Adv Syst Sci Appl 2020; 01:104–113
Published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/860

Comparison of Some Molecular Structures by Minimizing the
Comparison Function

Natalia Chernikova1*, Eugeniy Laneev1

1Peoples’ Friendship University of Russia, Moscow, Russia

Abstract: The geometric difference in the relative arrangement of atoms in molecules with
the same chemical formula significantly affects the properties of the crystal. For quantitative
comparison of the spatial geometric structure of two molecules, we used the method of optimal
superposition of molecules through the minimization of a certain comparison function by rotating
the molecules after superposing their centers of mass. The search for the minimum of the
comparison function over Euler rotation angles is performed by the Rosenbrock method. The
assumed criterion for comparing molecules allows us to quantify the proximity in the spatial
structure of molecules. The implementation of the method is shown by comparing the structure
of eight molecules in four crystalline substances.

Keywords: conformation, Euler angles, rotation matrix, optimization methods, Rosenbrock
method

1. INTRODUCTION

The paper deals with the mathematical problem that arises in structural chemistry, namely: the
numerical study of the spatial structure of molecules with the same chemical formula, which
may differ in the geometry of the mutual arrangement of atoms. The study of specific crystals
was carried out in the framework of a mathematical model for comparing two molecules,
studied in [1, 2]. The specified model is reduced to a comparison of two objects consisting
of N ordered points with rigid geometry that behave like rigid bodies in R3. The comparison
principle in this model is based on optimizing the alignment of these two objects by shifting
and rotating. The optimal superposition of these objects involves the minimization over
shifts and rotation angles of a certain comparison function of the geometry of the objects,
which is the sum of the squares of the distances between the points of two objects with
the same indices. It is proved that the minimum over shifts is achieved by superposing
some characteristic points, conventionally called centers of mass of molecules. To minimize
the [3–5] comparison function over rotation angles, the Rosenbrock zero-order method [6]
is used. The results obtained are used to study the geometry of molecules in real crystal
structures.

Four organic crystalline substances (I, II, III, IV) are investigated. Substances II-IV consist
of molecules with the same chemical formula and bond structure (same graphs), substance
I is only a part of this graph. In addition, substances II and III contain symmetrically
independent molecules (IIa, IIb, IIc, IId, and IIIa, IIIb) that may have the same graph, but
may differ in spatial structure, i.e., in the geometry of the arrangement of atoms. The problem
arises of objectively comparing the geometry in the space of all these eight molecules and

∗Corresponding author: cherni@list.ru



COMPARISON OF SOME MOLECULAR STRUCTURES... 105

quantifying the differences that exist in them, since the standard characteristics – bond lengths
(interatomic distances) and valence angles (angles between valence bonds) do not always
show differences in the geometry of molecules. In addition, it may be necessary to compare
fragments of different molecules. For this purpose, [1, 2] provides a method for pairwise
quantitative comparison of the geometry of molecules based on minimizing the comparison
function by shifting and rotating the molecules. Here is a brief description of the comparison
method.

2. ASSUMED MATHEMATICAL MODEL

The study of molecules with the same chemical formula by comparing their geometries uses
the concept of structure. Here is the definition of the structure adopted in [1, 2]

Definition 2.1:
A geometric structure, or in short – structure, will mean a rigid geometric construction of
N ordered points in R3 with coordinates (xi, yi, zi), i = 1, ..., N , whose movement in R3 is
performed like the motion of a rigid body.

We assume that each i−th point of the structure has a weight coefficient wi > 0, and∑N
i=1wi = W > 0. Let ik, k = 1, ..., K – all indices corresponding to non-zero weight

coefficients wi. Then

W =
N∑
i=1

wi =
K∑
k=1

wik > 0. (2.1)

In accordance with [1, 2], we will associate the optimal criterion for superimposing two
geometric structures with the minimum over shifts and rotation angles (Euler angles [7]) of
the comparison function of the form

U(r1,0, r2,0, ϕ, θ, ψ) =
N∑
i=1

wi|r1,i − r1,0 −Q(r2,i − r2,0)|2, (2.2)

where the vectors r1,i and r2,i determine the position of points in the first and second structure,
the vectors r1,0 and r2,0 determine the offsets of the corresponding points of the first and
second structure from the coordinate center, Q = Q(ϕ, θ, ψ) — the rotation matrix for Euler
angles (ψ — precession angle, θ — nutation angle, ϕ — self rotation angle):

Q =

(
cosψ cosϕ− sinψ sinϕ cos θ − cosψ sinϕ− sinψ cosϕ cos θ sinψ sin θ
sinψ cosϕ+ cosψ sinϕ cos θ − sinψ sinϕ+ cosψ cosϕ cos θ − cosψ sin θ
sinϕ sin θ cosϕ sin θ cos θ

)
(2.3)

Thus, the U function of the form (2.2) is the sum of the squares of the distances between
“eponymous” points of two geometric structures with weights wi after matching points
defined by vectors r1,0 and r2,0, with the coordinate center and rotation of the second structure
relative to the first.

By matching some characteristic points of two geometric structures, called “centers of
mass” you can reduce the task of minimizing the comparison function over a full set of
variables to finding the minimum of the U function over the rotation angles ϕ, θ, ψ.

The following theorem is proved in [1].

Theorem 2.1:
The minimum of the U function of the form (2.2) is achieved at the point corresponding to

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106 N. CHERNIKOVA, E. LANEEV

“center of mass” of two geometric structures and is defined by vectors

rj,0 =
1

W

N∑
i=1

wirj,i, j = 1, 2. (2.4)

So, the minimum of the function (2.2) corresponds to the shift of “centers of mass” of
geometric structures defined by the formula (2.4) to the origin. The function (2.2) can now be
considered as a function of rotation angles

U(ϕ, θ, ψ) =
N∑
i=1

wi|r1,i − r1,0 −Q(ϕ, θ, ψ)(r2,i − r2,0)|2, (2.5)

where r1,i, r2,i – given coordinates of geometric structures, and “centers of mass” r1,0 and r2,0
are calculated using the formula (2.4). The minimum of the function (2.5) is determined by
the rotation angles ϕ, θ, ψ.

Let the minimum of the function U = U(ϕ, θ, ψ) of the form (2.5) be reached at the point
(ϕ0, θ0, ψ0). In accordance with [1, 2], we will call the value

s =
(U(ϕ0, θ0, ψ0)

W

)1/2
. (2.6)

the measure of proximity of two geometric structures. This value can be considered as a
quantitative characteristic of the proximity of geometric structures, since it represents the
average distance between points with the same indices in two structures after “superposing”.

We will call two geometric structures equal if they can be superposed by points with the
same indices by various displacements of these structures as solids. As shown in [1], this
superposition is achieved by minimizing the comparison function (2.5) and corresponds to
s = 0.

We will call two structures approximately equal if

s =
( 1

W
min
ϕ,θ,ψ

U(ϕ, θ, ψ)
)1/2

=
(U(ϕ0, θ0, ψ0)

W

)1/2
6 s0, (2.7)

where s0 is the specified value (in applications, it is determined by the specific practical
situation). The inequality s 6 s0 will be called the criterion of proximity of structures.

Practical experience in studying the conformation of molecules, based on the results of
comparing a significant number of structures [8, 9], resulted in the following conditional
classification: s 6 s0 = 0, 1Å – molecules are practically equal, 0, 1Å < s 6 0, 2Å –
molecules are close, s > 0, 2Å – molecules are different.

Note that if the minimum point is not unique and there is a point (ϕ1, θ1, ψ1) such that
U(ϕ1, θ1, ψ1) = U(ϕ0, θ0, ψ0), then the value of s does not change and the non-uniqueness of
the minimum point does not affect the proximity criterion of structures.

The test for comparing the geometry of two geometric structures consists of three stages:
1) shifting the center of mass of each geometric structure to the coordinate center; 2)
minimizing the function (2.5) over angles; 3) calculating the value of s using the formula
(2.7) and inferring the proximity of the structures. Additional information is provided by
residual values ∆ri = |r1,i − r2,i|.

When comparing two structures in accordance with the algorithm 1)-3), determination of
the minimum of the comparison function U(ϕ, θ, ψ) over rotation angles after superposing
the centers of mass of structures is performed numerically.

Copyright © 2020 ASSA. Adv Syst Sci Appl (2020)



COMPARISON OF SOME MOLECULAR STRUCTURES... 107

Fig. 3.1. Molecule C14H13NO2 4,6-dimethyl-3-(phenylcarbonyl) pyridin-2 (1H) -one (I).

3. NUMERICAL STUDY OF CRYSTAL STRUCTURES

This section presents the results of the numerical study comparing chemically identical
molecules in a crystal by minimizing the comparison function (2.5). In this case, we will
consider molecules with an ordered structure ofN points-atoms as a geometric structure. The
coordinates of the molecules in the crystal are obtained from X-ray analysis. In this study, the
combined atoms in the molecules are assigned a weight of wi = 1. The remaining atoms are
assigned a weight of wi = 0. The coordinates of the hydrogen atoms were not included in
the calculations, since the accuracy of their determination is lower than that of other atoms
and their presence is insignificant in the present task of superposing molecules, so they are
assigned a weight of wi = 0.

When comparing molecules, we will be interested in differences in the spatial structure
of molecules with a single structural formula (graph). Since the comparison function (2.5)
is not convex, the Rosenbrock zero-order method [6] is used to numerically minimize it.
The program that implements the comparison algorithm 1)-3) uses the optimization library
module [10] to minimize the comparison function (2.5) using the Rosenbrock method.

In this paper, we set the task of comparing the spatial structure of 4,6-dimethyl-3-
(phenylcarbonyl) pyridin-2 (1H) -one (I) molecules with the chemical formula C14H13NO2

where N = 30 and in 3-benzoyl-4,6crystals-dimethyl-1-[2-(4-methylphenyl)-2-oxoethyl]
pyridin-2 (1H)-one (II, III, IV) with the chemical formula C23H21NO3, where N = 48. The
composition, bonds and geometry of these molecules are schematically depicted in Fig. 3.1
and Fig. 3.2.

The atomic coordinates were obtained by the method of X-ray analysis [11]. The
characteristics of these substances in crystallography are described as follows:

I C14H13NO2 space group P21/c, Z = 4 (molecule occupies 1 position);
II C23H21NO3 space group PI, Z = 8 (four independent molecules a, b, c, d);
III C23H21NO3 · 12H2O (two independent molecules a, b);
IV C23H21NO3 · 12H2O space group C 2/c, Z = 8 (molecule occupies one position).
Space group [12] describes all possible symmetries of an infinite number of periodically

located points in three-dimensional space. The number Z shows the number of molecules in
the unit cell. In crystals I and IV, the molecules occupy one system of positions, that is, they
are transformed into each other using symmetry operations of the space group. In crystals
II and III, the molecules occupy independent positions, respectively, four (IIa,IIb,IIc,IId)
and two (IIIa,IIIb), that is, the molecules are not transformed into each other by symmetry
operations of the space group of the crystal and therefore can be different in structure.

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108 N. CHERNIKOVA, E. LANEEV

Fig. 3.2. Molecule C23H21NO3,3-benzoyl-4,6crystals-dimethyl-1-[2-(4-methylphenyl)-2-oxoethyl] pyridin-2
(1H)-one (II, III, IV)

Table 3.1. Comparison of four independent molecules a, b, c, d in crystals II

The value of “residuals” ∆ri, characteristic s in crystals II
Atom a− b a− c a− d b− c b− c b− d c− d
N 1 0.20 0.06 0.03 0.23 0.05 0.18 0.08
C 2 0.19 0.16 0.04 0.34 0.08 0.15 0.20
C 3 0.06 0.15 0.05 0.20 0.02 0.03 0.19
C 4 0.05 0.10 0.06 0.09 0.03 0.10 0.16
C 5 0.02 0.08 0.05 0.08 0.08 0.07 0.11
C 7 0.05 0.06 0.01 0.04 0.04 0.04 0.04
C 8 0.19 0.03 0.03 0.17 0.13 0.16 0.02
C 9 0.03 0.10 0.03 0.08 0.06 0.08 0.08
O 10 0.27 0.22 0.04 0.47 0.07 0.29 0.23
C 11 0.07 0.24 0.04 0.30 0.14 0.04 0.26
O 12 0.22 0.35 0.08 0.56 0.30* 0.14 0.42
C 13 0.09 0.15 0.06 0.24 0.44* 0.13 0.12
C 14 0.30 0.35 0.06 0.65 0.69* 0.24 0.41
C 16 0.42 0.46 0.16 0.88 0.92* 0.26 0.63
C 18 0.37 0.41 0.15 0.78 0.96* 0.22 0.56
C 20 0.23 0.24 0.06 0.45 0.77* 0.18 0.30
C 22 0.12 0.09 0.07 0.19 0.53* 0.09 0.10
C 24 0.18 0.15 0.11 0.20 0.03 0.24 0.26
C 33 0.16 0.12 0.08 0.20 0.34* 0.15 0.12
O 34 0.31 0.28 0.15 0.39 0.56* 0.22 0.21
C 35 0.10 0.10 0.06 0.18 0.50* 0.14 0.14
C 36 0.04 0.29 0.04 0.27 0.40* 0.06 0.32
C 38 0.13 0.33 0.04 0.38 0.65* 0.12 0.30
C 40 0.24 0.14 0.06 0.37 0.91* 0.21 0.17
C 41 0.28 0.25 0.06 0.43 1.06* 0.24 0.26
C 43 0.19 0.24 0.08 0.35 0.88* 0.18 0.21
C 45 0.32 0.27 0.11 0.57 1.06* 0.35 0.22

s 0.21 0.23 0.08 0.40 0.08 0.18 0.27

Copyright © 2020 ASSA. Adv Syst Sci Appl (2020)



COMPARISON OF SOME MOLECULAR STRUCTURES... 109

Note that a comparison of bond lengths and bond angles does not reveal differences in
the spatial structure of molecules. In eponymic bonds, the differences in length are no more
than 0,02Å, and in bond angles no more than 5◦. Some differences are observed in individual
torsion angles - up to 19◦; however, how substantially this affects the general geometry of
the molecules is not always clear. Thus, it is not known in advance whether these molecules
are similar in structure or whether their differences are significant. In this situation, there is a
need for an objective, quantitative, comparison of the structure of molecules.

For this purpose, the technique described in the previous section was applied to the study
of these molecules. Comparison of molecules was carried out by minimizing the comparison
function (2.5). As already mentioned, the coordinates r1,i, r2,i, i = 1, ..., 48 (for substances II
- IV), i = 1, ..., 30 (for substance I) for each pair of compared molecules were obtained on
based on x-ray analysis data. All crystals (I - IV) are centrosymmetric (both left and right
molecules are present), therefore, to compare some pairs, one of the molecules was taken
with inversion. The centers of mass were determined by the formula (2.4). The comparison
function (2.5) was minimized numerically using the Rosenbrock method over three Euler
angles after superposing the centers of mass of the two molecules being compared.

The essential thing in our method of quantitative comparison of the spatial structure of
molecules is the variation in the values of the weight factors wi. The ability to superpose
molecules with separate fragments is achieved by setting the values of wi = 1 for compatible
atoms and wi = 0 for other atoms. This allows you to change the center of gravity of the
molecules, combine the molecules along the closest or most significant parts and thereby
more clearly visualise certain differences in the structure of the molecules and their parts. As
already mentioned, in our calculations we set the values wi = 0 for hydrogen atoms H.

In table 3.1 we present the results of applying the comparison algorithm 1) - 3) for
matching 4 symmetrically independent molecules (a, b, c, d) that coexist in one crystalline
substance II. Here, the residuals ∆ri = |r1,i − r2,i|, i = 1, ..., 48, i.e., the distances between
the eponymic atoms in the two molecules under comparison are calculated after matching (at
the point of minimum of the comparison function U ), as well as magnitude s – magnitude
measure of proximity. Results are shown for non-hydrogen atoms. The (*) indicates atoms
for which wi = 0 was taken in this calculation.

Let us analyze the data obtained. In practice, only the pair of a-d molecules (s = 0.08Å)
are practically equal in their spatial structure; another pair of b− dmolecules (s = 0.18Å) are
close. Other pairs of molecules have greater or lesser differences. The largest difference in the
structure of molecules, is in the b− c pair (s = 0.40Å). For this pair, another calculation was
performed in which only for atoms of the central pyridine ring and atoms directly connected
with it with indices i = 1, ..., 11, 24 and weight wi = 1. In this comparison, s = 0, 08Å,
which means the equality of these fragments and, in addition, this comparison highlighted
the differences in the rotation of the phenyl rings. The result of this comparison is presented
in Fig. 3.3.

As the calculations showed, two symmetrically independent molecules a and b in
crystal III (s = 0.04Å), as well as pairs of molecules IV and IIIa (s = 0.03Å), IV and
IIIb (s = 0.04Å) are almost identical. Calculations comparing molecules from IV with four
independent molecules from crystal II indicated their significant differences in geometry. To
identify differences for the same pairs of molecules, additional calculations were performed
with varying weight factors. The ∆ri values for atoms with wi = 0 are also marked with (*).
The calculation results are given in Table 3.2.

The results of the comparison point to parts of the molecules where there are differences
and make us look for geometric characteristics to visually describe these differences. These
characteristics were found as the differences in the angles between the planes (. 3.3) - plane 1:
through the atoms NC2C3C4C5C6, plane 2: through the atoms C13C14C16C18C20C22, plane
3: through the atoms (C35C36C38C40C41C43).

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110 N. CHERNIKOVA, E. LANEEV

Fig. 3.3. The best combination of molecules IIb and IIc with wi = 1 for i = 1, ..., 11, 24.

Table 3.2. Comparison of a molecule from crystal IV with a, b, c, d molecules in crystal II.

The value of “residuals” ∆ri, characteristic s
Atom IV &IIa IV &IIa IV &IIb IV &IIb IV &IIc IV &IIc IV &IId IV &IId
N 1 0.43 0.04 0.38 0.03 0.41 0.02 0.41 0.02
C 2 0.31 0.01 0.31 0.03 0.38 0.05 0.29 0.04
C 3 0.22 0.02 0.16 0.04 0.33 0.02 0.17 0.05
C 4 0.35 0.06 0,40 0.04 0.43 0.04 0.29 0.02
C 5 0.44 0.06 0.47 0.06 0.47 0.08 0.40 0.05
C 7 0.45 0.01 0.41 0.04 0.40 0.06 0.44 0.04
C 8 0.49 0.06 0.46 0.13 0.50 0.03 0.51 0.06
C 9 0.57 0.10 0.57 0.09 0.55 0.13 0.58 0.07
O 10 0.43 0.04 0.50 0.03 0.40 0.06 0.44 0.05
C 11 0.13 0.08 0.17 0.06 0.22 0.17 0.13 0.13
O 12 0.08 0.29* 0.21 0.21* 0.36 0.50* 0.08 0.29*
C 13 0.43 0.34* 0.34 0.29* 0.56 0.72* 0.45 0.34*
C 14 0.27 0.17* 0.40 0.10* 0.48 0.69* 0.27 0.20*
C 16 0.42 0.40* 0.27 0.27* 0.84 1.14* 0.32 0.26*
C 18 0.99 0.92* 0.62 0.85* 1.40 1.66* 0.84 0.72*
C 20 1.36 1.15* 1.22 1.21* 1.60 1.73* 1.32 1.04*
C 22 1.06 0.85* 1.05 0.90* 1.15 1.25* 1.11 0.84*
C 24 0.42 0.07 0.61 0.07 0.51 0.05 0.38 0.04
C 33 0.30 0.45* 0.31 0.54* 0.39 0.29* 0.36 0.37*
O 34 0.35 0.46* 0.35 0.54* 0.60 0.09* 0.40 0.30*
C 35 0.26 0.84* 0.20 1.05* 0.36 0.68* 0.27 0.80*
C 36 0.18 0.83* 0.20 2.38* 0.35 0.93* 0.16 0.84*
C 38 0.39 1.28* 0.35 2.54* 0.69 1.37* 0.41 1.37*
C 40 0.78 1.80* 0.64 2.07* 0.88 1.53* 0.79 1.87*
C 41 0.87 1.82* 0.67 3.37* 0.81 1.30* 0.83 1.81*
C 43 0.58 1.36* 0.44 3.02* 0.55 0.88* 0.51 1.29*
C 45 1.14 2.34* 0.96 2.65* 1.38 2.18* 1.22 2.50*

s 0,60 0,05 0,54 0,06 0,72 0,08 0,59 0,05

Table 3.3. Angles (in degrees) between planes in molecules IIa, IIb, IIc, IId and molecule in crystal IV

Planes IIa IIb IIc IId IV
Planes 1 2 84.4 85.9 78.2 89.4 86.7
Planes 1 3 86.5 87.8 71.7 85.7 71.2
Planes 2 3 55.4 51.8 61.8 52.5 84.1

In addition, we present in table 3.4 the distances between the far points of the C18 and C45

molecules.
The maximum distance is observed in molecules IIc and IId, the minimum - in IV.

This illustrates the largest difference s = 0, 72Å for the IIc - IV pair. Fig. 3.4 shows the

Copyright © 2020 ASSA. Adv Syst Sci Appl (2020)



COMPARISON OF SOME MOLECULAR STRUCTURES... 111

Table 3.4. Distances (in angstroms) between the “far points” C18 and C45 of the IIa, IIb, IIc, IId molecules
in crystals II and in the crystal IV molecule.

IIa IIb IIc IId IV
Distance C18C45 12.8 12.3 13.4 12.8 11.3

Fig. 3.4. The best combination of molecules IIb and IV with wi = 1 for i = 1, ..., 11, 24.

result of combining IIc - IV with wi = 1 for the pyridine nucleus and the nearest atoms
(NC2C3C4C5C7C9C11C24O2O10).

The application of the comparison algorithm 1) - 3) also works well for comparing
molecules from I with the corresponding fragments of molecules IIa, IIb, IIc, IId, IIIa,
IIIb, IV. The results are presented in Table 3.5. The table contains the calculated residuals
∆ri = |r1,i − r2,i|, i = 1, ..., 30. The results are shown for non-hydrogen atoms (except for
H8 from crystal I).

Comparison of molecule I with fragments of molecules - IV, IIIa, IIIb gave the same
results, s = 0, 27Å (there are some differences), which is explainable, since molecules IV,
IIIa, IIIb are practically identical to each other in their geometry.

The molecule in crystal I has the biggest differences with molecule IIc, s = 0, 54Å. To
identify these differences, we use the same technique - variation of weighting factors. The
calculation by combining the planes of the pyridine rings and the directly bonded C and O
atoms gave s = 0, 17Å (parts of the fragments are close). This result indicates differences
in the relative positions of the pyridine and phenyl rings. In this regard, the angles between
plane 1 s = 0, 17Å and plane 2 (C13C14C16C18C20C22) are calculated. For the molecule of
their crystal I, the angle is 81, 1◦; the angle in IIc is the most different from it, which is 78, 2◦.
In addition, differences in the orientation of C = O bonds were found: differences in the
torsion angles of C4C3C11O12, O12C11C13C22 and C3C11C13C22 are 19, 7◦, 19.9◦ and 12.5◦,
respectively.

4. CONCLUSION

The calculations presented here show the effectiveness of the technique used to study the
geometry of molecules in crystals using the comparison function. Interesting facts were

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112 N. CHERNIKOVA, E. LANEEV

Table 3.5. Comparison of molecule I with corresponding fragments of molecules IIa, IIb, IIc, IId, IIIa, IIIb, IV
((*) marked atoms for which wi = 0 is calculated)

The value of “residuals” ∆ri, characteristic s
Atom I&IIa I&IIb I&IIc I&IIc I&IId I&IIIc I&IIIb I&IV
N 1 0.28 0.27 0.46 0.05 0.24 0.19 0.20 0.20
C 2 0.37 0.39 0.45 0.07 0.36 0.30 0.31 0.31
C 3 0.20 0.23 0.26 0.04 0.19 0.15 0.13 0.13
C 4 0.33 0.33 0.45 0.04 0.29 0.15 0.14 0.15
C 5 0.37 0.37 0.45 0.08 0.34 0.24 0.24 0.24
C 7 0.17 0.24 0.21 0.10 0.18 0.13 0.15 0.14

() 8* 0.78* 0.68* 1.03* 0.52* 0.77* 0.77* 0.83* 0.69*
C 9 0.45 0.53 0.44 0.10 0.44 0.44 0.44 0.45
O 10 0.70 0.71 0.83 0.10 0.61 0.56 0.56 0.57
C 11 0.38 0.37 0.55 0.06 0.37 0.17 0.18 0.19
O 12 0.78 0.75 0.96 0.73* 0.75 0.37 0.37 0.36
C 13 0.10 0.13 0.03 0.97* 0.05 0.05 0.05 0.04
C 14 0.31 0.41 0.15 1.19* 0.33 0.14 0.13 0.12
C 16 0.16 0.23 0.17 1.96* 0.19 0.23 0.23 0.23
C 18 0.55 0.55 0.63 2.57* 0.46 0.31 0.31 0.31
C 20 0.73 0.83 0.78 2.39* 0.73 0.25 0.26 0.25
C 22 0.40 0.47 0.41 1,55* 0.45 0,08 0.09 0.09
C 24 0.62 0.56 0.88 0.08* 0.49 0.28 0.27 0.27
s 0.45 0.47 0.54 0.17 0.42 0.27 0.27 0.27

obtained - 3-benzoyl-4,6-dimethyl-1- [2- (4-methylphenyl) -2-oxoethyl] pyridin-2 (1H) -
one molecules have the same conformation in different crystals of III and IV, and different
conformation - in one crystal II.

ACKNOWLEDGEMENTS

The above calculations were carried out by the authors using the program COMPARISON
in the language C], written by N.V. Bakhtadze, to whom the authors express their sincere
gratitude.

The work was supported by the Russian Foundation for Basic Research (grant 17-51-
12064, grant 18-01-00106, grant 18-01-00590).

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	Introduction
	Assumed mathematical model
	Numerical study of crystal structures
	Conclusion