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PSCF v1.1
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Commands - pscf_pc (Up/Prev) Commands - pscf_pg (Next)
THE WRITE_GROUP command accepted by pscf_pc and pscf_pg programs writes a file containing a list of all the symmetry operations of the current space group. A discussion of symmetry operations and their representation is given as an appendix. A shorter overview is given here in a form sufficient to allow a user to understand the format of the file created by the WRITE_GROUP operation. The format used by the WRITE_GROUP command is the same as that used in the group files that are distributed with pscf, which are located in subdirectories of the data/groups directory.
A space group is defined by a set of crystallogrphic symmetry operations. Every space group symmetry operation \( A \) is a function that maps very position vector \( {\bf r} \) in a \( D \) dimensional space onto a transformed position
\[ A({\bf r}) = R{\bf r} + {\bf t} \quad, \]
in which \( R \) is a linear operator that performs a point group operation such as a rotation, reflection, or inversion on the vector to its right, and \( {\bf t} \) is a translation vector. Symmetry operations for which \( {\bf t} = 0 \) are referred to as symmorphic operations, while those for which \( {\bf t} \neq 0 \) are non-symmorphic . Operations involving glide planes and screw axes are non-symmorphic. The translation vectors for non-symmorphic operations generally always the position by a rational fraction of a unit cell (e.g., 1/4 or 1/2) along one or more Bravais lattice basis vectors.
Positions and symmetry operations are most conveniently expressed using a Bravais lattice basis vectors, which we refer to as a Bravais basis. Let
\[ {\bf a}_{0}, \ldots, {\bf a}_{D-1} \]
denote a list of the basis vectors used by PSCF to define the crystal Bravais lattice. Any position may be expanded in a Bravais basis as a superposition
\[ {\bf r} = \sum_{i=0}^{D-1} r_{i} {\bf a}_{i} \quad, \]
in which \((r_{0}, \ldots, r_{D-1} \) are dimensionless components, which we refer to as reduced coordinates.
Symmetry operations are also conveniently represented using a Bravais basis. Using such a basis, we may represent any point group operation \( R \) as a matrix with integer elements, and a translation \( {\bf t} \) as a vector with components given by rational numbers (i.e., fractions such as 1/2, 1/4, 2/3, etc.) If a symmetry operation \( A \) transforms a vector \( {\bf r} \) onto a transformed vector given by
\[ {\bf r}' = A({\bf r}) = \sum_{i=0}^{D-1} r_{i}' {\bf a}_{i} \quad. \]
then the components of \( {\bf r}' \) can be expressed as a sum
\[ r_{i}' = \sum_{j=0}^{D-1} R_{ij} r_{j} + t_{i} \]
for \( i = 0, \ldots, D-1 \). Here, \( R_{ij} \) is an element of a matrix representation of the point group operation \( R \) and \( t_{i} \) is a component of the translation vector \( {\bf t} \), defined in a Bravais basis. The matrix elements of \( R \) in this basis must be integers in order to guarantee that this point group operation maps the Bravais lattice (i.e., the set of all positions with integer reduced component values) onto itself. Components of the transalation vectors for non-symmorphic operations are always fractions such as multiples of 1/2, 1/4 or 1/3 or multiples thereof when defined in a Bravais basis.
The output of the WRITE_GROUP command displays a list of symmetry operations in which each operation for a three-dimensional crystal is listed as a block of the form
in which we use R(i,j) to denote \( R_{ij} \) and t(i) to denote \( t_{i} \). Nonzero components of the translation vector are written as rational fractions, in the form I/J, where i is an integer and J is an nonzero fractional integer, with J=2,3 or 4. The formats for one- and two-dimensional space group operations are analogous, and generally contain D columns and D+1 rows to display a symmetry operation of a D dimensional crystal.
The first two lines of each group file give the dimension of space and the number of symmetry operations in the group. The dimension of space is denoted by a label "dim". The number of symmetry operations in the group is denoted by a label "size". These two lines are followed by a list of all symmetry operations, each of which is output int the format shown above, with one empty line between successive symmetry operations. The order in which symmetry operations are listed has no significance - a group is a set of operations, defined without regard to order.
An example is shown below of the file created by the WRITE_GROUP command for a system with space group \( Ia\overline{3}d \). The resulting file is identical to that distributed with the PSCF package as the file pscfpp/data/groups/3/I_a_-3_d .
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1.9.5