Polarzation, kpoint grid set

[jinylee]

Dear ATK team,

I tried to calculate polarization which using Berry phase calculation.
Users only can put k-point sampling values in ATK script generator.
As I know, kpoint has typical format as "AxBxC", but in the polarization part of the script generator
there was 3x3 matrix.
At first I put my kpoint values in the  diagonal part, but all other part (off-diagonal terms) to be one.
as " 4  1   1
      1  12  1
      1   1   6 "
After that, I switched the off-terms to be 5, and calculated again.
as " 4  5  5
      5  12 5
      5   5  6 "

The final polarization values are different with respect to each off-diagonal term of kpoint matrix.
I wonder the meaning of off-diagonal term in kpoint 3x3 matrix.

How I should set the kpoints to calculate polarization?

Thank you in advance.

[Anders Blom]

First of all, it's easy enough to advise how to choose these 3x3 k-points. Then should be

M N N
N M N
N N M

where M is large and N can be smaller.

The reason it's not just 3 numbers can be seen from the equation defining P_e at http://quantumwise.com/documents/tutorials/latest/Polarization/index.html/chap.introduction.html#sect1.introduction.theory. P_e is a vector with 3 components (XYZ), but for each component you integrate in k-space, with very different kernels for k// and k_| (sorry for poor man's formulas). You need a very accurate integration along k// which for X is along A, and that's your M for "kpoints_a" (4 in your case).

So what you probably need is M=100 and N=3 or 5 or so.