= Solution
A scalar coupling linear in the <nematic order parameter> requires a second-rank anisotropy. A single <electric field> supplies no such tensor, while $E_iE_j$ does. Its isotropic part gives $E^2\operatorname{Tr}Q=0$. Hence the lowest joint order in $E$ and $Q$ of the <electric-field coupling to nematic order> is
$$
\boxed{f_E=-\chi E_iQ_{ij}E_j}.
$$
Terms depending only on $E$ do not affect minimization over $Q$. With $\vartheta$ the angle between the <nematic director> and the <electric field>,
$$
f_E=-\frac32\chi\lambda E^2\left(\cos^2\vartheta-\frac13\right).
$$
For $\chi\lambda>0$, the lowest <free energy> occurs at $\cos^2\vartheta=1$, so the <nematic director> aligns along either $+\mathbf E$ or $-\mathbf E$. Then $E_iQ_{ij}E_j=\lambda E^2$, giving
$$
\boxed{f_E=\bar d\lambda,\qquad \bar d=-\chi E^2<0\quad(E\ne0,\ \chi>0)}.
$$
Unlike the linear composition term in a conserved <binary fluid mixture>, this term changes the equilibrium <nematic order parameter>: its spatial integral is not fixed.
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