M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 47
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The out-of-plane vibration—that would make themolecule pyramidal—is of a2 symmetry. However, there is no irreducible representation with a2 symmetry among the irreducible representations that the direct product of E with itself reduces to (see,above). Therefore, a C3v -type of distortion is not possible—at leastnot in the Jahn–Teller manner.Coming back to the C2v -symmetry distortion of metal trihalides;the e -symmetry angle bending vibration can distort the molecule intwo ways. Either one of the bond angles decreases while the oppositebond lengthens, or the other way around, the bond angle increases andthe opposite bond shortens (Figure 6-43). In fact, both structures arereal.
Figure 6-44 shows the potential energy surface (see Chapter 7)of the AuCl3 molecule. This is a so-called Mexican-hat type potentialenergy surface. The high-energy point at the tip of the hat correspondsto an undistorted D3h -symmetry structure. The surface of the brim ofthe hat warps producing three wells separated by three humps of equalheight. The three wells correspond to the minima with a structure inwhich one of the bond angles opens and the opposite bond shortens(Figure 6-43 left; there are three of these structures because the samedistortion might happen involving each of the three bond angles andthe opposite bonds). The three humps correspond to the so-calledtransition-states (see Chapter 7 in more detail), and these structureshave one smaller bond angle with an elongated bond opposite to itas seen in Figure 6-43 right.
As Figure 6-44 shows, the Jahn–TellerFigure 6-43. The two types of structures that might result from an e -type vibrationof a metal trihalide molecule.3046 Electronic Structure of Atoms and MoleculesFigure 6-44. The “Mexican-hat” potential energy surface of the AuCl3molecule [84], copyright 2001, American Chemical Society.distortion stabilizes the lower-symmetry structures compared to thehigh-symmetry one.Quantum chemical calculations revealed that there are two possibleE symmetry electronic states for a D3h -symmetry gold trihalide. Inone of them (high-spin state) there is one electron on each of thetwo e orbitals with parallel spins (see Figure 6-45); this is the onethat should be the ground state according to Hund’s rule.
However,symmetry lowering would not bring about any energy gain; with oneorbital going up and the other down with no net change. The otherE -symmetry state (low-spin) has two electrons with opposite spinsin one of the two e orbitals and is less stable than the high-spinstate. However, the distortion of this low-spin state (see Figure 6-45)Figure 6-45. High-spin and low-spin electronic configuration of gold trihalides andthe splitting of the low-spin E state of gold trihalides.6.6. Jahn–Teller Effect305brings about a substantial energy gain.
In fact, the energy loweringis so large that the distortion of this higher energy state produces theoverall lowest energy structure that will be the ground state; this is theC2v -symmetry structure in Figure 6-43 left. The energy gain by theJahn–Teller distortion of the less stable trigonal planar structure is solarge that it can “pay the price” for the spin pairing and still producean overall lower-energy structure.At this point, it is of interest to quote Edward Teller about thediscovery of the Jahn–Teller effect (Figure 6-46) [85]:This effect had something to do with Lev Landau.I had a German student in Göttingen, R.
Renner,and he wrote a paper on degenerate electronicstates in the linear carbon dioxide molecule,assuming that the excited, degenerate state ofcarbon dioxide is linear.Figure 6-46. Edward Teller (courtesy of Lawrence Livermoore Nartional Laboratory) and Lev Landau (courtesy of Alexei Abrikosov, from A.A. Abrikosov:Academician L.
D. Landau: Short Biography and Review of his Scientific Work,Nauka, Moscow, 1965, in Russian).3066 Electronic Structure of Atoms and MoleculesIn the year 1934 both Landau and I were in NielsBohr’s Institute in Copenhagen and we had manydiscussions. He disagreed with Renner’s paper, hedisliked it. He said that if the molecule is in adegenerate electronic state then its symmetry willbe destroyed and the molecule will no longer belinear.
Landau was wrong. I managed to convincehim and he agreed with me. This was probably theonly case when I won an argument with Landau.A little later I went to London, and met Jahn.I told him about my discussion with Landau, andabout the problem in which I was convinced thatLandau was wrong. But it bothered me that he wasusually not wrong. So maybe he is always rightwith the exception of linear molecules. Jahn wasa good group-theorist, and we wrote this paper,the content of which you know, that if a moleculehas an electronic state that is degenerate, then thesymmetry of the molecule will be destroyed.
Thatis the Jahn–Teller theorem.The Jahn–Teller theorem has a footnote: thisis always true with the only exception of linearmolecules. So the amusing story of the Jahn–Teller effect is that I first worked with mystudent, Renner, on a paper that presented theonly general exception to the Jahn–Teller effect. Itreally should be the Landau–Jahn–Teller theorembecause Landau was the first one who expressedit, unfortunately using the only exception where itwas not valid.Linear molecules are the only exception to the Jahn–Teller effect.But linear molecules may also have instabilities in their degenerateelectronic states and this is called the Renner–Teller effect. It wasfirst described by Renner in a theoretical paper on the degeneratefirst excited electronic state of carbon dioxide [86].
It took more thantwenty years to find the first experimental evidence of this effect, inthe electronic absorption spectrum of the NH2 radical [87]. The NH2radical has one electron on a orbital and thus a ⌸ electronic state6.6. Jahn–Teller Effect307in its ground state, where it is bent; while it is linear in the excitednondegenerate state.Another, recently found, example of the Renner–Teller effect isthe chromium dichloride, CrCl2 , molecule [88].
Based on high-levelcomputations and electron diffraction experiments, it was found thatthe gas-phase molecule is not linear as could be expected for a transition metal dihalide but rather bent. According to the computations,among the different high-spin electronic states, the ⌸g twofold degenerate state is the lowest in energy. However, it is not stable andsplits into two nondegenerate states, of B2 and A2 symmetry, respectively, of which the B2 state is the ground state with a bond angle ofabout 147◦ .Chromium in its dichloride has a d4 electronic configuration and inits crystals is subject to the Jahn–Teller effect. Indeed, chromium hasa tetragonally distorted octahedral coordination in its crystals [89].CrCl2 is a fascinating molecule in that it displays both the Jahn–Tellerand the Renner–Teller effects.Another structural phenomenon related to the Jahn–Teller effectis the pseudo-Jahn–Teller effect.
This happens when two electronicstates of a molecule, the ground state and an excited state are closeenough in energy and thus, they can mix under nuclear displacements[90]. The pseudo-Jahn–Teller effect can appear separately from thereal Jahn–Teller effect, or together with it; their magnitude can alsobe either very small, or substancial.It is stressed that the physical bases for the Jahn–Teller effectand the pseudo-Jahn–Teller effect are quite different. Jahn–Tellerdistortion occurs due to the coupling of the electronic and vibrationalmotions of the molecule; i.e., the coupling of a degenerate electronicwave function with the vibrational wave function.
In case of thepseudo-Jahn–Teller effect, the vibronic interaction happens betweentwo electronic states that are close in energy and are not necessarilydegenerate (although because of their similar energies we mightconsider them “pseudo-generate”[91]); i.e., here the vibronic couplingis between two electronic wave functions. The effect pushes the twostates apart. The two states must belong to the same irreduciblerepresentation of the new point group as before and can continue tointeract, which, obviously, is not the case with the Jahn–Teller effect.In concluding this section, we mention that Bersuker points to avery general applicability of the Jahn–Teller effect, much beyond3086 Electronic Structure of Atoms and Moleculesmolecules [92].
Obviously, the Jahn–Teller effect, degeneracy, andsymmetry breaking are inherently related to each other. Thus, it musthave been phenomena triggered by a Jahn–Teller type coupling ofdegenerate states to the motions of particles that led to the symmetrybreaking after the Big Bang that, eventually, led to the formation ofthe Universe.References1.
A. L. Mackay, A Dictionary of Scientific Quotations, Adam Hilder, Bristol1991, p. 57/85.2. I. N. Levine, Quantum Chemistry, Sixth Edition, Prentice Hall, Upper SaddleRiver, New Jersey, 2008.3. P. Atkins, R. Friedman, Molecular Quantum Mechanics, Fourth Edition,Oxford University Press, New York, 2005.4. D. V. George, Principles of Quantum Chemistry, Pergamon Press, New York,1972.5. M. W. Hanna, Quantum Mechanics in Chemistry, Second Edition,W. A. Benjamin, New York, Amsterdam, 1969.6.
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