Insofar as the main question is concerned (that is, I am not concerned with various implementations of the DFT, and especially not with "codes"), to search for spin-polarised solutions, one will have to solve two coupled Kohn-Sham equations, one for each of the two spin species. The coupling between the two Kohn-Sham equations is both direct and indirect. The direct coupling comes about by the dependence of the effective potential in the Kohn-Sham equation corresponding to spin-up/down electrons on the number densities (or charge densities) of both spin-up and spin-down electrons (most evidently, the Hartree part of the effective potential depends on the total number density, which is the sum of the number densities of spin-up and spin-down electrons). The indirect  coupling comes about by the requirement that the total numbers of spin-up and spin-down electrons must add up to a given integer N, the total number of electrons in the system. Describing the effective self-consistent Kohn-Sham potentials corresponding to both spin species in terms of their Fourier components, and assuming that each potential is described accurately by M Fourier coefficients, at self-consistency a spin-polarised ground state is characterised by 2M Fourier coefficients, whereas a spin-unpolarised ground state by M Fourier coefficients, the effective potentials corresponding to spin-up and spin-down electrons being identical. A similar argument applies to the Kohn-Sham wave functions: in contrast to the spin-unpolarised case where each single-particle state is either unoccupied of doubly occupied, in the spin-polarised case (or in search for a possible spin-polarised ground state) each single-particle state is either unoccupied or singly occupied.

With regard to the question following the main question, basis functions cannot be relevant to the physics described by the DFT, provided that they form complete (or "sufficiently complete", by some to be adopted criterion) sets in the single-particle Hilbert space of the problem at hand.

Insofar as the main question is concerned (that is, I am not concerned with various implementations of the DFT, and especially not with "codes"), to search for spin-polarised solutions, one will have to solve two coupled Kohn-Sham equations, one for each of the two spin species. The coupling between the two Kohn-Sham equations is both direct and indirect. The direct coupling comes about by the dependence of the effective potential in the Kohn-Sham equation corresponding to spin-up/down electrons on the number densities (or charge densities) of both spin-up and spin-down electrons (most evidently, the Hartree part of the effective potential depends on the total number density, which is the sum of the number densities of spin-up and spin-down electrons). The indirect  coupling comes about by the requirement that the total numbers of spin-up and spin-down electrons must add up to a given integer N, the total number of electrons in the system. Describing the effective self-consistent Kohn-Sham potentials corresponding to both spin species in terms of their Fourier components, and assuming that each potential is described accurately by M Fourier coefficients, at self-consistency a spin-polarised ground state is characterised by 2M Fourier coefficients, whereas a spin-unpolarised ground state by M Fourier coefficients, the effective potentials corresponding to spin-up and spin-down electrons being identical. A similar argument applies to the Kohn-Sham wave functions: in contrast to the spin-unpolarised case where each single-particle state is either unoccupied of doubly occupied, in the spin-polarised case (or in search for a possible spin-polarised ground state) each single-particle state is either unoccupied or singly occupied.

With regard to the question following the main question, basis functions cannot be relevant to the physics described by the DFT, provided that they form complete (or "sufficiently complete", by some to be adopted criterion) sets in the single-particle Hilbert space of the problem at hand.

Thanks. I agree the physics should not depend on the choice of basis functions, since they should both approach a complete set in the limiting case.

I think your explanation helps me understand things better. In all implementations of Gaussian-based DFT that I'm familiar with, the total number of spin-up electrons Na and spin-down electrons Nb are set to fixed constants which are specified at the start of the calculation (in terms of the net charge and spin multiplicity).

By contrast, in a plane-wave implementation, it sounds like only Na + Nb = N is required, where N (the total number of electrons) is a fixed constant.  The number of spin-up and spin-down electrons is determined by filling the single-particle states up to the total number N.

If this is the case, the Gaussian-based implementations could in principle support the "Na + Nb = N" requirement, and the spin multiplicity would then become a dependent variable that is solved for.  

Let me know if this agrees with your understanding.

I agree. I should point out however that even when N↑ = N↓ = N/2 (assuming N even), one may still have n↑(r) ≠ n↓(r) over some finite region of r. In other words, while the system may not be macroscopically spin polarised, the spin polarisation density n↑(r) - n↓(r) need not be identically vanishing.

That sounds right.  I've encountered systems like that before.  

Thanks again for your help.  :)

You are welcome Lee-Ping.

Great question and answer, thanks to both of you for that.

As a follow up question, how is non-collinear spin magnetization handled? I've gathered that the FT becomes complex (complex Fourier coefficients, perhaps?), but I haven't fully grasped the connection.

You are welcome Christopher. As for your question, the issue is discussed in detail by Engel and Dreizler in their book Density Functional Theory - An Advanced Course (Springer, 2011). Essentially, for the purpose one must extend the Hamiltonian by a term that formally couples an external magnetic field with the spin magnetisation density. The theory then enables one to describe spiral spin-density wave. The formalism follows the general setting described by H.J.F. Jansen in his paper 'Many-body properties calculated from the Kohn-Sham equations in density-functional theory', whose link I attach below.

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.43.12025

Behnam--Thanks so much for the references and the explanation. Going to do some reading now...!

Best,

Chris

Thanks to Behnam who greatly answered the relevant questions. Just a comment on the aspect, if there are any quantum chemistry packages based on Gaussian-Type functions (or any other atomic orbitals), which may find the electronic ground state also with respect to spin. The Turbomole program for molecular systems uses so-called 'Fermi smearing' that creates partial occupations of low lying virtual levels determined by a parameter that is interpreted as 'temperature' during the SCF procedure (in a Hartree-Fock or KS-DFT calculation). This temperature parameter is decreased during the SCF iterations and usually leads (in the molecular case) to integer occupation numbers in the end. Though there is no guarantee, one may find 'automatically' the spin state which belongs to the electronic ground state. This method is described in the Turbomole manual of version 6.6 on page 287 in the PDF version (see http://www.turbomole-gmbh.com/turbomole-manuals.html). There is no citation given but it looks similar to smearing methods used in solid state calculations based on plane waves (and could be used in solid state calculations using Gaussian-type basis sets).

And so we are back at the main point: The choice of basis does not restrict the possibility to search for the most stable spin state.

I hope this example clarifies also the technical point of view.

Best regards.

You are welcome Asbjörn, and thank you for your kind words. Just for clarity, in my very first response on this page I emphasised (by writing "I am not concerned with various implementations of the DFT, and especially not with "codes"") that I explicitly did not go into various implementations of the DFT. This limitation partly reflects on my very personal approach to computation; in general, I write all the codes I need for my research from the scratch and thus know about the strengths and limitations of the various available codes only indirectly (for instance, through published reviews).

Asbjörn, thanks!  That's very helpful to know about.