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Local rotation matrices
Local rotation matrices are used to rotate the global coordinate
system (given by the definition of the Bravais matrix) to a local
coordinate system for each atomic site. They are used in the
program for two reasons:
- to minimize the number of LM combinations in the lattice
harmonics expansion (of potential and charge density according to
equ. 2.9). For example for point group mm2 one needs
for L=1 just LM=1,0 if the coordinate system is chosen such that the
z-axis coincides with the 2-fold rotation axis, while in an
arbitrary coordinate system the three terms 1,0; 1,1 and -1,1 are
needed (and so on for higher L).
- The interpretation e.g. of the partial charges requires a proper
orientation of the coordinate system. In the example given above,
the p orbitals split into 2 irreducible representations, but they
can be attributed to pz and px, py only if the z-axis is the
2-fold rotation axis.
It is of course possible to perform calculations without ``local
rotation matrices``, but in such a case the LM combinations given in
Table 7.2 (and by SYMMETRY) may not be
correct. (A program SYM written by G. Vielsack determines the LM
values for arbitrary orientations.)
Fortunately, the ``local rotation matrices`` are usually fairly simple
and are now automatically inserted into your
case.struct file. Nevertheless we recommend to check them in
order to be sure.
The most common coordinate transformations are
- interchanging of two axes (e.g. x and z)
- rotation by 45o
(e.g. in the xy-plane)
- rotation of z into the (111) direction
Inspection of the output of SYMMETRY tells you if the local rotation
matrix is the unit matrix or it gives you a clear indication
how to find the proper matrix.
The local rotation matrix R , which transforms the global coordinates
r to the rotated ones r', is defined by
R r = r'.
There are two simple ways to check the local rotation matrixes
together with the selected LM combinations:
- charge density plots generated with LAPW5 must be continuous
across the atomic sphere boundary (especially valence or difference
density plots are very sensitive, see 8.4)
- Perform a run of LAPW1 and LAPW2 using the GAMMA-point only
(or a complete star of another k point). In such a case, ``wrong``
LM combinations must vanish. Note that the latter is true only
in this case. For a k mesh in the IBZ ``wrong`` LM combinations
do not vanish and thus must be omitted!!
A first example for ``local rotation matrices`` is given for
the rutile TiO2, which has already been described as an example
in section 10.3. Also two other examples will be given
(see below).
Next: Rutile (TiO)
Up: Appendix
Previous: Appendix
2000-04-11