Jahn—Teller effects

Jahn-Teller distortion can only be expected if the energy integral 

Since MA is degenerate, its direct product with itself will always contain the totally symmetric irreducible representation and, at least, one other irreducible representation. For the integral to be nonzero, q must belong either to the totally symmetric irreducible representation or to one of the other irreducible representations contained in the direct product of vJ/ 0 with itself. A vibration belonging to the totally symmetric representation, however, does not decrease the symmetry 

Let us see an example, the H3 molecule, which has the shape of an equilateral triangle. Its symmetry is D3h, the electronic configuration is a V, and the symmetry of the ground electronic state is E. Thus, the electronic state of the molecule is degenerate and is subject to Jahn-Teller distortion. 

The symmetry of the normal mode of vibration that can take the molecule out of the degenerate electronic state will have to be such as to satisfy Eq. (6-7). The direct product of E with itself (see Table 6-11) reduces to A + A 2 + E. The molecule has three normal modes of vibration [(3 x 3) - 6 = 3], and their symmetry species are A + E. A totally symmetric normal mode, A, does not reduce the molecular symmetry (this is the symmetric stretching mode), and thus the only possibility is a vibration of E symmetry. This matches one of the irreducible representations of the direct product E E therefore, this normal mode of vibration is capable of reducing th eZ)3/, symmetry of the H3 molecule. These types of vibrations are called Jahn-Teller active vibrations. 

Obviously, only molecules with partially filled orbitals display Jahn-Teller distortion. As was shown in Section 6.3.2, the electronic ground state of molecules with completely filled orbitals is always totally symmetric, and thus cannot be degenerate. In comparison with the above-mentioned unstable H3 molecule, Hj has only two electrons in an a symmetry orbital therefore, its electronic ground state is totally symmetric, and the D3/,-symmetry triangular structure of this ion is stable (see, e.g., Reference [62]). On the other hand, take the benzene molecule, e.g., whose ground electronic state is of Alg symmetry and the molecule is stable and its structure is well understood. At the same time, in its cation, C6Hg, it loses one electron from an c -symmetry doubly-degenerate orbital, so that orbital is left with only one electron. The electronic state of the cation has E g symmetry and thus, it is subject to Jahn-Teller effect. Indeed, its vibrational spectrum is extremely complicated and can only be satisfactory explained if the Jahn-Teller distortion is taken into consideration (see, e.g., Reference [63]). 

Jahn-Teller type distortion, q must belong to one of the other irreducible representations. 

The second effect which has been neglected is that due to low-symmetry crystal fields. As the discussion in this chapter has been confined to octahedral MLg complexes one might assume that low-symmetry ligand fields could safely be ignored, but this is not so. A theorem due to Jahn and Teller states that any non-linear ion or molecule which is in an orbitally degenerate term will distort to relieve this degeneracy. This means that all Ey, and T2g terms of d configurations, in principle, are unstable with respect to some distortion which reduces the symmetry. Of course, as 

Remembering that the amplitude associated with the two ligands is twice that associated with the four, it can be seen that the inherent asymmetry in the potential energy surface indicates that the cost of (two short, four long) will be greater than that of the (two long, four short). This conclusion is in 

There is one result of the general theory which should be mentioned. It can be shown that in a regular octahedron the Jahn-Teller effect only operates by way of vibrations of symmetry when the electronic state is Eg (the value of n is irrelevant because the Jahn-Teller theorem applies only to space functions, not spin). For electronic states of either or T2g symmetries then the Jahn-Teller effect operates through vibrations of either 6g or t2g symmetries a vibration of the latter symmetry is shown in Fig. 8.11. Of course, the orbital degeneracy in an octahedral complex may be relieved by distortions other than those shown in Figs. 8.8 and 8.11. However, in such cases we may conclude that whatever is responsible for the distortion it is not the Jahn-Teller effect. In particular, all of the Jahn-Teller-active vibrations of an octahedron carry the g suffix and this means that they cannot give rise to a distortion which destroys the centre of symmetry of an octahedron. Distorted octahedral complexes which lack a centre of symmetry cannot owe their distortion to the operation of the Jahn-Teller effect. This account of the Jahn-Teller effect indicates why it is of little importance when the t2g orbitals are unequally occupied. Occupation of these orbitals 

If the Jahn-Teller effect operates in the electronic ground state of a complex it is unlikely to be apparent in the electronic spectrum. If the two ground states are split sufficiently far apart for transitions to the excited state 

Let us consider a pair of the electronic wave functions, i) and / 2), tending to crossover at the nuclear coordinate Qo (Fig. 114). A proper electronic wave function A) should be considered as a (quasi) degenerate, and then a linear combination of the respective pair is appropriate  

In terms of the linear variation method the secular equation is obeyed H 2 

The electronic matrix element Hfj can be analyzed in terms of the group theory it adopts a nonzero value only when the direct product of the corresponding IRs (which is a reducible representation) contains the fully symmet- 

When the wave functions A and 02) belong to different IRs (A r2) of the symmetry point group G(Qo), then the matrix element 

For example, the doubly degenerate electronic state implies that the proper molecular state function is a Unear combination of the degenerate trial functions 

If the off-diagonal matrix element is non-zero, the two roots are different they correspond to a lower (upper) sheet of the adiabatic potential surface for which a crossover is excluded. In order to show that Hx2 i1 0 let us apply the Taylor expansion of that matrix element with respect to symmetry coordinates Qs (nuclear displacement coordinates) around a reference nuclear configuration Q0, hence 

The linear vibronic matrix element is non-zero for any point group the direct product of the irreducible representations of the trial wave functions is a reducible representation which necessarily contains the irreducible representation of a symmetry coordinate 

Stereochemical consequences of the Jahn-Teller theorem are straightforward the higher symmetric polyhedra in electronic degenerate states should distort in order for their electronic ground state to become non-degenerate this is the static Jahn-Teller effect. Some possible distortions are exemplified by  

As an effect of the linear and quadratic vibronic constants the adiabatic potential surface no longer stays paraboloid-shaped. It exhibits an additional warping with several local minima out of the reference high-symmetry configuration Qq. 

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The co-ordination number in ionic compounds is determined by the radius ratio - a measure of the necessity to minimize cationic contacts. More subtle effects are the Jahn-Teller effect (distortions due to incomplete occupancy of degenerate orbitals) and metal-metal bonding. 

The stoi7 begins with studies of the molecular Jahn-Teller effect in the late 1950s [1-3]. The Jahn-Teller theorems themselves [4,5] are 20 years older and static Jahn-Teller distortions of elecbonically degenerate species were well known and understood. Geomebic phase is, however, a dynamic phenomenon, associated with nuclear motions in the vicinity of a so-called conical intersection between potential energy surfaces. 

The quadratic Jahn-Teller effect is switched on by including the ijiiadratic tenns in Hq. (7) thus, with the inclusion of the additional diagonal Hamiltonian iij. 

R. Englman, The Jahn-Teller effect in Molecules and Crystals, Wiley-1 nterscience. New York, 1972. 

I, B. Bersuker, The Jahn-Teller effect and Vibronic Interactions in Modern Chemistry, Plenum Press, New York, 1984. 

In molecular physics, the topological aspect has met its analogue in the Jahn-Teller effect [47,157] and, indeed, in any situation where a degeneracy of electronic states is encountered. The phase change was discussed from various viewpoints in [144,158-161] and [163]. 

C. C. Chancey and M. C, M. O Brien, The Jahn-Teller Effect in ttnd other Icosahedral Complexes, Princeton University Press, FYinceton, NJ, 1997. 

F. S. Ham, Jahn-Teller Effects in Electron Paramagnetic Spectra, Plenum Press, New York, 1971,... 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electi onic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164-168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

Now, we examine the effect of vibronic interactions on the two adiabatic potential energy surfaces of nonlinear molecules that belong to a degenerate electronic state, so-called static Jahn-Teller effect. 

The ti eatment of the Jahn-Teller effect for more complicated cases is similar. The general conclusion is that the appearance of a linear term in the off-diagonal matrix elements H+- and H-+ leads always to an instability at the most symmetric configuration due to the fact that integrals of the type do not vanish there when the product < / > / has the same species as a nontotally symmetiic vibration (see Appendix E). If T is the species of the degenerate electronic wave functions, the species of will be that of T, ... 

R, Englman, The Jahn-Teller Effect in Molecules and Crystals, John Wiley. Sons, Inc., Interscience, New York, 1972 I, B. Bersucker and V. Z. Polinger, Vibronic Interactions in Molecules and Crystab, Springer Verlag, 1989. 

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