Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies

The Riesz potential f s ( r ) = − $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones–type potentials n , m LJ : a b $f_{n,m}^{\rm {LJ}}(r):=a r^{-n}-b r^{-m}$ > $n>m$ that are widely used in molecular simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices Lennard-Jones energies. We discuss minimality body-centered-cubic (BCC) lattice, face-centered-cubic (FCC) simple hexagonal (SH) lattices, close-packing (HCP) structure, globally at fixed density. case, new evidence global density BCC lattice shown for < 0 $s<0$ HCP computed have higher energy than FCC (for 3 / 2 $s>3/2$ $s<3/2$ lattices. case with exponents $3<m<n$ ground state confirmed whereas phase occurs once added investigated structures. Furthermore, transitions type “FCC-SH” “FCC-HCP-SH” (when added) as inverse V increases observed large spectrum $(n,m)$ . SH phase, variation ratio Δ between interlayer distance d parameter studied increases. critical region $0<m<n<3$ extreme value anisotropy dominates. If one limits oneself rigid BCC-FCC-HCP diagram found. For $-2<m<n<0$ only minimizer. Choosing 4 $-4<m<n<-2$ latices become minimizers.