[Wien] Magnetic Moment of Nonmagnetic Atom

[Stefaan Cottenier]

> I have been watching this very interesting discussion and I found 
> myself that you brought me a bee in my mind. What would be "true 
> many-body ground state" for this isolated C atom?

Hmm, now I'll have to be careful... Anyone correct me if my 
understanding is too poor!

Imagine you apply an external magnetic field to a free C atom. This 
field will couple to the total magnetic moment of C, which is determined 
by the total angular momentum J (itself being a combination of an 
orbital L and spin S). As C has a 3P_0 ground state (where J=0), it's 
total magnetic moment is 0 and the external magnetic field will have no 
effect (apart from the much smaller diamagnetic effect caused by changes 
in the orbital motion induced by the external field).

Now try to calculate that total moment by wien2k. It will give you 2 
electrons in the same spin channel, hence S_z=1, as is true in nature as 
well. Lapwdm will show there is one of them in a m=-1 orbital, the other 
is in m=0, hence L_z=-1, which is true as well. Their sum yields 
J_z=S_z+L_z=0, again true. But you cannot proof that this J_z=0 is a 
component of J=0. The numbers wien2k gives us for S_z and L_z are in the 
assumption that S and L are parallel vectors. Such a vector picture is 
not sufficient to describe the true coupling of angular momenta.

This would still happen even if wien2k would use the exact 
XC-functional. But the latter should describe nature exactly, hence 
where is the catch? It is in the operators that are used to get S_z and 
L_z. They are single particle operators that act on the single particle 
(Kohn-Sham) wave functions. To be exact, one should use a functional 
that acts on the density, and this would return the true many body 
result. But we don't know how such a functional would look like, while 
the single particle operators are known. Therefore, even the exact 
density would give 'incomplete' results.

That's in a nutshell my understanding of the subject. No guarantee for 
correctness. If somebody can improve/correct this explanation, I would 
be happy to learn.

Stefaan


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