or the long wavelength limit ( − → q −→ 0), we found where ̂ h is the helix axis unit vector. We have assumed that q$_{z}$ = 0 since our calculation was performed on xy-plane. If one recognizes that the true meaning of ̂ e$_{12}$ is the spatial direction alone which the spins propagate, i.e., − → q , he can see how the form − → P ∼ $^{eI}$̂ e$_{12}$ × $^{(}$̂ e$_{1}$ × ̂ e$_{2}$ ) given by KNB can be transformed in the presence of magnetic orders. In order to see how magnetic orders and electric order are coupled, let us go back to the original work of Moriya[19]. He derived the following expression: where Here, J ( n, n $^{′}$, m, m$^{′}$) is the exchange interaction strength and − → l i denotes the angular momentum of the electron at site i . In our starting Hamiltonian, the exchange interaction comes from the charge transfer energy ε$_{p}$ − ε$_{d}$ and hybridization energy V [27]. Combined with the spin- orbit interaction in eq. (4b), DM interaction is clearly present in the system we considered. We can recast the wave functions we got previously in a form similar to Moriya’s by treating the hybridization energy and spin-orbit interaction as 12 H$_{DM}$ = ∑ N − → D$_{N,N}$$_{−}$$_{1}$ · ( − → S$_{N}$ × − → S$_{N}$$_{−}$$_{1}$ ) (20) − → D$_{N,N}$$_{−}$$_{1}$ = iλ ∑ J ( n, n $^{′}$, m, n$^{′}$) 〈 n | − → l i | m 〉 E$_{n}$ − E$_{m}$ − iλ ∑ J ( n, n $^{′}$, n, m$^{′}$) 〈 n $^{′}$| − → l j | m ′ 〉 E$_{n}$ ′ − E$_{m}$ ′ . (21) − → P − q ∼ 0 . 01 $^{eρa$_{0}$}$̂ e$_{z}$ ( q$_{x}$ cos φ + q$_{y}$ sin φ ) ∼ 0 . 01 eρa$_{0}$ − → q × ̂ h (19) 0.0.5 5. Discussion