetrahedral chemistry is impossible under every circumstance.

In a square plane with ligands on ±x and ±y, d(x²−y²) experiences especially strong destabilization and is usually the highest d orbital. A commonly used ordering below it is d(xy), then d(z²), then the degenerate d(xz)/d(yz) pair. Details of the middle levels can depend on the bonding situation; the large separation of the highest orbital is the central feature for many square-planar d⁸ examples.

Eight electrons can fill the four lower square-planar orbitals in pairs, leaving d(x²−y²) empty. The resulting state is diamagnetic in that model. Common comparisons include tetrahedral [NiCl₄]²⁻ with two unpaired electrons and square-planar [Ni(CN)₄]²⁻ with none. Both contain Ni(II), d⁸; the geometry and ligand-field pattern differ. Do not infer that every four-coordinate d⁸ species must be square planar.

07 / UNDERSTAND & APPLY

7. From unpaired electrons to a spin-only moment

An isolated species with unpaired electrons is generally paramagnetic: its electronic magnetic moments contribute an attraction in an applied field. A closed-shell species with all electrons paired has no permanent spin contribution and is ordinarily diamagnetic. Bulk magnetic ordering and interactions between centers require a more detailed treatment than counting one isolated ion.

The spin-only estimate is μ_so=√[n(n+2)] μ_B, where n is the number of unpaired electrons and μ_B is the Bohr magneton. It is equivalent to 2√[S(S+1)] μ_B when the n unpaired spins give S=n/2. This estimates a magnetic moment, not a count of electrons and not a molar susceptibility.

Useful benchmarks for n=1,2,3,4,5 are about 1.73, 2.83, 3.87, 4.90 and 5.92 μ_B. For n=0 the spin-only value is zero. The order is monotonic, but the square-root formula is not linear: doubling the number of unpaired electrons does not double the spin-only moment.

Measured moments can differ because of orbital angular momentum, spin–orbit coupling, temperature effects or coupling between metal centers. The approximation is especially useful for many first-row transition-metal textbook cases, but it should not be indiscriminately applied to lanthanides or used to claim exact agreement for every cobalt or heavier-metal complex.

08 / UNDERSTAND & APPLY

8. Ligand strength, absorption and observed colour

A short qualitative spectrochemical ordering is I⁻<Br⁻<Cl⁻<F⁻<H₂O<NH₃<en<CN⁻<CO for increasing field strength in common comparisons. The series is empirical and reflects bonding effects beyond a purely electrostatic model. A highly charged ligand is not automatically a stronger-field ligand; comparing fluoride with neutral carbon monoxide shows why charge alone is inadequate.

The splitting also depends on the metal, oxidation state, bond distances and transition series. For otherwise comparable cases, higher metal oxidation state often increases splitting, and 4d/5d complexes often have larger splittings than corresponding 3d complexes. Therefore a ligand label such as strong field is a useful comparison, not a substitute for the actual Δ/P balance of every complex.

If a specified absorption corresponds directly to an orbital energy separation in a simplified model, use Δ=hc/λ per absorbed photon and multiply by Avogadro’s constant for energy per mole. A shorter wavelength means a larger photon energy. In wavenumber units, the corresponding relation is Δ=hc times the wavenumber, with consistent length units.

The colour seen in transmitted or reflected light is associated with wavelengths left after absorption, often described approximately as complementary to the absorbed colour. It is not normally the colour that was absorbed. Real spectra can contain several electronic transitions and broad bands; assigning a single measured absorption to Δ requires the stated model or a justified spectral assignment.

Partly filled d levels permit many d–d transitions, but colour is not an infallible test for unpaired electrons. Some low-spin complexes are coloured despite having no unpaired electrons. d⁰ or d¹⁰ species can also be coloured through charge-transfer or ligand-centered transitions even though ordinary within-d-shell excitations are unavailable. Selection rules and covalency affect intensities, beyond this elementary ladder model.

09 / UNDERSTAND & APPLY

9. Solved practice and final checks

Unless stated otherwise, use ideal octahedral geometry and the spin-only magnetic model. E_CF always denotes the signed orbital contribution before pairing corrections. Explicitly check that the total occupancy equals the formal d count.

• Compute oxidation state before d count.
• Use the geometry-specific ordering and degeneracy.
• Apply Hund’s rule within each degenerate set.
• Count pairs explicitly when adding P.
• Distinguish the signed orbital energy from its positive stabilization magnitude.
• A low-spin label does not guarantee diamagnetism.
• A spin-only moment is an approximation, not an exact universal measurement.
• Observed colour and absorbed wavelength are different pieces of information.