  We have investigated the electronic states of CeB$_6$ and have directly
calculated the RKKY interaction on the basis of the 74-orbital effective
Wannier 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 theory
bandstructure calculation. By using not only the $c$-band dispersion but also
the $f$-$c$ mixing matrix elements of the Wannier model, the realistic
couplings for all 15 active multipole moments in $\Gamma_8$ quartet subspace
are 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 the
elastic softening of $C_{44}$. Also the couplings of the $\Gamma_{5u}$
octupoles $T_{x}^{\beta}$, $T_{y}^{\beta}$ and $T_{z}^{\beta}$ are quite large
for $q=(\pi,0,0)$, $(0,\pi,0)$ and $(0,0,\pi)$, which yields the
antiferro-octupolar ordering of a possible candidate for phase IV of
Ce$_{x}$La$_{1-x}$B$_6$. The intersite vector dependence of the RKKY couplings
exhibit different long-range, oscillating, isotropic and anisotropic behaviors
depending on the types of the multipole moments. The present approach enables
us to provide the information about the possible multipole ordering in an
unbiased way and is easily available for other localized $f$ electron materials
once the $c$ states and $f$-$c$ mixing elements are given from the
bandstructure calculation.
