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On chirality measures and chirality properties Noham Weinberg and Kurt Mislow Abstract: It is shown that chiral zeroes are integral to all pseudoscalar functions, and that these functions, and thus the chirality properties that are described by them, are therefore normally unsuitable as chirality measures. The multidi- mensional nature of chirality properties is explored. Chirality measures for nonrigid objects and stochastic systems are discussed. It is shown that if the chirality of a nonrigid object is described as a time average of the chirality measures of its instant configurations, this time average is nonzero not only for chiral but also for achiral molecules. This para- dox can be resolved if chirality measures are properly applied to nonrigid objects. Key words: chirality, chiral zeroes, chirality measures, chirality properties. Résumé : On démontre que les zéro chiraux font partie intégrale de toutes les fonctions pseudoscalaires et que ces fonctions, et en conséquence les propriétés de chiralité qu’elles décrivent, ne sont donc normalement pas des mesures adéquates de la chiralité. On a exploré la nature multidimensionnelle des propriétés de chiralité. On discute des mesures de chiralité pour les objets non rigides et les systèmes stochastiques. Si la chiralité d’un objet non rigide est décrite par une moyenne en fonction du temps de configurations instantanées, on démontre que cette moyenne en fonction du temps n’est pas égale à zéro non seulement pour les molécules chirales, mais aussi pour les molécules achirales. On peut résoudre ce paradoxe si les mesures de chiralité sont appliquées correctement aux objets non rigides. Mots clés : chiralité, zéro chiraux, mesures de chiralité, propriétés de chiralité. Weinberg and Mislow 45 Introduction More than a century ago, Guye (1) introduced a set of continuous functions designed to correlate a pseudoscalar property, i.e., optical rotation, with the structure of a chiral molecule. In effect, this was the first attempt to quantify chirality. Since then a number of other schemes have been suggested (see references in (2), as well as subsequent works (3)). Some of these approaches employ pseudoscalar func- tions that change sign under space inversion and thus would appear to lend themselves to the left–right classification of chiral objects. It has long been recognized, however, that left–right classifications are vitiated when such pseudoscalar functions as chirality polynomials assume a value of zero for chiral objects, the “chiral zero” (4). In the following two sections we show that chiral zeroes are integral to all pseudoscalar functions and that chirality properties therefore cannot normally be used as chirality measures. In the last section we analyze a paradoxical situation that arises when a chirality measure defined for a rigid object, such as a geo- metrical figure, is applied to a nonrigid one, such as a vibrat- ing molecule, and define a chirality measure that is suitable for nonrigid objects and stochastic systems. Chirality measures An object X (no matter whether physical or geometrical) is chiral if and only if it is not superposable on its mirror im- age X (X X ) by a combination of rotations and transla- tions. A chirality measure χ(X) is a real-valued function defined on a set of objects X. It ranks the members of in degrees of their chirality. The set can be either discrete or continuous. We assume that is a continuous set, any two elements of which can be connected by a continuous de- formation. The following would seem to be a set of natural require- ments for a chirality measure χ(X): 1. Any two close objects, X 1 and X 2 , are expected to have close degrees of chirality. Therefore, [1] χ(X) is a continuous function on 2. If X is superposable upon its mirror image X (X X ), i.e. if it is achiral, then its degree of chirality is zero. That is, [2] X = X ⇒χ(X)=0 3. Any two enantiomorphs, X and X , are chiral to the same degree. That is, [3a] χ( X )= χ(X) (scalar χ) or [3b] χ( X )=–χ(X) (pseudoscalar χ) Of the two types of real-valued functions, scalar and pseudoscalar, that satisfy conditions [1]–[3], the latter might appear to have certain advantages as a chirality measure. First, condition [2] follows directly from [3b]. Second, the opposite signs assigned to the degrees of chiralities of Can. J. Chem. 78: 41–45 (2000) © 2000 NRC Canada 41 Accepted October 28, 1999. N. Weinberg. 1 Department of Chemistry, University College of the Fraser Valley, Abbotsford, BC V2S 7M8, Canada. K. Mislow. 1 Department of Chemistry, Princeton University, Princeton, NJ 08544, U.S.A. 1 Authors to whom correspondence may be addressed. NW: Telephone: (604) 853-7441 x4493. Fax: (604) 855-7558. e-mail: [email protected]. KM: Telephone: (609) 258- 3941. Fax: (609) 258-6746. e-mail: [email protected]

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Page 1: On chirality measures and chirality properties

On chirality measures and chirality properties

Noham Weinberg and Kurt Mislow

Abstract: It is shown that chiral zeroes are integral to all pseudoscalar functions, and that these functions, and thus thechirality properties that are described by them, are therefore normally unsuitable as chirality measures. The multidi-mensional nature of chirality properties is explored. Chirality measures for nonrigid objects and stochastic systems arediscussed. It is shown that if the chirality of a nonrigid object is described as a time average of the chirality measuresof its instant configurations, this time average is nonzero not only for chiral but also for achiral molecules. This para-dox can be resolved if chirality measures are properly applied to nonrigid objects.

Key words: chirality, chiral zeroes, chirality measures, chirality properties.

Résumé: On démontre que les zéro chiraux font partie intégrale de toutes les fonctions pseudoscalaires et que cesfonctions, et en conséquence les propriétés de chiralité qu’elles décrivent, ne sont donc normalement pas des mesuresadéquates de la chiralité. On a exploré la nature multidimensionnelle des propriétés de chiralité. On discute desmesures de chiralité pour les objets non rigides et les systèmes stochastiques. Si la chiralité d’un objet non rigide estdécrite par une moyenne en fonction du temps de configurations instantanées, on démontre que cette moyenne enfonction du temps n’est pas égale à zéro non seulement pour les molécules chirales, mais aussi pour les moléculesachirales. On peut résoudre ce paradoxe si les mesures de chiralité sont appliquées correctement aux objets non rigides.

Mots clés: chiralité, zéro chiraux, mesures de chiralité, propriétés de chiralité.

Weinberg and Mislow 45

Introduction

More than a century ago, Guye (1) introduced a set ofcontinuous functions designed to correlate a pseudoscalarproperty, i.e., optical rotation, with the structure of a chiralmolecule. In effect, this was the first attempt to quantifychirality. Since then a number of other schemes have beensuggested (see references in (2), as well as subsequent works(3)). Some of these approaches employ pseudoscalar func-tions that change sign under space inversion and thus wouldappear to lend themselves to the left–right classification ofchiral objects. It has long been recognized, however, thatleft–right classifications are vitiated when such pseudoscalarfunctions as chirality polynomials assume a value of zero forchiral objects, the “chiral zero” (4). In the following twosections we show that chiral zeroes are integral toallpseudoscalar functions and that chirality properties thereforecannot normally be used as chirality measures. In the lastsection we analyze a paradoxical situation that arises when achirality measure defined for a rigid object, such as a geo-metrical figure, is applied to a nonrigid one, such as a vibrat-ing molecule, and define a chirality measure that is suitablefor nonrigid objects and stochastic systems.

Chirality measures

An object X (no matter whether physical or geometrical)is chiral if and only if it is not superposable on its mirror im-age X (X ≠ X) by a combination of rotations and transla-tions. A chirality measureχ(X) is a real-valued functiondefined on a setΩ of objectsX. It ranks the members ofΩin degrees of their chirality. The setΩ can be either discreteor continuous. We assume thatΩ is a continuous set, anytwo elements of which can be connected by a continuous de-formation.

The following would seem to be a set of natural require-ments for a chirality measureχ(X):

1. Any two close objects,X1 andX2, are expected to haveclose degrees of chirality. Therefore,

[1] χ(X) is a continuous function onΩ

2. If X is superposable upon its mirror imageX (X ≠ X),i.e. if it is achiral, then its degree of chirality is zero. That is,

[2] X = X ⇒ χ(X) = 0

3. Any two enantiomorphs,X and X, are chiral to thesame degree. That is,

[3a] χ(X) = χ(X) (scalarχ)

or

[3b] χ(X) = –χ(X) (pseudoscalarχ)

Of the two types of real-valued functions, scalar andpseudoscalar, that satisfy conditions [1]–[3], the latter mightappear to have certain advantages as a chirality measure.First, condition [2] follows directly from [3b]. Second, theopposite signs assigned to the degrees of chiralities of

Can. J. Chem.78: 41–45 (2000) © 2000 NRC Canada

41

Accepted October 28, 1999.

N. Weinberg.1 Department of Chemistry, University Collegeof the Fraser Valley, Abbotsford, BC V2S 7M8, Canada.K. Mislow. 1 Department of Chemistry, Princeton University,Princeton, NJ 08544, U.S.A.

1Authors to whom correspondence may be addressed. NW:Telephone: (604) 853-7441 x4493. Fax: (604) 855-7558.e-mail: [email protected]. KM: Telephone: (609) 258-3941. Fax: (609) 258-6746. e-mail: [email protected]

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enantiomorphs can be seen as a natural basis for a left–rightclassification of chiral objects. And, third, pseudoscalarfunctions [3b] are expected to have a direct connection toexperimentally observed pseudoscalar properties, such asoptical rotation.

There is, however, one more important constraint that hasto be applied toχ(X): in order to properly measure chirality,this function is allowed to assume a value of zero only forachiral objects. That is,

[4] χ( )X X X= ⇒ =0

Combining [2] and [4], it follows that a proper chiralitymeasureχ(X) equals zeroif and only if Xis achiral. That is,

[5] χ( )X X X= ⇔ =0

Accordingly, chiral zeroes, i.e., situations whereχ(X) = 0 forX ≠ X, are excluded.

Let us now return to the perceived advantages ofpseudoscalarχ(X). It has been demonstrated (5) that, withrare exceptions of low-dimensional sets, a chiral object canbe chirally connected to its enantiomorph, i.e., there alwaysexists a path (a parametrized curve)Λ Ω⊂ linking X = X0 toX = X1 such that allXλ e Λ , λ e [0,1] are chiral. Along thispath Λ , a pseudoscalarχ(X) can be considered as a real-valued continuous function defined on segment [0,1]. Itchanges its sign on this segment and therefore has to take azero value somewhere withinΛ , i.e., at some chiralXλ. Thusχ(X) possesses chiral zeroes. It follows that, as a rule,con-tinuous pseudoscalar functions cannot be used as chiralitymeasures in three and higher dimensions. This conclusionrenders questionable current and past attempts to formulate“chirality measures” on the basis of pseudoscalar functions.

An important exception obtains when a set of objects issplit by its achiral subset into two disjoint subsets of chiralobjects, such that any path connecting enantiomorphs mustpass through the achiral boundary. Examples of such sets arethe set of triangles of variable shape in two dimensions (6),the set of helices of variable pitch (7), and Ruch’s models ofclassa (4). In such rare cases, a pseudoscalarχ(X) changesits sign only at the achiral boundary and is therefore suitableas a chirality measure.

Chirality properties

Optical rotation was the first recognized manifestation ofchirality. Under identical conditions, enantiomers rotate theplane of polarization to the same degree but in opposite di-rections. The angle of rotation is therefore a pseudoscalarproperty. The situation is similar with respect to circulardichroism: ellipticity is a pseudoscalar property. Indeed,allmanifestations of chirality are described by pseudoscalars.But, as stated in the previous section, pseudoscalar functionscannot normally be used as chirality measures. It followsthat, as a rule,chirality properties are unsuited as chiralitymeasures.

In this connection, we need to point out a key distinctionbetween a chirality measure and a chirality property. Theformer is a mathematical construct that measures how far anobject is from being achiral; even though this measure de-pends on a particular defining scheme, it is always associ-ated with an isolated object and is independent of any

external object or field. The latter, on the other hand, is anobservable property that manifests itself through the differ-ence in the interactions of a chiral objectX and itsenantiomorphX with a reference chiral object or fieldH, or,alternatively, through the difference in the interactions ofXwith enantiomorphsH and H .

Let P(X B H) represent ascalar property of a chiral com-plex X B H or the result of the interaction ofX with a chiralobject or fieldH. Then an observed chirality propertyπH (X)can be expressed as a difference, either as

[6a] πH X P X H P X H( ) ( ) ( )= ° − °or as

[6b] πH X P X H P X H( ) ( ) ( )= ° − °It follows from eq. [6a] that πH(X) is a pseudoscalar prop-

erty. That is,

[7] π πH HX X( ) ( )= −

Equations [6a] and [6b] are equivalent becauseX H° andX H° are enantiomorphous. Their different appearance onlyreflects the difference in the operations used to determineπH(X). For example,πH(X) in [6a] might represent the dif-ference in retention times for enantiomersX and X on achiral columnH, while in [6b] it might represent the angleof rotation of plane-polarized light due to its interaction withX, found as a difference in phases of left and right circularlypolarized waves,H and H . The equivalence of definitions[6a] and [6b] implies that for each observable chiral prop-erty there should be two equivalent ways of experimental de-termination. For instance, the difference in retention timesmentioned above can be equivalently found by chromatogra-phy of the same compoundX through two enantiomeric col-umns,H andH , and the angle of rotation can be found as adifference of phases of a left (or right) circularly polarizedwaveH as a result of its interaction with enantiomersX andX.

Depending on the conditions of measurement (temperature,wavelength, etc.) an object and its enantiomorph may have ascalar property of the same magnitude. That is,P(X B H) andP X H( )° can beaccidentallyequal even for a chiralX. Thus,the fact thatπH(X) equals zero does not necessarily meanthat X is achiral. Chiral zeroes are therefore expected to oc-cur for chirality properties.

This does not mean, however, that pseudoscalar propertiesfail to characterize a chiral object as chiral. The reason isthat there are a virtually limitless number of such propertiesassociated with a given object, and its observed chirality istherefore expressed as an infinite-dimensional pseudovector.This pseudovector remains nonzero if one, or even several,of its components vanish. Only for a truly achiral object is itthe case thatall of its individual pseudoscalar properties arezero, and hence that the whole chirality pseudovector iszero.

This statement may be illustrated by the example of opti-cal rotation. Rotation angleδ and ellipticityψ are known tobe frequency-dependent, and at some frequencies becomezero. They are related to the refractive index and absorptioncoefficient, the real and imaginary parts of the same com-plex quantity (8), and are therefore not independent. As a re-sult, wheneverδ takes on a value of zero in an isolated

© 2000 NRC Canada

42 Can. J. Chem. Vol. 78, 2000

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transition, the absolute value ofψ is near or at its maximum(the Cotton effect) (9). Thus, a complex pseudoscalar

[8a] ϕ δ ψ= + i

or, equivalently, a two-dimensional pseudovector

[8b]r

ϕδψ

=

does not become zero for chiral molecules, but is identicallyzero for achiral molecules.

Similarly, if the entire functionδ(ν) that represents rota-tion angles for the whole range of frequenciesν is used,rather than a single value ofδ taken at a particular frequency,the problem of chiral zeroes is also avoided sinceδ(ν) isidentically zero only for achiral molecules.

The above discussion suggests that a pseudovectorr

χ( )Xcan be continuously transformed into

r r

χ χ( ) ( )X X= − withoutpassing through zero (Fig. 1), and that the problem of chiralzeroes can thus be avoided even though eq. [3b] is obeyed. Ithas to be added, however, that although a pseudovector

r

χ( )Xcan differentiate chiral objects from achiral ones, it cannotbe directly used to compare their degrees of chirality be-cause relations “greater than” or “less than” are not naturallydefined for any two vectors (10). In other words, if chiralityis described by

r

χ( )X , it becomes impossible in many cases tosay which of two objects is more chiral. To restore a linearordering, one needs to use scalar quantities associated withpseudovectors, such as Euclidean norms

[9a]r

ϕ δ ψ= +2 2

or

[9b] δ δ= ∫ [ ( )]min

maxv dv

v

v 2

Norms [9a] and [9b] are “well-behaved” scalar functionsthat obey conditions [1], [3a], and [5], and thus qualify aschirality measures.

These considerations may be illustrated with the referenceto Fig. 2. Three chiral objects,X1, X2, andX3, are character-ized by the pseudovector

r

ϕ (eq. [8b]). If δ were to be chosenas the sole measure of chirality, the ranking would beX3 >X2 > X1, with X1 being “achiral” (zero optical rotation). Sim-ilarly, ψ as the sole measure of chirality would produce adifferent ranking,X1 > X2 > X3, with X3 now being “achiral”(zero ellipticity). Both rankings are clearly unacceptable, notonly because they are mutually inconsistent but also becauseof the chiral zeroes. If, however, we use norm

r

ϕ (eq. [9a]),for example, then this problem can be overcome becausechiral zeroes no longer exist and the contributions of differ-ent chirality properties are integrated into a single chiralitymeasure.

Chirality measures for nonrigid objects andstochastic systems

If a molecule such as methane is considered in a state ofvibrational motion, most of its instantaneous configurationsare chiral (7). Each such configuration is as probable as itsenantiomorph, and the contributions from different configu-rations to the net observed chirality property will thereforecancel out on average. For this reason, no chirality proper-ties are observed for such molecules. At the same time, how-ever, in accordance with our earlier discussion, a chiralitymeasure is a sign-preserving function on a set of instantconfigurations, and the contributions from different configu-rations thereforecannot cancel each other. Hence, if the

© 2000 NRC Canada

Weinberg and Mislow 43

Fig. 1. A chiral (solid line) and an achiral (dashed line) pathconnect enantiomorphsX1 and X1. Pseudovector

r

ϕ( )X (eq. [8b])never turns zero on the chiral path even though one of its com-ponents turns zero for chiral objects:δ for X3 and ψ for X2. Foran achiral objectX4, bothδ and ψ are zero and

r

ϕ( )X = 0 .

Fig. 2. Chirality properties of three chiral objects,X1, X2, andX3, are characterized by pseudovectors

r

ϕ( )X1 ,r

ϕ( )X2 , andr

ϕ( )X3 ,respectively. Optical rotation angleδ identifies X3 as the mostchiral object but fails to detect the chirality ofX1. Similarly, el-lipticity ψ selectsX1 as the most chiral object but fails to detectthe chirality of X3. Whenδ and ψ are combined, however, andthe norm

r

ϕ (eq. [9a]) is used, all three objects are identified aschiral andX2 is found to be the most chiral of the three.

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chirality of a nonrigid object, such as a vibrating molecule,is described in terms of the time average of the chiralitymeasures of its instant rigid configurations,this time aver-age is nonzero not only for a chiral but, paradoxically, evenfor any achiral molecule.

This paradox can be avoided, however, if a chirality mea-sure is defined for a nonrigid object as a whole rather thanfor its instant configurations.

We consider a nonrigid objectX as a parametrized set ofits instant rigid configurationsX τ:

X = X τ; τ e T

Correspondingly, its enantiomorphX can be representedas a set of the enantiomorphsXτof rigid configurations Xτ.That is,

X = X Tτ τ; ∈

A congruency-based chirality measureχ (2b) is then de-fined for X as a minimum distance betweenX andX, for ex-ample as

[10] χτ

τ

ττ

τ τ

τ

( )

min ( , )

X =′∈

∈′

∫∫

TT

T

dist X X d

d

If X andX differ only by a permutation of their elements,by virtue of [10] they are achiral. The following two exam-ples serve as illustrations. Consider a vibrating water mole-cule in two-dimensional space (Fig. 3). Almost all instantconfigurationsXτ (exceptX0, X1/2, andX1) are chiral in 2D.The entire arrayX, however, is achiral becauseX Xτ τ= −1(e.g., X1/4 = X3/4, X3/4 = X1/4). Another example is acompound of the type 4-[(R)-sec-butyl]-4′-[(S)-sec-butyl]-2,2′,6,6′-tetramethylbiphenyl (Fig. 4) which consists entirelyof chiral conformations (11). Instant configurationsX τ ofthis molecule are characterized by triplesτ = (τ1,τ2,τ3),where τ1 and τ3 are torsion angles for phenyl-Cabc bondsand τ2 is the torsion angle for the central phenyl-phenylbond. Torsion anglesτ1, τ2, andτ3 can be changed to –τ1, –τ2,and –τ3 by internal rotations around the respective bonds.Although τ2 cannot assume a value of zero (the two phenylrings cannot become coplanar due to the presence of thebulky substituents R) it is nevertheless possible to transformτ2 into 180º –τ2, which, by virtue of the symmetry of therings and their R substituents, is equivalent to –τ2. As shownin Fig. 5, for anyconformationXτ of X with torsion pa-rameterτ = (α,β ,γ ) there exists another conformation ofX,X ′τ with τ′ = (–γ, –β, –α), such thatX Xτ τ= ′. Therefore, inaccordance with [10],χ(X) = 0, i.e. the discussed moleculeis achiral even though it does not have a single achiral con-formation.

A similar approach applies to nonrigid stochastic systems,such as ensembles of molecules. The simplest example ofthis kind is a system that contains a statistically significantnumberN of monoatomic molecules (12), e.g., a mole of he-lium gas in a cubic box. Such a system can be described byits dynamic trajectoryX in a 3N-dimensional configurationspace. Practically every instant configuration,X τ, acquiredby the system at timeτ, is bound to be chiral because ran-dom achiral arrangements of a large number of particles arehighly improbable. The enantiomorphous configurationsX ′τalso belong to the configuration space but do not necessarilybelong to the trajectoryX. In accordance with the ergodic

© 2000 NRC Canada

44 Can. J. Chem. Vol. 78, 2000

Fig. 3. An array of instant configurationsX of a vibrating watermolecule and its mirror-image arrayX.

Fig. 4. Schematic drawing of two enantiomorphous conformations of a biphenyl with bulky substituents (e.g., R = CH3) in the 2,2′,6,6′positions and enantiomorphous -Cabc groups (e.g., -CH(CH3)C2H5) in the 4,4′ positions.

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hypothesis (13), however, the trajectoryX thoroughly coversthe entire conformation space so that every point of thespace that represents a structure with the same energy is in-finitesimally closely approached by the trajectory. That is,for any configurationXτ of X there always exists an infini-tesimally close configurationX ′τ of X. Thus, it followsfrom [10] thatχ(X) = 0, i.e., that the system is achiral eventhough almost all of its instantaneous configurations are chiral.

Acknowledgments

We are indebted to a reviewer for drawing our attention tosome relevant publications. We thank the National ScienceFoundation for the support of this work.

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3. For example, see: (a) M.A. Osipov, B.T. Pickup, and D.A.Dunmur. Mol. Phys.84, 1193 (1995); (b) H. Zabrodsky and D.Avnir. J. Am. Chem. Soc.117, 462 (1995); (c) M. Randic andM. Razinger. J. Chem. Inf. Comput. Sci.36, 429 (1996);(d) A. Cossé-Barbi and M. Raji. Struct. Chem.8, 409 (1997);(e) D.J. Klein and D. Babic. J. Chem. Inf. Comput. Sci.37,656 (1997); (f) M. Petitjean. J. Math. Chem.22, 185 (1997);(g) B.B. Smirnov, A.V. Yevtushenko, and O.V. Lebedev. Zh.Org. Khim. 33, 1129 (1997); (h) B.B. Smirnov, O.V. Lebedev,and A.V. Yevtushenko. Zh. Org. Khim.33, 1134, 1326, 1328,1472 (1997); (i) P.G. Mezey. Int. J. Quantum Chem.63, 105(1997); (j) P.G. Mezey. Comp. Math. Appl.34, 105 (1997); (k)P.G. Mezey. THEOCHEM.455, 183 (1998); (l) P.G.Mezey.Chirality, 10, 173 (1998); (m) P.G. Mezey. J. Math. Chem.23,65 (1998); (n) M.A. Osipov, B.T. Pickup, M. Fehervari, andD.A. Dunmur. Mol. Phys.94, 283 (1998); (o) A. Ferrarini andP.L. Nordio. J. Chem. Soc. Perkin Trans. 2, 455 (1998); (p) P.Le Guennec. J. Math. Chem.23, 429 (1998); (q) V.M. Markov,V.A. Potyomkin, and A.V. Belik. J. Struct. Chem.39, 407(1998); (r) V.E. Kuzmin and A.G. Artemenko. J. Struct. Chem.39, 442 (1998); (s) S.É. Alikhanidi and V.E. Kuzmin, J. Struct.Chem. 39, 447 (1998); (t) V.E. Kuzmin, V.A. Chelombitko,I.V. Yudanova, I.B. Stelmakh, and I.S. Rublev. J. Struct. Chem.39, 452 (1998); (u) L. Coffey, J.A. Drapala, and T. Erber. J.Phys. A: Math. Gen.32, 2263 (1999); (v) M. Petitjean. C. R.Acad. Sci. Paris, Serie IIc,2, 25 (1999); (w) M. Petitjean. J.Math. Phys. (N.Y.),40, 4587 (1999); (x) B.B. Smirnov, O.V.Lebedev, and A.V. Evtushenko. Acta Crystallogr. Sect. A:Fundam. Crystallogr.A55, 790 (1999).

4. E. Ruch. Angew. Chem. Int. Ed. Engl.16, 65 (1977).5. (a) K. Mislow and P. Poggi-Corradini. J. Math. Chem.13, 209

(1993); (b) N. Weinberg and K. Mislow. Theor. Chim. Acta.95, 63 (1997).

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7. K. Mislow. Top. Stereochem.22, 1 (1999).8. D.J. Caldwell and H. Eyring. The theory of optical activity.

Wiley-Interscience, NewYork. 1971. p. 21.9. E. Charney. The molecular basis of optical activity. John Wiley

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also discussion in references (6, 7).12. K. Mislow and P. Bickart. Isr. J. Chem.15, 1 (1976/77).13. C. Truesdell.In Ergodic theories.Edited byP. Caldirola. Aca-

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© 2000 NRC Canada

Weinberg and Mislow 45

Fig. 5. Schematic side-view projection of conformations belong-ing to the molecular system of Fig. 4. Straight lines representthe edges of phenyl rings (substituents R are not shown); ori-ented circles represent -Cabc groups (the arrows point in the di-rections a→ b → c). Solid lines represent parts of the moleculethat are closer to the viewer; dashed lines represent the moredistant ones. ConformationsX ′τ (a) and Xτ (d) areinterconverted by internal rotationsτ i. ConformationX ′τ (a) isconverted toX ′τ (b) by reflectionσxz in the xz plane. A differ-ent view (c) of X ′τ is obtained by a rigid 180° rotation Rz aboutthe z axis. Conversion ofX ′τ (c) to X τ (d) is achieved by rigidrotation Rx about thex axis by an angleβ.

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