Jahn-Teller Theorem

1. Distortion of octahedral complexes
• The tetragonal distortions are illustrations of the Jahn-
Teller Effect.
• The effect can be summarized by a statement which
predicts which complexes will undergo distortion.
• It does not predict the extent of the distortion.
Statement
• If the ground electronic configuration of a non-linear
complex is orbitally degenerate, the complex will distort
so as to remove the degeneracy and achieve lower energy.
• 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.
• If the ground electronic configuration of a nonlinear complex is
orbitally degenerate, and asymmetrically filled, then the complex
distorts so as to remove the degeneracy and achieve a lower
energy.

2. Explanation
• The six-coordinated complexes in which all the six
distances between the ligand electron clouds and central
metal ion are the same are said to be regular (i.e.,
symmetrical) octahedral complexes.
• On the other hand the six-coordinated complexes in which
the distances are not equal are said to be distorted
octahedral complexes.
• Since their shape is changed (i.e. distorted).
• This phenomenon i.e., the change in shape is called
distortion.
Types of distorted octahedral complexes
• Distorted octahedral complexes may be of the following
three types:
Diagonally distorted octahedral complexes
• These are obtained when the distortion of a regular
octahedron takes place along a two-fold axis.

3. Trigonally distorted octahedral complexes
• In which the distortion takes place along a three-fold axis.
Tetragonally distorted octahedral complexes
• These are also known as tetragonal complexes.
• These are obtained when the distortion of a regular
octahedron takes place along a four-fold axis.
Figure.
(a) A tetragonally distorted complex
where two of the ligands have
moved further away from the central
ion.
(b) A tetragonally distorted complex
where two of the ligands have
moved closer towards the central
ion.

4. How to get tetragonal complexes
• Tetragonal complexes may be obtained by any of the
following two ways:
a) Elongation
• If the two trans-ligands lying on
the z-axis in an octahedron are
moved away from the central
metal cation so that their
distance from the metal cation is
slightly greater than it is for the
other four ligands lying in the
xy-plane, we get a tetragonal
structure.
• Quite obviously this structure
has two long bonds along the z-
axis and four short bonds in xy-
plane.
Figure. A tetragonally
distorted complex where two
of the ligands have moved
further away from the central
ion.

5. Examples
i) It has been shown that in tetra ammine of Cu2+ ion in
aqueous solution, [Cu(NH3)4(H2O)2]2+ two water
molecules are at larger distance from the central Cu2+ ion
than the four coplanar NH3 molecules and consequently
the complex has a tetragonal shape.
• The two water molecules are in a plane at right angles to
that containing Cu2+ ion and four NH3 molecules which
are at equal distances from Cu2+ion.
ii) Low-spin octahedral complexes of Ni2+, Pd2+ and Pt2+ (all
d8 ion) undergo strong distortion and assume square
planar geometry in which the two ligands along the z-axis
are at larger distance and four ligands in the xy-plane are
at shorter distance from M2+ ion.
• MIII(diars)2I2 is an example of such complexes.

6. iii) In [CuCl6]4- crystal each Cu2+ ion is surrounded by six Cl-
ions; four are at a distance of 2.30 A° and the other two are
2.95 A° away.
iv) In [CuF6]4- crystal four F- ions are 1.93 A° away from Cu2+
ion, while the two F- ions are 2.27 A° apart.
b) Compression
• If the two trans-ligands located at the z-axis are brought near
the central metal cation so that their distance from the metal
cation is smaller than it is for the other four ligands in the xy-
plane, we again get a tetragonal structure.
• This structure has two short bonds along the z-axis and four
long bonds in xy-plane.
Examples
i) In K2CuF6 the Cu2+ ion has two F- at 1.95 A° and four at 2.08
A°.
ii) In [FeF6]3- the Fe2+ ion has two F- at 1.99 A° and four at 3.12
A°.

7. Figure. A tetragonally distorted complex where
two of the ligands have moved closer towards the
central ion.

8. diars = 1,2-Bis(dimethylarsino)benzene →
C6H4(As(CH3)2)2
Diars is prepared by the reaction of ortho-dichlorobenzene and
sodium dimethylarsenide:
C6H4Cl2 + 2 NaAs(CH3)2 → C6H4(As(CH3)2)2 + 2 NaCl
It is a colorless liquid.
Oxygen converts diars to the dioxide, C6H4(As(CH3)2O)2.
o-Dichlorobenzene Sodium dimethylarsenide
1,2-Bis(dimethylarsino)benzene
What is diars?

9. Jahn-Teller Theorem (J-T Distortion)
• In 1937, Jahn-Teller put forward a theorem known as Jahn-Teller
theorem which can explain why certain six-coordinated complexes
undergo distortion to assume distorted octahedral (i.e., tetragonal)
geometry.
• That principle states that if a system has unequally populated
degenerate orbitals, the system will distort to remove the degeneracy.
• When the degeneracy is removed, the state of lower energy will be
more fully populated.
"This theorem states that any non-linear molecular system possessing
degenerate electronic state will be unstable and will undergo distortion
to form a system of lower symmetry and lower energy and thus will
remove degeneracy".
• If the undistorted configuration has a centre of symmetry the distorted
equilibrium configuration must have one too.
• Important: It should be noted that Jahn-Teller theorem only predicts
the occurrence of a distortion; it does not predict its nature or its
magnitude.

10. Splitting of the d-subshell by Jahn-Teller distortion
• Complexes of Cu2+ having a d9 configuration are among
the most common ones that exhibit such a distortion.

11. Symmetrical and unsymmetrical t2g- and eg-orbital
• The t2g and eg-orbitals which are empty (t2g0 and eg
0), half-filled
(t2g3 and e2
g) or completely filled (t2g
6 and eg
4) are said to be
symmetrical orbitals.
• Here it should be noted that in the strong field (i.e., LS-complexes)
eg-set having two electrons (i.e., eg
2 -set) is unsymmetrical orbital.
• Orbitals other than those mentioned above are called
unsymmetrical orbitals.
0 3 6
1 2 4 5
0 4
1 3
2
1 1
0 2

12. • In the following table symmetrical [abbreviated as
(sym)] and unsymmetrical [abbreviated as (unsym)] t2g -
and eg -orbitals are listed.
• It will be seen that t2g-eg pairs of orbitals that can exist is
t2g (unsym) - eg (unsym) pair.
Conditions for the prediction of distortion in octahedral
complexes
• There are following conditions for J-T distortion in
octahedral complexes:
– No J-T distortion
– Slight J-T distortion
– Strong J-T distortion

13. 0
1
2
3
4
5
6
7
8
9
10
0 0 0 0
1 0 1 0
2 0 2 0
3 0 3 0
3 1
4 0
3 2 5 0
4 2 6 0
5 2 6 1
6 2 6 2
6 3
6 4
6 3
6 4

14. No J-T distortion condition
• The d-orbitals which have both t2g and eg-sets as
symmetrical orbitals lead to perfectly symmetrical (i.e.,
regular) octahedral complexes.
• Thus it may be seen from above table that the d-orbitals:
– d0(t2g
0 eg
0)
– d3(t2g
3 eg
0)
– d5(t2g
3 eg
2)
– d8(t2g
6 eg
2)
– d10(t2g
6 eg
4)
of HS-octahedral complexes and d0(t2g
0 eg
0), d3(t2g
3 eg
0),
d6(t2g
6 eg
0) and d10(t2g
6 eg
4) of the octahedral complexes
give perfectly regular octahedral complexes.
• Thus in these cases there is no distortion.

15. Slight J-T distortion condition
• When d-orbitals of the central metal ion of an octahedral
complex have t2g-orbitals as unsymmetrical orbitals, there
occurs slight distortion in the complex, i.e., whenever the
t2g-orbitals, which do not come direct in the path of the
ligands disposed; octahedrally around the central metal
ion, but point between the ligands contain 1, 2, 4 or 5
electrons, we shall expect only slight distortions from the
regular octahedron.
• Thus, as is evident from above table, the HS-complexes
of d1(t2g
1 eg
0), d2(t2g
2 eg
0), d6(t2g
4 eg
2), and d7(t2g
5 eg
2) ions
and LS-complexes of d1(t2g
1 eg
0), d2(t2g
2 eg
0), d4(t2g
4 eg
0)
and d5(t2g
5 eg
0) ions undergo slight distortion (often not
experimentally detectable) from the octahedral shape.

16. Strong J-T distortion condition
• Whenever the eg orbitals which point directly towards the ligands,
are unsymmetrical i.e. contain 1, 3 or 2 (only in LS-complexes)
electrons, we shall expect strong distortions, leading to tetragonal
and even to square planar complexes.
• Distortion produced in an octahedral complex due to the presence
of unsymmetrical eg-orbitals is due to the fact that, since the
directions of the eg-orbitals of the central metal ion correspond with
the directions of the ligands disposed octahedrally around the metal
ion, the octahedral arrangement of the ligands is likely to be more
severely distorted by the dissymmetry of the eg-orbitals than that of
the t2g-orbitals.
• Thus from above table it can be seen that the configuration: d4(t2g
3 eg
1),
d9(t2g
6 eg
3) of HS-complexes and d7(t2g
6 eg
1), d8(t2g
6 eg
2), d9(t2g
6 eg
3) of LS-
complexes lead to strong distortion in octahedral complexes.

17. Summary of conditions for J-T distortion
• Conditions, for various types of distortions can be summarised as:
Cause of distortion with some complex
i) High spin octahedral complexes of d4 ion have any of the following
configurations:
• When the configuration has one electron in dz
2 orbital and the dx
2-y
2
orbital is empty (structure I), cation-anion interaction along the z-
axis is less than that along the x-and y-axes, leading to a larger
interionic distance along the z-axis and hence to a tetragonal
structure.
3
1 0 3 0 1

18. ii) Complexes of Cu2+ ion (d9 ion) such as aqueous solution of
[Cu(NH3)4]2+ in which the tetragonal distortion is so marked
that a square planar complex results.
• This ion has the configuration t2g
6 eg
3 in both the fields.
• Evidently in this ion the two possible arrangement of electrons in t2g- and
eg- orbitals are:
• Since in both the configurations t2g-orbitals are completely filled,
asymmetry (i.e., distortion) is caused by incomplete filling of eg-orbitals.
Configuration I → t2g
6. (dz
2)2 (dx
2-y
2)1
• Distortion arises mainly from the repulsion of ligands by the electrons
occupying eg-orbitals.
• If we consider configuration I namely t2g
6. (dz
2)2 (dx
2-y
2)1, the dz
2-orbital,
which is completely filled and points at the ligands on the z-axis, offers
greater shielding of the Cu2+ nucleus than the half-filled dx
2 –y
2-orbital,
which points towards the ligands in the xy-plane.
6
6
2
21
1

19. • Thus the ligands on the x- and y-axes experience a higher
effective nuclear charge, while those on the z-axis
experience a lower effective nuclear charge.
• Consequently the ligands on the x- and y-axes are drawn
in closer to the Cu2+ nucleus and those on the z-axis move
further out.
• We thus observe four-short and two long bond i.e., the
ligands L5 and L6 existing along the z-axis would be at a
greater distance from the central metal ion (Cu2+ ion) and
the remaining four coplanar ligands: L1, L2, L3 and L4
(coplanar with the centralized metal ion, Cu2+, in xy-
plane) would be at a shorter distance from Cu2+ ion.
• Thus:
5 6 1 2 3 4

20. where ML1......etc. indicate the metal ion-ligand distances
(following figure).
• Ligands L5 and L6 are trans-ligands while the ligands, L1,
L2, L3 and L4 are equatorial ligands.
Figure:
Distortion t in
an octahedral
complex.
Configuration I
→ (dz
2)2 (dx
2-
y
2)1. t2g
6 having
two long (along
the z-axis) and
four short (in
the xy-plane)
bonds, i.e. This configuration is more stable.

21. Configuration II → t2g
6 (dz
2)1 (dx
2-y
2)2
• If, on the other hand, we consider configuration II → t2g
6
(dz
2)1 (dx
2-y
2)2, we shall expect an exactly opposite
distortion i.e., the ligands in the xy-plane (i.e. L1, L2, L3
and L4 ligands) would move and those (i.e. L5 and L6
ligands) on the z-axis would move out in from their
equilibrium positions in the hypothetical regular
octahedron and we would expect two short bonds (along
the z-axis) and four long bonds (along x- and y- axes in
xy-plane).
• Thus:
(Following figure)
1 2 3 4 5 6

22. Figure: Distortion t in an octahedral complex. Configuration
II → (dz
2)1 (dx
2-y
2)2. t2g
6 having four long (co-planar in xy
plane) and two short (along the z-axis) bone. Two short bonds
are perpendicular to the xy-plane. Here:

23. Conclusion
• How then to decide which of the two possible octahedral
distortion configurations:
– Configuration I → t2g
6 (dz
2)2 (dx
2-y
2)1
– Configuration II → t2g
6 (dz
2)1 (dx
2-y
2)2
would yield the more stable complex.
• CFT offers no way of deciding it.
• Experimental results, however, show that it is octahedral
distortion configuration I namely t2g
6. (dz
2)2 (dx
2-y
2)1 with
two long and four short bonds which is more stable.
• There is no theoretical explanation for the instability of
structure corresponding to configuration II namely t2g
6.
(dz
2)1 (dx
2-y
2)2 having four long and two short bonds.
δ = Greek small letter delta

24. How the d orbital energy levels change in Cu2+ (d9 ion)
ion when the regular octahedron distorts
• Let us consider the manner in which the d-orbital energy
levels change in the Cu2+ ion (d9 ion) when there occurs a
small distortion of the type in which the regular
octahedron becomes stretched along z-axis.
• The splitting and energy level diagram caused by such
type of distortion are shown in following figure.
• In this diagram the various splitting are not drawn to
scale-in the interest of clarity.
• Here the splitting of the more stable octahedral distortion
corresponding to configuration I → t2g
6. (dz
2)2 (dx
2-y
2)1 has
been considered, δ1 and δ2, which represent the splitting of the
eg
3- and t2g
6-levels respectively, both are much smaller than Δo
and δ2 is much smaller than δ1, i.e.,
– Δo > > > δ1 > δ2

25. Octahedral
Free metal ion
Tetragonal

26. • The two eg-orbitals separate so that one (namely dx
2- y
2)
goes up as much as the other (namely dz
2) goes down; the
t2g-orbitals separate so that doubly degenerate pair (i.e.,
dyz and dzx) goes down only half as far as the single
orbital (namely dxy) goes up.
• Hence it can be seen that for t2g-electrons there is no net
energy change, since four electrons (namely dyz and dzx
electrons) are stabilised by:
while two electrons (namely dxy electrons) are destabilised
by:

27. • Thus in the splitting of t2g-levels:
• Thus the net lowering of the electronic energy in case of eg level is
δ1/2.
• This net energy equal to -δ1/2, might be called the Jahn-Teller
satabilisation energy and provides the driving force for the
distortion.

28. Why two electrons enter in dz
2 instead of singly occupied in
dx
2
-y
2 and dz
2
• We have already seen that the high-spin octahedral complexes
of d8 ion such as [Ni(H2O)6]2+, [Ni(NH3)6]2+, [Ni(N2H4)3]2+
etc. are not expected to show distortion, since both t2g and eg
orbitals in these complexes are symmetrically filled as is
evident from the arrangement of eight electrons in these
orbitals given below:
• However, low-spin octahedral complexes of d8 ion undergo
strong distortion and assume square planar geometry (see above
table).
• MII(diars)2I2 is a typical example of such complexes.
• Under the influence of strong ligand field of diarsine (diars), the
two electrons in eg level pair up in dz
2 orbital, leaving dx
2–y
2
orbital empty.
8
6 2 6 1 1

29. • Thus the arrangement of two electrons in eg orbitals is
unsymmetrical as shown below:
and due to the operation of Jahn-Teller effect those
orbitals split into a lower energy dz
2 orbital and a higher
energy dx
2 –y
2 orbital.
• The two ligands along z-axis move away from Ni2+ ion
while the four ligands in the xy-plane move closer and as
a result, the octahedral complex is considerably distorted
and assumes almost square planar geometry.
8
6 2 0

30. Assignment
• Explain how the d orbital energy levels change in d9 system
when the regular octahedron distorts in Cu2+ ion. Also
calculate the net gain in energy in the splitting of t2g and eg.
• What is the amount of Jahn-Teller satabilisation energy
which is the driving force for the distortion?
Ans.
• This net energy equal to -δ1/2, might be called the Jahn-Teller
satabilisation energy and provides the driving force for the
distortion

31. The End