Crystal field theory

Result

Crystal field theory

T-065Home CU-207Threads bonding · structure

Statement

d-orbital splitting in an octahedral ligand field.

Why it matters

coordination-geometry establishes what shape a complex adopts, but shape alone does not explain why coordination compounds are so often intensely coloured and why some are magnetic while others, containing the identical metal ion, are not. Crystal field theory supplies that explanation: placing a set of ligands around a metal ion in a specific geometry splits its previously degenerate d orbitals into groups of different energy, and the details of that splitting — its size, and how electrons fill the resulting levels — are what spectrochemical-colour and the complex's magnetic behaviour both derive from directly.

The theory also underlies chelate-effect and complex-isomerism indirectly, since the same splitting pattern that governs colour and magnetism also governs the relative stability of different geometric and electronic arrangements a given metal-ligand combination can adopt.

Hypotheses

Ligands are modelled as point negative charges (or point dipoles) that repel the metal's d electrons purely electrostatically, with no covalent metal–ligand orbital mixing considered.This is a deliberate simplification: crystal field theory ignores covalency entirely, treating the ligand-metal interaction as pure electrostatics. It gets the qualitative splitting pattern and its geometry-dependence right, but it cannot by itself explain features (like the specific ordering of some ligands in the spectrochemical series) that require partial covalent bonding, which the more complete ligand field theory restores. All five d orbitals are degenerate (equal in energy) in the free, ligand-free metal ion, and this degeneracy is broken only by the specific spatial arrangement of the surrounding ligands.Without ligands present there is no preferred direction in space, so nothing distinguishes one d orbital's energy from another's; splitting is entirely a consequence of the anisotropic (direction-dependent) electrostatic environment the ligands create once coordinated. In an octahedral field specifically, the five d orbitals split into only two energy levels (not five), because the orbital lobes fall into exactly two categories relative to the six ligand positions along the \(\pm x,\pm y,\pm z\) axes: those pointing directly at a ligand, and those pointing between ligands. Tetrahedral and square-planar fields, arising from different ligand geometries, produce genuinely different splitting patterns, covered by the same underlying logic applied to a different point-charge arrangement.

Proof

\text{Octahedral field: ligands lie along } \pm x, \pm y, \pm z \text{ axes.}

The \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals (collectively \(e_g\)) have lobes pointing directly along these axes, straight at the incoming ligand point charges, while the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals (collectively \(t_{2g}\)) have lobes pointing between the axes, away from the ligands. A

E(e_g) - E(t_{2g}) = \Delta_o \quad (\text{the octahedral crystal field splitting energy})

Because the \(e_g\) orbitals point directly at the ligand charges, electrons in them experience stronger electrostatic repulsion (higher energy) than electrons in the \(t_{2g}\) orbitals, which are shielded from the ligands by pointing between them; the two sets separate by an energy gap \(\Delta_o\), whose size depends on the specific ligand's field strength (spectrochemical-colour) and the metal's identity and charge. A

\text{Barycentre rule: } 2\,E(e_g) + 3\,E(t_{2g}) = 0 \ \text{ (relative to the pre-splitting, degenerate energy)}

Splitting redistributes but does not create or destroy total d-orbital energy, so, weighted by orbital count, the \(e_g\) and \(t_{2g}\) sets sit symmetrically above and below the original degenerate energy: standard convention places \(e_g\) at \(+0.6\,\Delta_o\) and \(t_{2g}\) at \(-0.4\,\Delta_o\), consistent with this weighted-average constraint. B

\text{For } d^4\text{–}d^7\text{: high-spin (small }\Delta_o\text{) vs. low-spin (large }\Delta_o\text{) filling, decided against the pairing energy } P

Electrons fill the lower \(t_{2g}\) level first, but whether a fourth (or fifth, sixth, seventh) electron pairs up in \(t_{2g}\) or instead occupies the higher, empty \(e_g\) level is decided by comparing \(\Delta_o\) against the electron-pairing energy \(P\): if \(\Delta_o < P\) (weak field), it is energetically cheaper to promote the electron to \(e_g\) (high-spin, more unpaired electrons); if \(\Delta_o > P\) (strong field), it is cheaper to pair it in \(t_{2g}\) (low-spin, fewer unpaired electrons). For \(d^1\)–\(d^3\) and \(d^8\)–\(d^{10}\) configurations no such ambiguity exists — only one filling pattern is possible. B

Result

\text{Octahedral field: } e_g \text{ (higher, } +0.6\Delta_o\text{)} \quad / \quad t_{2g} \text{ (lower, } -0.4\Delta_o\text{)}

Reading. A metal ion's five degenerate d orbitals split, in an octahedral ligand field, into two groups separated by an energy gap \(\Delta_o\): a lower, triply degenerate \(t_{2g}\) set and a higher, doubly degenerate \(e_g\) set, with the size of \(\Delta_o\) and the resulting electron filling pattern together determining the complex's colour and magnetism.

Scope. The specific splitting pattern given here is for octahedral geometry only; tetrahedral geometry inverts the pattern (a lower \(e\) set, higher \(t_2\) set, with a substantially smaller gap, \(\Delta_t \approx \tfrac{4}{9}\Delta_o\) for the same ligands and metal), and square-planar geometry splits the five orbitals into four distinct levels rather than two — both extensions of the identical point-charge logic applied to a different ligand arrangement.

Corollaries & converses

spectrochemical-colour follows directly: a complex absorbs visible light of energy matching \(\Delta_o\) to promote an electron from \(t_{2g}\) to \(e_g\), and the observed colour is the complement of the wavelength absorbed.
The high-spin/low-spin distinction (Step 4) directly determines a complex's magnetic behaviour (paramagnetic if unpaired electrons remain, diamagnetic if all are paired), a measurable property used experimentally to infer which regime, and hence roughly how large \(\Delta_o\) is, a given complex falls into.
Converse: measuring a complex's magnetic moment (number of unpaired electrons) for a \(d^4\)–\(d^7\) metal ion lets one infer, without any spectroscopic measurement, whether that complex's \(\Delta_o\) is large or small relative to the pairing energy \(P\) — magnetism as an indirect probe of ligand field strength.

Fails without

Apply the pure point-charge electrostatic model (Hypotheses) to a strong \(\pi\)-backbonding ligand like CO or CN−: the model cannot explain why these ligands, despite modest electronegativity, sit near the very top of the spectrochemical series — that behaviour requires the genuine covalent backbonding that ligand field theory, not bare crystal field theory, restores.
Assume the five d orbitals are non-degenerate even before any ligand field is applied: this contradicts the spherical symmetry of the free, ligand-free ion (Hypotheses), and without that starting degeneracy the entire premise — that splitting is caused specifically by the anisotropic ligand environment — has no defined baseline to split from.

Common errors

Reversing which set is higher in energy: in an octahedral field, \(e_g\) is higher and \(t_{2g}\) lower (Step 2), the opposite ordering to the tetrahedral case, and a frequent source of confusion when switching between geometries.
Assuming a high-spin or low-spin filling for every d-electron count; only \(d^4\)–\(d^7\) configurations admit a genuine choice (Step 4) — \(d^1\)–\(d^3\) and \(d^8\)–\(d^{10}\) have only one possible filling pattern regardless of field strength.
Forgetting the barycentre rule (Step 3) and placing \(e_g\) and \(t_{2g}\) symmetrically about zero (\(\pm0.5\,\Delta_o\)) rather than at the correct, orbital-count-weighted \(+0.6\,\Delta_o\) / \(-0.4\,\Delta_o\) split.
Treating \(\Delta_o\) as a fixed property of the metal alone, rather than a property of the specific metal–ligand combination that varies with both the ligand's position in the spectrochemical series and the metal's identity, charge, and row of the periodic table.

Discussion

Crystal field theory was originally developed by the physicist Hans Bethe in 1929 to describe electron energy levels in ionic crystal lattices, well before it was recognised, chiefly through work in the 1950s, as a useful (if electrostatically oversimplified) model for transition-metal coordination complexes specifically. Its later refinement into ligand field theory, which restores a degree of genuine covalent metal–ligand orbital mixing that pure electrostatics cannot capture, corrects several of crystal field theory's known quantitative shortcomings while preserving its qualitative splitting-diagram picture almost entirely.

The purely electrostatic model of the Hypotheses cannot, by itself, explain why some strong-field ligands (like CO or CN−) that are poor \(\sigma\)-donors on electronegativity grounds alone nonetheless sit near the top of the spectrochemical series; this requires invoking \(\pi\)-backbonding, a covalent effect entirely outside crystal field theory's electrostatic-only framework, and one reason ligand field theory is preferred for quantitatively accurate work.

Common misconception: that a larger \(\Delta_o\) simply means "a stronger bond" in some general sense. \(\Delta_o\) specifically measures the energy gap between the two d-orbital sets created by the ligand field, not metal–ligand bond strength directly, though the two are often loosely correlated for a given ligand across different metals.

Worked examples

[\text{Fe(H}_2\text{O)}_6]^{2+}\ (d^6,\ \text{weak-field H}_2\text{O}) \quad\text{vs.}\quad [\text{Fe(CN)}_6]^{4-}\ (d^6,\ \text{strong-field CN}^-)

Both complexes contain high-spin-capable Fe(II), a \(d^6\) ion. With weak-field water ligands, \(\Delta_o < P\), so the sixth electron is promoted to \(e_g\) rather than pairing in \(t_{2g}\): the high-spin configuration \(t_{2g}^4 e_g^2\) results, with four unpaired electrons, strongly paramagnetic. With strong-field cyanide ligands, \(\Delta_o > P\), so all six electrons pair up within \(t_{2g}\) alone: the low-spin configuration \(t_{2g}^6 e_g^0\) results, with zero unpaired electrons, diamagnetic. B

\text{Same metal, same charge, same d}^6\text{ count} \Rightarrow \text{opposite magnetic behaviour, by ligand field strength alone}

Reading. The identical d-electron count can give either four or zero unpaired electrons purely depending on where the attached ligand falls in the spectrochemical series, exactly the high-spin/low-spin choice of Step 4.

Scope. The same reasoning applies to any \(d^4\)–\(d^7\) metal ion; only the specific ligand field strength, read from the spectrochemical series, decides which regime a given complex falls into.

Problems

Give the \(t_{2g}\)/\(e_g\) electron configuration for a high-spin \(d^5\) octahedral complex, and state its number of unpaired electrons.
Solution
High-spin filling places one electron in each of the five d orbitals before any pairing occurs (maximising unpaired spins per Hund's rule, unconstrained by any pairing-energy penalty since one electron per orbital requires no pairing at all): \(t_{2g}^3 e_g^2\), with all five electrons unpaired — the maximum possible for any single d-electron count, which is why high-spin \(d^5\) complexes (e.g. many Fe(III) and Mn(II) complexes) are so strongly paramagnetic.
Explain why a low-spin \(d^5\) configuration instead has only one unpaired electron, contrasting directly with the previous problem.
Solution
Under a strong enough field (\(\Delta_o > P\)), it is energetically cheaper to pair electrons within the lower \(t_{2g}\) set than to promote any of them to the higher \(e_g\) set. All five electrons therefore fill \(t_{2g}\) first: two orbitals get a full pair each (four electrons, all paired) and the fifth electron occupies the third \(t_{2g}\) orbital singly, since only three \(t_{2g}\) orbitals exist: \(t_{2g}^5 e_g^0\), leaving exactly one unpaired electron, in sharp contrast to the five unpaired electrons of the high-spin case.
A \(d^8\) octahedral complex is found experimentally to have exactly two unpaired electrons regardless of which ligand is attached. Explain why no high-spin/low-spin ambiguity arises for this electron count.
Solution
With eight d electrons, filling \(t_{2g}\) completely (six electrons, all paired, since it holds a maximum of six) leaves exactly two electrons for the two \(e_g\) orbitals; per Hund's rule these two electrons occupy the two separate \(e_g\) orbitals singly rather than pairing in one, since no lower option remains available to promote from or demote to. This is the only possible filling for \(d^8\) in an octahedral field, independent of \(\Delta_o\), which is exactly why (per Step 4) only \(d^4\)–\(d^7\) configurations admit a genuine high-spin/low-spin choice at all.