A Brief Review on Results and Computational Algorithms for Minimizing the LennardJones Potential
Abstract
The LennardJones (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 $\tau_{ij} = (x_{3i2}  x_{3j2})^2 + (x_{3i1}  x_{3j1})^2 + (x_{3i}  x_{3j})^2$, $(x_{3i2},x_{3i1},x_{3i})$ 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2010
 arXiv:
 arXiv:1101.0039
 Bibcode:
 2011arXiv1101.0039Z
 Keywords:

 Physics  Computational Physics;
 Computer Science  Data Structures and Algorithms;
 Physics  Chemical Physics