scattering amplitude at site $j$: $$\begin{aligned}
 M_j(\mbox{\boldmath{$\epsilon$}}',\mbox{\boldmath{$\epsilon$}},\omega) 
 &=& \alpha_0(\omega)\mbox{\boldmath{$\epsilon$}}' \cdot
                     \mbox{\boldmath{$\epsilon$}} \nonumber\\
 &-& i\alpha_1(\omega)(\mbox{\boldmath{$\epsilon$}}' 
    \times \mbox{\boldmath{$\epsilon$}})
    \cdot\langle\psi_0|\textbf{J}|\psi_0\rangle \nonumber\\
 &+& \alpha_2(\omega)\sum_\nu 
 P_{\nu}(\mbox{\boldmath{$\epsilon$}}',\mbox{\boldmath{$\epsilon$}})
 \langle\psi_0|z_{\nu}|\psi_0\rangle,
\label{eq.Mtilde.2}\end{aligned}$$ where

\[eq.quadrupole1.def\] $$\begin{aligned}
 z_1 &\equiv& Q_{x^2-y^2} = \frac{\sqrt{3}}{2}(J_x^2-J_y^2), \\
 z_2 &\equiv& Q_{3z^2-r^2}= \frac{1}{2}(3J_z^2-J(J+1)), \\
 z_3 &\equiv& Q_{yz}      = \frac{\sqrt{3}}{2}(J_yJ_z+J_zJ_y), \\
 z_4 &\equiv& Q_{zx}      = \frac{\sqrt{3}}{2}(J_zJ_x+J_xJ_z), \\
 z_5 &\equiv& Q_{xy}      = \frac{\sqrt{3}}{2}(J_xJ_y+J_yJ_x), \end{aligned}$$

and

\[eq.quadrupole2.def\] $$\begin{aligned}
 P_1(\mbox{\boldmath{$\epsilon$}}',\mbox{\boldmath{$\epsilon$}})
 &=& \frac{\sqrt{3}}{2}(\epsilon'_x\epsilon_x-\epsilon'_y\epsilon_y), \\
 P_2(\mbox{\boldmath{$\epsilon$}}',\mbox{\boldmath{$\epsilon$}})
 &=& \frac{1}{2}(2\epsilon'_z\epsilon_z
                -\epsilon'_x\epsilon_x-\epsilon'_y\epsilon_y), \\
 P_3(\mbox{\boldmath{$\epsilon$}}',\mbox{\boldmath{$\epsilon$}})
 &=& \frac{\sqrt{3}}{2}(\epsilon'_y\epsilon_z+\epsilon'_z\epsilon_y), \\
 P_4(\mbox{\boldmath{$\epsilon$}}',\mbox{\boldmath{$\epsilon$}}) 
 &=& \frac{\sqrt{3}}{2}(\epsilon'_z\epsilon_x+\epsilon'_x\epsilon_z), \\
 P_5(\mbox{\boldmath{$\epsilon$}}',\mbox{\boldmath{$\epsilon$}}) 
 &=& \frac{\sqrt{3}}{2}(\epsilon'_x\epsilon_y+\epsilon'_y\epsilon_x).\end{aligned}$$

Here we have suppressed the dependence on $j$ in the right hand side of Eq. (\[eq.Mtilde.2\]). The energy profiles are given by only three functions, $\alpha_0(\omega)$, $\alpha_1(\omega)$, and $\alpha_2(\omega)$, whose expressions are explicitly given in Appendix.

Several facts are immediately deduced from Eq. (\[eq.Mtilde.2\]). First, since the scalar, dipole, and quadrupole terms exhaust the amplitude, the octupole ordering alone does not give rise to the RXS amplitude. Second, the choice of the CEF parameters in the ground state does not affect the shape of energy profiles $\alpha_0(\omega)$, $\alpha_1(\omega)$ and $\alpha_2(\omega)$, although it affects the expectation values of dipole and/or quadrupole operators. Third, $\alpha_0(\omega)$ has no contribution to the forbidden Bragg spots in the antiferro-type structure. In order to calculate the energy profiles, 