The Jahn-Teller Effect

The reducible representation and the application of the reduction formula for the 04h complex are set out in Table 7.14. This shows that the six a-donor ligand orbitals give SALCs with the following irreducible representations  

For the big representation we can use the projection operator method along with the 4 rotational subgroup of Ah as laid out in Table 7.15. The rotational subgroup does not contain the inversion centre, and so the gerade (g) and ungerade (u) labels do not appear. 

After the projection, we can assign these labels that are required by the full point group by considering the effect of the inversion operation. The fii projection in 4 with the S orbital as a generating function gives 

The projection based on the axial ligand orbital, 5, gives a zero result, and so the axial CT-orbitals cannot take part in a SALC with bi symmetry. Under the inversion operation of the full 4h point group, swaps with 3 and S2 with sc, this leaves the function in Equation (7.73) unchanged, so we assign gerade symmetry to give big. 

A quick check with the other operations in the full D41, group confirms that this function has the full set of characters required for the a2 representation. 

Further resolution of the 3d orbital energy levels takes place within a transition metal ion when it is located in a low-symmetry site, including non-cubic coordination environments listed in table 2.4 and polyhedra distorted from octahedral or cubic symmetries. As a result, the simple crystal field splitting parameter, A, loses some of its significance when more than one energy separation occurs between 3d orbitals of the cation. 

A converse situation exists whereby the two oxygen ions along the z axis may move closer to the Mn3+ ion (fig. 2.8 ). This results in the stabilization of the dx2 y2 orbital relative to the dz2 orbital, and shorter Mn-O distances along the z axis compared to the x-y plane. In either of the tetragonally distorted environments shown in fig. 2.8 the Mn3+ ion becomes more stable relative to a regular octahedral coordination site. In most minerals, however, the Mn3+ ion occurs in an axially elongated octahedron (see table 6.1). 

Transition metal ions most susceptible to large Jahn-Teller distortions in octahedral coordination in oxide structures are those with 3d4, 3d9 and low-spin 3(f configurations, in which one or three electrons occupy eg orbitals. Thus, the Cr2+ and Mn3+, Cu2+, and Ni3+ ions, respectively, are stabilized in distorted environments, with the result that compounds containing these cations are frequently distorted from type-structures. Conversely, these cations may be stabilized in distorted sites already existing in mineral structures. Examples include Cr2+ in olivine ( 8.6.4) and Mn3+ in epidote, andalusite and alkali amphiboles ( 4.4.2). These features are discussed further in chapter 6. 

Jahn-Teller distortions are also predicted for certain transition metal ions in tetrahedral coordination. The electronic configurations predicted to undergo no Jahn-Teller distortions are the high-spin 3d2, 3d5, 3d7 and low-spin 3d4 

10-5 The Jahn-Teller theorem states that there cannot be unequal occupation of orbitals with identical energies. To avoid such unequal occupation, the molecule distorts so 

Gerloch and R. C. Slade, Ligand Field Parameters, Cambridge University Press, London, 1973, p. 186. 

Using the usual d-orbital splitting diagrams, show that the Jahn-Teller effects in the table match the description in the preceding paragraph. 

Examples of significant Jahn-Teller effects are found in complexes of Cr(II) (X), high-spin Mn(III) (X), and Cu(II) (d ). Ni(III) (d ), and low-spin Co(II) should also show this effect, but NiFg is the only known example for these metal ions. It has a distorted structure consistent with the Jahn-Teller theorem. 

Low-spin Cr(II) complexes are octahedral with tetragonal distortion (distorted from Oi, to Z 4/, symmetry). They show two absorption bands, one in the visible and one in the near-infrared region, caused by this distortion. In a pure octahedral field, there should be only one d-d transition (see Chapter 11 for more details). Cr(II) also forms dimeric complexes with Cr — Cr bonds in many complexes. The acetate, Cr2(OAc)4, is an example in which the acetate ions bridge between the two chromiums, with significant Cr—Cr bonding resulting in a nearly diamagnetic complex. 

Besides the splitting of terms observed in the electronic transitions of coordination compounds, other forms of experimental verification of the Jahn-Teller effect include the ESR spectra of coordination compounds. X-ray crystallographic data showing different M-L bond lengths, and thermodynamic data. As an example, consider the stepwise formation constants in Table 16.28 for [Cu(NH3)6] +, which 

Tetragonal distortions from O, symmetry and their effects on the energies of the /-orbitals. The z-out tetragonal distortion is shown at left and z-in tetragonal distortion is shown at right. 

Stepwise stability constants for [M(en)3] coordination compounds of the first-row transition metals. [Reproduced from Figgis, B. N. Hitchman, 

Ligand Field Theory and Its Applications, Wiley-VCH New York, 2000. This material is reproduced with permission of John Wiley Sons, Inc.] 

Using the mathematical forms of the angular wavefunctions for the five d-orbitals and for the orbital given in Table 4.1, as well as the angular vravefunctions for the p, and pj, orbitals given in Equations (4.6) and (4.7) and ignoring the normalization constants, prove that Equations (16.1)-(16.5) are true. 

In high-spin species (to which the theorem is not restricted), perusal of the appropriate configurations for octahedral complexes across the transition period (see Fig. 5-1) shows that d and configurations are candidates for the Jahn-Teller 

Consider the local interactions between d orbitals, referred to a local frame, and various bond orbitals. In Fig. A, we represent such interactions for orbitals characterizing local o 

The driving force for Jahn-Teller distortions in transition-metal complexes is the open d shell. It is likely that explanations for them along the lines given above would have come about even if the theorem of Jahn and Teller had not been discovered. We make this remark not to denigrate that powerful piece of work, but as an attempt to defuse any mystery that might otherwise attach to OrgeTs application of that group-theoretical construction. 

Example The spectrum of [Ti(H20)6] ions, whose ground state geometry is nearly perfectly octahedral, is characterized by a large splitting. 

Only one band maximum is expected, of course, corresponding to the 7 2g— transition. 

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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. 

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. 

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. 

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. 

Vibrational stmcture is, at best, only partially resolved. The stmcture in the first and second systems is complex which could be due, in part, to there being a Jahn-Teller effect in each of them. Such an effect may arise when a molecule is in an orbitally degenerate E (or T) state. The Jahn-Teller effect involves a distortion of the molecule in order to destroy the... 

There are phenomena such as the Renner and the Jahn-Teller effects where the Bom-Oppenheimer approximation breaks down, hut for the vast majority of chemical applications the Born-Oppenheimer approximation is a vital one. It has a great conceptual importance in chemistry without it we could not speak of a molecular geometry. 

The complex ion (Figure 2.32) contains Rh2 bound cis to two phosphorus atoms (2.216 A) and more distantly to four oxygens (2.201—2.398 A), exhibiting a distortion ascribed to the Jahn-Teller effect it is paramagnetic (fi = 1.80 fiB) and exhibits an ESR spectrum (Figure 2.33) showing rhodium hyperfine coupling as the doublet for g. ... 

The Jahn-Teller effect in crystal chemistry and spectroscopy. I. B. Bersuker, Coord. Chem. Rev., 1975,14,357-412(105). 

A minor success is also seen in complexes of d and d" ions, in which the distorted octahedral geometries observed may be rationalized (and indeed predicted) in terms of the Jahn-Teller effect, and ultimately in terms of the steric activity of the open d shell. This is a common feature in copper(n) chemistry, and you will... 

Stable Mn(HI) compounds, Mn(R2r fc)3, have been known for a long time (42, 46). The structure of Mn(Et2C tc)3 is elucidated (47). The inner geometry of the Mn(CS2)3 core does not conform to the usual D3 point symmetry of transition metal complexes of this type, but shows a strong distortion attributed to the Jahn-Teller effect. The electronic spectrum (48, 49) and the magnetic properties of this type of complexes are well studied (50). 

Arcon D, Blinc R (2004) The Jahn-Teller Effect and Fullerene Ferromagnets 109 231-276 Arman HD, see Pennington WT (2007) 126 65-104... 

The Jahn-Teller effect is always to be expected when degenerate orbitals are unevenly occupied with electrons. In fact, it is observed for the following electronic configurations ... 

In Fig. 9.4 the additional stabilization by the Jahn-Teller effect has not been taken into account. Its inclusion brings the point for the (distorted) octahedral coordination for Cu2+ further down, thus rendering this arrangement more favorable. 

Distortions due to the Jahn-Teller effect. For example CdBr2 (Cdl2 type) —t CuBr2 (distorted Cdl2 type). 

Bersuker IB (1984) The jahn-teller effect and vibronic interactions in modern chemistry. Plenum Press, New York... 

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