and Eq. (\[eqn6\]) for a monochromatic wave having an $\omega$ angular frequency gives: $$\label{eqn3}
{\bf M}-{\bf M}_0 \simeq \frac{1}{c}{\dot{\bf G}}=-\frac{i\omega }{c}{\bf G}$$ in electrotoroidic systems.

Plugging this latter equation in Eq. (\[eq22\]) then gives: $$\label{eq24}
\chi _{ij}^{me} =\chi_{ij}^{me(0)}-\frac{i\omega }{c}\frac{dG_i }{dE_j }$$ where $\chi_{ij}^{me(0)}$ is the magnetoelectric tensor related to the derivative of ${\bf M}_0$ with respect to an electric field. Therefore $$\label{eq25}
\mathrm{Im}\,(\chi _{ij}^{me}-\chi_{ij}^{me(0)}) =-\frac{\omega }{c}\frac{dG_i }{dE_j }$$ This relation between the imaginary part of the magnetoelectric susceptibility and the field derivative of the electrical toroidal moment is reminiscent of the connection discussed in Ref. [@Spaldin] between the linear magnetoelectric response and the [*magnetic*]{} toroidal moment.

Inserting Eqs. (\[eq25\]) and (\[eq18\]) into Eq. (\[eq21\]) then provides: $$\begin{aligned}
\label{eq29}
\gamma _{ijk} =&&\gamma_{ijk}^{S} +\frac{4\pi c}{\omega} \left ( e_{jkl} {\rm Im} \chi_{li}^{me(0)}-e_{ikl}{\rm Im} \chi_{lj}^{me(0)} \right )  \nonumber \\
&&+4\pi \left[ {e_{ikl} \frac{dG_l }{dE_j }-e_{jkl} \frac{dG_l
}{dE_i }+\frac{1}{2}\left( {\frac{dQ_{kj} }{dE_i }-\frac{dQ_{ki} }{dE_j }}
\right)} \right]\end{aligned}$$

Combining this latter equation with Eq. (\[eq74\]), and recalling that $\gamma_{ijk}^{S}$ is a symmetric tensor while $e_{ijm}$ is antisymmetric (which makes their product vanishing), gives: $$\begin{aligned}
\label{eq33before}
g_{mk} =&&4\pi \left ( \delta_{mk} {\rm Im} \chi_{ll}^{me(0)}-{\rm Im}\chi_{mk}^{me(0)}  \right )  \nonumber \\
&& +\frac{4\pi \omega }{c}\left[ \left( {\frac{dG_m }{dE_k }-\frac{dG_l }{dE_l
