Octahedral Crystal Field Splitting

Ruby is red and emerald is green. Yet both are essentially the same ion, Cr³⁺, sitting in an octahedron of oxygen atoms. The difference is entirely a matter of geometry and electrostatics. It comes down to how six point charges, arranged at the corners of an octahedron, reach into a metal ion's five d-orbitals. Some orbitals get pushed up in energy while others get pulled down. That splitting pattern — two orbitals raised, three lowered, by a gap called Δₒ — is the single most useful diagram in inorganic chemistry. It explains color, magnetism, geometric preferences, and reaction rates across the entire transition series with one picture. This article builds that picture from first principles. It starts from the Crystal Field Theory: Basic Model and Assumptions treatment of a spherical field. Then it asks what happens when the field is concentrated at six specific points instead of smeared uniformly over a sphere. The answer is not arbitrary. The direction each d-orbital lobe points relative to the ligand positions determines exactly how much electrostatic repulsion it feels. Symmetry, via group theory, guarantees that the five orbitals fall into exactly two energy sets, t₂g and eg. Their average energy relative to the free ion is unchanged — the barycenter rule. The big picture is that this splitting, quantified by the single parameter Δₒ, is a knob nature and chemists both turn. Its size depends on the metal's identity, oxidation state, and period. It also depends on the ligand's field strength, captured empirically by the spectrochemical series. Once Δₒ is known, the placement of electrons among t₂g and eg decides several things. This placement obeys Hund's rule versus pairing energy. It decides whether a complex is High-Spin vs Low-Spin Complexes. It decides how much Crystal Field Stabilization Energy the complex gains. It also decides what color light the complex absorbs, and how labile or inert it is to ligand substitution. This article delivers a geometric and electrostatic derivation of the t₂g/eg splitting. It offers a quantitative statement and proof of the barycenter (center-of-gravity) rule. It includes worked numerical examples connecting Δₒ to absorbed wavelength and back. It provides a table of scope and common misconceptions. Finally, it looks at where this single splitting diagram shows up in real chemistry — from the color of gemstones to the design of spin-crossover materials.

Key takeaways

Six ligand point charges on the ±x, ±y, ±z axes turn the five degenerate d-orbitals into two sets: t₂g (dxy, dxz, dyz) and eg (dz², dx²-y²).
eg orbitals point straight at the ligands and are raised in energy; t₂g orbitals point between ligands and are lowered.
The splitting energy is called Δₒ (or 10Dq); by the barycenter rule, eg rises by +0.6Δₒ and t₂g falls by −0.4Δₒ, keeping the weighted average at zero.
Δₒ is set by ligand field strength (spectrochemical series) and metal charge/period, and is measured directly from the d–d absorption band, Δₒ = hc/λ_max.
The splitting pattern, not the crystal-field electrostatic picture itself, is what survives into the more accurate molecular-orbital treatment of metal–ligand bonding.
Key idea: an octahedral array of point charges lowers the symmetry from spherical to Oh, splitting the d-orbitals into a stabilized t₂g triplet and a destabilized eg doublet separated by Δₒ, with the energy-weighted average fixed by the barycenter rule.

Where this lives today

The t₂g/eg splitting diagram is not a historical curiosity confined to textbooks — it is the daily working tool of anyone who studies transition-metal chemistry. The color of gemstones (ruby's red, emerald's green, both from Cr³⁺ in slightly different octahedral oxide environments with different Δₒ) is explained directly by the position of the t₂g→eg absorption band. Hemoglobin and myoglobin, built around octahedral (or near-octahedral) Fe centers, rely on the Δₒ vs pairing-energy balance to switch between high-spin deoxy-states and low-spin oxy-states, a switch central to oxygen transport in blood — the biological embodiment of the same energy diagram derived here. Spin-crossover materials, an active area of molecular materials research for data storage and molecular switches, are engineered by tuning ligand field strength so that Δₒ sits right at the crossover point between high-spin and low-spin, letting temperature, pressure, or light flip the complex between magnetic states. Catalysis and industrial chemistry lean on the same diagram: the lability of octahedral complexes (how fast ligands exchange) correlates with CFSE loss on going to the five-coordinate transition state, itself computed from the t₂g/eg energies, explaining why some octahedral metal centers are kinetically inert (e.g., low-spin d⁶ Co³⁺, Cr³⁺) while others are extremely labile (e.g., high-spin d⁵ or d⁰ centers). Even instrumentation depends on it: UV-visible spectrophotometers used to characterize new coordination compounds are, in essence, direct measuring devices for Δₒ, and the diagram is the first thing any inorganic chemist sketches when interpreting a new spectrum.

Ask the tutor: How does the tetrahedral crystal field splitting pattern differ in sign and magnitude from the octahedral one derived here?

The physical picture

Start from the spherical crystal field already established: six negative point charges (or point dipoles, standing in for ligand lone pairs) smeared uniformly over a sphere around the metal ion raise all five d-orbitals by the same amount, since a spherically symmetric perturbation cannot distinguish one d-orbital from another. That spherical average is a bookkeeping device — a reference energy — not a physically realizable arrangement. The real octahedral complex localizes the six point charges at specific positions: along the +x, −x, +y, −y, +z, −z axes. The moment you fix the ligands on the axes, spherical symmetry is broken down to octahedral symmetry (point group Oh), and the five d-orbitals stop being degenerate. The reason is angular: each d-orbital has a specific shape and orientation, and some orbitals have lobes pointing directly at the incoming negative charges while others have lobes pointing into the gaps between them. Look at the orbital shapes. The dz² orbital has a large lobe along +z and −z (plus a doughnut in the xy-plane); the dx²-y² orbital has four lobes along ±x and ±y. Both of these orbitals therefore point straight at two or four of the six ligands respectively. Electron density in these orbitals is repelled strongly by the approaching ligand charges, so their energy rises. These two orbitals, transforming together as the eg irreducible representation of Oh, form the upper set. By contrast, dxy, dxz, and dyz all have their four lobes bisecting the axes — pointing, for example, into the regions between the +x and +y axes rather than along them. Electron density in these three orbitals is farther, on average, from the ligand point charges, so they are repelled less than in the spherical-field reference, and their energy falls relative to that average. These three orbitals transform together as t₂g and form the lower set. Group theory guarantees this 2+3 split is exact for any Oh field, regardless of the details of the ligand — only the size of the splitting (Δₒ) depends on chemistry, not the pattern of which orbitals group together. The gap between the eg and t₂g sets is defined as Δₒ, also historically written 10Dq (with Dq a convenient energy unit so that Δₒ = 10Dq).

Figure

The five d-orbitals relative to six octahedral ligands: eg orbitals point at ligands, t₂g orbitals point between them

Once this splitting exists, electrons fill the t₂g and eg sets following the aufbau principle modified by the size of Δₒ relative to the pairing energy P — the topic that leads directly into high-spin/low-spin behavior. But before electrons are placed, it is essential to fix the energy reference correctly, which is exactly what the barycenter rule does.

Figure

Energy level diagram: barycenter rule for octahedral splitting

The core ideas

The central quantitative statements are the splitting energies of the two sets and the constraint that ties them together.

E(e_g) - E(t_{2g}) = \Delta_o

where: Δₒ is the octahedral crystal field splitting energy, the energy gap between the eg and t₂g sets, measured in cm⁻¹, kJ mol⁻¹, or eV depending on context. Reading: this single number is the entire "field strength" content of the crystal field model for an octahedral complex; everything else (which orbitals are occupied, how much stabilization results, what color is absorbed) follows from knowing Δₒ and comparing it to the electron-pairing energy P.

E(e_g) = +0.6\,\Delta_o, \qquad E(t_{2g}) = -0.4\,\Delta_o

where: energies are measured relative to the barycenter (the hypothetical spherical-field energy, set to zero); +0.6Δₒ and −0.4Δₒ are the standard fractional shifts, sometimes written +3/5 Δₒ and −2/5 Δₒ. Reading: the eg set is destabilized by three-fifths of the total gap and the t₂g set is stabilized by two-fifths; the ratio 3:2 is fixed by orbital degeneracy and the barycenter constraint, not by any adjustable parameter.

2\,E(e_g) + 3\,E(t_{2g}) = 0

where: the coefficients 2 and 3 are the orbital degeneracies of eg and t₂g respectively, and the right-hand side is zero because energies are measured from the barycenter. Reading: this is the barycenter rule itself — a conservation law stating that splitting redistributes energy among the d-orbitals without changing their total (degeneracy-weighted) energy, exactly analogous to a lever balanced about a fixed fulcrum.

\Delta_o = \frac{hc}{\lambda_{\max}}

where: h is Planck's constant, c is the speed of light, and λ_max is the wavelength of maximum absorbance in the d–d electronic spectrum of the complex (valid directly for a d¹ or d⁹ ion; for other configurations λ_max must be matched to the correct spin-allowed transition). Reading: Δₒ is not a fitted or purely theoretical number — it is measured, directly, from the color of the compound, which is why crystal field theory is so easy to test against experiment.

Derivation AWhy eg rises by 0.6Δo and t2g falls by 0.4Δo

Goal. Starting only from (i) the fact that octahedral symmetry splits the five d-orbitals into a 2-fold eg set and a 3-fold t₂g set, and (ii) the barycenter rule that the total electrostatic energy summed over all five d-orbitals is unchanged by the splitting, derive the fractional shifts +0.6Δₒ and −0.4Δₒ.

Step 1 — set up the unknowns. Let the energy of each eg orbital, measured relative to the spherical-field barycenter, be $x$ , and let the energy of each t₂g orbital be $y$ . By definition of Δₒ as the gap between the sets,

x - y = \Delta_o .

There are two eg orbitals and three t₂g orbitals — this degeneracy count comes directly from the dimensions of the eg and t₂g irreducible representations of the Oh point group, which in turn follows from how dz², dx²-y² (basis for eg) and dxy, dxz, dyz (basis for t₂g) transform under the group's symmetry operations.

Step 2 — impose the barycenter condition. The barycenter rule states that placing six ligand charges exactly on the octahedral axes redistributes energy among the d-orbitals but does not change the degeneracy-weighted mean, because the first-order perturbation is purely angular (a redistribution of a fixed total repulsion), not a net change in the total charge or distance of the ligands from the metal center. Formally,

2x + 3y = 0 .

This is the same statement as "the average energy of all five d-orbitals in the octahedral field equals their average energy in the spherical reference field," weighted by how many orbitals sit in each set.

Step 3 — solve the two equations. From Step 1, $x = y + \Delta_o$ . Substitute into the barycenter equation:

2(y + \Delta_o) + 3y = 0

2y + 2\Delta_o + 3y = 0

5y = -2\Delta_o

y = -0.4\,\Delta_o .

Then

x = y + \Delta_o = -0.4\,\Delta_o + \Delta_o = 0.6\,\Delta_o .

Step 4 — check consistency. Substitute back: $2(0.6\Delta_o) + 3(-0.4\Delta_o) = 1.2\Delta_o - 1.2\Delta_o = 0$ , confirming the barycenter condition, and $x - y = 0.6\Delta_o - (-0.4\Delta_o) = 1.0\,\Delta_o$ , confirming the definition of Δₒ. The fractions 0.6 and 0.4 are therefore forced entirely by the 2:3 degeneracy ratio; they would be different (e.g., in a tetrahedral or square-planar field) only because the degeneracy pattern and geometric factor differ, not because the underlying logic changes.

Go deeper · Derivation B · 2 minElectrostatic origin of the splitting from orbital geometry

Derivation BElectrostatic origin of the splitting from orbital geometry

Goal. Show, using only the angular distribution of electron density in each d-orbital relative to six point charges on the Cartesian axes, why dz² and dx²-y² are raised while dxy, dxz, dyz are lowered — i.e., justify Step 1's premise from first principles rather than assuming it.

Step 1 — place the charges and orbitals. Put six identical negative point charges at $(\pm a, 0, 0)$ , $(0,\pm a,0)$ , $(0,0,\pm a)$ , representing the six ligand lone pairs at metal–ligand distance $a$ . The d-orbitals are standing electron-density waves with angular nodes; their energy in the presence of the ligand charges is the electrostatic repulsion integral,

E \propto \int |\psi_d(\mathbf r)|^2 \, V_{\text{lig}}(\mathbf r)\, d^3r,

where $V_{\text{lig}}(\mathbf r)$ is the electrostatic potential from the six point charges. This integral is large and positive (destabilizing) whenever the orbital's angular density is large along a ligand direction, and small when the orbital's density is concentrated in the gaps between ligand directions.

Step 2 — evaluate qualitatively for eg orbitals. The dz² orbital has its principal lobes along $\pm z$ , directly on top of two of the six point charges, plus a torus of density in the xy-plane which still points partway toward the four equatorial ligands. The dx²-y² orbital has four lobes exactly along $\pm x$ and $\pm y$ , directly on top of the remaining four point charges. In both cases the regions of highest electron density coincide with the regions of highest ligand charge density, so $V_{\text{lig}}$ is sampled at its largest (most repulsive) values — both orbitals are raised in energy.

Step 3 — evaluate qualitatively for t2g orbitals. The dxy orbital has four lobes along the directions $(\pm1,\pm1,0)/\sqrt2$ — that is, bisecting the x and y axes, 45° away from every ligand position in the xy-plane — and a node along the z-axis. By the same logic dxz and dyz have lobes bisecting the x–z and y–z axes respectively. In every case, the orbital's high-density regions sit in the angular gaps between ligand positions, where $V_{\text{lig}}$ is smaller (the point charges are farther away in angle even though the metal–ligand distance $a$ is the same). The repulsion sampled by these orbitals is therefore below the spherically-averaged value, and their energy drops relative to the barycenter.

Step 4 — connect to the barycenter numerically. Because the total number of d-electrons available to "feel" the ligand field is fixed and the six ligand charges are fixed, the sum of angular overlap between all five d-orbitals and the six ligand directions is a geometric constant — redistributing the splitting between sets cannot change this constant. This is the microscopic justification for the barycenter condition $2x+3y=0$ used as an axiom in the algebraic derivation: it is not an extra assumption but a consequence of the fact that the total angular electron density of a complete d-shell is spherically symmetric, so any redistribution among orbitals within the field must integrate to zero net change relative to the spherical reference.

Worked examples

Example 1 — Foundational: converting Δₒ between units and finding orbital energies. Suppose an octahedral Ni²⁺ complex has Δₒ = 8500 cm⁻¹. Find (a) Δₒ in kJ mol⁻¹, (b) the energy of the eg set and t₂g set relative to the barycenter in cm⁻¹, and (c) confirm the barycenter rule numerically. (a) Using 1 cm⁻¹ = 0.011963 kJ mol⁻¹,

\Delta_o = 8500 \times 0.011963 \ \text{kJ mol}^{-1} = 101.7 \ \text{kJ mol}^{-1}.

(b) Using the fractional shifts derived above,

E(e_g) = 0.6 \times 8500 = 5100 \ \text{cm}^{-1}, \qquad E(t_{2g}) = -0.4 \times 8500 = -3400 \ \text{cm}^{-1}.

2(5100) + 3(-3400) = 10200 - 10200 = 0 \ \text{cm}^{-1}. \checkmark

The gap is $5100-(-3400) = 8500$ cm⁻¹, matching Δₒ exactly. This example fixes the arithmetic pattern used in every subsequent case: multiply Δₒ by 0.6 for each eg orbital's rise and by 0.4 for each t₂g orbital's fall, and always verify the 2:3 weighted cancellation.

Example 2 — Applied: extracting Δₒ from an absorption spectrum. A dilute aqueous solution of [Ti(H₂O)₆]³⁺ (a d¹ ion, so crystal field theory applies with no interelectronic repulsion complications) shows a single broad d–d absorption band with λ_max = 493 nm. Find Δₒ in both cm⁻¹ and kJ mol⁻¹, and state which orbital the electron occupies before and after the transition. Convert wavelength to wavenumber:

\tilde\nu = \frac{1}{\lambda} = \frac{1}{493\times10^{-7}\,\text{cm}} = 20{,}284 \ \text{cm}^{-1}.

Since for a single d-electron the only spin-allowed transition is t₂g → eg, this wavenumber is Δₒ directly:

\Delta_o = 20{,}284 \ \text{cm}^{-1}.

Convert to kJ mol⁻¹:

\Delta_o = 20{,}284 \times 0.011963 = 242.6 \ \text{kJ mol}^{-1}.

Before absorption, the single d-electron sits in a t₂g orbital, 0.4Δₒ = 8114 cm⁻¹ below the barycenter. After absorbing a photon of energy Δₒ, the electron is promoted to an eg orbital, 0.6Δₒ = 12,170 cm⁻¹ above the barycenter — a net orbital energy change of exactly 20,284 cm⁻¹, consistent with the photon energy absorbed. The complex looks violet/purple because it absorbs strongly in the green-yellow region (~493–500 nm) and transmits the red and blue-violet ends of the visible spectrum.

Example 3 — Exam/competition level: comparing two octahedral fields with different geometric distortion parameters and back-solving from CFSE. An octahedral d⁶ high-spin Fe²⁺ complex, [Fe(H₂O)₆]²⁺, has a measured crystal field stabilization energy (CFSE) of −0.4Δₒ (a standard high-spin d⁶ result: four t₂g and two eg electrons, since one t₂g orbital must hold a pair). Independent spectroscopic measurement gives Δₒ = 10,400 cm⁻¹ for this ion. A second complex, [Fe(CN)₆]⁴⁻, is low-spin d⁶ (all six electrons in t₂g, none in eg) with Δₒ = 33,000 cm⁻¹. (a) Compute the CFSE (in cm⁻¹, ignoring pairing energy for now) for both complexes using the barycenter-derived fractional energies, and (b) compute the additional stabilization gained purely from the change of spin state and field strength between the two, to see why cyanide forces low spin. For high-spin d⁶: configuration is $t_{2g}^4 e_g^2$ . Using $E(t_{2g})=-0.4\Delta_o$ per electron and $E(e_g)=+0.6\Delta_o$ per electron,

\text{CFSE}_{\text{HS}} = 4(-0.4\Delta_o) + 2(+0.6\Delta_o) = -1.6\Delta_o + 1.2\Delta_o = -0.4\,\Delta_o.

With Δₒ = 10,400 cm⁻¹,

\text{CFSE}_{\text{HS}} = -0.4 \times 10{,}400 = -4160 \ \text{cm}^{-1}.

For low-spin d⁶: configuration is $t_{2g}^6 e_g^0$ ,

\text{CFSE}_{\text{LS}} = 6(-0.4\Delta_o) = -2.4\,\Delta_o.

With Δₒ = 33,000 cm⁻¹,

\text{CFSE}_{\text{LS}} = -2.4 \times 33{,}000 = -79{,}200 \ \text{cm}^{-1}.

The purely orbital stabilization gained by [Fe(CN)₆]⁴⁻ relative to [Fe(H₂O)₆]²⁺ is

\Delta(\text{CFSE}) = -79{,}200 - (-4160) = -75{,}040 \ \text{cm}^{-1},

an enormous number — but this comparison mixes two effects (different Δₒ and different spin state) and overstates the "true" energetic driving force, because pairing three extra electrons in cyanide's case costs pairing energy P that is not present in the high-spin count. A fair same-field comparison asks instead: for a single hypothetical field, would low-spin or high-spin be favored? That comparison is exactly the calculation that motivates High-Spin vs Low-Spin Complexes, where the criterion becomes Δₒ vs P rather than Δₒ alone; this worked example deliberately stops here to show why the barycenter-derived CFSE numbers alone are necessary but not sufficient for predicting spin state — pairing energy must be added as a second, independent ingredient.

Limiting cases, scope & common traps

Regime / limit	What happens to the t2g/eg picture	Why
Δₒ → 0 (very weak field, e.g., isolated ion, or hypothetical zero-charge ligands)	t₂g and eg collapse back to five degenerate d-orbitals	No angular perturbation exists; recovers the free-ion (spherical) limit used as the barycenter reference
Very strong field, Δₒ ≫ P	All electrons pair in t₂g before any occupy eg (low spin)	Pairing energy cost is smaller than the t₂g/eg energy gap; see High-Spin vs Low-Spin Complexes
Weak field, Δₒ ≪ P	Electrons singly fill t₂g and eg following Hund's rule before pairing (high spin)	Pairing cost exceeds the promotion energy to eg
d⁰, d¹⁰, and high-spin d⁵ configurations	CFSE = 0 regardless of Δₒ	Either no d-electrons, a full symmetric shell, or exactly one electron in every one of the five orbitals — barycenter cancellation is exact and configuration-independent
Jahn–Teller-active configurations (e.g., high-spin d⁴, low-spin d⁷, d⁹)	The simple two-level t₂g/eg diagram is distorted (further split)	Unequal occupation of degenerate orbitals within a set drives a geometric distortion that lowers symmetry below ideal Oh
Non-octahedral geometries (tetrahedral, square planar)	Different degeneracy pattern and different splitting fractions	The 2:3 degeneracy and 0.6/0.4 fractions are specific to Oh; tetrahedral splitting inverts the pattern (e below t₂), covered in Tetrahedral Crystal Field Splitting

The chemist behind it

The octahedral splitting diagram is the joint legacy of physicists who solved the group-theory problem of ions in crystal lattices and chemists who applied it to transition-metal complexes decades later. Hans Bethe, working in 1929 on the quantum mechanics of electrons in crystal lattices, first derived how a crystalline electric field of given symmetry splits atomic energy levels — the mathematical machinery behind t₂g/eg is his. John Hasbrouck Van Vleck, in the 1930s, connected this crystal-field mathematics to the magnetic and spectroscopic behavior of transition-metal ions in real compounds, effectively founding what would become ligand field theory. It was only in the 1950s, through the work of chemists including Leslie Orgel and Carl Ballhausen (extending Bethe and Van Vleck's physics into systematic chemical language, correlating it with the newly available UV-visible spectra of coordination compounds), that the t₂g/eg diagram became the everyday tool taught to every chemistry student today.

Origins · 1 minHans Albrecht Bethe and how this was discovered

Hans Albrecht Bethe

1906–2005 · Germany/United States; theoretical physics, nuclear astrophysics, and the group-theoretic foundations of crystal field theory

Bethe's 1929 paper "Termaufspaltung in Kristallen" ("Splitting of Terms in Crystals") solved, using group theory, how the electronic energy levels of an ion are split by the electric field of a surrounding crystal lattice of a given point-group symmetry — precisely the calculation that, applied to an Oh ligand arrangement around a transition-metal ion, produces the t₂g/eg splitting used throughout coordination chemistry. Bethe went on to a career centered on nuclear physics, most famously explaining the nuclear fusion reactions that power stars, for which he won the Nobel Prize in Physics in 1967; the crystal-field paper was an early, comparatively minor entry in an extraordinarily long research career, yet it became foundational to an entire subfield of inorganic chemistry that developed independently of Bethe's later work.

History

- 1929 — Hans Bethe derives the group-theoretic splitting of ionic energy levels in crystal fields of arbitrary symmetry.

- 1932 — John Van Vleck applies crystal-field ideas to the magnetic susceptibilities of transition-metal salts.

- 1935 — Van Vleck's theoretical treatments establish quantitative links between Δ-type splittings and observed magnetic moments.

- 1951–1957 — Leslie Orgel and Carl Ballhausen popularize crystal field theory among chemists, matching it to newly measured UV-visible spectra of octahedral complexes.

- 1960s — Crystal field theory is reconciled with molecular orbital theory (ligand field theory), explaining why Δo depends on covalent metal–ligand overlap, not electrostatics alone.

Why do only two energy levels (t2g and eg) appear instead of five distinct levels for five different orbitals?

Group theory says that in a field of Oh symmetry, orbitals that transform into one another under the symmetry operations of the group must remain degenerate — they cannot be pulled apart by a field of that exact symmetry. dxy, dxz, and dyz are related to one another by the C4 and C3 rotations of the octahedron (each maps onto a combination of the others), so they must stay degenerate as the triply-degenerate t₂g set. Similarly dz² and dx²-y² are related by these same symmetry operations and stay degenerate as the eg set. Only if the symmetry is lowered further (e.g., by axial elongation to D4h, as in Jahn–Teller distortion) do these sets split apart into more levels.

Is Δo the same thing as the crystal field stabilization energy, CFSE?

No. Δₒ is the gap between the t₂g and eg energy levels — a property of the ligand field and metal, independent of how many electrons are present. CFSE is the total stabilization gained by a particular electron configuration relative to the barycenter, computed by summing −0.4Δₒ for every t₂g electron and +0.6Δₒ for every eg electron (then subtracting any extra pairing energy cost). Two complexes can have identical Δₒ but very different CFSE if they have different d-electron counts, and conversely CFSE = 0 for d⁰, d¹⁰, and high-spin d⁵ regardless of how large Δₒ is.

Does the size of Δo depend on the metal, the ligand, or both?

Both, and the dependence is systematic. For a fixed ligand, Δₒ increases with metal oxidation state (a 3+ ion pulls ligands in closer and feels a stronger field than the corresponding 2+ ion) and increases down a group (4d and 5d metals have larger, more diffuse d-orbitals that overlap ligand orbitals more effectively, roughly 30–50% larger Δₒ per row). For a fixed metal, Δₒ depends on the ligand according to the empirical spectrochemical series, roughly I⁻ < Br⁻ < Cl⁻ < F⁻ < H₂O < NH₃ < en < CN⁻ < CO, with weak-field halides at one end and strong-field π-acceptors like CO and CN⁻ at the other.

Why does dz2 count as a single orbital in the eg set when it looks so different from dx2-y2?

Although dz² and dx²-y² look geometrically different (one has a lobe-plus-donut shape, the other four in-plane lobes), they belong to the same two-dimensional irreducible representation, eg, of the Oh point group — meaning that under the symmetry operations of the octahedron (90° rotations, etc.) these two orbitals mix into linear combinations of each other rather than transforming into orbitals outside the pair. Group theory, not visual resemblance, is what determines degeneracy; the mathematics of how each orbital's angular function transforms under the group's operations shows both belong to the same symmetry label and must remain degenerate in an ideal Oh field.

Does crystal field theory correctly explain why Δo is different for different ligands, given that it treats ligands as point charges?

Only partially, and this is a well-known limitation. Pure electrostatic crystal field theory predicts that Δₒ should track ligand charge and dipole strength, but the observed spectrochemical series does not follow ionic charge simply — for instance, neutral CO and CN⁻ produce far larger Δₒ than the anionic halides, which pure electrostatics would rank higher. The resolution requires molecular orbital / ligand field theory, where π-backbonding from filled metal t₂g orbitals into empty ligand π orbitals (as in CO and CN⁻) stabilizes t₂g further and increases the effective Δₒ beyond what point-charge electrostatics predicts. The t₂g/eg splitting pattern survives into the MO picture essentially unchanged; only the microscopic explanation of its size* is revised.

How does this splitting pattern connect to CFSE and to high-spin/low-spin behavior?

Once Δₒ is known, filling d-electrons into the t₂g (−0.4Δₒ each) and eg (+0.6Δₒ each) levels according to Hund's rule (if Δₒ is small relative to pairing energy P) or according to maximum t₂g filling first (if Δₒ is large relative to P) generates a specific electron configuration. Summing the individual orbital energies for that configuration, and subtracting any extra electron-pairing energy incurred, gives the crystal field stabilization energy (CFSE) — the net energetic payoff of forming the octahedral complex compared to a hypothetical spherical field with the same average d-orbital energy. The competition between Δₒ and P for configurations d⁴ through d⁷ is exactly the fork in the road that produces high-spin versus low-spin complexes.

Does every six-coordinate complex show the ideal t2g/eg pattern with equal metal-ligand distances?

Only idealized, perfectly symmetric Oh complexes show the clean two-level pattern derived here. Many real six-coordinate complexes are tetragonally distorted (elongated or compressed along one axis, lowering the symmetry to D4h), especially when the d-electron configuration is Jahn–Teller active (unevenly filled eg or, less strongly, unevenly filled t₂g). In those cases the eg level splits further into separate dz² and dx²-y² energies, and t₂g splits into dxy versus the degenerate dxz/dyz pair — a refinement built directly on top of the Oh diagram developed here, rather than a contradiction of it.