Entropic trust region for densest crystallographic symmetry group packings

TL;DR

This paper introduces an entropic trust region method on a geometric manifold to efficiently find densest packings within crystallographic symmetry groups, improving molecular crystal structure prediction.

Contribution

It develops a novel information-geometric optimization framework for densest packings restricted to crystallographic symmetry groups, extending trust region methods to a non-Euclidean toroidal space.

Findings

Successfully applied to convex polygon packings in 2D CSGs with known optimal solutions.

Demonstrated effectiveness in accelerating crystal structure prediction for pentacene thin-films.

Provides a new geometric approach for global optimization in molecular packing problems.

Abstract

Molecular crystal structure prediction (CSP) seeks the most stable periodic structure given the 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 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 the 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. Introducing an adaptive quantile reformulation of the fitness function into the optimization schedule provides the algorithm with 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 the behavior and performance of the method on various densest packings of convex polygons in $2$-dimensional CSGs for which optimal solutions are known, and demonstrate its application in the pentacene thin-film CSP.