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We have investigated the electronic states of CeB$_6$ and have directlycalculated the RKKY interaction on the basis of the 74-orbital effectiveWannier model which includes 14 Ce-$f$ orbitals and 60 conduction ($c$)orbitals of Ce-$d,s$ and B-$p,s$ derived from the density-functional theorybandstructure calculation. By using not only the $c$-band dispersion but alsothe $f$-$c$ mixing matrix elements of the Wannier model, the realisticcouplings for all 15 active multipole moments in $\Gamma_8$ quartet subspaceare obtained in the wavevector $q$-space and real-space. Both of the$\Gamma_{5g}$ quadrupoles $(O_{yz},O_{zx},O_{xy})$ and the $\Gamma_{2u}$octupole $T_{xyz}$ couplings are maximally enhanced with $q=(\pi,\pi,\pi)$which naturally explains the phase II of the antiferro-quadrupolar ordering at$T_{Q}=3.2$ K, and are also enhanced with $q=(0,0,0)$ corresponding to theelastic softening of $C_{44}$. Also the couplings of the $\Gamma_{5u}$octupoles $T_{x}^{\beta}$, $T_{y}^{\beta}$ and $T_{z}^{\beta}$ are quite largefor $q=(\pi,0,0)$, $(0,\pi,0)$ and $(0,0,\pi)$, which yields theantiferro-octupolar ordering of a possible candidate for phase IV ofCe$_{x}$La$_{1-x}$B$_6$. The intersite vector dependence of the RKKY couplingsexhibit different long-range, oscillating, isotropic and anisotropic behaviorsdepending on the types of the multipole moments. The present approach enablesus to provide the information about the possible multipole ordering in anunbiased way and is easily available for other localized $f$ electron materialsonce the $c$ states and $f$-$c$ mixing elements are given from thebandstructure calculation.

Derivation of RKKY Interaction between Multipole Moments in CeB$_6$ by the Effective Wannier Model based on the Bandstructure Calculation