cctbx - Multiple cell C or F and triple cell H settings

The multiple cell and triple cell symbols were first introduced in "Internationale Tabellen zur Bestimmung von Kristallstrukturen" in 1935 (IT-1935). They were dropped in the International Tables for Crystallography, Volume I from 1952 (IT-I1952) but kept in the comparative tables. In the International Tables for Crystallography, Volume A from 1983 (IT-A1983), the multiple cell and triple cell symbols reappear in the sub- and supergroup data, and are also included in the Synoptic Tables of Space-Group Symbols (Table 4.3.1).

IT-A1983 defines two transformation matrices for transforming from a P- or I-representation in the tetragonal system to C or F respectively. The matrices are shown in the table headers below. For most settings the choice of the transformation matrix does not effect the result of the transformation. However, there are a number of settings where the result of the transformation depends on the choice of the matrix. For these settings IT-A1983 does not unambiguously define a specific space group representation for the given Hermann-Mauguin symbol (see e.g. F41 below). The situation is similar for the transformation from a primitive trigonal or hexagonal setting to a triple cell H setting. Unfortunately there is no established notation for resolving these ambigous definitions of IT-1983.

IT-1935 defines unambigous transformations for the multiple cell and triple cell settings (p. 30):

The equation for transforming from a P- or I-representation (in the tetragonal system) to C or F respectively are:

• a(C,F) = a(P,I) + b(P,I); b(C,F) = -a(P,I) + b(P,I)

For the change from C to H in the hexagonal system we have:

• a(H) = a(C) + 2b(C); b(H) = -2a(C) - b(C)

c(C,F) = c(P,I) and c(H) = c(C) are implied.
Note that IT-1935 uses the symbol C for the primitive settings in the trigonal and hexagonal systems.

Unfortunately, some primitive C settings in IT-1935 are different from the primitive settings in IT-I1952 and IT-A1983 (Schoenflies symbols D3^3 and D3^5). Therefore it is not possible to derive all triple cell H settings of IT-1935 by applying a consistent transformation to the primitive settings in IT-A1983. It is currently not known to me (rwgk) if a similar problem exists for the tetragonal space groups.

It is beyond the scope of the sgtbx to provide a table of historic space group symbols that are only referred to in very old papers. A related problem is that the Protein Data Bank (PDB) sometimes uses the symbol H to distinguish between the hexagonal and the rhombohedral setting of trigonal space groups (e.g. the symbol H 3 in the PDB corresponds to R 3 with hexagonal axes in IT-A1983). It could therefore cause significant confusion if the sgtbx interprets the PDB symbols according to the IT-1935 conventions.

Considering the ambiguities and conflicting definitions, a direct support for multiple cell and triple cell settings is not included in the sgtbx. The tables below are provided for the rare cases where information about these symbols is required.

Tetragonal multiple cell C or F

The Hall symbols in this table were generated with the program Xtal 3.7.0.

Hermann-Mauguin symbol	Hall symbol a-b,a+b,c	Hall symbol a+b,-a+b,c
C 4	C 4
C 41	C 4w
C 42	C 4c
C 43	C 4cw
F 4	F 4
F 41	F 4ad	F 4d
C -4	C -4
F -4	F -4
C 4/m	-C 4
C 42/m	-C 4c
C 4/a :1	C 4a -1a
C 4/a :2	-C 4uv	-C 4auv
C 42/a :1	C 4ac -1ac
C 42/a :2	-C 4acuv	-C 4wd
F 4/m	-F 4
F 41/d :1	F 4ad -1ad	F 4d -1d
F 41/d :2	-F 4auw	-F 4vw
C 4 2 2	C 4 2
C 4 2 21	C 4a 2
C 41 2 2	C 4w 2w	C 4w 2cw
C 41 2 21	C 4aw 2c	C 4aw 2
C 42 2 2	C 4c 2c
C 42 2 21	C 4ac 2
C 43 2 2	C 4cw 2cw	C 4cw 2w
C 43 2 21	C 4acw 2c	C 4acw 2
F 4 2 2	F 4 2
F 41 2 2	F 4ad 2	F 4d 2
C 4 m m	C 4 -2
C 4 m g1	C 4 -2a
C 42 m c	C 4c -2
C 42 m g2	C 4ac -2
C 4 c c	C 4 -2c
C 4 c g2	C 4 -2ac
C 42 c m	C 4c -2c
C 42 c g1	C 4c -2ac
F 4 m m	F 4 -2
F 4 m c	F 4 -2a
F 41 d m	F 4ad -2ad	F 4d -2d
F 41 d c	F 4ad -2d	F 4d -2ad
C -4 m 2	C -4 -2
C -4 c 2	C -4 -2c
C -4 m 21	C -4 -2a
C -4 c 21	C -4 -2ac
C -4 2 m	C -4 2
C -4 2 c	C -4 2c
C -4 2 g1	C -4 2a
C -4 2 g2	C -4 2ac
F -4 2 m	F -4 2
F -4 2 c	F -4 2a
F -4 m 2	F -4 -2
F -4 d 2	F -4 -2ad	F -4 -2d
C 4/m m m	-C 4 2
C 4/m c c	-C 4 2c
C 4/a m g1 :1	C 4 2 -1a
C 4/a m g1 :2	-C 4uv 2a	-C 4auv 2
C 4/a c g2 :1	C 4 2 -1ac
C 4/a c g2 :2	-C 4uv 2ac	-C 4auv 2c
C 4/m m g1	-C 4 2a
C 4/m c g2	-C 4 2ac
C 4/a m m :1	C 4a 2 -1a
C 4/a m m :2	-C 4uv 2	-C 4auv 2a
C 4/a c c :1	C 4a 2c -1a
C 4/a c c :2	-C 4uv 2c	-C 4auv 2ac
C 42/m c m	-C 4c 2c
C 42/m m c	-C 4c 2
C 42/a c g1 :1	C 4ac 2a -1ac
C 42/a c g1 :2	-C 4wd 2ac	-C 4acuv 2c
C 42/a m g2 :1	C 4ac 2ac -1ac
C 42/a m g2 :2	-C 4wd 2a	-C 4acuv 2
C 42/m c g1	-C 4c 2ac
C 42/m m g2	-C 4ac 2
C 42/a c m :1	C 4ac 2 -1ac
C 42/a c m :2	-C 4wd 2c	-C 4acuv 2ac
C 42/a m c :1	C 4ac 2c -1ac
C 42/a m c :2	-C 4wd 2	-C 4acuv 2a
F 4/m m m	-F 4 2
F 4/m m c	-F 4 2a
F 41/d d m :1	F 4ad 2 -1ad	F 4d 2 -1d
F 41/d d m :2	-F 4uw 2ud	-F 4ud 2ud
F 41/d d c :1	F 4ad 2a -1ad	F 4d 2a -1d
F 41/d d c :2	-F 4uw 2vw	-F 4ud 2vw

Trigonal triple cell H

The Hall symbols in this table were generated manually.

Hermann-Mauguin symbol	Hall symbol a-b,a+2b,c	Hall symbol 2a+b,-a+b,c	Hall symbol a+2b,-2a-b,c
H 3	H 3
H 31	H 31
H 32	H 32
H -3	-H 3
H 3 2 1	H 3 2"
H 3 1 2	H 3 2
H 31 2 1	H 31 2"	H 31 2" (0 0 2)	H 31 2" (0 0 4)
H 31 1 2	H 31 2 (0 0 2)	H 31 2 (0 0 4)	H 31 2
H 32 2 1	H 32 2"	H 32 2" (0 0 4)	H 32 2" (0 0 2)
H 32 1 2	H 32 2 (0 0 4)	H 32 2 (0 0 2)	H 32 2
H 3 1 m	H 3 -2
H 3 m 1	H 3 -2"
H 3 1 c	H 3 -2c
H 3 c 1	H 3 -2"c
H -3 m 1	-H 3 2"
H -3 c 1	-H 3 2"c
H -3 1 m	-H 3 2
H -3 1 c	-H 3 2c

Hexagonal triple cell H

The Hall symbols in this table were generated manually.

Hermann-Mauguin symbol	Hall symbol a-b,a+2b,c	Hall symbol 2a+b,-a+b,c	Hall symbol a+2b,-2a-b,c
H 6	H 6
H 61	H 61
H 65	H 65
H 62	H 62
H 64	H 64
H 63	H 6c
H -6	H -6
H 6/m	-H 6
H 63/m	-H 6c
H 6 2 2	H 6 2
H 61 2 2	H 61 2 (0 0 4)	H 61 2	H 61 2 (0 0 2)
H 65 2 2	H 65 2 (0 0 2)	H 65 2	H 65 2 (0 0 4)
H 62 2 2	H 62 2 (0 0 2)	H 62 2	H 62 2 (0 0 4)
H 64 2 2	H 64 2 (0 0 4)	H 64 2	H 64 2 (0 0 2)
H 63 2 2	H 6c 2
H 6 m m	H 6 -2
H 6 c c	H 6 -2c
H 63 m c	H 6c -2c
H 63 c m	H 6c -2
H -6 2 m	H -6 -2
H -6 2 c	H -6c -2c
H -6 m 2	H -6 2
H -6 c 2	H -6c 2
H 6/m m m	-H 6 2
H 6/m c c	-H 6 2c
H 63/m m c	-H 6c 2c
H 63/m c m	-H 6c 2