International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.7, pp. 902-904

Section 9.7.4. Use of molecular symmetry

A. J. C. Wilson,a V. L. Karenb and A. Mighellb

aSt John's College, Cambridge CB2 1TP, England, and bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

9.7.4. Use of molecular symmetry

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It has long been recognized that in many crystal structures molecules with inherent symmetry occupy Wyckoff special positions, so that molecular and crystallographic symmetry elements coincide, but until recently systematic data have been lacking. Now the occurrence of molecules of particular symmetry in structures of various space-group types can be traced in the data of Belsky, Zorkaya & Zorky (1995[link]), and will be discussed briefly.

9.7.4.1. Positions with symmetry 1

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The empirical results for `homomolecular structures' with one molecule in the general position are given in Table 9.7.1.2[link]. The classification by arithmetic crystal class and degree of symmorphism follows Wilson (1993d[link]); the numerical data are taken from Belsky, Zorkaya & Zorky (1995[link]). Space groups symmorphic in the technical sense (Wilson, 1993d[link]) are prefixed by an asterisk (*), and in each arithmetic crystal class the space group most nearly antimorphic is followed by an obelus (†). The number of known structures having precisely one molecule in the general Wyckoff position is given as a superscript in brackets. It will be noticed immediately that structures with space groups `fully symmorphic' or `tending to symmorphism' are extremely rare. Most have no examples; three (P42, P4/n and [P\overline {3}]) are credited with a single example each. The frequency of space groups increases rapidly with increasing antimorphism. In the monoclinic system, the `fully symmorphic' space group P2/m has no examples with one molecule in the general position, the `equally balanced' P2/c has 11 examples, the `tending to antimorphism' C2/c has 587, and the `fully antimorphic' P21/c has 5951. Other systems have fewer examples, but the trend is the same; the really popular space groups are the `fully antimorphic' plus P1 and [P\overline {1}].

All space groups, of course, possess general positions of symmetry 1, and the data in Table 9.7.2.1[link] show that 116 of them exhibit structures of some kind, and that 57 exhibit structures in which one or more general positions are used. 13 space groups (P1, P21, Pc, Cc, P212121, Pca21, Pna21, P41,3, P31,2, P61,5) have no positions with symmetry higher than 1. These space groups contain no syntropic symmetry elements, and all are relatively popular.

9.7.4.2. Positions with symmetry [{\overline 1}]

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Many space groups are centrosymmetric (all those in the geometric classes [\overline {1}], 2/m, mmm, 4/m, 4/mmm, [\overline {3}], [\overline {3}m], 6/m, 6/mmm, [m\overline {3}], [m{\overline {3}}m]), but comparatively few of them possess special positions of symmetry [\overline {1}], as the centres of symmetry are often encumbered by other symmetry elements. All centres of symmetry in [P\overline {1}], P21/c and Pbca are free, as are some of those in [P\overline {3}] and [R\overline {3}]. When the encumbrance is an antitropic symmetry element, the special position can still be occupied by a molecule of symmetry [\overline {1}] only, but when the encumbrance is syntropic or atropic the position cannot accommodate such a molecule. Table 9.7.2.1[link] indicates that there are 38 space groups with special positions of symmetry [\overline {1}], that 28 of them have examples of structures of some kind, and that ten have structures in which the centre of symmetry is actually used by a molecule.

The three space groups with no special positions except those of symmetry [\overline {1}] are very popular, whether or not the centre of symmetry is actually used by the molecule. The single criterion `no special positions except possibly free centres of symmetry' thus selects the space groups favoured by structures in which inherent molecular symmetry is not used.

9.7.4.3. Other symmetries

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Table 9.7.2.1[link] gives statistics for the number of space groups possessing Wyckoff positions of symmetry [{\cal G}], where [{\cal G}] is one of the 32 point groups, the number exhibiting structures of some kind, and the number in which the special position of symmetry [{\cal G}] is actually used. It has to be remembered that this table represents the state of knowledge in 1994, that there may be small errors in the counts in the second column, and that new structures will gradually increase the numbers in the third and fourth columns. Nevertheless, some trends are clear. The arrangement of the point groups is in ascending order of their `order' (Hahn, 2005[link], Section 10.1.4.1[link] ), and all numbers show a general decrease with increasing order. When molecular symmetry is used, the favourite is the diad axis 2, closely followed by the mirror plane m, with the centre of symmetry [\overline {1}], the triad axis 3 and the tetrad inversion axis [\overline {4}] trailing. It must also be remembered that these data are for numbers of space groups, not numbers of structures.

9.7.4.4. Positions with the full symmetry of the geometric class

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The symmorphic space groups are in a one-to-one correspondence with the arithmetic crystal classes, and each has at least one Wyckoff position with the full symmetry of the geometric crystal class. It would thus be possible for each symmorphic space group to accommodate molecules with the full symmetry of the point group corresponding to the geometric crystal class. With the obvious exceptions of P1 and [P\overline {1}], there seem to be no symmorphic space groups with primitive cells and one molecule only in the cell that do so, but the data of Belsky, Zorkaya & Zorky (1995[link]) show that about half the possibilities are realized in symmorphic space groups with centred cells. The situation is set out in Table 9.7.4.1[link].

Table 9.7.4.1| top | pdf |
Occurrence of molecules with specified point group in centred symmorphic and other space groups, based on the statistics by Belsky, Zorkaya & Zorky (1995[link])

There is no entry in the `other space group' column if examples are found in the centred symmorphic group.

Point group Symmorphic space group Other space group Frequency
2 C2 [\cdots] 18
m Cm [\cdots ] 6
2/m C2/m [\cdots] 20
222 None Ccca 4
[\cdots] Fddd 2
[\cdots] P[\overline {4}]n2 2
[\cdots] P4/ncc 1
[\cdots] I41/acd 3
mm2 Fmm2 [\cdots] 2
mmm None P42/mnm 6
[\cdots] Im[\overline {3}] 1
4 I4 [\cdots] 1
[\overline {4}] None P4/n 1
[\cdots] P42/n 3
[\cdots] I41/a 12
[\cdots] P[\overline {4}]21c 17
[\cdots] I[\overline {4}]2d 1
[\cdots] I41/acd 1
4/m I4/m [\cdots] 1
422 None P4/nnc 1
4mm None None None
[\overline {4}]2m I[\overline {4}]2m [\cdots] 3
4/mmm I4/mmm [\cdots] 1
3 R3 [\cdots] 8
[\overline {3} ] R[\overline {3} ] [\cdots] 6
32 None R[\overline {3}]c 5
3m R3m [\cdots] 10
[\overline {3}]m R[\overline {3}]m [\cdots] 2
6 None None None
[\overline {6}] None P63/m 12
6/m None None None
622 None None None
6mm None None None
[\overline {6}]m2 None P63/mmc 1
6/mmm None None None
23 None F[\overline {4}]3c 1
m[\overline {3} ] Fm[\overline {3}] [\cdots] 2
432 None None None
[\overline {4}]3m I[\overline {4}]3m [\cdots] 4
m[\overline {3}]m Fm[\overline {3}]m [\cdots] 9
Im[\overline {3}]m [\cdots] 2

Although about half the point groups are not represented in symmorphic space groups with one molecule in the appropriate special position, it is interesting to look for molecules of these symmetries in space groups of higher symmetry. A few are in fact to be found in non-symmorphic space groups, but seven point groups have no established examples.

References

Belsky, V. K., Zorkaya, O. N. & Zorky, P. M. (1995). Structural classes and space groups of organic homomolecular crystals: new statistical data. Acta Cryst. A51, 473–481.
Hahn, Th. (2005). Editor. International tables for crystallography, Vol. A, Space-group symmetry, fifth edition. Heidelberg: Springer.
Wilson, A. J. C. (1993d). Space groups rare for organic structures. III. Symmorphism and inherent molecular symmetry. Acta Cryst. A49, 795–806.








































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