5. JAHN- TELLAR EFFECT

JAHN- TELLAR EFFECT

It was stated by Jahn and Teller that any non –linear molecular system in orbitally degenerate electronic state (i.e., a state which represents more than one electronic arrangement of the same energy) would be unstable and that it would get stabilised by undergoing distortion in its geometry and thus by causing a split in its orbitally degenerate electronic state. The above statement is known as Jahn-Tellar effect. The distortion in the geometry of the non-linear molecular system thus produced is known as Jahn-Tellar distortion. The lowering of symmetry of the non-linear system due to Jahn –Tellar effect always occurs in a manner which results in decrease in the energy of the system.

Let us consider an octahedral complex of Cu⁺² with unidentate ligands L^-. The Cu⁺² ion in a perfect octahedral ligand environment has the ground state electronic configuration t⁶_2ge³_g which represents two electronic arrangements t⁶_2gd²_z²d¹_x²_-y² of equal energy. Because of the Jahn-Tellar effect the octahedral symmetry of the complex will be automatically lowered to tetragonal symmetry to stabilise  the complex. This is explained as follows.

The t_2g orbitals of Cu⁺² in octahedral ligand environment are completely occupied in the ground state. Therefore, the electron charge denisty due to the six t_2g electrons will be uniformly distributed in all direction. The same is the case with the charge density of electrons in the inner shells which are completely filled.

The three electrons in e_g orbitals of an octahedral Cu⁺² complete can be placed in two equivalent ways as follows :

1- Two electrons in d_z² orbital and one electron in d_x²_-y²  orbital of Cu⁺² ion.

2-  Two electrons in d_x²_-y² orbital and one electron in d_z² orbital of Cu⁺² ion.

In possibility (1), the d electron charge density will obviously become higher in Z direction than in X or Y direction. The screening of the positive charge on Cu⁺² nucleus by d electron will, therefore, be more in Z direction than in X or Y direction. The negative charge on the ligand L- along the Z direction will thus be less attracted by the nuclear charge on Cu⁺² than the negative charge on the ligand along X and Y directions. Consequently, the Cu⁺² – ligand attraction along Z direction will be less than the Cu⁺² –ligand attraction along X and Y directions. As a result, the lignds along Z direction will move away from the metal ion whereas the ligands along X and Y directions will draw nearer to the metal ion. In other words, the octahedral geometry of the complex will get distorted to tetragonal geometry which is elongated along Z direction and compressed along X and Y directions, as shown in Fig.7

Because of the elongation of metal- ligand bonds along the Z direction, the electrostatic repulsion experienced by d electrons of the  metal ion from the negative charge along the Z direction becomes less than the similar repulsion experienced by electrons along the X and Y directions. Due to this the dz² obital becomes of lower energy and d x²-y² orbital becomes of higher energy. In this way, the degeneracy of the two orbitals gets lifted. This is illustrated in Fig.8

Since the distortion of octahedral geometry to tetragonal geometry is automatic, that is, without the supply of any energy from outside, the overall energy of the split orbitals remains equal to the overall energy of the unsplit orbitals. Thus, if we tentatively assume the overall energy of the unsplit dz² and dx²-y² orbitals to be zero, the energy of th lower energy d_z² orbital plus the energy of d_x²_-y² orbital after their split due to the lowering of symmetry must be equal to zero. This conclusion is expressed by the statement that the ‘ centre of gravity of the split orbitals must be maintained’. This is known as ‘center of gravity rule’ for the energies of the split orbitals relaative to the energies of the unsplit orbitals.

It can be safely concluded that, in general, whenever there are more electrons in d_z² orbital than in d_x²_{- y}² orbital of an octahedral complex of any metal ion, the distortion of octahedral geometry to tetragonal geometry would occur by elongation of its metal –ligand bonds along the Z direction and that the energy of the d_z² orbital becimes lower than the energy of d_x²-_y² orbital (Fig.8)

In possibility (2), in which the d_x²_-y² orbital has two electrons and the d_z² orbital one electron, these are the negative charges on the ligands along X and Y direction which are less attracted by the nuclear charge on Cu²⁺ because of more effective screening of the latter by d electrons in X and Y directions. This results in elongation of metal –ligand bonds along X and Y direction and a contraction of metal –ligand bonds along Z direction. This implies that the octahedral geometry of Cu⁺² complex gets distorted to tetragonal geometry which gets elongated along X and Y directions and compressed along Z direction, as shown in Fig.9

Further in possibility (2), the electrostatic replusion experienced by d electrons of Cu⁺² from the negative charge on the ligands along X and Y directions will be less than the similar repulsion experienced by d electrons from the negative charge on the lignds along Z direction. As a result, the d_x²_-y² orbital becomes of lower energy and d_z² orbital becomes of higher energy. In this way, the degeneracy of the two orbitals gets lifted. This is shown in Fig.10

It can be easily shown that, in general, whenever there are more electrons in d_x²_-y² orbital of an octahedral complex of any metal ion, the octahedral geometry would get distorted to tetragonal geometry by elongation of its metal–ligand bonds along the X and Y directions as result of which the energy of d_x²_-y² orbital becomes lower than the energy of d_z² orbital (Fig.10).

Apart from octahedral complexes of Cu²⁺ having t⁶_2ge³_g ground state configuration, the other octahedral complexes which show Jahn-Tellar effect due to enequal number of electrons in eg orbitals (i.e. d_z² and d_x²_-y² orbitals) comprise of metal ions with the ground state comfigurations, t⁶_2ge¹_g and t³_2g e¹_g. In both these configuration, the electronic charge due to t_2g electrons is uniformly distributed and, thereffore, the type of distortion in geometry depends upon the eg orbital (d_z² or d_x²_-y²) which the electron occupies.

Let us now consider an octahedral complex of a metal ion with t¹_2g configuration in ground state. The solitary electron can be present in any of the three t_2g orbitals. If the electron is present in d_xy orbital of the t_2g set, it would screen the nucleus of the metal ion more effectively in XY plane than in XZ or YZ plane. This would reduce the attraction between the nuclear charge on the metal ion and the negative charge onn the ligands in XY plane. As a result, the metal-ligand bonds in XY plane. In such a situation, the repulsion between d electrons of the metal ion and the negative charge on the ligands would become less in the XY plane compared to such a repulsion in the XZ or YZ plane, resulting in a decrease in the energy of the d_xy orbital and an increase in the energies of d_xz and d_yz orbitals, as illustrated in Fig11. The changes  in the energies of the orbitals would obviously be governed by the Centre of Gravity rule.

If, on the other hand, the electron is present in either of the d_xz and d_yz orbitals, it will screen the nucleus of the metal ion more effectively in the XZ or YZ plane. This would result in a decrease of attraction between the positive nuclear charge on the metal ion and the negative charge on the ligands aalong the Z direction and an increase in such attraction alonng the Z directtion whereby the octahedral geometry gets distorted to tetragonal geometry.

Since the ligands along Z axis move away from the metal ion while those along X and Y axes come nearer, the d electron would be repelled less by the negative charge on the ligands along Z direction whereas it will be repelled more by the negative charge on the ligands along X and Y directions. This increases the energy of dxy orbital an decreases the energies of d_xz and d_yz orbital, as illustrated in Fig.12

The change in energies of these orbitals will again obey the Center of Gravity rule.

It is pertinent to add here that the energies of both d_xy and d_x²-_y² orbitals (being functions of the same variable x and y) alter in a similar manner due to distortion of octahedral geometry by Jahn- Teller effect. The relative energies of all the five d orbitals in the two possible tetragonal distortions are given in below.

It may be noted that Jahn- Teller effect is operative not only in the case of complexes formed with negatively charged ligands but also in the case of complexes formed by neutral ligands such as NH_3, H2O, CO, NO etc., which are dipolar in character. The only difference in the two types of complexes is that while in the former category of complexes, the d electrons of the metal inon are repelled by the negative charges on the ligands, in the latter category, the d electrons are repelled by the negative charged poles of dipolar neutral ligands.

      It may be pertinent to point out that the Jahn- Teller effect shown by t_2g orbitals (designed as del₂) is much weaker than that shown by e_g orbitals (designed as del₁ ), i.e. del₂ < <del ₁ . The reason for this is that in t_2g orbitals the regions of maximum charge density lie in between the X,Y and Z directions and not directly in the X,Y and Z directions along which the ligands are placed whereas in e_g orbitals, the regions of maximum charge density lie directly in the directions along which the ligands aare placed in any of the t_2g orbitals would shield the positive charge on the nucleus of the metal ion much less effectively in the X,Y and Z directions i.e., along the directions of the ligands than what an electron placed in any of the e_g orbitals would do along these directions. Therefore, the magnitude of Jahn- Teller effect, which is related to the extent of screening of the nuclear charge of the metal ion by the d electron in the directions of the ligands, is smaller in octahedral complexes of metal ions with ground state configurations t¹_2g, t²_2g, t⁴_2gand t⁵_2g than in complexes of metal ions with ground state configurations t⁶_2ge¹_g, t⁶_2ge³_g, t³_2ge¹_g etc. In fact it has not been possible to detect Jahn- Teller distortion in octahedral complexes of metal ions with ground state configurationt¹_2g, t²_2g, t⁴_2g and t⁵_2g  (except from indirect sspectroscopic evidence) because the magnitude of the effect is comparatively very small. Nevertheless, these complexes do get somewhat more stabilised due to Jahn –Teller distrotion which these complexes would undergo depending upon the electronic configuration of the metal ion. This is illustrated in Table 1

As already stated, the predictions made in Table 1 cannot be verified experimentally as the magnitude of splitting of t_2g is very very small. Let us now see if we can predict the type of distortion that can be produced due to Jahn- Teller effect in octahedral complexes of metal ions with ground state configurations t⁶_2ge¹_g and t⁶_2ge³_g. We shall take into consideration the case of octahedral complex of Cu ⁺² ( t⁶_2ge¹_g); the low spin complex of Co ⁺² ( t⁶_2ge¹_g) would exihibit similar behaviour.