Local Energy Optimality of Periodic Sets

Abstract

We study the local optimality of periodic point sets in ${\mathbb{R}}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-cr}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family ${\mathsf{D}}_n^{+}$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n\ge 9$ we can hereby in particular show that ${\mathsf{D}}_n^{+}$ is locally $f_c$-optimal among periodic sets for all sufficiently large~$c$.

27 pages, 2 figures

Cited by in corpus (2)

• Minimizing lattice structures for Morse potential energy in two and three dimensions
• On Born's conjecture about optimal distribution of charges for an infinite ionic crystal