R. L. Flurry, Jr. University of New Orleans, New Orleans, LA 70148 Isomer counting has fascinated chemists for many years. Various approache; to the problem have been developed from the points of view of combinitorics, graph theory, and group theory. Polya's theorem (I) has heen "sed by a number of workers for isomer enumeration. For example, McDaniel(2) has used the theorem for enumerating the isomers of rather complicated inorganic structures which have various constraints placed on the allowed substitutions. This and most other work considers only parent molecules having a rigid ~ ~ l i the ed basic framework. Leonard and co-workers (3) . . a.. theorem to fluctional (nonrigid)molecules, molecules that can undergo spatial rearrangements (other than Euclidian rotations or translations) during the time scale of the experiment. Their arguments were based on Polya's cycle index (I),rather than on point symmetry and fluctional symmetry operations. Ruch and co-workers (4) have developed a very sophisticated enumeration scheme which works for rigid or fluctional molecules. The method involves the construction of double co-sets of the appropriate group of the parent system. The purDose of this work is to provide a simple technique, using nonrigid symmetry groups, for enumerating the isomers of fluctional molecules. The technique requires the concept of the covering group. This is defined for both rigid and fluctional molecules in the next section. The Full Covering Group McDaniel(2) has pointed out that the full covering group is the appropriate group for counting geometric isomers. The proper rotational suhgroup (i.e., the subgroup obtained when the improper operations, the reflections, inversion, and improper rotations are omitted) of this is appropriate for counting stereoisomers. If the two numbers are different, enantiomeric airs are present. Isomorohism of the full covering group fo; two strukures in which ihe positions available for substitution have the same site svmmetrv "i n d.i e s identical structure counts. The full coverine mouu differs from the point mouD only for t h w cases whe;Lall atoms lie in a symmltry e&ek (i.;, linear or planar molecules). The full covering group can be defined as the minimum set of operations which generates all mappings of a molecule onto itself. For example, in henzene the transformation
could he accomplished either by a Cz (= Cs3) or an inversion operation. Consequently,the full covering group is a suhgroup of Dsh. I t can he considered to he either Ds or Cs, (these are
isomorphic toearh other). For isomer counting, the D6group should be used since it is the maximal rotntional subgroupd U G ~This . is also the covrring group for a hypothetical allplanar cyclohexnne structure (with the hydrogens imd carbons all lying in the same plane) (5). 116 is both the full covering group and the properrotation group for both of these strucy tures. Thus, the number of geometric isomers equals the number of stereoisomers, and there are no chiral isomers. For the Haworth projection of cyclohexane (I)
(1)
all atoms are not in the same plane, consequently the full covering group D6h while the rotation suhgroup is D6. The number of geometric isomers does not equal the numhr of stereoisomers. As another example of the distinction between a point group and a covering group, acetylene, C2H2, has D,h point symmetry, hut the covering group is only Cz. I t should be pointed out that while the covering group is completely adequate for describing a molecular structure, it is not adequate for describing a wave function. Nonrlgld Symmetry Groups The enumeration of the maximum number of isomers for a given fluctional parent structure illustrates the use of the nonrigid symmetry group. Any of the several developments of the groups can be used. We will use that of Altmann (6). Altmann defines the nonrigid symmetry group of a molecule (S) as a product of the group of isodynamic operations (I)and the rigid point group ( G ) S=IXG (2) Isodynamic operations are operations, corresponding to fluctional motions, which convert a molecule into an isoenergetic conformation by some internal motion which is rapid on the time scale of the exoeriment under consideration. (Note the "time scale of the experiment." A given molecule may he fluctionalwith respect to one experiment, say isomer counting, while not with respect to another, say electronic spectroscopy. For isomer counting, all possible fluctional motions must be included, regardless of whether they contribute to a reasonable Hamiltmian for the molecule.) Thev are usuallv relativelv easy for a chemist to visualize.' his is not always the case (7)). The product may be a direct, semi-direct, or weak direct product. The semi-direct (or direct) product requires that the suhgroups be normal (and that one of them he invariant). The symmetric groups of degree 25 have no normal suhgroups
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other than the alternatine,... erouu. . For svstems havine five or more fluctionally equivalent atoms, the product structures must involve weak direct ~roduccs.For exam~le.PF, is such a structure. All five fluorkes are fluctionally eq"ivalkt. The appropriate group is isomorphic to S(5). This can he constructed as the weak direct product of the D3 covering group of the eouatorial fluorines of the rieid structure with cvclic " groups of depee four (corresponding to one axial fluorine held fixed and the other fuur cvclicallv . -uermuted) and five (all five cyclically permuted). The covering group for a fluctional molecule is the product of the covering group for the rigid structure and the isodynamic covering group. The isodynamic covering group contains all isodynamic operations except those corresponding to motions whose results can be interconverted by a Euclidian rotation. The rotation subgroup for a fluctional molecule is found by deleting from the nonrigid group the improper operations of the point group. The improper operations of the isodynamic group are those operations resulting from the product of the improper operations of one subgroup with a proper operation of the other. The improper operations of the isodynamic group correspond to motions which "average" all of the moving atoms into a plane which would he a plane of symmetry in the "averaged" structure. A maximum numher of stereoisomers that can arise from a given parent geometry with all substituents different, is (2) ~~~
~
n! number of stereoiaomers = -
hn
(3)
where n is the number of positions available for substitution and ha is the order of the rotation suherouu of the covering g r o u p 3 h e number of geometric isomek ic(2) n! number of geometric isomers = -
hc
(4)
where hc is the order of the full covering group. Ethane is a simple instructive example. The maximum point symmetry is D3d. The rotation of the CH3 groups relative to one another introduces an isodynamic C3. The nonrigid group is thus ('3) and has an order of 36. This is the full covering group, since the protons do not all lie on a line or a plane. The rotation subgroup is S n = @ 3A D3
(6)
and has the order 18. There are thus 6!/18 = 40 possible stereoisomers and 6!/36 = 20 possible geometric isomers, if all substituents are different. This leads to 20 enantiomeric pain, and no achiral isomers. The rigid D3d structure would have implied 120 stereoisomersand 60 geometric isomers. Benzene, with S c = G c = SR= GR = D6 (order 12),by way of contrast, has 60 stereoisomers, if all substituents are different, and same numher of eeometric isomers (with no chiral isomers). . Cv. clohexane, whose nonrigid covering group (81 is isomorphic to the 1)eh rovering group of the Hawonh projection ( I ) , has 39,916,800 swreoisomers and 19,958,400 geometric isomers (no chiral isomers~,if all substituents are different. Boron trimethyl presents a more complicated example. Startine from CR,rieid uoint svmmetrv. which results when one hy&ogen oikaLh i e t h y l group is pointing up, there are five isodvnamic svmmetrv elements. Three of these are the obvious internal & axes i f the three methyl groups. One of the others (Zz)corresponds to a simultaneous 180' rotation of the three methyl groups. The final one (W) involves a 180" rotation about one of the B-C bonds while holding the position of the protons on that carbon fixed and ~ i m u l l ~ n e o & ro-~ tatinp the methyl uoups on the other carhons hy 180" so that the final coufigur&ion has one carbon with its protons in the 664
Journal of Chemical Education
original position but with the other two carbons and their protons interchanged. The results of these last two operations are related by a 180' Euclidian rotation of the molecule; consequently, both cannot be in the covering group. The complete nonrigid group has the structure (8). and has an order of 648. The covering group lacks either the W or the Zz and has an order of 324. Its structure was first deduced by Longuet-Higgins (9). The rotation subgroup lacks the uv plnnes of the C3" maximum point symmetry and, consequently, has the order 162. Thus for all substituents different, number of stereoisomers = 9!1162 = 2240
(8)
and numher of geometric isomers = 9!/324 = 1120
(9) The examples we have presented are almost trivially simple. To take into account the full vower of Polva's theorem for isomer counts with restrictions, the permutational cycle structures corresponding to the symmetry operations will be needed. Even here, the use of the product structures for constructing the nonrigid group can offer simplifications for workers with a facility for constructing product groups. All that is needed is the class structure for the nonrigid group. The cycle structures can be obtained directly from these by using the techniques McDaniel(2) has outlined for rigid groups. Nonrigid Ethane Let us consider ethane as an example, since the product structure (6)and the character table (9) are in the literature. The product structure, working from Dad point symmetry (6) is where @3C3 is a subgroup having a simultaneous C 3 and C3 operation as its generator. The classes are These correspond to the cycle structures given by LonguetHiggins for the classes (9). Using standard partition notation (10) over only the protons, these are ((I6),2(3% 3(2% 2(3% 4(3,13),6(6),3(2% 60% 9(22,12)1 (12) McDaniel's permutation function (2) P,,m (which is equivalent to the f n m of Refs. (1) and (3))for these uses the cycle lengths and numbers of cycles as the indexes. They are, for ethane
(This is for the full covering group. The rotation subgroup would have the classes containing i or ud omitted.) The P,m are evaluated as
where m is the total numher of cycles in a partition, n ( A ) is the numherofcyclescontainingatumAonly,n~H)thenumher containing atom B only, etc. For a given structure, the numher of geometric isomers is 1 number ofgeometric isomers = -
hc a
gkPfk
(15)
where the summation is over the classes of Sc. The number of stereoisomers is 1 number of stereoisomers = -x g h P f k hs a
(16)
There are two stereoisomers, the same number of geometric isomers, and, thus, no optically active isomers. The structures are C A 3 - C B 3 and C A z B - C A B 2 . For the A 3 B 2 D substitution number of geometric isomers
where the summation is over the classes of SR.In both cases, gh is the order of the kth class and Pfk is the permutation function of that class. The permutation functions are the Pnm, evaluated for the permutations that would leave a given structure unchanged. If there is more than one grouping of atoms that is unchanged by a given permutation, the groupings must be summed. If there are no permutations of a given type that can leave a structure unchanged, Pf equals zero for that class. The procedure is more difficult to describe than i t is t o apply. Let us consider the number of stereoisomers and geometric isomers for ethanes having all possible structures for the substitutions A 3 B 3 and A 3 B 2 D . The allowed partitions are given in the table. Thus, for the A 3 B 3 substitution 1 number of geometric isomers = -IPl6 36
number of stereoisomers
1
=-{PIS
18
+ 2Ps2 + 3 P 2 3 + 2Pz2
+ 2P32 + 2 P 3 2
number of stereoisomers =
There are three geometric isomers; CA3-CB2D,CA2D-CB2A and CAzB-CABD, the last of which exists as a dl pair. Any other applications of Polya's theorem which involve the use of point groups can be extended to fluctional molecules in a similar manner. Obviously, the more complicated the molecule. the more difficult it will be t o be sure that all isod y n a m i ~ k ~ e r a t i ohave n s been found. However, the same caveat a p ~ l i e to s a~plicationsbased directlv on cvcle indexes generated in anyither manner. Acknowledgment I would like to thank D. H. McDaniel for helpful discussions. LRerature Cited Ill Polys, G., C. R. Hobd Seances Acod Sci., 201,1167 (1935);HeIu Chim. Aeto, 19.22 l1936):Z. Klisfollogr. Kristollgeom..Kr~lollphys..Klisfollchem..93,415 l1936); Acto Moth. (Uppaolol, 68,145 (1937). (2) MeDanie1,D.H.. Inorg. Cham., 11,2678(19721. (31 h n a r d , J.E., Hammond, G.S.,and Simmona, H. E., J Amer. Chem. Soe.,97,5052 119741. I 0 Rueh, E.. Hssselbarth. W., aod Richter. 6.. Tkohot. Chim. Ado IBerl.1, 19, 288 (19701. (51 h n a r d , J. E.,J. Phya. C k m , 81,2212 (19771. (61 Aitmann, S. L., Roc. Roy. Sm. London, Ser A..Z98,1&1(1967). (71 Altmsnn, S. L., Mol. Phys., 21,587 (1971). (8) F l w , R. L.,Jr., J Phys. Chem., 80.777 (19761. (9) hnguet-Higgina, H. C., Mol. Phys., 6,445 (1963). I101 Hamemesh, M., "Gmup Theory and it. Applications m Phyaicai Problm," Addison-Wesley Publishing Co., Ine., Reading, MA, 1962, p. 25. (la) aonesponds to sir atoms unpermuted, W ) to two setsof thrpeatoa. each being eyelieally permuted, ete.
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