= Solution
Use the supplied effective electrostatic term and enforce <charge neutrality> while varying the chain length: $R=Z/(2n_e)$, with $N$ and $Z$ fixed. The total <energy> per positive ion becomes
$$
\varepsilon(n_e)=-\frac{\alpha Z^2e^2}{4\pi\epsilon_0R}+\frac{ZE_F}{3}
=-\frac{\alpha Ze^2}{2\pi\epsilon_0}n_e
+\frac{Z\pi^2\hbar^2}{24m_e}n_e^2.
$$
The positive quadratic term supplies <degeneracy pressure> against compression. Differentiating gives the <neutral one-dimensional metal equilibrium density>,
$$
\boxed{n_e^*=\frac{6\alpha m_ee^2}{\pi^3\epsilon_0\hbar^2}
=\frac{24\alpha}{\pi^2a_B}\simeq6.37\times10^{10}\,\mathrm{m^{-1}},}
$$
where $a_B=4\pi\epsilon_0\hbar^2/(m_ee^2)$ is the <Bohr radius>. The second derivative $Z\pi^2\hbar^2/(12m_e)>0$ verifies a minimum. Within this specified toy <energy>, the minimizing <electron> density is independent of $Z$; instead the equilibrium ion spacing is $2R^*=Z/n_e^*$.
At the minimum the electrostatic contribution is minus twice the kinetic contribution, so $\varepsilon_*=-ZE_F^*/3$. These follow from the given electrostatic approximation, not from a full self-consistent calculation of a real one-dimensional metal. In particular, additional Coulomb terms cannot be appended without the corresponding neutral-background subtraction that defines the supplied finite <energy>.
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