Isomer count

FluR84 Flurry, R. L. Isomer counting for fluctional molecules. J. Chem. Educ. 61 (1984) 663-665. 

RouD75a Rouvray, D. H. Symmetry and isomer counts in the arenes. 

The Mills-Nixon hypothesis that small ring annelation on benzene would induce bond fixation (bond alternation) by trapping out one Kekul6 tautomer is a casualty of early twentieth century structural chemistry. Due to a lack of direct methods for analyzing molecular structure, structural postulates of that time were often supported by an analysis of product distributions. An experimental observable such as product selectivity or isomer count was correlated to an unobservable structural feature derived on the basis of a chemical model. Classical successes of this method are van t Hoff s proof of the tetrahedral carbon atom and Fischer s proof for the configuration of sugars. In the case of Mills and Nixon, however, the paradigm broke down. 

Let us now examine the means used by Werner to establish the configuration of cobalt ammines.S3 Hie technique of isomer counting that Werner used as a means of proving configuration... 

Table 2 Proof of Configuration for Coordination Number Six by Isomer Counting 11...

The concept of isomerism and isomer counts were used extensively in the development of Werner s theory. They are still important, but on a much more sophisticated level than Werner could have imagined some of the structures which he used are now known to be greatly distorted22 and others are not as simple as he believed. Geometric isomers may exist in one form or another, depending upon the solvent, temperature and other conditions, or they may change their coordination numbers. For example, it has been found that [Co(PEt3)2(NCS)2] is planar in the solid state,... 

The technique of isomer counting used by Wemer to prove the configuration of cobalt-ammines and other complexes did not originate with him, but had been used earlier by Wilhelm Kdmer for benzene derivatives in 1874, and was also suggested by Jacobus Henricus van t Hoff in 1875. However, the technique of comparing the number and type of isomers actually prepared with the number and type theoretically predicted for different configurations probably attained its zenith with Wemer. Not only did he use this method to discredit completely the chain theory bnt also to... 

Table 2 Proof of configuration for coordination number 6 by Isomer Counting . (Ref. 2b. Reproduced by permissionof Dover Publications, Inc.)... 

The octahedral coirfiguration of cobalt(lll) and plat-inum(lV) (coordination number six) was proven in three ways (1) chemical evidence snch as isomer counting and transformation reactions (see Section 4.3) (2) resolntion of selected compounds (see Section 4.3) and (3) X-ray diffraction studies (see Section 5.5). The first two methods were pursued successfiflly, largely by Werner. 

The time necessary to reach the photostationary c/s-state for the first time in LBK films of this azobenzene amphiphile transferred in the trans-state is considerably longer than for the subsequent cycles. This is explained by the time it takes to break up some aggregates in the dense packing of the LBK film. The extent of isomerization, determined by Brode s method,is given to be 30%. Reestimation employing electrochemical methods showed that the irradiation yielded a c/s-isomer count of only 19% ... 

This chapter deals with a number of applications of group theory to molecular properties and structure, all connected by the idea of symmetrized powers of representations. The GT calculator has a general facility for calculation of these powers, and specialised routines for their application to angular momentum, molecular electric properties and isomer counting. 

Isomer Counting using Point Group Symmetry... 

For isomer counting, the problem to be solved is to determine the number of distinct isomers corresponding to decoration of atoms in a parent molecule. Substitution of an atom by another isotope of the same element or by a different chemical species, addition of a structureless ligand or functional group oriented so as to preserve the local site symmetry, all are to be treated as aspects of the same decoration process. 

Although only two alkane isomers have four carbons, the number of possible isomers increases rapidly as carbon atoms are added to the parent chain. This is because longer chains provide more locations for branches to attach. A methyl branch on a six-carbon parent chain can be attached to either the second or the third carbon from the end of the chain. To make sure that these two compoimds are really isomers, count the number of carbon and hydrogen atoms in the structures formed by placing the methyl group at those two positions, and write the molecular formulas. 

Several methods of enumeration have been developed for the fullerene problem, and there is now general agreement on the isomer counts for n < 150, which is more than adequate for present chemical purposes. Two classes of method can be envisaged one that takes a vertical slice of fullerenes of some particular type or symmetry and one that takes a horizontal slice of all fullerenes at a given Both have advantages, and one method of each type will be summarized here. 

Finally, over the next few years it is likely that a similar systematic approach will pay dividends in the study of the chemical derivatives of the fullerenes, where the fullerene problems of isomer counting, electronic and steric effects all reappear in another form. A bibliography on fullerene research has recently been published. ... 

Now consider the symmetry point group G (or, more precisely, the framework group ) of the above ML coordination compound. This group has IGI operations of which lf l are proper rotations so that IGI/I/ I = 2if the compound is achiral and IGI/I I = 1 if the compound is chiral (i.e., has no improper rotations). The n distinct permutations of the n sites in the coordination compound or cluster are divided into nM R right cosets which represent the permutational isomers since the permutations corresponding to the IWI proper rotations of a given isomer do not change the isomer but merely rotate it in space. This leads naturally to the concept of isomer count, I, namely,... 

The earliest use of a combinatorial technique to determine isomer counts appears to be that of Flavitsky [76] who studied the alcohols. By devdoping sets of recursion relations for the saturated alcohols, Flavitsky was able to derive formulas which yielded the number of primary, secondary, and tertiary alcohols for a given number of carbon atoms. Since he also demonstrated that the number of primary alcohols having n atoms is equal to the total number of all alcohols having n — 1 carbon atoms, he was able to derive a table of values for primary, secondary, tertiary, and the total number of alcohols. His table, which lists results for the first ten alcohols in the series is reproduced in Figure 14. 

Figure 14. Reproduction of the first table of isomer counts enumerated by Flavitsky (1871). Moving from left to right this table presents for each of the first ten members of the alcohol series the chemical formula the total number of alcohols Um and isomer counts for the numbers of primary secondary, and tertiary alcohols.

J.E. Leonard, Studies in Isomerism Permutations, Point Group Symmetries, and Isomer Counting, Ph.D. Thesis, California Inst. Technol., California 1971. 

More recently, Ruch (a theoretical chemist formerly at West Berlin s Free University) developed a different proach based on group theory (double coset formalism) which can be successfully employed for isomer counting [48]. This approach entails less labour than the expansions of lengthy pdynomial required by Pdlya s theorem, but at the same time allows less flexibility in terms of figure wdghting. 

Philosophically, isomer counting offers also an insight into the dimensionality of the microcosmos down to the size of molecules, objects are tridimenrional just like macroscopic objects. Indeed, an asymmetrically substituted tetrahedral molecule known to afford stable enantiomers would change its configuration if it could be promoted into... 

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