defined in eq. (A-6), is of the order V ( pdσ ) sin β/V ( pdπ ) ∼ 1 for β ≈ π/ 12. Hence bond-bending gives larger polarization in many oxides. For example, α ≈ 5 π/ 6 in manganites[26] and in compounds such as Ni$_{3}$V$_{2}$O$_{8}$ the bond angle of Ni-O-Ni α can be as small as π/ 2[6]. Hence, α and β may both be large and the environment is favorable to ferroelectricity. Furthermore, ρ ∼ 0 . 1 ˚ A can very well be an underestimation because the O 2 − has a much larger radius than a neutral oxygen atom. All things considered, P can be an order of magnitude greater than previously estimated. Atomic displacement can also enhance electric polarization by destroy cancellation. However, if it has its own wave vector and it is not commensurate with − → q then there is no net polarization. 

In conclusion, we have analyzed the conditions for the emergence of fer- roelectricity due to magnetic orders. We found a simple relation between its wave vector − → q and polarization. Furthermore, the physical picture of the cou- pling between magnetic orders and ferroelectricity is made clear. Multiferroics is created by a generalized version of Dzyaloshinskii-Moriya interaction in the environment of certain spin orders, preferably helical. Above findings can also be applied to systems without bond-bending but with π -bond hybridization. The bond-bending tends to enhance polarization and it may be important for certain compounds. 

The author benefited from the activities of ”quantum novel phenomena in condensed matter” focus groups of NCTS, Taiwan and discussion with S. Maekawa. This work is supported in part by the National Science Council under the contract NSC 95-2112-M-002-048-MY3. 

## 0.0.6 Appendix 

In this Appendix, we present the eigen values and eigen vectors of the π -bonding systems. It is simpler because there are only one transition element ion (thus the dropping of the index m ) and two oxygen atoms in the basis. We consider 

