Minimum search space and efficient methods for structural cluster optimization

TL;DR

This paper introduces a unified approach using special lattices to efficiently find optimal clusters under pair potential functions, significantly reducing search space and discovering new optimal Lennard-Jones clusters.

Contribution

It develops a novel theoretical framework that unifies geometrical motifs and approximates the search space with special lattices, enabling efficient identification of optimal clusters.

Findings

Lattices include all optimal clusters up to 1000 particles.

Discovered new optimal Lennard-Jones clusters C*38 and C*98.

Provided data for reproducing optimal clusters as of 2005.

Abstract

A novel unification for the problem of search of optimal clusters under a well pair potential function is presented. My formulation introduces appropriate sets and lattices from where efficient methods can address this problem. First, as results of my propositions a discrete set is depicted such that the solution of a continuous and discrete search of an optimal cluster is the same. Then, this discrete set is approximated by a special lattice IF. IF stands for a lattice that combines lattices IC and FC together. In fact, two lattices IF with 9483 and 1739 particles are obtained with the property that they include all putative optimal clusters from 2 trough 1000 particles, even the difficult optimal Lennard-Jones clusters, C*38, C*98, and the Ino's decahedrons. C*98 is the only cluster where its initial configuration has a different geometry than the putative optimal cluster in term of the adjacency matrix stated by Hoare. My paper is not a benchmark, I develop a theory and a numerical experiment for the state of the art of the optimal Lennard-Jones clusters and even I found new optimal Lennard-Jones clusters with a greedy search method called Modified Peeling Method. The paper includes all the necessary data to allow the researchers reproduce the state of the art of the optimal Lennard-Jones clusters at April 8, 2005. This novel formulation unifies the geometrical motifs of the optimal Lennard-Jones clusters and gives new insight towards the understanding of the complexity of the NP problems.