Jack
2212
E. Leonard
Isomer Numbers of Nonrigid Molecules. The Cyclohexane Case Jack E. Leonard Department of Chemistry, Texas A & M University, College Station, Texas 77843 (Received May 2, 1977) Publication costs assisted by The Robert A. Welch Foundation
An extended Polya theorem method for computing the isomer number for any number of achiral ligands on an achiral nonrigid skeleton is discussed using cyclohexane as an example. It is shown that the primary advantage of the method is its relation to the chemist's notion of the structure's motions, removing the need for a nonrigid model system and also alleviating the possibility of error in selecting such a system on the basis of group isomorphism alone. With this extension of the Polya theorem, the number of diamutamers (isomers obtained by rearranging ligands on a skeleton) of any achiral skeleton can be computed, and the number of enantiomeric isomers distinguished without the need for inspection of physical models.
In a recent article my collaborators and I showed that the notion of point groups could be extended to include the rotations, inversions, ringflips, and fluxional motions of nonrigid molecules.' Furthermore, by combining this with the very powerful pattern counting method of Polya,2 it is possible to count by group theoretical methods the number of isomers possible for such nonrigid molecules. Thus, our extension renders it possible to count by the well-established Polya theorem method3 the number of possible isomers of an achiral skeleton substituted with any number of different achiral ligands. Furthermore, by employing the appropriate decomposition of the group (such as the removal of all rotation-reflection operators), it is possible to draw significant conclusions regarding the symmetries of the possible isomer^.^ We illustrated these points by considering a problem of long standing in organic chemistry: counting the number of isomers of a substituted cyclohexane skeleton (Figure la). This problem was shown very early1 to be reducible to the equivalent problem of counting the possible isomers of a planar projection cyclohexane (Haworth projection, Figure lb). The Polya method for rigid molecules yields the wrong cyclohexane isomer number when applied to chair form cyclohexane (which has D3d symmetry), but the correct answers for the D6h Haworth skeleton. Yet the theoretical justifications for selecting the Haworth projection have been rather muddled in the literature (see ref 2-5 in ref 1). In our paper we showed that combining the point group operation R6 (a ring flip from one chair form to another, followed by a 60" rotation to bring the final state in to alignment with the initial one; see Figure 2) with the D3d symmetry of the rigid skeleton leads to a Polya cycle index which is identical with that derived from the point group D6h operating on the same 12 substituent positions of the Haworth cyclohexane skeleton (Z(D6h;12)). Having obtained this result it is possible, from either cycle index, to calculate the number of cyclohexane isomers expected with any number of substituents. In passing it was mentioned that the permutation groups for the two groups D3& and D6h operating on appropriately labeled chair form and planar projection cyclohexane, respectively, were in fact identical. In a response published in this Journal,S Flurry showed that using the established group theoretical method of the semidirect product it is much easier to show that the symmetry group of the ring flipping cyclohexane (either boat or chair) is isomorphic to D6h and, further, that the method is simpler than the Polya cycle index. Indeed, it must be a simpler proof of isomorphism since, as we The Journal of Physical Chemistry, Vol. 8 1, No. 23, 7977
TABLE I: Cycle Indices for Calculating Cyclohexane Isomer Numbers Chairform Cyclohexane (Rigid DBd Symmetry) I
Z ( D 3 ; 1 2 ) = - [ f : *t 2 6 t 4f",l 6
1 Z ( D , d ; 1 2 ) = -[f:' 12
t 2 f : t 4 f ; 4- 2fz t 3 f : f : l
Planar-Projection (Haworth) Cyclohexane = Chairform Cyclohexane (Nonrigid D , , R , Symmetry) 1 Z ( D , ; 1 2 ) = - [ f i 2 t 2 f i + 7f6, t 2f,"] 12 1 Z ( D , h ; 1 2 ) s --[ti' + 4 f i t 12f6, t 4 f : t 3 f : f ' , ] 24 All Planar Cyclohexane 1 Z(D,; 1 2 ) = --[ti' + 2 f i t 7 f : t 2 f 3 12 1 Z(D,h; 2 4 ) -~[ 2 f : ' 4 f : t 1 4 f : + et,"]= 24 1 --[ti' t 2f:: t 7 f ; t 2 f i l 12
+
pointed out, the Polya cycle index cannot be used to prove isomorphism: the same cycle index may be derivable from very different point sets. If the point sets have a similar decompositionand if the generating symmetry groups are identical, then the cycle indexes are the same; the converse does not hold. However while misconstruing a minor point in our paper, Flurry then proceeds to obscure the major point: our method of nonrigid point groups allows a correct count of the isomers of any achiral skeleton containing any number of achiral substituents. If one were to accept his criterion of isomorphism as sufficient, then it would have to be admitted that the all planar structure (Figure IC)provides an equally acceptable basis for counting isomers.6 However, it can be seen immediately that no enantiomers are possible by substituting this skeleton with achiral ligands, since all isomers have a Uh mirror plane. Unlike the isomorphism criterion, the Polya cycle index clearly indicates the isomer number equivalence of the nonrigid chairform cyclohexane and the rigid Haworth structure and the nonequivalenceof either of these to the all planar cyclohexane structure (Table I). Notice that the cycle index for the rotations only group (We; 12)) is the same
Isomer Numbers of Nonrigid Molecules
2213
9 IJ
a
Haworth
Chairform
C All planar
--
Figure 1. Representations of cyclohexane.
8
&
ring
'
4 Figure 3. Numbering system for cyclohexane. TABLE 11: Isomer Numbers for the Cyclohexane Skeleton (Represented by C6) Substituted with up to Four Achiral Ligandsa EnantiDiamu- Diaster- omeric Achiral Formula tamers eomers pairs isomers 1 1 0 1
c6
rinq flip
Figure 2. The ring-flip-rotation operator, R6.
as the rotation-reflection cycle index (Z(DGh;12)) for the all planar structure. As we pointed out this identity means that all diamutamers (that is, structures obtained by permuting a fixed set of ligands on a given skeleton) are achiral, since the introduction of the nonphysical rotation-reflection operators does not reduce the number of isomers which are not interconvertible by the rotation operators alone. On the other hand, the cycle indices for the rotation group only and for the full group are not identical for the chair form and Haworth cyclohexanes, confirming that enantiomeric diamutamers (such as trans-1,2-dibromocyclohexane)are possible in this system. By selecting the model of a ring flipping cyclohexane molecule for the purposes of isomer counting, we obtained as a side result the observation that the Haworth rigid structure would have to work as well, since it generates the same cycle index. On the other hand, by focusing on isomorphism Flurry has opened the door to an obviously flawed skeleton for isomer counting. The final test of a self-consistent theoretical approach is its usefulness in actual application. From this standpoint both group isomorphism and Polya theorem calculations have been very useful and important within their own domains. The usefulness of Polya theorem isomer counts is not significantly enhanced by showing that it can be applied to a nonrigid cyclohexane because a rigid model for the system has long been available. However, the same method can be applied to many other systems such as ethane, propane, ferrocene, and dibenzenechromium for which no nonrigid isomer counting model exists.' Since there is no previous example of the actual use of these nonrigid point groups in the literature, an appendix illustrating their use in the cyclohexane case is included. This method's utility is not limited solely to providing the isomer number, since it can also be used to provide symmetry information not easily obtained without physical models. Indeed, this extension of the rigid point groups now provides a means of computing the isomer number
a
1
1
0
1
8 11
7 7
1 4
6 3
19 55
110
12 31 58
7 24 52
5 7 6
50 165 265 495
32 89 147 255
18 76 118 240
14 13 29 15
66 330 660 1320 1980
38 174 344 672 1008
28 156 316 648 972
28 24 36
90 462 1190 2310 1542 4620 7000
55 243 626 1170 793 2336 3558
35 219 564 1140 749 2284 3442
20 24 62 30 44 52 116
1386 2772 2310 6930 9240 13860
702 1404 1173 3492 4656 6984
684 1368 1137 3438 4584 6875
18
108
2940 11550 17430 23100
1491 5814 8784 11616
1449 5735 8646 11484
42 78 138 132
30804
15474
15330
144
10
18
36 36 54 72
See text for meaning of terms.
for any substitution of an achiral skeleton with achiral ligands.
Acknowledgment. Support from The Robert A. Welch Foundation is gratefully acknowledged. Appendix Calculation of Isomer Numbers for Nonrigid Molecules Using Extended Point Groups and Polya's Method. An Illustrative Example. S t e p 1. Generating the Nonrigid Chair form cyclohexane has DBd Point Groups: The Journal of Physical Chemistry, Vol. 81, No. 23, 1977
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Jack E. Leonard
symmetry. The new operation introduced to reflect the nonrigidity of this system, R6, has the following selfmultiplication:
R6, R:
(=C3),
R:, R: (=C:), R;, RZ ( = E )
Thus, multiplication of the group DBdby the R6 operator yields the group D3dR6with the operations: E , 2R2, 2C3, 3R6*C2,i, 2i*&, 2s6, b R 2 , 3OU*R6 S t e p 2. T h e Cycle Index. Polya's pattern counting method depends on developing a polynomial to represent the allowed permutations of ligands on the skeleton. Since the point group shows all allowed permutations on the skeleton, we can conveniently use it to generate this "cycle index". The operator C3 operating on the numbered cyclohexane in Figure 3 can be represented by the following permutation: C3 = (15 9)(2 6 10)(3 7 11)(4 8 12) where (1 5 9) is read "position 1 is permuted to position 5 which is permuted to position 9 which is permuted to position 1". Since this cycle of the permutation permutes 3 elements, it is an f 3cycle. Since four f3 cycles are required to permute all twelve elements, this entire permutation can be represented as f$ For a second example, the operation u, corresponds to the permutation (1)(2)(3 11)(4 w
5 9)(6 10)(7)(8) = fY':
Notice that (1 X 4) + (2 X 4) = 12 = number of points permuted. For the full rotation-reflection group, then, the cycle index (normalized by dividing by the number of operations) is
z ( D ~ R1~2 points) ; = I/
Iz[f:'
+ 2f': +
7f; + 2 f i l Notice that if we only employ the rotation operations (including &) the cycle index is
Z ( D 3 d R 612) ; = ' / ~ [ f f +' 4f': 4f: + 3f;'f':l
+ 12f;
t
S t e p 3. Generation of the Characteristic Polynomial. If we wish to find out how many isomers there are of the form, say, D6A4B4D2E2 (where A, B, D, and E represent achiral ligands), we need to substitute (a" + b" + d" + e"
The Journal of Physical Chemistry, Vol. 81, No. 23, 1977
=
Ex,")for each f, in the cycle index:
Z(D3R6;12) = 2(x4)4
I/
+2
12[(2xi)l2 + ~ ( Z X : )+~ ( ~ q ]
Step 4. Calculation of the Isomer Numbers. The total number of C6A4B4D2E2 isomers is the coefficient of the term a4b4d2e2in the expansion of the rotations-only cycle index, Z(D3R6;12). The isomer number is shown under the heading Diamutamers in Table 11. By allowing for the nonphysical symmetry operations of fhe rotationreflection group (using Z(D3dR6;12)) we can reduce the isomer number to only the diastereomers. By further manipulations, then, we can determine the number of enantiomeric pairs (Z(D3R6;12) - Z(D3,; 12)) and the number of achiral isomers (diastereomers-enantiomeric pairs). Incidentally, the table does not show it, but for a cyclohexane skeleton substituted with 12 different achiral ligands there are a total of 39 916 800 diamutamers, all of which are chiral, so that there are 19958 400 enantiomeric pairs. The most complex substitution of the cyclohexane skeleton which still has achiral isomers is C6A2B2D2E2FGHI.This substitution produces a total of 2 494 800 diamutamers of which 144 are achiral. Using other nonphysical permutation operators it is possible to count positional isomers separately4b and probably other significant classes of molecules. We are currently attempting to computerize these calculations.
References and Notes J. E. Leonard, G. S. Hammond, and H. E. Simmons, J. Am. Chem. Soc., 97,5052 (1975). (a) G. Poiya, C . R . Hebd. Seances Acad. Scl., 201, 1167 (1935); (b) G. Poiya, Helv. Chim. Acta, l B , 22 (1936); (c) 0. Poiya, 2. Kristalkgr., K&ta&eom., Wstalibhys., kitstakhsm.,93, 415 (1936); (d) G. Poiya, Acfa Math., 68, 145 (1937). B. A. Kennedy, D. A. McQuarrie, and C. H. Brubaker, Jr., Inorg. Chem., 3, 265 (1964); J. %la-Pala and J. E. Guerchais, C . R . MM. Seances Acad. Sci., Ser. C , 268, 2192 (1969); A. J. Menez, J. Sala-Pala, and J. E. Guerchais, Bull. Soc. Chlm. Fr., 46 (1970). (a) T. Hill, J . Phys. Chem., 11, 294 (1943); (b) W. J. Taylor, lbM., 11, 532 (1943). R. L. Flurry, Jr., J . Phys. Chem., 80, 77 (1976). I am indebted to Dr. D. Tavernier, RijksuniversiteR-Gent, Belgium, for having brought this representation to my attention. J. E. Leonard, Thesis, California Instihite of Technology (1971): Dlss. Absir., 328,5659 (1972). An Barser study on isomerbnof fenocenes (K. ScMgi, Tcp. Stereochem., 1, 39 (1967)) has the isomer numbers for the following formulas in error: XA,B2D and XA,BDE, where X is the ferrocene skeleton.
