M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 70
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At leastone of the intramolecular distances of an atom in the molecule issignificantly smaller than its distances to the adjacent molecules.Every molecule in the molecular crystal may be assigned a certainwell-defined space in the crystal. In terms of interactions, there arethe much stronger intramolecular interactions and the much weakerintermolecular interactions. Of course, even among the intramolecular interactions, there is a range of interactions of various energies.Bond stretching, for instance, requires a proportionately higher energythan angular deformation, and the weakest are those interactions thatdetermine the conformational behavior of the molecule [62].
On theother hand, there are differences among the intermolecular interactions as well. For example, intermolecular hydrogen bond energiesmay be equal to or even greater than the conformational energy differences. Thus, there may be some overlap in the energy ranges of theintramolecular and intermolecular interactions.The majority of molecular crystals are organic compounds.
Thereis usually little electronic interaction between the molecules in these9.6. Molecular Crystals457crystals, although even small interactions may have appreciable structural consequences. The physical properties of the molecular crystalsare primarily determined by the packing of the molecules.9.6.1. Geometrical ModelAs structural information for large numbers of molecular crystalshas become available, general observations and conclusions haveappeared. An interesting observation was that there are characteristic shortest distances between the molecules in molecular crystals. The intermolecular distances of a given type of interactionare fairly constant.
From this observation a geometrical model hasbeen developed for describing the molecular crystals [63]. First, theshortest intermolecular distances were found, and then the so-called“intermolecular atomic radii” were postulated. Using these quantities, spatial models of the molecules were built. Fitting togetherthese models, the densest packing could be found empirically.
Anexample of a packing arrangement is shown in Figure 9-43. Themolecules are packed together in such a way as to minimize the emptyspace among them. The concave part of one molecule accommodatesthe convex part of the other molecule. The example is the packingof 1,3,5-triphenylbenzene molecules in their crystal structure. Thearrangement of the areas designated to the molecules is analogous toa characteristic decoration pattern, an example of which is also shownin Figure 9-43. The analogy is not quite superficial. The decoration isfrom the metal-net dress of a Chinese warrior.
The dress was made ofsmall units to maintain flexibility, the small units were identical foreconomy, and they covered the whole surface without gaps to ensuremaximum protection.The importance of symmetry in structure does not mean that thehighest symmetry is the most advantageous. This can be illustratedbeautifully in molecular crystals.
Lucretius proclaimed two millenniaago in his De rerum natura [65]:Things whose fabrics show opposites that match,one concave where the other is convex, and viceversa, will form the closest union.Lucretius could have meant this as a fundamental principle of thebest packing arrangements for molecules in crystals had he known4589 Crystals(a)(b)(c)Figure 9-43. (a) Dove-tail dense packing of 1,3,5-triphenylbenzene molecules [64];(b) Chinese decoration on a sculpture in the sculpture garden of the Ming tombs,near Beijing; and (c) Detail of the armor (photographs by on the authors).about molecules of arbitrary shape.
At the dawn of the 20th century,Lord Kelvin (William Thomson) returned to Lucretius’s observation.Lord Kelvin’s geometry [66] was mostly forgotten, but we find itinstructive in understanding the development of crystal chemistry andthe teachings of symmetry in crystallography [67]. As Lord Kelvinwas building up the arrangement of molecular shapes, he examinedtwo basic variations (Figure 9-44). In one, the molecules are alloriented in the same way, while, in the other, the rows of moleculesare alternately oriented in two different ways. Lord Kelvin consideredthe puzzle of the boundary of each molecule as a purely geometricalproblem.
He used nearly rectilinear shapes for partitioning the planebut he did not let the molecules touch each other. Apart from this, hecreated a modern representation of molecular packing in the plane,9.6. Molecular Crystals459Figure 9-44. Arrangements of molecular shapes by Lord Kelvin (1904) [68].including the recognition of the most important complementariness inpacking.Lord Kelvin then came to extending the division of continuous twodimensional space into the third dimension. However, he restrictedhis examinations to polyhedra and found one of the five space-fillingparallelohedra, which were discovered by E. S.
Fedorov as capableof filling the space in parallel orientation without gaps or overlaps. The Fedorov polyhedra are the cube, the hexagonal prism, therhombic dodecahedron, an elongated rhombic dodecahedron witheight rhombic and four hexagonal faces, and the truncated octahedron.The complementary character of molecular packing is wellexpressed by the term of dove-tail packing [69].
The arrangementof the molecules in Figure 9-45a can be called head-to-tail. On theother hand, the molecules of a similar compound are arranged headto-head as seen in Figure 9-45b. The head-to-head arrangement is lessadvantageous for packing. This is well seen in the arrangement of themolecules in the crystal displayed in the lower part of Figure 9-45b.Many of Escher’s periodic drawings with interlocking motifs are alsoexcellent illustrations for the dove-tail packing principle.
Figure 9-46reproduces one of them. Note how the toes of the black dogs are theteeth of the white dogs and vice versa in this drawing.An important contribution appeared in 1940 by the structuralchemist Linus Pauling and the physicist turned biologist MaxDelbrück. They titled their note in Science, “The Nature of the4609 Crystals(a)(b)Figure 9-45. (a) Head-to-tail [70]; and (b) Head-to-head [71] arrangement ofmolecules and the crystal structure [70].9.6. Molecular Crystals461Figure 9-46. Escher’s periodic drawing of dogs from MacGillavry’s book [72].Reproduced with permission from the International Union of Crystallography.Intermolecular Forces Operative in Biological Processes” [73].
Thenote was prepared in response to a series of papers by the physicistPascual Jordan, who had suggested that a quantum mechanical stabilizing interaction operates preferentially between identical or nearlyidentical molecules or parts of molecules. The suggestion came up inconnection with the process of biological molecular synthesis, leadingto replicas of molecules present in the cell. Pauling and Delbrücksuggested precedence for interaction between complementary parts,instead of the importance of interaction between identical parts.
Theyargued that the intermolecular interactions of van der Waals attraction and repulsion, electrostatic interaction, hydrogen bond formation,and so forth give stability to a system of two molecules with complementary structures in juxtaposition, rather than two molecules withidentical structures. Accordingly, they argued that complementarinessshould be given primary consideration in discussing intermolecularinteractions. They summarize their general argument as follows [74]:Attractive forces between molecules vary inverselywith a power of the distance, and maximumstability of a complex is achieved by bringing themolecules as close together as possible, in sucha way that positively charged groups are brought4629 Crystalsnear to negatively charged groups, electric dipolesare brought into suitable mutual orientation, etc.The minimum distances of approach of atoms aredetermined by their repulsive potentials, whichmay be expressed in terms of van der Waals radii;in order to achieve maximum stability, the twomolecules must have complementary surfaces, likedie and coin, and also complementary distributionof active groups.The case might occur in which the two complementary structures happened to be identical;however, in this case also the stability of thecomplex of two molecules would be due to theircomplementariness rather than their identity.Complementariness remained on Pauling’s mind, and in 1948, hediscussed molecular replication [75]:The detailed mechanism by means of which a geneor a virus molecule produces replicas of itself isnot yet known.
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