$\hat{D}^{\alpha\beta}_{ij}$, however they are calculated only within atom sphere. In the calculations of real material, they adopt the values in the calculations of isolate atom and don’t update in Kohn-Sham self-consistent loop. More information on non-collinear density functional theory within PAW methods can be found in the reference [@Hobbs1].

The atoms can have their initial magnetic moments by constructing the magnetic electronic density and the magnetic moments around each of atoms are equal to their desired values. If the initial total magnetic moments point $\vec{n}$ direction, generally the direction of total moment changes in Kohn-Sham self-consistent loop. We can use the constrained DFT calculations to fix the direction of magnetic moments. Our self-consistent calculations show that the directions of magnetic moments change very small in ferromagnetic materials although using the unconstrained DFT calculations (see the table.(\[table2\]) in section \[Ni\_chain\_15\] ). In real ferromagnetic materials, the directions of magnetic moments are hard to change, once the magnetic domains have formed. This is because the change of magnetization direction must include the changes of magnetic moments of all magnetic atoms in the materials. An external magnetic field can force to change the direction of magnetic moments. Of course there exists an easy 