Physical condition and spin-resolved exchange correlation kernels in an inhomogeneous many electron system

We first exploit the spin symmetry relation $f^{\rm xc}_{s\bar s} (\zeta)=f^{\rm xc}_{\bar s s}(-\zeta)$ for the exact exchange correlation kernel $f^{\rm xc}_{s\bar s}(\zeta)$ in an inhomogeneous many electron system with arbitrary spin polarization $\zeta$. The physical condition required to satisfy the specific symmetry relation $f^{\rm xc}_{s\bar s}(\zeta) = f^{\rm xc}_{\bar s s}(\zeta)$ is derived and examined for simple ferromagnetic-nonmagnetic structure by taking the electrochemical potential into account. The condition is then applied to several composite systems useful in spintronics applications such as the magnetic system with net spin polarization.