Symmetry relations for spin-resolved exchange correlation kernels in soft magnetic layered systems

We first exploit the physical condition satisfying the symmetry relation of the exact'' spin-resolved exchange correlation kernel based on the mixed scheme’’ in soft magnetic layered systems. The conditions are derived and examined by taking into account the field gradients of the magnetic moment as well as that of the electric moment. We also exploit the physical condition by means of deviation distribution function suitable for complex electronic structures with arbitrary $\zeta$