ON THE ORIGIN OF TRANSITIONAL SPACE GROUPS IN BETWEEN TRICLINIC AND PSEUDO-MONOCLINIC CRYSTALS

Alajos Kálmán

Institute of Chemistry, Chemical Research Center, Hungarian Academy of Sciences, Budapest P.O.B. 17., H-1525, Hungary, E-mail: akalman@cric.chemres.hu

Keywords: asymmetric translation, pseudosymmetries, symmetry migration

Symmetry operators with translation of 1/4->3/4->1/4 type have been discovered in the course of a symmetry analysis [1] of triclinic crystals reported with space group P and Z=8 (four molecules in the asymmetric unit). More than a half dozen triclinic crystals (e.g. EXCHLN, KUGWUE, YIHHOM, etc.) have been found among the structures retrieved from CSD (January 1997 release) to exhibit pseudosymmetries: *21 and *c with regular translation:

but mixed frequently with unusual screw axes and glide planes showing irregular translation:

They are labelled with *2q and *cq, and termed as secondary pseudosymmetries [1]. These symmetries were also revealed in four structures which had beeen shown to be centred monoclinic by a compulsory transformation test [2] seeking for higher crystal systems. Remarkably, in CIJSAP, DILFEJ, and VARCOG with relatively high figure of merit: (fm = 0.267, 0.114 and 0.490, respectively) the eight molecules pertaining to the original triclinic unit cell are located on two orbits formed solely by these secondary pseudosymmetries, which topologically can be described by a symmetry triplet:

2q(1/8,Y,1/4) <--> nq(Xq,1/8, Z) = (0,0,0) (1)

where q = 1/4->3/4->1/4, while X,Y,Z indicate the infinite extension(s) of the commutative (<->) operators and (0,0,0) is one of the crystallographic inversion centers, located in the origin. In contrast, only DIYMED with the lowest fm = 0.025 could be transformed into a unit cell to which a centred crystallographic space group C2/c (No. 15) could be assigned. However, in this unit cell, parallel with the regular orbit 2/c, a simpler irregular pair 21/cq, is also retained. These two orbits are separated by y = 1/8. From this and Eq. (1) it follows that the origin of asymmetric translations along with and/or perpendicular to the orthogonal b axis is the unusual (2n-1)1/8 position(s) occupied occasionally by symmetry operators, either regular or irregular or both. Topologically three irregular orbits

(1): 2₁/cq, (2): 2q/c and (3): 2q/cq

can be inferred from the most popular space group P2₁/c (No. 14) if parameters a, b, g are altered between 0 and 1/4 (±1/2) in the generalized triplet 2_t(a,Y,b) <--> c_t(X,g,Z)=(0,0,0), where the translation t is either regular (0, or 1/2) or irregular (1/4 ->3/4), while X,Y,Z are as given above. When parameter b or g = 1 is alternatively shifted to 0 then two other standard space groups P2₁/m (No. 11) and P2/c (No. 13) are defined. Their triplets

indicate that (0,0,0) is located either on 2₁ or c. However, when the migration of these symmetries stops at the midpoint between 0 and 1/4 then two new transitional space groups

are generated. They are equal to the 1-st and 2-nd combinations of the irregular symmetries recognized in the above mentioned triclinic and pseudo-monoclinic crystals. Naturally, the 3-rd pair of these symmetries can also be regarded as a transitional space group between P2₁/c and P2/m (No 10). These symmmetry migrations can be summarized as follows:

As observed, molecules cannot develope such space groups in perfect form. At least partly, they are always formed by pseudosymmetries. However, the transitions between triclinic and pseudo-monoclinic forms are well described by such irregular symmetries in centred unit cells approaching either to the canonical space group C2/m and or C2/c. By all means these transitional space groups (two of them are revealed in the crystals investigated) create a novel subdivision of the 230 space groups.

The advantage of transitional space groups in the structures mentioned above can be attributed to the duplication of the secondary pseudosymmetries with one or two coordinates of (2n-1)1/8. They give rise simultaneously to the loss of the half of their function within and around the unit cell. In other words, the function of a glide plane g at 1/4, is divided symmetrically by its twin form located at 1/4±1/8, respectively. This seems to help the rigid and bulky molecules against their diminished self-complementarity to reach an optimum of close packing.

This work has been sponsored by the Hungarian Res. Fund Grant No. OTKA T023212.

[1] Kálmán, A.& Argay, Gy. (1998). Acta Cryst B54. in the press.
[2] Spek, A. L. (1988). J. Appl. Cryst. 21, 578-579.