Molecular graph. The variety of structure and shapes of molecules of organic compounds Molecular graph examples

To create automated program complexes. synthesis optimal. highly reliable production (including resource-saving) along with the principles of the arts. intelligence, they use oriented semantic, or semantic, graphs of CTS solution options. 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 the ordered selection among them of a scheme that is optimal according to a certain criterion system efficiency (see Optimization).

Graph theory is also used to develop algorithms for optimizing time schedules for the operation of multi-product flexible equipment, optimization algorithms. placement of equipment and routing of pipeline systems, optimal algorithms. management of chemical technology processes and production, during network planning of their work, etc.

Lit.. Zykov A. A., Theory of finite graphs, [in. 1], Novosibirsk, 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 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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Often chemical bonds are formed by electrons located in different atomic orbitals (for example,s - And R– orbitals). Despite this, the connections turn out to be of equal value and are located symmetrically, which is ensured by hybridization atomic orbitals.

Orbital hybridization is a change in the shape of some orbitals during the formation of a covalent bond to achieve more efficient orbital overlap.

As a result of hybridization, new hybrid orbitals, which are oriented in space in such a way that after their overlap with the orbitals of other atoms, the resulting electron pairs are as far apart as possible. This minimizes the repulsion energy between electrons in the molecule.

Hybridization is not a real process. This concept was introduced to describe the geometric structure of a molecule. Shape of particles resulting from formation covalent bonds in which hybrid atomic orbitals participate depends on the number and type of these orbitals. In this case, σ-bonds create a rigid “skeleton” of the particle:

Orbitals involved in hybridization	Hybridization type	Spatial shape of the molecule	Examples
s, p	sp – hybridization	Linear	BeCl2 CO2 C2H2 ZnCl2 BeH 2 Twosp - orbitals can form two σ - bonds ( BeH 2 , ZnCl 2 ). Two morep- connections can be formed if two p - orbitals not involved in hybridization contain electrons (acetylene C 2 H 2 ).
s, p, p	sp 2 – hybridization	Triangular (flat trigonal)	BH 3 BF 3 C2H4 AlCl3 If a bond is formed by overlapping orbitals along a line connecting the atomic nuclei, it 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 ( B.F. 3 , AlCl 3 ). Another bond (π - bond) can be formed if p- the orbital not participating in hybridization contains 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, the geometric structure of the molecule is first established experimentally, 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 water molecule NH 3 – 107.3˚.

How can this be explained?

Ammonia NH 3

The ammonia molecule has the shape trigonal pyramid with a nitrogen atom at the apex . The nitrogen atom is in sp 3 - hybrid state; Of the four hybrid orbitals of nitrogen, three are involved in the formation of single N–H bonds, and the fourth sp 3 - the hybrid orbital is occupied by a lone 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

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

VARIABILITY DIVERSITY OF STRUCTURES AND FORMS AND FORMS OF MOLECULES OF ORGANIC COMPOUND OF ORGANIC COMPOUNDS MOLECULES L. P. OLEKHNOVICH g. and. ygTspzyZau KUTU‚TNLI „UTY‰‡ TЪ‚VMM˚I YML‚V TLVI, KUTU‚-M‡-SUMY The question of genesis and the variety of the types of mirror configurational isomerism of organic com- The chemistry of carbon – organic chemistry – is distinguished by the variety of structure and extreme pounds are discussed with the multiplicity of individual compounds the application of some. The total number of known organic compounds - elements of graph theory. tions (over ten million) is annually replenished with tens of thousands of new substances synthesized in laboratories. Organic chemistry surprises the analysis of molecular with the variety of classes of molecules, in the structure of which, at first glance, no logic is visible. The main reason for the emergence of a set of organizations that cannot be easily enumerated (>107) of organizations are enlightened. Different compounds are the unique properties of ferences of achiral and central element – ​​carbon. chiral compounds are The world of carbon compounds is an inexhaustible combinatorics of options and methods for constructing classified. molecules of n C atoms, m O atoms, k–N, l–S, h–P, etc. k‡TTPUЪ VM˚ ‚UF UT˚ schgTseZnkh ntsikaa YkDoyZ F ULTıUK‰VMLfl PMU„U-Z abyEkDZaip eigTsdmg U· ‡BLfl ‚L‰U‚ BV N‡O¸- Similar to physicists use economical ˆLUMMUI in a new but capacious language mathematical formulas and LBUPV LL U „‡ML˜VTNLı calculations, chemists use a special language for recording the structure of compounds. This language is especially not - TUV‰LMVMLI T F L‚OV˜V- bypassed in organic chemistry to organize the ideas about numerous subclasses of “SCHU”. d ‡ЪNU UT‚В˘В- a gigantic variety of molecules. In order to spend less time and space when depicting structural formulas, organic chemists often do not bother themselves with the designations of atoms. This technique M‡ UTMU‚V F V‰ТЪ‡‚OV- is especially convenient when one considers not any properties of a particular compound, but the general patterns of the structure and shape of the series of mo- b LL, BUT‡TTLSHLˆL U‚‡- lecules. So, instead of drawing the letter M˚ UTU·VMMUTL TJUV- designations of carbon and hydrogen atoms in all structural isomers, for example, the limiting hydrocarbon hexane – С6Н14, ‡ТЛППВЪ Л˜М˚ı TUV‰ЛМВ-graphs (scheme 1) MLI, ‰‡MU UV V‰VOVMLV NL ‡O¸MUI TLPPV LL are depicted. Scheme 1 44 lykyljZldav jEkDbjZDnTsg'zhv LmkzDg, No. 2, 1997 The vertices of the graphs (points) are carbon atoms, compounds, also depict complex transformations, and the lines (edges) connecting them are C–C bonds. molecules (reactions) and understand each other. Since carbon is tetravalent, and hydrogen is monovalent, it is clear that at the terminal (free) vertices of the graph there should be three H atoms, with graphs . In this theory, a graph G of order n is determined by the average vertices of the type - two each, and is tertiary as a non-empty set of vertices V1, V2, ..., Vn. hydrogen atoms at quaternary vertices calling different vertices. The theory of graphs began with the famous arguments of L. Euler. The above graphs, therefore, are not (1736) about the Königsberg bridges, where the formulas were complete, but they are sufficient to represent the criteria for traversing all the edges of the graph without cross-structural isomers of hydrocarbons. Below are greetings, as well as his other works related to Madena graphs of unsaturated hydrocarbon molecules with thematic puzzles and entertainment. double (C=C) and triple (C≡C) bonds, and so- Important Milestones on the path of development of this theory are compiled by graphs of some cyclic and frame carbons of the work of G. Kirchhoff (1847) and W. Hamilton of hydrocarbons (Table 1). (1859). A. Cayley (1857, 1874–1875) was the first to generalize the trigonal graph (Scheme 2) to use graph representations (enumerations of the figures of very different molecules. graph “trees”) in connection with counting the number of isomers of the first terms a number of saturated hydrocarbons. Thus, graphical (graphic) forms of alkanes. Indeed, only with the help of la connection is an economical representation of the important apparatus of graph theory (Pólya’s theorem, 1937) of its possible particular and most general patterns but to solve the problem of enumeration (enumeration) of all structures and forms. For chemists, similar structural isomers of molecules CnH2n + 2, CnH2n, graphs are enough so that, without using long names CnH2n - 2, CnH2n - 4, etc. (see graphs of hexane isomers), Table 1 Z,E-isomers of butene-2 ​​Z E H3C CH CH CH3 .R . R,L-isomers of 1,3-dimethyl-3-cumulene L. Z. . E. Z,E-isomers of 1,4-dimethyl-4-cumulene. . . . . . Dimethylacetylene 1,4-dimethylbiacetylene Xylenes Benzene Toluene ortho-meta-para- Cyclic saturated hydrocarbons Cyclopropane Cyclobutane Cyclopentane Cyclohexane and so on Framework hydrocarbons Tetrahedran Prizman Kuban ygTspzyZau g.i. ezyYYYEKDBATS lnkyTszaa oike eigTsdmg ykYDzauTsldap lyTSSazTszav 45 O 2− O − CH2 + F B C N C = F F O O O O H2C CH2 Trifluoride Anion of acidic residue Trimethylenemethane cation boron carbonic nitric acid acid Scheme 2 as well as isomers of derivatives (substituted) carbons nkaa DlaeeTsnka eigTsdmg, hydrogens, when “manually”, for large n, this takes a lot of time. Currently, the theory Let us now turn to another feature of our graphs that naturally enters many consciousnesses - attention. When we consider branches of modern mathematics, such as topology of some of the surrounding objects (including logic and combinatorics, linear algebra and molecular graph theory), then often realizu- groups, probability theory and numerical analysis. There are also consciously uncontrolled operations. It is successfully used in physics, chemistry, genetics, which note the correspondence of parts of an object to each other. Ancient Greeks in computer science, architecture, sociology and linguistics. the term “commensurable” (σιеёετροσ) was used to designate. It is necessary to keep in mind the features of the “language” such features of mutual arrangement, relational graphs: parts of an object, which determine its symmetrical appearance, shape - strict symmetry; molecular graphs formalize the connections of buildings, crystals minerals, two-sided symmetry, including, as a rule, several (two and geometry of plant leaves, rotational symmetry more) varieties of atoms-vertices; flowers, etc. If general theory graphs allow the production of Objects are symmetrical, if the proportionality and free number of edges emanating from one ver- the relative arrangement of their parts allow such buses (including isolated vertices during the operation of rotations, internal reflections, complete absence of edges), then the vertices of chemical versions (combination of rotations and reflections), hographs must have exactly as many edges (the connection of which leaves them (objects) unchanged), what is the valence (coordination number) of them, transfers them into themselves. The structure of a symmetry-given atom in a chemical compound; ric objects is such that it is characterized by the presence of at least one of the following elements of the vertex of the chemical graph along with symmetry: the directions of the edges must be clearly oriented, the planes of mirror reflection σ (S1) - because they represent the relative position of the speed of symmetry, having it objects consist of atoms in molecules, as well as the angles between the bonds of identical, mirror-identical halves of atoms: for a tetrahedral carbon atom these (see graphs in diagrams 1, 2 and in table 1); the angles are usually equal to 109.5°, for trigonal planar - 120°, for digonal, acetylene - symmetry axes Cn, n = 2, 3, 4, ..., - parts of the object - 180°, but there may be exceptions ( see the graphs of the map are combined, like the object as a whole, with its pokasal hydrocarbons in Table 1), and three-dimensional (turned around at angles 2π / n (see Table 1 and Scheme 2); large) graph projections are necessary for the pre-mirror-rotational axis Sn, S2 = i is the center of arrangement of molecular configurations. inversion - is a combination of C2 + S1, S4 = com- Experimental chemists design, combinations of C4 + S1 (see E-isomers of butene-2, even like engineers, graphs of new, previously unknown cumulenes, tetrahedron and cubane in Table. 1). connections, think through and implement methods for them. The object is asymmetrical if its internal synthesis. Theoretical chemists compare the structure and the external form; it is impossible to characterize the characteristic analysis in quantum chemical calculations with any of the listed elements sym- sometimes very different structures in order to reveal the properties (see the 2nd and 4th isomers of hexane in Scheme 1, limits of changes in interatomic distances and ras-alanine in Scheme 3). For such objects there is a common distribution of electrons in ions and molecules, the previous trivial symmetry operation is C1. put in one graph (see diagram 2). Graphic- Rotating C1 by 360° (2π) combines the asymmetrical formulas that have become commonplace little more than the object with itself. Of course, the action of the operation - 100 years ago, and the graph communication language of chemistry C1 are combined with themselves and everything is symmetrically - continuously improved. ny objects, since this rotation is trivial. 46 lykyl Zldav yEkDbjZDnTsgzhv LmkzDg, ‹2, 1997 3 3 H H H H H COOH H3C COOH HOOC CH3 C 2 C C C 2 C 1 4 4 1 H H H2N H H2N H H2N H H NH2 Methane Methylamine Glycine l-alanine r-alanine Scheme 3 Spheres s, balls - examples objects that have an asymmetric molecule (alanine) have a mirror - infinite sets of all symmetry elements - a double - a double (see diagram 3). S1(σ), Cn , Sn . The ball is aligned with itself at any rotation, any orientation of the mirror In the 60–70s of our century, scientists of stereoplanes and axes of rotation passing through it, chemists R. Kahn, K. Ingold and V. Prelog developed a center. Therefore, the correct convex polyhedral general rules for assigning duplicate components (tetrahedron, cube, octahedron, dodecahedron, icosahedron - similar types to left (l) and right (r) forms: ideal Platonic solids), into which substituents are inscribed (atoms) associated with the asymmetry of the sphere, although they have finite sets of elemental carbon or other atomic centers of symmetry, but their number and diversity are always sorted according to their hierarchy, and the oldest (but larger in comparison with others polyhedrons. measure 1) is the one that has the largest It has long been noted that if the asymmetric atomic mass: in alanine (Scheme 3) 14N is older than 12C, and this figure is reflected in a mirror plane, located among the carbon atoms of the methyl and carboxyl groups the latter is older: it is associated with heavy 16O, placed outside this object, then a figure is obtained, while the first is associated with light 1H; accuracy similar to the original one, but incompatible with the first one for any shifts and rotations. The follower-observer is oriented (of course, mentally, all asymmetric objects can be salted) towards the molecule, or the molecule is oriented towards placing mirror-like twins. It is commonplace for an observer to see that carbon examples of this are our shoes and gloves, the left center is “overshadowed” by the youngest substituent (N), and the right pairs of figures of which fit correspondingly, and if at the same time the trajectory of the successive song left and right are mirrored -doublet finite transition from the oldest to the youngest (unobscured by our generally planar-symmetrical) substituents (that is, from the first number to the figures. Crystallographers several centuries ago to the next) is similar to the movement of clock hands, noted the prevalence of mirror-like then the configuration is absolutely right (r), if the two enantiomorphic forms in the inorganic world are the mouth, then it is absolutely left (l). left and right crystals of quartz, tourmaline, calcite (Iceland spar). Having introduced ideas about absolutely left and right configurations, we must warn about Mirror isomerism, enantiomerism, in the organic relativity of this absoluteness. Zerce chemical operations are a very common phenomenon. cal reflection corresponds to P – inversion of the co- The priority of its discovery in the middle of the past table- ordinates of all atomic and subatomic parts of the object. tion belongs to the outstanding Frenchman Louis Pas- However, since the internal structure of Theur, who drew attention to the mirror similarity of atomic (electrons) and subatomic (quarks, gluoforms of crystals of potassium-ammonium tartaric salts) particles, is unknown, the operation P of physics is complemented by phoric acids. The name of Pasteur is associated with the formation of the small operation of charge conjugation C - stereochemistry, based on the problems of sim- change to opposite signs of charges and all the geometry and asymmetry of molecules, their structure (shape) and other antipodeal quantum characteristics of atoms. three-dimensional space . An important milestone in developmental (protons, neutrons, electrons) and subatomic stereochemistry was the proposed in 1874 of particles (quarks, gluons), as well as the operation of J. Van't Hoff and J. Le Bel tetrahedral inversion of the directions of motion (momentums and mo - model of the carbon atom. If in the simplest carbon-momentum pulses) of all components of the object, co-rode, the figure of which is similar to the high-momentum corresponds to the reversal of time T. Poetometric tetrahedron - methane hydrogen atoms indeed the limiting inversion is to successively replace (replace ) others - combined CRT surgery. From this it follows, between atomic groups and groups, that the symmetry that is the absolute antipode of the original one, for example, of the resulting molecules quickly decreases. After the r-molecule there must be its l-partner, but consisting of three such procedures, four different substituents are already connected to the tetrahedral carbon of the antimatter and the center moving in time, and in reverse. Ideas for combining P-, C- and T-operators ygTspzyZau g.i. ezyYyyEkDbaTs lnkyTsza a oike eigTsdmg ykYDzauTsldap lyTSSazTszav 47 symmetries belong to G. Lüders and W. Pauli to unite to an infinitely symmetric sphere, then everything (1954–1955). the symmetry elements of the original object are degraded due to the gigantic possibilities vary completely, that is, the asymmetric “adding” of atoms and atomic groups capable of bonding transforms a perfectly symmetrical (singlet) with carbon, an infinitely realizable in principle object into the class of enantiomeric doublets. However, one should not assume that enantiomerism of molecules with asymmetric carbon centers is impossible among symmetrical figures (molecules). mi. Let us note their fundamental feature: let us remember a simple regularity: whether a mirror asymmetric carbon or another atom, configurational isomerism, a truly impossible center can be placed as a substituent in the rows of objects (molecules) that have at each of the vertices of a highly symmetrical object (on - the quality of internal elements of symmetry plane - for example, tetrahedron, cuban; Table 1) and even specular reflection suction σ (S1) and/or mirror reflection - Table 2 C2 C2 R L . . (CH 2) n (CH 2) n R, L-trans-cyclooctenes R, L-trans-cycloethylenes C2 C2 C2 C2 C2 C2 Twistan R L Z Z -biphenyls of symmetry C2 Z Z R L -triphenylmethanes of symmetry C3 L R C2 R Hexagelicene L Spirals, springs, screws, screws, nuts, bolts 48 lykylZldav jEkDbyZDnTsgzhv LmkzDg, ‹2, 1997 rotary axes i (S2, 3, 4, ...). When such d are reflected, they are topologically chiral molecules (their shapes by the outer mirror plane are nans, nodes in Scheme 4). copy objects identical to the original ones (see graph However, the convention of partitioning is obvious in the light of diagrams 1, 2 and Table 1). On the contrary, if the structure developed by R. Kahn, K. Ingold and V. Prelonie objects (molecules) is characterized by the absence of rules for assigning enantiomeric configurations (σ, i), supplemented by their successors internal mirror symmetry elements Sn, but they are symmetrical relative to rotations of molecules to R- or L-rows, these are circular, spi- Cn (n = 2, 3, 4, ...), then such figures are always R, L-dual (chiral) movements along (R) or against flying. The simplest example is 1,3-dimethyl-3-cumu-(L) clock hands with sequential distribution (Table 1) and all its homologues with an odd number depending on the “seniority” (weight) of substituents, carbon ra- atoms in linear circuit. In table 2 shown (scheme 3) around the atomic center - we have some R, L-doublets from a large set of a, selected plane - b (trans-cycloethylenes, molecules symmetric with respect to rotations. Table 2), when going around the contours of the propellers - c , vin- Note that they do not have asymmetries at all - r, nodes - d in the table. 2, in the diagram 4. there are many carbon centers. In technology, the molecules of biphenyls and triphenylmethyls are similar to the shapes of fan blades, propellers, and turbine rotors; The figures of helicene molecules are similar to spirals, springs, screws, left and right threading of screws. For brief description of the discussed phenomenon at the turn of the 19th and 20th centuries, Lord Kelvin Trefoiled knot (CH2)m, with oriented and non-minimum m = 66, proposed the term “chirality” (from the Greek χειρ - hand). identical rings In Russian, two variants of pronunciation and spelling of this term are used: chirality and Scheme 4 chirality. The author, together with physicists, gives preference to the first. Conjugated by the operation of mirror reflection (coordinate inversion P) can- Therefore, strictly speaking, there are no molecules - components of enantiomeric doublets - qualitatively different kinds chirality of molecules. differ only in one property - pro- For example, called topological chiralty opposite signs +(R) and −(L) of the rotation angle d in diagram 4 is just a reflection of that structure of the plane of polarization of light. Similar to the features of the depicted molecules, that antipodal (+, −) relationships are also characteristic; their individual parts are held together not by chemical - for the poles of magnets, charges and other quantum bonds, but by the topology of the structure of the chains (as - characteristics of atomic and subatomic particles. Such tenanes ), closed spirals and knots; their chiral relationship is called by physicists the chiral sym- (R, L) form is quite similar to the propellemetric form. ditch – in and spirals – g. Therefore, all of the above types of chirality of molecules are quantitatively, thanks to the efforts of synthetic chemists, uniformly: the sign (+, −) and quantified by the problems of stereostructure, over the last rank of the angle of rotation of the plane of polarization for decades have become known and available at a very wide variety of wavelengths of light. numerous, including exotic, types of cy- However, it is also known that at polyral molecules (see Table 2 and Scheme 4). It is accepted that the condensation of centrally chiral (r or l) ami- consider that the diversity of chiral chemical acids, ribonucleotides, the total chirality of rest compounds is divided into five types in the corresponding polymer (protein, DNA) cannot be assessed with symmetrical structural features : trivial summation of individual chia – molecules with a chiral center do not have unit ralities: Σrn(ln) . This sum “volume- no symmetry elements, except for the element letsya” spiral (helical) chirality identity C1 (examples - amino acids (ala- macromolecules, which have their own sign (+R h, −Lh) and nin in Scheme 3), sugars-carbohydrates ); absolute value, b – planar-chiral molecules of symmetry Nr (l) ∑ l (r) ⊂ R (L). h h C1 and/or C2 (the selected structural element is the plane n n (1) bone, examples are trans-cycloethylenes in Table 2); The fact that the regular ortho-condensation of Akiva – axially chiral molecules from symmetrical benzene rings also leads to spirium Cn (have the shape of propellers or swastikas, ral helicenes (Table 2), only confirmed examples - twistane, biphenyl, triphenylmethane in gives general rule : and circular union of the table. 2, etc.); achiral monomers of a suitable structure, and d – helical-chiral molecules of symmetry linear polycondensation of chiral (only r C2 (characteristic shape is a helix, examples are hexa- or only l) units automatically lead to spigelicene in Table 2, proteins, DNA) ; ral form of the polymer. It can be assumed that in ygTspzyZau g.i. In 49 rows of such macromolecules, a certain symmetry of chiralities is realized, which corresponds to the configuration Sn, configurationally unambiguous (singlet), hierarchy of levels of stereostructure. For example, per- since their internal structure is P-even. Objective, secondary, tertiary and quaternary levels of the structure of hemoglobin, which do not have internal P-parity of structure, are obviously character- (not having symmetry elements Sn), are always inter- terized by the sequences “nested-figurationally two-valued (doublet, left + chiralities” type (1) sum of individual chiralities). In order to obtain from a P-even object its co-amino acids into the helical chirality of polypeppia, one Pσ(i)-operation is sufficient, but in order to copy these two into the “globular” chirality of a P-odd object, two tertiary levels are needed, finally, these three - into “super-sequential P-operations: lecular” chirality of a quartet (tetrahedron) of united globules. From here, by the way, it follows that the stereochemistry of polymers and their associates should take into account in addition to those listed. Note, however, that not everyone around us also has the “globular” - e and “supramolecular” us in living and inanimate nature P-odd” – types of chirality. It is the following objects that you can easily find the left or right primary (structural) upper configurations of twin partners, for example, the selected tree levels of organization of macromolecules play in the forest or a stone from a pile of rubble. Let us further note that the decisive role in their functioning in the body is that chiral symmetry is absolutely (100%) important. Thus, biochemical reactions with the participation of P-odd molecules of organic enzymes are effectively carried out only in compounds that are part of all living organ- cases when previously realized com- isms on our planet. If these are amino acids, breeding, that is, “recognition,” selection of those molecules is only left (l); if sugars are carbohydrates, then only a cool of reagents and substrates, features config-right (r); if these are biopolymers, then they are spirals of which (“figures” of which) are ideally but twisted only to the right (proteins, DNA). This is consistent with the contours and shapes of a corresponding pattern called chiral asymmetric cavities in enzyme globules. The everyday anaria of the biosphere was also the first to draw attention to the log of such complementation by D. Koshland, who proposed L. Pasteur. lived to consider the correspondence of the key and the lock. ganTskDnmkD dakDguzD DlaeeTsnka 1. General organic chemistry: Trans. from English M.: Chemistry, Let us summarize the above. This article is pre- 1981–1986. T. 1–12. the goal should be to show that in the boundless on the per- 2. Zhdanov Yu.A. Carbon and life. Rostov n/d: Publishing house view on the material of organic chemistry easier than the Russian State University, 1968. 131 p. to navigate if you master the principles of graph- 3. Tatt U. Graph theory. M.: Mir, 1988. most images general characteristics structure of molecules, as well as principles of their evaluation 4. Sokolov V.I. Introduction to theoretical stereo-configurations – shapes in three-dimensional spatial chemistry. M.: Nauka, 1982; Advances in chemistry. 1973. T. 42. ve - based on the ideas of symmetry and asymmetry. Po- pp. 1037–1051. the latter include ideas about the main 5. Nogradi M. Stereochemistry. M.: Mir, 1984. symmetry speakers: planes, axes and mirror- 6. Hargittai I., Hargittai M. Symmetry through the eyes of rotary axes used in identifying those chemists. M.: Mir, 1989. internal features of the structure of molecules, which 7. Filippovich I.V., Sorokina N.I. // Let's make progress. others define them appearance, form and ultimately biology. 1983. T. 95. pp. 163–178. their most important properties. When “sorting” molecules into symmetric and * * * asymmetric, a special role belongs to the mirror reflection operator – coordinate inversion Lev Petrovich Olekhnovich, Doctor of Chemical Sciences R. Operator Pσ coordinates of all parts (atouk, professor, head of department chemistry of natural and mov) object located to the left of the selected high-molecular compounds of the Rostov go-plane, puts it in unambiguous correspondence of the cooperative university, head. laboratory of the dinata of the inverted (reflected) object of the internal dynamics of molecules of the Faculty of Chemistry and the Research Institute of Phy- to the right of this plane. The operator Pi carries out the sic and organic chemistry of the Russian State University, the corresponding term is a similar coordinate inversion of the relative-pondent Russian Academy natural sciences. but a point chosen outside the object (it’s easy to figure out Area of ​​scientific interests: organic synthesis, and check that under the action of the Pi operator, also the kinetics and mechanisms of molecular rearrangements, a mirror double of the object is obtained, but verified, stereochemistry and stereodynamics. Co-author 180°). Objects (molecules), two monographs and author of more than 370 scientific articles. 50 likes Zldav yEkDbyZDnTsg'zkhv LmkzDg, No. 2, 1997

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 use 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 the ironic entry of graphs into theoretical chemistry is associated with quantum chemical calculation methods electronic structure molecules.
The main section discusses methods for analyzing molecular structures in terms of graphs, which are then used to construct topological indices and based on structure-property correlations, and also outlines 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 condition of minimum free energy and depends on temperature and pressure. Every 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 preserved for quite a long time, although its shape and size may change. With a decrease in temperature or an increase in pressure, the mobility of these subsystems decreases, but the movement of nuclei (zero oscillations) does not stop at absolute zero temperature. Such stable coherent formations, consisting of a small number of molecules, can exist in a liquid, in a bunk or in a solid and are called molecular systems.
MG in a perspective projection reflects the main features of the geometry of the molecule and gives visual representation about its structure. Let us discuss some types of molecular structures in MG terms. Let us consider molecules for which it is convenient to use planar graph implementations to describe their structure. 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 for such compounds is CH,g+2. CH+2 molecules in the ground state are usually flat. 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 tg 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 paraffin hydrocarbons, there are molecules whose structures can be described in terms of graphs having only vertices of degree four and one. The simplest example of such a system is cyclohexane (see Fig. 1.3,6). Typically, the structure of cyclohexane is described as MG in a perspective image, while omitting the vertices of degree one. For cyclohexape, the existence of three rotary isomers is possible (Fig. 1.7).

Moreover, for the last 12 years of his life, Euler was seriously ill, went blind, and, despite his serious illness, continued to work and create. Statistical calculations show that Euler made on average one discovery per week. Difficult to find math problem, which would not have been touched upon in Euler's works. All mathematicians of subsequent generations studied with Euler in one way or another, and it was not without reason that the famous French scientist P.S. Laplace said: “Read Euler, he is the teacher of us all.” Lagrange says: "If you really love mathematics, read Euler; the presentation of his works is remarkable for its amazing clarity and accuracy." Indeed, the elegance of his calculations was brought to the highest degree. Condorcet concluded his speech at the Academy in memory of Euler in the following words: “So, Euler stopped living and calculating!” Living to calculate - how boring it seems from the outside! It is customary to imagine a mathematician as dry and deaf to everything everyday, to what interests ordinary people. Named after Euler, is the problem of three houses and three wells.

GRAPH THEORY

One of the branches of the topology. A graph is a geometric diagram that is a system of lines connecting certain points. The points are called vertices, and the lines connecting them are called edges (or arcs). All graph theory problems can be solved both in graphical and matrix form. In the case of writing in matrix form, the possibility of transmitting a message from a given vertex to another is denoted by one, and its absence is denoted by zero.

The origin of Graph Theory in the 18th century. associated with mathematical puzzles, but a particularly strong impetus for its development was given in the 19th century. and mainly in the 20th century, when the possibilities of its practical applications were discovered: for calculating radio-electronic circuits, solving the so-called. transport tasks, etc. Since the 50s. Graph theory is increasingly used in social psychology and sociology.

In the field of Graph Theory, one should mention the works of F. Harry, J. Kemeny, K. Flament, J. Snell, J. French, R. Norman, O. Oyser, A. Beivelas, R. Weiss, etc. In the USSR, according to T. g. work Φ. M. Borodkin et al.

The Graph Theory language is well suited for analyzing various kinds of structures and transferring states. In accordance with this, we can distinguish following types sociological and socio-psychological problems solved using Graph Theory.

1) Formalization and construction of a general structural model of a social object at different levels of its complexity. For example, a structural diagram of an organization, sociograms, comparison of kinship systems in different societies, analysis of the role structure of groups, etc. We can consider that the role structure includes three components: persons, positions (in a simplified version - positions) and tasks performed in a given position. Each component can be represented as a graph:

It is possible to combine all three graphs for all positions or only for one, and as a result we get a clear idea of ​​the specific structure of the c.l. this role. Thus, for the role of position P5 we have a graph (Fig.). Weaving informal relations into the specified formal structure will significantly complicate the graph, but it will be a more accurate copy of reality.

2) Analysis of the resulting model, identification of structural units (subsystems) in it and study of their connections. In this way, for example, subsystems in large organizations can be distinguished.

3) Studying the levels of the structure of hierarchical organizations: the number of levels, the number of connections going from one level to another and from one person to another. Based on this, the following tasks are solved:

a) quantities. assessing the weight (status) of an individual in a hierarchical organization. One of the possible options for determining status is the formula:

where r (p) is the status of a certain person p, k is the value of the level of subordination, defined as the smallest number of steps from a given person to his subordinate, nk is the number of persons at a given level k. For example, in the organization represented by the following. Count:

weight a=1·2+2·7+3·4=28; 6=1·3+2·3=9, etc.

b) determination of the group leader. The leader is usually characterized by greater connectedness with the rest of the group compared to others. As in the previous task, various methods can also be used here to identify the leader.

The simplest method is given by the formula: r=Σdxy/Σdqx, i.e. the quotient of dividing the sum of all distances of each person to all others by the sum of the distances of a given individual to all others.

4) Analysis of the effectiveness of this system, which also includes tasks such as searching for the optimal structure of the organization, increasing group cohesion, analysis social system in terms of its sustainability; study of information flows (transmission of messages when solving problems, the influence of group members on each other in the process of uniting the group); with the help of technology, they solve the problem of finding an optimal communication network.

When applied to Graph Theory, as well as to any mathematical apparatus, it is true that the basic principles for solving a problem are set by a substantive theory (in this case, sociology).

Task : Three neighbors have three common wells. Is it possible to build non-intersecting paths from each house to each well? Paths cannot pass through wells and houses (Fig. 1).

Rice. 1. To the problem of houses and wells.

To solve this problem, we will use a theorem proven by Euler in 1752, which is one of the main ones in graph theory. The first work on graph theory belongs to Leonhard Euler (1736), although the term “graph” was first introduced in 1936 by the Hungarian mathematician Dénes König. Graphs were called diagrams consisting of points and segments of straight lines or curves connecting these points.

Theorem. If a polygon is divided into a finite number of polygons such that any two polygons have no or no partition common points, or have common vertices, or have common edges, then the equality holds

B - P + G = 1, (*)

where B is the total number of vertices, P is the total number of edges, G is the number of polygons (faces).

Proof. Let us prove that the equality does not change if a diagonal is drawn in some polygon of a given partition (Fig. 2, a).

A) b)

Indeed, after drawing such a diagonal, the new partition will have B vertices, P+1 edges, and the number of polygons will increase by one. Therefore, we have

B - (P + 1) + (G+1) = B – P + G.

Using this property, we draw diagonals that split the incoming polygons into triangles, and for the resulting partition we show the feasibility of the relation.

To do this, we will sequentially remove external edges, reducing the number of triangles. In this case, two cases are possible:

to remove triangle ABC, you need to remove two edges, in our case AB and BC;

To remove triangle MKN, you need to remove one edge, in our case MN.

In both cases the equality will not change. For example, in the first case, after removing the triangle, the graph will consist of B-1 vertices, P-2 edges and G-1 polygon:

(B - 1) - (P + 2) + (G -1) = B – P + G.

Thus, removing one triangle does not change the equality.

Continuing this process of removing triangles, we will eventually arrive at a partition consisting of a single triangle. For such a partition B = 3, P = 3, G = 1 and, therefore,

This means that equality also holds for the original partition, from which we finally obtain that the relation is valid for this partition of the polygon.

Note that Euler's relation does not depend on the shape of the polygons. Polygons can be deformed, enlarged, reduced, or even their sides bent, as long as the sides do not break. Euler's relation will not change.

Let us now proceed to solving the problem of three houses and three wells.

Solution. Let's assume that this can be done. Let us mark the houses with points D1, D2, D3, and the wells with points K1, K2, K3 (Fig. 1). We connect each house point with each well point. We get nine edges that do not intersect in pairs.

These edges form a polygon on the plane, divided into smaller polygons. Therefore, for this partition the Euler relation B - P + G = 1 must be satisfied.

Let's add one more face to the faces under consideration - the outer part of the plane in relation to the polygon. Then the Euler relation will take the form B - P + G = 2, with B = 6 and P = 9.

Therefore, Г = 5. Each of the five faces has at least four edges, since, according to the conditions of the problem, none of the paths should directly connect two houses or two wells. Since each edge lies on exactly two faces, the number of edges must be at least (5 4)/2 = 10, which contradicts the condition that their number is 9.

The resulting contradiction shows that the answer to the problem is negative - it is impossible to draw non-intersecting paths from each house to each village

Graph Theory in Chemistry

Application of graph theory to the construction and analysis of various classes of chemical and chemical-technological graphs, which are also called topology, models, i.e. models that take into account only the nature of the connections between the vertices. The arcs (edges) and vertices of these graphs reflect chemical and chemical-technological concepts, phenomena, processes or objects and, accordingly, qualitative and quantitative relationships or certain relationships between them.

Theoretical problems. Chemical graphs make it possible to predict chemical transformations, explain the essence and systematize some basic concepts of chemistry: structure, configuration, confirmations, quantum mechanical and statistical-mechanical interactions of molecules, isomerism, etc. Chemical graphs include molecular, bipartite and signal graphs of kinetic reaction equations. Molecular graphs, used in stereochemistry and structural topology, chemistry of clusters, polymers, etc., are undirected graphs that display the structure of molecules. The vertices and edges of these graphs correspond to the corresponding atoms and chemical bonds between them.

In stereochemistry org. c-c, molecular trees are most often used - spanning trees of molecular graphs that contain only all the vertices corresponding to atoms. Compiling sets of molecular trees and establishing their isomorphism makes it possible to determine molecular structures and find full number isomers of alkanes, alkenes and alkynes. Molecular graphs make it possible to reduce problems related to coding, nomenclature and structural features(branching, cyclicity, etc.) of molecules of various compounds, to the analysis and comparison of purely mathematical features and properties of molecular graphs and their trees, as well as their corresponding matrices. To identify the number of correlations between the structure of molecules and the physicochemical (including pharmacological) properties of compounds, more than 20 so-called ones have been developed. Topological indices of molecules (Wiener, Balaban, Hosoya, Plata, Randic, etc.), which are determined using matrices and numerical characteristics molecular trees. For example, the Wiener index W = (m3 + m)/6, where m is the number of vertices corresponding to C atoms, correlates with molecular volumes and refractions, enthalpies of formation, viscosity, surface tension, chromatographic constants of compounds, octane numbers of hydrocarbons and even physiol. activity of drugs. Important parameters of molecular graphs used to determine the tautomeric forms of a given substance and their reactivity, as well as in the classification of amino acids, nucleic acids, carbohydrates and other complex natural compounds are medium and full (H) information capacities. Analysis of molecular graphs of polymers, the vertices of which correspond to monomer units, and the edges correspond to chemical bonds between them, makes it possible to explain, for example, the effects of excluded volume leading to qualities. changes in the predicted properties of polymers. Using Graph Theory and artificial intelligence principles, it has been developed software information retrieval systems in chemistry, as well as automated systems for identifying molecular structures and rational planning of organic synthesis. For the practical implementation on a computer of operations for selecting rational chemical paths. transformations based on the retrosynthetic and syntonic principles use multi-level branched search graphs for solution options, the vertices of which correspond to the molecular graphs of reagents and products, and the arcs depict transformations.

To solve multidimensional problems of analysis and optimization of chemical technological systems (CTS), the following chemical technological graphs are used: flow, information flow, signal and reliability graphs. For study in chemistry. physics of disturbances in systems consisting of large number particles, use the so-called. Feynman diagrams are graphs, the vertices of which correspond to the elementary interactions of physical particles, the edges of their paths after collisions. In particular, these graphs make it possible to study the mechanisms of oscillatory reactions and determine the stability of reaction systems. Material flow graphs display changes expenses in-in in CTS. Thermal flow graphs display heat balances in CTS; the vertices of the graphs correspond to devices in which the heat consumption of physical flows changes, and, in addition, to the sources and sinks of thermal energy of the system; arcs correspond to physical and fictitious (physical-chemical energy conversion in devices) heat flows, and the weights of the arcs are equal to the enthalpies of the flows. Material and thermal graphs are used to compile programs for the automated development of algorithms for solving systems of equations for material and heat balances of complex chemical systems. Information flow graphs display the logical information structure of systems of mathematical equations. XTS models; are used to develop optimal algorithms for calculating these systems. A bipartite information graph is an undirected or directed graph whose vertices correspond respectively. equations fl -f6 and variables q1 – V, and the branches reflect their relationship. Information graph – a digraph depicting the order of solving equations; the vertices of the graph correspond to these equations, sources and receivers of XTS information, and the branches correspond to information. variables. Signal graphs correspond linear systems equations mathematical models chemical-technological processes and systems. Reliability graphs are used to calculate various reliability indicators X.

References :

1.Berge K., T. g. and its application, translation from French, M., 1962;

2. Kemeny J., Snell J., Thompson J., Introduction to Finite Mathematics, trans. from English, 2nd ed., M., 1963;

3.Ope O., Graphs and their application, trans. from English, M., 1965;

4. Belykh O.V., Belyaev E.V., Possibilities of using technology in sociology, in: Man and Society, vol. 1, [L.], 1966;

5. Quantitative methods in sociological research, M., 1966; Belyaev E.V., Problems of sociological measurements, "VF", 1967, No. 7; Bavelas. Communication patterns in task oriented groups, in the book. Lerner D., Lass well H., Political sciences, Stanford, 1951;

6. Kemeny J. G., Snell J., Mathematical models in the social sciences, N. Y., 1962; Filament C., Applications of graph theory to group structure, N. Y., 1963; Оeser Ο. A., Hararu F., Role structures and description in terms of graph theory, in the book: Biddle V., Thomas E. J., Role theory: concepts and research, N. Y., 1966. E. Belyaev. Leningrad.