A symmetric shape that won't sit still
The previous guides in this rung handed you the central picture: in an octahedral complex the five d orbitals split into a lower set of three — the t2g, pointing into the gaps between the ligands — and an upper set of two — the eg, aimed straight down the metal-ligand axes and so shoved up by the ligand charge. The energy gap between them is delta-o, and filling the cheap t2g before the expensive eg buys you a discount called the crystal field stabilization energy (CFSE). That whole machinery assumed a tidy, perfectly symmetric octahedron with all six metal-ligand bonds the same length. This guide is about what happens when the electrons themselves refuse to allow that tidiness.
Here is the trigger. Take an octahedral copper(II) complex, a d9 ion such as [Cu(H2O)6]2+. Fill the orbitals: the three t2g hold six electrons, full; the two eg must share the remaining three. But two orbitals cannot hold three electrons evenly. One eg orbital — say dz2 — gets two electrons, the other — dx2-y2 — gets just one, or you make the opposite choice. Either way the two eg orbitals, which point along different axes, are now unequally filled. The electron cloud around the copper is lopsided: denser along one direction than another. A ground state like this, where the electrons could be arranged in more than one equally-good way, is called orbitally degenerate, and it is inherently unstable.
The Jahn-Teller theorem: degeneracy buys distortion
In 1937 Hermann Jahn and Edward Teller proved a strikingly general theorem: any non-linear molecule sitting in an orbitally degenerate electronic ground state cannot be stable in its symmetric shape. It will spontaneously distort — lower its symmetry — until the degeneracy is lifted and one arrangement becomes genuinely lower in energy than the rest. The [[jahn-teller-distortion|Jahn-Teller effect]] is not a special force; it is the molecule taking an energy discount that symmetry was forbidding it to take. The theorem tells you *that* a distortion must happen, but, honestly, not its size or exact direction — those depend on which orbitals are unevenly filled.
Back to copper(II). Suppose the two electrons sit in dz2 (the orbital aimed up and down the z-axis) and only one electron sits in dx2-y2 (aimed along x and y). There is then more electron density pointing at the two ligands on the z-axis than at the four in the xy-plane. Those two axial ligands feel extra repulsion from the crowded dz2 cloud, so they ease away — the complex stretches along z. As the axial bonds lengthen, the orbitals with a z-component (dz2, and the t2g pair dxz, dyz) feel less repulsion and drop in energy; the purely in-plane orbitals (dx2-y2, dxy) rise. The doubly-occupied dz2 is now lower than the singly-occupied dx2-y2, and the molecule has paid less total energy. The degeneracy is gone, and the distortion has paid for itself.
perfect octahedron tetragonal elongation (z-out)
------------------ -----------------------------
___ ___ ___ dx2-y2 (1 e-)
eg | | eg splits --> /
(raised by 0.6 do) \___ dz2 (2 e-)
(this pair drops)
___ ___ ___ ___ ___ dxy
t2g | | t2g splits --> \___ ___ dxz,dyz
(lowered by 0.4 do) (this pair drops)
net: occupied orbitals sink, empty/half-occupied rise -> energy savedWhich complexes distort — and how much
The size of the effect depends entirely on *which* set is unevenly filled. Uneven occupation of the eg orbitals causes a strong, easily measured distortion, because eg orbitals point straight at the ligands and so couple powerfully to bond lengths. The classic strong cases are d9 (copper(II)), and high-spin d4 (chromium(II), manganese(III)) — both have an eg set holding an odd number of electrons. Uneven occupation of only the t2g orbitals causes a weak distortion you usually cannot spot, because those orbitals point into the gaps and barely touch the bonds.
Run the checklist. A configuration is Jahn-Teller-degenerate when a sub-shell is unevenly filled: count 0, 1, or 2 electrons in the eg pair as uneven unless it is empty (0) or exactly full (2 means one each, even). So for octahedral high-spin ions, t2g^3 eg^1 (d4) and t2g^6 eg^3 (d9) are strongly distorting; t2g^6 eg^4 (d10) and t2g^3 eg^2 (high-spin d5) are symmetric, because every orbital in the open set is filled the same. Whether you are in the high-spin or low-spin regime changes the count: low-spin d7, for instance, reaches t2g^6 eg^1 and so also distorts. The honest punchline is that Jahn-Teller is a consequence, not a separate rule — it falls straight out of the d-orbital splitting you already know.
The double hump: CFSE written across the periodic table
Jahn-Teller is the most vivid local fingerprint of d-electron energetics, but CFSE leaves a broader signature across the whole first transition row. If d-electrons were energetically blind to their surroundings, several properties would change smoothly from calcium across to zinc, tracking only the rising nuclear charge. Instead they trace a gently rising background with two dips — a double-humped curve. That extra wiggle, the deviation from the smooth line, is CFSE doing its quiet bookkeeping.
Take the ionic radius of the divalent ions, M2+, all in high-spin octahedral surroundings. The smooth expectation is a steady shrink from Ca2+ to Zn2+ as the growing nuclear charge pulls the shell in tighter. The measured radii instead fall, bulge back up, fall again, and bulge once more — a W-shaped, double-humped trace. The reason is *where* the electrons sit. From d0 to d3 electrons go into t2g, which point away from the ligands and shield them poorly, so the bonds tighten faster than charge alone predicts and the radius dips below the smooth line. At d4 and d5 the eg orbitals — pointing right at the ligands — begin to fill, pushing the ligands back out, so the radius pops up toward the smooth line. The cycle repeats over the second half of the row.
The same double hump shows up in two energies. Hydration enthalpy — the energy released when a gaseous M2+ ion is swallowed by water and dressed in six water ligands — gets *more* exothermic than the smooth line wherever CFSE is large (d3 and d8 sit at the peaks of extra stabilization) and falls back to the line at d0, d5, and d10, where CFSE is exactly zero. Lattice enthalpy of the simple halides and oxides traces the same shape, for the same reason: in the solid each metal ion sits in an octahedron of anions, so the same CFSE bonus stiffens the lattice. The trick chemists use is to *subtract* the CFSE, calculated from delta-o and the d-electron count, and check that the corrected points fall back onto a clean smooth curve. They do — which is strong, quantitative support for the whole crystal field picture.
Reading the trend, end to end
It is worth walking the row once, slowly, to see how a single ledger explains so many curves at once. Use high-spin octahedral M2+ ions and remember CFSE = (number in t2g) x 0.4 delta-o stabilization minus (number in eg) x 0.6 delta-o penalty.
- Start at the zero points. Ca2+ (d0), Mn2+ (high-spin d5, t2g^3 eg^2), and Zn2+ (d10) all have CFSE of exactly zero — for d5 the discount on three t2g electrons exactly cancels the penalty on two eg electrons. These three ions sit right on the smooth background line in every plot.
- Climb to the first peak. From d1 to d3 every electron lands in cheap t2g, so CFSE grows steadily and tops out at d3 (V2+, t2g^3) with 1.2 delta-o. Here radii dip most below the line, and hydration and lattice energies bulge most above it. This is the first hump.
- Fall into the valley. At d4 (Cr2+) and d5 (Mn2+, high-spin) the eg orbitals start filling, paying the 0.6 delta-o penalty and pushing CFSE back down to zero at d5. The radius pops back out, the energies sag back to the line. Note that Cr2+, a strong Jahn-Teller ion, also distorts here.
- Climb the second hump and finish. From d6 to d8 electrons again pour into t2g, rebuilding CFSE to a second peak at d8 (Ni2+, t2g^6 eg^2) with 1.2 delta-o, before d9 (Cu2+, distorting) and d10 (Zn2+, zero again) bring the curve home. Two t2g-filling climbs, two eg-filling valleys: that is the whole double hump.
Honest limits and what comes next
Keep the model honest. The double-hump analysis assumed high-spin octahedral ions throughout; switch to low-spin and the t2g and eg occupancies change, so the peaks and zeros move — the same ledger, different entries. The point-charge picture of crystal field theory ignores covalency entirely, yet it reproduces these thermodynamic trends remarkably well, which tells you the electrostatic skeleton is sound even though the flesh of real bonding is partly covalent. And remember a tetrahedral field would invert the whole scheme — there the lower set holds the e orbitals and the upper set the t2, with a much smaller splitting — so a tetrahedral row would hump in different places.
Step back and notice the unity. The Jahn-Teller distortion of a single copper complex and the gentle double hump running across an entire row of the periodic table are the *same* phenomenon at two scales. Both are the d-electrons declining to be energetically indifferent to where the ligands sit; both are the molecule or the ion cashing in the CFSE that an uneven, lopsided occupation makes available. Once you see d-electrons as active accountants rather than passive passengers, a surprising amount of transition-metal behaviour stops being a list to memorize and becomes something you can predict.
From here the rung turns to the most visible payoff of all. CFSE explained shapes and energies; the next move is colour and light. We will see why a d-d transition across the delta-o gap absorbs exactly the photon whose energy matches the gap, why a complex then displays the *complementary* colour to the light it swallows, and why the size of delta-o — and so the colour — slides predictably as you change the ligand. The lopsided electron bookkeeping you have just met is about to start painting.