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

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 $f(x) = 4\sum_{i=1}^N \sum_{j=1,j<i}^N (\frac{1}{\tau_{ij}^6} - \frac{1}{\tau_{ij}^3} {subject to} x\in \mathbb{R}^n$, 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 $\mathbb{R}^3$, $i,j=1,2,...,N(\geq 2 \quad \text{integer})$, 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.