Discrete Optimal Global Convergence of an Evolutionary Algorithm for Clusters under the Potential of Lennard Jones

TL;DR

This paper introduces a new evolutionary algorithm utilizing a novel CB lattice for optimizing Lennard Jones clusters, demonstrating theoretical convergence and the ability to find both known and new optimal clusters.

Contribution

The paper presents a new CB lattice-based evolutionary algorithm with proven convergence properties and the capability to discover optimal Lennard Jones clusters from 13 to 1612 particles.

Findings

Successfully replicates known Lennard Jones clusters from CCD

Finds new optimal clusters beyond existing databases

Provides theoretical proofs of convergence and optimality

Abstract

A review of the properties that bond the particles under Lennard Jones Potential allow to states properties and conditions for building evolutive algorithms using the CB lattice with other different lattices. The new lattice is called CB lattice and it is based on small cubes. A set of propositions states convergence and optimal conditions over the CB lattice for an evolutionary algorithm. The evolutionary algorithm is a reload version of previous genetic algorithms based in phenotypes. The novelty using CB lattice, together with the other lattices, and ad-hoc cluster segmentation and enumeration, is to allow the combination of genotype (DNA coding for cluster using their particle's number) and phenotype (geometrical shapes using particle's coordinates in 3D). A parallel version of an evolutionary algorithm for determining the global optimality is depicted. The results presented are from a standalone program for a personal computer of the evolutionary algorithm, which can estimate all putative optimal Lennard Jones Clusters from 13 to 1612 particles. The novelty are the theoretical results for the evolutionary algorithm's efficiency, the strategies with phenotype or genotype, and the classification of the clusters based in an ad-hoc geometric algorithm for segmenting a cluster into its nucleus and layers. Also, the standalone program is not only capable to replicate the optimal Lennard Jones clusters in The Cambridge Cluster Database (CCD), but to find new ones.