A Brief Review on Results and Computational Algorithms for Minimizing the Lennard-Jones Potential

Abstract

The Lennard-Jones (LJ) Potential Energy Problem is to construct the most stable form of $N$ atoms of a molecule with the minimal LJ potential energy. This problem has a simple mathematical form $f(x)=4{\sum }_{i=1}^N{\sum }_{j=1,j<i}^N(\frac{1}{{\tau }_{ij}^6}-\frac{1}{{\tau }_{ij}^3}subjecttox\in {\mathbb{R}}^n$, where ${\tau }_{ij}=(x_{3i-2}-x_{3j-2}{)}^2+(x_{3i-1}-x_{3j-1}{)}^2+(x_{3i}-x_{3j}{)}^2$, $(x_{3i-2},x_{3i-1},x_{3i})$ is the coordinates of atom $i$ in ${\mathbb{R}}^3$, $i,j=1,2,...,N(\ge 2\text{integer})$, and $n=3N$; however it is a challenging and difficult problem for many optimization methods when $N$ is larger. In this paper, a brief review and a bibliography of important computational algorithms on minimizing the LJ potential energy are introduced in Sections 1 and 2. Section 3 of this paper illuminates many beautiful graphs (gotten by the author nearly 10 years ago) for the three dimensional structures of molecules with minimal LJ potential.