The Jahn-Teller Theorem

The Jahn-Teller Theorem (named after Hermann Arthur Jahn and Edward Teller), was published in 1937 and essentially means that:
"any non-linear molecular system in a degenerate electronic state will be unstable and will undergo distortion to form a system of lower symmetry and lower energy thereby removing the degeneracy"
In an octahedral crystal field, the t2g orbitals occur at lower energy than the eg orbitals. This is a reflection of the orientation of the orbitals since the t2g are directed between bond axes while the eg point along bond axes. The shielding effect this has on the electrons is used to explain why the Jahn-Teller effect is generally only important for odd number occupancy of the eg level.
The effect of Jahn-Teller distortions is best documented for Cu(II) complexes (with 3 electrons in the eg level) where the result is that most complexes are found to have elongation along the z-axis.
Jahn-Teller distortions

The Jahn-Teller Theorem does NOT say how large a distortion should occur.

Take for example, the 3 copper complexes shown below:

tris(octamethylmethylenediphosphonic diamide)copper(II) perchlorate (cuomdd)
hexakis(pyridine N-oxide)copper(II) dinitrate dihydrate (cupyno6)
tris(octamethylpyrophosphoramide)copper(II) perchlorate (cuompp)

The distance between atoms is displayed in response to mouse clicks, click on the Cu(II) then drag to a coordinating atom to see the distance.

Determine the 6 bond lengths around each Cu(II) ion, I've started you off by showing 1 bond length for each. Having found the others, determine whether the molecules correspond to elongation or contraction of 2 bonds (the z-direction).

Apparent exceptions to the theorem are probably examples of what has been called the "dynamic Jahn-Teller effect". In these cases either the time frame of the measurement does not allow the distortion to be seen because of the molecule randomly undergoing movement or else the distortion is so small as to be negligible.

For one of the copper complexes above, the bond lengths are apparently identical. If the X-ray structure of the sample is redone at varying temperatures it is sometimes possible to "freeze" a molecule into a static position showing the distortions.
A well documented example includes complexes of the type M 2PbCu(NO 2) 6.
For M=Cs, below 285K the molecule shows tetragonal symmetry, for M=K this occurs at below 273K, for M=Rb at less than 276K and for M=Tl at temperatures less than 245K.
Above these temperatures the molecules appear octahedral due to the dynamic Jahn-Teller effect.
Some examples of Jahn-Teller distorted complexes
CuBr 2 4 Br at 240pm 2 Br at 318pm
CuCl 2 4 Cl at 230pm 2 Cl at 295pm
CuCl 2.2H 2O 2 O at 193pm 2 Cl at 228pm 2 Cl at 295pm
CsCuCl 3 4 Cl at 230pm 2 Cl at 265pm
CuF 2 4 F at 193pm 2 F at 227pm
CuSO 4.4NH 3.H 2O 4 N at 205pm 1 O at 259pm 1 O at 337pm
K 2CuF 4 4 F at 191pm 2 F at 237pm
KCuAlF 6 2 F at 188pm 4 F at 220pm
CrF 2 4 F at 200pm 2 F at 243pm
KCrF 3 4 F at 214pm 2 F at 200pm
MnF 3 2 F at 209pm 2 F at 191pm 2 F at 179pm
The Jahn-Teller Theorem predicts that distortions should occur for any degenerate state, including degeneracy of the t 2g level, however distortions in bond lengths are much more distinctive when the degenerate electrons are in the e glevel.

In the electronic spectrum of an aqueous solution of Ti(III), a d 1octahedral system, the absorption band is not symmetric but rather shows a distinct broad shoulder. This has been interpreted in terms of a lowering of the degeneracy of the t 2g level and promotion to the excited state occurring to either of the two orbitals, the d z2 and d x2-y2, which will no longer be degenerate. Thus, two transitions are possible but because the energy difference is small, a shoulder appears rather than 2 distinct peaks.


References.
H.A. Jahn and E. Teller, Proc. R. Soc. London A, 161, 220-235, 1937.
H.A. Jahn, Proc. R. Soc. Lond. A 164, 117-131, 1938.
Jahn Teller on Wikipedia
Structures from:
P.T.Miller, P.G.Lenhert and M.D.Joesten, Inorg. Chem., 11, 2221, 1972.
J.S.Wood, C.P.Keijzers and R.O.Day, Acta Crystallogr., Sect.C (Cr. Str. Comm.), 40, 404, 1984.
M.D.Joesten, M.S.Hussain and P.G.Lenhert, Inorg. Chem., 9, 151, 1970.
http://www.mapr.ucl.ac.be/~crystal/jahnteller.html
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