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der certain conditions the helical spin configuration is stable. Thus, we shall take that as our starting point. The transition metal ion, with position vector − → R$_{j,m}$ = − → R$_{j}$ + − → r m where − → R$_{j}$ is the position vector of the j-th lattice point and − → r m is the position vector of the m-th ion in the basis, has the following form for its spin (presumably those of t$_{2}$$_{g}$ electrons) 

The helical spin order has a wave vector − → q . As shown in Fig. 1, the projection of spins on xy-plane makes a fixed angle φ with x-axis. 

Hund’s coupling in the transition metal ions, − J$_{H}$ Σ − → S$_{j,m}$ · − → s $_{j,m}$, is the dom- inant mechanism in the system. If the local spins in eq. (1) are treated as classical spins, their effect on the spins of hybridizing electrons, denoted by − → s $_{j,m}$, is equivalent to an effective magnetic field. Thus, the eigen state of the spin of a hybridizing electron on the m-th transition metal ion with lower energy is 

where θ$_{j,m}$ = − → q · − → R$_{j,m}$/ 2. This will affect the hybridization of orbitals. 

Next, we consider the hybridization. KNB studies a TM-O-TM three-atom triad. Due to the symmetries of orbitals, the hybridization arises from π - bonding. If the TM-O-TM triad is not linear (illustrated in Fig. 2a where α is the bond angle), the p$_{x}$ orbitals of oxygen (if the bond is approximately in the x-direction) can also take part in electron transfer, σ -bonding can be realized and the hybridization energy can be greater. In fact, the bond-bending occurs quite often in transition metal oxides. See, for example, reference 24. The ab plane of the crystal is shown in Fig. 2b where the solid dots and circles denote the transition metal ions and oxygen atoms respectively. There are two distinct 

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