M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 46
Текст из файла (страница 46)
In both cases the octahedral arrangement undergoes tetragonal distortion along the z axis, in the former by elongation, while inthe latter by compression. The original Oh symmetry reduces to D4h .The symmetry-reducing vibrational mode here is of eg symmetry andhas the form shown in Figure 6-40. The splitting of d orbitals in bothenvironments is given in Table 6-10 and is also shown here:Oheg(dx 2 −y 2 , dz 2 )t2g(dx z , d yz , dx y )→→→a1g(dz 2 )eg(dx z , d yz )D4h++b1g(dx 2 −y 2 )b2g(dx y )Figure 6-41 illustrates the tetragonal elongation and compressionof an octahedron. For the Cu2+ ion the relative energies of the dz2 anddx 2 −y 2 orbitals depend on the location of the unpaired electron.Consider now a qualitative picture of the splitting of the t2gorbitals.
If the ligands are somewhat further away along the zaxis, their interaction with the dxz and dyz orbitals will decrease,and so will their energy compared with that of the dxy orbital.This is illustrated by the left-hand side of Figure 6-41. TetragonalFigure 6-40. The symmetry-reducing vibrational mode of eg symmetry for anoctahedron.6.6. Jahn–Teller Effect299Figure 6-41. Tetragonal distortion of the regular octahedral arrangement arounda d 9 ion.compression can be accounted for by similar reasoning (cf. right-handside of Figure 6-41).The splitting of the d orbitals in Figure 6-41 shows the validity ofthe “center of gravity rule.” One of the eg orbitals goes up in energy asmuch as the other goes down.
From among the t2g orbitals, the doublydegenerate pair goes up (or down) in energy half as much as the nondegenerate orbital goes down (or up). Thus, for the Cu(II) compoundsthe splitting of the fully occupied t2g orbitals does not bring about a netenergy change. The same is true for all other symmetrically occupieddegenerate orbitals, such as t32g , e4g , or e2g . On the other hand, the occupancy of the eg orbitals of Cu2+ is unsymmetrical, since two electronsgo down and only one goes up in energy, and here there is a net energy3006 Electronic Structure of Atoms and Moleculesgain in the tetragonal distortion. This energy gain is the Jahn–Tellerstabilization energy.The above example referred to an octahedral configuration. Otherhighly symmetrical systems, for example, tetrahedral arrangements,can also display this effect.
For general discussion, see, e.g. References [65–67].The Jahn–Teller effect enhances the structural diversity of Cu(II)compounds [68]. Most of the octahedral complexes of Cu2+ , forexample, show elongated tetragonally distorted geometry. Crystallinecupric fluoride and cupric chloride both have four shorter and twolonger copper–halogen interatomic distances, 1.93 vs. 2.27 Å and 2.30vs. 2.95 Å, respectively [69].The square planar arrangement can be regarded as a limiting case ofthe elongated octahedral configuration. The four oxygen atoms are at1.96 Å from the copper atom in a square configuration in crystallinecupric oxide, whereas the next nearest neighbors, two other oxygenatoms, are at 2.78 Å. The ratio of the two distances is much largerthan in the usual distorted octahedral configuration [70].Tetragonal compression around the central Cu2+ ion is muchrarer; K2 CuF4 is an example with two shorter and four longer Cu–Fdistances, viz., 1.95 vs.
2.08 Å [71].There are also numerous cases when experimental investigationfailes to provide evidence for Jahn–Teller distortion. For example,several chelate compounds of Cu(II), as well as some compoundscontaining the [Cu(NO2 )6 ]4– ion, show no detectable distortion fromthe regular octahedral structure (see Reference [72] and referencestherein).Bersuker [73–75] has shown the need for a more sophisticatedapproach to account for such phenomena. We attempt to convey atleast the flavor of his ideas here.
Jahn–Teller distortions are of adynamic nature in systems under no external influence. This meansthat there may be many minimum-energy distorted structures in suchsystems. Whether an experiment will or will not detect such a dynamicJahn–Teller effect, depends on the relationship between the time scaleof the physical measurement used for the investigation, and the meanlifetime of the distorted configurations. If the time period of themeasurement is longer than the mean lifetime of the distorted configurations, only an average structure, corresponding to the undistorted6.6. Jahn–Teller Effect301high-symmetry configuration will be detected. Since different physical techniques have different time scales, one technique may detect adistortion which appears to be undetected by another.The static Jahn–Teller effect can be observed only in the presenceof an external influence. Bersuker [76, 77] stresses this point as theopposite statement is found often in the literature.
According to thestatement criticized, the effect is not to be expected in systems wherelow-symmetry perturbations remove electronic degeneracy. However,it is exactly the low-symmetry perturbations that make the Jahn–Tellerdistortions static and thus observable. Such a low-symmetry perturbation can be the substitution of one ligand by another. In this case one ofthe previously equivalent minimum-energy structures, or a new one,will become energetically more favored than the others.The so-called cooperative Jahn–Teller effect is another occuranceof the static distortions. Here, interaction, that is, cooperation betweendifferent crystal centers, make the phenomenon observable. Withoutinteraction, the nuclear motion around each center would be independent and of a dynamic character.Lattice vibrations tend to destroy the correlation among Jahn–Teller centers.
Thus, with increasing temperature, these centers maybecome independent of each other at a certain point, and their staticJahn–Teller effects convert to dynamic ones. At this point the crystalas a whole becomes more symmetric. This temperature-dependentstatic ⇔ dynamic transition is called a Jahn–Teller phase transition.
Below the temperature of the phase transition, the cooperativeJahn–Teller effect governs the situation providing static distortion; theoverall structure of the crystal is of a lower symmetry. Above thistemperature, the cooperation breaks down, the Jahn–Teller distortionbecomes dynamic and the crystal itself becomes more symmetric.The temperature of the Jahn–Teller phase transition is very highfor CuF2 , CuCl2 , and K2 CuF4 among the examples mentioned above[78]. Therefore, at room temperature their crystal structures displaydistortions.
Other compounds have symmetric crystal structures atroom temperature as their Jahn–Teller phase transition occurs atlower temperatures. Cupric chelate compounds and [Cu(NO2 )6 ]4–compounds, such as K2 PbCu(NO2 )6 and Tl2 PbCu(NO2 )6 , can bementioned as examples [79]. Further cooling, however, may makeeven these structures distorted.3026 Electronic Structure of Atoms and MoleculesOur last examples of Jahn–Teller distortion are from among gasphase metal halides, manganese trifluoride and the gold trihalides.Both manganese and gold have partially filled d orbitals in theirtrihalides; manganese has four and gold eight d electrons.
Manganesetrifluoride is one of the typical Jahn–Teller cases in its crystal witha strongly elongated octahedral arrangement of the fluorine atomsaround manganese. Without the ramifications of the Jahn–Tellereffect, we would expect a tigonal planar geometry of D3h symmetryfor the gas-phase molecules of these trihalides. However, both experiments and computations found that these gas-phase molecules have alower, C2v symmetry structure [80–83].
In the assumed D3h -symmetrystructure, these molecules have a partially filled e -symmetry orbitaland an E electronic state. Using the same line of thought as for the H3molecule, the direct product of E with itself reduces to A1 + A2 + E .The four-atomic metal trihalides have 6 normal modes of vibration[(3 × 4) – 6 = 6] with symmetries A1 + A2 + 2E (as we discussedearlier, each doubly-degenerate vibration, E , counts as two). We cansee that there is no A2 -symmetry irreducible representation among theones that the direct product of the ground electronic state symmetry,E , with itself reduces to. There are two matches between the irreducible representations and the normal modes of vibration: A1 and E .The totally symmetric A1 vibration cannot take out the molecules fromthe D3h -symmetry structure, so the only possibility is one of the two Evibrations (see Figure 6.42), just as was the case with the H3 molecule.However, while for the H3 molecule both of these vibrations resultedin dissociation, for these metal trihalides, the angle bending vibrationresults in a stable structure of C2v symmetry that is of lower energyFigure 6-42.
The two e symmetry normal modes of vibration of a metal trihalidemolecule.6.6. Jahn–Teller Effect303than the higher symmetry D3h structure. This is why C2v symmetry isobserved as the ground-state structure of these metal trihalides.The question might arise whether it would not be more “straightforward” for the MnF3 molecule to distort into a structure of C3vsymmetry? The D3h character table (Table 6-11) helps us to understand the reason.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
