[QE-users] How to extract the px, py, pz, dz2, ....contributions with SOC from projdos out

[Thomas Brumme]

Dear Elie,


The short answer is: You can't!


Spin-orbit interaction couples the orbital angular momentum with the 
spin momentum and thus neither l or s are good quantum number anymore. 
You now have to use the total angular momentum J. Sure, a lot of people 
still use the nomenclature from the non-SOC calculations, such as 
speaking in the case of 2D TMDCs of dz² states at the K point, but 
strictly speaking this is not correct. If you really want to understand 
the relation between J and L in detail, I can recommend the paper by 
Andrea Dal Corso and Adriano Mosca Conte, Phys. Rev. B 71, 115106 (2005) 
(https://link.aps.org/doi/10.1103/PhysRevB.71.115106) which nicely shows 
which spherical harmonics are included in which spin-angle functions for 
which total angular momentum j. If I remember this correctly (some time 
ago that I did this and I can't find the table anymore) the states with 
higher mj (+-3/2 and +- 5/2) have more in-plane characterfor both j=3/5 
and j=5/2 (i.e., the contain terms with spherical harmonics which are 
usually combined such that the result is in the xy-plane) while the 
states with mj=+-1/2 have more out-of-plane character (z direction). But 
I could be wrong here since I don't have the details anymore.


The only option for you - if you don't want to check the paper or if 
this is too much and if nobody else comments here - is to do a non-SOC 
calculation and then hope that SOC is not mixing states too much and you 
can still call the result, e.g., dz² like. OR you plot the corresponding 
wave function in real space and decide by "looking" at the form of |psi|².


Regards


Thomas



P.S.: another detail concerning, e.g., the states at K in a WS2 
monolayer - is this your system? :)

The states are not simple split into spin up and spin down even if a lot 
of people use this nomenclature. For the valence band the two SOC-split 
bands are to nearly 100% spin up or down but not for the conduction band 
where you won't have states which are 100% up or down, even if there is 
no in-plane contribution... Some details can also be found here: 
https://link.aps.org/doi/10.1103/PhysRevB.101.235408


On 7/4/21 6:52 AM, Elio Physics wrote:
> Dear all,
>
> I am studying the contribution of the orbitals to the bands of a 
> structure, in the presence of spin orbit coupling. At the beginning of 
> the projwfc.x output file, I got:
>
> state #   1: atom   1 (S  ), wfc  1 (l=0 j=0.5 m_j=-0.5)
>      state #   2: atom   1 (S  ), wfc  1 (l=0 j=0.5 m_j= 0.5)
>      state #   3: atom   1 (S  ), wfc  2 (l=1 j=0.5 m_j=-0.5)
>      state #   4: atom   1 (S  ), wfc  2 (l=1 j=0.5 m_j= 0.5)
>      state #   5: atom   1 (S  ), wfc  3 (l=1 j=1.5 m_j=-1.5)
>      state #   6: atom   1 (S  ), wfc  3 (l=1 j=1.5 m_j=-0.5)
>      state #   7: atom   1 (S  ), wfc  3 (l=1 j=1.5 m_j= 0.5)
>      state #   8: atom   1 (S  ), wfc  3 (l=1 j=1.5 m_j= 1.5)
> .
> .
> .
> state #  39: atom   5 (W ), wfc  3 (l=2 j=1.5 m_j=-1.5)
>      state #  40: atom   5 (W), wfc  3 (l=2 j=1.5 m_j=-0.5)
>      state #  41: atom   5 (W), wfc  3 (l=2 j=1.5 m_j= 0.5)
>      state #  42: atom   5 (W), wfc  3 (l=2 j=1.5 m_j= 1.5)
>      state #  43: atom   5 (W ), wfc  4 (l=2 j=2.5 m_j=-2.5)
>      state #  44: atom   5 (W ), wfc  4 (l=2 j=2.5 m_j=-1.5)
>      state #  45: atom   5 (W), wfc  4 (l=2 j=2.5 m_j=-0.5)
>      state #  46: atom   5 (W), wfc  4 (l=2 j=2.5 m_j= 0.5)
>      state #  47: atom   5 (W), wfc  4 (l=2 j=2.5 m_j= 1.5)
>
> The l=1 wavefunctions are the p contributions. But How can we 
> specifically identify which one is px, py and pz?
> Similary, how to identify which ones of the 10 d orbitals are the dz^2 
> for example
>
> regards
>
> Elie
> Federal Universiy of Rondonia
> Brazil
>
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-- 
Dr. rer. nat. Thomas Brumme
Theoretical chemistry
TU Dresden - BAR / II49
Helmholtzstr. 18
01069 Dresden

Tel:  +49 (0)351 463 40844

email: thomas.brumme at tu-dresden.de

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