The quasi-harmonic model proposes that a crystal can be modeled as atoms
connected by springs. We demonstrate how this viewpoint can be misleading: a
simple application of Gauss' law shows that the ion-ion potential for a cubic
Coulomb system can have no diagonal harmonic contribution and so cannot
necessarily be modeled by springs. We investigate the repercussions of this
observation by examining three illustrative regimes: the bare ionic, density
tight-binding, and density nearly-free electron models. For the bare ionic
model, we demonstrate the zero elements in the force constants matrix and
explain this phenomenon as a natural consequence of Poisson's law. In the
density tight-binding model, we confirm that the inclusion of localized
electrons stabilizes all major crystal structures at harmonic order and we
construct a phase diagram of preferred structures with respect to core and
valence electron radii. In the density nearly-free electron model, we verify
that the inclusion of delocalized electrons, in the form of a background
jellium, is enough to counterbalance the diagonal force constants matrix from
the ion-ion potential in all cases and we show that a first-order perturbation
to the jellium does not have a destabilizing effect. We discuss our results in
connection to Wigner crystals in condensed matter, Yukawa crystals in plasma
physics, as well as the elemental solids.