Periodic ground states for the Lennard-Jones potential in 3D
In plain words
Prove that many particles interacting through the standard Lennard-Jones force (repulsive at short range, attractive at long range) have a lowest-energy arrangement that is a periodic crystal. Computations suggest the hexagonal close-packed crystal.
Precise statement
For $V(r)=4\epsilon ((s/r)^{12}-(s/r)^6)$, let $x_1..x_N \text{ in } R^3$ minimize $E_N=\sum_{i<j} V(\mid x_i-x_j\mid)$. Prove that $\min E_N/N$ converges to the energy per particle of one periodic configuration (expected hcp, nearly degenerate with fcc) and that minimizers, after a rigid motion, coincide with it outside a boundary layer of $o(N)$ particles. A proof for any pair potential with a repulsive core and an attractive tail in 3D would also answer the qualitative question.
What would settle it
A proof of convergence of $N$-particle minimizers to a periodic lattice for Lennard-Jones or a comparable two-body potential in $R^{3}$.
Status in the literature
Proved in 3D only for a tailored potential with a three-body term and fcc ground state (Flatley, Theil, ARMA 2015).