[Wien] Doubt regarding the inter-band velocity term from the OPTIC module.

[VIVEK PANDEY]

Dear Wien2K community,
             I am currently using the latest version of the WIEN2k
package (*Version:
WIEN2k_23.2*). At certain point of my work, I am in need of velocity term (
*<nk|vx|mk>* ), where *nk* and *m*k are the indices of Bloch states. Here,
*n* & *m* denotes the band-indices and *k* stands for the particular *k*-point
of the Brillouin zone. I found from here (
https://triqs.github.io/dft_tools/latest/faqs/faqs.html#x-optic-does-not-write-a-case-pmat-file)
that one can obtain the velocity term using the *x optic* module. For this,
I need to modify the *case.inop* file as mentioned in the above link. The
velocity term will be contained in the *case.pmat* file.

I have performed the calculations on TaAs. My *TaAs.inop *file look-like:

100  1       number of k-points, first k-point
-0.2 0.2 9999 Emin, Emax for matrix elements, NBvalMAX
2             number of choices (columns in *outmat): 2: hex or tetrag. case
1             Re xx
3             Re zz
ON            ON/OFF   writes MME to unit 4
1

So, as mentioned in the *TaAs.inop *file, I want to calculate the velocity
term for 100 *k*-points. These *k*-points are already contained in my
*TaAs.klist* file.

After running the command: *x optic -so*, the obtained *TaAs.pmat* file
look-like (I am providing the data of initial 2 *k*-points):

       1      67      68
  0.000000000000  0.000000000000  0.000000000000         1
 (-1.617771001625472E-004,1.226564556964964E-008)
 (-1.439040550334954E-004,-1.379984625073599E-008)
 (3.785649513346538E-011,-1.299045946281420E-005)
 (-3.656362974619897E-003,4.416165542701399E-003)
 (4.419121064225817E-003,3.653538690785011E-003)
 (2.453482640439684E-007,-8.805238041136547E-009)
 (1.617766828253836E-004,-1.226567120783516E-008)
 (1.439043262718125E-004,1.378900067924048E-008)
 (7.662998780467998E-012,8.582197920130107E-009)
       2      67      68
  0.100000000000  0.100000000000  0.000000000000         2
 (1.395573059329797E-002,3.799018979369327E-007)
 (1.395573021772884E-002,-3.799076130893441E-007)
 (5.130244989109358E-010,-5.404839562596613E-006)
 (1.657834175714330E-002,3.964486338191684E-003)
 (1.657834294376005E-002,-3.964481019261168E-003)
 (-2.893793368690930E-010,1.776652555720962E-003)
 (1.918588062557931E-002,-2.109479429269941E-007)
 (1.918588071334665E-002,2.109422247967537E-007)
 (-5.090469730277025E-010,-5.689825446102657E-006)

I can see that at both these *k*-points, there falls two bands (indices: 67
& 68) within the energy window considered in the *TaAs.inop* file.

To understand the data in the *TaAs.pmat* file, I analysed another file
that was produced, namely *TaAs.symmat. *The data for the first two *k*-point
in this file look-like:

SO        2  Re <x><x>    Re <z><z>          ALL

   KP:     1 NEMIN NEMAX :    67   68 dE:  -0.2   0.2 K:         1

     67  67 0.123929E-18 0.168529E-09
     67  68 0.000000E+00 0.000000E+00
     68  68 0.123929E-18 0.168529E-09

   KP:     2 NEMIN NEMAX :    67   68 dE:  -0.2   0.2 K:         2

     67  67 0.109837E-02 0.123092E-09
     67  68 0.000000E+00 0.000000E+00
     68  68 0.109837E-02 0.123092E-09

It is seen from the *TaAs.symmat *file that at each *k*-point, there are
data for three pairs of bands (67-67, 67-68 & 68-68). If the *TaAs.pmat *file
corresponds to these combinations of bands, then at each *k*-point, there
should be 9 lines. This is because,  corresponding to each pair of bands,
there will be three components of velocity (vx, vy & vz). The results
obtained in *TaAs.pmat* file seems to be consistent with this.

Here, I want to ask my question. In the *TaAs.pmat* file, I am getting the
data for *<67 k|vx|68 k> *but not the data for *<68 k|vx|67 k>. Why? Will
they both be same or anyway related to each other?*

After going through the WIEN2k userguide, I found a line mentioning: "*The
program optic generates the symmetrized squared momentum matrix elements*".
Does it mean that for the above given case, the code is making a 2x2
velocity matrix (for the two band-indices 67 & 68 ) and then it is
symmetrizing the matrix by following the formula (A+A_Transpose)/2 ? If
this is the case, then the inter-band (*n* not equal to *m*) velocity will
not be correct in the *TaAs.pmat* file. How can I obtain the correct value
of the inter-band velocity?

Kindly help me with this. Also, please correct me if I am misinterpreting
something at some point.

*Thanks and Regards*

*Yours Sincerely*
Vivek Pandey
Research Scholar
Indian Institute of Technology, Mandi
Himachal Pradesh, India
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