Section 9.7.4.4. Positions with the full symmetry of the geometric class

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 , 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) 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.

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) 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 ⋯ 18 m Cm 6 2/m C2/m ⋯ 20 222 None Ccca 4 ⋯ Fddd 2 ⋯ Pn2 2 ⋯ P4/ncc 1 ⋯ I41/acd 3 mm2 Fmm2 ⋯ 2 mmm None P42/mnm 6 ⋯ Im 1 4 I4 ⋯ 1 None P4/n 1 ⋯ P42/n 3 ⋯ I41/a 12 ⋯ P21c 17 ⋯ I2d 1 ⋯ I41/acd 1 4/m I4/m ⋯ 1 422 None P4/nnc 1 4mm None None None 2m I2m ⋯ 3 4/mmm I4/mmm ⋯ 1 3 R3 ⋯ 8 R ⋯ 6 32 None Rc 5 3m R3m ⋯ 10 m Rm ⋯ 2 6 None None None None P63/m 12 6/m None None None 622 None None None 6mm None None None m2 None P63/mmc 1 6/mmm None None None 23 None F3c 1 m Fm ⋯ 2 432 None None None 3m I3m ⋯ 4 mm Fmm ⋯ 9 Imm ⋯ 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.