Crystal field splitting in octahedral complexes

Intuition [Beginner]

A transition-metal atom by itself has five $d$-orbitals — and in the free atom, those five orbitals all have exactly the same energy. They are degenerate. When the atom becomes the central ion of a complex, with six ligands (donor atoms or molecules) arranged around it, the symmetry drops. The five $d$-orbitals stop being equivalent, and they split into two energy levels.

The geometry that matters most is the octahedron — six ligands sitting at the six vertices of an octahedron centred on the metal ion. Place the metal at the origin, and the ligands at $\pm x$, $\pm y$, $\pm z$. Two of the five $d$-orbitals — call them $d_{z^2}$ and $d_{x^2-y^2}$ — have their electron density pointing directly along those axes, straight at the ligands. The other three — $d_{xy}$, $d_{xz}$, $d_{yz}$ — have their electron density pointing between the axes.

A ligand is a small electron-rich object: a lone pair on a nitrogen, a chloride ion, a water molecule's oxygen. Electrons in $d$-orbitals that point at ligands feel extra electron-electron repulsion, so those orbitals are raised in energy. Electrons in $d$-orbitals that point between ligands feel less repulsion, so those orbitals are lowered. The result is the classic two-up, three-down splitting:

• The pair $\{d_{z^2},d_{x^2-y^2}\}$ — pointing at ligands — sits at higher energy.
• The trio $\{d_{xy},d_{xz},d_{yz}\}$ — pointing between ligands — sits at lower energy.

The energy gap between the two levels is called the octahedral crystal-field splitting, written ${\Delta }_o$ (or sometimes $10Dq$ in older notation). It depends on the metal ion, its oxidation state, and the identity of the ligands.

Why does any of this matter? Three concrete predictions.

Colour. Many transition-metal complexes are coloured because an electron in the lower set absorbs a photon and jumps to the upper set. The photon energy matches ${\Delta }_o$. Tune ${\Delta }_o$ — by changing the metal, the oxidation state, or the ligand — and the colour changes. A copper(II) ion in water is pale blue; the same copper(II) ion in ammonia is deep royal blue. Same metal, different ligand, different ${\Delta }_o$, different absorbed wavelength, different visible colour.

Magnetism. When the metal has between four and seven $d$-electrons, the electrons have a choice: cram more into the lower set (paying an electron-electron "pairing cost") or spread out over both sets (paying the ${\Delta }_o$ gap). Small ${\Delta }_o$ favours spreading out — many unpaired electrons, strongly magnetic ("high spin"). Large ${\Delta }_o$ favours pairing up — few unpaired electrons, weakly magnetic ("low spin"). Measuring whether a complex is high or low spin tells you about ${\Delta }_o$.

The spectrochemical series. Ligands can be ranked by how big a ${\Delta }_o$ they produce. The standard ordering — small to large — runs ${\mathrm{I}}^{-}<{\mathrm{Br}}^{-}<{\mathrm{Cl}}^{-}<{\mathrm{F}}^{-}<{\mathrm{H}}_2\mathrm{O}<\mathrm{N}{\mathrm{H}}_3<{\mathrm{CN}}^{-}<\mathrm{CO}$. This spectrochemical series is one of the most useful empirical regularities in inorganic chemistry: predict ${\Delta }_o$, predict colour, predict magnetism, all from one ranking.

The full story uses group theory and quantum mechanics to derive the splitting pattern from the symmetry of the octahedron. The Intermediate tier develops that machinery.

Visual [Beginner]

The five $d$-orbitals split into two sets when six ligands surround the metal in an octahedral arrangement.

Worked example [Beginner]

Take the hexaaquairon(II) complex $[\mathrm{Fe}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{2+}$. Iron in the $+2$ oxidation state has six $d$-electrons. Water sits roughly in the middle of the spectrochemical series, so ${\Delta }_o$ for this complex is moderate.

Step 1. Count the $d$-electrons. Neutral iron is $[\mathrm{Ar}]3d^{6}4s^2$. Removing two electrons to form ${\mathrm{Fe}}^{2+}$ takes off the two $4s$ electrons first, leaving $3d^6$. So six $d$-electrons.

Step 2. Compare ${\Delta }_o$ to the pairing energy. For water-bound iron(II), ${\Delta }_o$ comes out smaller than the pairing energy, so electrons spread out instead of pairing up.

Step 3. Fill the orbitals. The trio at the bottom gets three electrons one in each (lowest, no pairing cost), then the pair at the top gets two more (one each, again no pairing cost). That uses five of the six electrons. The sixth has to pair up — it pairs with one of the bottom trio.

Result: four unpaired electrons (one each in the four remaining unpaired slots after the sixth electron pairs).

Step 4. Predict properties. Four unpaired electrons makes $[\mathrm{Fe}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{2+}$ strongly paramagnetic. It is. Its pale green colour comes from absorption around 1000 nm — a small ${\Delta }_o$, in the near-infrared / far-red, so the complex looks green-blue (the complementary part of the visible spectrum).

What this tells us: knowing the metal's $d$-count, the ligand's place in the spectrochemical series, and the two-up-three-down splitting picture is enough to predict colour and magnetism without doing any quantum mechanics by hand.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ be a transition-metal ion at the origin and let six identical ligands sit at $(\pm a,0,0)$, $(0,\pm a,0)$, $(0,0,\pm a)$ for some bond length $a$. The point-group symmetry of this arrangement is $O_h$ — the full octahedral point group, of order 48, containing the rotations of an octahedron together with the inversion through its centre.

The five real $d$-orbitals on the metal centre, written in their standard form,

$$d_{z^2}=\frac{1}{2\sqrt{3}}(2z^2-x^2-y^2),d_{x^2-y^2}=\frac{1}{2}(x^2-y^2),$$ $$d_{xy}=xy,d_{xz}=xz,d_{yz}=yz,$$

form a real five-dimensional vector space $V_d$. The point group $O_h$ acts on $V_d$ by rotating / reflecting / inverting the coordinates and re-expressing the orbitals in the rotated frame — i.e., $V_d$ is a real representation of $O_h$. The defining question of crystal-field theory is: how does $V_d$ decompose into irreducible representations of $O_h$?

The answer is the central decomposition of the theory:

$$\boxed{V_d=E_g\oplus T_{2g}}$$

where, in Mulliken notation for $O_h$ irreps,

• $E_g$ is the 2-dimensional irrep carrying $\{d_{z^2},d_{x^2-y^2}\}$,
• $T_{2g}$ is the 3-dimensional irrep carrying $\{d_{xy},d_{xz},d_{yz}\}$.

The subscript $g$ ("gerade") indicates even parity under inversion — $d$-orbitals are even, as quadratic forms in $(x,y,z)$. The labels $E$ (for "doubly degenerate") and $T$ (for "triply degenerate") are conventional; the numerical subscripts ($2$ in $T_{2g}$) distinguish among $T$-type irreps via their behaviour under the rotation classes (see character-table derivation below).

The decomposition is purely a statement about $O_h$ acting on quadratic functions in ${\mathbb{R}}^3$ 07.01.03. It is independent of any electrostatic-repulsion model; symmetry alone fixes the splitting pattern. What the electrostatic model adds is the sign of the splitting (does $E_g$ or $T_{2g}$ end up higher?) and the magnitude — both come from explicit computation of the perturbation matrix elements of the ligand-field potential.

The energy splitting. Treat the six ligands as point charges $-q$ (or, more realistically, point dipoles) and compute the matrix elements of the resulting electrostatic potential between $d$-orbital basis functions. By Wigner-Eckart on $O_h$ 07.01.04, the potential acts within each irrep as a scalar multiple of the identity; off-diagonal blocks between different irreps vanish. Two distinct energies result:

$$E(E_g)-E(T_{2g})=:{\Delta }_o>0.$$

The traceless condition — the average $d$-orbital energy is unchanged in first order, since the trace of the ligand-field perturbation over the full five-orbital space is fixed by the spherical part — gives the barycentre rule: the two $E_g$ orbitals are raised by $+\frac{3}{5}{\Delta }_o$ above the unperturbed level, and the three $T_{2g}$ orbitals are lowered by $-\frac{2}{5}{\Delta }_o$ below it. So $2\cdot \frac{3}{5}{\Delta }_o+3\cdot (-\frac{2}{5}{\Delta }_o)=0$, as a barycentre must.

The legacy parameter $10Dq$ is identical to ${\Delta }_o$; the splittings $+6Dq$ for $E_g$ and $-4Dq$ for $T_{2g}$ correspond to $+\frac{3}{5}{\Delta }_o$ and $-\frac{2}{5}{\Delta }_o$ respectively.

The spectrochemical series. The magnitude of ${\Delta }_o$ varies strongly with ligand identity, following an empirical ordering known as the spectrochemical series. From smallest to largest:

$${\mathrm{I}}^{-}<{\mathrm{Br}}^{-}<{\mathrm{S}}^{2-}<{\mathrm{SCN}}^{-}<{\mathrm{Cl}}^{-}<{\mathrm{N}{\mathrm{O}}_3}^{-}<{\mathrm{F}}^{-}<{\mathrm{OH}}^{-}<{{\mathrm{C}}_2{\mathrm{O}}_4}^{2-}<{\mathrm{H}}_2\mathrm{O}<{\mathrm{NCS}}^{-}<\mathrm{N}{\mathrm{H}}_3<\mathrm{en}<{\mathrm{N}{\mathrm{O}}_2}^{-}<\mathrm{phen}<{\mathrm{CN}}^{-}<\mathrm{CO}.$$

A complementary spectrochemical series ranks metal ions: ${\mathrm{Mn}}^{2+}<{\mathrm{Ni}}^{2+}<{\mathrm{Co}}^{2+}<{\mathrm{Fe}}^{2+}<{\mathrm{V}}^{2+}<{\mathrm{Fe}}^{3+}<{\mathrm{Cr}}^{3+}<{\mathrm{V}}^{3+}<{\mathrm{Co}}^{3+}<{\mathrm{Mn}}^{4+}<{\mathrm{Mo}}^{3+}<{\mathrm{Rh}}^{3+}<{\mathrm{Ir}}^{3+}<{\mathrm{Pt}}^{4+}$. Higher oxidation states and heavier metals give larger ${\Delta }_o$ at fixed ligand.

The empirical ordering is explained at the master-tier level by going beyond the point-charge picture to ligand-field theory and identifying $\sigma$-donor, $\pi$-donor, and $\pi$-acceptor contributions.

High-spin vs low-spin. For a $d^n$ configuration with $4\le n\le 7$, two filling patterns compete:

Configuration	High-spin filling	Low-spin filling
$d^4$	$t_{2g}^3{e}_g^1$ (4 unpaired)	$t_{2g}^4{e}_g^0$ (2 unpaired)
$d^5$	$t_{2g}^3{e}_g^2$ (5 unpaired)	$t_{2g}^5{e}_g^0$ (1 unpaired)
$d^6$	$t_{2g}^4{e}_g^2$ (4 unpaired)	$t_{2g}^6{e}_g^0$ (0 unpaired)
$d^7$	$t_{2g}^5{e}_g^2$ (3 unpaired)	$t_{2g}^6{e}_g^1$ (1 unpaired)

The crossover is governed by comparing ${\Delta }_o$ to the pairing energy $P$, the energy cost of placing two electrons in the same spatial orbital (electron-electron Coulomb repulsion plus the loss of exchange stabilisation). The rule is:

• ${\Delta }_o>P$ → low-spin (electrons prefer to pair in $t_{2g}$ rather than occupy $e_g$),
• ${\Delta }_o<P$ → high-spin (electrons prefer to occupy $e_g$ rather than pair).

The crossover region is narrow; many complexes sit close to it and can be tuned between spin states by small changes in temperature, pressure, or ligand environment (the spin-crossover phenomenon).

Magnetic moment. The number of unpaired electrons $n$ determines the spin-only effective magnetic moment

$${\mu }_{s.o.}=g_s\sqrt{S(S+1)}{\mu }_B\approx \sqrt{n(n+2)}{\mu }_B,$$

with $S=n/2$, $g_s\approx 2$, and ${\mu }_B$ the Bohr magneton. Spin-only is a good first approximation for first-row TM complexes 12.01.02 pending whenever orbital angular momentum is quenched by the ligand field; deviations grow for heavier elements where spin-orbit coupling competes with the ligand field (master tier).

Counterexamples to common slips

• Tetrahedral geometry inverts the splitting. For a tetrahedral ${\mathrm{T}}_{\mathrm{d}}$ complex (four ligands at alternate vertices of a cube), the $e$ (= analogue of $E_g$, no $g$ subscript because ${\mathrm{T}}_{\mathrm{d}}$ has no inversion centre) orbitals sit below the $t_2$ (= analogue of $T_{2g}$), and ${\Delta }_t\approx \frac{4}{9}{\Delta }_o$. The same group-theoretic reasoning applies with the ${\mathrm{T}}_{\mathrm{d}}$ character table substituted; the inversion of order follows from which orbitals now point at the ligands.

• The barycentre rule holds only in first-order perturbation. Configuration interaction between $d$-orbitals and ligand orbitals (the LFT-vs-CFT extension) modifies the levels asymmetrically, and the average can shift. The barycentre is an idealisation, useful but not exact.

• The point-charge model gets the sign right but the magnitude wrong. Computed ${\Delta }_o$ from electrostatic point charges underpredicts the observed values by roughly an order of magnitude. The correction comes from including covalent overlap between metal $d$ and ligand orbitals — i.e., from extending CFT to LFT. This does not mean CFT's predictions are wrong; the ordering (spectrochemical series, high-vs-low-spin pattern, splitting topology) is qualitatively correct, which is why CFT survives as a useful effective theory.

• $T_{2g}$ vs $T_{1g}$. In $O_h$ there are two distinct triplet irreps, $T_{1g}$ and $T_{2g}$, differing in their character under the $C_3$ axes through the cube diagonals. The d-orbital basis spans $T_{2g}$, not $T_{1g}$; $T_{1g}$ would be carried by, e.g., the three Cartesian components of an axial vector. The subscript is load-bearing — a common slip is to write $T_{1g}$ where $T_{2g}$ is meant.

Key theorem with proof [Intermediate+]

Theorem (Crystal-field splitting in $O_h$). Let $V_d$ be the five-dimensional real vector space of $d$-orbital basis functions on a transition-metal centre at the origin, acted on by the octahedral group $O_h$. Then $V_d$ decomposes as a real representation of $O_h$ into

$$V_d\cong E_g\oplus T_{2g}$$

with $E_g$ the two-dimensional and $T_{2g}$ the three-dimensional irreducible representations of $O_h$. The two subspaces are precisely $\{d_{z^2},d_{x^2-y^2}\}$ and $\{d_{xy},d_{xz},d_{yz}\}$ respectively.

Proof. The proof uses the character-orthogonality machinery of 07.01.04 applied to the explicit character table of $O_h$.

The group $O_h\cong O\times \mathbb{Z}/2$ where $O$ is the rotation subgroup (order 24) and the $\mathbb{Z}/2$ factor is generated by inversion $i$. The ten conjugacy classes of $O_h$ are

$$\{E,8C_3,6C_2,6C_4,3C_2=C_4^2,i,6S_4,8S_6,3{\sigma }_h,6{\sigma }_d\}.$$

The character table for $O_h$ contains ten irreducible representations; the relevant rows are

Irrep	$E$	$8C_3$	$6C_2^{\prime }$	$6C_4$	$3C_2$	$i$	$6S_4$	$8S_6$	$3{\sigma }_h$	$6{\sigma }_d$
$A_{1g}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$E_g$	$2$	$-1$	$0$	$0$	$2$	$2$	$0$	$-1$	$2$	$0$
$T_{1g}$	$3$	$0$	$-1$	$1$	$-1$	$3$	$1$	$0$	$-1$	$-1$
$T_{2g}$	$3$	$0$	$1$	$-1$	$-1$	$3$	$-1$	$0$	$-1$	$1$

(The $u$-counterparts $A_{1u},E_u,T_{1u},T_{2u}$ carry the opposite sign under inversion and the four remaining conjugacy classes involving $i$, $S_4$, ${\sigma }_h$, ${\sigma }_d$.)

Step 1. Compute the character ${\chi }_d$ of $O_h$ acting on $V_d$.

The $d$-orbitals transform as the $\ell =2$ representation of $\mathrm{SO}(3)$. Restricted to $O_h\subset \mathrm{O}(3)$, the character of the rotation by angle $\theta$ in the $\ell =2$ representation is

$${\chi }^{(2)}(\theta )=\frac{\sin (5\theta /2)}{\sin (\theta /2)}=1+2\cos \theta +2\cos 2\theta .$$

Evaluated at the rotation angles of the conjugacy classes of $O$ (and combined with inversion-parity $+1$ for $d$-orbitals, since they are even functions of position):

• $E$: $\theta =0$, ${\chi }_d(E)=5$.
• $8C_3$: $\theta =2\pi /3$, $\cos \theta =-\frac{1}{2}$, $\cos 2\theta =-\frac{1}{2}$, ${\chi }_d(C_3)=1-1-1=-1$.
• $6C_2^{\prime }$ (rotations about face-diagonals of the octahedron's circumscribed cube): $\theta =\pi$, $\cos \theta =-1$, $\cos 2\theta =1$, ${\chi }_d(C_2^{\prime })=1-2+2=1$.
• $6C_4$: $\theta =\pi /2$, $\cos \theta =0$, $\cos 2\theta =-1$, ${\chi }_d(C_4)=1+0-2=-1$.
• $3C_2$ (about the four-fold axes squared): $\theta =\pi$, same as $C_2^{\prime }$: ${\chi }_d(C_2)=1$.
• $i$: ${\chi }_d(i)=+{\chi }_d(E)=5$ (d-orbitals are gerade).
• $6S_4$, $8S_6$, $3{\sigma }_h$, $6{\sigma }_d$: improper-rotation characters are obtained from rotation-by-corresponding-$\theta$ times $+1$ for d-orbitals (parity factor for $\ell =2$ under inversion is $+1$):
• $6S_4=i\cdot C_4$: ${\chi }_d(S_4)=-1$;
• $8S_6=i\cdot C_3$: ${\chi }_d(S_6)=-1$;
• $3{\sigma }_h=i\cdot C_2$: ${\chi }_d({\sigma }_h)=1$;
• $6{\sigma }_d=i\cdot C_2^{\prime }$: ${\chi }_d({\sigma }_d)=1$.

So ${\chi }_d=(5,-1,1,-1,1,5,-1,-1,1,1)$ as a length-10 row of values, one per conjugacy class.

Step 2. Decompose using the orthogonality formula.

The multiplicity of irrep $\Gamma$ in $V_d$ is

$$n_{\Gamma }=\frac{1}{|O_h|}\sum _C|C|{\chi }_d(C)\overline{{\chi }_{\Gamma }(C)},$$

where the sum is over conjugacy classes $C$, $|C|$ is the class size, $|O_h|=48$, and the bar denotes complex conjugation (vacuous here, since all characters are real).

For $\Gamma =E_g$, ${\chi }_{E_g}=(2,-1,0,0,2,2,0,-1,2,0)$:

$$n_{E_g}=\frac{1}{48}[1\cdot 5\cdot 2+8\cdot (-1)(-1)+6\cdot 1\cdot 0+6\cdot (-1)\cdot 0+3\cdot 1\cdot 2$$ $$+1\cdot 5\cdot 2+6\cdot (-1)\cdot 0+8\cdot (-1)(-1)+3\cdot 1\cdot 2+6\cdot 1\cdot 0]$$ $$=\frac{1}{48}[10+8+0+0+6+10+0+8+6+0]=\frac{48}{48}=1.$$

For $\Gamma =T_{2g}$, ${\chi }_{T_{2g}}=(3,0,1,-1,-1,3,-1,0,-1,1)$:

$$n_{T_{2g}}=\frac{1}{48}[1\cdot 5\cdot 3+8\cdot (-1)\cdot 0+6\cdot 1\cdot 1+6\cdot (-1)(-1)+3\cdot 1\cdot (-1)$$ $$+1\cdot 5\cdot 3+6\cdot (-1)(-1)+8\cdot (-1)\cdot 0+3\cdot 1\cdot (-1)+6\cdot 1\cdot 1]$$ $$=\frac{1}{48}[15+0+6+6-3+15+6+0-3+6]=\frac{48}{48}=1.$$

For every other irrep $\Gamma$ in the $g$-block ($A_{1g},A_{2g},T_{1g}$) the analogous sum yields $0$; for the $u$-block all sums vanish identically because ${\chi }_d(i)=+5$ while ${\chi }_{\Gamma }(i)=-\dim \Gamma$ for any $u$-irrep, which mismatches the necessary positive contribution.

So $V_d=E_g\oplus T_{2g}$, with dimensions $2+3=5$, accounting for all of $V_d$.

Step 3. Identify the basis vectors.

The projection operator onto $E_g$ is

$$P_{E_g}=\frac{\dim E_g}{|O_h|}\sum _{g\in O_h}{\chi }_{E_g}(g{)}^{\ast }\rho (g)=\frac{2}{48}\sum _g{\chi }_{E_g}(g)\rho (g).$$

Acting $P_{E_g}$ on $d_{z^2}$ returns $d_{z^2}$ (a routine check using the explicit $O_h$ action on quadratics); acting it on $d_{xy}$ returns $0$. So $\{d_{z^2},d_{x^2-y^2}\}$ span $E_g$ and $\{d_{xy},d_{xz},d_{yz}\}$ span $T_{2g}$.

The sign of the splitting — $E_g$ above $T_{2g}$, not the reverse — is not determined by symmetry alone. It comes from the explicit perturbative matrix element of the ligand-field potential within each irrep block, computed in the master tier below. ∎

Corollary (Barycentre). Let $\bar{E}$ be the average $d$-orbital energy in the free ion (no ligand field). To first order in the ligand-field perturbation, the energies satisfy $E(E_g)=\bar{E}+\frac{3}{5}{\Delta }_o$ and $E(T_{2g})=\bar{E}-\frac{2}{5}{\Delta }_o$. Consequently the average $\frac{1}{5}(2E(E_g)+3E(T_{2g}))=\bar{E}$ is unchanged: the centre of mass of the five orbitals is preserved.

The corollary follows from the tracelessness of the traceless part of the ligand-field perturbation (the trace shifts only the spherically symmetric average and does not enter ${\Delta }_o$). ^{[Cotton-Wilkinson Ch. 16; see also TODO_REF Housecroft-Sharpe Ch. 21.]}

Worked example: rationalising the colour of $[\mathrm{Ti}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{3+}$

Titanium(III) has the electron configuration $[\mathrm{Ar}]3d^1$ — a single d-electron. In the hexaaquatitanium(III) ion, the d-electron occupies the lower $t_{2g}$ set; the single visible-light transition is $t_{2g}^1\to e_g^1$, depositing one quantum of ${\Delta }_o$ into the electronic excitation.

Experimentally, $[\mathrm{Ti}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{3+}$ shows a single absorption band centred at ${\lambda }_{\max }\approx 500$ nm (in the green). Convert: ${\Delta }_o=hc/\lambda =(1240\text{eV nm})/(500\text{nm})\approx 2.5$ eV, or about $20,000$ cm${}^{-1}$ in wavenumber units typical of the chem literature.

The complex absorbs green and transmits the complementary colours (red + violet) — the observed pale purple. With one d-electron there is no high-/low-spin issue; the magnetism is straightforwardly one unpaired electron, ${\mu }_{s.o.}\approx 1.73{\mu }_B$.

A second prediction the framework makes: the band is slightly broad and slightly asymmetric. The asymmetry — a low-energy shoulder — is the signature of a Jahn-Teller distortion in the excited $e_g^1$ state (master tier), since an unevenly occupied $E_g$ pair is symmetry-unstable and lifts the orbital degeneracy by a small Q-mode distortion.

Exercises [Intermediate+]

Ligand field theory and the covalent extension [Master]

Crystal-field theory treats the ligands as point charges (or point dipoles) and computes the splitting from pure electrostatics. The qualitative ordering — sign of ${\Delta }_o$, the spectrochemical series direction, high-vs-low-spin patterns — comes out correctly. The quantitative magnitudes do not: a serious point-charge computation underestimates ${\Delta }_o$ for typical complexes by roughly an order of magnitude. The deficit comes from covalency: the metal $d$-orbitals overlap appreciably with ligand orbitals, and the true molecular orbitals are admixtures, not pure metal d.

Ligand field theory (LFT) is the MO-theoretic extension of CFT. Build the symmetry-adapted linear combinations (SALCs) of ligand orbitals using the same character-table machinery 07.01.04; classify them by their irrep content under $O_h$; combine each SALC with the metal $d$-orbitals of the same irrep symmetry. The five $d$-orbitals span $E_g\oplus T_{2g}$; the six ligand $\sigma$-donor orbitals span $A_{1g}\oplus E_g\oplus T_{1u}$ in $O_h$. The $\sigma$-bonding (and the corresponding ${\sigma }^{\ast }$ antibonding) MOs form by combining metal $E_g$ ($d_{z^2},d_{x^2-y^2}$) with ligand $E_g$ SALCs; the ligand $A_{1g}$ and $T_{1u}$ SALCs combine with metal $s$ and $p$ orbitals, which sit higher than $d$ in energy.

The metal $T_{2g}$ orbitals ($d_{xy},d_{xz},d_{yz}$) have no $\sigma$-donor partner among the ligand $\sigma$ basis — they are non-bonding in $\sigma$-only LFT. But the ligand $\pi$ basis (orbitals perpendicular to the M–L axis) spans $T_{1g}\oplus T_{2g}\oplus T_{1u}\oplus T_{2u}$ in $O_h$; the $T_{2g}$ portion mixes with metal $d_{xy},d_{xz},d_{yz}$.

Three ligand classes follow:

• $\sigma$-donors only (saturated amines like NH${}_3$): metal $T_{2g}$ stays non-bonding, ${\sigma }^{\ast }$-antibonding $E_g$ sits high. Splitting ${\Delta }_o=E(e_g^{\ast })-E(t_{2g}^{\text{nb}})$ is moderate.
• $\sigma$-donor + $\pi$-donor (halides, oxide, sulfide): ligand $\pi$ orbitals are occupied and lie below metal $T_{2g}$; the resulting $\pi$-bonding pushes metal $T_{2g}$ up in energy, reducing ${\Delta }_o$. Halides therefore sit low in the spectrochemical series.
• $\sigma$-donor + $\pi$-acceptor (CO, CN${}^{-}$, phen, NO${}_2^{-}$): ligand ${\pi }^{\ast }$ orbitals are empty and lie above metal $T_{2g}$; the resulting back-bonding pushes metal $T_{2g}$ down, increasing ${\Delta }_o$. CO and CN${}^{-}$ therefore sit at the top of the spectrochemical series and are the strongest-field ligands known.

The spectrochemical series, qualitatively just an empirical ordering in CFT, is explained by LFT: it tracks the $\pi$-donor → $\pi$-non-interacting → $\pi$-acceptor progression of the ligand $\pi$ basis.

A quantitative refinement is the angular overlap model (AOM) of Schäffer and Jørgensen ^{[Bersuker Ch. 4]}: parameterise each metal-ligand interaction by a $\sigma$-overlap parameter $e_{\sigma }$ and a $\pi$-overlap parameter $e_{\pi }$ (one per metal-ligand pair); the molecular-orbital energies are linear combinations of these parameters with coefficients fixed by the geometry. AOM reduces ${\Delta }_o$ to ${\Delta }_o=3e_{\sigma }-4e_{\pi }$ for an octahedral $\sigma +\pi$ complex; $e_{\sigma }>0$ always, $e_{\pi }>0$ for donors and $<0$ (formally) for acceptors. AOM transfers cleanly across geometries (square planar, trigonal bipyramidal) and is the workhorse method for quantitative ligand-field analysis short of full DFT.

Term symbols, Russell-Saunders coupling, and Tanabe-Sugano diagrams [Master]

Beyond single-electron orbital filling, the inter-electron Coulomb repulsion within the partly-filled d-shell organises configurations into term symbols ${}^{2S+1}{L}_J$. For a free ion with $d^n$ configuration:

• $L$ is the total orbital angular momentum (with $L=0,1,2,3,4\leftrightarrow S,P,D,F,G$ in spectroscopic notation),
• $S$ is the total spin,
• $J$ is the total angular momentum after spin-orbit coupling.

The lowest term of $d^n$ is selected by Hund's rules: maximise $S$, then maximise $L$, then ($n<5$) minimise $J$ or ($n>5$) maximise $J$. For first-row TM ions (where spin-orbit is small relative to inter-electron repulsion and the ligand field), the Russell-Saunders coupling scheme $L\otimes S\to J$ is the right organisational framework.

In an octahedral ligand field, each free-ion term ${}^{2S+1}L$ splits according to the reduction of the $\ell =L$ representation of $\mathrm{SO}(3)$ under $O_h$:

• $L=0$ ($S$ term): one $A_{1g}$ component (no splitting).
• $L=1$ ($P$ term): one $T_{1g}$ component (no splitting).
• $L=2$ ($D$ term): $E_g\oplus T_{2g}$ (splits — same pattern as the orbitals themselves).
• $L=3$ ($F$ term): $A_{2g}\oplus T_{1g}\oplus T_{2g}$ (splits into three components; see Exercise 9).
• $L=4$ ($G$ term): $A_{1g}\oplus E_g\oplus T_{1g}\oplus T_{2g}$ (four components).

A Tanabe-Sugano diagram for a $d^n$ configuration plots the energies of all the resulting sub-terms (with their octahedral symmetry labels) as a function of the dimensionless ratio ${\Delta }_o/B$, where $B$ is the Racah parameter measuring inter-electron repulsion. The ground term is chosen as the energy zero. At ${\Delta }_o/B=0$ (free ion limit), only the free-ion terms appear; as ${\Delta }_o/B$ grows, the field splits each term, and term crossings occur where the high-spin/low-spin transition happens (typically around ${\Delta }_o/B\sim 25$ for $d^5,d^6,d^7$).

The original Tanabe-Sugano diagrams ^{[Tanabe-Sugano 1954]} cover the seven configurations $d^2$ through $d^8$ and remain the standard tool for assigning electronic spectra of first-row transition-metal complexes.

Spin-orbit coupling. For first-row TM ions, the spin-orbit coupling constant ${\zeta }_{3d}\approx 100\text{–}800$ cm${}^{-1}$ is smaller than the ligand-field splitting (${\Delta }_o\approx 10,000\text{–}30,000$ cm${}^{-1}$) and the inter-electron-repulsion scale ($B\approx 1000\text{–}1100$ cm${}^{-1}$ for first-row). Russell-Saunders coupling is valid; spin-orbit acts as a perturbation lifting fine-structure degeneracies. For 4d and 5d series ions, ${\zeta }_{4d,5d}$ is an order of magnitude larger and competes with the ligand field — the j-j coupling scheme or intermediate-coupling schemes are needed. Heavy-element complexes display strong spin-orbit-induced mixing of states; the orbital angular momentum is no longer "quenched" and the spin-only magnetic-moment formula fails.

The bridge to physics §12 spin angular momentum 12.01.02 pending is the underlying $\mathrm{SU}(2)$ Lie-algebra structure: spin couplings $S_1\otimes S_2\to S_{\mathrm{total}}$ in the term-symbol formalism are Clebsch-Gordan decompositions of $\mathrm{SU}(2)$ representations, the same construction used in atomic and nuclear physics.

Distortions, square-planar, and back-bonding [Master]

The Jahn-Teller theorem [Jahn-Teller 1937] generalises the $[\mathrm{Cu}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{2+}$ analysis: any orbitally degenerate electronic ground state in a non-linear molecule drives a symmetry-lowering distortion along a normal mode whose symmetric square contains the degenerate irrep. For $E_g$ ground terms ($d^4$ HS, $d^7$ LS, $d^9$ in $O_h$) the distortion is strong; for $T_{2g}$ ground terms ($d^1$, $d^2$, $d^4$ LS, $d^5$ LS, $d^6$ LS — when the partial-filling fraction in $T_{2g}$ creates orbital degeneracy) the distortion is weaker because $T_{2g}$ orbitals point between the ligands.

The strong-J-T limit of $d^8$ low-spin (Ni(II), Pd(II), Pt(II)) is the square-planar geometry: two axial ligands are removed entirely, leaving a four-coordinate square. The $d$-orbital splitting in $D_{4h}$ becomes a four-level pattern $a_{1g}(d_{z^2})<e_g(d_{xz},d_{yz})<b_{2g}(d_{xy})<b_{1g}(d_{x^2-y^2})$, with $b_{1g}$ separated by a large gap. The 8 d-electrons fill the lower four levels, giving a diamagnetic ground state with a large HOMO-LUMO gap — explaining the structural preference of $d^8$ low-spin for square planar and the family of catalytically important square-planar complexes (Vaska's complex, Wilkinson's catalyst, the Pd cross-coupling cycle).

Back-bonding in $\pi$-acceptor ligands is the mechanism that makes CO and CN${}^{-}$ the strongest-field ligands in the spectrochemical series. In $[\mathrm{Fe}(\mathrm{CN}{)}_6{]}^{4-}$, the filled metal $T_{2g}$ donates electron density into the empty ${\pi }^{\ast }$ orbitals of CN${}^{-}$; this M→L $\pi$-donation strengthens the M-C bond (synergic bonding with the C→M $\sigma$-donation) and lowers the metal $T_{2g}$ energy, increasing ${\Delta }_o$. Carbonyl complexes show the characteristic IR signature of back-bonding: the C-O stretch frequency drops by $50\text{–}200$ cm${}^{-1}$ on coordination (free CO at $2143$ cm${}^{-1}$; $\mathrm{Cr}(\mathrm{CO}{)}_6$ at $\approx 2000$ cm${}^{-1}$), reflecting the loss of C-O bond order as the ${\pi }^{\ast }$ becomes partially occupied. The 18-electron rule for organometallics — that stable complexes prefer 18 valence electrons (filled $T_{2g}$ + filled ${\sigma }^{\ast }$-bonding $E_g$ + filled $A_{1g}$ + filled $T_{1u}$) — is the closed-shell criterion in this MO framework.

In a $d^6$ low-spin $[\mathrm{Co}(\mathrm{N}{\mathrm{H}}_3{)}_6{]}^{3+}$ complex, by contrast, NH${}_3$ has no $\pi$-orbitals; the M-L bond is $\sigma$-only, metal $T_{2g}$ stays roughly non-bonding, and ${\Delta }_o$ is set entirely by the ${\sigma }^{\ast }$ position of $E_g$. The combination Co(III) + strong $\sigma$-donor NH${}_3$ still gives ${\Delta }_o>P$, hence low-spin and diamagnetic — but the ${\Delta }_o$ value sits below the cyanide complex's, consistent with NH${}_3$ lying below CN${}^{-}$ in the spectrochemical series.

Computational extensions and where CFT/LFT sit in the hierarchy [Master]

Density functional theory (DFT) calculations on transition-metal complexes can reproduce ${\Delta }_o$ values, spin-state energies, and Jahn-Teller distortions to chemical accuracy ($\sim 0.1$ eV) for many systems. The relationship to CFT/LFT is one of refinement, not replacement: the qualitative organisation provided by the $E_g\oplus T_{2g}$ decomposition, the spectrochemical series, and the high-spin/low-spin framework remains the conceptual scaffolding within which DFT results are interpreted. A DFT calculation that reports a ${\Delta }_o$ value is implicitly using the CFT/LFT vocabulary; without that vocabulary the energies would lack interpretive structure.

Multiconfigurational methods (CASSCF, CASPT2, NEVPT2, density-matrix renormalisation group adaptations) become necessary for systems with near-degenerate states — spin-crossover regimes, intermediate-field configurations near term crossings, heavy elements with strong spin-orbit. The state-of-the-art for first-row TM electronic structure is CASSCF/NEVPT2 with active spaces that include the metal d-orbitals plus relevant ligand $\sigma /\pi$ orbitals. For 4d/5d series, relativistic two-component or four-component methods become essential to capture the strong spin-orbit splitting; in the heaviest elements (5f actinides), the spin-orbit splitting can rival or exceed the ligand-field splitting and the L-S coupling scheme breaks down.

The relationship to physics-side many-body theory is direct: a lattice of TM ions with on-site $E_g/T_{2g}$ splitting and inter-site hopping is exactly the multi-orbital Hubbard model on a regular lattice. The CFT-derived single-ion picture is the non-interacting (in the inter-site sense) starting point of the lattice problem; adding kinetic exchange ($J=4t^2/U$ in the strong-coupling limit) generates the Heisenberg / Kugel-Khomskii Hamiltonians whose study is condensed-matter magnetism 12.17.01 pending.

Lean formalization [Intermediate+]

Mathlib does not yet contain a formalisation of $O_h$-symmetry chemistry. The path to one builds on the existing finite-group representation-theory layer:

• Mathlib.RepresentationTheory.Basic, Mathlib.RepresentationTheory.FdRep: generic finite-group representations on a field.
• Mathlib.RepresentationTheory.Character: characters as class functions, orthogonality (over $\mathbb{C}$).
• Mathlib.GroupTheory.SpecificGroups: named finite groups including some symmetric and dihedral groups.

What's absent: $O_h$ as a named Mathlib group with its conjugacy structure pre-computed; the character table of $O_h$ as a formal object; the d-orbital basis as the spin-2 representation of $\mathrm{SO}(3)$ restricted to $O_h$; the $V_d\cong E_g\oplus T_{2g}$ decomposition theorem of this unit. See lean_mathlib_gap in frontmatter for the contribution roadmap. lean_status: none; reviewer-attested.

Connections [Master]

• Character of a representation 07.01.03 is the central tool: the character of the d-orbital representation of $O_h$ is computed from the spherical-harmonic character formula and projected against the $O_h$ irreducible characters. Without character theory the splitting derivation would have to proceed by explicit basis-by-basis diagonalisation.

• Character orthogonality 07.01.04 is what makes the multiplicity formula $n_{\Gamma }=\frac{1}{|G|}{\sum }_C|C|{\chi }_V(C)\overline{{\chi }_{\Gamma }(C)}$ work; that formula is used directly in the proof of the central theorem of this unit.

• Stern-Gerlach and spin-1/2 12.01.02 pending establishes spin angular momentum and the $\mathrm{SU}(2)$ Lie-algebra structure; the magnetic-moment computation in CFT borrows the spin-only formula $\mu =g_s\sqrt{S(S+1)}{\mu }_B$ from there. At master tier, the spin-orbit coupling discussion uses the full $L\otimes S$ Clebsch-Gordan decomposition of $\mathrm{SU}(2)$ developed in the physics treatment.

• Atomic orbitals from H-atom QM 14.04.01 pending (pending) supplies the d-orbital basis itself — the five real $d_{z^2},d_{x^2-y^2},d_{xy},d_{xz},d_{yz}$ orbitals come from solving the hydrogen-atom Schrödinger equation, classifying solutions by $(n,\ell ,m_{\ell })$, and taking real linear combinations of $m_{\ell }=\pm 2,\pm 1,0$ states.

• MO theory for homonuclear diatomics 14.05.02 pending (Wave 1 chem seed) supplies the molecular-orbital framework that LFT extends to coordination compounds; the LFT picture treats the metal-ligand bond as a multi-centre MO problem with the metal d-orbitals as the strongly-correlated active space.

• Coordination chemistry [16.04] (pending) takes the splitting pattern of this unit as input and builds the structural / kinetic / thermodynamic theory of coordination compounds on top.

• Organometallic 18-electron rule [16.05] (pending) is the closed-shell criterion in the LFT MO framework developed at master tier here.

• Metalloenzymes 17.04.01 (pending bio unit, target of an outbound hook): the redox potentials of hemoglobin, cytochromes, and electron-transfer copper proteins follow from the d-orbital splittings of their TM centres.

• Transition-metal magnetism in solids 12.17.01 pending (pending physics unit, target of an outbound hook): the lattice version of the single-ion CFT picture, with inter-site hopping turned on.

• Philosophy of chemistry: reduction 20.04.01 pending (pending phil unit, target of an outbound hook): CFT as a test case for the chem-reduces-to-physics question.

Historical & philosophical context [Master]

Crystal-field theory was developed by Hans Bethe in 1929 ^{[Bethe 1929]} to explain the splittings of atomic spectral terms in ionic crystals, using the group-theoretic methods that had been worked out by Wigner and others through the 1920s. Bethe's paper introduced the systematic application of point-group character tables to electronic-structure problems — what is now standard physical-chemistry practice was at the time a genuinely new methodological move.

Van Vleck extended the theory through the 1930s to magnetic susceptibilities ^{[Van Vleck 1932]}, establishing the connection between crystal-field splittings and the observed paramagnetism of d- and f-block salts. The next major advance was Tanabe and Sugano's 1954 paper ^{[Tanabe-Sugano 1954]}, which produced the systematic energy diagrams that now bear their name and made electronic-spectroscopy assignments tractable.

Ligand field theory, the MO-theoretic extension covering covalency, was developed in parallel by Griffith and Orgel in the late 1950s and consolidated by Ballhausen's 1962 monograph ^{[Ballhausen 1962]} and Griffith's 1961 reference ^{[Griffith 1961]}. The angular overlap model emerged in the 1960s and 70s from Schäffer and Jørgensen's collaboration in Copenhagen.

The Jahn-Teller theorem appeared in Jahn and Teller's 1937 paper, motivated by Teller's earlier interest in vibrational coupling in degenerate electronic states. The full group-theoretic classification of which configurations are Jahn-Teller-active — and which mode of distortion they select — is in Bersuker's monograph ^{[Bersuker 2010]}.

Cotton and Wilkinson's Advanced Inorganic Chemistry (first edition 1962; sixth edition 1999) ^{[Cotton-Wilkinson]} consolidated the CFT/LFT framework into the standard graduate textbook treatment that has held for half a century. The first-year-graduate inorganic curriculum at most universities still organises around its chapter sequence.

Bibliography [Master]

Primary literature (most not yet in reference/; see pending: true flags in frontmatter):

• Bethe, H. "Termaufspaltung in Kristallen." Annalen der Physik 3 (1929), 133–208. [Need to source — originating CFT paper.]

• Van Vleck, J. H. The Theory of Electric and Magnetic Susceptibilities. Oxford: Clarendon, 1932.

• Jahn, H. A. & Teller, E. "Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy." Proc. Roy. Soc. A 161 (1937), 220–235.

• Tanabe, Y. & Sugano, S. "On the absorption spectra of complex ions I, II." J. Phys. Soc. Japan 9 (1954), 753–766, 766–779.

• Mulliken, R. S. "Electronic structures of polyatomic molecules and valence. IV. Electronic states, quantum theory of the double bond." Phys. Rev. 43 (1933), 279–302. (Mulliken-notation origin.)

• Racah, G. "Theory of complex spectra II." Phys. Rev. 62 (1942), 438–462. (Racah parameters $B$, $C$.)

• Griffith, J. S. The Theory of Transition-Metal Ions. Cambridge University Press, 1961.

• Ballhausen, C. J. Introduction to Ligand Field Theory. McGraw-Hill, 1962.

• Schäffer, C. E. & Jørgensen, C. K. "The angular overlap model: an attempt to revive the ligand field approaches." Mol. Phys. 9 (1965), 401–412.

• Cotton, F. A. & Wilkinson, G. Advanced Inorganic Chemistry, 6th ed. Wiley-Interscience, 1999.

• Crabtree, R. H. The Organometallic Chemistry of the Transition Metals, 7th ed. Wiley, 2019.

• Bersuker, I. B. Electronic Structure and Properties of Transition Metal Compounds, 2nd ed. Wiley, 2010.

• Housecroft, C. E. & Sharpe, A. G. Inorganic Chemistry, 5th ed. Pearson, 2018.

• Miessler, G. L., Fischer, P. J. & Tarr, D. A. Inorganic Chemistry, 5th ed. Pearson, 2014.

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Wave 1 chem seed unit, agent-drafted 2026-05-18 per docs/plans/CHEMISTRY_PLAN.md §6 — third of three seed chem units (after 14.05.02 MO theory and 15.04.02 SN1/SN2). Status remains draft pending Tyler's review and external inorganic-chemistry reviewer per CHEMISTRY_PLAN §7. The math §07 cross-cite into character-table machinery is the load-bearing dependency this unit was designed to validate; the physics §12 cross-cite into spin-orbit is structural at master tier; chem §14.04 (pending) is the only intra-chem prereq.