Use the supplied effective electrostatic term and enforce charge neutrality while varying the chain length:

R = Z / (2 n_{e})

, with

N

and

Z

fixed. The total energy per positive ion becomes

ε (n_{e}) = - \frac{α Z ^{2} e ^{2}}{4 π ϵ _{0} R} + \frac{Z E _{F}}{3} = - \frac{α Z e ^{2}}{2 π ϵ _{0}} n_{e} + \frac{Z π ^{2} ℏ ^{2}}{24 m _{e}} n_{e}^{2} .

The positive quadratic term supplies degeneracy pressure against compression. Differentiating gives the neutral one-dimensional metal equilibrium density,

n_{e}^{*} = \frac{6 α m _{e} e ^{2}}{π ^{3} ϵ _{0} ℏ ^{2}} = \frac{24 α}{π ^{2} a _{B}} ≃ 6.37 \times 1 0^{10} m^{- 1},

where

a_{B} = 4 π ϵ_{0} ℏ^{2} / (m_{e} e^{2})

Z π^{2} ℏ^{2} / (12 m_{e}) > 0

verifies a minimum. Within this specified toy energy, the minimizing electron density is independent of

Z

; instead the equilibrium ion spacing is

2 R^{*} = Z / n_{e}^{*}

At the minimum the electrostatic contribution is minus twice the kinetic contribution, so

ε_{*} = - Z E_{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.