M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 45
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The orientation of the different symmetry d orbitals in a cubicenvironment.6.5. Influence of Environmental Symmetry293Figure 6-36. Relative energies of the d orbitals in octahedral and cubic ligandenvironment.Figure 6-37. The d orbital splittings in different ligand environments.2946 Electronic Structure of Atoms and MoleculesFigure 6-38. The variation of octahedral M2+ ionic radii [51].non-uniform distribution of d electrons around the nuclei is the originof this phenomenon.
In the octahedral environment the d orbitals splitinto orbitals with t2g and eg symmetry. The electrons, added gradually, occupy t2g orbitals in Sc2+ , Ti2+ , and V2+ as well as in Fe2+ , Co2+ ,and Ni2+ , if only high-spin configurations are considered. Since theseorbitals are not oriented towards the ligands, the degree of shieldingbetween the ligands and the positively charged atomic cores decreasesalong with the ionic radius. The fourth electron in Cr2+ as well as theninth electron in Cu2+ occupy eg symmetry orbitals. The degree ofshielding thus somewhat increases and, accordingly, there is a smallerrelative decrease in the ionic radii.6.6. Jahn–Teller Effect“Somewhat paradoxically, symmetry is seen to play an importantrole in the understanding of the Jahn–Teller effect, the very natureof which is symmetry destruction” [52].
In a recent review the original paper published by Jahn and Teller [53] was called “one of themost seminal papers in chemical physics” [54]. Only a brief discussion of this effect will be given here; for more detail we refer thereader to References [55–59]. Bersuker says that all structural instabilities and distortions of high-symmetry configurations of polyatomicsystems are of Jahn–Teller origin (here he also refers to other relatedeffects, such as the Renner–Teller effect and the pseudo-Jahn–Tellereffect—they will be mentioned later). Bersuker likes to call this the6.6.
Jahn–Teller Effect295“Jahn–Teller approach” and he considers the usual formulation ofthe effect somewhat obsolete [60]. Here, we will mostly discuss theJahn–Teller effect according to its conventional meaning, but willalso discuss briefly Bersuker’s approach. According to the originalformulation of the Jahn–Teller effect, [61] a non-linear symmetricalnuclear configuration in a degenerate electronic state is unstable andgets distorted, thereby removing the electronic degeneracy until anon-degenerate ground state is achieved. This formulation indicatesthe strong relevance of this effect to orbital splitting and generallyto the relationship of symmetry and electronic structure discussed inprevious sections.
Owing to the coupling of the electronic and vibrational motions of the molecule, the ground-state orbital degeneracy isremoved by distorting the highly symmetrical molecular structure toa lower-symmetry structure. An important aspect of the Jahn–Tellereffect is that it represents an exception to the Born-Oppenheimerapproximation (see in Section 6.3.1) since it involves the couplingof the electronic and nuclear motions in the molecule.
Due to thismixing, a Jahn–Teller distorted molecule is expected to be inherentlydynamic.Jahn–Teller distortion can only be expected if the energy integral ⭸E (6-6)⌿0 ⌿0⭸qhas nonzero value (⌿0 is the ground-state electronic wave function ofthe high-symmetry nuclear configuration, and q is a normal mode ofvibration). According to what has already been said about the valueof an energy integral (Section 4.9.2), this can only happen if the directproduct of ⌿0 with itself is, or contains, the irreducible representationof the q normal mode of vibration:⌫⌿0 · ⌫⌿0 ⊂ ⌫q(6-7)Since ⌿0 is degenerate, its direct product with itself will alwayscontain 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 thedirect product of ⌿0 with itself. A vibration belonging to the totallysymmetric representation, however, does not decrease the symmetry2966 Electronic Structure of Atoms and Moleculesof the molecule. Accordingly, in order to have a Jahn–Teller typedistortion, q must belong to one of the other irreducible representations.Let us see an example, the H3 molecule, which has the shape of anequilateral triangle. Its symmetry is D3h , the electronic configurationis a12 e , and the symmetry of the ground electronic state is E .
Thus,the electronic state of the molecule is degenerate and is subject toJahn–Teller distortion.The symmetry of the normal mode of vibration that can take themolecule out of the degenerate electronic state will have to be suchas to satisfy Eq. (6-7). The direct product of E with itself (seeTable 6-11) reduces to A1 + A2 + E . The molecule has three normalmodes of vibration [(3 × 3) – 6 = 3], and their symmetry species areA1 + E .
A totally symmetric normal mode, A1 , does not reduce themolecular symmetry (this is the symmetric stretching mode), and thusthe only possibility is a vibration of E symmetry. This matches one ofthe irreducible representations of the direct product E · E ; therefore,this normal mode of vibration is capable of reducing the D3h symmetryof the H3 molecule. These types of vibrations are called Jahn–Telleractive vibrations.The two E symmetry vibrations of the H3 molecule are the anglebending and the asymmetric stretching modes (see Figure 6-39). Theylead to the dissociation of the molecule into H2 and H.
Indeed, H3 is sounstable that it cannot be observed as it would immediately dissociateinto H2 and H. This is one of the reasons why it has been so difficultto find experimental evidence of the Jahn–Teller effect for quite sometime. The structures that are predicted to be unstable are often notfound, and the observed structures are so different from them that theTable 6-11. The D3h Character Table and the Reducible Representation E · ED3hE2C33C2h2S33A1A2EA1A2E11211211–111–11–101–10112–1–1–211–1–1–111–10–110E ·E410410x2 +y2 , z2Rz(x, y)(x2 –y2 , xy)z(Rx , Ry )(xz, yz)= A1 + A2 + E 6.6.
Jahn–Teller Effect297Figure 6-39. The two E symmetry normal modes of vibration of the H3 moleculeleading to dissotiation.connection is not obvious (other reasons of the difficulty encounteredin observing the Jahn–Teller effect will be given later).Obviously, only molecules with partially filled orbitals displayJahn–Teller distortion. As was shown in Section 6.3.2, the electronicground state of molecules with completely filled orbitals is alwaystotally symmetric, and thus cannot be degenerate. In comparison withthe above-mentioned unstable H3 molecule, H+3 has only two electrons in an a1 symmetry orbital; therefore, its electronic ground stateis totally symmetric, and the D3h -symmetry triangular structure ofthis ion is stable (see, e.g., Reference [62]). On the other hand, takethe benzene molecule, e.g., whose ground electronic state is of A1gsymmetry and the molecule is stable and its structure is well understood.
At the same time, in its cation, C6 H+6 , it loses one electronfrom an e1g -symmetry doubly-degenerate orbital, so that orbital isleft with only one electron. The electronic state of the cation has E1gsymmetry and thus, it is subject to Jahn–Teller effect. Indeed, its vibrational spectrum is extremely complicated and can only be satisfactoryexplained if the Jahn–Teller distortion is taken into consideration (see,e.g., Reference [63]).Transition metals have partially filled d orbitals, and therefore theircompounds are obvious candidates for Jahn–Teller systems.
Let usconsider an example from among the much studied cupric compounds[64]. Suppose that the Cu2+ ion with its d9 electronic configurationis surrounded by six ligands in an octahedral arrangement. We havealready seen (Table 6-10 and Figure 6-36) that the d orbitals split intoa triply (t2g ) and a doubly (eg ) degenerate level in an octahedral environment. For Cu2+ the only possible electronic configuration is t2g6 eg3 .2986 Electronic Structure of Atoms and MoleculesSuppose now that of the two eg orbitals, dz 2 is doubly while dx 2 −y 2is only singly occupied. Thus, the two ligands along the z axis arebetter screened from the electrostatic attraction of the central ion,and will move farther away from it, than the four ligands in the xyplane. The opposite happens if the unpaired electron occupies the dz2orbital.
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