The Jahn-Teller theorem

SpectrochimicaActa, 1962, Vol. 18, pp. 775 to 789. PergamonPress Ltd. Printedin NorthernIreland

The J&n-Teller theorem* NORMAN S. HAM Division of Chemical Physics, C.S.I.R.O. Chemical Research Laboratories Melbourne, Australia

(Received

10 November1961)

INTRODUCTION

Jahn-Teller theorem states that a symmetrical polyatomic molecule in an orbitally degenerate electronic state will distort to a conformation of lower energy, unless the molecule is linear [l]. A molecule in which the total electronic state of orbital and spin motion is degenerate is also unstable, unless the spin degeneracy is two-fold as in the case of an odd number of electrons [S]. These general statements have considerable relevance for- the structural parameters, magnetic properties and spectra of such systems. JAH-N and TELLER did not give any examples to illustrate their theorem and VAN VLECK in 1939 was the first to obtain numerical estimates of the theorem’s effects [3]. The theorem was occasionally mentioned over the next 15 years but since 1955 there has been renewed theoretical and experimental interest in it.? This review will give a short account of the theorem’s predictions and will discuss some of the more clear-cut examples in the light of these predictions.

THE

PREDICTIONS

OF THE

THEOREM

The static problem The origin of the distorting forces can be understood in terms of the following simple electrostatic picture [5]. Consider a molecule XY,, in which the central atom or ion X is co-ordinated octahedrally via its d-electrons with the six ligands Y. Fig. 1 shows the angular dependence, with respect to an octahedron, of the five independent charge distributions of a d-electron. The five-fold degeneracy of the d-orbitals is divided into a two-fold degeneracy, an e, group, with charge distribution directed towards the corners of the octahedron or towards the ligands, and a three-fold degeneracy, a t,, group, with the charge distribution directed between the ligands. Now consider a central ion in which there is a hole in the eg. group, as for example the cl9 system in Cu2+. With one of the e, orbitals unfilled, * An invited review paper presented in Sydney at the Third Australian Spectroscopy Conference, 22-24 August 1961. t A good background discussionto the Jahn-Teller theorem is given in the introduction to ~PIK and PRYCE'S survey of the static problem in Ref. 4. [l] [S] [3] [4] [5]

H. A. Jm and E. TEUER, Proc. Roy. Sot. A161, 220 (1937). H. A. JAEN, PTOC. Roy.Soc. Alf34, 117 (1938). J. H. VAN VLECK, J. Chem. Phys. 7, 61 (1939); 7, 72 (1939). U. ~PIK and M. H. L. PRYCE, Proc. Roy. Sot. A238, 425 (1957). L. E. ORGEL, An Introduction to Transition-Metal Chemidry: Ligancl-Field p. 57. Methuen, London (1960). 775

Theory

(1st Ed.)

‘776

N. S. HB

there is less shielding between the negatively charged ligands and the positively charged central ion and the ligands will move therefore towards the central atom. Either two of them will move towards the central atom or four of them will move towards the central atom. It is impossible with this simple argument to say which arrangement is preferred but it is expected that in those ions in which the charge distribution is directed towards the ligands, the octahedron will be distorted so that it ends up with two long and four short bonds or four long and two short bonds. If however the central ion has a charge distribution of the tss type the

Fig. 1. Three-dimensional polar diagram showing the combined 0 and + probability distribution for a single S&electron. The six ligands at the corners of an octahedron define the 2, y, z axes. The total electron-cloud density is obtained by multiplying the polar factor by the radial probability factor at each point. The upper two distributions form the e, orbitals and the lower three distributions the tzv orbitals. The tzs orbitals are more stable than the e, ones.

distorting effect will be much smaller since this charge distribution does not point directly at the ligands. The conclusions drawn from. this picture are that an octahedral molecule or complex with a doubly degenerate charge distribution as in an E state should show a tetragonal distortion, while a molecule in a triply degenerate state will distort to a very much smaller extent. The forces tending to remove spin degeneracy will be much smaller again since these can only act via spin-orbit coupling. However they might be larger for the elements higher in the periodic table. JAHN and TELLER gave a group-theoretical proof of their theorem. The energy levels of the distorted conformation were obtained from a perturbation matrix, whose elements were expressed as a power series in the nuclear displacements. These elements were evaluated up to terms linear in the displacements using electronic wave-functions in which the nuclear co-ordinates appeared as parameters, and it was shown that for all molecules, except linear ones, there is always at least one non-totally symmetric displacement which will lower the energy of a degenerate electronic state. Fig. 2 shows the symmetry of the appropriate displacements for molecules with the symmetries 0, and D,,. In 0, symmetry a

The Jahn-Teller

711

theorem

doubly degenerate displacement e, will distort a doubly degenerate or a triply degenerate state, whereas a triply degenerate displacement tzg will only distort a triply degenerate state. For D,, symmetry two eZ8 displacements will distort either an El or an E, state, the only doubly degenerate states in D,,. However as the distortions increase towards the zero-point amplitude of a typical vibration and the distortion energy approaches the zero-point vibrational energy, coupling between the electronic and vibrational motions will occur [6, 71.

129

e9

distorts

distorts E,T states

T stotes

Dsll

e2, distorts

Fig. 2. Jahn-Teller distortions Only one component

E, , E, stotes

for octahedral (0,) and hexagonal (II,& molecules. of each degenerate displacement is shown.

The dy?zasnical problem The effects in this coupled or dynamic situation can be shown by examining an ideal case [8]. Consider an electron moving on a circle with one unit of orbital angular momentum; its motion can be either in a clockwise or anti-clockwise direction, so its electronic state, neglecting spin, is two-fold degenerate. Then consider the nuclear framework undergoing two-dimensional elliptical distortions in a plane, i.e. the nuclei move as a two-dimensional harmonic oscillator. The vibrational energy values are determined by the parabolic energy sheet shown in Fig. 3(a). With complete independence between the electronic and vibrational motions, the total energy is then merely the sum of the electronic and the vibrational energy. However, as the amplitude of the distortions or the vibrations increases, the electronic and vibrational motions may mix and in this situation the [6] W. MOFFITT and W. THORSON, Phys. Rev. 108,1251 (1957). [7] W. MOFFITT and A. D. LIEHR, Phye. Rev. 8, 1195 (1957). [8] H. C. LONGUET-HIGGINS, U. ~PIK, M. H. L. PRYCE and R. A. SACK, Proc. Roy. Abe. AM, 1 (1958).

N. S. HAM

778

combined motion is determined by the two potential surfaces shown in Fig. 3(b). The coupled motion, or the coupled potential determining the motion, still has circular symmetry and the whole system is still two-fold degenerate. Higher approximations for representing the coupling terms add periodic bumps to both of these sheets [9]. Now various cases arise depending on the relative magnitudes of (a) the distortion energy representing the separation of the sheets, (b) the vibrational zero-point energy or its height above the bottom of the potential sheets, (c) the height of the bumps. If the masses are large the motion is restricted to the bottom

FIG 3a.

FIG 3b

Fig. 3(a). Nuclear potential energy surface for two-dimensional harmonic oscillator. Fig. 3(b). Nuclear potential energy surfaces for two-dimensional harmonic oscillator with the electronic motion coupled to the vibrational motion.

potential sheet and below the bumps, so that this case corresponds to a distorted conformation. We expect to see this distortion in X-ray structures or spectral properties. If however the nuclei can tunnel through the bumps by virtue of their vibrational energy, then any measured property may still show isotropy, or symmetry, at high temperatures but perhaps may show anisotropy, or distortions, at lower temperatures. If the zero-point energy is about the same as the distortion energy, then obviously the motion is not restricted to either sheet. It is no longer possible to write the total wave-function as a product of an electronic and a vibrational wave-function and the possibility arises that peculiar vibrational effects will be seen in infra-red and Raman spectra, and in the vibrational selection rules for an allowed electronic transition involving a degenerate electronic state. Normally the vibrational selection rules for an allowed electronic transition A t) E are that totally symmetrical vibrations may appear in single quanta but doubly degenerate vibrations may only appear with double quanta. This latter selection rule is changed in the coupled situation, so that odd quanta of doubly degenerate vibrations may appear [S, lo]. [9] A. D. LIERR, Revs. Modem Phys. 32, 436 (1960). [lo] A. D. LIEHR and W. E. MOFFITT, J. Chem. Phy. 25, 1074 (1956).

The Jahn-Teller

theorem

719

EXAMPLES OF DEGENERATE GROUND STATES In discussing examples it will be convenient to distinguish degenerate ground states and degenerate excited states. For the ground states we can look at such properties as X-ray structural data, paramagnetic resonance spectra, and infrared and Raman spectra. 3d transition-metal

ions

Much information is available for the 3d transition metals in octahedral environments, e.g. the central ion surrounded by six ligands such as fluoride or oxide ions or water molecules. For these three ligands their octahedral complexes are Table 1. High-spin ground states of 3d transitionmetal ions in octahedral co-ordination

Symmetry

;: d3 d4

d5 de d7 ds dQ

2T28 3TIg 4A2s 5E,

6A1, 5T2, 4T 18 3A2s 2-%

Expected distortion via orbital degeneracy

small small 0

large 0 small small 0 large

generally of the high-spin type, that is the central ion generally has a maximum number of unpaired electrons. In Table 1 the high-spin ground states for the d” configurations in octahedral co-ordination are listed. The doubly degenerate E, state is the ground state for d4 and dg configurations only. This is the charge distribution which points towards the ligands and is more susceptible to tetragonal distortions than the triply degenerate T,, or T,, charge distributions. Therefore large distortions are expected for cl4 and da configurations, smaller ones for dl, d2, 8 and d7 and none for the non-degenerate ds and ds systems [ 111. Structural data. Table 2 summarizes some of the information for the transition metal ions surrounded octahedrally by fluoride ions, as deduced from X-ray diffraction studies. The first column summarizes the information for the difluorides, MF,. These have the rutile structure in which there are two sets of fluorines, four in one set in one plane and two in the other set in a perpendicular plane [lS, 13, 14, 151. The figures indicate the difference in M-F bond lengths for the set of four fluorines minus the set of two fluorines. For magnesium (CaF, does not have the rutile structure), manganese, cobalt, nickel and zinc this difference is [ 111 [ 121 [13] [14] [15]

J. K. C. W. J.

D. DUNITZ and L. E. ORGEL, J. Phys. Chem. Sobida 3, 20 (1957). H. JACK and R. MAITLAND, Proc. Chew. Sot. 232 (1957). BILLY and H. M. HAENDLER, J. Am. Chem. Sot. 79, 1049 (1957). H. BAXJR,Acta Cry&. 9, 515 (1956); 11, 488 (1958). W. STOUT and R. 0. SRULMAN, Phys. Rev. 118, 1136 (1960).

d'O

0.016

FeFs CoF,

*s CrF,

.

(2.09(2)

MnF bond lengths

structure

TiF,

VF,

KZnF,

J-*3

KFeF, KCoF, KNiF,

Ideal perovskite structure CuF bond lengths

KsNiF,

A[(4) -

0.03

WI

Fluorospinel structure

comnounds

A[(4) - (2)] is the difference in MF bond length for the set of four fluorines minus the set of two fluorines.

Zfi,

(CuF,I -0.34

PI

0.01 0.13 0.004 0.01

-0.43

0.015

MnF, FeF, CoF, NiF,

ICrF,I

WFz

WI

OF d6 d' d6

pi

cl3

;:

do

A[(4) -

Rutile structure with pn79”

Table 2. Structural data for some 3d transition-metal

The Jahn-Teller theorem

781

small and positive, but for the d4 and dQ difluorides, chromous and cupric, it is very definitely negative, and in these two cases there are two long bonds and four short bonds. For the trifluorides [16] the octahedron is regular for titanium, chromium, iron and cobalt, but for manganese there are three pairs of metal fluorine distances, that is, the d4 configuration shows anomalies again. Similarly for the KMF, systems, which crystallize in the cubic ideal perovskite structure, single crystal studies on KCuF, show that the dQ configuration has an elongated octahedron of fluoride ions [17, 181. For the K,MF, systems, the fluorospinels, the dQ configuration has four long bonds and two short bonds, a distortion in the opposite direction from that observed in the difluorides [19]. This pattern of structures seems to fit the predictions that the distortions should be large for d4 and dQ configurations and small for the others. However there are probably other effects to be considered also as shown by the figures for FeF, and CoF,, (both expected to show small distortions), and MnF, and KCuF,, where three pairs of lengths are unexpected. For transition-metal ions surrounded octahedrally by ligands other than fluoride there is not much structural information on d4 complexes but ORGEL and DUWITZ concluded that for a large number of Cu 2+ dQ complexes the more usual situation was four short and two long bonds, rather than the four square co-planar arrangement. Again the oxides MO of the transition metals all have the rock salt structure CrO chromium oxide is not at high temperature except Cu2+. Unfortunately known [20]. Paramagnetic resonance spectra. The transition-metal ions and their complexes, with unfilled d-shells and incompletely compensated angular momentum, are often and measurement of the paramagnetic resonance spectra on paramagnetic, single crystals, or effectively the Zeeman splitting of the ground state in an external magnetic field, gives a very sensitive indication of the local symmetry of the paramagnetic complex. There is a particularly interesting set of measurements by Low [21] on the paramagnetic resonance spectra of the transition-metal ions incorporated substitutionally into a single crystal of magnesium oxide, that is, the transition-metal ion is in an octahedral arrangement surrounded by oxide ions. To discuss the paramagnetic resonance spectra, the states of the complex must be calculated with the inclusion of the spin-orbit coupling, since for many of the ions the spin angular momentum is the only angular momentum present in the ground state. Table 3 shows the ions that have been studied in this manner. The ground states for the configurations d3, d5, d6, cl? and d8 have the degeneracies 4, 6, 3, 2 and 3 respectively. The only ones of this set which have orbital degeneracy are the d6 and d7 systems; the others, d3, d5, and d8 have only spin degeneracy, so that [16] R. D. PEACOCK, Progress in Inorganic Chemistry Vol. 2, p. 193. Interscience, New York (1960). [17] A. OEAZAKI and Y. SUEMUNE, J. Phys. Sot. Japan 16,176 (1961). [ 181 KERRO KNOX, Acta Cry&. (To be published). [19] KERRO KNOX, J. Chem. Phys. 39, 991 (1959). [20] L. E. ORGEL and J. D. DUNITZ, Nature 179, 462 (1957). [21] W. Low, Phys. Rev. 195, 793,801 (1957); 109, 247, 256 (1958); 118 1130 (1960).

N.S.

782

HAM

with these three, distortion is expected to be very, very small. For Co2f d7 with an odd number of electrons, the lowest level with spin-orbit coupling included is a doublet and this system is not expected to distort at all since, by Kramers theorem, an electrostatic force cannot remove its two-fold spin degeneracy. In the Fez+ case a small distortion is expected due to the orbital degeneracy and the spin degeneracy. However the paramagnetic resonance spectra on single crystals show that g is isotropic in all these cases, that is, the local symmetry of the paramagnetic ion is accurately cubic. There is a slight difficulty with the Fe2+ case in that a forbidden line is observed at half field, but its g-value is isotropic so that if there are any distortions these are completely at random. The cause of this is not clear. Table 3. Ground-state

properties of 3~2transition-metal MgO single crystals

d3 Crw : MgO Ground-state symmetry and J-value J-degeneracy Predicted distortion g-value

4A !2s,3lz

4 v. v. small 1.98000 isotropic

d5 Mn2+

: MgO

6A 18,5/Z

6 0 2.0014 isotropic

ions in

da Fe*

: MgO

5T3,1

3’ v. small 3.4277 isotropic

d7 Cow

: MgO

4T lS,l/Z 2 0 4.2785 isotropic

dB Niw

: MgO

3A 2891 3 v. v. small 2.221 isotropic

In the transition-metal hexahydrates, with the ions surrounded by six water molecules in octahedral arrangement, the paramagnetic resonance spectra are more complicated because there are small distortions of the octahedra in the crystals. Perhaps the most interesting hexahydrate is Cu2+(H,0), which is found when mixed crystals of copper and zinc fluosilicate hexahydrate are prepared. Here the g-value is isotropic above 90”K, but becomes anisotropic below about 20’K [22]. This is unexpected, since the orbitally doubly degenerate ground state of Cuzf is expected to split into two spin-doublets under the combined action of Then an anisotropic g value would the ligand field and the spin-orbit coupling. be obtained. ABRACJAMand PRYCE explained the observed isotropy in terms of the coupled situation-the octahedron interconverts freely between three identical tetragonally distorted conformations [23] and so the observed g-value is isotropic. At low temperatures the distortions may be trapped so that the q then becomes anisotropic [24]. Kidtransition-metal

hex@uorides

The hexafluorides of tungsten, rhenium, osmium, iridium and platinum provide interesting examples of the effect of ground state degeneracy on vibrational spectra [%I. The spectra of these molecules indicate an octahedral geometry but there is no evidence of permanent distortions from 0, symmetry. However, Fig. 4 [22] B. BLEANEY and D. J. E. INGRAM, Proc. Phys. Sot. (London) A63,408 (1950); B. BLEANEY and K. D. BOWERS,%~. Af35, 667 (1952). [23] A. ABRAGAM and M. H. L. PRYCE, Proc. Phys. Sot. (London) A63, 409 (1950). [24] A. D. LIEHR and C. J. BALLHAUSEN, Ann. Phys. 3, 304 (1958). [25] B. WEINSTOCK and H. H. CLAASSEN, J. Chem. Phys. 31, 262 (1959); 33, 436 (1960).

The Jahn-Teller

783

theorem

shows two peculiarities observed in the infrared and Raman spectra. Firstly the doubly degenerate (e,), Raman-active vibration y2 is observed easily for WF,, weakly for ReF, and easily for IrF,. This frequency has not been observed for OsF, under comparable conditions, and no Raman spectra for PtF, have been reported because that compound is dark red. The second peculiarity is that the 1200

1600

cm-l

1200 cm-1

1600

ReF6 J’=%‘P’l

OsF, J’=Z(‘P’)

1500

I

1300 cm-l

I

I

Ir F6

izoocm-I

1400 1

I

Pi F6 J’=ot’P’)

Fig. 4. Infrared and Raman spectra of the 5d transition-metal hexafluorides. The left-hand side of the Fig. shows part of the infrared spectra and on the right, tracings of two of the Raman spectra. Only the Raman lines r1 and Yehave been observed for OsF,. The Raman line vshas been observed for IrF, but not for OsF, under comparable conditions. Xo Raman spectrum of PtFs has been obtained. The ground state J’ values are given by MOFFI[TT et al [NJ. [26] W. MOFFITT, G. L. GOODMAN, M. FRED and B. WEINSTOCK, MoZ. Phys.

8

2, 109 (1959).

N. S. HAM

784

infra-red combination bands vZ + vs and vi + vg are easily observed and equally sharp for tungsten, iridium and platinum, but for osmium and rhenium the vg + vs band is much broader than v1 + vg. Note that it is the e, vibration v2 which is involved here in rhenium and osmium. These hexafluorides have zero, one, two, three and four 5d-electrons outside a closed shell and the spin-orbit coupling parameters for these atoms are sufficiently large, about 3000 cm- I, that the ground state degeneracies must be calculated with their inclusion [26]. The effective J values are then 0, 3/2, 2, 3/2 and 0 respectively, so that rhenium, osmium and iridium hexafluorides have 4-, 5- and 4-fold degenerate ground states respectively. The orbital part of the iridium ground state is predominantly an S state, a totally symmetric one, whereas rhenium and osmium have orbital P states, so that one expects rhenium and osmium to manifest degeneracy effects more than iridium, and such is observed. In the corresponding 4d transition-metal hexafluorides [27] a similar broadening of the v2 + vs combination band is observed for RuF,(4d2) but not for RhF,(4$). Benzene monon*egative ion A degenerate ground state is also expected for the benzene negative ion. With a D,, framework the ion would be in a 2El, state. Fig. 5 shows the electron spin resonance spectrum of the benzene negative ion. There are seven lines with approximately the binomial intensity distribution of 1: 6 : 15 : 21: 15 : 6 : 1, which is interpreted as showing that the six protons around the ring are equivalent [28]. If the system were distorted then one would expect a different distribution of intensities. However it should be noticed that the lines are very much broader than in the naphthalene and anthracene negative ions. Similar observations have been made for the symmetrical negative ions of triphenylene (D,, framework) and coronene (D,, framework) both of which would have degenerate ground states [28] but not for the perinaphthenyl radical (D,, framework) which is expected to have a non-degenerate ground state [29]. The broadness is obviously associated with degeneracy. The question arises whether it is due to Jahn-Teller forces removing Calculations would suggest that such forces do not remove the the degeneracy. degeneracy. For the benzene ion the gain in stability by distortion is somewhere about 450 cm-l with bumps at the bottom of the potential sheet of between 50 and 100 cm-l [30-331. Comparing these with the zero-point energy of about 500 cm-l it seems that the ion can easily interconvert between distorted conformations so one expects to observe D 6h symmetry and the electron spin resonance spectra certainly show that any distortions which are occurring are happening faster than 10-v sec. Some other mechanism such as solvent-solute interaction or perhaps [27]

H. H. CUSSEN, H. SELIC, J. G. MALM, L. L. CHERMICK and B. WEINSTOCK, J. Am. Chem. 83, 2390 (1961); C. L. CHERNICK, H. H. CLAASSEN and B. WEINSTOCK, &d. 83, 3165 (1961). M. G. TOIVNSEND and S. I. WEISSMAIT, J. Chem. Phys. 32, 309 (1960). P. B. Soao, M. NAKAZAKI and M. CALVIN, J. Chem. Phys. 26, 1343 (1957). A. D. LIEHR, 2. physik. Chem. (F~um@ti) 9, 338 (1956). L. C. SNYDER, J. Chem. Phya. 33, 619 (1960). W. D. HOBEY and A. D. MCLACEILAN, J. Chem. Phys. 33, 1695 (1960). H. M. MCCONNELL and A. D. MCLACHLAN, J. Chem. Phys. 34, 1 (1961).

SOC.

[28] [29] [30] [31] [32] [33]

The Jahn-Teller

theorem

785

spin-orbit coupling must be causing the spin densities to fluctuate the hyperfine lines [33]. DEGENERATE EXCITED STATES

and broaden

Proceeding now to degenerate excited states, here the effect of the Jahn-Teller distorting forces may give rise to split bands in the spectra if the distortion is large enough. More probably it will give rise to changed vibrational selection rules in an allowed electronic transition, that is, single quanta of the doubly degenerate vibrations might be observed instead of double quanta. A.

O-

cd

Fig. 5. Electron spin resonance spectra of the negative ions of benzene, coronene, triphenylene, naphthalene and anthracene and the peri-naphthenyl radical. 3d

Transition-metal

ions

In the d-d spectra of the transition-metal ions there is no clear spectral evidence of distortions for the ions embedded in the magnesium oxide lattice, since the bands are so broad. In Fig. 6 there is some indication for the hexahydrates of manganic, copper, titanium and iron, the d4, I$‘, dl and da configurations, where the transition occurs between doubly and triply degenerate levels, that the spectra are broader than for the other ions [34]. [34] 0. G. HOLMES and D. S. MCCLURE, J. Chem. Phya. 26, 1686 (1957).

N.S.

786

HAM

For Cr3+, which has degenerate excited states, there information available on the energy levels in MgO and chromic ion in magnesium oxide Fig. 7 shows the energy of the Zeeman splitting of the narrow red fluorescence line

is considerable spectr&l A&.0, lattices. For the levels. A careful study from the 2E level to the

10

Lid d2 v”

5

0

iD 15 d9 clI** 10

6

r-7-l

IL!Ldu 6

IO

14

I6

cm-’

22 L

26

30

34

36

42

t

ds

NI’*

I

$Ilc4L4 6

IO

14

IO“

I6

22

26

30

34

38

cm-lx IO-'

Fig. 6. The crystal absorption

spectra of hydrated

transition-metal

ions (after

HOLMES and McCLURE[~~]).

ground state indicated that no distortion was needed to explain completely the Zeeman splitting [35]. For CI3+ in Al,O,, the ion is in a trigonal field and the 4T states are split into non-degenerate and doubly degenerate levels. In the absorption transition to the lower 4E level single quanta of the totally symmetric lattice vibration and, significantly, of the doubly degenerate lattice vibrations have been identified [36]. For absorption to the lower doublet levels there are two interpretations for the observed bands. One analyses the bands in terms of the doublets [35] S.SUGANO,A.L.SCHAWLOW andF.vuw~N~~, Phys. Rev.l20,2045 [36] R. A. FORD and 0. F. HILL, Spectrochim. Acta 16, 493 (1960).

(1960).

The Jahn-Teller theorem

787

R,, R, of the 2E level with single quanta of the totally symmetric and doubly degenerate lattice vibrations [37], while the other identifies three of the vibronic bands as the components S, S, S, of the 2T, level,[38]. Benzene alzd bemene monopositive

ion

Among the excited states of benzene there are quite a number of degenerate ones which are susceptible to Jahn-Teller distorting forces and Fig. 8 shows some of the lower benzene energy levels. The 2000-A band (50,000 cm-l) is a forbidden

25,000

l6,760 4A

RI SlS3 e12 R2 S2

61 ‘33

Fig. 7. Energy levels, in cm-l, of Cra+ (d3) in MgO (cubic) and A120, (trigonal) lattices. For the lower doublet levels in Also, there are two assignments: one [37] identifies the bands R,R,, while the other [38] identifies the bands R,R, and S&$S,. to either a lBlu or a lEzs level. DUNN and INGOLD concluded that the upper state was 1Ez8 and that it showed a splitting of about 150 cm-l 11391.For the allowed transition at 1800 A to the lElu level there are two peaks observed, 703 cm-1 apart [40], but it is not clear whether this separation represents vibrational structure or is actually two separate transitions. Beyond 1800 A there are four Rydberg series converging to the first ionization potential producing the ion, C,H,+ in the 2Els state. There is no evidence of any large splitting of the ground state of this ion as judged by the values of the ionization potentials derived from the four Rydberg series. However in the Rydberg series themselves a vibrational analysis of some progressions in the bands does show some peculiarities. In one group of bands of the R series, the 3R system, there is a progression of 695, 642, 628, 599 cm-l. This R series is probably an allowed transition from a ground state to doubly degenerate E,, Rydberg level. If the usual selection rules for this kind of a transition hold transition

R. A. FORD, Spectrochim. Acta 16, 582 (1960). [381 W. Low, J. Chem. Phya. 33, 1162 (1960). [39] T. M. DUNN and C. K. INGOLD, Nature 176, 65 (1955).

[37]

[40]

P. G. WILKINSON, Can. J. Phya. 34, 596 (1956).

N.S.

788

HAM

then this progression represents two quanta of a doubly degenerate vibration eSg. On the other hand if the modified selection rules for the coupled electronic and vibrational motions operate, then this progression would be one quantum of ezs. WILKINSON, in his analysis of his Rydberg bands, favours the double quanta assignments since he identifies in the 3 R c lA,, sy stem the vibrational transitions 1 t 1 and 3 t 1 in eSs quanta (see Fig. 9) [40]. However the 3 t 1 band, which is quite clearly observed, can only be assigned that way if he identifies the band at 65451 cm-l as being two superposed bands, and his analysis gives a strange

Fig. 8. Energy levels and conf?gur&ions for the lower n* and Rydberg states of benzene. lf&,:

Colzfcgurations a:, et,

1R2u, ‘Blur lElu: lE2,: aL 6, azg nR: a:, ef, a;,

a:, 4, e2u

nR;, nR;, nR’“: a:, e:, e;, 2E,,: a:, e”,, vibrational spacing for the 3R state, viz. 340, 355, back to 340, down to 302, then perhaps 314 twice followed by 300 cm-l twice, compared with 606 cm-l in the ground state. On the other hand LIEHR and MOFFITT, using the selection rules for the coupled situation, prefer the single quantum assignment and they obtain the spacings 695, 624, 628, 599 cm-l compared with 606 cm-l in the ground state [lo]. However, in support of WILKINSON’S assignment, this same pattern of vibrations is observed four separate times, that is in two systems (3R and 3R’) for both benzene and hexadeuterobenzene. CONCLUSION The experimental evidence for the predictions of the Jahn-Teller theorem seems to be that the crystal structures of transition-metal complexes generally fit

The Jahn-Teller theorem

789

the predictions. There is no sign of degeneracy including distortions in paramagnetic resonance spectra of the transition-metal ions in magnesium oxide. The paramagnetic resonance. spectrum of the copper hexahydrate seems to be a clear case of the coupled situation, and the effect of degeneracy and the ezp vibration for the MF, vibrational spectra also seems clear. For the aromatic negative ions, the ground state degeneracy remains when the electronic and vibrational motions are e29

spacings

W

LM

-8-4 599 599 -6-3 628 628 -4-2 $i$ 3 642 -2-l $j& I 692 o-o

3R

‘A,

606

Fig. 9. Energy level diagram for the 1521.65 A (65,718 cm-l) 3R Rydberg band of benzene. The vibrational analysis of this band shown here gives only the spacings attributed to the ezovibrations. The vibrational transitions 1 + 1 and 3 c 1 in es8 quanta are identi6ed by WILXINSON [40] with the bands at 65,451 and 66,147 vibrational spacingsfor the cm-l resp. On the right-hand side of the figure the eStr 3R level rareshown for the Wi~on analysis (W) and the Liehr ~offitt analysis (Lx) [lOI.

coupled. In optical spectra the evidence for the degeneracy effects is suggestive, but there are often two alternative interpretations of the same data, so that more careful investigations are needed. The investigations of the effects of the Jahn-Teller theorem have followed the course of many other theoretical studies. The first qualitative examples seem to fit predictions, but on closer examination the position always turns out to be more complicated. The theoretician and experimentalist need to combine forces to devise, perform and interpret definite experiments. -4c~nu~~edge~~n~The author wishesto thankDrs. C. K. COOGAN, B. DAwsoxand A. C. H~CLEY for many helpful discussionsduring the course of preparation of this review,