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y −→ − y . In the previous section, we calculated the polarization of a planar crystal. Now we consider a more general case of orthorhombic structure and helical spin configuration. It is easier to catch the essence if one considers the (pd π ) part of E$_{y,xy}$ or the (pd σ ) part of E$_{x,x}$ $^{2}$− y 2 of H$_{V}$ , which is applicable to KNB’s original work and to the x 2 − y 2 orbital part of our work respectively. Assuming the bond direction of H$_{V}$ is in the m -direction then polarization is finite for either k = h , m = i or k = i , m = h . As a result, eq. (23) can be simplified as 

<img src='content_image/310606.jpg'>

where H$_{V,i}$ denotes the hybridization bond along i − direction and i , j and k are cyclic. 

The spin part needs more attention. 〈 σ$_{n}$ | s$_{j}$ | σ$_{l}$ 〉 give rise to a spin state | σ$_{l}$ 〉 different from the original state | σ$_{n}$ 〉 and 〈 σ$_{l}$ | σ$_{m}$ 〉 gives rise to a interesting contribution. Only the imaginary part needs to be considered because 〈 l | l$_{j}$ | n 〉 is imaginary: 

<img src='content_image/310607.jpg'>

<img src='content_image/310608.jpg'>

for the spin configuration in eq. (1). Here the additional subscript j of | σ$_{l,j}$ 〉 denotes that it comes from 〈 σ$_{n}$ | s$_{j}$ | σ$_{l}$ 〉 . The right hand sides of eqs. (24) are very similar to respective components of − → S N × − → S N − 1 which appears in H$_{DM}$ . The difference is in the arguments of sin functions. The factor 1 / 2 arises because in our model it is the hybridization electrons that mediate the exchange interaction while in the original DM interaction it is direct exchange. Substituting eqs. (25) 

