Isometric groups and chirality of nonrigid molecules: a generalization of Kelvin's theorem

‘Chemical Physics X,(1976) 155-165 0 North-Holland Publishing Company.

.-

ISOMETRIC GROUPS AND CRIRALXTY OF NONRSGID MOLECULES: A GENERALIZATION

OF KELVIN’S THEOREM

H. FREE and I-Is-H. GijrJTHARD

Received 21 January 1976 A ~ene&lization of Kelvin’s theorem for the chklity of rigid molecules to nonrigid molecules is given. It is based on the concept of the isometric group of nonrigid systems which allows a formulation of the chirality criterion of nonrigid systems in close analogy with Kelvin’s original formulation. A number of examptes, in particular such 9s investigated by Mislow, are given.

subs consistent -with experimental data it has been remarked by Mislow [3--61 that this recipe may lead to erroneous results. Characteristic examples are given by molecules of the general type

1. Introduction In 1884 Lord Kelvin [1,2] formulated the theorem that chirality of a quasi rigid molecule is a sufficient and necessary condition for optical activity. A quasi rigid molecule is called chin& if its re structure is improperly congruent with its mirror image. This theorem has since been considered as a fundamental basis for numerous applications of the optical activity to problems of molecular structure. Straightforward application of Kelvin’s theorem is restricted to rigid molecules since the concept of geometrical congruence requires rigidity of the molecular structure invalved. Nevertheless, it has been used in an intuitive way for the discussion of optical activity of nonrigid molecules. By far the largest part of organic molecules possesses a certain flexibility of one kind or another and therefore may be considered as semi-rigid molecules. Molecules with internal rotations, inversion, ring puckering or ring interconversions belong to this class, which is typified with manifolds of nuclear configurations described by finite variations of structural parameters (dihedral an- ,gles, inversion and pucker angles, etc.). Application of KelVis theorem to nonrigid molecules traditionally has been made by investigating the manifold of nuclear configurationS for specific configurations with sym-’ metry elements of the second kind. If such contigura‘tions exist for a nonrigid molecule, it is assumed to be Bchiral. Though in most cases this procedure led to re--

(1)

For sufficiently large substituents X(implying non-coplanarity of the two phenyl rings) there exist no vahtes of the internal rotation angles of the- two endgroups CABC leading to a nuclear coniiguration with nontrival covering symmetry. Consequently according to the recipe mentioned above these molecules should be chiral. However, Mislow has shown a specific example to-be opti&lly inactive. This fading initiated.him to search for more general Criteria for the chirality.of nonrigidmolecules r3,51. . In this paper we wish to show that Kelvin’s theorem may be generalized to,nonrigi.d molecules if an appropriate-.concept for the symmetry groups for-such systems is used. Sy$mktr& concepts for nonrigid molecules have Rrst been.developedb$Hou.gen’[7] and Longuef Higgins [8]._‘.Recently Bauder et al. [9] introduced-a new method for the description of th& symmetry of

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156

H. Fret, Hs.H. Giinthard/Isomctric groups a,zd chirality of nonrigidm~lecules

nonrigid molecules by. the isometric group. The isometric group concept Will be used in this work for a generalizatien of Kelvin’s theorem. In section 2 a brief recapitulation of the definitions will be given which define the isometric group. Then (section 3) a generalized definition of the chirality of nonrigid molecules and a sufficient and necessary symmetry criterion for chirality is presented. In section 4 the new criterion will be demonstrated by a few examples including MisIow’s molecule. Since emphasis in this work is put on nonrigid systerns, it appears difficult to establish a clear cut relation to Ruth’s theory of optical activity [IO], In a forthcoming paper the Rosenfeld-Born theory will be generalized to the case of nonrigid molecules again based on the isometric group concept.

this function is periodic insome (or all) of the internal coordinates, depending on their geometrical nature. If it is possibieto find substitutions of the internal coordinates .5’= F(t)

(2)

such that the two sets of distances &(U,

. - - , ~K_l,K(~N and I&F’(t)),

... ,

d,_,

K(F-1(.$))} are identical, we will call the substitution isometric. The interrelation between the two sets of distances may be considered as a mapping of the set of distances onto itself and may be expressed as a linear transformation:

= &&+=1(F), 2. The isometric group of nonrigid molecules The isometric group is based on the concept of the relative nuclear configuration of a molecular system with K nuclei [9], defined by a set of 3K-6 internal structural parameters El, t2, . . . , ijgKd, iike bond lengths, bond angles, etc. Molecules whose nuclear configurations form a continuous set described by finite variations of 1, 7_, . . . ,3K-6 internal structural parameters will be called nonrigid molecules, in contrast to quasirigid molecules, where the structural parameters vary only infinitesimally. For most nonrigid molecules only a few (1 ,2 ,3 , . . . ,4 3K-6) of the structural parameters vary over finite domains. Many aspects of molecular properties of nonrigid molecules withf= 1,2,3, _. . finite internal coordinates.may approximately be described by a semirigid model for which all other structural pirameters are assumed to be constant. The variable structural parameters wili be calied (dynamical) internal coordinates of the semirigid model, cf. sect. 4 for exsrnples. In general the internuclear distances of a nuclear configutition of a semirigid model will depend on the internal coordinates. i.e.,

By this equation the set of distances {d&$)} becomes : a function of the internal coordinates. In many cases -. .:.

(3)

where the operator jF is associated with thesubstitution (2). It is obvious that the set of isometric substitutions {E, F2, . ;--rXformsa group, the internal isometric groupY(t).).g (@is~d&ned as theabstract group $7, represented faithfully by either the set of substitutions (2). which often may be written as linear transformations (4) or the set of permutation matrices

rtne,19);

=

{r rWe)(E), rQ=)(F&

... )

(4’)

defined by eq. (3). It should be pointed out that nuclear configurations interrelated by isometric substitutions are either properly or improperly congruent. In all cases where the distances are periodic functions of some (or all) internal coordinates, eq. (1) defmes the primitive period of the distance set. The substitutions (4) are to be subject to the primitive periods of each internal coordinate. The coordinate vectors of the nuclear configuration are referred to a molecule futed system (frame system) and in general will be functions of the internal coordinates ,5_, i.e., X, =X,(f), k = 1,2, . . . ,K. Let.the coop dinate vectors be ordered-in a row and then apply the isometric substitutions (2). It-then may be shown that in thii way a representation i’O of the isometric

H. Frei. Hs.H. Gihhard/Isometric groups and chiraliry of nonrigid molecu~les

157

2.1. Fixed poirlts

group is generated *:

If the equation

= {Tk (5)) l-l(F) @P3’(F) ,

&’= A(F) 5 + u(F) = t (5)

where lT(J’) and rc3)(F) denote a K-dimensional permutation matrix and a properly or improperly orthogonal 3 by 3 matrix, respectively. Eq. (5J expresses the fact that the nuclear configuration PF{Xk($)} may be obtained from the original one by the proper or irnproper rotation r(‘)(F) up to permutation of identical nuclei. Besides internal symmetry nuclear configurations may have covering symmetry in the sense that nontrivial covering operations exist for arbitrarily chosen but fixed values of the internal coordinates. The set of such covering operations wiil be called the covering group ). If the covering operations are apg(g),)= &W2,-.plied to the nuclear coordinate vectors X,(.$), one finds & &(~)~

= rx,(#

B(G) @C(3)(E)

= r;i;,($)} B(E) 8 l+(C)

(6)

In eq. (6) B(G) and I’c3)(G) denote K dimensional permutation matrices and 3-dimensional (proper or irnproper) rotation matrices, respectively. Eq. (6) expresses the fact that covering operations imply permutations of equivalent nuclei and their coordinate vectors. Each of the proper sets rc3) {‘IX},rt3) (($1 of matrices, defined by

rt3)fxj: = {r(3)(F);r(3)(~) E r-ma{9-j] rc3){Q):= {r(3)(c);r(3)(c)E rRJQgjq} defines an orthogonal transformation group which must be isomorphic to one of.the point symmetry groups. E(3) {‘X) should be interpreted as the group of (properly or improperly) orthogonal transformations which map the reference NC onto NC’s, which are generated from it by isometric substitutions. The complete isometric group 5%!(g)of a.nonrigid molecule is generated by the two groups 9(l) and g(E) represented by the two representations (5) and (6).

perceptualway to contiuct the.matricesrmc)(F) is illustrated in the appendix.

* A

has a solution &., it defines a fxed point of the isometric substitution (4). It may be shown that then the rotational part T’t3)(F) in eq. (5) is an additional covering. symmetry which is not contained in the coveming group S(t) of the nuclear configuration NC{X,,Z,,M,} [9] .’ Typical examples will be given below. Such nuclear con_ figurations will be called fiied point configurations; they play an important role in the conventional criteria for optical activity of nonrigid molecules. In Mislow’s terminology they are called symmetric conformations.

3. Chiratity of nonrigid molecules For a definition of the chirality of nonrigid molecules we start from Kelvin’s criterion of rigid molecules: Molecules with rigid nuclear configuration are chiral if the covering group of the nuclear configuration does not contain a covering operation of the second kind L&111. Now we put forward the definition: A nonrigid molecule approximated by a semirigid model with finite internal coordinates will be called chiral if both conditions (i) and (ii) are fulfdled: (i) no NC in the set may be mapped onto its mirror image by translations and rotations. (ii) no NC in the set may be transformed into its mirror image by isometric substitutions. This definition is consistent with the conventional defmition for rigid molecules. It forms a necessary and sufficient condition for optical activity of nonrigid molecules. Basing on this detiiition a necessary and sufficient criterion for chirality (optical activity) of nonrigid molecules may be formulated as follows (generalization of Kelvin’s theorem): A nonrigid molecule is chiral if the rotational part I’(31of the representations II(F) @I’r3’(F) and B(L?)@ rc3)(G) does not contain operations of the second kind. Before giving examples some statements seem to be in order: (i) if rt3)(G) contains operations of the second kind, then the set of NC’s contains to any nuclear configuration a properly congruent mirror image.

158’.

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‘H. FM, Hs.H. Giinthordfhometric groups ond chirality of &u&id molecules

..-. ! .$)-if a r&$onal ~perat&~(~)@‘) is of the second .kind and if the corr&po&.ling isometric substituiion : has a ftied point tF, then the nuclear configuration NC {X&J) has covering symmetry of the second kind and.therefore is achiral. As stated above the fRed point configurations represent the~symmetric cotifolmations. The sortlng.&t of symmetric confo&ations, i.e., of fxed poht NC’s with.covering symmetry of the second .kind forms the basis of the conventional procedure to ch&cterize the achirality of n&rigid molecules. Thus, this conventional procedure is included in and rationalized~by the isometric group concept. (iii) if the isometric group {Ii(F) 63r’3’(F)) contains improper. rotational parts, which however are not associated with isdmetric substitutions with fixed points, the ConventiOnal approach fails. Mislow’s molecules(i) represent typical examples for this case which will be further illustrated in the next section. Occurrence of improper rotational parts violates the second condition for chirajity. Isometric substitutions of this type gener.ate to.any NC contained in the continuous set of nuclear configurations.of a semirigid system an improperly congruent NC, Le., with any NC the manifold of NC’s also contains its improperly congruent mapping.

z[, , f “’

Fig. 1. Newman projection of D,bF(C~TR)2 system (type-l molecule), -x/2 4 T < + n/i/z. Key: The origin of the frame ccordinate system ef is situated in the center of the C-C bond; the origin of the local top coordinate system et lies in the nucleus CO.

(S,S)(RS)).

Molecules belonging to this type of systims are symbolized by figs. 1 and 2. (2) Molecules of the Mislow type (I), furtheron denoted by D2(7)F(CC1TR)2and DZ(r)F(CITR)(CITS) (type-2 systems). The latter symbol is used to describe a semirigid system consisting of a semirigid frame with covering symmetry D,, to which two rigid tops of local symmetry C, are attached. The latter two subsystems may either be properly or improperly congruent. A molecule represented by this system is scetched in figs. 4a and b (Mislow-type molecules).

4. Examples

4.1. Type-l molecules

In this section we wish to illustrate the statements made in section 3 by typical examples of semirigid sys‘terns: (1) Molecules of the type CHFCI-CHFCl, furtheron denoted by D_,F(C,TW~TS) and D_hF(C,TR), -(hype-I systems).. This symbolism indicates a system consisting of a rigid frame of symmetry D,,, to which respectively two improperly congruent and two properIy congruent tops of local symmetry C, are attached. By the phrase congruent ticj improperly congruent the fact should be expressed, that the partial NC’s CHFCi repr,e$e&ng the two tops may be mapped onto each other by an operation.of the group P(3j *. If the roti+onaI part of this mapping is properly (impioperiy) orthogonal, the system is to be classified as (R,R) or 1 ;

The isometric groups are obtained in a straightforward manner from the nuclear coordinate vectors given in table I, which refer to the frame system shown in fig. 1. Tabies 2 and 3 g@gthe representations I”3Lc) {YJ, rcnz)(S}, d{7), PC’{7), I@‘){g? of the isometric groups of the two cases D_hF(C1TR)2 and DWhF(C,TR)(C,TS) symbolized by fig. 1 arid fig. 2,

t

*J?‘(3)

denotes.ihe gioup of inhomogeneous isometric tran!+ ix& 3; ia, ‘he set of m&ices ifi‘_$b where-l; and ~_.t stahd foi a properly or imprqerl? orihqonal matr,k and a +slationvect& tE?Z& respectively..-‘.

i:. fo&Gio~

Fig.2. Newman projection of D;hF(ClTR)(61TI?i)

system, -7712 c ; < + n/2. Key: Two isometric ni~clear confmrationi

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H. Frei.HsH. Gijnthardllsonietr~~ric’groccps and chirality of nonrigid molecules

Table 1 Coordinate vectors of equivalent nuclei of D-h FtCrTR)a and D,hFKsTR)KsTS)

159 _.

systems

vector ”

Set frame

;h = 0,l

DmhF(C1W2 top x

system

A =O,l DmhF(CITR)(CLTS)

system a Coordinates refer to the frame coordinate system ef indicated in figs. 1 and 2; the structural parameter r denotes the C-C bond length. b Coordinate vector of rep!esentative nucleus w&t. tocal top coordinate system et indiezted in fig. 1.

respectively. The two groups may be commented upon as follows: In both cases the isometric groups are isomorphic to the 2group V2_ For the (R,R) system it is realized by the covering operation C*(ef) and there exists no nontrivial isometric substitution. The rotational part of this operation is properly orthogonal, therefore the system is chiral. In the case of the (R,S) system there exists no nontrivial covering operation and the only nontrivial internal isometric substitution may be chosen either as

Table 2 isome.tric group of the D,hF(C,TR)a system (CHFCI-CHFCI) ~Wb?l2,%r)~C,

(i) 7’ = -T with the primitive domain -n/2 < r < + G, (ii) 7’ = n---7 with the primitive domain 0 G r < P. The two choices lead to the two forms of I’tNC){9) given in table 3. The rotational part of both of these contains improper operations, hence the system is achiml. Furthermore, the two choices(i) and {ii) imply fmed points as follows: (i) 7’ = r = 0, corresponding to the symmetry plane Si2 as a covering symmetry (cf. section 2.1). In the Mislow sence this is a symmetric conformation witt covering symmetry of the second kind. (ii) T’ = r = n/2, correspondiig to a center of symmetry Z. This again represents a symmetric confonation with an improper symmetry element. Both fared point miclear configurations are visualized in fig. 3. 4.2. Type-2 molecules

a Representationgeneratedby the vectors %!to(~1.-?t, (7) of a set of equivalentnuclei originating from the two equivalent tops.(seefig. 1).

b Representation by the set of distances dcb,~*(r), d~,He(r)

1).

originating from the.two equivalent tops (see fii.

In table 4 coordinate vectors of representatives of typical sets of frame and top nuclei are collected, referred to the frame coordinate system indicated in figs. 4a; 4b. The results of the constr&tion of isometric groups are collected in table 5 (R,S) and 6 (RR). only the groups resulting from the choice -a/2 f r G.+ $2 are given. -1tmay be shown, that the choice 0 d T <:n. leads to an outer automorphism of thegroups giiren. .. 1. The following comments should be made. .. .’ ., ;: .: ... _’ : -. .. . :

laole,. ..

‘.

Isometric &otip of the D

a Represeittation ofTby substitutions of the internal coordinate: 1; ] = 1°F’ “‘F ‘1 [;] _ b Representation generated by the vectors ft,,(r), .f?tr(~) of a set of equivalent nuclei originating from the two equivalent tops (see t-ii. 2). ’ Representation by the set of distancest.Q,~, (r), dcl,,~,,(T) originating from the two equivalent tops (see fig. 2).

4.2.1. D,(7)F(C,TR)(C,

Is) system

(i)Qf7,uo,u;)=e,: (ii) For the internal isometric group the following group relations hold (59, of the index n):

Fig. 3. Fked point nuclear configurations of D,hF(CrTR)(CrTS) system.

denotes the dihedral group

where the group S T6 is a group of order 16 with the following structure: (a) generators: C and U, C4=r+=~, c%J= uc3, T = (CU)2 commutes with both C and I/; (b) subgrpups of index 2 Ik = II (mod 4)] :

Table 4 Coordinate wctors of equivalent nuclei of Dz(~)F(C,TR)~ and D2(r)F(CrTR)(CrTS) systems 1 set

" COpr#flh2S

Vector =

r&fer.to the frXW Coordiiatc syste; gf indicated in fig. 4;~.ti the.distance between the c&on nuclei of the two MS& fig. 4a). .; .,

I stituted rnethii &up(see

17. Frei, Hs.H. Gibtthord/lsometricgroups and chimlity of nonrigid molecules

Table 5 isometric group of theD&)F(CITR)(C1TS) system.G$r, vo,v& g &,~(~,uwI)

lperator

i -i I... iI F ii ...1n i I I i _i .I -: .-i*:. i i i -i ‘t I i i i i 22 I

9 16

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‘I . i .@-l..,..i II.

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a Representation of 7 by substitutions of the internal coordinates:

.

.

1 1

.-1 .

..@

l.i

II

.

1

..I 1

‘:

~[#‘4;.yl

-n/2 < 7 < + d2

1 ;

i

ljq:[j

--a
dt,,f&q,r).

dtl,fol(q,r)

(see

c Representation generated by the v&tors~t&,,r), &(ul,r), f&& $fe1(7), .?fi&), .?fl,(r) of two sets of equfyalesitnuclei :’ or,iginatingfrom the two.equivalent tops and the frame, respectively (see $43. .,. _.

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H. F?ei,es H. Giinthnrdikometric groupiacd chira@y of nonrigid molecliles.

-fable6 ._. .~ ‘. : Isotietric groupof the DT(T)FKITR)z system$j(r,vo,&) ~el,%,vo,vl)

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L a Representa:iori of 9 by substitutions of the internal coordinates: -nS b

’ [yflg,

-n/ZCr<+n/Z

d;l];[;:]~

V&U, <+i

Representationby permutationsgeneratedby the set of distancesdto,f~o(uo,7Ldfb,f&o,r), : .-.

49).

dtl;fooCu,,z),dtl,faI(u,,+)

(see f%

’ Repres;ntation'~neratedbL thevectors%~o(uo,s),~~l(ul,~),~~eO(r),~~Ol(~)r~~;IO~~),~~,i(~) of ~WO’&S ofequi&entnucIei. ,.: ~r@din~ from the two equivalent iops and the frame, respectively (see fii 4a). : -.‘.-. .I.-

: .::-

.1..

1

17, Frei, f&H. C~nthardlisometricgro;ps

{Ck,TCk]%&, _ UCT,cu,CW @EC23zC,,

{E,C2,C2T,C2u) c &2,2,2) g 9$ eq?, (c) center bf $?I~: {E, T, C2 TC2} 1 V 49. (d) cokutator group: [G161 = ETC21; (e) important homomorohisms:

$l&ti

%J4,

9&TC2~

and chiraiity

of nom&id mblecules

g D2,.,;

-

a.=,,

c&6/{E,TC2}%4hz e4 eq,. (iii) The rotational parts of the group I’cNc){5?3) form a .group: _ r(3) {31) !z CICI c,,. Since this group is improperly orthogonal the molecule is achiral. As may be derived from table 5, the isometric substitution group SQ{Y) has two (nontrivial) fixed points, namely Fs:

q)=-ul,

7=0,

with covering symmetry S:2, and F7:

uO=n-vl,

T=O,

with covering symmetry 2, i.e. , the symmetric conformations have improper symmetry. Tables 5 and 6 show the remarkable facithat each element of SQe} is represented by two elements of I’ma (7). This phenomenon is related to the topology of the internal coordinate space, since the domain --n/2 < T < + n/2 has to be considered as toroidal. A more detailed discussion of it will be given elsewhere. 4.2.1.1. &$(CI TR)(CI TS) system - An interesting special case arises if the angle T of the foregoing system is frozen to the value 7 = x/4. This value is a fixed point of the D,(T)F subsystem, leading to a frame of D,, covering symmetry. Thii case may be realized for molecules of the type shown in fig. 4 by introducing bulky substituents in positions o,o’ of the biphenyl group. It may be considered as a prototype of Mislow’s molecule. For this special case the isometric group is shown in table 7, which may be commented upon as follows *: * It should be pointed out that these results hold for any DzdF(CrTR)(CrTS) system, irrkspective of the chemical na- : ture of its rigid parts, e.& the groupsCHFCiare repIaceable by arbitrary iigid groups of lo&Cl-symmetry.

Fig. 4. D2(r)F(CLTR)(CITS) system (Mislow-type molecules). (a) Pictorial view; (b) Newman projection (symbolic), -n/2 4 T < + 42, --H 4 vo,u, < + 77.Key: The origin of the local top axxdinate system et is situated> the nucIe~s Co; the origii of the local coordinate system ef coincides with the one bf the frame system ef.

(i) S(v,,vl).=e1,3(v,,u,)~aC~$e,. -. (ii) r(3){cK) g S,, hence the system is achhai. Hcwever, in this case there exist no fmed points of the group SQ(9). As a consequence there exist no NC’s with improper covering symmetry. This fact formed the background for the use of molecules of this type by Mislow to demonstrate that the absence of “syminetriq’f conformations tith second kind covering operations is a necessary but not a suffi&nt condition for dpticaLactivity of no&id molecules. 4.2.2. D2(r)F(Cl TkJ2 system This case follows closely the treaiment given for the.. (R,S) system.. R e1evtit isometric group r&.&s .are given. ...

-.

.:: -.

..: rable-7

‘-,

~.

- -. ..

H. krci: Hs.H.Gii&h~rd}Isomeiric groupsand~hirglity .ofnon&i mohki& I

lsonietricgroup of the &~F(C~TR)(CITS) EE!,,?(“&) %!,

systekg

(tio,uJ

,(NC),,, b

.c

1

“I I 1.-l.. 8 .:

.

1

.'

..I

.

1.

'.

.-1 @

1.

. -1 I

'.I.

[: i]@

.

1..

I’.-; . L:]

:1+]

.[;

.l.

.

.

.-1

coordi-

bi !epresent&ion

generated by the vector .%?t&o), ztl(u1) of nuclei originating from the twoequiva11 ent tops (see fii. 4~3).

aset of equivalent

in table 6. Comments may be made as follows:

$&, denotes the group characterized in section 4.2.1, (iii) the rotational part f‘c3){Cx) of I’ya {c(, obeys: $ D4.

The Iatter group contains only properly orthogonal

elements, therefore the molecule should be chiral. From table 6 fixed points of the groups2 {% are derived as follows: F5: vo=ul, ..r E (-n/2, . _. ) k/2) arbitrary, F,; u. = u1 + K, r arbitrary. The coveiing symmetries of the’NC’s correspoading to the iwo fried $oints are tht rotations C$,ei) and Cz(eP, respxiively. :

. . 5. Conchrsioti

dees the cover&g.g;oup for the, case of rigid molecules. 1 Thii fact showsagain that the isometric group of nonrigid molecules may be consideied g the nature! generaL&ion of the covering symmetrygroup of rigid mclecules to no&gid systems. Second it may be pointedout that in the isometric group approach fsed poi+ NC’s are a consequence of the semirigid model chosen to represent a nonrigid molecule. However, they should not be considered essen: tial for the chirality criterion; much more important is the occurrence of improperly orthogonal operations in the group rt3) {‘K }.

Acknowledgement

.

xpresentation of LTby sub:jti ,iutions of the internal

j’(3) {3(} 2 9,

:

We wish to thank the Swiss National Foundation (Projects Nr. 2.110-0.74,2,808:73) for fmancial support of this work.

Appendix: Construction of the representation I@0 (7) by mappings With each isometric_substitution E’= F(t) we associate a linear operator PF with the property FF rQu>

= c~;kcF-‘(E))> = @&)PC)(F) = {Z#)

rt(~) 0

r(3)(fy.

(A-1)

With this operator jF there may be associated a mapping (x;) = II(F) @l+)(F>(X,),

(A-2)

by which &matrices II(F) and rc3)(F) can be determined by geometrical reasoning. To demonstrate this, we shall determine the representation @Q(F) o f one of the isometric substitutions of a type-2 system. Fig. 5 shows a reference NC A and an NC B, obtained from NC A by the isometric transformation Fx

._

: - (A-3). I& it .ihohld.be stated &at the groti& rt3) {‘X}, which originate from. the repiesentation l?(Nc) ET}, al- : common to both the l$?)F(C,TR)(CiTS) rindthe wayi &g isomorphic to one of the wellknown point ; ,Di(?)F(C,TF); system (cf..tables 4,5 and 6).. It is .; sytimetry‘groups: The g!oUp rt3) {X.$ays.tbe:same. easily shown by fig.-5 .that ry3)iF3), the operaticn- _. .&for the chi;ality.cri~eerion,o~nbnrigid molecules & ~ : .._‘. _._. ‘.L :.I ;..‘. ‘y :. ‘:~~.::./‘~~:~~,:..I.... ‘..I ,- _;-: -.I .. ;_.:._ y. _._.‘; ._.., _. 1 _I. : ;.-y:. -._. :_. ., , 1:. _ .: (_ -. ,. : _,.‘. :; :;;_‘-.,. i -, _:_: ,: i., .. -.-. -.-.1._= ‘.._’ :- _ ,_:.,-. ,, ‘;_’ : .: ‘j ,. ., ,_.~I-. .~‘-‘..::_.“_---...__,-.:,__;.:.::_ .-i .,_,.:.7...- .. :-.._. : ..,. :; :_’,:: _-.,I: ., r .,_:._ ;. :;..:_,’..:: .. /. .f :,(_ . : _,.:_; --. . : i- _ ::. ., -~_.,.r.-.: -., .____‘.....;‘, -,

H. Frei, Hs.H. GiinthnrdfI~ometricgroups and chirality of nonrigid molecules’

165.. ,.

By construct‘ing the representation T’(Nc){‘$y in this wai, a molecular model may be helpfuL However; in moxe complicated examples, determination of.r(NC)(7) by (A-2) will become difficult. Then direct solution of eq. (A-l) may be preferable.

References

lid A

El

Fig. 5. Isometric nuclear configurations of a type-2 system. Key: NC B is obtained from the isometric NC A by the isometric transormation F3.

[l] Lord Kelvin, Baltimore Lectures, 1884 [published in: Baltimore Lectures (C.J. Clay znd Sons, London 1904)

pp.,436,619]. 121 L.L. Whytc, Nature (London) 180 (1957) 513; 182 (1958) 198. [3] K. Misiow, Science 120 (1954) 232. [4] K. Mislow and R. B&tad, J. Am. Chem. Sot 77 (1955) 6712: [SJ K. Mtslow,Trans. N.Y. Acad. Sci. 19 (1957) 298. [6j K. Misiow, Introduction to Stereochemistry (Benjamin. New York 1966) p. 93. [7] J.T. Hougen, J. Chem. Phys 37 (1962) 1433. IS] H.C. Longuet-Higgii, MoL Phys 6 (1963) 445. [9] A. Bauder, R. Meyer and Hs.H. Giinthard, Mol. Phys. 28 (1974) 1305; for a detaited discussion the reader is referred to this paper. [lo] C.A. Mead, Symmetry and Chirality, in: Topis in Current Chemistry, Vol. 49, ed. F. Boschke (Springer, Berlin, 1974). [ 111 R. McWeeny, Symmetry (Pergamon Press, Oxford, 1963).

which maps the reference NC A onto the transformed

NC B, is a rotation about thee: axis through a:

c -! ; J -1

r’3’(F3)=

.

i :

.I

1

The permutation matrix lI(F,) may be obtained by identifying the mapped position vectors X6, Xx, (NC B) with the vectors rc3)(F3) Xtx, f’(3)(F3)Xh, respectively, i.e., with the position vectors Xn,Xh of the NC A, transformed by I’(3)(F3):

(A-5)

., .

_‘.

..

.‘_.

_:...

_.:. I,

:

:

,..

.:

: -. _.-

.,

..

y-

:.-

:

:

.,..

G

-\

._:....