[QE-users] How to compute spin projections?

[Hongyi Zhao]

On Wed, Mar 30, 2022 at 11:51 PM Luiz Gustavo Davanse da Silveira
<lgsilveira at fisica.ufpr.br> wrote:
>
> Dear Hongyi Zhao,
>
> Thanks for your replys. I am answering them all at once. I tried projwfc.x
> but got only orbital projections. I took a look at the inout documentation
> and couldn't find an apropriate variable to compute spin projections.
> Perhaps I am missing something but I can't realize what it would be.
>
> I also tried pyprocar but it doesn't support QE non-colinear calculations
> right now.

Then, the only feasible pathway may be to do the job manually
according to the description on PROCAR file specification shown on
vasp wiki [1]:

```
For LNONCOLLINEAR=.TRUE. three additional projections for each ion are
printed and the output is similar to

ion      s     py     pz     px    dxy    dyz    dz2    dxz  x2-y2    tot
1  0.144  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.145
2  0.291  0.000  0.006  0.000  0.000  0.000  0.000  0.000  0.000  0.298
3  0.291  0.000  0.006  0.000  0.000  0.000  0.000  0.000  0.000  0.298
tot    0.727  0.000  0.013  0.000  0.000  0.000  0.000  0.000  0.000  0.740
1 -0.011 -0.000 -0.000 -0.000 -0.000 -0.000 -0.000  0.000 -0.000 -0.011
2 -0.023 -0.000 -0.000  0.000  0.000 -0.000 -0.000  0.000 -0.000 -0.023
3 -0.023 -0.000 -0.000  0.000  0.000 -0.000 -0.000  0.000 -0.000 -0.023
tot   -0.057 -0.000 -0.001  0.000  0.000 -0.000 -0.000  0.000 -0.000 -0.058
1 -0.142 -0.000  0.000  0.000  0.000  0.000 -0.000 -0.000 -0.000 -0.142
2 -0.286  0.000 -0.006 -0.000 -0.000  0.000 -0.000 -0.000  0.000 -0.293
3 -0.286  0.000 -0.006 -0.000 -0.000  0.000 -0.000 -0.000  0.000 -0.293
tot   -0.715  0.000 -0.012 -0.000 -0.000  0.000 -0.000 -0.000  0.000 -0.727
1 -0.024 -0.000  0.000 -0.000 -0.000  0.000 -0.000  0.000 -0.000 -0.024
2 -0.048  0.000 -0.001  0.000  0.000  0.000 -0.000  0.000  0.000 -0.049
3 -0.048  0.000 -0.001  0.000  0.000  0.000 -0.000  0.000  0.000 -0.049
tot   -0.119  0.000 -0.002  0.000  0.000  0.000 -0.000  0.000  0.000 -0.121

Here the entries correspond to the projected magnetizations


1 / 2 \sum_{\mu, \nu=1}^{2} \sigma_{\mu \nu}^{j}\left\langle\chi_{n
\mathbf{k}}^{\mu} \mid Y_{l m}^{\alpha}\right\rangle\left\langle Y_{l
m}^{\alpha} \mid \chi_{n \mathbf{k}}^{\nu}\right\rangle

and are calculated for the spinor of the spinor

\left|\Psi_{n \mathbf{k}}\right\rangle=\left(\begin{array}{c}\chi_{n
\mathbf{k}}^{\uparrow} \\ \chi_{n
\mathbf{k}}^{\downarrow}\end{array}\right)

and the Pauli matrices:


\sigma^{x}=\left(\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right),
\quad \sigma^{y}=\left(\begin{array}{cc}0 & -i \\ i &
0\end{array}\right), \quad \sigma^{z}=\left(\begin{array}{cc}1 & 0 \\
0 & -1\end{array}\right)

The first set is the total (absolute) magnetization, while the
remaining three sets of entries correspond to the three directions j =
1 , 2 , 3.
```

[1] https://www.vasp.at/wiki/index.php/PROCAR

Yours,
Hongyi