Non colinear magnetism¶
Notations and theoretical considerations¶
We will denote the spinor by , being the two spin indexes. The magnetic properties are well represented by introducing the spin density matrix:
where the sum runs over all states and is the occupation of state .
With , we can express the scalar density by
and the magnetization density (in units of ) whose components are:
where the are the Pauli matrices.
In general, is a functional of , or equivalently of and . It is therefore denoted as .
The expression of taking into account the above expression of is:
In the LDA approximation, due to its rotational invariance, is indeed a functional of and only. In the GGA approximation, on the contrary, we assume that it is a functional of and and their gradients. (This is not the most general functional of dependent upon first order derivatives, and rotationally invariant.) We therefore use exactly the same functional as in the spin polarized situation, using the local direction of as polarization direction.
We then have
where . Now, in the LDA-GGA formulations, and and therefore, if we set and , we have:
This makes the connection with the more usual spin polarized case.
Expression of in LDA-GGA
One would like to use the routine mkrho which does precisely this but this routine transforms only real quantities, whereas is hermitian and can have complex elements. The trick is to use only the real quantities:
and compute and with the help of:
Note that only the forurier transform are performed in mkrho.f, while the final transformation to , is performed in symrhg.f.
The computation of is performed in rhohxc.f. The only transformation to this routine, is to compute and yield back the four component , from the expression of .