Correction to Fuchs' calculation of the electrostatic energy of a Wigner solid

In 1959 Plaskett found an error in Fuchs' calculation of the electrostatic energy of a Wigner solid and calculated the additive, lattice-dependent, correction term for the three cubic lattices. We extend his theory to treat any Bravais lattice in one, two, or three dimensions, showing that there is no correction term in one and two dimensions. For the simple hexagonal lattice, we calculate the correction term as a function of arbitrary $r=\frac{c}{a}$ ratio and find it has a minimum value at $r=r_m={\left(\frac{5}{6}\right)}^{\frac{1}{2}}$. The magnitude of the correction $A$ to Fuchs' value $S$ can be 20% or more of $\mathrm{}\left|S\right|\mathrm{}$, and $|A_{\mathrm{f}\dot{\mathrm{c}}\mathrm{c}}-A_{\mathrm{bcc}}|$ is much larger than $|S_{\mathrm{fcc}}-S_{\mathrm{bcc}}|$. If sph refers to a spherical approximation, we find $A_{\mathrm{sph}}<A_{\mathrm{bcc}}<A_{\mathrm{fcc}}<A_{\mathrm{sh}}\left(min\right) <A_{\mathrm{sc}}<A_{\mathrm{sh}}\left(\mathrm{ideal}\right)$, where ideal refers to $r_i={\left(\frac{8}{3}\right)}^{\frac{1}{2}}$ and sh and sc mean simple hexagonal and simple cubic, respectively. The $A_{\mathrm{bcc}}$ reported by Callaway in another connection is incorrect.