O$_{1}$O$_{3}$. According to Goodenough, 11 the predominant interactions between neighboring cations whose cation- ccupied octahedra share an edge, are assumed to be di- rect cation-cation exchange interaction. Thus the inter- action J$_{2}$ is antiferromagnetic (Heitler-London type) for β -MnO$_{2}$. 

## C. Origin of superexchange interaction J$_{1}$ 

We consider the interaction between Mn 4+ at the point B$_{1}$ and the Mn 4+ at the point C$_{1}$ as shown in Figs. 20 (a) and (b), where the distance C$_{1}$B$_{1}$ is 3 . 424 ˚ A . Accord- ing to Goodenough, 11 this interaction ( J$_{1}$ ) is a superex- change one since the cation-occupied octahedra share a common corner (point O$_{1}$). The angle C$_{1}$O$_{1}$B$_{1}$ is equal to 130 . 62 $^{◦}$, this interaction is antiferromagnetic for β - MnO$_{2}$. According to Goodenough, 11 when the cation- occupied octahedral share a common corner, there can be no direct overlap of neighboring cation orbitals and therefore there is no cation-cation interactions. 

What is the origin of J$_{1}$ ? The point C$_{1}$ is the center of the neighboring octahedron. The direction of C$_{1}$O$_{1}$ is the z axis of this octahedron. A p orbital of O$_{1}$ is expected to be directed toward C$_{1}$ ( p$_{σ}$ orbital) so as to overlap the d (3 z $^{2}$− r $^{2}$) orbital of the point C$_{1}$ , where the p orbital ( p$_{x}$ , p$_{y}$ , p$_{z}$ states) is called as p$_{σ}$ orbital when the principal axis of the p orbital coincides with the di- rection of the bond. A partial covalent bond between the d (3 z 2 − r $^{2}$) orbital and the p$_{σ}$ orbital can be formed. Then the charge transfer occurs from the p$_{σ}$ orbital with the spin-up state |↑〉 to the ϵ$_{g}$ state. Consequently, the spin of Mn 4+ at the point C$_{1}$ is still in the spin-up state, while the resulting spin of O 2 − at the point O$_{1}$ is in the spin-down state. The remaining p$_{σ}$ state at the point O$_{1}$ is magnetically coupled with the t$_{2}$$_{g}$ state of the Mn 4+ at the point B$_{1}$. When this coupling (denoted as K$_{0}$ ) is antiferromagnetic, then the suprexchange interaction between Mn 4+ at the point C$_{1}$ and Mn 4+ at the point B$_{1}$ is ferromagnetic. On the other hand, when this coupling K$_{0}$ is ferromagnetic, then the superexchange interaction between Mn 4+ at the point C$_{1}$ and Mn 4+ at the point B$_{1}$ is antiferromagnetic. Here we note that the angle α = ∠ C$_{1}$O$_{1}$B$_{1}$ is equal to 130 . 62 ◦ for β -MnO$_{2}$, which is very different from 90 $^{◦}$. If α = 90 $^{◦}$, the p$_{σ}$ orbital on the bond O$_{1}$C$_{1}$ coincides with the p$_{π}$ orbital on the bond O$_{1}$B$_{1}$ (the y axis), where the p orbital ( p$_{x}$ , p$_{y}$ , p$_{z}$ states) is called as p$_{π}$ orbital when the principal axis of the p or- bital is perpendicular to the direction of the bond. The p$_{π}$ orbital is coupled with the t$_{2}$$_{g}$ state of the Mn 4+ at the point B$_{1}$, since the ϵ$_{g}$ state is empty. According to the Goodenough-Kamamori-Anderson rule, 11–14 the in- teraction K$_{0}$ is antiferromagnetic, since the p$_{π}$ orbital is not orthogonal to the t$_{2}$$_{g}$ orbital. If α = 180 $^{◦}$, the p$_{σ}$ orbital is coupled with the t$_{2}$$_{g}$ orbital of the Mn 4+ at the point B$_{1}$ the interaction K$_{0}$ is ferromagnetic, since the 

p$_{σ}$ orbital is orthogonal to the t$_{2}$$_{g}$ orbital. 

The sign of K$_{0}$ is dependent on the value of α . There may be a critical angle α$_{c}$ . K$_{0}$ is ferromagnetic for α > α$_{c}$ and K$_{0}$ is antiferromagnetic for α < α$_{c}$ . Experimentally, the superexchange interaction J$_{1}$ is antiferromagnetic for β -MnO$_{2}$, which means that K$_{0}$ is ferromagnetic. The critical angle α$_{c}$ is lower than 130 . 62 $^{◦}$. 

The direct cation-cation exchange interaction ( J$_{2}$ ) is expected to be stronger than the superexchange in- teractions J$_{1}$ . The competition between J$_{1}$ and J$_{2}$ can lead to a complicated compromise magnetic order; J$_{2}$/J$_{1}$ = 1 . 68469 for β -MnO$_{2}$. Note that the discussion of Osmond 15 on the nature of J$_{2}$ may be inappropriate. 

## D. Origin of superexchange interaction J$_{3}$ 

We consider the interaction between Mn 4+ at the point A$_{1}$ and the Mn 4+ at the point A$_{2}$, as shown in Figs. 20(a) and (b), where the distance A$_{1}$A$_{2}$ is 4 . 396 ˚ A . Accord- ing to Goodenough, 11,12 this interaction ( J$_{3}$ ) is a su- perexchange one (cation-anion-cation) since the cation- occupied octahedra share a common corner (point O$_{4}$). The angle ∠ A$_{1}$O$_{4}$A$_{2}$ is equal to α = 111 . 60 $^{◦}$, the dis- tance O$_{4}$A$_{2}$ is 3 . 343 ˚ A , and the distance O$_{4}$A$_{1}$ is 1 . 878 ˚ A . This interaction is experimentally antiferromagnetic for β -MnO$_{2}$; p$_{3}$ = J$_{3}$/J$_{1}$ = 0 . 537. This means that the crit- ical angle α$_{c}$ is between 90 ◦ and 111 . 60 $^{◦}$. 

We have studied the phase diagram of ( J$_{2}$ vs J$_{3}$ ) with J$_{1}$ = − 1 in the rutile type β -MnO$_{2}$ by using the equi- energy contour plot. The distribution of the magnetic Bragg peaks can be clearly visualized. The magnetic phase diagram consists of the multricritical point (the intersection J$_{2}$J$_{3}$ = J 2 1 and J$_{2}$ + J$_{3}$ = 2 J$_{1}$ ), the helical order along the c axis, the ( h = 1 / 2 , k = 0 , l = 1 / 2) phase, the helical order along the a axis, and the phase ( h = 0 , k = 0 , l = 1). The phase transition is of the first order between the ( h = 1 / 2 , k = 0 , l = 1 / 2) phase and the helical order along the c axis, and between the ( h = 1 / 2 , k = 0 , l = 1 / 2) phase and the helical order along the a axis. The phase transition is of the second order between the phase ( h = 0 , k = 0 , l = 1) and the helical order along the c axis, and between the phase ( h = 0 , k = 0 , l = 1) and the helical order along the a axis. 

We are grateful to Prof. H. Sato for useful discussions on the itinerant nature of e$_{g}$ electrons in β -MnO$_{2}$. 

## XI. CONCLUSION 

## Acknowledgments 

