[Pw_forum] question about spinors

[Alexey Akimov]

Dear Gabriele,

Thank you for your great comments! So as i understand the key for orbital overlap is additional summation over spin components. 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). 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). Also if i understand it correctly we should not consider spin-up and spin-down functions separately. Is this right? I would appreciate if someone could give some reference (apart from wikipedia and QE tutorials/presentations) for the spinor algebra.  

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?


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 




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-- 
Dr. Alexey V. Akimov 

Postdoctoral Research Associate 
Department of Chemistry 
University of Rochester 

aakimov at z.rochester.edu 
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§ 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