From Hard to Soft Dense Sphere Packings: The Cohesive Energy of Barlow Structures Using Exact Lattice Summations for a General Lennard-Jones Potential.

There are infinitely many distinguishable dense sphere packings in dimension three (Barlow structures) with maximum packing density of ρ = π/√18. While Hales [Hales, T. C. Ann. Math. 2005, 162, 1065-1185.] proved that the packing density of the most dense cubic packing cannot be surpassed, it is currently not known how the infinite possibilities of densely packed Barlow structures with different sequences of hexagonal close packed layers are energetically related compared to the well-known face centered cubic (fcc) and hexagonal close packed (hcp) structures. We demonstrate that, for a general Lennard-Jones potential, which includes the hard-sphere model as a limiting case, Barlow packings lie energetically between fcc and hcp. Exceptions to this energy sequence only occur when fcc and hcp are energetically quasi-degenerate.