content_image/320201.jpg content_image/320200.jpg
perturbations, 

<img src='content_image/320201.jpg'>

where H$_{V}$ is the hybridization energy (the third and fourth terms of eq. (4a)). | l 〉 and | σ$_{l}$ 〉 are respectively the spatial and spin part of an intermediate state. | n 〉 and | l 〉 are states of d-orbitals and | m 〉 p-orbitals. 〈 n | H$_{V}$ | m 〉 / ( ε$_{d}$ − ε$_{p}$ ) is the exchange part with H$_{V}$ also containing information of spins. λ 〈 l | → l | n 〉 · 〈 σ$_{l}$ | − → s | σ$_{n}$ 〉 is the spin-orbit coupling. Eq. (22) is also applicable to the situations without bond-bending. In that case, the π -bonding will be considered as they were in KNB’s original work and H$_{V}$ is the third term of eq. (A-1). One can see that eq. (22) has the same origin as H$_{DM}$ in eqs. (20) and (21). However, we have to note that the helical spin configuration is not caused by DM interaction whose strength is too small. Rather, it can be due to the next-near-neighbor hybridization as shown in ref. 23. 

It is easier to analyze with the following form: 

<img src='content_image/320200.jpg'>

where ε$_{hij}$ is the antisymmetric Levi-Cevita symbol and r$_{k}$ is the k -th compo- nent of − → r in space. We consider the mirror symmetry of above equation. H$_{V}$ may change sign under mirror reflection operation because of the orbital wave functions involved. It is also related to the direction of the displacements of oxygen atoms. For example, the (pd π ) part of E$_{y,xy}$ and (pd σ ) of E$_{x,x}$ $^{2}$− y 2 ( in Slater-Koster notation) with bond along x-direction change sign if one makes the operation x −→ − x . On the other hand, the (pd σ ) part of E$_{x,xy}$ (due to bond-bending) does not change sign if one makes the operation x −→ − x or 

