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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5Tr 3 (8) (b) Fig. l a and 1b.-Schematic representation of the permissible nuclear-electronic potential energy surfaces of trigonal (51.2), hexagonal (§1.5), cubical (52.1 and §2.3), etc., chemicals in orbitally non-degenerate electronic states [confer also Fig. 4a,b and 5a,b]. The case depicted is for clj, positive. Figure la illustrates the surface modulation for small c j j j and Figure I b for large cjjj. The radial minima of Fig. I b lie a t 60, 180, and 300' and the saddles a t 0,120, and 240'.
PRIMARY ATOM ORBITALS
MOLECULAR O R B I T A L S OF T H E COMPOUND
LIGAND MOLECULAR ORBITALS
Fig. 2.-Molecular orbitals for a planar equilateral geometry Note that the o-bond structure is closely approximated by the valence bond hybrids slpZdSand not sp2, dp2, dsa, or dS electively. (D3h).
I. Precedent
It is usual in stereochemical theory to start with a n experimentally given geometry and argue rearwards. However, a procedure of this sort is in many respects self-debilitory. First, this procedure beclouds the proper visualization of the fashion in which conformations of various imaginable shapes are related one to another. Second, it bemasks the realization that conformational reciprocation is feasible. Third, it conceals the substantivity of isotopic intramolecular transposition. Fourth, it suggests archetypes for chemical ligature which are invalid. Fifth, it circumscribes and obfuscates kinetical considerations which are based upon the shape of electronic potential energy surfaces. Sixth, it renders obscure intramolecular metastatic processes which occur in kinetic transitional forms. Numerous other objections to this procedure doubtless exist. We do not here pretend to have compiled a complete catalog. With this prolog we wish only to emphasize the inherent inexpedience of disputations of this kind. To eliminate a portion of the theoretical deficiencies registered above, we have developed a view of stereochemical theory alternate to that currently prevalent. (1) Various portions of this paper were presented a t several conferences. (1) the X V I I I International Congress of Pure and Applied 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 Codrdination Chemistry, Stockholm, Sweden, June 25-29, 1962; ( 5 ) Symposium in Quantum Chemistry a n d Solid State Physics, RBttvik, Dalarna, Sweden, August 27-September 1, 1962; (6) the Informal hfeeting on Recent Developments in Quantum Chemistry, Hakone National Park, Japan, September 7-8, 1962; (7) the International Symposium on Molecular Structure and Spectroscopy, Tokyo, Japan, September 10-16, 1962. An amplified account of these lectures may be found in (a) Progr. Inorg. Chem., 3, 4, and 6 (1961, 1962, and 1963); (b) Ann. Rev. Phys. Chern., 13 (1962); a n d (e) Proceeding8 of the Symposium on the Theory and Structure of Complex Compounds, Wroclaw, Poland, to be published, 1963. (d) The author's address is Mellon Institute, Pittsburgh 13, Pennsylvania.
Fig. 3.-Pictorial definition of the threefold permutation operator, e,. Witness the contragredience of permutational and physical rotations.
I n the paragraphs which succeed, we demonstrate that a group theoretical contemplation of various hypothetical nuclear structures for polyatomic agglomerates allouTs a complete specification of the stereochemical problem. We furthermore show that the utilization of permutational group theoretic concepts, in conjunction with $n explicit nuclear symmetry coordinate base, permits a complete particularization of kinetical stereochemical posers. Although the hypothetical nuclear structures enlisted in the controversions of this work will startle some by their exotic shapes, these structures were not chosen for this end, but were selected to exemplify, with clarity, the dialectics employed. Indeed, it is manifested that the sometimes perverse view of chemical tectonics here adopted frequently leads to fertile, productive, and suggestive new approaches to old stereochemical inquiries. 11. Account Much has been said of late about conformational stability of electronic systems in degenerate electronic states.2 However, despite this circumstance, little has been said of the correspondent topic for non-degenerate electronic states, This occurrence is most unfortunate : the stereochemical phenomena descriptive of degenerate electronic situations are quite universal and in no way different from those of non-degenerate electronic incidents. To demonstrate this fact we review for the moment the theory of stereochemical stability for degenerate electronic configurations. (2) See as a n instance Part I, A. D. Liehr, J . Phya. Chem., 67, 389 (1963).
Feb., 1863
?;ON-DEGENIERATE
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Fig. 4a and 4b.-Plot of the conceivable radial dependencies of the nuclear-electronic energy for an orbitally non-degenerate electronic state. Figure 4a graphs the possible nuclear-electronic energ,y variations for nuclear displacements of species (Y and 9, for ci,ii and c , , , , , ~positive and negative, and Fig. 4b those for nuclear displacements of species E , 7 , y, and g for c;ii negative. In Fig. 4b, cases 1 and 3 pertain to a-space nuclear displacements, and cases 2, 4, and 5 to 7 , y, and ?-space displacements. The integers j and I measure the angular extrema1 separations in E- and 7 , y, ?-nuclear coordinate space, serially, in units of 2r/n (n = 3,5) [for cubic geometries, the contingency painted, cases 1, 2, and 3 correspond to situations (i) and (iii) of Part I, eq. 3.2-10 and 18, and cases 2 and 4 to situations (ii) and (iv) of Part I, eq. 3.2-11 and 191.
A non-linear chemical system may exhibit degenerahe electronic states if, and only if, its atomic framework is comprised of numbers of identical nuclei sufficient to form a threefold or higher axis of permutational symmetry la linear chemical, of course, always can display degenerate possibilities by virtue of its intrinsic cylindrical regularity]. Let us suppose we have such a system. Let 6 be any permutation operation which transposes identical nuclei. Then, since the Hamiltonian function, X, is an invariant under transpositions of identical nuclei, the commutator of 6 and X vanishes [@,XI= 0
(1)
Therefore, the characteristic total energy, W , and the adiabatic electronic energy, E, are constant with respect to all permutations, 6, of identical nuclei within a givlen chemical structure. This statement is true regardless of whether tbese permutations be actual symmetry operations permitted the structure or not. Hence, the characteristic total energies, W , are specified by the eigenvalues, p , of 6, and the adiabatic electronic energies, E, are singularized by the actual operations of 6
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If the electronic configuration under entertainment is
degenerate, these assertions may be made: (1) the ,JahnTeller theorem applies and thus the electronic potential energy surface varies linearly near the origin of nuclear coordinate space in certain special directions [i.e., in those directions which remove the electronic degeneracy] of asymmetric nuclear displacement; and (2) the permutational symmetry theorem relates [cf. eq. 1 and 21 and hence the electronic potential energy surface varies angularly, throughout nuclear coordinate space, in all those directions of nuclear displacement which permit polar and azimuthal movement. In Part I,2 Fig. IC,2c, et seg., the consequences of these two declarations [which there were derived by an alternate path] have been graphed. If, on the contrary, the electronic configuration under attendance is non-degenerate, the succeeding pronouncements may be made: (1) the harmonic theorem pertains and the electronic potential energy surface changes quadraticqlly, close to the origin of nuclear coordinate space, in all possible directions of nuclear diisplacement; and (2) the permutational symmetry theorem, eq. 1 and 2, corresponds and the electronic potential energy surface changes angularly, throughout nuclear coordinate space, in all those directions of nuclear displacement which allow polar and azimuthal movement. In Part 11, Fig. la,b, the issue of these statements for trigonal nuclear aggregates is shown. The attendant analytical forms are given by the (diagonal)
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D. LIEHR
Yol. 67
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(a) Fig. 5a and 5b.--IClaterial display of the allowed nuclear-electronic potential energy surfaces of a planar equilateral compound in an orbitally non-degenerate electronic state [look too a t Fig. la,b and 4b]. Figure 5a traces the surface contour for the Xa molecule [case of ozone, cllI large and negative], and Fig. 5b for the YX3 molecule [case of chlorine trifluoride, czzz large and positive and caa3large and negative]. In Fig. 5b we show the triplex of two and one dimensional cross-cuts, S P a , b , S a a , b , and SO,respectively, of a five-dimensional space, (So, SZa,b, 8 3 a , b ) . Motion in the five-dimensional space, (So,SZit.b, S 3 a , b ) , can best be viewed S z a , b , and S d a , b , sepas a coupled motion in the disjoint spaces 80, arately. I n each of these smaller spaces the figure shows, solely for specificity, minima a t negative values of SOand Sza [Le., a t (02 equals 60, 180, and 300°] and a t positive values of Sa, [pa equals 0, 120, 240'1. The saddles in S 2 s . b ( j = 2,3) space lie a t the obverse positions. The triple of one-dimensional parabolas drawn for Soare not meant to imply added dimensionality; they are only meant to aid correlatively.
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111. Employment $1. The Regular Polygonal and Irregular Polyhedral Molecular Systems. 1.1 The Digonal Geometries.As no linear asymmetric nuclear displacements entered the off -diagonal matrix elements for the degenerate linear arrangement of Part I [by dint of Renner's theorem], and as the degenerate and non-degenerate diagonal (3) Since the degenerate electronic representation, $*, chosen in Part I produoea peifectly symmetrical, and equal, diagonal nuclear-electronic energy matrix elements, these diagonal matrix elements also describe the nuclear-electronio energy variation with nuclear displacement for nondegenerate electronic representations.
matrix elements are homologous,3 the chorographical mold of the sanctionable degenerate electronic potential energy sheets, individually, is congruous with that of the warrantable non-degenerate surfaces. Therefore, the discussion of Part I pertains, and no further development is needed; Fig. lb,c,d,e of Part I apply directly [the designations V+and V - of Part I, Fig. IC,must now be regarded as separate and unrelated though, and of different origin and character; both designations are now single examples of conceivable non-degenerate nuclear-electronic behavior]. Note especially in Part I, Fig. ICand e, that whenever surface maxima are low, pseudo-linearity ~ e s u l t sand , ~ that true non-linearity ap-
Feb., 1963
NON-DEGENERATE ELECTIZOXIC
pears only when these maxima are high [for truly ncinlinear conformations the vibrational phase angles p3 and p4of Part I, Fig. lb,c,d, become angles of torsion and rigid rotation, either singularly or combinatorially, dependent upon the final aspect of the equilibrium nonlinearity]. 1.2 The Trigonal Geometries.-New topologilcal behaviors are feasible for polyatomic systems of permutational symmet'ry three or greater. T o visualize this truth consider for the moment the simplest of all trigonal arrangements, the X3 and YX3 arrangement 03O,*, FS01*, c13°'*, [for example Hao,*, Ca03*, E&',*, Br30,*, 1303*, BF30,*, CH30,*, NHaol*, N3Ho,*, 03S0'*, F3Cl0**,etc.]. I n Fig. 2 we sketch schematically t,he expected molecular orbital energy level diagram for su.ch equilateral, triangular (and pyramidal) arrangement>sj; the appropriate nuclear displacements already liaxe been drawn in Part I, Fig. 2b and 41a,b. It is evident from eq. 1 and 2 that whatever the sha,pe of the electronic potential energy surface of an X3 or YX3 aggregate this shape is invariant of all permutations, 6, of the identical nuclei, X, which construct the compound. Specifically, it is invariant of the permutation 6 equals e3,where e3is as defined in Fig. 3. [We do not wish to imp1.y by Fig. 3 that the compound is truly equilateral, but only that an equilateral conformation is conceivable. The equilateral arrangement drawn in Fig. 3 may be fictitous (as is the case for O3 and Cll?a,for example), in the same sense that the Riimer diagrams6 of valence bond theory are fictitous. ] From Fig. 3 it is manifest that the permutation e3there defined keeps physical attributes of the nuclei [designated as a, b, and c in Fig. 31 such as charge densities, displacement vectors, etc., fixed; only the arbitrazy nuclear labels 0, 1, and 2 are regarded as variable. I n terms of these latter labels it is seen that eJis defined as that permutation which cyclically renames the nuclei (0,1,2) as (2,0,1). [The physical consequence of this renomination in truly equilateral molecules is a rigid counterclockwise rotation of nuclear framework by 2a/3 (see, e.g., Part I, Fig. 2a,b, and Part 11,Fig. 3).'] As the permutation e3replaces the X3 nuclear displacements SOby SO,SIa by - (1/2)S1a - (2/3/2)Slb, and 8 l b by (d3/2)Xla - (l/2)S1b [cf. Part I, Fig. 2b] in the rectilinear representation, in the polar representaequals 40, S 1 a equals q1 cos 91, and equals q1 tion SO sin 91, jt therefore replaces qj by gj, (j = O , l ) , and p1by
475
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Fig. 6.-The structure of chlorine trifluoride in its experimenta122bZe (upper figure) and theoretical (lower figure) coxnposures. An exact theoretics1 fit is practicable; the fit given is but indicatory. Intramolecular intwconversion proceeds as in l'art I, Fig. 41b, and Part 11, Fig. lb, 4k1, and 5a,b. The symmetry coordinate labels in Fig. 6 relate to the planar equilateral basis of Part I, Fig. 41b.
+ +
(vi 2n/3). Hence because of eq. 1 and 2, we then may conclude that eBE(q0,ql, pl), which equals E(q0,ql, p1 ( 2 ~ / 3 ) by ) the precedent sentence, identically equals itself again. Thus, the nuclear phase angle pi may appear in E(qo,ql, pl) solely as a multiple of 3pl, that is, solely as 31p1 ( I = 0,1,2,---) E(@,41,Pi) = E(@,Qi, 3lPi), ( I = 0,1,2,---) (:3) Then, to the fourth order in the iiuclear displacements, E(qo,g1, pl) has the structure
+ + coooqo2 + + ---)SO + + coo~lqo+ ---)yoylz + + + + + colllqo + ---)q? >
