Yuriy Kruglyak. Quantum Chemistry_Kiev_1963-1991

The canonical description which would correspond uniquely to a polyhex can be obtained in this way. The polyhex is oriented according to the nomenclature rules of IUPAC [19], but these rules sometimes do not ensure unambiguity of orientation, as for example in the case of the benzenoid

For these cases we have made an algorithmic modification of the nomenclature rule A22, which ensures unambiguity of orientation [7]. The first hexagon on the left in the main horizontal row is chosen as the root. First of all, to obtain the covering tree, the adjacent hexagons of the main horizontal row are numbered and subsequently connected by branches oriented from a lower to a higher number. We then number, and subsequently connect with the first hexagon, those hexagons which are adjacent to it and are not yet numbered (in the lexicographical order of connecting sides). Following this the procedure is continued for the second hexagon, then the third one etc., although all the hexagons will not be joined to the tree. The branches are numbered in the opposite order to the lexicographical one.

It should be noted that the construction of a covering tree for a polyhex for the purpose of coding is currently under investigation [20]. Our method of construction of the covering tree differs from the one mentioned in [20]. Examples of canonical descriptions (codes) of some benzenoids are given in the table 1.

11.1. 2. Coding of substituted benzenoids

We propose to code substituted benzenoids in the following way. Let us mark the vertices of the regular hexagon which corresponds to the benzene ring by numbers 1 – 6 as follows

In the case of homosubstitution (only one type of substituent), the code for substituted benzenoids may be obtained by putting the numbers enumerating the

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sights of substitution in the corresponding fragments of the code for unsubstituted benzenoids. This is carried out only on condition that the type of substituent is known. Otherwise one can proceed as in the case of heterosubstitution (different types of substituents). In this case we suggest putting a mark indicating the substituted atom (or group) after every number in the code of the substituted compound. Examples of codes for substituted benzenoids are given in Table 2.

We suggest coding PCS, which contain cycles with lengths differing from six in this way. We shall first consider the relation of a complete ordering which operates on the set of polyhexes.

Table 1

Codes of some benzenoids

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Lexicographical comparison of the codes can be used as such as a complete ordering. We then enlarge the coding alphabet by the use of square brackets and the symbols “+” and “–” which will be placed in order of priority after the letters, figures and parentheses. The method of unambiguous construction of a polyhex corresponding to a given PCS is as follows.

(1)The longest linear row of cycles of the future polyhex should be established.

(2)This row should be oriented so that it starts from the left-hand side with the largest number of hexagons.

Table 2

Codes of some substituted and heteroderivatives of benzenoids

(3)An initial mixed polyomino (with cycles of different length) should be transformed into a polyhex by the addition of vertices (in the case of 3-, 4- and 5-cycles) or by the elimination of vertices (in the case of ≥ 7-cycles) using, if necessary, for unambiguity, the IUPAC nomenclature rule A22. Then the initial PCS should be coded using the obtained polyhex with the modification of the algorithm described previously: added (eliminated) vertices should be described at the appropriate place in the code by their numbers in square brackets. Added vertices should be described by the “–” sign and eliminated vertices by the “+” sign.

(4)If, however, the procedure described in points (1) – (3) does not ensure unambiguity, one should choose a polyhex which gives the smallest code in the lexicographical sence.

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It should be noted that points (1) – (3) were introduced only for simplicity and routine of algorithm in most cases; an algorithm consisting only of point (4) should not admit manual coding.

Some examples of PCS codes containing cycles of lengths differing from six and their substituted and heteroanalogues are given in Table 3.

We shall now consider compounds with structures which can be represented by any connected part of the hexagonal lattice (i. e. by a polyhex missing some of its edges). Such a compound should be coded in the following way. The edges of a fragment of the hexagonal lattice should be dichotomized: (i) the edges that belong to some complete hexagonal cycle (an elementary cell of a polyhex); (ii) the rest of the fragments’s edges.

Table 3 Codes of some PCS, their substituted compounds and heteroderivatives

Let us construct two hexagons on each edge of the second group and add them to the initial polyomino. Let us canonically orient the obtained polyhex according to the rules of orientation. Note that if we make a choice we prefer the orientation with the maximum quantity of edges of the first group, or, as a second choice, that of the second group. After renumbering the hexagons of the polyhex as in the case of the benzenoids, place each edge of the second group next to the hexagon of the two

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hexagons incident to this edge, which has the lower number. Then construct a tree which should cover the polyhex as in the case of the benzenoid. Remove from the obtained polyhex the hexagons meeting the following conditions simultaneously:

(i)The hexagon does not wholly consist of edges belonging to the first group;

(ii)No edge of the second group is placed next to this hexagon; (iii) It is impossible to reach any hexagon, which does not meet conditions (i) or (ii), when moving from the given hexagon along our oriented covering tree.

We then enlarge our alphabet by the use of braces, placing them in order of priority after the square brackets. We then code the initial compound by the code of the transformed polyhex added to the description of the edges of the second group. This description should be enclosed by the braces and put in the appropriate place of the code. The abovementioned description is constructed from letters a – f being preceded by the “+” sign. The only exception is the case when the quantity of edges of the given hexagon which belongs to the set-theoretical complement of the union of the first and second groups of edges is less than the quantity of edges of this hexagon, which belongs to the second group. Then this complement should be marked in the description, being preceded by the ”–“ sign. For example, consider the carbon skeleton of papaverine

Mark the edges of the first group by thick lines and the edges of the second group by thin lines, namely:

Construct the covering polyhex (the edges which do not belong to the first group or to the second one are marked by broken lines), namely:

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Its canonical orientation is

The transformed covering polyhex is

Thus, the code of the initial compound is (bc{+a})a{–b}(b{+ac})aa{+c}.

Now consider the next class of quasi-PCS that may be coded: PCS joined with one or several fragments of a hexagonal lattice. For these compounds we suggest the following coding method.

The mixed polyomino that corresponds to the given PCS should be oriented at the plane and replaced by a polyhex with marked vertices according to the rules of PCS coding. The obtained polyhex should be completed by the given fragments of the hexagonal lattice.

If the fragments of the hexagonal lattice joined to the PCS reduce the symmetry of the polyhex, the vertices of the latter should be marked for subsequent lexicographical regulation by the code of the fragment joined with this vertex.

Consider, as an example, a fragment of a frame of 6-aminopenicillic acid

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The polyhex corresponding to the PCS is

The completed polyhex is

Its transformation is

And finally, the code is {+f}(f{+a})a[–36](b[–6])a{+ab}a{+c}.

The next class of compounds to be considered is formed by quasi-polycyclic structures which are the result of joining several PCS by fragments of the hexagonal lattice. In this case, fragments of hexagonal lattice mentioned should be completed to form covering polyhexes; the latter should be joined with the initial mixed polyomino. The resultant mixed polyomino should be coded according to the rules of PCS coding. Consider as an example the following compound

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The corresponding PCS is

This can be developed into a polyhex in two ways, namely:

with code a[–4]a{+c}b[–1]a and

with code a[–4]a{+c}b[–3]b. As the first code precedes the second one in the lexicographical order, the first code is the canonical one for the compound.

The final class of compounds considered includes PCS which have pairs of cycles with more than one common edge. For coding compounds of this class one should cut the unions of the cycles with two or more common edges along surplus common edges. The obtained, mixed polyominos should be coded according to the general rules of PCS with special marking of the separated vertices.

Let us now enlarge our alphabet by use of the symbols “ < ” and “ = ”, and placing them after the “ – ” sign. We shall code “divorced” vertices in this way. On reaching the last hexagon in a code which contains a divorced vertex, we open the square brackets and enter the number of the divorced vertex in the hexagon. Following this we code a path from the given hexagon to the hexagon containing the vertex divorced from the given one. This path must be along the tree which covers

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the polyhex. On reverting by one step we place the symbol “ < ” (one should revert to a minimum number of steps). After completing the reverse movement we describe the motion along the tree as usual. On arrival at the appropriate hexagon we place the symbol “ = ” and a code for the divorced vertex in this hexagon.

Each method of cutting has its own code. We choose the smallest of these codes in the lexicographical order as a canonical code for the initial compound.

Consider as an example a fragment of strychnine

which gives rise to polyominos

with code ab[3 < = 3] and

with code aa[1 < = 1]. The latter code is the canonical one.

11.2.Analytical formulae for the enumeration of substitutional isomers of planar molecules

11.2.1. Introduction

Polya’s theorem [21] is a powerful means of enumerating the substitutional isomers of chemical compounds [22 – 37]. However, its direct use of calculating the number of isomers leads to an opening of brackets and a reduction in the number of terms expressing the generating function of the number of isomers. This procedure may be rather cumbersome for complicated compounds and, moreover, when passing from one compound to another it requires that the entire procedure be repeated [3].

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In the present study we derived formulae for the direct calculation of the number of substitutional isomers that avoid the need for computations using the corresponding generating functions. The derived formulae may easily be programmed into a computer. For small parameter values the formulae allow one to make manual calculations. The use of these formulae for the analytical investigation of the correlations between the number of isomers is illustrated.

In the present paragraph we use the terms “homosubstitution” and “heterosubstitution” to designate the cases of one or several substituents respectively. In addition, we introduce the following notation: Z – the searching number of substitutional isomers; N – the general number of real substitutions; n – the number of types of substituent; Ni – the number of substituents of type i used; G – the number of atoms or groups of atoms which can substitute formally (number of vacancies); G´ – the number of vacancies which lie on the axes of symmetry of the molecule; S – the symmetry group of the molecule.

The following formula for calculating Z shows that Z is a function of {Ni} (or, in the case of homosubstitution, N), G, G´ and S only:

Z = Z({Ni},G,G′,S).

11.2. 2. The case when G′ = 0

Let us consider the case of homosubstitution. In this case it is not difficult to calculate the cycle index which is equal to

where the summation should be carried out over all divisors K of the natural number m, and φ(K) is Euler’s function which is equal to the number of positive integers which are less than K and relatively prime to k.

According to Polya’s theorem, the searching number of isomers Z(N,G,0, S) is

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