[Pw_forum] question about spinors

[Gabriele Sclauzero]

> 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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