Permutational isomers

Figure 2-80. The permutation matrix oFthe reference isomer the second line gives the indices of the sites of the skeleton and the first line the indices of the ligands (e.g.. the ligand with index 3 is on skeleton site 3).

Figure 2-81. The permutation matrices oftwo structures that differ through rotation by 120 T The permutation matrix of the rotated Isomer can be brought Into correspondence with the permutation matrix of the reference isomer by two Interchanges of two ligands (transpositions),...

Figure 2-85. The permutation matrix of the reference isomer for double bonds.

One of the segments represents the reference isomer, and the permutation matrix of this one gets the descriptor +1) (Figure 2-85). 

Figure 2-86. The permutation matrices of the fragments of a o s isomer, One transposition is necessary to bring these matrices into correspondence. The overall descriptor Is obtained from those of the two separate units by multiplication = (-1).

The same reference unit of Figure 2-86 is also used here for the determination of the permutation descriptor of the trans isomer. In the same manner as above, wc write down the permutation matrices of the two structures, and then determine the transpositions (Figure 2-87),... 

Figure 2-88. The permutation matrices of the fragments of the rotated trans Isomers. The rotated structure (right-hand side) has two descriptors of (-1) whereas the initial structure (left-hand side) had two values of (-r 1). The overall descriptor of both sides is obtained by multiplication (+1)(+1) = (41) and (-1)(-1) = (41),...

Cayley s extensive computations have been checked and, where necessary, adjusted. Real progress has been achieved by two American chemists, Henze and Blair Not only did the two authors expand Cayley s computations, but they also improved the method and introduced more classes into the compound. Lunn and Senior , on the other hand, discovered independently of Cayley s problems that certain numbers of isomers are closely related to permutation groups. In the present paper, I will extend Cayley s problems in various ways, expose their relationship with the theory of permutation groups and with certain functional equations, and determine the asymptotic values of the numbers in question. The results are described in the next four chapters. More detailed summaries of these chapters are given below. Some of the results presented here in detail have been outlined before ... 

Similarly, the increase in the number of isomers in other homologous series (e.g., in the series starting with naphthalene and anthazene) is asymptotically proportional to the number of isomers of the alcohol series. The proportionality factor can easily be derived from the cycle index of the permutation group of the replaceable bonds of the basic compound. 

GarJ85 Garavelli, J. S., Leonard, J. E. Improvements in the computer enumeration of permutation isomers. Computers and Chemistry 9 (1985) 133-147. 

TapR78 Tapscott, R. E. Enumeration of permutational isomers. The porphyrins. J. Chem. Educ. 55 (1978) 446-447. 

Valence Isomerism. Restricted Rotation and Permutational Isomer i sm. - The variable temperature spectra of bicyclic tetraphosphines (29) has been analysed in terms of valence isomerism involving the corresponding two coordinate open-chain isomers.62 A report on ab initio MO calculations of the inversion barriers for a series of methyl phosphines includes a discussion of the electronic consequences of steric effects.63 Inversion barriers of 1,2-diphosphinobenzenes are in the range 100 - 110 kJ mol". 66 The ring inversion barrier for a dibenzophosphorin has also been measured.63... 

We shall be applying permutations and reflections to molecules to give other molecules (isomers), but also to functions of the properties of ligands on the sites. A simple example will suffice to show how permutations may be interpreted in this way. 

Execution of the number of pairwise ligand exchanges, as given by Wmin, can lead to the mirror image via permutation isomers (see below) which are constitutionally different from the considered system. The conversion of 13 into its antipode 13 via 19 or 19 serves as an illustration. 

Mapping of the ligand indices Z= 1... m of permutation isomers onto their skeletal indices s = l,. .., m) indicates which ligand occupies which skeletal site. The standard mapping with matching pairs of indices... 

As stated earlier, ligand permutations by which a given configuration is obtained from a reference isomer are accomplished by a permutation of... 

In the case of permutation isomers with some indistinguishable ligands and/or with skeletal symmetry there are different but equivalent ligand permutations leading to the reference isomer of the configuration. Among the numerically ordered equivalent ligand index permutations that one is chosen for the descriptor which contains the lowest number of transpositions ). 

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