TOPOLOGICAL ASPECTS OF THE CONFORMATIONAL STABILITY

DEGENERATE ELECTRONIC STATES

Feb., 1963

( 2 6 0 A,) which reduce the attractive potential to negligible values (Table IV). I n the latter case, even with very little entropic repulsion, stabilization is possible when 2(r f d ) 2 2.4.

389

Acknowledgment.-The author wishes to express his appreciation to Messrs. T. S. Wollenberg, E. D. Lewis and W. D. Ross for their encouragement during the course of this inwstigation.

TOPOLOGICAL ASPECTS OF THE CONFORMATIONAL STABILITY PROBLEM. PART I. DEGENERA.TE ELECTRONIC STATES1 BY ANDREW D. L I E H R ~ ~ Bell Telephone Laboratories, Inc., Murray Hzll, New Jersey Received March 30,1966 If it is not true it io very well invented-Bruno, The topography of Jahn-Teller-Renner electronic potential energy surfaces of symmetric polyatomic systems is derived by means of group theoretic and permutational symmetry principles. Degenerate perturbation theory is utilized t o obtain itn explicit analytical expression of this topography. The dependence of the nuclear-electronic interaction matrix elements, which occur in the perturbation development, on the nuclear coordinates is shown t o be precisely determinable by group theoretic and permutational symmetry techniques. The numerical coefficients which prefix the nuclear coordinates which appear in these interaction matrix arrays are regarded as phenomenological parameters in their subsequent topographical applications. It is demonstrated t h a t the magnitudes of these parameters determine only the relative placements of the Jahn-Teller-Renner surface reliefs and not their basic character (in this sense, minimal and maximal domains are taken t o be of the same character). The formulas derived by means of the perturbation theory (with and without the complication of spinorbit interactions) are eqployed t o describe the topology of the multiply degenerate electronic energy surfaces of the regular polygons (including t h e digon) and the regular polyhedra, as well a13of a number of selected highly symmetric irregular polygons and polyhedra. The results obtained reveal a number of important maxims: (1) the Jahn-Teller-Renner behavior of a polyatomic group theoretic system is completely determined by that of its elemental subgroups (principle of mathematical inheritance); (2) isomorphous point groups exhibit isomorphous Jahn-Teller deportments (formation of Jahn-Teller families); (3) whereas the prime number groups produce a single unique Jahn-Teller-Renner topography, the non-prime numbe,r groups produce all those topographies which are required by the principle of mathematical inheritance (1) (law of prime numbers); (4) the dynamical quantization and the topography of the Jahn-Teller problem are completely specified by group theoretic and permutational symmetry precepts (symmetrical transcendence); ( 5 ) although not mathematically required, the symmetry of the stable Jahn-Teller conformation is always the highest symmetry which is yet compatible with the loss of the initial inherent degeneracy (mimimax rule); ( 6 ) the Jahn-Teller-€tenner energy resolvants factor only a t locations of high nuclear symmetry (factorization theorem); ( 7 ) certain nublear structures can never be Jahn-Teller stabilized (exclusion principle); (8) it is impossible t o distinguish between ordinary anharmonic elasfic distortions and Jahn-Teller distortions in non-homologous series of compounds (indistinguishability theorem), (9) spin-orbit forces remove cuspidal Jahn-Teller radial electronic energy singularities for geometries less reghlar than the cube (spin-orbit law); (10) ionic and covalent bonding forces always produce complementary Jahn-Teller deformatiofis,. whereas conjugate hole-electron electronic configurations always produce the identical deformation, bonding forces being equal (complementarity rules); (11) pseudo-Jahn-Teller interactions make conformational isomerisms possible even for dispositions of low regularity (conformational isomerization tenet); (12) doubly degenerate electronic states possess continuoua Jahn-Teller-Renner electronic energy surfaces, but triply degenerate electronic states possess disjoint electronic energy surfaces in certain directions (canon of dimensional variability); (13) tetragonal Jahn-Teller conformational interconversion takes place in two dimensions for doubly degenerate elegtronic states, and five dimensicns for triply degenefate electronic states (doctrine of dimen8ional impenetrability); (1.4) experimental proofs of Jahn-Teller-Renner consequences can only be obtained from studies of homologous series of compounds (criterion of probity). Particular care has been taken t o graphically illustrate all the important consequences of the mabhematical derivations, so that their chief predictions should be available t,o theoretician and experimentalist alike. A detailed discussion of the theoretical and experimental status of the Jahm-Teller-Renner problem is presented, and courses for future advance are signposted. A master plan is sketched for the solution of the numerous as yet unsolved theoretical and experimental Jahn-Teller-Renner puzzl@.

COXTESTS I. Introduction.

,

.

..............................

5ii. Preface.. . . . . . . . . . . . . .

A. THEJAHN-TrELLER MECHANICS: 11. The One and Two Dimensional §I. The Digonal Molecular 52.

The Regular Polygonal 2.1 The Trigonal C 2.2 The Tetragonal

........................

391 . . . . . . . . . 392 . . . . . . . . . . . . . . . . 392 ......................... ....................... 392 .............................. 394 ..................................... 394 .............................. ........................ . . . . . . . . . . . 398 ........................ .................... 398 ......................... ............................ 399

.....................

2.7 The n-Tagonal 2.8 Some Cases of

.......................

ICs . . . . . . . . . . . . . . . . . . ..........

.................

................................ ............................ . . . . . . . . 401

. . . . . . . . . . . . . . 401

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........................................... 111. The Three Dimensional Case ...... ................................... $3. The Regular Polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Cubic Molecular Systems: The Doubly Degenerate Electronic States with Zero Spin-Orbit Forc 3.2 Cubic' Molecular Systems: The Triply Degenerate Electronic States with Zero Spin-Orbit Forces 3.3 Cubic Molecular Systems: The General Interaction Matrix with Zero Spin-Orbit Forces. . . . . . . . . . . . . . . . . 3.4 Cubic Molecular Systems: The Triply Degenerate Electronic States with Non-zero Spin-Orbit Forces. . . . . 3.5 Cubic Molecular Systems: The Doubly Degenerate Electronic States in-orbit Forces., , , 3.6 Dodecahedral and Icosahedral Molecular Systems. . . . . . . . . . . . . . . . . . . . . . . . . $4. The Irregular Polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Regular Polygonal Monopyraniids . . . . . . . . . . . . . . . . , , , , , , , , , , , , , , , , , , 4.2 The Regular Polygonal Bipyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................. 4.3 The Perturbed Cubic Structures. . . . . . . . . . . . . . . . . . . . . . .................................... B. THEJAHN-TELLER MECHAKICS: DYNAMICS. ...................................................... IV. Vibrational and Electronic Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $5. The Theory of Weak Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................... 5.1 General Considerations. . 5.2 Quantization of the Solut , ,. , . . , . , .,,,, , , . , . , . . , , . , . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . , . , , $6. The Theory of Strong Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................... 6.1 General Considerations. . 6.2 Quantization of the Solut ..................................... , ,

, ,

,

,

,

, , ,

,

,

402 402 425 426 436 441 445 446

448 448 448 450 450 451

............................................................................................... 452 V. Connection with Experiment. . . . . . . . . . . . . .............................. $7. Generic Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Motions on the Jahn-Teller Electronic Potential Energy Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 7.2 Symmetry Properties of the Jahn-Teller Electronic Pot'ential Energy Sufaces. . . . . . . . . . $8. Specific Surveyal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................... 8.1 Struct,ural and Thermodynamic Manifestations of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 8.2 Microwave Frequency, Radio Frequency, Electric, and ssions of the Jahn-Teller Theorem, , 462 8.3 Infrared and Raman Demonstrations of the Jahn-Tell ................................ 464 8.4 Optical Exhibitions of the Jahn-Teller Theorem, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 VI. Fantasy and Speculation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................... 469 §9. The Future Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 D. ACKKOWLEDGMEKTS. ,. .................................................... 471

I. Introduction Si. Foreword.-It has become increasingly apparent' in recent years that one cannot adopt with absolute impunity the view that a molecular aggregate's rotational, vibrational, and electronic motions are independent and separable.2 The inadequacy of this traditional concept is nowhere as dramatically demonstrable as in the case of molecules in degenerate electronic state^.^ Although this fact has been recognized by a number of authors, some confusion still exists as to the proper means of correcting this inadequacy. This confusion is the compound result of two circumstances : (1)the overt (deceptive) simplicity of the original initial pronouncement's of Jahn and Teller4 and Rennerj5and (2) the inherent complexity of the problem itself. What me wish to do in this paper is to examine more closely (1) Various portions of thib paper were presented a t several conferences. (1) the XVIII International Congress of Pure and dpplied Chemistry, Montreal, Canada, August 6-12, 1961; (2) Symposium on Molecular Structure and Spectroscopy, Ohio State University, Columbus, Ohio, June 11- 15, 1962; (3) Symposium on the Theory and Structure of Complex Compounds, Wroclaw, Poland, June 15-19, 1962; (4) the VI1 International Conference on Coordination Chemistry, Stockholm, Sweden, June 25-29, 1962; ( 5 ) Symposium in Quantum Chemistry and Solid State Physics, Rettvik, Dalarna, Sweden, August 27-September l, 1962; (6) the Informal Meeting o n Recent Developments in Quantum Chemistry, Hakone National Park, Japan, September 7-8, 1962; (7) the International Symposium on Molecrlar Structure and Spectroscopy, Tokyo, Japan, September 10-15, 1962. An amplified account of these lectures may be found in (a) P ~ o g rI.n o r g . Chem., 3, 4, and 6 (1961, 1962, and 1963); (b) Ann. Reo. Phys. Chem., 13 (1962); and (0) Proceedings of the Symposium on the Theory and Soructure of Complex Compounds, Wroolaw, Poland, t o be published, 1963. (d) Mellon Institute, 4400 Fifth hvenue, Pittsburgh 13, Pennsylvania. (2) A. D. Liehr, Ann. Phys. (N. Y . ) ,1, 221 (1957). (3) A similar inadequacy may also. occur for non-degenerate electronic states. See Part 11, J . Phys. Chem., 67, 471 (1963). Jahn and E. Teller, Proc. Roy. Soc. (London), A161, 220 (1937); (b) H. 4 . Jahn, ibid., A164, 117 (1938). (5) (a) R. Renner, Z. Physik, 9'2, 172 (1934): (b) J. A. Pople and H. C. Longuet-Higgins, MoZ. Phys., 1, 372 (1938); (c) J. A. Pople, i b i d . , 3, 16 (1960); (d) J. T. Houaen, Symposium on Molecular Structure and Spectroscopy, Ohio State University, Columbus, Ohio, June, 1961; (e) J. T. H o w e n , J . Chem. Phys., 36, 519, 1874 (l9G2), and to be published.

the subtleties of the Jahn-Teller-Renner theorems and delve more deeply into the intricacies of their insinuations and consequences. It is hoped that this examination will resolve some of the disorder which a t present still clouds the issues. Our attack is divided into two parts: the treatment of systems with negligible spin-orbit forces and the treatment of systems with strong spin-orbit forces. The intermediary cases are quite complex,6and will not be considered in any detail. They can, in any event, be understood qualitatively in terms of their limiting behaviors. We develop a theory based solely on symmetry principles and independent of any specific model of molecular binding. I n this spirit we derive the potential energy surfaces and discuss the permissible nuclear and electronic motions for molecular geometries whose n-fold rotational (or rotational-reflectional) symmetry axes (n = 3,4, 5 , 6 , 7 , 8 , -- -) require the existence of degenerate electronic states (conformations of these sorts are linear, polygonal, or polyhedral), and discuss a number of concrete chemicaI examples which possess such geometries. V e assume that we are given the initial (degenerate) nuclear and electronic dispositions whose stability me are to assess. We then proceed to test for conformational stability. This procedure entails the determination of the vanishing or non-vanishing of the nuclear +

gradients, (V&)v, ( r = 1, 2,---), of the electronic matrix element, (Xo- Eont)np,nt [equals AVnp,nt]of the Hamiltonian operator, x0 ( x k , b/bzk; &), minus the ($xed) electronic energy eigen-value, E o n t ( l O i ), between the pth and tthcomponents of the degenerate (either accidentally or essentially) electronic state, n, a t the $xed nuclear conformation { 60, ) ; that is, we wish to determine (6) U. 6 p i k and BI. H. L. Pryce, Proc. Roy. Soc. (London), A238, 423 (1957).

Feb,, 1963

DEGEXERATE ELECTRONIC

ST.4TES

39 1

the explicit values of the various derivatives of this electronic matrix element, with respect to the nuclear coordinates, evaluated a t the assumed initial nuclear conformation.2 The actual determination of the exact mathematical structure of this matrix element is accomplished by the method of (‘descentin ~ y m r n e t r y . ” ~ } ~ This method utilizes the symmetry properties, Le., the transformational properties, of the electronic and nuclear displacements under coordinate transformations to specify the detailed mathematical form of the requisite interaction matrix elements. Once this form has been obtained, we classify the stability of the system as succeeds: if all the nuclear derivatives of the matrix elements with respect to the asymmetric nuclear displacements vanish identically (by symmetry or / fortuitously) when p # t, the initial nuclear conformation { t0i ] is a stable one; if the even derivatives vanish z but the odd ones do not, the initial conformation is said to exhibit Jahn-Teller instability; if the odd derivatives vanish but the even ones do not, it is said to maniY fest Renner instability; if neither the even nor the odd (b) terms vanish, it is said to shorn a simultaneous JahnFig. la.-The real ( a ) and imaginary ( b ) parts of the complex Real molecular Teller and Renner instability. n, wave function of linear acetylene. These functions transform in D2h,respectively. as B2, and systems which are non-linear fall into the last category, and those that are linear into the next to last category. Both cases will be amply exemplified in what follows. $ii. Preface.-In this preface we shall attempt to present a detailed outline of the topical range covered by this paper. It is hoped that this outline will enhance the lucidity of the article. We are concerned in this work with the topology of electronic potential energy surfaces of degenerate electronic states. We deal with the analogous problem for non-degenerate electronic states in the article which follow^.^ In Division A, sections I1 and 111,we begin our study of Jahn-Teller topography with an investigation of the statistical problem. Section I1 deals with the algebraic and geometric description of the statistical JahnTeller potential energy surfaces of one ($1) and two %b(nul S d W ($2) dimensional molecular systems; section I11 with that of three dimensional systems ($3 and $4). For Fig. 1b.-The symmetry displacements of linear acetylene. the case of orbitally doubly degenerate electronic states The designations S3,.( T,) and S4+( T,) represent the complex ($1, $2, $3.1, and $4) described by complex conjugate displacements dt/i/2[& + i&b] and d $ [ S ? f iSlb], each. wave functions $+ and $-, it is shown that, regardless These displacements possess one unit of vibrational angular momentum apiece. of the dimensionality of the compound, the resultant of the product charge densities Gnp*Gnt, only those nuclear coordinates energy surfaces and w7ave functions are described by the which possess the same Bymmetries and group theoretic classifications as formg lt2

e charge densities

AEn* = AV++

IAV+-l

(ii-1)

Similar equations are found to describe spin-orbitally quadruply degenerate electronic states (53.4, $3.5, and (7) H. A. Bethe, Ann Physzk, [51 3, 133 (192R). (8) (a) A. D. Liehr, 2. Phys. Chem. (Frankfurt), 8, 338 ( l 9 5 6 ) , ( b j J. Chem. Phys., 27, 476 (1957); (0) Rev. M o d . Phys., 32, 436 (1960); (d) 0. X. Davtyan, Zh. Fzz. Kham., 34, 108, 295 (1960). (9) These relations are easlly obtained upon noting the reality and equality of AT’** and the complex oonjugacy of AV=, [The reflectional permutation of $* under a vertical planar reflectlon uy (or ud In some cases) also requires these same forms. Notice that this reflectional permutation implies that the phase angles pi of a nuclear symmetry coordinate, 8,= p , e W , change sign under uy (01 ud), % . e , , upA = -9, when u$* = $?I. The coiresponding equations for real wave functlons are not so neat and compact [see 63 1 a n d $4.1a n d ref. 1 and 8a,c, for example]. The specific structure of the energy matrix elements of AVn’np,nt = [KO Eno(60J]np,nt a l e obtained by a combination of group theoretic and symmetry arguments [Since these matrix elements must reflect the permutational symmetries

-

$L‘np*$nt may appear in the analytic expressions of ,nt.] Therefore, these matrix elements, as here determined, are comy general and independent of any specific model of uhemical bonding; we leave the explicit numerical evaluation of the matrix elements to future workers. [Of couree, predictions as to precise nuclear locations within given chemical compounds are highly dependent upon the bonding model assumed (this is amply demonstrated in references l a , b ) : however, many general statements of great stereochemical significance can be derived from symmetry considerations alane.] An a n example of their generality, the energy matrix so derived for an orbital TI or Tz electronic etate is identical in form with t h a t derived for a T4 or r6 spin-orbital electronic state. [Thus if one wished t o compute the very small Jahn-Teller interactions within the ground state of tetrahedral trivalent vanadium (a F s ( 3 . b ) state) one could readily do so by’utilizing the results derived in 63.2 for a n orbital Tz state. I n this regard compare (a) J. €€. Van T’leck, Ph?ls. Rea., 67, 426. 1052 [Erra,tal 11940), and V. I. Avvak.umov, Optik Speklrosk., 13, 588 (1962).] Similarly, the Jahn-Teller and! pseudo Jahn-Teller interactions within the spinorbital Fs(aT1,) and rs(aTigj ground electronic states of octahedral trivalent vanadium are readily described by the theory of 63.3 for the orbital d Tzg states. [Recognize, however, that the energy expressions a n d functions for triply and (accidentally) quintuply degenerate elecc states are much more complex than those of eq. ii-1 and 2. For example, look a t the nasty cubic resolvants tabulated in (b) R. S. Burington. “Handbook of Mathematical Tables and Formulas,” Third Ed., Handbook Publishers, Inc., Sandusk,y, Ohio, 1949. I t is such resolvants which must be.used t o discuss the Ti and T2 electronic states of $3.1

ANDREWD. LIEHR

392 tE

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4x

tE

Fig. lc.-Schematic representation of the permissible electronic potential energy surfaces of a linear molecule in a degenerate electronic state. Only the cut in &-space is shown; that in 8,space is similar. $2,

CASEI

$b3=0

CASE 2

$3-77

CASE 4

#4

S2b

QyQy CASE 3

6,

-0

:V

s2a +%b

CASE 5

&=

0

, d4;'+-

CASE 6

$4=

0

, d 3=

Fig. Id.-Distorted geometries possible for a linear acetylene molecule in a degenerate electronic state. Cases 1 and 2 apply to a equals zero; cases 3 and 4 for Sazero; an extremum for which S and cases 5 and 6 t o an extremum for which neither is zero. In all cases vibrational angular momentum has been partially (or completely) quenched and replaced by over-all rotational angular momentum of the distorted non-linear molecule.

$4.3). Orbitally triply degenerate electronic states ($2.8 and S3.2) and higher ($3.3 and $3.6) have more complex energy and wave function resolvants which are factorable only a t specific nuclear conformations.sb It is shown, however, that despite this limitation, the requirement of energy invariance under permutations of identical nuclei allows a fairly complete particularization of the energy surfaces to be made. This requirement leads to a complete characterization of the extrema of the potential energy surfaces in a manner quite reminiscent of the classical electrostatic method of determining equipotential surfaces by the method of images [e.g., recall the potential problem of the wedge of dihedral angle 2?r/n]. I n addition, it produces a factorization of (ii-1) and its higher analogs ($2.8, $3.2, $3.3, and $3.6) a t nuclear field points of high symmetry [e.g., at nuclear field points which are oriented along an n-fold axis of symmetry]. The analysis of Jahn-Teller topography is generalized in Division B, section IV, to include dynamical con-

S,a F0l.l OA'F"

9Y

e

Fig. le.-The symmetry displacements of a linear triatomic molecule, e.g., a hypothetical linear water molecule. There are also indicated the two ways in which the SZ(ru)displacements can distort the molecule: by pseudo-rotation (X2& X2b) and by double inversion ( Szafollowed by S2t,).

+

siderations. It is demonstrated that regardless of whether the vibronjc interactions are weak ($5), strong ($6), or intermediate ($5 and $6) the solutions of the Jahn-Teller problem may be rigorously classified according to their permutational symmetry properties under permutations of identical nuclei. Comments also are made to indicate how the general n-dimensional dynamical Jahn-Teller problem may be solved ($5.2). With the conclusion of the formal theoretical development, we present in Division C, section V, a discussion of the present theoretical and experimental ($7, $8, and $9) status of the Jahn-Teller problem. A number of concrete experimental situations are discussed ($8) in the light of the theory derived in Divisions A and B, sections 11-IV, and the future prospects for research in this area are considered ($9). Acknowledgments are given in Division D to our many friends and colleagues who have aided us directly or indirectly in our work. Division A The Jahn-Teller Mechanics : Statics 11. The One and Two Dimensional Cases $1. The Digonal Molecular Systems.-To exemplify the treatment of digonal molecular systems we shall

DEGENERATE ELECTRONIC STATES

Feb., 1963

Ox

-c

OX

%OX

Ci‘

m u

(my)

(ax)

393

Fig. 2a.-The real and imaginary parts oi the complex E” wave function of a planar triatomic trigonal molecule (e.g., “CsHa”). These functions transform as B1 and Az in Cz, individually.

consider a linear molecular in a II electronic state-for specificity linear acetylene. We introduce no essential restriction in so doing; the theory applies equally well to other chemical compounds. A concrete example is picked for illustrative clarity only. We obtain the form of the perturbation matrix, AT’,, 5 [XO- E,O(&O)]np,nt, by consideration of the electronic and nuclear transformation properties under a rotation, em(a), by an arbitrary angle, CY, about the infinitefold rotation axis, a. Such a rotation multiplies r-type electronic (Fig. la) andnuclear (Fig. lb) functions, X , say, by a phase factor ekia and the a-type functions by unity.’O Now as the energy terms XO- Eno(&’) are invariant under all symmetry operations (permutational or otherwise) permitted the molecule, the matrix element AVpt (a function of the nuclear coordinates only) must transform as the product #p*#t, (p, t = .t). Hence, only those nuclear coordinates which transform exactly as do the products #p*#t, (p, t = A),occur in AVpt. If we designate the permissible nuclear dis2a,+ a,+ r, T“,as So, SI, SZ, placements,lO~ll and (Fig. lb), serially we may then write [letting Xj = qj, (j = 0, 1, 2) and Xkrt = q k e * i q k , and noting that a,#+ = #-, a v q k = - q k , (k = 3,411

+

AV++

=

4

i = O

AV-- =

cop0

+ +

+ clql +

4 ciipiz

i = O

+

+ higher order terms in

(10) E. B. Wilson, Jr., J. C. Deoius, and P. C. Cross, “Molecular Vibrations,” McGraw-Hill Book Co., Inc., New York, W. Y., 1955. (11) G. Hersberg, “Infrared and Raman Spectra of Polyatomic Molecules,” D. Van Nostrand Co., Inc., Nev York, N. Y., 1945.

Fig. 2b.-The symmetry displacements of a planar triatomic trigonal molecule. The displacements 81, and 8 l b are, respectively, the real and imaginary parts of the complex codrdinate SI,, which physically can be depicted as the superposition of Sls,b 90” out of phase (yielding circular rotatory atomic displacements of unit vibrational angular momentum), as shown above in the diagram labeled 81, 8lb. When Jahn-Teller or non-quadratic elastic forces are introduced, saddle points and minima make their appearance every 60” along the original 81, 8 l b circular trajectory. These extrema then tend to quench the “pseudo” rotatory behavior of SI, Sib, and hence tend to quench ita associated vibrational angular momentum.

+

+

+

The solution of the resulting 2 X 2 secular equation2 yields the variation of the electronic energy, AE, = En(Ei) E n o ( & ) , of the nthelectronic state with nuclear displacements as (the equality of AV++ and AVfollows not only from reflection symmetry under a, but also from the reality and Hermiticity of the electronic Hamiltonian, Eo)

-

AEn,

=

l/z(AV++

l/zl/=+-

or AE,,

=

AV++

+ AV--)

f

- AV--)2

I AV+-I 1

and

+ (21 AV+-j #n*

=

A V+-

)2

(1-2)

ANDREWD. LIEHR

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AEn* = AB++ f

I

+ -

COS ~ ( ( P S

+ -----)”’

2ff(~i)g(~i)q3~~ X4 ’ (~4)

+

(1-5)

y’(qi)~4~

The extremization of eq. (1-5) with respect to $Dk, ‘p3 - ‘p4 equalsjn/2, ( j = 0, 1, 2, 3) [the higher order terms give extrema a t ’p3 - 9 4 = limjn/2n, ( j = 0, 1,---,4n - 1). Hence, ‘pa - ‘p4 = 0,

(k = 3, 4), yields

n+ m

n/2, nl and 3n/2 are the only true extrema for the general case also]. Therefore, the extrema of an, in the quartic approximation assume the guise Sje

AEn* =

Av++*

+ (-1)’

[f(qi)~3~

g(yi)Q421

(1-6)

which has radial extrema in g k space at 0 and

{ -(Ckk

+ (-1)’

Ckk)/(Ckkkk

Ekkkk)}1’2,

(k = 3 ~ 4 )

The absolute extrema in the complete radial space of gk, (k = 0, 1, 2, 3,4), are more difficult to determine and Fig. 2d.-Transverse secFig. 2c.-Pictorial display will not be further discussed here. Suffice it to say tions of the trigonal potential of the allowed electronic that quartic (and higher) terms in the dis,placements energy surfaces (Fig. 2c) in potential energy surfaces of the planes (01 equals j ~ / 3 , provide the .mechanism for non-linear and non-planar a planar trigonal (or hexagonal-see $2.4) molecule in a ( j = 0, 1, 2, 3, 4,5 ) . minimm, a few of which are pictured in absbrraict [Fig. degenerate electronic state. IC] and physical [Fig. Id] space in Fig. ICand Id. Here the minima are chosen Indeed, even asymmetric carbon-hydrogen distances are to be a t 0, 120, and 240’ provided by minima due to such terms in g2 [i.e., and the saddles a t 60, 180, and 300’. XZ(a,+)] space. It is possible that the bizarre conformations of some of the excited states of acetylene have their origin. in this manner.lb The electronic &potentialenergy suriaces of a linear triatomic molecule are easily extracted from (1-5 and 6,) by setting q l and g3 [for &heX Y 2 type system] OT 92 and q3 [for the XYZ type system] equal to zero. In Fig. le, we show the appropriate nuclear modes for a linear XY2 type compound. $2. The Regular Polygonal Molecular Systems. 2.1. The Trigonal Case.-A planar moleoule with a threefold axis may exhibit only two different types of degenerate electronic states, the E’ and the E”, which differ solely in their evenness (’) or oddnms (”) under C 4 x reflection in the molecular plane ( a h ) . In both of these degenerate electronic states, the wave functions #* may be chosen to transform so as to gain the phase factor e*2ai/3 upon a dookwise permutation, e,(z), of the nuclei about the threefold ,z axis $by the angle -c 0 2n/3 [Fig. 2aI.l’ Therefore, as in $1, only those nuclear coordinates which transform exactly as do the products #p*#t, (p,t = f),may occur in the vibronic matrix elements, AV,,. For simplicity, we restrict our discussion to triatomic equilateral nuclear conformations; no essential loss in generality is introduced by this restriction. As the only nuclear deformations allowed a three atomic equilateral molecule [the four atomic case is mathematically isomorphous to the trigonal pyramid 9x*011 ($4.1)] are S~(cq’)and Xlk(e’) (Fig. 2b), the vibronic matrix AVpt, (p, t = ,&), assumes the form [letiting Fig. 3a.-The real (BX) and imaginary (By) parts of the complex E, wave function of a planar tetraatomic square molecule So = go and SI* = qle*$pl, and noting that a,#+ = #(e.g., “CdH4*”). These functions transform as Bag and Bz, in and uVq1= -p1p3 D2b, separately.

If we call the coefficient ,of the q3’ term in (1-4) f ( q i ) and that of -the q2‘ term g(yi), we finally obtain (the constants E in (1-4) are real because of the equality of uvAV+-, AV+-*, and AB-+)

(12) I n this paper we shall always choose our symmetry operations t o be of the replacement type; e.g., the clockwise permutation described above replaces atom r b y atom r-1, a n d so nuclear dmplacement Ep by &-I

and orbital pr by v ? - ~ , Please note t h a t although the a t o m s &re mathematzcally permuted in a clockwise direotion b y this convention, the molecule is physically rotated in a counter-cloclcwiae direction.”

Feb., 1963

DEGENERATE ELECTRONIC STATES

395

E

6 S4b(Eu)

754d(EU)

s 4 a+s4 b

Fig. 3b.-The in plane symmetry displacements of a planar tetraatomic square molecule. The displacements 84% and &b are, individually, the real and imaginary parts of the complex coordinate S4*,which physically can be depicted as the superposition of &arb 90' out of phase (producing circular rotatory atomic displacements of unit vibrational angular momentum). The rotatory motions So 81and SI S3are obtained by adding, with equal amplitudes, the appropriate non-degenerate modes (hence no vibrational angular momentum is produced) 90" out of phase. When Jahn-Teller or non-quadratic elastic forces are introduced, saddles and minima appear every 90" along these circular trajectories. These extrema tend to quench the pseudo rotatory demeanor (vide, Fig. 3d).

+

+

1

nV++ = AV--

= CO~O

(Cii

Coii40)pi2

1 = 0

C l ~ l @COS ~

AV+-

=

(El

-tEo1qo)q&"

+

(E11

3p1

+ ------

+ Eo11qo)q12e--2'rp1 4+ -------

~ 1 1 1 q 1 ~ d ~ ~

(2.1-1)

Again, as in 4 1, the equality of AV++ and AV-- follows from the reality and Hermiticity of the electronic Hamiltonian, XO. Therefore, eq. 1-3 applies here also [and applies for all other doubly degenerate states regardless of molecular geometry (Oii) 1, and we have AEn* = AV++

p E 1

+

+ e-"9'

f

I

Eo140

q12(

Fig. 3c and 3d.--Portrait of the electronic potential energy surfaces of a planar square (or octagonal-note $2.6) molecule in a degenerate electronic state (specifically an Ezs or state for the octagon). Figure 30 shows a one dimensional cut of the surface along the SI(@,,) axiis (Eo represents the original unperturbed parabolic potential curve). Figure 3d manifests two dimensional Sa(@,,),and sa(ollB),sl(P1,)-space. cuts of the surface in S1(@lS), Below the respective potential sheets are shown the geometries described by the minima ( t h a t a t SI > 0 yields the darkened circle distortions, while that a t XI < 0 produces the open circle deformations-compare with Fig. 3b).

+ C;ll4l2 + ---)

cll +

Eo1140

+ ---I + ----, I

When the coefficient of the q1 term is denoted by f(m) and that of the q12 term by g ( q i ) > we find [the constants Eare real as we chose u,AV+ = AV+-* = AV-+] (13) For a discussion of the construction of complex symmetry coordinates see (a) A. D. Liehr, Thesia, Harvard University, 1955 [corrected copies of this thesis are available from the author upon request]; (b) A. U. Liehr, 2. Naturforsch., 16a, 641 (1961), and ref. IC.

and if quadratic terms are neglected in AV+-, the wave functions +n2t may be written

The extremixation of this energy expression with respect to cpl yields extrema a t pl equals 7rj/3, ( j = 0, 1, 2, 3, 4, 5 ) . The minima will lie at either the even [ f ( d * d g i i > > 01 or odd L f k j I ) . g ( d < 01 values of j dependent upon the relative signs of f(gJ and g@) a t their radial extrema1 points ql. These minima correspond to an isosceles triangular conformation [see

ANDREW D. LIEHR

396

Vol. 67

Fig. 2b, c, d]. An example of a system which might exhibit such a potential surface in its ground state is CaH3 [a2E” ground electronic configuration-see Fig. 2a]. At the extrema1values of 9 1 , eq. 2.1-3 factors and assumes the simple structure AEn,

* I qlj(qi) + q12g(qi)(-1)’l

AV++

(2,l-s)

whose extrema in q1 space lie at [-(Cii

- 511)

f

+ 4Ci(3Sii(-l)’

{ (Cii -

”‘I/

-

[3ciii(-l)’ -

~111))

tY

2.2 The Tetragonal Case.-A planar molecule with a fourfold axis may display but two species of degenerate electronic state, E, or E,, which differ only in their evenness (g) or oddness (u) under inversion in the center of symmetry (i). These two species of electronic state are described by wave functions $, which obtain a phase factor of ef2uk/4 under the clockwise atom replacement permutation, e4(z), about the fourfold z axis [Fig. 3a].I2 As before, the structure of the vibronic matrix AVpt, (p, t = A),for each of these species is completely determined by the transformational properties of &. In the interest of economy we shall explicitly discuss, in what succeeds, four atomic square (quadrate) nuclear arrangements [the five atomic case is mathematically isomorphous to the tetragonal pyramid ($4.1)1a]. Then the appropriate nuclear symmetry coordinates are So(Oclg>, &(Pig) ; 82@1,), 8 3 @ 2 g ) , and S4,(eu) [Fig. 3b]. The vibronic matrix must, therefore, have the structure [letting LY~ = qj, ( j = 0, 1, 2, 3) and = and noting that ad$+ = $- and Ud94 = - p 4 n/2, where Ud is an apical plane (the yz plane of Fig. Sa-see Fig. 3b)I

+

AV++ = AV--

=

coqo

4

+

i ( ~ ~ 4~

~

~

~

cos

q

+

E i - 0

(cii

2

4 ~344~394’

+ i(Ei -I- EOIQO)PI-I-(Ea + C144qlq4’

AV+-

+

~

coiiqo)pi2

sin

EO3qO)q3

+ + -----

294

+

+ i(Cd4 + C044q0)q4ze-2i(04+ ~

)

q

~

2

e

~

~

~

0

h 1 1

~

Thus [the constants c‘ are real because of the equality U d A v + - = AV-+ = AV+-*]

c co

SIN?

1

(my)

Fig. 4a.-The real (6.and ) imaginary parts of the complex E” wave function of a planar pentaatomic pentagonal molecule (e.g., “CgH6)’). These functions transform as BI and A2 in CZ,, respectively.

ticular cases lead to a trapezoidal (pa = 2j?r/4) or “arrowhead” (1p4 = (2j l)a/4) type structure [Fig. 3b]. The trapezoidal structure may ultimately dissociate into a cis-bent four atomic chain,’* the arrow-

+

(14) Such behavior has been observed for C4H4 in the simple Htlckel ficheme when configurational mixing due t o nuclear deformations is not neglected. See (a) L. C. Snyder, J. Chem. Phys., 88, 619 (1960); (b) Symposium on Molecular Structure and Spectroscopy, Ohio State University, Columbus. Ohio, June, 1960 and 1961; (c) BdE. Am. Phy8. Soc., 6, 165 (1961); (d) Symp. on Quantum Mechanics, 141st National Meeting, American Ch’emical Society, Washington, D. C., March, 1962; (e) J . Phys. Chem., 68, 2299 (1962); (f) A. D. McLachlan and L. C. Snyder, J. Chem. Phys., 86, 1159 (1962). (Some of Snyder’s results are tabulated in ref. lb.) (For an exhaustive and thorough discussion of Htlckel theory there is no better reference than the original literature: (n) E. Htlckel, Z.PhvsiB, 60, 423 (1930); 70,204 (1931); 73, 310 (1931); 76,628(1932); 88, 632 (1933); “Grundztlge der Theorie der ungesgttigen und Aromatischen Verbindungen ,” Verlag Chemie, Berlin, 1938.) Mark t h a t whenever h + ( q J equal8 & h - ( q d , eq. 2.2-3 simplifies and the extrema at ( ~ 4a 2jn/4 or (2’ f l ) s / 4 I

A

(2.2-2)

When extremiaed with respect to 94, the energy expression 2‘2-3 yields extrema at (F4 = ja/4 in 4‘ ‘pace, at (p4 = 2 j n / 4 ill 81,84-SpaCe, and at (p4 = ( 2 j f l)n/4 in 83,X4-~pade, ( j = 0, 11-2,3,4,5, 6,7). These extrema usually will be of a secondary nature, but may in par-

become extrema within the more complete qi,qs,q4-space. Note also t h a t i t is the mode S4a.t,(su) of Fig. 42 ( $ 4 . 1 ) which describes the arrowhead geometries (pr (2j l ) s / 4 ] . This mode is related to the analogous mode of Fig. 3b b y the linear transformation =i

+

(42)

S“(6)

=

-1 4 2

[&a(3b)

f S4b‘abi]

(contd.)

DEGERERATE ELECTRONIC STATES

Feb., 1963

397 2 gCO.5

L

5

SI N

e':'.. (b) (C) Fig. 4b and 4c.-Symmetry displacements for a planar pentaatomic pentagonal molecule. The displacements S,. and S ) b are, respectively, the real and imaginary parts of the complex coordinates s,,, ( j = 1, 2, 3), which physically can be depicted as the superposition of sla,b 90"out of phase. Figure 4b illustrates the Heraberg-type" symmetry displacements and Fig. 4c the internal ( s ' , a , h ) and Cartesian (P1,,t)-type.11 Note t h a t while the Herzberg type coordinates (by construction) yield identical circular rotatory modes a t each atom, the Cartesian and internal type yiold non-identical circular and elliptical rotatory modes a t these same atoms (by necessity, the sfvcl,.b modes are of the same form as the corresponding Herzberg &a,b mode, us only one e', mode is symmetry allowed). Although any of theso three types of coordinate basis will serve for a statical or dynamical pentagonal vibronic problem, the Herzberg type coordinates form the most physically illuminating set. I n terms of these latter coordinates, the interconversion of the Jahn-Teller or nonquadratic elastically distorted structures (obtained by joining the appropriately phased set of dots or dashes in Fig. 4b) is most easily seen.

head structure into two distinct diatomic fragments. The extrema in So,& and Srspace are as pictdred in Fig. 3c and 3d. An exemplary molecule which might show such minima is C4H4*.l5 At the (p4 extrema in S&-space A E n & factorizes to yield A&*

=

AV++

* IS(Qi)Qi

(-1>'q4'[h+(qi)

A

r

I

h-(q*)1 (2.2-4) from which the g1,94 extrema1 values are easily determined. [A similar expression may be derived for

-

+

(thus pd0) -(pd(ab) s/4), where the parenthesized superscripts indicate the figure which defines the S4..b(eJ displacements. [Please keep firmly in mind in what followa that whenever two (or more) displaoementa of the aame symmetry are poseible, we have consistently chosen these 80 t h a t one will have Sj+ Sjr, iSJb and the other Sk+ P Sk& iSkb attain the phsee 2 d / n under a clockwiae rotation of the oo6rdinatea b y 2 r / n [in Fir. 41a of $4.1 we have deviated from this convention and chosen both a modes t o have Si+ equal to Sj, iSjbl.'I This aomed.hat perverse choice of symmetry coilrdinate orientation waa made so t h a t the motion of atom 0 would always be direoted outward in the a mode and direoted toward the right in the b mode (except in Fig. 2b of 02.1)]. (15) In the Hockel approximation,1* the C4H4* Jahn-Teller prohlem beaomes mathematically identical with the C4H4 pseudo Jahn-Teller problem,"*'* exoept for a reduction of the C4H4 off-diagonal matrix element,,,'TA (see $2.8), b y a factor of -1/s. Hence, in this approximation. where T I equals b [for the definition of b see ref. 131, the stationary wave functions, $n*, become [recall (ii-2). (1-3), and (2.1-2)]

+

-

+

Thus. aside from a n irrelevant phaae factor, the wave functions, t,bn+, 1 1 become [@= 7 @ y ] if81> 0 and [& st @,I if SI < 0 [view Fig, 4 2 .\/z

-

ao,d 1.

-

Fig. 4d.-Sketch of the electronic potential energy surfrtces of a planar pentagonal molecule in a degenerate electronic state. The minima were chosen to be a t 0, 72, 144, 216. and 288" and the saddles at 36, 108, 180, 252, &nd324'.

Fig. 4e.-Cross sections of the pentagonal potential energy surfares (Fig. 4d) in the directions pj ( j = 1, 2 , 3 ) equals k x / 5 ( k = 0, 1, . . ., 9).

ANDREWD. LIEHR

398

S3,Sq-space. The expression for S4-space alone is readily obtained from (2.2-4) by setting q1 equal to zero.] Notice that according to (2.2-1 and 2), out of plane deformations can only arise from quartic and higher order terms in S2(plu). This situation is to be contrasted with the odd in plane eu type deformations which occur already in the second order [see (2.2-1 and 4) 1. 2.3 The Pentagonal Case.-Planar molecular aggregates with fivefold axes may be characterized by four types of degenerate electronic states, Ej’l’’, ( j = 1,2). These types, #*j! exhibit phase factors of eiZaij/j, ( j = 1,2), under a clockwise atomic replacement permutation, %(z) about the fivefold z axis [Fig. 4a]. Thus in accordance with our previous procedures, AV,,, (p, t = fj), may be put down as follows [the nuclear symmetry deformations (Fig. 4b,c) So(al’), Slf(cl’), SZ*(EZ’), s3,t(EZ’ and ), &*(E~”)have been symbolized as qo and qke*iwk, (k = 1, 2, 3, 4), respectively. We describe here only the five atomic pentagonal nuclear disposition; the six atomic arrangement is mathematically isomorphous to the pentagonal pyramid (54.111

Vol. 6’3”

which extremizes angularly at multiples of ~ / 5 . The factorization of e-iq’ and eiql out of AV+I-, and AV+2-2, respectively, in (2.3-1) shows immediately that the radicands of (2.3-3 and 4) are perfect squares at the angular extrema, and so the energy expressions are factorable a t these points. The nature of the extrema in Xj, ( j = 1 , 2 , 3 ) ,space are displayed in Fig. 4d and 4e. A sample compound in which such behavior might be found is CBHG.~~’l8>149 Observe that, although out of plane distortions of the five atomic pentagonal system can already arise from quadratic (and higher) terms in Ez’e”electronic states, they can only arise in the third (and higher} orders for El‘,‘’electronic states. 2.4 The Hexagonal Case.-The method for the determination of the structure of the vibronic matrix AVPt, (p, t = *j)! is by now obvious and need not be reiterated here for the Ejg or u, (j = 1, 2), electronic states of the D 6 h point group. Suffice it to say that these electronic states, #&j, transform as [Fig. ja], and that the permissible nuclear modes for the six atomic hexagon [the seven atomic hexagon is mathematically isomorphous to the hexagonal pyramid ($4.111 are [Fig. 5bl S O ( ~ ISl(Pd, ,), S2@zg),fl3(pZ&

(2.3-3)

I 1j

z

(2.3-4)

Feb., 1963

DEGENERATE ELECTRONIC STATES

The energy surfaces AEj, are given by eq. ii-1 anld 1-3 as before, and are pictured in Fig. 5c and 2d in the individual &-spaces, (k = 5,6). The angular extrema lie a t $Ok = d / 3 , (k = 4,5,6), and a t P k = d / 6 , (k = 71, (I = 0, 1, 2,---,ll). Specific chemical entities which might manifest such extrema are neutral benzene and its postive and negative ions.Ib?8 a ~cp 14,16, 1, Again symmetry requirements force the radicand in lAV+j-i to factorize, and thus allow a straightforward determinimtion of the g k extrema. Mark that out of plane distortions may occur as a result of cubic (and higher) order terms which contain S2(p2,)and S,(ezu).

1

399

The Heptagonal Case.-A molecule which falls into the D7h point group18 is characterized by the degenerate electronic states, #*j, of symmetry species Ej"", ( j = 1, 2, 3), which transform as e*2azj/7 under e&) [Fig. 6a]. If for illustrative purposes we restrict ourselves to a seven atomic heptagon [the eight atomic heptagon is mathematically isomorphous to the heptagonal pyramid ($4.1) 1, the non-trivial nuclear motions are [Fig. 6bl SO~CYI'), S ~ Q ' )XZ~(EZ'), , 8 3 + ( ~ 2 ' ) , SrJe3'), 8 6 & ( € 3 ' ) , Xsl(eZN), and S7+(e3").Our previous techniques then yield 2.5

2-

Ck45qkq496e --i(q4

+ rp5 - Vk)

+ ------_

(2.5-3)

k - 2 (16) (a) W. D. Hobey and A. D. McLaehlan, J . Chem. Phiis., 98, 1696 (1960): (b) A. D. McLachlan. Mol. Phus.. 4. 417 (lgG1): (e) W. D. Hobev.

"Vibronio Interaotions in Conjugated Systems," Thesis, California Institute of Technology, 1962.

( 1 7 ) H. M. McConnell and A. D. McLachlan, J . Chern. Phys., 34, 1 61961). (18) W . G. Fateley, B. Chrnutte, and E. R. Lippincott, ibid., a6, 1471 ~I

(1967).

ANDREW D. LIEHR

400

Vol. 67

zklrnqkq1qrne

--i(cok

+ + am) at

+ ________

(2.5-4)

k. 1. m = 2. 3 and k'+ 1 = m = 6

This energy surfaces A& are, as previously, described by eq. ii-l and 1-3, and are pictured in Fig. 6c and lid in the individual Sk-spaces, (k = 1, 2, 3, 4,5). The angular extrema lie a t P k = d / 7 , (I = 0,1,2,----13). Particular chemical specimens which might exhibit such extrema are neutral tropylium and its positive and negative ions. The radicand of /AV+j,-j/ is faotorized a t the angular extrema by symmetry restrictions as usual, and therefore, a straightforward determination of the q k extrema is again possible. Out of plane deformations may happen already from quadratic (and higher) terms in SB(e2") and &(E?,") for Ej"", (j = 1, 2, 3) electronic states [for the E2'l" Rtates S7(e3") appears asymmetrically first in the third order].

2.6 The Octagonal Case.-A planar octagonal chemical system [point group D 8 h (equals D 4 d X i)] is individualized by the degenerate electronic states, $*I,of symmetry species E j g u , (j = 1, 2, 3), which transform as e*2rij/8 under e8(2) [Fig. 7a]. When, for descriptive convenience, we restrict our considerations to an eight atomic octagon [the nine atomic case is mathematically isomorphous to the octagonal pyramid ($4.1) 1, we have the non-trivial nuclear modes given by (Fig. 7b and 7c) So(qJ, SI(&),S2(Pd, S 3 ( b g ) , S4+(eu) > &&t(EZg)

fh(E2g)

,

S7,t(e2u!',

'%zt

IO

6

i = 5 k-8 fi

cikkqiqk2cos( P i f 2pk) f

(e3d

flP+(E3u)

,

and SIo&(eQu). Therefore, our standard methodology produces the energy matrices as

i = 9, 10; k = 7,Z = 8 i = 5 , 6; k = 9, 2 = 10

CikZqiqkqZ

cos

(Pi

f

Pk

+

Pl)

(2.6-1)

Feb., 1963

DEGENER,ATE ELECTRONIC STATEEI

4

4

0

0X.OY

(ox)

Fig. 5a.-The real and imaginary (0,) parts of the complex El, wave function of a planar hexaatomic hexagonal inoleciile (e.g., “C6H&’). These functions transform as Bag and Rzn in D Z h , singly.

The energy sheets, AB,*, are generated as previously by eq. ii-1 and 1-3, and are depicted in Fig. 3c,d and 7d,e in the individual Sk-spaces, (k = 1,3, 5,6). The angular extrema lie at Pk = O,n,(k = 1,3); Pk = d/4, (k = 5, 6, 7); and $Ok = ~ 1 / 8(k , = 4,8, 9, lo), where (I = 0, 1, 2,--7). Sample molecular aggregates which might display such extrema are planar cyclooctatetraene and its positive and negative ions. The radicand of lAV+j-j] is factorized a t the angular extrema by the symmetry restrictions, as of old [note that for j = 2, the “angular” extrema of SIand S3 are artificial as q,

401

in general, can have only two values, 0 and T,for these coordinates. The extrema are at ql = gl, g3 = 0 and s7; = 0, q3 = q 3 (Fig. 3d) so the radicand does factor despite appearances-recall eq. 2.2-3 and 41. The determination of the qk extrema is straightforward. Deformations out of the molecular plane may appear &(eZu), and Ss(eag),which contribute due to 52(/31n), perturbation terms in AT‘+,-, already in the second order. 2.7 The n-Tagonal Case.-By now it is obvious that the method here outlined is readily extensible to planar systems of arbitrarily high rotational symmetry, and that in all cases eq. ii-1 and 2 and 1-3 apply and are factorable at the angular extrema. The form of the resultant potential energy surfaces also is easily visualized by analogy with what has been described and drawn. It is apparent that only the prime number rotational groups produce a single unique Jahn-TellerRenner energy surface per group, and that the non-prime number systems reproduce, for the most part, the behavior of their prime progenitors [witness, however, Fig. 7d, which proves that non-prime systems can also produce new sorts of Jahrt-Teller energy surfaces]. Therefore, we need not pursue our detailed study of the topological implications of the Jahn-Teller theorem further for planar systems : we have already amply demonstrated the issuant consequences. 2.8 Some Cases of Planar Pseudo Jahn-Teller Instability.-In the Huckel appro~imationl~g the ground electronic states of cyclobutadieiie and cyclooctatetraene are accidentally triply degenerate. They thus can exhibit pseudo Jahn-Teller instabilities.2 To see this fact wemust first note that tthethree fortuitously degen@Iyl erate states of C4H4 [Fig. 8al and CsHs [Fig. 8b1, and are of species BZg,Blg, and AI,, respectively, in their separate point groups D4hand Dsh. Therefore, by our symmetry arguments of the previous paragraphs, we may write the requisite vibronic matrix elements as [using the nuclear coordinates of $2.2 and $2.61

a,,

me,

(2.8-1)

c

.k=5,6;i=k=7 i = 4, k = 9, 10

C’likqlpiqk

sin (pi

+ Vk) +

C

,E

i = 5, 6; IC = 7 i=4;k=.8

cp2ikq2qipk sin

(pi

+ pk) +

c

i, k = 5 , 6 ; i = k = 7 i = 4, k = 9, 10

cP3ikq3qiqk

x

ANDREW D. LIEHR

402

cxy3ikq3qiqk i = k = 5, G , 7 i = 4; k = 9, 10

sin

(pi

+ + pk)

1'01. 67

cxy4ikq4qiqk i = 4 ; k = 5 , 8 = 9, 10

sin

(pk

- pi

-

+

(~4)

i = 5. ti: k

(2.8-2)

It is obvious that the three by three secular equation which results from eq. 2.8-1 and 2 is, in general, quite messy to solve.9 Therefore, to determine some of the qualitative features of the issuant potential sheets, we shall introduce several simplifications. First, we shall take all the diagonal elements AVp, (P = x, y, z), to be equal, and second we shall set AV,, equal to zero. [This is the situation which is obtained within the Hiickel theoryI4gof conjugated systems. Without this approximation the cubic seeular equation mill factor only a t the ql- and q3-space extrema]. The latter restriction has been introduced in order to factorbe the cubic energy equation; it is certainly sensible as this particular matrix element is of a higher order in the nuclear disturbances [of coursc, pathological instances exist where AV,,, cannot be neglected]. We then find [I' = x, y, z ] A& = A V p

In gk-space, (k = 0, 1, 3), the deformed potential surfaces, AEl,z are as pictured in Fig. 3c,d in which we have chosen the qo origin such that cpO vanishes. The linear approximation produces a simple analytical description of the surfaces of Fig. 3c,d (we have here dropped the now redundant suprrscripts, I', xz, and yz) AE1,z =

cl1ql2

+

c33qa2

*

___--

-I-

~

3

(2.8-4) ~ ~

It is interesting to note in passing that the quadratic and higher order off-diagonal terms in

S4*(eu) can lead to the dissociation of the quadrate form of cyclobutadiene (indeed, the quadratic terms alone could do this in certain morbid cases-recall $2.2),14and that the can quadratic and higher off-diagonal terms in &*(e&)

lead to a stabilization of tho boat form of cyclooctaS 7 b produces this tetracne [the displacement -Sra particular noli-planar deformation of cyclooctatetraene ---we Fig. 7f].

+

111. The Three Dimensional Case $3. The Regular Polyhedra. 3.1 Cubic Molecular Systems : The Doubly Degenerate Electronic States with Zero Spin-Orbit Forces.-The cubic (Oh) class of chemicals is comprised of hexacoordinated octahedra [Fig. gal, tetracoordinatrd tetrahrdra [Fig. 9b], octacoordinated cubes [Fig. 9c], and dodecacoordinated cubes [Fig. 9d]. We shall discuss each of these classes in turn, starting with the octahedroii. Rut first a few preliminaries. To illustrate the origins of doubly and triply degenerate electronic states for cubical molecules, we draw in Fig. lOa,b,c,d,e,f schematic energy level patterns for an exemplary group of cubical chemicals, the transition metal Werner complexes. I n Fig. 10a,c, me show their levels in the ionic approximation, and in Fig, lOb,d,e,f, in the covalent approximation. From these figures it is evident that electroiiic double and triple degeneracy arises from the occupancy of molrcular orbitals of e, tl, or tz symmetry. The geometrical disposition of these particular orbitals will br of cardinal concern to us in all that follows. Thus, in 1;ig. lla,b, we graph their tectonic idiosyncrasies. 3 ~ [We shall later demonstrate that the tl and t 2 orbitals lead to mathematically isomorphous Jahn-Teller topologies. Hence, in Fig. lla,b, we illustrate only one of these two sets of orbitals, the tzset.] Finally, since the coulombic interaction of the e, tl, and tz electronic charge distributions with randomly disposed nuclear charge centers will be our prime interest in what suc-

Feb., 1963

DEGENERATE ELECTRONIC PTATES

403

Fig. 5b.-The algand ~2~ symmetry displacements of a planar hexaatomic hexagonal molecule. The displacements S,, and S,b are individually the real and imaginary parts of the complex coordinates S,,, ( j = 5, 6), which physically can be depicted as the superposition of S l a , b 90" out of phase. The coordinates S,a,b, S:a,bt and SE,b,( j = 5, 6), are serially, the Heraberg,llinternal,10 and Cartesian10 type symmetry displacements. Perceive that while the Herzberg type coordinates (by construction)yield identical circular rotatory modes a t each atom, the Cartesian and internal type yield hon-identical circular and elliptical rotatory modes a t these same atoms. Although it is mathematically immaterial which type of coordinate is utilized for statical or dynamical computations, the Herzberg type is the most phyeically appealing. I n terms of these latter digplacements the interchange of the Jahn-Teller or non-quadratic elastically deformed structures (procured by connecting the correctly phased set of dots or dashes) is most readily visualized.

ceeds, we sketch in Fig. 12a,b,c,d the local Cartesian nuclear coordinate systems exploited. [Considerable mathematical advantage is gotten if instead of these local sets of Cartesian nuclear coordinates, delocalized sets of Cartesian symmetry coordinates are used. Such a system of eurythmic coordinates (tetragonally oriented) id pictured in Fig. 13a,b,c,d,e. This system of symmetry coordinates is not unique, however. In Fig. 14 and 15a,b,c,d are shown an alternative set (trigonally oriented) of T ~ ,r2 (Fig. 14a,b,c,d), and e (Fig. 15a,b,c,d) symmetry displacements,] The permissible seven atomic octahedral symmetry displacements are [Fig. gal 8,( a l g )82a,b(Bg), , 83a,b,c(Tlu), 84a,b,c(71u), SSa,b,o(T2g), and 86a,b,c(T2u). The doubly degenerate electronic charge distributions [Fig. 1l a ] , are of symmetry species E, [when A = x2 y'.

-

%3'

B = 3z2 - r2] or E,, and they transform as

4 5 tc. B *2 (A)

--l/2$

12 0 ' :

Fig. 5c.-Visual representation of the electronic potential energy surfaces of a planar hexagonal molecule in a degenerate eIectronic state (we show the duplex of two dimensional crosscuts, S b a , b and SGa,b, of a four dimensional space, S j a , b , ( j = 5, 6)). Motion in the four dimensional space of S J a , b , ( j = 5, a), can best be pictured as a coupled motion in the two disjoint two dimensional spaces &n,b and &a,b. In each of these smaller spaces the figure shows (for specificity only) minima a t 0, 120, and 240" and saddles a t 60, 180, and 300" (the cross cuts of these potential snrfaces at these anguular locations are shown in Fig. 2d).

A (B)

under the threefold rotation operator,

e3 (2')

[the z axis is taken to be the fourfold axis-observe Fig, Qa-16a].18

Vol. 67

ANDREWD. LIEHR

404

Under the counterclockwise rotation eg(z') which replaces x by z, y by x, and z by y, atom 0 by 5 , 1 by 0, 2 by 4, 3 by 2, 4 by 3, and 5 by 1 (see Fig. sa), the nuclear coordinates Sz a are replaced by -'/zS'(~) =k

G / 2

+

(b)

SZ(;,.~~

Thus, as e3(AV,4

4- AVB) equals

AVA AVB and ~ ~ A Vequals A B 4 3 / 4 ( A V ~- Ab,) l/zAV~B, by the definitions of AV, $A, and $G, the operation of e3onto the right and left hand sides of (3.1-1) yields two mutually independent expressions for e3 (AVA AVB) and e 3 A V ~which, ~ , when equated, de, = -cA2, cA1l = ~ ~ 1cA22 1 , = cB22, mand that cB1 = C ~ I cBz

-

+

EB22 =

-&,

CAB22 = 2FA22, CAB2

=

-CAz,

CB12 = -CA12,

and cAB12= -cAB12. Therefore, setting '/2a equal to cA2, l12pequal to and 1/2y equal to cA12, we may write our vibronic matrix array as '/~(AVAd- AVB) (AVA - AVB)

~ A V A= B -$.'

4 C 2 SIN?

To simplify the presentation, we shall, in what follows, truncate the power series expansion of the vibronic matrix elements, AVPQ, (P, Q = A, B), a t degree two. I n addition, we shall quadratically include only those nuclear coordinates which may theoretically appear both in first and second order. This latter restriction eliminates from possible consideration all higher order instabilities [which may be quite sizable]. These higher order instabilities are, however, of a similar form as those which occur in the more usual non-degenerate situation,3 and so will not be further pursued here. With these reservations, we may, by the methods of section 11,write the needful vibronic interaction elements A V ~ Q(P, , Q = A, B) as [here we have used the "descent in symmetry" t e ~ h n i q u e ~ * * ] ~ ~

+

~ ' 1 ~ 1

+ cP11Sl2 + cpZZ(SZa2 + +

(cp2

+

AVAB

(cAB2

+

+

EP22(S2a2

- S2b2)

~ ~ ~ 1 2 S i ) S 2 b~ * ~ 2 2 S 2 ~ S z (3.1-1) b

(ID) For the non-planar geometries i t is more convenient to use a real set of electronic and nuclear bssia functions throughout. This choice is not necessary, however, a s we could just as well define ib* as

2-'/2

(+A

rSl)X2a

')'Si)XZb

C22(&a2

-

p(x2a2

+ -

- 2bS2aSlb

&b2)

SZb2)

(i.1-2)

2-'l2((t)[$+ h I,!-],*~ the energy formula for AEn is changed to (in the linear approximation to (3.1-2) the phase angle w equals the nuclear angle (FZ) A&+

=

'/Z(AVA

-

+ AVB) * +

'/~V'(AVA AVB)' (~AVAB)' '/z(AVA f AVB) f '/2(Av~ - AVB) SeC w (3.1-2a) and the wave function formula for Gn is changed to cos 1/20$A- sin sin ' / 2 ~ $ ~ cos

+

1/*w$B '/2w$B

tanw =

-2Avn~ (3.1-2 b) (AVA - AVB)

Therefore, the energy surfaces charactdristic of an octahedral molecule in a doubly degenerate electronic state are given by [setting 91 = SI, q 2 cos 'p2 = S2a, 92 sin cp2 = s 2 b l

+

CP1ZS1)S2a

52b2)

+

-(CY

+

+

(B)

Fig. 6a.-The real (@J and iwaginary ( B y parts ) of the complex Ez"wave function of a planar heptaatomic heptagonal molecule (e.g., "C7HI)'). These functions transform as B1 and A2 in CzV,distributively.

=

= (a

Cii&2

When the complex wave functions $* of eq. ii-1 and 2 and 1-3 are expressed in a real basis $ A equals

s l N y'~ ' " 0 X . O Y

AVP

f

= Cis'

F i ~ s )Sz,(eg) , [equals q z e ~ f i m las 2 - ' / 2 (Sz, F iSzb),

since the symmetric direct product [ ESg or a ] equals Ai, Eg in Oh, we further know t h a t the terms which are allowed to appear because of their a p t behavior under the (restricted) symmetry operation$ of the D4h subgroup of O h can be sequestered into linear combines which transform as AI, or E,; hence, the terms Sz&2 Szb* [equals Q Z (Alg) ~ I, Spaz 8 z b Z [equals 912 cos 292 (Eg)], s n d 2SZs&b [equals qz1 sin 292 (Eg)]rather than the simple products Sza', &b*, a n d SbQZb. Note t h a t eq. 3.1-1 is essentially the same as t h a t derived previously by C. J. Ballhausen and the author b y use of the electrostatic crystalline field theory [animadvert, (a) A. D. Liehr and C. J. Ballhausen, Ann. Phys. [N. Y.], 8 , 304 (1958), a n d (b) C. J. BallD. Liehr, Acta Chem. Soand., 16, 775 A V ~ Q(P,Q , = A,B), here derived are enOs AV+* and AV+lO b y the relations A AVB) and AV+- = ~ / z ( A V A AVB) ~ A V A B . Note that in terms of the tracelegs matrix A v o f 13.2, A V = AVPQ l/zPdpg, (P,Q = A,B), where I/& 9 pV** = ~ / s ( A V A AYB). we have t h a t A P p '/z(AVA AVB), (P = A , B ) , a n d AVAB = AVAB. Hence, for the traceless matrix A P , the general energy expressions (ii-1 a n d 2), (1-3), and (3.1-2) simplify to read ABna = z/L\Bp2 AVABZ, (P = A.B), where A B = A E '/zP,

-

-

where S'ko = qk cos Bk', S ' k i = rn sin ek'efick'. With these definitions, vibronic expressions analogous to those for the planar geometries may readily be set up, and the requisite energy surfaces and wave functions obtained b y substitution of these expressions into eq. ii-1 a n d 2 [or equivalently (1-3) 1. (20) The detailed structure of eq. 3.1-1 is readily discerned b y noting that $A,B span the irreducible representations Big and Aig, each, in the D& subgroup of the octahedral group Oh; a n d therefore, that the matrix elements. AVp, (P = A, B), may only contain those nuclear displacements whioh span the irreducible representation Ai, of D4hr while the matrix element AVAB may contain only those which span Big in Dih. Moreover.

-

+

a n d tan

+

w =

APAB AVB'

+

+

+

*

-

+

(21) I n contrast to our convention of 02.1 we now take e, to be a counterclockwise rotation rather than a clockwise one a s before. This occurrence will be general: for planar geometries en (n = 3, 4, 5 - - - ) will be a clockwise rotation and for non-planar geometries i t will be a counterclockwise one. This choice is dictated b y past convention, rather than obstinate perversity. [For additional discussion, with applications, of symmetry operations and conventions read A. D. Liehr, J. Phys. Chem., 64, 43 (1960).1

405

DEGENERATE ELECTRONIC STATES

Feb., 1963

Q 1;;; 277

-

477 7

277 7

377 7

7

I,d 5

47T 7

Fig. 6b.-The n'l and dj, (j = 1,2,3), Herzberg type" symmetry displacemente of a planar heptaatomic heptagonal molecule. The displacements S j a and S j b are seriately the real and imaginary parte of the complex coordinates S j , , ( j = 1,2,3,4,5),which physically can be depicted as the superposition of S j e . b 90' out of phase. The reciprocation of the Jahn-Teller or non-quadratically elastically distorted forms (gotten by joining the correctly phased set of dots or dashes) is nicely seen from these latter superpositions.

ABn& = '/2Q2d(a

C141

f

Cllql2 ?'Q1)2

+

*

C Z Z ~ ~ ~

3% (3.1-3)

P2PJ2- 2 ( a f YQI)PQ~

I n glancing a t eq. 3.1-3 we are immediately struck by its resemblance to the analogous formulas for the planar cquilatcral [§2.1, eq. 2.1-31 and hexagonal 182.4, eq. 2.4-1 J molecular architectures. Indeed, eq. 3.1-3 is

ANDREWD. LIEHR

406

vof. 67

i

Fig. 6c.-Schematic representation of a two dimensional cut in SJa,b, ( j = 1,2,3,4,5), space of the electronic potential energy surface of a planar heptagonal molecule in a degenerate electronic state. Motion in the actual four dimensional space &a,b, ( j = 2, 3 for an El’?’’ state; j = 4,5 for an Ez’,” state), can best be viewed as a coupled motion in the two disjoint two dimensional spaces SJa.b, ( j = 2,3; or 4,5); that in t h e two dimensional space SI,, (for an E3’~’’electronicstate) is as drawn. I n each of these smaller spaces the figure k e s (for illustrative c1arit.y solely) the minima a t jn/7, ( j = 0,2,4,6,8,10,12), and the saddles a t jn/7, ( j = 1,3,5,7,9,11,13). (The required rapid passage from saddles t o minima implies an almost smooth potential well here.) Fig. 6d.-Azimuthal cuts of the heptagonal potential energy surfaces (Fig. 6c) along the lines pl, ( j = 1,2,3,4,5), equals k ~ / 7 , ( k= 0,1, . . .,13).

c

OQO

and elg Herzberg type” symmetry Fig. 7b.-The alg,pia, displacements of a planar octaatomic octagonal molecule. The displacements Si, and S j b are, consecutively, the real and imaginary parts of the complex coordinates Sj*, ( j = 5,6), which physically can be depicted as the superposition of Sja,b 90’ out of phase. The interconversion of the Jahn-Teller (for an El, or Eap or , electronic state) or non-quadratically elastically distorted forms (obtained by joining the correctly phased set of dots or dashes) is readily seen from these latter superpositions.

i

sl(Plg)+ s3 ( P 2 g ) symmetry displacements SO(alg),.S1(plg) and

so(a,g)+sl(P,g) 0-x

-C

Fig. 7c.-The &(&), &(&J added 90” out of phase. These combinations give the physical movements which accompany the nuclear motions of an El,,, electronic state on the Jahn-Teller potential surfaces of Fig. 30 and 3d. The dots and dashes locate the minima and saddles, sequentially.

nately] which is derivable from (2.4-1). Furthermore, the symmetry arguments of this paragraph, when applied to tetrahedral and hexahedral architectures lead to energy expressions identical in structure to (3.1-3) .22

0x.Oy

(ax)

(ay)

Fig. 7a.-The real and imaginary parte of the COMplex Ezu wave function of a planar octaatomic octagonal molecule (e.g., “C8H~”). These functions transform as EL, and BI, in Dah, serially.

mathematically isomorphous to eq. 2.1-3 as well as to its related expression tin qo,q5- and qo,q6-space,alter-

(22) As the tetrahedron has nuclear displacements [Fig. 13bl Si(ad. SZs,b(e), S3a.b,o(72), and S&,,B.~(T~),and as the point group Td and its subgroups Dzd are isomorphous t o the 0 and D4 subgroups of the hexahedral a n d octahedral point group Oh,me have t h a t isomorphio arguments require t h a t eq. 3.1-1, 2, a n d 3 also describe vibronic interactions in the doubly degenerate electronic state $A,B (species E) of tetrahedral m6hUleS (with &(ai) and Sza,b(e) now playing the predominant roles). Similarly, since t h e hexahedron has nuclear movements [Fig. 13cl &(aig),& ( r r 2 d , 8aa,b(€g). S4n, b(fuu), SSa,b,o( nu), SGa,b,o(Tlu),S7a.blo(T2g) SBa,b,o(rzp), a n d SSa, bvo(r2u) v we see that only the coordinates SIand Ssa,b may perturb the E g 01 u cubically (Oh) degenerate electronic state $A,B. Hence, when the identical &rguments of this paragraph are applied In the cubzc Oh pomt group and its D411 rubproup. eq. 3.1-1 to 3 are again reproduced, The cubic tetradecahedron

DEGENERATE ELECTRONIC STATES

Feb., 1963

407

E

f 3

AE

2

tY

tY X

A.ib C

0

0

n

n

I (a) (e) Fig. 7d.-Diagram of a two dimensional cross section in S , c , b , ( j = 5,6), space of the electronic potential energy surface of a planar octagonal molecule in the degenerate electronic states El, or and Egg or ,. Nuclear movements in t h e actual four dimensional space S l a , b can best be visualized as a coupled motion in the two disjoint spaces &a,b and SB.,t,. In these lower spaces, the drawing places (for demonstration purposes only) the minima a t 0,90, 180, and 270' and the saddles at 45, 135,225, and 315". Fig. '7e.-Longitudinal sections of $he octagonal potential energy surfaces (Fig. 7d) at the angles pj, ( j = 5,6), equals kr/4, ( k = 0, 1, ., 7).

..

0

0

0

o x

--

ox *0,

0, 43,

(4

e I

0

0

'0,

.

tY

S2(PlU)

--x

0-x

0

ox. o y

+as

0

0

4

0

0,.

Oy.

Ot

(b) Fig. "8a and 8b.-The accidentally degenerate (in D4h and Dsh) Btg( Big( and Ale( 6.) electronic wave functions of cyclobutadiene (Fig. Sa) and cyclooctatetraene (Fig. Sb).

ax), aY),

Sea(f3g)

Seb(cjg)

%a+Seb

Fig. 7f.-Typical out-of-plane Jahn-Teller or elastic deformations possible for an octagon due t o second order- (and higher) terms in &(pi,), &(e& and S*(egg). The displacements &o,b and &&,b are here added in phase t o produce the sums &b and &a

+

+

&b.

has nuclear modes [Fig. 13d,el Sl(alg),Sdazp), Sa(azu), a a , b ( f g ) , S58.b(q), S6a,b(%), &e.b.a(rlg), 88a,b,c(Tlu), &s.b.c(Tlu), &oa.b,c(Tlu), 8lla.b8o(Tzp), &ee,b,o(T2g)l ~ 1 3 a , b , o ( T 2 ~ )and , S 1 1 & , b , ~ ( 7 2 ~ )Therefore, . its potential sur-

face for doubly degenerate electronic states strongly depends upon two eg modes, S4a.b and Sra.b, and is thus mathematically isomorphic with that of the hexagon, eq. 2.4-1 and 1-3. [This statement may be readily proved either by usc of complex wave functions and nuclear oo6rdinatesls (working in the c8 subgroup of Oh) or by use of real wave functions and nuclear eoordinates working in the D4h subgroup of Oh as in the text.]

This occurrence is not happenstance, but, on the contrary, of deep mathematical significance. [We shall again encounter this situation in $4 when we discuss the irregular polyhedra. ] The phenomenon is most aptly termed mathematical inheritance. The "mathematical gene" which passes on these inherited Jahn-Teller traits is the threefold axises(z'). I n Fig. 16a we illustrate this point by demonstrating the geometrical relationship which exists among the simple polygons and polyhedra which possess doubly degenerate electronic states by virtue of their threefold axis. All the characteristics of inolecu'les which occur already in

408

Vol. 67

ANDREW D. LIEHR

doubly degenerate electronic states are pictured in physical space in Figures 15a, Ha, 19 [the octahedron], E'ig. 15b, 18b [the tetrahedron], Fig. l5c, 18c [the hexahedron], and Fig. lsd, 18d, e [the tetradecahedron], and in abstract symmetry coordinate space in Fig. 17, 20a, and 20b. It is important to observe here, just as it was for the polygonal dispositions, that the only diference betueen normal nuclear motions and the JahnTeller motions i s that in the latter the nuclei cannot return 20 the conformation of m a x i m u m symmetry, and that the subsidiary extrema1 geometries df lower symmetry of the normal non-Jahn-Teller molecule3 now become the primary geometries. 3.2 Cubic Molecular Systems: The Triply Degenerate Electronic States with Zero Spin-Orbit Forces.The triply degenerate electronic charge distributions +A,B,C are of symmetry Tl(Td) or TI, or " ( o h ) and TdTd) or Tz, or .(Oh) for cubic aggregates. As an instance, the charge distributions $A,B,C resemble the polynomial distributions xy, xz, and yz, seriately, for T,(Td) or T2,(Oh) states; z, y, and x for Tl.(Oh) states; and l,, E,, and I, [the symbol lt, (t = x, y, z) signifies the tth component of the angular momentum operator, I] for T1(Td) and Tl,(Oh) states. Under the threefold rotation (permutation) e3(z'),the three component wave functions $A,B,C permute to $ B , c , ~ (cf. Fig. lla,b). This selfsame operation also replaces the nuclear coordinates Xka,b by S k b , c , a [ C f . Fig. 13a,b,c,d,e, 14a,b,c,d]. The comments of $3.119 as to the use of the complex wave functions

t'

Fig. Qa-Sc.-The

octahedral (Fig. Qa),tetrahedral (Fig. Qb), and hexahedral (Fig. Qc)dispositions.

Fig. 9d.-The

cubic tetradecahedral and trigonal dodecahedral arrangements.

their C3 subgroup, also appear in their over-all group. Hence, in general, all molecules can be segregated into Jahn-Teller families, which are generated by the simplest subgroups contained in the over-all group which still preserve the electronic degeneracy in question. For example, d [or Cdv, b z d , or D4d] subgroup of D 4 h and Dsh the c completely specifies the Jahn-Teller behavior of Ez,: or electronic states [see Fig. 16bI. The Jahn-Teller nuclear motions of cubic systems in

.

(.i= 0,

* 11,

_-

214

with ea$]= e $j, and the complex nuclear coordinates apply equally well here. The termination of the expansion of the matrix elements A V ~ Q(P, , Q = A, B, C), a t degree two together with the exclusion of those nuclear coordinates from the expansion which do not appear both linearly and quadratically yields, in the manner of $3.1 [setting p = '/3 trace AV = '/3 (AVA AVB AVc)]

X'kj

+

+

-"/pa

(3.2-1)

40'3

DEGEKERATE ELECTIZONIC STATES

k'eb., 19G3

(c) Hexahedron.-Equations thc same as (3.2-2) with Sj. (j = 2; 3, 4) rcplaccd by S k (k = 3; 7, S), respectively. (d) Tetradecahedron.-Equations the same as (3.2-2) with S; (j = 2; 3, 4) replaced by S k (k = 4, 5 ; 11, 12), respectively. An additional linear term, S7, is also introduced together with supplemental and Sk, (k = 4, 5 , 11, 12). quadratic term$ in Sz,8.1, Equations 3.2-1 and 2 are readily derived upon ~ from descent in symmetry from o h to D41, and D z and T d to D2d and C',, by thc methods outlined in 53.1. It is to be espccially noted that thc pliasc angles, cp, x , and u dcfiiicd in (3.1-1 and 2) obcy the relationships e3(p,x,u) = ( p - 2 ~ / 3 ,x - 2 ~ , ' 3 , u - 2 ~ / 3 ) . [Thc expressions m2 cos (2x - 27r/3), ctc., in t y . 3.1-1 and 2 m 2 sin 2x. are a shorthand for m 2 cos 2x cos 27r/3 sin 27r/3, where m2 cos 2x, m2 sin 2x, etc., are as defined under the subheadings (a) and (b). J The mathcmatical isomorphism of the direct products TI X TI and Tz X Tzrequires that eq. 3.2-1 and 2 hold equally well for both TI and Tztype electronic states. we define thc traceless (a) The 0ctahedron.-If matrix A 7 as ( A V ~ Q l / 3 p G p ~1 , we find that thc 3 X 3 secular equation for our triply degenerate set of clectronic states takes on thc form

+

+

AE3 + [ A ~ A A+~ B APAAVC A ~ * A B- APZ*c ]AI!?

AI!?

=

AE

+ A v B A v c - A P A C- det A 7 = 0,

+

'/3p

A P ~ A - ~~ Z 7 ~~ ~-~A 9~ 2~ ~ A(3.2-3) ~ v I

~

This cubic equation has as its solutions AE = -

l/3p

+ AE = -

f2

'/3p

with cos ip

=

4

rI

cos

k = 0 , 1, 2

4;[$]

(" +32=/C)

=

-___

3=

upper sign b

> 0, lower sign b < 0 (3.2-4)

where a denotcs the coefficient of the linear term in AB and -6 the constant term in eq. 3.2-3.9 The extremization of eq. 3.2-4 with respect to the polar and azimuthal nuclear phase angles Oj and pj, separately, shows that the angular energy extrema are completely determined by the angular extrema of the quantities a and b defined above. When expressed in terms of the radial amplitudes gj and the nuclear phase angles 6j and q,the octahedral a and b factors assume the guise [defining7 ~ cos 5 ~21,h = 75' 0, the covalent case, the angular positions of the mininia and saddles are reveriled).

orbital basis fuiictioiis, with I / ~ , (P = A,B,C,D,E), as defined in $3.3 and $PI, as their trigonal analogs [A' = x'Y', B' = x ' ~ ' ' ,C' = y'z', D' = x'2 - y'2, E' = 32'2 - 1 ' 2 1 . '*A'-

*B'

For later utilization we shall also tabulate here the matrix elements for a trigonally oriented cubic basis. These elements are obtained by use of the functions listed in eq. 16 of ref. 21, and by use of the following matrix connection between the quadrate and trigonal

ANDREWD. LIEHR

428

+, = 0,

Vol. 67

x,

= T SADDLE

$2

MINIMUM

tz

X

-c

4

sb2

= 2 Ty , MINIMUM

#2

=IT, SADDLE

tz

4

I

#2

=q,

MINIMUM

+,= Figure 18a (see Fig. Be).

We then find (b)

Trigonal Orientation AV,; =

I?

f z‘, (S

=

G,b)

y,

SADDLE

429

DEGENERATE ELECTRONIC STATES

Feb., 1963

(a) Quadrate Orientation

+ K')

A E k = I/Z(K

(b)

f. ~ / z { ( K

-

Ip1

-k 4 ( ( X 1 2 -k

K')'

]

(twice) (8.4-9)

2) I ' '

Trigonal Orientation A E k = ri

f \A/'

f (2"

+

\pi

(twice)

2)1"

(3.4-10)

Although eq. 3.4-9 and 10 look quite different, the substitution of the appropriate copulative relationships from (3.4-5 and 6) shows them to be mathematically identical, as required. It is interesting to notice a t this juncture, that the placement of a finite energy separation 6 (the quadrate and r8c,d case) or 8 (the trigonal case) between the states, corresponding to the physical imposition of a tetragonal or trigonal electrostatic field, into the matrix elements of (3.4-1 and 5 ) , and hence, into the vibronic secular determinant,, leads to energy expressions but slightly different from eq. 3.4-9 and 10.

where

Tetragonal Field AE, = ~ / z ( K K' 4- 6) f ' / z { ( K (a)

+

4(/A\2

+

K'

- 6)'

l p \ 2 ) ) 1 "(twice)

4(3.4-11)

(b) Trigonal Field A E + = (2

+ '/zF)

{ ( i '-

( ] XI

'/ZJ)~

+ 1 pi

2))"z

(twice) (3.4-12)

The extremizLtion of (3.4-9) [or its equivalent (3.4-lo)] is quite similar to that performed in $3.2. To visualize this falct we need but observe that, by (3.4- 1) (K -

The matrix elements AV7i+, (E = ii,z,E,a),are readily determined by the application of the symmetry operation u(x' - z'), defined in ref. 21, to the right and left, hand sides of eq. 3.4-8 [recall that X and p are symbolic representations of AVsa,sE and AVsa,8& serially]. The null value of the matrix elements which connect the rkaand r k b , (k = 6, 7 , 8), wave functions or the I'scm and rRd wave functions is required by the Kramers theorem. 1 v 36 The 4 X 4 secular equations formed by the r8s and r8S matrix elements of (3.4-1 and 5), separately, yield as their roots

( 3 6 ) This null value may be derived easily by use of the momentum reversal operator As, by definition [(a) E. Fick, Z. Physik, 147, 307 (1967); (b) ,J. S. Griffith, "The Theory of Transition Metal Ions," Cambridge University Press, Cambridge (1961)1,X(AV,t) equals AT',$*, and sinceat X(rks*I'kt) equals -I'kt*I'ka, (9 = a ; c and t = b; d), and X A V equals A V , we espy t h a t AV.t* equals -AT'.$*, and so AV,t vanishes, (s = a: o and t == b; d). This circumstance has as its consequence t h a t Fa and I'r type electronic states are Jahn-Teller stable. [This does not imply, however, that they are also stable with respect to psuedo-Jahn-Teller forces ($2.8). For further discussion of these points see ( c ) A. D. Liehr, Bell System Tech. J . , 89, 1617 (1960), and references 4b and 21.1

x.

K')'

+ 4 ( 1 ~ /+' I p l ' )

=

+ A P B ' 4- AvBAvc] +

4/9[Av.4z

Avc2 - AvAAvB - AvaAvc

4 / 3 [ A p ~ ~f2 APAc' f APpc2] (3.4-13)

and so, whatever has angularly extremized the cubic coefficient a of eq. 3.2-3, likewise will extremize the radicands of eq. 3.4-9 and 10. (a) The Octahedron.-If, for the moment, we confine ourselves to octahedral rs electronic states, we may explicitly display the nuclear dependencies of eq. 3.4-9 and 10. The substitution of the octahedral matrix elements, (3.2-l), into (3.4-9) yields the (Kramers) energy surfactis

Vol. G7

I

SIN

1

92

r=?;r5?

42= 0, MINIMUM T

-

Figum 18h (aec Fig. 1%).

The singular points of the rnrrgy shrrts (3.4-14) are most emily uncovered, as in $3.2, by t,raiisforming to the trigonal coordinate system (:ridenwith situations (i) 6) Of ('3'2-'o 11), was already foreseen from (3.4-13).3'

+

Vobr (i), ms', m2 cos (2xs m), and ml*ros (2xs - 202) take on the valurs 4qbq,2 ( - 1)Jqs2,and 2qb2,separately, v5 0)are rendily obtained from these by the simple interchange of the labels saddle and minimum. The angle e is 88 defined in Fig. li. (In Fig. IXa,b,c,d,e, the quantities pi and a symbolize the equilibrium amplitude of S,;.,,,,( j = 2,3,4,5), and the equilibrium metal-ligand bond distance. This latter quantity ia unchanged by asymmetric Jahn-Teller deformations of the tetrahedral and hexahedral doubly degenerate electronic states).

--+-

VOl. ti7

!

I

$b-

--_

a d

t 4.SADDLE

-+ OI

e--

_.

C

b = 1.76

d

1.N

e =2.00 0 '2.09 C '2.13

(b) Fig. I9a and 19b.-Two plats of the various points of NU octahedral molecule in s doubly degenerate (E) electronic state. The numerical vnluea given arc for Cu(H,O).++ in the ionic approximation ( 8 < Ohmo. For this case the energy minima are a t the dots, the saddles a t the full lines, and the cusps at the dashed lines in Fig. l9b (recall Fig. l i ) . When 8 > 0, the covalent case, the minima and saddles of Fig. I(Ja,I) (LE reversed.

Thereforc, tho energy cquatioti (3.4-14) may bc factorized to read4n

= 42COS42 S2b 5 92s" 4 2 320

(3.4-18)

and thc radial extrema located at

(b) Fig. 2IIa and 201,.--Correlat,ion of the niirlcur motions of an oekthrclron (Fig. 2Ilrt) imtl tctrahrrlron (or hi:raherlron and tetrah a h r r l r o n ) (Fig. 2Illr) in a b s l c c t (Fig. l i )and physical (Fig. 15, IR, and I!))spwe. Thc E, &., nntl Y. :W:B d l lii. in t h r mmc ~ilrtnc. In this cvmt, the ~ ~ ~ c l imotion ~ i i i in :distract spnec, whcre w pamw I n n 0 to 2r/3 to 4r/3,is readily wcn to earrespond to R simultmeous motion in rcirl spuec, whrrc thc unique tctrngonxl axis of thc octnhcdron or tetrahedron (or hexahedron and tetmdecahcclron) pnssos from the z-axis (e= 0 ) to the y-axis (M = 2 r / 3 ) to the x-ask (e= .lr/:%).

There is a cusp at ql,qs equals naught i n all situations. The nature of the angular rxtrema of situations (i) and (ii) are illustrated in Vig. 38a,b.

DEGENERATE ELECTRONIC STATES'

B'eb., 1.963

To plot eq. 3.4-14 in all its generality is impossible, since we live in a three-dimensional world. However, some insight into the nature of these energy sheets can be obtained by the projection of eq. 3.4-14 into three-dimensional space. First, we espy that in yl,yzspace (3.4-14) is mathematically identical to (3.1-3), and hence, that it is pictorially identical to Fig. 17, with concomitant iiuclear motions as given in Fig. Ga,b,o,d, 18a,b,c,d,e, 19, and 20a,b. Second, we descry that, in y6-space,the energy surface AE- has a set of tetrahedrally disposed minima and maxima at the SBa,b,r, cube corners (trigonal distortions-see Fig. 29, 32a,b, and 36a,b), and saddles a t the cube face centers (rhombic distortions-see Fig. 28, 31, 33, and 34)) as is shown in Fig. 38b. The energy surface AE, bears the same relation to the surface AE- in ys-space as it doea in yn-space; that is, its sole radial minimum is a cusp at the origin, and it is angularly modulated in a manner complemeiitary to AE- [recall Fig. 171. I n the linear approximation, the surfaces AE, become cylindrically symmetric in gz-space (Fig. 17) and spherically symmetric in y6-space [Fig. 38a]. It i s especially to be emphasized that the s'I energy surfaces, A E i , are invariant under all the nuclear permutations permitted a cubically symmetric system, just as were the corresponding E and T electronic state surfaces. This point will be discussed further in 56. The solution of the secular array generated by (3.4-1) 8a AE K

8b

8c

8d

0

x

p 1

K'

- AB

x

(b) y6 v a n i s h e ~ ~ ~ v ~ ~

(3.4-23)

(b) The Tetrahedron.-From eq. 3.2-2 we observe that, in the event q3 or y4 (or both) equals zero, the tetrahedral problem reduces to the octahedral oiie discussed above. Thus situations (i) and (ii) of eq. 3.2-10, 11, 18, and 19 apply here as well, and are summarized in eq. 3.4-16-19. For situations (iii) aiid (iv) of eq. 3.2-20 aiid 21, we obtain a factorization of (3.4-9 or 10) which is a logical extension of eq. 3.4-16-19 to the tetrahedral case. (iii)

L

+

4/3

I

- AE

-P

435

0

- AE

K'

(3.4-24)

(3.4-20)

(j

for the eigen functions, $*(j), forrn:ilas [IAlz

$*(U

$*(')

+

== [ I A / ' + [-X$8a

1pI2+

1, 2 ) , produces the

AE*t)21-1/2X

+

[ - P h

-

+

-

AE*)2]-1/2X

(K

- AE,)$8c]

lpi2

+

-

(K

=

P*$8b

(K

+

A*$8b

(K

- AE*:)lC/adI

(3.4-21)

Since, by eq. 3.4-1, X depends linearly upon Sfaaiid SS&, and p upon i&b, we have that $c/Lt(.lj is modulated by both the e and 7 iiuclear modes, through the phase angles ( P Z , &, and (PS. To see this better let us examine the linearized form of eq. 3.4-21 in the I w o extremes, y2 equals zero and q 6 equals zero.

+

(a) ys canishes4I $&(l)

$*(2)

= =

2-'/2[siii 0 d q S$sa - cos 06

2 - '1%[cos 06

+ sin

$8b

$8b

F

i$8dI

=F i4bgC]

(3.4-22) (41) As in $3.228 the eleotronlc paramagnetic g-factors of a rs electionic state are determined by the electronic Mave functions of (3.4-22 atrd 23). The g-factor expressions are then put in final form by an averageToagb of the phasia angles qpe, 86, and p5.

(3.4-25) The radial extrema for situation (iii) are as given in eq. 3.4-17 with g3,@equal to naught J those for situation (iv) are determined' from the solution of a set of three simultaneous linear equations in gl, 8, and g4. There are n o extrema for situation (iv) when qz and g3 or g4 are zero, and g4 or @ i s not zero. A cusp lies at the origin [y2, y3, and 94 all naught] in all situations. The energy surface cross-cuts are similar to those of the octahedron [Fig. 23 and 241, and the extrema1 nuclear dispositions are identical to those described for the tetrahedral T electronic states [Fig. 18b, 31, and 32a,b].

*

(42) If the definitions ++'*" = 2-'/2(+sarb z+to.d) are entered into eq. 3.4-23, the expressions for +*Us2) assume the alternate form Z-'/2($4''v'T e%'Z+-''! ') ,

AKDREW D. LIEHR

436

It is interesting to view the variation of the energy with nuclear displacements for a tetrahedral Tb(T) electronic state in all its gory detail. This energy variation displays many of the same features as the more complicated pure T electronic states of $3.2, and i s yet of a simple enough form to exhibit explicitly.

+

Vol. 67

author,21and, therefore, need not be rederived in this place. If the wave functions #p, (P = A,B; A = x2 - y2, B = 3z2 - r2),of $3.1 are substituted into eq. 15 of ref. 21, and the vibronic matrix elements computed, we obtain the results of eq. 3.5-1. Similarly, if the orthonormal transformation of eq. 3.4-4 is intro-

"8

(3.4-26)

duced into eq. 16 of ref. 21, we obtain the vibronic array given in eq. 3.5-2.

where we have defined [j,k = 3,41 hj2hk cos (2Vj hj2hk

+

sin ( ~ V J

+

+

Vk)

Vk)

=

= 2517~lk-

fi (tk73{l

+

5'lvklJ

- EJVkrj)

-

EkvilJ

(3-4-27)

with e3(2vl vk) equal to 21q Y k - 4a/3. Theenorm:ty of the task of fully plotting a surface such as eq. 3.4-26 is immediately evident. It is thus easily understood why we have restricted our considerations to simplified one, two, and three dimensional projections of this surface. The general behavior of the wave functions associated with eq. 3.4-26 has been given in eq. 3.4-21, which, as before, again simplifies to eq. 3.4-22 and 23 in the appropriate limiting cases. (c) The Hexahedron.-As previously, the octacoordiiiated cube is mathematically isomorphous to the tetrahedron, and hence eq. 3.4-25-27 give the desired results. All that need be done is to replace yj, (j = 2; 3, 4), serially by qk, (k = 3; 7, 8). Witness Fig. 18c, 33, and 35a,b. (d) The Tetradecahedron.-Agailz, the dodecacoordinated cube is mathematically similar to the tetrahedron-see $3.2and view Fig. 18d,e, 34, and 36a,b. 3.5 Cubic Molecular Systems: The Doubly Degenerate Electronic States with Non-zero Spin-Orbit Forces.-As in $3.4, whenever spin-orbit interactions are introduced, wave functions of the proper spinorbital symmetry characteristics must be employed. Such functions have been tabulated earlier by the

(a) Quadrate Orientation AVss = AVA, (S = a, b), AVsa,sb AVst = AVB, (t

= C,

d), AVss,sO=

AVsa,sd =

A V 8 b , 8 ~=

= AV80,sd E AVsb,sd

= AVAB

(3.5-1)

0

(b) Trigonal Orientation AVs; = 7 = '/2(AVa f AVB), (s

0

-

5,b, E , d)

=

AVs&,si; = AVse,&j e 0 ip i_a

e4 ilVsisa,sc= e 4 u = -(AVB - AVA 2d2

- -i P AVse,s;l = e

+ 2iAVa~)

!? U,

AVsE;,sc = ie4

AV&,s;i = -ie

_ i_P U

U

(3.5-2)

The solutions of the 4 X 4 secular determinants generated by the Tg9(E)and T3i(E)matrix elements of eq. 3.5-1 and 2 individually turn out to be identical to eq. 3.1-2.

(a) Quadrate Orientation Same as eq. 3.1-2, repeated twice.

DEGENERATE ELECTROXIC STATES

Feb., :1963 (b) Trigonal Orientation A E = ~ 7 =t4

2

1 c[ (twice)

(3.5-3)

This result is not surprising, as a pure l?8(2E)electronic state must behave exactly as does a spinless E electronic state, since spin-orbit forces are not operative within a pure orbital E configuration. If our only purpose in this paragraph were to examine analytically the nature of a cubic F8(%) electronic potential energy surface, we would have been able t o dispense with this examination with the single sentence just written. However, for future reference ($4) we wish to note here the effects of electrostatic fields of lower symmetry on a I'gPE) electronic state, and for this purpose eq. 3.5-1-3 need be derived. Let us denote, in eq. 3.5-1 and 2 a finite separation of the r s & , b and r & , d Kramers states by the symbol 6 (the quadrate case) or 6 (the trigonal case), in anticipation of the effect of imposing a tetragonal or trigonal electrostatic field on our initially cubic system. Then the consequent energy expressions become

437

240T SE

120°

2

/

(a) Tetragonal Field A E h = l/z(AV* AVB 6 ) f ' / ~ ~ ( A V B (3*5-4) AVA 6)' ~ A V A B ~ (twice) )"~

+ + + +

(b) Trigonal Field A E ~ = (7 l / z ~ )=t 1 / ~ { 6 2

+

+ 81

CASE 1 g [2 }

'I2 (twice), or

(3.5-5)

+

+

+

AE, = '/z(Al/~ AVg 6) f ' / 2 { 6 ' (AVB - Av*)*f ~ A ~ A B (twice) ~ } ~ /(3.5-6) ~ which are but slightly altered forms of eq. 3.1-2. We shall discuss the nature of the solutions (3.5-4-6) a t length in $4. Observe now, however, that the unitary transformation

(3.5-7)

converts eq. 3.5-2 into two identical 2 X 2 subdeterminants whose bases are $88',8c' and $gp,Sa~,separately. 8i' or 86' r

86' or 8a'

- AE (3.5-8)

In the linear displacement approximation the quantity u equals - 5 3 q2e-iq2 [compare eq. 3.1-2 and 3.5-2, so that eq. 3.5-8 reduces to r

-

- AE 1/2aq2e2q2

]

1/2aq2e-qq2 r - AE -l- 8

-

(3.5-9)

Hence, the statics and dynamics of the trigonal cubic electronic states may be described by the same

C2< 0

Fig. 21.-Two dimensional cross section in S ~ ~ , b ( space s) of the electronic potential energy surface of a cubic molecule (oh or T d ) in a triply degenerate (T) electronic state. The surfaces consist of three intersecting disjoint parabolas, whose minima lie a t 0, 120, and 240°, when cz is negative, and a t 60, 180, and 300" when cBis positive. The case of negative c2 is shown in the figure.

techniques as were utilized t o describe trigonal E's" and hexagonal E1,2$'. or electronic ~ t a t e ~ . ~ ~ , ~ ~ , ~ ~ , 3.6 Dodecahedral and Icosahedral Molecular Systems.-The last of the regular polyhedra are the group theoret'ically isomorphous dodecahedron and icosahedron [Fig. 39a].:'4 The twelve coordinated dodecahedral arrangement bears the same relationship t o the twenty coordinated icosahedron as t'he six coordinated octahedral arrangement does to the eight coordinated hexahedron. In Fig. 39b we shorn, after Lipscomb and B r i t t ~ n the , ~ ~relationship ~ of the regular icosahedron ( I h ) to the previously discussed irregular tetradecahedron ( O h ) and the regular hexahedron ( O h ) . 4 4 The (43) (a) W. E . bloffitt, and A. D. Liehr, Phys. Rev., 106, 1195 (1957); (b) W. E . Moffitt and W. R. Thorson, "Calcul des Fonctions d'Onde Moleculaire," R. Daudel, Ed., Paris (C.N.R.S.), 1958, pp. 141-156; (0) H. C. Longuet-Higgins, U. o p i k , M. H. L. Pryce, and R. A. Sack, Proc. Roy. Soc. (London), A244, 1 (1958); (d) W. E . Moffitt and W. R. Thorson, Phys. Rev., 108, 1251 (1957); (e) bl. S. Child, MoZ. Phys., 8, 601 (1960); ( f ) .4. Witkoivski and W. E. Moffitt, J . Chem. Phys., 33, 872 (1960); (9)R . L . Fulton and M. Gouterman, i b i d . , 86, 1059 (1961); (h) A. Witkonski, Roczniki Chem., 35, 1398, 1409 (1961), and Bull. Acad. Polon. Sci., Ser. Math. Astron. Phys., 9, 1997 (1961); (i) R. Englman, Phys. Letters, 2 , 227 (1962); (j) J. C. Slonczewski, Bull. A m . Phys. SOC.Ser. 11, 8, 19 (1963). (44) A discussion of a number of molecular entities which might possess tetradecahedral or icosahedral geometries, among other geometries equally as exotic, may be found in (a) B. R. Judd, Proc. Roy. SOC.(London), Aa41, 122 (1957); (b) W. N. Lipscomb and D. Britton, J . Chem. Phys., 83, 275 (1960): (c) J. H. Macek and G. H. Duffey, ibid., 84, 288 (1961); (d) J. R. Canon and G. H. Duffey, ibid., 36, 1657 (1961); (e) R. Hoffmann a n d W. N. Lipscomb, ibid., 86, 2179 (1962); (f) R. Hoffmann and M. Gouterman, ibid., 86,2189 (1962). Xotice that the group theoretical species representations of the stretching and bending modes given in eq. 3.6-1 a n d 2 may be derived readily from the tabulations of c- and n-bonding atomic orbital species printed in the above references b y observing t h a t the group theoretical species representations of the stretching modes must be the same as those for u-bonding, and that those of the bending modes must be the same as those for r-bonding, mr:nus the rotational mode species.

F'ol. 67

438

--s 5 b ts5c

Figure 22a (see Fig. 22d).

nuclear displacements associated with the dodecahedron and icosahedron are [Fig. 40 1 (a) Dodecahedron rdisplaoements

alg

+

+ + + yu + 211, + + + + + + + + (3.6-1)

=

r b e n d = 71u

(b) Icosahedron rdisgiacements = aig

+

r b e n d = 71u

=

alg

x

71u

Yr

27iU

Tu

+ + 7zg

TZU

qig

2721,

y =

=

TZU

Yg

?g

Tu

+

Yu

+ + vr,

a1g

alg

72

X

=

119

7211

(3.6-2)

For the icosahedral group, I h , me have the multiplicatioii table for g or u representations (we denote the symmetrical direct product by the symbol { 1)

{ y2)

=

w g

alg

722)

7g1

7lg

72

7%

rg,

rlg, { 1 2 f

=

alg

alg

rl

all3

72g

n g

+ 2% + 3% + 2% + + f + + 71u

+ ng+ rIg { = + vet + + + + yg + x rl = + + + 2 Y g + 2% + yg + + + 2% (3.6-3)

= alg

vu

+ + + yg + y u + 27, + 27u 72g

71

{ 7121 y

27,

rstretoh

X

271u

y g

ratretch

71

Yg

and hence, that only the a l g and 2qg nuclear coordinates may contribute linear Jahn-Teller terms to a dodecahedral rl or 2 electronic state, and that only the alg, y p , and 2rg nuclear coordinates may contribute such terms to a dodecahedral y or electronic state. The icosahedral T~ or 2, y, or 7 electronic state behaves exactly as does its dodecahedral twin, but with an enhanced Jahn-Teller dimensionality, as these states have linear terms in the algand 3 q g ; and alp, 2yg, and 317, coordinates, seriately. ,4s the atomic term symbols P, D, F, G, and H generate the icosahedral (Ih) species 71, v, 7 2 y, y 7, and 71 72 7 , consecutively, we see, for example, that rare earth, actinide, or transition metal compounds44which possess such atomic term

+

+

+ +

439

li’eb., 1983 b

a + s3c) case;- (s -38, 4; =

477

- (s 4 a -+ $4 b)

case; =-E,+i=+T Figure 22b (see Fig. 22d).

symbols as progenitors, may exhibit quite complex vibronic demeanor. The icosahedral double g r ~ ~ contains p ~ ~the~ additional Jahn-Teller electronic states r & , b , c , d and r s a , b , o , d , e , f . For example, the 2D8/2and 2D6,2wa,ve

functions tabulated in ref. 21 form a basis for the icosahedral group rS and rs electronic states, apiece. ,(Since ~ ~the~Pa, r7,TS,and r9group theoretical species have the direct product relationships, we denote the antisymmetrical direct product by the symbol [I.

ANDREWD. LIEHR

440

s8bs s 8 C coseL= +;=

&,

Vol. 67

T

- (S7a+S7cI cos e;

The change from the symmetrical to the antisymmetrical direct product for the determination of the perturbatiifie Jahii-Teller nuclear displacements for double group representations is discussed in ref. 3, 4b, and 36b.)

I’g

8‘

X

=-4, &=

re = ~ l i g+ 7 1 % ) [rg21= alg,

r7

=

alg

+

72y

i r =~ alg

+ + + TQ+ x I’9 = a l v + 2718 +

r8= a l ~ 71g

I’Y

I’i X

4T

rlgr 272g

+ 27, +

317p

DEGENERATE ELECTRONIC STATES

Feb., 1963

441

THE POSITIVE RHOMBIC DISTORTIONS

THE NEGATIVE RHOMBIC DISTORTIONS Figure 22d. Fig. 22a-22d.-Some exemplary nmfdionaty geometries of the octahedron (Fig. 22a), tetrahedron (Fig. 22b), hexahedron (Fig. 22c), and tetradecahedron (Fig. 22d) (the complete set of tetradecahedral geometriea analogous to thoseof the oetahedron, tetrahedron, and hexahedron are readily obtained by comparison of Fig. 22a,bLc and Fig. 22d). The octahedral conformations (Fig. 228) are generated hy the coardinates qsq. # 0; sine6’ = d%,eo8 &’ = - da/a( -l)’+*; M = j w / 3 ; and a’= (3k - j ) r / 3 . I n the figure the case p~ equals zero is plotted. The condition k odd ( j k if qr # 0) produces the first line of the figures (e.g., ( S b b S,, etc.) and k even (j kifq, # 0) thesecond(e.g., -Shb -S,,etc.). [Whenq2,q5# 0, theoctahedral “axid” groups are cornpremedand theplanargroups elongated for j even and vice versa for j odd (recall Fig. 188). Simultaneously, the “axial” and planar (shaded) systems are canted one with respect to the other (the angle of cantation equals sigma, I). Commenta of a similar nature apply to the tetrahedron, hexahedron, and tetradecahedron I.

+

+

[rd =

+

‘Ir, [roil = W.

+ yr + 21.

+

(3.6-4)

we %e that the r, and roelectronic states are perturbed by nuclear displacements in a manner similar to the T,.P. or and ys, . vsoru electronic states, individually, and that the r e a n d rrstatesremainundisturbed. (This latter result is a consequence of the Kramers theorem.4-a6b) The extrema of the icosahedral (Ih) potential energy surfaces lie a t the six equivalent DMgeometries, the ten equivalent D3d geometries, and the fifteen equivalent c 2 h geometries and their negatives. [It will be shown in $4 that the Dsd and Dad electronic degeneracies are lifted in a fashion identical to the DP and Dah electronic degeneracies of $2.3 and $2.1, respectively.] This fact is readily verified by means of tho adiabatic eorrelation principle.sb As the orbital degeneracy is not

completely removed for any component of a y electronic state in a Da arrangement, such a conformation cannot be a relative energy minimum, but can only be, a t most, a n energy saddle. [Further discussion as to the relative stability of the sundry extrema is relegated to future works.] $4. The Irregular Polyhedra 4.1 The Regular Polygonal Monopyramids.-The polygonal monopyramid class of compounds is composed of those compounds which are built from an n-atom polyponal base and a central apex atom, and hence, all belong to, a t most, the point group classification Cmv. Their nuclear modes are identical with those of their n-polygon ancestor [see liig. 2b, 3b, 4b,c, 5b, 6b, 7b,c,f], plus the addition of a mode in which the apex atom translates in a direction opposite to the rigid translation of the basal npolygon [comparc Fig. 21, and 4la,b and Fig. 3b and 421.

442

ANDREW D. LIEHH

Vol. 67 AE

CASE 1 c2,

cs < 0

Fig. 2?.-Plot of the radial dependencies of the energy of n triply degenerate electronic state in S U , ~and ( ~ S6..,..( ) 7 ) space for B cubic molecular system with zero spin-orbit forces [eq. 3:2-12]. The c * mdcpicted is for a and c6 negative. When c? is pasitive the lnbelsj add a n d j even muat be reversed; when cs is positive the lrihels EOand E* mllst be reversed [view eq. 3.2-121. The symbol 1 defines the trigonal azimuthal angle *sal of (3.2-10 and 11) as 1 ~ 1 3(,1 = ( ] , I , . . .). Mark that the points olkuined for 9%eqiid to zero in (: 0 the angular positions of the minima and saddles are reversed).

spectrum is still unsettled. Only two distinct experimental claims of Jahn-Teller phenomena have been put foward: the one concerns the anomalous decrease in intensity of the eg vibrational mode and its binary harmonics in the infrared and Raman spectra of the second and third row transition metal hexafluoride^,^^ and the other, the broadening of the infrared a vibrational modes in manganio trisacetylacetonate relative to those in the aluminum, chromic, and ferric trisacetylace ton ate^.^^^^^ Atr yet there exists no concrete theoretical basis for such observations. The theoretical calculations to date [with the exception of twogg]have

Fig. &.-Two examples of a distorted cube: the irregular dodecahedron (upper figure) and the square antiprism (lowcr figure).

(96) (a) B. Weinstock and H. H. Claassen, J . Cham. Phgs., 31, 262 (1959); (b) B. Weinstock, H. H. Claassen, and J. G. Malm, ibid., 32, 181 (1960); ( 0 ) H. H. Claassen and B. Weinstock, ibid., 33, 436 (1960); (d) H. H. Claassen, Symposium on Molecular Structure and Spectrbscopy, Ohio State University, Columbus, Ohio, June, 1961; (e) B. Weinstock, H. H. Claassen, and C. L. Chernick, Bull. Am. Phys. Soc. Ser. ZI,8, 50 (1963); b u t see also (f) J. A. Creighton, L. A. Woodward, and M. F. A. Dove, Spectrochim. Acta 18, 267 (1962). (97) A. Forman a n d L. E. Orgel. M o l . Phus., 2, 362 (1959). (98) We are a t present investigating complexes of nickel, copper, and zinc trisethylenediamine to aee whether a similar broadening appears here also. This experiment will provide a nice test of the Forman and Orgel hypothesis. 97 (99) (a) M. 9. Child and H. C. Longnet-Higgins, PhE. Trans. Roy. SOC. (London), 8254, 259 (1961); (b) M. S. Child, ibid., 8266, 31 (1962).

466

ANDREWD. LIEHR

7 5cos

lo

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scose

cose 5 C O S 3 0 + 1.25 cose 5COS 3 0 t 0.5 cose

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2.5

MAX

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demonstration of the ~ O bias Z a t 0 and 180' for a tetragonally distorted octahedron, tetrahedron, hexahedron, or tetradecahedron in a doubly degenerate (E) electronic state with non-zero spin-orbit forces.

concerned themselves with the more negative aspects of the Jahn-Teller theorem: a proof that there should be no breakdown of the usual infrared selection rules for fundamental vibrations due to weak Jahn-Teller interactions,loOand a proof that r8 configurations which arise from non-degenerate orbital states would show no Jahn-Teller phenomena. lol We wish to establish here some more positive theoretical expectations of Jahn-Teller implications for the infrared-Raman frequency spectrum. If the Jahn-Teller linear forces are large, the vibronic wave functions of the electronic state M assume the form -

+

-+

Q & ~ , K = +i+I(ri,se)NRM(se)

(8.3-1)

where, by eq. 2.1-2, 3.2-15, and 3.4-21 (100) W.R. Thorson, J . Chem. Phys., 29, 938 (1958); see also ref. 99b. (101) M. S. Child, Mol. Phys., 3 , GO5 (1960); see also ref. 9a and 99b.

Feb., 1963

DEGENER$TE

+

ELECTRONIC STATES

467

and N k n f ( S a ) are non-harmonic oscillator functions whose angular dependence in E and T space [denoted here by cpz and ( O ~ , ( F S ) , seriately], for example, would be given as in eq. 5.1-4. The electronic wave functions $(ri,sa) on the right hand side of eq. 8.3-2 depend on the nuclear coordinates only in an adiabatic sense.2 Now the fundamental point to catch sight of is that with functions such as (8.3-2), which contain the nuclear pz and (06, $06) angular variables parametrically, the usual infrared and Raman selection rules will be relaxed. To see this fact, consider the infrared selection rules for an E electronic state. The expectation of the dipole moment operator Fm is then given by [for the lowest electronic state]

dependencies in eq. 8.3-2 and 3.1°3 Also, it is to be remarked that the Jahn-Teller vibronic couplings lift the fourfold [for an E nuclear mode in an E electronic state], sixfold [for an E nuclear mode in a T electronic state], etc. degeneracy of the first excited "vibrational" state. Hence, infrared and Raman fundamentals lo4 involving the Jahn-Teller modes should be Of course, we should also observe [in a homologous series of compounds] unusual shifts in the vibrational frequencies of Jahii-Teller molecules as compared with their normal homomorphic neighbors. The results derived above are not new with the author, but have been derived independently by an alternate route by Child and Longuet-Higgin~.~~ Their discussion is by far the most satisfactory and complete

The expansion of the electronic dipole moment matrix elements, mhk(sa)and %&+(sa), in a power series in the normal coordinates Q, [which are linear combinations of the symmetry coordinates X,], yields

4*

-

-1

4

-

4

3

-

3

where the superscripts /I and I indicate that we are to take the components which transform as vectors parallel and perpendicular to the molecular n-fold axis about --+ which the electronic functions $*(q,sa) are oriented. When this expansion is carried out for the various geometries we have considered in this paper, one finds that (i) a Jahn-Teller molecule can exhibit a dipole moment whenever the symmetrical product of their group theoretic species classification contains the species of the dipole moment vector [hence, the comments of $8.2. Of course, if a non-centrosymmetric molecule is permanently distorted, it will always have a dipole (ii) the substitution of eq. 8.3-4 into 8.3-3 predicts the appearance of new fundamentalslo2 anid of strong overtone and combination bands; and (iii) anomalous rotational and vibrational patterns should appear. Indeed, in certain cases [those satisfying condition (i)1, the totally symmetric vibration ma,y become infrared active.99 The appearance of strong overtones and combination bands is a direct consequence of the explicit appearance of the nuclear angular (102) The considerations of Thorson'oo indicate that these new fundamentals should in the main be excessively weak, b u t perchance this circumstance might not be universally true. Mark that the arguments utilized here for the breakdown of vibrations1 selection rules are quite similar to that already given for the analogous breakdown of certain electronic selection rules.'Sb

W Fig. 48.-The non-Jahn-Teller distorted conformation of the nickel-germanium solute-solvent system. (This geometry is a non-Jahn-Teller one as it is obtained from a SS,(7 2 ) displacement (Fig. 13b), which displacement, as we have seen (Fig. 31), yields only Jahn-Teller energy saddles, not minima. Hence, it can only be stabilized by anharmonic elastic forces, and not JahnTeller forces.) Note the resemblance of this molecular disposition t o that of ordinary triatomic nickel compounds ( e . q . , the nickel dihalides).

[indeed, the present author had missed noticing t'he breakdown of the selection rule which prohibits the appearance of totally symmetric vibrations in . the infrared for highly symmetric molecules, until it was privately pointed out to him by Professor Longuet(103) I t is conceivable that in some situations, these electrical anharmonicities could comptnsate the ordinary mechaaicai anharmonicities a n d lead t o a diminution of the binary combination bands. This ciroumstance might be the explanation of Weinstoak, Claassen. a n d Malm's results. 96 (104) Perhaps the broadness of certain infrared and Raman bands observed@%Q' for Jahn-Teller molecules has this as its cause.

468

ANDREW D. LIEHR

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Fig. 49.-The spectrum of copper in zinc oxide (after Pappalardo and Dietz118). The divalent copper cations reside in a trigonally distorted tetrahedral site (three of the metal-oxygen distances in zinc oxide are 1.95 A. and the third is 1.98 A . 8 1 ) , hence their upper electronic state is of the r@E) variety discussed in $4.3. As the ground electronic state of a divalent copper cation in zinc oxide is a Kramers doublet,*' and is thus a non-Jahn-Teller state,21,aeC the complex appearance of the Cu:ZnO spectrum must arise, in part, from Jahn-Teller motions in the rs( 2E)excited electronic state (see text).

Higgins in February, 19611. lo6 Experimental tests of all these theoretical predictions would be most gratifying. It is to be hoped that these confirmations (or denials!) will appear shortly.10e~107 8.4 Optical Exhibitions of the Jahn-Teller Theorem. -Of late an increasing number of reflections of the Jahn-Teller theorem in optical spectra have been uncovered. These observations naturally divide themselves into two classes: (i) the discovery of uniquantal progressions of asymmetric vibrations in electronic specta, and (ii) the detection of unusual cleavages of electronic spectral bands. Of these two observations only the first is an unambiguous indication of JahnTeller funny business; the latter is also a possibility for non-Jahn-Teller systems. Let us initially concentrate on class (i). Following the initial observation of asymmetric parades of vibrational lines in the ultraviolet spectrum of methyl iodide by Mulliken and Teller, lo8a number of other workers have noted similar occurrences in other systems. Wilkinsonlog discovered an asymmetric pro-

gression of eg vibrations in the benzene Rydberg spectrum [which were subsequently interpreted in terms of the Jahn-Teller theorem by Liehr and Moffitt13*llo]; Grechushnikov and Feofilovlll uncovered an asymmetric series of E nuclear modes in the spin-allowed visible spectrum of ruby, Cr+3:A1203[which were latterly re-examined and explained by Ford and Hi11112]; and Pryce and Runcimanlla found a similar series in the visible spectrum of V+*:A1203; Deutschbeinll4 detected an asymmetric procession of E nuclear modes in the spin-forbidden visible spectrum of ruby [which were recently re-investigated and interpreted by F ~ r d ' l ~ ] . Sayrell' '~~ noted an asymmetric row of E' nuclear movements in the ultraviolet spectrum of potassium nitrate and sodium nitrate; and Pappalardo and Dietzlis and Weakliemllg have seen uniquantal asymmetric E nuclear excursions in C U +:~ ZnO [vide, Fig. 491 which can be interpreted as Jahn-Teller trains.*20 Probably a number of other such observations have been made which have unfortunately slipped the author's notice, since usually there is no mention

(105) Child a n d Longuet-HigginsQn have also shown, in addition t o the points enumerated above, that the usual Raman depolarization ratios may be altered for Jahn-Teller chemicals. (106) Care must be exercised in this procedure, however, as certain anomalies appear in the infrared and Raman spectroscopy of ordinary molecules alao. As a n instance, in the series of non-degenerate XYa-2 molecules, the platinum and palladium compounds exhibit a Raman cg intensity which is greater than that of the aig mode, in sharp contrast t o the usual case [e.g., in tin]-see L. A. Woodward and J. 9.Creighton, Spectrochim. Acta, 17, 594 (1961). (107) Another extremely interesting experimental proof of Jahn-Teller interactions would be the observation of a n infrared transition between the t w o separate electronic potential sheets of Fig. 20. 3e,d, 4d, 5 0 , 6c, 7d, 17, 2 5 , 26, 38a,b, 46. (108) (a) R. 9. Mulliken and E. Teller, Phys. Rea., 61, 283 (1942): (b) A. D. Walsh, J . Chem. Soe., 2321 (1951). It is interesting to witness that a similar such series is not seen in trifluoromethyl iodide-peruse L. H. Sutcliffe a n d A. I).Walsh, Trans. Faraday Soc., 57, 873 (1961).

(109) (a) P. 0.Wilkinson, J. Chem. Phys.. 24, 917 (1056): (b) Can. J. Phys., 34, 596 (1956). (110) A. D. Liehr and W. E. Moffitt, J. Chem. Phys., 25, 1074 (1956). (111) B. N. Grechushnikov and P. P. Feofilov, Zh. Ekspenm. Teor. Fzz , 20, 384 (1955). (112) R. 4. Ford and 0. F. Hill, Spedrochzm. Acta, 16, 493 (1960). (113) M. H. L. Pryce and W. A. Runoiman, Dzscusszons Faraday Soc., 26, 34 (1958). (114) 0. Deutschbein, Ann. Physik., 14, 712, 719 (1932). (115) R. A. Ford, Spectrochzm. Acta, 16, 582 (1960). (116) Recently McClure'l has restudied the spectra of transition metal ions in corundum (AhOa) and discovered added instances of asymmetric e nuclear progressions; see D. 8. MoClure, J. Chem. Phys., 36, 2757 (1962). (117) E. V. Sayre, J. Chem. Phys., 31, 73 (1959). (118) R. Pappalardo and R. E. Dietz, Phys. Rev., 123, 1188 (1961). (119) H. A. Weakliem, Jr., Progress Report, R. C. A. Laboratories, Princeton, New Jersey, and J . Chem. Phys., 36, 2117 (1962). (120) A. D. Liehr, to be published.

Feb., 1963

DEGENERATE ELECTROKIC STATEE

in an article's title or abstract t o indicate the observation of such anomalies. Many speculations based on class(ii) observatio.ns have been made. Again most of these have probably eluded the author's attention due to inadequate advertisement. A few instances of which the author is aware will be listed. Hartmann, Schlafer, and Hansen121have suggested that the doubling of the absorption bands of trivalent titanium complexes is due .to Jahn-Teller forces, and Cotton and Meyers1222 have proposed the same for the analogous bands of the hexaaquoferrous(I1) cation and the hexafluorocobaltate(II1) anion; calculations20aseem to bear these workers out. Dunn and Ing01dl~~ have summarized arguments for Jahn-Teller antics in the benzene ultraviolet spectrum, and Brinen and Goodman124have done the same for s-triazine ; computations on benzene13bappear to lend support to these authors. Kamimura and Sugano lZ5 have rationalized the luminescence processes in the T1: KC1 phosphor on the basis of Jahn-Teller deformations, and Sugano126has discussed the spin-forbidden fluorescent spectrum of ruby in terms of Jahn-Teller reciprocations. Blankenship and B e l f ~ r d have ' ~ ~ postulated Jahn-Teller motions in the visible spectrum of vanadium tetrachloride and attempted to assess their magnitudes theoretically. Gruen and McBethlZshave assigned the spectrally observed distortions of the vanadium oxytrichloride anion to Jahn-Teller causes, and Mc,Clure71has attributed the unusual breadth of the spin-orbit multiplet structure in the visible spectrum of nickel hexafluorosilicate hexahydrate to pseudoJahn-Teller interactions. Caution must be exercised in rationalizations of the breadth and the numbers of optical lines which appear in spectra in terms of the Jahn-Teller theorem as one can easily be led astray. I n this regard, attention must be drawn to the paper of Kemeny and HaakelZ9where other sources of spectrral cleavage, which are of equal or greater importance in delineating the spectral features of Mn+4:MgGeP6, have been neglected, and only Jahn-Teller actions considered. Clearly, reasoning of this type leaves much to be desired, and should be avoided where possible.

VI. Fantasy and Speculation 59. The Future Outlook.-It is obvious that there exist innumerable systems in which Jahn-Teller phenomena might be uncovered: any molecule or crystal possessing a threefold rotation axis, a fourfold rotation,-reflection axis, or any higher axis of rotational (121) (a) H. Hartmann, H. L. SchlBfer, and K. 13. Hansen, 2. anorg. allgem. Chem., 289, 40 (1957). This work recently has been extended b y H. L. SchgAer and R. Gotz, ibid., 309, 104 (1961). (122) F. A. Cotton a n d M. D. Meyers, J . Am. Chem. Soc., 82, 5023 (1960).

(123) (a) T. M. Dunn a n d C. K. Ingold, Nature, 176, 65 (1955); (b) 0. ,J. Hoijtink,, Mol. Plays., 2, 85 (1959) has done the same for the benzene-like systema of corenene (and its positive a n d negative ions) a n d triphenylene (and its positive and negative ions). (124) J. H. Brinenand L. Goodman, J . Chem. Phys., 35, 1219 (1961). (125) H. Kamimura and S. Sugano, J. Phys. Soc. Japan, 14, 1612 (lQ5Ul; see also N. .Kristofel, Optik Spektrosk., 9, 615 (1960), Trudy Inst. Fiz. Astron. Akad. N a u k Eston. SSR, 12, 20 (19601, Izvest. Akad. N a u k SSR Ser. Fiz., 2!L, 553 (1961), G. S. Zart and N. N. Kristofel, Optik. Spektrosk., 13, 229 (1962), and related works; ( 0 ) W. B. Fowler and D. L. Dexter, Phys. Rev., 128, 2154 (1962). (126) S. Bugano, Progr. Thaor. Phys. Suppl., 14, 66 (1960). (127) F. A. Blankenship and R. L. Belford, J. Chem. Phys., 36,633 (19621, and Errata, ibid., 37, 675 (1962). (128) D. M. Gruen a n d R. L. McBeth, J . Phys. Chem., 66, 57 (1962). (129) G. Kemeny a n d C. H. Haake, J . Chsm. Phys., 3S, 783 (1960).

4:69

symmetry will possess degenerate electronic states, and hence, possibly be Jahn-Teller active. To cite a few examples, we list the following abstract systems which have either degenerate ground states or degenerate excited states: Y X 3 (e.g., "ao*, X O Q ~ ~etc.)130; *, YX4 (e.g., CHIo,*, SiF4O~*, CrC14, PtC14-2, Mn04-", Fe04-n, Ni(CN)4-2,etc.) 131-133; YXS(e.g., SF8Ot* etc.) 134: XnYn (e.g., CnHno*,C n O n - 2 , 3 , 4 , n = 3,4---)l36; X n (e.g., Hno,*,etc., n = 3, 4, 5---)lC; XY3Z (e.g., POF303*, etc.) ; symmetric crystals [e.g., ionic crystals (sodium chloride), valence crystals (diamond), molecular crystals (silicon tetrafluoride), metallic crystals (ahminum), etc.]791l36, 137; solidsolutions of ions in symmetric solvent matrices [e.g., M+k: CaF2, ZnO, MgO, KaF, etc.] ; ad injhhrn. Experimental investigation of such systems should produce exciting and illuminating results. Thus the future experimental prospects are rosy and unbounded. Theoretically, th.e future bodes a rocky road. I n this paper we have derived the analytical form of the Jah:nTeller surfaces for most geometries of common interest and have characterized them as t o their singularities. With this major hurdle overcome, the next step is the explicit evaluation of the phenomenological constants which appear and which will particularize the detailed nature of the Jahn-Teller interactions in specFfic instances. Several attempts already have been made at this task,6, EL, 13, 14, 16, 20, 5 3 69, 61, 71, 119, 127, 138 and (130) D. A. Ramsay (private communication, February, 1961, a n d M a y , 1962) has seen no evidence of Jahn-Teller interactions in the gas phase spectra of ammonia and nitrogen trioxide to date. (131) The chemical and physical properties of chromium tetrachloride a n d other tetrahedral in0rgani.c compounds have been cited b y T. L. Allen, J . Chem. Phys., 26, 1644 (1957). (132) I n this connection it is interesting to observe t h a t tetrahedral clusters of spin-free trivalent cobalt in the heteropolytungstocobaltate anion are distorted [L. C. W. Baker and V. E. Simmons, private communications, August and December, 19I3ll. However, before one can assign these distortions to Jahn-Teller causes, isc-structural, undistorted, non-Jahn-Tellhr systems must be prepared 8150. (133) R. M. Hexter [prviate communication, January. 19611 is seeking indications of Jahn-Teller phenomena in the gas phase spectra of tha silicon tetrafluoride molecule. To date this search has been unsuccessful. (134) G. L. Goodman, Ninth Annual Conference of the Western Spectroscopy Association, Asilomar, California, January, 1962, a n d privste communications February, 1982, has given a partial explanation of the low temperature vibronic spectra of metal hexafluoride molecules in terms of t h e Jahn-Teller i,heorem [remember 18.3g61. (135) The lower excited states of these systems recently have been reparticularized b y (a) N. Rouman, J . Chem. Phys., 35, 1661 (1961) CnH$,*I and (b) R. West, H.-Y. Niu, D. L. Powell, and M. V. Evans, J . Am. Chem. SOC.,82, 6204 (1960); (0) R. West and H.-Y. Niu, ibid., 84, 1324 (1962); a n d (d) D. L. Powell, M. Ito, and R. West, t o be published [CgOn-%84]. (136) An interesting application of the Jahn-Teller theorem which should be investigated, but has not been as yet [but see ref. 1371, is t h a t to the problem of the oonformational (or phase) stability of various crystalline and metallic lattices [recall g8.16g*'O]. As a n illustration, we show in Fig. 50 the Brillouin zones for the facecentered cubic [e,g. aluminum] and the bodycentered cubic [e.@, barium] lattices. The calculation of the vibronic Jahn-Teller energy stateti [Fermi surfaces] of electrons moving within such zones should lead to intriguing results [a conventional discussion of $he electronic Fermi surfaces associated with the Brillouin zones of Fig. 50 (as well aa with the hexagonal.close packed Brillouin zones characteristic of t h e zinc, cadmium, a n d thallium lattices) may be found in (a) W. A. Harrison, Phys. Rev., 118,1182, 1190 (1960) and (b) B. Segall, ibid., 124, 1797 (196l)l. I n this regard i t might be mentioned that a tentative step in this direction has beenmade by (c) R. K. Nesbet [ibid., 126, 2014, and 128, 139 (1962), but see R. Englman. PFys. Rev., Jan. 15, 19631 in his attempt to correlate vibronic interactions in the metallic superconducting state with JahnTeller forces. (137) J. L. Birman [(a) Bull. Am. Phys. Soc., 7, 65 (1961); (b) Phys. Rev., 125, 1959 (1962)l has latterly extended J a h n and Teller's consideralions to include the space groups. (138) (a) C. A. Coulson, Chem. Soe. (London) Special Publ., 12, 85 (1958); (b) Tetrahedron, 12, 193 (1961); (e) A. Golebiewski, Progress Report, Wave Mechanics Group, Mathematical Institute, University of Oxford, 1960-1961; (d) H. L. Strauss, ibid.; (e) C. A. Coulson and A. Golebiewski, Mol. Phys., 6, 71 (1962); (f) H. L. Strauss and C. A. Coulson, Bull. Am. Phys. Soc., 7, 44 (1962), a n d C. A. Coulson and H. L. Strauss, Proc. Roy. SOC.

470

ANDREW D. L I E H ~ ~

2

\rol. 67

(approximate) dynamical theories of motion which 14fs 17, 4a Those theories have been erected latterl~.l3~* which turn out to be most promising in explaining the vibrational and vibronic spectra of Jahn-Teller systems can then be extended to embrace the more precise field of rotational and rotational-vibronic spectroscopy. The date at which these chores will be accomplished is very dependent upon the genius of the experimentalisthe must present the theoretician with an unfudgeable body of facts and thereby nail him fast. Needless to say this is one of the more important problems to solve in quantum chemistry. I n termination, the author wishes t o cite a few recent papers and reviews which concern themselves with the Jahn-Teller problem [a list of earlier reviews may be found in ref. 1 and 13bl and have newly appeared. Goodman134has presented a Jahn-Teller analysis of the vibronic spectrum of the gaseous transition metal hexafluorides. Nesbet 136c has utilized the Jahn-Teller theorem to derive a novel approach to the theory of superconductivity, and Birmanl3’ has extended the original theorem t o also cover the space groups. Clin-. ton139has given a classical description of the motion of the electronic charge density in a Jahn-Teller molecule, and amplified the author’s di~cussion~3~ of the relationship of permutational symmetry to the JahnTeller theorem. Weyhmaiin and Pipkin140 have considered the possibility of Jahn-Teller forces in alkali metal-inert gas matrices, and (tentatively) decided against their presence. Kamimura and Koidel4I have reviewed the theory and applications of the JahnTeller theorem in relatively non-mathematical terms for chemical physicists; Dunn14’ has done the same for inorganic chemists; and Ballhausen143 has given a mathematical survey of the problem as it stands today.144 It is hoped that these articles together with the present one [and its simplified versions1] will give interested chemists and physicists an insight into the frustrations and gratifications which the Jahn-Teller theorem adds t o life on the molecular level.145 (139) (a) W. L. Clinton, J . Chem. Phys., 32, 626 (1960); (b) ibid., to be published. (140) W. Weyhmann and F. M. Pipkin, Bull. A m . Phys. Soc., I , 84 (1962).

Fig. 5O.-Two instances of cubically symmetric ( O h ) Brillouin zones. The upper figure illustrates the Brillouin zone structure for a face-centered cubic lattice, and the lower that for a body-centered cubic lattice. Such highly symmetric Brillouin architectures lead to inherently degenerate electronic crystalline states, and hence. t o possible Jahn-Teller interactions.1a6 [Notice that these figures could also equally well represent the true geometries of complex polyhedral ( O h ) molecules such as those discussed by Hoffman, Lipscomb, and G ~ u t e r m a n . ~(These ~~ latter polyhedral molecules present extremely interesting JahnTeller possibilities.)]

undoubtedly more will come in the future. This job will not be an easy one; certainly the primitive efforts in this direction of the past and the present are only valid as to orders of magnitude (if that!). When the Jahn-Teller vibronic constants have been properly evaluated, tests can be made of the several (London), A269, 443 (1962); (8)C. J. Ballhausen and E. T. Christenson, Acta. Chem. Scand., to be published; (h) J. Colpa, to be published. [References a, b, c, e, a n d h are on aromatic Jahn-Teller and pseudo Jahn-Teller distortions; (d) and (f) on distortions in C&+, CI.r+, “a+, and NHa; and (n) on distortions in VClal.

(141) H. Kainimura and S. Koide, N i p p o n Bulsuri Gakkaisi, 16, 436 (1961). (142) T. M. Dunn, “Modern Coordination Chemistry,” Ed. J. Lewis a n d R. G. Wilkins, Interscience, Xew York a n d London, 1960, pp. 229ff. (143) C. J. Ballhausen, Advan. Chem. Phys., 6 (19621, and “Introduction to Ligand Field Theory,” McGraw-Hill Book Co., New York, N. Y., 1962, Chapter 8. (144) Since the completion of this article a number of other accounts of the Jahn-Teller theorem have come t o the authors attention: (a) D. d. Ramsay, Advan. Spectr., 1, 1 (1959); (b) Ann. Re% Phys. Chem., 12, 255 (1961); (e) H. C. Longuet-Higgins, Advan. Spectr., 2, 429 (1961); (d) D. A. Ramsay, “Determination of Organic Structures by Physical Methods,” Vol. 2, F. C. Nachod and W. D. Phillips, Ed., Academic Press, Inc., New York and London, 1962, pp. 245-338; ( e ) G. W. Robinson, “Methods of Experimental Physics,“ in “RiIolecular Physics,” D. Williams, Ed., Academic Press, Inc., New York and London, 1962, pp. 155-265; (f) N. 8. Ham, Spectrochim. Acta, 16, 775 (1962). (145) Some papers connected with the main thesis of this dissertation which have appeared too late for a convenient placement in the text are listed here: (a) M. S. Child, X o l . Phys., 6, 391 (l9G2); (b) R. Englman, Quarterly Progress Report No. 45, Solid State a n d Molecular Theory Group, Massachnaetts Institute of Technology, July 15, 1962, pp. 114-118, No. 46, October 15, 1962, pp. 158-162 and 163-173, and No. 47, January 15, 1963, pp, 74-80; (e) R. Englman, Bull. A m . Phys. Soc., Ser. IT, 7, 530 (1962); (d) R. Englman and D. Horn, to be published [the author is grateful t o Dr. Englman for the opportunity to read a preprint of this latter work prior t o its publication]; (e) M. W. Hanna, J . Chem. Phys., 37, 686 (1962); 9. Arai, S. Shida, K. Yamaguchi, and 2. Kuri, ibid., 37, 1885 (1962); R. Dingle, J . Mol. Spectry., 9, 426 (19G2).

NON-DEGENERBTE ELECTRONIC STATES

Feb., 1963

Division D Acknowledgments.-The author wishes to take this opportunity to thank a number of his friends who were kind enough to provide him with preprints or have enhanced his appreciation of their own work via private conversation or other foreknowledge of their work prior to publication: C. J. Ballhausen, L. C. W. Baker, R. L. Belford, W. L. Clinton, F. A. Cotton, C. A. Coulson, R. E. Dietz, S. Geschwind, M. Gouterman, G. L. Goodman, J. S. Griffith, D. M. Gmen, W. Hayes, R. M. Hexter, W. D. Hobey, R. Hoffmann, H. Kamimura, K. Knox, H. C. Longuet-Higgins, G. W. Ludwig, D. S. iLlcClure, H. 111. McConnell, A. D. McLachlan, R. K. Kesbet, L. J. Oosterhoff, R. Pappalardo, D. A.

47 1

Ramsay, R. G. Shulman, V. E. Simmons, L. C. Snyder, H. L. Strauss, S. Sugano, W. R. Thorson, G. D. Watkins, H. A. Weakliem, Jr., B. Weinstock, J. E. Wertz, R. West, P. G. Wilkinson, and P. J. Wotowicz. This paper has benefited immensely from their thoughtful kindness. Especial thanks are due the Bell Telephone Laboratories research drafting department, supervised by H. M. Yates, for the many beautiful illustrations with which they embellished and clarified this complex article. The vast, majority of the illustrations were furnished by C. J. Jernstedt, who was aided at times by H. J. Seubert, W. R. Brown, and F. M. Thayer. Without their gracious cooperation, this dissertation would have been impossible.

TOPOLOGICAL ASPECTS OF THE CONFORMATIONAL STABILITY PROBLEIM. PART 11. NON-DEGENERATE ELECTRONIC STATES' BY ANDREWD. L I E H R ~ ~ Melton Institute, Pittsburgh 13, Pennsylvania,'d and Bell Telephone Laboratories, Inc., Murray Hill, New Jersey Received M a y 18, 1962 To be, or not to be: that is the question.-Hamlet The topography of electronic potential energy surfaces of polyatomic systems is derived by means of group theoretic and permutational symmetry principles. It is demonstrated that the relative placement of the surface reliefs may be perfectly specified by such principles for systems which contain sufficient numbers of identical nuclei. The results obtained reveal a number of important maxims: (I) t h e topography of a polyatornic electronic potential energy surface is wholly determined by t h a t of its most symmetric hypothetical geometry (law of eurythmy); (2) given a hypothetical geometry of high group theoretic symmetry, the topographical behavior of a polyatomic electronic potential energy surface is determined entireIy by that of its elemental symmetrical subgroups (principle of mathematical inheritance); (3) isomorphous ;grouptheoretical conformations exhibit isomorphous topological deportment5 (formation of topological families); (4) whereas the prime number group theoretic constellations produce only two distinct electronic potential energy topographies, the non-prime number constellations produce all those topographies which are required bg the principle of mathematical inheritance (2) (law of prime numbers); (5) th.e dynamical quantization and the topography of the nuclear-electronic problem are fully specified by group theoretic and permutational symmetry precepts (symmetrical transcendance); (6) certain nuclear structures can never be topologically stabilized without external aid (exclusion principle); (7) i t is impossible t o distinguish between ordinary anharmonic elastic distortions and Jahn-Teller distortions in non-homologous series of compounds (indistinguishability theorem); (8) experimental proofs of Jahn-Teller-Renner consequences can be procured only from studies of homologous series of conipounds (criterion of probity). Particular care has been taken t o illustrate graphically all important consequences of the mathematical derivations, so that their chief predictions should be available to theoretician and experimentalist alike. A detailed discussion of the theoretical and experimental status of the nuclear,-electronic problem is presented, and courses for future advance are signposted. A master plan is sketched for the solution of the numerous as yet unsolved theoretical and experimental nuclear-electronic puzzles, C'ONTESTS

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