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MOLECULAR PACKING IN CRYSTALS OF ORGANIC COMPOUNDS
A. I. Kitaigorodskii
1. IONIC AND MOLECULAR LATTICES
Ten to fifteen years after the study of X-ray diffraction patterns became a method for determining the structure of a crystal, it proved possible to divide all crystalline structures into two classes: the first, in which atoms or radicals had to be regarded as the building elements, and the second, for which it was natural to choose the molecule of the substance as the structural unit. The necessity for such a division was dictated by the entirely dissimilar spatial relationships of the atoms of different molecules in the solid state.
Fig. 1.
In such compounds as, for example, NaCl or CaCO₃ (Fig. 1, a and b), it is not possible to single out in the solid state a separate group of atoms forming the gaseous molecule. In an NaCl crystal, each Cl atom is surrounded by six neighbors, as is each Na atom. All the neighbors are completely equivalent, i.e., they are at equal distances from the given atom; there is no such pair of atoms that would be bound especially strongly and could be regarded as forming a separate molecule.
as a certain whole. In the crystal CaCO₃ the oxygen atoms are connected with a definite C atom—this means that each C atom has three nearest oxygen atoms in accordance with the chemical formula, and that an oxygen atom comes especially close to only one carbon atom. Thus the CO₃ radical exists in the crystal as a certain whole. However, there do not exist two particularly close Ca atoms and CO₃ groups; each Ca atom has, at equal distances and with the same orientation relative to it, 6 CO₃ groups, and correspondingly each CO₃ group is surrounded by the same number of Ca atoms situated at equal distances from it. In the crystal there is no CaCO₃ molecule.
Lattices of the opposite type are called molecular. Fig. 2 shows the structure of a crystal of solid CO₂. Each carbon atom has two especially close oxygen neighbors; each oxygen atom is bonded (i.e. is especially close) to a definite C atom. Thus the CO₂ group is naturally singled out, and it is expedient to regard the molecule as the element of the crystal structure. In all crystals of organic compounds (not to speak of salts of organic bases) we encounter analogous structures, i.e. the presence of interatomic distances of two types—intramolecular and intermolecular. The group of atoms belonging to a given molecule is singled out very sharply. If the interatomic distances in organic compounds vary within the limits 1.3–1.7 Å, then the distances between atoms of different molecules are of the order of 2–4 Å.
Fig. 2.
2. TASKS AND ACCURACY OF METHODS OF STRUCTURAL ANALYSIS OF ORGANIC COMPOUNDS
At the present time the structure of a very large number (several thousand) of organic compounds has been investigated. There are two principal methods for this purpose—electron diffraction in the gas phase and X-ray diffraction on a crystal. The first method has the advantage that its experimental data are directly connected with the structure of an individual molecule. The curve of the intensity distribution of scattered electron radiation is connected with the structure of the molecule by the formula
\[ I \sim \sum_{i=1}^{m}\sum_{j=1}^{m} f_i f_j \frac{\sin x_{ij}}{x_{ij}}, \]
where \(x_{ij}=4\pi l_{ij}\dfrac{\sin\theta}{\lambda}\), \(l_{ij}\) is the distance between atoms \(i\) and \(j\), \(2\theta\) is the angle of deflection, \(\lambda\) is the wavelength, and \(f\) is the atomic factor.
As is evident from the formula, it is possible to calculate, for various models, the theoretical intensity curve and to compare it with experiment. By this method it is sometimes possible to select, from among several probable structures, one as certain. However, the results of investigations of this type may often be called into question. First, as always in any trial-and-error method, there is the question whether the true model of the molecule has remained untested; and second—and this is the main point—the experimental results are too meager to permit the unique model to be chosen with confidence. Indeed, as a rule, 4–5 not very sharply expressed maxima are observed on a photograph, whereas the number of parameters determining the structure of a more or less complex molecule (valence angles and interatomic distances) may amount to several tens. The electron-diffraction method gives satisfactory results only in the case when only one or two structural parameters are in doubt. The very rich experimental material obtained in this field should therefore be treated with great caution.
The method of X-ray diffraction, as applied to a crystal, gives quite different results. The experimental data are the angles at which the diffracted rays arise, and the intensities of these rays. The geometry of the diffraction pattern almost always makes it possible to determine the size and shape of the elementary cell of the crystal with an accuracy not worse than \(0.01\ \text{Å}\) (for the usual period dimensions of \(10\)–\(20\ \text{Å}\)) and the angles with an accuracy not worse than \(0.1^\circ\). These data acquire especially great significance in those cases in which the lengths of the cell periods are at the same time the distances between atoms. Interatomic distances obtained in such a direct manner may be regarded as an indisputable experimental result.
Of course, in molecular lattices there are no periods whose dimensions would be equal to interatomic distances. If, in the crystal chemistry of inorganic compounds, almost the entire scheme of ionic radii is constructed on the basis of indisputable measurements of lattice periods, then in the crystal chemistry of organic compounds we can rely only on two limiting figures with respect to the carbon—carbon distance. Measurements of the lattices of diamond and graphite lead to the following data—the foundation of organic structural chemistry: in diamond the carbon atoms are arranged tetrahedrally at a distance of \(1.54\ \text{Å}\) (Fig. 3, \(a\)); in graphite (Fig. 3, \(b\)) they are arranged trigonally, with a distance of \(1.42\ \text{Å}\) within the layer and \(3.40\ \text{Å}\) between the atomic layers.
All other data concerning the structure of organic molecules can in no way claim analogous reliability. They are based
on the results of analysis of the intensity of the scattered radiation. We have already discussed above why these data leave one with a feeling of uncertainty in the case of the electron-diffraction method. Analysis of the intensity of X-ray diffracted radiation is much more complicated and suffers from other shortcomings. The experimental material in this case is incomparably richer. Depending on the size of the crystal that can be obtained and on the proximity of the temperature of the investigation to the melting temperature, the number of diffraction rays (reflections) that can practically be measured ranges from a hundred to a thousand. The measured intensities are related to the crystal structure as follows: the electron density \(\rho(x,y,z)\) is expressed by a three-dimensional Fourier series
Fig. 3.
\[ \rho(x,y,z)=\frac{1}{V}\sum_{h=-\infty}^{+\infty}\sum_{k=-\infty}^{+\infty}\sum_{l=-\infty}^{+\infty} F(hkl)\exp 2\pi i(hx+ky+lz), \]
where \(F^2(hkl)\) is proportional to the intensity \(I(hkl)\), and \(h\), \(k\), and \(l\) are the orders of the three-dimensional diffraction.
Thus, the experiment does not permit the structure to be determined directly—the phases or signs of the coefficients of the series must be guessed or determined from other indirect considerations.
There are a number of methods that make it possible to a considerable extent to circumvent the indicated difficulty; nevertheless, the correct structure can only be guessed, or, just as in the electron-diffraction method, with the aid of experience a proposed model can be tested or rejected (for a given model, here too one can calculate the theoretical intensity). The trial method in the study of the structure of a crystal is immeasurably more complicated than the same method applied to gases. If in the latter case one can sometimes be satisfied with a dozen molecular models, then in the case of a crystal it is necessary, for each molecule, to try an enormous number of possibilities for the mutual arrangement of the molecules. Thus the cen-
A. I. KITAYGORODSKII
the usefulness of the trial method for organic compounds, and still more the conclusiveness of the solutions found with its aid, becomes doubtful.
If one reviews studies of the structure of organic single crystals,* then we shall see that in all cases 2–3 models of the molecule with variation of one parameter were tried, and by very laborious and lengthy work—by trials of all possible orientations of the molecules relative to the axes of the cell—the true structure was established. English investigators usually assert that in this way they have determined the structure of the molecule with great accuracy (of the order of \(0.01 \ \text{\AA}\) and \(0.1^\circ\)); this assertion seems to us incorrect. As a result of investigations of this type one can only assert that the proposed model is not in contradiction with experiment. It can be proved that many models of molecules with interatomic distances varying within \(\pm 0.03 \ \text{\AA}\), and with orientations varying within several degrees, satisfy the experiment to the extent to which it was carried out by these authors.
There is a very important case when a quite objective analysis of the structure is possible (without the trial method), namely, when among the atoms of the molecule there are one or two possessing a considerably large weight. Since the scattering power of a heavy atom is considerably higher than that of light atoms, it will almost always be a valid assumption that the signs of the coefficients of the Fourier series are determined by this atom. The localization of one heavy atom is easily carried out with the aid of the Patterson series,*) and thus the structure is established objectively without the trial method.
The Patterson method is incomparably simpler and more convincing than the trial method, but the accuracy of both methods is ultimately the same. In both cases the coordinates of the atom are found as the coordinates of a maximum of electron density. If \(\varepsilon\) is the mean value of the derivative of the electron density near the maximum with respect to distance,**) then the accuracy \(\Delta x\) in determining the coordinate of an atom in the elementary cell is expressed, as was shown by us,\({}^{4}\) by the following formula:
\[ \Delta x = \frac{0.2}{\varepsilon} \cdot bnZ^{2}\sum f. \]
Here \(n\) is the number of independent atoms (atoms of the microtwin), \(Z\)—
* The Patterson series is constructed on the basis of experiment according to the formula
\[ A(u,v,w)=\sum_{hkl=-\infty}^{+\infty}\sum\sum F^{2}(hkl)\exp 2\pi i(hu+kv+lw). \]
The maximum of this series with coordinates \(u, v, w\) indicates, as was shown by Patterson,\({}^{3}\) the presence in the cell of an interatomic vector with projections \(u, v, w\). In this case the magnitude of the maximum is proportional to the product of the atomic numbers of the atoms between which the vector has been drawn.
** If \(\Delta \bar{\rho}\) is the mean difference between the value of the electron density at the maximum and at neighboring points and \(d_{0}\) is the size of the intervals into which the period of the cell is divided when calculating the series, then \(\varepsilon=\Delta \bar{\rho}/d_{0}\).
multiplicity of position (the number of microdynes in the cell), \(f\) is the square of the atomic factor, whose value must be summed for all measured reflections from the crystal, and \(b\) is the error of the intensity measurement. Analysis of this formula and investigation of particular cases lead to the conclusion that, in the best case, i.e., at the smallest values of \(b\) and at the attainable sharpness of the maximum (large \(s\)), an accuracy of \(0.01\ \text{Å}\) can be achieved only with exceptionally careful work using Mo radiation, making it possible to draw upon, for the construction of a series of electron-density maps, experimental material at least 2–3 times larger than has been done hitherto in most investigations.
Thus an analysis of all the material obtained concerning the structure of organic molecules by the X-ray structural method, and an analysis of the accuracy of this method, permit the following conclusions to be drawn: 1) in almost all investigations carried out up to the present, the accuracy in determining interatomic distances does not exceed \(\pm 0.03\ \text{Å}\), and for the most part is even less; 2) obtaining even such results by the trial-and-error method is, for organic compounds, associated with enormous labor. There is a considerable percentage of erroneous investigations; the majority of investigations leave a feeling of uncertainty as to the correctness of the results obtained; 3) the Patterson method gives objective results and is, strictly speaking, the only way of establishing the structure of a molecule.
Organic chemistry sets before the physicist investigating the structure of the molecule two types of problems. The first is the exact measurement of interatomic distance for characterizing the strength of a chemical bond; the second is the determination of the form of the molecule, the position in the nucleus to which a substituent atom is attached, etc. It should be emphasized that the first problem requires: 1) an exceptionally accurate experiment, 2) the possibility of applying the objective Patterson method. The trial-and-error method, in the form in which it has been used by English authors, is unsuitable for this purpose. For the second problem there is no need to determine the distances between atoms with particular accuracy.
The existing intensity method does not allow one to make a methodological distinction in solving these quite different problems. The cumbersome nature and duration of work by the intensity method have greatly hindered the wide application of structural analysis in organic chemistry.
3. THE PRINCIPLE OF CLOSE PACKING FOR AN ORGANIC CRYSTAL
The difficulties that existed in the structural analysis of organic compounds are explained, as it seems to us, by the complete absence until recently of any data on the rules according to which molecules fill crystalline space. Knowledge
crystallochemical rules greatly facilitates the determination of structure. In the crystallochemistry of inorganic compounds one can find many examples where the structure of a substance is predicted if the general nature of the construction of the corresponding class of compounds is known.
One of the most important guiding ideas in theoretical prediction of structure of this kind is the principle of closest packing.
In the works of N. V. Belov,^5 who developed this principle in detail, the fundamental significance of this idea for the crystallochemistry of inorganic compounds is shown.
Fig. 4.
We have recently shown^6 that for organic compounds as well it is possible to establish certain general rules, the knowledge of which greatly facilitates structural analysis; above all, the principle of closest packing can be extended to organic compounds. Of course, the packing of organic molecules cannot be considered as the packing of spheres, since the distances between atoms within a molecule are much smaller than intermolecular ones. Experience shows, however, that a definite “form” can be assigned to a molecule in the following way. To each atom there must be assigned, in addition to the atomic “radius,” which determines the distance between chemically bonded atoms, also an intermolecular radius. It determines the distance to which two atoms belonging to different molecules can approach one another. If \(R_C\) is the intermolecular radius of a carbon atom, and \(R_H\) that of hydrogen, then it is impossible for these two atoms to approach to a distance smaller than \(R_H + R_C\); if this distance is equal to \(R_H + R_C\) or greater than it, then we shall say that the molecules are, or are not, in contact by the corresponding atoms.
With the aid of intermolecular radii the external contours of a molecule can be constructed, as is shown for naphthalene in Fig. 4.
The unconditional overlap of the spheres of the intermolecular radii of two adjacent chemically bonded atoms has no physical meaning. The overlap of these spheres of two chemically unbonded atoms,
belonging to one molecule may occur. The latter assumption is based on the unquestionable fact that intermolecular forces are considerably smaller in magnitude than the forces of the chemical bond. It is also possible that this overlap has a reverse influence on the chemical forces (the steric factor). Fig. 5 shows the shape of a paradiphenylbenzene molecule, illustrating this situation.
Fig. 5.
As can be seen from Figs. 5 and 4, the shape of the molecule is bounded by a surface having hollows and protrusions. Dense packing of molecules therefore consists in the protrusions of one molecule entering the hollows of another. In the example of the structure of the dimethylnaphthalene crystal (Fig. 6) this is especially clearly visible.
Fig. 6.
On the basis of the idea of dense packing it proved possible to develop a geometrical method of structural analysis^7, which makes it possible to find the orientation of a molecule in a crystal, provided that the elementary cell and the symmetry are known. This same method can be used to find the values of intermolecular radii. For this purpose spatial models of molecules are made and arranged, with the aid of a special device, so that their centers are located at distances following from the dimensions found and from the symmetry of the cell. Usually only one molecule in the cell is in an independent position. It has 3 degrees of freedom,
if its center is located at the center of symmetry of the crystal, or, in the general case, by 6 degrees of freedom. Assigning arbitrary values to these three (or six) parameters, we determine the mutual arrangement of the molecules (since the remaining molecules of the cell are related to the first by symmetry operations). Only very few values of the molecule’s degrees of freedom will prove compatible with the dimensions and shape of the molecule. And from these positions, if there are several of them, we shall choose the one in which the maximum number of contacts between molecules is realized (the idea of closest packing). In this way the orientation of the molecules can be found if their shape is known (in general outline).
Fig. 7a.
In carrying out the geometrical analysis we use models whose dimensions are established with an accuracy up to 0.1 Å. Even with a still rougher implementation of the method it is always clear, as experience shows, from geometrical considerations which atom of one molecule is in contact with the given atom of another. Therefore one can find the values of the intermolecular radii with an accuracy up to 0.01 Å, if all equations of contact are solved jointly. In hydrocarbon molecules possessing a center of symmetry, we shall encounter 5 unknowns (three degrees of freedom of the molecule and two intermolecular radii \(R_C\) and \(R_H\)); if the number of contacts is 5, then the problem is solved, and if it is greater than 5, then it has an independent check. In this way, for a large number of compounds, the following values of the intermolecular radii have been established with an accuracy of the order of 0.02 Å:
\[ R_C = 1.72,\qquad R_H = 1.17. \]
It seems to us that proof of the correctness of the ideas expressed above is already provided by the very fact that it is possible to solve these equations. However, the validity of the structures found was also checked and confirmed by the intensity method.
Let us describe the course of the geometrical analysis on models. In Fig. 7a the “structure finder” is shown in its initial position. Let us assume, for definiteness, that we are dealing with a rhombic crystal in whose cell there are 4 molecules. Along the main rod of the instrument
the molecule holders are arranged so that the distance between the centers of the molecules is equal to the smallest period. First of all
Fig. 7б.
Fig. 7в.
all possible positions of two molecules connected by this translation are considered. There will, however, be a continuous multitude of such arrangements; by bringing up to the first two molecules a third (Fig. 7б),
associated with the first symmetry operation, we find that only in a single (or a few) cases can the molecules be arranged densely. Bringing the fourth molecule (Fig. 7c) to an experimental distance (third period), we check the correctness of the arrangement found—there must be close contact.
It remains only to refine the Euler angles. For this purpose two coordinate systems are introduced into consideration: \(X_0Y_0Z_0\), associated with the molecule, and \(XYZ\), associated with the crystal. For each atom of the molecule there are the equations, given in any course of analytic geometry:
\[ \begin{aligned} X &= X_0(\cos\varphi \cos\psi-\sin\varphi \sin\psi \cos\vartheta)+{}\\ &\quad +Y_0(-\cos\varphi \sin\psi-\sin\varphi \cos\psi \cos\vartheta)+Z_0\sin\varphi \sin\vartheta,\\ Y &= X_0(\sin\varphi \cos\psi+\cos\varphi \sin\psi \cos\vartheta)+{}\\ &\quad +Y_0(-\sin\varphi \sin\psi+\cos\varphi \cos\psi \cos\vartheta)-Z_0\cos\varphi \sin\vartheta,\\ Z &= X_0\sin\psi \sin\vartheta+Y_0\cos\psi \sin\vartheta+Z_0\cos\vartheta . \end{aligned} \]
Considering the arrangement of the molecules on the apparatus, we note which atoms are in contact, and we set up an equation of the type
\[ (X_1-X_2)^2+(Y_1-Y_2)^2+(Z_1-Z_2)^2=\text{const}. \]
Solving these equations, as indicated above, we find the values of the Euler angles, the values of the constants, and can even refine the values of the coordinates \(X_0Y_0Z_0\). The idea of close packing is well confirmed by the study of a series of \(\beta\)-derivatives and 2,6-derivatives of naphthalene. All the compounds may be regarded as packings of layers of naphthalene nuclei. Depending on the character of the substituent, the orientation of the molecules in the layer, as well as the character of the superposition of the layers, may change somewhat. The molecules in the layer are packed densely, and in such a way that the atomic groups attached to the nucleus project out of the layer. All \(\beta\)-derivatives possess cells very close in shape and nearly coincident sections in the plane of the layer. The \(\alpha\)-derivatives and 1,5-derivatives have cells of an entirely different form. This is quite natural, since the position of the groups in the \(\alpha\)-position compels the molecules in the layer to pack with an entirely different orientation, determined again by the requirement that the atomic groups attached to the nucleus project out of the layer.
The significance of geometrical analysis for organic crystal chemistry is very great. This method makes it possible, on the basis of cell measurements alone, to find the mutual arrangement of the molecules. If the form of the molecule has been chosen incorrectly, then geometrical analysis will show its impossibility. In many cases the solution of problems of the second type (see the end of the preceding paragraph) becomes possible solely on the basis of measurements of the size and symmetry of the cell. If
...a desirably exact determination of interatomic distances, here too geometrical analysis plays a major role, replacing, at the preliminary stage of the investigation, the trial-and-error method and freeing the investigator from enormous and wearisome labor.
4. CONSTRUCTION OF AN IDEAL CRYSTAL
There are various ways of representing the internal symmetry (structure) of crystals. It seems to us that, for organic compounds, the division of crystalline space into islands of zero, one, and two dimensions (properly speaking, islands, chains, and nets), proposed by Weissenberg and developed by us,^8 is especially expedient.
One may mentally construct a crystal in the following way: a certain element of the substance (part of a molecule, a molecule, or several molecules)—let us call it a microdynade—is repeated by a symmetry operation along a line. A chain arises. In turn, the chain is translated to form a net. And, finally, by means of a third translation, a crystal arises.
The basic idea of the structure of a crystal, confirmed by all the many years of experience in X-ray structural analysis and by an enormous number of other facts, is “order at long range.” This means that the position of the microdynade with respect to three translating symmetry elements uniquely determines the position of all atoms of the crystal. The atoms of the microdynade itself, by definition, are not connected with one another by symmetry operations belonging to the crystal (see the end of the paragraph).
Since we do not set ourselves the task of systematically deriving the space groups of crystals, but only wish to set forth the foundations of the doctrine of symmetry in a form suitable for the problem under consideration, we shall confine ourselves to considering the symmetry elements encountered in the lower crystal systems, i.e., in the triclinic, monoclinic, and rhombic systems. As will be seen from what follows, this restriction corresponds also to the essence of the subject, since among organic compounds crystals of the higher systems occur only as exceptions and are not very typical.
By introducing this restriction, we reduce the number of translating symmetry elements to three: these are simple translation, a glide plane, and a twofold screw axis. If \(x, y, z\) are the coordinates of points of the microdynade and the translation occurs along the \(Y\) axis, then under simple translation the points of each subsequent microdynade have coordinates \(x, y + a, z\), while under translation by a glide plane they have coordinates \(\bar{x}, y + \dfrac{a}{2}, z\) (in this case the direction \(y\) is called the glide axis), and under translation by a screw axis \(\bar{x}, y + \dfrac{a}{2}, \bar{z}\). The origin of coordinates
is assumed to lie on the translation axis. The quantity \(a\) is the period of repetition. In Fig. 8 these three types of translations are shown; a wedge is a plane triangle, shaded on one side; the translation axes lie in the plane of the drawing, while the glide plane is perpendicular to the plane of the drawing.
Fig. 8.
We shall denote these three simplest chains by the symbols \(a\), \(\tilde a\), and \(2_1\).
Translating the chains at an angle to their axes, we obtain various nets. It is not difficult to see that the angle between two translations will be arbitrary only for the following combinations: \(aa\) (two simple translations), \(a\tilde a\) (a simple translation and a glide axis). In the second case, the simple translation must lie in the glide plane. The remaining translations can lie only at right angles to one another. Thus the nets \(2_1a\), \(2_1\tilde a\), \(2_12_1\), \(\tilde a\tilde a\) have rectangular cells. All six types of nets are shown in Fig. 9. We shall not rigorously prove the impossibility of oblique-angled nets of the last four types. These proofs are very simple—
Fig. 9.
...; we shall confine ourselves only to Fig. 10, which shows that the translations \(2_1\) and \(a\), proceeding at an arbitrary angle, do not form an infinite lattice in which a finite number of elements falls per unit volume: the second screw axis is carried in the first row from the third triangle \(3_1\), in the fifth row by the third screw axis \(5_1\), and so on.
Attention must be paid to one more detail, important for what follows. In forming a network from chains, additional symmetry elements may arise. Thus, for example, Fig. 9 shows that, by shifting the chain \(2_1\) by a simple translation at an angle of ninety degrees, we create a new twofold screw axis in the middle between the principal chains.
Fig. 10.
degrees, we create a new twofold screw axis in the middle between the principal chains.
In forming a three-dimensional lattice, certain requirements are likewise imposed on the direction of the third translation. There exists only one lattice in which all three translations may proceed at arbitrary angles: this is the simple triclinic lattice \(aaa\). An oblique network \(aa\) may, in addition, be superposed by a twofold screw axis proceeding at a right angle to the network, which gives the monoclinic lattice \(aa2_1\). Monoclinic lattices are also obtained if oblique networks \(aa\) are superposed: by a simple translation—the lattice \(aa\tilde a\), and by a twofold screw axis—the lattice \(aa\tilde 2_1\).
Figure 11 depicts these lattices, the simplest in symmetry. The accepted symbol for the simple triclinic lattice is \(P1\) (\(P\)—primitive, \(1\)—absence of symmetry). A monoclinic lattice with glide planes is denoted \(Pa\), with screw axes \(P2_1\), and with both these symmetry elements \(P2_1/a\). In a triclinic lattice one microdiad enters the unit cell; in lattices \(Pa\) and \(P2_1\) there are two, and in the lattice \(P2_1/a\)—four. The unit cell is, as is known, that elementary volume by whose simple translation
from which the entire crystal is constructed. In the cells under consideration new symmetry elements have formed: in the lattice \(Pa\), additional glide planes (midway between the “basic” ones), and in the lattice \(P2_1\), additional axes. The presence of additional symmetry elements is not difficult to detect analytically. In the cell \(Pa\) there are
Fig. 11.
two microdynads, the points of which are related by the relation
\(x,y,z \longrightarrow x+\dfrac{a}{2}, \bar y,z\). The basic glide plane has the equation \(y=0\). Taking into account the existence of the point \(x, y-b, z\), we see that the point \(x+\dfrac{a}{2}, \bar y, z\) is related to it by a glide operation in the plane whose equation is \(y=\dfrac{b}{2}\).
In the group \(P2_1/a\), in addition to the additional axes and planes, yet another new symmetry element arises. In order to make this circumstance evident, let us choose the origin at a point lying midway between the screw axes and the glide planes. We may construct the lattice either starting from the oblique net \(\tilde a a\), or from the rectangular \(2_1a\). Let us follow, at least, the first path. The coordinates of the points of the microdynads of the net \(\tilde a a\), with the indicated choice of origin, will be related by the relation
\(x,y,z \longrightarrow \dfrac{a}{2}+x,\dfrac{b}{2}-y,z\)
(the equation of the glide plane is \(y=\dfrac{b}{4}\)). With the aid of a screw axis parallel to the \(y\)-axis and passing through the point \(x=\dfrac{a}{4}\) and \(z=0\),
we shall obtain two from each point by carrying out the operation \(\dfrac{a}{2}\) (...),
\[ (\ldots)-\frac{b}{2},\ (-1)\times(\ldots). \]
The result is as follows: \(x,y,z;\ \dfrac{a}{2}+x,\dfrac{b}{2}-y,z;\ \dfrac{a}{2}-x,\ y-\dfrac{b}{2},\ \bar z;\ \bar x,\bar y,\bar z.\) We see that the point lying between the axes and planes, chosen by us as the origin of coordinates, is a center of inversion (the relation \(x,y,z \to \bar x,\bar y,\bar z\)).
We now proceed to consider space lattices constructed on rectangular nets. As we showed above, there are four such nets: \(2_{1}a,\ 2_{1}\tilde a,\ 2_{1}2_{1}\), and \(\tilde a\tilde a\).
A rectangular net \(2_{1}a\) leads either to the same monoclinic lattices or forms rhombic cells (all three translations have different periods and are directed at right angles to one another). The remaining rectangular nets can lead only to rhombic lattices. It is not difficult to prove that the third translation can be directed only in the indicated way by making constructions analogous to Fig. 10.
Fig. 12.
When translating the net, the place where the plane of the net is intersected by the third translating element becomes essential. Let us explain this by an example. Suppose we wish to construct a crystal on the net \(2_{1}a\), displacing them in the direction normal to the net with the aid of one more screw axis \(2_{1}\). This screw axis cannot intersect the plane of the net at an arbitrary point: if this were so, we would not be able to construct cells with a finite number of elements (a proof analogous to Fig. 10). Two cases are possible, shown in Fig. 12: a) the third axis intersects the axes of the net; b) the third axis passes midway between the axes of the net. With the choice of the origin of coordinates indicated in Fig. 5, in the first case the third axis passes through the point \(y=0,\ z=0\), and in the second—through the point
\[ y=0,\quad x=\frac{1}{4}. \]
The coordinates of the equivalent points of the basic net \(2_{1}a\) are \(x,y,z;\ \bar x, y+\dfrac{1}{2}, \bar z.\) In the first variant of the passage of the third axis, from two points the following two will arise:
\[ \begin{aligned} &1)\quad x,y,z \longrightarrow 3)\quad \bar x,\bar y,z+\frac{1}{2},\\ &2)\quad \bar x,y+\frac{1}{2},\bar z \longrightarrow 4)\quad x,\frac{1}{2}-y,\frac{1}{2}-z; \end{aligned} \]
in the second variant we obtain the set:
\[ \begin{aligned} &1)\ x,y,z \quad \longrightarrow \quad 3)\ \frac{1}{2}-x,\ \bar y,\ z+\frac{1}{2},\\ &2)\ \bar x,\ y+\frac{1}{2},\bar z \quad \longrightarrow \quad 4)\ \frac{1}{2}+x,\ \frac{1}{2}-y,\ \frac{1}{2}-z. \end{aligned} \]
It is not difficult to see that the combinations obtained possess different symmetry properties. Indeed, in the first set of points a new symmetry element has arisen—a simple (not screw) twofold axis passing through the point \(y=\frac{1}{4}\), \(z=\frac{1}{4}\) and parallel to the \(X\) axis. This axis relates points 1 and 4, and also 2 and 3. In
Fig. 13.
the second set of points there has also arisen a new symmetry element, relating points 1 and 4, and also 2 and 3; but this is not a simple twofold axis, but a screw axis. Thus, depending on the place where the lattice is intersected by a third translating element, two different space groups have arisen. The first is denoted \(P2_1 2_1 2\), and the second \(P2_1 2_1 2_1\).
It is thought that what has been said is sufficient for the reader, if desired, to derive also the remaining space groups constructed with the aid of the basic translating elements of a simple translation, a glide plane, and a screw axis.
These, however, do not exhaust all the space groups, since besides these three methods of translation there exists the possibility of filling crystalline space by operations which we have called\(^8\) pseudotranslations. Within the lower syngonies we may encounter three types of pseudotranslations: these are—a linear regular sequence of centers of inversion \((\bar I)\), a row of simple twofold axes \((2)\), and a system of parallel reflection planes \((m)\) (Fig. 13).
In order to derive all possible groups, it is necessary to consider chains and lattices constructed with the aid of pseudotranslations,
and also use these operations for constructing a crystal from nets. The space groups obtained must be carefully compared with one another, since very often by different paths we arrive at the same combinations. For example, the group \(P2_1/a\), which we obtained starting from the symmetry elements \(2_1\) and \(\tilde a\) (the inversion center was a derived symmetry element), is not difficult to obtain from a net constructed on the translation \(2_1\) and the pseudotranslation \(\bar 1\) (the glide plane will be a derived element).
In order that our exposition should have a completed character at least in some part, let us carry out the derivation of all space groups of the triclinic and monoclinic systems. We shall start from oblique nets. As we saw above, the “true” translations give nets of two types, \(aa\) and \(a\tilde a\). A pseudotranslation \((m)\) cannot lead to the formation of an oblique net. Therefore the complete list of oblique nets will be as follows: \(aa,\ a\tilde a,\ a\bar 1,\ a2,\ a\tilde{\bar 1},\ \tilde a2\). The nets \(\bar 1\bar 1\) and \(22\) are identical with the nets \(a\bar 1\) and \(a2\). Superposing these nets with a simple translation, we obtain the following space groups:
\[ \begin{array}{ccccccc} aaa & a\bar 1a & a\tilde a a & a2a & a\tilde{\bar 1}a & \tilde a2a & 2\bar 1a\\ P1 & P\bar 1 & P\tilde a & P2 & P2_1/a & P2/a & P2/m\\ C_1 & C_i & C_s^2 & C_2^1 & C_{2h}^4 & C_{2h}^1 & C_{2h}^4 \end{array} \]
In the first row our designations are given, in the second—the international symbols, and in the third—the Schoenflies symbols.
It should be noted that the mutual arrangement of the center of symmetry and the twofold axis in the net \(2\bar 1\) gives the group \(P2/m\) only when the center of symmetry and the twofold axis coincide; in the remaining possible cases the net \(2\bar 1\) is identical with the net \(\tilde a2\).
The superposition of oblique nets cannot occur with the aid of a glide plane (the glide axis is perpendicular to the net) and with the aid of a pseudotranslation of a twofold screw axis; such a superposition is possible only under rhombic symmetry. Therefore only the following additional groups are possible:
\[ \begin{array}{lll} aa\bar 1=P\bar 1=C_i; & a\bar 1\bar 1=P\bar 1=C_i; & \tilde a a\bar 1=P2_1/a=C_{2h}^5;\\ a2\bar 1=P2/a=C_{2h}^4; & a\bar 1\bar 1=P2_1/a=C_{2h}^5; & \tilde a2\bar 1=A2_1/a=C_{2h}^6;\\ aa2_1=P2_1=C_2^2; & a\bar 1 2_1=P2_1/a=C_{2h}^5; & \tilde a a2_1=P2_1/a=C_{2h}^5;\\ a22_1=A2=C_2^3; & \tilde a12_1=A2/a=C_{2h}^6; & \tilde a22_1=A2/a=C_{2h}^6;\\ aam=Pm=C_s^1; & a\bar 1m=P2_1/m=C_{2h}^2; & \tilde a am=Am=C_s^4;\\ a2m=P2m=C_{2h}^1; & a\bar 1m=A2/m=C_{2h}^3; & \tilde a2m=A2/m=C_{2h}^3. \end{array} \]
Let us emphasize that to each of our symbols there corresponds not one, but several space groups, depending on the mutual arrangement of the symmetry elements. In the table only one of the variants is given; for example, the symbol \(\tilde a2_1\) may lead to the groups \(C^1_{2h}\), \(C^3_{2h}\), \(C^4_{2h}\), \(C^6_{2h}\), depending on the manner in which the centers of inversion are arranged.
From this table we see that the same space groups are obtained by different routes. The proofs reduce to finding additional symmetry elements by the method illustrated above; we therefore leave them to the reader. In addition to the two triclinic syngony groups and the four monoclinic groups given above, we find another nine groups. Thus it has been shown that in the monoclinic syngony there exist three groups containing only axes of symmetry (the dieudric axial class), four containing only planes (the diedric anaxial class), and six groups containing both kinds of elements (the monoclinic-prismatic class). Let us note that in groups containing axes and symmetry planes perpendicular to them, centers of symmetry arise.
Attention should be paid to the symbol \(A\) in the international notation: it denotes the presence of a centered face constructed on the vectors \(b\) and \(c\) of the space lattice. Looking at the table, one can discover that centering arises in the case when the group contains alternating rows of axes of two types (\(2\) and \(2_1\)) or planes of two types (\(m\) and \(a\)). By definition, the face \(bc\) is centered if in the cell, to the point \(xyz\), there corresponds the point \(x, y+\frac{1}{2}, z+\frac{1}{2}\). Take the net \(a2\)—it consists of two points \(x,y,z\) and \(\bar x, y, \bar z\) (the axis \(2\) at the origin of coordinates). Superpose the following net by means of the screw axis \(2_1\)—the only possible place for this axis is midway between the axes \(2\). Then we obtain two more points
\[ 1)\ x,y,z \longrightarrow 2)\ \bar x,\, y+\frac{1}{2},\, \frac{1}{2}-z, \]
\[ 3)\ \bar x,y,\bar z \longrightarrow 4)\ x,\, y+\frac{1}{2},\, z+\frac{1}{2}. \]
Points 1 and 4 are indeed connected by the centering operation.
Up to now we have said nothing about the symmetry of the molecule in the crystal. From what has been said above it is obvious that formally a crystal may be constructed by means of translations and pseudotranslations applied to some group of atoms called a microdynad. In this case the atoms of the microdynad are not connected with one another by symmetry operations. This means that, knowing the coordinates of one atom \(x,y,z\) (the axes \(xyz\) being chosen in the directions of the translations), nothing can be said
say about the coordinates of the other atoms of the microdynade. This does not mean that the microdynade has no symmetry, but means only that it does not possess symmetry in the crystal. In Fig. 14 an oblique symmetry net \(a\bar{1}\) is shown, constructed from trapezia shaded on one side. Each little figure has its own plane of symmetry, but this symmetry element does not belong to the net. In what follows we shall speak of such cases as the “loss by a figure of its symmetry upon formation of a crystal.”
Fig. 14.
If the microdynade is a molecule or, still more, consists of several molecules, then the molecule does not possess symmetry in the crystal. The microdynade may be a part of a molecule. This is possible only in the case when the space group contains, besides the translational elements \(a\), \(\tilde{a}\), \(2_1\), also other symmetry elements. Indeed, parts of a molecule can be connected with one another only by symmetry elements characteristic of finite figures. Such symmetry elements are the center of inversion \(\bar{1}\), a simple twofold axis \(2\), the mirror plane of symmetry \(m\), and their combinations. In crystals of the lower systems, to the consideration of which we have limited ourselves, an arrangement without loss of symmetry is in principle possible for molecules possessing the following symmetry:
\[ \text{orthorhombic system } mmm,\ mm,\ 222,\ 2/m,\ 2,\ m,\ \bar{1}, \]
\[ \text{monoclinic system } 2/m,\ 2,\ m,\ \bar{1}, \]
\[ \text{triclinic system } \bar{1}. \]
A molecule possessing the symmetry \(mmm\), i.e. of three mutually perpendicular planes, consists of eight microdynades; the point \(x, y, z\) of the first microdynade is transformed into the points of the other seven according to the rule
\[ x,y,z \to \bar{x},y,z,\ x,\bar{y},z,\ x,y,\bar{z},\ \bar{x},\bar{y},\bar{z},\ x,\bar{y},\bar{z},\ \bar{x},y,\bar{z},\ \bar{x},\bar{y},z. \]
The orthorhombic dipyramid possesses this symmetry. The symmetry \(mm\) is the symmetry of an orthorhombic pyramid. The orthorhombic tetrahedron possesses the symmetry \(222\); three mutually perpendicular axes of the second order produce from the point \(x,y,z\) the transformations
\[ \to \bar{x},y,\bar{z},\ -x,\bar{y},\bar{z},\ x,\bar{y},\bar{z}; \]
the molecule consists of four microdynades. A molecule possessing sym-
with the symmetry of a rhombic prism \(2/m\) (a twofold axis and a plane of symmetry perpendicular to it), is possible in principle in the monoclinic and rhombic systems; the point \(x, y, z\) is transformed into \(\bar{x}, y, z\), \(\bar{x}, \bar{y}, \bar{z}\), \(x, \bar{y}, \bar{z}\).
Attention should be drawn to the fact that molecules with the symmetries \(mmm\) and \(2/m\) contain, among other elements, a center of symmetry. Such a molecule may crystallize, for example, in the triclinic system with the loss of part of its symmetry. We shall see below what experience leads to.
5. DENSE LAYERS
The idea of dense packing makes it possible to find certain general regularities in the structure of an organic crystal.
At first glance the problem seems very complicated, since it concerns the packing of bodies of arbitrary shape. However, below it will become obvious that, without any limitation of the generality of the conclusions, one may approximate a molecule by an ellipsoid and consider their packing in space. One may proceed still more simply and first consider the two-dimensional problem of arranging elliptical plates.
Fig. 15a. Fig. 15b. Fig. 15c.
It will become obvious from what follows why, for organic compounds, only the three lower systems are important. Therefore it is sufficient to consider sections of space lattices with oblique and rectangular cells. The former is realized in the monoclinic and triclinic systems. Figure 15 shows the packing of elliptical plates, blackened on one side, in such a cell; the packing is maximally dense, with coordination number 6, i.e. each ellipse is in contact with six neighbors. Centrosymmetric plates are colored black on each side over one half (on the opposite side, opposite the black color, there is white).
The same arrangement is realized with different symmetry: in groups \(1\), \(Pm\) the molecules are related by a translation (Fig. 15a, lattice \(aa\)); in groups \(\bar{1}\), \(P2_1/a\)—by centers of symmetry (lattice \(a\bar{1}\), Fig. 15b) and in groups \(P2\), \(P2_1m\), \(A2/a\)—by twofold axes (lattice \(a2\)). In projection, the operations of translation and of a twofold axis look the same (Fig. 15a). If the molecule possesses a center of symmetry, then close packing in an oblique cell is possible in the groups \(\bar{1}\), \(P2_1/a\), and \(P2_1/m\) (Fig. 15b).
The second type of closest packing with coordination 6 is realized in a rectangular cell constructed by applying to the ellipse the translations \(2_1a\) or \(ba\), where \(2_1\) is a twofold screw axis lying in the section, \(a\) is a glide plane along \(a\), perpendicular to the section, and \(a\) and \(b\) are simple translations. A packing of this kind is shown in Fig. 16.
If the ellipses are centrosymmetric (Fig. 16b), then the filling of the plane occurs identically in both ways: \(2_1a\) and \(ba\).
The following space groups possess symmetry allowing close packing of this type *): \(2_1a\)—\(P2_1\), \(P2_1/c\), \(P2_12_12_1\), \(P22_12_1\), \(Pbca\), \(Pnma\), \(Pna\), \(ba\)—\(Pa\), \(Aa\), \(P2_1/a\), \(A2/a\), \(Pbcn\), \(Pcc\). For centrosymmetric figures the groups \(P2_1/a\), \(Pbca\), \(A2/a\) are suitable.
It is not difficult to see that all the remaining symmetry possibilities in constructing a layer of molecules lead to less dense packing. Figures 17–19 show packings constructed by applying to the ellipse the translations \(bn\) (Fig. 17), \(mm\) (Fig. 18), and \(ab\) (Fig. 19).
In all cases coordination 4 is realized instead of 6.
The packing density of ellipses for coordination number 6 can be calculated. It is equal to 0.905, i.e. to the same number as the coefficient of closest sphere packing, shown for comparison in Fig. 20. For a rectangular cell, as is easy to verify for the limiting cases,
\[ ab\sqrt{3}=AB, \]
where \(a\) and \(b\) are the semi-axes of the ellipse, and \(A\) and \(B\) are the semi-axes of the cell. Thus the area of the cell uniquely determines the area of the ellipse.
*) \(a, b, c\)—glide planes along the axes, \(n\)—along a diagonal, \(m\)—mirror plane, \(2_1\)—screw axis, \(2\)—rotation axis, \(P\)—primitive lattice, \(A\)—face-centered on \(bc\). The symbol \(Pbca\) means that there is a glide plane along \(b \perp a\), along \(c \perp b\), and along \(a \perp c\).
Exactly the same consequences will, of course, be obtained for plane figures of arbitrary shape.
Fig. 17.
Fig. 18.
Fig. 19.
What, then, will be true for three-dimensional figures? Undoubtedly, the considerations that led us to exclude from consideration a number of layers,
as sparse ones, remain in force also for such figures. On the contrary, for such figures the number of dense packings becomes still smaller, since layers possessing polarity perpendicular to the layer must be excluded from consideration. Of the oblique-angled nets only \(\widetilde{a1}\) is dense, and of the rectangular ones only \(2_1a\). This is illustrated in Fig. 21, which shows the view along the layer \(\widetilde{ba}\) (the axis \(b\) is perpendicular to the drawing) and the layer \(2_1\widetilde{a}\), composed of cones. For the packing of ellipsoids the symmetry layers \(\widetilde{ba}\) and \(2_1\widetilde{a}\) are equivalent; any deviation from this form makes the layer \(2_1\widetilde{a}\) more advantageous from the point of view of the possibilities of dense packing.
Fig. 20.
Fig. 21.
6. SELECTIVE DISTRIBUTION AMONG SPACE GROUPS
The idea of dense packing for organic crystals may be formulated as follows: the crystal must be constructed from layers of molecules that do not possess polarity perpendicular to the layer and have coordination number 6; moreover, the layers must also be packed as densely as possible with one another.
The second part of this rule leads to important consequences. Of the space groups mentioned above, convenient for the formation of a crystal are only those in which the layers are superposed in the densest manner. Layers cannot be superposed upon one another by an operation of mirror symmetry (and only as an exception—by means of a twofold rotation axis); in this case the protrusions of one layer would have to fall on the protrusions of another.
Selecting the corresponding space groups, we find that the probable groups in which the majority of organic compounds should crystallize are the groups: \(\overline{1}\), \(P2_1\), \(P2_12_12_1\),
$P2_1/a$, $Pbca$, $Pna$ and $Pba$. For centrosymmetric molecules the groups $P2_1/a$ and $Pbca$ are probable.
The consequence obtained is in excellent agreement with experiment. Table I gives data on the distribution among space groups of cyclic organic compounds investigated during the period 1928–1938 (the data were taken without any selection
Table I
| Group | Number of components | Group | Number of components | Group | Number of components | Group | Number of components |
|---|---|---|---|---|---|---|---|
| $C_1$ | 5 | $C_{2v}^{5}$ | 1 | $V^{1}$ | 0 | $V_h^{14}$ | 0 |
| $C_i$ | 8 | $C_{2v}^{6}$ | 0 | $V^{2}$ | 0 | $V_h^{15}$ | 9 |
| $C_2^{1}$ | 2 | $C_{2v}^{7}$ | 0 | $V^{3}$ | 2 | $V_h^{16}$ | 3 |
| $C_2^{2}$ | 14 | $C_{2v}^{8}$ | 1 | $V^{4}$ | 10 | $V_h^{17}$ | 0 |
| $C_2^{3}$ | 0 | $C_{2v}^{9}$ | 3 | $V^{5}$ | 2 | $V_h^{18}$ | 0 |
| $C_s^{1}$ | 0 | $C_{2v}^{10}$ | 0 | $V^{6}$ | 1 | $V_h^{19}$ | 1 |
| $C_s^{2}$ | 2 | $C_{2v}^{11}$ | 0 | $V^{7--9}$ | 0 | $V_h^{20}$ | 0 |
| $C_s^{3}$ | 0 | $C_{2v}^{12}$ | 1 | $V_h^{1}$ | 0 | $V_h^{21}$ | 0 |
| $C_s^{4}$ | 1 | $C_{2v}^{13}$ | 0 | $V_h^{2}$ | 1 | $V_h^{22}$ | 0 |
| $C_{2h}^{1}$ | 0 | $C_{2v}^{14}$ | 0 | $V_h^{3}$ | 0 | $V_h^{23}$ | 0 |
| $C_{2h}^{2}$ | 2(?) | $C_{2v}^{15}$ | 0 | $V_h^{4}$ | 0 | $V_h^{24}$ | 0 |
| $C_{2h}^{3}$ | 1 | $C_{2v}^{16}$ | 0 | $V_h^{5}$ | 1 | $V_h^{25}$ | 0 |
| $C_{2h}^{4}$ | 1 | $C_{2v}^{17}$ | 0 | $V_h^{6}$ | 0 | $V_h^{26}$ | 0 |
| $C_{2h}^{5}$ | 79 | $C_{2v}^{18}$ | 0 | $V_h^{7}$ | 0 | $V_h^{27}$ | 0 |
| $C_{2h}^{6}$ | 4 | $C_{2v}^{19}$ | 1 | $V_h^{8}$ | 0 | $V_h^{28}$ | 0 |
| $C_{2v}^{1}$ | 0 | $C_{2v}^{20}$ | 0 | $V_h^{9}$ | 0 | All remaining space groups | 5 |
| $C_{2v}^{2}$ | 0 | $C_{2v}^{21}$ | 0 | $V_h^{10}$ | 0 | ||
| $C_{2v}^{3}$ | 0 | $C_{2v}^{22}$ | 0 | $V_h^{11}$ | 4 | ||
| $C_{2v}^{4}$ | 0 | $V_h^{12}$ | 0 | ||||
| $V_h^{13}$ | 1 |
from volumes I–V of the Strukturbericht). Experiment shows that about 85% of compounds crystallize in the closest-packed space groups, although there are only 6 of them out of 230. Eight percent of the compounds fall into groups in which close-packed layers are possible, and only 7% fall into all the remaining groups.
The order in which the space groups are arranged according to the number of representatives is also quite natural from the point of view being developed. The chief representative of organic crystals is the group $P2_1/a$ (in Schoenflies notation $C_{2h}^5$), or $P2_1/c$, depending on the choice of axes.
In this group: 1) all three types of close-packed layers are possible, 2) the stacking of layers $2_1\tilde a$ or $\tilde b a$ satisfies the least stringent conditions, 3) crystallization of both symmetric and asymmetric molecules is possible. The groups $P2_1$, $P2_12_12_1$, and the triclinic groups come next and occur approximately equally often.
A more complete classification, carried out by Nowacki[^9] for all organic compounds, obscures certain regularities because this author did not single out molecular crystals separately. Our main conclusions, however, are fully confirmed by these statistics as well, as is evident from the enumeration of the groups represented in 1940 by more than 20 compounds: $P2_1/a$—200, $P2_12_12_1$—116, $P2_1$—109, $Pnma$—31, $A2/a$—31, $P2_12_12_1$—26, $\bar I$—22, $C2$—22, $Pbca$—19.
In our latest work the selection rule is given in an extended form, and it is shown that exceptions to the rule are due to experimental errors[^10].
7. INTRINSIC SYMMETRY OF THE MOLECULE AND ITS SYMMETRY IN THE CRYSTAL
In a space group one can always single out a group of points (islands) of a definite symmetry, by applying to which the symmetry elements of the group the crystal is constructed. The number of these islands is quite definite and depends on their symmetry. Thus, for example, in the group $C_{2h}^5$ the existence of twofold islands of symmetry $\bar I$ is possible; the number of pairs of such islands is four. In addition, of course, asymmetric islands, which exist in all lattices, are possible and, in the case of the group $C_{2h}^5$, have multiplicity 4. Thus a priori it may be asserted that the cell of the group $C_{2h}^5$ contains an even number of molecules. If there are 2 of them, then their centers occupy centers of symmetry. If there are 4 of them, two cases are possible: 1) 4 molecules in a general position, 2) molecules connected in pairs are arranged with their centers in
centers of inversion of the cell. If there are 6 of them, then either they are divided into a group of 4 and a group of 2 molecules, or into three groups of 2 molecules, etc.
At first glance it would seem natural to relate the position occupied by a molecule in a crystal exclusively to its own symmetry. A closer examination of the experimental material shows that this is not so. Indeed, benzene crystallizes in the group \(Pbca\), and not in hexagonal syngony; naphthalene and anthracene crystallize in the group \(P2_1/a\), which does not possess the mirror planes of symmetry characteristic of the molecules, etc.
In what cases, then, does a molecule “lose” its symmetry in the crystal, and why? Or perhaps the chemical formulas do not take into account the real asymmetry of the molecule? The principle of close packing answers this question in the following way\(^{12}\): the closest packings include only those groups in which either unsymmetrical islands or centrosymmetrical islands (symmetry with respect to the elements of the crystal) are possible. A packing of molecules with the preservation of some symmetry element, except for a center of symmetry, will for the most part not at the same time possess maximum density. Therefore we shall say that a “normally symmetrical” packing occurs if a molecule not possessing a center of symmetry occupies a general position in the crystal, while a molecule possessing the symmetry \(\bar{1}\) preserves it in the crystal. Packing with “increased symmetry” takes place when a molecule preserves its axis of symmetry or plane of symmetry in the crystal. Finally, a packing possesses “reduced symmetry” if a centrosymmetrical molecule occupies a general position in the crystal, or also if the crystal contains 2 molecules not connected by a symmetry operation.
Considering the experimental material (see Table II), we see that the case which is normal from the point of view of the idea of close packing is chiefly the one realized in nature.
Table II
| Symmetry in crystal / Symmetry of molecule | \(C_i\) and other elements | \(C_i\) | Symmetry without \(C_i\) |
|---|---|---|---|
| No symmetry | 4 | 1 | 33 |
| \(C_i\) | 22 | 9 | — |
| \(C_s\) | — | — | 4 |
One can imagine a case in which the symmetry of the molecule in the gas and in the crystal is different. The difference in the form of the free molecule and of the molecule in the crystal can easily be imagined on account of the free
of the rotation of atomic groups joined by a single bond. The arrest of free rotation in a crystal will be determined by the advantage of a denser packing of the molecule with immobile groups. However, it must be borne in mind that free rotation increases the symmetry of the molecule. The gain in symmetry may compel the molecules to sacrifice dense packing.
8. PACKING COEFFICIENT OF MOLECULES IN A CRYSTAL
It is of interest to introduce a parameter by the numerical value of which the packing of a molecule can be characterized. Since the concept of the shape of a molecule has been defined (see § 4), its volume \(V\) can be calculated if the atomic and intermolecular radii are known. The density with which molecules are packed in a crystal will then be characterized by the value
\[ k=\frac{V}{v/Z}, \]
where \(v\) is the volume of the cell, and \(Z\) is the number of molecules in the cell.
The coefficient \(k\) may be called the packing coefficient of molecules in a crystal. Table III gives the values of the packing coefficients for certain crystals. A whole series of general conclusions and hypotheses can be made on the basis of an analysis of this table.
The packing coefficients of aromatic molecules in a crystal range from 0.6 to 0.8. The packing coefficient of 0.595 found in the crystal of dioctylnaphthalene is evidently the smallest value for the organic crystals investigated. It must be supposed that at still smaller values of the packing coefficient the formation of a crystal is impossible. It is interesting to note that compounds of the dioctylnaphthalene type, as a rule, do not crystallize, but give glasses. Evidently, the packing of such inconveniently constructed compounds into a periodic structure is possible only with a very small packing coefficient, insufficient to create an energetic advantage of the crystal over the amorphous state. One may suggest the complete impossibility of obtaining crystals from compounds possessing a large nucleus and long chains. It may be predicted that derivatives of perylene or coronene in which hydrogen is replaced by an aliphatic chain of 8–10 links will necessarily be glassy.
The packing coefficient is determined above all by the shape of the molecule. The less regular the shape of the molecule, the, generally speaking, smaller its packing coefficient in the crystal. It is quite understandable that for phenanthrene
\(k\) is smaller than for anthracene
,
naturally—
Table III
Packing coefficients for crystals of aromatic compounds
| No. | Compound | k |
|---|---|---|
| Benzene derivatives | ||
| 1 | Benzene | 0.681 |
| 2 | Resorcinol α | 0.665 |
| 3 | Resorcinol β | 0.678 |
| 4 | p-Toluidine | 0.677 |
| 5 | p-Quinone | 0.693 |
| 6 | p-Dibromobenzene | 0.740 |
| 7 | p-Bromochlorobenzene | 0.714 |
| 8 | p-Dichlorobenzene | 0.687 |
| 9 | Durol | 0.704 |
| Polyphenyls | ||
| 10 | Diphenyl | 0.740 |
| 11 | Paradiphenylbenzene | 0.730 |
| 12 | Quaterphenyl | 0.746 |
| 13 | Triphenylmethane | 0.638 |
| 14 | Triphenylbenzene | 0.716 |
| 15 | Dibenzyl | 0.705 |
| 16 | Stilbene | 0.720 |
| 17 | Tolane | 0.685 |
| Naphthalene derivatives | ||
| 18 | Naphthalene | 0.702 |
| 19 | 1,2-Naphthaquinone | 0.760 |
| 20 | 1,4-Naphthaquinone | 0.753 |
| 21 | 2,6-Dimethylnaphthalene | 0.740 |
| 22 | β-Methylnaphthalene | 0.712 |
| 23 | β-Naphthol | 0.710 |
Continuation of Table III
| No. | Compound | \(k\) |
|---|---|---|
| 24 | α-naphthol | 0.714 |
| 25 | β-naphthylamine | 0.680 |
| 26 | α-naphthylamine | 0.705 |
| 27 | 2,6-diphenylnaphthalene | 0.668 |
| 28 | 2,6-dicyclohexyl naphthalene | 0.690 |
| 29 | Normal dioctyl naphthalene | 0.595 |
| 30 | Di—2,2,4,4—tetramethylbutylnaphthalene | 0.627 |
| Anthracene derivatives | ||
| 31 | Anthracene | 0.722 |
| 32 | Dibromoanthracene | 0.773 |
| 33 | Dichloroanthracene | 0.800 |
| 34 | Mesanthracene | 0.765 |
| 35 | 1,2-anthraquinone | 0.781 |
| 36 | 1,4-anthraquinone α | 0.778 |
| 37 | 1,4-anthraquinone β | 0.773 |
| 38 | Dianthracene | 0.708 |
| Other polynuclear compounds | ||
| 39 | Phenanthrene | 0.684 |
| 40 | Chrysene | 0.737 |
| 41 | Naphthacene | 0.735 |
| 42 | Retene | 0.760 |
| 43 | 1,2-benzpyrene | 0.745 |
| 44 | Perylene | 0.805 |
| 45 | Dimethyldibenzphenanthrene | 0.765 |
| 46 | 1, 2, 5, 6-dibenzanthracene | 0.708 |
| 47 | 1,2-dibenzanthracene | 0.713 |
| 48 | Coronene | 0.726 |
| 49 | Graphite | 0.887 |
...an extremely small value of the coefficient $k$ for triphenylmethane, whose molecule has the form of a flower with three petals, etc. The closer the shape of the molecule is to that of an ellipsoid, the closer $k$ is to the value 0.74, which obtains for the densest spherical and ellipsoidal packings (see above); examples are 2,6-dimethylnaphthalene, naphthacene, and diphenyl.
The limiting value of $k$ for aromatic compounds is the value 0.887, calculated for graphite. It should be emphasized that 0.887 is the coefficient of molecular packing (and not atomic). Therefore the quoted figures for packing coefficients cannot be compared with the coefficients of atomic packing*)—which have no physical meaning for organic compounds—values of which are given in some books and articles.
It is quite clear that, as the number of nuclei in condensed compounds increases, the packing coefficient generally increases. Thus, for benzene derivatives the average coefficient is less than 0.69, for naphthalene derivatives less than 0.71, and for anthracene derivatives and polynuclear compounds it is of the order of 0.75. The reason for such an increase is quite clear: the superposition of carbon planes one upon another occurs with the density of graphite; the least dense are hydrogen and carbon–hydrogen contacts, whose “specific weight” becomes smaller as the aromatic nucleus increases.
Noteworthy is the equality of the packing coefficients in two crystalline modifications of one and the same compound: for resorcinol, 0.665 and 0.678; for anthraquinone, 0.778 and 0.773. This important circumstance indicates that packing density is an important energetic factor in the construction of the crystal. When a number of possible packings exist, evidently the densest one is realized. Unfortunately, among aromatic compounds the number of known isomers on which this assertion could be checked is extremely small.
Also noteworthy is the sameness of the packing coefficients of molecules that are similar in shape: in particular, four different anthraquinones have $k$ in the range 0.76–0.78; naphthaquinones are packed somewhat less densely (0.75–0.76); linear polyphenyls are packed in the same type of manner.
This, however, is not always the case. The packing coefficients of the very similar mononuclear molecules of $p$-dibromobenzene, $p$-bromochlorobenzene, and $p$-dichlorobenzene give different values, $k = 0.74$, 0.71, and 0.69. At the same time, the cells of these compounds are almost identical not only in volume but also in shape. Evidently, in the structure of dichlorobenzene the determining contacts are contacts of carbon
*) The coefficient of atomic packing for graphite is calculated from the assumption that the space of the graphite cell is filled with spheres of radius 0.71 (1.42 is the distance between chemically bonded atoms of graphite); such a coefficient is equal to 0.17.
atoms, and introducing a bromine atom into the molecule instead of chlorine does not require any great separation of the molecules.
It seems to us that measurements of packing coefficients shed some light on the reason for the crystallization of molecules with “lowered symmetry” (see § 7). If it is assumed that, in addition to the tendency toward dense packing, there exists a tendency toward the formation of a maximally symmetrical crystal, then one should expect an increased value of the coefficient \(k\) for all compounds crystallizing with “lowered symmetry.” Since compounds whose molecules are similar in shape must be compared, there are unfortunately very few such examples. All the examples that could be found confirm the point of view expressed.
Dichloroanthracene crystallizes with a double lowering of symmetry: four non-equivalent molecules in the cell. Its packing coefficient, 0.800, is almost maximal. The perylene molecule, crystallizing with two non-equivalent molecules in the cell, has an even larger and already maximal coefficient, 0.805. At the same time coronene (a seven-ring compound) has a relatively small packing coefficient and occupies a position in the crystal with a center of symmetry.
It would be of considerable interest to find and analyze other similar cases, which could demonstrate the reality of the principle of a peculiar competition between packing density and symmetry.
The values of the packing coefficients are very sensitive to the correctness in determining the cell parameters. An error of 0.5–0.6% in determining a parameter changes the second digit of the packing coefficient by one unit. Although such accuracy is quite attainable, it is not always achieved in studies. Therefore the selection of material used for calculating \(k\) must be treated very cautiously. In Table III the values of \(k\) are given to an accuracy of 0.001. However, it must be assumed that in most cases only the second significant digit can be guaranteed.
CONCLUSION
Structural investigations in the field of organic chemistry solve two problems: 1) determination of the structure of the molecule, 2) determination of how the crystal is constructed from molecules. Until recently investigators set themselves only the first goal. The comparatively small successes achieved in this respect are explained in large measure by neglect of the second problem. Only knowledge of the rules governing the arrangement of molecules in the crystal gives the trial-and-error method the necessary reliability and makes it possible to develop a rapid and effective geometrical analysis. The limited accuracy of the method of intensity analysis in the general case requires that structural problems concerning the molecule be solved not on random objects, but on specially
synthesized compounds containing heavy atoms. The question of the accuracy of the analysis for organic compounds, however, must be investigated specially.
Researchers in the field of the structure of organic compounds face two tasks: 1) the accumulation of systematic material on the geometry and symmetry of crystal lattices in order to refine and establish the rules of organic crystal chemistry—with their aid it will be possible to find the form of the molecule and the mutual arrangement of molecules; 2) the study of specially synthesized organic compounds suitable for the precise solution of structural problems by the method of analyzing the intensities of the maximum possible number of X-ray interferences—on the basis of these works it will be possible to determine valence angles and interatomic distances in molecules.
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