Solid-state Symmetry Operations

Solid-state symmetry operations are used in two ways in Program BZ.

(A)  To symmetrize the energy matrix so that the band-structure is more symmetric.  If symmetry operations are not performed on a band-structure, then small artifacts can appear in band structures.  For example, bands of different symmetry should be able to cross.  If the energy matrix is not symmetrized, small errors in symmetry, due to errors in geometry or insufficiently tight criteria, can result in small gaps between the bands, resulting in the appearance of a non-crossing interaction.

(B)  To generate the little group for a point in k-space, i.e., symmetry theory can be used in analyzing the properties of individual points in the Brillouin zone.

Description and construction of the symmetry-operations file <file>.ops

 Because of the usefulness of symmetry, the option exists to read in all the space-group operations for the system.  For a file containing energy band information <file>.brz, the symmetry operations would be provided in a file named <file>.ops.  If <file>.ops is  present, these operations will be automatically used in constructing little groups and in symmetrizing the band structure.

The order of translation vectors is important. Within BZ, k-space is orientated so that the first k-direction is along the "x" axis, the second k-direction is in the "xy" plane, and the third k-direction is out-of-plane.  If the axes are all at 90° to each other, no problems arise. If the axes are not all at 90° to each other, then for the symmetry operations to work, the order of the axes is important.   If, as would normally be the case, internal coordinates are used, then the first translation direction should be along the unique axis, if one exists.  For a worked example see graphite.

Symmetry operations are represented by the product of a point-group operation and a translation in the crystallographic unit cell. Each symmetry operation is defined using 12 data, these are, in order:

Datum Value Purpose
1 0 Pure rotation or identity operation
1 1 Inversion operation
1 2 End of symmetry operations
2-4 0 No translation
2-4 2 or larger The reciprocal of the fraction non-trivial translation vector in each
of the three directions.  For example, "2 0 0" would represent a translation
of 0.5, 0.0, 0.0 of the crystallographic unit cell or half a translation vector
representing motion through one unit cell in the first translation direction.
5 n.nn The rotation angle expressed as a fraction of a circle.
Example: 0.1666 represents a rotation of 60º, i.e.,  360 times 1/6.
6-8 a.aa The axis for the rotation.  This can be un-normalized.  If no rotation is involved, then use (0.0 0.0 1.0)
9-11 a.aa The location of the center of the point-group operation, in Ångstroms. 
In most systems, the first atom will be at the origin, i.e., at coordinates (0.0,0.0,0.0) 
A useful strategy is to make sure that atom 1 is at the center of symmetry. 
If the center of the point-group operation is between atoms, then use the MOPAC
data set, output, or arc file to work out its coordinates.

12 text A brief (max: 12 characters) description of the operation. 
The text must be preceded and followed by either an inverted comma (') or a quotation symbol (").


Examples of such data are:

0 0 0 0 0.5       1.0       0.0       0.0  0.0 0.0 0.0  'C2=C4(2)'
0 2 2 2 0.25      0.0       1.0       0.0  0.0 0.0 0.0  'C4'
1 2 2 2 0.5       0.0       1.0       0.0  0.0 0.0 0.0  'Sigma-h'
2 0 0 0 0.0       0.0       0.0       1.0  0.0 0.0 0.0  'End'