condition, $\partial F / \partial G_n = 0$, implies that $$\label{eq35}
\partial F_0 / \partial G_n+(\zeta_{njkl}+\zeta_{jnkl})G_j\eta_{kl}+(\lambda_{njkl}+\lambda_{jnkl})G_jP_kP_l
=h_n$$ which indicates that $h_n$ depends on both the polarization and strain.

As a result, the change in electrical toroidal moment with electric field can be separated into the following two contributions:

$$\label{eq34b}
\frac{d{\rm { G_i}}}{d{\rm { E_j}}}= \left( {\frac{dG_i }{dE_j }} \right)^{(1)}+ \left( {\frac{dG_i }{dE_j }} \right)^{(2)}$$

with $$\label{eq35a}
\left( \frac{dG_i }{dE_j } \right)^{(1)}=
 \frac{dG_i}{d h_n}  \frac{\partial h_n}{\partial  P_l}  \frac{d P_l}{d E_j} = \chi^{(G)}_{in}\frac{\partial h_n}{\partial
P_l}\chi^{(P)}_{lj}$$ and $$\label{eq35b}
\left( {\frac{dG_i }{dE_j }} \right)^{(2)}= \frac{d G_i}{\partial h_n} \frac{\partial h_n}{\partial \eta_{kl}} \frac{d \eta_{kl}}{d E_j} = \chi^{(G)}_{in}
\frac{\partial h_n}{\partial \eta_{kl} } d_{klj}$$ Here $$\chi^{(G)}_{in}=\frac{dG_i}{dh_n}$$ is the response of the electrical toroidal moment to its conjugate field, $$\chi_{ij} ^{(P)}=\frac{dP_i}{dE_j}$$ is the electric susceptibility, and $$d_{ijk}=\frac{d \eta_{ij}}{dE_k}$$ is a piezoelectric tensor.

The remaining derivatives appearing in Eqs. (\[eq35a\]) and (\[eq35b\]) can be found from Eq. (\[eq35\]): $$\label{eq38}
\left( {\frac{\partial h_n }{\partial P_l }} \right)= (\lambda_{njlm} +\lambda _{njml}+\lambda_{jnlm} +\lambda _{jnml})G_jP_m$$ and $$\label{eq39}
\left( {\frac{\partial h_n }{\partial \eta_{kl} }} \right)=(\zeta_{njkl}+\zeta _{jnkl}) G_j$$

Equations (\[eq34b\])-(\[eq39\]) reveal that there are two scenario for the occurence of natural optical activity in electrotoroidic systems. In the first scenario, the system possesses a finite polarization 