M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 72
Текст из файла (страница 72)
The term densest-packedmeant six-coordination with any orientation of the molecules withrespect to the unit cell axes. The term maximum density was usedfor the packing if six-coordination was possible at any orientation ofthe molecules with respect to the unit cell axes while the moleculesretained their symmetry.For the plane group p1 it is possible to achieve densest packing withany molecular form if the translation periods t1 and t2 and the anglebetween them are chosen appropriately as illustrated in Figure 9-48.The same is true for the plane group p2, shown also in Figure 9-48.On the other hand, the plane groups pm and pmm are not suitablefor densest packing.
As is seen in Figure 9-49, the molecules areoriented in such a way that their convex parts face the convex partsof other molecules. This arrangement, of course, counteracts densepacking. The plane groups pg and pgg may be suitable for sixcoordination as an example shows it in Figure 9-50a. This layer is notof maximum density and in a different orientation of the molecules9.6. Molecular Crystals467(a)(b)Figure 9-48.
Densest packing with space groups (a) p1; and (b) p2 afterKitaigorodsky [89].(a)(b)Figure 9-49. The symmetry planes in the space groups (a) pm; and (b) pmm preventdense packing; after Kitaigorodsky [90].(a)(b)Figure 9-50. Two forms of packing with pgg space groups after Kitaigorodsky [91].(a) Densest packing of molecules with arbitrary shape; (b) Another orientation ofthe molecules which reduces the coordination number to four.4689 Crystalsonly four-coordination is achieved as seen in Figure 9-50b. For theplane groups cm, cmm, and pmg, six-coordination cannot be achievedfor a molecule with arbitrary shape.
For higher symmetry groups, forexample, tetragonal p4 or hexagonal p6, the axes of the unit cell areequivalent, and the packing of the molecules is not possible withoutoverlaps. This is illustrated for group p4 in Figure 9-51.Figure 9-51. Molecules of arbitrary shape cannot be packed in space group p4without overlaps after Kitaigorodsky [92].If the molecule, however, retains a symmetry plane, then it maybe packed with six-coordination in at least one of the plane groups,pm, pmg, or cm. The form shown in Figure 9-52 is suitable for suchpacking in pmg and cm, though not in pm. Thus, depending on themolecular shape, various plane groups may be applicable in differentcases.The criteria for the suitability as well as incompatibility of planegroups for achieving molecular six-coordination have been considered.
The next step is to apply the geometrical model to the examination of the suitability of three-dimensional space groups for densest(a)(b)Figure 9-52. Molecules with a symmetry plane achieve 6-coordination in the spacegroups (a) cm; and (b) pmg after Kitaigorodsky [93].9.6. Molecular Crystals469packing. The task in this case is to select those space groups in whichlayers can be packed allowing the greatest possible coordinationnumber. Mirror planes, for instance, would not be applicable forrepeating the layers.Low-symmetry crystal classes are typical for organic compounds.Densest packing of the layers may be achieved either by translation atan arbitrary angle formed with the layer plane, or by inversion, glideplane, or by screw-axis rotation.
In rare cases closest packing mayalso be achieved by twofold rotation.Kitaigorodskii [94] analyzed all 230 three-dimensional spacegroups from the point of view of densest packing. Only the followingspace groups were found to be available for the densest packing ofmolecules of arbitrary form:P1, P21 , P21 /c, Pca, Pna, P21 21 21For molecules with symmetry centers, there are even fewer suitablethree-dimensional space groups, namely:P1, P21 /c, C2/c, PbcaIn these cases all mutual orientations of the molecules are possiblewithout losing the six-coordination.The space group P21 /c occupies a strikingly special position amongthe organic crystals. This space group has the unique feature that itallows the formation of layers of densest packing in all three coordinate planes of the unit cell.The space groups P21 and P21 21 21 are also among those providingdensest packing.
However, their possibilities are more limited thanthose of the space group P21 /c, and these space groups occur onlyfor molecules that take either left-handed or right-handed forms.According to statistical examinations performed some time ago, thesethree groups are the first three in frequency of occurrence.An interesting and fundamental question is the conservation ofmolecular symmetry in the crystal structure. Densest packing mayoften be well facilitated by partial or complete loss of molecularsymmetry in the crystal structure. There are, however, space groupsin which some molecular symmetry may “survive” densest packingwhen building the crystal. Preserving higher symmetry though usually4709 Crystalscosts too great a sacrifice of packing density.
On the other hand, theremay be certain energy advantage of some well-defined symmetricalarrangements. The alternative to the geometrical model for discussingand establishing molecular packing in organic crystals has been thecalculations of energy, based on carefully constructed potential energyfunctions [95].9.6.3. Energy Calculations and Structure PredictionsIt is important to be able to determine a priori the arrangement ofmolecules in crystals. The correctness of such predictions is a test forour understanding of how crystals are built.
A further benefit is thepossibility of calculating even those structures whose determination isnot amenable to experimental analysis. But even as part of an experimental study, it is instructive to build good models, which can thenbe refined. The main advantages of the geometrical model have beenseen above. Its main limitations are the following: It cannot accountfor the structural variations in a series of analogous compounds. It isrestricted in correlating structural features with various other physical properties. Finally, it is unable to make detailed predictions forunknown structures. Calculations seeking the spatial arrangement ofmolecules in the crystal corresponding to the minimum of free energyhave become a much used tool. If the system is considered completelyrigid, the molecular packing may be determined by minimizing thepotential energy of intermolecular interactions.Considering the molecules to be rigid, i.e., ignoring the vibrationalcontribution, the energy of the crystal structure is expressed as afunction of geometrical parameters including the cell parameters, thecoordinates of the centers of gravity of the symmetrically independent molecules, and parameters characterizing the orientation of thesemolecules.
In particular cases the number of independent parameterscan be reduced. On the other hand, considerations for the non-rigidityof the molecules necessitate additional parameters. Minimizing thecrystal structure energy leads to structural parameters correspondingto optimal molecular packing. Then it is of great interest to comparethese findings with those from experiment.To determine the deepest minimum on the multidimensional energysurface as a function of many structural parameters is a formidablemathematical task. Usually, simplifications and assumptions are9.6.
Molecular Crystals471introduced concerning, for example, the space-group symmetry.Accordingly, the conclusions from these theoretical calculationscannot be considered to be entirely a priori.The considerations on the intermolecular interactions can beconveniently reduced to considerations of atom–atom nonbondedinteractions. Although these interactions can be treated by nonempirical quantum mechanical calculations, empirical and semi-empiricalapproaches have also proved useful in dealing with them. In thedescription of the atom–atom nonbonded interactions it is supposedthat the van der Waals forces originate from a variety of sources.In addition to the intermolecular interactions, the intramolecularinteractions may also be taken into account in a similar way.
Thisrather limited approach may nevertheless be useful for calculatingmolecular conformation and even molecular symmetry. Deviationsfrom the ideal conformations and symmetries may also be estimatedthis way, provided they are due to steric effects.By summation over the interaction energies of the molecular pairs,the total potential energy of the molecular crystal may be obtained inan atom–atom potential approximation.
The result is expected to beapproximately the same as the heat of sublimation extrapolated to 0 Kprovided that no changes take place in the molecular conformationand vibrational interactions during evaporation.In many of the molecular packing studies, the crystal classes aretaken from the experimental X-ray diffraction determinations. Theoptimal packing is then determined for the assumed crystal class. Inother cases, the crystal classes have also been established in the optimization calculations.Ideally, it should be possible to predict molecular packing, and thusthe crystal structure, from the knowledge of the composition of acompound and the symmetry and geometry of its molecule. It hasproved, however, a rather elusive task. Two decades ago, the editorof Nature expressed the frustration over the difficulty in predictingcrystal structures in the following words: “One of the continuing scandals in the physical sciences is that it remains impossible to predictthe structure of even the simplest crystalline solids from a knowledge of their chemical composition” [96].
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
