[Pw_forum] Orbital order in the non-collinear relativistic case... again!

[Andrea Dal Corso]

On Fri, 2012-06-08 at 10:30 +0200, Rodrigo Neumann wrote:
> Dear QE users and developers,
> 
> 
> This issue has already been raised before, but no matter how many
> times I read the available info I cannot conclude what is the final
> answer to it. So, I kindly ask you to go back to this subject, once
> and for all.
> 
> 
> In a previous post to the pw_forum
> (http://www.democritos.it/pipermail/pw_forum/2011-February/019281.html) it was suggested that the several columns present in the "X.pdos_atm#1(X)_wfc#Y(d_j2.5)" file were directly related to the eigenvalues of J^2 and Jz, that is, the total angular momentum quantum number "j" and its projection "m_j" (j, m_j). So the header line could be interpreted as:
> 
> 
> # E  ldos  (5/2,-5/2)  (5/2,-3/2) (5/2,-1/2)  (5/2,+1/2)  (5/2,+3/2)
>  (5/2,+5/2)
> 
> 
> 
> 
> However, the in-place documentation on
> espresso-5.0/PP/src/atomic_wfc_nc_proj.f90
> (http://qe-forge.org/scm/viewvc.php/trunk/espresso/PP/src/atomic_wfc_nc_proj.f90?view=markup&root=q-e&pathrev=8404), suggests otherwise:
>   ! If lspinorb=.TRUE. it makes linear combinations of eigenstates of
>   ! the atomic total angular momenta j and j_z; otherwise, of
> eigenstates of
>   ! the orbital angular momenta l, l_z and of s_z (the z-component of
> the spin).
> 
Here linear combinations means tight binding linear combinations, not
combinations of different eigenstates of j^2 and j_z to obtain
eigenstates of other operators.


> 
> 
> 
> The "Orbital Order" entry on the INPUT_PROJWFC documentation
> (http://www.quantum-espresso.org/wp-content/uploads/Doc/INPUT_PROJWFC.html#id485086, by the way, there's a minor typo regarding dxy) further strengthens the "linear combination hypothesis". From where I stand, it seems to suggest that the order is actually something like:
> 
> 
> # E   ldos   "(5/2,+1/2)+(5/2,-1/2)"   "(5/2,+1/2)-(5/2,-1/2)"
> "(5/2,+3/2)+(5/2,-3/2)"   "(5/2,+3/2)-(5/2,-3/2)" ...the same for m_j=
> +-5/2...
> where the + and - operation between eigenstates means (loosely)
> "linear combinations".
> 

This is wrong. Nowhere in the manual there are formulas like these. It
is true that in QE we are using real spherical harmonics, but they are
combined with the correct complex coefficients to obtain complex
spherical harmonics that are combined to obtain eigenstates of j^2 and
j_z. See the transformation matrix (Eq. A2 in Phys. Rev. B 71, 115106
(2005)).

HTH,
Andrea

> 
> 
> 
> I hope I was clear enough while presenting the issue. Maybe, if I
> could read FORTRAN, I'd be able to figure it out for myself. But I
> can't, and that's why I'm asking for you help. I also suggest that
> whatever answer we come up to in this thread should be written
> explicitly on INPUT_PROJWFC in order to prevent this issue from
> surfacing again in a few months.
> 
> 
> So, for once and for all, are those the columns related to the
> eigenstates of J^2 and Jz or to their linear combinations? 
> 
> 
> Thanks in advance!
> 
> 
> Best regards,
> Rodrigo Neumann
> 
> 
> --------------------------------------------------------
> Rodrigo Neumann Barros Ferreira
> PhD Student
> Departamento de Física dos Sólidos - Instituto de Física
> Universidade Federal do Rio de Janeiro
> Cx.Postal 68.528
> 21945-972, Rio de Janeiro - RJ - BRASIL
> 
> Tel: [+55] (21) 2562-7341
> http://www.if.ufrj.br/~neumann/
> http://tmc.if.ufrj.br/
> --------------------------------------------------------
> 
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-- 
Andrea Dal Corso                    Tel. 0039-040-3787428
SISSA, Via Bonomea 265              Fax. 0039-040-3787249
I-34136 Trieste (Italy)             e-mail: dalcorso at sissa.it