and φ . We now made the following transformation where − → R$_{i,l}$ = − → R$_{i}$ + − → r l with − → r l being the position vector of the l -th oxygen atom in the basis without bond-bending. The Hamiltonian becomes where d$_{jm}$ = c$_{djm}$ , δθ$_{l}$ = a$_{0}$q$_{x}$/ 2, η$_{l}$ = 1 for l = 3 , 4 and δθ$_{l}$ = b$_{0}$q$_{y}$/ 2, η$_{l}$ = − 1 for l = 5 , 6. a$_{0}$ and b$_{0}$ are, respectively, the distance between transition metal ions along x and y direction. The Hamiltonian in momentum space is p$_{il,}$$_{↑}$ = cos( − → q · − → R$_{i,l}$/ 2) c$_{pil,}$$_{↑}$ + e $^{iφ}$sin( − → q · − → R$_{i,l}$/ 2) c$_{pil,}$$_{↓}$ (5a) p$_{il,}$$_{↓}$ = cos( − → q · − → R$_{i,l}$/ 2) c$_{pil,}$$_{↓}$ − e − $^{iφ}$sin( − → q · − → R$_{i,l}$/ 2) c$_{pil,}$$_{↑}$ (5b) H$_{0}$ = ∑ ε$_{p}$p † $_{il,σ}$p$_{il,σ}$ + ∑ ε$_{d}$d † $_{jm}$d$_{jm}$ − ∑ n.n. V [( − 1) $^{l}$cos δθ$_{l}$d † $_{jm}$p$_{il,}$$_{↑}$ − η$_{l}$ ( − 1) $^{m}$e$^{iφ}$sin δθ$_{l}$d † $_{jm}$p$_{il,}$$_{↓}$ ] + H.c. (6) H$_{0}$ = ∑ ε$_{p}$p † $_{lk,σ}$p$_{lk,σ}$ + ∑ ε$_{d}$d † $_{mk}$d$_{mk}$ − ∑ n.n. V [( − 1) $^{l}$cos δθ$_{l}$e i − → k · ( − → R$_{i,l}$ − − → R$_{j,m}$ $^{)}$d † $_{mk}$p$_{lk,}$$_{↑}$ − η$_{l}$ ( − 1) $^{m}$e$^{iφ}$sin δθ$_{l}$e i − → k · ( − → R$_{i,l}$ − − → R$_{j,m}$ $^{)}$d † $_{mk}$p$_{lk,}$$_{↓}$ ] + H.c.. (7) 7