Copper is never quite octahedral

Copper(II) is d⁹. Put it in an octahedral field and the count is forced: six electrons fill the lower set of three, and three electrons are left for the upper pair. Two orbitals, three electrons, and no way to divide them evenly.

That is the condition for the Jahn–Teller theorem, and the theorem’s conclusion is not that such a complex is unusual. It is that such a complex cannot be octahedral at all.

The same argument works at two other sizes. Delocalisation is not always stabilising made it for cyclobutadiene, whose square geometry gives a degenerate non-bonding pair holding two electrons. A chain cannot stay even made it for a hundred-site chain, where the degeneracy is replaced by a continuum of levels at the Fermi energy. The present essay is the molecular case with a metal in the middle, and it is the one where the consequence is measured in bond lengths.

The arithmetic of the distortion #

Elongate two ligands along z and pull the other four in. The upper pair splits: dz², which points at the ligands that moved away, goes down; dx²−y², which points at the ones that came closer, goes up. The lower set splits as well, more weakly, and in the opposite sense.

For a symmetric occupation the two shifts cancel. Six electrons in the lower set, or an even number in the upper pair, means every level that went down is matched by one that went up with the same weight, and the total is unchanged to first order — and, as it turns out, to arithmetic precision.

For d⁹ they do not cancel. Three electrons in the upper pair means two in the lower of the two new orbitals and one in the higher, so the occupation weights the two shifts unequally and the sum falls linearly in the distortion.

The elastic cost of moving atoms from equilibrium is quadratic, because the leading term in a displacement from a minimum is always the square. A linear gain against a quadratic cost has a minimum at a non-zero distortion for any stiffness whatever — divide both by the distortion and one is a constant while the other goes to zero.

The computed slope is $-7.32$ in units of $e_\sigma$ per unit fractional elongation. At a stiffness of 14 the best distortion is 0.141 of a bond length and is worth 0.537 $e_\sigma$.

The zeroes are exact, and they were not free #

The interesting half of this computation is the configurations that gain nothing.

Run the same scan for d³, for high-spin d⁵, for low-spin d⁶ and for d¹⁰, and the initial slopes come back at 0, $-4\times10^{-14}$, 0 and $-9\times10^{-14}$. Those are arithmetic zeroes, of the kind this site treats as a different sort of statement from a small number — the same kind as the symmetry-forbidden overlaps of exactly zero.

Getting those zeroes requires a correction, and the obvious scan does not supply it. Let the interaction strengths follow the bond distances directly — the standard convention that $e_\sigma$ falls as $R^{-5}$ — with the axial ligands moving out and the equatorial ones moving in. Under that arrangement a half-filled shell appears to gain energy from the distortion, which symmetry says it cannot.

The check was right and the scan was wrong. Moving two ligands out and four in changes the total metal–ligand interaction as well as its anisotropy, and the total is not what the Jahn–Teller argument is about: it is a smooth, symmetric function of the distortion which every configuration feels identically, and it belongs with the elastic term rather than with the electronic one.

The repair uses the angular-overlap sum rule. The sum rule says the trace of the angular overlap matrix is $N e_\sigma$ for any geometry whatever, so renormalising the six strengths to hold their sum fixed is exactly the operation that separates the isotropic part from the anisotropic one. With the trace held, every symmetric occupation gives zero to fourteen decimal places and the two unequal ones give $-7.32$.

That is a small piece of arithmetic with a general moral. A model that mixes two effects will attribute one to the other, and the way to tell them apart is to find the conserved quantity that one of them cannot change.

Why linear beats quadratic, in one line #

The argument’s strength is easy to underestimate, so it is worth putting the comparison as starkly as it deserves.

A gain that is linear in the distortion is $-a\delta$ for some positive $a$. A cost that is quadratic is $K\delta^2$. Their sum is $K\delta^2 - a\delta$, which is negative for every $\delta$ between zero and $a/K$, however large $K$ is. The minimum sits at $\delta = a/2K$ and is worth $-a^2/4K$.

So a stiff structure distorts less and still distorts. Doubling the stiffness halves the distortion and quarters the energy gained, and never reaches zero. There is no stiffness at which the symmetric arrangement becomes stable, and that is what makes this a theorem rather than a tendency.

The same three lines appear in a chain cannot stay even with $\delta^2\log(1/\delta)$ in place of $a\delta$, where the conclusion survives because a logarithm diverges — slowly, and without limit — so the ratio of gain to $\delta^2$ still exceeds any fixed $K$ once $\delta$ is small enough. The molecular case is the easier one: no limit needs taking, because linear beats quadratic at the origin outright.

What the distortion does to the symmetry #

An octahedron elongated along one axis is no longer octahedral. Its group drops from Oh to D4h, and what that does to every level can be worked out exactly.

The doubly degenerate $E_g$ pair splits into two non-degenerate levels, $A_{1g}$ and $B_{1g}$ — and once they are non-degenerate the argument that forced the distortion no longer applies. The distortion removed its own cause, which is the general shape of a Jahn–Teller effect and the reason it stops somewhere rather than running away.

Where it stops is decided by the elastic term, and that is the one quantity here that is not computed: the stiffness is a parameter. What the model does say without a parameter is that the stopping point is not zero.

Descent in symmetry works that out, and it is worth noting that it was worked out for a different purpose entirely — working out what happens to a spectrum when a substituent lowers a molecule’s symmetry — and applies here unchanged.

Carried far enough, it is a square plane #

Elongating two axial bonds does not have a natural stopping point in the model beyond the balance with the stiffness. Push it further and the two axial ligands leave altogether, and what is left is a square-planar complex.

Taken far enough, a tetragonal elongation removes the axial ligands altogether and the square planar field is what is left. Copper(II) complexes sit somewhere along that path, and where they sit is what fixes the two splittings the argument above divides by.

That continuity explains a pattern in the shape data. The configurations that gain most from a tetragonal distortion are d⁹ and high-spin d⁴, and the configuration that gains most from going all the way to square planar is d⁸ — and the three are neighbours for a reason. All three have an uneven occupation of the upper pair or benefit from splitting it, and they differ in how far the ion is willing to go.

Which shape a given d count prefers is a competition computed separately, and it is a different question from how far a fixed shape distorts. The first chooses between arrangements; the second asks what one arrangement does when its own symmetry does not suit its electrons.

What is measured #

Copper(II) complexes are the standard demonstration and the numbers are unambiguous.

The hexaaqua ion has four short bonds of about 1.97 Å and two long ones of about 2.30 — a difference of 0.33 Å, or seventeen per cent, which is far larger than the precision of any structure determination. Copper(II) fluoride, copper(II) chloride and a long list of other copper compounds show the same pattern in the solid, with four near neighbours and two further ones.

High-spin d⁴ shows it too, and the classic case is manganese(III): the same elongation, in a compound where the metal has one electron in the upper pair rather than three.

The prediction that would falsify the argument is the negative one, and it holds: chromium(III), which is d³ with a half-filled lower set, forms regular octahedral complexes with six equal bonds; so does low-spin cobalt(III), which is d⁶ with a full lower set; and so does zinc(II), which is d¹⁰. Those are the four configurations whose computed slopes are zero.

The same argument at the other size #

The same theorem begins with a chain of a hundred atoms, and the difference between the chain and the complex is instructive.

A molecule with an exactly degenerate pair gains linearly in the distortion. A chain has no exact degeneracy — only a continuum of levels arbitrarily close together at the Fermi energy — and its gain is $\delta^2\log(1/\delta)$, which is weaker than linear and still stronger than any quadratic, so the conclusion survives with a different exponent.

Three systems, one theorem, three exponents that depend only on whether the degeneracy is exact and how the levels crowd around it. That is the kind of unity worth following through three systems.

The molecular twin of this is cyclobutadiene’s half-filled degenerate pair, whose gain is also linear in the distortion and for the same reason. That case needs no ligand field at all, which is worth saying: the two are one theorem met twice.

What it does to the spectrum #

A distortion that splits two levels into four has a spectroscopic consequence, and it is the reason copper(II) spectra look the way they do.

An undistorted d⁹ complex would have one d–d transition, at the splitting, and a band that is one band. The distorted complex has three one-electron excitations available instead of one, at energies that differ by the size of the tetragonal splitting — so its absorption is broad and visibly asymmetric, often with a shoulder, and the shoulder is a measurement of the distortion.

That is the same relation between structure and band count this site established for vibrational spectra in two structures, two spectra: the count of features is decided by the symmetry before any of them is measured, so a count that does not match the assumed structure is evidence about the structure. A copper(II) band with a shoulder is not a badly resolved single transition; it is what a tetragonally distorted d⁹ ion is supposed to look like.

The magnitudes work out consistently. The tetragonal splitting inferred from the band shape agrees with what the measured bond lengths imply through the $R^{-5}$ convention, to the accuracy such a convention deserves — which is a check of the whole picture against two instruments that have nothing in common.

What the model cannot see #

The stiffness is a parameter. How far the distortion goes is a balance between a computed electronic term and an assumed elastic one, so the 0.141 in the hero figure is not a prediction of 0.33 ångströms in the hexaaqua ion.

The distortion may be dynamic. The theorem says the symmetric structure is not a minimum; it does not say the molecule sits still in one of the distorted ones. There are three equivalent elongations, one per axis, and at ordinary temperatures a complex may hop between them fast enough that a slow measurement sees an average. Copper(II) complexes are a standard example: X-ray structures at room temperature sometimes show an apparently regular octahedron with unusually large thermal ellipsoids, which is three structures being averaged.

The direction is not decided. Compression is as good as elongation at the level of this model — both split the degenerate pair — and which occurs is settled by higher-order terms the model does not contain. Elongation is what is usually observed, and this calculation does not explain why.

There is no electron repulsion. The occupations here are one-electron occupations, and whether a d⁴ ion is high-spin at all is a question the pairing energy decides rather than the splitting, as the pairing energy decides the moment computes.

The measurement that separates the two explanations #

There is an alternative account of copper’s unequal bond lengths that has to be dealt with, because it is the one a crystallographer reaches for first: packing. A solid places a lot of constraints on a coordination sphere, and unequal bond lengths in a crystal can perfectly well be a consequence of what the neighbouring ions are doing.

Three observations separate the accounts, and none of them requires a calculation.

The distortion follows the configuration, not the compound. Every d⁹ and every high-spin d⁴ complex shows it and the neighbouring configurations do not, across compounds with entirely different packing. Chromium(III) and copper(II) form isostructural salts in which only the copper one is distorted.

It survives in solution. The elongation is visible in the electronic spectra and in the electron paramagnetic resonance of copper(II) in solution, where there is no lattice to blame.

Its magnitude tracks the ligand field. Stronger-field ligands, which increase the splitting the distortion is exploiting, produce a larger elongation — which is a correlation between two independently measurable quantities and is not something a packing argument predicts.

That style of argument — find the observable that the two explanations disagree about, then measure it — is the same one the spectrochemical series is not electrostatics uses to separate a charge argument from a π-bonding one. Two accounts that agree everywhere are not two accounts; the work is in finding where they part.

The distortion that only holds still when the sample is cold #

The theorem says the symmetric arrangement cannot survive, and it says nothing about which of the equivalent distorted arrangements a given complex adopts — because there are three, related by the octahedron’s own symmetry, and the molecule has no reason to prefer one.

That degeneracy of the distorted structures produces a phenomenon the static picture misses. A copper complex can move between its three elongations, and whether it does so slowly or quickly compared with the timescale of a measurement decides what the measurement returns.

Warm, and the three interconvert faster than the instrument can resolve. A magnetic resonance spectrum then shows one averaged, nearly isotropic signal — the average of three axially distorted arrangements is isotropic, because the three axes are equivalent — and a diffraction measurement shows an apparently undistorted octahedron with unusually large thermal parameters.

Cold, and the motion freezes. The same complex gives a resolved spectrum with a unique axis, and a structure with four short bonds and two long ones.

So a compound can be reported as octahedral and as distorted, from the same sample, depending only on the temperature — and neither report is wrong. The static distortion is real at all temperatures; what changes is whether the measurement is fast enough to catch one arrangement before the molecule moves to the next.

That is worth knowing before reading any structure of a d⁹ complex. An undistorted octahedral copper is almost always a time average, and the way to tell is to cool the sample: a genuine absence of distortion stays absent, and an averaged one resolves.

It also says what the theorem does and does not promise. It promises that the symmetric arrangement is not a minimum. It does not promise that a molecule sits in one minimum long enough to be seen there, and the three minima being equivalent is exactly the condition under which it may not.

Who found it, and when #

Hermann Jahn and Edward Teller proved the theorem in 1937, and the proof is a survey: they went through every point group and every degenerate electronic state and showed that in each case there exists a non-totally-symmetric vibration whose linear coupling is non-zero — with the single exception of linear molecules, where no such vibration exists.

That structure is worth noticing. It is not an argument from a mechanism but an exhaustive check of a finite list, which is a style of proof this site meets again in the crystallographic restriction and in the classification of point groups. The conclusion is that the exception is a theorem about lists rather than about physics: linear molecules escape because they have too few vibrations, not because anything protects them.

The chemical consequences took another twenty years to be recognised as the reason copper(II) behaves as it does — the structural evidence had been accumulating and being explained away as packing effects, which is what a systematic deviation usually looks like before somebody has a reason to expect it.