[Pw_forum] question about spinors

[Alexey Akimov]

Dear Gabriele,

Thank you for additional clarifications.

Best,
Alexey 

----- Original Message -----
From: "Gabriele Sclauzero" <gabriele.sclauzero at epfl.ch>
To: "PWSCF Forum" <pw_forum at pwscf.org>
Sent: Monday, August 27, 2012 1:04:17 AM
Subject: Re: [Pw_forum] question about spinors


Hello, 



Il giorno 25/ago/2012, alle ore 04.06, Alexey Akimov ha scritto: 


Dear Gabriele, 

Thank you for your great comments! So as i understand the key for orbital overlap is additional summation over spin components. 


Just to be a bit more pedantic... I would not call them as that, but rather "spinor components", to avoid confusion. In general, when magnetization is non collinear you cannot identify the first component with "spin-up" and the second with "spin-down", unless the magnetization is collinear and oriented along z. 

>>> ok. thank you for pointing this out

This makes a lot sense to what i see - the odd-even alteration of 1 or 0 overlaps. In other words Psi_1 = psi_1,1 , Psi_2 = psi_1,2, ... Psi_2n+1 = psi_n,1, Psi_2n+2 = psi_n,2 and then <Psi_1|Psi_1> = <psi_1,1|psi_1,1> = 1, <Psi_2|Psi_2> = <psi_1,2|psi_1,2> = 0 (alteration), but <Psi_1|Psi_1> + <Psi_2|Psi_2> = <psi_1,1|psi_1,1> + <psi_1,2|psi_1,2> = 1, ... (no alteration). 


I've got a bit lost here with your notation, but it seems that you got the point. I think you meant "alternation" here, am I correct? 

>>> right, alternation


That is the orbitals written in the output are actually not 1, 2, 3, ... etc, but rather 1,1, 1,2, 2,1, 2,2, ... etc. (here the second index is the spin-state or more precisely the spinor component). 

>>> i see. this is the point from the previous sentence. 


I don't know of which output you are talking about here (pp.x?). Anyway, I think that the wave function coefficients for the first and the second component are stored consecutively in the same array (as said by Paolo in the first reply). 

>>> yes, i meant the output produced by pp.x. So, as you say the orbitals are written as 1,1, 1,2, 2,1, 2,2, ... etc. Do you mean the first band (and hence its planewave expansion) is 1,1 ("spin-up"), the second band (and its coefficients in output) is 1,2, etc. ? 


Also if i understand it correctly we should not consider spin-up and spin-down functions separately. Is this right? 


I'm not sure I understand this point correctly, maybe it is related to what I've written above? 

>>> yes, you already answered this in the first part.


I would appreciate if someone could give some reference (apart from wikipedia and QE tutorials/presentations) for the spinor algebra. 



I do not have any references at hand, but I'm pretty sure that if you look up at "Pauli equation" in any good quantum mechanics book (maybe advanced ones) you'll find what you need. Indeed, the fully-relativistic non collinear KS equation is pretty similar to that one. 


Ok. I see the point with the npwx vs. npw. However i was using a single k-point (gamma), so one would expect that this should not introduce additional complications. You also mentioned that the parallelization can effect this - would storing the entire wavefunction in one file solve such issue (wf_collect = .true.) instead of storing it in different files - one per process? 



It should, but I'm not sure... you can make a small test. 




GS 




Thank you, 
Alexey 

----- Original Message ----- 
From: "Gabriele Sclauzero" < gabriele.sclauzero at epfl.ch > 
To: "PWSCF Forum" < pw_forum at pwscf.org > 
Sent: Friday, August 24, 2012 3:57:54 AM 
Subject: Re: [Pw_forum] question about spinors 







Dear Paolo, 

I'm not sure that i completely understand what you mean by empty coefficients. Also what is npwx, how is it different from npw? 


I think this is due to the fact that wave functions at different k-points can have different number of plane waves if the basis set cut-off is expressed in terms of the kinetic energy of the plane wave ~ |k+G|^2. Still, it's more practical to use the same array (of size npwx>npw) for storing the wave function (one k-point at a time). Also G-vector parallelization might introduce this kind of issue. 




Also am i correct that the spin-up and spin-down orbitals are orthogonal not because of the artificial convention <alpha|beta> = 0, but rather by construction of the corresponding plane wave expansions (given by coefficients c_gi )? 


I would not call this is an artificial convention... it's the way you write the wavefunction (space+spin components) that allows you to do this, which is turn depends on the Hamiltonian that you consider. 
Anyway I think this is correct, although you should be aware that when you take the norm of a two-component spinor you need to sum over the two components, i.e. 
< Psi_i | Psi_j > = < Psi_i,1 | Psi_j,1 > + < Psi_i,2 | Psi_j,2 >, where 1 and 2 denote first and second component, resp. 




or the orthogonality is already included in "spatial" part (so <phi_i|phi_j> = 0 for alpha and beta spin-orbitals)? 



Not really in the 3D spatial part, but rather in the "relations" between the first and second component. I mean, the overlap between first components and that between the second components can both be nonzero, but the sum might be zero. This is the most general case, when you have spin-orbit and/or non-collinear magnetization. If the ground state has collinear magnetization you can always rotate the magnetic axis such that each wave function has Psi_1 or Psi_2 which is zero everywhere (and you get the same result, which should be the same as in LSDA). 


HTH 




GS 





Thank you, 
Alexey 


----- Original Message ----- 
From: "Paolo Giannozzi" < giannozz at democritos.it > 
To: "PWSCF Forum" < pw_forum at pwscf.org > 
Sent: Wednesday, August 22, 2012 8:27:59 AM 
Subject: Re: [Pw_forum] question about spinors 


On Aug 21, 2012, at 23:55 , Alexey Akimov wrote: 



I try to understand the format of the wavefunction in case of spin- 


polarization 


(nspin=4, spinorb=.true. (or something similar)) 

KS orbitals for the spin-orbit case have coefficient on a basis of 
NPW plane 
waves with spin up, NPW plane waves with spin down. The dimension of the 
orbitals is 2*NPWX >= 2*NPW, so there can be empty coefficients in the 
middle. 

P. 
--- 
Paolo Giannozzi, Dept of Chemistry&Physics&Environment, 
Univ. Udine, via delle Scienze 208, 33100 Udine, Italy 
Phone +39-0432-558216, fax +39-0432-558222 










§ Gabriele Sclauzero, EPFL SB ITP CSEA 
PH H2 462, Station 3, CH-1015 Lausanne 







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-- 
Dr. Alexey V. Akimov

Postdoctoral Research Associate
Department of Chemistry
University of Rochester

aakimov at z.rochester.edu