Summary

In the DFT, for both theoretical and practical reasons, it is important to know what general forms of densities are which obey SCS of interest. In the case of the space symmetries, such general forms can be established by means of methods of constructing the isotropic tensor fields.

For the spherical symmetry, the local densities are the isotropic scalar, vector, or (the second rank) tensor fields, depending on the position vector $\bm{r}$. The form of an isotropic field with given rank is unique and determined through one arbitrary scalar function. In particular, the parity of the field is unique for a given rank. Pseudoscalar, pseudovector, and pseudotensor fields do not exist. This is why in the case of the rotational and mirror symmetry, the pseudoscalar (spin-divergence), pseudovector (spin, spin-kinetic and tensor-kinetic), and pseudotensor (symmetric spin-current) local densities vanish.

For the axial symmetry, the local densities are isotropic fields depending on two components of the position vector: $\bm{r}_z$ and $\bm{r}_{\perp}$. The case of SO $^{\perp }(2)$ is interesting as it allows us to better understand the assumption of the isotropy of a field. Indeed, in this case it might seem that the local densities are fields, depending on the SO $^{\perp }(2)$ vector $\bm{r}_{\perp}$. However, such fields are not isotropic. There is another vector, $\bm{r}_z$, which plays the role of a material vector fixed by the direction of the symmetry axis of the system.