[Wien] Local rotation matrix

[Stefano Rubino]

Hi all,
I have some questions about the local rotation matrix. I apologize if they have 
been already asked many times...
In the examples at pages 147-148 the LRM is defined as R so that r'=Rr where 
r'= (x',y',z') is the reference system of the atom (that defines the px, py and 
pz orbitals, for example) and r = (x, y, z) is the reference system of the 
crystal. It is not clear to me if the reference system should be represented as 
single row or single column.
Assuming the row representation I have

(x',y',z') = (x,y,z) |R11  R12  R13|
                     |R21  R22  R23|
                     |R31  R32  R33|

while in the column representation I have


(x')    |R11  R12  R13|(x)
(y') =  |R21  R22  R23|(y)
(z')    |R31  R32  R33|(z)



In the example of the Ti in Rutile I therefore obtain

x' = - 1/√2 x + 1/√2 y
Y' = + 1/√2 x + 1/√2 y
z' = z

For both representation. However, that is not a rotation of 45° around the z 
axis, that is
x' = + cos 45°x + sin 45° y
y' = - sin 45°x + cos 45° y
z' = z

That difference in the sign makes the "rotated" atomic reference system left-
handed.

For the Oxygen I have, in the row representation:

x' = z
y' = - 1/√2 x + 1/√2 y
z' = + 1/√2 x + 1/√2 y

In the column representation:

x' = - 1/√2 y + 1/√2 z
y' = + 1/√2 y + 1/√2 z
z' = x

Therefore I suppose that the row representation is used, since it is the one 
that makes the z' axis parallel to the (1,1,0) direction and y' parallel to
(-1,1,0) as required by the symmetry operations. However, here too, the 
resulting "rotated" atomic reference system is left-handed.

Moreover, the delta of both matrices is not 1 (it is -1).

To correct the matrix for Ti it is sufficient to make x' parallel to (1,1,0) 
and y' to (-1,1,0), while for the Oxygen a change in the sign of x' is enough.
I don't know how the first column of the rotation matrix of the oxygen has been 
calculated, since the requirements given by the output of symmetry are z' 
parallel to (1,1,0) and y' parallel to (-1,1,0).
The same applies for the other example, the one on Si. The z' axis is parallel 
to (1,1,1) and the y' to (-1,1,0), but nothing is said for the x' (by the way, 
there is a typo in the version of the matrix with the square rootes, R32 should 
be zero).
For that matrix, delta is zero and the resulting "rotated" atomic reference 
system is right-handed.

Where can I find more info about how the symmetry assignments are made?

Thanks for your attention,
best regards

-- 
D-I Stefano Rubino
Institut für Festkörperphysik, Technische Universität Wien
Wiedner Hauptstrasse 8-10/137, A-1040 Vienna, Austria
Tel.: (+43) [01] 58801-13720
Mobil: (+43) [0] 650 40 70 968

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