in passing that since α is close to π , if | β − α $^{′}$|  ≈ π/ $_{12, then (}$√$_{3}$$_{/}$$_{2) sin(}$$_{α}$ ′ − β ) sin( α/ 2) V ( pdσ ) ≈ 0 . 2 V ( pdσ ), which is often greater than V ( pdπ ). Due to the Hund’s coupling, the spins of the hybridizing electron are projected to the local spins and the hybridization energy is modified. It is the main part of our Hamiltonian. We considered σ bond only. Hence, the orbitals involved are that in (3) of transition metal ions, p$_{x}$ − orbitals of oxygen atom 3 and 4 and p$_{y}$ − orbitals of oxygen atom 5 and 6 where the numerals are shown in Fig. 2b. Later, we take the spin-orbit interaction, and more orbitals, into account. It will be treated as a perturbation. Our Hamiltonian thus has two parts, H = H$_{0}$ + H$_{1}$ : and where the hybridizations energy V $_{= (}$√$_{3}$$_{/}$$_{2) sin(}$$_{β}$ − α $^{′}$) sin( α/ 2) V ( pdσ ), is only among nearest neighbors. λ is the strength of spin-orbit coupling. ε$_{p}$ , ε$_{d}$ and c$_{pil,}$$_{↑}$$_{(}$$_{↓}$$_{)}$ , c$_{djm}$ are the energies and field operators of the p − orbitals in oxygen atoms and d − orbitals in transition metal ions. l and m are the indices of oxygen atoms and transition metal ions in the basis. As shown in Fig. 3, m = 1 , 2 for manganese ions and l = 3 − 6 for oxygen atoms. As previously mentioned, the localized spins were approximated as an effective magnetic field of angles θ$_{j,m}$ 6 H$_{0}$ = ∑ ε$_{p}$c † $_{pil,σ}$c$_{pil,σ}$ + ∑ ε$_{d}$c † $_{djm}$c$_{djm}$ − ∑ n.n. V ( − 1) $^{l}$[cos θ$_{j,m}$c † $_{djm}$c$_{pil,}$$_{↑}$ + e $^{iφ}$sin θ$_{j,m}$c † $_{djm}$c$_{pil,}$$_{↓}$ ]+ H.c. (4a) H$_{1}$ = λ ∑ j,m − → l j,m · − → s j,m (4b) 0.0.3 3. Calculation