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Subsections

# A. 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. 2.10). 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 and , 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 7.42 (and by SYMMETRY) may not be correct. (The LM values for arbitrary orientations may be obtained from a procedure described in Singh 94.)

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 (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 , which transforms the global coordinates to the rotated ones , is defined by .

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 8.6)

• 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).

# 1 Rutile ( )

Examine the output from symmetry. It should be obvious that you need local rotation matrices for both, Ti and O:

   ....
Titanium   operation #  1     1
Titanium   operation #  2     -1
Titanium   operation #  5     2 || z
Titanium   operation #  6     m n z
Titanium   operation # 12     m n 110
Titanium   operation # 13     m n -110
Titanium   operation # 18     2 || 110
Titanium   operation # 19     2 || -110
pointgroup is mmm (neg. iatnr!!)
axes should be: m n z, m n y, m n x


This output tells you, that for Ti a mirror plan normal to z is present, but the mirror planes normal to x and y are missing. Instead, they are normal to the (110) plane and thus you need to rotate x, y by around the z axis. (The required choice of the coordinate system for mmm symmetry is also given in Table 7.42)

   ....
Oxygen     operation #  1     1
Oxygen     operation #  6     m n z
Oxygen     operation # 13     m n -110
Oxygen     operation # 18     2 || 110
pointgroup is mm2 (neg. iatnr!!)
axes should be: 2 || z, m n y


For O the 2-fold symmetry axes points into the (110) direction instead of z. The appropriate rotation matrices for Ti and O are: # 2 Si -phonon

Si possesses a face-centered cubic structure with two equivalent atoms per unit cell, at ( 0.125, 0.125, 0.125). The site symmetry is -43m. For the -phnon the two atoms are displaced in opposite direction along the (111) direction and cubic symmetry is lost. The output of SYMMETRY gives the following information:

Si         operation #  1     1
Si         operation # 13     m n -110
Si         operation # 16     m n -101
Si         operation # 17     m n 0-11
Si         operation # 24     3 || 111
Si         operation # 38     3 || 111
pointgroup is 3m (neg. iatnr!!)
axis should be: 3 || z, m n y
lm: 0 0  1 0  2 0  3 0  3 3  4 0  4 3  5 0  5 3  6 0  6 3  6


Therefore the required local rotation matrix should rotate z into the (111) direction and thus the matrix in the struct file should be:

 0.4082483 -.7071068 0.5773503   0.4082483 0.7071068 0.5773503   -.8164966 0.0000000 0.5773503   # 3 Trigonal Selenium

Selenium possesses space group P3121 with the following struct file:

H    LATTICE,NONEQUIV.ATOMS: 1
MODE OF CALC=RELA  POINTGROUP:32
8.2500000 8.2500000  9.369000
ATOM= -1: X= .7746000  Y= .7746000  Z= 0.0000000
MULT= 3          ISPLIT= 8
ATOM= -1: X= .2254000  Y= .0000000  Z= 0.3333333
ATOM= -1: X= .0000000  Y= .2254000  Z= 0.6666667
Se         NPT=  381  R0=.000100000 RMT=2.100000000  Z:34.0
LOCAL ROT.MATRIX:    0.0       0.5000000 0.8660254
0.0000000 -.8660254 0.5000000
1.0000000 0.0000000 0.0
6    IORD OF GROUP G0
......


Se         operation #  1     1
Se         operation #  9     2 $|$$|$ 110
pointgroup is 2 (neg. iatnr!!)
axis should be: 2 || z
lm: 0 0  1 0  2 0  2 2 -2 2  3 0  3 2 -3 2  4 0  4 2 -4 2 ......


Point group 2 should have its 2-fold rotation axis along z, so the local rotation matrix can be constructed in two steps: firstly interchange x and z (that leads to z (011) ) and secondly rotate from (011) into (001) (see the struct file given above). Since this is a hexagonal lattice, SYMMETRY uses the hexagonal axes, but the local rotation matrix must be given in cartesian coordinates.    Next: B. Periodic Table Up: 4 Appendix Previous: 4 Appendix   Contents
pblaha 2011-03-22