Spectrochimlca Aera. Vol. 49A. No. 13114, pp. 1935-1946. 1993Printed in Great Britain

0584-8539193 $6.00+0.00© 1993PergamonPressLtd

Non-rigid symmetry groups of molecular trimers and three-rotormolecules

PETER GRONER

Department of Chemistry and Biochemistry, University of South Carolina, Columbia, SC 29208, U.S ,A.

(Received 13 September 1991; accepted 27 September 1991)

Abstract-A simple geometrical model is used to derive symmetry operators and symmetry groups formolecules with three internal rotors and for homomolecular and heteromolecular trimers. Operators aredefined in terms of isometric substitutions and permutation-inversion operations. The non-rigid molecularsymmetry groups were derived explicitly for molecules in which any of three internal rotors has one-, two-, orthreefold periodicity. These groups are applicable also to molecular trimers in which none of the monomerunits has molecular symmetry other than Cn' C" or enu (n"'i>3). The symmetry groups are characterized assemi direct products and, where possible, by factorization into direct products. Among those which cannot befactored are two whose character tables are given, two which are already known, and a few for whichno simpleexamples exist. Symmetry groups of molecules in which internal motions are restricted to concerted gearing orantigearing motions are derived for some examples but not investigated systematically.

INTRODUCfION

DURING the last few years, many homomolecular and heteromolecular trimers have beenproduced in molecular beams and investigated by various spectroscopic techniques [1­10]. Some of these molecules are quite floppy and have several equivalent potentialminima in often shallow potential surfaces. Spectra of such molecules may be extremelycomplex. The use of group theoretical methods facilitates the assignment of highresolution spectra because qualitative splitting patterns and spin statistical weights can bepredicted if the appropriate molecular symmetry group is known. The order of suchgroups may increase rapidly with the addition of monomer units particularly if amonomer has symmetry on its own. Molecular symmetry groups of molecular trimershave been investigated only for highly symmetric molecules like (H20 )3[11, 12], (NHJ)3[13] and (CH4)3 [13]. Related to the symmetry groups of molecular trimers are thenon-rigid symmetry groups of molecules with three internal rotors. A few such groupshave been analyzed, among them the group for B(CH3)3 in the fundamental paper byLONGUET-HIGOINS [14] and for HC(CH3)3 [15J.

No systematic analysis has been made of symmetry groups of molecular trimers ormolecules with three internal rotors; however, such investigations have been made formolecular dimers [11] and molecules with two internal rotors [16-18]. It may beconcluded from the results that many molecular dimers have molecular symmetry groupsisomorphic or, in terms of permutation-inversion (PI) operations, even identical togroups found for molecules with two internal rotors. In many cases, identical PI groupswere obtained for completely different physical models [19].

The current experimental and theoretical interest in homomolecular and heteromole­cular trimers justifies a systematic analysis of potential molecular symmetry groups forsuch molecules. General methods for such an undertaking have been described [11,20]but not actually carried out for trimers or general three-rotor molecules. This paperpresents the results of such an investigation for a class of these molecules. The methoddescribed here can be used to generate all possible symmetry groups of molecules withthree equivalent or non-equivalent internal rotors. The method applies equally well toany homomolecular or heteromolecular trimer provided that the molecular symmetrygroup of any monomer unit is only Cn, C" Cn• (n";;3). For example, trimers withnon-inverting NH3 monomers are included whereas trimers with CH4 or inverting NH3

are not.

1935

1936 PETER GRONER

z

Fig. 1. Geometric model for a molecule with three internal rotors. The origin of the moleculefixed Cartesian axes system ex, y, z) is at the position of nucleus C. Aois in the x-z plane.jf, 1"0, 1"\

and 1"2are the large amplitude coordinates.

RESULTS

Geometric model

The non-rigid molecular symmetry groups of some molecular trimers and of moleculeswith three internal rotors were derived from a simple geometric model (Fig. 1). Theorigin of the molecule-fixed axes system is at the central nucleus C of a simple moleculewith three internal rotors. The jth internal rotor consists of the anchor nucleus Aj and aset of nfequivalent nuclei Bfk(j=0, 1, 2; k =0, 1, ... nf- 1). The coordinates of Af in themolecule-fixed axes system are given by

Xf=D(j2n/3,~, 0) Rf (1)

where the rotation matrix D(Q?, ~, X) is defined as

D(Q?'~'X)=(:~:: ~:~: ~)(Co~~ ~ Si~~)(:~:: ~:~nxx ~1).(2)o 0 1 -sin ~ 0 cos ~ 0 0

The only non-zero component of Rf, the third, is equal to the CAf distance, Rf. Thecoordinates of the nuclei Bfk in the molecule-fixed axes system are defined by

x;=D(j2n/3, o, O)[Rj+D(rf+ kpj' pj, O)rj]. (3)

The variable if is the coordinate of internal rotation of the jth internal rotor and

pf=2nlnf. (4)

The polar angles Pf (angles between the directions of CAf and AfBfk) are constant. Onlythe third component of the vector rj is different from zero. It is equal to the constantdistance from Af to Bfb if. The polar angle ~ in Eqns (1) and (3) may also be variable toallow for the inversion at the central atom C. The position vectors in an axes systemparallel to the space fixed axes are defined as

Xj= D(a,p, y)Xj ,

Xjk=D(a,p,y)Xfb (5)

where a, pand yare the Eulerian angles. For model calculations, the origin of the axessystem needs to be shifted to the molecular center of mass. This shift is not necessary forthe investigation of just the symmetry properties.

Non-rigid symmetry groups 1937

The molecular symmetry groups will be defined for a number of different models. Aparticular model will be referred to as [no nl n2] G where G is the symbol for a point groupto be defined later. Each nj may assume any positive integer value.

Symmetry operations

The symmetry operations and symmetry groups can be worked out in terms of the PIoperations of Cartesian coordinates in the space-fixed axes system [14, 21]. If one wishesto generate systematically all possible symmetry groups of three-rotor molecules in whicheach rotor can have one-, two-, or threefold or even n-fold periodicity, the methodsdeveloped for the isometric group approach [22]seem to be more effective (at least in theeyes of the author). However, the selection of the method does not matter because aone-to-one correspondence of symmetry operations between these methods has beenestablished at least in the absence of primitive period transformations [23]. In the presentwork, the symmetry operations will be defined in terms of isometric substitutions but theequivalent PI operations will also be defined.

For further reference, a few definitions and selected results of the isometric groupapproach are summarized here. An internal isometric substitution is defined as thesubstitution of internal coordinates which preserves, apart from permutations, the set ofinternuclear distances between identical pairs of nuclei. It is written as an inhomo­geneous substitution [22]

(6)

where ~ is the vector of the variable internal coordinates. This equation is oftenabbreviated symbolically as

~'=F~.

The associated function operator PF is defined according to WIGNER [24]

PF!(~') == PF!(F~) =f(~)

from which one obtains, by replacing ~ with F-1t

The "multiplication law" for two successive operators holds [24]

(7)

(8)

(9)

A function operator PF associated with the substitution F may be applied to the positionvector expressed in the molecule-fixed axes system with the result [22]

(11)

The subscripts nand n' which mayor may not be equal refer to labels of identical nuclei.1'3)( F) is a 3 x 3 orthogonal matrix representing a proper or improper rotation. For aparticular operator PF, the same matrix f(3)( F) is obtained regardless of the positionvector Xn(s) to which PF is applied. This matrix indicates a transformation of the

1938 PETER GRONER

molecular axes induced by PF• The associated transformation of functions of the Eulerianangles can be written as [23]

PpD(a, {3, y) = D(a,{3, y)R(F),

R(F) =r(3)(F)lr(3)(F) I. (12)

The determinant If(3)( F) Iis +1 or -1 if 1"3)(F) represents a proper or improper rotation,respectively. Applying PF now to position vectors in the space-fixed axes system, oneobtains [23]

PFX~(~, a, {3 , y) =PpD(a, {3, y)Xn(g) =D(a, {3, y)R(F)t(3)(F)Xn'(~)

=If(3)(F)IX~.(g,a,{3,y). (13)

This equation establishes the connection between the isometric substitution F with itsassociated function operator PF and a PI operation because PF applied to X~ results in apermuted vector X~, which is multiplied by +1 or -1.

The isometric substitutions were derived for the geometrical model described in theprevious section by the alternative method from the molecule-fixed position vectors [19].The substitution matrices f(F) for the operations F= Co, Ch C2, T, S, TS and Paretabulated in Table 1 together with the results of the application of the function operatorsr, to the position vectors in the molecule-fixed and space-fixed axes systems. Also listedare the conditions under which a given substitution is an allowed transformation.However, it is always assumed that 1'0, 1'1 and 1'2 are finite large amplitude internalrotation coordinates, each with the domain [0, 27r].

The PI operations equivalent to the substitutions listed in Table 1 are tabulated inTable 2 for all models [no nl n2]G with 1:G;nj:G;3, j=O, 1, 2. In order to simplify thenotation for the PI operations, the nuclei A o• Boo, BOh B02, AI> B IO, B II , B 12, A 2, B10, Bl 1

and B22 are identified by the symbols a, la, 2a, 3., b, 1b , 2b, 3b, C, Ie, 2e, and 3e,respectively. The PI operations corresponding to Co, CI and C2 are always treated asfeasible operations. The PI operations equivalent to T are feasible if the inversion at thecentral atom C is over a sufficiently low barrier or if the four nuclei C, A o, AI and A z arepermanently coplanar. The PI operations corresponding to S are feasible if the internalrotors 1 and 2 are equivalent (e.g. nl=n2, A I=A2, Blk=B2b {31={3z, .. .). The PIoperations equivalent to TS are feasible if the internal rotors 1 and 2 are equivalent andthe inversion barrier at the central atom is sufficiently low or the four nuclei C, AD, Aland A 2 are coplanar. The PI operation belonging to the substitution P are feasible if allthree internal rotors are equivalent (e.g . no = nl = n2, A o= A 1=A 2, ...).

Molecular symmetry group

The isometric substitutions listed in Table 1 or their equivalent PI operations listed inTable 2 can be considered to be generators of non-rigid molecular symmetry groups.However , T, Sand TS are not independent because TS = T· S = S . T. The properties ofthe group generators are as follows:

E= C30= ql = qz= T2=S2= (TS)2=p3,

CqP= PCq+ 1> CqS=SC=~, CqT= TC~t, CqC,= C,Cq,q, r= 0,1,2 (mod 3)

TP=PT, TS=ST, Sp=p-1S. (14)

It should be emphasized again that the operation P is allowed (feasible) only if all threeinternal rotors or monomer units are equivalent. Similarly, S or TS are allowed only ifthe internal rotors or monomers with the labels 1 and 2 are equivalent. Many differentgroups can be constructed from these generators. The notation for these groups is thesame as the notation for the models, that is [no n\ 1l1)G. The point group G is the groupgenerated by the matrices f(3)( F) associated with the allowed or feasible operations T, S,TS, or P. The point group symbols Cn Cl and C20 imply that the internal rotors ormonomer units belonging to nj and n2 are equivalent whereas the symbols C3, C3" , C3u,

F

Non-rigid symmetry groups

Table 1. Isometric substitutions for molecules with three internal rotors"

reF)

1939

-D(.7!', 0, O)XI

-D(D, n, O)X_ j

D(.7!', n, O)X_ j

D(2Jr/3, 0, D)Xj _ l •k

-D(n, 0, O)Xj . _ k

-D(D, n, D)X_I. _k

D(n,.7!',O)X_I.k

D(2Jr/3, 0, O)X/_l•k

X'I

X'I

-Xi

-Xi-I

Xi.H/I

Xi.k-6j2

-Xi.-k

-Xj-I.k

• The subscripts j and k are always taken mod 3 and mod nl' respectively.t {}o; large amplitude coordinate or {}=nl2 (constant).:j: nl = nz, AI = A z, BIk = BZk' R1= Rz, PI = f3z, 'I = 'z.§ no=nl = nz, Ao=A j =Az, BOk= B1k= B2k , Ro= R1= Rz, Po= f31 =f3z, '0='1 = '2'

D3 and D3h imply that all three internal rotors or monomer units are equivalent. On theother hand, the point group symbols C; and C\ imply that none of the rotors ormonomers are equivalent even if nl =n2 or no =n\ ::::: n2'

The group [nonl n2]C1constructed from the generators Co, CI and C2 is isomorphic tothe direct product of three cyclic groups of order no, nil and n2' It is easy to show that thisgroup is an invariant subgroup of all groups [no nl n2]G which can be generated from[no nl n2]C\ and the generators T, S, TS and P. Because any combination of thesegenerators forms a group <§ isomorphic to the point group G, the general group [no n\n2]G can always be written in the form of a semidirect product

[no 11\n2]G= [nOn\ n2]C/\<§, (15)

The decomposition of many molecular symmetry groups into semi-direct products hasalready been recognized by WOODMAN [20] who used the terms "torsional subgroup"

1940 PETER GRONER

Table 2. Equivalence of isometric sustitutions and PI operations for models [no nl nz]G

[no nl nz]1St PI* [333] [233] [133] [322] [222] [122] [311] [211] [111] [321]

Co (1.2.3.) x x x x(1.2.) x x xE x x X

C1 (l b2b3b) x x x(l b2b) X X X xE x x X

Cz (1e2e3e) x x x(1e 2e) x x xE x x x x

T (2.3.)(2b3b)(2e3e)' x(2b3b)(2e3e)* x x(2.3.)' x x xE' x x x x

S (be)(2.3.)(1ble)(2b3e)(3b2e)' x(be)(lble)(2b3e)(3b2c)' x x(be)(2.3.)(lble)(2~e)· x(be)(lble)(2~e)' x x(be)(2.3.)(l ble)· x(be)(l ble)* x x

TS (be)(lble)(2b2e)(3b3C> x x x(be)(lb1c)(2b2e) x x x(be)(l ble) x x x

r (abe)(I.lbIe)(2.2b2e)(3.3b3e) X

(abe)(I.lblc)(2.2~c) x(abc)(I.lble) x

t IS, Isometric substitution.* PI, Permutation inversion operation.

and "frame subgroup" for the invariant subgroup and the group si, respectively. All thegroups [no nt n2]G (nj";;;3) are listed in Table 3 which can be constructed from thegenerators Co, CI> C2, T, S, TS and P, as well as their order and, if possible, theirfactorization into direct products. The script symbols denoting abstract groups aredefined as follows:

~"(A): cyclic group of order n, generating element A; A"=E~",m(A,B) =~"(A) x ~m(B)

~",m.p(A, B, C) =~"(A) x~m(B) x~p(C)

~",m.p.q(A, B, C, D)=~"(A)x~m(B)x~p(C)X~q(D)eLJ"(A, D): dihedral group of order 2n, generating elements A, D; A"=D2=E,

AD=DA- t

eLJ".m(A, B, D): generalized dihedral group of order 2nm, generating elements A, B,D; A"=Bm=D2=E, AB=BA, AD=DA- 1

, BD=DB- 1

eLJ",m,p(A, B, C, D): generalized dihedral group of order 2nmp, generating elements A,B, C, D; An==Bm=cP=D2=E, AB=BA, AC=CA, BC=CB, AD=DA- 1,

BD=DB-t, CD=DC- 1

.stl4(A , B, C, P): alternating group of order 12, isomorphic to point group T,A2=B2= C2=p3= E, AB=BA, AC= CA, BC= CB

9'4(A, B, C, P, S): symmetric group of order 24, isomorphic to point groups Td and 0,properties of A, B, C, P same as for .stl4, S2=E.

The groups ~4' ~8t' ~162' ~62' ~62' and ~324 cannot be defined as direct products ofsmaller groups. '9324 and its character table has been derived by LONGUET-HIGGINS [14]for trimethylboron, whereas ~f62 and its character table have been investigated twice[15]. Groups isomorphic to many of the other groups in Table 3 have been derived

Non-rigid symmetry groups 1941

Table 3. Molecularsymmetrygroupsof molecular trimers and three-rotor molecules

Group' Order Abstract group IGt Three rotor; Trimer§

[333]CI 27 '€",,3(c" C" C2) HC(CF,)(CD,)(CH,)(333IC; 54 !2Il,,3,3(Co• c, C2,n B(CF3)(CD3)(CH3)[333IC, 54 '€,(C,C~) !8!21l,,3(Co. CIC2• S) HC(CD,)(CH')2[3331C2 54 '€dCo• C,C2)!8!21l,(CIC!. TS)[3331Clv 108 !2Il3(C,C~. TS)!8!21l3,3(CO' CIC,.S) B(CD3)(CH,h[3331C, 81 '981

[333IC311 162 '9162[3331C3, 162 '9162 HC(CH3)3[333]D, 162 '9l.2[333JD3Il 324 '§'2' B(CH,),. N(CH,M (NH,),1[222]CI 8 ~2,2,2(CO' C" C2) Dv.[222]C; 16 '€2.2,2.2(CO• C" C2• T) CO2,D,O .H,O[222JC, 16 %(c")!8!21l.(SCoC,,S) D..[222)C2 16 ~2(Co)!8!21l,(TSCoC" TS) D..[222]C2• 32 '€dCo,n!8!21l.(SCOC2,S) COdH2O)2[222JC, 24 (~h(CoCIC.)!8~,(CIC" COC2• COCh P) T.[222IC,. 48 '€dCOC,C2, T)!8~,(C,C2' CoC2• COCh P)[2221C,u 48 '€2(COCtC.)!8ff,(C,C2,COC2• COCh P,S) O. HC(NO')3[2221D, 48 '€2(COCtC2)!8ff,(C,C2,COC2,CDC" p. TS) O.[222IDJh 96 %,2(COC,C2• T)!8ff.(C,C2• COC2, COCh p. S) B(CoHl), (H2O)3[l11]C, 1 "I, C,[1l1]C; 2 '€,<n C,[lll]C, 2 '€.(S) C,[1l1]C, 2 '€2(TS) C2[lllJC2, 4 '€2.ZCT. S) Clv B(OD)(OHh OF· (HF)2[1l1IC, 3 %(P) C,[lll]C,. 6 'fI.3,2(p, n C3Io[l11JC,. 6 !2Il,(P. S) C,.[1l1JD, 6 !2Il,(p. TS) D3

[1l1J0 3ll 12 ~2(T)!8(')J,(P. S) D,. B(OHh (HFh[233)C, 18 c~',3,2(Ch C2• Co)[233)C; 36 '€.(Co)!8!21ldC" C" T)[233)C, 36 'fI.',2(CIC~. Co)!8!21l,(C,C2• S) (N02)CH(CH,h[2331C2 36 'fI.,,2(C,C2• Co)®!2Il,(C,c!. TS)[233]C2, 72 'fI.2(CO)®!2Il,(C,C2,S)!8!21l,(C,c~. TS) H20. (NH,)~I[1331CI 9 'fI.',3(Ch C2)[133]C; 18 !2Il,,3(C,,C2,T)[133]C, 18 '€,(C,G1)!8(')J,(CIC2• S) HC(OH)(CH')2(133)C2 18 '€3(CtC2)!8!21l3(CtC~. TS)[133)C.. 36 !2Il,(C,C" S)®(')J,(C,q. TS)[322)C, 12 'fI.3,2.2( Co. c, C2) C6Io(322)C; 24 ,€2.2CCh C2)®!2Il,(Co• T) D611 NH,' H20. 0 2011[322]C, 24 '9" (CH3)CH(N02h[322]C2 24 '€,(Co) !8(')J.(TSC., TS)[322]C2• 48 ~3(CO. T)!8(')J,(TSC2, TS) (CH,)B(C.,H')2 NH3'(H20)~1[I221CI 4 '€2,2(Ch C2) O2[122]C; 8 '€2.,,2(c, C2• T) Ov.[122]C, 8 !2Il,(SC2• S) D,[122]C2 8 !2Il,(TSC2,TS) 0,[122]C2u 16 'fI.2(T) ®!2Il.(SC2, S) 0 .. HF'(H2Oh(321)CI 6 ~',2(Co, CI) C.[321IC; 12 '€,(C,)®(')J3(CO, T) O.(311)CI 3 'fI.,(Co) C,[31lIC; 6 'fI.',2(Co,T) C.[31lIC, 6 !2Il,(Co• S) 0,(31l1C2 6 (')J,(Co• TS) D3

[31lIC2• 12 '€2<n®(')J,(Co, S) D.[211)C, 2 'fI.2(CO) C2[211)C; 4 'fI.dco• T) O2[211]C, 4 '€2.2CCo, S) O2(211)C2 4 %,2(CO• TS) O2[211)C2u 8 '€2,dCo, T. S) Ov.

• In the notation [no nl n2]G where G is one of the point group symbols Cn C2 , or C2.. the internal rotors ormonomeric units belonging to nl and n2 are always equivalent. If the point group symbol G is written as C; theinternal rotors or monomer units belonging to nl and n2 are not equivalent even if n, =n2.

t Isomorphic point group.~ Examples of molecules with three internal rotors.§ Examples of molecular trimers. Only the highest possible symmetry is indicated. In the case of some high

barriers. a subgroup may be sufficient.~ Inversion motion at N is feasible.I Inversion at N of the NH, monomers is not feasible.

previously for other molecular models. For instance, the groups [133]G are isomorphicto groups obtained for molecules with two methyl groups [16, 17]. The group [122]Czuis

1942 PETER GRONER

d)

Fig. 2. The groups [no nl n2lG and their subgroups. (a) G = C2u, C2, C" C;, Cli (b) G= C;, c;the broken lines imply a permutation of the generators Co, Cl> C2 (see text); (c) G =D3h , D3, C3u ,

C3h , C3 , c.; C2 , C" C;, CI ; (d), (e), (f) valid for any n, n', n",

isomorphic to the molecular symmetry group obtained for the large amplitude motions ofNH2NH2 [14] and (H20 )2 [25]. The group 9J3, 3 is isomorphic to group GIS [26]. Thegroups [nnn]C3 and [nnn]D 3 are wreath product groups [27] whereas the groups [n'nn]C2are generalized wreath product groups [27].

Many of the symmery groups [no nl n2]~ are related to each other as groups andsubgroups. It can be easily shown that (the symbol c: means "is a subgroup of")

[no nl n2]G'c [nonl n2]G if G' c G,

[no n; n2] G c [no nl n2]G if 'f6n,j c 'f61l() and/or 'f6nj c: 'f6"1 and/or 'f6n:l c: 'f6n2•

If the point group symbol G is C; or CI the isomorphic groups

Is is[n'nn]G = [nn'n]G = [nnn']G

can be converted into each other by relabelling the generators Co, C. and C2appropria­tely. Therefore, if one implies such a permutation of labels,

[321]G c [233] G

is also valid for G =C; and G =Cl . Many of these group-subgroup relationships areillustrated in Fig. 2.

Character tables and the matrices of degenerate irreducible representations belongingto the group generators were derived for all groups listed in Table 3 by the method ofinduced representations [281because all groups [no nl n2]G are semidirect products. Thecharacter tables for [322]Cs ~ ~24 and [333]C; ~ 9J 3•3•3 are reproduced in Tables 4 and 5,respectively. In these tables, the notation for the irreducible representations consists ofthe symbol for an orbit and the symbol of an irreducible representation of the associatedlittle cogroup [20,28]. The basis functions exp(i(ao't'0+at't"1+a2'r2» and its conjugatepartners transform like one of the irreducible representations labeled by the orbit {ao alaJ. A similar notation has been used by BALASUBRAMANIAN [27]. The character tablesof the new groups ~8h ~162 and ~t62 are not reproduced here because it is difficult toimagine molecular examples requiring these groups.

DISCUSSION

The geometric model used to derive molecular symmetry groups may represent areasonable physical description of the internal motions in molecules with three internal

Non-rigid symmetry groups 1943

rotors. There is no doubt that the same model is entirely inadequate to describe internalmotions in molecular trimers. In this case, the model was used merely as a tool togenerate symmetry (PI) operations and symmetry groups quickly and systematicallybecause for molecular trimers, internal motions corresponding to feasible PI operationsmay not be expressible in terms of a few simple internal coordinates. The fact that thesymmetry group for (H 20h obtained by the simple model, [222]D3h ~ ~2~@?4' agreeswith the group from the direct approach with PI operators [11]is evidence for the validityof this procedure.

The method is applicable directly to systems where the orders of the generators Cj , nj ,

are larger than three. The relations between all group generators are still given byEqn (14); however, the groups will have more classes and irreducible representations. Asomewhat different approach is necessary only if a monomer unit has an effectivesymmetry larger than Cnv' BALASUBRAMANIAN [27] has shown that many non-rigidmolecular symmetry groups can be formulated in terms of generalized wreath productgroups. Unfortunately, not all non-rigid symmetry groups are generalized wreathproducts; however, from the experience with the groups [no nl n2]G derived in this paperand with other molecular symmetry groups, it seems that pure permutation groups (PIgroups without starred PI operations) are generalized wreath products. In this case, if atleast one starred PI operation is of order two, any general PI group could be written asthe semidirect product of the invariant wreath product subgroup consisting of all purepermutation operations and a group of order two whose elements are the identity and astarred PI operation of order two.

Sample molecules are listed in Table 3 for some non-rigid molecular symmetrygroups.For each example given, the group indicated is the largest possible symmetry group aparticular molecule with three internal rotors or molecular trimer may require under thepresent constraints (e.g. inversion of NH3 monomer units is not feasible). It is possiblethat a subgroup of the group listed for a molecule may be sufficient to interpret amolecular spectrum. In some of these cases, the group-subgroup relations illustrated inFig. 2 may be useful. In other cases, only concerted gearing or antigearing motions maybe feasible. However, the subgroup resulting in this instance is usually not contained inthe group-subgroup trees shown in Fig. 2. The subgroups resulting from restrictions togearing or antigearing motions were not investigated systematically. As an example ofthe method to derive such groups, the model [333]0 is discussed briefly.

For molecules like B(CH3)3, only two types of gearing or antigearing motions lead tosmaller molecular symmetry groups. In the first case, the simultaneous rotation of allthree methyl groups in the same direction and by the same amount can be considered tobe the only feasible internal rotation. The corresponding symmetry operator COCIC2 =Cgenerates a group of order three, %(C). If this group is combined with some or all of thegenerators P, T, Sand TS, the following groups are obtained: 21J 3(C, T), ~3(C, S),

Table 4. Character table of [322]C, ~ 'fJ24t*E C2 C3 C4 c, C6 C, Cs C9 Class§ Elements'[

{OOO}A' 1 1 1 1 1 1 I I 1 CI(I,I) E{OOO}A" 1 I 1 1 1 1 1 -1 -1 C2(1,2) 011{OII}A' 1 1 -1 1 1 -1 -1 1 -1 C3(2,2) 001,010{Oll}A" 1 1 -1 1 1 -1 -1 -I 1 C4(2,3) 100,200{DOl} 2 -2 0 2 -2 0 0 0 0 Cs(2,6) 111,211{IDO} 2 2 2 -1 -1 -1 -1 0 0 C6(2,6) 101,210{lll} 2 2 -2 -1 -1 1 1 0 0 C,(2,6) 110,201{101} 2 -2 0 -1 1 w w· 0 0 Cs(6,2) Sjkk;j=O, 1,2; k=O,1{201} 2 -2 0 -1 1 w· w 0 0 C9(6,4) Sjkk';j=O, 1,2; k=O,l; k' == l-k

t The little cogroup of the orbits {OOO} and {OIl} is C,.*w=iV3.§ The first and second number in parentheses indicate the number of elements in the class and the order of

each element, respectively.~ The element C!Jctcq' is abbreviated by the string jkm.

~

Table 5. Character table of [333]C; ~ ~,3,3"

E C2 C3 C. Cs C6 C7 Cs C9 C10 Cn C12 C13 C14 C\5 Classt Elementst

{OOO}A' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Cl(I,I) E{OOO}A" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 C2(2,3) 111,222{lll} 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 C](2,3) 012,021{OI2} 2 2 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 0 C.(2,3) 120,210{I20} 2 2 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 0 Cs(2,3) 201,102{lOI} 2 2 -1 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 0 C~2,3) 100,200 "tl

{Ioo} 2 -1 2 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 0 C~2,3) 001,002 ~"{ooI} 2 -1 -1 2 -1 2 -1 2 -1 2 -1 -1 -1 -1 0 Cs(2,3) 010,020 0

{OIO} 2 -1 -1 -1 2 2 2 -1 -1 -1 2 -1 -1 -1 0 C9l:2,3) 011,022 "0rOll} 2 -1 2 -1 -1 2 -1 -1 -1 -1 -1 -1 2 2 0 CJ[l(2,3) 110,220 z

t!l

{lIO} 2 -1 -1 2 -1 -1 2 -1 -1 -1 -1 2 -1 2 0 CI1(2,3) 101,202 "{IOI} 2 -1 -1 -1 2 -1 --I 2 -1 -1 -1 2 2 -1 0 Cn(2,3) 211,122{211} 2 -1 2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 0 Cn(2,3) 112,221{112} 2 -1 -1 2 -1 -1 -1 -1 2 -1 2 -1 2 -1 0 Cl.(2,3) 121,212{I2I} 2 -1 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 2 0 C\5(27,2) Tjkm;j, k,m=O, 1,2

.. The little cogroup of the omit {OOO} is Cs•

t The first and second nwnber in parentheses indicate the number of elements and the order of each element, respectively.:j: The element qCfCi is abbreviated by the string jkm.

Non-rigid symmetry groups 1945

Table 6. Charac ter table of 'B~t:j:

E C2 CJ C. Cs C6 C, c, C9 CIO Class§ Elements'

{OO}A 1 1 1 1 1 1 1 1 1 1 1 CI(I ,I ) E{OO}A2 1 1 1 1 1 1 1 -1 - 1 -1 Cl (2,3) Coo{OO}E 2 2 2 2 -1 -1 -1 0 0 0 Cs(3,3) DCk

{02}A t 1 1 e· e 1 e· e 1 e· e Cl3,3 ) D2Ck

{02}A2 1 1 e' e 1 e' e -1 - e' - £ Cs(6,3) P'"Ck

{02}E 2 2 2e" 2e -1 - e' -e 0 0 0 C6(6,3) P'"DCk

{Ol}A I 1 1 e e' 1 e e' 1 e e' C,(6,3) P'"DlCk

{OI}A2 1 1 s e' 1 E e' -1 -e -e' C8(9 ,2) SplCk

{OI}E 2 2 2e 2e' -1 -e r-e" 0 0 0 C9(9 ,6) snoc:{ll} 6 -3 0 0 0 0 0 0 0 0 Cui9,6) SPID2C k

t The little cogroup of the orbits {OO}, {02} and {OI} is Cs••:j: e 0= exp(-iln/3) .§ The first and second number in parentheses indicate the number of elements and the order of each element,

respectively.~j,k=O,I,2 ; rno=I ,2.

Table 7. Character table of 'B1os

E C2 CJ C. c, C6 C, c, C9 CIO CII Classt Elements:j:

{OO}A; 1 1 1 1 1 1 1 1 1 1 1 CI (I ,I) E{OO}A i 1 1 1 1 1 1 1 -1 -1 -1 -1 C2(2,3) c"{OO}Ai' 1 1 1 1 1 -1 -1 1 1 -1 -1 CJ(6 ,3) D'"Ck

{OO}A;' 1 1 1 1 1 -1 -1 -1 -1 1 1 C.(6,3) P"'Ck

{OO}E' 2 2 2 -1 -1 2 -1 0 0 0 0 Cs(12,3) P"'D"Ck

{DO}E" 2 2 2 -1 -1 -2 1 0 0 0 0 C6(9 ,2) TDlCk

{OI}A! 2 2 - 1 2 - 1 0 0 2 -1 0 0 C,(18,6) TP"'Dlck

{OI}A2 2 2 - 1 2 -1 0 0 -2 1 0 0 Cs(9,2) SP'C k

{OI}E 4 4 -2 - 2 1 0 0 0 0 0 0 Cg(18,6) SP'D'"Ck{10}A 6 -3 0 0 0 0 0 0 0 2 -1 C1o(9,2) TSP'YClk{lO}B 6 -3 0 0 0 0 0 0 0 -2 1 C ll(18,6) TSpiDkclk+m

• The little cogroups of the orbits {OO}, {OI}, and {10}are D:!h , CJ• and Cl , respectively.t The first and second number in parentheses indicate the number of elements and the order of each element,

respectively.:j:j, k=O,I,2; rn, n = I , 2.

C(63.2(C, TS), ~z(TS) l8l Q.MC, T), ~3.3(C, P), ~3(P)®9)3(C, T), ~dC, P, S),~3(C)l8l~3(P, TS) and ~3(C, T)®C!iJ3(P, TS). Another type offeasible gearing motion isobtained if one methyl group is kept still whereas the other two are rotated simul­taneously in opposite directions by equal amounts. A feasible symmetry operatorcorresponding to such a motion is clq= D. Because all methyl groups are treatedequally in this example, the operat ions ('5Cz and CoG also need to be included. Theseoperations form a group of order nine , ~3.3 ( C, D) , where C is the element COC1CZ asbefore. The combination of this group with some or all of the generators T, Sand TSyields the groups 2iJdC, D, T), '€3(D)®~3(C, S) , '€3(C)®~3(D, TS) and~3(C, S)l8l~3(D , TS ). The addition of the operator P produces five new groups, one oforder 27, three of order 54 (two of them isomorphic) and one of order 108. They cannotbe factored into direct products. The groups ~S4 (generated by the operators C, D, P andS) and ~108 (generators C, D , P, Sand T) may have applications in the spectroscopy of-rnolecules like HCCCH3) 3 and B(CH3h, respectively . Their character tables are shown inTables 6 and 7.

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