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To create software complexes avtomatizir. synthesis optim. highly reliable products (including resource-saving ones) along with the principles of the arts. intelligence, oriented semantic, or semantic, graphs of CTS decision options are used. These graphs, which in a particular case are trees, depict procedures for generating a set of rational alternative CTS schemes (for example, 14 possible when separating a five-component mixture of target products by rectification) and procedures for orderly choosing among them a scheme that is optimal according to some criterion system efficiency (see Optimization).

Graph theory is also used to develop algorithms for optimizing the time schedules for the functioning of equipment for multi-assortment flexible production, algorithms for optimizing. placement of equipment and tracing of pipeline systems, optimal algorithms. chemical-technological management. processes and productions, with network planning of their work, etc.

Lit.. Zykov A. A., Theory of finite graphs, [v. 1], Novosib., 1969; Yatsimirsky K. B., Application of graph theory in chemistry, Kyiv, 1973; Kafarov V. V., Perov V. L., Meshalkin V. P., Principles of mathematical modeling of chemical-technological systems, M., 1974; Christofides N., Graph Theory. Algorithmic approach, trans. from English, M., 1978; Kafarov V. V., Perov V. L., Meshalkin V. P., Mathematical foundations of computer-aided design of chemical production, M., 1979; Chemical applications of topology and graph theory, ed. R. King, trans. from English, M., 1987; Chemical Applications of Graph Theory, Balaban A.T. (Ed.), N.Y.-L., 1976. V. V. Kafarov, V. P. Meshalkin.
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1. Graphical representation of molecules and their properties - graph theory in chemistry

The study of the relationship between the properties of substances and their structure is one of the main tasks of chemistry. A great contribution to its solution was made by the structural theory of organic compounds, among the founders of which is the great Russian chemist Alexander Mikhailovich Butlerov (1828-1886). It was he who first established that the properties of a substance depend not only on its composition (molecular formula), but also on the order in which the atoms in the molecule are interconnected. This order was called "chemical structure". Butlerov predicted that two substances with different structures, butane and isobutane, could correspond to the composition of C 4 H 10, and confirmed this by synthesizing the latter substance.

The idea that the order in which atoms are connected is of key importance for the properties of matter has proved to be very fruitful. It is based on the representation of molecules using graphs, in which atoms play the role of vertices, and the chemical bonds between them - the edges connecting the vertices. In the graphical representation, the lengths of the bonds and the angles between them are ignored. The C 4 H 10 molecules described above are represented by the following graphs:

Hydrogen atoms are not indicated in such graphs, since their location can be unambiguously determined from the structure of the carbon skeleton. Recall that carbon in organic compounds is tetravalent, therefore, in the corresponding graphs, no more than four edges can depart from each vertex.

Graphs are mathematical objects, so they can be characterized using numbers. From this came the idea to express the structure of molecules by numbers that are associated with the structure of molecular graphs. These numbers are called "topological indices" in chemistry. By calculating some topological index for a large number of molecules, one can establish a relationship between its values ​​and the properties of substances, and then use this relationship to predict the properties of new, not yet synthesized substances. To date, chemists and mathematicians have proposed hundreds of various indices characterizing certain properties of molecules.

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Molecular graphs and types of molecular structures

from "Application of Graph Theory in Chemistry"

Chemistry is one of those areas of science that are difficult to formalize. Therefore, the informal application of mathematical methods in chemical research is mainly associated with those areas in which it is possible to construct meaningful mathematical models of chemical phenomena.
Another way of ironic graphs in theoretical chemistry is associated with quantum chemical methods for calculating the electronic structure of molecules.
The first section discusses ways to analyze molecular structures in terms of graphs, which are then used to build topological indices and on the basis of structure-property correlations, as well as the elements of molecular design.
As you know, a substance can be in a solid, liquid or gaseous state. The stability of each of these phases is determined by the free energy minimum condition and depends on temperature and pressure. Any substance consists of atoms or ions, which, under certain conditions, can form stable subsystems. The elemental composition and relative arrangement of atoms (short-range order) in such a subsystem are retained for quite a long time, although its shape and size may change. With a decrease in temperature or with an increase in pressure, the mobility of these subsystems decreases, but the motion of the nuclei (zero oscillations) does not stop at absolute zero temperature. Such stable connected formations, consisting of a finite number of atoluves, can exist in a liquid, in a bed, or in a solid, and are called molecular systems.
MG in a perspective projection reflects the main features of the molecular geometry and gives a visual representation of its structure. Let us discuss in terms of MG some types of molecular structures. Let us consider molecules whose structure is conveniently described using planar implementations of graphs. The simplest systems of this type correspond to tree-like MGs.
In the case of molecules of the ethylene series, MGs contain only vertices of degree three (carbon) and degree one (hydrogen). The general formula of such compounds is C H, r + 2. C H +2 molecules in the ground state are usually planar. Each carbon atom is characterized by a trigonal environment. In this case, the existence of cis- and trans-type isomers is possible. In the case of r 1 the structure of the isomers can be quite complex.
Let us now consider some molecular systems containing cyclic fragments. As in the case of hydrocarbons of the paraffinic series, there are molecules whose structures can be described in terms of graphs that have only vertices of degree four and one. The simplest example of such a system is cyclohexane (see Fig. 1.3.6). Usually, the structure of cyclohexane is described as MG in a perspective image, while omitting the vertices of degree one. For cyclohexane, the existence of three rotational isomers is possible (Fig. 1.7).

Often chemical bonds are formed by electrons located in different atomic orbitals (for example,s - and R are orbitals). Despite this, the bonds are equivalent and are arranged symmetrically, which is ensured by the hybridization of atomic orbitals.

Hybridization of orbitals - this is a change in the shape of some orbitals during the formation of a covalent bond in order to achieve a more efficient overlap of orbitals.

Hybridization results in new hybrid orbitals, which are oriented in space in such a way that after they overlap with the orbitals of other atoms, the resulting electron pairs are as far apart as possible from each other. This minimizes the repulsive energy of electrons in the molecule.

Hybridization is not a real process. This concept was introduced to describe the geometric structure of a molecule. The shape of particles arising from the formation of covalent bonds in which hybrid atomic orbitals participate depends on the number and type of these orbitals. At the same time, σ-bonds create a rigid "skeleton" of the particle:

Orbitals involved in hybridization	Type of hybridization	Spatial shape of a molecule	Examples
s, p	sp - hybridization	Linear	BeCl2 CO2 C 2 H 2 ZnCl 2 BeH2 Twosp - orbitals can form two σ - bonds ( BeH 2 , ZnCl 2 ). Two morep- bonds can be formed if on two p - orbitals not participating in hybridization are electrons (acetylene C 2 H 2 ).
s, p, p	sp 2 - hybridization	Triangular (flat trigonal)	BH 3 BF 3 C 2 H 4 AlCl 3 If a bond is formed by overlapping orbitals along a line connecting the nuclei of atoms, it is called σ-bond. If the orbitals overlap outside the line connecting the nuclei, then a π bond is formed. Three sp 2 - orbitals can form three σ - bonds ( bf 3 , AlCl 3 ). Another bond (π - bond) can be formed if on p- the orbital not participating in hybridization is an electron (ethylene C 2 H 4 ).
s, p, p, p	sp 3 - hybridization	tetrahedral	C H 4 NH4+ PO 4 3- BF 4-

In practice, first, the geometric structure of the molecule is experimentally established, after which the type and shape of the atomic orbitals involved in its formation are described. For example, the spatial structure of ammonia and water molecules is close to tetrahedral, but the angle between bonds in a water molecule is 104.5˚, and in a molecule NH 3 - 107.3˚.

How can this be explained?

Ammonia NH3

The ammonia molecule has the form trigonal pyramid with a nitrogen atom at the top . The nitrogen atom is in the sp 3 hybrid state; Of the four nitrogen hybrid orbitals, three are involved in the formation of single N–H bonds, and the fourth sp 3 - the hybrid orbital is occupied by an unshared electron pair, it can form a donor-acceptor bond with a hydrogen ion, forming an ammonium ion NH 4 +, and also causes a deviation from the tetrahedral angle in the structure

Water H2O

The water molecule has angular structure: is an isosceles triangle with an apex angle of 104.5°. The oxygen atom is in the sp 3 hybrid state; of the four oxygen hybrid orbitals, two are involved in the formation of single O–H bonds, and the other two sp 3 - hybrid orbitals are occupied by unshared electron pairs, their action causes the angle to decrease from 109.28˚ to 104.5°.

The study of the relationship between the properties of substances and their structure is one of the main tasks of chemistry. A great contribution to its solution was made by the structural theory of organic compounds, among the founders of which is the great Russian chemist Alexander Mikhailovich Butlerov (1828-1886). It was he who first established that the properties of a substance depend not only on its composition (molecular formula), but also on the order in which the atoms in the molecule are interconnected. This order was called "chemical structure". Butlerov predicted that the composition C 4 H 10 can correspond to two substances having a different structure - butane and isobutane, and confirmed this by synthesizing the latter substance.

The idea that the order in which atoms are connected is of key importance for the properties of matter has proved to be very fruitful. It is based on the representation of molecules using graphs, in which atoms play the role of vertices, and the chemical bonds between them are the edges connecting the vertices. In the graphical representation, the lengths of the bonds and the angles between them are ignored. The C molecules described above 4 H 10 are shown in the following columns:

Hydrogen atoms are not indicated in such graphs, since their location can be unambiguously determined from the structure of the carbon skeleton. Recall that carbon in organic compounds is tetravalent, therefore, in the corresponding graphs, no more than four edges can depart from each vertex.

Graphs are mathematical objects, so they can be characterized using numbers. From this came the idea to express the structure of molecules by numbers that are associated with the structure of molecular graphs. These numbers are called "topological indices" in chemistry. By calculating some topological index for a large number of molecules, one can establish a relationship between its values ​​and the properties of substances, and then use this relationship to predict the properties of new, not yet synthesized substances. To date, chemists and mathematicians have proposed hundreds of various indices characterizing certain properties of molecules.

1. Methods for calculating topological indices

Methods for calculating topological indices can be very diverse, but all of them must satisfy quite natural requirements:

1) each molecule has its own, individual index;

2) Molecules with similar properties have similar indices.

Let's see how this idea is implemented using the example of saturated hydrocarbons - alkanes. The key to constructing many indices is the concept of the "distance matrix" D. This is the name of the matrix whose elements show the number of edges separating the corresponding vertices of the molecular graph. Let us construct this matrix for three isomeric hydrocarbons of composition C 5 H 12 . To do this, we draw their molecular graphs and renumber the vertices (in an arbitrary order):

The diagonal elements of the distance matrix for hydrocarbons are equal to 0. In the first column, vertex 1 is connected to vertex 2 by one edge, so the matrix element d 12 = 1. Similarly, d 13 = 2, d 14 = 3, d 15 = 4. The first row in the distance matrix of normal pentane is: (0 1 2 3 4). Complete distance matrices for three graphs:

molecule chemistry topological index

The distance between vertices does not depend on the order of their enumeration, so the distance matrices are symmetrical with respect to the diagonal.

The first topological index reflecting the structure of a molecular graph (G) was proposed in 1947 by Wiener. It is defined as the sum of the diagonal elements of the distance matrix plus half the sum of its off-diagonal elements:

(1)

For the above graphs corresponding to pentanes C 5 H 12 , the Wiener index takes values ​​of 20, 18, and 16. It can be assumed that it describes the degree of hydrocarbon branching: the largest values ​​correspond to the least branched hydrocarbons. With an increase in the length of the carbon skeleton, the Wiener index increases, as there are more elements in the distance matrix. Statistical analysis on the example of several hundred hydrocarbons showed that the Wiener index correlates with some physical properties of alkanes: boiling points, heats of vaporization, molar volume.

Another type of index is not based on distances between vertices, but on the number of nearest neighbors for each vertex. As an example, let's calculate the Randic index, which is defined as follows:

(2)

where vi- the degree of the i-th vertex, that is, the number of edges extending from it. For the graphs above, the Randic index is:

(3)

(4)

(5)

This index also decreases with increasing degree of branching of the carbon skeleton and can be used to describe the physical properties of alkanes.

Alkanes are the most boring type of organic molecules from a chemical point of view, since they do not contain any "features" - double and triple bonds or atoms of elements other than hydrogen and carbon (such elements are called heteroatoms). The introduction of heteroatoms into the composition of a molecule can radically change the properties of a substance. Thus, the addition of just one oxygen atom converts the rather inert gaseous ethane C 2 H 6 to liquid ethanol C 2 H 5 OH, which exhibits rather high chemical and biological activity.

Consequently, in the topological indices of molecules more complex than alkanes, the presence of multiple bonds and heteroatoms must be taken into account. This is done by assigning certain numerical coefficients - "weights" to the vertices and edges of the graphs. For example, in the distance matrix, the diagonal elements can be defined in terms of the nuclear charge Zi(recall that for carbon Z = 6):

(6)

Off-diagonal elements are determined by summation over edges, and each edge connecting atoms with charges Ziand Zj, weight is assigned

(7)

where b is equal to the bond order between the atoms (1 for a single bond, 2 for a double bond, 3 for a triple bond). For ordinary carbon-carbon single bonds, k = 1. Compare propane Wiener indices C 3 H 8 and three oxygen-containing substances similar in composition: propyl alcohol C 3 H 8 O, its isomeric isopropyl alcohol C 3 H 8 O and acetone C 3 H 6 Oh

To do this, we calculate the distance matrices according to the indicated rules. In molecular graphs, we indicate all atoms, except for hydrogen atoms. 1) Propane

2) In the propyl alcohol molecule, oxygen is bonded to the extreme carbon atom:

For a single C–O bond, the weighting factor is 36/(68) = 0.75. Diagonal element of the matrix corresponding to oxygen:

d 44 = 1 – 6/8 = 0.25.

For molecules containing heteroatoms, the Wiener index ceases to be an integer. 3) In the isopropyl alcohol molecule, oxygen is bonded to the middle carbon atom:

4) In acetone, the order of connection of atoms is the same as in isopropyl alcohol, but the bond between carbon and oxygen is double:

For the C=O double bond, the weighting factor is 36/(268) = 0.375

As can be seen, the addition of a heteroatom to the structure of alkanes leads to an increase in the Wiener index due to an increase in the size of the distance matrix. Adding multiple bonds and increasing the degree of branching of the molecule reduces this index. These rules also hold for more complex molecules. Initially, topological indices were developed only for the purpose of predicting the physicochemical properties of substances. However, later they began to be used to solve other problems. Let's consider some of them. One of the applications of topological indices is related to the classification of organic compounds and the creation of organic databases. The problem is to find such an index that one-to-one characterizes the chemical structure and from which this structure can be restored. The required index must have a good discriminating ability, that is, to distinguish among themselves even molecules that are close in structure. This task is daunting, since more than 20 million organic structures are already known. Its solution, apparently, will be found as a result of using composite topological indices.

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