In this part, we aim to discuss the symmetry of the spin potential. General atomic potential owns the Euclidean symmetry that keeps invariant under operations from the E(3) group on the position of the structure, including three-dimensional rotations, translations, and inversion. For spin potential, the symmetry of the spin should also be considered. Under the time-reversal operation, the positions of atoms and the spin potential of the system keep invariant while the spin is flipped, while under spatial inversion operation, the spin of atoms and the spin potential keep invariant while the positions of atoms are reversed. Here, we use 𝐏 and 𝐓 to denote the operation of inversion and time-reversal respectively, and use 𝑟 ⃗ and 𝑠 ⃗ to denote the position and spin vectors of the spin atomic system while 𝐸(𝑟 ⃗, 𝑠 ⃗) represent its energy. So we have: