Entropic trust region for densest crystallographic symmetry group packings

Abstract

Molecular crystal structure prediction (CSP) seeks the most stable periodic structure given a chemical composition of a molecule and pressure-temperature conditions. Modern CSP solvers use global optimization methods to search for structures with minimal free energy within a complex energy landscape induced by intermolecular potentials. A major caveat of these methods is that initial configurations are random, making thus the search susceptible to the convergence at local minima. Providing initial configurations that are densely packed with respect to the geometric representation of a molecule can significantly accelerate CSP. Motivated by these observations we define a class of periodic packings restricted to crystallographic symmetry groups (CSG) and design a search method for densest CSG packings in an information geometric framework. Since the CSG induce a toroidal topology on the configuration space, a non-euclidean trust region method is performed on a statistical manifold consisting of probability distributions defined on an n-dimensional flat unit torus by extending the multivariate von Mises distribution. By introducing an adaptive quantile reformulation of the fitness function into the optimization schedule we provide the algorithm a geometric characterization through local dual geodesic flows. Moreover, we examine the geometry of the adaptive selection quantile defined trust region and show that the algorithm performs a maximization of stochastic dependence among elements of the extended multivariate von Mises distributed random vector. We experimentally evaluate its behavior and performance on various densest packings of convex polygons in 2-dimensional CSG for which optimal solutions are known.

Densest plane group packings of regular polygons

“…Using the entropic trust region packing algorithm [14] we obtained the densest packings of various convex regular polygons in all 17 plane groups including a disc as a limiting n-gon when the number of vertices n approaches infinity. For the purposes of brevity, we name a n vertex regular convex polygon, simply by n-gon.…”

“…Using the entropic trust region method [14] we obtained and analyzed the densest packings in all 17 plane groups of various regular convex polygons including a disc as a limiting regular polygon when the number of vertices approaches infinity. Since we restricted our packings to 2-dimensional crystallographic symmetry groups, which are inherently periodic and combined with the fact that the p2 plane group is a local optimum in the space of all packings [24], and further confirmed by our results, it can be stated that densest plane group packings for all regular convex polygons among all 17 plane group are realized by the p2 plane group.…”

“…It has to be noted that the entropic trust region is not a physical simulation since during the search particles in the system are allowed to overlap. A detailed treatment of the search method we used is presented in [14].…”

“…We consider finding the densest plane group packing purely as an optimization problem. Using the Entropic trust region packing algorithm [14], developed specifically to search for densest crystallographic symmetry group packings of arbitrary dimension, we successfully recovered approximations of known densest lattice and double-lattice packing configurations including a disc as a limiting n-gon when the number of vertices approaches infinity. Further, we obtained previously unknown densest packing configurations of n-gons, as well as densest configurations subject to plane groups restrictions.…”

Densest plane group packings of regular polygons

“…We consider finding the densest plane group packing as a nonlinear, constrained, and bounded optimization problem. Using the Entropic Trust Region Packing Algorithm [17], developed specifically to search for densest crystal- 1: The colored rank table of plane groups in relation to the number of vertices n of an n-gon. For every n = 3, .…”