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transition metal ions in the basis, and thus m = 1 , 2, due to bond-bending. As a result, two transition metal ions and four oxygen atoms form the basis of the crystal which is enlarged by bond-bending. The atoms are labeled so as to facilitate later deduction. As mentioned above, bond-bending will affect the hybridization energies. They can be found in Slater and Koster[25]. 

Bond-bending has other profound effects. It changes the symmetry of the surrounding of atoms and thus the symmetry of crystal field. In an orthorhombic crystal ( a ̸ = b ) with bond-bending, the xy -orbitals of the transition metal ion will mix with the x 2 − y 2 orbitals. The resulting orbital has the form 

where the ± sign is determined by the direction of the displacement of the oxygen atom away from the line joining two transition metal ions. β and α are of the same order of magnitudes. Ideally, if the crystal field is determined by the four nearest oxygen ions on the ab plane, then β = α ′ + δα ′ where α ′ = π − α . The deviation δα ′ is due to the fact that the symmetry of crystal field is not determined by bond-bending alone. We expect δα ′ to be finite but smaller than α in orthorhombic crystals of manganites but large in compounds like Ni$_{3}$V$_{2}$O$_{8}$. However, to get its magnitude, one need to carry out a very precise first-principle calculation which is beyond the scope of the current paper. 

The hybridization of the p-orbital of oxygen atoms and the xy-orbitals of transition ions is equal to ± $_{(}$√$_{3}$$_{/}$$_{2) sin}$$_{α}$$_{sin(}$$_{α/}$$_{2)}$$_{V}$$_{(}$$_{pdσ}$$_{) where the sign is again}$ determined by the direction of the displacements of oxygen atoms. That of the p-orbital of oxygen atoms and the x $^{2}$-y$^{2}$-orbitals of transition ions is equal to − $_{(}$√$_{3}$$_{/}$$_{2) cos}$$_{α}$$_{sin(}$$_{α/}$$_{2)}$$_{V}$$_{(}$$_{pdσ}$$_{). Although the x}$ $^{2}$-y$^{2}$-orbitals have greater hy- bridization energy, its amplitude is smaller as one can see from (3). The re- sulting hybridization energy is ± $_{(}$√$_{3}$$_{/}$$_{2) sin(}$$_{α}$ ′ − β ) sin( α/ 2) V ( pdσ ). We note 

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