Minimum inter-particle distance at global minimizers of Lennard-Jones clusters;
Guoliang Xue
Journal of Global Optimization;
Vol. 11(1997), pp. 83--90.


ABSTRACT:
In computer simulations of molecular conformation and protein folding, a
significant part of computing time is spent on the evaluation of potential
energy functions and force fields. Therefore many algorithms for fast
evaluation of potential energy functions and force fields are proposed in
the literature. However, most of these algorithms assume that the particles
are uniformly distributed in order to guarantee good performance.
In this paper, we prove that the minimum inter-particle distance at any
global minimizer of Lennard-Jones clusters is bounded away from zero
by a positive constant which is independent of the number of particles.
As a by-product, we also prove that the global minimum of an n particle
Lennard-Jones cluster is bounded between two linear functions.
Our first result is useful in the design of fast algorithms for potential
function and force field evaluation. Our second result can be used to decide
how good a local minimizer is.


NOTE: This is the first result on minimum inter-particle distance.
Extensive simulation results seem to indicate that the minimum inter-particle
distance at ground states of Lennard-Jones clusters is bounded from below by
a positive lower bound that is independent of the number of particles in the
cluster. Many scientists had hoped this to be true, but were not able to prove
it. In fact, simulation results indicate that the minimum inter-particle
distance at ground states decreases when the cluster size increases. Therefore
it is significant to be able to prove it mathematically. The paper not only
settles a long standing open problem, but also opened a new research direction,
as indicated by the follow-up papers that either improve this lower-bound or
prove similar lower-bound for other clusters.