From symmetry to syntopy : a fuzzy-set approach to

J Chim Phys (1990) 87, 1025-1047 ©Elsevier, Paris

From symmetry to syntopy : a fuzzy-set approach to quasi-symmetric systems* J Maruani 1, PG Mezey 2 1Laboratoire de Chimie Physique, Université Paris-VI and CNRS, 11, rue Pierre-et-Marie-Curie, 75005 Paris, France; 2 Department of Chemistry and Department of Mathematics, University of Saskatchewan, Saskatoon, S7N 0W0 Canada (Received 28 February 1990; accepted 27 March 1990)

ABSTRACT - Point symmetry is essentially a discrete concept, whereas molecular structures and theirconformational changes display continuous features. Using the formalism of fuzzy-set theory, we propose a continuous extension of the symmetry concept for quasi-symmetric systems. For each point kof the reduced nuclear configuration space Mof all N-atom systems having a specified stoi­ chiometry, and for an energy criterion e suggested by the statics or the dynamics of these systems, we define membership functions |ii(k, e) which generate 'syntopy' fuzzy subsets Sj in the metric space M; these latter are used to obtain a continuous extension s(k, e) of the symmetry point groups gp Illustrations of the procedures used arc given on simple examples, and applications of the new concept are outlined. RF.SUME - La symétrie ponctuelle est essentiellement un concept discret, alors que les structures moléculaires et leurs changements de conformations présentent des caractères continus. Utilisant le fonnalisme de la théorie des ensembles flous, nous proposons une extension continue du concept de symétrie pour des systèmes quasi-symétriques. Pour chaque point k de l'espace de configuration nucléaire réduit Mde tous les systèmes N-atomiques ayant une stoechiométric définie, et pour un cri­ tère énergétique e suggéré par la statique ou la dynamique de ces systèmes, nous définissons des fonctions d'appartenance Pi(k, e) qui engendrent des sous-ensembles flous Sj, dits de 'syntopie', dans l'espace métrique M ; ceux-ci sont utilisés pour réaliser une extension continue s(k, e) des groupes de symétrie ponctuels g;. Les procédures employées sont illustrées sur des exemples sim­ ples, et les applications du nouveau concept sont évoquées.

Apreliminary account of this work was published in (a) J Maruani and PG Mezey; (1987)

Compt Rend Acad Sci

(Paris), II 305, 1051; errata in II 306, 1141. More details on the mathematical treatment and an explicit illustration on

thewater molecule can be found in (b) PG Mezey, J Maruani; (1990) Mol Phys 69, 97.

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1. IDEAL SYMMETRY AND REAL SYMMETRY The concept of symmetry plays a fundamental role in the description and study of all natural objects, from elementary particles to living beings. Every nat­ ural pattern is situated somewhere between 'perfect chaos' and 'perfect order' : on one hand, even in the most turbulent motion there emerge some typical regular­ ities ; on the other hand, even in the most perfect crystal there subsist various typical defects. Macroscopic complex structures often display some overall approx­ imate symmetry. Usually, immobile beings possess a quasi symmetry determined by the gravitational field (which defines a vertical, achiral axis, leading to Cnv), whereas mobile beings have a quasi symmetry determined by the vertical axis and the direction of the motion (which together define a vertical, reflection plane, lead­ ing to Cs). Planar art works such as church stained-glass windows or persian car­ pets also show some approximate symmetry - usually close to Dn]v As a matter of fact, it can be said that life and art are essentially a harmonious compromise bet­ ween order and fancy, chance and necessity [1-31. Only with 'rigid' molecules do we meet 'perfect' symmetry. The set of symmetry elements of a given rigid molecule forms one of 34 point groups, which help classifying its electronic and vibrational quantum states. But complex, substi­ tuted or perturbed molecules may also exhibit a quasi symmetry close some ideal symmetry : one may quote aggregates, supermolecules, polymers and macromolccules, including proteins and nucleic acids, which exhibit various overall periodici­ ties ; molecular central frames, which retain permanent features through chemical substitutions ; molecules in a homogeneous or inhomogeneous, chiral or achiral, electromagnetic or solid-state, external field, etc.. Point symmetry is essentially a discrete concept, based on the classical mechanical idea of spatial localization of the nuclei : a formal nuclear geometry of a given molecular system either docs or docs not possess a specified symmetry ele­ ment. In a quantum mechanical model, the concept of spatial localization in a geo­ metrical sense, hence also that of point symmetry for a nuclear configuration, arc not strictly valid - and in some recent studies the emphasis has been placed on a topological reinterpretation of molecular structures and reactions [4-6], Nonethe­ less, methods based on the approximate validity of the semiclassical model have been of great practical utility in the study of molecular properties. For instance, in spectroscopic applications, it is customary to use local internal coordinates defined in terms of normal coordinates or symmetry coordinates [7], with reference to some critical point of the potential energy surface. Within this geometrical model, the discrete nature of the symmetry concept is especially evident in conformational

-1 0 2 7 changes : during a nuclear rearrangement - such as, for instance, an internal rota­ tion or inversion in a nonrigid molecule -, the appearance or disappearance of certain symmetry elements occurs in a sudden way, at precise values of the con­ formational angles. However, molecular structures and their conformational changes generally exhibit continuous features. For instance, external fields or molecular interactions may lead to reductions of symmetry in the perturbed units, even though these lat­ ter may retain strong similarities with the unperturbed units of higher symmetry. In fact, there is a continuous range of formal, semiclassical molecular shapes that may resemble the shape of a formal, more symmetric configuration. Similarly, dynamical properties of peculiar molecules may require the description of a con­ tinuous range of quasi-symmetric structures : for instance, in spectroscopic mea­ surements, a molecule may behave as a C3V structure within a certain frequency range and as a D3h structure in a different range. Escher’s continuously changing patterns [2bJ are good art illustrations of such continuous changes. The. problems considered in this paper should be distinguished from those arising from the quantum mechanical behaviour of the nuclei [8-10], Lathouwcrs el al. [11J have proposed a non-adiabatic theory of molecular dynamics, using a 'gen­ erator coordinate' variational wavefunction. Fuzziness in the positions of the nuclei themselves, not in the resulting symmetry pattern, is introduced, by replacing the discontinuous distribution 8, which leads to clamped nuclei in the Bom-Oppenhcimer approximation, by a continuous distribution of finite width, centered about a given molecular shape. The approach developed in our paper, on the other hand, starts with the Born-Oppenhcimcr model and takes into account possible fuzziness in the symmetry pattern. However, the symmetry pattern of the above distribution could also be investigated using a similar approach. Tire problems raised in this paper should also be distinguished from those raised by Longuet-IIiggins in his pioneer paper [12] which initiated the develop­ ment of the theory of symmetry for nonrigid molecules 113-15]. Some examples may help understand the differences of the two approaches. (i) The theory of symmetry for nonrigid molecules helps understand why one must use D3h for classifying the quantum states involved in the inversion spec­ trum of NH3, even though this molecule has symmetry C 3V in its most stable configuration. However, there would be no inversion spectrum at all if this latter strictly had symmetry D3h. Now, how should one call a pyramidal structure 'so close' to planar as not to yield any detectable inversion spectrum ? (ii) Considering again the ammonia molecule, substitution of a hydrogen by a deuterium nucleus would greatly change its vibrational, but not its electronic levels, whereas substitution by a fluorine atom would also change its electronic

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states. How should then one formally call the symmetry of NH2D, 'intermediate' between those of NH3 (C3V) and NH2F (Cs) ; more precisely, how should one for­ mally express the 'departure' of the symmetry of NH2D or NH2F, respectively, from that of NH3 (Fig. la) ? Similarly, one may ask oneself how to define the 'chi­ rality departure' of structures NHDT, NHDF and NH(CH3)F, respectively, from the above achiral structures [16]. (iii) The molecule C2Hg has symmetry D3 and Cs (minimal symmetry). If A * B = C (e.g. H2O), the point group D3h (which corresponds to equilateral triangles) is removed from this set. If A * B ^ C (c.g. HCN), the only possible point groups arc K (ponctual), Coov (linear) and Cs (planar). The II2O case has been fully dis­ cussed in a previous paper (# b) and we shall only recall its main features here. In Figure 3 all possible nuclear configurations of a general triatomic sys­ tem ABC arc represented by the three Cartesian coordinates Xu, Yuand Xc.w'1'1

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the choice of reference axes shown in the upper left box. Only a formal quarter Q of the usual Euclidean space is required to represent the internal configurations. Each configuration corresponds to a single point of Q as long as the system remains trialomic. Coincidence of two nuclei occurs within the plane D - B (A+C) - and along the lines D' -,(A+B) C - and D" - A (B+C) -. The intersection point U

Figure 3. An illustration of the ¡menial configuration space M for a iriatomic system ABC. The conventions used arc shown in the upper left box. The apparent distances in this representation do not correspond to the actual distances in the metric of space M. Plane D and lines D' and D" correspond to diatomic 'nuclear reaction' configurations, and point U to the 'united atom' configuration. Within plane D all points on each circle centered at the origin represent identical confi­ gurations. In all oilier subdomains of this space the correspondence of points to configurations is one-to-one. Plane L contains all linear trialomic systems, and line I all equidistant BAC arrangements. In the lower box we have shown the de­ composition of the space M for all possible systems with 112O stoichiometry into symmetry invariance domains Gj corresponding to llic symmetry point groups g; (adapted from a previous paper H b ) .

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represents the 'united atom' configuration (A+B+C). All the points of a given cir­ cle centered at the origin U in plane D represent the same internal configuration. Another feature of interest is the plane L of all linear triatomic systems, which contains, in addition to the lines D' and D", the special line I of linear BAC sys­ tems having equal AB and AC distances. In the example of H2O, the index assignment is 0 AHBHc. In Figure 3, we have displayed the decomposition of the internal configuration space M into sym­ metry invariance domains Gj, as indicated above ; that is :

where M designates space M deprived of its subsets having nontrivial symmetries. The origin point U has the highest symmetry, described by the spherical group K. Along the line I one finds all linear MOH systems having point symmetry Dooh- All II2O systems with maximal point symmetry Coov arc found within the planes L (including lines D' and D") and D. The rest of the conical surface C contains all formal nuclear geometries with maximal point symmetry C2v- The point E on the surface C represents the equilibrium conformation of the water molecule. All the points of space M not within the planes L and D and not within the surface C correspond to the lowest possible symmetry, Cs. For tctratomic molecules, one may have the following formal chemical possibilities : AAAA (e.g. S4), AAAB (c.g. NH3), AABB (c.g. II2O2), AABC (c.g. I-I2CO), and ABCD (c.g. NCOl I). As usual, for all systems, the 'united atom' configuration (A+B+C+D) has the spherical symmetry point group K, and the linear configurations - including the special 'nuclear reaction' configurations (A+B+C) D, (A+B+D) C, (A+C+D) B, and (B+C+D) A - may have the symmetry group Dooh (cxccpL for ABCD) or Cooy The planar configurations - including the nonlinear occurences of the special 'nuclear reaction' configurations (A+B) CD, (A+C) BD, (A+D) BC, (B+C) AD, (B+D) AC, and (C+D) AB - may have the symmetry group D4h (only for AAAA), D3h (only for AAAB), D2h (for AAAA and AABB), C2v (except for ABCD), and Cs (the molecular plane). The spatial configurations have as maximal symmetry point group T ( o n l y for AAAA), which may be reduced to C3V(loss of the C2 awes, 3 or 4 atoms being identical), f*2d (loss °f hie C3 axes, 2 or 4 atoms being identical), C2v (nonplanar ease), Cs (nonplanar ease), and C 1 (no symmetry element). For the ammonia molecule, for instance, the 6-climension internal configuration space can be decomposed into symmetry invariance domains as follows :

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A similar syslcmatics can be used for molecules wilh a larger number of atoms. For Cl I4, for instance, the decomposition of the 9-dimcnsion space M into group domains Gj is as follows :

In Eqs (7) and (8) we have gathered into brackets the symmetry invariance do­ mains involving only 'nuclear reaction' configurations (but similar configurations may occur also in other domains). As in the case of H2O, some of the subsets in Eqs (7) and (8) result from the union of the hypcrspacc subdomains having the same symmetry invariance. 3. AN ENERGY CRITERION FOR DEFINING CONTINUOUS MEMBERSHIP FUNCTIONS We shall replace the family G of the maximum connected components Gy of the internal configuration space M by a family S of fuzzy subsets Sj : S = ( Si } ,

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expressing the 'symmetry resemblance' of each point k of space M to points in the various sets Gy having the point symmetry gj. These fuzzy subsets S; will be re­ ferred to as syntopy sets. We require the following conditions to be met for each syntopy set Sj. (i) The symmetry resemblance of a nuclear configuration k to a point having symmetry gy as expressed by the membership function pj relative to the syntopy set Sy changes continuously in terms of the metric of M. (ii) The accessibility of any formal nuclear configuration from any given symmetric configuration depends on the energy variation along a transformation path in M, the membership function of a nuclear configuration k relative to a synlopy set S; depending on an energy parameter e : pj(k, e).

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(iii) The membership function relative to S; of each nuclear configuration k having exactly symmetry gi is equal to unity : p;(k, e) = 1, for every value of e . (iv) The ordering relation p;(k, e) > p.j(k, e) holds, that is, Sj is a subset of Sj, whenever gj is a subgroup of gj. (v) In the limit case e = 0, the fuzzy subsets Sj give back the original struc­ ture of space M in terms of the invariance domains G,. Since an energy criterion e is invoked, a fuzzy-set structure on M must refer to a particular potential energy surface E(k) for the actual N-alom system. The syntopy concept as defined depends on the particular electronic slate of the considered family of chemical species. The energy condition is specified as follows. First we introduce a relation /;(e) for each ordered pair of points ki and k2 and energy value e : kj /;(e) k2 if and only if the easiest path P in M leading from k] to k2 is such that for every point k e P (ki, k2) the energy value E(k) falls within the open energy interval E(ki) - e, E(ki) + e :

Intuitively, k2 can be reached from kj by a path, on the potential surface, invol­ ving an increase or decrease of energy with respect to E(kj) that is less than the value e. For each point k e M, a subset B(k, e) is then introduced, consisting of the set containing point k and the collection of all points of M that arc accessible from k by paths requiring an energy change less than e, that is :

Note that in the limit case e = 0 this set is not empty : B(k, 0) = {k). Our next step is to introduce an auxiliary function t (k, k', c) that des­ cribes the energetic accessibility of configurations k and k' relatively to e. In some systems we may require that this function takes a maximum value of 1 if the ener­ gies of k and k' are equal and decreases monotonically when the energy difference between k and k' increases. Several possible choices have been discussed in a pre­ vious paper (# b) ; their features can be exploited in various ways when one de­ fines membership functions for fuzzy sets in terms of them. As an example, let us consider the following choice :

— 1039 — where a is some positive scaling parameter that allows control over the rate of monotonic decrease. For each point k and symmetry domain G;, one can define the intersection set

that is the subset of points which both have symmetry g; and arc e-acccssiblc from k. Then, for each point k e M, the membership function ^¡(k, e) relative to syntopy set S; can be defined, in terms of the above function t (k, k', e), as :

If no point having symmetry g; is e-acccssible from k, then the membership func­ tion of k relative to Si is 0 ; if, on the other hand, such points k' do exist, jq(k, c) is the largest of their t values. This definition is similar to that of taking as the dis­ tance of a point to a line the shortest distance of this point to all points of the line [16a], The membership function pj(k, e) constructed according to Eqs (12)-( 14) fulfills all requirements (i)-(v), except for the fact that it changes discontinuously from the minimum nonzero value of exp (-a) 'within' Sj to the value 0 in the rest of space M. This choice may be made when one intends to use a threshold, exp (-a), for significant symmetry resemblance, below which such resemblance is ignored. By choosing an appropriate value for the parameter a, this threshold can be adjusted as required. The syntopy fuzzy sets Sj defined in Eqs (10)-( 14) represent a continuous extension of the symmetry point groups g; with reference to a specified electronic slate (and potential energy surface) and to an energy parameter e which expresses the accessibility of configurations from one another. In the e = 0 limit die above fuzzy set decomposition of space M reduces to the previous point group model : for c = 0

Sj = Gi

for every gj ,

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no other configuration being then accessible from a given one. As a simple example, consider a projected nuclear configuration space spanned by a reduced set of internal coordinates, such as the single-dimension configuration space spanned by the reaction coordinate for the internal motions of

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CIl4+ [17b] or for those of C5II 1o+ between C2 structures (Fig. lc) ; or by the torsion angle around the C3 axis in the ethane molecule (Fig. lb) evolving bet­ ween D3d (S) and D3h (E) through D3 (I) structures with all other internal coordinates remaining unchanged ; or by the bending angle about the ah plane in the ammonia molecule (Fig. la) evolving between Cooy and D3h through C3V structures ; of by the bond angle of a water molecule with C2v symmetry and fixed bond lengths. In this latter ease the motion of the representative point in the configuration space is restricted to a defined circle on the conical surface C shown in Figure 3. In Figure 4 we have drawn the variation of the potential energy V as a function of the bond angle 0 for such a system. When 0 ~> 0 (point 0) V ~> « ('nuclear reaction') and G --> CooV; when 0 = tc (point L) V = 0 (by definition)

figure 4. Potential-energy curve for a water-like molecule with C2v symmetry and fixed bond lengths. For a bond angle of 0 (formal 'nuclear reaction' configu­ ration 0) the symmetry is CooV, whereas for a bond angle of it (linear configura­ tion L) the symmetry is Dooh- This latter conformation corresponds to a maxi­ mum in the energy, which is used to define the energy parameter e. Two oilier remarquable points arc F (equilibrium conformation) and K (isocncrgctic with L). The membership function giJk, c < Vc) is represented by the upper curve which presents a maximum of 1 and a sudden drop below exp (- ex) -, and the synlopy set S| by the linear cloud on the 0 axis.

— 1041 — and G = Dooh- The equilibrium angle 0C (point E) corresponds to the energy mi­ nimum Vc. We arc looking for the membership functions |J.l of 'negative energy' configuration points relative to the syntopy set Sl corresponding to point group Dooh- 0nc always has here BL(k, e) = (L) or 0. If the energy parameter e = 0, = 0 for every point of the curve except for L, where p.L = 1, according to Eqs (10)-(14). If e —> °°, Pl = 1 f°r every accessible point of the curve (the symme­ tric configuration being energetically undistinguishablc from this point). The ease s < Vc, which is the one relevant here for quasi-symmctric structures, is illustra­ ted in the Eigurc. The corresponding fuzzy set Sl is represented by a linear cloud of small circles with decreasing density in the configuration space spanned by 0. 4. SYNTOPY FUZZY SETS AND SYNTOPY FUZZY GROUPS FOR NUCLEAR CONFIGURATIONS Within the syntopy model, the discrete groups of symmetry operations of individual configurations will be extended to larger groups, as well as augmented with continuous features, as a result of the replacement of the binary characteristic functions of ordinary sets with the continuous membership functions of fuzzy sets. Consider a representative point k in the configuration space M, and the fa­ mily S(k) of all syntopy sets Si for which the membership function of point k is nonzero, with the energy parameter e :

The set generated by all operations Rj of all point symmetry groups gi associated with the above syntopy sets Sj is denoted by g(k, c) :

where J is the family of all possible j indices of the relevant symmetry operators Rj, and all possible ordered subsets J of J are considered, hence the elements Rj of set g(k, e) arc all possible ordered products of these operators. The product Rj' • Rj" of two elements of g(k, e) is defined as the appli­ cation (as symmetry operations) of Rj' after Rj".Thc fact that the above set g(k, e) is a group is easy to demonstrate b) by noting that all operators Rj and all their products above arc symmetry operators in the group K of the united atom. Closure is ensured by the very definition of g(k, c) ; note that this set contains all operators Rj for which the membership function at point k is positive, and also all symmetry

— 1042 — operators that can be generated as products of the above, even those for which the membership function at point k is zero. Associativity follows from the fact that all elements of g(k, e) are symmetry operators for the united atom. A unit element E necessarily occurs among the elements Rj of g(k, e), for example as a product of unit elements of groups gi involved in the definition of g(k, e). For every element Rj of g(k, e), the inverse Rj‘l, constructed as the reversed product of the inverses Rj -1 of the operators Rj occurring in Rj , is also present in g(k, e). With respect to a representative point k, each symmetry operator Rj is assigned a positive real number g, defined as the maximum value the membership function of point k takes in those syntopy sets Sj for which Rj e gi :

The 0 in the bracket is intended to cover the ease where none of the groups gj contains Rj, leading to g = 0. Such a ease may occur, for instance, if the confi­ guration k belongs to the syntopy sets of symmetry groups C \, C 2 and Cs, with positive membership values, but has zero membership value in the syntopy set of the product group C2y I’1 this case, a product of elements from C 2 and Cs is a second reflection operation that is present only in C2v> hence its g value is 0. The family of pairs (Rj, g), where Rj e g(k, e) and g is defined by Eq. (18), forms the syntopy group s(k, c) at point k :

where the group operation in s(k, c) is defined as follows :

with Rc = Ra • Rg from the group properties of g(k, e) and gc being defined for Rc in Eq. (18). The scalar g associated with each symmetry operation R is a mea­ sure of the relevance of the corresponding symmetry element for configuration k. The product defined above provides an isomorphism between the algebra­ ic structure of g(k, c) and that of s(k, e) ; hence s(k, e) is also a group. As abstract groups (in the sense of pure algebraic structures), g(k, c) and s(k, e) arc identical. However, the same symmetry operator R may be associated with a different value of g for a different configuration k or a different energy parameter e. As a result, as concrete realizations of g(k, e), the syntopy groups s(k, e) arc generally dif­ ferent for different values of k or c.

— 1043 — 5. POSSIBLE APPLICATIONS AND DEVELOPMENTS OF THE SYNTOPY FUZZY GROUPS As the problems mentioned in the beginning suggested, the concept of syntopy could help to rationalize the statics or the dynamics of perturbed or interme­ diate quasi-symmetric systems. For instance, in the static problem evoked in Fig. la, one may consider the three systems, NH3 + D + F, NH2D + H + F, and NH2F + H + D, in a common nuclear configuration space, with atoms and molecules far apart. If the energy criterion e is chosen so as to make these structures pairwise accessible from each other, there will exist nuclear configuration changes which may interconvert them within this space. Similarly, in the dynamic problem re­ called in Fig. lc, one may define a syntopy fuzzy-set structure in the internal con­ figuration space allowing to attribute membership function values to each obser­ vable system, defined by a set of internal motion rates (including the extreme cases where the system displays either C2 or C5h point symmetry as effective symme­ try), if one knows the potential energy surface for the considered state. A straightforward application of the syntopy model is related to the temp­ erature dependence of structural features of molecular systems. By choosing the energy parameter e as a simple function (e.g., a multiple, with the Boltzmann fac­ tor k) of the absolute temperature T, e (D = f (¿T) ,

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the syntopy sets S| and membership functions gi(k, e) represent all the significant accessible symmetries for a chemical species that is classically described by the for­ mal nuclear configuration k. With an appropriate choice of the auxiliary function t, the syntopy set Sj may be regarded as the collection of all those formal nuclear configurations which are expected to exhibit symmetry properties characterized by symmetry group gj at temperature T, and the membership function pj(k, e) pro­ vides a measure of the relative importance of this particular symmetry at the said temperature. In particular, the syntopy model lends itself to an analysis of the relevance of molecular chirality at various temperatures. Take all point symmetry groups gj and the respective syntopy sets Sj that contain a symmetry operator corresponding to a reflection plane or an inversion point. For any (chiral) nuclear configuration k having no such symmetry element, the greatest value gj(k, e) among the corres­ ponding membership functions provides a measure of the achiral behaviour dis­ played at temperature T. However, one may wish to use a geometric [16] rather than energetic criterion to define membership functions for fuzzy sets. In the inter-

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nal nuclear configuration space M, a degree of chirality could be associated with each configuration k, based upon the configuration space geometric (rather than energetic) distances : d(k, 1), d(k, 2), d(k, m), of the configuration k from a series rj, r2, rm of chiral reference structures. These distances could be used to define fuzzy set membership functions for configurations in the chirality refer­ ence sets. The overall degree of chirality can then be obtained by some weighted sum of the membership function values in the relevant fuzzy sets. In spectroscopic applications the energy parameter e can be taken as a function of the frequency v applied to the probe : e = f(v) .

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With respect to the formal nuclear configurations k routinely used in spectro­ scopic studies to represent 'equilibrium' or 'distorted' molecular structures, the family of membership functions {pi(k, e)} provides relative measures of the spec­ troscopic features expected for configurations of the various point symmetries described by groups g[. Syntopy sets and the associated membership functions could also provide a measure of the relevance of local internal coordinates defined in terms of symme­ try coordinates [7] for configurations not strictly belonging to the corresponding symmetry. It may be useful to consider different local coordinate systems within different local regions of the internal configuration space, tailor-made for the con­ figurations occurring within each region. It seems natural to define such coordi­ nate systems with respect to a symmetry group relevant to the considered region. In general, manifold theory provides the standard description of the families of compatible local coordinate systems within a space, and various manifold struc­ tures have been introduced in earlier studies [4, 5]. Syntopy sets also generate a manifold structure in the configuration space M. Take the ordinary set Fj of all points k having positive membership value in syntopy set Sj :

By generating the coordinate neighborhoods Fjj defined as the maximum con­ nected components of the sets Fj, the entire space M is covered with such coordi­ nate neighborhoods, as required by a manifold model ;

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Within each neighborhood Fjj, the symmetry coordinates obtained for symmetry group gi are applicable, and are, in fact, the natural choice for local coordinates. Hence the entire family of internal configurations for all molecules with a given stoichiometry can be represented within a single framework of coordinate sys­ tems, where syntopy sets justify the choice of the coordinate domains. One may note an interesting contrast between our syntopy model and the general theory of sudden changes, commonly called catastrophe theory [29, 30], When introducing the syntopy groups, our goal was to replace a discrete model (point symmetry) with a continuous one (syntopy). The sudden appearence of a given symmetry at a specific configuration is replaced by a smooth, continuous change of a membership function in a syntopy set. By contrast, in catastrophe theory, the smooth, continuous changes of a set of variables, called the control variables, are the primary events, that lead to sudden, discontinuous changes. In the above sense, the introduction of syntopy sets may be regarded as the inverse pro­ cess of introducing catastrophes : the sudden, catastrophic changes of point sym­ metries are replaced by smooth, non-catastrophic changes of membership functions relative to syntopy sets. This may lead to the following formulation. Starting with the syntopy model and regarding the internal coordinates of configuration k in the (3N-6)dimension space as the control variables of catastrophe theory, one may study the sudden, catastrophic appearance or disappearance of discrete point groups, char­ acterized by values g = 1 or 0 of the respective membership functions. A com­ plete classification of all possible catastrophes in a general, n-dimension space is not known. However, simplifications in our problem are possible if one considers only a subset of internal coordinates, such as puckering, torsion, inversion or ben­ ding angles (as in the case considered in Figure 4), thereby reducing the number of control variables. Finally, it may be worthwhile to recall that our fuzzy set representation of molecular point symmetry takes into account the non-localized nature of nuclei while salvaging as much as possible from the classical nuclear configuration con­ cept. This may provide a general treatment for all families of formal configura­ tions, noting that a probability distribution of nuclear configurations has physical reality even if the individual configurations have not. Our continuous extension of the symmetry concept, with an energy criterion to define membership functions, recognizes this fact. However, while this criterion might be best suited for analys­ ing quantum-mechanical molecular problems, a geometric criterion may be more appropriate to treat the more classical quasi-symmetric systems.

— 1046 — Acknowledgements

The authors are honored to contribute to this volume dedicated to the me­ mory of their late colleague Pierre Claverie. J. M. is thankful to Claude Gaudcau for introducing him to fuzzy-set theory. P.G. M. acknowledges a research gram from the Natural Sciences and Engineering Research Council of Canada. REFERENCES [1] I. HARGITTAI, ed., vol. 1, 1986, vol. 2, 1989, Symmetry : Unifying Human Understanding, Pergamon Press, New York. [2] (a) D.R. HOFSTADTER, 1979, Gödel, Escher, Bach, Random House, New York, (b) M. C. ESCHER, 1967, The Graphic Work, Bal lant ine Books, New York. [3] E. NOEL and G. MINOT, eds, 1989, La Symétrie Aujourd'hui, Editions du Seuil and Radio France, Paris. [4] P.G. MEZEY, 1987, Potential Energy Hypersurfaces, Elsevier, Amster­ dam, and references therein. [5] P.G. MEZEY, in R. DAUDEL et al., eds, 1985, Structure and Dynamics of Molecular Systems, Reidel, Dordrecht, vol. 1, p. 41. [6] P.G. MEZEY, in J. MARUANI, ed., 1988, Molecular Organization and Engineering, Kluwer, Dordrecht, vol. 2, p. 61. [7] E.B. WILSON Jr, J.C. DECIUS and P.C. CROSS, 1955, Molecular Vibra­ tions, McGraw-Hill, New York. [8] H. PRIMAS, 1981, Chemistry, Quantum Mechanics, and Rcductionism, Springer Verlag, Berlin, Chapter 6. [9] R.S. BERRY, in R.G. WOOLLEY, ed., 1980, Quantum Dynamics of Mo­ lecules, Plenum Press, New York. [10] R.G. WOOLLEY, in J. MARUANI, ed., 1988, Molecular Organization and Engineering, Kluwer, Dordrecht, vol. 1, p. 45. [11] (a) L. LATHOUWERS, P. VAN LEUVEN, and M. BOUTEN, 1977, Chem. Phys. Lett., 52, 439. (b) L. LATHOUWERS and P. VAN LEUVEN, 1978, Int. J. Quantum Chem. Symp., 12, 371. (c) L. LATHOUWERS, 1978, Phys. Rev. A, 18, 2150. [12] (a) H.C. LONGUET-HIGGINS, 1963, Mol. Phys., 6, 445. See also : (b) E.B. WILSON Jr, C.C. LIN and D.R. LIDE, 1955, J. Chem. Phys., 23, 136 ; (c) R.E. MILLEN and J.C. DECIUS, 1973, J. Chem. Phys., 59, 4871. [13] (a) S.L. ALTMANN, 1971, Mol. Phys., 21, 587. (b) A. BAUDER, R. MEYER and Hs H. GÜNTHARD, 1974, Mol. Phys., 28, 1305. (c) G.S. EZRA, 1979, Mol. Phys., 38, 863, and references therein.

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