[Wien] Definition of directions in QTL band character calculations

[pluto]

Dear All,

I am interested in calculating different types of orbital characters.

When calculating a TMDC of 1T type (space group 164) I experience the 
following:

(A). When using regular commands (x lapw2 -band -qtl -up/dn -so) I am 
having x swapped with z. Perhaps also other things happen, according to 
Loc-Rot matrices in case.struct. This is explained in UG, page 277.

(B). When using QTL to calculate characters, with loro=0 I have the same 
result as in point (A).

(C). When using QTL to calculate characters with loro=1, and setting the 
"new-z-axis" to "0 0 1", I am having the pz orbital facing out of plane 
of the layers (i.e. along the hexagonal c-axis). This is of course a 
good starting point for anyone who compares bands to ARPES data.

I just want to make sure, if I understand the logic of this. I 
understand that with "lapw2 -qtl" and with "qtl, loro=0" the coordinate 
system is rotated according to the Loc-Rot matrices from the 
case.struct. I also understand that these Loc-Rot matrices can be in 
principle different for each inequivalent atom, so in principle setting 
"pz" for atom 1 can be physically a different direction than "pz" for 
atom 2.

Now I am not sure to which axis the program QTL refers when loro=1 (or 
loro=2). It seems to me that it somehow refers to the hexagonal 
coordinate system defined in the case.struct. For a layered hexagonal 
system, conventionally this would mean a-b-c, with c being out-of-plane 
of the layers, and with 120 deg between a and b. Lets say that a-b-c 
transforms into h-k-l in the reciprocal space. Then l (and c) are 
out-of-plane, so in Cartesian system it is parallel to z=(001). But what 
are x and y of the Cartesian system? For "qtl, loro=2", "new z-axis 0 0 
1", "new x axis 1 0 0" is x=(100) direction parallel to a? Or maybe 
parallel to h? (y would be determined this way).

In other words, with "qtl loro=2', "new z-axis 0 0 1", "new x axis 1 0 
0", would the new x-axis be along Gamma-M or along Gamma-K?

How would this work for the rhombohedral system (layered 2D system)? 
Because for the k-list of the rhombohedral system one needs to use the 
rhombohedral reciprocal vectors, while in the case.struct file one 
provides hexagonal lattice parameters (this was discussed many times in 
the forum), and for real orbitals we need Cartesian system... Would QTL 
with loro=1 again set the z-axis out of plane of the layers according to 
the hexagonal abc (or hkl)?

Best,
Lukasz